Citation |

- Permanent Link:
- https://ufdc.ufl.edu/UF00082469/00001
## Material Information- Title:
- Realization of invariant system descriptions from Markov sequences
- Creator:
- Candy, James Vincent, 1944-
- Publication Date:
- 1976
- Language:
- English
- Physical Description:
- viii, 124 leaves : ; 28 cm.
## Subjects- Subjects / Keywords:
- Algebra ( jstor )
Canonical forms ( jstor ) Covariance ( jstor ) Factorization ( jstor ) Integers ( jstor ) Linear systems ( jstor ) Mathematical vectors ( jstor ) Matrices ( jstor ) Stochastic models ( jstor ) Transfer functions ( jstor ) Dissertations, Academic -- Electrical Engineering -- UF Electrical Engineering thesis Ph. D Invariant subspaces ( lcsh ) Markov processes ( lcsh ) City of Gainesville ( local ) - Genre:
- bibliography ( marcgt )
non-fiction ( marcgt )
## Notes- Thesis:
- Thesis--University of Florida.
- Bibliography:
- Bibliography: leaves 114-123.
- General Note:
- Typescript.
- General Note:
- Vita.
- Statement of Responsibility:
- by James Vincent Candy.
## Record Information- Source Institution:
- University of Florida
- Holding Location:
- University of Florida
- Rights Management:
- Copyright [name of dissertation author]. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
- Resource Identifier:
- 025692890 ( ALEPH )
03197863 ( OCLC ) AAU7374 ( NOTIS )
## UFDC Membership |

Downloads |

## This item has the following downloads: |

Full Text |

REALIZATION OF INVARIANT SYSTEM DESCRIPTIONS FROM MARKOV SEQUENCES By JAMES VINCENT CANDY A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR TH DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1976 To my wife, Patricia,and daughter, Kirstin,for unending faith, encouragement and understanding. To my mother, Anne, for her constant support and my mother-in-law, Ruth, for her encouragement. To "big" Ed, my father-in-law, whose sense of humor often lifted my sometimes low spirit. :' .. ~'~~-~P1"*-~qnq8-.s~-rrarrm~mtin*a-s--- --il----~---~i~a~rCI~AI~~-I- -. --~- ACKNOWLEDGMENTS I would like to express my sincere appreciation to the members of my supervisory committee: Dr. Thomas E. Bullock, Chairman, and Dr. Michael E. Warren, Cochairman, Dr. Donald G. Childers, Dr. Z.R. Pop-Stojanovic and Dr. V.M. Popov. A special thanks to Dr. Thomas E. Bullock and Dr. Michael E. Warren for their constant encouragement, unending patience, and invaluable suggestions in the course of this research. I would also like to thank my fellow students and friends, Zuonhua Luo, Arun Majumdar, Jose DeQueiroz, and Jaime Roman, for many fruitful discussions and suggestions. -- iii TABLE OF CONTENTS ACKNOWLEDGMENTS ........................................... iii LIST OF SYMBOLS .......... .................. ... ....... ... vi ABSTRACT .............................. ..... ..... ............. vii CHAPTER 1: INTRODUCTION ................................. ...... 1 1.1 Survey of Previous Work in Canonical Forms for Linear Systems ............................ 2 1.2 Survey of Previous Work in Realization Theory.. 5 1.3 Purpose and Chapter Outline ................... 10 1.4 Notation ..................................... 11 CHAPTER 2: REALIZATION OF INVARIANT SYSTEM DESCRIPTIONS ...... 12 2.1 Realization Theory ............................ 12 2.2 Invariant System Descriptions ................. 18 2.3 Canonical Realization Theory .................. 33 2.4 Some New Realization Algorithms .............. 45 CHAPTER 3: PARTIAL REALIZATIONS ............................. 54 3.1 Nested Algorithm ............................. 54 3.2 Minimal Extension Sequences ................... 64 3.3 Characteristic Polynomial Determination by Coordinate Transformation ..................... 74 CHAPTER 4: STOCHASTIC REALIZATION VIA INVARIANT SYSTEM DESCRIPTIONS ...................................... 78 4.1 Stochastic Realization Theory ................ 81 4.2 Invariant System Description of the Stochastic Realization ........................ 87 4.3 Stochastic Realizations Via Trial and Error ... 97 4.4 Stochastic Realization Via the Kalman Filter .. 104 CHAPTER 5: CONCLUSIONS ....................................... 111 5.1 Summary ....................................... 111 5.2 Suggestions for Future Research ............... 112 ~1_1__1 __~~__ j TABLE OF CONTENTS (Continued) REFERENCES ................................................ 114 BIOGRAPHICAL SKETCH ................................. ....... 124 _ ____r___ll___ Cr ~_I_ _~__ 1__1_1 _ LIST OF MATHEMATICAL SYMBOLS Usage A aT A-1 A-T p(A) IA| or det A diag A xty x>y XcY xeX X-+Y X:= xoy {.) dim X XCA). /x max(.) Z K Meaning First Usage Symbol T -1 -T P I I Transpose of A, a Inverse of A Inverse of AT Rank of A Determinant of A Diagonal elements of A x is not equal to y x is greater than y X is contained in or a subset of Y x is an element of X Map (set X into set Y) x is defined by Abstract group operation Sequence or set with elements Summation Infinity Footnote End of proof Group action operator Dimension of vector space X if and only if Eigenvalues ,of A Square root of x Maximum value of Positive integers Field pg. pg. pg. pg. pg. pg. pg. pg. pg. pg. pg. pg. pg. pg. pg. pg. pg. pg. pg. pg. pg. pg. pg. pg. pg. pg. 13 13 89 13 30,21 103 30 16 18 12 20 13 19 13 13 13 2 34 21 15 14 96 99 23 12 12 iff X / ~~1~1_ _I_ I Abstract of Dissertation Presented to the Graduate Council of the university of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy REALIZATION OF INVARIANT SYSTEM DESCRIPTIONS FROM MARKOV SEQUENCES By James Vincent Candy March, 1976 Chairman: Dr. Thomas E. Bullock Cochairman: Dr. Michael E. Warren Major Department: Electrical Engineering The realization of infinite and finite Markov sequences for multi- dimensional systems is considered, and an efficient algorithm to extract the invariants of the sequence under a change of basis in the state space is developed. Knowledge of these invariants enables the deter- mination of the corresponding canonical form, and an invariant system description under this transformation group. For the partial realization problem, it is shown that this algorithm possesses some attractive nesting properties. If the realization is not unique, the class of all possible solutions is found. The stochastic version of the realization problem is also examined. It is shown that the transformation group which must be considered is richer than the general linear group of the deterministic problem. The invariants under this group are specified and it is shown that they can be determined from a realization of the measurement covariance sequence. Knowledge of these invariants is sufficient to specify an invariant system description for the stochastic problem. The link between the realization from the measurement covariance sequence, the white noise model and the steady state Kalman filter is established. viii CHAPTER 1 INTRODUCTION Special state space representations of linear dynamic systems have long been the motivation for extensive research. These models are generally used to simplify a problem, such as pole placement, by introducing arrays which require the fewest number of parameters while exhibiting the most pertinent information. In general, system represen- tations have been studied in literature as the problem of determining canonical forms; Canonical forms have been used in observer design, exact model matching methods, feedback system design, and Kalman filtering techniques. In realization theory, canonical forms for linear multi- variable systems are important. Since it is only possible to model a system within an equivalence class, the ability to characterize the class by a unique element is beneficial. The problem of determining a canonical form has its roots in invariant theory. Over the past decade many so-called "canonical" system representations have evolved in the literature, but unfortunately these representations were obtained from a particular application or from computational considerations and not derived from the invariant theory point of view. Generally, these representations are not even unique and therefore cannot be called a canonical form. Representations derived in this manner have generally been a source of confusion as evidenced by the ambiguity surrounding the word canonical itself. In this dissertation we follow an algebraic procedure to obtain unique system representations, i.e., we insure that these representations dre in fact canonical forms. In simple terms this approach seeks the determination of certain entities called invariants obtained by applying particular transformation groups *(e.g., change of basis in the state space) to a well-defined set representing a system parameterization. The invariants are the basic structural elements of a system which do not change under this trans- formation and are used to specify the corresponding canonical form. This approach insures that the ambiguities prevalent in earlier work are removed. Initially, we develop a simple solution to the problem of determining a state space model from the unit pulse response of a given linear system and then extend these results to the stochastic case where the system is driven by a random input. The technique developed to extract the invariants from this (response) sequence not only provides a simple solution to the realization problem, but also gives more insight into the system structure. 1.1 Survey of Previous Work in Canonical Forms for Linear Systems The study of canonical forms for linear dynamic systems evolved slowly in the sixties. The main impetus of investigation was initiated by Kalman (1962,1963) when he compared two different methods for describing linear dynamic systems: (1) internally by the state space representation denoted by the triple (F,G,H), or (2) externally by the transfer function--the input/output description. Development over the past decade in such areas as optimal control, decoupling theory, estimation and filtering, identification theory, etc., have relied heavily on the tThis defines (simply) the realization problem. j state space representation for analysis and design. In early literature, however, transfer function representations were used. For highly complex systems it is much easier to determine external behavior rather than internal,since the state variables are normally not available for measurement. As pointed out by Kalman (1963) the language of these representations may be different, but both describe the same problems and are related. Many researchers have investigated the relationship between both representations, but always with one common goal--to obtain a state space model which specifies the external description directly by inspection. Kalman (1963) and later Johnson and Wonham (1964), Silverman (1966) have shown that there exists a canonical form (under change of basis in the state space) in the scalar case for the triple (F,g,h) where F is in companion matrix form (see Hoffman and Kunze (1971)) and g is a unit column vector. It was shown that there exists a one to one correspondence between the non-zero/non-unit elements of the triple and the transfer function. This representation was used by Bass and Gura (1965) to solve the pole-positioning problem and recently by Wolovich (1972b)in solving the exact model matching problem. The progress in determining a canonical form for the internal description of multivariable systems came more slowly. The earliest work appears to be that of Langenhop (1964) in which he develops a representation to study system stability. Brunovsky (1966,1970) was probably the first to recognize the invariant properties of the canonical form for the controllable pair (F,G). Tuel (1966,1967) developed canonical forms for multivariable systems in his investigation of the quadratic optimization problem. Subsequently, Luenberger (1967) proposed certain sets of canonical forms for controllable pairs; however, his development allowed the possibility of nonuniqueness of these representations. Bucy (1968) extended the results of Langenhop and Luenberger when he developed a canonical form for certain subclasses of observable systems, but he too was unaware of its invariant properties. Proceeding from the external system description many researchers began to realize the usefulness in the development of canonical forms. Popov (1969) developed a canonical form for the transfer function in his investigation of irreducible system representations. Gilbert (1969) examined the invariant properties of a system with feedback applied to solve the decoupling problem. Dickinson et al. (1974a) discuss the construction and appli- cation of these canonical forms for the transfer function matrix in a recent survey. The properties of canonical forms were not fully understood initially. In fact, the basic question of their uniqueness posed many doubts as to their usefulness. This issue wasn't resolved until the work of Rosenbrock, Kalman, and Popov in the early seventies. The properties of the Luenberger forms were clarified by the results of Rosenbrock (1970) and Kalman (1971a) in their studies of the minimal column indices (or Kronecker indices) of the matrix pencil [Iz-F,G], or more commonly, the indices of the pair (F,G). These indices were shown to be invariants under the following transformations: change of basis in the state space, input change of basis, and state feedback. These results precisely resolve the question of what can (or cannot) be altered by applying feedback to a linear multivariable system. At the same time Popov (1972) examined the properties of the controllable pair (F,G) under the same transformations in a very precise, step-by-step, algebraic procedure todetermine the corresponding invariants. He shows clearly that obtaining the invariants under a particular transformation group is the only information required to specify the corresponding canonical form. Wonham and Morse (1972) obtained the feedback invariants of the controllable pair from the not as lucid geometric viewpoint. Their results were identical to those of Brunovsky and-4osenbrock. Morse (1973) examined the invariants of the triple (F,G,H) under a lars group of transformations which includes output change of basis. A complete set of feedback invariants of this triple still remains an open problem, but some fragmentary results were presented by Wang and Davison (1972) when they investigated certain sets of restricted triples. Along these lines Rissanen (1974), Caines and Rissanen (1974), . Mayne (1972a,b),Weinert and Anton (1972), Tse and Weinert (1973,1975), Glover and Willems' (1974) examined the identification problem from the invariant theory viewpoint and obtained some rather interesting results. Recent results in decoupling theory were obtained by Warren and Eckberg (1973), Concheiro (1973), and Forney (1975) using the Kronecker invariants. Probably the most extensive survey of these results has been compiled by Denham (1974) and we refer the interested reader to this paper. We temporarily leave this area to consider one specific application of these results--the realization problem. 1.2 Survey of Previous Work in Realization Theory The first realization problem proposed for control systems was the determination of a state space model (internal description) from a given transfer function (external description). Gilbert (1963) and Zadeh and Desoer (1963) describe realization procedures based on the determination of the rank of the residue matrices of the given transfer function matrix, but unfortunately these procedures only apply to the I case of simple poles. Kalman (1963) proposed an algorithm whereby the given transfer function is realized as a parallel combination of single input, controllable subsystems in companion form, and then applied the "canonical structure theory" (Kalman (1962)) to delete the uncontrollable dynamics. This technique handles simple as well as multiple transfer function poles. Later Kalman (1965) showed the equivalence of the realization problem of control theory to the corresponding network theory formulation. A significant advance in realization theory was given by Ho and Kalman (1966). They showed that the state space model could be found from the impulse response sequence provided the system under investi- gation is finite dimensional. They also developed an algorithm based on forming the generalized Hankel array from the given sequence and then extracted the state space triple from it. Shortly after the pub- lication of Ho's algorithm, Youla and Tissi (1966) working in network synthesis and Silverman and Meadows (1966) in control theory developed similar realization techniques again based on the impulse response sequence. Ho's algorithm gave new impetus to realization theory. Several authors have provided alternate or improved realization algorithms based on the Hankel array formulation. Mayne (1968), Panda and Chen (1969), Roveda and Schmid (1970), Rosenbrock (1970), Lal et al. (1972) and even more recently Huang (1974), Rozsa and Sinha (1975) among others, considered the older transfer function matrix approach, while Rissanen (1971,1974), Silverman (1971), Ackermannand Bucy (1971), Chen and Mital (1972), Mital and Chen (1973), and Bonivento et al. (1973) approached the problem from the Hankel array formulation. Rissanen (1974), Furata and Paquet (1975), Roman (1975), Dickinson et al. (1974a,b) have recently considered the problem of realizing a given infinite impulse response matrix sequence with a polynomial matrix pair. Such a pair is referred to as a matrix- fraction description of the system and is becoming well known in control literature largely due to the ground work established by Popov (1969), Rosenbrock (1970), Wolovich (1972a,b, 1973a,b) and others. Kalman (1971b), Tether (1970), and Godbole (1972) later considered the more realistic case where only a finite number of terms of the impulse response sequence are specified. This is commonly known as the partial realization problem and corresponds in the scalar case to the classical Pade approximation problem. Generally most realization altorithms can be used to process partial data, but usually at a loss of efficiency and even more seriously the possibility of yielding misleading results. A wealth of new techniques have recently been published to handle this very special, yet realistic variant of the realization problem. Rissanen (1972a,b), Ackermann (1972), Dickinson et al. (1974a), Roman and Bullock (1975a), Anderson et al. (1975) published some efficient and improved algorithms to solve this problem. Also of recent interest is the development of algorithms which realize a system directly in a canonical form (under a change of basis in the state space), i.e., algorithms which solve the canonical realiza- tion problem. The algorithms of Ackermann (1972), Bonivento et al. (1973), Rissanen (1974), Dickinson et al. (1974a), Rozsa and Sinha (1975), Luo (1975), and Roman and Bullock (1975a) solve this problem. One of the main contributions of this dissertation is to use the results developed from invariant theory to solve the realization and partial realization problems in the deterministic as well as stochastic cases. The realization of a system directly in a canonical form actually reduces to first determining which transformation groups are present, specifying the corresponding invariants, and then developing a method to extract these invariants from the given unit pulse response sequence. This philosophy is basic to any canonical realization scheme and actually provides an explicit formula which is applied throughout this dissertation. In the last few years, several interesting extensions have emerged from the original concept of realization theory. The major motivation evolved just after the development of the Kalman filter (see Kalman (1961)) in estimation theory because a priori knowledge of the state space model and noise statistics are required to begin data processing. The link between the filtering and realization problem was established by Kalman (1965) just prior to the advent of Ho's algorithm. The work of Gopinath (1969), Budin (1971,1972), Bonivento et al. (1973), and Audley and Rugh (1973,1975) were concerned with the more general problem of. obtaining a state space representation given a general input/output sequence of the system in both deterministic and stochastic cases. The stochastic version of the realization problem has not received quite as much attention as the deterministic case mainly due to its g-eater complexity and high dependence on the adequacy of covariance estimators. The realization of stochastic systems was studied by Faurre (1967,1970) and more recently by Rissanen and Kailath (1972), Gupta and Fairman (1974) tThe Hankel array formulation is used exclusively in this dissertation; and Akaike (1974a,b). From the transfer function viewpoint this problem has been solved using spectral factorization as originally introduced by Wiener (1955,1959) and studied by others such as Gokhberg and Krein (1960), Youla (1961), Davis (1963), Motyka and Cadizow (1967), and Strintzis (1972). The link between the stochastic realization problem and spectral factorization evolved from the work in stability theory by Popov (1961,1964), Yakubovich (1963), Kalman (1963), Szeg6 and Kalman (1963). The equations establishing this link were derived in the Kalman-Yakubovich-Popov lemma for continuous systems and the Kalman- Szego-Popov lemma for discrete time systems. Newcomb (1966), Anderson (1967a,b,1969), and Denham (1975) extended these results and provided techniques to solve these equations. Defining the invariants of these problems is still an area of active research as evidenced by the recent work of Denham (1974), Glover (1973), and Dickinson et al. (1974b). This is one area developed in this dissertation. It will be shown that the invariants of the stochastic realization problem not only lends more insight into the structure of the problem, but also yields some new results. Research in realization theory and its applications continues as evidenced by the recent results of Rissanen (1975) in estimation theory, Ackermann (1975) in feedback system design,De Jong (1975) in the numerical aspects of the problem and Roman and Bullock (1975b) in observer theory. The results presented in this dissertation tie together some previously well-known results in stochastic realization and filtering theory as well as provide a technique which can be used to study other problems. 1.3 Statement of Purpose and Chapter Outline It is the purpose of this dissertation to provide an extensive discussion of the realization problem in both the deterministic and stochastic cases as well as specify the invariants under particular transformation groups in each case. It is also desired to develop a simple and efficient algorithm to solve the canonical realization problem. This algorithm is to be modified to process data sequentially such that only the pertinent information--the invariants, are extracted from the given sequence. In the case of a fixed finite unit pulse response sequence (the partial realization problem), the solution is to be obtained such that all possible degrees of freedom are specified. The relationship between the stochastic realization and steady state Kalman filtering problems are discussed by again examining the corresponding invariants. In so doing, a considerable amount of knowledge about the existence and structure of realizations and the steady state filter is gained. The basic theoretical essentials of realization and invariant theory are reviewed in Chapter 2. A "formula" essentially outlined in Popov (1973) and Kalman (1974) is developed which will be applied to various realization problems throughout the text. Some new theoretical results in canonical realization theory are established and used to develop a new canonical realization algorithm. In Chapter 3 the algorithm is modified to handle sequentially the case of partial data and also that of a fixed finite sequence. New results evolve which completely characterize the class of all minimal partial realizations and extension sequences as well as determining the characteristic equation in a simple manner. The stochastic case of the canonical realization problem is in- vestigated in Chapter 4. A complete set of independent invariants is found to characterize the corresponding solution. Equivalent solutions to this problem as well as to the steady state Kalman filtering problem are studied and it is shown that the filter parameters can be specified by solving an analogous realization problem. The specific contributions of this research and further research possibilities are outlined in Chapter 5. Examples are used generously throughout this work to illustrate the various algorithms discussed and to point out significant details that are otherwise difficult to see. A comment on notation to be used through- out this dissertation closesdthis chapter. 1.4 Notation Uppercase letters denote matrices, and vectors are represented by underlined lowercase letters. Lowercase letters are used to represent scalars and integers. All matrices and vectors appearing in this work are assumed to be real and constant. A = [a..] is an nxm matrix with m ijmis an nxm matrix with elements a.i; On is the nxm null matrix with row and column vectors given by T and 0; In represents the nxn identity matrix, and eT or stands for its j-th row or j-th column; jem means j=1,2,...,m. ~J CHAPTER 2 REALIZATION OF INVARIANT SYSTEM DESCRIPTIONS In this chapter we present a brief review of the major results in realization theory. We establish a basic "formula" and apply it to various system representations. It is shown that this approach greatly simplifies the realization problem. Two new algorithms for realization are developed which appear to be more efficient than previous techniques because they extract only the minimal information necessary to specify a system from the given input/output sequence in an extremely simple fashion. All of the essential theory is developed and a multivariable example is presented. 2.1 Realization Theory A real finite dimensional linear constant dynamic system has internal description given by the state variable equations in discrete time as, x+l = Fx + Guk Y-= Hk (2.1-1) where keZ xeKn=X, ueKm=U, eKP=Y and F, G, H are nxn, nxm, pxn matrices over the field K. X,U,Y are the state, input, and output spaces, respectively. The external system description may be given either in rational form as, T(z) = H(Iz-F)-IG (z complex) (2.1-2) or equivalently as an infinite matrix power series kk T(z) = E Azk (2.1-3) k=l where the sequence {Ak} is the unit pulse response or Markov sequence of (2.1-1). The Markov parameters are Ak = HFkG k=1,2,... (2.1-4) The problem of determining the internal description (F,G,H) from the external description (T(z) or {Ak}) is the realization problem. Out of all possible realizations, E:=(F,G,H) having the same Markov parameters, those of smallest dimension are minimal realizations. Prior to stating some of the significant results from realization theory several useful definitions will be given. The j-controllability and j-observability matrices are the nxmj and pjxn arrays, W = [G j1G] and = [HT (HF'-1)T]. The pair (F,G) is completely controllable if p(W )=n and the pair (F,H) is completely observable if p(V )=n. Throughout this dissertation we will only be concerned with systems possessing these properties. For a completely controllable and observable system, the controllability index, u, and the observability index, v, are the least positive integers such that the rank of W and V is n. 1I v I If two minimal realizations E, 7 are equivalent under a change of basis in X, then there exists a nonsingular T such that (Fj,,~,)t = (TFT-1,TG,HT-1). It also follows by direct substitution that the controllability and observability indices of these realizations are identical and W = TW. for j = 1,2,... j J Vi = ViT-1 for i = 1,2,... The generalized NxN' block submatrix of the doubly infinite Hankel array is given by rAl SN,N' : AN ... AN' .. AN+N'-1 Implicit in the realization problem is determining when a finite dimensional realization exists and, if so, its corresponding minimal dimension. The following proposition by Silverman gives the necessary and sufficient conditions for {Ak} to have finite dimensional realiza- tion. Proposition. (2.1-5) An infinite sequence {Ak} is realizable iff there exist positive integers p,v,n such that p(S ) = (S + ) = (S +j, +1)=n. or j=0,1,... Further, if {Ak} is realizable, then p,v are the controllability and observability indices and n is the dimension of the minimal realization. tThis notation means F = TFT", G = TG, and H = HT-1 Proof. See Silverman (1971). Note that the essential point established in Ho and Kalman (1966), which is used in the proof of the above proposition is that Z is a minimal realization iff it is completely controllable and observable. Since S =V W it follows for dim: = n that: p(S ) min[p(V ),p(W )]=n. This result is essential to construct any realization algorithm. In (2.1-5) the crucial point of finite dimensionality is carefully woven into necessary and sufficient conditions for an infinite sequence to be realizable. What if only partial information about the system is available in the form of a finite Markov sequence? Is this sequence realizable? What is the relationship between the minimal realization and one based only on partial data? These are only a few of the questions which must be resolved when we are limited to partial data. Intrinsic in the realization from a finite Markov sequence is the fact that enough data are contained in S to recover the infinite sequence, i.e., knowledge of {A1,...,Av,-1} is sufficient to determine {Ak}, k=1,2,.... But in reality the only way to be sure of this is knowledge of the actual system dimension (or at least an upper bound). A minimal partial realization is a realization of smallest dimension determined from a finite Markov sequence {Ak},keM which realizes the sequence up to M terms. The order of the partial realization is M and the realization is denoted by Z(M). The realization induces an extension k-1 of {Ak}, i.e., Ak=HF k-G for k>M. The following basic result analogous to (2.1-5) answers the realizability question when only partial data are given. For a proof, see Kalman (1971). Proposition. (2.1-6) (Realizability Criterion) The minimal partial realization problem of order M possesses a solution, E(M) iff there exist positive integers.. v,v, M = v~-pM, such that (R) p( 5~) = p(SV+1,) = p(Sv,+1) where dimE(M) = p(S ) = n. v,11 In this proposition (R) is designated the rank condition. Also, it is important to note that when (R) is satisfied the minimal extension (of S(M)) is unique (see Tether (1970) for proof), but E(M) is not unique because there exist other minimal partial realizations equiv- alent to E(M) under a change of basis in X. We must consider three possible cases when only partial data is available. In the first case enough data is available such that M>M for known n; thus, a minimal realization is found. Second, v and p are available such that (R) is satisfied. In this case a minimal par- tial realization can be found, but this in no way insures it is also a minimal realization of the infinite sequence, since the rank of S may increase as v,u increase. Third, the rank condition does not hold. How can a realization be found when no more data is available? The only possibility in this case is to extend the sequence until (R) is satisfied, but there can exist many extensions satisfying (R) while giving nonminimal realizations. For this reason define a minimal extension as any that corresponds to a minimal (partial) realization. To obtain minimality we must somehow select the right extension among the many possible. Prior to summarizing the main results of Kalman (1971) and Tether (1970), define the incomplete Hankel array associated with a given partial sequence {Ak}, keM as A AA 1 A2 .. AM A2 A3 A.... * S(M,M) := . AM * where the asterisks denote positions where no data is available. The rank of S(M,M) is the number of linearly independent rows (columns) determined by comparing only the data specified elements in each row (column) with the preceding rows (columns) with the cognizance that upon the availability of more data this number can only remain the same or increase. Thus, the rank is a lower bound for any extension when the * are filled in-consistent with the preservation of the Hankel pattern. Both Kalman and Tether show that there are three pertinent integers associated with the incomplete Hankel array. They are defined as: n(M), v(M), p(M) and correspond to the rank of S(M,M), the observability index, and the controllability index of the given data. The latter two are lower bounds (separately) for v and p. Knowledge of either v(M), or y(M) enables us to construct extensions, since they are the least integers such that (R) holds for all minimal extensions. It should also be noted that the integers n,v,p,... are actually non-decreasing functions of the amount of data available, M, and should be written, n(M), v(M), p(M) etc. to be precise. However, the argument tIt also follows from this that the p(S(M,M)) is a lower bound for dim.E (see Kalman (1971)). M will be understood throughout this dissertation in order to maintain notational simplicity. There is one more variant of the partial realization problem that must be considered. A sequence of minimal partial realizations such that each lower order realization is contained in one of higher order will be called a nested realization. Symbolically, this is given by ...S(M)EE(M)I... for M appear as submatrices of the corresponding matrices in Z(M). The solution to this problem is most desirable from the computational viewpoint, since each higher order model can be realized by calculating just a few new elements in the corresponding realization. Rissanen (1971) has given an efficient recursive algorithm to determine this solution. Another related problem of interest is determining a unique member of equivalent systems under similarity and is discussed in the following section. 2.2 Invariant System Descriptions In this section we review some of the fundamental ideas encountered when examining the invariants of multivariable linear systems. The framework developed here will be used throughout this dissertation in formulating and solving various realization problems. Not only does this formulation enable the determination of unique system representations under some well-known transformations, but it also provides insight into the structure of the systems considered. First, we briefly define the essential terminology and then use it to describe some of the more common sets of canonical forms employed in many recent applications (e.g., Roman and Bullock (1975a,b), Tse and Weinert (1975)). For any two sets X and Y, a subset R = X x Yt is called a binary relation on X to Y (or, a relation "between" X and Y). Then (x,y)eR is usually written as xRy and is read: "x stands in the relation R to y". If for X=Y this relation is reflexive, symmetric, and transitive, then it is an equivalence relation E on X given by xEy for x,ysX. The set of all elements z equivalent to x is denoted by E(x).= {zeXfxEz} and is called the equivalence class or orbit of x for the equivalence relation E. The set of all such equivalence classes is called the quotient set or orbit space and is given by X/E. Thus, the relation E of X partitions the set X into a family of mutually disjoint subsets or orbits by sending elements which are related into the same equivalence class. Consider a fixed group Gtt of transformations acting on a set X. Then the elements xl,x2 of X are equivalent under the action of G iff there exists a transformation TeG which maps x1 into x2. This is basically the "formula" we will apply throughout, i.e., we first formulate the set of elements (generally the internal system description), then define a transformation group; and finally determine the orbits under the action of G. To be more precise, let us first define the function f mapping a set X into Y as an invariant for E if for x1,x2 X, x1Ex2 implies f(x i)f(x2). In addition if f(xl)=f(x2) implies xlEx2, then f is a This is the standard Cartesian product, XxY = {(x,y)|xeX, yeY} tHere we mean "group" in the standard algebraic sense, i.e., (Go) where G is a closed set of elements each possessing an inverse and the identify element; o is an associative binary operation. When o is understood, the group is merely denoted by G. tNote that an invariant is actually a function, but common usage refers to its image as the invariant. We will also use this terminology throughout this dissertation. complete invariant. In general we will be interested in a complete system of invariants for E given by the.set of invariants {fi} where f:t X + Y1xY2x...xY fi is an invariant for E, and fl(x,)= (x2),...' fn(xl)=fn(x2) imply xEx2. Completeness of this set of invariants means that the set is sufficient to specify the orbit of x, i.e., there is a one to one correspondence between the equivalence classes in X and the image of f. If the set of complete invariants is independent, then the map f: X-YlX...xYn is surjective. This property means that corresponding to every set of values of the invariants there always exists an n-tuple in Y specified by this set. A complete system of independent invariants will be called an algebraic basis. Generally, we consider a subset of X (e.g., in system theory a controllable system). Correspondingly, let fo be a function mapping the subset X of X into set Y, then fo is a restriction of f if f (x)=f(x) for each xeXo. We can uniquely characterize an equivalence class E(x) by means of the set of values of the functions fi(x), ien where the {fi} constitute a complete set of invariants for E on X. If the corresponding complete invariant f is restricted such that its image is itself a subset of X, then we have specified a set of canonical forms C for E on X. To be more precise, a canonical form C for X under E is a member of a subset CcX such that: (1) for every xeX there exists one and only one cEC for which xEc, and since C is the image of a complete invariant f, then (2) for any xeX and cl, c2EC, xEcl,and xEc2 implies f(x)=f(cl)=f(c2)=cl=c2 (invariance); (3) for any ceC if f(x1)=c and f(x2)=c, then x1Ex2 (completeness). Thus, c=f(x) is a unique member of This notation is actually f=(fl,...,f ):x+Y1...xYn but it is shortened when the set {fi} is clearly understood. FR~-"YI i E(x) for every xeX. With these definitions in mind, our "formula" becomes (i) Formulate the set of elements; (ii) Define the transformation group; (iii) Determine a set of complete invariants under this transformation group; and (iv) Develop the canonical form in terms of the corresponding invariants. (2.2-1) We now apply (2.2-1) to various restricted sets related to multivariable systems. This approach is essentially given in Kalman (1971a),Popov (1972), Rissanen (1974), or Denham (1974). In this sequel we review the main results of Popov. First, define the set of matrix pairs (F,G) as X = {(F,G)IFeKnxn, GeKnxm; (F,G)controllable} The general linear group, which corresponds to a change of basis in the state space, is specified by the set GL(n):= {TITeKnxn; det TO} (2.2-2) with the group operation standard matrix multiplication, i.e., To T = T T. In order to determine the orbits of X under the action of GL(n), it is first necessary to specify the action operator "+" T + (F,G):= (TFT-1,TG) In general the problem of determining a canonical form is quite difficult. However in this dissertation we consider restricted sets which make the problem much simpler. For a thorough discussion of this problem see Kalman (1973). 1 or alternately we can say that the action of GL(n) on Xo induces F + TFT-1 G + TG The action of GL(n) induces an equivalence relation on Xo. We indicate (F,G)ET(F,G) if there exists TeGL(n) such that (T,G)=T+(F,G). Dual results are defined for the observable pair (F,H) and the analogous set denoted by X . The third step of (2.2-1) is established in Popov (1972), but first consider the following definitions. For a controllable pair (F,G) define the j-th controllability index p ., jEm as the smallest positive integer such that the vector F 3gj is a linear combination of its predecessors, where a predecessor of Figj is any vector Frgs where rm+s we have assumed p(G) = m. Throughout this dissertation we use the following definition of predecessor independence: a row or column vec- tor of a given array is independent if it is not a linear combination of its regular predecessors. The following results were established by Popov (1972) Proposition. (2.2-3) (1) The regular vectors are linearly independent; (2) The controllability indices satisfy the m following relationship, E j = n; (3) Ti.ere exists n=1 exactly one set of ordered scalars, ajkscK defined for jem, kej-l, s = O,l,...,min(pj ,k-1) and for jEm, k = j,...,m,s = 0,l,...,min(pj,pk) 1 such that tThroughout this dissertation we use the overbar on a set to denote the dual set. ttThese indices are also called the Kronecker indices. j-1 min(sj.,k-l) m min(,ljpk)-1 F jg. Z Z ajks gk + E ajks Fk' 3 k=1 s=O k=j s=O This proposition follows directly from the controllability of (F,G) and indicates that the regular vectors forma basis where the a's are the coefficients of linear dependencies. The set [{Pj},{ajks}], j,kem, s=0,...,j.-I are defined as the controllability invariants of (F,G), and v=max(pj). The main result of Popov is: Proposition. (2.2-4) The controllability invariants are a complete set of independent invariants for (F,G)eX under the action of GL(n). The proof of this proposition is given in Popov (1972) and consists of verifying the invariance, completeness, and independence of [{1j},{ajks}]. Invariance follows directly from Proposition (2.2-3), since (F,G)ET(F,G), then T'gk can be replaced by TFSgk in the given recursion and the controllability invariants remain unchanged. Completeness is shown by constructing a TeGL(n) such that for two pairs of matrices (F,G), (F,G)eX with identical controllability invariants, (F,G) = (TFT1, TG) or (F,G)ET(F,G). Independence of the controllability invariants is obtained by constructing a canonical form determined only in terms of these invariants. Thus, by introducing a finite set of indices {pj}, Popov shows that this set along with the {ajks} are invariants under the action of GL(n). The main reason for specifying a set of complete and in- dependent invariants is that it enables us to uniquely characterize the orbit of (F,G). It should also be noted that dual results hold for the observ- able pair (F,H), and it follows that the observability invariants are the ~~I __L 24 set [{vi},{ ist}], i,sep, t=O,...,v.i- where the {vi}are the observa- bility indices. The last step of (2.2-1) is to specify the corresponding canonical forms under GL(n). These forms are commonly called the Luenberger forms and are specified by the controllability and observability invariants. They are defined by the pairs (FC,Gc), (FR,HR) where the subscripts C,R reference the fact that the regular vectors span either the columns of W+,1 or the rows of V +l' p + 1 H+ 1 FC = [t2 l1 2 .. emm] (2.2-5) GC = el e ql+l1 e l+l J qj = s s jem Ss=1 s ~ T -2 T eT e1 T T -J e i FR R = (2.2-6) T T er P-+2 __r 0+1 T erp T ri = Vs iE2 s=l1 -i---------*-------i--_~-_ -- where aj, i are n column, n row vectors containing {aI {Bi s} respectively over appropriate indices and zeros in the other places. Luenberger (1967) shows that the transformation, TC, required to obtain the pair (FC,Gc) is determined from the columns of W as TC =[T1 T2 ... Tm (2.2-7) where Tj =[gj Fg lF1j] jFm pCG) = m, and gj is the j-th column of G. Similar results hold for the pair (FR,HR) and is specified by TR constructed from the rows of V . Unfortunately Luenberger (1967) in attempting to develop multi- variable system representations did not determine the invariants under GL(n). It is essential to use the approach outlined in (2.2-1) in order to obtain the corresponding canonical forms or else it is possible to obtain erroneous results. The following example due to Denham (1974), shows that the Luenberger form, as originally stated is not canonical. If we are given the pair (F,G) as 0 0 1 1 1 0 ----,------- 1 0 2 1 0 0 F = G 0 1 2 1 0 0 0 0 1 1_ 0 1 These matrices are in the form of (2.2-5), but it is easliy verified by constructing TC that the controllability invariants are in fact pl=2, P2=2 and 21 = [-1 -1 -1 1], a2 = [-2 0 -2 4T1 The problem with the Luenberger forms is that the maps 7: X0+ XO/E are not well defined. Thus, the image of the maps are indeed canonical forms, but as shown here for (F,G)eXO/E, we need not have rT(F,G)=(F,G), i.e., the mapping does not leave the canonical forms unchanged. The point to remember is that the invariants are the necessary entities of interest which must be determined. The procedure to construct the transformation matrix TC of (2.2-7) is called the Luenberger second plan. The first Luenberger plan consists of examining the columns of ,n' given by n = Wn = [gl ... Fn-g1 "' gm Fn- m] (2.2-8) where Y is an nmxnm permutation matrix, for predecessor independence. Thus, we can define a new set of invariants (under GL(n)) [{f }, {ajks}] completely analogous to the controllability invariants. The canonical forms associated with the invariants obtained in this fashion have tThis procedure amounts to examining the column vectors of Wn for predecessor independence, i.e., examine g1 ... g Fg1 ... Fg . Fn-191 ... Fn-lgm. i I become known as the Bucy forms which were derived directly from the results of Langenhop (1964), Luenberger (1967), and Bucy (1968). We refer the interested reader to these references as well as the recent survey by Denham (1974). Here we will be satisfied to note that the procedure of (2.2-1) applies with the set of controllable pairs (F,G) restricted to the {j} invariants rather than {pj}. Analogous to the Luenberger forms, we define the row and column Bucy forms as (FBR,HBR), (FBC,GBC) respectively. The row form is given by T L eT 11 2-1 0 L21 L22 He +1 FBR BR (2.2-9) Lpl Lp L eT + +..+p+1 where I ..-1 V L.. = ---- ; L.. = ] 11 ST 13 -T" L Bii ij I P v. > 0 and satisfy E v =n ; s=l B, j are v.,vj row vectors containing ist } invariants. The transformation, TBR,required to obtain the pair (FBR,HBR) is T T T VlT BR = [T T F B T (2.2-10) B i i The importance of the Bucy form is that the characteristic equation can be found by inspection of the block diagonal arrays of FBRt. Since FB FBR is block lower triangular, the characteristic equation is given as XFBR(z) = det(Iz-FBR) = (z) ... XL (z) (2.2-11) FBR 1BR Lpp where the Lii are the companion matrices of (2.2-9). Similar results hold for the pair (FBCGBc) and the transformation is specified by TBC constructed from the columns of Wn. This completes the discussion of invariants and canonical forms for controllable or observable pairs. To extend these results to matrix triples (internal system description), it is more convenient to examine an alternate characterization of the corresponding equivalence class-- the Markov sequence of (2.1-4). This approach was used by Mayne (1972b) and Rissanen (1974), in order to determine the orbits of Z under GL(n). It is obvious that the sequence is invariant under this group action Aj = (HT-1)(TFT-1)j-1(TG) = HFj-1G (2.2-12) Consequently every element of Aj can be considered an invariant of Z with respect to GL(n); therefore, two systems which are equivalent under GL(n) possess identical Markov sequences. The converse is also true, i.e., any two systems with identical Markov sequences are equivalent. The standard approach to investigate a system characterized by its Markov sequence is to form the Hankel array, SN,N, where we define S , It should be noted that the Bucy form is not a canonical form if the transformation group includes a change of basis in either input or output spaces, while the Luenberger form is still a canonical form. iAN and S., jeN' as the block rows and columns of SN,N, and the block column and row vectors, a or aT denote the r-th column of S .r s. ,. or the s-th row of S,1 for remN', spN. Rissanen (1974) has shown that by examining the set X1 = {2 I E controllable and observable with {ii} invariants} under the action of GL(n) that Proposition. (2.2-13) The set of controllability invariants and block column vectors, [{1j},{ajks},{a.i}] for the appropriate indices constitute an algebraic basis for any SeX1 under the action of GL(n). The proof of this proposition is given in Rissanen (1974) and consists of showing that any two members of X1 with identical Markov sequences are equivalent under GL(n). Thus, invariance follows by showing that a dependent column vector of the Hankel array can be uniquely represented in terms of the set [{CI-},{acjks}]. These parameters remain unchanged under GL(n); therefore, they are invariants. The block column vectors, a.t satisfy a recursion analogous to (2.2-3), i.e., n-I min(j,Pk-1) m min(lj,Pk)-l a.j+m. = aksa.j+ms + zE jksa.j+ms k=l s=O k=j s=O Thus, all dependent block columns can be generated directly from the set, {a.t} of regular block column vectors. These vectors are invariants under GL(n), since every column vector of Aj is an invariant as shown in (2.2-12). Completeness follows immediately from the above recursion, since any two members of X1 possessing identical invariants satisfy the above recursion and therefore have identical Markov sequences. Independence is shown by constructing the Luenberger form of (2.2-5) and (2.2-14) below* directly from these invariants. The dual result yields another basis on X1, [{i},{B.ist},{a ]. The corresponding canonical forms for ZeX1 or X1 are given by the Luenberger pairs of (2.2-5,2.2-6) and HC = [a.1 ... a.(l )m+ll.. la.m .. a. mm (2.2-14) and T al aT a(l-1)p+l. GR a P* T Vpp. and the canonical triples are denoted by EC and ZR respectively. Rissanen (1974) also shows that a canonical form for the transfer function can be constructed from the invariants of (2.2-13). This is possible because the determination of canonical forms for Z based on the Markov parameters is independent of the origin of Ak's. Rissanen defines the (left) matrix fraction description (MFD) as T(z) := B-1(z)D(z) (2.2-15) where B(z) = z Bi.z for IB \f 0 i=O ' v-1 D(z) E Diz . i=0 I The relationship of the MFD to the Hankel array, S +1,+l ,follows by writing (2.2-15) as B(z)T(z) = D(z) (2.2-16) and equating coefficients of the negative powers of z to obtain the recursion BAj + B1Aj+I + + BA =j j=1 ,2... expanding over j gives the relation over the block Hankel rows as [B ... B] B S T p ) (2.2-17) T v+ ,. where the pxp(v+l) matrix of Bi's is called the coefficient matrix of B(z). Similarly equating coefficients of the positive powers of z gives the recursion Dk = Bk+A + Bk+A + ... + BA-k k=0,l,...,v-1 or expanding over k gives the relation in terms of the first block Hankel column as v-1 Bv 0 D2 B B Dv-2 v- v S (2.2-18) DO Bl B2 ... B The canonical forms for both left and right MFD's are defined by the polynomial pairs (BR(z),DR(z)) and (BC(z),DC(z)) respectively, where R and C have the same meaning as in (2.2-5,2.2-6) and the former is given by T b -1 BR(z) := : bT -p p Izv ; bT K ki Kp (2.2-19) for T T b = OTk where k=i+pvi and Bkj ar S {ist Bkj : 0 1 *T 1 k2 *"'* k(i+pvi) l0-i] i e given by j=i,i+p,...,i+p(vi-1) jfi,i+p,...,i+p(vi-1) j=i+pvi and DR(z) is determined from (2.2-18). Dual results hold for the corresponding column vectors, bj, jem of the -J coefficient array of IC,(z) in terms of the controllability invariants. This completes the discussion of canonical forms for Z or T(z). Note that analogous forms can easily be determined for the Bucy forms if X1 is restricted to {vj}. Henceforth, when we refer to an invariant system description, we will mean any representation completely specified by an algebraic basis. In the next section we develop the theory necessary to realize these representations directly from the Markov sequence. 2.3 Canonical Realization Theory In this section we develop the theory necessary to solve the canonical realization problem, i.e., the determination of a minimal realization from an infinite Markov sequence, directly in a canonical form for the action of GL(n). Obviously from the previous discussion, this solution has an advantage over other techniques which do not obtain Z in any specific form. From the computational viewpoint, the simplest realization technique would be to extract only the most essential information from the Markov sequence--the invariants under GL(n). Not only do the invariants provide the minimal information required to completely specify the orbit of E, but they simultaneously specify a unique representation of this orbit--the corresponding canonical form. Thus, subsequent theory is developed with one goal in mind--to extract the invariants from the given sequence. The following lemma provides the theoretical core of the subsequent algorithms. Lemma. (2.3-1) Let VN and WN, be any full rank factors of SN,N, = VN WN' Then each row (column) of SNN,, is dependent iff it is a dependent row (column) of VN (WN,). Proof. From the factorization SN,N, = VN WN, it follows if the j-th row of SN,N, is dependent, then there exists an eKpN, iaj0 such that T T SN,N' = OmN' Since p(WN,)=n, i.e., WN, is of full row rank, it follows that TSN T = SS,NIWN' -mN' or aTVN(WN' N) = OT N' but det (WNWN' ) 0; thus, a = Om0N, i.e., a dependent row of SN,N, is a dependent row of VN. Conversely assume that there. T exists a nonzero a as before such that T T aV = 0 T VN -mN' Since p(WN,)=n, it follows that this expression remains unaltered if post-multiplied by WN,, i.e., T T aVNN, = iN' and the desired result follows immediately.V The significance of this lemma is that examining the Hankel rows (columns) for dependencies is equivalent to examining the rows (columns) of the observability (controllability) matrix. When these rows (columns) are examined for predecessor independence, then the corresponding indices and coefficients of linear dependence have special meaning-- they are the observability (controllability) invariants. Thus, the obvious corrollary to this lemma is Corollary. (2.3-2) If the rows of the Hankel array are examined for predecessor independence, then the j-th (dependent) row, where j=i+pvi, iep is given by i-l min(vi,vs-1) p min(v,v s)-l T T T -J = ist-s+pt + s istys+pt s=1 t=0 s=l t=0 where{Bist an:d{vi} are the observability invariants and -, kcpN is the k-th row vector of S N,N' Proof. The proof is immediate from Proposition (2.2-3) and Lemma (2.3-1).V Note that similar results hold for the columns of the Hankel array when examined for predecessor independence. In the solution to some problems knowledge of both controllability and observability indices are required. Moore and Silverman (1972) require both indices to design dynamic compensators in order to solve the exact model matching problem. Similarly the requirement exists in the design of pole placement compensators and also stable observers as indicated in Brausch and Pearson (1970) and more recently Roman and Bullock (1975b). In an on-line application Saridis and Lobbia (1972) require the controllability invariants as well as the observability indices to solve the problem of parameter identification and control. The latter case exemplifies the fact that in some instances it is first necessary to determine the structural properties of a system from its external description prior to compensation. The need for an algorithm which determines both sets of controllability and observability invariants from an external system description is apparent. Computationally the simplest and most efficient technique to determine these invariants would be some type of Gaussian elimination scheme which utilizes elementary operations (e.g., see Faddeeva (1959)). If we perform elementary row operations on VN such that the predecessor dependencies of PVN are identical to those of VN and perform column operations on WN, so that WNE and WN, have the same dependencies then examination of SN,N' = PSN,NE is equivalent to the examination of SN,N' We define SN,N as the structural array of SN,N,. This array is specified by the indices {vi} and {p.} which are the least integers such that the row and column vectors of SNN are respectively, ii **i rnonzero fa=Q,...,v-1 zero J la=, ..,N- J Snonzero b=0,...,p -1 a = for 1 S zero b= ..,N- These results follow since SN,N has identical as SN,N,, then SNN' T -1 T -pN for s=j+mb predecessor dependencies where T= 0 if it depends on its predecessors. To find the observability indices, let a be the index of the last nonzero row of 6i+p t=0,...,N-. T T Then if T6 = 0 v = 0 otherwise v. = (a-i)/p+l. Similar results * follow when SNN is expressed in terms of the g The following theorem * specifies the matrices P and E required to obtain SN,N. Theorem. (2.3-3) There exist elementary matrices P and E, respectively lower and upper triangular with unit diagonal elements, such that SN,N=PSN,N.E has identical predecessor dependencies as SNN,N Proof. Let PS N,N=Q where Q is row equivalent to SN,N, and P=[prs]. If the j-th row of SN,N, is dependent on its predecessors, i.e., T j-1 T T T. + Z a = 0 3 k=l jk-k - then selecting P lower triangular such that I 0O r ajk r>s gives this relation. From this choice of P it follows that dependent rows of SN,N, are zero rows of Q. If the j-th row of SN,N, is regular, then P unit diagonal-lower triangular insures that the corresponding row of Q is nonzero and regular. Similar results hold for the columns of SNN with E unit diagonal- upper triangular. This choice of P does not alter the column dependencies of SN,N; for if the i-th column of SN,N, is dependent on its predecessors, then from Corollary (2.3-2) .i is uniquely represented as a linear combination of regular vectors in terms of the control- lability invariants. Since P is unit diagonal-lower triangular, it is the matrix representation of a nonsingular linear transformation, Pir=q. where _q is the i-th column vector of Q. Thus, multiplying on the left every vector Li in (2.3-2) with this P gives for i=j+mpj j-1 min(P ,k"-1) m min(j ,pk)-1 =i = Z Z ajksq.k+ms + E Z j ksqk+ms k=l s=O k=j s=O Thus, we have shown that selecting P with the given structure does not alter the predecessor column dependencies of SN,N' or equivalently Q. Since the column vectors of Q satisfy the above recursion, SN,N, and Q have identical predecessor column dependencies, therefore, performing column operations on Q is analogous to performing them on SN,N, and so we have S = (PSN,N,)E = QE or the predecessor dependencies of SN,N, and SN,N, are identical.V This theorem shows that the indices can be found by performing a sequence of elementary lower triangular row and upper triangular column operations in a specified manner on the Hankel array and examining the nonzero rows and columns of SNN., the structural array of SN,N,. The {ajks} and ({ist} are also easily found by inspection from the proper rows of P and columns of E as given by Corollary. (2.3-4) The sets of invariants {8ist',{jiks} or more compactly the sets of n vectors {. )},{a} are given by the rows of P and columns of E in (2.3-3) respectively as i = [qrqr+p "' Pqr+p(v-l)' 1 i+i ir a. [estes+mt ... es+m(.j. )tT t=mtj+j, j,sm -JJ where 1 q=r, r=s Pqr'est q Proof. The proof of this corollary is immediate from Theorem (2.3-3).V We can also easily extract the set of invariant block row or column vectors, {a },{a k from the Hankel array and therefore, we have a solution to the canonical realization problem. Theorem. (2.3-5) If the generalized Hankel submatrix of rank n is transformed by elementary row operations to obtain a row equivalent array, then by proper choice of P the matrix Q is given by: TG I TFG ... TF TH,,, S= --------------------- __ -- nPn-N I pn-N LmN' L mN' where (F,G) is a controllable pair and det TO. Proof. .If x is a minimal realization, then it is well-known that p(VN)=P(WN,)=n. Since P is an elementary array, then it follows [PVN]-min[p(P),p(VN)]=n; thus P can be chosen such that PV -- and det TfO. N pN-n Sn Post multiplication by WN, gives "VN"''N' pRn N' 1 PVN N' =FG] PSN,N' := Q n Multiplication of the arrays gives the desired results.V Corollary. (2.3-6) If P is selected such that Q is as in (2.3-5) with the pair (F,G) in Luenberger column form, then the set of invariants {a.}, jem is given by the columns of -J W N wk, kemN' with -k = k=pjm+j Proof. If P is selected in Theorem (2.3-5) such that T=T then it follows that each column of WN, corresponding to the (j+mpj)-th for each jem contains the {ajks} invariants.V J ks The method of selecting P is given in the ensuing algorithm. Theorem. (2.3-10) Given the infinite realizable Markov sequence from an unknown system, then EC=(FC,GC,H)n is a minimal canonical realization of {Ak} with Fc = [W* I W* 1 ... I W] C 1 2 Wm GC is a submatrix of (W+1i)C given by the first m columns HC = [a.1 ... al+m(ll) I ... I a .. a m and W = [wj+ ... w jm], jsm, w is a column vector of (Wy+1)C. Proof. Since the sequence is realizable, there exist integers, n,v,p, satisfying Proposition (2.1-5). If Q is given as in Corollary (2.3-6), then (W k)C GC ... FclGC Q = -------- = -------------------- for k>p+l 0Pv-n pv-n L mk Omk Thus, GC is obtained immediately from the first m columns of (Wk)c. Form two nxn arrays, A and A each constructed by selecting n regular columns of (Wk)C starting with the first column for A and the (l+m) column for A The independent columns of (Wk)C are indexed by the pj and satisfy (2.3-8); thus, they are unit columns and A is a permutation matrix, i.e., A = [w ... w I Wl+m ". 2m I .. I .. +m(p.-) ...], jem 3 Theorem. (2.3-10) Given the infinite realizable Markov sequence from an unknown system, then EC=(FC,GC,HC)n is a minimal canonical realization of {Ak) with FC [ W I W I ... I Wm GC is a submatrix of (W.+1)C given by the first m columns HC = [a.l ... a.l+m(Pll ) I ... a ... am ] and W = [j+m ... j+m], jm, w is a column vector of (W +1)C. Proof. Since the sequence is realizable, there exist integers, n,v,p, satisfying Proposition (2.1-5). If Q is given as in Corollary (2.3-6), then F(Wk)CGC | ... | FC GC Q = ------- = ----------------- for k>i+l mk mk Thus, GC is obtained immediately from the first m columns of (Wk)C. Form two nxn arrays, A and A each constructed by selecting n regular columns of (Wk)C starting with the first column for A and the (l+m) column for A The independent columns of (Wk)C are indexed by the pj and satisfy (2.3-8); thus, they are unit columns and A is a permutation matrix, i.e., A = [w1 ... w~ l+m 2m ... j+m( 1) jem where it follows from (2.3-8) that the columns of A form chains satisfying m [w. ...w. j = [eq 1 ... e ] for qj= E P.. 3 -34+m1( y1) -qj-1+ll -s=1 Since A is A shifted m columns to the right, each chain of A is given by [wAj+ ... w J+m ] and again each column is unit -j- -j+mp. 3 T except wj+m = aj from Corollary (2.3-6). Thus, FC := A gives the matrix of (2.2-5). HC is obtained directly from H G I ... I F G = [a.1 ... a.m I .. a.l+m ... am(k+l)] since multiplication by the unit columns of (Fc,Gc) select the n columns of H .V Analogous results hold for the dual ZR. It should also be noted that if the Hankel array is transformed to SN,N' and both rows and columns examined for predecessor independence as before, i.e., NN := N,N' U N (2.3-11) where WN is given in (2.2-8) and T is a permutation array, then all of the previous theory is applicable. The only exception in this case is that the Bucy invariants and forms given by IBR and IBC are obtained instead of the Luenberger forms. These results follow directly from (2.2-1). In many applications the characteristic polynomial XF(z) is required. Many efficient classical methods (e.g., see Faddeeva (1959)) exist to determine XF(z) from the system matrix. Even more recently some techniques have been developed to extract the characteristic polynomial 43 from the Markov sequence, but in general they are only valid in the cyclic case (see Candy et al. (1975)). An alternate solution to this problem is to obtain the Bucy form and use (2.2-11) to find XF(Z) by inspection. It is possible to realize the system directly in Bucy form as mentioned in the previous paragraph, but in this dissertation we prefer to take advantage of the structure of the Luenberger form to construct TBR or TBC. Superficially, this method does not appear simple because the transformation matrix and its inverse must be constructed, but the following lemma shows that TBR can almost entirely be written by inspection from the observability invariants after the {I.} are known. Lemma. (2.3-12) The transformation matrix TBR, such that -1 -1 (FBRGBRHBR) = (TBRFBR' BRG, HTBR is given by T = T T BR B= [T .. B I 1 p If the given triple is in Luenberger form, ZR' then the (.ixn) submatrices TB are T i-l+l T eT ei-+1 S-r+' +2 1FR T T > i-V T-i> R where v'.~ are the observability invariants of ER vi are the invariants associated with EBR and i recall ri = E v ro=O. 1 s s o s=1 Proof. This lemma is proved by direct construction of the T 's. B i Since each T satisfies for v.'v. F 1 1 TB. = F R-1 i h1F' then analogous to and therefore v hiFR = R h.F 'i h FR i-l T property (2.3-8), it follows that h.F =e i^R RR -r (hiFRi )FR = eFR = i v v.-V.-l1 v.-v.-l' = (hiFR )FR = ~.FR . In order to construct TBR it is first necessary to find the {ui} from the rows of [V ]R, but in this case the i 's can generally be found by inspection while simultaneously building TBR. Also, TBR is generally a sparse matrix with unit row vectors; therefore, the inverse can easily be found by solving T RT-B = n directly for the unknown elements of TB . BR BR n BR' In the next section we develop some new algorithms which utilize the theory developed here. 2.4 Some New Realization Algorithms In this section we present two new algorithms which can be used to extract both observability and controllability invariants from the given Markov sequence. Recall from Theorem (2.3-3) that performing row operations on the Hankel array does not alter the column dependencies, however, it is possible to obtain the row equivalent array, Q in a form such that the controllability invariants can easily be found. The first part of the algorithm consists of performing a restricted Gaussian elimination (see Faddeeva (1959) for details) procedure on the Hankel array. This procedure is restricted because there is no row or column interchange and the leading element or first nonzero element of each row is not necessarily a one. Define the natural order as 1,2,.... Algorithm. (2.4-1) (1) Form the augmented array: [IpN I SN,N' I ImN (2) Perform the following row operations on SN,N. to obtain EP Q I ImN,: (i) Set the first row of Q equal to the first Hankel row. (ii) Search the first column of SN,N, by examining the rows in their natural order to obtain the first leading element. This element is qjl. (iii) Perform row operations (with interchange) to obtain qkl=0,k>j. tAlternately it is possible to extract the Bucy invariants from Q by reordering the columns as in (2.2-8) to obtain (=QU and examining the columns for predecessor dependencies. (iv) Repeat (ii) and (iii) by searching the columns in their natural order for leading elements. (v) Terminate the procedure after all the leading elements have been determined. (vi) Check that at least the last p rows of Q are zero. This assures that the rank condition, (R) is satisfied. (3) Obtain the observability and controllability indicest as in Theorem (2.3-3). (4) Obtain T iep from the appropriate rows of P as in Corollary (2.3-4) T * and bi as in (2.2-19) where Bij-pij (5) Perform the following column operations on Q to obtain [P SN,, E]: (i) Select the leading element in the first column of Q, qjl. (ii) Perform column operations (with interchange) to obtain qjs=0 for s>l. (iii) Repeat (i) and (ii) until the only nonzero elements in each row are leading elements. (6) Obtain cj, jem from the appropriate columns of E as in Corollary (2.3-4) and -b from the dual of (2.2-19). -J (7) From the invariants construct the Luenberger and MFD forms as in Section (2.2). If we also require the characteristic polynomial, then we must include: tt. (8) Determine the {.}, iep and (simultaneously) construct TBR as in Lemma (2.3-12). -1 1 (9) Find TBR by solving for the non unit rows in TT = In BR BRTBR n Note that the leading elements have been selected from the rows by examining the columns in their natural order; therefore, the dependent columns are not zero as in (2.3-3), but are easily found from this form of Q by inspection. It should also be noted that the leading elements could have been selected in the j, (j+m), (j+2m)... columns; therefore, facilitating the determination of the Bucy invariants and forms. Alternately the {(3v}, jcm and TBC could be used. These indices can be found easily from the columns of Q. I If we consider the alternate method implied in Corollary (2.3-6), then the following modifications to the preceding steps are required: (1)* Start with the following augmented array: [IpN I SN,N.] (2)* Obtain [P I Q] as before. (5)* Perform additional row operations on Q to obtain unit columns for each column possessing a leading row element, and perform row interchanges such that (2.3-8) is satisfied for each jem, i.e., obtain Q :------n OpN-n kmn, (6)* Obtain the aj, jem, as in (2.3-6). It should be noted that these algorithms are directly related to those developed by Ho and Kalman (1966), Silverman (1971), or Rissanen (1971). As in Ho's algorithm, the basis of the first technique is performing the special equivalence transformation of Theorem (2.3-3) * on SN,N, to obtain SN,N,. The second technique accomplishes the same objectives by restricting the operations to only the rows of SN,N, which is analogous to either the Silverman or Rissanen method. The initial storage requirements in the first method are greater than the second if mN'>pN, since P and E can be stored in the same locations due to their lower and upper triangular structure; and (2) P will be altered in the second method, since row interchanges must be performed in (5)*; whereas, it remains unaltered in the first method which may be important in some applications. Consider the following example which is solved using both techniques. I Example. (2.4-2) Let m=2, p=3, and the Hankel array be given as, S4,4 1 2 2 4 4 8 8 16 1 2 2 4 6 10 13 22 1 0 1 0 3 2 6 6 2 4 4 8 8 16 16 32 2 4 6 10 13 22 28 48 1 0 3 2 6 6 13 16 S = -------------------------- S 4,4 4 8 8 16 16 32 32 64 6 10 13 22 28 48 58 102 3 2 6 6 13 16 27 38 8 16 16 32 32 64 64 128 13 22 28 48 58 102 19 214 6 6 13 16 27 38 56 86 (1) [112 I S4,4 I 8 (2) Performing the row operations as in (2.4-1), obtain [P I Q I 8 where the leading elements are circled, i] O 2 2 4 4 88 16: -1 .1 00 00 2 5 6 1 0 -1 -4 0 -5 -7 -2 0 1 0 0 0 0 0 0 0 0 S0 D 1 I 0( 2, 0 1 1 11 1 0 -] 1 -4 0 0 0 0 0 1 -3 0 -1 0 -1 0 0 1 0 1 -2 0 -1 0 D 0 1 07 -8 0 ,0 0 0 0 0 0 0 1 -8 1 -2 0 -2 0 0 0 0 0 1 -1 2 -3 0 -2 0 0 0 0 0 0 1 (3) The indices are obtained by inspection from the independent rows and columns of Q in accordance with Theorem (2.3-3) as: v1 : 1 v2 2 v3 =1 P1 = 3 112 = 1 and p(S2,3) = p(S3,3) = p(S2,4) = 4 satisfying (R). T T (4) The Ti and bi are determined from the appropriate rows and columns of P as: P45 P43] = P85I P83 = P65 P633 = [2 I 0 0] [3 I 0 1 1] (-1 -1 1 I 1] b = 1 P41 P42 P43 P44 I 0 [0 1-2 0 0 1 ] - = Cp81 P82 P83 P84 P85 P86 P87 P881 01] -3 0-1 0 -1 0 0110] T = [ IP61 62 P63 P64 P65 P66] = [1 0 -1 1] -i" 3 i6 p6 p6 1 ] (5) Performing the column operations, obtain the structural array SN,N and E as: [P I S4,4 ) E] where the leading elements are circled,, = -EP41 = -[P81 = -[p61 P42 P82 I P62 0 000 0 0 0 0 0 0 0 0 0 0 0 0 o0 ( 0 0 0 0 0 7 8 1 -2 -1 1 -4 7 S- 2 4 4 1 -1 T - 1 0 0 0 0 1 -1 -- -3 0 1 0 0 1 0 1 (6) The a. and b. are determined from the appropriate rows and columns of E as: e17 e37 e57 e27 el7 "27 e37 e47 e57 e67 e77 0 5 4 1 1 4 0 5 2 0 1 0 5- 2 1 8 ' 2 L2 -4 e14 e24 e34 e44 e 14 e34 e54 e24 0 3 -1 1 -1 1 0 3 7 I 1 = N 51 (7) The canonical forms of ZR, BR(z), DR(z) and EC, BC(z), iC(Z) are: FR R T f3 T T " -3 T a 1. a 2. T a5 5. T a 3. 1 2 1 2 2 4 1 0 z2-2z 0 -3 z2+z z -z2+z 0 1 z2-zj ; DR() = z 2z z+1 2z+2 0 -2z FC = e2 e3 -1 a 2 GC = [el 4] HC = [a.1 a.3 a5 1 a 2] C a1. z.- [ z2 + 'Z + 4 BC(z) = i. 8 1 -1 1 _Z3, 2 -z +z 2 4 2 2 6 2 1 3 0 ; c(z) = (8) The {vi} and TBR are determined simultaneously as: and TBR BR T T T eT -1 a{. 1 0 0 0 0 1 0 0 0 0 1 0 3 0 1 1 3011 BR(z) [ - I- + i- + 3 IT z2 z2 1 j2 z2- Iz z2- 2z 22-z V1 2=3 I (9) TB1 is given by solving 1 0 T-1 0 1 -1 BR 0 o0 -3 0 (10) Find FBR and XF(z) as BR BRR-BR FBR-TBRFRTBR 2 0 0 2 the equations for the last row as: 0 0 0 0 1 0 -1 -1 0 0 1 2 and XF(z) = (z-2)(z3-2z2+1) = z4_4z3+4z2+z-2 This then completes the first method. If the second method is used instead, only (5)*, (6)*, and (8)* differ. (5)* Performing the additional row operations and interchanges to satisfy (2.3-8) gives: 5 1 1 0 5 7 -- 1 0 0 0 -I 0 -1 ~ 5 3 1 1 1 -- 1 0 0 0 0 0 0 1 0 1 1 3 0V O- o- o o o o -$ 0 -4 X 08 are determined - i; =w- ' from the appropriate 1 "-Il 1 0 -4 3 columns of Q as: (6)* The a.'s a = -1 I 53 This completes the algorithms. In the next chapter the first method is modified to develop a nested algorithm from finite Markov sequences. CHAPTER 3 PARTIAL REALIZATIONS One of the main objectives of this research is to provide an efficient algorithm to solve the realization problem when only partial data is given. As new data is made available (e.g., an on-line application, Mehra (1971)), it must be concatenated with the old (previous) data and the entire algorithm re-run. What if the rank of the Hankel array does not change? Effort is wasted, since the previous solution remains valid. An algorithm which processes only the new data and augments these results (when required) to the solution is desirable. Algorithms of this type are nested algorithms. In this chapter we show how to modify the algorithm of (2.4-1) to construct a nested algorithm which processes data sequentially. The more complex case of determining a partial realization from a fixed number of Markov parameters arises when the rank' condition, abbreviated (R), is not satisfied. It is shown not only how to determine the minimal partial realization in this case, but also how to describe the entire class of partial realizations. In addition, a new recursive technique is presented to obtain the corresponding class of minimal extensions and the determination of the characteristic equation is also considered. 3.1 Nested Algorithm Prior to the work of Rissanen (1971) no earlier recursive methods appeared in the realization theory literature. Rissanen uses a I factorization technique to solve the partial realization problem when (R) is satisfied. His algorithm not only solves the problem in a simple manner, but also provides a method for checking (R) simultaneously. In the scalar case, Rissanen obtains the partial realizations, Z(K), K=1,2,... imbedded in the nested problem of (2.1), but unfortunately this is not true in the multivariable case. Also, neither set of invariants is obtained. The development of a nested algorithm to solve the partial realization problem given in this dissertation follows directly from (2.4-1) with minor modification. There are two cases of interest when only a finite Markov sequence is available. Case I. (R) is satisfied assuring that a unique partial realization exists; or Case II. (R) is not satisfied and an extension sequence must be constructed. The nested algorithm will be given under the assumption that Case I holds in order to avoid the unnecessary complications introduced in the second case. The modified algorithm is given below. The corresponding row or column operations are performed only on the data specified elements. Partial Realization Algorithm. (3.1-1) (1) Same as (1) and (2) of Algorithm (2.4-1) except (iv) is qkjfO k>j if k is a row whose leading element has been specified. (2) If (R) is satisfied for some M*=v+p, obtain the invariants as before in (3), (4) of (2.4-1) and go to (5).If not, continue. (3) Add the next piece of data, AM+1 and form S(M+1,M+1). (4) Multiply S(M+1,M+1) by P. Perform row operations (if necessary) using old leading elements to obtain Q (M+1,M+1). If (R) is satisfied, continue. If not, go to 3. (5) Perform column operations as in (5) of (2.4-1) and obtain the invariants and canonical forms as in (6), (7). Go to 3. Example (2.4-2) will be processed to demonstrate the modified algorithm for comparison. Assume that the Markov parameters are sequentially available at discrete times, i.e., Al is received, then A2, etc., and the system is to be realized. Example. (3.1-2) Let the Markov sequence be given 1 2] 2 4 [4 8 A = 1 -2 A = 2 4 A3 6 10 A4 _1 0 1 0 3 2- by S8 161 [16 32- 13 22 A5= 28 48 6 6 13 16 and apply the algorithm of (3.1-1). It is found that the rank condition is first satisfied when A1, A2 are processed, i.e., (1) [I6 I S(2,2) 1 14 (2) Performing first row and then obtain [P I S*(2,2) J E] or 1 0 0 -l 1 O -2 0 0 1 0 0 -2 0 0 0 1 0 0 0 0 -1 0 0 1 0 0 column operation as in (3.1-1), 1 -2 -1 1 - 0 1 I _ (3) Indices are: v = 1 = 1 V2 =0 2 = 1 V =1 (4) :Invariants are: = -[P41 P43] = [2 ) 0] __ -1'P 4 = [0 1 0] S=-[P61 63 = 1 and -N = [P41 P42 P43 P44 0] [-2 0 0 1 0 0] T T T b2 = [0 P21 P22 ] = [0 0 0-1 1 0] S[P61 P62 P63 P64 P65 P66]= 0 0 -1 0 0 1] e --1 e4 0 -i l13 114 3 1 e23 e24 -2 22 = ~ e 1 e S24 2 33 34 -0 O_ e 44 1 (5) Canonical forms are: 1 2 FC I.1 2 GC = [ 1 I e2 HC = [a.1 I 2] = 1 2 z-1 0 1 2 Fc(z) = f C(z) = -2" z-2 1 0 and T P T -] - FR =R IR 1 T 3 . 3 ~ ~_eT2_"3- - where wTt = -[P 21 P23] =[1 0] z-2 0 0 1 2 BR(z) = -2 z 0; DR(Z) = 0 0 0 0 z-1- 1 0 The rank condition is next satisfied when A1,...,A5 are processed, .e, M =5 and we obtain [P (5,5) E as: i.e., M =5 and we obtain [P | S (5,5) | E] as: [P I Q(5,5)] = 1 -1 -1 -2 -2 1 -4 -3 0 -8 -8 -1 -16 -24 -8 () 2 2 4 4 8 8 16 16.32 0 0. 0 ( 2 5 61216 0 -1 -4 -1 -6 -2 -10 -3 -16 0 0 0 0 0 0 0 0 0 0 (J) 2 5 6 12 16 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 and performing the column operations give [S (5,5) I E~ wT is found easily from HRGR=A1 or solving for the second row of Hp, wTGR [1 2]. GR 1 0 1 0 0 1 0 0 0 1 0 0 0 0 1 0 1 0 0 0 0 1 -1 0 0 0 -1 1 -2 0 0 1 -2 2 -3 0 0 0 -4 0 -5 1 0 1 0 -1 0 0 0 -1 0 -1 0 0 0 -2 0 -2 0 0 0 0 0 0 ) 0 0 0 0 0 0 0 0 0 1 -2 -1 -1 -i- 4 10 -12 4 2 0 0 0 0 @ 0 0 0 0 0 1 i 1 -L' -6 -14 0o( 0 0 0 0 0 0 0 1 -1 -- -3 -- -- 15 20 0 0 0 0 0 0 0 01 0 0 0 0 0 0 0 0 () 0 0 0 0 0 1 -1 -3 -6 -8 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0. 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 a 0 a a The results in this case are identical to those of Example (2.4-2). Let us examine the nesting properties of this realization algorithm. Temporarily, we resort to using data dependent notation for this discussion with the same symbols as defined previously in the previous sections, e.g., the minimal partial realization of order M is given by Z(M) :=- (F(M),G(M),H(M)). Thus, Z(M+k) is a (M+k)-order partial realization. We also assume for this discussion that E(M) is in row canonical form; therefore, it can be expressed in terms of the set of invariants, [{vi(M)},{ (M)},{aT(M)}]. If E(M) is an n dimensional, minimal partial realization specified by these invariants, then there are n regular vectors, -+pt(M) spanning the rows of S(M,M). Furthermore, Ss+pt each dependent row vector, J.(M) is uniquely represented as a linear combination of regular vectors in terms of the observability invariants and it can be generated from the recursion of Corollary. (2.3-2). Similarly, it follows from Proposition (2.2-13) that the dependent block row vectors, aj (M) satisfy an analogous recursion. The following lemma j. describes the nesting properties of minimal partial canonical realizations. Recall that M is the integer of Proposition (2.1-6) given by M =v+p. Lemma. (3.1-3) Let there exist an integer M*(M)IM such that the rank condition is satisfied and let Z(M) be the corresponding minimal partial canonical realization of {Ar}, reM specified by the set of invariants [{v.(M)},{ist(M)}, {a (M)}]. Then vi(M) = vi(M+k) Bist(M) = ... = ist(M+k) aT (M) = = aT (M+k) j. J. iff p(S(M,M))=p(S(M+1,M+1) = ... = p(S(M+k,M+k)) for the given k. Proof. If v.(M) = ... = v.(M+k), etc., then the minimal canonical 1 1 partial realizations are identical, E(M)=E(M+1)= ... =E(M+k). It follows that p(S(M,M))=dimE(M)=p(S(M+1,M+I))=dimE(M+I)= p(S(M+k,M+k)). Conversely, P(S(M,M))=P(S(M+1,M+1))= ... =P(S(M+k,M+k)) implies dimZ(M)=dimE(M+1)=... =dimE(M+k). Since E(M) is a unique minimal canonical partial realization, so is Z(M*). Furthermore, since each realization has the same dimension, each realization has has M*(M)=M*(M+1)= ... = M*(M+k) so that each canonical realization is equal to Z(M*); therefore, Z(M)=Z(M+1)= ... =Z(M+k).V Next we examine the case where E(M) and z(M+k) are of different dimension. The nesting properties are given in the following lemma. Lemma. (3.1-4) Let there exist integers, M*(M)M, M (M+k)-M+k such that the rank condition is satisfied (separately) and E(M), Z(M+k) are minimal partial canonical realizations of {Ar} whenreM andrsM+k, respectively, for given k. If p(S(M+k,M+k))>p(S(M,M)), then vi(M+k)-vi(M), iep. T T Furthermore, a .(M+k)=a. (M), j=i,i+p,...,i+p(vi(M)-l). j. j. 1 Proof. Since p(S(M+k,M+k))>p(S(M,M)), M*(M+k)>M*(M) and therefore, Sv(M),(M) is a submatrix of Sv(M+k),,(M+k). If the j-th row of Sv(M),p(M) is regular, it follows that the j-th row of Sv(M+k),i(M+k) is also regular by the nature of the Hankel pattern, i.e., the rows of Sv(M),p(M) are subrows of Sv(M+k),p(M+k). The addition of more data (AM+1,...,AM+k) to S(M,M) makes previously dependent rows become independent rows but previously independent rows remain independent; thus, the v.(M) can only increase or remain the same, i.e., v.(M+k) - vi(M), iep. The set of regular {a (M+k)} are specified by the vi(M+k)'s; therefore aT (M+k)=a (M), j=i,i+p,...,i+p(v.(M)-1), ( k'. 1 since vi(M+k)Qvi(M), iep.V The results of these two lemmas are directly related to the nesting properties of the partial realization algorithm. First, define JM as the set of indices of regular Hankel row vectors based on M Markov parameters 1 available, i.e., J = {1,1+p,... ,1+p( .(M)-l),...,p,2p,... ,pv (M)} and similarly denote the row vectors of P1 the elementary row matrix of the previous chapter, by T(M). From Lemma (3.1-3), it follows S* T T that JM J+k and i+p(M)(M) =... i+ (M+k)(M+k) since the observability invariants are identical. The vi specify the elements in J and along with the Bist, they specify the elements of T Pi+p. (M)(M) (see Corollary (2.3-4)). From Lemma (3.1-4) it is clear that JMJM+k since vi(M+k)Zvi(M). Reconsider Example (3.1-2), to see these properties. In this case we have M=2, k=3, M (2)=2, M*(5)=5, and p(S(5,5))>p(S(2,2)) as in Lemma (3.1-4); therefore, J*cJ*, since J2 = {1,3} and J* = {1,3,2,5}. The observability indices are identical except for v2(5)>v2(2); thus, {a (2),a, (2)}c{aT (5),aT (5),aT (5),aT (5)} since aT (2) = aj (5) 1 3. I 3. 2. 5. J. j. for j=1,3. We also know from Example (2.4-2) that Z(5) is the solution to the realization problem and therefore the properties of Lemma (3.1-3) will hold for {AM}, M>5. Table (3.1-5) summarizes these properties. The results for the dual case also follow directly. We now proceed to the case of constructing minimal partial realizations when.(R) is not satisfied, i.e., the construction of minimal extensions. 63 Table. (3.1-5) Nesting Properties of Algorithm (3.1-1) Augment M-4+k JMJM+k n(M+k)=n(M) n(M+k)>n(M) T pi+pvi Vi Bist where (R) is satisfied for some k and C means that the corresponding invariants, vectors, or indices are nested or contained in a set of higher order. I 3.2 Minimal Extension Sequences In this section we discuss the more common and difficult problem of obtaining a minimal partial realization from a finite Markov sequence when (R) is not satisfied. Two different approaches for the solution of this problem have evolved. The first is based on constructing an extension sequence so that (R) is satisfied and the second is based on extracting a set of invariants from the given data. We will show that these methods are equivalent in the sense that they may both lead to the same solution. In order to do this the existing algorithm is extended to obtain the more general results of Roman and Bullock (1975a). Also a new recursive method for obtaining the entire class of minimal extensions is presented. It is shown that the existing algorithm does in fact yield a particular solution to this problem which is valuable in many modeling applications. In the first approach, Kalman (1971b),Tether (1970), and subsequently Godbole (1972) examine the incomplete Hankel array to determine if (R) is satisfied. If so, the corresponding minimal partial realization is found. If not, a minimal extension is con- structed such that (R) holdsand a realization is found as before. They show that a minimal extension can always be found, but in general it will be arbitrary. They also show that this extension must be constructed so that the rank of S(M,M) remains constant and the existing row or column dependencies are unaltered. Considerable confusion has resulted from the degrees of freedom available in the choice of minimal extensions. In fact, initially, the major motivation for construction an extension was that it was necessary in order to be able to apply Ho's algorithm. Un- fortunately, these approaches obscure the possible degrees of freedom and may lead to the construction of non-minimal extensions as shown by Godbole (1972). Roman and Bullock (1975a)developed the second approach to the solution of this problem. They show that examining the columns or rows of the Hankel array for predecessor independence yields a systematic procedure for extracting either set of invariants imbedded in the data. They also show that some of these would-be invariants are actually free parameters which can be used to describe the entire class of minimal partial realizations. These results precisely specify the number and intrinsic relationship between these free parameters. Unfortunately Roman and Bullock did not attempt to connect their results precisely with those in Kalman (1971b),Tether (1970). It will be shown that this connection offers further insight into the problem as well as new results which completely describe the corresponding class of minimal extensions. Before we state the algorithm to extract all invariants available in the data, let us first motivate the technique. When operating on the incomplete Hankel array, only the elements specified by the data are used. It is assumed that the as yet unspecified elements will not alter the existing predecessor dependencies when they are specified by an extension sequence. Since the predecessor dependencies are found by examining only the data in S(M,M), we must examine complete submatrices of S(M,M) in order to extract the invariants associated with a particular chain (see Roman and Bullock (1975a)). Therefore, it is possible that a dependent vector, say !T of a sub- -1 matrix of S(M,M) later corresponds to an independent vector in S(M,M). When representing any other dependent vector in this submatrix in terms of regular predecessors, T. must be included, since it is -1I a regular vector of S(M,M) under the above assumption. In this represen- tation the coefficient of linear dependence corresponding to Y -1 is arbitrary. Reconsider Example (3.1-2) for{Ai}, i=1,2,3 where we only consider the (row) map P. Example. (3.2-1) For A1, A2, A3 of (3.1-2) we have P: S(3,3)+Q(3,3) or 1 2 2 414 8~ -@2 2 414 8 1 2 2 416 10 0 0 0 0)2 1.0 1 03 2 .0_1-4-1-6 2448 0 0 0 0 P 2 4 6 10 -- 0 0 2 1 0 3 2 0 0 0 0 48 00 6 10 0 0 3 2 _0 0 The indices are {vl,v2,v3} = {1,2,1}. Since v1=1, the fourth row of S(3,3) (or equivalently Q(3,3) ) is dependent on its regular predecessors as shown in the corresponding 3x4 submatrix (in dashed lines) of S(3,3) ^T (or Q(3,3) ). The second row, say 2 in this submatrix is dependent, yet it is an independent row of S(3,3) (or Q(3,3) ). Now, expand T of this submatrix, i.e., ^T =T + T ^T 4 110 1 120 130 3 121'0) or [2 4 4 8] = B110 [1 2 2 4] + B120 [1 2 2 4] + B130[1 0 1 0] The solution is B110 = 2 120' g130=0; thus, the coefficient 8120 is ah arbitrary parameter. Note that this recursion is essentially the technique given in Roman and Bullock (1975a). Clearly, if (R) is satisfied as in the previous section, then there exists a complete submatrix (data is available for each element) of S(M*,M*) in which every regular vector of S(M,M) is always a regular vector of the submatrix corresponding to a particular chain; thus, there will be no arbitrary or free parameters. The algorithm for the case when (R) is not satisfied may be illustrated by considering row operations on S(M,M) to obtain Q(M,M), since the identical technique can be applied to obtain S*(M,M). The arbitrary (column) parameters are found by performing additional column operations to Q(M,M). As in Example (3.2-1), we must find the largest submatrix of Q(M,M) for each chain, i.e., if we define k as the index of the block row of S(M,M) containing the: vector , - -1i+p vi then the largest submatrix of data specified elements corresponding to the i-th chain is given by the first (i+pvi-l) rows and m(M+1-ki) columns of Q(M,M). Also, we define Ji,iep as the sets of Hankel row indices corresponding to each dependent (zero) row of the (i+pu-1l)x (m(M+1-ki) submatrix of Q(M,M) which becomes independent, i.e., it contains a leading element. In Example (3.2-1) for i=l, we have (l+pv-l)3 and k1=2; thus, m(M+l-kl)=4 and the corresponding submatrix is given by the first 3 rows and columns of Q(3,3), and. of course, J1={2}. I_ I Arbitrary Parameter Partial Realization Algorithm. (3.2-2) (1) Perform (1) of Algorithm (3.1-1) to obtain [P I Q(M,M)] (2) For each iep, determine the largest (i+pwl)xm(M+1-k.) sub- 1 1 matrix of Q(M,M) of data specified elements and form the set Ji T T T (3) For each iep, replace by + b, ba scalar. (4) Determine the corresponding canonical forms incorporating these free parameters. Dual results hold for the columns. The free-parameters are fund in analogous fashion by examining the zero columns of the submatrices of S*(M,M). Example. (3.2-3) The following example is from Tether (1970). For m=p=2 and 1 1 2 4 3 10 7 22 15] A 0 0 0 0 1 3 3 (1) [ P r Q(4,4) ] = 1 ( 1 4 3 10 7 22 15 0 1 0 0 0, 0 ( 3 -4 0 1 0o o-6 -5-18-13 0 0 0 1 0 0 06 1 0 0 2 0 -3 0 1 0 01 0 -1 0 0 1 0 1 0 0o 0o 6 0 -7 0 0 0 1 0 0 -3 0 0 0 0 0 0 1 0 0 It should be noted that when (R) is not satisfied, some of the v. may not be defined, i.e., the last independent row of a.chain is in the last block Hankel row. In this case all would-be invariants are arbitrary. 69 The indices are: v1=2, v2=3 (2) For i=1, (1+p -1)=4, kl=3, m(M+1-k,)=4; thus, the corresponding submatrix is constructed from the first 4 rows and columns of Q(4,4) (small dashes). J1={2}. For i=2, (2+pv2-1)=7, k2=4, m(M+1-k2)=2; thus, the corresponding submatrix of Q(4,4) is given by the first 7 rows and 2 columns (large ra'shes). J2={2,4,6}. (3) Replacing the fifth and eighth rows of P with p + bPj2 and T T T T p + cP2 + dp + ePg where b,c,d,e are real scalars gives T S= [ 2 b -3 0 1 0 0 0] S[-3-e c 0 d+e 0 e 0 1 ] p8 = [-3-e c 0 d+e 0 e 0 1 The T are: -1 _-= [ -2 T = [ 3+e 3 -b 0 0] -c -(d+e) -e ] (4) The canonical form is T -2 T F T1 FR= eT T -4 4 T e -1 HR= GR= T eL Corresponding to these realizations is a minimal extension sequence which can be found by determining the Markov parameters. These parameters T a Ta 3. a a4. T 6, 0 0 0 0 0 0 are cumbersome to obtain due to the general complexity of the expressions in zR or ZC; therefore, a technique to determine these extensions without forming the Markov parameters directly (or the realization) was developed. This method consists of recursively solving simple linear equations (one unknown) to obtain the minimal extension. Extensions constructed in this manner not only eliminate the possibility of non- minimality as expressed in Godbole (1972), but also describe the entire class of minimal extensions. The method of constructing the minimal extension sequence evolves easily from the lower triangular-unit diagonal structure of P. Since a dependent row of Q(M,M) is a zero row, it follows from Theorem (2.3-3) that T +pvi j qi+pi j 0 for jemM (3.2-4) where recall that p i+i=0 for j>i+pv.. Thus, by inserting the 1ipq,j 1 unknown extension parameters, x..(r) for rx11(r) *** xm(r) Ar x pl(r) *.. pm(r) into S(M,M) a system of linear equations is established in terms of the xi (r)'s by (3.2-4). Due to the structure of P, this system of equations is decoupled and therefore easily solved. Example. (3.2-5) Reconsider (3.1-2) for Al, A2. Since (R) is satisfied, the extension A., j>2 is unique. We would like to obtain 3 x11(3) x12(3) x21(3) x22(3) x31(3) x32(3) Since P maps S(2,2) into Q(2,2), we have 1 2 2 4 1 2 2 4 1 0 1 0 r-" - 2 4 x11(3) 2 4 x21(3) _1 0 x31(3) X - x12(3) x22(3) x32(3) P -jo 0 2 2 4 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 I01 -4 0 0 0 and in this case,{vl,v2,v3} ={1,0,1}. Thus, using (3.2-4), we solving 0 = P4T 13 = [-2 0 0 1 o x11(3) x21(3) x31(3) for x11(3) gives x11(3)=4 Similarily solving: T p44 = 0 T T for x12(3) gives x12(3) = 8 for x31(3) gives x31(3) = 1 for x32(3) gives x32(3) = 0 In this example, x21(3)=x11(3) and x22(3)=x12(3), since v2=0. have Thus, this example shows that the minimal extension sequence can be found recursively due to the structure of P. Of course, the problem of real interest is when (R) is not satisfied and (as in Ho's algorithm) a minimal extension with arbitrary parameters must be constructed. Minimal Extension Algorithm. (3.2-6) (1) Perform (1), (2), (3) of Algorithm (3.2-2). (2) Determine M* = v+-p. (The values of v,p are determined by the partial data) (3) Recursively construct the minimal extension {Ar}, r = M+1, ... ,M* where Ar [xij(r)] P by solving the set of equations for xij(r) given by +v = 0, j = m(M+l-k)+l, ... ,m(M*+1-ki), for each iep. p-i +pv 1i) and recall that ki is the index of the block row of S(M,M) containing T the row vector, Y -i+pvvi. Example. (3.2-7) Reconsider (3.2-3) for illustrative purposes. (1) These results are given in Example (3.2-3) (2) M*=6; thus, find A5 = Fxll(5) x12(5)1 x21(5) x22(5) (3) Recursively solve: Pi2vi r = . i=2, j=3,4,5,6. p5 5 = 0 gives x11(5); p T3 = 0 gives x21(5); = 0 gives = 0 gives A6 = xl'(6) x 2(6)- x21(6) x22(6) for i=l, j=5,6,7,8 and for T = 0 gives x12(5) = 0 gives x22(5) = 0 gives = 0 gives' xl1(6); x21(6) ; x12(6) x22(6) and therefore 46-b 31-b 94-6b 63-6b A5 = A6 12-d 9-d-e 30-c-3d-5e+de 21-c-3d-5e+de-e2 By solving for the x i's in A5, A6 we obtain the extension as A x11(5) x11(5)-15~ A 6x11(5)-182 6x11(5)-213 x21(5) x22(5) x21(6) x21(6)+(21(5)- (5-3)-9 The number of degrees of freedom is 4,i.e.,{x11(5),x21 (5),x22(5),x2(6)}. The technique used to solve the partial realization problem when (R) is not satisfied was to extract the most pertinent information from the given data in the form of the invariants, which completely described the class of minimal partial realizations. A recursive method to obtain the corresponding class of minimal extensions was also presented in (3.2-6). This method is equivalent to that of Kalman (1971b) or Tether (1970) for if the minimal extension is recursively constructed and Ho's algorithm is applied to the resulting Hankel array the corresponding partial real- ization will belong to the same class. Note that if the extension is not constructed in this fashion, it is possible that all degrees of freedom available may not be found (see Roman (1975)). It should be noted that the integers v and i are determined from the given data,i.e., knowledge of the invariants enables the construction of a minimal extension such that v and p can be found. The approach completely resolves the ambiguity pointed out by Godbole (1972) arising in the Kalman or Tether technique. The results given above correspond directly to those presented in Kalman (1971b) and Tether (1970). They have shown, when (R) is satisfied, there exists no arbitrary parameters in the minimal partial realization or corresponding extension. Therefore, the existence of arbitrary parameters can be used as a check to see if the rank condition holds. Although it is not essential to construct both sets of invariants, it is necessary to determine M* which requires v and i; thus, the algorithm presented has definite advantages over others, since these integers are simultaneously determined. In practical modeling applications, the prediction of model performance is normally necessary; therefore, knowledge of a minimal extension is required. Also in some of the applications the number of degrees of freedom may not be of interest, if only one partial realization is required rather than the entire class. In this case such a model is easily found by setting all free parameters to zero which corresponds to merely applying the Algorithm (3.1-1) directly to the data and obtaining the corresponding canonical forms as before. Describing the class of minimal extensions offers some advantages over the state space representation in that it is coordinate free and indicates the number of degrees of freedom available without compensation. 3.3 Characteristic Polynomial Determination by Coordinate Transformation In this section we obtain the characteristic equation of the entire class of minimal.partial realizations described by FR or FC of the previous section. It is easily obtained by transforming the realized FR or FC into the Bucy form as before. Recall that the advantage of this representation over the Luenberger form is that it is possible to find the characteristic polynomial directly by inspection of FBR in (2.2-11). I Even though it is possible to realize the system directly in Bucy form as implied in the discussion of (2.3-12), it has been found that this method has serious deficiencies when dealing with finite Markov sequences. If (R) is satisfied, the partial realization is unique. When (R) is not satisfied, this technique does not yield all degrees of freedom. For example, reconsider the arbitrary parameter realization of Example (3.2-3). This realization is given in Ackermann (1972) as -0 1 0 0 0 0 1 0 0 0 -2 3 -b 0 0 -2 3 0 0 0 FR = 0 0 0 1 0 FAck = 0 0 0 1 0 0 0 0 0 1 0 0 0 0 1 3+e 0 -c -(d+e) -e 3+e 0 -c -(d+e) -e Note that one degree of freedom (b=0) has been lost. Similarily Ledwich and Fortmann (1974) have shown by example that this technique can also lead to non-minimal realizations. These deficiencies arise due to the procedure used for the determination of the Bucy invariants. This procedure does not account for the possibility that an independent row vector of a particular chain may actually be dependent if it is compared with portions of the same length of vectors in different chains. To cir- cumvent the problem, the previous technique will be used,i.e., the system is realized directly in Luenberger form and transformed to Bucy form. Not only does this assure minimality as well as the determination of all possible degrees of freedom, but TBR is almost found by inspection as shown in (2.3-12). Reconsider the example of the previous section. Example. (3.3-1) Recall that in (3.2-3) m=p=2, n=5, and v1=2, v2=3, T = [ -2 3 -b 0 0 .k = [ -2 3 -b 0 0 ] (1) Simultaneously construct TBR for predecessor independence I -1 T e -2 T TBR - BTFR -1 R BTF -1 R. (2) Determine T -1 from BR -1 BR 1 0 -2/b 0 0 T j2= [3+e 0 -c -(d+e) -e i from (3.3-4) while examining the rows 3 -b 7 -3b -14 15 -7b -3b' -b TR R= I which gives BRBR n 1 3/b -2/b 0 -1/b 3/b -2/b 0 -1/b 3/b -1/b (3) Determine FBR: -1BR BR RBR FBRTBRFRTBR= (4) Find the characteristic polynomial by inspection. 0 0 0 0 -3b-2c-ce 1 0 0 0 3c-2d-2e 0 1 0 0 -c+3d+e 0 0 1 0 -d+2e-2 0 0 0 1 -e+3 XFBR(z) = z+(e-3)z4+(d-2e+2)z'+(c-3d-e)z2+(-3c+2d+2e)z+(b+2c+be) This example points out some very interesting points. When this technique is combined with the algorithm of (3.2-2), it offers a method which can be used to obtain the solution to the stable realization problem developed in Roman and Bullock (1975b). Also, if the system were realized directly in Bucy form, then b=O and a degree of freedom is lost; thus, in Ackermann's example 91=1, while ours is Z1=5. It is critical that all degrees of freedom are obtained as shown in this case, since the system is observable from a single output. This section concludes the discussion of the deterministic case of the realization problem. In the next chapter we examine the stochastic version of the realization problem. CHAPTER 4 STOCHASTIC REALIZATION VIA INVARIANT SYSTEMS DESCRIPTIONS In this chapter the stochastic realization problem is examined by specifying an invariant system description under suitable trans- formation groups for the realization. Superficially, this may appear to be a direct extension of results previously developed, but this is not the case. It will be shown that the general linear group used in the deterministic case is not the only group action which must be considered when examining the Markov sequence for the corresponding stochastic case. Analogous to the deterministic realization problem there are basically two approaches to consider (see Figure 1): (1) realization from the matrix power spectral density (frequency domain) by performing the classical spectral factorization; or (2) realization from the measurement covariance sequence (time domain) and the solution of a set of algebraic equations. Direct factorization of the power spectral density (PSD) matrix is inefficient and may not be very accurate. Recently developed methods of factoring Toeplitz matrices by using fast algorithms offer some hope, but are quite tedious. Alternately, realization from the covariance sequence is facilitated by efficient realization algorithms and solutions of the Kalman-Szeg6-Popov equations. REALIZATION FROM FACTO T COVARIANCE SEQUENCE -PSD -- MTOR TIO AND ALGEBRAIC METHODS METHODS STOCHASTIC REALIZATION Figure 1. Techniques of Solution to the Stochastic Realization Problem. The problem considered in this chapter is the determination of a minimal realization from the output sequence of a linear constant system driven by white noise. The solution to this problem is well known (e.g.. see Mehra (1971)) as diagrammed below in Figure 2. The output sequence of an assumed linear system driven by white noise is correlated and a realization algorithm is applied to obtain a model whose unit pulse response is the measurement covariance sequence. A set of algebraic equations is solved in order to determine the remaining parameters of the white-noise system. This problem is further complicated by the fact that the covariance sequence must be estimated from the measurements. From the practical viewpoint, the realization is highly dependent on the adequacy of the estimates. Although in realistic situations the covariance-estimation problem cannot be ignored, it will be assumed throughout this chapter that perfect estimates are made in order to concentrate on the realization portion of the problem.t In this chapter we present a brief review of the major results necessary to solve the stochastic realization problem. We use the Majumdar (1976) has shown in the scalar case that even if imperfect estimates are made realization theory can successfully be applied. *White Noise Input Linear Constant System Solve Algebraic Equations Output Sequence Measurement Covariance Sequence Stochastic Realization Figure 2. A Solution to the Stochastic Realization Problem Correlation Techniques Realization Algorithm m __ ___ algebraic structure of a transformation group acting on a set to obtain an invariant system description for this problem. A new realization algorithm is developed to extract this description from the covariance sequence. Recently published results establishing an alternate approach to the solution of this problem are also considered. 4.1 Stochastic Realization Theory Analogous to the deterministic model of (2.1-1) consider a white- noise (WN) model given by SFx + w (4.1-1) Xk =Hx where x and y are the real, zero mean, n state and p output vectors, and w is a real, zero mean, white Gaussian noise sequence. The noise is uncorrelated with the state vector, A., j 5 k and Cov(w ,w.):=E[(w.-Ew.)(w.-Ewj)] = Q6i where 6ij is the Kronecker delta. This model is defined by the triple, WN :=(F,I,H) of compatible dimensions with (F,H) observable and F a nonsingular,t stability matrix, i.e., the eigenvalues of F have magnitude less than 1. The transfer function.of (4.1-1) is denoted by TWN(z). tIn the discussion that follows the WN model parameters will be used to obtain a solution to the stochastic realization problem. Denham (1975) has shown that if the spectral factors of the PSD are of least degree, i.e., they possess no poles at the origin, then F is a nonsingular matrix. The corresponding measurement process is given by -k -k + Y-k (4.1-2) where z is the p measurement vector and v is a zero mean, white Gaussian noise sequence, uncorrelated with x., j 5 k with Cov(vi,.j) = R6ij Cov(wA,.) = S6ij for R a pxp positive definite, covariance matrix and S a nxp cross covariance matrix. Thus, a model of this measurement process is completely specified by the quintuplet, (F,H,Q,R,S). When a correlation technique is applied to the measurement process, it is necessary to consider the state covariance defined by Sk: =Cov(x, ) We assume that the processes are wide sense stationary; therefore, k = T, a constant here. It is easily shown from (4.1-1) that the state covariance satisfies the Lyapunov equation (LE) I = FIFT + Q (4.1-3) It is well known (e.g. see Faurre (1967)) that since F is a stability matrix, corresponding to any positive semidefinite covariancee) matrix Q, there exists a unique, positive semidefinite solution I to the (LE). The measurement covariance is given (in terms of lag j) by Cj:= Cov(k+j'k) = Cov(yk+j,yk)+Cov(yk+j'vk)+Cov!(vk+j'yk)+Cov(k+j ',k) (4.1-4) and from (4.1-1) it may be shown that C. = HFj-I(F~HT+S) j > 0 (4.1-5) C = HnHT + R 0 The PSD matrix of the measurement process is obtained by taking the bilateral z-transform of the sequence C. defined in (4.1-4) which gives 3 DZ(z) = H(Iz-F) Q(Iz1-FT)l HT+H(Iz-F)-1S+ST(Iz-1-FT) -HT+R (4.1-6) It is important to note that this expression is the frequency domain representation of the measurement process which can alternately be expressed directly in terms of the measurement covariance sequence as 00 - 4Z(Z) = Z C.z j=-o, J Since the measurement process is stationary and z is real, C =CT and -k k therefore the PSD can be decomposed as 00 .z (z) = Z C.z" + C + Z CTz. (4.1-7) Sj=l J j=l J Note that {C.} is analogous to the Markov sequence of the deterministic realization problem. We define the problem of determining a quintuplet, (F,H,Q,R,S) in (4.1-6) from Z(z) or {C,} as the stochastic realization problem. In this chapter we are only concerned with the realization from the measurement covariance sequence. When a realization algorithm is applied to the covariance sequence, we define the resulting realization as the Kalman-Szegi-Popov (KSP) model because of the parameter constraints (to follow) which evolve from the generalized Kalman-Szegb-Popov lemma (see Popov (1973)t). Thus, we specify the KSP model as the realization of {C .}t defined by the quadruple, KSP:=(A,B,C,D) of appropriate dimension with transfer function, TKSP(z)=C(Iz-A) -B+D. Note that since the unit pulse response of the KSP model is simply related to the measurement covariance sequence, then (4.1-7) can be written as the sum decomposition. S(z) = TKSP(z)+TKp(z-1) = C(Iz-A)-1B+D+DT+BT(Iz'1-AT)-1CT (4.1-8) The relationship between the KSP model and the stochastic realization of the measurement process is shown in the following proposition by Glover (1973). Proposition (4.1-9) Let ZKSP=(A,B,C,D) be a minimal realization of {Cj}. Then the quintuplet (F,H,Q,R,S) is a minimal stochastic realization of the measurement process specified by (4.1-1) and (4.1-2), if there exists a positive definite, symmetric matrix H and TeGL(n) such that the following KSP equations are satisfied: n-AlIAT =Q D+DT-ICT = R B-AHCT = S where A=T-1FT and C=HT. The proof of this proposition is given in Glover (1973) and This book was published in Romanian in 1966, but the English version became available in 1973. tNote that the sequence, {C.}1 is related to the measurement covariance sequence as Co =Co and C.=C. for j > 0. o o 33 corresponds directly to the results presented by Anderson (1969) in the continuous case. The proof follows by comparing the two distinct representations of 0Z(z) given by (4.1-6) and (4.1-8). Minimality of (F,H,Q,R,S) is obtained directly from Theorem (3.7-2) of Rosenbrock (1970). The KSP equations are obtained by equating the sum decomposition of (4.1-8) to (4.1-6). This proposition gives an indirect method to check whether a given EKSP and stochastic realization, (F,H,Q,R,S) correspond to the same covariance sequence. Attempts to use the KSP equations to construct all realizations, (F,H,Q,R,S) with identical {C.} from ZKSP and T by choice of all possible symmetric, positive definite matrices, I will not work in general because all I's do not correspond to Q,R,S matrices that have the properties of a covariance matrix, i.e., A:= Cov( [w v) Q S (4.1-10) T '6 Li R First, it is necessary to question if the stochastic realization problem always has a solution, or equivalently, when is there a i so that (4.1-10) holds. Fortunately, the well-known PSD property, is sufficient to insure the existence of a solution. This result is available in the generalized Kalman-Szegb-Popov lemma (see Popov (1973)). Proposition (4.1-11) If (F,H) is completely observable, then DZ(Z) Z 0 on the unit circle is equivalent to the existence of a quintuplet, ( ,,R,,) such that z(z) = [f(Iz-?)-1 I p[Q (Iz-1-FT)-1"T i" Rj- p where STV [VT T > The proof of this proposition is given in Popov (1973) and essentially consists of showing there exists a spectral factorization of the given PSD. Thus, this proposition assures us that there exists at least one solution to the stochastic realization problem. Proposition (4.1-9) shows that once ZKSP, T, and n are determined then a stochastic realization, (F,H,Q,R,S) may be specified; however, it does not show how to determine I. Recently many researchers (e.g. Glover (1973), Denham (1974,1975), Tse and Weinert (1975)) have studied this problem. They were interested in obtaining only those solutions to the KSP equations of (4.1-9) which correspond to a stochastic realization such that AO of (4.1-11). Denham (1975) has shown that any solution, n*, of the KSP equations which corresponds to a factorization as in (4.1-11) with V=KN, W=N for K=Knxp, NeKPxP, K full rank and N symmetric positive definite, is in fact a solution of a discrete Riccati equation. V 'V TT T T This can readily be seen by substituting, (Q,R,S) =(KNN KNN KNN) of (4.1-11) into (4.1-9) I*-AI*AT = KNNTKT (4.1-12) D+DT-CT*CT = NNT B-AJ*CT = KNNT for A = T- FT, C=HT, TeGL(n). Solving the last equation for K and substituting for NNT yields K = (B-AI*CT )(D+DT-C*CT)-1 (4.1-13) Now substituting (4.1-13) and NNT in the first equation shows that I* satisfies = AI*AT-(B-An*cT)(D+DT-_*cT) (B-AH*CT)T (4.1-14) a discrete Riccati equation. Thus, in this case the stochastic realization problem can be solved by (1) obtaining a realization, EKSP from {C.}; (2) solving (4.1-14) for 1*; (3) determining NNT from (4.1-12) and K from (4.1-13); and (4) determining Q,R,S from K and NN. A quintuplet specified by T and I* obtained in this manner is guaranteed to be a stochastic realization, but at the computational expense of solving a discrete Riccati equation. Note that solutions of the Riccati equation are well known and it has been shown that there exists a unique, I*, which gives a stable, minimum phase, spectral factor (e.g. see Faurre (1970), Willems (1971), Denham (1975), Tse and Weinert (1975)). We will examine this approach more closely in a subsequent section, but first we must find an invariant system description for the stochastic realization. 4.2 Invariant System Description of the Stochastic Realization Suppose we obtain two stochastic realizations by different methods from the same PSD. We would like to know whether or not there is any way to distinguish between these realizations. To be more precise, we would like to know whether or not it is possible to uniquely characterize the class of all realizations possessing the same PSD. We first approach this problem from a purely algebraic viewpoint. We define a set of quintuplets more general than the stochastic realizations, then consider only those transformation groups acting on this set which leave the PSD or equivalently {C invariant, and finally specify various invariant system descriptions under these groups which subsequently prove useful in specifying a stochastic realization algorithm. The groups employed were first presented by Popov (1973) in his study of hyperstability. The results we obtain are analogous to those of Popov as well as those obtained in the quadratic optimization problem (e.g. see Willems (1971)). Define the set X2 = {(F,H,Q,R,S)I FeKnxn,HeKPxn,QeKnxnReKPxPSeKnxp; Q,R symmetric} and consider the following transformation group specified by the set GK := {L ILEKnxn; L symmetric} and the operation of matrix addition. Let the action of GK on X2 be n b defined by L (F,H,Q,R,S) := (F,H,Q-FLFT+L,R-HLHT,S-FLHT) (4.2-1) This action induces an equivalence relation on X2 written for each pair (F,H,Q,-R,S), (F,H,Q,R,S)eX2 as (F,H,Q,R,S)EL(F,H,Q,R,S) iff there exists a LEGKn such that (F,H,Q,-R,S) = L +(F,H,Q,R,S). This group and GL(n) are essential to this discussion, but we must consider their composite action. Therefore, we define the transformation group, GRn which is the cartesian product of GL(n) and GK , GRn := GL(n)xGK The following proposition specifies GR . nn n i Proposition. (4.2-2) The closed set GRn and operation o form a group where GRn = {(T,L) IT GL(n);LeGK n and the group operation is given by (T,)o(T,L) = (TT,L+T- LTT). Proof. This proof of this proposition follows by verifying the standard group axioms with respective identity and inverse elements (In,0O) and (T-1,-TLTT).v Let the action of GRn on X2 be defined by (T,L) + (F,H,Q,R,S):=(TFT-1,HT-',T(Q-FLFT+L)TT,R-HLHT,T(S-FLHT)) (4.2-3) An element (F,HQ,R,<) of the set X2 is said to be equivalent to the element (F,H,Q,R,S) of X2 if there exists a (T,L)eGR such that (F,H,Q,R,S)=(T,L)+(F,H,Q,R,S). This relation is reflexive (F,H,Q,R,S) = (InOn) +(F,H,Q,R,S) and symmetric (T-1,TLTT)+(F,H,Q,R,S)=(T-1,-TLTT)+((T,L)+(F,H,Q,R,S)) = ((T-1,TLTT)o(T,L))+(F,H,Q,R,S)=(nn0 )+(F,H,Q,R,S) . Transitivity follows from (F,H,Q,R,S)=(T,L)+(F,H,Q,R,S) and (F,H,Q,R,S)= T,T)+(T,H,Qs,,S)=(T,f)+((T,L)+(F,H,Q,R,S))=(f,L)+ (F,H,Q,R,S). Thus, GRn induces an equivalence relation on X2 which we denote by ETL and (4.2-3) defines the partitioning of X2 into classes. Note that our first objective has been satisifed, i.e., two ETL-equivalent quin- tuplets have the same PSD; for if we let the pair (F,H,Q,R,S), (F,H,Q,R,S)EX2 then if (T,L)eGRn _ i __ z(z)=H(Iz-F)-Q(Iz-FT)- +H(IzF) S+S(Iz-- -H+R =(HT-')T(Iz-F)-1T-1(T(Q-FLFT+LT)TT)TT(Iz -FT )ITT(HT1 )T +(HT-I)T(Iz-F)-1T-1T(S-FLHT)+(ST-HLFT)TTT-T(Iz-1-FT)1 TT(HT 1)T +R-HLHT (4.2-4) or D (z)=Z (z)+H(Iz-F)I [L-FLFT-FL(Iz-1-FT)-(Iz-F)LFT-(Iz-F)L -FT)(Iz I1-FT)-1HT which gives QZ(z) = DZ(z). The measurement covariance sequence is also invariant under the action of GRn on X2 because the PSD is also given by DZ(z) = Z C.z-J. Thus, we will call any two systems represented by the j=- j quintuplet of X2 covariance equivalent, if they are ETL-equivalent. Clearly, any two covariance equivalent systems have identical PSD's (or measurement covariance sequences). Conversely, any two systems with identical PSD's are covariance equivalent (see Popov (1973) for proof). In order to uniquely characterize the class of covariance equi- valent quintuplets we must determine an invariant system description for X2 under the action of GR The number of invariants may be found by counting the parameters. If we define, M :=dim(F,H,Q,R,S) and M2:=dim(T,L), then there are M1=n2+np+n(n+l)+p(p+l)+np parameters specifying this quintuplet and GR acts on M2=n2 +n(n+l) of them; thus, there exist M1-M2=2np+'p(p+l) invariants. If we consider the transfor- mation, (TR,L)EGR, to the Luenberger row coordinates, then np of these invariants specify the canonical pair (FR,HR) of (2.2-6). The action of (TR,L) on Q,R,S is given by a_ _~ __ ~I ~ R = TR(Q-(FTR -)(TRLTR )(FTR- ) +L)TR Q-FRLRFR +L (4.2-5) RR = R-(HTR-1)TRLTRT)(HTR T = R-HRLRHRT (4.2-6) R = TR(S-(FTR-1)(TRLTRT)(HTR-1)) = SR-FRLRHRT (4.2-7) where LR = TRLTRT, FR = TRFTR-, HR = HTR-, QR = TRQTRT SR = TRS. The transformation LR acts on n(n+l) parameters of the total n(n+l)+'-p(p+l)+np parameters available in QRR,SR as shown above for the given (FR,HR). Once this action is completed the remaining np+p(p+l) parameters are invariants. There are only four possible ways that LR can act on the triple, (Q,R,S): (i) LR acts only on QR; (4.2-8) (ii) LR acts first to specify SR with the remaining elements of LR acting on QR; (iii) LR acts first to specify R with the remaining elements of LR free to act on QR or SR or both; and (iv) LR acts on any combination of elements in Q,R,S. If we choose to restrict the action of LR to only the kn(n+l) elements of QR, then for any choice of QR (given (FR,HR) and any QR)' the transformation LR is uniquely defined. Since FR is a nonsingular stability matrix, then it is well -known (Gantmacher (1959)) from (4.2-5) that LR is the unique solution of L-FL F = Q* for Q* = RQ It is important to note that the elements of QR are completely free, but once they are selected, LR is fixed by (4.2-5) for any QR and therefore the np elements of SR and the p(p+l) elements of RR are the invariants. Thus, for a particular choice of QR we can uniquely specify the equivalence _I_ ______ ___I I _ class of X2 under the action of GR i.e., (FRHR'R,RR,SR) is a canonical form for ETL-equivalence on X2. On the other hand, if we choose to let LR act on the np elements of SR, then from (4.2-7) only np-p(p-l) elements of LR are uniquely specified, i.e., since LR is symmetric and LRH R rl+ l rp-1+1 there are p(p+l) redundant elements in the R.'s. Thus, for any choice of SR (given (FR,HR) and any SR), np-p(p-l) elements of LR are uniquely defined by (4.2-7) and the remaining elements of LR are free to act on QR. In other words np-p(p-l) elements of QR are invariants,t as well as the elements of RR, since any choice of SR specifies the elements of LR in (4.2-6). Similarly restricting the action of LR to act on the elements of R specifies ;p(p+l) elements of LR from (4.2-6) and we are free to allow the remaining elements of LR to act exclusively on QR or SR or both. Clearly, there are many choices available to distribute the action of LR on QR,R,SR; however, the important point is that once the choice is made, the invariants are specified. Any choice of symmetric QR is acceptable, since LR is uniquely determined from (4.2-5) for given QRFR, but this is not the case when an SR is selected. First recall that FR is nonsingular (see footnote p.81). Then if we define SR:=SR-'R it follows from (4.2-7) that FR-1S LRHRT (4.2-9) and then tThis was pointed out by Luo (1975) and Majumdar (1975). _I |

Full Text |

91 Qr = ^(Qrn^bCTRLVjtFt^^+LjT^QR-F^F^+L^ (4.2-5) Rr = (HTr_1)(TrLTrT)(HTR~1)T = R-HrLrHrT (4.2-6) SR = Tr(S(FTr""^ ) (TrLTrT) (HTr_1 )T) = Sr-FrLrHrT (4.2-7) where LR = TRLTRT, Fr = TrFTr*1, Hr = HTR-1, QR = TRQTRT, SR = TRS. The transformation LR acts on %n(n+l) parameters of the total %n(n+l)+%p(p+l)+np parameters available in Qr,R,Sr as shown above for the giveh (Fr,Hr). Once this action is completed the remaining np+*sp(p+l) parameters are invariants. There are only four possible ways that LR can act on the triple, (Q,R,S): (i) Lr acts only on QR; (4.2-8) (ii) Lr acts first to specify SR with the remaining elements of Lr acting on QR; (iii) L_ acts first to specify R with the remaining elements of K Lr free to act on QR or SR or both; and (iv) Lr acts on any combination of elements in Q,R,S. If we choose to restrict the action of LR to only the %n(n+l) elements of Qr, then for any choice of QR (given (Fr,Hr) and any QR), the transformation Lr is uniquely defined. Since FR is a nonsingular stability matrix, then it is well known (Gantmacher (1959)) from (4.2-5) that Lr is the unique solution of Lr-FrLrFrT = Q* for Q* = Qr-Qr* is important to note that the elements of QR are completely free, but once they are selected, LR is fixed by (4.2-5) for any QR and therefore the np elements of Â¥r and the Jsptp+l) elements of RR are the invariants. Thus, for a particular choice of QR we can uniquely specify the equivalence 116 C. T. Chen and D. P. Mital [1972] "A Simplified Irreducible Realization Algorithm," IEEE Trans, on Auto. Contr., Vol. AC-17, pp. 535-537. A. Alonso-Concheiro [1973] "A Complete Set of Independent Decoupling Invariants," Int. J. Contr., Vol. 18, pp. 1211-1220. M. C. Davis [1963] "Factoring the Spectral Matrix," IEEE Trans, on Auto. Contr., Vol. AC-8, pp. 296-305. L. S. De Jong [1.975] Numerical Aspects of Realization Algorithms in Linear Systems Theory, Ph.D. Dissertation, Tech. Univ. of Eindhoven, Netherlands. M. J. Denham [1974] "Canonical Forms for the Identification of Multivariable Linear Systems," IEEE Trans, on Auto. Contr., Vol. AC-19, pp. 646-656. [1975] "On the Factorization of Discrete-Time Rational Spectral Density Matrices," IEEE Trans, on Auto. Contr., Vol. AC-20, pp. 535-536. B. W. Dickinson, M. Morf, and T. Kailath [1974a] "A Minimal Realization Algorithm for Matrix Sequences," IEEE Trans, on Auto. Contr,, Vol. AC-19, pp. 31-38. B. W. Dickinson, T. Kailath, and M. Morf [1974b] "Canonical Matrix Fraction and State-Space Descriptions for Deterministic and Stochastic Linear Systems," IEEE Trans, on Auto. Contr., Vol. AC-19, pp. 656-667. V. N. Faddeeva [1959] Computational Methods in Linear Algebra, Dover Pubs.,. - New ..York. P. Faurre [1967] Representation of Stochastic Processes, Ph.D. Dissertation, Stanford University, California. [1970] "Identification par Minimisation d'une Representation Markovienne-de Processus Aleatoire," Lecture Notes in Mathematics, Vol. 132, Springer, Mew 'York,/ pp- 85-106. E. E. Fisher . [1965] "The Identification of Linear Systems," Joint Auto. Contr. Conf, Preprints, pp. 473-475. 90 z(z)=K(Iz-?)-1^(Iz-1-?r)-1(Iz-?)-^+ST(I2-1-?T)-1K =(HT"1)T(Iz-F)'1T"1(T(Q-FLFT+L)TT)T"T(Iz"1-FT)"1TT(HT1)T + (HT"1)T(Iz-F)"1T"1T($-FLHT)+(ST-HLFT)TVT(Iz'1-FT)1TT(Ht1)T +R-HLH1 (4.2-4) or $z(z)=$z(z)+H(Iz-F)'1[L-FLFT-FL(Iz"1-FT)-(Iz-F)LFT-(Iz-F)L(Iz"1-FT)](Iz1-FT)~1HT % which gives $z(z) = $z(z). The measurement covariance sequence is also invariant under the action of GR on X0 because the PSD is also given by n 2 GO $z(z) = 2 C.z"'5. Thus, we will call any two systems represented by the j=- 3 quintuplet of Xp covariance equivalent, if they are E-^-equi valent. Clearly, any two covariance equivalent systems have identical PSD's (or measurement covariance sequences). Conversely, any two systems with identical PSD's are covariance equivalent (see Popov (1973) for proof). In order to uniquely characterize the class of covariance equi valent quintuplets we must determine an invariant system description for X0 under the action of GR The number of invariants may be found by counting the parameters. If we define, :=dim(F,H,Q,R,S) and p Mz:=dim(T,L), then there are M^=n +np+%n(n+l)+%p(p+l)+np parameters 2 specifying this quintuplet and GRn acts on l^n +J^n(n+1) of them; thus, there exist M^-Mz=2np+i2p(p+l) invariants. If we consider the transfor mation, (TD,L)eGRn to the Luenberger row coordinates, then np of these k n invariants specify the canonical pair (FR,H^) of (2.2-6). The action of (TR,L) on Q,R,S is given by 11 The stochastic case of the canonical realization problem is in vestigated in Chapter 4. A complete set of independent invariants is found to characterize the corresponding solution. Equivalent solutions to this problem as well as to the steady state Kalman filtering problem are studied and it is shown that the filter parameters can be specified by solving an analogous realization problem. The specific contributions of this research and further research possibilities are outlined in Chapter 5. Examples are used generously throughout this work to illustrate the various algorithms discussed and to point out significant details that are otherwise difficult to see. A comment on notation to be used through out this dissertation cl oses0this chapter. 1.4 Notation Uppercase letters denote matrices, and vectors are represented by underlined lowercase letters. Lowercase letters are used to represent scalars and integers. All matrices and vectors appearing in this work are assumed to be real and constant. An = [a. is an nxm matrix with m L lj m elements a.. .; 0^ is the nxm null matrix with row and column vectors T T given by 0^ and 0^; In represents the nxn identity matrix, and e. or e. stands for its j-th row or j-th column; jqn means j=T,2,...,m. vi \ 4 the possibility of nonuniqueness of these representations. Buey (1968) extended the results of Langenhop and Luenberger when he developed a canonical form for certain subclasses of observable systems, but he too was unaware of its invariant properties. Proceeding from the external system description many researchers began to realize the usefulness in the development of canonical forms. Popov (1969) developed a canonical form for the transfer function in his investigation of irreducible system representations. Gilbert (1969) examined the invariant properties of a system with feedback applied to solve the decoupling problem. Dickinson et al. (1974a) discuss the construction and appli cation of these canonical forms for the transfer function matrix in a recent survey. The properties of canonical forms were not fully understood initially. In fact, the basic question of their uniqueness posed many doubts as .to their usefulness. This issue wasn't resolved until the work of Rosenbrock, Kalman, and Popov in the early seventies. The properties of the Luenberger forms were clarified by the results of Rosenbrock (1970) and Kalman (1971a) in their studies of the minimal column indices (or Kronecker indices) of the matrix pencil [Iz-F,G], or more commonly, the indices of the pair (F,G). These indices were shown to be invariants under the following transformations: change of basis in the state space, input change of basis, and state feedback. These results precisely resolve the question of what can (or cannot) be altered by applying feedback to a linear multivariable system. At the same time Popov (1972) examined the properties of the controllable pair (F,G) under the same transformations in a very precise, step-by-step, algebraic procedure to.determine the corresponding invariants. He shows clearly that obtaining the invariants under a particular transformation 48 Example. (2.4-2) Let m=2, p=3, and the Hankel array be given as, ^ - 1 2 2 4 4 8 8 16 1 2 2 4 6 10 13 22 1 0 1 0 3 2 6 6 2 4 4 8 8 16 16 32 2 4 6 10 13 22 28 48 1 0 3 2 6 6 13 16 4 8 8 16 16 32 32 64 6 10 13 22 28 48 58 102 3 2 6 6 13 16 27 38 8 16 16 32 32 64 64 128 13 22 28 48 58 102 119 214 6 6 13 16 27 38 56 86 ^ ^12 I S4,4 I V . Y (2) Performing the row operations as in (2.4-1), obtain [P | Q I I0], O where the leading elements are circled, ] 2 2 4 4 8 8 16: -1 1 0 o o o ro 5 6 Jl 2 i T 1 0 @-1 -4 0 -5 _i_ 2 -7 -2 b 0 1 0 0 0 0 0 0 0 0 0 1 2 _ 5 . 2 0 0 1 0 0 0 201 1 2 1 1 1 -1 0 -] 1 -4 0 b 0 0 0 1 r8 -3 0 -i -1 0 0 1 0 1 -2 0 -1 0 0 0 1 o 00 -^1 -8 0 .0 0 . 0 0 0 0 0 1 -8 1 -2 0 -2 0 0 0 0 0 1 -1 2 -3 0 -2 0 0 0 0 0 0 1 25 where g^., 3.j are n column, n row vectors containing {cu^}, {3..^} respectively over appropriate indices and zeros in the other places. Luenberger (1967) shows that the transformation, T^, required to obtain the pair (F^.G^) is determined from the columns of Wn, as where TC*T1 T2 <2-2-7> Tj =t9j Fgj F J_1gj] jera p(G) = m, and g. is the j-th column of G. vl Similar results hold for the pair and is specified by constructed from the rows of V . n Unfortunately Luenberger (1967) in attempting to develop multi- variable system representations did not determine the invariants under GL(n). It is essential to use the approach outlined in (2.2-1) in order to obtain the corresponding canonical forms or else it is possible to obtain erroneous results. The following example due to Denham (1974), shows that the Luenberger form, as originally stated is not canonical. If we are given the pair (F,G) as "o 0 1 1 1 Â¡ 1 ~i o" 1 1 0 2 1 ! 1 0 0 F = G = 0 1 2 S 1 1 l 0 0 _0 0 1 1 l ! 1j _0 1_ 118 R. D. Gupta and F. W. Fairman [1974] "Parameter Estimation for Multivariable Systems," IEEE Trans, on Auto. Contr., Vol.'AC-19, pp. 546-549. B. L. Ho and R. E. Kalman [1966] "Contruction of Linear State Variable Models from Input/ Output Functions," Regelungstechnik, VoT. 14, pp. 545-548. K. Hoffman and R. Kunze f1971] Linear Algebra, Prentice-Hall Pubs., Second Edition, New Jersey H. L. Huang [1974] "A Generalized-Jordan-Form-Approach One-Step Irreducible Realization of Matrices," IEEE Trans, on Auto. Contr., Vol. AC-19, pp. 271-272. C. D. Johnson and W. M. Wonham [1964] "A Note on the Transformation to Canonical (Phase-Variable) Form," IEEE Trans, on Auto. Contr., Vol. AC-9, pp. 312-313. R. E. Kalman [1960] "Control System Analysis and Design Via the Second Method of Lyapunov," J. of Basic Engr., Vol. 82D, pp. 394-499. [1961] "A New Approach to Linear Filtering and Prediction Problems," J. of Basic Engr., Vol. 82D, pp. 35-43. [1962] "Canonical Structure of Linear Dynamical Systems," Proc. Nat. Acad, of Sci. (USA), Vol. 48, pp. 596-600. [1963] "Mathematical Description of Linear Dynamical Systems," SIAM J. on Contr., Vol. 1, pp. 152-192. [1964] "Lyapunov Functions for the Problem of Lure in Automatic Control," Proc. Nat. Acad. Sci. (USA), Vol. 49, pp. 201-205. [1965] "Linear Stochastic Filtering Theory-Reappraisal and Outlook," Proc. of Symposium of System Theory, Brooklyn, N. Y., pp. 197-205. [1971a] "Kronecker Invariants and Feedback," Proc. 1971 Conf. on Ord. Diff. Eg., National Research Laboratory Mathematics Research Center, Washington, D. C. [1971b] "On Minimal Partial Realizations of a Linear Input/Output Map," in Aspects of Network and System Theory, R. E. Kalman and N. De Claris (eds.), pp. 385-407, Holt, Rinehart and Winston., N.Y. [1973] "Global Structure Classes of Linear Dynamical Systems," presented at the NATO ASI Geometric and Algebraic Methods for Nonlinear Systems, London, England. [1974] Class Notes on System TheoryCourse by R. E. Kalman at the University of Florida, Gainesville, Fla. * 92 class of under the action of GRn, i.e., (FR,HR,QR,RR,SR) is a canonical form for Ej^-equivalence on X2. On the other hand, if we choose to let LD act on the np elements K of SR, then from (4.2-7) only np-%p(p-l) elements of LR are uniquely specified, i.e., since LR is symmetric and LRHR f-l 4^+1 * +i ] Vi 1 there are *sp(p+l) redundant elements in the Â£.'s. Thus, for any choice J of (given (FR,HR) and any SR), np-Jgp(p-l) elements of LR are uniquely defined by (4.2-7) and the remaining elements of LR are free to act on QR In other words np-^p(p-l) elements of QR are invariants, as well as the elements of RR, since any choice of ifR specifies the elements of Lr in (4.2-6). Similarly restricting the action of LR to act on the elements of R specifies %p(p+l) elements of LR from (4-2-6) and we are free to allow the remaining elements of LR to act exclusively on QR or SR or both. Clearly, there are many choices available to distribute the action of LR on Qr,R,Sr; however, the important point is that once the choice is made, the invariants are specified. Any choice of symmetric (IR is acceptable, since LR is uniquely determined from (4.2-5) for given QR,FR, but this is not the case when an SR is selected. First recall that FR is nonsingular (see footnote p.81). Then if we define SR: =SR-S"R it follows from (4.2-7) that FR 1sR = LRHR (4.2-9) and then ^This was pointed out by Luo (1975) and Majumdar (1975). 13 The external system description may be given either in rational form as, T(z) = H(Iz-F)~^G (z complex) (2.1-2) or equivalently as an infinite matrix power series T(z) = Z A zk (2.1-3) k=l K where the sequence {A^} is the unit pulse response or Markov sequence of (2.1-1). The Markov parameters are Ar> HF^G k=l ,2,... (2.1-4) The problem of determining the internal description (F,G,H) from the external description (T(z) or {A^>) is the realization problem. Out of all possible realizations, Z:=(F,G,H) having the same Markov parameters, those of smallest dimension are minimal realizations. Prior to stating some of the significant results from realization theory several useful definitions will be given. The j-control!ability and j-observability matrices are the nxmj and pjxn arrays, W, [G | ... | Fj_1G] and v! = [HT | ... | (HFj"1)T]. The pair (F,G) J J is completely controllable if p(Wn)=n and the pair (F,H) is completely observable if p(Vn)=n. Throughout this dissertation we will only be concerned with systems possessing these properties. For a completely controllable and observable system, the controllability index, y, and the observability index, v, are the least positive integers such that the rank of W and V is n. y v REFERENCES J. E. Ackermann [1972] "On Partial Realizations," IEEE Trans, on Auto. Contr., Vol. AC-17, pg. 381. [1975] "On the Synthesis of Linear Control Systems with Specified Characteristics," Proc. 1975 IFAC Congress, Boston, Mass., pp. 88-92. J. E. Ackermann and R. S. Buey [1971] "Canonical Minimal Realization of a Matrix of Impulse Response Sequences," Inf, and Contr., Vol. 19, pp. 224-231. H. Akaike [1974a] "Stochastic Theory of Minimal Realization," IEEE Trans, on Auto. Contr., Vol. AC-19, pp. 667-674. [1974b] "A New Look at the Statistical Model Identification," IEEE Trans, on Auto. Contr., Vol. AC-19, pp. 716-723. [1975] "Markovian Representation of Stochastic Processes by Canonical Variables," SIAM J. on Contr., Vol. 13, pp. 162-173. > B. D. 0. Anderson [1967a] "A System Theory Criterion for Positive Real Matrices," SIAM J. on Contr., Vol. 5, pp. 171-182. [1967b] "An Algebraic Solution to the Spectral Factorization Problem," IEEE Trans, on Auto. Contr., Vol. AC-12, pp. 410-414. [1969] "The Inverse Problem of Stationary Covariance Generation," J. of Statistical Physics, Vol. 1, pp. 133-147. B. D. 0. Anderson and S. Vongpanitlerd [1973] Network Analysis and Synthesis, Prentice-Hall Inc., New Jersey. B. M. Anderson, F. M. Brasch, and P. V. Lopresti [1975] "The Sequential Construction of Minimal Partial Realizations from Finite Input Output Data," SIAM J. on Contr., Vol. 13, pp. 552-570. D. R. Audley and W.J. Rugh [1973] "On the H-Matrix System Representation," IEEE Trans, on Auto. Contr., Vol. AC-18, pp. 235-242. D. R. Audley, S. L. Baumgartner and W. J. Rugh [1975] "Linear System Realization Based on Data Set Representations," IEEE Trans, on Auto Contr., Vol. AC-20, pg. 432. 134 76 Example. (3.3-1) Recall that in (3.2-3) m=p=2, n=5, and Vj=2, V2=3, = [ -2 3-bOO] Â§J2 = [3+e 0 -c -(d+e) -e ] (1)Simultaneously construct TBR from (3.3-4) while examining the rows for predecessor independence -1 1 0 0 0 o" 4 0 1 0 0 0 4 s -2 3 -b 0 0 4fr -6 7 -3b -b 0 /. -14 15 -7b -3 b' -b 1 1 (2)Determine TBR from TBR TBR = In which gives f1 'BR 1 0 -2/b 0 0 (3)Determine Fbr: F =T F T"^ = rBR BRrR BR 0 0 0 0 1 3/b 2/b 0 -1/b 3/b -2/b 0 -1/b 3/b -1/b 1 0 0 0 -3b-2c-ce 3c-2d-2e 0 1 0 0 -c+3d+e 0 0 1 0 -d+2e-2 0 0 0 1 -e+3 (4)Find the characteristic polynomial by inspection. 24 set [{v.-LB.j^}], i,seÂ£,. where the {v..} are the observa- bi1ity indices. The last step of (2.2-1) is to specify the corresponding canonical forms under GL(n). These forms are commonly called the Luenberger forms and are specified by the controllability and observability invariants. They are defined by the pairs (Fq.Gq), (FrjHr) where the subscripts C,R reference the fact that the regular vectors span either the columns of W P+1 or the rows of V v+1 (2.2-5) 3 l P- Jem s=l s F R J H R (2.2-6) P T r. = Z v ieÂ£ 1 s=l s 10 1.3 Statement of Purpose and Chapter Outline It is the purpose of this dissertation to provide an extensive discussion of the realization problem in both the deterministic and stochastic cases as well as specify the invariants under particular transformation groups in each case. It is also desired to develop a simple and efficient algorithm to solve the canonical realization problem This algorithm is to be modified to process data sequentially such that only the pertinent informationr-the invariants, are extracted from the given sequence. In the case of a fixed finite unit pulse response sequence (the partial realization problem), the solution is to be obtained such that all possible degrees of freedom are specified. The relationship between the stochastic realization and steady state Kalman filtering problems are discussed by again examining the corresponding invariants. In so doing, a considerable amount of knowledge about the existence and structure of realizations and the steady state filter is gained. The basic theoretical essentials of realization and invariant theory are reviewed in Chapter 2. A "formula" essentially outlined in Popov (1973) and Kalman (1974) is developed which will be applied to various realization problems throughout the text. Some new theoretical results in canonical realization theory are established and used to develop a new canonical realization algorithm. In Chapter 3 the algorithm is modified to handle sequentially the case of partial data and also that of a fixed finite sequence. New results evolve which completely characterize the class of all minimal partial realizations and extension sequences as well as determining the characteristic equation in a simple manner. REALIZATION OF INVARIANT SYSTEM DESCRIPTIONS FROM MARKOV SEQUENCES By JAMES VINCENT CANDY A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1976 To my wife, Patricia, and daughter, Kirs tin,, for unending faith, encouragement and understanding. To my mother, Anne, for her constant support and my mother-in-law, Ruth, for her encouragement. To "big" Ed my father-in-law, whose sense of humor often lifted my sometimes low ACKNOWLEDGMENTS I would like to express my sincere appreciation to the members of my supervisory committee: Dr. Thomas E. Bullock, Chairman, and Dr. Michael E. Warren, Cochairman, Dr. Donald G. Childers, Dr. Z.R. Pop-Stojanovic and Dr. V.M. Popov. A special thanks to Dr. Thomas E. Bullock and Dr. Michael E. Warren for their constant encouragement, unending patience, and invaluable suggestions in the course of this research. I would also like to thank my fellow students and friends, Zuonhua Luo, Arun Majumdar, Jos DeQueiroz, and Jaime Roman, for many fruitful discussions and suggestions. TABLE OF CONTENTS ACKNOWLEDGMENTS .. iii LIST OF SYMBOLS vi ABSTRACT vii CHAPTER 1: INTRODUCTION 1 1.1 Survey of Previous Work in Canonical Forms for Linear Systems .. ... 2 1.2 Survey of Previous Work in Realization Theory.. 5 1.3 Purpose and Chapter Outline 10 1.4 Notation 11 CHAPTER 2: REALIZATION OF INVARIANT SYSTEM DESCRIPTIONS 12 2.1 Realization Theory ........ 12 2.2 Invariant System Descriptions .18 2.3 Canonical Realization Theory 33 2.4 Some New Realization Algorithms 45 CHAPTER 3: PARTIAL REALIZATIONS 54 3.1 Nested Algorithm ..... ........ 54 3.2 Minimal Extension Sequences ... 64 3.3 Characteristic Polynomial Determination by Coordinate Transformation ........ ..... 74 CHAPTER 4: STOCHASTIC REALIZATION VIA INVARIANT SYSTEM DESCRIPTIONS 78 4.1 Stochastic Realization Theory 81 4.2 Invariant System Description of the Stochastic Realization ......... ........... 87 4.3 Stochastic Realizations Via Trial and Error ... 97 4.4 Stochastic Realization Via the Kalman Filter .. 104 CHAPTER 5: CONCLUSIONS Ill 5.1 Summary Ill 5.2 Suggestions for Future Research 112 iv TABLE OF CONTENTS (Continued) REFERENCES .....114 BIOGRAPHICAL SKETCH .....124 V \ V LIST OF MATHEMATICAL SYMBOLS Symbol Usage Meaning First Usage T AT, aT Transpose of A, ai pg. 13 -1 A1 Inverse of A pg. 13 -T at Inverse of AT pg. 89 P p(A) Rank of A pg. 13 |.| |A| or det A Determinant of A pg. 30,21 diag A Diagonal elements of A pg. 103 \ x^y x is not equal to y pg. 30 > x>y x is greater than y pg. 16 c XcY X is contained in or a subset of Y pg. 18 e xeX x is an element of X pg. 12 X->Y Map (set X into set Y) pg. 20 : = x: = x is defined by pg. 13 0 xoy Abstract group operation pg. 19 { } {.> Sequence or set with elements pg. 13 Z Summation pg. 13 00 Infinity pg. 13 t Footnote pg. 2 V End of proof pg. 34 4 Group action operator pg. 2i dim X Dimension of vector space X . pg. 15 iff if and only if pg. 14 X X(A). , Eigenval u'es of A pg- 9e / /x Square root of x pg. 99 max() Maximum value of pg. 23 Z+ Positive integers pg. 12' K Field pg. 12 vi Abstract of Dissertation Presented to the Graduate Council of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy REALIZATION OF INVARIANT SYSTEM DESCRIPTIONS FROM MARKOV SEQUENCES By James Vincent Candy March, 1976 Chairman: Dr. Thomas E. Bullock Cochairman: Dr. Michael E. Warren Major Department: Electrical Engineering The realization of infinite and finite Markov sequences for multi dimensional systems is considered, and an efficient algorithm to extract the invariants of the sequence under a change of basis in the state space is developed. Knowledge of these invariants enables the deter mination of the corresponding canonical form, and an invariant system description under this transformation group. For the partial realization problem, it is shown that this algorithm possesses some attractive nesting properties. If the realization is not unique, the class of all possible solutions is found. The stochastic version of the realization problem is also examined. It is shown that the transformation group which must be considered is richer than the general linear group of the deterministic problem. The invariants under this group are specified and it is shown that they can be determined from a realization of the measurement covariance sequence. Knowledge of these invariants is sufficient to specify an invariant system description for the stochastic problem. The link between the vii realization from the measurement covariance sequence, the white noise model and the steady state Kalman filter is established. vm CHAPTER 1 INTRODUCTION Special state space representations of linear dynamic systems have long been the motivation for extensive research. These models are generally used to simplify a problem, such as pole placement, by introducing arrays which require the fewest number of parameters while exhibiting the most pertinent information. In general, system represen- . \ ' tations have been studied in literature as the problem of determining canonical forms; Canonical forms have been used in observer design, exact model matching methods, feedback system design, and Kalman filtering techniques. In realization theory, canonical forms for linear multi- variable systems are important. Since it is only possible to model a system within an equivalence class, the ability to characterize the class by a unique element is beneficial. The problem of determining a canonical form has its roots in invariant theory. Over the past decade many so-called "canonical" system representations have evolved in the literature, but unfortunately these representations were obtained from a particular application or from computational considerations and not derived from the invariant theory point of view. Generally, these representations are not even unique and therefore cannot be called a canonical form. Representations derived in this manner have generally been a source of confusion as evidenced by the ambiguity surrounding the word canonical itself. In this dissertation 1 2 we follow an algebraic procedure to obtain unique system representations, i.e., we insure that these representations re in fact canonical forms. In simple terms this approach seeks the determination of certain entities called invariants obtained by applying particular transformation groups (e.g., change of basis in the state space) to a well-defined set representing a system parameterization. The invariants are the basic structural elements of a system which do not change under this trans formation and are used to specify the corresponding canonical form. This approach insures that the ambiguities prevalent in earlier work are removed. Initially, we develop a simple solution to the problem of determining a state space model from the unit pulse response of a given linear systerr and then extend these results to the stochastic case where the system is driven by a random input. The technique developed to extract the invariants from this (response) sequence not only provides a simple solution to the realization problem, but also gives more insight into the system structure. 1.1 Survey of Previous Work in Canonical Forms for Linear Systems The study of canonical forms for linear dynamic systems evolved slowly in the Sixties. The main impetus of investigation was initiated by Kalman (1962,1963) when he compared two different methods for describing linear dynamic systems: (1) internally by the state space representation denoted by the triple (F,G,H), or (2) externally by the transfer function--the input/output description. Development over the past decade in such areas as optimal control, decoupling theory, estimation and filtering, identification theory, etc., have relied heavily on the _ This defines (simply) the realization problem. 3 state space representation for analysis and design. In early literature, however, transfer function representations were used. For highly complex systems it is much easier to determine external behavior rather than internal, since the state variables are normally not available for measurement. As pointed out by Kalman (1963) the language of these representations may be different, but both describe the same problems and are related. Many researchers have investigated the relationship between both representations, but always with one common goalto obtain a state space model which specifies the external description directly by inspection. Kalman (1963) and later Johnson and Wonham (1964), Silverman (1966) have shown that there exists a canonical form (under change of basis in the state space) in the scalar case for the triple (F,g,h) where F is in companion matrix form (see Hoffman and Kunze (1971)) and g is a unit column vector. It was shown that there exists a one to one correspondence between the non-zero/non-unit elements of the triple and the transfer function. This representation was used by Bass and Gura (1965) to solve the pole-positioning problem and recently by Wolovich (1972b)in solving the exact model matching problem. , The progress in determining a canonical form for the internal description of multivariable systems came more slowly. The earliest work appears to be that of Langenhop (1964) in which he develops a representation to study system stability. Brunovsky (1966,1970) was probably the first to recognize the invariant properties of the canonical form for the controllable pair (F,G). Tuel (1966,1967) developed canonical forms for multivariable systems in his investigation of the quadratic optimization problem. Subsequently, Luenberger (1967) proposed certain sets of canonical forms for controllable pairs; however, his development allowed 4 the possibility of nonuniqueness of these representations. Buey (1968) extended the results of Langenhop and Luenberger when he developed a canonical form for certain subclasses of observable systems, but he too was unaware of its invariant properties. Proceeding from the external system description many researchers began to realize the usefulness in the development of canonical forms. Popov (1969) developed a canonical form for the transfer function in his investigation of irreducible system representations. Gilbert (1969) examined the invariant properties of a system with feedback applied to solve the decoupling problem. Dickinson et al. (1974a) discuss the construction and appli cation of these canonical forms for the transfer function matrix in a recent survey. The properties of canonical forms were not fully understood initially. In fact, the basic question of their uniqueness posed many doubts as .to their usefulness. This issue wasn't resolved until the work of Rosenbrock, Kalman, and Popov in the early seventies. The properties of the Luenberger forms were clarified by the results of Rosenbrock (1970) and Kalman (1971a) in their studies of the minimal column indices (or Kronecker indices) of the matrix pencil [Iz-F,G], or more commonly, the indices of the pair (F,G). These indices were shown to be invariants under the following transformations: change of basis in the state space, input change of basis, and state feedback. These results precisely resolve the question of what can (or cannot) be altered by applying feedback to a linear multivariable system. At the same time Popov (1972) examined the properties of the controllable pair (F,G) under the same transformations in a very precise, step-by-step, algebraic procedure to.determine the corresponding invariants. He shows clearly that obtaining the invariants under a particular transformation 5 group is the only information required to specify the corresponding canonical form. Wonham and Morse (1972) obtained the feedback invariants of the controllable pair from the not as lucid geometric viewpoint. Their results were identical to those of Brunovsky and ftosenbrock. : Morse (1973) examined the invariants of the triple (F,G,H) under a larg group of transformations which includes output change of basis. A complete set of feedback invariants of this triple still remains an open problem, but some fragmentary results were presented by Wang and Davison (1972) when they investigated certain sets of restricted triples. Along these lines Rissanen (1974), Caines and Rissanen (1974), . Mayne (1972a,b),Weinert and Anton (1972), Tse and Weinert (1973,1975), Glover and Willems' (1974) examined the identification problem from the invariant theory viewpoint and obtained some rather interesting results. Recent results in decoupling theory were obtained by Warren and Eckberg (1973), Concheiro (1973), and Forney (1975) using the Kronecker invariants Probably the most extensive survey of these results has been compiled by Denham (1974) and we refer the interested reader to this paper. We temporarily leave this area to consider one specific application of these resultsthe realization problem. 1.2 Survey of Previous Work in Realization Theory The first realization problem proposed for control systems was the determination of a state space model (internal description) from a given transfer function (external description). Gilbert (1963) and Zadeh and Desoer (1963) describe realization procedures based on the determination of the rank of the residue matrices of the given transfer function matrix, but unfortunately these procedures only apply to the 6 case of simple poles. Kalman (1963) proposed an algorithm whereby the given transfer function is realized as a parallel combination of single input, controllable subsystems in companion form, and then applied the "canonical structure theory" (Kalman (1962)) to delete the uncontrollable dynamics. This technique handles simple as well as multiple transfer function poles. Later Kalman (1965) showed the equivalence of the realization problem of control theory to the corresponding network theory formulation. A significant advance in realization theory was given by Ho and Kalman (1966). They showed that the state space model could be found from the impulse response sequence provided the system under investi gation is finite dimensional. They also developed an algorithm based on forming the generalized Hankel array from the given sequence and then extracted the state space triple from it. Shortly after the pub lication of Ho's algorithm, Youla and Tissi (1966) working in network synthesis and Silverman and Meadows (1966) in control theory developed similar realization techniques again based on the impulse response sequence. Ho's algorithm gave new impetus to realization theory. Several authors have provided alternate or improved realization algorithms based on the Hankel array formulation. Mayne (1968), Panda and Chen (1969), Roveda and Schmid (1970), Rosenbrock (1970), Lai et al. (1972) and even more recently Huang (1974), Rozsa and Sinha (1975) among others, considered the older transfer function matrix approach, while Rissanen (1971,1974), Silverman (1971), Ackermannand Buey (1971), Chen and Mita! (1972), Mita! and Chen (1973), and Bonivento et al. (1973) approached the problem from the Hankel array formulation. 7 Rissanen (1974), Furata and Paquet (1975), Roman (1975), Dickinson et al. (1974a,b) have recently considered the problem of realizing a given infinite impulse response matrix sequence with a polynomial matrix pair. Such a pair is referred to as a matrix- fraction description of the system and is becoming well known in control literature largely due to the ground work established by Popov (1969), Rosenbrock (1970), Wolovich (1972a,b, 1973a,b) and others. Kalman (1971b), Tether (1970), and Godbole (1972) later considered the more realistic case where only a finite number of terms of the impulse response sequence are specified. This is commonly known as the partial realization problem and corresponds in the scalar case to the classical Pad approximation problem. Generally most realization altorithms can be used to process partial data, but usually at a loss of efficiency and even more seriously the possibility of yielding misleading results. A wealth of new techniques have recently been published to handle this very special, yet realistic variant of the realization problem. Rissanen (1972a,b), Ackermann (1972), Dickinson et al. (1974a), Roman and Bullock (1975a), Anderson et al. (1975) published some efficient and improved algorithms to solve this problem. Also of recent interest is the development of algorithms which realize a system directly in a canonical form (under a change of basis in the state space), i.e., algorithms which solve the canonical realiza ti on problem. The algorithms of Ackermann (1972), Bonivento et al. (1973), Rissanen (1974), Dickinson et al. (1974a), Rozsa and Sinha (1975), Luo (1975), and Roman and .Bullock (1975a) solve this problem. 8 One of the main contributions of this dissertation is to use the results developed from invariant theory to solve the realization and partial realization problems in the deterministic as well as stochastic cases. The realization of a system directly in a canonical form actually reduces to first determining which transformation groups are present, specifying the corresponding invariants, and then developing a method to 4* extract these invariants from the given unit pulse response sequence. This philosophy is basic to any canonical realization scheme and actually provides an explicit formula which is applied throughout this dissertation. In the last few years, several interesting extensions have emerged from the original concept of realization theory. The major motivation evolved just after the development of the Kalman filter (see Kalman (1961)) in estimation theory because a priori knowledge of the state space model and noise statistics are required to begin data processing. The link between the filtering and realization problem was established by Kalman (1965) just prior to the advent of Ho's algorithm. The work of Gopinath (1969), Budin (1971,1972), Bonivento et al. (1973), and Audley and Rugh (1973,1975) were concerned with the more general problem of, obtaining a state space representation given a general input/output sequence of the system in both deterministic and stochastic cases. The stochastic version of the realization problem has not received quite as much attention as the deterministic case mainly due to its greater complexity and high dependence on the adequacy of covariance estimators. The realization of stochastic systems was studied by Faurre (1967,1970) and more recently by Rissanen and Kailath (1972), Gupta and Fairman (1974) 4* * The Hankel array formulation is used exclusively in this dissertation.- 9 and Akaike (1974a,b). From the transfer function viewpoint this problem has been solved using spectral factorization as originally introduced by Wiener (1955,1959) and studied by others such as Gokhberg and Krein (1960), Youla (1961), Davis (1963), Motyka and Cadzow (1967), and Strintzis (1972). The link between the stochastic realization problem and spectral factorization evolved from the work in stability theory by Popov (1961,1964), Yakubovich (1963), Kalman (1963), Szego and Kalman (1963). The equations establishing this link were derived in the Kalman-Yakubovich-Popov lemma for continuous systems and the Kalman- Szego-Popov lemma for discrete time systems. Newcomb (1966), Anderson (1967a,b,1969), and Denham (1975) extended these results and provided techniques to solve these equations. Defining the invariants of these problems is still an area of active research as evidenced by the recent work of Denham (1974), Glover (1973), and Dickinson et al. (1974b). This is one area developed in this dissertation. It will be shown that the invariants of the stochastic realization problem not only lends more insight into the structure of the problem, but also yields some new results. Research in realization theory and its applications continues as evidenced by the recent results of Rissanen (1975) in estimation theory, Ackermann (1975) in feedback system design,De Jong (1975) in the numerical aspects of the problem and Roman and Bullock (1975b) in observer theory. The results presented in this dissertation tie together some previously well-known results in stochastic realization and filtering theory as well as provide a technique which can be used to study other problems. 10 1.3 Statement of Purpose and Chapter Outline It is the purpose of this dissertation to provide an extensive discussion of the realization problem in both the deterministic and stochastic cases as well as specify the invariants under particular transformation groups in each case. It is also desired to develop a simple and efficient algorithm to solve the canonical realization problem This algorithm is to be modified to process data sequentially such that only the pertinent informationr-the invariants, are extracted from the given sequence. In the case of a fixed finite unit pulse response sequence (the partial realization problem), the solution is to be obtained such that all possible degrees of freedom are specified. The relationship between the stochastic realization and steady state Kalman filtering problems are discussed by again examining the corresponding invariants. In so doing, a considerable amount of knowledge about the existence and structure of realizations and the steady state filter is gained. The basic theoretical essentials of realization and invariant theory are reviewed in Chapter 2. A "formula" essentially outlined in Popov (1973) and Kalman (1974) is developed which will be applied to various realization problems throughout the text. Some new theoretical results in canonical realization theory are established and used to develop a new canonical realization algorithm. In Chapter 3 the algorithm is modified to handle sequentially the case of partial data and also that of a fixed finite sequence. New results evolve which completely characterize the class of all minimal partial realizations and extension sequences as well as determining the characteristic equation in a simple manner. 11 The stochastic case of the canonical realization problem is in vestigated in Chapter 4. A complete set of independent invariants is found to characterize the corresponding solution. Equivalent solutions to this problem as well as to the steady state Kalman filtering problem are studied and it is shown that the filter parameters can be specified by solving an analogous realization problem. The specific contributions of this research and further research possibilities are outlined in Chapter 5. Examples are used generously throughout this work to illustrate the various algorithms discussed and to point out significant details that are otherwise difficult to see. A comment on notation to be used through out this dissertation cl oses0this chapter. 1.4 Notation Uppercase letters denote matrices, and vectors are represented by underlined lowercase letters. Lowercase letters are used to represent scalars and integers. All matrices and vectors appearing in this work are assumed to be real and constant. An = [a. is an nxm matrix with m L lj m elements a.. .; 0^ is the nxm null matrix with row and column vectors T T given by 0^ and 0^; In represents the nxn identity matrix, and e. or e. stands for its j-th row or j-th column; jqn means j=T,2,...,m. vi \ CHAPTER 2 REALIZATION OF INVARIANT SYSTEM DESCRIPTIONS In this chapter we present a brief review of the major results in realization theory. We establish a basic "formula" and apply it to various system representations. It is shown that this approach greatly simplifies the realization problem. Two new algorithms for realization are developed which appear to be more efficient than previous techniques because they extract only the minimal information necessary to specify a system from the given input/output sequence in an extremely simple fashion. All of the essential theory is developed and a multivariable example is presented. 2.1 Realization Theory A real finite dimensional linear constant dynamic system has internal description given by the state variable equations in discrete time as, ^<+1 + GJk (2.1-1) where keZ+, xeKn=X, uÂ£Km=U, yeK^Y and F, G, H are nxn, nxm, pxn matrices over the field K. X,U,Y are the state, input, and output spaces, respectively. 12 13 The external system description may be given either in rational form as, T(z) = H(Iz-F)~^G (z complex) (2.1-2) or equivalently as an infinite matrix power series T(z) = Z A zk (2.1-3) k=l K where the sequence {A^} is the unit pulse response or Markov sequence of (2.1-1). The Markov parameters are Ar> HF^G k=l ,2,... (2.1-4) The problem of determining the internal description (F,G,H) from the external description (T(z) or {A^>) is the realization problem. Out of all possible realizations, Z:=(F,G,H) having the same Markov parameters, those of smallest dimension are minimal realizations. Prior to stating some of the significant results from realization theory several useful definitions will be given. The j-control!ability and j-observability matrices are the nxmj and pjxn arrays, W, [G | ... | Fj_1G] and v! = [HT | ... | (HFj"1)T]. The pair (F,G) J J is completely controllable if p(Wn)=n and the pair (F,H) is completely observable if p(Vn)=n. Throughout this dissertation we will only be concerned with systems possessing these properties. For a completely controllable and observable system, the controllability index, y, and the observability index, v, are the least positive integers such that the rank of W and V is n. y v 14 If two minimal realizations Z, t are equivalent under a change of basis in X, then there exists a nonsingular T such that (F,G,H)^ = (TFT\tG,HT_1 ). It also follows by direct substitution that the controllability and observability indices of these realizations are identical and W. = TW. for j = 1,2,... J J V. = V.T"1 for i = 1,2,... The generalized NxN' block submatrix of the doubly infinite Hankel array is given by SN,N' = Implicit in the realization problem is determining when a finite dimensional realization exists and, if so, its corresponding minimal dimension. The following proposition by Silverman gives the necessary and sufficient conditions for {A^} to have finite dimensional realiza tion. Proposition. (2.1-5) An infinite sequence {A^} is realizable iff there exist positive integers y,v,n such that otVuu+j) Further, if {A^} is realizable, then p,v are the controllability and observability indices and n is the dimension of the minimal realization. ^This notation means F = TFT"\ G = TG, and H = HT ^. AN' W-l 15 Proof. See Silverman (1971). Note that the essential point established in Ho and Kalman (1966), which is used in the proof of the above proposition is that Z is a minimal realization iff it is completely controllable and observable. Since $ *V W it follows for dims n that: p(S ) min[p(V ),p(W )]=n. v,p v y v,y v y This result is essential to construct any realization algorithm. In (2.1-5) the crucial point of finite dimensionality is carefully woven into necessary and sufficient conditions for an infinite sequence to be realizable. What if only partial information about the system is available in the form of a finite Markov sequence? Is this sequence realizable? What is the relationship between the minimal realization and one based only on partial data? These are only a few of the questions which must be resolved when we are limited to partial data. Intrinsic in the realization from a finite Markov sequence is the fact that enough data are contained in S to recover the infinite a v,y sequence, i.e., knowledge of (A^,...,A is sufficient to determine {Ak>, k-1,2,... But in reality the only way to be sure of this is knowledge of the actual system dimension (or at least an upper bound). A minimal partial realization is a realization of smallest dimension determined from a finite Markov sequence {A^},keM_ which realizes the sequence up to M terms. The order of the partial realization is M and the realization is denoted by Z(M). The realization induces an extension k-1 of {Ak>, i.e., Ak=HF G for k>M. The following basic result analogous to (2.1-5) answers the realizability question when only partial data are given. For a proof, see Kalman (1971). 16 Proposition. (2.1-6) (Realizability Criterion) The minimal partial realization problem of order M possesses a solution, Â£(M) iff there exist positive integers * v,y, M = \H-y where dimE(M) = p(S^ ) = n. In this proposition (R) is designated the rank condition. Also, it is important to note that when (R) is satisfied the minimal extension (of 2(M)) is unique (see Tether (1970) for proof), but S(M) is not unique because there exist other minimal partial realizations equiv alent to S(M) under a change of basis in X. We must consider three possible cases when only partial data is * available. In the first case enough data is available such that M>M for known n; thus, a minimal realization is found. Second, v and y are available such that (R) is satisfied. In this case a minimal par tial realization can be found, but this in no way insures it is also a minimal realization of the infinite sequence, since the rank of S v may increase as v,y increase. Third, the rank condition does not hold How can a realization be found when no more data is available? The only possibility in this case is to extend the sequence until (R) is satisfied, but there can exist many extensions satisfying (R) while giving nonminimal realizations. For this reason define a minimal extension as any that corresponds to a minimal (partial) realization. To obtain minimality we must somehow select the right extension among the many possible. 17 Prior to summarizing the main results of Kalman (1971) and Tether (1970), define the incomplete Hankel array associated with a given partial sequence {A^}, keM. as where the asterisks denote positions where no data is available. The rank of S(M,M) is the number of linearly independent rows (columns) determined by comparing only the data specified elements in each row (column) with the preceding rows (columns) with the cognizance that upon the availability of more data this number can only remain the same 4* or increase. Thus, the rank is a lower bound for any extension when the * are filled in-consistent with the preservation of the Hankel pattern. Both Kalman and Tether show that there are three pertinent integers associated with the incomplete Hankel array. They are defined as: n(M), v(M), y(M) and correspond to the rank of S(M,M), the observability index, and the controllability index of the given data. The latter two are lower bounds (separately) for v and y. Knowledge of either v(M), or y(M) enables us to construct extensions, since they are the least integers such that (R) holds for all minimal extensions. It should also be noted that the integers n,v,y,... are actually non-decreasing functions of the amount of data available, M, and should be written, n(M), v(M), y(M) etc. to be precise. However, the argument ^It also follows from this that the p(S(M,M)) is a lower bound for dim 2 (see Kalman (1971)). 18 M will be understood throughout this dissertation in order to maintain notational simplicity. There is one more variant of the partial realization problem that must be considered. A sequence of minimal partial realizations such that each lower order realization is contained in one of higher order will be called a nested realization. Symbolically, this is given by ...-E(M)-S(M)-... for M to this problem is most desirable from the computational viewpoint, since each higher order model can be realized by calculating just a few new elements in the corresponding realization. Rissanen (1971) has given an efficient recursive algorithm to determine this solution. Another related problem of interest is determining a unique member of equivalent systems under similarity and is discussed in the following section. 2.2 Invariant System Descriptions In this section we review some of the fundamental ideas encountered when examining the invariants of multivariable 1 inear systems. The framework developed here will be used throughout this dissertation in formulating and solving various realization problems. Not only does this formulation enable the determination of unique system representations under some well-known transformations, but it also provides insight into the structure of the systems considered. First, we briefly define the essential terminology and then use it to describe some of the more common sets of canonical forms employed in many recent applications (e.g., Roman and Bullock (1975a,b), Tse and Weinert (1975)). 4 For any two sets X and Y, a subset R c X x Y is called a binary relation on X to Y (or, a relation "between" X and Y). Then (x,y)eR is usually written as xRy and is read: "x stands in the relation R to y". If for X=Y this relation is reflexive, symmetric, and transitive, then it is an equivalence relation E on X given by xEy for x,yeX. The set of all elements z equivalent to x is denoted by E(x),= {zeXjxEz} and is called the equivalence class or orbit of x for the equivalence relation E. The set of all such equivalence classes is called the quotient set or orbit space and is given by X/E. Thus, the relation E of X partitions the set X into a family of mutually disjoint subsets or orbits by sending elements which are related into the same equivalence class. ff Consider a fixed group G of transformations acting on a set X. Then the elements Xj,Xg of X are equivalent under the action of G iff there exists a transformation TeG which maps x-j into.Xg.. This is basically the "formula" we will apply throughout, i.e., we first formulate the set of elements (generally the internal system description), then define a transformation group; and finally determine the orbits under the action of G. To be more precise, let us first define the function f mapping a set X into Y as an invariant71^for E if for x-j^eX, x^Ex^ implies f(Xi)=f(x2). In addition if f.(xi)asf(Xg) implies x-jExg, then f is a X ^ ~ This is the standard Cartesian product, XxY = {(x,y)|xeX, yeY} . .Here we mean "group" in the standard algebraic sense, i.e., (G) where G is a closed set of elements each possessing an inverse and the identify element; 0 is an associative binary operation. When o is understood, the group is merely denoted by G. 4.4-4. Note that an invariant is actually a function, but common usage refers to its image as the invariant. We will also use this terminology throughout this dissertation. 20 complete invariant. In general we will be interested in a complete system of invariants for E given by the,set of invariants (f^} where 4* f : X + Y1xY2X. xYn> f. is an invariant for E, and f-j (x^i (xg) * > ffi(X1)~fn(x2) imPly x]Ex2* Completeness of this set of invariants means that the set is sufficient to specify the orbit of x, i.e., there is a one to one correspondence between the equivalence classes in X and the image of f. If the set of complete invariants is independent, then the map f: X+Y-jX.. ,xYn is surjective. This property means that corresponding to every set of values of the invariants there always exists an n-tuple in Y specified by this set. A complete system of independent invariants will be called an algebraic basis. Generally, we consider a subset of X (e.g., in system theory a controllable system). Correspondingly, let f be a function mapping the subset XQ of X into set Y, then f is a restriction of f if fQ(x)=f(x) for each xeXQ. We can uniquely characterize an equivalence class E(x) by means of the set of values of the functions f.(x), ien. where the {f..} constitute a complete set of invariants for E on X. If the corresponding complete invariant f is restricted such that its image is itself a subset of X, then we have specified a set of canonical forms C for E on X. To be more precise, a canonical form C for X under E is a member of a subset C<=X such that: (1) for every xeX there exists one and only one ceC for which xEc, and since C is the image of a complete invariant f, then (2) for any xeX and c-j, C2eC, xEc^, and xEc2 implies f(x)=f(c-|)=f(c2)=c-j=C2 (invariance); (3) for any ceC if f(x-|)=c and f(x2)=c, then x-|Ex2 (completeness). Thus, c=f(x) is a unique member of ^This notation is actually f=(f-|,... ,fn) :x-^Y^x.. .xYn, but it is shortened when the set {f.} is clearly understood. 21 E(x) for every xeX. With these definitions in mind, our "formula" becomes (i)Formulate the set of elements; (ii)Define the transformation group; (iii)Determine a set of complete invariants under this transformation group; and (iv)Develop the canonical form in terms of the corresponding invariants. (2.2-1) We now apply (2.2-1) to various restricted sets related to multivariable systems. This approach is essentially given in Kalman (1971a), Popov (1972), Rissanen (1974), or Denham (1974). In this sequel we review the main results of Popov. First, define the set of matrix pairs (F,G) as XQ = i(F,G)|FeKnxn, GeKnxm; (F,G)controllable} The general linear group, which corresponds to a change of basis in the state space, is specified by the set GL(n):= {T|TeKnxn; det T?0} (2.2-2) with the group operation standard matrix multiplication, i.e., T o T = T T. In order to determine the orbits of XQ under the action of GL(n), it is first necessary to specify the action operator T + (F,G):= (TFT-1,TG) 4* In general the problem of determining a canonical form is quite difficult. However in this dissertation we consider restricted sets which make the problem much simpler. For a thorough discussion of this problem see Kalman (1973). 22 or alternately we can say that the action of GL(n) on XQ induces F TFT"1 G + TG The action of GL(n) induces an equivalence relation on XQ. We indicate (F,G)Ej(F,G) if there exists TcGL(n) such that (F,G)=Tt(F,G). Dual results are defined for the observable pair (F,H) and the A. analogous set denoted by XQ. The third step of (2.2-1) is established in Popov (1972), but first consider the following definitions. For a controllable pair (F,G) 4*J* define the j-th controllability index y., jem as the smallest positive integer such that the vector F Jg. is a linear combination of J 4 y its predecessors, where a predecessor of F g. is any vector F g^ where J * rm+s we have assumed p(G) = m. Throughout this dissertation we use the following definition of predecessor independence: a row or column vec tor of- a given array is independent if it is not a linear combination of its regular predecessors. The following results were established by Popov (1972) Proposition. (2.2-3) (1) The regular vectors are linearly independent; (2) The controllability indices satisfy the m following relationship, E y. = n; (3) There exists n=l J exactly one set of ordered scalars, a^cK defined for jem, kej-1, s = 0,1,...,min(pj,pk-T) and for jem, k = j,...,m,s = 0,l,...,min(y.,y(c) 1 such that + Throughout this dissertation we use the overbar on a set to denote the dual set. ^These indices are also called the Kronecker indices. 23 yi F Jg. *= E J k=l J-l nnn(y ,y.-1) " 1 % + z s=0 JKS K k=j m min(y.,yk)-l JE s=0 ajksF gi This proposition follows directly from the controllability of (F,G) and indicates that the regular vectors form a basis where the a's are the coefficients of linear dependencies. The set [{yj},{ctjks}], j,kem, s*0,...,y.-1 are defined as the controllability invariants of (F,G), J and y=max(y.). The main result of Popov is: J Proposition. (2.2-4) The controllability invariants are a complete set of independent invariants for (F,G)eX under the action of GL(n). The proof of this proposition is given in Popov (1972) and consists of verifying the invariance, completeness, and independence of [(y^},-Cctjj then can be replaced by TFsgk in the given recursion and the controllability invariants remain unchanged. Completeness is shown by constructing a TeGL(n) such that for two pairs of matrices (F,G), (F,G)eX0 with identical controllability invariants, (F,G) = (TFT""^, TG) or (F,G)Ej(F,G). Independence of the controllability invariants is obtained by constructing a canonical form determined only in terms of these invariants. Thus, by introducing a finite set of indices (y^}, Popov shows that this set along with the {ajks} are invariants under the action of GL(n). The main reason for specifying a set of complete and in dependent invariants is that it enables us to uniquely characterize the orbit of (F,G). It should also be noted that dual results hold for the observ able pair (F,H), and it follows that the observabi1 ity invariants are the 24 set [{v.-LB.j^}], i,seÂ£,. where the {v..} are the observa- bi1ity indices. The last step of (2.2-1) is to specify the corresponding canonical forms under GL(n). These forms are commonly called the Luenberger forms and are specified by the controllability and observability invariants. They are defined by the pairs (Fq.Gq), (FrjHr) where the subscripts C,R reference the fact that the regular vectors span either the columns of W P+1 or the rows of V v+1 (2.2-5) 3 l P- Jem s=l s F R J H R (2.2-6) P T r. = Z v ieÂ£ 1 s=l s 25 where g^., 3.j are n column, n row vectors containing {cu^}, {3..^} respectively over appropriate indices and zeros in the other places. Luenberger (1967) shows that the transformation, T^, required to obtain the pair (F^.G^) is determined from the columns of Wn, as where TC*T1 T2 <2-2-7> Tj =t9j Fgj F J_1gj] jera p(G) = m, and g. is the j-th column of G. vl Similar results hold for the pair and is specified by constructed from the rows of V . n Unfortunately Luenberger (1967) in attempting to develop multi- variable system representations did not determine the invariants under GL(n). It is essential to use the approach outlined in (2.2-1) in order to obtain the corresponding canonical forms or else it is possible to obtain erroneous results. The following example due to Denham (1974), shows that the Luenberger form, as originally stated is not canonical. If we are given the pair (F,G) as "o 0 1 1 1 Â¡ 1 ~i o" 1 1 0 2 1 ! 1 0 0 F = G = 0 1 2 S 1 1 l 0 0 _0 0 1 1 l ! 1j _0 1_ 26 These matrices are in the form of (2.2-5), but it is easliy verified by constructing that the controllability invariants are in fact Pl=2, ^=2 and a-j = [-1-1 -1 1]I ol, = [-2 0 -2 4]I The problem with the Luenberger forms is that the maps it: Xq-* Xq/E are not well defined. Thus, the image of the maps are indeed canonical forms, but as shown here for (F,G)eXq/E, we need not have tt(F,G)=(F,G), i.e., the mapping does not leave the canonical forms unchanged. The point to remember is that the invariants are the necessary entities of interest which must be determined. 4* The procedure to construct the transformation matrix Tq of (2.2-7) is called the Luenberger second plan. The first Luenberger plan consists of examining the columns of i^n, given by V ... F--'g, F"'\l '2-2-8> where is an nmxnm permutation matrix, for predecessor independence. Thus, we can define a new set of invariants (under GL(n)) [{f^.}, ^jks^ completely analogous to the controllability invariants. The canonical forms associated with the invariants obtained in this fashion have 4* This procedure amounts to examining the column vectors of Wft for predecessor independence, i.e., examine g-j ... gm Fg^ ... Fgm . . 27 become known as the Buey forms which were derived directly from the results of Langenhop (1964), Luenberger (1967), and Buey (1968). We refer the interested reader to these references as well as the recent survey by Denham (1974). Here we will be satisfied to note that the procedure of (2.2-1) applies with the set of controllable pairs (F,G) restricted to the {y.} invariants rather than {y.}. Analogous to the J J Luenberger forms, we define the row and column Buey forms as (^dr^br)* (Fbc>GBc) respectively. The row form is given by L11 fbr= L21 L22 0 > HBR = T v +1 V] + l Lpl Lp2 ... L PP + +% +i VI -^Vp.f (2.2-9) where L.. n I V T. U 'Xi 'o>r 'vr 8.. v. > 0 and satisfy E v =n ; 1 s=l 5 a. ^ KI.. are v.,v-, row vectors containing (3. invariants. 1 J! 1 SC n ij The transformation, TRB,required to obtain the pair (FgR>HBR) is ' BR 'BR [T 1 (2.2-10) where tI = [hT(h_.F)T D.j I I Vl T (h,F 1 ) ], ieÂ£ 28 The importance of the Buey form is that the characteristic equation can JL be found by inspection of the block diagonal arrays of FBR Since FBR is block lower triangular, the characteristic equation is given as Xp (z) = det(Iz-FpR) = Xj (z)...x, (z) (2.2-11) hBR bK L11 Lpp where the L.. are the companion matrices of (2.2-9). Similar results hold for the pair (Fg^.Gg^,) and the transformation is specified by TB(, constructed from the columns of W . n This completes the discussion of invariants and canonical forms for controllable or observable pairs. To extend these results to matrix triples (internal system description), it is more convenient to examine ft an alternate characterization of the corresponding equivalence class the Markov sequence of (2.1-4). This approach was used by Mayne (1972b) and Rissanen (1974), in order to determine the orbits of Z under GL(n). It is obvious that the sequence is invariant under this group action A. = (HT^MTFT-VVtG) = HF'3'1G (2.2-12) Consequently every element of A. can be considered an invariant of Z J with respect to GL(n); therefore, two systems which are equivalent under GL(n) possess identical Markov sequences. The converse is also true, i.e., any two systems with identical Markov sequences are equivalent. The standard approach to investigate a system characterized by its Markov sequence is to form the Hankel array, N, where we define sT , + It should be noted that the Buey form is not a canonical form if the transformation group includes a change of basis in either input or output spaces, while the Luenberger form is still a canonical form. 29 ieN and S jeN1 as the block rows and columns of SM and the J IN5I1 block column and row vectors, a or a^ denote the r-th column of si or the s-th row of S for remN1, sepN. Rissanen (1974) has shown 9 that by examining the set X, = {Â£ I I controllable and observable with {y.} invariants} I J under the action of GL(n) that Proposition. (2.2-13) The set of controllability invariants and block column vectors, [{yj>,-Caj| appropriate indices constitute an algebraic basis for any ZeX^ under the action of GL(n). The proof of this proposition is given in Rissanen (1974) and consists of showing that any two members of X^ with identical Markov sequences are equivalent under GL(n). Thus, invariance follows by showing that a dependent column vector of the Hankel array can be uniquely represented in terms of the set [{y.},{a.. _}]. These parameters remain unchanged J JKS under GL(n); therefore, they are invariants. The block column vectors, a t satisfy a recursion analogous to (2.2-3), i.e., n-1 min(yJ.,y[<-l) m min(y.,uk)-l z k=l Z o .1 a , + Z jks .j+ms s=0 k=j ajksa.j+ms s=0 Thus, all dependent block columns can be generated directly from the set, {a of regular block column vectors. These vectors are invariants under GL(n), since every column vector of A. is an invariant as shown in J (2.2-12). Completeness follows immediately from the above recursion, since any two members of X^ possessing identical invariants satisfy the above recursion and therefore have identical Markov sequences. 30 Independence is shown by constructing the Luenberger form of (2.2-5) and (2.2-14) below^ directly from these invariants. The dual result yields another basis on X,, [{v.},{g. .},{aT }]. I I I S L J The corresponding canonical forms for EeX-j or are given by the Luenberger pairs of (2.2-5,2.2-6) and and a' (ia]-1 )m+ l I (v^ljp+l. ] (2.2-14) and the canonical triples are denoted by and respectively. Rissanen (1974) also shows that a canonical form for the transfer function can be constructed from the invariants of (2.2-13). This is possible because the determination of canonical forms for Â£ based on the Markov parameters is independent of the origin of A^'s. Rissanen defines the (left) matrix fraction description (MFD) as T(z) := B"1(z)D(z) (2.2-15) v , where B(z) = z B.z for |B \f 0 i=0 1 v V-l 4 D(z) = I D.z1 . i=0 1 31 The relationship of the MFD to the Hankel array, Sv+^ ^follows by writing (2.2-15) as B(z)T(z) = D(z) (2.2-16) and equating coefficients of the negative powers of z to obtain the recursion BoAj + BlVl 'f + BvAj+v = mj j=1>2 expanding over j gives the relation over the block Hankel rows as [B0 B, ... Bu] where the pxp(v+l) matrix of B.'s is called the coefficient matrix of B(z). Similarly equating coefficients of the positive powers of z gives the recursion \ 1 1 5 * t Vi,. = 0. m(y+l) (2.2-17) D, = B, ,A, + Bli0A0 + ... + B A . k k+1 1 k+2 2 v v-k k=0,l,...,v-l or expanding over k gives the relation in terms of the first block Hankel column as Dv-1 Bv O 1 Dv-2 II Bv-1 Bv * o 1 _B1 Bn ... B 2 v (2.2-18) 32 The canonical forms for both left and right MFD's are defined by the polynomial pairs (BR(z),DR(z)) and ("Bc(z) ,Dc(z)) respectively, where R and C have the same meaning as in (2.2-5,2.2-6) and the former is given by 11 CL Izv ; bT e K^v (2.2-19) for 4 = [4(V-V,.) * * 6ki ek2 ... ek(i+pv.) I * where k=i+pvi and Bkj are given by {6ist} j=i,i+p,.. .i+P.(V|-l) * Bkjs< 0 j^i,i+p,.. .*i+p(v.-T) . 1 j=i+pvi and DR(z) is determined from (2.2-18). Dual results hold for the corresponding column vectors, b., jem of the J coefficient array of !q(z) in terms of the controllability invariants. This completes the discussion of canonical forms for Â£ or T(z). Note that analogous forms can easily be determined for the Buey forms if X, is restricted to {v.}. Henceforth, when we refer to an invariant . J system description, we will mean any representation completely specified by an algebraic basis. In the next section we develop the theory necessary to realize these representations directly from the Markov sequence 33 2.3 Canonical Realization Theory . * In this section we develop the theory necessary to solve the canonical realization problem, i.e., the determination of a minimal realization from an infinite Markov sequence, directly in a canonical form for the action of GL(n). Obviously from the previous discussion, this solution has an advantage over other techniques which do not obtain E in any specific form. From the computational viewpoint, the simplest realization technique would be to extract only the most essential information from the Markov sequence--the invariants under GL(n). Not only do the invariants provide the minimal information required to completely specify the orbit of Z, but they simultaneously specify a unique representation of this orbitthe corresponding canonical form. Thus, subsequent theory is developed with one goal in mind--to extract the invariants from the given sequence. The following lemma provides the theoretical core of the subsequent algorithms. Lemma. (2.3-1) Let and W^, be any full rank factors of = V^, Then each row (column) of is dependent iff it is a dependent row (column) of (W^,). Proof. From the factorization = V^, it follows if the j-th row of is dependent, then there exists an aTe:KpN, a^O such that T_ nT ~ N,N' -W Since p(WN,)=n, i.e., W^, is of full row rank, it follows that -TsN N' WN' = 34 or .V^-r T T T but det (W^, WN,} t 0; thus, a = 0^, i.e., a dependent row of is a dependent row of V^. Conversely assume that there . exists a nonzero aT as before such that v nT VN ^tnN' Since p(W^,)=n, it follows that this expression remains unaltered if post-multiplied by W^,, i.e., A/V 4- and the desired result follows immediately.V The significance of this lemma is that examining the Hankel rows (columns) for dependencies is equivalent to examining the rows (columns) of the observability (controllability) matrix. When these rows (columns) are examined for predecessor independence, then the corresponding indices and coefficients of linear dependence have special meaning-- they are the observability (controllability) invariants. Thus, the obvious corro!ary to this lemma is \ Corollary. (2.3-2) If the rows of the Hankel array are examined for predecessor independence, then the j-th (dependent) row, where j=i+pv., ieÂ£ is given by P + Z i-l min(v.,v -1) T 1 5 T Ij = 2 z s-1 t=0 p min(vi,vs)-l isre+pt s=l Z t=0 ^ist-s+pt whereig^^} an.d{v/} are the observability invariants and kepN is the k-th row vector of . 35 Proof. The proof is immediate from Proposition (2.2-3) and Lemma (2.3-1).V Note that similar results hold for the columns of the Hankel array when examined for predecessor independence. In the solution to some problems knowledge of both controllability and observability indices are required. Moore and Silverman (1972) require both indices to design dynamic compensators in order to solve the exact model matching problem. Similarly the requirement exists in the design of pole placement compensators and also stable observers as indicated in Brausch and Pearson (1970) and more recently Roman and Bullock (1975b). In an on-line application Saridis and Lobbia (1972) require the controllability invariants as well as the observability indices to solve the problem of parameter identification and control. The latter case exemplifies the fact that in some instances it is first necessary to determine the structural properties of a system from its external description prior to compensation. The need for an algorithm which determines both sets of controllability and observability invariants from an external system description is apparent. Computationally the simplest and most efficient technique to determine these invariants would be some type of Gaussian elimination scheme which utilizes elementary operations (e.g., see Faddeeva (1959)). If we perform elementary row operations on such that the predecessor dependencies of PV^ are identical to those of and perform column operations on W^, so that W^,E and W^, have the same dependencies then a examination of = PS^ niE is equivalent to the examination of . * We define ^ as the structural array of ^,. This array is specified by the indices {v^} and {y^} which are the least integers such that the row and column vectors of ^ are respectively, 36 nonzero i zero for ra=0;,... .a=Vj..N-1 for k=i+pa 'nonzero' for 4 b-0,... ,y -1 _ zero [b=y .,... ,N-1 J for s=j+mb These results follow since ^ has identical predecessor dependencies as SNjN,, then N.N' %>N IT where ^. = 0 if it depends on its predecessors. To find the observability indices, let a be the index of the last nonzero row of Â¡+pt t=0,l,...,N-1. T T Then if 6_. = jD v- = 0 otherwise = (a-i)/p+l. Similar results follow when is expressed in terms of the c^. The following theorem * specifies the matrices P and E required to obtain Theorem. (2.3-3) There exist elementary matrices P and E, respectively lower and upper triangular with unit diagonal elements, * such that N=PS^ ^(E has identical predecessor dependencies as Proof. Let PS^ M,=Q where Q is row equivalent to Â¡^i and P=[pr$]. If the j-th row of ^, is dependent on its predecessors, i.e., T T T V yi ;* then selecting P lower triangular such that 37 P rs gives this relation. From this choice of P it follows that dependent rows of are zero rows of Q. If the j-th row of SN is regular, then P unit diagonal-lower triangular insures that the corresponding row of Q is nonzero and regular. Similar results hold for the columns of ^, with E unit diagonal upper triangular. This choice of P does not alter the column dependencies of for if the i-th column of is dependent on its predecessors, then from Corollary (2.3-2) Â£. is uniquely represented as a linear combination of regular vectors in terms of the control- . lability invariants. Since P is unit diagonal-lower triangular, it is the matrix representation of a nonsingular linear transformation, Pr^q^. where q. is the i-th column vector of Q. Thus, multiplying on the left every vector Â£. in (2.3-2) with this P gives for i-j+mp. J 3-1 minivyy^l) q = Z Z k=l s=0 m min(u.,u.)-l Thus, we have shown that selecting P with the given structure does not alter the predecessor column dependencies of S^ or equivalently Q. Since the column vectors of Q satisfy the above recursion, Sf^ and Q have identical predecessor column dependencies, therefore, performing column operations on Q is * analogous to performing them on S^ and so we have SfJ . * (PS^ n,)E = QE or the predecessor dependencies of S^ N, and S^ M are identical.V 38 This theorem shows that the indices can be found by performing a sequence of elementary lower triangular row and upper triangular column operations in a specified manner on the Hankel array and examining the nonzero rows and columns of S^,, the structural array of The {cu^} and {BTjsare also easily found by inspection from the proper rows of P and columns of E as given by Corollary. (2.3-4) The sets of invariants or more compactly j the sets of n vectors {8.},{a.} are given by the rows of P and columns of E in (2.3-3) respectively as 4 CVV+P Pqr+P(vrl)] q'pV1 1>reÂ£ a. [e .e . -j st s+mt es+m(prl)t] j-5Â£a where Pqrest : q=r, r=s q Proof. The proof of this corollary is immediate from Theorem (2.3-3).V We can also easily extract the set of invariant block row or column vectors, {a! },{a } from the Hankel array and therefore, we have a J - > solution to the canonical realization problem. Theorem. (2.3-5) If the generalized Hankel submatrix of rank n is transformed by elementary row operations to obtain a row equivalent array, then by proper choice of P the matrix Q is given by: 39 TG | TFG ... j TFN'_1G 'V pn-M mN 0Pn-N _umN _ Q = where (F,G) is a controllable pair and det TVO. Proof. .If z is a minimal realization, then it is well-known that p(VN)=p(W^,)=n. Since P is an elementary array, then it follows [PV^] r N -I.. p= and det T^O. Post multiplication by W^, gives [G | FG | ... | PVV = _T__ pN- FN'"1G] = PS N,N1 Multiplication of the arrays gives the desired results.V Corollary. (2.3-6) If P*is selected such that Q is as in (2.3-5) with the pair (F,G) in Luenberger column form, then the set of invariants {a-}, jem is given by the columns of J Wl\|> > w^ kemN1 with ak = k=pjm+j Proof. If P is selected in Theorem (2.3-5) such that T=T^, then it follows that each column of W.,, corresponding to the (j+mp.)-th '* u for each jem contains the {a^} invariants.V t. The method of selecting P is given in the ensuing algorithm. 41 Theorem. (2.3-10) Given the infinite realizable Markov sequence from an unknown system, then SQ=(,rQ>GQ>H(.)n is a minimal canonical realization of {Ar} with 7C X Fc = [W, | H2 *_ w ] nr Gc is a submatrix of (Wu+i)r given by the first m u columns y HC ta.l a.l+m(y^-l) a ... a_ 1 . m mu ' Km and j fcj+m j+m vector of Vt ], jqn, Wr is a col umn Proof. Since the sequence is realizable, there exist, integers, n,v,y, satisfying Proposition (2.1-5). If Q is given as in Corollary (2.3-6), then Q = "Wk>c.r 0pv-" L mk _ i1""' for k>y+l Thus, Gc is obtained immediately from the first m columns of * (Wr)c. Form two nxn arrays, A and A each constructed by selecting n regular columns of (Wr)q starting with the first * column for A and the (1+m) column for A The independent columns of (Wr)q are indexed by the y. and satisfy (2.3-8); thus, they are unit columns and A is a permutation matrix, i.e,, A = [w, ... | w -1+m w 2m *' -j+miyj-l) ], jem 41 Theorem. (2.3-10) Given the infinite realizable Markov sequence from an unknown system, then SQ=([:Q>GcH(,)n is a minimal canonical realization of A^} with Fc [, | W2 WJ nr Gc is a submatrix of (W +1)c given by the first . columns . m HC = t\i 1+m(u1-1) ,m am ] ^m umn and Uj = C%+m Sj+rapj]- k fs a 1 vector of (W^+^)c. Proof. Since the sequence is realizable, there exist integers, n,v,y, satisfying Proposition (2.1-5). If Q is given as in Corollary (2.3-6), then Q = Gc 1 1 Fc Gc jr\ I! /"c" for k>y+l Thus, Gc is obtained immediately from the first m columns of (Wk)c* ^orm two nxn arrays ^ and A each constructed by selecting n regular columns of (W^)c starting with the first column for A and the (1+m) column for A The independent columns of (\)c are indexed by the y. and satisfy (2.3-8); thus, they are unit columns and A is a permutation matrix, i.e,, A = [w-, . I 1+m 2m w. j+m(yj-l) * 42 where it follows from (2.3-8) that the columns of A form chains satisfying -j+miUj-l)-* ~ e ] m for q .= Eu. . 3 s=l 3 * * Since A is A shifted m columns to the right, each chain of A is given by [w.j+m ... Wj+m^ ] and again each column is unit J T except Wj+miJ = aj from Corollary (2.3-6). Thus, := A A gives the matrix of (2.2-5). is obtained directly from HcCGe k-1 Fq Gq] = [a | ... a m a.l+m a.m(k+l) L since multiplication by the unit columns of (F^.G^J select the n columns of H^.V Analogous results hold for the dual ER. It should also be noted that if the Hankel array is transformed to and both rows and columns examined for predecessor independence as before, i.e., ?S N,N' % U = 'b V vnV (2.3-11) Of where is given in (2.2-8) and T is a permutation array, then all of the previous theory is applicable. The only exception in this case is that the Buey invariants and forms given by 3igR and liBC are obtained instead of the Luenberger forms. These results follow directly from (2.2-1). In many applications the characteristic polynomial xR(z) is required. Many efficient classical methods (e.g., see Faddeeva (1959)) exist to determine XR(z) from the system matrix. Even more recently some techniques have been developed to extract the characteristic polynomial 43 from the Markov sequence, but in general they are only valid in the cyclic case (see Candy et al. (1975)). An alternate solution to this problem is to obtain the Buey form and use (2.2-11) to find xp(z) by inspection. It is possible to realize the system directly in Buey form as mentioned in the previous paragraph, but in this dissertation we prefer to take advantage of the structure of the Luenberger form to construct Tg^ or Tgg. Superficially, this method does not appear simple because the transformation matrix and its inverse must be constructed, but the following lemma shows that Tgg can almost entirely be written by inspection from the observability invariants after the {v.} are known. is given by \ If the given triple is in Luenberger form, ZD, then the (v^xn) submatrices Tg are 'V > w v.-v. or T. v.>v. B V .-VI 44 where v.,jg are the observability invariants of ZR i\, v.Â¡ are the invariants associated with ZRR and recall r. = Z v, ro=0> Proof. This lemma is proved by direct construction of the TD 's, Since each T0 satisfies for v.*v. i i hi Bi* h.F v.-l i' R hiFR 'V , Vi-1 v.-l T then analogous to property (2.3-8), it follows that h.FD =e 1 K Â¥ 1 and therefore v.-l V- V.-l j TP hiFR - V1 vi Vv1 Â¥r ' iFR In order to construct TRR it is first necessary to find the {v..} from the rows of [Vn]R, but in this case the v.'s can generally be found by inspection while simultaneously building TRR. Also, TRR is generally a sparse matrix with unit row vectors; therefore, the inverse can easily be -1 1 found by solving M >n directly for the unknown elements of TRR. 45 jr\ In the next section we develop some new algorithms which utilize the theory developed here. 2.4 Some New Realization Algorithms In this section we present two new algorithms which can be used to extract both observability and controllability invariants from the given Markov sequence. Recall from Theorem (2.3-3) that performing row operations on the Hankel array does not alter the column dependencies, however, it is possible to obtain the row equivalent array, Q in a form such that the controllability invariants can easily be found. The first part of the algorithm consists of performing a restricted Gaussian elimination (see Faddeeva (1959) for details) procedure on the Hankel array. This procedure is restricted because there is no row or column interchange and the leading element or first nonzero element of each row is not necessarily a one. Define the natural order as 1,2,... . Algorithm. (2.4-1) (1) Form the augmented array: [IpN | S^, | ImN)] . (2) Perform the following row operations on N, to obtain Cp I Q I ImN'3: (i) Set the first row of Q equal to the first Hankel row. (ii) Search the first column of S^ ^, by examining the rows in their natural order to obtain the first leading element. This element is q^. (iii) Perform row operations (with interchange) to obtain q^-j =0,k>j. 4* Alternately it is possible to extract the Buey invariants from Q by reordering the columns as in (2.2-8) to obtain ()=QU and examining the columns for predecessor dependencies. 46 (iv) Repeat (ii) and (i i i) by searching the columns in their natural order for leading elements. (v) Terminate the procedure after all the leading elements have been determined. (vi) Check that at least the last p rows of Q are zero. This assures that the rank condition, (R) is satisfied. (3) Obtain the observability and controllability indices^ as in Theorem (2.3-3). T (4) Obtain Â£. iejj from the appropriate rows of P as in Corollary (2.3-4) T * and jb. as in (2.2-19) where $...=p... * J J (5) Perform the following column operations on Q to obtain [P | S* Nl | E]: (i) Select the leading element in the first column of Q, . (ii) Perform column operations (with interchange) to obtain qjs=0 for s>l. (iii) Repeat (i) and (if) until the only nonzero elements in each row are leading elements. (6) Obtain a., jem from the appropriate columns of E as in Corollary J (2.3-4) and Â¥. from the dual of (2.2-19). (7) From the invariants construct the Luenberger and MFD forms as in Section (2.2). If we also require the characteristic polynomial, then we must include: 4*4* (8) Determine the v.}, cjd and (simultaneously) construct TgR as in Lemma (2.3-12). (9) Find Til by solving for the non unit rows in TnnTÂ¡i I . BR BR BR n ^Note that the leading elements have been selected from the rows by examining the columns in their natural order; therefore, the dependent columns are not zero as in (2.3-3), but are easily found from this form of Q by inspection. It should also be noted that the leading elements could have been selected in the j, (j+m), (j+2m)... columns; therefore, facilitating the determination of the Buey invariants and forms. 4.4* r\j Alternately the {vj}, jem and Tjjc could be used. These indices can be found easily from the columns of Q. 47 If we consider the alternate method implied in Corollary (2.3-6), then the following modifications to the preceding steps are required: (1)* Start with the following augmented array: ^pN I SN,N'l (2)* Obtain [P | Q] as before. (5)* Perform additional row operations on Q to obtain unit columns for each column possessing a leading row element, and perform row interchanges such that (2.3-8) is satisfied for each jem, i.e., obtain (6)* Obtain the a., jem, as in (2.3-6). J It should be noted that these algorithms are directly related to those developed by Ho and Kalman (1966), Silverman (1971), or Rissanen (1971). As in Ho's algorithm, the basis of the first technique is performing the special' equivalence transformation of Theorem (2.3-3) rk on S^ to obtain S^ The second technique accomplishes the same objectives by restricting the operations to only the rows of S^ which is analogous to either the Silverman or Rissanen method. The initial storage requirements in the first method are greater than the second if mN'>pN, since P and E can be stored in the same locations due to their lower and upper triangular structure; and (2) P will be altered in the second method, since row interchanges must be performed in (5)*; whereas, it remains unaltered in the first method which may be important in some applications. Consider the following example which is solved using both techniques. 48 Example. (2.4-2) Let m=2, p=3, and the Hankel array be given as, ^ - 1 2 2 4 4 8 8 16 1 2 2 4 6 10 13 22 1 0 1 0 3 2 6 6 2 4 4 8 8 16 16 32 2 4 6 10 13 22 28 48 1 0 3 2 6 6 13 16 4 8 8 16 16 32 32 64 6 10 13 22 28 48 58 102 3 2 6 6 13 16 27 38 8 16 16 32 32 64 64 128 13 22 28 48 58 102 119 214 6 6 13 16 27 38 56 86 ^ ^12 I S4,4 I V . Y (2) Performing the row operations as in (2.4-1), obtain [P | Q I I0], O where the leading elements are circled, ] 2 2 4 4 8 8 16: -1 1 0 o o o ro 5 6 Jl 2 i T 1 0 @-1 -4 0 -5 _i_ 2 -7 -2 b 0 1 0 0 0 0 0 0 0 0 0 1 2 _ 5 . 2 0 0 1 0 0 0 201 1 2 1 1 1 -1 0 -] 1 -4 0 b 0 0 0 1 r8 -3 0 -i -1 0 0 1 0 1 -2 0 -1 0 0 0 1 o 00 -^1 -8 0 .0 0 . 0 0 0 0 0 1 -8 1 -2 0 -2 0 0 0 0 0 1 -1 2 -3 0 -2 0 0 0 0 0 0 1 49 (3)The indices are obtained by inspection from the independent rows and columns of Q in accordance with Theorem (2.3-3) as: v1 1 u1 = 3 ^2 2 u2 ^ and p(S2s3) = p(S3 3) p(S2 = 4 satisfying (R). T T (4)The jfj and bjj are determined from the appropriate rows and columns of P as: 1 ~ -CP41 I P42 P45 I P433 = [2 I 0 0 I 0] -2 = "^p81 I p82 p85 I p83-* = I 0 V'M3 -3 = "^p61 I p62 p65 I p63^ = C"1 I 1 Ml -1 = tO-3 I P4i P42 p43 p44 I 5^3 = [O3 I -2 0 0 1 I ^ 2 = ^-p81 P82 P83 P84 P85 P86 P87 p88 I 3=E"3 O*1 0-1 0 011 3 -3 = % IP61 P62 p63 P64 P65 P66"* = % Â¡ 1 1 _1 0 _1 13 (5)Performing the column operations, obtain the structural array k N, and E as: ... Jc [P I 4 I E] where the leading elements are circled,. 50 0 0 0 0 0 0 0 0 0 0 00 o o 000 0 0 0 0 0 0000 0 000 0 000.0.0 o o i _jl _jl n JL i. 13 2 2 U 4 8 4 10 o o o 1 -1 -4 -3 0 10 0 1 o The a. and b. are determined from the appropriate rows and columns J J of E as: e17 r 5-i -T e14 ~ -1 e37 , 1 4 e34 1 e57 .5 2 ao ~ "* e54 0 e27 1 8 e24 3 T el 7 e27 - _5 - 4 1 8 ^4 S4 e37 1 4 e14 1 e47 = 0 I -2 = e24 s 3 " ~T e57 5 2 e34 -1 e67 0 e44 1 e77 1 if 0 0 51 (7) The canonical forms of zR, BR(z), DR(z) and Eg, Bg(z),. DG(z) are z2-2z 0 0 z 2z ~ Br(z) = -3 z2+z 2 z -z +z 1 z2-z_ ' VZ> = z+1 0 2z+2 -2z Fq = [eg Â£3 ] I 2-1 GC = [^1 ^4] HC = *-a.l a.3 a.5 a>2] * 1 1 LI 2 4 I 2 2 6,2 1 3 0 _ Bc(z) = TZ + fz + i. 8 3. _2 -z +z 3 3 2 z fz ; Dc(z) z fz Â£ z -z + Â£ z2- fz + f (8) The {v.} and TgR are determined simultaneously as: 1Â¡ 1 1 1 0 0 0 and tbr 4 0 1 0 0 4 0 0 1 0 1 1 3 0 1 1 52 (9) T"1 is given by solving the equations for the last row as: r1 1BR 1 0 0 3 0 1 .0 0 0 0 0 0 1 0 -1 1 (10) Find FBR and Xp(z) as F =T F BR VBRVbR 2 0 0 2 0 0 0 1 0 0 1 0 0 0 1 2 and XF(z) = (z-2)(z3-2z2+l) = z4-4z3+4z2+z-2 This completes the first method. If the second method is used instead, then only (5)*, (6)*, and (8)* differ. (5)* Performing the additional row operations and interchanges to satisfy (2.3-8) gives: 5 1 T 1 -T 1 l T * -T 0 0 1 T l T 3 8 0 0 0 0 -$- (D 0 0-1 0 --z- -f T o 0 Q 1 o i * 0 0 0 0 1 Â£ 3 0 o I- 0 9 1 T -T 1 3 0 (6)* The a.'s are determined from the appropriate columns of Q as: J -T "-1 -i 1 -1 = 5 T = W7 2 = 0 I L_ T _ - - = W4 53 This completes the algorithms. In the next chapter the first method is modified to develop a nested algorithm from finite Markov sequences. CHAPTER 3 PARTIAL REALIZATIONS One of the main objectives of this research is to provide an efficient algorithm to solve the realization problem when only partial data is given. As new data is made available (e.g., an on-line application, Mehra (1971)), it must be concatenated with the old (previous) data and the entire algorithm re-run. What if the rank of the Hankel array does not change? Effort is wasted, since the previous solution remains valid. An algorithm which processes only the new data and augments these results (when required) to the solution is desirable. Algorithms of this type are nested algorithms. In this chapter we show how to modify the algorithm of (2.4-1) to construct a nested algorithm which processes data sequentially. The more complex case of determining a partial realization from a fixed number of Markov parameters arises when the rank condition, abbreviated (R), is not satisfied. It is shown not only how to determine the minimal partial realization in this case, but also how to describe the entire class of partial realizations. In addition, a new recursive technique is presented to obtain the corresponding class of minimal extensions and the determination of the characteristic equation is also considered. 3.1 Nested Alqorithm Prior to the work of Rissanen (1971) no earlier recursive methods appeared in the realization theory literature. Rissanen uses a 54 55 factorization technique to solve the partial realization problem when (R) is satisfied. His algorithm not only solves the problem in a simple manner, but also provides a method for checking (R) simultaneously. In the scalar case, Rissanen obtains the partial realizations, Z(K), K=l,2,... imbedded in the nested problem of (2.1), but unfortunately this is not true in the multivariable case. Also, neither set of invariants is obtained. The development of a nested algorithm to solve the partial realization problem given in this dissertation follows directly from (2.4-1) with minor modification. There are two cases of interest when only a finite Markov sequence is available. Case I. (R) is satisfied assuring that a unique partial realization exists; or Case II. (R) is not satisfied and an extension sequence must be constructed. The nested algorithm will be given under the assumption that Case I holds in order to avoid the unnecessary complications introduced in the second case. The modified algorithm is given below. The corresponding row or column operations are performed only on the data specified elements. Partial Realization Algorithm. (3.1-1) (1) Same as (1) and (2) of Algorithm (2.4-1) except (iv) is q^O k>j if k is a row whose leading element has been specified. (2) If (R) is satisfied for some M*=v+y, obtain the invariants as before in (3), (4) of (2.4-1) and go to (5). If not, continue. 56 (3) Add the next piece of data, Am+-j and form S(M+1,M+1). (4) Multiply S(M+1,M+1) by P. Perform row operations (if necessary) using old leading elements to obtain Q (M+1,M+1). If (R) is satisfied, continue. If not, go to 3. (5) Perform column operations as in (5) of (2.4-1) and obtain the invariants and canonical forms as in (6), (7). Go to 3. Example (2.4-2) will be processed to demonstrate the modified algorithm for comparison. Assume that the Markov parameters are sequentially available at discrete times, i.e., A^ is received, then Ag, etc., and the system is to be realized. Example. (3.1-2) Let the Markov sequence be given by "l 2' C\i 1 4 "4 8 8 16 '16 32~ A1 * 1 -2 J _ ~ ,2 _1 4 0_ , a3 = 6 10 _3 2_ ii 13 22 6 6 V 28 48 J3 16_ and apply the algorithm of (3.1-1). It is found that the rank condition is first satisfied when A^, Ag are processed, i.e., v (1) [I6 | S(2,2) | I4] (2) Performing first row and then column operation as in (3.1-1), obtain [P ) S*(2,2) Â¡ E] or l 0 0 0 1 -2 -1 0 -1 1 0 0 0 0 0 1 -i -2 -1 0 1 0 0 0 0 1 0 -2 0 0 1 0 0 1 -2 0 0 0 1 0 0 __0 0 -1 0 0 1 0 0 57 (3) Indices are: v-j = 1 v2 = 0 h = 1 y2 = 1 oo n (4) ^Invariants are: Is- tP41 1 P43] [2 1 0] 4-- CP61 O | 1 II 1 1 OO *Â£> a. 1 1] and -1 = ^P41 P42 P K43 P44 1 $ [-2 0 0 10 0] 2 = *--3 1 P21 P 2 1 o{] = [0 0 0-1 i 0] -3 = ^P61 P62 P 63 P P P 64 K65 0-1001] el 3 -1 ' e14 *0 " eT3 1 a 2 ' e14 _0 e23 1 ~T e24 -2 T *." e23 v 1 T e24 2 . b, = e33 1 5 bo~ e34 0 tm- _0 _ 0- -e44- _1 _ 58 where wT+ = -[P2] | P^] = [1 | o] z-2 o o J ~i 2 -Z 2 0 ; dr(z) = 0 0 _ 0 0 2-l_ _ i 0_ The rank condition is next satisfied when A1,Ag.are processed, i.e., M =5 and we obtain [P | S (5,5) | E] as: [P I Q(5,5)] = 1 2 2 4 4 8 8 16 16 .32 -1 1 0 0. 0 0 2 5 6 12 16 ^1 0 1 0( -1 -4 -1 -6 -2 -10 -3 -16 -2 0 0 1 0 0 0 0 0 0 0 0 -2 0 0 0 1 0 0 2 5 6 12 16 1 1 -1 0 -1 1 0 0 0 0 0 0 0 0 -4 0 0 0 0 0 1 0 0 0 0 0 0 -3 0 -1 0 -1 0 0 1 0 0 0 0 0 0 0 1 -2 0 -1 0 0 0 1 0 0 0 0 0 0 -8 0 0 0 0 0 0 0 0 1 0 0 0 0 -8 1 -2 0 -2 0 0 0 0 0 1 0 0 0 0 -1 2 -3 0 -2 0 0 0 0 0 0 1 0 0 0 0 -16 0 0 0 0 0 0 0 0 0 0 0 1 0 0 -24 0 -4 0 0 0 0 0 0 0 0 0 0 1 0 0 -8 0 -5 0 0 0 0 0 0 0 0 0 0 0 1 0 0 and performing the column operations give [S (5,5) J E] +W is found easily from HrGr=A-j or solving for the second row of HR, wTGr [1 2]. 59 XD 0 0 0 0 0 0 0 0 1 -2 -1 -1 - T 4 _5 4 7 2 -10 -12"- 0 0 0 0 0 0 0 0 0 1 4 * -1 8 1.3 4 -6 -14 0 0 0 0 0 0 0 0 0 1 -1 __5. 2 -3 a. 4 2 15 20 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 -1 5 ~T -3 -6 -8 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0. 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 a 0 0 0 a a a a _o: a The results in this case are identical to those of Example (2.4-2). Let us examine the nesting properties of this realization algorithm. Temporarily, we resort to using data dependent notation for this discussion with the same symbols as defined previously in the previous sections, e.g., the minimal partial realization of order M is given by S(M) (F(M),G(M),H(M)). Thus, I(M+k) is a (M+k)-order partial realization. We also assume for this discussion that Â£(M) is in row canonical form; therefore, it can be expressed in terms of the set of invariants, [{v.j(M)},{B.Â¡st(M)},{aT(M)}]. If S(M) is an n dimensional, minimal partial realization specified by these invariants, then there T arc n regular vectors, Â¥g+pt(M) spanning the rows of S(M,M). Furthermore, 60 T each dependent row vector, f.(M) is uniquely represented as a linear J combination of regular vectors in terms of the observability invariants and it can be generated from the recursion of Corollary (2)3-2); Similarly it follows from Proposition (2.2-13) that the dependent block row T vectors, a. (M) satisfy an analogous recursion. The following lemma J * describes the nesting properties of minimal partial canonical realizations Recall that M is the integer of Proposition (2.1-6) given by M =v+y. Lemma. (3.1-3) Let there exist an integer M (M)-M such that the rank condition is satisfied and let Â£(M) be the corresponding minimal partial canonical realization of {Ar>, rcM specified by the set of invariants [v.(M)},{fi. .(M)}, I l U ia^(M)}]. Then J ' v.(M) = ... = v.(M+k) 6-st(M) = ... = 3ist(M+k) al (M) = .... = aT (M+k) J J ' . ' A. , iff p(S(M,M))=p(S(M+l,M+l) = ... = p(S(M+k,M+k)) for the given k. Proof. If v. (M) = ... = v. (M+k), etc., then the minimal canonical partial realizations are identical, Â£(M) = Â£(M+1)= ... = Â£(M+k). It follows that p(S(M,M))=dimÂ£(M)=p(S(M+l,M+l))-dimÂ£(M+l) = p(S(M+k,M+k)). Conversely, P(S(M,M))=P(S(M+1,M+1))= ... =P(S(M+k,M+k)) implies dimÂ£(M)=dimÂ£(M+l)=... =dimÂ£(M+k). Since Â£(M) is a unique minimal canonical partial realization, so is Â£(M ). Furthermore, since each realization has the same dimension, each realization has 61 has M (M)=M (M+l)= ... = M (M+k) so that each canonical realization is equal to Z(M*); therefore, Â£(M)=Â£(M+1)= ... =Z(M+k).V Next we examine the case where Â£(M) and z(M+k) are of different dimension. The nesting properties are given in the following lemma. ^ "At Lemma. (3.1-4) Let there exist integers, M (M)^M, M (M+k)^M+k such that the rank condition is satisfied (separately) and Â£(M), Z(M+k) are minimal partial canonical realizations of (Ar> when reM and reM+k, respectively, for given k. If p(S(M+k,M+k))>p(S(M,M)), then v.(M+k)*v.(M), ieÂ£. Furthermore, a. (M+k)=a. (M), j=i,i+p,...,i+p(v.(M)-l). J J Proof. Since p(S(M+k,M+k))>p(S(M,M)), M*(M+k)>M*(M) and therefore, Sv(M),y(M) is a submatHx of Sv(M+k).uCM+k)* If the row of is regular, it follows that the j-th row of Sv(M+k) y(M+k) ls also re9u1ar by the nature of the Hankel pattern, i.e., the rows of Sv^ are subrows of Sv(M+k) ,y(M+k) The addition of more data (AM+],... ,AM+|<) to S(M,M) makes previously dependent rows become independent rows but previously independent rows remain independent; thus, the v..(M) can only increase or remain the same, i.e., v..(M+k) ^ T v.(M), icÂ£. The set of regular {a.(M+k)} are specified by the ' J v.(M+k)'s; therefore ai (M+k)=aT (M), j=i,i+p,.. .,i+p(v.(M)-l), 1 J J 1 since vi(M+k)-vi(M), ieÂ£.V The results of these two lemmas are directly related to the nesting 4 properties of the partial realization algorithm. First, define JM as the set of indices of regular Hankel row vectors based on M Markov parameters 62 available, i.e., {1,1+p,...,l+p(v1(M)-l),...,p,2p,...,pvi(M)} and similarly denote the row vectors of the elementary row matrix of the previous chapter, by (M). From Lemma (3.1-3), it follows that jJ J*+k and 4+pv.(H)(M) Â£.T+pv1 (M+k)(M+k) si"ce the observability invariants are identical. The specify the elements in and along with the 3ist they specify the elements of pi+pv.(M)(M) (see Coro11ary (2-3-4)). From Lemma (3.1-4) it is clear that JÂ¡Â¡jcjjÂ¡Â¡+k since v. (M+k^v^M). Reconsider Example (3.1-2), to see these properties. In this case we have M=2, k=3, M*(2>2, M*(5)=5, and p(S(5,5))>p(S(2,2)) as in Lemma (3.1-4); therefore, since 3-? = and J5 = d*3,2,5}. The observability indices are identical except for v2(5)>v2(2); thus, iaj (2) ,a2> (2)}<={aj^ (5) ,a^ (5) ,a2>(5) ,a^ (5)} since aT (2) = a! (5) for j=l,3. We also know from Example (2.4-2) that Â£(5) is the solution to the realization problem and therefore the properties of Lemma (3.1-3) will hold for {A|yj}, M>5. Table (3.1-5) summarizes these properties. The results for the dual case also follow directly. We now proceed to the case of constructing minimal partial realizations when (R) is not satisfied, i.e., the construction of minimal extensions. 63 Table. (3.1-5) Nesting Properties of Algorithm (3.1-1) Augment M^M+k n(M+k)=n(M) n(M+k)>n(M) JM"^M+k " fi-i+pv. > c where (R) is satisfied for some k and means that the corresponding invariants, vectors, or indices are nested or contained in a set of higher order. Vj 5ist J. 64 3.2 Minimal Extension Sequences In this section we discuss the more common and difficult problem of obtaining a minimal partial realization from a finite Markov sequence when (R) is not satisfied. Two different approaches for the solution of this problem have evolved. The first is based on constructing an extension sequence so that (R) is satisfied and the second is based on extracting a set of invariants from the given data. We will show that these methods are equivalent in the sense that they may both lead to the same solution. In order to do this the existing algorithm is extended to obtain the more general results of Roman and Bullock (1975a). Also a new recursive method for obtaining the entire class of minimal extensions is presented. It is shown that the existing algorithm does in fact yield a particular solution to this problem which is valuable in many modeling applications. In the first approach, Kalman (1971b), Tether (1970) and subsequently Godbole (1972) examine the incomplete Hankel array to determine if (R) is satisfied. If so, the corresponding minimal partial realization is found. If not, a minimal extension is con structed such that (R) holdsand a realization is found as before. They show that a minimal extension can always be found, but in general it will be arbitrary. They also show that this extension must be constructed so that the rank of S(M,M) remains constant and the existing row or column dependencies are unaltered. Considerable confusion has resulted from the degrees of freedom 65 available in the choice of minimal extensions. In fact, initially, the major motivation for constructing an extension was that it was necessary in order to be able to apply Ho's algorithm. Un fortunately, these approaches obscure the possible degrees of freedom and may lead to the construction of non-minimal extensions as shown by Godbole (1972). Roman and Bullock (1975a)developed the second approach to the solution of this problem. They show that examining the columns or rows of the Hankel array for predecessor independence yields a systematic procedure for extracting either set of invariants imbedded in the data. They also show that some of these would-be invariants are actually free parameters which can be used to describe the entire class of minimal partial realizations. These results precisely specify the number and intrinsic relationship between these free parameters. Unfortunately Roman and Bullock did not attempt to connect their results precisely with those in Kalman (1971b),Tether (1970). It will be shown that this connection offers further insight into the problem as well as new results which completely describe the corresponding class of minimal extensions. Before we state the algorithm to extract all invariants available in the data, let us first motivate the technique. When operating on the incomplete Hankel array, only the elements specified by the data are used. It is assumed that the as yet unspecified elements will not alter the existing predecessor dependencies when they are specified by an extension sequence. Since the predecessor dependencies are found by examining only the data in S(M,M), we must examine complete submatrices of S(M,M) in order to extract the invariants 66 associated with a particular chain (see Roman and Bullock (1975a)). Therefore, it is possible that a dependent vector, say ^ of a sub- matrix of S(M,M) later corresponds to an independent vector in S(M,M). When representing any other dependent vector in this submatrix m terms of regular predcessors, Â¥. must be included, since it is a regular vector of S(M,M) under the above assumption. In this represen- tation the coefficient of linear dependence corresponding to is arbitrary. Reconsider Example (3.1-2) for{A..}, i =1,2,3 where we only consider the (row) map P. Example. (3.2-1) For A^, Ag, A3 of (3.1-2) we have P: S(3,3)-K)(3,3) or " 1 2 2 414 8 2 2 414 8* 1 2 2 416 10 1 000 0(2)2 IJ_J3 2 0^|)-1-4jl-6 2 4 4 8 0 0 0 0 2 4 6 10 P 0 0(2)2 10 3 2 0 0 0 0 4 8 0 0 6 TO 0 0 .3 2 _ 0 0 The indices are = {1,2,1}. Since v^l, the fourth row of S(3,3) (or equivalently Q(3,3) ) is dependent on its regular predecessors as shown in the corresponding 3x4 submatrix (in dashed lines) of S(3,3) (or Q(3,3) ). The second row, say ^ > in this submatrix is dependent, yet it is an independent row of S(3,3) (or Q(3,3) ). Now, expand of this submatrix, i.e., 67 = 3110 + e!20 -2 +f3130 -3 {312r0) or [2 4 4 8] = 3110 [1 2 2 4]+ g]20 [1 2 2 4]+ g^H 01 0] . The solution is = 2 g^o ^130= t*1us* t*ie coefficient B-^O 1s ah arbitrary parameter. Note that this recursion is essentially the technique given in Roman and Bullock (1975a). Clearly, if (R) is satisfied as in the previous section, then there exists a complete submatrix (data is available for each element) of S(M*,M*) in which every regular vector of S(M,M) is always a regular vector of the submatrix corresponding to a particular chain; thus, there will be no arbitrary or free parameters. The algorithm for the case when (R) is not satisfied may be illustrated by considering row operations on S(M,M) to obtain Q(M,M), since the identical technique can be applied to obtain S*(M,M). The arbitrary (column) parameters are found by performing additional column operations to Q(M,M). As in Example (3.2-1), we must find the largest submatrix of Q(M,M) for each chain, i.e., if we define k_. as the index of the block row of S(M,M) containlirig the vector. , then the largest submatrix of data specified elements corresponding to the i-th chain is given by the first (i+pv^-1) rows and m(M+l-k.) columns of Q(M,M). Also, we define J|ieÂ£ as the sets of Hankel row indices corresppnding to each dependent (zero) row of the (i+pMj-l)x (m(M+l-kj) submatrix of Q(M,M) which becomes independent, i.e., it contains a leading element. In Example (3.2-1) for i=l, we have (1 +pv1-1)=3 and k-j =2; thus, nKM+l-k-^A and the corresponding submatrix is given by the first 3 rows and 4 columns of Q(3,3), and.of course, J^={2}. 68 Arbitrary Parameter Partial Realization Algorithm. (3.2-2) (1) Perform (1) of Algorithm (3.1-1) to obtain [P | Q(M,M)]t. (2) For each ieÂ£, determine the largest (i+pYl)xm(M+l-k..) sub matrix of Q(M,M) of data specified elements and form the set J.. (3) For each ieÂ£, replace pi by J + z bJ, b a scalar. H i H i seJ^ (4) Determine the corresponding canonical forms incorporating these free parameters. Dual results hold for the columns. The fre-parameters are fpund in analogous fashion by examining the zero columns of the submatrices of S*(M,M). Example. (3.2-3) The following example is from Tether (1970). For m=p=2 and "1 f 4 3 10 1 22 15 Ar _0 0_ _0 0_ ,A3- _ 1 1_ 4* II 3 3 _ (1) [ P f Q(4,4) ] = _1 1 4 3 j 10 7 22 15 0 1 0 0 0 ! 1 3 3 -4 0 1 0 -6 5i 18- 13 0 0 0 1 0 0 0 ! Jj 0 0 2 0 -3 0 1 0 0 0 0 -1 0 0 1 0 1 0 0 0 6 0 -7 0 0 0 1 0 0 -3 0 0 0 0 0 0 1 0 0 4* It should be noted that when (R) is not satisfied, some of thev. may not be defined, i.e., the last independent row of a.chain is in the last block Hankel row. In this case all_ would-be invariants are arbitrary. 69 The indices are: v-|=2, V2=3 (2) For 1=1, (1+p ^-1)=4, k.j=3, m'(M+T-k1 )=4;' thus, the corresponding submatrix is constructed from the first 4 rows and columns of Q(4,4) (small, dashes). J-j={2}. For i=2, (2+pv2-l)=7, k2=4, m(M+l-k2)=2; thus, the corresponding submatrix of Q(4,4) is given by the first 7 rows and 2 columns (large r'a'shes). J2={2,4,6}. I T (3) Replacing the fifth and eighth rows of P with jDj. + bÂ£2 and Â£Â¡ + c4 + T T + ej^. where b,c,d,e are real scalars gives = [2 b -3 0 1 0 0 0 ] = [-3-e c 0 d+e 0 e 0 1 ] The Â§1 are: g_{ = [ -2 3 -b 0 0] g_2 = [ 3+e 0 -c -(d+e) -e ] (4) The canonical form is Corresponding to ttiese realizations is a minimal extension sequence which can be found by determining the Markov parameters. These parameters 4 70 are cumbersome to obtain due to the general complexity of the expressions in Er or therefore, a technique to determine these extensions without forming the Markov parameters directly (or the realization) was developed. This method consists of recursively solving simple linear equations (one unknown) to obtain the minimal extension. Extensions constructed in this manner not only eliminate the possibility of non minimality as expressed in Godbole (1972), but also describe the entire class of minimal extensions. The method of constructing the minimal extension sequence evolves easily from the lower triangular-unit diagonal structure of P. Since a dependent row of Q(M,M) is a zero row, it follows from Theorem (2.3-3) that for jemM (3.2-4) where recall that p.^ .=0 for j>i+pv.. i+pv^,j r 1 unkndwn extension parameters, x.y(r) fr Thus, by inserting the 'lm (r) into S(M,M) a system of linear equations is established in terms of the x..(r)s by (3.2-4). Due to the structure of P, this system of equations i J is decoupled and therefore easily solved. Example. (3.2-5) Reconsider (3.1-2) for Since (R) is satisfied, the extension A., j>2 is unique. We would like to obtain 0 71 A- *11(3) x21(3) x31(3) Since P maps S(2,2) into 0(2,2), we x12(3) x22(3) x32(3) have 2 2 4 2 2 4~ 12 2 4 0 0 0 0 1 0 1 0 P 0 -1 -4 0 0 ^0 0 2 A J x-ji (3) x]2(3) 2 4 j x2i(3) x22(3) 0 0 j 0 0 1 0 | x3i (3) x32(3)_ 0 0 | 0 0 and in this case,{v.j ,v2,v3> ={1,0,1}. Thus, using (3.2-4), we have solving 0 = Â£4 Â£3 = C-2 0 0 1 0 Q] 2 2 1 for x11(3) gives x11(3) X1 -j (3) x2i(3) X31(3) Similarily solving: Â¡^ = 0 for x-jg(3) gives x12(3) = 8 Â£3 = 0 for x3i(3) gives x^(3) =1 Â£^1^ = 0 for x32(3) gives x32(3) = 0 In this example, x2-j(3)=x^(3) and x22(3)=x-|2(3), since v2=0. 72 Thus, this example shows that the minimal extension sequence can be found recursively due to the structure of P. Of course, the problem of real interest is when (R) is not satisfied and (as in Ho's algorithm) a minimal extension with arbitrary parameters must be constructed. Minimal Extension Algorithm. (3.2-6) (1) Perform (1), (2), (3) of Algorithm (3.2-2). (2) Determine M* = v+y. (The values of v,y are determined by the partial data) (3) Recursively construct the minimal extension {A^,}, r = M+l, ... ,M* where Ar = [x^.(r)] by solving the set of equations for j(r) given by j+pv. Lj = 0 j m(M+l-k.)+l, ... ,m(M*+l-k.), for each iej>. and recall that k. is the index of the block row of S(M,M) containing the row vector, i+pv-. Example. (3.2-7) Reconsider (3.2-3) for illustrative purposes. (1) These results are given in Example (3.2-3) (2) M*=6; thus, find A5 = x-j 1 (5) x-j 2(5) A6 x-j 1 (6) x-j2(6) x21(5) x22(5) X21(6) x22(6) (3) Recursively solve: pT+2v Â£j 0 for i=1> j=5,6,7,8 and for i=2, j=3,4,5,6. Â£5 I5 = 0 gives x^iB); 1^ = 0 gives x12(5) Â£^ r3 =0 gives x21(5); ^ = 0 gives x22(5) Â£5 I7 = 0 gives xn(6); ^ = 0 gives Xj2(6) Â£g Z5 = 0 gives x21(6); 1^ = 0 gives x22(6) 73 and therefore A5" 46-b 31 -b 94-6b 63-6b 12-d 9-d-e 30-c-3d-5e+de 21-c-3d-5e+de-e2_ By solving for the x^.'s in Ag, Ag we obtain the extension as x^B) Xll(5)-15" A, = ~6x11(5)-182 6x11 (5)-213 x2i(5) x22 ^ 6 X21 X21 (6)+(x21 (5)-X22(5)-3)2-9 The number of degrees of freedom is 4,i .e. .{x^(5),x21 (5),x22(5),x2-| (6)}. The technique used to solve the parcial realization problem when (R) is not satisfied was to extract the most pertinent information from the given data in the form of the invariants, which completely described the class of minimal partial realizations. A recursive method to obtain the corresponding class of minimal extensions was also presented in {3.2-6) This method is equivalent to that of Kalman (1971b) or Tether (1970) for if the minimal extension is recursively constructed and Ho's algorithm is applied to the resulting Hankel array the corresponding partial real ization will belong to the same class. Note that if the extension is not constructed in this fashion, it is possible that all degrees of freedom available may not be found (see Roman (1975)). It should be noted that the integers v and y are determined from the given data,i.e., knowledge of the invariants enables the construction of a minimal extension such that v and y can be found. The approach completely resolves the ambiguity pointed out by Godbole (1972) arising in the Kalman or Tether technique. The results given above correspond directly to those presented in 74 Kalman (1971b) and Tether (1970). They have shown, when (R) is satisfied, there exists no arbitrary parameters in the minimal partial realization or corresponding extension. Therefore, the existence of arbitrary parameters can be used as a check to see if the rank condition holds. Although it is not essential to construct both sets of invariants, it is necessary to determine M* which requires v and y; thus, the algorithm presented has definite advantages over others, since these integers are simultaneously determined. In practical modeling applications, the prediction of model performance is normally necessary; therefore, knowledge of a minimal extension is required. Also in some of the applications the number of degrees of freedom may not be of interest, if only one partial realization is required rather than the entire class. In this case such a model is easily found by setting all free parameters to zero which corresponds to merely applying the Algorithm (3.1-1) directly to the data and obtaining the corresponding canonical forms as before. Describing the class of minimal extensions offers some advantages over the state space representation in that it is coordinate free and indicates the number of degrees of freedom available without compensation. 3.3 Characteristic Polynomial Determination by Coordinate Transformation In this section we obtain the characteristic equation of the entire class of minimal partial realizations described by Fr or F^, of the previous section. It is easily obtained by transforming the realized Fr or Fc into the Buey form as before. Recall that the advantage of this representation over the Luenberger form is that it is possible to find the characteristic polynomial directly by inspection of FgR in (2.2-11). 75 Even though it is possible to realize the system directly in Buey form as implied in the discussion of (2.3-12), it has been found that this method has serious deficiencies when dealing with finite Markov sequences. If (R) is satisfied, the partial realization is unique. When (R) is not satisfied, this technique does not yield all degrees of freedom. For example, reconsider the arbitrary parameter realization of Example (3.2-3). This realization is given in Ackermann (1972) as Q 1 0 0 O' "0 1 0 0 0 -2 3 -b 0 0 -2 3 0 0 0 0 0 0 1 0 n . 11 0 0 0 1 0 0 0 0 0 1 0 0 0 0 1 3+e 0 -c -(d+e) -e_ _3+e 0 -c -(d+e) -e_ Note that one degree of freedom (b=0) has been lost. Similarity Ledwich and Fortmann (1974) have shown by example that this technique can also lead to non-minima! realizations. These deficiencies arise due to the procedure used for the determination of the Buey invariants. This procedure does not account for the possibility that an independent row vector of a particular chain may actually be dependent if it is compared with portions of the same length of vectors in different chains. To cir cumvent the problem, the previous technique will be used,i .e., the system is realized directly in Luenberger form and transformed to Buey form. Not only does this assure minimality as well as the determination of all possible degrees of freedom, but Tg^ is almost found by inspection as shown in (2.3-12). Reconsider the example of the previous section. 76 Example. (3.3-1) Recall that in (3.2-3) m=p=2, n=5, and Vj=2, V2=3, = [ -2 3-bOO] Â§J2 = [3+e 0 -c -(d+e) -e ] (1)Simultaneously construct TBR from (3.3-4) while examining the rows for predecessor independence -1 1 0 0 0 o" 4 0 1 0 0 0 4 s -2 3 -b 0 0 4fr -6 7 -3b -b 0 /. -14 15 -7b -3 b' -b 1 1 (2)Determine TBR from TBR TBR = In which gives f1 'BR 1 0 -2/b 0 0 (3)Determine Fbr: F =T F T"^ = rBR BRrR BR 0 0 0 0 1 3/b 2/b 0 -1/b 3/b -2/b 0 -1/b 3/b -1/b 1 0 0 0 -3b-2c-ce 3c-2d-2e 0 1 0 0 -c+3d+e 0 0 1 0 -d+2e-2 0 0 0 1 -e+3 (4)Find the characteristic polynomial by inspection. 77 Xr (z) = z5+(e-3)z4+(d-2e+2)z3+(c-3d-e)z2+(-3c+2d+2e)z+(b+2c+be) rBR This example points out some very interesting points. When this technique is combined with the algorithm of (3.2-2), it offers a method which can be used to obtain the solution to the stable realization problem developed in Roman and Bullock (1975b). Also, if the system were realized directly in Buey form, then b=0 and a degree of freedom is lost; thus, in Ackermann's example Vj=l, while ours is v^=5. It is critical that al_]_ degrees of freedom are obtained as shown in this case, since the system is observable from a single output. This section concludes the discussion of the deterministic case of the realization problem. In the next chapter we examine the stochastic version of the realization problem. CHAPTER 4 STOCHASTIC REALIZATION VIA INVARIANT SYSTEMS DESCRIPTIONS, In this chapter the stochastic realization problem is examined by specifying an invariant system description under suitable trans formation groups for the realization. Superficially, this may appear to be a direct extension of results previously developed, but this is not the case. It will be shown that the general linear group used in the deterministic case is not the only group action which must be considered when examining the Markov sequence for the corresponding stochastic case. . Analogous to the deterministic realization problem there are basically two approaches to consider (see Figure 1): (1) realization from the matrix power spectral density (frequency domain) by performing the classical spectral factorization; or (2) realization from the measurement covariance sequence (time domain) and the solution of a set of algebraic equations. Direct factorization of the power spectral density (PSD) matrix is inefficient and may not be very accurate. Recently developed methods of factoring Toeplitz matrices by using fast algorithms offer some hope, but are quite tedious. Alternately, realization from the covariance sequence is facilitated by efficient realization algorithms and solutions of the Kalman-Szego-Popov equations. 78 79 REALIZATION FROM COVARIANCE SEQUENCE ^ PSD AND ALGEBRAIC METHODS STOCHASTIC REALIZATION Figure 1. Techniques of Solution to the Stochastic Realization Problem. The problem considered in this chapter is the determination of a minimal realization from the output sequence of a linear constant system driven by white noise. The solution to this problem is well known (e.g.. see Mehra (1971)) as diagrammed below in Figure 2. The output sequence of an assumed linear system driven by white noise is correlated and a realization algorithm is applied to obtain a model whose unit pulse response is the measurement covariance sequence. A set of algebraic equations is solved in order to determine the remaining parameters of the white-noise system.. This problem is further complicated by the fact that the covariance sequence must be estimated from the measurements. From the practical viewpoint, the realization is highly dependent on the adequacy of the estimates. Although in realistic situations the covariance-estimation problem cannot be ignored, it will be assumed throughout this chapter that perfect estimates are made in order to concentrate on the realization -J* portion of the problem. In this chapter we present a brief review of the major results necessary to solve the stochastic realization problem. We use the 4-' Majumdar (1976) has shown in the scalar case that even if imperfect estimates are made realization theory can successfully be applied. FACTORIZATION METHODS 80 White Noise Input Stochastic Realization Figure 2. A Solution to the Stochastic Realization Problem 81 algebraic structure of a transformation group acting on a set to obtain an invariant system description for this problem. A new realization algorithm is developed to extract this description from the covariance sequence. Recently published results establishing an alternate approach to the solution of this problem are also considered. 4.1 Stochastic Realization Theory Analogous to the deterministic model of (2.1-1 ) consider a white- noise (WN) model given by Vi = F*k + *k (4.1 4 =H4 where and ^ are the real, zero mean, n state and p output vectors, and Wj, is a real, zero mean, white Gaussian noise sequence. The noise is uncorrelated with the state vector, X., j k and J Cov(wi,wj.):=E[(wi-Ewi)(Wj-Ew[.)T] = x. where 6.. is the Kronecker delta. This model is defined by the triple, ij ZWN:=(F,In*H) compatible dimensions with (F,H) observable and F a nonsingular,^ stability matrix, i.e., the eigenvalues of F have magnitude less than 1. The transfer function of (4.1-1) is denoted by TWN(z). +In the discussion that follows the WN model parameters will be used to obtain a solution to the stochastic realization problem. Denham (1975) has shown that if the spectral factors of the PSD are of least degree, i.e., they possess no poles at the origin, then F is a nonsingular matrix 82 The corresponding measurement process is given by h = h + \ (4.1-2) where is the p measurement vector and v^ is a zero mean, white Gaussian noise sequence, uncorrelated with x.., j k with J Covtvj.Vj) = Covfw^Vj) = S5, j for R a pxp positive definite, covariance matrix and S a nxp cross covariance matrix. Thus, a model of this measurement process is completely specified by the quintuplet, (F,H,Q,R,S). When a correlation technique is applied to the measurement process, it is necessary to consider the state covariance defined by n^Covtx^,)^) We assume that the processes are wide sense stationary; therefore, \ nk = n, a constant here. It is easily shown from (4.1-1) that the state covariance satisfies the Lyapunov equation (LE) n = FIIFT + Q (4.1-3) It is well known (e.g. see Faurre (1967)) that since F is a stability matrix, corresponding to any positive semidefinite (covariance): matrix Q, there exists a unique, positive semidefinite solution n to the (LE). The measurement covariance is given (in terms of lag j) by cj:= Gov(%j-4) = Cov(^+j4,+Cov(Vj^)+Co'' 83 and from (4.1-1) it may be shown that C. = HFJ_1(FnHT+S) j > 0 (4.1-5) J Co = HnHT + R The PSD matrix of the measurement process is obtained by taking the bilateral z-transform of the sequence C. defined in (4,1-4) which gives 3 $z(z) = H(Iz-F)'1Q(Iz"1-FT)"1HT+H(Iz-F)1S+ST(Iz"1-FT)'1HT+R (4.1-6) It is important to note that this expression is the frequency domain representation of the measurement process which can alternately be expressed directly in terms of the measurement covariance sequence as 00 $z(z) = E C.Z_J j=-3 T Since the measurement process is stationary and z is real, C and therefore the PSD can be decomposed as 00 00 . Mz) = 2 c Z*J + C + E C z3 (4.1-7) L j=1 J 0 j=1 J Note that {C.> is analogous to the Markov sequence of the deterministic J realization problem. We define the problem of determining a quintuplet, (F,H,Q,R,S) in (4.1-6) from ^(z) or {Cj> as the stochastic realization problem. In this chapter we are only concerned with the realization from the measurement covariance sequence. When a realization algorithm is applied to the covariance sequence, we define the resulting realization as the Kalman-Szegb-Popov (KSP) model because of the parameter constraints 84 (to follow) which evolve from the generalized Kalman-Szego-Popov lemma a. (see Popov (1973) ). Thus, we specify the KSf3 model as the realization of {C.l defined by the quadruple, E^cd:=(A,B,C,D) of appropriate J 0 Iw r dimension with transfer function, TK<.p(z)=C(Iz-A)"^B+D. Note that since the unit pulse response of the KSP model is simply related to the measurement covariance sequence, then (4.1-7) can be written as the sum decomposition. *z(z) = T^pUJ+T^pU"1) = C(Iz-A)1B+D+DT+BT(Iz"1-AT)_1CT (4.1-8) The relationship between the KSP model and the stochastic realization of the measurement process is shown in the following proposition by Glover (1973). Proposition (4.1-9) Let zKSp=(A,B,C,D) be a minimal realization of {Cj}. Then the quintuplet (F,H,Q,R,S) is a minimal stochasti realization of the measurement process specified by (4.1-1) and (4.1-2), if there exists a positive definite, symmetric matrix n and TeGL(n) such that the following KSP equations are satisfied: n-AnAT * Q D+DT-CnCT = R B-AHC1" s where A=T-1FT and C=HT. The proof of this proposition is given in Glover (1973) and > ; -J- This book was published in Romanian ini966, but the English version became available in 1973. 4j* oo Note that the sequence, -CC^>Q is related to the measurement covariance sequence as Cn-hC and C-=C. for j > 0. 0 0 J J 85 corresponds directly to the results presented by Anderson (1969) in the continuous case. The proof follows by comparing the two distinct representations of $z(z) given by (4.1-6) and (4.1-8). Minimality of (F,H,Q,R,S) is obtained directly from Theorem (3.7-2) of Rosenbrock (1970). The KSP equations are obtained by equating the sum decomposition of (4.1-8) to (4.1-6). This proposition gives an indirect method to check whether a given KSP anc* stoc^ast'lc realization, (F,H,Q,R,S) correspond to the same covariance sequence. Attempts to use the KSP equations to construct all realizations, (F,H,Q,R,S) with identical {CL} from and T by choice of all possible symmetric, positive definite matrices, H will not work in general because all n's do not correspond to Q,R,S matrices that have the properties of a covariance matrix, i.e., A:= Cov( pw. V [wj vÂ¡]) . ST R. 6,. 0 (4.1-10) First, it is necessary to question if the stochastic realization problem always has a solution, or equivalently, when is there a n so that (4.1-10) holds. Fortunately, the well-known PSD property, 4>z(z) 0 on the unit circle (see e.g. Gokhberg and Krein (I960) and Youla (1961)) is sufficient to insure the existence of a solution. This result is available in the generalized Kalman-Szego-Popov lemma (see Popov (1973)). Proposition (4.1-11) If (F,H) is completely observable, then $z(z) 0 on the unit circle is equivalent to the existence of a quintuplet, (?,fr,$,$,^Â¡) such that 86 $z(z) [F(iz-f')'1 ip] 'V Q '(i2-1-f'T)-19r where a, Oj- Q s V O.T -W_ Ls RJ yj a. S R [VT WT] ^ 0 The proof of this proposition is given in Popov (1973) and essentially consists of showing there exists a spectral factorization of the given PSD. Thus, this proposition assures us that there exists at least one solution to the stochastic realization problem. Proposition (4.1-9) shows that once T, and n are determined then a stochastic realization, (F,H,Q,R,S) may be specified; however, it does not show how to determine n. Recently many researchers (e.g. Glover (1973), Denham (1974,1975), Tse and Weinert (1975)) have studied this problem. They were interested in obtaining only those solutions to the KSP equations of (4.1-9) which correspond to a stochastic realization such that A^O of (4.1-11). Denham (1975) has shown that any solution, n*, of the KSP equations which corresponds to a factorization as in (4.1-11) with V=KN, W=N for K=Knxp, NeKpxp, K full rank and N symmetric positive definite, is in fact a solution of a discrete Riccati equation. This can readily be seen by substituting, (Q,R,S) = (KNNTKT,NNT,KNNT) of (4.1-11) into (4.1-9) n*-An*AT = knnV d+dt-cji*ct = nnt (4.1-12) B-An*CT = KNNT for A = T~]FT, C=HT, TeGL(n) 87 T Solving the last equation for K and substituting for NN yields K = (B-An*CT) (D+DT-CII*CT)"1 (4.1-13) Now substituting (4.1-13) and NN^ in the first equation shows that n* satisfies n* = An*AT-(B-An*cT)(D+DT-cn*cT)"1(B-An*cT)T (4.1-14) a discrete Riccati equation. Thus, in this case the stochastic realization problem can be solved by (1) obtaining a realization, ^KSP ^rom'{Cj}.; (2) solving (4.1-14) for n*; (3) determining NN^ from (4.1-12) and K from (4.1-13); and (4) determining Q,R,S from K and NN1. A quintuplet specified by T and n* obtained in this' manner is guaranteed to be a stochastic realization, but at the computational expense of solving a discrete Riccati equation. Note that solutions of the Riccati equation are well known and it has been shown thpt there exists a unique, n*, which gives a stable, minimum phase, spectral factor (e.g. see Faurre (1970), Willems (1971), Denham (1975), Tse and Weinert (1975)). We will examine this approach more closely in a subsequent section, but first we must find an invariant system description for the stochastic realization. 4.2 Invariant 'System Description of the Stochastic Realization Suppose we obtain two stochastic realizations by different methods from the same PSD. We would like to know whether or not there is any way to distinguish between these realizations. To be more precise, we would like to know whether or not it is possible to uniquely characterize the class of all realizations possessing the same PSD. We first approach this problem from a purely algebraic viewpoint. 88 We define a set of quintuplets more general than the stochastic realizations, then consider only those transformation groups acting on this set which leave the PSD or equivalently (C .} invariant, and *3 finally specify various invariant system descriptions under these groups which subsequently prove useful in specifying a stochastic realization algorithm. The groups employed were first presented by Popov (1973) in his study of hyperstability. The results we obtain are analogous to those of Popov as well as those obtained in the quadratic optimization problem (e.g. see Willems (1971)). Define the set X2 = ((F,H,Q,R,S)| FeKnxn,HeKpxn,QeKnxri,ReKpxp,SeKnxp; Q,R symmetric} and consider the following transformation group specified by the set GKn := {L | LxKnxn; L symmetric} and the operation of matrix addition. Let the action of GK^ on X2 be defined by L t (F,H,Q,R,S) := (F,H,Q-FLFT+L,R-HLHT,S-FLHT) V (4.2-1) This action induces an equivalence relation on X2 written for each pair (F,H,Q,R,S), (F,H,Q,R,S)eX2 as (F,H,Q,R,S)EL(F,H,Q,R,S) iff there exists a LeGKn such that (F,H,Q,R,S) = L T(F,H,Q,R,S). This group and GL(n) are essential to this discussion, but we must consider their composite action. Therefore, we define the transformation group, GRn which is the cartesian product of GL(n) and GKn, GRn := GL(n)xGKn. The following proposition specifies GRn* 89 Proposition. (4.2-2) The closed set GRn and operation form a group where GRn = {(T,L) | TeGL(n);LeGKn} and the group operation is given by (T,L)o(T,L) = (TTfL+T"1LT"T). Proof. This proof of this proposition follows by verifying the standard group axioms with respective identity and inverse elements (In.0n) and (T_1,-TLTT).V Let the action of GR on X0 be defined by (T,L) 4- (F,H,Q,R,S) : = (TFT_1 ,HT~\t(Q-FLFT+L)TT,R-HLHT,T(S-.FLHT) ) (4.2-3) An element (F,h7q7R,1>) of the set is said to be equivalent to the element (F,H,Q,R,S) of X2 if there exists a (T,L)eGRn such that (F,H>'Q,R,S) = (T,L)4'(F,H,Q,R>S). This relation is reflexive (F,H,Q,R,S) = (InOn) + (F.H,Q,R,S) \ and symmetric (r1iTLTT)T(F,H,Q,RiS) = (r1>-TLTT)f((T,L)T(FsHsQ>R>S)) = ((T"1JLTT)o(T,L))+(F,H,Q,R,S)=(In,On)T(F,H,Q,R,S) . Transitivity follows from (F,H,Q,R,S)=(T,L)4'(F,HsQ,R,S) and (F,H,Q,R,S)=(T,Lj 4'(Tr,lT,^,R,S^) = (T,r)T((T ,L)+(F,H,Q,R,S.)) = (f ,L)4-(F,H,Q,R,S). Thus, GRn induces an equivalence relation on X2 which we denote by ETL and (4.2-3) defines the partitioning of X2 into classes. Note that our first objective has been satisifed, i.e., two EyL-equivalent quin tuplets have the same PSD; for if we let the pair (F,H,QRS), (F,H,Q,R,S)eX2 then if (T,L)eGRn 90 z(z)=K(Iz-?)-1^(Iz-1-?r)-1(Iz-?)-^+ST(I2-1-?T)-1K =(HT"1)T(Iz-F)'1T"1(T(Q-FLFT+L)TT)T"T(Iz"1-FT)"1TT(HT1)T + (HT"1)T(Iz-F)"1T"1T($-FLHT)+(ST-HLFT)TVT(Iz'1-FT)1TT(Ht1)T +R-HLH1 (4.2-4) or $z(z)=$z(z)+H(Iz-F)'1[L-FLFT-FL(Iz"1-FT)-(Iz-F)LFT-(Iz-F)L(Iz"1-FT)](Iz1-FT)~1HT % which gives $z(z) = $z(z). The measurement covariance sequence is also invariant under the action of GR on X0 because the PSD is also given by n 2 GO $z(z) = 2 C.z"'5. Thus, we will call any two systems represented by the j=- 3 quintuplet of Xp covariance equivalent, if they are E-^-equi valent. Clearly, any two covariance equivalent systems have identical PSD's (or measurement covariance sequences). Conversely, any two systems with identical PSD's are covariance equivalent (see Popov (1973) for proof). In order to uniquely characterize the class of covariance equi valent quintuplets we must determine an invariant system description for X0 under the action of GR The number of invariants may be found by counting the parameters. If we define, :=dim(F,H,Q,R,S) and p Mz:=dim(T,L), then there are M^=n +np+%n(n+l)+%p(p+l)+np parameters 2 specifying this quintuplet and GRn acts on l^n +J^n(n+1) of them; thus, there exist M^-Mz=2np+i2p(p+l) invariants. If we consider the transfor mation, (TD,L)eGRn to the Luenberger row coordinates, then np of these k n invariants specify the canonical pair (FR,H^) of (2.2-6). The action of (TR,L) on Q,R,S is given by 91 Qr = ^(Qrn^bCTRLVjtFt^^+LjT^QR-F^F^+L^ (4.2-5) Rr = (HTr_1)(TrLTrT)(HTR~1)T = R-HrLrHrT (4.2-6) SR = Tr(S(FTr""^ ) (TrLTrT) (HTr_1 )T) = Sr-FrLrHrT (4.2-7) where LR = TRLTRT, Fr = TrFTr*1, Hr = HTR-1, QR = TRQTRT, SR = TRS. The transformation LR acts on %n(n+l) parameters of the total %n(n+l)+%p(p+l)+np parameters available in Qr,R,Sr as shown above for the giveh (Fr,Hr). Once this action is completed the remaining np+*sp(p+l) parameters are invariants. There are only four possible ways that LR can act on the triple, (Q,R,S): (i) Lr acts only on QR; (4.2-8) (ii) Lr acts first to specify SR with the remaining elements of Lr acting on QR; (iii) L_ acts first to specify R with the remaining elements of K Lr free to act on QR or SR or both; and (iv) Lr acts on any combination of elements in Q,R,S. If we choose to restrict the action of LR to only the %n(n+l) elements of Qr, then for any choice of QR (given (Fr,Hr) and any QR), the transformation Lr is uniquely defined. Since FR is a nonsingular stability matrix, then it is well known (Gantmacher (1959)) from (4.2-5) that Lr is the unique solution of Lr-FrLrFrT = Q* for Q* = Qr-Qr* is important to note that the elements of QR are completely free, but once they are selected, LR is fixed by (4.2-5) for any QR and therefore the np elements of Â¥r and the Jsptp+l) elements of RR are the invariants. Thus, for a particular choice of QR we can uniquely specify the equivalence 92 class of under the action of GRn, i.e., (FR,HR,QR,RR,SR) is a canonical form for Ej^-equivalence on X2. On the other hand, if we choose to let LD act on the np elements K of SR, then from (4.2-7) only np-%p(p-l) elements of LR are uniquely specified, i.e., since LR is symmetric and LRHR f-l 4^+1 * +i ] Vi 1 there are *sp(p+l) redundant elements in the Â£.'s. Thus, for any choice J of (given (FR,HR) and any SR), np-Jgp(p-l) elements of LR are uniquely defined by (4.2-7) and the remaining elements of LR are free to act on QR In other words np-^p(p-l) elements of QR are invariants, as well as the elements of RR, since any choice of ifR specifies the elements of Lr in (4.2-6). Similarly restricting the action of LR to act on the elements of R specifies %p(p+l) elements of LR from (4-2-6) and we are free to allow the remaining elements of LR to act exclusively on QR or SR or both. Clearly, there are many choices available to distribute the action of LR on Qr,R,Sr; however, the important point is that once the choice is made, the invariants are specified. Any choice of symmetric (IR is acceptable, since LR is uniquely determined from (4.2-5) for given QR,FR, but this is not the case when an SR is selected. First recall that FR is nonsingular (see footnote p.81). Then if we define SR: =SR-S"R it follows from (4.2-7) that FR 1sR = LRHR (4.2-9) and then ^This was pointed out by Luo (1975) and Majumdar (1975). 93 HRFR_1S* = HRLRHRT (4.2-10) -1 * In general, HRFR SR is not symmetric; therefore, the set of acceptable S is restricted by (4.2-10). Since any square matrix can be decomposed as the sum of a symmetric and skew-symmetric matrix, i.e., hrfr 1sr = (hrfr 1sr^sym + (hrfr sr)skw A then from (4.2-10) ^sp(p-1) elements of SR are constrained to satisfy (for given (FR,HR)) (Hrfr Sr)$kw = 0p (4.2-11) We limit our discussion to only cases (i) and (ii) of (4.2-8) because the techniques employed to obtain the invariant system description will be used in the next section to determine a solution to the stochastic realization problem. Thus, we have satisfied our second objective, i.e., we have specified a unique characterization of covariance equivalent systemsan invariant system description for X2 under the action of GR \ n It is important to note that when QR is selected corresponding to case (i) of (4.2-8), then SR and R are uniquely specified, but when S"R is selected as in case (ii), R is again uniquely specified; however, this is not true for Q. There is a family of Q 1s which correspond to this S"R and R because only np-%p(p-l) elements of QR are fixed. Consider the following example which not only illuminates this point, but also shows how to uniquely characterize the class of covariance equivalent systems by determining an invariant system description corresponding to both (i) and (ii) of (4.2-8). 94 Example. (4.2-12) Suppose we are given the stochastic realization, (F,H,Q,R,S) as F = 0 1 o' ro i 01 rjin 5 -7 5" 0 0 1 , H = V JO) il -7 1 -1 _ 24 3 8 1 2 Lo i 1J _ 5 -1 2_ R = '2 i" " 1 1" J 4. * S = 0 0 _-l _1_ and we would like to obtain (FR,HR,QR,RR,SR) corresponding to (i) of (4.2-8) (1)Use the transformation, (TR,I) to obtain, (FR,HR,QR,R,SR) as -1 1 0 "l 0 0 "1 0 0 o o" 0 1 0 7 1 1 hr 0 1 0_ * qr = 0 o 1 0 0 3 6 6 7 SR -1 -1 1 L 4 24 2880-J 24 24 V (2)First, select a QR as 0 0 J-i u u 24 0 1 12 13 1 3051 2 4 12 ,144 0 then solving (4.2-5), we obtain and therefore "l f ' /V " 1 -T J 3_ li -1 -i 2 4 3 (3)If we choose to select an SR instead, corresponding to (ii) of (4.2-8) we must first determine the constraint imposed in (4.2-1). The choice 5^2 trivially satisfies this constraint; thus,we solve (4.2-5) for the (np-%p(p-l)) elements of LR 95 L R 1 1 -1 1 1 -1 -1 -1 33 and therefore 2 -1 0 RR 0 3 Or 1 ^2_Â£33^ (-It^33) This example illustrates two methods of specifying an invariant system description of the given stochastic realization. It also points out that selecting Q in (2) uniquely specifies R and S; however, selecting S in (3) uniquely fixes R, but not Q. Thus, there is an entire family of Q's which have the same R and S and each particular Q specifies a canonical form for (F,H,Q,R,S) on X_ under the action of GR . c n We must place these results into the proper perspective, since we are primarily concerned with the stochastic realization problem. Suppose Clearly, we are free to choose any coordinate system, (T,I_)eGR Once the coordinates are selected F and H are fixed from (4.1-9), since -1-1 F=TAT H=CT but the major problem of finding an not only such that the KSP equations are satisfied, but also so that A-0 still remains. The above methods of specifying an invariant system description partially resolve this problem. The first method shows that for given (F,H), the matrices R and S are fixed once a Q is specified; therefore, this quintuplet is an invariant system description, but whether or not it is a stochastic realization corresponding to the same PSD or {Cj} as can only be resolved by first determining if there exists a T and n such that the KSP 96 equations are satisfied. Clearly any choice of Q uniquely specifies a n for given F; for if, there exist two solutions and corresponding to identical Q,F, then n^=FH^F^+Q and n^FH^F^+Q. Subtract these equations to obtain n*-FII*F"I"=On for -n^. It is well known (e.g. see Gantmacher (1959)) that n*=0 is a unique solution of FH*-H*F*"^=0 , n n sinc^ A(F)^X(-F~T) in this case. Therefore, selecting a Q uniquely specifies a n and of course fixes R and S which can now be obtained from the remaining KSP equations. Practical considerations in selecting a Q which yields a positive definite II will be discussed in the next section. Here the point is for given F and H (modulo TcGKn)) are obtained from (4.1-9); moreover, selecting Q uniquely specifies a n which fixes R and S such that the KSP equations are satisfied. The resulting model, (F,H,Q,R,S) has the same PSD or equivalently {C.} as SKSP but ^ stl"^ may not satisfy A^O. Obtaining stochastic realizations such that the latter condition is satisfied is the subject of the next section. Similar results can be obtained by using the second method of specifying an invariant system description; however, recall in this case that only np-%p(p-l) elements of Q are uniquely specified. Since n is linearly related to Q for given F through the (LE), then the same number of elements are uniquely specified in n. Thus, when we select T=TR, the observability invariants of (2.2-4) uniquely specify the pair (FR,HR) and for any choice of QR (or alternately SR) we specify an invariant system description for the stochastic realization by (F^Hj^Qj^RpjS^). In the next section we develop an algorithm based on these techniques to obtain a stochastic realization. 97 4.3 Stochastic Realization Via Trial and Error In this section, we develop an algorithm-to obtain a stochastic realization from the measurement covariance sequence. We would like to find this realization directly in a form which uniquely characterizes the class of covariance equivalent quintuplets, i.e., quintuplets which have the same PSD or equivalently {C.}. From Section (4.1) we already know that one way to obtain a stochastic realization is to solve a discrete Riccati equation; however, this technique can become computa tionally burdensome when system order is large. Therefore, we would like to develop an algorithm to directly extract an invariant system description (under GR^) of the stochastic realization from the measurement covariance sequence which does not require a solution of the Riccati equation. We briefly recall the results of earlier chapters to obtain a canonical realization of the KSP model.. We show how constraints which evolve from the stochastic nature of the problem can be used to obtain a stochastic realization. The canonical realization of from {Cj> follows by recalling that the Hankel arrayadmits the factorization. SN,N' = VNWN* where Vj^jW^, are the corresponding observability and controllability matrices. Thus, by applying the canonical realization algorithm of (2.4-1) to {C.} we obtain the set [{v^}{6.-c+.}{a. }] which uniquely j 1 1ST J . specifies, (A,C)=(FR,HR) of (2.2-6) and B=GR of (2.2-14). ^Note that in realizing ien from (C.} we start with C, and not Cn; l\b r j y I U therefore, R is uniquely determined once ITH is found. 98 From Proposition (4.1-9), the observable pair.(A,C) of the KSP model and (F,H) of the WN model are E^-equivalent; therefore, the invariants are identical. The link between the canonical realization of ZKSp and the stochastic realization is provided by the KSP equations of (4.1-9), i.e., the (LE) and (A.B.C) = (FR,FRnRHRT+sR,HR) (4.3-1) D+DT = HRnRHRT+R Recall that under the action of GR^ on X^, there are 2np+^p(p+l) invariants--np specifying (FR,HR) and np+%p(p+l) specifying QR,R,SR. It is possible to extract these np+%p(p+l) invariants from the measurement covariance sequence using the KSP model realization, (A,B,C,D). As before, if we assume the action of LR is restricted to only the elements of QR, then for any choice of QR a unique nR is specified by the (LE) and therefore R and SR are uniquely obtained from (4.3-2) T R = W-DT-HRnRHRT SR B-FRnRHR On the other hand, suppose the action of LR is restricted to SR and Qr, then for any choice of SR, R and np-%p(p-l) elements of QR are fixed. Since nR is linearly related to QR through the (LE), the same number of elements are uniquely specified in nR. We are free to select the remaining elements in QR and n^. The realization invariants, }, of the previous chapters allow us to uniquely specify the invariants of (F,H,Q,R,S) from the KSP equations. Therefore, using EKSp and (4.1-9) we are able to extract the 2np+%p(p+l) invariants of (F,H,Q,R,S) from the measurement covariance sequence. 99 In many practical situations, it is known a priori that the system and measurement noise sequences are uncorrelated. This case has been considered by many researchers (e.g. Faurre (1967), Anderson (1969), Mehra (1971), Rissanen and Kailath (1972), etc.) and it corresponds to setting S.-o in the WN model of (4.1-1). It is crucial to note that with this choice of S, it appears that the only trans formation group which leaves the PSD invariant is GL(n). From (4.2-4) it is clear that GRn not just GL(n) must be considered; therefore, there are %n(n+l) fewer invariants when GRn rather than GL(n) acts on X2. Recall the first technique outlined in Section (4.2) to obtain (F,H,Q,R,S) from {Cj}: realize S^p, select a Q, specify a n from the (LE), and then find R and 5 from the KSP equations. The selection of a proper Q is essential to obtain a quintuplet of X2 that is a stochastic realization. Therefore, it is useful to consider constraints which evolve from the fact that Q and II,R,S are stationary covariance matrices. For given (FR,HR)+, each choice of QR uniquely specifies a nR and hence R,SR as in (4.3-2), i.e., (FR,HR,QR,R,SR) is a canonical form on X^ for E-^-equivalence. Since F is a stability matrix, it follows from stability theory that if (F,/Q)^ is completely controllable, corresponding to each Q^O there exists a unique positive definite solution n to the (LE). Therefore, restricting the choice of QR to be non-negative definite simultaneously satisfies this "stability constraint" as well as the fact that Qr must be a covariance matrix. The results of the generalized Kalman-Szego-Popov lemma of (4.1-11) assures us that there exists at least one realization such that A is a +Here we assume the action of Gl(n) is completed with T=TR. ++/q is any full rank factor of Q, i.e., QVQ/Q"*" 100 covariance matrix; thus, the condition A-0 reduces to det(Q-SR_1ST) ^ 0 (4.3-3) since R is a positive definite covariance matrix. On the other hand, if we consider the special case S=0p, then this constraint reduces the condition A-0 to det(Q) 0 for R > 0 (4.3-4) Thus, the choice of admissible Q,R,S must be restricted such that these constraints are satisfied. Recall that one possible choice is (Q,R,S) = (KNnV,NN^,KNNT) where K and NN^"'are specified by IT*, the unique solution of the discrete Riccati equation. Of course, if a canonical realization algorithm is applied to {C^}, then is found with T=TR, the Luenberger row coordinates, and (A,B,C)=(FR,GR,HR). If a positive semidefinite is selected, then a nR>0 is uniquely specified and therefore R and SR are found from the KSP equations. The quintuplet, (FR,HR,QR,R,'R) is an invariant system description (under GR ) and also a stochastic realization, if the above constraints are n satisfied. Note that the Riccati equation need not be solved. The following algorithm summarizes this technique as well as the alternate method discussed in Section (4.2). Stochastic Realization Algorithm (4.3-5) Step 1. Obtain from {Cj} as in (2.4-1). Step 2. Select a positive semidefinite Qp and solve the (LE) for nR. Step 3. Solve (4.3-2) for R and SR. TOT Step 4. Check that (4.3-3) is satisfied. If so, stop. If not, choose another QR-0 and go to 2. If numerous choices of simple Q^O do not yield a stochastic realization, solve the discrete Riccati equation of (4.1-13) and go to 3. Or Step 2* Select an SR satisfying (4.2-10). Step 3* Solve (4.3-2) for R and Check that det (R)>0. If so, continue. If not, go to 2*. Step 4* Determine QR from the (LE) and select its free elements to satisfy (4.3-4) if ^=0p or (4.3-3). If so, stop. If not possible select another simple SR, i.e., go to 2*, or try the first procedure, i.e., go to 2. Consider the following example which illustrates this algorithm. Example (4.3-6) For m=3, p=2 the measurement covariance sequence is 20 59 iT5T 3 13 0 9 " 1,5 0 1 9 13 150 4 1 TT 22 6 1 - TO T " 79 1 6W tl o o 3 7 w O II 66 25 ,1 6 ~ TT S* O ro H 13 9 1 6 0 0 " 1 9 3 6 0 0 c3 5 115 55 720 0 3 3 3 7 720 0 u. U 1 5577 6155 ~ 275717 561964 720 0 7200 C5 1036800 1036800 1 043 1 285588 21434293 4057079 _1 728 0 - 311040 12441600 " 124416 00. Applying the algorithm of (2.4-1) we obtain (1) The observability invariants are: v-|=l, V2=2 and 4 = t -i i i o] J r i i __7_ Li 2 L 4 j 2 4 12*^ 102 T T T T (2)The {a. } invariants are {a^ ,a2 ,a4 }; therefore, the KSP model is A=Fr 4 13 0 9 150 1 4 , B = 66 _ ~ _ 1 6 T5 4 1391 1 9 3 ' . 600 6 0 0 , C=Hn -1 J (3)Using the first approach of selecting a QR-0: (i) 0^=1^ and solving the (LE) gives n R 67787 1261 82 9 9' 6 6 0 0 6 6 0 3 3 0 0 126 1 9 3 1 997 660 3 3 0 1650 8299 997 6 0 1 3 30 0 16 50 3 3 0 (it) Solving (4.2-14) for R and SR gives ' * * 76 0 3 719 2419 59 6 6 0 0 66 0 22 0 0 660 and SR = 41 3 59 719 1379 ~ 3 3 0 0 _ 1 6 50 660 3 3 0 . 59 1 77 nsa : - 6 0 0 66 0 0 (4) det(I3-SRR"1SR^)>0 or using the second approach (5) Let SR=02> then B=FRnRHRT and - 1 6 09 150 = Cu-, n2] = 2 3 6 6 16 ~25 ~ *25 and 103 where 7 -2 -24" T 1609 "TFT 2 66 - nr 1 -6 -2 -24 and therefore nR = 2 3 16 - TF 1 0 6 8 1 6 0 -TT ~TT n22_ (i) From (4.2-2) R is R = C0-H^irH-r 3 1 1 4 and det R >0 (ii) Using the (LE) and (i) we have 0 0 i 3-n R 1 0 22 T2H22- 0 12n22" 6 143 il 1 T 2053 144 22 2880 (iii) Choose q22=1 so that (4.3-4) is satisfied, then n22~2 and Qr = diag(1 ,1 28 8 o) This example points out some very important facts. First, it follows that the measurement covariance sequence contains all of the essential information necessary to specify a stochastic realization. By choosing QR, nR is determined from the (LE) and the matrices R,SR are uniquely found from (4.3-2) for given (FR,HR). Alternately, selecting SD=0n, which satisfies (4.2-10), R is uniquely specified k p and np-%p(p-l) elements of QR and nR are invariants. The remaining elements of QR and nR are free. This example also shows that there may exist a measurement process with uncorrelated system and measurement noise (S=0p) equivalent (under GRn) to a model with correlated noise (S/0p), i.e., they both have identical PSD's or {C-}. r w 104 Thus, the Riccati equation solution has essentially been circumvented by this algorithm. However, if one not only desires a stochastic realization, but also a stable minimum phase, spectral factor, then the Riccati equation solution should be investigated. This the subject of our next section. 4.4 Stochastic Realization Via the Kalman Filter In this section we present a special case of the Riccati equation approach to solving the stochastic realization problem. This approach is a special case of the factorization (Denham (1975)) discussed in Section (4.1) because we require the unique, steady state solution to the discrete Riccati equation. It is well known (e.g. see Tse and Weinert (1975)) that the steady state solution uniquely specifies the optimal or Kalman gain. The significance of obtaining a stochastic realization via Kalman gain is twofold.,, First, since the Kalman gain is unique (modulo GL(n)), so is the corresponding stochastic realization. Second and even more important, knowledge of this gain specifies a stable, minimum phase spectral factor (e.g. see Faurre (1970) or Willems (1971)). The importance of this approach compared to that of the last section is that once the gain is specified, a stochastic realization is guaranteed immediately, while this is not true using the trial and error technique. However, the price paid for so elegant a solution to the stochastic realization problem is the computational burden of solving the Riccati equation. We use the innovations representation of the optimal filter and briefly develop it in the standard manner--from the estimation theory viewpoint. We then examine the realization of this model from the 105 measurement covariance sequence in the steady state, stationary case and show how this realization can be used to Represent the measurement process of (4.1-2). Care is taken to formulate this realization problem in precisely the same manner as the WN model of Section (4.1) in order to emphasize the striking similarity between these two distinctly different models of the same measurement process. Finally, we present the algorithm to solve the stochastic realization problem using the innovations representation. It should be emphasized that this technique was presented in Mehra (1971) and improved in Carew and Belanger (1973) and Tse and Weinert (1975). The basic filtering problem is to find the best minimum error covariance estimate of the state vector of the WN model in'terms of the currently available measurement sequence, z^. A convenient model used in the Kalman theory is the innovations (INV) representation given by ^k+l|k = F4|k-1 + Kk% (4.4-1) . /y- 4|k-l = H-k| k-1 \ -k = ^k | k-1 + ~k where _x, z, e_ are n state estimate vector, p measurement vector, and p innovation (of z) vector and is the optimal estimate of given z^.z^ ,... ,z^ . The innovations sequence, {e^} is a zero mean, white Gaussian process which is related to the WN model by % ~ H^kIk-1 + -k % where x^.j jjC_i is the error in the estimate of x^, given zQ,z^,... ,z^^ defined by 106 k| k-1 = h 4| k-1 and is the measurement noise. Note that the'{z^} of (4.4-1) is precisely the measurement process of (4.1-2); for if, we substitute the above expressions of ancl 2ik||<-l int0 (4*4-1) we obtain (4.1-2). The respective covariances of _x, x_, Â£ are denoted by the nxn % matrices, n, E, n and (R )^ is the innovation covariance which satisfies It is well known that the Kalman gain, K^, satisfies Kk= (FV|1+s)(Re>"1k (4*4-3) O/ ' where satisfies a discrete Riccati equation *k fVifT +1 Kk-i(Vk-iKk-i (4-4-4) The standard solution to the estimation problem is to solve (4.4-4) and (4.4-2) for nk and then to calculate the corresponding Kalman gain, K^ from (4.4-3). Since the observable pair (F,H) is known, the state estimate is updated using the INV model. If we consider the stationary, steady state case, then K^ = K^ = ... = K, 11^ = n^-j = ... = H, and therefore, (R ). = (R ). = ... = R The stationary, steady state, S K Â£ K** I Â£ INV model is given by the quadruple, E^y = (F,K,H,Ip) and RÂ£ with K the steady state Kalman gain and RÂ£ the innovation covariance. The transfer function is given by T^U) = H(Iz-F)\ +. I The Kalman filter accepts as inputs the current measurement sequence of the WN model and has as its.state the best minimum error covariance estimate of the corresponding state vector. 107 The stochastic realization problem can be reformulated in terms of the INV model in precisely the same manner as the WN model of (4.1-1) and (4.1-2). Thus, expressions analogous to (4.1-4) and (4.1-5) can be derived and therefore the measurement covariance sequence is alternately given (in terms of lag j) by C. = HFJ_1(FnHT+KR ) j>0 J Â£ Cn = HilHT+R 0 e where the state estimate covariance matrix n satisfies /s ^ T T . n-FnF = kr.k' e analogous to the (LE) of (4.1-3) in n. The PSD matrix of the measurement process in terms of the INV model is obtained in precisely the same way as (4.1-6); therefore, we have , $7(z)=H(Iz-F)_1KR KT(Iz1-FT)1HT+H(Iz-F)"1KR +R KT(Iz"1-FT)_1HT+R L e e e e \ (4.4-7) It is also possible to express the PSD in factored form; thus, by simple manipulation (4.4-7) can be written as $z(z)=CH(Iz-F)'1K+Ip]Re[Ip+KT(Iz*1-FT)"1HT>TINV(z)ie(z)T]NV(z"1) (4.4-8) Since R >0, the PSD is positive definite and (4.1-11) is always satisfied; Â£ therefore, we can specify a stochastic realization immediately, once F,H,K,R are determined. This stochastic realization is defined by Â£ e e Ze := (FH>QINVsRINV>SINV) = (F>H*KReKTRe>KRe) (4*4-9) (4.4-5) (4.4-6) where 108 Note that this realization is just a special case of the factorization of Denham (1975) discussed in Section (4.1) with n*=n in (4.1-13), K the Kalman gain in (4.1-12) and RÂ£=NN^. Clearly, the relationship between the canonical realization of E^p and EÂ£ of (4.4-9) is provided by the KSP equations, i.e., (4.4-6) and (A,B,C) (FR,FRHRHR +(Sinv)rHr) (4.4-10) D+D = HRnRHR+RINV Note that since K and R are unique, then E is unique (e.g. see Tse and Weinert (1975) or Denham (1975) for proof). Therefore, it is futile to attempt to determine EÂ£ from the trial and error algorithm of (4.3-5) because this quintuplet is a unique stochastic realization (modulo GL(n)). Recall from Section (4.2), if we let T=TR, then np of the total 2np+%p(p+l) invariants specify the pair (FR,HR) and it follows that np specify Kr and %p(p+l) specify RÂ£. Thus, the canonical realization of 'Xj 'X/ ^ the INV model is analogous to the WN model; however, unlike the QR,RR,SR obtained in the WN case by trial and error, (QINV)R> RjNV> ^Sinv^R are uniquely specified by KD and R in (4.4-9). The following diagram summarizes the relationship between these two distinct approaches to obtaining a stochastic realization. REALIZATION FROM {C^ J PSD KSP (Trial and Error) FACTORIZATION METHODS E INV (Riccati Equation) (F,H,Q,R,S) Figure 3, Solutions of the Stochastic Realization Problem 109 As a matter of completeness, we would like to briefly present an algorithm to obtain the stochastic realization using the Riccati equation approach. Mehra (1970,1971), Carew and Belanger (1973) and even more recently Tse and Weinert (1975) have proposed iterative schemes to obtain n,K,RÂ£, but the theoretical connection to the KSP equations and the stochastic realization was never established. Their results are summarized below and we refer the interested reader to these references for a detailed discussion of convergence properties and simulation results. Iterative Solution to the Riccati Equation (4.4-11) Step 1. Set nQ = 0 Step 2. (RÂ£).-. = D+D^-Hn.HT, where i is the i-th iteration step. Step 3. K. = (B-Fn.HT)(R )T] 1 1 Â£ 1 Step 4. n.,, = Fn-FT+K.(R ).kI K i+l i i e i i Once K and R are found in this manner, then the stochastic realization e follows from (4.4-9). The following algorithm: summarizes the realization technique using the INV model. < Stochastic Realization Algorithm via INV Model (4.4-12) Step 1. Obtain ZKSp from {C^} as in (2.4-1). Step 2. Use the iterative technique of (4.4-11) to obtain K, RÂ£. Step 3. Determine QINV,RjNy,SINV from K and RÂ£ as in (4.4-10). Thus, we have two algorithms to obtain an invariant system description for the stochastic realization using either the lsiN model or the INV model. The following figure summarizes these techniques. * no Figure 4. Stochastic Realization Algorithms, CHAPTER 5 CONCLUSIONS 5.1 Summary This dissertation has contributed results in realization theory for both deterministic and stochastic cases. It was shown that by carefully specifying the invariants of the realization problem under a change of basis in the state space that a simple and efficient algorithm to extract these entities from the Markov sequence could be developed. This technique provides a solution to the realization problem directly in a canonical form, and an invariant system description under this transformation group is.'specified. The partial realization problem was solved by modifying this technique to develop a nested algorithm. It was shown that this method specifies the class of minimal partial canonical realizations. A new recursive technique to determine the corresponding class of minimal extensions while conserving all degrees of freedom available was developed. These results bridge the gap between the more classical approach of constructing a minimal extension and that of extracting the realization invariants. The characteristic equation is determined from the transition matrix in a convenient coordinate system by inspection. These coordinates were easily obtained from the given solution to the partial realization problem. In the stochastic realization problem it was shown that the transformation group which must be considered is richer than the general 111 112 linear group of the deterministic problem. The equivalence class under this group was specified and it was shown how the additional constraints imposed by the stochastic realization further restrict the selection of free parameters available in the corresponding noise covariance matrices. Specifying the invariants under this transformation group enabled the development of a trial and error algorithm to obtain a stochastic realization without requiring a Riccati equation solution. The link between the KSP, WN and steady state Kalman filter was presented. It was shown that realization of the KSP model allowed both representations to be determined. It was shown that determination of the filter parameters uniquely specifies a stochastic realization. An algorithm requiring the solution of a Riccati equation was also presented. 5.2 Suggestions for Future Research The results given in this dissertation open several interesting * possibilities for future research. Applying the algebraic framework of a transformation group acting on a set offers definite advantages over unstructured approaches. Simple equivalent solutions which confirm physical intuition may evolve. It may be possible to specify a set of invariants under the action of this transformation group which yields considerable insight into the problem structure. If the problem possesses additional constraints, it may be possible to utilize this information to influence the choice of free parameters available. Many problems of current interest can be examined in this framework (e.g. identification, exact model matching, and stable observer design problems). Efficient covariance estimators should be examined in order to facilitate the development of realization algorithms which yield useful 113 results with realistic noisy data. Along these lines the use of maximum likelihood estimators by Caines and Rissanen (1974) and the least squares estimates in the technique of Majumdar (1976) should be investigated further. The use of Markov sequences to design controllers to solve the model following problem (e.g. see Moore and Silverman (1972)) should be examined by first defining the problem invariants and then inves tigating the possibility of using the realization algorithms of Chapters 2 or 3 to extract them. The use of the class (under GL(n)) of minimal extension sequences developed directly from a given finite sequence may prove instrumental in this technique and should be studied. An efficient technique to factor Toeplitz matrices (see Rissanen and Kailath (1972)) should be developed by extracting the invariants of the stochastic realization specified in Chapter 4. Analogous techniques for the equivalent frequency domain representation of this problem should also be investigated. REFERENCES J. E. Ackermann [1972] "On Partial Realizations," IEEE Trans, on Auto. Contr., Vol. AC-17, pg. 381. [1975] "On the Synthesis of Linear Control Systems with Specified Characteristics," Proc. 1975 IFAC Congress, Boston, Mass., pp. 88-92. J. E. Ackermann and R. S. Buey [1971] "Canonical Minimal Realization of a Matrix of Impulse Response Sequences," Inf, and Contr., Vol. 19, pp. 224-231. H. Akaike [1974a] "Stochastic Theory of Minimal Realization," IEEE Trans, on Auto. Contr., Vol. AC-19, pp. 667-674. [1974b] "A New Look at the Statistical Model Identification," IEEE Trans, on Auto. Contr., Vol. AC-19, pp. 716-723. [1975] "Markovian Representation of Stochastic Processes by Canonical Variables," SIAM J. on Contr., Vol. 13, pp. 162-173. > B. D. 0. Anderson [1967a] "A System Theory Criterion for Positive Real Matrices," SIAM J. on Contr., Vol. 5, pp. 171-182. [1967b] "An Algebraic Solution to the Spectral Factorization Problem," IEEE Trans, on Auto. Contr., Vol. AC-12, pp. 410-414. [1969] "The Inverse Problem of Stationary Covariance Generation," J. of Statistical Physics, Vol. 1, pp. 133-147. B. D. 0. Anderson and S. Vongpanitlerd [1973] Network Analysis and Synthesis, Prentice-Hall Inc., New Jersey. B. M. Anderson, F. M. Brasch, and P. V. Lopresti [1975] "The Sequential Construction of Minimal Partial Realizations from Finite Input Output Data," SIAM J. on Contr., Vol. 13, pp. 552-570. D. R. Audley and W.J. Rugh [1973] "On the H-Matrix System Representation," IEEE Trans, on Auto. Contr., Vol. AC-18, pp. 235-242. D. R. Audley, S. L. Baumgartner and W. J. Rugh [1975] "Linear System Realization Based on Data Set Representations," IEEE Trans, on Auto Contr., Vol. AC-20, pg. 432. 134 115 R. W. Bass and I. Gura [1965] "High Order System Design via S.tate-Space Considerations," Proc. Joint Auto. Contr. Conf., Rensselaer, N. Y., pp. 311-318. C. Bonivento, R. Guidorzi, and G. Marro [1973] "Irreducible Canonical Realizations from External Data Sequences," Int. J. Contr., Vol. 17, pp. 553-563. F. M. Brausch and J. B. Pearson [1970] "Pole Placement Using Dynamic Compensators,"IEEE Trans, on Auto. Contr., Vol. AC-15, pp. 34-43. P. Brunovsky [1966] "On Stabilization of Linear Systems Under a Certain Class of Persistent Perturbations," Differential Equations, Vol. 2, pp. 401-406. [1970] "A Classification of Linear Controllable Systems," Kibern., Vol. 3, pp. 173-187. R. S. Buey [1968] "Canonical Forms for Mutivariable Systems," IEEE Trans, on Auto. Contr., Vol. AC-13, pp. 567-569. M. A. Budin [1971] "Minimal Realization of Discrete Linear Systems from Input-Output Observations," IEEE Trans, on Auto. Contr., Vol. AC-16, pp. 305-401. [1972] "Minimal Realization of Cohtinuous Linear Systems from Input-Output Observations," IEEE Trans, on Auto. Contr., Vol.. AC-17, pp. 252-253. T. E. Bullock and J. V. Candy [1974] "Modeling of Wind Tunnel Noise Using Spectral Factorization and Realization Theory," Proc. IEEE Southeastcon, Orlando, Florida. P. E. Caines and J. Rissanen [1974] "Maximum Likelihood Estimation of Parameters in Multi- variable Gaussian Stochastic Processes," IEEE Trans. Inform. Theory, Vol. IT-20, pp. 102-104. J. V. Candy, M. E. Warren and T. E. Bullock [1975] "An Algorithm for the Determination of System Invariants and Canonical Forms," Proc. 1975 Southeastcon, Auburn, Alabama. B. Carew and P. R. Belanger [19731 "identification of Optimum Filter Steady-State Gain for Systems with Unknown Noise Covariances," IEEE Trans, on Auto. Contr., Vol. AC-18, pp. 582-587. 116 C. T. Chen and D. P. Mital [1972] "A Simplified Irreducible Realization Algorithm," IEEE Trans, on Auto. Contr., Vol. AC-17, pp. 535-537. A. Alonso-Concheiro [1973] "A Complete Set of Independent Decoupling Invariants," Int. J. Contr., Vol. 18, pp. 1211-1220. M. C. Davis [1963] "Factoring the Spectral Matrix," IEEE Trans, on Auto. Contr., Vol. AC-8, pp. 296-305. L. S. De Jong [1.975] Numerical Aspects of Realization Algorithms in Linear Systems Theory, Ph.D. Dissertation, Tech. Univ. of Eindhoven, Netherlands. M. J. Denham [1974] "Canonical Forms for the Identification of Multivariable Linear Systems," IEEE Trans, on Auto. Contr., Vol. AC-19, pp. 646-656. [1975] "On the Factorization of Discrete-Time Rational Spectral Density Matrices," IEEE Trans, on Auto. Contr., Vol. AC-20, pp. 535-536. B. W. Dickinson, M. Morf, and T. Kailath [1974a] "A Minimal Realization Algorithm for Matrix Sequences," IEEE Trans, on Auto. Contr,, Vol. AC-19, pp. 31-38. B. W. Dickinson, T. Kailath, and M. Morf [1974b] "Canonical Matrix Fraction and State-Space Descriptions for Deterministic and Stochastic Linear Systems," IEEE Trans, on Auto. Contr., Vol. AC-19, pp. 656-667. V. N. Faddeeva [1959] Computational Methods in Linear Algebra, Dover Pubs.,. - New ..York. P. Faurre [1967] Representation of Stochastic Processes, Ph.D. Dissertation, Stanford University, California. [1970] "Identification par Minimisation d'une Representation Markovienne-de Processus Aleatoire," Lecture Notes in Mathematics, Vol. 132, Springer, Mew 'York,/ pp- 85-106. E. E. Fisher . [1965] "The Identification of Linear Systems," Joint Auto. Contr. Conf, Preprints, pp. 473-475. G. D. Forney [1975] "Minimal Bases of Rational Vector Spaces with Applications to Multivariable Linear Systems," SIAM J. on Contr., Vol. 17, pp. 192-212. K. Furuta and J. G. Paquet [1975] "Determination of Matrix Transfer Function in the Form of Matrix Fraction from Input-Output Observations," IEEE Trans, on Auto. Contr., Vol. AC-20, pp 392-396. F. R. Gantmacher [1959] The Theory of Matrices, Vols.T and 2, Chelsea Publishing Co.,N.Y M. R. Gevers and T. Kailath [1973] "An Innovations Approach to Least Squares Estimation- Part VI: Discrete-Time Innovations Representation and Recursive Estimation," IEEE Trans, on Auto. Contr., Vol. AC-18, pp. 588-600. E. G. Gilbert [1963] "Controllability and Observability in Multivariable Control Systems," SIAM J. on Contr., Vol. 1, pp. 128-151. [1969] "The Decoupling of Multivariable Systems by State Feedback," SIAM J. on Contr., Vol.7, pp. 51-63. K. Glover [1973] "Structural Aspects of System Identification," Rep. ESL-R-516, Electronic Systems Laboratory, M.I.T., Cambridge, Mass. K. Glover and J. Willems [1974] "Parameterizations of Linear Dynamical Systems: Canonical Forms and Identifability," IEEE Trans, on Auto. Contr., Vol. AC-19, pp. 640-646. S. S. Godbole [1972] "Comments on 'Contruction of Minimal Linear State- Variable Models from Finite Input-Output Data,'" IEEE Trans, on Auto. Contr., Vol. AC-17, pp. 173-175. I. Gokhberg and M. G. Krein [1960] "Systems of Integral Equations on a Half Line with Kernels Depending on the Difference of Arguments," Uspekh: Mat. Naut. 13, pp. 217-287. B. Gopinath [1969] "On the Identifcation of Linear Time Invariant Systems from Input-Output Data," Bell Syst. Tech. J., Vol. 48, pp. 1101-1113. 118 R. D. Gupta and F. W. Fairman [1974] "Parameter Estimation for Multivariable Systems," IEEE Trans, on Auto. Contr., Vol.'AC-19, pp. 546-549. B. L. Ho and R. E. Kalman [1966] "Contruction of Linear State Variable Models from Input/ Output Functions," Regelungstechnik, VoT. 14, pp. 545-548. K. Hoffman and R. Kunze f1971] Linear Algebra, Prentice-Hall Pubs., Second Edition, New Jersey H. L. Huang [1974] "A Generalized-Jordan-Form-Approach One-Step Irreducible Realization of Matrices," IEEE Trans, on Auto. Contr., Vol. AC-19, pp. 271-272. C. D. Johnson and W. M. Wonham [1964] "A Note on the Transformation to Canonical (Phase-Variable) Form," IEEE Trans, on Auto. Contr., Vol. AC-9, pp. 312-313. R. E. Kalman [1960] "Control System Analysis and Design Via the Second Method of Lyapunov," J. of Basic Engr., Vol. 82D, pp. 394-499. [1961] "A New Approach to Linear Filtering and Prediction Problems," J. of Basic Engr., Vol. 82D, pp. 35-43. [1962] "Canonical Structure of Linear Dynamical Systems," Proc. Nat. Acad, of Sci. (USA), Vol. 48, pp. 596-600. [1963] "Mathematical Description of Linear Dynamical Systems," SIAM J. on Contr., Vol. 1, pp. 152-192. [1964] "Lyapunov Functions for the Problem of Lure in Automatic Control," Proc. Nat. Acad. Sci. (USA), Vol. 49, pp. 201-205. [1965] "Linear Stochastic Filtering Theory-Reappraisal and Outlook," Proc. of Symposium of System Theory, Brooklyn, N. Y., pp. 197-205. [1971a] "Kronecker Invariants and Feedback," Proc. 1971 Conf. on Ord. Diff. Eg., National Research Laboratory Mathematics Research Center, Washington, D. C. [1971b] "On Minimal Partial Realizations of a Linear Input/Output Map," in Aspects of Network and System Theory, R. E. Kalman and N. De Claris (eds.), pp. 385-407, Holt, Rinehart and Winston., N.Y. [1973] "Global Structure Classes of Linear Dynamical Systems," presented at the NATO ASI Geometric and Algebraic Methods for Nonlinear Systems, London, England. [1974] Class Notes on System TheoryCourse by R. E. Kalman at the University of Florida, Gainesville, Fla. * 119 R. E. Kalman and R. S. Buey [1961] "New Results in Linear Filtering and Prediction Theory," J. of Basic Engr. Vol. 83, pp. 95-108. R. E. Kalman, P. L. Falb, and M. A. Arbib [1969] Topics in Mathematical System Theory, McGraw-Hill, Inc. N.Y M. Lai, S. C. Puri, and H. Singh [1972] "On the Realization of Linear Time-Invariant Dynamical Systems," IEEE Trans, on Auto. Contr., Vol. AC-17, pp. 251-252: C. E. Langenhop [1964] "On the Stabilization of Linear Systems," Proc. Amer. Math. Soc., Vol. 15, pp. 735-742. G. Ledwich and T.E. Fortmann [1974] "Comments on 'On Partial Realizations'," IEEE Trans, on Auto. Contr., Vol. AC-19, pp. 625-626. D. G. Luenberger [1966] "Observers for Multivariable Systems," IEEE Trans, on Auto. Contr., Vol. AC-11, pp. 190-197. [1967] "Canonical Forms for Linear Multivariable Systems," IEEE Trans, on Auto. Contr., Vol. AC-12, pp. 290-293. Z. Luo [1975] Discrete Kalman Filtering and Stochastic Identification Using a Generalized Companion Form, Ph.D. Dissertation, Univ. of Florida, Gainesville, Florida. Z. Luo and T.E. Bullock [1975] "Discrete Kalman Filtering Using a Generalized Companion Form," IEEE Trans. Auto. Contr., Vol. AC-20, pp. 227-230. A. Majumdar 1975] Private communication. 1976] Modeling and Identification of the Nerve Excitation Phenomena, Ph.D. Dissertation, Univ. of Florida, Gainesville, Fla. D. Q. Mayne [1968] "Computational Procedure for the Minimal Realization of Transfer-Function Matrices," Proc. IEEE, Vol. 115, pp. 1363-1368. [1972a] "Parameterization and Identification of Linear Multi- variable Systems," Lecture Notes in Mathematics, Springer, Berlin, Vol. 294, pp. 56-61. [1972b] "A Canonical Model for the Identification of Multi- variable Linear Systems," IEEE Trans, on Auto. Contr., Vol. AC-17, pp. 728-729. 120 R. K. Mehra [1970] "On the Identification of Variances and Adaptive Kalman Filtering," IEEE Trans. Auto. Contr., Vol. AC-15, pp. 175-184. [1971] "On-Line Identification of Linear Dynamic Systems with Applications to Kalman Filtering," IEEE Trans, on Auto. Contr., Vol. AC-16, pp. 12-20. D. P. Mi tal and C. T. Chen [1973] "Irreducible Canonical Form Realization of a Rational Matrix," Int. J. Contr., Vol. 18, pp. 881-887. B. C. Moore and L. M. Silverman [1972] "Model Matching by State Feedback and Dynamic Compen sation," IEEE Trans on Auto. Contr,,Vol. AC-17, pp. 491. A. S. Morse [1973] "Structural Invariants of Linear Multivariable Systems," SIAM J. on Contr., Vol. IT, pp 446-465. P. R. Motyka and J. A. Cadzow [1967] "The Factorization of Discrete-Process Spectral Matrices," IEEE Trans, on Auto. Contr., Vol. AC-12, pp 698-706. R. W. Newcomb [1966] Linear Multiport Synthesis, Me Graw-Hill, New York. ' S. P. Panda and C. T. Chen - [1969] "Irreducible Jordan Form Realization of a Rational Matrix," IEEE Trans, on Auto. Contr., Vol.AC-14, pp. 66-69. V. M. Popov [1961] "Absolute Stability of Nonlinear Systems of Automatic Control," Avt. i Telemekh., Vol. 22, pp. 961-979. [1964] "Hyperstability and Optimality of Automatic Systems With Several Control Functions," Rev. Roum. Sci. Tech. Electr. Enerqetique, Vol. 10, pp. 629-690. [1969] "Some Properties of the Control Systems with Irreducible Matrix Transfer Functions," in Seminar on Pifferentia! Equations and Dynamical Systems-II. Springer,; Berlin, pp. 169-180. [1972] "Invariant Description of Linear, Time-Invariant Controllable Systems," SIAM J. On Contr., Vol. 10, pp. 252-264 [1973] Hyperstability of Control Systems, Springer-Verlag, N.Y. J. Rissanen [1971] "Recursive Identification of Linear Systems," SIAM J. on Contr., Vol. 9, pp 420-430. [1972a] "Realization of Matrix Sequences," IBM Research Report, RJ 1032, San Jose, Calif. [1972b] "Recursive Evaluation of Pade Approximants for Matrix Sequences," IBM J. df Res. and Dev., Vol. 16, pp. 401-406. 127 [1974] "Basis of Invariants and Canonical Forms for Linear Dynamic Systems," Automtica, Vol. 10, pp. 175-182. [1975] "Canonical Markovian Representations and Linear Prediction," Proc. 1975 IFAC Congress, Boston, Mass. J. Rissanen and T. Kailath [1972] "Partial Realizations of Random Systems," Automtica, Vol. 8, pp. 389-396. J. Roman [1975] Low Order Observer Design Via Realization Theory,: Ph.D. Dissertation, Univ. of Florida, Gainesville, Florida. J. Roman and T. E. Bullock [1975a] "Minimal Parital Realizations in a Canonical Form," IEEE Trans, on Auto. Contr., Vol. AC-20, pp. 529-533. [1975b] "Design of Minimal Order Stable Observers to Estimate Linear Functions of the State via Realization Theory," IEEE Trans, on Auto. Contr., Vol. AC-20, pp. 613-623. H. H. Rosenbrock [1970] "State-Space and Multivariable Theory, John Wiley and Sons, Inc. N.Y. C. A. Roveda and R. M. Schmid [1970] "Upper Bound on the Dimension of Minimal Realizations of Linear Time Invariant Dynamical Systems," IEEE Trans, on Auto. Contr., Vol. AC-15, pp. 639-644. P. Rozsa and N. Sinha [1975] "Minimal Realization of a Transfer Function Matrix in Canonical Forms," Int. J. Contr., Vol. 21, pp. 273-284. G. N. Saridis and R. Lobbia [1972] "Parameter Identification and Control of Linear Discrete Time Systems," IEEE Trans, on Auto. Contr., Vol. AC-17, pg. 491. L. M. Silverman [1966] "Transformation of Time Variable Systems to Canonical Form," IEEE Trans, on Auto. Contr.., Vol. AC-11, pp. 300- 303. [197T] "Realization of Linear Dynamical Systems," IEEE Trans, on Auto. Contr,, Vol. AC-16, pp. 554-567. L. M. Silverman and H. E. Meadows [1966] "Equivalence and Synthesis of Time Variable Linear Systems," Proc 4-th Allerton Confr. Circuit and System Theory, pp. 776-784. 122' M. G. Strintzis [1972] "A Solution to the Matrix Factorization Problem," IEEE Trans, on Info. Th,y., Vol. IT-18, pp. 225-232. G. Szeg'd and R. E. Kalman [1963] "Sur la Stabilite Absolve d'un Systeme D'Equations aux Differences Finies," Compte Rendus a L'Academie des Sciences, pp. 388-390. A. J. Tether [1970] "Construction of Minimal Linear State-Variable Models from Finite Input-Output Data," IEEE Trans, on Auto. Contr., Vol. AC-15, pp. 427-436. E. Tse and H. L. Weinert [1973] "Extension of 'On the Identifiability of Parameters'," IEEE Trans, on Auto. Contr., Vol. AC-18, pp. 687-688. [1975] "Structure Determination and Parameter Identification for Multivariable Stochastic Linear Systems," IEEE Trans, on Auto. Contr., Vol. AC-20, pp. 596-603. W. Tuel [1966] "Canonical Forms for Linear Systems-Pt. 1," IBM Res. Lab., RJ 375, San Jose, Calif. [1967] "An Improved Algorithm for the solution of Discrete Regulation Problems," IEEE Trans, on Auto. Contr., Vol. AC-12, pp. 522-528. S. H. Wang and E. J. Davison [1972] "Canonical Forms of Linear Multivariable Systems," Univ. Toronto, Toronto, Canada, Control Syst. Rept. 7203. M. E. Warren arid A. E. Eckberg [1973] "On the Dimensions of Controllability Subspaces: A Characterization via Polynomial Matrices and Kronecker Invariants," 1974 JACC Preprints, Austin, Texas, pp. 157- 163. H. Weinert and J. Anton [1972] "Canonical Forms for Multivariable System Identification," Proc. Conf. Decision and Contr., New Orleans, La. N. Wiener [1955] "On the Factorization of Matrices," Comment. Math. Helv., Vol. 29, pp. 97-111. [1959] "The Prediction Theory of Multivariate Stochastic Processes: I-The Regularity Condition," Acta. Math., Vol. 98, pp. 111-150. J. C. Willems [1971] "Least Squares Stationary Optimal Control and the Algebraic Ricatti Equation," IEEE Trans, on Auto. Contr., Vol. AC-16, pp. 621-634. W. A. Wolovich [1972a] "On the Synthesis of Multivariable Systems," 1972 JACC Pre prints, Stanford, Calif, pp. 158-165. [1972b] "The Use of State Feedback for Exact Model Matching," SIAM J. on Contr., Vol. 10, pp. 512-523. [1973a] "Frequency Domain State Feedback and Estimation," Int. J. Contr., Vol. 17, pp. 447-428. [1973b] "Multivariable System Synthesis with Step Disturbance Rejection," Proc. 1973 IEEE CDC, San Diego, Calif., pp 320- 325. W. M. Wonham and A. S. Morse [1972] "Feedback Invariants of Linear Multivariable Systems," Automtica, Vol. 8, pp. 93-100. V. A. Yakubovich [1963] "Absolute-Stability of Nonlinear Control Systems in Critical Cases," Avt. i Telem., Vol. 24, pp. 293-303. D. C. Youla [1961] "On the Factorization of Rational Matrices," IRE Trans, Info. Thy., Vol. IT-7, pp 172-189. D. C. Youla and P. Tissi [1966] "N-port Synthesis via Reactance Extraction," IEEE Int. Conv. Rec., Vol. 14, pp. 183-205. L. A. Zadeh and C. A. Desoer F19631 Linear System Theory: The State Space Approach., McGraw- BIOGRAPHICAL SKETCH James Vincent Candy was born in Astoria, New York on January 21, 1944 He graduated from Holy Cross High School, Flushing, New York in June, 1961 He received the degree of Bachelor of Science in Electrical Engineering in June, 1966 from the University of Cincinnati, Cincinnati, Ohio. Upon graduating he worked with the General Electric Company for 9 months. Then he enlisted in the Air Force of the United States in April, 1967. He received a commission as a Second Lieutenant in June, 1967 after completion of Officers Training School at Lackland AFB, Texas. He spent the majority of his four years' active duty at Eglin AFB, Fla. as a Threat Systems Engineer and Test Director until separated in June, 1971 as a Captain. In January 1968, he began study at the University of Florida Extension School (GENEYSIS) for a Master of Science Degree in Electrical Engineering. He completed his residency requirements in March, 1972 and received the M.S.E. from the University of Florida. From March, 1972, until the present time he has done work toward the degree of Doctor of Philosophy. James Vincent Candy is married to the former Patricia Meyers and they have one lovely daughter, Kirstin Patrice. He is a member of Phi Kappa Theta, Phi Kappa Phi, Eta Kappa Nu and the Institute of Electrical and Electronics Engineers. 124 I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Thomas E. Bullock, Chairman Professor of Electrical Engineering I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as/a dissertation for the-^degree of Doctor of Philosophy. MTchael E. Warrenchairman Assistant Professor of Electrical Engineering I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Donald G. Childers Professor of Electrical Engineering I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Zoran R. Pop/Stojanovic Professor of Mathematics I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. 'i ufa opov Vasile M. Popov Professor of Mathematics This dissertation was submitted to the Graduate Faculty of the College of Engineering and to the Graduate Council, and was accepted as partial fulfillment of the requirements for the degree of Doctor of Philosophy. Dean, Graduate School Internet Distribution Consent Agreement In reference to the following dissertation: AUTHOR: Candy, James TITLE: Realization of Invariant system descriptions from markov (record number: 180838) PUBLICATION DATE: 1976 I, \i V O f\C>l as copyright holder for the aforementioned dissertation, hereby grant specific and limited archive and distribution rights to the Board of Trustees of the University of Florida and its agents. I authorize the University of Florida to digitize and distribute the dissertation described above for nonprofit, educational purposes via the Internet or successive technologies. This is a non-exclusive grant of permissions for specific off-line and on-line uses for an indefinite term. Off-line uses shall be limited to those specifically allowed by "Fair Use" as prescribed by the terms of United States copyright legislation (cf, Title 17, U.S. Code) as well as to the maintenance and preservation of a digital archive copy. Digitization allows the University of Florida to generate imageT .and..text-based-ves.ions-as-approprftiand to provide and enhance access using search software This gr-apt of permissions prohibits use of the xfigrtKedwersitmsfof commercial use or profit. Printed or Typed Name of Cpvfi obTTTrnar/Ocm Mfat - Printed or Typed Address of Copyright Holder/Licensee Printed or Typed Phpne Number and Email Address of Copyright Holder/Licensee / Date of Signature Please print, sign and return to: Cathleen Martyniak UF Dissertation Project Preservation Department University of Florida Libraries P.O. Box 117007 Gainesville, FL 32611-7007 5/28/2008 103 where 7 -2 -24" T 1609 "TFT 2 66 - nr 1 -6 -2 -24 and therefore nR = 2 3 16 - TF 1 0 6 8 1 6 0 -TT ~TT n22_ (i) From (4.2-2) R is R = C0-H^irH-r 3 1 1 4 and det R >0 (ii) Using the (LE) and (i) we have 0 0 i 3-n R 1 0 22 T2H22- 0 12n22" 6 143 il 1 T 2053 144 22 2880 (iii) Choose q22=1 so that (4.3-4) is satisfied, then n22~2 and Qr = diag(1 ,1 28 8 o) This example points out some very important facts. First, it follows that the measurement covariance sequence contains all of the essential information necessary to specify a stochastic realization. By choosing QR, nR is determined from the (LE) and the matrices R,SR are uniquely found from (4.3-2) for given (FR,HR). Alternately, selecting SD=0n, which satisfies (4.2-10), R is uniquely specified k p and np-%p(p-l) elements of QR and nR are invariants. The remaining elements of QR and nR are free. This example also shows that there may exist a measurement process with uncorrelated system and measurement noise (S=0p) equivalent (under GRn) to a model with correlated noise (S/0p), i.e., they both have identical PSD's or {C-}. r w LIST OF MATHEMATICAL SYMBOLS Symbol Usage Meaning First Usage T AT, aT Transpose of A, ai pg. 13 -1 A1 Inverse of A pg. 13 -T at Inverse of AT pg. 89 P p(A) Rank of A pg. 13 |.| |A| or det A Determinant of A pg. 30,21 diag A Diagonal elements of A pg. 103 \ x^y x is not equal to y pg. 30 > x>y x is greater than y pg. 16 c XcY X is contained in or a subset of Y pg. 18 e xeX x is an element of X pg. 12 X->Y Map (set X into set Y) pg. 20 : = x: = x is defined by pg. 13 0 xoy Abstract group operation pg. 19 { } {.> Sequence or set with elements pg. 13 Z Summation pg. 13 00 Infinity pg. 13 t Footnote pg. 2 V End of proof pg. 34 4 Group action operator pg. 2i dim X Dimension of vector space X . pg. 15 iff if and only if pg. 14 X X(A). , Eigenval u'es of A pg- 9e / /x Square root of x pg. 99 max() Maximum value of pg. 23 Z+ Positive integers pg. 12' K Field pg. 12 vi 94 Example. (4.2-12) Suppose we are given the stochastic realization, (F,H,Q,R,S) as F = 0 1 o' ro i 01 rjin 5 -7 5" 0 0 1 , H = V JO) il -7 1 -1 _ 24 3 8 1 2 Lo i 1J _ 5 -1 2_ R = '2 i" " 1 1" J 4. * S = 0 0 _-l _1_ and we would like to obtain (FR,HR,QR,RR,SR) corresponding to (i) of (4.2-8) (1)Use the transformation, (TR,I) to obtain, (FR,HR,QR,R,SR) as -1 1 0 "l 0 0 "1 0 0 o o" 0 1 0 7 1 1 hr 0 1 0_ * qr = 0 o 1 0 0 3 6 6 7 SR -1 -1 1 L 4 24 2880-J 24 24 V (2)First, select a QR as 0 0 J-i u u 24 0 1 12 13 1 3051 2 4 12 ,144 0 then solving (4.2-5), we obtain and therefore "l f ' /V " 1 -T J 3_ li -1 -i 2 4 3 (3)If we choose to select an SR instead, corresponding to (ii) of (4.2-8) we must first determine the constraint imposed in (4.2-1). The choice 5^2 trivially satisfies this constraint; thus,we solve (4.2-5) for the (np-%p(p-l)) elements of LR 42 where it follows from (2.3-8) that the columns of A form chains satisfying -j+miUj-l)-* ~ e ] m for q .= Eu. . 3 s=l 3 * * Since A is A shifted m columns to the right, each chain of A is given by [w.j+m ... Wj+m^ ] and again each column is unit J T except Wj+miJ = aj from Corollary (2.3-6). Thus, := A A gives the matrix of (2.2-5). is obtained directly from HcCGe k-1 Fq Gq] = [a | ... a m a.l+m a.m(k+l) L since multiplication by the unit columns of (F^.G^J select the n columns of H^.V Analogous results hold for the dual ER. It should also be noted that if the Hankel array is transformed to and both rows and columns examined for predecessor independence as before, i.e., ?S N,N' % U = 'b V vnV (2.3-11) Of where is given in (2.2-8) and T is a permutation array, then all of the previous theory is applicable. The only exception in this case is that the Buey invariants and forms given by 3igR and liBC are obtained instead of the Luenberger forms. These results follow directly from (2.2-1). In many applications the characteristic polynomial xR(z) is required. Many efficient classical methods (e.g., see Faddeeva (1959)) exist to determine XR(z) from the system matrix. Even more recently some techniques have been developed to extract the characteristic polynomial 4 For any two sets X and Y, a subset R c X x Y is called a binary relation on X to Y (or, a relation "between" X and Y). Then (x,y)eR is usually written as xRy and is read: "x stands in the relation R to y". If for X=Y this relation is reflexive, symmetric, and transitive, then it is an equivalence relation E on X given by xEy for x,yeX. The set of all elements z equivalent to x is denoted by E(x),= {zeXjxEz} and is called the equivalence class or orbit of x for the equivalence relation E. The set of all such equivalence classes is called the quotient set or orbit space and is given by X/E. Thus, the relation E of X partitions the set X into a family of mutually disjoint subsets or orbits by sending elements which are related into the same equivalence class. ff Consider a fixed group G of transformations acting on a set X. Then the elements Xj,Xg of X are equivalent under the action of G iff there exists a transformation TeG which maps x-j into.Xg.. This is basically the "formula" we will apply throughout, i.e., we first formulate the set of elements (generally the internal system description), then define a transformation group; and finally determine the orbits under the action of G. To be more precise, let us first define the function f mapping a set X into Y as an invariant71^for E if for x-j^eX, x^Ex^ implies f(Xi)=f(x2). In addition if f.(xi)asf(Xg) implies x-jExg, then f is a X ^ ~ This is the standard Cartesian product, XxY = {(x,y)|xeX, yeY} . .Here we mean "group" in the standard algebraic sense, i.e., (G) where G is a closed set of elements each possessing an inverse and the identify element; 0 is an associative binary operation. When o is understood, the group is merely denoted by G. 4.4-4. Note that an invariant is actually a function, but common usage refers to its image as the invariant. We will also use this terminology throughout this dissertation. 64 3.2 Minimal Extension Sequences In this section we discuss the more common and difficult problem of obtaining a minimal partial realization from a finite Markov sequence when (R) is not satisfied. Two different approaches for the solution of this problem have evolved. The first is based on constructing an extension sequence so that (R) is satisfied and the second is based on extracting a set of invariants from the given data. We will show that these methods are equivalent in the sense that they may both lead to the same solution. In order to do this the existing algorithm is extended to obtain the more general results of Roman and Bullock (1975a). Also a new recursive method for obtaining the entire class of minimal extensions is presented. It is shown that the existing algorithm does in fact yield a particular solution to this problem which is valuable in many modeling applications. In the first approach, Kalman (1971b), Tether (1970) and subsequently Godbole (1972) examine the incomplete Hankel array to determine if (R) is satisfied. If so, the corresponding minimal partial realization is found. If not, a minimal extension is con structed such that (R) holdsand a realization is found as before. They show that a minimal extension can always be found, but in general it will be arbitrary. They also show that this extension must be constructed so that the rank of S(M,M) remains constant and the existing row or column dependencies are unaltered. Considerable confusion has resulted from the degrees of freedom 106 k| k-1 = h 4| k-1 and is the measurement noise. Note that the'{z^} of (4.4-1) is precisely the measurement process of (4.1-2); for if, we substitute the above expressions of ancl 2ik||<-l int0 (4*4-1) we obtain (4.1-2). The respective covariances of _x, x_, Â£ are denoted by the nxn % matrices, n, E, n and (R )^ is the innovation covariance which satisfies It is well known that the Kalman gain, K^, satisfies Kk= (FV|1+s)(Re>"1k (4*4-3) O/ ' where satisfies a discrete Riccati equation *k fVifT +1 Kk-i(Vk-iKk-i (4-4-4) The standard solution to the estimation problem is to solve (4.4-4) and (4.4-2) for nk and then to calculate the corresponding Kalman gain, K^ from (4.4-3). Since the observable pair (F,H) is known, the state estimate is updated using the INV model. If we consider the stationary, steady state case, then K^ = K^ = ... = K, 11^ = n^-j = ... = H, and therefore, (R ). = (R ). = ... = R The stationary, steady state, S K Â£ K** I Â£ INV model is given by the quadruple, E^y = (F,K,H,Ip) and RÂ£ with K the steady state Kalman gain and RÂ£ the innovation covariance. The transfer function is given by T^U) = H(Iz-F)\ +. I The Kalman filter accepts as inputs the current measurement sequence of the WN model and has as its.state the best minimum error covariance estimate of the corresponding state vector. 86 $z(z) [F(iz-f')'1 ip] 'V Q '(i2-1-f'T)-19r where a, Oj- Q s V O.T -W_ Ls RJ yj a. S R [VT WT] ^ 0 The proof of this proposition is given in Popov (1973) and essentially consists of showing there exists a spectral factorization of the given PSD. Thus, this proposition assures us that there exists at least one solution to the stochastic realization problem. Proposition (4.1-9) shows that once T, and n are determined then a stochastic realization, (F,H,Q,R,S) may be specified; however, it does not show how to determine n. Recently many researchers (e.g. Glover (1973), Denham (1974,1975), Tse and Weinert (1975)) have studied this problem. They were interested in obtaining only those solutions to the KSP equations of (4.1-9) which correspond to a stochastic realization such that A^O of (4.1-11). Denham (1975) has shown that any solution, n*, of the KSP equations which corresponds to a factorization as in (4.1-11) with V=KN, W=N for K=Knxp, NeKpxp, K full rank and N symmetric positive definite, is in fact a solution of a discrete Riccati equation. This can readily be seen by substituting, (Q,R,S) = (KNNTKT,NNT,KNNT) of (4.1-11) into (4.1-9) n*-An*AT = knnV d+dt-cji*ct = nnt (4.1-12) B-An*CT = KNNT for A = T~]FT, C=HT, TeGL(n) This dissertation was submitted to the Graduate Faculty of the College of Engineering and to the Graduate Council, and was accepted as partial fulfillment of the requirements for the degree of Doctor of Philosophy. Dean, Graduate School 56 (3) Add the next piece of data, Am+-j and form S(M+1,M+1). (4) Multiply S(M+1,M+1) by P. Perform row operations (if necessary) using old leading elements to obtain Q (M+1,M+1). If (R) is satisfied, continue. If not, go to 3. (5) Perform column operations as in (5) of (2.4-1) and obtain the invariants and canonical forms as in (6), (7). Go to 3. Example (2.4-2) will be processed to demonstrate the modified algorithm for comparison. Assume that the Markov parameters are sequentially available at discrete times, i.e., A^ is received, then Ag, etc., and the system is to be realized. Example. (3.1-2) Let the Markov sequence be given by "l 2' C\i 1 4 "4 8 8 16 '16 32~ A1 * 1 -2 J _ ~ ,2 _1 4 0_ , a3 = 6 10 _3 2_ ii 13 22 6 6 V 28 48 J3 16_ and apply the algorithm of (3.1-1). It is found that the rank condition is first satisfied when A^, Ag are processed, i.e., v (1) [I6 | S(2,2) | I4] (2) Performing first row and then column operation as in (3.1-1), obtain [P ) S*(2,2) Â¡ E] or l 0 0 0 1 -2 -1 0 -1 1 0 0 0 0 0 1 -i -2 -1 0 1 0 0 0 0 1 0 -2 0 0 1 0 0 1 -2 0 0 0 1 0 0 __0 0 -1 0 0 1 0 0 35 Proof. The proof is immediate from Proposition (2.2-3) and Lemma (2.3-1).V Note that similar results hold for the columns of the Hankel array when examined for predecessor independence. In the solution to some problems knowledge of both controllability and observability indices are required. Moore and Silverman (1972) require both indices to design dynamic compensators in order to solve the exact model matching problem. Similarly the requirement exists in the design of pole placement compensators and also stable observers as indicated in Brausch and Pearson (1970) and more recently Roman and Bullock (1975b). In an on-line application Saridis and Lobbia (1972) require the controllability invariants as well as the observability indices to solve the problem of parameter identification and control. The latter case exemplifies the fact that in some instances it is first necessary to determine the structural properties of a system from its external description prior to compensation. The need for an algorithm which determines both sets of controllability and observability invariants from an external system description is apparent. Computationally the simplest and most efficient technique to determine these invariants would be some type of Gaussian elimination scheme which utilizes elementary operations (e.g., see Faddeeva (1959)). If we perform elementary row operations on such that the predecessor dependencies of PV^ are identical to those of and perform column operations on W^, so that W^,E and W^, have the same dependencies then a examination of = PS^ niE is equivalent to the examination of . * We define ^ as the structural array of ^,. This array is specified by the indices {v^} and {y^} which are the least integers such that the row and column vectors of ^ are respectively, 5 group is the only information required to specify the corresponding canonical form. Wonham and Morse (1972) obtained the feedback invariants of the controllable pair from the not as lucid geometric viewpoint. Their results were identical to those of Brunovsky and ftosenbrock. : Morse (1973) examined the invariants of the triple (F,G,H) under a larg group of transformations which includes output change of basis. A complete set of feedback invariants of this triple still remains an open problem, but some fragmentary results were presented by Wang and Davison (1972) when they investigated certain sets of restricted triples. Along these lines Rissanen (1974), Caines and Rissanen (1974), . Mayne (1972a,b),Weinert and Anton (1972), Tse and Weinert (1973,1975), Glover and Willems' (1974) examined the identification problem from the invariant theory viewpoint and obtained some rather interesting results. Recent results in decoupling theory were obtained by Warren and Eckberg (1973), Concheiro (1973), and Forney (1975) using the Kronecker invariants Probably the most extensive survey of these results has been compiled by Denham (1974) and we refer the interested reader to this paper. We temporarily leave this area to consider one specific application of these resultsthe realization problem. 1.2 Survey of Previous Work in Realization Theory The first realization problem proposed for control systems was the determination of a state space model (internal description) from a given transfer function (external description). Gilbert (1963) and Zadeh and Desoer (1963) describe realization procedures based on the determination of the rank of the residue matrices of the given transfer function matrix, but unfortunately these procedures only apply to the 6 case of simple poles. Kalman (1963) proposed an algorithm whereby the given transfer function is realized as a parallel combination of single input, controllable subsystems in companion form, and then applied the "canonical structure theory" (Kalman (1962)) to delete the uncontrollable dynamics. This technique handles simple as well as multiple transfer function poles. Later Kalman (1965) showed the equivalence of the realization problem of control theory to the corresponding network theory formulation. A significant advance in realization theory was given by Ho and Kalman (1966). They showed that the state space model could be found from the impulse response sequence provided the system under investi gation is finite dimensional. They also developed an algorithm based on forming the generalized Hankel array from the given sequence and then extracted the state space triple from it. Shortly after the pub lication of Ho's algorithm, Youla and Tissi (1966) working in network synthesis and Silverman and Meadows (1966) in control theory developed similar realization techniques again based on the impulse response sequence. Ho's algorithm gave new impetus to realization theory. Several authors have provided alternate or improved realization algorithms based on the Hankel array formulation. Mayne (1968), Panda and Chen (1969), Roveda and Schmid (1970), Rosenbrock (1970), Lai et al. (1972) and even more recently Huang (1974), Rozsa and Sinha (1975) among others, considered the older transfer function matrix approach, while Rissanen (1971,1974), Silverman (1971), Ackermannand Buey (1971), Chen and Mita! (1972), Mita! and Chen (1973), and Bonivento et al. (1973) approached the problem from the Hankel array formulation. Internet Distribution Consent Agreement In reference to the following dissertation: AUTHOR: Candy, James TITLE: Realization of Invariant system descriptions from markov (record number: 180838) PUBLICATION DATE: 1976 I, \i V O f\C>l as copyright holder for the aforementioned dissertation, hereby grant specific and limited archive and distribution rights to the Board of Trustees of the University of Florida and its agents. I authorize the University of Florida to digitize and distribute the dissertation described above for nonprofit, educational purposes via the Internet or successive technologies. This is a non-exclusive grant of permissions for specific off-line and on-line uses for an indefinite term. Off-line uses shall be limited to those specifically allowed by "Fair Use" as prescribed by the terms of United States copyright legislation (cf, Title 17, U.S. Code) as well as to the maintenance and preservation of a digital archive copy. Digitization allows the University of Florida to generate imageT .and..text-based-ves.ions-as-approprftiand to provide and enhance access using search software This gr-apt of permissions prohibits use of the xfigrtKedwersitmsfof commercial use or profit. Printed or Typed Name of Cpvfi obTTTrnar/Ocm Mfat - Printed or Typed Address of Copyright Holder/Licensee Printed or Typed Phpne Number and Email Address of Copyright Holder/Licensee / Date of Signature Please print, sign and return to: Cathleen Martyniak UF Dissertation Project Preservation Department University of Florida Libraries P.O. Box 117007 Gainesville, FL 32611-7007 5/28/2008 80 White Noise Input Stochastic Realization Figure 2. A Solution to the Stochastic Realization Problem 68 Arbitrary Parameter Partial Realization Algorithm. (3.2-2) (1) Perform (1) of Algorithm (3.1-1) to obtain [P | Q(M,M)]t. (2) For each ieÂ£, determine the largest (i+pYl)xm(M+l-k..) sub matrix of Q(M,M) of data specified elements and form the set J.. (3) For each ieÂ£, replace pi by J + z bJ, b a scalar. H i H i seJ^ (4) Determine the corresponding canonical forms incorporating these free parameters. Dual results hold for the columns. The fre-parameters are fpund in analogous fashion by examining the zero columns of the submatrices of S*(M,M). Example. (3.2-3) The following example is from Tether (1970). For m=p=2 and "1 f 4 3 10 1 22 15 Ar _0 0_ _0 0_ ,A3- _ 1 1_ 4* II 3 3 _ (1) [ P f Q(4,4) ] = _1 1 4 3 j 10 7 22 15 0 1 0 0 0 ! 1 3 3 -4 0 1 0 -6 5i 18- 13 0 0 0 1 0 0 0 ! Jj 0 0 2 0 -3 0 1 0 0 0 0 -1 0 0 1 0 1 0 0 0 6 0 -7 0 0 0 1 0 0 -3 0 0 0 0 0 0 1 0 0 4* It should be noted that when (R) is not satisfied, some of thev. may not be defined, i.e., the last independent row of a.chain is in the last block Hankel row. In this case all_ would-be invariants are arbitrary. 120 R. K. Mehra [1970] "On the Identification of Variances and Adaptive Kalman Filtering," IEEE Trans. Auto. Contr., Vol. AC-15, pp. 175-184. [1971] "On-Line Identification of Linear Dynamic Systems with Applications to Kalman Filtering," IEEE Trans, on Auto. Contr., Vol. AC-16, pp. 12-20. D. P. Mi tal and C. T. Chen [1973] "Irreducible Canonical Form Realization of a Rational Matrix," Int. J. Contr., Vol. 18, pp. 881-887. B. C. Moore and L. M. Silverman [1972] "Model Matching by State Feedback and Dynamic Compen sation," IEEE Trans on Auto. Contr,,Vol. AC-17, pp. 491. A. S. Morse [1973] "Structural Invariants of Linear Multivariable Systems," SIAM J. on Contr., Vol. IT, pp 446-465. P. R. Motyka and J. A. Cadzow [1967] "The Factorization of Discrete-Process Spectral Matrices," IEEE Trans, on Auto. Contr., Vol. AC-12, pp 698-706. R. W. Newcomb [1966] Linear Multiport Synthesis, Me Graw-Hill, New York. ' S. P. Panda and C. T. Chen - [1969] "Irreducible Jordan Form Realization of a Rational Matrix," IEEE Trans, on Auto. Contr., Vol.AC-14, pp. 66-69. V. M. Popov [1961] "Absolute Stability of Nonlinear Systems of Automatic Control," Avt. i Telemekh., Vol. 22, pp. 961-979. [1964] "Hyperstability and Optimality of Automatic Systems With Several Control Functions," Rev. Roum. Sci. Tech. Electr. Enerqetique, Vol. 10, pp. 629-690. [1969] "Some Properties of the Control Systems with Irreducible Matrix Transfer Functions," in Seminar on Pifferentia! Equations and Dynamical Systems-II. Springer,; Berlin, pp. 169-180. [1972] "Invariant Description of Linear, Time-Invariant Controllable Systems," SIAM J. On Contr., Vol. 10, pp. 252-264 [1973] Hyperstability of Control Systems, Springer-Verlag, N.Y. J. Rissanen [1971] "Recursive Identification of Linear Systems," SIAM J. on Contr., Vol. 9, pp 420-430. [1972a] "Realization of Matrix Sequences," IBM Research Report, RJ 1032, San Jose, Calif. [1972b] "Recursive Evaluation of Pade Approximants for Matrix Sequences," IBM J. df Res. and Dev., Vol. 16, pp. 401-406. 83 and from (4.1-1) it may be shown that C. = HFJ_1(FnHT+S) j > 0 (4.1-5) J Co = HnHT + R The PSD matrix of the measurement process is obtained by taking the bilateral z-transform of the sequence C. defined in (4,1-4) which gives 3 $z(z) = H(Iz-F)'1Q(Iz"1-FT)"1HT+H(Iz-F)1S+ST(Iz"1-FT)'1HT+R (4.1-6) It is important to note that this expression is the frequency domain representation of the measurement process which can alternately be expressed directly in terms of the measurement covariance sequence as 00 $z(z) = E C.Z_J j=-3 T Since the measurement process is stationary and z is real, C and therefore the PSD can be decomposed as 00 00 . Mz) = 2 c Z*J + C + E C z3 (4.1-7) L j=1 J 0 j=1 J Note that {C.> is analogous to the Markov sequence of the deterministic J realization problem. We define the problem of determining a quintuplet, (F,H,Q,R,S) in (4.1-6) from ^(z) or {Cj> as the stochastic realization problem. In this chapter we are only concerned with the realization from the measurement covariance sequence. When a realization algorithm is applied to the covariance sequence, we define the resulting realization as the Kalman-Szegb-Popov (KSP) model because of the parameter constraints 41 Theorem. (2.3-10) Given the infinite realizable Markov sequence from an unknown system, then SQ=(,rQ>GQ>H(.)n is a minimal canonical realization of {Ar} with 7C X Fc = [W, | H2 *_ w ] nr Gc is a submatrix of (Wu+i)r given by the first m u columns y HC ta.l a.l+m(y^-l) a ... a_ 1 . m mu ' Km and j fcj+m j+m vector of Vt ], jqn, Wr is a col umn Proof. Since the sequence is realizable, there exist, integers, n,v,y, satisfying Proposition (2.1-5). If Q is given as in Corollary (2.3-6), then Q = "Wk>c.r 0pv-" L mk _ i1""' for k>y+l Thus, Gc is obtained immediately from the first m columns of * (Wr)c. Form two nxn arrays, A and A each constructed by selecting n regular columns of (Wr)q starting with the first * column for A and the (1+m) column for A The independent columns of (Wr)q are indexed by the y. and satisfy (2.3-8); thus, they are unit columns and A is a permutation matrix, i.e,, A = [w, ... | w -1+m w 2m *' -j+miyj-l) ], jem 61 has M (M)=M (M+l)= ... = M (M+k) so that each canonical realization is equal to Z(M*); therefore, Â£(M)=Â£(M+1)= ... =Z(M+k).V Next we examine the case where Â£(M) and z(M+k) are of different dimension. The nesting properties are given in the following lemma. ^ "At Lemma. (3.1-4) Let there exist integers, M (M)^M, M (M+k)^M+k such that the rank condition is satisfied (separately) and Â£(M), Z(M+k) are minimal partial canonical realizations of (Ar> when reM and reM+k, respectively, for given k. If p(S(M+k,M+k))>p(S(M,M)), then v.(M+k)*v.(M), ieÂ£. Furthermore, a. (M+k)=a. (M), j=i,i+p,...,i+p(v.(M)-l). J J Proof. Since p(S(M+k,M+k))>p(S(M,M)), M*(M+k)>M*(M) and therefore, Sv(M),y(M) is a submatHx of Sv(M+k).uCM+k)* If the row of is regular, it follows that the j-th row of Sv(M+k) y(M+k) ls also re9u1ar by the nature of the Hankel pattern, i.e., the rows of Sv^ are subrows of Sv(M+k) ,y(M+k) The addition of more data (AM+],... ,AM+|<) to S(M,M) makes previously dependent rows become independent rows but previously independent rows remain independent; thus, the v..(M) can only increase or remain the same, i.e., v..(M+k) ^ T v.(M), icÂ£. The set of regular {a.(M+k)} are specified by the ' J v.(M+k)'s; therefore ai (M+k)=aT (M), j=i,i+p,.. .,i+p(v.(M)-l), 1 J J 1 since vi(M+k)-vi(M), ieÂ£.V The results of these two lemmas are directly related to the nesting 4 properties of the partial realization algorithm. First, define JM as the set of indices of regular Hankel row vectors based on M Markov parameters 105 measurement covariance sequence in the steady state, stationary case and show how this realization can be used to Represent the measurement process of (4.1-2). Care is taken to formulate this realization problem in precisely the same manner as the WN model of Section (4.1) in order to emphasize the striking similarity between these two distinctly different models of the same measurement process. Finally, we present the algorithm to solve the stochastic realization problem using the innovations representation. It should be emphasized that this technique was presented in Mehra (1971) and improved in Carew and Belanger (1973) and Tse and Weinert (1975). The basic filtering problem is to find the best minimum error covariance estimate of the state vector of the WN model in'terms of the currently available measurement sequence, z^. A convenient model used in the Kalman theory is the innovations (INV) representation given by ^k+l|k = F4|k-1 + Kk% (4.4-1) . /y- 4|k-l = H-k| k-1 \ -k = ^k | k-1 + ~k where _x, z, e_ are n state estimate vector, p measurement vector, and p innovation (of z) vector and is the optimal estimate of given z^.z^ ,... ,z^ . The innovations sequence, {e^} is a zero mean, white Gaussian process which is related to the WN model by % ~ H^kIk-1 + -k % where x^.j jjC_i is the error in the estimate of x^, given zQ,z^,... ,z^^ defined by 97 4.3 Stochastic Realization Via Trial and Error In this section, we develop an algorithm-to obtain a stochastic realization from the measurement covariance sequence. We would like to find this realization directly in a form which uniquely characterizes the class of covariance equivalent quintuplets, i.e., quintuplets which have the same PSD or equivalently {C.}. From Section (4.1) we already know that one way to obtain a stochastic realization is to solve a discrete Riccati equation; however, this technique can become computa tionally burdensome when system order is large. Therefore, we would like to develop an algorithm to directly extract an invariant system description (under GR^) of the stochastic realization from the measurement covariance sequence which does not require a solution of the Riccati equation. We briefly recall the results of earlier chapters to obtain a canonical realization of the KSP model.. We show how constraints which evolve from the stochastic nature of the problem can be used to obtain a stochastic realization. The canonical realization of from {Cj> follows by recalling that the Hankel arrayadmits the factorization. SN,N' = VNWN* where Vj^jW^, are the corresponding observability and controllability matrices. Thus, by applying the canonical realization algorithm of (2.4-1) to {C.} we obtain the set [{v^}{6.-c+.}{a. }] which uniquely j 1 1ST J . specifies, (A,C)=(FR,HR) of (2.2-6) and B=GR of (2.2-14). ^Note that in realizing ien from (C.} we start with C, and not Cn; l\b r j y I U therefore, R is uniquely determined once ITH is found. 37 P rs gives this relation. From this choice of P it follows that dependent rows of are zero rows of Q. If the j-th row of SN is regular, then P unit diagonal-lower triangular insures that the corresponding row of Q is nonzero and regular. Similar results hold for the columns of ^, with E unit diagonal upper triangular. This choice of P does not alter the column dependencies of for if the i-th column of is dependent on its predecessors, then from Corollary (2.3-2) Â£. is uniquely represented as a linear combination of regular vectors in terms of the control- . lability invariants. Since P is unit diagonal-lower triangular, it is the matrix representation of a nonsingular linear transformation, Pr^q^. where q. is the i-th column vector of Q. Thus, multiplying on the left every vector Â£. in (2.3-2) with this P gives for i-j+mp. J 3-1 minivyy^l) q = Z Z k=l s=0 m min(u.,u.)-l Thus, we have shown that selecting P with the given structure does not alter the predecessor column dependencies of S^ or equivalently Q. Since the column vectors of Q satisfy the above recursion, Sf^ and Q have identical predecessor column dependencies, therefore, performing column operations on Q is * analogous to performing them on S^ and so we have SfJ . * (PS^ n,)E = QE or the predecessor dependencies of S^ N, and S^ M are identical.V 15 Proof. See Silverman (1971). Note that the essential point established in Ho and Kalman (1966), which is used in the proof of the above proposition is that Z is a minimal realization iff it is completely controllable and observable. Since $ *V W it follows for dims n that: p(S ) min[p(V ),p(W )]=n. v,p v y v,y v y This result is essential to construct any realization algorithm. In (2.1-5) the crucial point of finite dimensionality is carefully woven into necessary and sufficient conditions for an infinite sequence to be realizable. What if only partial information about the system is available in the form of a finite Markov sequence? Is this sequence realizable? What is the relationship between the minimal realization and one based only on partial data? These are only a few of the questions which must be resolved when we are limited to partial data. Intrinsic in the realization from a finite Markov sequence is the fact that enough data are contained in S to recover the infinite a v,y sequence, i.e., knowledge of (A^,...,A is sufficient to determine {Ak>, k-1,2,... But in reality the only way to be sure of this is knowledge of the actual system dimension (or at least an upper bound). A minimal partial realization is a realization of smallest dimension determined from a finite Markov sequence {A^},keM_ which realizes the sequence up to M terms. The order of the partial realization is M and the realization is denoted by Z(M). The realization induces an extension k-1 of {Ak>, i.e., Ak=HF G for k>M. The following basic result analogous to (2.1-5) answers the realizability question when only partial data are given. For a proof, see Kalman (1971). 21 E(x) for every xeX. With these definitions in mind, our "formula" becomes (i)Formulate the set of elements; (ii)Define the transformation group; (iii)Determine a set of complete invariants under this transformation group; and (iv)Develop the canonical form in terms of the corresponding invariants. (2.2-1) We now apply (2.2-1) to various restricted sets related to multivariable systems. This approach is essentially given in Kalman (1971a), Popov (1972), Rissanen (1974), or Denham (1974). In this sequel we review the main results of Popov. First, define the set of matrix pairs (F,G) as XQ = i(F,G)|FeKnxn, GeKnxm; (F,G)controllable} The general linear group, which corresponds to a change of basis in the state space, is specified by the set GL(n):= {T|TeKnxn; det T?0} (2.2-2) with the group operation standard matrix multiplication, i.e., T o T = T T. In order to determine the orbits of XQ under the action of GL(n), it is first necessary to specify the action operator T + (F,G):= (TFT-1,TG) 4* In general the problem of determining a canonical form is quite difficult. However in this dissertation we consider restricted sets which make the problem much simpler. For a thorough discussion of this problem see Kalman (1973). 7 Rissanen (1974), Furata and Paquet (1975), Roman (1975), Dickinson et al. (1974a,b) have recently considered the problem of realizing a given infinite impulse response matrix sequence with a polynomial matrix pair. Such a pair is referred to as a matrix- fraction description of the system and is becoming well known in control literature largely due to the ground work established by Popov (1969), Rosenbrock (1970), Wolovich (1972a,b, 1973a,b) and others. Kalman (1971b), Tether (1970), and Godbole (1972) later considered the more realistic case where only a finite number of terms of the impulse response sequence are specified. This is commonly known as the partial realization problem and corresponds in the scalar case to the classical Pad approximation problem. Generally most realization altorithms can be used to process partial data, but usually at a loss of efficiency and even more seriously the possibility of yielding misleading results. A wealth of new techniques have recently been published to handle this very special, yet realistic variant of the realization problem. Rissanen (1972a,b), Ackermann (1972), Dickinson et al. (1974a), Roman and Bullock (1975a), Anderson et al. (1975) published some efficient and improved algorithms to solve this problem. Also of recent interest is the development of algorithms which realize a system directly in a canonical form (under a change of basis in the state space), i.e., algorithms which solve the canonical realiza ti on problem. The algorithms of Ackermann (1972), Bonivento et al. (1973), Rissanen (1974), Dickinson et al. (1974a), Rozsa and Sinha (1975), Luo (1975), and Roman and .Bullock (1975a) solve this problem. G. D. Forney [1975] "Minimal Bases of Rational Vector Spaces with Applications to Multivariable Linear Systems," SIAM J. on Contr., Vol. 17, pp. 192-212. K. Furuta and J. G. Paquet [1975] "Determination of Matrix Transfer Function in the Form of Matrix Fraction from Input-Output Observations," IEEE Trans, on Auto. Contr., Vol. AC-20, pp 392-396. F. R. Gantmacher [1959] The Theory of Matrices, Vols.T and 2, Chelsea Publishing Co.,N.Y M. R. Gevers and T. Kailath [1973] "An Innovations Approach to Least Squares Estimation- Part VI: Discrete-Time Innovations Representation and Recursive Estimation," IEEE Trans, on Auto. Contr., Vol. AC-18, pp. 588-600. E. G. Gilbert [1963] "Controllability and Observability in Multivariable Control Systems," SIAM J. on Contr., Vol. 1, pp. 128-151. [1969] "The Decoupling of Multivariable Systems by State Feedback," SIAM J. on Contr., Vol.7, pp. 51-63. K. Glover [1973] "Structural Aspects of System Identification," Rep. ESL-R-516, Electronic Systems Laboratory, M.I.T., Cambridge, Mass. K. Glover and J. Willems [1974] "Parameterizations of Linear Dynamical Systems: Canonical Forms and Identifability," IEEE Trans, on Auto. Contr., Vol. AC-19, pp. 640-646. S. S. Godbole [1972] "Comments on 'Contruction of Minimal Linear State- Variable Models from Finite Input-Output Data,'" IEEE Trans, on Auto. Contr., Vol. AC-17, pp. 173-175. I. Gokhberg and M. G. Krein [1960] "Systems of Integral Equations on a Half Line with Kernels Depending on the Difference of Arguments," Uspekh: Mat. Naut. 13, pp. 217-287. B. Gopinath [1969] "On the Identifcation of Linear Time Invariant Systems from Input-Output Data," Bell Syst. Tech. J., Vol. 48, pp. 1101-1113. 102 T T T T (2)The {a. } invariants are {a^ ,a2 ,a4 }; therefore, the KSP model is A=Fr 4 13 0 9 150 1 4 , B = 66 _ ~ _ 1 6 T5 4 1391 1 9 3 ' . 600 6 0 0 , C=Hn -1 J (3)Using the first approach of selecting a QR-0: (i) 0^=1^ and solving the (LE) gives n R 67787 1261 82 9 9' 6 6 0 0 6 6 0 3 3 0 0 126 1 9 3 1 997 660 3 3 0 1650 8299 997 6 0 1 3 30 0 16 50 3 3 0 (it) Solving (4.2-14) for R and SR gives ' * * 76 0 3 719 2419 59 6 6 0 0 66 0 22 0 0 660 and SR = 41 3 59 719 1379 ~ 3 3 0 0 _ 1 6 50 660 3 3 0 . 59 1 77 nsa : - 6 0 0 66 0 0 (4) det(I3-SRR"1SR^)>0 or using the second approach (5) Let SR=02> then B=FRnRHRT and - 1 6 09 150 = Cu-, n2] = 2 3 6 6 16 ~25 ~ *25 and 107 The stochastic realization problem can be reformulated in terms of the INV model in precisely the same manner as the WN model of (4.1-1) and (4.1-2). Thus, expressions analogous to (4.1-4) and (4.1-5) can be derived and therefore the measurement covariance sequence is alternately given (in terms of lag j) by C. = HFJ_1(FnHT+KR ) j>0 J Â£ Cn = HilHT+R 0 e where the state estimate covariance matrix n satisfies /s ^ T T . n-FnF = kr.k' e analogous to the (LE) of (4.1-3) in n. The PSD matrix of the measurement process in terms of the INV model is obtained in precisely the same way as (4.1-6); therefore, we have , $7(z)=H(Iz-F)_1KR KT(Iz1-FT)1HT+H(Iz-F)"1KR +R KT(Iz"1-FT)_1HT+R L e e e e \ (4.4-7) It is also possible to express the PSD in factored form; thus, by simple manipulation (4.4-7) can be written as $z(z)=CH(Iz-F)'1K+Ip]Re[Ip+KT(Iz*1-FT)"1HT>TINV(z)ie(z)T]NV(z"1) (4.4-8) Since R >0, the PSD is positive definite and (4.1-11) is always satisfied; Â£ therefore, we can specify a stochastic realization immediately, once F,H,K,R are determined. This stochastic realization is defined by Â£ e e Ze := (FH>QINVsRINV>SINV) = (F>H*KReKTRe>KRe) (4*4-9) (4.4-5) (4.4-6) where 73 and therefore A5" 46-b 31 -b 94-6b 63-6b 12-d 9-d-e 30-c-3d-5e+de 21-c-3d-5e+de-e2_ By solving for the x^.'s in Ag, Ag we obtain the extension as x^B) Xll(5)-15" A, = ~6x11(5)-182 6x11 (5)-213 x2i(5) x22 ^ 6 X21 X21 (6)+(x21 (5)-X22(5)-3)2-9 The number of degrees of freedom is 4,i .e. .{x^(5),x21 (5),x22(5),x2-| (6)}. The technique used to solve the parcial realization problem when (R) is not satisfied was to extract the most pertinent information from the given data in the form of the invariants, which completely described the class of minimal partial realizations. A recursive method to obtain the corresponding class of minimal extensions was also presented in {3.2-6) This method is equivalent to that of Kalman (1971b) or Tether (1970) for if the minimal extension is recursively constructed and Ho's algorithm is applied to the resulting Hankel array the corresponding partial real ization will belong to the same class. Note that if the extension is not constructed in this fashion, it is possible that all degrees of freedom available may not be found (see Roman (1975)). It should be noted that the integers v and y are determined from the given data,i.e., knowledge of the invariants enables the construction of a minimal extension such that v and y can be found. The approach completely resolves the ambiguity pointed out by Godbole (1972) arising in the Kalman or Tether technique. The results given above correspond directly to those presented in REALIZATION OF INVARIANT SYSTEM DESCRIPTIONS FROM MARKOV SEQUENCES By JAMES VINCENT CANDY A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1976 58 where wT+ = -[P2] | P^] = [1 | o] z-2 o o J ~i 2 -Z 2 0 ; dr(z) = 0 0 _ 0 0 2-l_ _ i 0_ The rank condition is next satisfied when A1,Ag.are processed, i.e., M =5 and we obtain [P | S (5,5) | E] as: [P I Q(5,5)] = 1 2 2 4 4 8 8 16 16 .32 -1 1 0 0. 0 0 2 5 6 12 16 ^1 0 1 0( -1 -4 -1 -6 -2 -10 -3 -16 -2 0 0 1 0 0 0 0 0 0 0 0 -2 0 0 0 1 0 0 2 5 6 12 16 1 1 -1 0 -1 1 0 0 0 0 0 0 0 0 -4 0 0 0 0 0 1 0 0 0 0 0 0 -3 0 -1 0 -1 0 0 1 0 0 0 0 0 0 0 1 -2 0 -1 0 0 0 1 0 0 0 0 0 0 -8 0 0 0 0 0 0 0 0 1 0 0 0 0 -8 1 -2 0 -2 0 0 0 0 0 1 0 0 0 0 -1 2 -3 0 -2 0 0 0 0 0 0 1 0 0 0 0 -16 0 0 0 0 0 0 0 0 0 0 0 1 0 0 -24 0 -4 0 0 0 0 0 0 0 0 0 0 1 0 0 -8 0 -5 0 0 0 0 0 0 0 0 0 0 0 1 0 0 and performing the column operations give [S (5,5) J E] +W is found easily from HrGr=A-j or solving for the second row of HR, wTGr [1 2]. ACKNOWLEDGMENTS I would like to express my sincere appreciation to the members of my supervisory committee: Dr. Thomas E. Bullock, Chairman, and Dr. Michael E. Warren, Cochairman, Dr. Donald G. Childers, Dr. Z.R. Pop-Stojanovic and Dr. V.M. Popov. A special thanks to Dr. Thomas E. Bullock and Dr. Michael E. Warren for their constant encouragement, unending patience, and invaluable suggestions in the course of this research. I would also like to thank my fellow students and friends, Zuonhua Luo, Arun Majumdar, Jos DeQueiroz, and Jaime Roman, for many fruitful discussions and suggestions. 9 and Akaike (1974a,b). From the transfer function viewpoint this problem has been solved using spectral factorization as originally introduced by Wiener (1955,1959) and studied by others such as Gokhberg and Krein (1960), Youla (1961), Davis (1963), Motyka and Cadzow (1967), and Strintzis (1972). The link between the stochastic realization problem and spectral factorization evolved from the work in stability theory by Popov (1961,1964), Yakubovich (1963), Kalman (1963), Szego and Kalman (1963). The equations establishing this link were derived in the Kalman-Yakubovich-Popov lemma for continuous systems and the Kalman- Szego-Popov lemma for discrete time systems. Newcomb (1966), Anderson (1967a,b,1969), and Denham (1975) extended these results and provided techniques to solve these equations. Defining the invariants of these problems is still an area of active research as evidenced by the recent work of Denham (1974), Glover (1973), and Dickinson et al. (1974b). This is one area developed in this dissertation. It will be shown that the invariants of the stochastic realization problem not only lends more insight into the structure of the problem, but also yields some new results. Research in realization theory and its applications continues as evidenced by the recent results of Rissanen (1975) in estimation theory, Ackermann (1975) in feedback system design,De Jong (1975) in the numerical aspects of the problem and Roman and Bullock (1975b) in observer theory. The results presented in this dissertation tie together some previously well-known results in stochastic realization and filtering theory as well as provide a technique which can be used to study other problems. W. A. Wolovich [1972a] "On the Synthesis of Multivariable Systems," 1972 JACC Pre prints, Stanford, Calif, pp. 158-165. [1972b] "The Use of State Feedback for Exact Model Matching," SIAM J. on Contr., Vol. 10, pp. 512-523. [1973a] "Frequency Domain State Feedback and Estimation," Int. J. Contr., Vol. 17, pp. 447-428. [1973b] "Multivariable System Synthesis with Step Disturbance Rejection," Proc. 1973 IEEE CDC, San Diego, Calif., pp 320- 325. W. M. Wonham and A. S. Morse [1972] "Feedback Invariants of Linear Multivariable Systems," Automtica, Vol. 8, pp. 93-100. V. A. Yakubovich [1963] "Absolute-Stability of Nonlinear Control Systems in Critical Cases," Avt. i Telem., Vol. 24, pp. 293-303. D. C. Youla [1961] "On the Factorization of Rational Matrices," IRE Trans, Info. Thy., Vol. IT-7, pp 172-189. D. C. Youla and P. Tissi [1966] "N-port Synthesis via Reactance Extraction," IEEE Int. Conv. Rec., Vol. 14, pp. 183-205. L. A. Zadeh and C. A. Desoer F19631 Linear System Theory: The State Space Approach., McGraw- 33 2.3 Canonical Realization Theory . * In this section we develop the theory necessary to solve the canonical realization problem, i.e., the determination of a minimal realization from an infinite Markov sequence, directly in a canonical form for the action of GL(n). Obviously from the previous discussion, this solution has an advantage over other techniques which do not obtain E in any specific form. From the computational viewpoint, the simplest realization technique would be to extract only the most essential information from the Markov sequence--the invariants under GL(n). Not only do the invariants provide the minimal information required to completely specify the orbit of Z, but they simultaneously specify a unique representation of this orbitthe corresponding canonical form. Thus, subsequent theory is developed with one goal in mind--to extract the invariants from the given sequence. The following lemma provides the theoretical core of the subsequent algorithms. Lemma. (2.3-1) Let and W^, be any full rank factors of = V^, Then each row (column) of is dependent iff it is a dependent row (column) of (W^,). Proof. From the factorization = V^, it follows if the j-th row of is dependent, then there exists an aTe:KpN, a^O such that T_ nT ~ N,N' -W Since p(WN,)=n, i.e., W^, is of full row rank, it follows that -TsN N' WN' = 32 The canonical forms for both left and right MFD's are defined by the polynomial pairs (BR(z),DR(z)) and ("Bc(z) ,Dc(z)) respectively, where R and C have the same meaning as in (2.2-5,2.2-6) and the former is given by 11 CL Izv ; bT e K^v (2.2-19) for 4 = [4(V-V,.) * * 6ki ek2 ... ek(i+pv.) I * where k=i+pvi and Bkj are given by {6ist} j=i,i+p,.. .i+P.(V|-l) * Bkjs< 0 j^i,i+p,.. .*i+p(v.-T) . 1 j=i+pvi and DR(z) is determined from (2.2-18). Dual results hold for the corresponding column vectors, b., jem of the J coefficient array of !q(z) in terms of the controllability invariants. This completes the discussion of canonical forms for Â£ or T(z). Note that analogous forms can easily be determined for the Buey forms if X, is restricted to {v.}. Henceforth, when we refer to an invariant . J system description, we will mean any representation completely specified by an algebraic basis. In the next section we develop the theory necessary to realize these representations directly from the Markov sequence 43 from the Markov sequence, but in general they are only valid in the cyclic case (see Candy et al. (1975)). An alternate solution to this problem is to obtain the Buey form and use (2.2-11) to find xp(z) by inspection. It is possible to realize the system directly in Buey form as mentioned in the previous paragraph, but in this dissertation we prefer to take advantage of the structure of the Luenberger form to construct Tg^ or Tgg. Superficially, this method does not appear simple because the transformation matrix and its inverse must be constructed, but the following lemma shows that Tgg can almost entirely be written by inspection from the observability invariants after the {v.} are known. is given by \ If the given triple is in Luenberger form, ZD, then the (v^xn) submatrices Tg are 'V > w v.-v. or T. v.>v. B V .-VI 82 The corresponding measurement process is given by h = h + \ (4.1-2) where is the p measurement vector and v^ is a zero mean, white Gaussian noise sequence, uncorrelated with x.., j k with J Covtvj.Vj) = Covfw^Vj) = S5, j for R a pxp positive definite, covariance matrix and S a nxp cross covariance matrix. Thus, a model of this measurement process is completely specified by the quintuplet, (F,H,Q,R,S). When a correlation technique is applied to the measurement process, it is necessary to consider the state covariance defined by n^Covtx^,)^) We assume that the processes are wide sense stationary; therefore, \ nk = n, a constant here. It is easily shown from (4.1-1) that the state covariance satisfies the Lyapunov equation (LE) n = FIIFT + Q (4.1-3) It is well known (e.g. see Faurre (1967)) that since F is a stability matrix, corresponding to any positive semidefinite (covariance): matrix Q, there exists a unique, positive semidefinite solution n to the (LE). The measurement covariance is given (in terms of lag j) by cj:= Gov(%j-4) = Cov(^+j4,+Cov(Vj^)+Co'' 70 are cumbersome to obtain due to the general complexity of the expressions in Er or therefore, a technique to determine these extensions without forming the Markov parameters directly (or the realization) was developed. This method consists of recursively solving simple linear equations (one unknown) to obtain the minimal extension. Extensions constructed in this manner not only eliminate the possibility of non minimality as expressed in Godbole (1972), but also describe the entire class of minimal extensions. The method of constructing the minimal extension sequence evolves easily from the lower triangular-unit diagonal structure of P. Since a dependent row of Q(M,M) is a zero row, it follows from Theorem (2.3-3) that for jemM (3.2-4) where recall that p.^ .=0 for j>i+pv.. i+pv^,j r 1 unkndwn extension parameters, x.y(r) fr Thus, by inserting the 'lm (r) into S(M,M) a system of linear equations is established in terms of the x..(r)s by (3.2-4). Due to the structure of P, this system of equations i J is decoupled and therefore easily solved. Example. (3.2-5) Reconsider (3.1-2) for Since (R) is satisfied, the extension A., j>2 is unique. We would like to obtain 0 74 Kalman (1971b) and Tether (1970). They have shown, when (R) is satisfied, there exists no arbitrary parameters in the minimal partial realization or corresponding extension. Therefore, the existence of arbitrary parameters can be used as a check to see if the rank condition holds. Although it is not essential to construct both sets of invariants, it is necessary to determine M* which requires v and y; thus, the algorithm presented has definite advantages over others, since these integers are simultaneously determined. In practical modeling applications, the prediction of model performance is normally necessary; therefore, knowledge of a minimal extension is required. Also in some of the applications the number of degrees of freedom may not be of interest, if only one partial realization is required rather than the entire class. In this case such a model is easily found by setting all free parameters to zero which corresponds to merely applying the Algorithm (3.1-1) directly to the data and obtaining the corresponding canonical forms as before. Describing the class of minimal extensions offers some advantages over the state space representation in that it is coordinate free and indicates the number of degrees of freedom available without compensation. 3.3 Characteristic Polynomial Determination by Coordinate Transformation In this section we obtain the characteristic equation of the entire class of minimal partial realizations described by Fr or F^, of the previous section. It is easily obtained by transforming the realized Fr or Fc into the Buey form as before. Recall that the advantage of this representation over the Luenberger form is that it is possible to find the characteristic polynomial directly by inspection of FgR in (2.2-11). 26 These matrices are in the form of (2.2-5), but it is easliy verified by constructing that the controllability invariants are in fact Pl=2, ^=2 and a-j = [-1-1 -1 1]I ol, = [-2 0 -2 4]I The problem with the Luenberger forms is that the maps it: Xq-* Xq/E are not well defined. Thus, the image of the maps are indeed canonical forms, but as shown here for (F,G)eXq/E, we need not have tt(F,G)=(F,G), i.e., the mapping does not leave the canonical forms unchanged. The point to remember is that the invariants are the necessary entities of interest which must be determined. 4* The procedure to construct the transformation matrix Tq of (2.2-7) is called the Luenberger second plan. The first Luenberger plan consists of examining the columns of i^n, given by V ... F--'g, F"'\l '2-2-8> where is an nmxnm permutation matrix, for predecessor independence. Thus, we can define a new set of invariants (under GL(n)) [{f^.}, ^jks^ completely analogous to the controllability invariants. The canonical forms associated with the invariants obtained in this fashion have 4* This procedure amounts to examining the column vectors of Wft for predecessor independence, i.e., examine g-j ... gm Fg^ ... Fgm . . 104 Thus, the Riccati equation solution has essentially been circumvented by this algorithm. However, if one not only desires a stochastic realization, but also a stable minimum phase, spectral factor, then the Riccati equation solution should be investigated. This the subject of our next section. 4.4 Stochastic Realization Via the Kalman Filter In this section we present a special case of the Riccati equation approach to solving the stochastic realization problem. This approach is a special case of the factorization (Denham (1975)) discussed in Section (4.1) because we require the unique, steady state solution to the discrete Riccati equation. It is well known (e.g. see Tse and Weinert (1975)) that the steady state solution uniquely specifies the optimal or Kalman gain. The significance of obtaining a stochastic realization via Kalman gain is twofold.,, First, since the Kalman gain is unique (modulo GL(n)), so is the corresponding stochastic realization. Second and even more important, knowledge of this gain specifies a stable, minimum phase spectral factor (e.g. see Faurre (1970) or Willems (1971)). The importance of this approach compared to that of the last section is that once the gain is specified, a stochastic realization is guaranteed immediately, while this is not true using the trial and error technique. However, the price paid for so elegant a solution to the stochastic realization problem is the computational burden of solving the Riccati equation. We use the innovations representation of the optimal filter and briefly develop it in the standard manner--from the estimation theory viewpoint. We then examine the realization of this model from the 27 become known as the Buey forms which were derived directly from the results of Langenhop (1964), Luenberger (1967), and Buey (1968). We refer the interested reader to these references as well as the recent survey by Denham (1974). Here we will be satisfied to note that the procedure of (2.2-1) applies with the set of controllable pairs (F,G) restricted to the {y.} invariants rather than {y.}. Analogous to the J J Luenberger forms, we define the row and column Buey forms as (^dr^br)* (Fbc>GBc) respectively. The row form is given by L11 fbr= L21 L22 0 > HBR = T v +1 V] + l Lpl Lp2 ... L PP + +% +i VI -^Vp.f (2.2-9) where L.. n I V T. U 'Xi 'o>r 'vr 8.. v. > 0 and satisfy E v =n ; 1 s=l 5 a. ^ KI.. are v.,v-, row vectors containing (3. invariants. 1 J! 1 SC n ij The transformation, TRB,required to obtain the pair (FgR>HBR) is ' BR 'BR [T 1 (2.2-10) where tI = [hT(h_.F)T D.j I I Vl T (h,F 1 ) ], ieÂ£ 95 L R 1 1 -1 1 1 -1 -1 -1 33 and therefore 2 -1 0 RR 0 3 Or 1 ^2_Â£33^ (-It^33) This example illustrates two methods of specifying an invariant system description of the given stochastic realization. It also points out that selecting Q in (2) uniquely specifies R and S; however, selecting S in (3) uniquely fixes R, but not Q. Thus, there is an entire family of Q's which have the same R and S and each particular Q specifies a canonical form for (F,H,Q,R,S) on X_ under the action of GR . c n We must place these results into the proper perspective, since we are primarily concerned with the stochastic realization problem. Suppose Clearly, we are free to choose any coordinate system, (T,I_)eGR Once the coordinates are selected F and H are fixed from (4.1-9), since -1-1 F=TAT H=CT but the major problem of finding an not only such that the KSP equations are satisfied, but also so that A-0 still remains. The above methods of specifying an invariant system description partially resolve this problem. The first method shows that for given (F,H), the matrices R and S are fixed once a Q is specified; therefore, this quintuplet is an invariant system description, but whether or not it is a stochastic realization corresponding to the same PSD or {Cj} as can only be resolved by first determining if there exists a T and n such that the KSP 100 covariance matrix; thus, the condition A-0 reduces to det(Q-SR_1ST) ^ 0 (4.3-3) since R is a positive definite covariance matrix. On the other hand, if we consider the special case S=0p, then this constraint reduces the condition A-0 to det(Q) 0 for R > 0 (4.3-4) Thus, the choice of admissible Q,R,S must be restricted such that these constraints are satisfied. Recall that one possible choice is (Q,R,S) = (KNnV,NN^,KNNT) where K and NN^"'are specified by IT*, the unique solution of the discrete Riccati equation. Of course, if a canonical realization algorithm is applied to {C^}, then is found with T=TR, the Luenberger row coordinates, and (A,B,C)=(FR,GR,HR). If a positive semidefinite is selected, then a nR>0 is uniquely specified and therefore R and SR are found from the KSP equations. The quintuplet, (FR,HR,QR,R,'R) is an invariant system description (under GR ) and also a stochastic realization, if the above constraints are n satisfied. Note that the Riccati equation need not be solved. The following algorithm summarizes this technique as well as the alternate method discussed in Section (4.2). Stochastic Realization Algorithm (4.3-5) Step 1. Obtain from {Cj} as in (2.4-1). Step 2. Select a positive semidefinite Qp and solve the (LE) for nR. Step 3. Solve (4.3-2) for R and SR. 77 Xr (z) = z5+(e-3)z4+(d-2e+2)z3+(c-3d-e)z2+(-3c+2d+2e)z+(b+2c+be) rBR This example points out some very interesting points. When this technique is combined with the algorithm of (3.2-2), it offers a method which can be used to obtain the solution to the stable realization problem developed in Roman and Bullock (1975b). Also, if the system were realized directly in Buey form, then b=0 and a degree of freedom is lost; thus, in Ackermann's example Vj=l, while ours is v^=5. It is critical that al_]_ degrees of freedom are obtained as shown in this case, since the system is observable from a single output. This section concludes the discussion of the deterministic case of the realization problem. In the next chapter we examine the stochastic version of the realization problem. Abstract of Dissertation Presented to the Graduate Council of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy REALIZATION OF INVARIANT SYSTEM DESCRIPTIONS FROM MARKOV SEQUENCES By James Vincent Candy March, 1976 Chairman: Dr. Thomas E. Bullock Cochairman: Dr. Michael E. Warren Major Department: Electrical Engineering The realization of infinite and finite Markov sequences for multi dimensional systems is considered, and an efficient algorithm to extract the invariants of the sequence under a change of basis in the state space is developed. Knowledge of these invariants enables the deter mination of the corresponding canonical form, and an invariant system description under this transformation group. For the partial realization problem, it is shown that this algorithm possesses some attractive nesting properties. If the realization is not unique, the class of all possible solutions is found. The stochastic version of the realization problem is also examined. It is shown that the transformation group which must be considered is richer than the general linear group of the deterministic problem. The invariants under this group are specified and it is shown that they can be determined from a realization of the measurement covariance sequence. Knowledge of these invariants is sufficient to specify an invariant system description for the stochastic problem. The link between the vii 47 If we consider the alternate method implied in Corollary (2.3-6), then the following modifications to the preceding steps are required: (1)* Start with the following augmented array: ^pN I SN,N'l (2)* Obtain [P | Q] as before. (5)* Perform additional row operations on Q to obtain unit columns for each column possessing a leading row element, and perform row interchanges such that (2.3-8) is satisfied for each jem, i.e., obtain (6)* Obtain the a., jem, as in (2.3-6). J It should be noted that these algorithms are directly related to those developed by Ho and Kalman (1966), Silverman (1971), or Rissanen (1971). As in Ho's algorithm, the basis of the first technique is performing the special' equivalence transformation of Theorem (2.3-3) rk on S^ to obtain S^ The second technique accomplishes the same objectives by restricting the operations to only the rows of S^ which is analogous to either the Silverman or Rissanen method. The initial storage requirements in the first method are greater than the second if mN'>pN, since P and E can be stored in the same locations due to their lower and upper triangular structure; and (2) P will be altered in the second method, since row interchanges must be performed in (5)*; whereas, it remains unaltered in the first method which may be important in some applications. Consider the following example which is solved using both techniques. 36 nonzero i zero for ra=0;,... .a=Vj..N-1 for k=i+pa 'nonzero' for 4 b-0,... ,y -1 _ zero [b=y .,... ,N-1 J for s=j+mb These results follow since ^ has identical predecessor dependencies as SNjN,, then N.N' %>N IT where ^. = 0 if it depends on its predecessors. To find the observability indices, let a be the index of the last nonzero row of Â¡+pt t=0,l,...,N-1. T T Then if 6_. = jD v- = 0 otherwise = (a-i)/p+l. Similar results follow when is expressed in terms of the c^. The following theorem * specifies the matrices P and E required to obtain Theorem. (2.3-3) There exist elementary matrices P and E, respectively lower and upper triangular with unit diagonal elements, * such that N=PS^ ^(E has identical predecessor dependencies as Proof. Let PS^ M,=Q where Q is row equivalent to Â¡^i and P=[pr$]. If the j-th row of ^, is dependent on its predecessors, i.e., T T T V yi ;* then selecting P lower triangular such that TABLE OF CONTENTS ACKNOWLEDGMENTS .. iii LIST OF SYMBOLS vi ABSTRACT vii CHAPTER 1: INTRODUCTION 1 1.1 Survey of Previous Work in Canonical Forms for Linear Systems .. ... 2 1.2 Survey of Previous Work in Realization Theory.. 5 1.3 Purpose and Chapter Outline 10 1.4 Notation 11 CHAPTER 2: REALIZATION OF INVARIANT SYSTEM DESCRIPTIONS 12 2.1 Realization Theory ........ 12 2.2 Invariant System Descriptions .18 2.3 Canonical Realization Theory 33 2.4 Some New Realization Algorithms 45 CHAPTER 3: PARTIAL REALIZATIONS 54 3.1 Nested Algorithm ..... ........ 54 3.2 Minimal Extension Sequences ... 64 3.3 Characteristic Polynomial Determination by Coordinate Transformation ........ ..... 74 CHAPTER 4: STOCHASTIC REALIZATION VIA INVARIANT SYSTEM DESCRIPTIONS 78 4.1 Stochastic Realization Theory 81 4.2 Invariant System Description of the Stochastic Realization ......... ........... 87 4.3 Stochastic Realizations Via Trial and Error ... 97 4.4 Stochastic Realization Via the Kalman Filter .. 104 CHAPTER 5: CONCLUSIONS Ill 5.1 Summary Ill 5.2 Suggestions for Future Research 112 iv CHAPTER 4 STOCHASTIC REALIZATION VIA INVARIANT SYSTEMS DESCRIPTIONS, In this chapter the stochastic realization problem is examined by specifying an invariant system description under suitable trans formation groups for the realization. Superficially, this may appear to be a direct extension of results previously developed, but this is not the case. It will be shown that the general linear group used in the deterministic case is not the only group action which must be considered when examining the Markov sequence for the corresponding stochastic case. . Analogous to the deterministic realization problem there are basically two approaches to consider (see Figure 1): (1) realization from the matrix power spectral density (frequency domain) by performing the classical spectral factorization; or (2) realization from the measurement covariance sequence (time domain) and the solution of a set of algebraic equations. Direct factorization of the power spectral density (PSD) matrix is inefficient and may not be very accurate. Recently developed methods of factoring Toeplitz matrices by using fast algorithms offer some hope, but are quite tedious. Alternately, realization from the covariance sequence is facilitated by efficient realization algorithms and solutions of the Kalman-Szego-Popov equations. 78 45 jr\ In the next section we develop some new algorithms which utilize the theory developed here. 2.4 Some New Realization Algorithms In this section we present two new algorithms which can be used to extract both observability and controllability invariants from the given Markov sequence. Recall from Theorem (2.3-3) that performing row operations on the Hankel array does not alter the column dependencies, however, it is possible to obtain the row equivalent array, Q in a form such that the controllability invariants can easily be found. The first part of the algorithm consists of performing a restricted Gaussian elimination (see Faddeeva (1959) for details) procedure on the Hankel array. This procedure is restricted because there is no row or column interchange and the leading element or first nonzero element of each row is not necessarily a one. Define the natural order as 1,2,... . Algorithm. (2.4-1) (1) Form the augmented array: [IpN | S^, | ImN)] . (2) Perform the following row operations on N, to obtain Cp I Q I ImN'3: (i) Set the first row of Q equal to the first Hankel row. (ii) Search the first column of S^ ^, by examining the rows in their natural order to obtain the first leading element. This element is q^. (iii) Perform row operations (with interchange) to obtain q^-j =0,k>j. 4* Alternately it is possible to extract the Buey invariants from Q by reordering the columns as in (2.2-8) to obtain ()=QU and examining the columns for predecessor dependencies. 93 HRFR_1S* = HRLRHRT (4.2-10) -1 * In general, HRFR SR is not symmetric; therefore, the set of acceptable S is restricted by (4.2-10). Since any square matrix can be decomposed as the sum of a symmetric and skew-symmetric matrix, i.e., hrfr 1sr = (hrfr 1sr^sym + (hrfr sr)skw A then from (4.2-10) ^sp(p-1) elements of SR are constrained to satisfy (for given (FR,HR)) (Hrfr Sr)$kw = 0p (4.2-11) We limit our discussion to only cases (i) and (ii) of (4.2-8) because the techniques employed to obtain the invariant system description will be used in the next section to determine a solution to the stochastic realization problem. Thus, we have satisfied our second objective, i.e., we have specified a unique characterization of covariance equivalent systemsan invariant system description for X2 under the action of GR \ n It is important to note that when QR is selected corresponding to case (i) of (4.2-8), then SR and R are uniquely specified, but when S"R is selected as in case (ii), R is again uniquely specified; however, this is not true for Q. There is a family of Q 1s which correspond to this S"R and R because only np-%p(p-l) elements of QR are fixed. Consider the following example which not only illuminates this point, but also shows how to uniquely characterize the class of covariance equivalent systems by determining an invariant system description corresponding to both (i) and (ii) of (4.2-8). 115 R. W. Bass and I. Gura [1965] "High Order System Design via S.tate-Space Considerations," Proc. Joint Auto. Contr. Conf., Rensselaer, N. Y., pp. 311-318. C. Bonivento, R. Guidorzi, and G. Marro [1973] "Irreducible Canonical Realizations from External Data Sequences," Int. J. Contr., Vol. 17, pp. 553-563. F. M. Brausch and J. B. Pearson [1970] "Pole Placement Using Dynamic Compensators,"IEEE Trans, on Auto. Contr., Vol. AC-15, pp. 34-43. P. Brunovsky [1966] "On Stabilization of Linear Systems Under a Certain Class of Persistent Perturbations," Differential Equations, Vol. 2, pp. 401-406. [1970] "A Classification of Linear Controllable Systems," Kibern., Vol. 3, pp. 173-187. R. S. Buey [1968] "Canonical Forms for Mutivariable Systems," IEEE Trans, on Auto. Contr., Vol. AC-13, pp. 567-569. M. A. Budin [1971] "Minimal Realization of Discrete Linear Systems from Input-Output Observations," IEEE Trans, on Auto. Contr., Vol. AC-16, pp. 305-401. [1972] "Minimal Realization of Cohtinuous Linear Systems from Input-Output Observations," IEEE Trans, on Auto. Contr., Vol.. AC-17, pp. 252-253. T. E. Bullock and J. V. Candy [1974] "Modeling of Wind Tunnel Noise Using Spectral Factorization and Realization Theory," Proc. IEEE Southeastcon, Orlando, Florida. P. E. Caines and J. Rissanen [1974] "Maximum Likelihood Estimation of Parameters in Multi- variable Gaussian Stochastic Processes," IEEE Trans. Inform. Theory, Vol. IT-20, pp. 102-104. J. V. Candy, M. E. Warren and T. E. Bullock [1975] "An Algorithm for the Determination of System Invariants and Canonical Forms," Proc. 1975 Southeastcon, Auburn, Alabama. B. Carew and P. R. Belanger [19731 "identification of Optimum Filter Steady-State Gain for Systems with Unknown Noise Covariances," IEEE Trans, on Auto. Contr., Vol. AC-18, pp. 582-587. 22 or alternately we can say that the action of GL(n) on XQ induces F TFT"1 G + TG The action of GL(n) induces an equivalence relation on XQ. We indicate (F,G)Ej(F,G) if there exists TcGL(n) such that (F,G)=Tt(F,G). Dual results are defined for the observable pair (F,H) and the A. analogous set denoted by XQ. The third step of (2.2-1) is established in Popov (1972), but first consider the following definitions. For a controllable pair (F,G) 4*J* define the j-th controllability index y., jem as the smallest positive integer such that the vector F Jg. is a linear combination of J 4 y its predecessors, where a predecessor of F g. is any vector F g^ where J * rm+s we have assumed p(G) = m. Throughout this dissertation we use the following definition of predecessor independence: a row or column vec tor of- a given array is independent if it is not a linear combination of its regular predecessors. The following results were established by Popov (1972) Proposition. (2.2-3) (1) The regular vectors are linearly independent; (2) The controllability indices satisfy the m following relationship, E y. = n; (3) There exists n=l J exactly one set of ordered scalars, a^cK defined for jem, kej-1, s = 0,1,...,min(pj,pk-T) and for jem, k = j,...,m,s = 0,l,...,min(y.,y(c) 1 such that + Throughout this dissertation we use the overbar on a set to denote the dual set. ^These indices are also called the Kronecker indices. 65 available in the choice of minimal extensions. In fact, initially, the major motivation for constructing an extension was that it was necessary in order to be able to apply Ho's algorithm. Un fortunately, these approaches obscure the possible degrees of freedom and may lead to the construction of non-minimal extensions as shown by Godbole (1972). Roman and Bullock (1975a)developed the second approach to the solution of this problem. They show that examining the columns or rows of the Hankel array for predecessor independence yields a systematic procedure for extracting either set of invariants imbedded in the data. They also show that some of these would-be invariants are actually free parameters which can be used to describe the entire class of minimal partial realizations. These results precisely specify the number and intrinsic relationship between these free parameters. Unfortunately Roman and Bullock did not attempt to connect their results precisely with those in Kalman (1971b),Tether (1970). It will be shown that this connection offers further insight into the problem as well as new results which completely describe the corresponding class of minimal extensions. Before we state the algorithm to extract all invariants available in the data, let us first motivate the technique. When operating on the incomplete Hankel array, only the elements specified by the data are used. It is assumed that the as yet unspecified elements will not alter the existing predecessor dependencies when they are specified by an extension sequence. Since the predecessor dependencies are found by examining only the data in S(M,M), we must examine complete submatrices of S(M,M) in order to extract the invariants 28 The importance of the Buey form is that the characteristic equation can JL be found by inspection of the block diagonal arrays of FBR Since FBR is block lower triangular, the characteristic equation is given as Xp (z) = det(Iz-FpR) = Xj (z)...x, (z) (2.2-11) hBR bK L11 Lpp where the L.. are the companion matrices of (2.2-9). Similar results hold for the pair (Fg^.Gg^,) and the transformation is specified by TB(, constructed from the columns of W . n This completes the discussion of invariants and canonical forms for controllable or observable pairs. To extend these results to matrix triples (internal system description), it is more convenient to examine ft an alternate characterization of the corresponding equivalence class the Markov sequence of (2.1-4). This approach was used by Mayne (1972b) and Rissanen (1974), in order to determine the orbits of Z under GL(n). It is obvious that the sequence is invariant under this group action A. = (HT^MTFT-VVtG) = HF'3'1G (2.2-12) Consequently every element of A. can be considered an invariant of Z J with respect to GL(n); therefore, two systems which are equivalent under GL(n) possess identical Markov sequences. The converse is also true, i.e., any two systems with identical Markov sequences are equivalent. The standard approach to investigate a system characterized by its Markov sequence is to form the Hankel array, N, where we define sT , + It should be noted that the Buey form is not a canonical form if the transformation group includes a change of basis in either input or output spaces, while the Luenberger form is still a canonical form. 51 (7) The canonical forms of zR, BR(z), DR(z) and Eg, Bg(z),. DG(z) are z2-2z 0 0 z 2z ~ Br(z) = -3 z2+z 2 z -z +z 1 z2-z_ ' VZ> = z+1 0 2z+2 -2z Fq = [eg Â£3 ] I 2-1 GC = [^1 ^4] HC = *-a.l a.3 a.5 a>2] * 1 1 LI 2 4 I 2 2 6,2 1 3 0 _ Bc(z) = TZ + fz + i. 8 3. _2 -z +z 3 3 2 z fz ; Dc(z) z fz Â£ z -z + Â£ z2- fz + f (8) The {v.} and TgR are determined simultaneously as: 1Â¡ 1 1 1 0 0 0 and tbr 4 0 1 0 0 4 0 0 1 0 1 1 3 0 1 1 TABLE OF CONTENTS (Continued) REFERENCES .....114 BIOGRAPHICAL SKETCH .....124 V \ V 55 factorization technique to solve the partial realization problem when (R) is satisfied. His algorithm not only solves the problem in a simple manner, but also provides a method for checking (R) simultaneously. In the scalar case, Rissanen obtains the partial realizations, Z(K), K=l,2,... imbedded in the nested problem of (2.1), but unfortunately this is not true in the multivariable case. Also, neither set of invariants is obtained. The development of a nested algorithm to solve the partial realization problem given in this dissertation follows directly from (2.4-1) with minor modification. There are two cases of interest when only a finite Markov sequence is available. Case I. (R) is satisfied assuring that a unique partial realization exists; or Case II. (R) is not satisfied and an extension sequence must be constructed. The nested algorithm will be given under the assumption that Case I holds in order to avoid the unnecessary complications introduced in the second case. The modified algorithm is given below. The corresponding row or column operations are performed only on the data specified elements. Partial Realization Algorithm. (3.1-1) (1) Same as (1) and (2) of Algorithm (2.4-1) except (iv) is q^O k>j if k is a row whose leading element has been specified. (2) If (R) is satisfied for some M*=v+y, obtain the invariants as before in (3), (4) of (2.4-1) and go to (5). If not, continue. 127 [1974] "Basis of Invariants and Canonical Forms for Linear Dynamic Systems," Automtica, Vol. 10, pp. 175-182. [1975] "Canonical Markovian Representations and Linear Prediction," Proc. 1975 IFAC Congress, Boston, Mass. J. Rissanen and T. Kailath [1972] "Partial Realizations of Random Systems," Automtica, Vol. 8, pp. 389-396. J. Roman [1975] Low Order Observer Design Via Realization Theory,: Ph.D. Dissertation, Univ. of Florida, Gainesville, Florida. J. Roman and T. E. Bullock [1975a] "Minimal Parital Realizations in a Canonical Form," IEEE Trans, on Auto. Contr., Vol. AC-20, pp. 529-533. [1975b] "Design of Minimal Order Stable Observers to Estimate Linear Functions of the State via Realization Theory," IEEE Trans, on Auto. Contr., Vol. AC-20, pp. 613-623. H. H. Rosenbrock [1970] "State-Space and Multivariable Theory, John Wiley and Sons, Inc. N.Y. C. A. Roveda and R. M. Schmid [1970] "Upper Bound on the Dimension of Minimal Realizations of Linear Time Invariant Dynamical Systems," IEEE Trans, on Auto. Contr., Vol. AC-15, pp. 639-644. P. Rozsa and N. Sinha [1975] "Minimal Realization of a Transfer Function Matrix in Canonical Forms," Int. J. Contr., Vol. 21, pp. 273-284. G. N. Saridis and R. Lobbia [1972] "Parameter Identification and Control of Linear Discrete Time Systems," IEEE Trans, on Auto. Contr., Vol. AC-17, pg. 491. L. M. Silverman [1966] "Transformation of Time Variable Systems to Canonical Form," IEEE Trans, on Auto. Contr.., Vol. AC-11, pp. 300- 303. [197T] "Realization of Linear Dynamical Systems," IEEE Trans, on Auto. Contr,, Vol. AC-16, pp. 554-567. L. M. Silverman and H. E. Meadows [1966] "Equivalence and Synthesis of Time Variable Linear Systems," Proc 4-th Allerton Confr. Circuit and System Theory, pp. 776-784. 20 complete invariant. In general we will be interested in a complete system of invariants for E given by the,set of invariants (f^} where 4* f : X + Y1xY2X. xYn> f. is an invariant for E, and f-j (x^i (xg) * > ffi(X1)~fn(x2) imPly x]Ex2* Completeness of this set of invariants means that the set is sufficient to specify the orbit of x, i.e., there is a one to one correspondence between the equivalence classes in X and the image of f. If the set of complete invariants is independent, then the map f: X+Y-jX.. ,xYn is surjective. This property means that corresponding to every set of values of the invariants there always exists an n-tuple in Y specified by this set. A complete system of independent invariants will be called an algebraic basis. Generally, we consider a subset of X (e.g., in system theory a controllable system). Correspondingly, let f be a function mapping the subset XQ of X into set Y, then f is a restriction of f if fQ(x)=f(x) for each xeXQ. We can uniquely characterize an equivalence class E(x) by means of the set of values of the functions f.(x), ien. where the {f..} constitute a complete set of invariants for E on X. If the corresponding complete invariant f is restricted such that its image is itself a subset of X, then we have specified a set of canonical forms C for E on X. To be more precise, a canonical form C for X under E is a member of a subset C<=X such that: (1) for every xeX there exists one and only one ceC for which xEc, and since C is the image of a complete invariant f, then (2) for any xeX and c-j, C2eC, xEc^, and xEc2 implies f(x)=f(c-|)=f(c2)=c-j=C2 (invariance); (3) for any ceC if f(x-|)=c and f(x2)=c, then x-|Ex2 (completeness). Thus, c=f(x) is a unique member of ^This notation is actually f=(f-|,... ,fn) :x-^Y^x.. .xYn, but it is shortened when the set {f.} is clearly understood. 79 REALIZATION FROM COVARIANCE SEQUENCE ^ PSD AND ALGEBRAIC METHODS STOCHASTIC REALIZATION Figure 1. Techniques of Solution to the Stochastic Realization Problem. The problem considered in this chapter is the determination of a minimal realization from the output sequence of a linear constant system driven by white noise. The solution to this problem is well known (e.g.. see Mehra (1971)) as diagrammed below in Figure 2. The output sequence of an assumed linear system driven by white noise is correlated and a realization algorithm is applied to obtain a model whose unit pulse response is the measurement covariance sequence. A set of algebraic equations is solved in order to determine the remaining parameters of the white-noise system.. This problem is further complicated by the fact that the covariance sequence must be estimated from the measurements. From the practical viewpoint, the realization is highly dependent on the adequacy of the estimates. Although in realistic situations the covariance-estimation problem cannot be ignored, it will be assumed throughout this chapter that perfect estimates are made in order to concentrate on the realization -J* portion of the problem. In this chapter we present a brief review of the major results necessary to solve the stochastic realization problem. We use the 4-' Majumdar (1976) has shown in the scalar case that even if imperfect estimates are made realization theory can successfully be applied. FACTORIZATION METHODS 52 (9) T"1 is given by solving the equations for the last row as: r1 1BR 1 0 0 3 0 1 .0 0 0 0 0 0 1 0 -1 1 (10) Find FBR and Xp(z) as F =T F BR VBRVbR 2 0 0 2 0 0 0 1 0 0 1 0 0 0 1 2 and XF(z) = (z-2)(z3-2z2+l) = z4-4z3+4z2+z-2 This completes the first method. If the second method is used instead, then only (5)*, (6)*, and (8)* differ. (5)* Performing the additional row operations and interchanges to satisfy (2.3-8) gives: 5 1 T 1 -T 1 l T * -T 0 0 1 T l T 3 8 0 0 0 0 -$- (D 0 0-1 0 --z- -f T o 0 Q 1 o i * 0 0 0 0 1 Â£ 3 0 o I- 0 9 1 T -T 1 3 0 (6)* The a.'s are determined from the appropriate columns of Q as: J -T "-1 -i 1 -1 = 5 T = W7 2 = 0 I L_ T _ - - = W4 xml record header identifier oai:www.uflib.ufl.edu.ufdc:UF0008246900001datestamp 2009-02-09setSpec [UFDC_OAI_SET]metadata oai_dc:dc xmlns:oai_dc http:www.openarchives.orgOAI2.0oai_dc xmlns:dc http:purl.orgdcelements1.1 xmlns:xsi http:www.w3.org2001XMLSchema-instance xsi:schemaLocation http:www.openarchives.orgOAI2.0oai_dc.xsd dc:title Realization of invariant system descriptions from Markov sequencesdc:creator Candy, James Vincentdc:publisher James Vincent Candydc:date 1976dc:type Bookdc:identifier http://www.uflib.ufl.edu/ufdc/?b=UF00082469&v=0000103197863 (oclc)000180838 (alephbibnum)dc:source University of Floridadc:language English 34 or .V^-r T T T but det (W^, WN,} t 0; thus, a = 0^, i.e., a dependent row of is a dependent row of V^. Conversely assume that there . exists a nonzero aT as before such that v nT VN ^tnN' Since p(W^,)=n, it follows that this expression remains unaltered if post-multiplied by W^,, i.e., A/V 4- and the desired result follows immediately.V The significance of this lemma is that examining the Hankel rows (columns) for dependencies is equivalent to examining the rows (columns) of the observability (controllability) matrix. When these rows (columns) are examined for predecessor independence, then the corresponding indices and coefficients of linear dependence have special meaning-- they are the observability (controllability) invariants. Thus, the obvious corro!ary to this lemma is \ Corollary. (2.3-2) If the rows of the Hankel array are examined for predecessor independence, then the j-th (dependent) row, where j=i+pv., ieÂ£ is given by P + Z i-l min(v.,v -1) T 1 5 T Ij = 2 z s-1 t=0 p min(vi,vs)-l isre+pt s=l Z t=0 ^ist-s+pt whereig^^} an.d{v/} are the observability invariants and kepN is the k-th row vector of . 69 The indices are: v-|=2, V2=3 (2) For 1=1, (1+p ^-1)=4, k.j=3, m'(M+T-k1 )=4;' thus, the corresponding submatrix is constructed from the first 4 rows and columns of Q(4,4) (small, dashes). J-j={2}. For i=2, (2+pv2-l)=7, k2=4, m(M+l-k2)=2; thus, the corresponding submatrix of Q(4,4) is given by the first 7 rows and 2 columns (large r'a'shes). J2={2,4,6}. I T (3) Replacing the fifth and eighth rows of P with jDj. + bÂ£2 and Â£Â¡ + c4 + T T + ej^. where b,c,d,e are real scalars gives = [2 b -3 0 1 0 0 0 ] = [-3-e c 0 d+e 0 e 0 1 ] The Â§1 are: g_{ = [ -2 3 -b 0 0] g_2 = [ 3+e 0 -c -(d+e) -e ] (4) The canonical form is Corresponding to ttiese realizations is a minimal extension sequence which can be found by determining the Markov parameters. These parameters 4 30 Independence is shown by constructing the Luenberger form of (2.2-5) and (2.2-14) below^ directly from these invariants. The dual result yields another basis on X,, [{v.},{g. .},{aT }]. I I I S L J The corresponding canonical forms for EeX-j or are given by the Luenberger pairs of (2.2-5,2.2-6) and and a' (ia]-1 )m+ l I (v^ljp+l. ] (2.2-14) and the canonical triples are denoted by and respectively. Rissanen (1974) also shows that a canonical form for the transfer function can be constructed from the invariants of (2.2-13). This is possible because the determination of canonical forms for Â£ based on the Markov parameters is independent of the origin of A^'s. Rissanen defines the (left) matrix fraction description (MFD) as T(z) := B"1(z)D(z) (2.2-15) v , where B(z) = z B.z for |B \f 0 i=0 1 v V-l 4 D(z) = I D.z1 . i=0 1 3 state space representation for analysis and design. In early literature, however, transfer function representations were used. For highly complex systems it is much easier to determine external behavior rather than internal, since the state variables are normally not available for measurement. As pointed out by Kalman (1963) the language of these representations may be different, but both describe the same problems and are related. Many researchers have investigated the relationship between both representations, but always with one common goalto obtain a state space model which specifies the external description directly by inspection. Kalman (1963) and later Johnson and Wonham (1964), Silverman (1966) have shown that there exists a canonical form (under change of basis in the state space) in the scalar case for the triple (F,g,h) where F is in companion matrix form (see Hoffman and Kunze (1971)) and g is a unit column vector. It was shown that there exists a one to one correspondence between the non-zero/non-unit elements of the triple and the transfer function. This representation was used by Bass and Gura (1965) to solve the pole-positioning problem and recently by Wolovich (1972b)in solving the exact model matching problem. , The progress in determining a canonical form for the internal description of multivariable systems came more slowly. The earliest work appears to be that of Langenhop (1964) in which he develops a representation to study system stability. Brunovsky (1966,1970) was probably the first to recognize the invariant properties of the canonical form for the controllable pair (F,G). Tuel (1966,1967) developed canonical forms for multivariable systems in his investigation of the quadratic optimization problem. Subsequently, Luenberger (1967) proposed certain sets of canonical forms for controllable pairs; however, his development allowed realization from the measurement covariance sequence, the white noise model and the steady state Kalman filter is established. vm 62 available, i.e., {1,1+p,...,l+p(v1(M)-l),...,p,2p,...,pvi(M)} and similarly denote the row vectors of the elementary row matrix of the previous chapter, by (M). From Lemma (3.1-3), it follows that jJ J*+k and 4+pv.(H)(M) Â£.T+pv1 (M+k)(M+k) si"ce the observability invariants are identical. The specify the elements in and along with the 3ist they specify the elements of pi+pv.(M)(M) (see Coro11ary (2-3-4)). From Lemma (3.1-4) it is clear that JÂ¡Â¡jcjjÂ¡Â¡+k since v. (M+k^v^M). Reconsider Example (3.1-2), to see these properties. In this case we have M=2, k=3, M*(2>2, M*(5)=5, and p(S(5,5))>p(S(2,2)) as in Lemma (3.1-4); therefore, since 3-? = and J5 = d*3,2,5}. The observability indices are identical except for v2(5)>v2(2); thus, iaj (2) ,a2> (2)}<={aj^ (5) ,a^ (5) ,a2>(5) ,a^ (5)} since aT (2) = a! (5) for j=l,3. We also know from Example (2.4-2) that Â£(5) is the solution to the realization problem and therefore the properties of Lemma (3.1-3) will hold for {A|yj}, M>5. Table (3.1-5) summarizes these properties. The results for the dual case also follow directly. We now proceed to the case of constructing minimal partial realizations when (R) is not satisfied, i.e., the construction of minimal extensions. 29 ieN and S jeN1 as the block rows and columns of SM and the J IN5I1 block column and row vectors, a or a^ denote the r-th column of si or the s-th row of S for remN1, sepN. Rissanen (1974) has shown 9 that by examining the set X, = {Â£ I I controllable and observable with {y.} invariants} I J under the action of GL(n) that Proposition. (2.2-13) The set of controllability invariants and block column vectors, [{yj>,-Caj| appropriate indices constitute an algebraic basis for any ZeX^ under the action of GL(n). The proof of this proposition is given in Rissanen (1974) and consists of showing that any two members of X^ with identical Markov sequences are equivalent under GL(n). Thus, invariance follows by showing that a dependent column vector of the Hankel array can be uniquely represented in terms of the set [{y.},{a.. _}]. These parameters remain unchanged J JKS under GL(n); therefore, they are invariants. The block column vectors, a t satisfy a recursion analogous to (2.2-3), i.e., n-1 min(yJ.,y[<-l) m min(y.,uk)-l z k=l Z o .1 a , + Z jks .j+ms s=0 k=j ajksa.j+ms s=0 Thus, all dependent block columns can be generated directly from the set, {a of regular block column vectors. These vectors are invariants under GL(n), since every column vector of A. is an invariant as shown in J (2.2-12). Completeness follows immediately from the above recursion, since any two members of X^ possessing identical invariants satisfy the above recursion and therefore have identical Markov sequences. 119 R. E. Kalman and R. S. Buey [1961] "New Results in Linear Filtering and Prediction Theory," J. of Basic Engr. Vol. 83, pp. 95-108. R. E. Kalman, P. L. Falb, and M. A. Arbib [1969] Topics in Mathematical System Theory, McGraw-Hill, Inc. N.Y M. Lai, S. C. Puri, and H. Singh [1972] "On the Realization of Linear Time-Invariant Dynamical Systems," IEEE Trans, on Auto. Contr., Vol. AC-17, pp. 251-252: C. E. Langenhop [1964] "On the Stabilization of Linear Systems," Proc. Amer. Math. Soc., Vol. 15, pp. 735-742. G. Ledwich and T.E. Fortmann [1974] "Comments on 'On Partial Realizations'," IEEE Trans, on Auto. Contr., Vol. AC-19, pp. 625-626. D. G. Luenberger [1966] "Observers for Multivariable Systems," IEEE Trans, on Auto. Contr., Vol. AC-11, pp. 190-197. [1967] "Canonical Forms for Linear Multivariable Systems," IEEE Trans, on Auto. Contr., Vol. AC-12, pp. 290-293. Z. Luo [1975] Discrete Kalman Filtering and Stochastic Identification Using a Generalized Companion Form, Ph.D. Dissertation, Univ. of Florida, Gainesville, Florida. Z. Luo and T.E. Bullock [1975] "Discrete Kalman Filtering Using a Generalized Companion Form," IEEE Trans. Auto. Contr., Vol. AC-20, pp. 227-230. A. Majumdar 1975] Private communication. 1976] Modeling and Identification of the Nerve Excitation Phenomena, Ph.D. Dissertation, Univ. of Florida, Gainesville, Fla. D. Q. Mayne [1968] "Computational Procedure for the Minimal Realization of Transfer-Function Matrices," Proc. IEEE, Vol. 115, pp. 1363-1368. [1972a] "Parameterization and Identification of Linear Multi- variable Systems," Lecture Notes in Mathematics, Springer, Berlin, Vol. 294, pp. 56-61. [1972b] "A Canonical Model for the Identification of Multi- variable Linear Systems," IEEE Trans, on Auto. Contr., Vol. AC-17, pp. 728-729. CHAPTER 5 CONCLUSIONS 5.1 Summary This dissertation has contributed results in realization theory for both deterministic and stochastic cases. It was shown that by carefully specifying the invariants of the realization problem under a change of basis in the state space that a simple and efficient algorithm to extract these entities from the Markov sequence could be developed. This technique provides a solution to the realization problem directly in a canonical form, and an invariant system description under this transformation group is.'specified. The partial realization problem was solved by modifying this technique to develop a nested algorithm. It was shown that this method specifies the class of minimal partial canonical realizations. A new recursive technique to determine the corresponding class of minimal extensions while conserving all degrees of freedom available was developed. These results bridge the gap between the more classical approach of constructing a minimal extension and that of extracting the realization invariants. The characteristic equation is determined from the transition matrix in a convenient coordinate system by inspection. These coordinates were easily obtained from the given solution to the partial realization problem. In the stochastic realization problem it was shown that the transformation group which must be considered is richer than the general 111 99 In many practical situations, it is known a priori that the system and measurement noise sequences are uncorrelated. This case has been considered by many researchers (e.g. Faurre (1967), Anderson (1969), Mehra (1971), Rissanen and Kailath (1972), etc.) and it corresponds to setting S.-o in the WN model of (4.1-1). It is crucial to note that with this choice of S, it appears that the only trans formation group which leaves the PSD invariant is GL(n). From (4.2-4) it is clear that GRn not just GL(n) must be considered; therefore, there are %n(n+l) fewer invariants when GRn rather than GL(n) acts on X2. Recall the first technique outlined in Section (4.2) to obtain (F,H,Q,R,S) from {Cj}: realize S^p, select a Q, specify a n from the (LE), and then find R and 5 from the KSP equations. The selection of a proper Q is essential to obtain a quintuplet of X2 that is a stochastic realization. Therefore, it is useful to consider constraints which evolve from the fact that Q and II,R,S are stationary covariance matrices. For given (FR,HR)+, each choice of QR uniquely specifies a nR and hence R,SR as in (4.3-2), i.e., (FR,HR,QR,R,SR) is a canonical form on X^ for E-^-equivalence. Since F is a stability matrix, it follows from stability theory that if (F,/Q)^ is completely controllable, corresponding to each Q^O there exists a unique positive definite solution n to the (LE). Therefore, restricting the choice of QR to be non-negative definite simultaneously satisfies this "stability constraint" as well as the fact that Qr must be a covariance matrix. The results of the generalized Kalman-Szego-Popov lemma of (4.1-11) assures us that there exists at least one realization such that A is a +Here we assume the action of Gl(n) is completed with T=TR. ++/q is any full rank factor of Q, i.e., QVQ/Q"*" 108 Note that this realization is just a special case of the factorization of Denham (1975) discussed in Section (4.1) with n*=n in (4.1-13), K the Kalman gain in (4.1-12) and RÂ£=NN^. Clearly, the relationship between the canonical realization of E^p and EÂ£ of (4.4-9) is provided by the KSP equations, i.e., (4.4-6) and (A,B,C) (FR,FRHRHR +(Sinv)rHr) (4.4-10) D+D = HRnRHR+RINV Note that since K and R are unique, then E is unique (e.g. see Tse and Weinert (1975) or Denham (1975) for proof). Therefore, it is futile to attempt to determine EÂ£ from the trial and error algorithm of (4.3-5) because this quintuplet is a unique stochastic realization (modulo GL(n)). Recall from Section (4.2), if we let T=TR, then np of the total 2np+%p(p+l) invariants specify the pair (FR,HR) and it follows that np specify Kr and %p(p+l) specify RÂ£. Thus, the canonical realization of 'Xj 'X/ ^ the INV model is analogous to the WN model; however, unlike the QR,RR,SR obtained in the WN case by trial and error, (QINV)R> RjNV> ^Sinv^R are uniquely specified by KD and R in (4.4-9). The following diagram summarizes the relationship between these two distinct approaches to obtaining a stochastic realization. REALIZATION FROM {C^ J PSD KSP (Trial and Error) FACTORIZATION METHODS E INV (Riccati Equation) (F,H,Q,R,S) Figure 3, Solutions of the Stochastic Realization Problem 67 = 3110 + e!20 -2 +f3130 -3 {312r0) or [2 4 4 8] = 3110 [1 2 2 4]+ g]20 [1 2 2 4]+ g^H 01 0] . The solution is = 2 g^o ^130= t*1us* t*ie coefficient B-^O 1s ah arbitrary parameter. Note that this recursion is essentially the technique given in Roman and Bullock (1975a). Clearly, if (R) is satisfied as in the previous section, then there exists a complete submatrix (data is available for each element) of S(M*,M*) in which every regular vector of S(M,M) is always a regular vector of the submatrix corresponding to a particular chain; thus, there will be no arbitrary or free parameters. The algorithm for the case when (R) is not satisfied may be illustrated by considering row operations on S(M,M) to obtain Q(M,M), since the identical technique can be applied to obtain S*(M,M). The arbitrary (column) parameters are found by performing additional column operations to Q(M,M). As in Example (3.2-1), we must find the largest submatrix of Q(M,M) for each chain, i.e., if we define k_. as the index of the block row of S(M,M) containlirig the vector. , then the largest submatrix of data specified elements corresponding to the i-th chain is given by the first (i+pv^-1) rows and m(M+l-k.) columns of Q(M,M). Also, we define J|ieÂ£ as the sets of Hankel row indices corresppnding to each dependent (zero) row of the (i+pMj-l)x (m(M+l-kj) submatrix of Q(M,M) which becomes independent, i.e., it contains a leading element. In Example (3.2-1) for i=l, we have (1 +pv1-1)=3 and k-j =2; thus, nKM+l-k-^A and the corresponding submatrix is given by the first 3 rows and 4 columns of Q(3,3), and.of course, J^={2}. CHAPTER 1 INTRODUCTION Special state space representations of linear dynamic systems have long been the motivation for extensive research. These models are generally used to simplify a problem, such as pole placement, by introducing arrays which require the fewest number of parameters while exhibiting the most pertinent information. In general, system represen- . \ ' tations have been studied in literature as the problem of determining canonical forms; Canonical forms have been used in observer design, exact model matching methods, feedback system design, and Kalman filtering techniques. In realization theory, canonical forms for linear multi- variable systems are important. Since it is only possible to model a system within an equivalence class, the ability to characterize the class by a unique element is beneficial. The problem of determining a canonical form has its roots in invariant theory. Over the past decade many so-called "canonical" system representations have evolved in the literature, but unfortunately these representations were obtained from a particular application or from computational considerations and not derived from the invariant theory point of view. Generally, these representations are not even unique and therefore cannot be called a canonical form. Representations derived in this manner have generally been a source of confusion as evidenced by the ambiguity surrounding the word canonical itself. In this dissertation 1 CHAPTER 3 PARTIAL REALIZATIONS One of the main objectives of this research is to provide an efficient algorithm to solve the realization problem when only partial data is given. As new data is made available (e.g., an on-line application, Mehra (1971)), it must be concatenated with the old (previous) data and the entire algorithm re-run. What if the rank of the Hankel array does not change? Effort is wasted, since the previous solution remains valid. An algorithm which processes only the new data and augments these results (when required) to the solution is desirable. Algorithms of this type are nested algorithms. In this chapter we show how to modify the algorithm of (2.4-1) to construct a nested algorithm which processes data sequentially. The more complex case of determining a partial realization from a fixed number of Markov parameters arises when the rank condition, abbreviated (R), is not satisfied. It is shown not only how to determine the minimal partial realization in this case, but also how to describe the entire class of partial realizations. In addition, a new recursive technique is presented to obtain the corresponding class of minimal extensions and the determination of the characteristic equation is also considered. 3.1 Nested Alqorithm Prior to the work of Rissanen (1971) no earlier recursive methods appeared in the realization theory literature. Rissanen uses a 54 122' M. G. Strintzis [1972] "A Solution to the Matrix Factorization Problem," IEEE Trans, on Info. Th,y., Vol. IT-18, pp. 225-232. G. Szeg'd and R. E. Kalman [1963] "Sur la Stabilite Absolve d'un Systeme D'Equations aux Differences Finies," Compte Rendus a L'Academie des Sciences, pp. 388-390. A. J. Tether [1970] "Construction of Minimal Linear State-Variable Models from Finite Input-Output Data," IEEE Trans, on Auto. Contr., Vol. AC-15, pp. 427-436. E. Tse and H. L. Weinert [1973] "Extension of 'On the Identifiability of Parameters'," IEEE Trans, on Auto. Contr., Vol. AC-18, pp. 687-688. [1975] "Structure Determination and Parameter Identification for Multivariable Stochastic Linear Systems," IEEE Trans, on Auto. Contr., Vol. AC-20, pp. 596-603. W. Tuel [1966] "Canonical Forms for Linear Systems-Pt. 1," IBM Res. Lab., RJ 375, San Jose, Calif. [1967] "An Improved Algorithm for the solution of Discrete Regulation Problems," IEEE Trans, on Auto. Contr., Vol. AC-12, pp. 522-528. S. H. Wang and E. J. Davison [1972] "Canonical Forms of Linear Multivariable Systems," Univ. Toronto, Toronto, Canada, Control Syst. Rept. 7203. M. E. Warren arid A. E. Eckberg [1973] "On the Dimensions of Controllability Subspaces: A Characterization via Polynomial Matrices and Kronecker Invariants," 1974 JACC Preprints, Austin, Texas, pp. 157- 163. H. Weinert and J. Anton [1972] "Canonical Forms for Multivariable System Identification," Proc. Conf. Decision and Contr., New Orleans, La. N. Wiener [1955] "On the Factorization of Matrices," Comment. Math. Helv., Vol. 29, pp. 97-111. [1959] "The Prediction Theory of Multivariate Stochastic Processes: I-The Regularity Condition," Acta. Math., Vol. 98, pp. 111-150. J. C. Willems [1971] "Least Squares Stationary Optimal Control and the Algebraic Ricatti Equation," IEEE Trans, on Auto. Contr., Vol. AC-16, pp. 621-634. 14 If two minimal realizations Z, t are equivalent under a change of basis in X, then there exists a nonsingular T such that (F,G,H)^ = (TFT\tG,HT_1 ). It also follows by direct substitution that the controllability and observability indices of these realizations are identical and W. = TW. for j = 1,2,... J J V. = V.T"1 for i = 1,2,... The generalized NxN' block submatrix of the doubly infinite Hankel array is given by SN,N' = Implicit in the realization problem is determining when a finite dimensional realization exists and, if so, its corresponding minimal dimension. The following proposition by Silverman gives the necessary and sufficient conditions for {A^} to have finite dimensional realiza tion. Proposition. (2.1-5) An infinite sequence {A^} is realizable iff there exist positive integers y,v,n such that otVuu+j) Further, if {A^} is realizable, then p,v are the controllability and observability indices and n is the dimension of the minimal realization. ^This notation means F = TFT"\ G = TG, and H = HT ^. AN' W-l I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Thomas E. Bullock, Chairman Professor of Electrical Engineering I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as/a dissertation for the-^degree of Doctor of Philosophy. MTchael E. Warrenchairman Assistant Professor of Electrical Engineering I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Donald G. Childers Professor of Electrical Engineering I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Zoran R. Pop/Stojanovic Professor of Mathematics I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. 'i ufa opov Vasile M. Popov Professor of Mathematics 84 (to follow) which evolve from the generalized Kalman-Szego-Popov lemma a. (see Popov (1973) ). Thus, we specify the KSf3 model as the realization of {C.l defined by the quadruple, E^cd:=(A,B,C,D) of appropriate J 0 Iw r dimension with transfer function, TK<.p(z)=C(Iz-A)"^B+D. Note that since the unit pulse response of the KSP model is simply related to the measurement covariance sequence, then (4.1-7) can be written as the sum decomposition. *z(z) = T^pUJ+T^pU"1) = C(Iz-A)1B+D+DT+BT(Iz"1-AT)_1CT (4.1-8) The relationship between the KSP model and the stochastic realization of the measurement process is shown in the following proposition by Glover (1973). Proposition (4.1-9) Let zKSp=(A,B,C,D) be a minimal realization of {Cj}. Then the quintuplet (F,H,Q,R,S) is a minimal stochasti realization of the measurement process specified by (4.1-1) and (4.1-2), if there exists a positive definite, symmetric matrix n and TeGL(n) such that the following KSP equations are satisfied: n-AnAT * Q D+DT-CnCT = R B-AHC1" s where A=T-1FT and C=HT. The proof of this proposition is given in Glover (1973) and > ; -J- This book was published in Romanian ini966, but the English version became available in 1973. 4j* oo Note that the sequence, -CC^>Q is related to the measurement covariance sequence as Cn-hC and C-=C. for j > 0. 0 0 J J 57 (3) Indices are: v-j = 1 v2 = 0 h = 1 y2 = 1 oo n (4) ^Invariants are: Is- tP41 1 P43] [2 1 0] 4-- CP61 O | 1 II 1 1 OO *Â£> a. 1 1] and -1 = ^P41 P42 P K43 P44 1 $ [-2 0 0 10 0] 2 = *--3 1 P21 P 2 1 o{] = [0 0 0-1 i 0] -3 = ^P61 P62 P 63 P P P 64 K65 0-1001] el 3 -1 ' e14 *0 " eT3 1 a 2 ' e14 _0 e23 1 ~T e24 -2 T *." e23 v 1 T e24 2 . b, = e33 1 5 bo~ e34 0 tm- _0 _ 0- -e44- _1 _ To my wife, Patricia, and daughter, Kirs tin,, for unending faith, encouragement and understanding. To my mother, Anne, for her constant support and my mother-in-law, Ruth, for her encouragement. To "big" Ed my father-in-law, whose sense of humor often lifted my sometimes low 38 This theorem shows that the indices can be found by performing a sequence of elementary lower triangular row and upper triangular column operations in a specified manner on the Hankel array and examining the nonzero rows and columns of S^,, the structural array of The {cu^} and {BTjsare also easily found by inspection from the proper rows of P and columns of E as given by Corollary. (2.3-4) The sets of invariants or more compactly j the sets of n vectors {8.},{a.} are given by the rows of P and columns of E in (2.3-3) respectively as 4 CVV+P Pqr+P(vrl)] q'pV1 1>reÂ£ a. [e .e . -j st s+mt es+m(prl)t] j-5Â£a where Pqrest : q=r, r=s q Proof. The proof of this corollary is immediate from Theorem (2.3-3).V We can also easily extract the set of invariant block row or column vectors, {a! },{a } from the Hankel array and therefore, we have a J - > solution to the canonical realization problem. Theorem. (2.3-5) If the generalized Hankel submatrix of rank n is transformed by elementary row operations to obtain a row equivalent array, then by proper choice of P the matrix Q is given by: 39 TG | TFG ... j TFN'_1G 'V pn-M mN 0Pn-N _umN _ Q = where (F,G) is a controllable pair and det TVO. Proof. .If z is a minimal realization, then it is well-known that p(VN)=p(W^,)=n. Since P is an elementary array, then it follows [PV^] r N -I.. p= and det T^O. Post multiplication by W^, gives [G | FG | ... | PVV = _T__ pN- FN'"1G] = PS N,N1 Multiplication of the arrays gives the desired results.V Corollary. (2.3-6) If P*is selected such that Q is as in (2.3-5) with the pair (F,G) in Luenberger column form, then the set of invariants {a-}, jem is given by the columns of J Wl\|> > w^ kemN1 with ak = k=pjm+j Proof. If P is selected in Theorem (2.3-5) such that T=T^, then it follows that each column of W.,, corresponding to the (j+mp.)-th '* u for each jem contains the {a^} invariants.V t. The method of selecting P is given in the ensuing algorithm. 113 results with realistic noisy data. Along these lines the use of maximum likelihood estimators by Caines and Rissanen (1974) and the least squares estimates in the technique of Majumdar (1976) should be investigated further. The use of Markov sequences to design controllers to solve the model following problem (e.g. see Moore and Silverman (1972)) should be examined by first defining the problem invariants and then inves tigating the possibility of using the realization algorithms of Chapters 2 or 3 to extract them. The use of the class (under GL(n)) of minimal extension sequences developed directly from a given finite sequence may prove instrumental in this technique and should be studied. An efficient technique to factor Toeplitz matrices (see Rissanen and Kailath (1972)) should be developed by extracting the invariants of the stochastic realization specified in Chapter 4. Analogous techniques for the equivalent frequency domain representation of this problem should also be investigated. 49 (3)The indices are obtained by inspection from the independent rows and columns of Q in accordance with Theorem (2.3-3) as: v1 1 u1 = 3 ^2 2 u2 ^ and p(S2s3) = p(S3 3) p(S2 = 4 satisfying (R). T T (4)The jfj and bjj are determined from the appropriate rows and columns of P as: 1 ~ -CP41 I P42 P45 I P433 = [2 I 0 0 I 0] -2 = "^p81 I p82 p85 I p83-* = I 0 V'M3 -3 = "^p61 I p62 p65 I p63^ = C"1 I 1 Ml -1 = tO-3 I P4i P42 p43 p44 I 5^3 = [O3 I -2 0 0 1 I ^ 2 = ^-p81 P82 P83 P84 P85 P86 P87 p88 I 3=E"3 O*1 0-1 0 011 3 -3 = % IP61 P62 p63 P64 P65 P66"* = % Â¡ 1 1 _1 0 _1 13 (5)Performing the column operations, obtain the structural array k N, and E as: ... Jc [P I 4 I E] where the leading elements are circled,. 81 algebraic structure of a transformation group acting on a set to obtain an invariant system description for this problem. A new realization algorithm is developed to extract this description from the covariance sequence. Recently published results establishing an alternate approach to the solution of this problem are also considered. 4.1 Stochastic Realization Theory Analogous to the deterministic model of (2.1-1 ) consider a white- noise (WN) model given by Vi = F*k + *k (4.1 4 =H4 where and ^ are the real, zero mean, n state and p output vectors, and Wj, is a real, zero mean, white Gaussian noise sequence. The noise is uncorrelated with the state vector, X., j k and J Cov(wi,wj.):=E[(wi-Ewi)(Wj-Ew[.)T] = x. where 6.. is the Kronecker delta. This model is defined by the triple, ij ZWN:=(F,In*H) compatible dimensions with (F,H) observable and F a nonsingular,^ stability matrix, i.e., the eigenvalues of F have magnitude less than 1. The transfer function of (4.1-1) is denoted by TWN(z). +In the discussion that follows the WN model parameters will be used to obtain a solution to the stochastic realization problem. Denham (1975) has shown that if the spectral factors of the PSD are of least degree, i.e., they possess no poles at the origin, then F is a nonsingular matrix 46 (iv) Repeat (ii) and (i i i) by searching the columns in their natural order for leading elements. (v) Terminate the procedure after all the leading elements have been determined. (vi) Check that at least the last p rows of Q are zero. This assures that the rank condition, (R) is satisfied. (3) Obtain the observability and controllability indices^ as in Theorem (2.3-3). T (4) Obtain Â£. iejj from the appropriate rows of P as in Corollary (2.3-4) T * and jb. as in (2.2-19) where $...=p... * J J (5) Perform the following column operations on Q to obtain [P | S* Nl | E]: (i) Select the leading element in the first column of Q, . (ii) Perform column operations (with interchange) to obtain qjs=0 for s>l. (iii) Repeat (i) and (if) until the only nonzero elements in each row are leading elements. (6) Obtain a., jem from the appropriate columns of E as in Corollary J (2.3-4) and Â¥. from the dual of (2.2-19). (7) From the invariants construct the Luenberger and MFD forms as in Section (2.2). If we also require the characteristic polynomial, then we must include: 4*4* (8) Determine the v.}, cjd and (simultaneously) construct TgR as in Lemma (2.3-12). (9) Find Til by solving for the non unit rows in TnnTÂ¡i I . BR BR BR n ^Note that the leading elements have been selected from the rows by examining the columns in their natural order; therefore, the dependent columns are not zero as in (2.3-3), but are easily found from this form of Q by inspection. It should also be noted that the leading elements could have been selected in the j, (j+m), (j+2m)... columns; therefore, facilitating the determination of the Buey invariants and forms. 4.4* r\j Alternately the {vj}, jem and Tjjc could be used. These indices can be found easily from the columns of Q. 8 One of the main contributions of this dissertation is to use the results developed from invariant theory to solve the realization and partial realization problems in the deterministic as well as stochastic cases. The realization of a system directly in a canonical form actually reduces to first determining which transformation groups are present, specifying the corresponding invariants, and then developing a method to 4* extract these invariants from the given unit pulse response sequence. This philosophy is basic to any canonical realization scheme and actually provides an explicit formula which is applied throughout this dissertation. In the last few years, several interesting extensions have emerged from the original concept of realization theory. The major motivation evolved just after the development of the Kalman filter (see Kalman (1961)) in estimation theory because a priori knowledge of the state space model and noise statistics are required to begin data processing. The link between the filtering and realization problem was established by Kalman (1965) just prior to the advent of Ho's algorithm. The work of Gopinath (1969), Budin (1971,1972), Bonivento et al. (1973), and Audley and Rugh (1973,1975) were concerned with the more general problem of, obtaining a state space representation given a general input/output sequence of the system in both deterministic and stochastic cases. The stochastic version of the realization problem has not received quite as much attention as the deterministic case mainly due to its greater complexity and high dependence on the adequacy of covariance estimators. The realization of stochastic systems was studied by Faurre (1967,1970) and more recently by Rissanen and Kailath (1972), Gupta and Fairman (1974) 4* * The Hankel array formulation is used exclusively in this dissertation.- 87 T Solving the last equation for K and substituting for NN yields K = (B-An*CT) (D+DT-CII*CT)"1 (4.1-13) Now substituting (4.1-13) and NN^ in the first equation shows that n* satisfies n* = An*AT-(B-An*cT)(D+DT-cn*cT)"1(B-An*cT)T (4.1-14) a discrete Riccati equation. Thus, in this case the stochastic realization problem can be solved by (1) obtaining a realization, ^KSP ^rom'{Cj}.; (2) solving (4.1-14) for n*; (3) determining NN^ from (4.1-12) and K from (4.1-13); and (4) determining Q,R,S from K and NN1. A quintuplet specified by T and n* obtained in this' manner is guaranteed to be a stochastic realization, but at the computational expense of solving a discrete Riccati equation. Note that solutions of the Riccati equation are well known and it has been shown thpt there exists a unique, n*, which gives a stable, minimum phase, spectral factor (e.g. see Faurre (1970), Willems (1971), Denham (1975), Tse and Weinert (1975)). We will examine this approach more closely in a subsequent section, but first we must find an invariant system description for the stochastic realization. 4.2 Invariant 'System Description of the Stochastic Realization Suppose we obtain two stochastic realizations by different methods from the same PSD. We would like to know whether or not there is any way to distinguish between these realizations. To be more precise, we would like to know whether or not it is possible to uniquely characterize the class of all realizations possessing the same PSD. We first approach this problem from a purely algebraic viewpoint. 63 Table. (3.1-5) Nesting Properties of Algorithm (3.1-1) Augment M^M+k n(M+k)=n(M) n(M+k)>n(M) JM"^M+k " fi-i+pv. > c where (R) is satisfied for some k and means that the corresponding invariants, vectors, or indices are nested or contained in a set of higher order. Vj 5ist J. 85 corresponds directly to the results presented by Anderson (1969) in the continuous case. The proof follows by comparing the two distinct representations of $z(z) given by (4.1-6) and (4.1-8). Minimality of (F,H,Q,R,S) is obtained directly from Theorem (3.7-2) of Rosenbrock (1970). The KSP equations are obtained by equating the sum decomposition of (4.1-8) to (4.1-6). This proposition gives an indirect method to check whether a given KSP anc* stoc^ast'lc realization, (F,H,Q,R,S) correspond to the same covariance sequence. Attempts to use the KSP equations to construct all realizations, (F,H,Q,R,S) with identical {CL} from and T by choice of all possible symmetric, positive definite matrices, H will not work in general because all n's do not correspond to Q,R,S matrices that have the properties of a covariance matrix, i.e., A:= Cov( pw. V [wj vÂ¡]) . ST R. 6,. 0 (4.1-10) First, it is necessary to question if the stochastic realization problem always has a solution, or equivalently, when is there a n so that (4.1-10) holds. Fortunately, the well-known PSD property, 4>z(z) 0 on the unit circle (see e.g. Gokhberg and Krein (I960) and Youla (1961)) is sufficient to insure the existence of a solution. This result is available in the generalized Kalman-Szego-Popov lemma (see Popov (1973)). Proposition (4.1-11) If (F,H) is completely observable, then $z(z) 0 on the unit circle is equivalent to the existence of a quintuplet, (?,fr,$,$,^Â¡) such that 60 T each dependent row vector, f.(M) is uniquely represented as a linear J combination of regular vectors in terms of the observability invariants and it can be generated from the recursion of Corollary (2)3-2); Similarly it follows from Proposition (2.2-13) that the dependent block row T vectors, a. (M) satisfy an analogous recursion. The following lemma J * describes the nesting properties of minimal partial canonical realizations Recall that M is the integer of Proposition (2.1-6) given by M =v+y. Lemma. (3.1-3) Let there exist an integer M (M)-M such that the rank condition is satisfied and let Â£(M) be the corresponding minimal partial canonical realization of {Ar>, rcM specified by the set of invariants [v.(M)},{fi. .(M)}, I l U ia^(M)}]. Then J ' v.(M) = ... = v.(M+k) 6-st(M) = ... = 3ist(M+k) al (M) = .... = aT (M+k) J J ' . ' A. , iff p(S(M,M))=p(S(M+l,M+l) = ... = p(S(M+k,M+k)) for the given k. Proof. If v. (M) = ... = v. (M+k), etc., then the minimal canonical partial realizations are identical, Â£(M) = Â£(M+1)= ... = Â£(M+k). It follows that p(S(M,M))=dimÂ£(M)=p(S(M+l,M+l))-dimÂ£(M+l) = p(S(M+k,M+k)). Conversely, P(S(M,M))=P(S(M+1,M+1))= ... =P(S(M+k,M+k)) implies dimÂ£(M)=dimÂ£(M+l)=... =dimÂ£(M+k). Since Â£(M) is a unique minimal canonical partial realization, so is Â£(M ). Furthermore, since each realization has the same dimension, each realization has 2 we follow an algebraic procedure to obtain unique system representations, i.e., we insure that these representations re in fact canonical forms. In simple terms this approach seeks the determination of certain entities called invariants obtained by applying particular transformation groups (e.g., change of basis in the state space) to a well-defined set representing a system parameterization. The invariants are the basic structural elements of a system which do not change under this trans formation and are used to specify the corresponding canonical form. This approach insures that the ambiguities prevalent in earlier work are removed. Initially, we develop a simple solution to the problem of determining a state space model from the unit pulse response of a given linear systerr and then extend these results to the stochastic case where the system is driven by a random input. The technique developed to extract the invariants from this (response) sequence not only provides a simple solution to the realization problem, but also gives more insight into the system structure. 1.1 Survey of Previous Work in Canonical Forms for Linear Systems The study of canonical forms for linear dynamic systems evolved slowly in the Sixties. The main impetus of investigation was initiated by Kalman (1962,1963) when he compared two different methods for describing linear dynamic systems: (1) internally by the state space representation denoted by the triple (F,G,H), or (2) externally by the transfer function--the input/output description. Development over the past decade in such areas as optimal control, decoupling theory, estimation and filtering, identification theory, etc., have relied heavily on the _ This defines (simply) the realization problem. 96 equations are satisfied. Clearly any choice of Q uniquely specifies a n for given F; for if, there exist two solutions and corresponding to identical Q,F, then n^=FH^F^+Q and n^FH^F^+Q. Subtract these equations to obtain n*-FII*F"I"=On for -n^. It is well known (e.g. see Gantmacher (1959)) that n*=0 is a unique solution of FH*-H*F*"^=0 , n n sinc^ A(F)^X(-F~T) in this case. Therefore, selecting a Q uniquely specifies a n and of course fixes R and S which can now be obtained from the remaining KSP equations. Practical considerations in selecting a Q which yields a positive definite II will be discussed in the next section. Here the point is for given F and H (modulo TcGKn)) are obtained from (4.1-9); moreover, selecting Q uniquely specifies a n which fixes R and S such that the KSP equations are satisfied. The resulting model, (F,H,Q,R,S) has the same PSD or equivalently {C.} as SKSP but ^ stl"^ may not satisfy A^O. Obtaining stochastic realizations such that the latter condition is satisfied is the subject of the next section. Similar results can be obtained by using the second method of specifying an invariant system description; however, recall in this case that only np-%p(p-l) elements of Q are uniquely specified. Since n is linearly related to Q for given F through the (LE), then the same number of elements are uniquely specified in n. Thus, when we select T=TR, the observability invariants of (2.2-4) uniquely specify the pair (FR,HR) and for any choice of QR (or alternately SR) we specify an invariant system description for the stochastic realization by (F^Hj^Qj^RpjS^). In the next section we develop an algorithm based on these techniques to obtain a stochastic realization. TOT Step 4. Check that (4.3-3) is satisfied. If so, stop. If not, choose another QR-0 and go to 2. If numerous choices of simple Q^O do not yield a stochastic realization, solve the discrete Riccati equation of (4.1-13) and go to 3. Or Step 2* Select an SR satisfying (4.2-10). Step 3* Solve (4.3-2) for R and Check that det (R)>0. If so, continue. If not, go to 2*. Step 4* Determine QR from the (LE) and select its free elements to satisfy (4.3-4) if ^=0p or (4.3-3). If so, stop. If not possible select another simple SR, i.e., go to 2*, or try the first procedure, i.e., go to 2. Consider the following example which illustrates this algorithm. Example (4.3-6) For m=3, p=2 the measurement covariance sequence is 20 59 iT5T 3 13 0 9 " 1,5 0 1 9 13 150 4 1 TT 22 6 1 - TO T " 79 1 6W tl o o 3 7 w O II 66 25 ,1 6 ~ TT S* O ro H 13 9 1 6 0 0 " 1 9 3 6 0 0 c3 5 115 55 720 0 3 3 3 7 720 0 u. U 1 5577 6155 ~ 275717 561964 720 0 7200 C5 1036800 1036800 1 043 1 285588 21434293 4057079 _1 728 0 - 311040 12441600 " 124416 00. Applying the algorithm of (2.4-1) we obtain (1) The observability invariants are: v-|=l, V2=2 and 4 = t -i i i o] J r i i __7_ Li 2 L 4 j 2 4 12*^ 31 The relationship of the MFD to the Hankel array, Sv+^ ^follows by writing (2.2-15) as B(z)T(z) = D(z) (2.2-16) and equating coefficients of the negative powers of z to obtain the recursion BoAj + BlVl 'f + BvAj+v = mj j=1>2 expanding over j gives the relation over the block Hankel rows as [B0 B, ... Bu] where the pxp(v+l) matrix of B.'s is called the coefficient matrix of B(z). Similarly equating coefficients of the positive powers of z gives the recursion \ 1 1 5 * t Vi,. = 0. m(y+l) (2.2-17) D, = B, ,A, + Bli0A0 + ... + B A . k k+1 1 k+2 2 v v-k k=0,l,...,v-l or expanding over k gives the relation in terms of the first block Hankel column as Dv-1 Bv O 1 Dv-2 II Bv-1 Bv * o 1 _B1 Bn ... B 2 v (2.2-18) 53 This completes the algorithms. In the next chapter the first method is modified to develop a nested algorithm from finite Markov sequences. 44 where v.,jg are the observability invariants of ZR i\, v.Â¡ are the invariants associated with ZRR and recall r. = Z v, ro=0> Proof. This lemma is proved by direct construction of the TD 's, Since each T0 satisfies for v.*v. i i hi Bi* h.F v.-l i' R hiFR 'V , Vi-1 v.-l T then analogous to property (2.3-8), it follows that h.FD =e 1 K Â¥ 1 and therefore v.-l V- V.-l j TP hiFR - V1 vi Vv1 Â¥r ' iFR In order to construct TRR it is first necessary to find the {v..} from the rows of [Vn]R, but in this case the v.'s can generally be found by inspection while simultaneously building TRR. Also, TRR is generally a sparse matrix with unit row vectors; therefore, the inverse can easily be -1 1 found by solving M >n directly for the unknown elements of TRR. 98 From Proposition (4.1-9), the observable pair.(A,C) of the KSP model and (F,H) of the WN model are E^-equivalent; therefore, the invariants are identical. The link between the canonical realization of ZKSp and the stochastic realization is provided by the KSP equations of (4.1-9), i.e., the (LE) and (A.B.C) = (FR,FRnRHRT+sR,HR) (4.3-1) D+DT = HRnRHRT+R Recall that under the action of GR^ on X^, there are 2np+^p(p+l) invariants--np specifying (FR,HR) and np+%p(p+l) specifying QR,R,SR. It is possible to extract these np+%p(p+l) invariants from the measurement covariance sequence using the KSP model realization, (A,B,C,D). As before, if we assume the action of LR is restricted to only the elements of QR, then for any choice of QR a unique nR is specified by the (LE) and therefore R and SR are uniquely obtained from (4.3-2) T R = W-DT-HRnRHRT SR B-FRnRHR On the other hand, suppose the action of LR is restricted to SR and Qr, then for any choice of SR, R and np-%p(p-l) elements of QR are fixed. Since nR is linearly related to QR through the (LE), the same number of elements are uniquely specified in nR. We are free to select the remaining elements in QR and n^. The realization invariants, }, of the previous chapters allow us to uniquely specify the invariants of (F,H,Q,R,S) from the KSP equations. Therefore, using EKSp and (4.1-9) we are able to extract the 2np+%p(p+l) invariants of (F,H,Q,R,S) from the measurement covariance sequence. 88 We define a set of quintuplets more general than the stochastic realizations, then consider only those transformation groups acting on this set which leave the PSD or equivalently (C .} invariant, and *3 finally specify various invariant system descriptions under these groups which subsequently prove useful in specifying a stochastic realization algorithm. The groups employed were first presented by Popov (1973) in his study of hyperstability. The results we obtain are analogous to those of Popov as well as those obtained in the quadratic optimization problem (e.g. see Willems (1971)). Define the set X2 = ((F,H,Q,R,S)| FeKnxn,HeKpxn,QeKnxri,ReKpxp,SeKnxp; Q,R symmetric} and consider the following transformation group specified by the set GKn := {L | LxKnxn; L symmetric} and the operation of matrix addition. Let the action of GK^ on X2 be defined by L t (F,H,Q,R,S) := (F,H,Q-FLFT+L,R-HLHT,S-FLHT) V (4.2-1) This action induces an equivalence relation on X2 written for each pair (F,H,Q,R,S), (F,H,Q,R,S)eX2 as (F,H,Q,R,S)EL(F,H,Q,R,S) iff there exists a LeGKn such that (F,H,Q,R,S) = L T(F,H,Q,R,S). This group and GL(n) are essential to this discussion, but we must consider their composite action. Therefore, we define the transformation group, GRn which is the cartesian product of GL(n) and GKn, GRn := GL(n)xGKn. The following proposition specifies GRn* 59 XD 0 0 0 0 0 0 0 0 1 -2 -1 -1 - T 4 _5 4 7 2 -10 -12"- 0 0 0 0 0 0 0 0 0 1 4 * -1 8 1.3 4 -6 -14 0 0 0 0 0 0 0 0 0 1 -1 __5. 2 -3 a. 4 2 15 20 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 -1 5 ~T -3 -6 -8 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0. 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 a 0 0 0 a a a a _o: a The results in this case are identical to those of Example (2.4-2). Let us examine the nesting properties of this realization algorithm. Temporarily, we resort to using data dependent notation for this discussion with the same symbols as defined previously in the previous sections, e.g., the minimal partial realization of order M is given by S(M) (F(M),G(M),H(M)). Thus, I(M+k) is a (M+k)-order partial realization. We also assume for this discussion that Â£(M) is in row canonical form; therefore, it can be expressed in terms of the set of invariants, [{v.j(M)},{B.Â¡st(M)},{aT(M)}]. If S(M) is an n dimensional, minimal partial realization specified by these invariants, then there T arc n regular vectors, Â¥g+pt(M) spanning the rows of S(M,M). Furthermore, 16 Proposition. (2.1-6) (Realizability Criterion) The minimal partial realization problem of order M possesses a solution, Â£(M) iff there exist positive integers * v,y, M = \H-y where dimE(M) = p(S^ ) = n. In this proposition (R) is designated the rank condition. Also, it is important to note that when (R) is satisfied the minimal extension (of 2(M)) is unique (see Tether (1970) for proof), but S(M) is not unique because there exist other minimal partial realizations equiv alent to S(M) under a change of basis in X. We must consider three possible cases when only partial data is * available. In the first case enough data is available such that M>M for known n; thus, a minimal realization is found. Second, v and y are available such that (R) is satisfied. In this case a minimal par tial realization can be found, but this in no way insures it is also a minimal realization of the infinite sequence, since the rank of S v may increase as v,y increase. Third, the rank condition does not hold How can a realization be found when no more data is available? The only possibility in this case is to extend the sequence until (R) is satisfied, but there can exist many extensions satisfying (R) while giving nonminimal realizations. For this reason define a minimal extension as any that corresponds to a minimal (partial) realization. To obtain minimality we must somehow select the right extension among the many possible. CHAPTER 2 REALIZATION OF INVARIANT SYSTEM DESCRIPTIONS In this chapter we present a brief review of the major results in realization theory. We establish a basic "formula" and apply it to various system representations. It is shown that this approach greatly simplifies the realization problem. Two new algorithms for realization are developed which appear to be more efficient than previous techniques because they extract only the minimal information necessary to specify a system from the given input/output sequence in an extremely simple fashion. All of the essential theory is developed and a multivariable example is presented. 2.1 Realization Theory A real finite dimensional linear constant dynamic system has internal description given by the state variable equations in discrete time as, ^<+1 + GJk (2.1-1) where keZ+, xeKn=X, uÂ£Km=U, yeK^Y and F, G, H are nxn, nxm, pxn matrices over the field K. X,U,Y are the state, input, and output spaces, respectively. 12 50 0 0 0 0 0 0 0 0 0 0 00 o o 000 0 0 0 0 0 0000 0 000 0 000.0.0 o o i _jl _jl n JL i. 13 2 2 U 4 8 4 10 o o o 1 -1 -4 -3 0 10 0 1 o The a. and b. are determined from the appropriate rows and columns J J of E as: e17 r 5-i -T e14 ~ -1 e37 , 1 4 e34 1 e57 .5 2 ao ~ "* e54 0 e27 1 8 e24 3 T el 7 e27 - _5 - 4 1 8 ^4 S4 e37 1 4 e14 1 e47 = 0 I -2 = e24 s 3 " ~T e57 5 2 e34 -1 e67 0 e44 1 e77 1 if 0 0 23 yi F Jg. *= E J k=l J-l nnn(y ,y.-1) " 1 % + z s=0 JKS K k=j m min(y.,yk)-l JE s=0 ajksF gi This proposition follows directly from the controllability of (F,G) and indicates that the regular vectors form a basis where the a's are the coefficients of linear dependencies. The set [{yj},{ctjks}], j,kem, s*0,...,y.-1 are defined as the controllability invariants of (F,G), J and y=max(y.). The main result of Popov is: J Proposition. (2.2-4) The controllability invariants are a complete set of independent invariants for (F,G)eX under the action of GL(n). The proof of this proposition is given in Popov (1972) and consists of verifying the invariance, completeness, and independence of [(y^},-Cctjj then can be replaced by TFsgk in the given recursion and the controllability invariants remain unchanged. Completeness is shown by constructing a TeGL(n) such that for two pairs of matrices (F,G), (F,G)eX0 with identical controllability invariants, (F,G) = (TFT""^, TG) or (F,G)Ej(F,G). Independence of the controllability invariants is obtained by constructing a canonical form determined only in terms of these invariants. Thus, by introducing a finite set of indices (y^}, Popov shows that this set along with the {ajks} are invariants under the action of GL(n). The main reason for specifying a set of complete and in dependent invariants is that it enables us to uniquely characterize the orbit of (F,G). It should also be noted that dual results hold for the observ able pair (F,H), and it follows that the observabi1 ity invariants are the 109 As a matter of completeness, we would like to briefly present an algorithm to obtain the stochastic realization using the Riccati equation approach. Mehra (1970,1971), Carew and Belanger (1973) and even more recently Tse and Weinert (1975) have proposed iterative schemes to obtain n,K,RÂ£, but the theoretical connection to the KSP equations and the stochastic realization was never established. Their results are summarized below and we refer the interested reader to these references for a detailed discussion of convergence properties and simulation results. Iterative Solution to the Riccati Equation (4.4-11) Step 1. Set nQ = 0 Step 2. (RÂ£).-. = D+D^-Hn.HT, where i is the i-th iteration step. Step 3. K. = (B-Fn.HT)(R )T] 1 1 Â£ 1 Step 4. n.,, = Fn-FT+K.(R ).kI K i+l i i e i i Once K and R are found in this manner, then the stochastic realization e follows from (4.4-9). The following algorithm: summarizes the realization technique using the INV model. < Stochastic Realization Algorithm via INV Model (4.4-12) Step 1. Obtain ZKSp from {C^} as in (2.4-1). Step 2. Use the iterative technique of (4.4-11) to obtain K, RÂ£. Step 3. Determine QINV,RjNy,SINV from K and RÂ£ as in (4.4-10). Thus, we have two algorithms to obtain an invariant system description for the stochastic realization using either the lsiN model or the INV model. The following figure summarizes these techniques. * 17 Prior to summarizing the main results of Kalman (1971) and Tether (1970), define the incomplete Hankel array associated with a given partial sequence {A^}, keM. as where the asterisks denote positions where no data is available. The rank of S(M,M) is the number of linearly independent rows (columns) determined by comparing only the data specified elements in each row (column) with the preceding rows (columns) with the cognizance that upon the availability of more data this number can only remain the same 4* or increase. Thus, the rank is a lower bound for any extension when the * are filled in-consistent with the preservation of the Hankel pattern. Both Kalman and Tether show that there are three pertinent integers associated with the incomplete Hankel array. They are defined as: n(M), v(M), y(M) and correspond to the rank of S(M,M), the observability index, and the controllability index of the given data. The latter two are lower bounds (separately) for v and y. Knowledge of either v(M), or y(M) enables us to construct extensions, since they are the least integers such that (R) holds for all minimal extensions. It should also be noted that the integers n,v,y,... are actually non-decreasing functions of the amount of data available, M, and should be written, n(M), v(M), y(M) etc. to be precise. However, the argument ^It also follows from this that the p(S(M,M)) is a lower bound for dim 2 (see Kalman (1971)). 66 associated with a particular chain (see Roman and Bullock (1975a)). Therefore, it is possible that a dependent vector, say ^ of a sub- matrix of S(M,M) later corresponds to an independent vector in S(M,M). When representing any other dependent vector in this submatrix m terms of regular predcessors, Â¥. must be included, since it is a regular vector of S(M,M) under the above assumption. In this represen- tation the coefficient of linear dependence corresponding to is arbitrary. Reconsider Example (3.1-2) for{A..}, i =1,2,3 where we only consider the (row) map P. Example. (3.2-1) For A^, Ag, A3 of (3.1-2) we have P: S(3,3)-K)(3,3) or " 1 2 2 414 8 2 2 414 8* 1 2 2 416 10 1 000 0(2)2 IJ_J3 2 0^|)-1-4jl-6 2 4 4 8 0 0 0 0 2 4 6 10 P 0 0(2)2 10 3 2 0 0 0 0 4 8 0 0 6 TO 0 0 .3 2 _ 0 0 The indices are = {1,2,1}. Since v^l, the fourth row of S(3,3) (or equivalently Q(3,3) ) is dependent on its regular predecessors as shown in the corresponding 3x4 submatrix (in dashed lines) of S(3,3) (or Q(3,3) ). The second row, say ^ > in this submatrix is dependent, yet it is an independent row of S(3,3) (or Q(3,3) ). Now, expand of this submatrix, i.e., 41 Theorem. (2.3-10) Given the infinite realizable Markov sequence from an unknown system, then SQ=([:Q>GcH(,)n is a minimal canonical realization of A^} with Fc [, | W2 WJ nr Gc is a submatrix of (W +1)c given by the first . columns . m HC = t\i 1+m(u1-1) ,m am ] ^m umn and Uj = C%+m Sj+rapj]- k fs a 1 vector of (W^+^)c. Proof. Since the sequence is realizable, there exist integers, n,v,y, satisfying Proposition (2.1-5). If Q is given as in Corollary (2.3-6), then Q = Gc 1 1 Fc Gc jr\ I! /"c" for k>y+l Thus, Gc is obtained immediately from the first m columns of (Wk)c* ^orm two nxn arrays ^ and A each constructed by selecting n regular columns of (W^)c starting with the first column for A and the (1+m) column for A The independent columns of (\)c are indexed by the y. and satisfy (2.3-8); thus, they are unit columns and A is a permutation matrix, i.e,, A = [w-, . I 1+m 2m w. j+m(yj-l) * 72 Thus, this example shows that the minimal extension sequence can be found recursively due to the structure of P. Of course, the problem of real interest is when (R) is not satisfied and (as in Ho's algorithm) a minimal extension with arbitrary parameters must be constructed. Minimal Extension Algorithm. (3.2-6) (1) Perform (1), (2), (3) of Algorithm (3.2-2). (2) Determine M* = v+y. (The values of v,y are determined by the partial data) (3) Recursively construct the minimal extension {A^,}, r = M+l, ... ,M* where Ar = [x^.(r)] by solving the set of equations for j(r) given by j+pv. Lj = 0 j m(M+l-k.)+l, ... ,m(M*+l-k.), for each iej>. and recall that k. is the index of the block row of S(M,M) containing the row vector, i+pv-. Example. (3.2-7) Reconsider (3.2-3) for illustrative purposes. (1) These results are given in Example (3.2-3) (2) M*=6; thus, find A5 = x-j 1 (5) x-j 2(5) A6 x-j 1 (6) x-j2(6) x21(5) x22(5) X21(6) x22(6) (3) Recursively solve: pT+2v Â£j 0 for i=1> j=5,6,7,8 and for i=2, j=3,4,5,6. Â£5 I5 = 0 gives x^iB); 1^ = 0 gives x12(5) Â£^ r3 =0 gives x21(5); ^ = 0 gives x22(5) Â£5 I7 = 0 gives xn(6); ^ = 0 gives Xj2(6) Â£g Z5 = 0 gives x21(6); 1^ = 0 gives x22(6) no Figure 4. Stochastic Realization Algorithms, BIOGRAPHICAL SKETCH James Vincent Candy was born in Astoria, New York on January 21, 1944 He graduated from Holy Cross High School, Flushing, New York in June, 1961 He received the degree of Bachelor of Science in Electrical Engineering in June, 1966 from the University of Cincinnati, Cincinnati, Ohio. Upon graduating he worked with the General Electric Company for 9 months. Then he enlisted in the Air Force of the United States in April, 1967. He received a commission as a Second Lieutenant in June, 1967 after completion of Officers Training School at Lackland AFB, Texas. He spent the majority of his four years' active duty at Eglin AFB, Fla. as a Threat Systems Engineer and Test Director until separated in June, 1971 as a Captain. In January 1968, he began study at the University of Florida Extension School (GENEYSIS) for a Master of Science Degree in Electrical Engineering. He completed his residency requirements in March, 1972 and received the M.S.E. from the University of Florida. From March, 1972, until the present time he has done work toward the degree of Doctor of Philosophy. James Vincent Candy is married to the former Patricia Meyers and they have one lovely daughter, Kirstin Patrice. He is a member of Phi Kappa Theta, Phi Kappa Phi, Eta Kappa Nu and the Institute of Electrical and Electronics Engineers. 124 112 linear group of the deterministic problem. The equivalence class under this group was specified and it was shown how the additional constraints imposed by the stochastic realization further restrict the selection of free parameters available in the corresponding noise covariance matrices. Specifying the invariants under this transformation group enabled the development of a trial and error algorithm to obtain a stochastic realization without requiring a Riccati equation solution. The link between the KSP, WN and steady state Kalman filter was presented. It was shown that realization of the KSP model allowed both representations to be determined. It was shown that determination of the filter parameters uniquely specifies a stochastic realization. An algorithm requiring the solution of a Riccati equation was also presented. 5.2 Suggestions for Future Research The results given in this dissertation open several interesting * possibilities for future research. Applying the algebraic framework of a transformation group acting on a set offers definite advantages over unstructured approaches. Simple equivalent solutions which confirm physical intuition may evolve. It may be possible to specify a set of invariants under the action of this transformation group which yields considerable insight into the problem structure. If the problem possesses additional constraints, it may be possible to utilize this information to influence the choice of free parameters available. Many problems of current interest can be examined in this framework (e.g. identification, exact model matching, and stable observer design problems). Efficient covariance estimators should be examined in order to facilitate the development of realization algorithms which yield useful 89 Proposition. (4.2-2) The closed set GRn and operation form a group where GRn = {(T,L) | TeGL(n);LeGKn} and the group operation is given by (T,L)o(T,L) = (TTfL+T"1LT"T). Proof. This proof of this proposition follows by verifying the standard group axioms with respective identity and inverse elements (In.0n) and (T_1,-TLTT).V Let the action of GR on X0 be defined by (T,L) 4- (F,H,Q,R,S) : = (TFT_1 ,HT~\t(Q-FLFT+L)TT,R-HLHT,T(S-.FLHT) ) (4.2-3) An element (F,h7q7R,1>) of the set is said to be equivalent to the element (F,H,Q,R,S) of X2 if there exists a (T,L)eGRn such that (F,H>'Q,R,S) = (T,L)4'(F,H,Q,R>S). This relation is reflexive (F,H,Q,R,S) = (InOn) + (F.H,Q,R,S) \ and symmetric (r1iTLTT)T(F,H,Q,RiS) = (r1>-TLTT)f((T,L)T(FsHsQ>R>S)) = ((T"1JLTT)o(T,L))+(F,H,Q,R,S)=(In,On)T(F,H,Q,R,S) . Transitivity follows from (F,H,Q,R,S)=(T,L)4'(F,HsQ,R,S) and (F,H,Q,R,S)=(T,Lj 4'(Tr,lT,^,R,S^) = (T,r)T((T ,L)+(F,H,Q,R,S.)) = (f ,L)4-(F,H,Q,R,S). Thus, GRn induces an equivalence relation on X2 which we denote by ETL and (4.2-3) defines the partitioning of X2 into classes. Note that our first objective has been satisifed, i.e., two EyL-equivalent quin tuplets have the same PSD; for if we let the pair (F,H,QRS), (F,H,Q,R,S)eX2 then if (T,L)eGRn 75 Even though it is possible to realize the system directly in Buey form as implied in the discussion of (2.3-12), it has been found that this method has serious deficiencies when dealing with finite Markov sequences. If (R) is satisfied, the partial realization is unique. When (R) is not satisfied, this technique does not yield all degrees of freedom. For example, reconsider the arbitrary parameter realization of Example (3.2-3). This realization is given in Ackermann (1972) as Q 1 0 0 O' "0 1 0 0 0 -2 3 -b 0 0 -2 3 0 0 0 0 0 0 1 0 n . 11 0 0 0 1 0 0 0 0 0 1 0 0 0 0 1 3+e 0 -c -(d+e) -e_ _3+e 0 -c -(d+e) -e_ Note that one degree of freedom (b=0) has been lost. Similarity Ledwich and Fortmann (1974) have shown by example that this technique can also lead to non-minima! realizations. These deficiencies arise due to the procedure used for the determination of the Buey invariants. This procedure does not account for the possibility that an independent row vector of a particular chain may actually be dependent if it is compared with portions of the same length of vectors in different chains. To cir cumvent the problem, the previous technique will be used,i .e., the system is realized directly in Luenberger form and transformed to Buey form. Not only does this assure minimality as well as the determination of all possible degrees of freedom, but Tg^ is almost found by inspection as shown in (2.3-12). Reconsider the example of the previous section. 71 A- *11(3) x21(3) x31(3) Since P maps S(2,2) into 0(2,2), we x12(3) x22(3) x32(3) have 2 2 4 2 2 4~ 12 2 4 0 0 0 0 1 0 1 0 P 0 -1 -4 0 0 ^0 0 2 A J x-ji (3) x]2(3) 2 4 j x2i(3) x22(3) 0 0 j 0 0 1 0 | x3i (3) x32(3)_ 0 0 | 0 0 and in this case,{v.j ,v2,v3> ={1,0,1}. Thus, using (3.2-4), we have solving 0 = Â£4 Â£3 = C-2 0 0 1 0 Q] 2 2 1 for x11(3) gives x11(3) X1 -j (3) x2i(3) X31(3) Similarily solving: Â¡^ = 0 for x-jg(3) gives x12(3) = 8 Â£3 = 0 for x3i(3) gives x^(3) =1 Â£^1^ = 0 for x32(3) gives x32(3) = 0 In this example, x2-j(3)=x^(3) and x22(3)=x-|2(3), since v2=0. 18 M will be understood throughout this dissertation in order to maintain notational simplicity. There is one more variant of the partial realization problem that must be considered. A sequence of minimal partial realizations such that each lower order realization is contained in one of higher order will be called a nested realization. Symbolically, this is given by ...-E(M)-S(M)-... for M to this problem is most desirable from the computational viewpoint, since each higher order model can be realized by calculating just a few new elements in the corresponding realization. Rissanen (1971) has given an efficient recursive algorithm to determine this solution. Another related problem of interest is determining a unique member of equivalent systems under similarity and is discussed in the following section. 2.2 Invariant System Descriptions In this section we review some of the fundamental ideas encountered when examining the invariants of multivariable 1 inear systems. The framework developed here will be used throughout this dissertation in formulating and solving various realization problems. Not only does this formulation enable the determination of unique system representations under some well-known transformations, but it also provides insight into the structure of the systems considered. First, we briefly define the essential terminology and then use it to describe some of the more common sets of canonical forms employed in many recent applications (e.g., Roman and Bullock (1975a,b), Tse and Weinert (1975)). |