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- On some structural properties of linear control systems--reachability from W in N and controllability to W in N
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- Hamano, Fumio, 1949-
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- 1979
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ON SOME STRUCTURAL PROPERTIES OF LINEAR CONTROL SYSTEMS-REACHABILITY FROM W IN N AND CONTROLLABILITY TO W IN N By FUMIO HAMANO A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILI4ENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1979 ACKNOWLEDGED NTS I wish to express my sincere appreciation to all those who contributed in various degrees toward the fulfillment of this work. I am particularly grateful to Professor R. E. KALMAN, the chairman of my supervisory committee, for his constant encouragement in seeking a concrete understanding of system theory and other areas in science. His guidance has been of a great help in establishing an organized view in this dissertation. Without the financial support which he arranged for me during the past four years and without the stimulating environment of the CENTER FOR MATHEMATICAL SYSTEM THEORY, this work may not have existed. I am thankful to Professor C. V. SHAFFER, co-chairman of my supervisory committee, who has not only given me valuable comments concerning my dissertation, but also made favorable arrangements for me during my personal emergency. The specific motivation for the research reported here was provided by the stimulating discussions with Professor G. BASILE relating to his earlier works concerning "geometric" views on various control and system problems. I appreciate his friendship and his deep interest in this work. A dissertation is only a part of a doctorate. In the educational process during the past four years the influence of Professors E. EMREE. D. SONTAG, Y. YAMAMOTO, G. SONNEVEND, T. E. BULLOCK, M. E. WARREN, C. A. BURNAP, M. HEYMANN, M. L. J. HAUTUS, V. KUCERA and others was essential in the preparation for a doctoral degree. Of course, no research would be made were it not for the long-term love and encouragement of a few close people. My parents and my wife, Shoko, have been constant sources of encouragement. To them I dedicate this work. Needless to say, I am grateful to Ms. Karen Todd for her understanding and patience not only as a typist but as a friend. This research was supported in part by US Army Grant DAAG 29-77-GO225 and US Air Force Grant AFOSR 76-3034 through the Center for Mathematical System Theory, University of Florida, Gainesville, FL 32611, USA. The research was also supported in part by the Foundation for International Information Processing Education, 2-6-1 Marunouchi, Chiyoda-ku, Tokyo 100, JAPAN. TABLE OF CONTENTS ACKNOLEDGEMENTS . . ii ABSTRACT . . v CHAPTER I. INTRODUCTION . . 1 II. THE r-STEP REACHABLE SUBSPACE FROM W IN N . . 7 1. F mod G Invariant Subspaces . . 7 2. Reachability From W in N . . 8 3. Nonrecursive Characterization of the r-Step Reachable Subspace from W in N . . . 12 4. Concluding Remarks . .14 III. THE r-STEP CONTROLLABLE SUBSPACE TO W IN N . . 16 5. Controllability to W in N . . .16 6. Nonrecursive Characterization of the r-Step Controllable Subspace to W in N . . . 25 7. Concluding Remarks . . . 27 IV. UNKNOWN INPUT OBSERVABILITY . . 28 8. Unknown Input Final State Observability . . 28 9. Unknown Input Final State Observability-Part 2 (Special Cases) . . 37 10. Unknown Input Initial State Observability . . 39 11. Concluding Remarks . . .43 V. STABILIZABILITY, OUTPUT ZEROING AND DISTURBANCE DECOUPLIN . . . . . 45 12. Stabilizability . . . 45 13. Output Zeroing . . 46 14. Disturbance Decoupling . . .48 15. Concluding Remarks . . . 51 VI. CONCLUSION . . .52 APPENDIX . 3 REFERENCES . . . . . 54 BIOGRAPHICAL SKETCH . . 57 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 ON SOME STRUCTURAL PROPERTIES OF LINEAR CONTROL SYSTEMS- REACHABILITY FROM W IN N AND CONTROLLABILITY TO W IN N By FUMIO HAMANO August, 1979 Chairman: Dr. R. E. Kalman Major Department: Electrical Engineering In the "geometric approach" to the study of linear systems two important notions have been successfully used; namely, the maximal reachability subspace Xrac (N) contained in a given subspace N and the maximal F mod G invariant subspace contained in N. However, the definition of X ach (N) is not as natural in discrete time systems as in continuous time systems. It loses an important meaning when it is applied to discrete time systems, i.e., Xreach (N) is not the set of states reachable from 0 via trajectmax onies in N. In this work similar notions which are suitable to treat discrete time systems are developed. In general the study is concerned with ttreachabilityf and "controllability" internal to the subspace N in discrete time systems. More specifically, the notions of the r-step reachable subspace from W in N and the r-step controllable subspace to W in N are introduced for given subspaces W and N (satisfying W C N). These are respectively defined to be the set of states reachable from W via trajectories in N in r steps and the set of states that can reach W in r steps via trajectories in N. Algebraic characterizations and sequential and "feedback" properties of the above notions are first given, and then applications to the unknown-input observability and other problems in control systems are given to show the significance of the results. CHAPTER I. INTRODUCTION This dissertation discusses some structural properties of finitedimensional, discrete time, constant, linear dynamical systems. Great efforts have been made to study "geometric" properties in the continuous time systems as well as those which are common to both discrete and continuous time systems. (See for instance BASIIE and MARRO [1968a and b], WONHAM and MORSE [19701, WONHAM [1974, Chapters 1 through 51 and SILVERMAN [1976, Section III].) However, distinctive features of differentiating discrete and continuous time systems have not received much attention. This work intends to point out that there are important differences between the two kinds of systems and that discrete time systems, therefore, should be treated separately in such cases. The following notation will be used in the sequel: "im", "ker" and "dim" respectively stand for "the image of", "the kernel of" and "the dimension of". "E" and ":=" mean "is an element of" and "is defined to be", respectively. Let us now turn to the definition of systems which is pertinent to the discussion of this dissertation. Let k be an arbitrary field, and let m, n and p be positive integers. A finite-dimensional, constant (coefficient), discrete time, linear dynamical system is a triple (F, G, H) whose dynamical interpretation is given by (0.1) x(t + 1) = Fx(t) + Gu(t), t = 0, 1, (0.2) y(t) = Hx(t), t = 0, 1, . where x(t) E X := kn, u(t) G U := ki, y(t) E Y :- kp for t = 0, 1, . F: X -4X, G: U -*X and H: X -*Y are k-homomorphisms (or matrices) independent of time. The vector spaces X, U and Y are called the state, the input (value) and the output (value) spaces, respectively. The elements of X, U and Y are called states, inputs and outputs, respectively. We shall refer to the triple defined above as the system (F, G, H). However, when the output is of no interest, we shall simply say the pair (F, G) disregarding (0.2). In this chapter we shall sometimes refer to continuous time interpretation for comparision. We choose the continuous time and set k to be complex numbers. Then we replace (0.1) by a pair (F, G) with the This is defined as follows: either the set of real or ('0.1)' i*(t) = Fx(t) + Gu(t), t > 0, in the definitions of (discrete time) pair we are interested in, and the pair (F, G) the (discrete time) system (F, G, H) and the (F, G). Since continuous time systems are not what unless otherwise specified the system (F, G, H) will always be in discrete time. Our main concerns are placed in the structural properties of the trajectories governed by (0.1). Due to the algebraic nature of difference equations, these properties can be studied in a purely algebraic way without losing the intuition of the original dynamical nature of (0.1). At this stage it seems appropriate to give a quick review of related concepts which have been treated in the literature: Mi The reachable subspace (of the pair (F, G)), reach .is defined to be the set of states which can be reached from the zero state (via some trajectories) in a finite number of steps. It is known that Xreach is equal to (0.3) im G +Fim G + . +F nl im G. The pair (F, G) is said to be reachable if and only if xreac (ii) The controllable subspace (of the pair (F, G)), Xcontr ,i the set of states from which the zero state can be reached (via some trajectories) in a finite number of steps. It is characterized by (0.4) X nr = (F n)- (imG +Fim G+ .+ F'1 im G) where (F n)lX = x E: X: F n x E: X 3 for a subspace X sof X. Note that we have Xr hC X cnrin general. The pair (F, G) is called controllable if and only if Xcontr = X. (In the continuous time case Xreach and Xcntr can be defined similarly. The characterization of Xcntr , however, is different from the one in the discrete time case, cntr namely, we have for the continuous time case Xc = im G + F im G + .+ Fn-1 im G = Xreach.) unob (iii) The unobservable subspace, Xu , is the set of initial states which can not be distinguished from the zero state by any input/output experiment. The subspace Xunmb is given by H (0.5) Xunob = ker . ï¿½ : n-1 L. The system (F, G, H) (or simply the pair (H, F)) is said to be observable if and only if Xunob = 0. The condition Xub = 0 is necessary and sufficient for the initial state x(O) to be uniquely determined based on a sufficiently long interval of input/output measurement. For more details about reachability, controllability and observability the reader should refer to KAIMAN [1968], FURUTA [1973, Chapter 2, Section 7 through 101, MARRO [1975, Chapter 61 and WONHAM [1974, Chapters 1, 2 and 3]. (iv) A subspace V is called an F mod G invariant subspace if and only if 0.6) FVCV+imG. It is well-known that (0.6) holds if and only if there exists a k-homomorphism ("state feedback") K: X -4U satisfying (0.7) (F + GK)V C V. The set of F mod G invariant subspaces is closed under subspace addition. Therefore, for a given subspace N of X there is an F mod G invariant subspace which contains any other F mod G invariant subspace. This is called the maximal F mod G invariant subspace contained in N, and we shall denote it by V mx(N). (v) Closely related to V a(N) is the subspace called the maximal reachability subspace contained in N which is denoted by X () It is defined as follows: Let K: X -+U be a k-homomorphism such that (F + GK)V a(N) C V(N) and let L: U -4 U be another k-homomorphism satisfying im. GL = im G n Vm=(N). m1~ah(N), then, is defined to be the reachable subspace of the pair (F + GK, GL). It is evident that reach Fe, (F + GK)Xmax (N) MA ~ec(N) if one uses (0.3) and the Cayley-Hamilton theorem. Consequently, Xrec (N) is an F mod G invariant subspace max contained in N (or equivalently in V M(N). Also, since V Max V V N) riillitfolwsta Fe~(N) = Xreach(V mx(.N)). m~ax( max () trvalifoowtht max max max The definition of -Zeach (N) does not depend on whether the pair max (F, G) represents a discrete time system or a continuous time system. However, there is an important difference in interpretation of Xreac (N). each max In the continuous time case )F (N) can be interpreted as the set of max states that can be reached from the zero state in a finite time via trajectories contained (at each time) in N. On the contrary this interpretation fails in the discrete time case. This point has not been clarified in the literature, not to mention its importance. The above observation then raises the following questions: What is the set of states reachable from the zero state via trajectories contained in N, XN ec in the (discrete time) system (F, G, H)? How is it characterized? Are there any interesting properties? A natural question which comes next is: What is then the set of states controllable to the zero state via trajectories in N? What are its properties? In Chapters II and III we shall attempt to answer the above questions in more general contexts. We shall introduce the following new notion in Chapter II. Let r be a positive integer, and let W and N be subspaces of X satisfying W C N. The r-step reachable subspace from W in N, denoted by rah(r), is the set of states which can be reached from some states in W in r steps via trajectories contained (at each instant of time) reach. in N. We shall characterize re N (r) and study the properties of the each X sequence of subspaces 2w^eN (i), i = 1, 2, . with respect to W and N. What appears to be intriguing is the fact that the properties of reach. ,N (i), i = 1, 2, . change drastically depending on W. _reach( In Chapter III we shall introduce the natural counterpart of XWN (r) which we shall call r-step controllable subspace to W in N, contr, ,WN (r). This subspace is defined to be the set of states in N from which some state in W can be reached in r steps via trajectories contained in N. It should be noted that, except for the special cases of X ntr) (MARRO [1975, Chapter 4]) and ï¿½Ntr (r) (which we shall isu shor , contr discuss shortly), the more general subspace XWN rr) has not been contr, studied as such. It is this generality that makes coN (r) an interesting object. reach, The significance of introducing aN (r) will become clear when we consider the unknown input observability at the final time r in Chapter IV (Sections 8 and 9). The problem is stated as follows: Given (F, G, H), a priori information about the initial state v(O) = Jx(O) and the output sequence y(l), y(2), ., y(r), find the state x(r) at the final time r. We shall see that the best we can do to identify x(r) is to determine the coset x(r) + keach (r) which in fact can be determined. Thus, X'kerJ ,kerH for an in-depth understanding of the unknown input observability at the final time r, it is essential to study the properties of reach r) \erJ,kerH~r which depend on J, H and r. The results in Sections 8 and 9 appear to be new. contrt contr, A special case of Xo N (r), namely, XNN (r) has been discussed in the literature in relation to the unknown input initial state observability, and it is known that, if we are given (F, G, H) and the output c ont r~r a sequence y(O), y(l), ., y(r), only the coset x(O) + XNN (r) can be recovered based on the above data. (See, for example, BASILE and MARRO [1973, Theorem 1 and Corollary 11 and SILVERMAN [1976, Definition 2 in Section III],) We shall include this problem with more generality in Chapter IV, Section 10 to implement our knowledge of unknown input observability. The method used here to recover x(O) (or its coset) is less complex to understand than the ones which have been used in the literature so far. The subspace cWNt (r) is closely related to "state feedback" K: X --)U. Those results in Chapter III which are related to state feedback will be applied in Chapter V in which we consider stabilizability, output zeroing and disturbance decoupling. Problems of these kinds have been treated in the literature in different fashions. (See WONHAM [1974, Theorem (2.3) for stabilizability, Theorem (4.4) for output stablization and Theorems (4.2) and (5.8) for disturbance decoupling].) The contents of Chapter V will also serve to exemplify the significance of oNtr (r) in the cases where W N. The theorem (12.1 (i), (iii)) concerning etabilility is of interest in the sense that it gives a new interpretation of stabilizability. CHAPTER II. THE r-STEP REACHA.BIE SUBSPACE FROM W IN N We study a finite-dimensional, constant (coefficient) discrete time, linear dynamical system (F, G, H) over an arbitrary field k; in this chapter we shall be interested in properties of the pair (F, G). We shall define a new notion of "the r-step reachable subspace from W in N", denoted by each (r), and study the properties of the sequence of subspaces each (i5 i = 1, 2, . The subspace defined here will find its application in Chapter IV. We begin this chapter with a well-known notion of F mod G invariant. subspaces. 1.F mod G Invariant Subspaces Consider a finite dimensional, constant, discrete time, linear dynamical system given by (1.1) x(t + 1) Fx(t) + Gu(t), t = 0, 1, where x(t) eX:=kn u(t)EU :=kI?) t =O0 1, .,and k is an arbitrary field. We call X the state space and U the input (value) space. Since the output is of no interest at the moment, we shall refer to (1.1) by the pair (F, G). (1.2) DEFINITION. A subspace V of X is an F mod G invariant subspace (or simply F mod G invariant) iff (1.3) FV C V + i G. The most important properties of F mod G invariant subspaces are expressed by the following (1.4) LEM. Let V be a subspace of X. Then the following statements are equivalent: (i) V satisfies (1.3). (ii) For any x,, E V there exists an input u E U such that Fx. + Gu E V. (iii) There exists a k-homomorphism K: X -.U ("state feedback") such that (1.5) (F + GK)V C V. PROOF. The proof for the equivalence between (i) and (iii) can be found in BASILE and MARRO [1968a, Theorem 31, WONHAM and MORSE 11970. Le na (3.2)] and WONHAM [1974, Lemma (4.2)]. For convenience we shall give the proof here. Suppose that (1.5) holds. Let x. G V. Then (F + GK)x* = Fx. + GKx. C V. So Fx. v + GKx. for some v E V, which implies (1.3). Conversely, assume that (1.3) holds. Let (vl, ., Vq ) be a basis of V where q := dim V. Then (1.3) implies that for each i 1, ., q there exist wi C V and ui C U satisfying Fv = wi + Gui. Now define a k-homomorphism K: X -4U by Kvi = -ui, i =1, ., q. Then we have (F + GK)vi = wi C V, i = 1, ., q, which implies (1.5). The equivalence of (i) and (ii) is easily proved and omitted here. 0 2. Reachability from W in N We consider a finite dimensional, constant, discrete time, linear dynamical system (F, G) given by (1.1). From here on, W and N will be subspaces of X satisfying W C N C X, and r will be a positive integer. (2.1) DEFINITION. A state x. C N is r-step reachable from W in N iff there exist another state xo* C W and an input sequence u(t) C U, t = 0, ., r - 1 such that x(O) = xo., x(t) C N, t = 0, ., r and x(r) = x. The set of states x.'s satisfying the above requirement form a (linear) subspace. reach, (2.2) DEFINITION. The r-step reachable subspace ,N (r) from W in N is the set of states r-step reachable from W in N. We show that the sequence eaNch(i), i = 1, 2, . can be recursively computed. Let (2.3) XW,N(0) = W (2.4) XW,N(i) = (FXW,N(i - 1) + im G) iN, i = 1, 2, . Then we have (2.5) THEOREM. each(r) = X, N(r). PROOF. The proof is done by induction. (i) X,N(1) = (EW + im G) N = (x E N: x = Fw + Gu for some w E W and u E km = each . = JW,N () (ii) Assume that XW,N(j) is equal to the j-step reachable subspace from W in N, J > 0. Xw,N(J + 1) = (FXW,N(J) + im G) ON = (Xj+1 E N: xJ+ = Fxj + GuJ for some xj E X N(j)' and uj E km]. By induction assumption any xj E XWIN(j) has at least one pair of sequences xt E N, t = O, ., j - 1, and ut. kn," t=. , ., j - 1 such that xo 0 W, xt =Fxt + Gt-, t = 1, ., j. Therefore XW,N(j + 1) = (xj+l E N: There exist sequences xt E N, t = 0, ., j and ut E k, t = 0,., J satisfying xt+1 = Fxt + Gut, t = 0, ., j and x E W) o Sreachj ). O We now study properties of the sequence W, XWN (), each (2), . by looking at some properties of the sequence XW,N(i), i = 0, , . The sequence XW,N(i), i = 0, 1, . (equivalently, W, reacWN ), each(2), .) has conditional monotone properties. (2.6) LEMMA. (i) If X,N(i) CXWN(i + 1) for some integer i > 0, then X,N(j) C XW,N( + 1) for all integer j > i. (ii) If XW,N(i) D W,N(i + 1) for some integer i > 0, then XW,N(J) D XW,N(j + 1) for all integer j > i. (iii) If x,N(i) = XW,N(i + 1) for some integer i > 0, then X,N(i) = XW,N(i + j) for all j = 0, 1, . PROOF. (i) Let XW,N(i) C X,N(i + 1) for some integer i > 0. Assume that XW,N(s) C XW,N(s + 1) holds for some integer s > i. Then xW,N(s + 2) = (FXW,N(s + 1) + im G) N D (FXW,N(s) + im G) N = XW,N(s + 1). Therefore, by induction XW,N(J) C W,N( + 1) for all integer j > i. (ii) Similar to the proof of (i). (iii) Suppose X, N(i) = XW,N(i + 1) for some integer i > 0. If we assume that XW,N() = XW,N(S + 1) for some integer s > i, then we have XW,N(s + 2) = (FXWN(s + 1) + im G)C N = (FXW, N(s) + im G) nhN = XW,N(s + 1). Therefore, by induction XW,N(j) = XW,-N(j + 1) for all j =i i i + 1, . (2.7) REMARK. It is possible that none of three conditions in Lemma (2.6) may hold. For instance, let F :=[0 1], G :=[, N := X and W := span 1 , and let i be any positive integer. However, if either XW,N(i) C XW,N(i + 1) or XW,N(j)D XW,N(j + 1) holds for some i, j = 0, 1, ., the sequence XW,N(), A = 0, 1, . will stop increasing or decreasing in a finite number of steps since X is finite dimensional. If it is the case, let v be the least integer i > 0 such that XWN(i) = xN(i + 1). (2.8) LEMMA. (i) If W C XW, N(1), then v < dim N - dim W < n. (ii) If W D X~WN(1), then V < dim W < n. PROOF. Immediate from Lemma (2.6) by using the finite dimensionality of X. Ol Let v be as in the paragraph prior to Lemma (2.8). XWN(v) is not an F mod G invariant in general. However, we have (2.9) LEMMA. If FN C N + im G, then FXWN(v) C XWN(v) + im G holds. PROOF. Since XWN(V) = (FXWN(v) + im G)iN, it follows that FXWN(V) C FN C N + im G. Therefore, we have FXW,N((v) C (Fxw,N(v) + im G) n (N + im G) C (FXWN(v) + im G) nTN + im G = XN(v) + im G. 3. Nonrecursive Characterization of thy, r-Step Reachable Subspace from W in N. The sequence defined by (2.3) and (2.4) determines reach(r) recursively. contr " We now give a nonrecursive characterization of XWN (r). Let B be a matrix having n rows. Let C and A be matrices with n columns. Define fo(B, A, C) := AB rCB 0l fl(B, A, C) := CAFB AG 1 [F G] CB 0 0 f2(B, A, C) := CFB CG 0 , AF2B AFG At_ where 0 is the zero matrix of appropriate size. procedure, we define for r = 1, 2, . f (B, A, C) := r CB CFB CF2B CF B CFr--B AFrB 0 CG CFG CFr-2G AFr-1G Extending the above . . . CG 0 . . . AFG AG In this section we only use a special case where B = In, n X n identity matrix. (Another case where B j In will be used in Chapter III, Section 6.) As before, W C N C X. Let N be a matrix with n columns satisfying . . . O . . . O . Denote by BW a basis matrix of W. Then we have 0 0~ 0 MFBW M G. MFBw r-lG InrF~ VrlG1 gr(Bw) := [IFBW; Fr-1G . FG G], r > 0. (3.1) THEOREM. -reach(r) = gr(B) ker fr( , , ), PROOF. Let x. E X. x E gr(BW) ker fr (, M;, M;) r > 0. iff there exist E E kdiM (3.2) uo E ker fr(Bw, U _ r-. u (3.3) x = gr(BW) o r-1 and u , ., u E k such that o r-1 M MN), The conditions (3.2) and (3.3) are equivalent to 0 . 0 0 . O . 0 . 0N Define ker * = N. 0 MqBW + 1A Guo = 0, MFrBW + FG-u + + M GUr_1 0, x. = Fr + F r-Gu + .+ Gu o r-l' which in turn is equivalent to the conditions FBW + Gu 0C N r r-1 x=FrBW + Fr 1Gu 0+ . + Gu 1 CN. The last set of relations hold for some , C kdimW and uo, U, ., Ckm reach, ur1 c iff x reach (r) (by Definition (2.2)). (3.4) REMARK. A set of vectors B, ut C km, t = 0, ., r -1 satisfying (3.2) and (3-3) are seen to be an initial state and a sequence of inputs satisfying the conditions of Definition (2.1) for the final state x*. 4. Concluding Remarks. In our discussion of this chapter we have assumed W C N. This assumption has been made since we are interested in structural properties of the system (F, G) inside the subspace N. Technically speaking, however, the above assumption is not essential. With slight modification all the statements still hold without assuming W C N. Theorem (3.1) can be easily modified to yield the corresponding result: reachA (4.1) WN (r) = gr(EW) ker fr(Bw, MN, M ), r > 0 where f r(B, A, C) is the matrix obtained by eliminating the first block row in fr(B, A, C) (hence, fr(B, A, C) consists of r block rows). From Lemma (2.6) it follows that X,,N(i) CXWN(i + i), i = 0, , ., iff and that XW, IN(i) D X,N(i + 1), i = 0, 1, ., iff (4.3) W :XWN(l). The natural question is then what is the significance of the conditions (4.2) and (4.3). The significance of (4.2) will be partially answered in Chapter III, Section 5. However, the implication of (4.3) is yet to be clarified. _reach, There are other questions to ask: when does the sequence XWN (i), I reach , reach ,X , i = 1, 2, . "oscillate", i.e., a,N rJ) XcN hj + 1) for every j > 0?; when does the sequence "oscillate" at the outset and stop its oscillation some time later?; etc. These are open problems. CHAPTER III. THE r-STEP CONTROLLABLE SUBSPACE TO W IN N. In the previous chapter we have studied the r-step reachable subspace from W in N. In this chapter we introduce another new notion which we call "the r-step controllable subspace to W in N". Its applications will be found in Chapters IV and V. 5. Controllability to W in N. Consider a finite dimensional, constant discrete time, linear dynamical pair (F, G) represented by (5.1) x(t + 1) = Fx(t) + Gu(t), t = 0, 1, where x(t) C X := kn, u(t) e U := km, t = 0, 1, ., and k is an arbitrary field. We denote the system (5.1) by the pair (F, G). As before W and N denote subspaces of X satisfying W C N C X, and r is a positive integer. (5.2) DEFINITION. A state x. C N is r-step controllable to W in N iff there exists an input sequence u(t) C U, t = 0, ., r - 1 such that x(O) = x., x(r) ( W and x(t) C N, t = 0, ., r. The set of states x.'s satisfying the above conditions form a (linear) subspace. .contr, (.53) DEFINITION. The r-step controllable subspace X CN (r) to W in N is the set of states r-step controllable to W in N. contrt We show that XWN (r) can be computed recursively. Let (5 WN (O) = w (5.5) ,N() = F-1(WN(i - 1) + im G) rT)N, i = 1, 2, where F-X := (x C X: Fx C X ) for a subspace X of X. (5.6) THEOREM. X ontr(r) = RW,N(r), r > 0. PROOF. The proof is by induction. For r = 1, XW,N(1) = F-1(W + im G) n = (xI E X: FX1 = - Gul for some wE W and ul km, and xlE N) = {xl E N: Fxl + Gul E W for some ul e km) reach1) contr Now let WN(j) = XW,N (j), j > 0. Then we have W,N(j+1) = (xj+1 E N: Fx+1 = xj - Guj+1 for some x E j,N(j) u+l E km. ) By assumption of ),N N (J) this is equal to {Xj+1 G N: Fx1 + Guj+ = x, Fx + Gu = Xl, ., Fx1 + Gu1 = w contr m for some w E W, x, 2 XW,N (a), u k, a = , ., j and uj+l 6 km1 Scontr,. =),N (0 + ). O contr contr We now study properties of the sequence W, X,N (1), WN (2), by examining properties of the sequence (5.4), (5.5). contr contr The sequence (5.4), (5.5) (equivalently WW,N tr(1), (2),.) has conditional monotone properties. (5.7) LEMMA. (i) If N(i) ,N(i + 1) for some integer i > 0, then XW,(j) CXW,N(j + 1) for all integer j > i. (ii) If (i) D X.,N(i + 1) for some integer i > 0, then ,N(i) D ,(J+ I for all integer j > i. (ii) If ,N (i) = W,N(i + i1) for some integer i > 0, then X,N(J) = X,N(i + j) for all integer j > 0. PROOF. (i) The proof is by induction. Let XW,N(i) C X,N(i + 1). Assume XW,N() C XW,N( + 1) where 2 is a nonnegative integer. Then kW,N(I + 2) = F-I (W,N( + 1) + im G) (IN D F-I(kW,N(I) + im G)(-hN = ,N( + i). Therefore, by induction we get XW,N(j) C i,N(j + 1) for all integer j > i. (ii) The proof is similar to that of (i). (iii) This is again proved by induction. We assume w,N(i) = XW,N(i+l) for some i > 0. If XW,N() = XW,N(i + 1) for some integer A > 0, then we have WN(I + 2) = F-WN(4 + 1) + im G) AN = F-( ,N(2) + im G) TnN = XW,N(A + 1). Therefore, XWN() = ,N(i + j) for all integer j > 0. 0 (5.8) REMARK. It is possible that neither one of three conditions in Lemma (5.7) may hold (e.g., let F, G, N, W be as in Remark (2.7), and consider any positive integer i). However, if either XW,N(i) C WN(i + 1) or W,N(J)D W,N(j + 1) happens for some i, j = O, 1, ., the sequence XWNW' 9 = O, 1, . will stop increasing or decreasing in a finite number of steps since X is finite dimensional. If it is the case, let P be the least integer i > 0 such that XWN(i) = WN(i + 1). (5.9) LEMMA. (i) If W C X (1), then 0 < dim N - dim W < n. (ii) If W D X (1), then p < dim W < n. ~~,N -PROOF. Immediate from Lemma (5.7) and the above comment. O (5.10) LEMMA. Let 0 be as in the paragraph preceeding Lemma (5.9). Then (5.11) FX~W, ' ) C X wYN() + im G. (5.12) WN() c N. Scontr So XWN (-) is an F mod G invariant subspace in N. PROOF. By the definition of XWN(P) = XWN( + 1). Therefore WN WN( + 1) - F-1(XW I() + im G) nN. Hence, FXN(R) C XW(9) + im G, XWN(ii) C N. rn (5.13) REMARK. Lemma (5.10) guarantees the existence of a feedback contr contr K: X -~U such that (F + GK) XWN (0) C XWN i). (See Lemma (1.4).) The set of F mod G invariant subspaces is closed under subspace addition. (See BASILE and MARRO [1968a, Section 2, Assertion 11 and WONHAM [1974, Lemma (5.3)].) Therefore, the following is well defined: (5.14) Vmax(N) := max[V C N: FV CV + im G). The subspace V max(N) is called the maximal F mod G invariant subspace max in N. It has been known (SILVERMAN [1976, Section III, A]) that Vmax(N) is equal to the set of states in N for which there are input sequences such that the corresponding trajectories remain in N for v units of time. It should be noted that this statement is a special case of Theorem (5.6). It should also be noted that the algorithm for computing V max(N) is a max special case of (5.4), (5.5) where W = N. (See BASILE and MARRO [1968a, Section 3, Corollary 1] and WONHAM [1974, Theorem (4.3)].) Summarizing, we have (5.15) COROLLARY OF THEOREM (5.6). Let p be as in Lemma (5.9), and let Vmax(n) be as used above. Then maax X~ontr(, =, mxN 2, Scontr The sequence XWN (i), i = 1, 2, . has a conditional monotone property. (See Theorem (5.6) and Lemma (5.7).) The natural question to ask is then when it is monotonically nondecreasing or nonincreasing. ( .J) LEMMA. (i) W C XWN(1) iff FW CW + im G. (ii) If W D XN(1), then N(0) = Vmax(W) where 9 is as in Lemma (2.9) and V max(W) is as in (5.14). max PROOF. (i) If W C XWN(1), then W CF-1 (w + im G) N. Hence -1 CxN') W CF (W +im G). So we have FW CW + im G. Conversely, if FW CW + im G, then W C F (W + im G). Since W CN by assumption, we get W C F-1(W + im G)(- )'N, i.e., W C XwN(1). (ii) If W D XWN(1), by Lemma (5 .7 ii) we have XWN(i) CW for all i = 0, 1, . . Then obviously XWN(i) = XW(i), i = 0, 1,. Therefore, by Corollary (5.15) ,N(p) = Vmax(W). 11 contr For the nondecreasing sequence XN (i), i = 1, 2, ., we have the following as the immediate consequence of Lemma (5.7 i), Lemma (5.16 i) and Theorem (5.6). contr contr (5.17) PROPOSITION. X (i) C X (i + 1) for all i = l, 2, iff FW CW + im G. contr contr (5.18) COROLLARY. If FW C W + im G, then FeNr (r) CWN (r) + im G for each r = 1, 2. PROOF. Suppose FW CW + im G. Then by Proposition (5.17) we contr contr have XW,N (i - 1) C X,N (i), i = 1, 2, . By the definitions of the i and (i - l)-step controllable subspaces to W in N, for every contr v. c r i) there must exist u. EU such that Fv. + Gu. XWN(i - 1) C contr contr contr ' XWN (i). Therefore, FXN (i) C XN (i) + im G, i = 1, 2, . As for the nonincreasing sequence we just note a special case of Lemma contr contr (5.7), i.e., XWN i) WN i + ), i = i, 2, . iff 1(i) : XW,N ( + 1) W CF (W + im G) 'N. contr XWN (r) has the following properties in relation to state feedback. c contr, By the definition of contr (r) it is clear that the subspace is state feedback invariant, i.e., the r-step controllable subspace to W contr in N of the system (F + GK, G) is equal to coN (r) of (F, G). Under some conditions, the input sequence u(t) E U, t = O, ., r - 1 given in Definition (5.2) can be implemented by a suitably chosen state feedback u(t) = K rx(t), t = 0, ., r- 1. contr contr (5.19) THEOREM. (i) If XW,N (i) C WN + ), i = 1, 2, ., i.e., if FW CW + im G, then for each r = , 2, . there is a feedback K : X -+V such that r (5.20) (F + GKr contr(r) C N, j = 0, ., r -, r XN(r) , j contr (5.21) (F + GKr) XWN (r) CW, j = r, r + 1,. Scontr contrr (ii) If YDN (i) DXWN (i + 1), i = 1, 2, ., then for each r = 1, 2, ., there is a feedback K : X -4U such that r (5.22) (F + GKr) contr(r) CN, j = 0, ., r - 1 r contr (5.23) (F + GKr) r cont (r) CW. Scontr ontr PROOF. (i) By Proposition (5.17) ,ontr () ontr(i + 1), i = 1, 2, ., iff W is an F mod G invariant. contr contr Assume XWN i) C otr(i + ), i = 1, 2, . For each r = 1, 2, . we:choose a basis of X as-follows. Let (eol, ., eolo be a basis of W. Extend this basis to get the basis (eol, ., eoo; ell, ., el) of coN tr(1). Repeat the extension until we get no more vectors to add to, say r' times where r' < r, and we obtain the basis e1, ., e~o e, ., er'- l,r ; er'l, ., e rA of oN tr(r') = XcoN (r). We further extend this basis arbitrarily to get the basis of X, (e , ., e., e r. Here := contr Ontr r dim Xk N nt) - dim X N (j - 1), j ., r' and A= ' contr ' contr n - dim XW,N (r). Since ejsj E IN (J), J = O, ., r and s = 1 s = 1, ., AI , there exists an input ujsj such that (5.24) Fe + Gu E ontr - 1) jsy jsj XWN where j = 1, ., r' and sj = 1, ., Aj. (See Definitions (5.2) and (5.3)). Define K : X -U so that it satisfies the following r conditions: (a) Ujsj = Kes for j = , ., r and s = l, ., Aj, (b) (F + GK )eos E W for s = 1, ., ao (such a Kr always exists since FW CW + im G. See Lemma (1.4) and Proposition (5.17).) (c) Krej, j = 1, ., qr are arbitrary. Then (5.25) (F + GKr)eos E W, s = 1, ., o' (5.26) (F + GKr)ejsj C otr(j - 1), j = 1, ., r; s= , . It is easy to check that the following relations hold: (5.27) (F + GKr)ejsj C N for j = O, ., r'; s. = i, ., e.; i = , .,r - 1 and (5.28) (F + GK ) ejsj C W for j = 0, ., r contr contr (ii) Suppose WN ,N i + ), i = i, 2, . For each r = 1, 2, . we choose a basis of X in the following way. contr . contr Let (erl, ., erer) be a basis of XoWN (r) where := dim XcoNtr). Extend the basis to get the new basis (e rl, ., err; erl,l, e ) contr r-l,r-l of XW,N (r - 1). Repeat the procedure, and we obtain the basis ferl, ., erB; er-1,,., elï¿½; 01, ., eo) of W where S:= dim cont (j) - dim Xontr(j + ), j = , ., r - 1. Note that if ï¿½. = 0 we do not extend the basis at this step and go to the next step. By adding linearly independent vectors el' . qr we complete the basis of X as erl, ., el l; o, ., eoAo; ell ., eqr where r := n - dim W. Since ejsj c XWN tr), = 1, ., r and sj = , ., ,there must be an input u.jsj such that contr (5.29) Fe. + Gu. ntr( - 1) asj jsj XWN where j = 1, ., r and s =1,., . It is straightforward to check (5.31) (F + GK )re rs W, s = 1, ., r' r rsr rr (5.32) (F + GKr) rsre N, J = 0, .' r s 1 r hold. 5 (5.33) REMARK. The choice of K : X -)U r This (limited) freedom in choosing Kr (r application. (See Chapter V.) (r > O) is by >0) is rather no means unique. useful in (5.34) REMARK. The inputs u. 's (s. = 1, ., ; .Jsj or r) based on which K is defined, can be determined r (but not necessarily uniquely). See Remark (6.3). = 0, ., r' explicitly Recall the definition of reach(r). Given x c w there may not " ,N Go be any input sequence producing x(O)= X , x(t) C N, t = 1, ., r reach,0 and x(r) XWN (r), r > 0. The condition that x must satisfy to have such an input sequence is as follows. Let x C W. Then there is an 0 input sequence u(t) C U, t = 0, ., r - 1 such that the state trajectory satisfies x(O) = X, x(t) C N, t = 1, ., r and x(r) C reach, rec (r) iff ontr x CW , (r) o 7, - reach where X :=WN (r). We now turn to the question of what is the implication of has been posed at the end of Chapter II. (5.35) i > 0, (4.2) which contr LEMMA. Let W(i) := X W ri), i > 0. Then for each integer (4.2) holds iff .reach (5.36) W .rah() Reach PROOF. Suppose (4.2) holds. Trivially, W D (i),Wi). To show the converse inclusion let w ( W. Then by (4.2) there exist x1 T W, u C U such that w = Fx + Gu N. Since w ( W we obtain w = Fx + Gu 1 W. Repeating the above argument i times, we see that there also exist xt W, ut C U, t - 2, ., i satisfying xt-1 = Fxt + Gut, t 2 *., .contr,.+ Uï¿½ t = 2 ,., i. Then x contr i). Hence, noting that xt C W, t = 1, ;ach E 0 .,, i, we get w-6 XW1(i).W() Conversely, if (5.36) holds, then there exists a subspace W C W reach. satisfying each (1) = W, which implies (4.2). 0 Consequently, each (ï¿½ reach~i+i ,2 f (5.37) PROPOSITION. reach(i) 0~each WN(i + 1), i = 1, 2, ., r each (5.38) w = rmax ( ) where V is as in Lemma (2.8). PROOF. Immediate by using the above lemma and Lemma (2.8). 6. Nonrecursive Characterization of the r-step Controllable Subspace to W in N. The sequence defined by (5.4) and (5.5) determines the r-step controlcontr lable subspace to W in N, X N (r), recursively. We now give a '' contr, nonrecursive characterization of XWN (r). As before, W CN CX and r = 1, 2, . Recall the definition of fr(B, A, C) in Section 3 of Chapter II. Denote by I the n X n identity matrix, and let 0 be the zero matrix n of suitable dimension. Let M be a matrix with n columns such that ker = W, and define M similarly. Then frIn, MMMi)= 0 M 0 .2 M r-1 M Fr 0 NFG Mr-2 M FrlG r- 1 0 . . . . 0. . M 0 . G MG r = 2, 3, ., and M F M G Write P := [I 0. 0] r n where we have r blocks of n X m zero matrices. contr (6.1) THEOREM. Xc,N (r) = Pr ker fr(In, Mw, N), r > 0. PROOF. The condition x 0 (6.2) Uo C ker fr(In , - r-1. x 0 X, ut k , t = 0, ., r - 1, is equivalent to Xo = 0O xo + MGuo = 0, r-i o r-2u x F r 2Gu + . + MGUr-2 = 0, S+ r-Guo + . + Gur-1 = 0, which in turn is equivalent to the conditions x E N, Fx + Gu ( N, o o F x + F Gu +.+ Gu r-2 N, Frx + Fr-1Gu + . + Gu E W. o o r-1 c contr, The last statement is equivalent to x G WN (r) (by Definitions (5.2) and (5.3)). 0 (6.3) REMARK. Vectors xï¿½ E X, ut C kc , t = 0, ., r - 1 satisfying (6.2) are viewed as an initial state and a sequence of inputs satisfying the conditions of Definition (5.2). (6.4) REMARK. Methods of computing the maximal F mod G invariant subspace in N have received considerable attention. (See BASILE and MARRO [1968a, Section 3, Corollary 11, WONHAM [1974, Theorem (4.3)], SILVERMAN [1976, Lemma 6 in Section III, C1 and MOORE and LAUB[1978. Also recall Corollary (5.15).) Corollary (5.15) and Theorem (6.1) yield (6.5) Vmax(N) = P ker f (I , This gives a new nonrecursive method of computing Vma(N). 7. Concluding Remarks The remarks similar to those in Section 4 of Chapter II apply to the results of this chapter. .r each, As is the case with X ,N Cr) the assumption W C N is not .' contr, essential in the discussion of XW,N (r) from the technical point of view; the statements in this chapter can be modified straightforwardly to fit the case where W C N is not assumed. (See also Section 4 of the previous chapter.) . contr contr From Lemma (5.7) we know that XW,N (i) D XWN r' + 1), i = 0, 1, ., iff (7.1) WDF' (W+im.G) nTN. The significance of the condition (7.1), however, is not yet clear. CHAPTER IV. UNKNOWN INPUT OBSERVABILITT The notions of the r-step reachable subspace from W in N and of the r-step controllable subspace to W in N as developed in Chapters II and III are now applied to the study of unknown input observability of a system (F, G, H). Sections 8 and 9 discuss the unknown input final state observability. Section 10 treats the unknown input initial state observability. Hereafter Y denotes the output (value) space and is defined by Y := kp. Also p := dim Y is used throughout Chapters IV and V. 8. Unknown Input Final State Observability We consider a finite-dimensional, constant, discrete time linear dynamical system (F, G, H) given by (8.1) x(t + 1) = Fx(t) + Gu(t), t = 0, 1, (8.2) y(t) = Hx(t), t = 0, 1, . where x(t) E X := kn, u(t)CU:=km, y(t)Y-kp (t = 0, 1, .) and k is a field. (Recall the definition in the second paragraph of Chapter I.) It is assumed that we have some degree of a priori information about the initial state given by (8.3) W(x(O)) = x(O) where X: X -kq is a k-homomorphism and q is a positive integer. In particular, if J is an isomorphism where q = n, then the initial state is a priori known. If J is zero, then the initial state is a priori unknown. Depending on an initial state x(O) = x C X and an input sequence 0 u(t) = ut G U, t = 0, 1, ., the system produces the corresponding state and output sequences (or trajectories) x(t) C X, t = 0, 1, . and y(t) e Y, t = 0, 1, ., respectively. As a result of (8.3) we also have v(xo) = Jx0 which we call v0 Considering the initial state, we have v(x(o)) J (8.4) - x(o). y(x(o)) H Let v(O) j y(x(0)) 1 For each r = 1, 2, . the sequence [v(0), y(l), ., y(r)) is called an initially modified output sequence (till time r). When we discuss two initially modified output sequences we prefer to denote them by (vol Yll' -' Yrl) and (vo2) Yl21 .' Yr2)" (8.5) PROBLEM. Given an initially modified output sequence (v(O), y(l), ., y(r)) of (F, G, H), find the corresponding final state x(r) C X for r = 1, 2l. Note that the definition of v(O) and J and (8.4) clearly implies (8.6) ker J C ker H. (See Section 11 for the remark on the condition (8.6).) We say that the system (F, G, H) is unknown-input observable at the final time r iff Problem (8.5) has a unique solution. It is not always the case that this problem has a unique solution. In fact, it will be seen that rank HG = rank G is necessary for the final state x(r) to be uniquely determinable. (See Theorem (9.4).) So the important question is "to what extent can we recover x(r)?" (8.7) DEFINITION. x. E X is unknown input indistinguishable from 0 at the final time r iff thereexist an input sequence u(t) G U, t = 0, ., r - 1, and an initial state x C ker J such that the 0 resulting x(t) C ker H, t = 1, ., r and x(r) = x. The set of states unknown-input indistinguishable from 0 at the final time r is a subspace of X. (8.8) DEFINITION. The unknown-input unobservable subspace at the final UO time r, denoted as XerJ (r), is the set of states that are unknown- ierJa input indistinguishable from 0 at the final time r. (8.9) DEFINITION. Two states x*1, x.2 guishable at the final time r iff there and ( tr-1, x ) of input sequences ut2 t=o 02) j = 1, 2 and initial ataees x(0) = x , conditions hold: conditions hold: C X are unknown-input indistinr-1 exist pairs ({utl)t=o, xol) u(t) = ut ; t = 0, ., r - 1; j = 1, 2, such that the following (i) x . is the final state (at the time r) the pair (( tr-1, x2) where j = 1, 2, utj t=o' 0o2 (ii) the initially modified output sequences the pairs coincide. corresponding to corresponding to The significance of X UO(r) is clarified by kerJ (8,10) PROPOSITION. Two states x*1, x*2 7 X are unknown-input indistinguishable at the final time r iff U0 (8.11) x.1 - x2 krJ(r). PROOF] [Necessity.1 Let (v., Ylj, ., y .) be the 3 lj rj . initially modified output sequence corresponding to the pair (fut t=o' tzro' xoj) where j = 1, 2. Suppose v1 = v2 and ytl Yt2, t = 1, ., r. Let the new initial state of (F, G, H) be x(O) := xol - xo2. Since vl = v2, we have Jx(0) = Jxol - Jxo2 = 1 - v2 = 0O, i.e., x(0) C ker J. Apply the input u(t) = utl - ut2, t = O, ., r - 1 to the system (F, G, H). Then it is straightforward to show x(r) = x.1 - x*2 y(t) = ytl - Yt2 = 0, t = 0, ., r. Therefore, x1 - x2 X O(r). x*1 - *2 CXerJ [Sufficiency.] Suppose (8.11) holds. UO Then clearly x E XkerJ(r). Therefore, there r-1 fut)r- 1 and an initial state x C ker J of t=o o corresponding trajectory satisfies x(t) C ker x(r) = x*. Write x := X1 - x2. exist an input sequence (F, G, H) such that the H, t = O, ., r and Define a new input sequence and a new initial state of (F, G, H) by u(t) := ut2 + ut, t = O, ., r - 1 and x(0) := xo2 + x where r-1 xo2 and (ut2 t=o are the initial state and input sequence, respectively, giving rise to the final state x*2. It is easy to show that v(0) = Jx(0) = Jxo2, y(t) = Yt2 t = 0, ., r, x(r) = x*2 + x. = xl' where (yt2 rt=o is the output sequence produced by ut )r-t t2 t=o N2 t=o and xo2. It follows from Proposition (8.10) that (8.11) defines equivalence classes, each of which consists of the states unknown-input indistinguishable at the final time r and the collections of these classes are denoted by X/XU0 (r). kerJ X/ rj(r) is a linear space over k where addition and sealar multiplication are defined in the obvious ways. X rj(r) is characterized as follows: (8.12) THEOREM. (8.13) (8.13) X rJ(r) = XkerJ ker(r) YlierJ~r Xker,kerH~r where kerJkerH (r) is defined by (2.1) and-(2.2). XrerJ,kerH-PROOF. Immediate by using Definitions (2.1) and (2.2) and Definitions (8.7) and (8.8). 0 Proposition (8.10) tells us that to recover x(r) we cannot do better than identifying the equivalence class x(r) + X rJ(r). Then the kerJ question is, "can we really find the equivalence class containing x(r)?" (8.14) THEOREM. If x(r) is the final state at the time r, then x(r) + X~ rj(r) can be uniquely determined based on the knowledge of the corresponding initially modified output sequence (v(O), y(1), ., y(r)). PROOF. Given the initially modified output sequence (v(O), y(1), ., y(r)) of (F, G, H). Choose an x 0 X satisfying (8.15) v(O) = Jx , e.g., on a fixed coordinate basis, x may be taken to be (8.16) o : tv(0) where J is the pseudo-inverse of J. (See Appendix for the definition of the pseudo-inverse.) Then the initial state x(0) is written as (8.17) x(O) = + x 0 0 for some xo E ker J. 9(t) := y(t) - HFtx , are given by (8.1) and Then it is easy to see t Define 2(t) := x(t) - F x0, t = 0, 1, ., and t = 0, 1, . where x(t), y(t), t = 0, 1, . (8.2) with the initial condition x(O) = Xo + Xo. that 2(t), 9(t), t = 0, 1, ., satisfy (8.18) 2(t + 1) = F2(t) + Gu(t), t = 0, 1, ., (8.19) 9(t) = HR(t), t = 0, 1, . with the initial condition 2(0) = x0 E ker J. Since o rUO U (8.20) x(r)+ r(r) = (r)+ FrX) X (r), +XkeJr + o + erJ it suffices to determine 2(r) + Xrj(r). "Notice that xo is known. The next lemma provides a way to obtain 2(r) + XrJ(r). We now consider the system (F, G, H) with the assumption that the initial state x(O) is in ker J. Let x(0) C ker J. Then (8.21) x(O) = BerJ 5 for some t G kn where B,e n' := dim ker J. For each r = be an input sequence (till time t = 0, ., r corresponding to is given by r(0)(8.22) y(1) y(r) is a basis matrix of ker J and 1, 2, ., let u(t) C U, t = 0, ., rr - 1). The output sequence y(t) G Y, the above initial state and input sequence = f(BkerJ, H, 1) where fr(BkerJ, H, H): kn'+rm -k(r+1l)p is the matrix given in Section 3 (in which BW is replaced by BkerJ and M by H). Note that y(O) = 0 since x(O) C ker J C ker H. The corresponding final state x(r) C X is obtained by u(O) (8.23) x(r) = gr(BkerJ : u(r - 1) where gr(BkerJ) is defined in Section 3 (Chapter II). Our problem is to determine x(r) + rJ(r) from the knowledge of y(1), ., y(r). (v(O) = Jx(O) = 0 since x(O) C ker J. Also y(O) = 0.) Define u o n' +rm: k" , r (BkerJ) := " i , Ur-l where n' := dim ker J and BkerJ is a basis r(ker J) := inm fr(BkerJ, H, H). (8.24) LEMMa. There is a unique homomorphism X/r (BkerJ) ker fr(NerJ, H, H) such that the f (BkerJ, H, H) Sr (BkerJ) - gr(BkerJ) u km, j = 0, .,r- 1) matrix of ker J. Write 0kerJ: P (ker J) 4 r r following diagram co-mutes: rr (ker J) I kerJ r X/r (BkerJ) ker fr(BkerJ, H, H) where f r(,', ) and gr(.) are as in Section 3 and p: X X/r(BkerJ) ker fr(BkerJ, H, H) is the canonical projection. -.1p Furthermore, (8.25) gr(BkerJ) ker fr(BkerJ H, H) = X (r). Therefore, kerJ r (ker J) -X/ rJ(r). PROOF. Let yr E r (ker J). such that There is an element Ur C 1r(BkerJ) (8.26) fr(BkerJ, H, H) (Ur) = yr. Define -kerJ r(ker J) -X/gr(BkerJ) ker fr(BkerJ, H, H) (8.27) 'erJ r) = Pgr(BkerJ) (r) kerJ r is well-defined. satisfying (8.26). (8.28) pgr(BkerJ) (-u) r = pgr(BkerJ) (ur + 7) Indeed, let "u be another element in r0(BkerJ) r r kerJ (for some 7 E ker fr(BkerJ, H, H)) = P(gr(BterJ) (ur) + gr(BkerJ)()) = gr(BkerJ) ( + gr(BkerJ (Y) = gr kerJ) (ur To show the linearity of kerJ, let yrl' Yr2 e r(ker J). Then there exist Ur, Ur2 E nr(BkerJ) such that rj = f (B kerJ, H, H)(u rj), j = 1i, 2. Therefore, y1 + = f(Bk ,rJ H, H) (rl + r2) and we have kerJ + + U r l +r2)= Pr(BkerJ) (u r + ur2 = Pgr(BkerJ) (url) + gr(BkerJ) (r2) kerJ - kerJ,=r Yrl) + r Jr2 kerJ,Similarly, for each a E k, 7r E r (ker J) we have rr (r) = SerJ,r r To show the uniqueness of the map suppose kerJ: r (ker J) -+ r r X/gr(BkerJ) ker fr(BkerJ, H, H) is another map for which the diagram commutes. Then for each ur E Sr(ker J) e r(BkerJ , H) r )= (BkerJ (r kerJ , r r(BkerJ H, H) r Hence, noting that fr(BkerJ, H, H) is onto, we obtain kerJ = kerJ (8.25) follows from Theorems (3.1) and (8.12). 0 (8.29) REMARK. ker r(ker J) - rX/X rj(r) does not depend on the choice of BkerJ' (8.30) REMARK. For a fixed coordinate basis 0kerJ Pr(ker J) -s X/Xr(r) may be defined by (8.31) erJ r(BkerJ) kerJ H, H) where fr(BkerJ, H, H) is the pseudo-inverse of fr(BkerJ, H, H). (See Appendix.) As an immediate consequence of Proposition (8.10) and Theorems (8.12) and (8.14) we have (8.32) THEOREM. A system (F, G, H) is unknown-input observable at the final time r iff (8.33) erJch kr(r) = 0. 4erJ,kerHf 9. Unknown Input Final State Observability--Part 2 (Special cases). Let r = 1, 2, .R. ecall Theorem (8.12). The unknown input unobservable subspace Xerj(r) at the final time r depends on the time r in general. However, in some cases the time dependence disappears in a finite time. (See Lemma (2.6) and the paragraph prior to Lemma (2.8).) And if (9.1) ker J C (F ker J +im G) ker H holds, more can be said, namely, the sequence DOjM) i = 1, 2, ., is monotonically nondecreasing (Lemma (2.6 i)) as well as it stops increasing in at most n steps, i.e., (9.2) Xerj(n) = Xerj(n + 1) = (See Lemma (2.8 i).) Also, if (9.3) ker J D (F ker J + im G) nker H is true, then the sequence Xerj(i), i = 1, 2, ., is monotonically nonincreasing (Lemma (2.6 1i)) and (9.2) holds. (See Lemma (2.8 ii).) Note that if ker J = 0, i.e., the initial state is known a priori, then (9.1) holds, and that if ker J = ker H, i.e., if the initial state is not known a priori, then (9.3) holds. Therefore, for these cases the above statements are true. Let us consider the first situation where ker J = 0 (i.e., the initial state is a priori known). The condition under which Problem (8.5) has a unique solution becomes particularly-simple. (9.4) THEOREM. If ker J = 0, then the system (F, G, H) is unknown input observable at the final time r iff (9.5) dim im HG = dim im G, i.e., (9.6) rank HG = rank G. PROOF. By Theorem (8.32) and Theorem (2.5) (F, G, H) is unknown input observable at the final time r iff (9.7) XokerH(r) = 0. o,kr Since Lemma X ker(i) C X (i + ), i = 0, ., it follows from O,kerH o,kerH (2.6) that (9.7) holds iff (9.8) o = Xo,kerH(1) = im Gn ker H. It remains to show that (9.8) holds iff (9.5) holds. Suppose im G fker H = 0. Let m' := rank G = dim im G. Denote by (g g2' . gm, a basis of im G. m'e claim that Hg1, Hg2, ., Hgm, are linearly independent. In fact, let . , jHg. =0 for some a.j k, j = 1, m j=l 3 2,., m, not all zero. Then H jI gj 0, which implies j.Zajgj = 0 since im Gnker H = 0. Now since gl' ". gm, are linearly independent, we get a. = 0 for all j = 1, ., m'. Contradiction. Hence, dim H im G = m'. Conversely, suppose im G^) ker H / (0). Let Trivially el is a basis of Span el C im GE(ker H. a basis (el, e2' ., em,) of im G. Then 0 j el G im G()ker H. Extend el to form (9.9) im HG = Span (Hel, ., Hem') = Span (He2, ., Hem,). Therefore, dim im HG < m'. OE (9.10) COROLLARY. Let m = p = 1 and define polynomials H[(z) and X(z) by I(z)/X(z) = H(zI - F)-IG ( 0). Suppose the initial state is known. Then the system (F, G, H) is unknown-input observable at the final time r iff (9.11) deg X(z) - deg 11(z) 1. PROOF. Since m = p 1, it follows that rank G = 1 iff G / 0 and that rank HG = 1 iff HG / 0. Hence, (9.6) holds iff HG / 0 (since G 1 0 by assumption). El (9.12) COROLLARY. Assume that the initial state is known, i.e., ker J 0. Then there is a unique state trajectory x(t) G X, t =, 1, ., r corresponding to an initially modified output sequence (v(O), y(l), ., y(r)) iff (9.5) (i.e., (9.6)) is satisfied. (9.13) REMARK. Assume that the initial state is known and that rank G m (i.e., dim im G = m). It can be shown that the input sequence corresponding to an initially modified output sequence (v(O), y(O), ., y(r)) is unique iff (9.5) (or (9.6)) is true. 10. Unknown Input Initial State Observability. Consider the system (F, G, H) given by (8.1) and (8.2). As before r = l, 2,. We wish to find x(O) based on the measurement of the output sequence y(O), ., y(r). Suppose that we are allowed to make an extra measurement at the final time r. (Suppose for example, that the "motion" of the system stops at the end (at the time r) and that we can examine the system more carefully to get an extra information at the time.) We will denote this extra information by (10.1) Ve(r) = e x(r), where v' (r) C ks and J : X -)k is a k-homomorphism (for some integer e e s > 0). Then (10.2) [e x(r). LY(r)- H Denote v (r) J V (r) := e e : e y(r) e H Now we shall call the sequence (y(O), ., y(r - i), ve(r)) r-modified output sequence. Consequently, we have (10.3) PROBLEM. Given an r-modified output sequence [y(O), ., y(r - 1), ve(r)), find the corresponding initial state x(O). By (10.2), clearly, (10.4) ker J C ker H. e (See the remark on this condition in Section 11.) We say that the system (F, G, H) is unknown-input initial state observable at the time r iff Problem (10.3) has a unique solution. A state x. E ker H is said to be unknown-input initial state indistinguishable from 0 at the time r iff there is an input sequence u(t) E U, t = 0, 1, ., r - 1 such that the corresponding state sequence satisfies x(0) = x,, x(t) E ker H, t = 0, ., r - 1 and x(r) E ker Je. The set of states that are unknown-input initial state indistinguishable from 0 at the time r is a subspace of X. (10.5) DEFINITION. The unknown-innut initial state unobservable subspace at the time r, denoted by XrJe(r), is the set of states that are unknowneinput initial state indistinguishable from 0 at the time r. From Definitions (10.5) and (5.5) we obtain (10.6) THEOREM. XrIU e(r) = Xkeontr (r). TCerJe TierJe,kerH Recall the results given in Chapter III. We know the characterizations contr and the properties of erjekerH(r). The following result given by SILVERMAN [1976, Section III, A, therJekerH The following result given by SILVERMAN (1976, Section III, A) the paragraph after theorem 5] is a special case of the above theorem. (10.7) COROLLARY. X rU (n) = V (ker H) where Vmax(ker H) is defined XerH maxma lr(5.14). PROOF. Let J = H, and then use Corollary (5.15). O e Two states x.1, x*2 E X are unknown-input initial state indistinguishable at the time r iff there are two input sequences utj; t = 0, 1, ., r - 1; j = 1, 2, such that the r-modified output sequences corresponding ï¿½ r-1r-1 to the pairs (x*1, (utl)r o) and (x*2, (ut2)t=O) coincide, (10.8) PROPOSITION. Two states x.1, x42 E X are unknown-input initial state indistinguishable at the time r iff IU () (10.9) x.1 - x2 e irJe(r). PROOF. Similar to the proof of Proposition (8.10). Therefore, we omit the proof here. 0 XI rJe(r) defines the set of the equivalence classes X/XU rJe(r). By Proposition (10.8) the best that we can recover of the initial state x(O) is the equivalence class x(O) + XUr(r). What remains to be IU + aere shown is how to determine x(O) + XkerJe(r) from (y(O), ., y(r - 1), ve(r)). Suppose we have the initial state x(O) E X u(t) E U, t = 0, ., r - 1. The corresponding sequence y(t) 6 Y, t = 0, ., r - 1, ve(r) C and input sequence r-modified output kq is given by y(O) - x(O) : = fr(I n, J e H) u(0) , y(r - 1) ev (r) u(i -e where In is the n X n identity matrix (or identity map kn -4 kn) and f (., *, *) is as in Section 3 of Chapter II and Section 6 of Chapter III. Define (In) := kn+rm and r (ker J ) := im f (In, Je' H). IU (10.11) THEOREM. There is a unique homomorphism : P (r) such that the following diagram commutes: f (In, J e H) Sr(I ) e - r(ker J ) r n r e P I X X/XrJe(r) wher- I is the n x n identity matrix (or the identity map kn - kn), fr(., ., ') and Pr are as in Sections 3 and 5, U: X -X/XUrJe(r) is the canonical projection. PROOF. By Theorems (6.1) and (10.6) we have (10.11) XIU (r) = P ker f (I , J , H). Let y E r(ker Je). There is an Ur e r(I ) such that (10.12) 7~ = f (I , J H)(- ). r n e Define Wr: Pr(ker J) - X/xrJe(r) = X/Pr ker fr(In, Je' H) by kr e (10.13) r r ) = O-P ur We claim that rr is well-defined. In fact, let u' C Q (In) be another r r r n vector satisfying (10.12). Then u - u ker f (In, J e H). Sr n e Therefore, (io.14) fr(In, Je' H)(-) = fr (In, Je' H)(T + U) (for some C ker f (I, Je , H)) -- fr(I n Jer H)(e The linearity and the uniqueness can be shown similarly to Lemma (8.24). By Proposition (10.8) (or the remark after the proposition) and Theorem (10.12), we get the necessary and sufficient condition for (F, G, H) to be unknown input initial state observable at the time r. (10.15) THEOREM. The system (F, G, H) is unknown input initial state observable at the time r iff (10.16) count (r ) = 0. 0 (I016)XierJe,kerH (10.17) REMARK. If r > n and if J = H, then (10.16) is equivalent to (10.18) Vmax(ker H) = 0, where Vmax(ker H) is the maximal F mod G invariant subspace in ker H. 11. Concluding Remarks If one does not go through the arguments at the beginnings of Sections 8 and 10, i.e., if one does not want such conditions as (8.6) and (10.4), one can easily accomodate the discussion in this chapter to the new reach c r) (without situations. The modified versions of XW,N r) and WNt (r) the assumption of W C N) should be used accordingly. (See Section 4 of Chapter II and Section 7 of Chapter III.) Techniques similar to those used in this chapter can be applied to study unknown input observability problems in non-constant dynamical systems (F(t), G(t), H(t)). (The problem statements (8.5) and (10.3) should be modified in the obvious ways.) The unknown input unobservable subspace X UO(s, s + r) at the final time based on the observation over the time interval [s, s + r] is defined similarly to (8.8). XUO(s, s + r) UO can be characterized by X UO(s, s + r) = X(r) where X(O) = ker H(s), X(i) = (F(i - l)X(i - 1) + im G(i - l))Qf^ker H(s + i), i = 1, 2,. The unknown input initial state unobservable subspace XIU(s, s + r) based on the measurement over the time interval [s, s + r] is defined similarly to (10.5) and it is characterized by XIU(s, s + r) = X(r) where X(O) = ker H(s + r), x(i) = F-1(s + r - i)(X(i - 1) + im G(s + r - i + l))n ker H(s + r - i), i = 1, 2, . Non-recursive characterizations can also be given in analogous ways to those in constant dynamical systems. CHAPTER V. STABILIZABILITY, OUTPUT ZEROING AND DISTURBANCE DECOUPLING reachh contr In the previous chapter we have applied WeN (r) and XN. (r) to the problems of unknown input observability. contr( In this chapter we shall demonstrate the use of XWN tr(r) in the discussions of stabilizability, output zeroing and disturbance decoupling. 12. Stabilizability We consider the system (F, G) given by (8.1). In this section we assume that k := R or C where R is the field of real numbers and C is the field of complex numbers. The pair (F, G) or a map F (over k = R or C) is said to be asymptotically stable if Jim I Ftx.i = 0 (t = 0, 1, 2, .) for all x E X where 11*11: X -R+ is a norm defined on X and R is the set of nonnegative real numbers. It is known that the pair (F, G) is asymptotically stable iff jij < 1, i = 1, 2, ., n where ?i', i = 1, 2, ., n are the eigenvalues of F. (See, for example, FREEMAN [1965, Chapter 7, Section 9], MARRO [1975, Chapter 5, Section 4].) If there exists a feedback K: X -)U such that %imJ (F + GK)tx*I = 0 (t 0, 1, 2, .) for all x* G X, then the pair (F, G) is said to be asymptotically stabilizable. Let i(A) = ous(?)) s(N) be the minimal polynomial of X with respect to F where the roots of Ous(P) have the magnitudes greater than or equal to unity and the roots of s(A) have the magnitude less than unity. Then we have (12.1) THEOREM. The following statements are equivalent: (i) The pair . (F, G) is asymptotically stabilizable. (ii) ker $us(F) C Xcontr(n). o,X (iii) contr , Xii k(F) X(n) = X. PROOF. (iii) = (i). Suppose (iii) holds. Since F ker ae(F) c ker $s(F), by applying Theorem (5.19) we see that there is a feedback K: X -4U satisfying (12.2) (F + GK) X C ker i(F), i = n, n + 1, . K can be chosen so that, in addition, (12.3) (F + GK)Iker S(F) = FIker s(F). (See Remark (5.33).) Therefore, for any state x* C X we have (12.4) im(F + GK)txII = 1im llFt n(F + GK)nxl1 = 0 where t takes on the values n, n + 1, . (Note that ker s(F) is the stable mode.) (i) = (ii). It can be shown that (i) implies ker US(F) C im G + F im G + ,,, + Fn-l im G. (See Theorem 2.2 in WONHAM [1974] .) n-1 contr Since im G + F im G + . + F'l im G CoXtr (n), the implication follows. (ii) (iii). Let xCE X. Write x = xs + x where xs 6 ker $S(F) and xu E ker Us(F). Since xu E ker (F) C X ontr (n) there exist u(O), u(1), ., U(n - 1) E U satisfying (12.5) Fnxu + Fn-Gu(O) + . + FGu(n - 2) + Gu(n - 1) = 0. Therefore, for the initial state x(O) = x = xs + xu we have (12.6) x(n) = Fn(xs + x ) + Fn-Gu(O) + . + Gu(n - 1) = Fnxs C ker OS(F). 13. Output Zeroing We are interested in the system (F, G, H) given by (8.1) and (8.2) over an arbitrary field k. (13.1) PROBLEM. (Output Zeroing by State Feedback) Given (F, G, H), find a k-homomorphism K: X -4U ("state feedback") for which there exists an integer i > 0 such that, for every x(O) E X, we have y(t) = O, t > i where y(t) is the output of the system (F + GK, 0, H) due to the initial state x(O). (13.2) THEOREM. Let x. e X. Then there exists a state feedback K: X ->U for which there is an integer i > 0 such that y(t) = 0, t > i for x(O) = x. where y(t) is the output of the system (F + GK, 0, H) due to the initial state x. iff (13) contr (1.5) x. E max(kerH),X(n) where V (ker H) is defined by (5.14). PROOF. ["if"] Since V max(ker H) is an F mod G invariant subspace in N, by Theorem (5.19) there is a feedback K : X -U such that n contr (F + GK) x(kerH),(n) C Vmax(ker H). This implies that, if x(O) = x., then x(1) = (F + GK)x. x(2) = (F + GK)x(1) = (F + GK)2x x(n) = (F + GK)x(n - 1) = . = (F + GK)nx E V (ker H). If K is so chosen as to satisfy (F + GK) V (ker H) C V (ker H) max (see the proof of Theorem (5.19)), we have x(t) = (F + GK) -nx(n) C V (ker H) for all t > n. Thus, y(t) = Hx(t) = 0 for all t > n. ["only if"] Let x(t) C X, t = 0, 1, ., be the trajectory of the system (F + GK, 0, H) with the initial state x(O) = x. Then by assumption we have for some integer i > 0 x(i + j) e ker H, j = 0, ., n. Therefore, by Corollary (5.15) we know x(i) C Vmax(ker H), which implies that x(O) = x. r Vi(V max(ker H), X) by Definition (5.2) and (5.3). Appealing to Proposition (5.17) we conclude contr contr x Vax(kerH),X XVmax(kerH),X (15.4) COROLLARY. Let x EC X. Then there exists a feedback K : X -- U such that y(t) = 0, for all t = r, r + 1, . where y(t) is the output of the system (F + GK , 0, H) for x(O) = x, iff contr (13.5) x, e V (kerH),X(r). PROOF. Similar to the proof of Theorem (13.2). O (13.6) COROLLARY. The output zeroing problem by state feedback has a solution iff contr (13.7) XV (kerH),(n) = X. max PROOF. Immediate from Theorem (13.2). O] 14. Disturbance Decoupling Consider a finite-dimensional, constant coefficient, discrete time, linear dynamical system with disturbance given by (14.1) x(t + 1) = Fx(t) + Gu(t) + Dv(t), t = 0, 1, (14.2) y(t) = Hx(t), t = 0, 1, . where X := kn, U := kim, Y := k, V:= ks (v(t) e V, t = 0, 1, .) and k is a field. We may denote the system (14.1), (14.2) as (F, G, H, D). As before r denotes a positive integer. (14.3) PROBLEM. (r-step Disturbance Decoupling Problem) Given (F, G, H, D) with x(O) = 0, find (if possible) a feedback Krl: X -4U such that y(t) = O, t = 0, ., r for any v(t) C ks, t = 0, 1, ., where y(t) is the output of the system (F + GKr-l, O, H, D) with x(O) = 0. (14.4) THEOREM. The r-step disturbance decoupling problem has a solution iff (14.5) im D c: xontr r - 1). PROOF. [Sufficiency] By assumption x(O) = 0. let Kl: X -4U r-1 be as in Theorem (5.19 ii) where we assume W = N = ker H. Replacing u(t) in (14.1) by Kr_1x(t), we get (14.6) x(t + 1) = (F + GKr_ 1)x(t) +- Dv(t). Consider the system (14.6), (14.2). Let v ( ks, j 0, 1, . . For each j = 0, ., r - 1 suppose the disturbance v(t), t = 0, 1, ., be such that v(j) = vj(f 0 possibly) and that v(t) = 0 if t J j. Then we have x(t) = O, t = 0, ., j, x(t) = (F + GK r_)tiDv., t = j+ , j + 2,. . Therefore, by Theorem (5.19 ii) with W and N both replaced by ker H, y (t) := Hx(t) = 0, t = 0, ., r + j. Now we superpose all the disturbance used above, i.e., we use v(t) such that v(t) = vt for t = 0, ., r - 1. Then the output y(t), t = 0, 1, ., satisfies r-1 y(t) = jO j(t) = 0, t = 0, ., r. [Necessity] Suppose that the problem has a solution but that im D keH kerH(r - 1). Since x(O) = 0, we have x(1) = Dv(O). s contr There must be v E k such that Dv ontr, - 1). Then by So erH,kerH Definitions (2.2), (2.3) there does not exist an input sequence u(t) C U, t = 1, 2, ., such that the corresponding state trajectory (sequence) satisfies x(t) C ker H for t - 2, ., r. In particular there is no input sequence of the form Kx(t) satisfying the above requirement where K: X --U. Thus, we have contradiction. CI The limit case of this problem ( r -)-) is the well-known Disturbance Decoupling Problem. It is known that the problem has a solution iff im D C V max(ker H) where V max(ker H) is the maximal F mod G invariant subspace in ker H. (Refer to WONTAM and MORSE [1970, Theorem (3.1) and WONHAM [1974, Theorem (4.2)].) Let us now add a constraint to the above problem so that the state of the resulting system (F + GK, 0, D, H) will eventually reach zero when the disturbance becomes zero. (14.7) PROBLEM. (Disturbance Decoupling Problem with Reset). Given (F, G, H, D) with x(O) = 0, find (if possible) a state feedback K: X -4U such that the resulting system (F + GK, 0, D, H) (with x(O) 0) has zero output sequence and that (F + GK)i = 0 for some i > 0. (14.8) THEOREM. Problem (14.7) has a solution iff the following conditions hold: (14.9) im ID C X contrH(n), contr (14.) imD C o, kerHn, (14.10) X contr) = X. 0 Z, X n= PROOF. [Sufficiency] Assume (14.9) is true. By Theorem (5.19 i) there exists K: X -4U such that (14.11) (F + GK)JXcntr (n) C ker H, j = 0, . n- 1, o, kerH (14.12) (F + GK)JXb, (n) = 0, j = n, n + 1. b, kerh (14.9), (14.11) and (14.12) then imply (14.15) (F + GK)j im D C ker H, j = 0, 1, . which means that the output y(t), t = 0, 1, . of the system (F + GK, O, D, H) with x(0) = 0 is zero for all t = 0, 1, . It remains to show that there is an integer i > 0 satisfying i contr eontr (F + GK) = 0. If (4.10) holds, then X = Xo X (n) C Xc X (n) for contr 6ontr any subspace W of X. So X =X X (n) where W := X kerH(n). cotr WX o kerH contr Noticing that 0 and X kerH(n) are F mod G invariant subspaces and repeatedly using Theorem (5.19 i), we see that there is a state feedback K: X ->U satisfying (14.11), (14.12) and n contr (14.14) (F + GK)nX C Xcontr (n). okerH Hence, (14.15) (F + GK)2nX = 0. [Necessity] Let K: X ->U be a solution of the problem. Then n contrt .Nwdfn clearly (F + GK) = 0. Hence Xco (n) = X. Now define o,X(n V := im D + (F + GK) im D + . + (F GK)imD.(FGK)n-1 im D. Then clearly im DC V, (F + GK)V CV and V C ker H. Since (F + GK)n = O, contr we have (F + GK)nV = 0. Therefore, V C X .r Thus im D C V C Xcontr (n). okerH o,kerH contrr contr. (14.16) REMARK. Recall (0.4). We have Xcontr (n) = Xcontr oX 15. Concluding Remarks reach xreach(N There is an important difference between X N each(n) and Xreach(N). neach max By choosing K: X -)U the spectrum of (F + GK)jXrax (N) can be assigned max reach arbitrarily, while to discuss the spectrum of (F +- GK)IX eN (n) may reach reach not make sense since (F + GK) XoN (n) C Xo e (n) may not hold (unless _reach, N is an F mod G invariant subspace). However, though X each(r) loses contr the arbitrary pole assignability, XcN (r) can treat stabilizability by state feedback (Theorem (12.1)). The usefulness of Theorem (12.1) is yet to be clarified. CHAPTER VI. CONCLUSION We have introduced the r-step reachable subspace each (r) from contr),N W in N and the r-step controllable subspace xi;,N (r) to W in N of the finite-dimensional, constant, discrete-time, linear dynamical system (F, G, H) over a field k. We have characterized these subspaces and discovered several interesting properties pertaining to them. The notions eN ,r). (Ntr) are natural generalizations of reachable and controllable subspaces when we are interested in structural properties of state trajectories contained in the subspace N and when initial and final states of the trajectories may not be zero. reach ( a contr, The significance of xWN (r) and XWN r) in control problems has been demonstrated in Chapters IV and V. Among the applications presented in these chapters, the unknown-input observability at the final time r is the most important in the sense that it has motivated the author to study the subjects treated here. As has been mentioned in the concluding remarks of each chapter, .reach, ic there are several other interesting topics concerning WN (r) which are still open to further research. The relation between the transfer function M(zI - F)'IG and the subspaces each r), cont r(r) should also be studied. APPENDIX Al. Pseudo-inverse Given an r X r nonsingular matrix M in a field k. There is -1 a unique r X r matrix M satisfying (Al.l) MM- = M- 1 = I. -i The matrix M is called the inverse of M. Now let M be r1 X r2 matrix in k (which may be singular). The inverse of M does not exist in general. The idea of inverse, however, can be generalized as follows: Let s := rank M, and let M be an s X r2 matrix consisting of a collection of s linearly independent rows of M. Then the pseudo-inverse M of M is defined by (Al.2) M:t M M (MT) where A' denotes the transpose of a matrix A. The meaning of M is as follows: Using trdom" to denote "the domain of", we have (Al.3) LEMMA. Let x ( dom M, and write x = x1 + x2 where x1 C im M" and x2 G ker M. Then M Mx =x. PROOF. If xl r im M then xI M x1 for some s-vector x1. Therefore M Mx M f(MM( 1 Mx X1* If x2 C ker M, then trivially MtMx2 -O. L If M is of full column rank, Mt is called the left inverse of M; if M is of full row rank, M is called the right inverse of M; if M i 7 n n s i n 7 u l a r , t h e n T.' t - 7A s ,e_.1S ~ i ] O < 1 2 REFERENCES B. 0. ANDERSON [1976] "A note on transmission zeros of a transfer function matrix", IEEE Trans. Automatic Control, AC-21: 589-591. G. BASILE [19691 "Some remarks on the pseudoinverse of a nonsquare matrix", Rendiconti Serie XII - Tomo VI, Atti della Accademia delle Scienze dell'Istituto di Bologna. G. BASILE and G. MARRO [1968&] "Controlled and conditioned invariant subspaces in linear system theory", Report No. AM-68-7, University of California, Berkeley. [1968b] "On the observability of linear time-invariant systems with unknown inputs", Report No. AM-68-8, University of California, Berkeley. [1969] "L'invarianza rispetto ai disturbi studiata nello spazio degli stati", Rendiconti Della LXX Riunione Annuale AEI. [1973] "A new characterization of some structural properties of linear systems: unknown input observability, invertibility and functional controllability", Int. J. Control, 17: 951-945. S. P. BHATTACHARYYA, J. B. PEARSON and W. M. WONHAM [19721 "On zeroing the output of a linear system", Information and Control, 20: 135-142. R. W. BROCKETT and M. D. MESAROVIC [19651] "The reproducibility of multi-variable systems", J. Math. Analysis and Applications, 11: 548-563. E. EMRE and M. L. J. HAUTUS [1978] "A polynomial characterization of (A, B)-invariant and reachability subspaces", Eindhoven Univ. of Technology, Memorandum COSOR 78-19. H. FREEMAN I oali3 Son2. New -: K. FURUTA [1973] Senkei Shistem Seigyo Riron (Theory on Linear Systems and Control), Shohkodoh, Tokyo. F. HAMANO and K. FURUTA [1975] "Localization of disturbance and output decomposition in decentralized linear multi-variable systems", Int. J. Control, 22: 551-562. B. HARTLEY and T. 0. HAWKES [1970] Rings, Modules and Linear Algebra, Chapman and Hall, London. D. G. LUENBERGER [1966] "Observers for multivariable systems", IEEE Trans. Aut. Control, AC-11: 190-197. R. E. KALMAN [1963] "Mathematical description of linear dynamical systems", SIAM J. Control, 1: 152-192. [1968] "Lectures on controllability and observability", Proc. C.I.M.E. Summer School, Edizioni, Cremonese, Roma, 1-149. R. E. KAIMAN, P. L. FALB and M. A. ARBIB [1969] Topics in Mathematical System Theory, McGraw-Hill, New York. R. LASCHI and G. MARRO [1969] "Alcune considerazioni sull'osservabilith dei sistemi dinamici con ingressi inaccesibility", Rendiconti Della LXX Riunione Annuale AEI. G. MARRO [19751 Fondamenti di Theoria dei Systemi, Patron, Bologna. B. C. MOORE and A. LAUB [1978] "Computation of supremal (A, B)-invariant and controllability subspaces", IEEE Trans. Auto. Control, AC-32: 783-792. M. K. SAIN and J. L. MASSEY (1969] "Invertibility of linear time-invariant dynamical systems", IEEK Trans. Auto. Control, AC-14: 141-149. L. M. SILVERMAN [1976] "Discrete Reccati equations: alternative algorithms, asymptotic properties and system theoretic interpretations", in Control and Dynamical Systems: Advances in Theory and Applications, Vol. 12 (edited by C. T. Leondes), Academic Press, pages 313-386. L. M. SILVERMAN and H. J. PAYNE [1971] "Input-output structure of linear systems with application to the decoupling problem", SIAM J. Control, 2: 199-233. E. D. SONTAG [1979] "On the observability of polynomial systems, I: finite-time problems", SIAM J. Control and Optimization, 17: 139-151. L. WEISS and R. E. KALMAN [1965) "Contributions to linear system theory", Int. J. Engineering Science, 3: 141-171. W. M. WONHAM [1974] Linear Multivariable Control: A .Geometric Approach, Springer, New York. W. M. WONHAM and A. S. MORSE [1970] "Decoupling and pole assignment in linear multivariable systems: A geometric approach", MIAM J. Control, 8: ' 1-18. BIOGRAPHICAL SKETCH Fumio HAMANO was born on August 28, 1949,. in Wikayama, JAPAN, to Zenichi HAMANO and Mitsuko HAMANO. He received his Bachelor of Engineering from Tokyo Institute of Technology in 1973 and his Master of Science and Engineering from the same institute in 1975. II 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 Thil6sophy. Rudolf E. Kalman, Chairman Graduate Research Professor 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 Thilisophy. Charles V. Shaffer, Co-cha 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. Thomas E. Bullock Professor of Electrical Engineering I certify that I have read this study and that in my opinion it conforms to ceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Assistant Pr fe~sor 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. Charles ABmnap Assistant 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. August, 1979 Dean, College of Engineering Dean Graduate School |

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CHAPTER I. INTRODUCTION
This dissertation discusses some structural properties of finite dimensional, discrete time, constant, linear dynamical systems. Great efforts have been made to study "geometric" properties in the continuous time systems as well as those which are common to both discrete and continuous time systems. (See for instance BASILE and MARRO [1968a and b], WONHAM and MORSE [1970], WONHAM [1974, Chapters 1 through 5] and SILVERMAN [1976, Section IH].) However, distinctive features of differentiating discrete and continuous time systems have not received much attention. This work intends to point out that there are important differences between the two kinds of systems and that discrete time systems, therefore, should be treated separately in such cases. The following notation will be used in the sequel: "im", "ker" and "dim" respectively stand for "the image of", "the kernel of" and "the dimension of". "" and ":=" mean "is an element of" and "is defined to be", respectively. Let us now turn to the definition of systems which is pertinent to the discussion of this dissertation. Let k be an arbitrary field, and let m, n and p be positive integers. A finite-dimensional, constant (coefficient), discrete time, linear dynamical system is a triple (F, G, H) whose dynamical interpretation is given by (0.1) x(t + 1) = Fx(t) + Gu(t), t = 0, 1, ... , (0.2) y(t) = Hx(t), t 0, 1, ... , where x(t) X := kn, u(t) U := km, y(t) Y := kP for t = 0, 1, ... ; F: X -X, G: U ->X and H: X -* Y are k-homomorphisms (or matrices) independent of time. The vector spaces X, U and Y are called the state, the input (value) and the output (value) spaces, respectively. The elements of X, U and Y are called states, Inputs and outputs, respectively. We shall refer to the triple defined above as the system (F, G, H). However, when the output is of no interest, we shall simply say the pair (F, G) disregarding (0.2). 1 APPENDIX Al. Pseudo-inverse Given an r X r nonsingular matrix M in a field k. There is -1 a unique r X r matrix M satisfying (Al.l) MM_1 = M_1M = I. The matrix M ^ is called the inverse of M. Now let M he r^ x r^ matrix in k (which may be singular). The inverse of M does not exist in general. The idea of inverse, however, can be generalized as follows: Let s := rank M, and let M be an s X r matrix consisting of a collection of s linearly independent rows t of M. Then the pseudo-inverse M of M is defined by (A1.2) M+ := m'{(Mm')'(Mm'))'1(MM/)' ' t where A denotes the transpose of a matrix A. The meaning of M is as follows : Using "dom" to denote "the domain of", we have (A1.3) LEMMA. Let x Â£ dom M, and write / x = x^ + x^ where x^ Â£ im M and Xg ker M. Then M Mx = x . PROOF. If x^ Â£ im M then x^ = : M x^ for some s-vector x . Therefore M+Mx1 = T{(Mm')'(MM'))"1(MM')'mM/x1 = M x = xr If x0 Â£ ker M, then trivially M^Mx^ =0. t If M is of full column rank, M is called the left inverse of M; if M is of full row rank, M is called the right inverse of M; if M is nonsingular, then M = M 1. (Also see G. BASILE [1969].) 53 time interval [s, s + r] is defined similarly to (8.8). X^(s, s + r) can be characterized by X (s, s + r) = X(r) where X(0) = ker H(s), X(i) = (F( i l)X(i l) + imG(i 1))P ker H(s + i), i = 1, 2, ... . The unknown input initial state unobservable subspace X^(s, s + r) based on the measurement over the time interval [s, s + r] is defined similarly to (10.5) and it is characterized by X^(s3 s + r) = X(r) where X(0) = ker H(s + r), x(i) = F ^(s + r i)(x(i l) + ; im G(s + r i + l))Oker H(s + r i), i = 1, 2, ... Non-recursive characterizations can also be given in analogous ways to those in constant dynamical systems. CHAPTER II. THE r-STEP REACHABLE SUBSPACE FROM W IN N We study a finite-dimensional, constant (coefficient) discrete time, linear dynamical system (F, G, H) over an arbitrary field k; in this chapter we shall be interested in properties of the pair (F, G). We shall define a new notion of "the r-step reachable subspace from W in N'f, denoted by XL. (r), and study the properties of the sequence of .reach/ < subspaces ^ (i), i = 1, 2, ... The subspace defined here will find its application in Chapter IV. We begin this chapter with a well-known notion of F mod G invariant subspaces. 1. F mod G Invariant Subspaces Consider a finite dimensional, constant, discrete time, linear dynamical system given by (1.1) x(t + l) = Fx(t) + Gu(t), t = 0, 1, ... , where x(t) X := kn, u(t) U := km, t = 0, 1, ..., and k is an arbitrary field. We call X the state space and U the input (value) space. Since the output is of no interest at the moment, we shall refer to (1.1) by the pair (F, G). (1.2) DEFINITION. A subspace V of X is an F mod G invariant subspace (or simply F mod G invariant) iff (1.3) FV C V + im G. The most important properties of F mod G invariant subspaces are expressed by the following (1.4) LEMMA. Let V be a subspace of X. Then the following statements are equivalent: (i) V satisfies (1.3). (ii) For any x* Â£ V there exists an input uÂ£U such that 7 46 (12.2) (F + Gk/x C ker /(F), i = n, n + 1, ... . K can be chosen so that, in addition, (12.3) (F + GK) Jker /(F) = Fjker /(F). (See Remark (5.33).) Therefore, for any state x^ G X we have (12.4) Â£imJ|(F + GK/xJ = Jimj||Ft"n(F + GK)\\\ = 0 where t takes on the values n, n + 1, ... (Note that ker /(F) is the stable mode.) (i) = (ii). It can be shown that (i) implies ker /S(f) C im G + F im G + ,,, + F*1"1 im G. (See Theorem 2.2 in WONHAM [1974].) Since im G + F im G + ... + / ^ im G C XC0^r(n), the implication follows O y A (ii) =5- (iii). Let x# G X. Write x_^ = xg + x where xg G ker /(F) and x^ ker /S(f). Since Xy^ ker ^S(f) C X;Cn^r (n) there exist u(0), u(l), ..., U(n l) G U satisfying (12.5) Fnxu + Fn_1Gu(0) + ... + FGu(n 2) + Gu(n l) = 0. Therefore, for the initial state x(0) = x = x + x we have (12.6) x(n) = Fn(x + x ) + Fn~1Gu(0) + ... + Gu(n l) = Fnx G ker /(F) s u s 13. Output Zeroing We are interested in the system (F, G, H) given by (8.l) and (8.2) over an arbitrary field k. (13.1) PROBLEM. (Output Zeroing by State Feedback) Given (F, G, H), find a k-homomorphism K: X -U ("state feedback") for which there exists an integer i > 0 such that, for every x(0) G X, we have y(t) = 0, t > i where y(t) is the output of the system (F + GK, 0, H) due to paragraph after theorem 5] is a special case of the above theorem. (10.7)COROLLARY. X*Y._(n) V..___(ker H) where V (ker H) is defined xCQITn THELX 1 luclX by (5.14). PROOF. Let Jg = H, and then use Corollary (5.I5). Two states able at the time x#1, x#2 Â£ X are unknown-input initial state indistinguish r iff there are two input sequences u r 1; 3 = 1, 2, to the pairs (x. t y t 0, 1, such that the r-modified output sequences corresponding (u+ilb and (x#2> coincide. *1 ltl't=oJ (10.8)PROPOSITION. Two states x^, x*g e X are unknown-input initial state indistinguishable at the time r iff (10.9)**! x*2e3We(r>- PROOF. Similar to the proof of Proposition (8.10). Therefore, we omit the proof here. xj^erJe(r) defines the set of the equivalence classes X/X^^Je(r). By Proposition (10.8) the best that we can recover of the initial state x(0) is the equivalence class x(0) + xjj-grje(r)* 'What remains to be shown is how to determine x(0) + Xj^rje(r) fro (y(0), y(r l), ve(r)). Suppose we have the initial state x(0) Â£ X and input sequence u(t) Â£ U, t = 0, ..., r 1. The corresponding r-raodified output sequence y(t) Â£ Y, t = 0, ..., r 1, v (r) Â£ k^ is given by 6 fy() 1 t (I J H) 0 0 1 y(r 1) r n e v (r) L- ev - u(i il where 1^ is the n X n identity matrix (or identity map k11 kn) 33 for some xJ ker J. Define Â£(t) := x(t) F^X-, t = 0, 1, ..., and $(t) := y(t) HF x t = 0, 1, .... where x(t), y(t), t = 0, 1, ... ~ j are given by (8.1) and (8.2) with the initial condition x(0) = xq + xq. Then it is easy to see that Â£(t), Â£(t), t = 0, 1, ..., satisfy (8.18) Â£(t + 1) = Fifc(t) + Gu(t), t = 0, 1, ..., (8.19) $(t) = Hfc(t), t = 0, 1, ... J with the initial condition Â£(0) = x ker J. Since o (8.20) *(r) + xÂ£rJW = (S(r) + + 3Â¡2rJ> TO it suffices to determine Â£(r) + ^erj(r) 'Notice that xq is known. uo The next lemma provides a way to obtain &(r) + j(r). We now consider the system (F, G, H) with the assumption that the initial state x(0) is in ker J. Let x(o) f ker J. Then (8.21) x(0) t for some f kn where B, T is a basis matrix of ker J and ker-J n" := dim ker J. For each r = 1, 2, ..., let u(t) U, t = 0, ..., r 1 be an input sequence (till time r l). The output sequence y(t) f. Y, t = 0, ..., r corresponding to the above initial state and input sequence is given by y(o) "l (8.22) y(i) i'r^BkerJ> H, H) u(0) y(r) u(r 1). where H, H): kn +rm the matrix given in Section 3 18 then X^ N(j) C ^N(d + 1) for all integer j > i. (ii) If X^ (i) D X^. N(i + 1) for some integer i > 0, then X^) D X^N(j + l5 for all integer j > i. (iii) If X^. N(i) = X^. ^(i + l) for some integer i > 0, then X^N(j) = N(i + J) for all integer > 0. PROOF, (i) The proof is by induction. Let X^ N(i) C X^. N(i + l). Assume X^ N(Â£) N( + l) where Z is a nonnegative integer. Then yN(i + 2) = + i) + im a) On 3^ ,(0 + In S)Hn =vu + 1)- Therefore, by induction we get 3^. ^(j) CX^ + l) for all integer i > i. (ii) The proof is similar to that of (i). (V . (iii) This is again proved by induction. We assume X^ ^(i) = X^ N(i + 1) for some i > 0. If X^ N() = X^ ^( + l) for some integer Z > 0, then we have 3^n( + 2) = + l) + im G) PlN = i,1(5^ N(J) + im G)flH = V + 1) Therefore, ^(i) = ^(i + j) for all integer j > 0. (5.8) RIMARK. It is possible that neither one of three conditions in Lemma (5.7) may hold (e.g., let F, G, N, W be as in Remark (2.7), and consider any positive integer i). However, if either X^N(i) c^jN(i + 1) or D X^N(A + 1) Internet Distribution Consent Agreement In reference to the following dissertation: AUTHOR: Hamano, Fumio TITLE: On Some Structural Properties of Linear Control Systems... (record number: 98007) PUBLICATION DATE: 1979 I, ~Pu. 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. 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Box 117007 Gainesville, FL 32611-7007 5/28/2008 27 The last statement is equivalent to xq X^tr(r) (by Definitions (5.2) and (50)). (6.3) REMARK. Vectors x X, u, Â£ t = 0, ..., r 1 satisfying O X (6.2) are viewed as an initial state and a sequence of inputs satisfying the conditions of Definition (5*2). (6.4) REMARK. Methods of computing the maximal F mod G invariant subspace in N have received considerable attention. (See BASILE and MARRO [1968a, Section 3, Corollary!], WONHAM [1974, Theorem (4.3)], SILVERMAN [1976, Lemma 6 in Section III, C] and MOORE and LAUB>.:[1978] Also recall Corollary (5.15).) Corollary (5.15) and Theorem (6.1) yield <6-5) W> = 3. W V V- This gives a new nonrecursive method of computing VnnY(N). 7. Concluding Remarks The remarks similar to those in Section 4 of Chapter II apply to the results of this chapter. As is the case with X^e^C^(r) the assumption WCI is not contr essential in the discussion of X^. ^ (r) from the technical point of view; the statements in this chapter can be modified straightforwardly to fit the case where W C N is not assumed. (See also Section 4 of the previous chapter.) iff From Lemma (5.7) we know that X^^r(i) D X^^r(i + l), i = 0, 1, (7.1) W D F_1(W + im G) PlE. The significance of the condition (7.1), however, is not yet clear. 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(5-17) PROPOSITION. X^JJtr(i) C X^JJtr(i + 1) for all i = 1, 2, ... iff iÂ¥ C W + im G. (5.18)COROLLARY. If FW C W + im G, then FX^JJtr(r) C X^JJtr(r) + im G for each r = 1, 2, ... . PROOF. Suppose FW C W + im G. Then by Proposition (5.17) we have X^. (x 1; C X^. (1), 1 = 1, 2, ... By the definitions of the i and (i l)-step controllable subspaces to W in N, for every f* OTVt" T* v. G XL, (i) there must exist u. U such that Fv. + Gu 6 X,, (i l) C ^tr(i). Therefore, FX^Ri) C xÂ£Â¡fr(i) + im G, i = 1, 2, .. . n As for the nonincreasing sequence we just note a special case of Lemma (5-7), i.e., ^tr(i) D X^JJtr(i + 1), i = 1, 2, ... iff W C F_1(W + im G) f>N. rcontr X^. (r) has the following properties in relation to state feedback. By rcontr the definition of X^ ^ (r) it is clear that the subspace is state feedback invariant, i.e., the r-step controllable subspace to W in N of the system (F + GK, G) is equal to X^^r(r) of (F, G). Under some conditions, the input sequence u(t) U, t = 0, ..., r 1 given in Definition (5*2) can be implemented by a suitably chosen state feedback u(t) = K^x(t), t = 0, ..., r 1. (5.19) THEOREM. (i) If X^^tr(i) C X^JÂ¡tr(i + 1), i = 1, 2, ..., i.e., if FW C W + im G, then for each r =1, 2, ... there is a feedback K : X -> V such that r (5.20) (F + GKr)JX^JJtr(r) CN, j = 0, ..., r 1, (5.21)(F + GKr)JX^JJtr(r) CW, j = r, r + 1, CHAPTER V. STABILIZABILITY, OUTPUT ZEROING AND DISTURBANCE DECOUPLING In the previous chapter we have applied X^.e^Cil(r) and X^^tr(r) to the problems of unknown input observability. In this chapter we shall demonstrate the use of X^^ r(r) in the discussions of stabilizability, output zeroing and disturbance decoupling. 12. Stabilizability We consider the system (F, G) given by (8.1). In this section we assume that k := R or C where R is the field of real numbers and C is the field of complex numbers. The pair (F, G) or a map F (over k = R or C) is said to be asymptotically stable if lim )| F^x !| =0 (t = 0, 1, 2, ...) for all > 00 ^ x# G X where || |Â¡: X > R is a norm defined on X and R is the set of nonnegative real numbers. It is known that the pair (F, G) is asymptotically stable iff Â¡AÂ¡ <1, i = 1, 2, ..., n where A, i = 1, 2, ..., n are the eigenvalues of F. (See, for example, FREEMAN [1965, Chapter 7, Section 9l, MARRO [1975, Chapter 5, Section 4].) If there exists a feedback K: X U such that Â£imj| (F + GK)^x*|| = 0 (t =0, 1, 2, ...) for all x# X, then the pair (F, G) is said to be asymptotically stabilizable. Let $((A) = ^US(A)^S(A) be the minimal polynomial of X with respect to F where the roots of ^US(A) have the magnitudes greater than or equal to unity and the roots of (A) have the magnitude less than unity. Then we have (12.1) THEOREM. The following statements are equivalent: (i) The pair (F, G) is asymptotically stabilizable. (ii) ker jUS(F) C Xtr(n). x- PROOF, (iii) =* (i). Suppose (iii) holds. Since F ker tfS(F) C ker $S(F), by applying Theorem (5.19) we see that there is a feedback K: X -> U satisfying 45 35 Furthermore, to (8.25) g^j) ker f^B^, H, H) = ^(r). Therefore. (^erJ: r.(ker J) X/X^rJ(r). PROOF. Let yr G r^(ker j). There is an element G nr(BkerJ) such that (8.26) f^B^j, H, H) (i.) = yr. Define rr(ker J) X/gr(BkerJ) ker f^B^,, H, H) by (8.27) <6rJ(5r) (\). 5^er^ is well-defined. Indeed, let he another element in ^r(Bkerj) satisfying (8.26). (8.28) pg^j) (;) = pgr(\erj) (\ + y) (for some 7 ker H, H)) = p(sr(\erJ) (ur) + gr(BkerJ^7^ = PSr(\erj) + ^r^erJ^ ^ PgpC^grj) (ur)* To show the linearity of ^erJ, let yrl, yr2 G Tr(ker J). Then there exist rl, rg G -Qr(BkerJ) such that = ^^kerJ^ H 3 =1, 2. Therefore, yrl + yy2 = fr(BkerJ> H, H) (url + u^) and we have ^kerJ( + y ) = pg (B _) ( + c) rr wrl r2 r kerJ rl r2' 13 ker = N. Denote by a basis matrix of W. Then we have 0 ... 0 ^ ^G. 0 0 MjFCS LV\ M?r'1(5 0 Define gr(Bfrr) := [Y\; Fr-1G ... FG G], r > 0. (3.1) THEOREM. xJÂ¡**Ch(r) = gr(y ker f^, Mj, Mj), r > 0. PROOF. Let x* G X. ** gr(V ker fr(^, Kj, Hj) iff there exist | G and u ..... u .. G km such that o r-1 (3-2) u u i L r-U Â£ ker W k- *9' (3-3) X = 8r(B) u u L r-1 The conditions (3*2) and (3*3) are equivalent to REFERENCES B. 0. ANDERSON [1976] "A note on transmission zeros of a transfer function matrix", l Kb!hi Trans. Automatic Control, AC-21: 589-591* G. BASILE [1969] ,fSome remarks on the pseudoinverse of a nonsquare matrix", Rendiconti Serie XII Tomo VT, Atti della Accademia delle Scienze dell'1stituto di Bologna. G. BAS HE and G. MARRO [I968&] "Controlled and conditioned invariant subspaces in linear system theory", Report No. AM-68-7, University of California, Berkeley. [1968b] "On the observability of linear time-invariant systems with unknown inputs", Report No. AM-68-8, University of California, Berkeley. [1969] "L'invarianza rispetto ai disturb! studiata nello spazio degli stati", Rendiconti Della LXX Riunione Annuale AEI. [1973] "A new characterization of some structural properties of linear systems: unknown input observability, invertibility and functional controllability", Int. J. Control, 17: 931"9^3 S. P. BHATTACHARYYA, J. B. PEARSON and W. M. WONHAM [1972] "On zeroing the output of a linear system", Information and Control, 20: 135-lte. R. W. BROCKET! and M. D. MESAROVIC [1965] "The reproducibility of multi-variable systems", J. Math. Analysis and Applications, 11: 5^8-563. E. EMRE and M. L. J. HAUTU3 [1978] "A polynomial characterization of (A, B)-invariant and reachability subspaees", Eindhoven Univ. of Technology, Memorandum C0S0R 78-19 H. FREEMAN [1965] Discrete-time Systems; An Introduction to the Theory, John Wiley and Sons, New york. TABLE OP CONTENTS ACKNOWLEDGEMENTS ABSTRACT v CHAPTER I. INTRODUCTION 1 II. THE r-STEP REACHABLE SUBSPACE FROM W IN N 7 1. F mod G Invariant Subspaces 7 2. Reachability From W in N 8 3. Nonrecursive Characterization of the r-Step Reachable Subspace from W in N 12 4. Concluding Remarks 14 HI. THE r-STEP CONTROLLABLE SUBSPACE TO W IN N l6 5. Controllability to W in N 16 6. Nonrecursive Characterization of the r-Step Controllable Subspace to W in N 25 7. Concluding Remarks 27 IV. UNKNOWN INPUT OBSERVABILITY 28 8. Unknown Input Final State Observability 28 9. Unknown Input Final State Observability Part 2 (Special. Cases) 37 10. Unknown Input Initial State Observability 39 11. Concluding Remarks 43 V.STABILTZABILITY, OUTPUT ZEROING AND DISTURBANCE DECOUPLING 45 12. Stabilizability 45 13. Output Zeroing 46 14. Disturbance Decoupling 48 15. Concluding Remarks 51 VI. CONCLUSION 52 APPENDIX 53 REFERENCES 54 BIOGRAPHICAL SKETCH 57 iv 26 y = 2, 3, ,,,, and fi *1 o' o where we have r blocks of n X m zero matrices. (6.1) THEOREM. xÂ£jjtr(r) = Pr ker fr(ln, Mj), r > 0. PROOF. The condition (6.2) u u n L r-1 ker fr(ln, M, Mj), x X, u, Â£ km, t = 0, ..., r 1, is equivalent to O tl fro - ^ = 0, + + " + V2 = VS + + ' + y Vl which in turn is equivalent to the conditions x N, o * Fx + Gu N, o o Fr_1x + Fr2Gu + ... + Gu N, o o r-2 Frx + Fr-1Gu + ... + Gu , o o r-1 e w. (5.25)(F + GK )e GW, s = 1, ..., Z , \ r OS o (5.26)(F + GKr)e.s. G X^tr(j 1), g=l, r; s.. = 1, ..., It is easy to check that the following relations hold: (5.27)(F + GK )1e. G N ry jsj for 0, ..., r'; s = } ...} .i i = 0, ..., r 1 and J J (5.28)(F + GK )Xe. GW r jsj for j =0, ..., r ^contr/. ^ ,rcontr,. (11) Suppose 3^ N (x) N (1 + 1), 1 = 1, 2, ... For each r = 1, 2, ... we choose a basis of X in the following way. Let ierl, e^ ) be a basis of X^JJtr(r) where : = dim X^jJtr(r), Extend the basis to get the new basis {e .... e : e _ , rl rlr r-1,1,... er-l,ir_i^ of X^^r(r l). Repeat the procedure, and we obtain the basis ierl, e^; er-:L,:L,..., ei1 ''ol ^ := dim r(o) dim Xjjjtr(j + l), J = 0, r 1. Note that if i = 0 we do not extend the basis at this step and go to the next step. D e we complete the basis Or ev,_n i e, ; ertl, ..., e^ ) of W where By adding linearly independent vectors e.., of X as {e .,..., e. .; ,, ..., e ; en, ..., ) where rl 1^1 01 Contr Â£ 1 qr q := n dim W. Since e. G X (j) j = 1, r and s. = r jsj w,N j 1, ..., ., there must be an input u. such that 0 Jsj (5.29) Fe.s. Gu.s. C 1) where 3 = 1, ..., r and s. = 1, ..., Â£.. It is straightforward to J d check (5.31) (F + GKr)rers GW, s^ = 1, J 12 fxw,n(v) C (fxw,n(v) + im G) n (N + im Q) C (FX^. n(v) + im G) pin + im G = + im G. 3. Nonrecursive Characterization of the, r-Step Reachable Subspace from W in N. X*6&cll The sequence defined by (2.3) and (2.4) determines X-. (r) recursively. We now give a nonrecursive characterization of Let B be a matrix having n rows. Let C and A be matrices with n columns. Define f (B, A, C) := AB f (B A C) 4CB " V > '~|afb AGJ f2(B, A, C) CB 0 0 CFB CG 0 af2b AFG AQ_ where 0 is the zero matrix of appropriate size. Extending the above procedure, we define for r = 1, 2, ... CB 0 0 . . 0 CFB CG 0 . . 0 fr(B, A, C) : = cf2b CFG CG CFr-1B CFr"2G CG 0 AFrB AFr_1G . . AFG AG In this section we only use a special case where B I n X n rr identity matrix. (Another case where B 4 I T n will be used in Chapter IH, Section 6.) As before, W C N C X. Let be a matrix with n columns satisfying 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. .^Wxv\ Rudolf E. Kalman, Chairman Graduate Research Professor 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 Philisophy. C&aJIqa 2/. Charles V. Shaffer, 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. Thomas E. Bullock Professor of Electrical Engineering The subspace V (N) is called the maximal F mod G invariant subspace max in N. It has been known (SILVERMAN [1976, Section III, A]) that V (n) max is equal to the set of states in N for which there are input sequences such that the corresponding trajectories remain in N for p units of time. It should be noted that this statement is a special case of Theorem (5.6). It should also be noted that the algorithm for computing V (n) is a max special case of (5.4), (5-5) where W = N. (See BASILE and MARRO [1968a, Section 3, Corollary 1] and WONHAM [1974, Theorem (4.3)1*) Summarizing, we have (5.I5) COROLLARY OF THEOREM (5.6). Let p be as in Lemma (5.9)* and let V (n) be as used above. Then max ''max D The sequence X^^r(i), i = 1, 2, ... has a conditional monotone property. (See Theorem (5*6) and Lemma (5*7)*) The natural question to ask is then when it is monotonically nondecreasing or nonincreasing. ( 5 .6) LEMMA, (i) W C X^ N(l) iff FW C W + im G. (ii) If WDX^N(l), then X^ n(p) =vmax(w) where P is as in Lemma (2.9) and V (w) is as in (3.14). " 1 max , PROOF. (i) If W C X^ (1), then Â¥ C F_1(W + im G)On. Hence W C F ^(W + im G). So we have FW C W + im G. Conversely, if FW C W + im G, then W C F ^(W + im G). Since W C N by assumption, we get W C F ^(W + im G) ^ K, i.e., W CX^ .^(l). (ii) If W D w(l), by Lemma (5.7 ii) we have ^(i) C W for all i = 0, 1, ... Then obviously X^ N(i) = w(i), i = 0, 1, ... . Therefore, by Corollary (5-15) ^ N(p) = V^W). For the nondecreasing sequence X^^r(i), i = 1, 2, > we have 22 each r = 1, 2, (ii) If X^tr(i) D Xjjjtr(i + 1), i = 1, 2, then for there is a feedback K : X r U such that (5.22) (F + GKr)jxJJ}tr(r) CH, 3=0, (5 .23) (F + GKr)rxÂ£"tr(r) C W. PROOF, (i) By Proposition (5.17) xJJÂ¡tr(i.) C X^j}tr(i + l), i = 1, 2, ..., iff W is an F mod G invariant. Assume xÂ£jjtr(i) C xÂ£JJtr(i + l), i = 1, 2, ... For each r = 1, 2, ... we'choose a basis of X as follows. Let {e^, ... o^o ) 'oV be a basis of W. Extend this basis to get the basis {e ,, ... contr ol ell> *> eL?i^ N ^ePea^ the extension until we get no more vectors to add to, say r' times where r' < r, and we obtain the J of basis (eQ1, > e^S e^, ., X^ r(r') = X^0^ r(r). We further extend this basis arbitrarily to get 6 ^ ^ 5 6 /<. r -l,r'_i* r V / 0 r e ). Here l and 3 \ : = the basis of X, {e ..., ^ - dim Xw N (j) dim X^ ^ (3-1), 3 = 2, ..., r n dii xÂ£*tr(r). Since e^ 3 = 0, ..., r' and 8^=1 s, =1, ..., there exists an input u^ such that (5.24) Fe. + Gu. sj )sj e i) where 3=1, ..., r' and s^ = 1, ..., y (See Definitions (5.2) and (5*3)). Define Kp: X ->U so that it satisfies the following conditions: (a) u. = K e. for 3 =1, ..., r' and s. = 1, Js3 r jsj ' 0 ' A' (b) (F + GKr)eQs 6 W for s = 1, ..., Z (such a always exists since FW C W + im G. See Lemma (1.4) and Proposition (5.17).) (c) 3=1, ..., are arbitrary. Then 25 exist xtPM, ut e U, t 2, ..., i satisfying x^ = Fxt + Gut, t = 2, i. Then x Â£ X^^tr(i). Hence, noting that x f. W, t = 1, ria.i'h Z i, we get w X^^i). Conversely, if (5.36) holds, then there exists a subspace t Cl satisfying X~e^Ch(l) = W, which implies (4.2). Consequently, (5.3?) PROiosmoN. X^h(i) c + 1), 1 = 1,2,..., Iff <5-38> ff = CÂ£(W),W PROOF. Immediate by using the above lemma and Lemma (2.8). 6. Nonrecursive Characterization of the r-step Controllable Subspace to W in N. The sequence defined by (5.4) and (5.5) determines the r-step control lable subspace to W in N, X^^tr(r), recursively. We now give a nonrecursive characterization of X^^r(r). As before, W C N C X and N.c< *N r = 1, 2, Recall the definition of fr(B, A, C) in Section 3 of Chapter II. Denote by 1^ the n X n identity matrix, and let 0 be the zero matrix of suitable dimension. Let be a matrix with n columns such that ker W , and define M similarly. Then *4 0 0 . . 0 >4 0 . . 0 W V - * *4. : 0 0 r1 Vr'2s \ 0 . . MJFG M^G 1. i Vn(s + 2) = (fxh,n(s + 1} + im G)nH = (F\ Therefore, by induction Vn(J) = Vn< + 1) for all j = i, i + 1, ... (2.7) REMARK. It is possible that none of_ three conditions in (2.6) may hold. For instance, let F := '0 W := span 0 1 1 0 G : = N := X and let i be any positive integer. Lemma and However, if either X^N(i) C X^N(i + l) or => ^jjC + l) holds for some i, 3 = 0, 1, ..., the sequence X^. ^(.0)> & = 0, 1, ... will stop increasing or decreasing in a finite number of steps since X is finite dimensional. If it is the case, let v be the least integer i > 0 such that X^N(i) = X^N(i + 1). (2.8) LEMMA. (i) If W C Xw N(l), then v < dim N dim W < n. (ii) _If WDX^^(l), then V < dim W < n. PROOF. Immediate from Lemma (2.6) by using the finite dimension ality of X. Let V be as in the paragraph prior to Lemma (2.8). X^ ^(v) is not an F mod G invariant in general. However, we have (2.9) LEMMA. If FN C N + im G, then FXW N(v) C N(v) + im G holds. PROOF. Since X^. ^(v) (FX^. ^(v) + im G)flN, it follows that FX^ N( v) C FN C N + im G. Therefore, we have CHAPTER IV. UNKNOWN INPUT OBSERVABILITY The notions of the r-step reachable subspace from W in N and of the r-step controllable subspace to W in N as developed in Chapters II and III are now applied to the study of unknown input observability of a system (F, G, H). Sections 8 and 9 discuss the unknown input final state observability. Section 10 treats the unknown input initial state observability. Hereafter Y denotes the output (value) space and is defined by Y := k^. Also p := dim Y is used throughout Chapters IV and V. 8. Unknown Input Final State Observability We consider a finite-dimensional, constant, discrete time linear dynamical system (F, G, H) given by (8.1) x(t + 1) = Fx(t) + Gu(t), t = 0, 1, .... (8.2) y(t) = Hx(t), t = 0, 1, ... where x(t) F X : = kn, u(t) F U := k, y(t) Y = k^ (t = 0, 1, ...) and k is a field. (Recall the definition in the second paragraph of Chapter I.) It is assumed that we have some degree of a priori information about the initial state given by (8.3) v(x(0)) = Jx(0) where J: X -k^ is a k-homomorphism and q is a positive integer. In particular, if J is an isomorphism where q = n, then the initial state is a priori known. If J is zero, then the initial state is a priori unknown. Depending on an initial state x(0) = xq F X and an input sequence u(t) = Uj_ F u, t 0, 1, ..., the system produces the corresponding state and output sequences (or trajectories) x(t) F X, t = 0, 1, ... 28 (sequence) satisfies x(t) G ker H for t 2, .., r. In particular there is no input sequence of the form Kx(t) satisfying the above requirement where K: X ->U. Thus, we have contradiction. The limit case of this problem ( r -<>) is the well-known Disturbance Decoupling Problem. It is known that the problem has a solution iff im D C V (ker H) where V (ker H) is the maximal F mod G invariant max max subspace in ker H. (Refer to WONHAM and MORSE [1970, Theorem (3.1) and WONHAM [1974, Theorem (4.2)].) Let us now add a constraint to the above problem so that the state of the resulting system (F + GK, 0, D, H) will eventually reach zero when the disturbance becomes zero. (l4.7)PROBLEM. (Disturbance Decoupling Problem with Reset). Given (F, G, H, D) with x(0) = 0, find (if possible) a state feedback K: X -4 U such that the resulting system (F + GK, 0, D, H) (with x(0) = 0) has zero output sequence and that (F + GK)1 = 0 for some i > 0. (14.8) THEOREM. Problem (l4.7) has a solution iff the following condi tions hold: (14.9) 4.Cl*(n), (14.10)XGOJtr(n) = X. Oj A PROOF. [Sufficiency] Assume (l4.9) is true. By Theorem (5.19 i) there exists K: X -> U such that (14.11) (F + GK)jXGÂ£^H(n) C ker H, j = 0, ..., n 1, (14.12) (F + GK)JXGÂ¡^H(n) =0, j = n, n + 1, . . (l4.9), (l4.ll) and (14.12) then imply (14.13)(F + GK)J im D C ker H, j = 0, 1, ..., research was also supported in part by the Foundation for International Information Processing Education, 2-6-1 Marunouchi, Chiyoda-ku, Tokyo 100, JAPAN. iii in N. ^.reach/ N V> (r) and study the properties of the 1, 2, with respect to W and We shall characterize sequence of subspaces X^e^|C^(i), N. What appears to be intriguing is the fact that the properties of X^. jy (i), i = 1, 2, ... change drastically depending on W. In Chapter III we shall introduce the natural counterpart of X^ ^ (r) which we shall call r-step controllable subspace to W in N, X^^r(r). This subspace is defined to be the set of states in N from which some state in W can be reached in r steps via trajectories contained in N. It should be noted that, except for the special cases of X^^r(r) (MARRO [1975> Chapter 4]) and X^^r(r) (which we shall discuss shortly), the more general subspace X^^r(r) has not been WjN contr studied as such. It is this generality that makes X^ ^ (r) an inter esting object. The significance of introducing X^ (r) will become clear when we consider the unknown input observability at the final time r in Chapter TV (Sections 8 and 9)* The problem is stated as follows: Given (F, G, H), a priori information about the initial state v(o) = Jx(0) and the output sequence y(l), y(2), ..., y(r), find the state x(r) at the final time r. We shall see that the best we can do to identify x(r) is to determine (r) which in fact can be determined. Thus, the coset x(r) + xfea^1. v herJ,kerH for an in-depth understanding of the unknown input observability at the final time r, it is essential to study the properties of X^ j kerirr^ which depend on J, H and r. The results in Sections 8 and 9 appear to be new. A special case of X^oatr(r), namely, X^atr(r) has been discussed in the literature in relation to the unknown input initial state observa bility, and it is known that, if we are given (F, G, H) and the output sequence y(0), y(l), ..., y(r), only the coset x(0) + Xaatr(r) can be recovered based on the above data. (See, for example, RASILE and MARRO [1973, Theorem 1 and Corollary 1] and SILVERMAN [1976, Definition 2 in Section III].') We shall include this problem with more generality in Chapter IV, Section 10 to implement our knowledge of unknown input observability. The method used here to recover x(0) (or its coset) is less complex to understand than the ones which have been used in the 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 ON SOME STRUCTURAL PROPERTIES OF LINEAR CONTROL SYSTEMS REACHABILITY FROM W IN N AND CONTROLLABILITY TO W IN N By FUMIO HAMANO August, 1979 Chairman: Dr. R. E. Kalman Major Department: Electrical Engineering In the "geometric approach" to the study of linear systems two important notions have been successfully used; namely, the maximal reachability sub space Xf*ck(N) contained in a given subspace N and the maximal F mod G ulQJC . invariant subspace contained in N. However, the definition of X (N) max is not as natural in discrete time systems as in continuous time systems. It loses an important meaning when it is applied to discrete time systems, i.e., Xch(N) is not the set of states reachable from 0 via traject- max ories in N. In this work similar notions which are suitable to treat discrete time systems are developed. In general the study is concerned with "reachability" and "controllability" internal to the subspace N in dis crete time systems. More specifically, the notions of the r-step reachable subspace from W in N and the r-step controllable subspace to W in N are introduced for given subspaces W and N (satisfying WCN), These are respectively defined to be the set of states reachable from W via trajectories in N in r steps and the set of states that can reach W in r steps via trajectories in N. Algebraic characterizations and sequential v ACKNOWLEDGEMENTS I wish to express my sincere appreciation to all those who contributed in various degrees toward the fulfillment of this work. I am particularly grateful to Professor R. E. KALMAN, the chairman of my supervisory committee, for his constant encouragement in seeking a concrete understanding of system theory and other areas in science. His guidance has been of a great help in establishing an organized view in this dissertation. Without the financial support which he arranged for me during the past four years and without the stimulating environment of the CENTER FOR MATHEMATICAL SYSTEM THEORY, this work may not have existed. I am thankful to Professor C. V. SHAFFER, co-chairman of my supervisory committee, who has not only given me valuable comments concerning my dissertation, but also made favorable arrangements for me during my personal emergency. The specific motivation for the research reported here was provided by the stimulating discussions with Professor G. BASILE relating to his earlier works concerning ''geometric" views on various control and system problems. I appreciate his friendship and his deep interest in this work. A dissertation is only a part of a doctorate. In the educational process during the past four years the influence of Professors E. EMRE, E. D. SONTAG, Y. YAMAMOTO, G. SONNEVEND, T. E. BULLOCK, M. E. WARREN, C. A. BURNAP, M. HEYMANN, M. L. J. HAUTUS, V. KUCERA and others was essential in the preparation for a doctoral degree. Of course, no research would be made were it not for the long-term love and encouragement of a few close people. My parents and my wife, Shoko, have been constant sources of encouragement. To them I dedicate this work. Needless to say, I am grateful to Ms. Karen Todd for her understanding and patience not only as a typist but as a friend. This research was supported in part by US Army Grant DAAG 29-77-G0225 and US Air Force Grant AFOSR 76-3034 through the Center for Mathematical System Theory, University of Florida, Gainesville, FL 326II, USA. The ii 9 (2.2) DEFINITION. The r-stex reachable subspace X^e^ch(r) from W in N is the set of states r-step reachable from W in N. We show that the sequence X^e^ch(i), i = 1, 2, ... can he recursively computed. Let (2.3) Xw>N(0)=W (2.4) ^ (i) = (FX^jjd 1) + im G)Hn, i = 1, 2, ... . Then we have (2.5) THEOREM. xÂ£**Ch(r) = X^r). PROOF. The proof is done by induction. (i) ^ N(l) = (IW + im G)ON = {x, Â£ N: x. = Fw + Gu for some w W and u Â£ km) 1 l o o = Ch(1)- (ii) Assume that ^(d) is equal to the j-step reachable subspace from W in N, d > 0. YjjJ + 1) = (FX^J) + im G)Hn = {xJ+l Â£ N: xi+1 = FXj + Gu. for some Xj ^^(d)' and u. Â£ km). d By induction assumption any X. Â£ X^^(d) has at least one pair of o > HI sequences N, t = 0, ..., j 1, and u^. Â£ k, t.- 0, ..., d 1 such that Xq Â£ W, ]_ + ^ t = 1, ..., j. Therefore y(j +1) (x3+l 6 N: ii 0, There exist sequences x^ N, d and u km, t = 0, ..., d t = FjCj. + Gu^., t = 0, d satisfying x 6 literature so far. The subspace X^^r(r) is closely related to "state feedback" K: X -U. Those results in Chapter III which are related to state feedback will be applied in Chapter V in which we consider stabilizability, output zeroing and disturbance decoupling. Problems of these kinds have been treated in the literature in different fashions. (See WONHAM [1974, Theorem (2.3) for stabilizability, Theorem (4.4) for output stablization and Theorems (4.2) and (5.8) for disturbance decoupling].) The contents of Chapter V will also serve to exemplify the significance of X^^r(r) in the cases where W j N. The theorem (12.1 (i), (iii)) concerning etabilility is of interest in the sense that it gives a new interpretation of stabilizability. CHAPTER III. THE r-STEP CONTROLLABLE SUBSPACE TO W IN N. In the previous chapter we have studied the r-step reachable subspace from W in N. In this chapter we introduce another new notion which we call ,rthe r-step controllable subspace to W in N,r. Its applications will be found in Chapters IV and V. 5. Controllability to W in N. Consider a finite dimensional, constant discrete time, linear dynamical pair (F, G) represented by (5.1) x(t + l) = Fx(t) + Gu(t), t = 0, 1, ... where x(t) 6 X := kn, u(t) Â£ U := km, t = 0, 1, ..., and k is an arbitrary field. We denote the system (5.1) by the pair (F, G). As before W and N denote subspaces of X satisfying W C N C X, and r is a positive integer. (5-2) DEFINITION. A state x^ Â£ N jLs r-step controllable to W in N iff there exists an input sequence u(t) Â£ u, t = 0, ..., r 1 such that x(0) = x^, x(r) Â£ W and x(t) Â£ N, t = 0, r. The set of states x^'s satisfying the above conditions form a (linear) subspace. (5.3) DEFINITION. The r-step controllable subspace X^^r(r) to. W in N is the set of states r-step controllable to W _in N. We show that X^j^r(r) can be computed recursively. Let (5*If) Vn(0)=w (5.-5) 3^ N(i.) = F_1(^^N(i 1) + im G) On, i = 1, 2, ..., where F "Si := (x Â£ X: Fx Â£ X ] for a subspace X of X. s s s 16 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 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. Chtos d- ftuAna# Charles A. Burnap Assistant 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. August, 1979 Dean, College of Engineering Dean Graduate School BIOGRAPHICAL SKETCH Fundo HAMANO was "born on August 28, 1949, in Wakayama, JAPAN, to Zenichi HAMANO and Mitsuko HAMANO. He received his Bachelor of Engineering from Tokyo Institute of Technology in 1973 and his Master of Science and Engineering from the same institute in 1975. 57 6 + ^Guo = 0, Â£+ Vr"lGuo+ '' + ^GVl = > x* = F1^ i + Fr 1Guq + ... + Gur_1, which in turn is equivalent to the conditions % e G N F% I + Gu0 G N = F ^ + F Gu^ + + Gu^ ^ Â£ N. d. imW The last set of relations hold for some | Â£ k and u u^, ..., ur_1 km iff x# G X^G^Ch(r) (by Definition (2.2)). (3-4) REMARK. A set of vectors i^|, u^_ Â£ k, t = 0, ..., r 1 satisfying (3-2) and (3-3) are seen to be an initial state and a sequence of inputs satisfying the conditions of Definition (2.1) for the final state x#. 4. Concluding Remarks. In our discussion of this chapter we have assumed Â¥ C N. This assumption has been made since we are interested in structural properties of the system (F, G) inside the subspace N. Technically speaking, however, the above assumption is not essential. With slight modification all the statements still hold without assuming W C N. Theorem (3.1) can be easily modified to yield the corresponding result: (4.1) x^ch(r) = gr(y 4er ?r(^, Mj, Mj), r > 0 xml record header identifier oai:www.uflib.ufl.edu.ufdc:UF0008245600001datestamp 2009-02-16setSpec [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 On some structural properties of linear control systems--reachability from W in N and controllability to W in Ndc:creator Hamano, Fumiodc:publisher Fumio Hamanodc:date 1979dc:type Bookdc:identifier http://www.uflib.ufl.edu/ufdc/?b=UF00082456&v=0000106655526 (oclc)000098007 (alephbibnum)dc:source University of Floridadc:language English 17 (5.6) THEOREM. Xjjjtr(r) = J^N(r), r > 0. PROOF. The proof is by induction. For r = 1, 3^ N(l) = F (W + in G) O N = {x^ X: Fx^ = w Gu^ for some w G W and m \ k , and xx G N) (x^ G N: Fx^ + Gu^ G W for some G km) each/ -cr- Now ,contr let ^>N(d) = N (j), d > 0. Then we have Vn(3 + 1) ix3+l N: *Vi xj GVl for some *3 e Vh(j)' ,m > Vie k By assumption of ^(3) = X^^rf3) this is equal to (*J+1 E B> F*3+1 + GUJ+1 *3, FXj + GU3 = FXj_ + GUX a for some _ vcontr/v .m t . w G W, x^ G Xy jj \Â£), Ug G k Â£ ~ 2.f > J and VI e ^ -*Â£?'<* + l5- We now study properties of the sequence W, X^^r(l), X^^r(2), by examining properties of the sequence (5.4), (5.5). The sequence (5.4), (5*5) (equivalently Wtr(l), X^jJtr(2), ...) has conditional monotone properties. (5.7) LEMMA, (i) If X^ N(i) C Xw N(i + l) for some Integer i > 0, time r iff ')'-J (9.11) deg X(z) deg ]l(z) = 1. PROOF- Since m = p = 1, it follows that rank G = 1 iff G / 0 and that rank HG = 1 iff HG ^ 0. Hence, (9*6) holds iff HG 4 0 (since G 4 0 by assumption). (9*12) COROLLARY. Assume that the initial state is known, i.e., ker J = 0. Then there is a unique state trajectory x(t) G X, t = 0, 1, ..., r corresponding to an initially modified output sequence (v(0), y(l), ..., y(r)} iff (9*5) (i.e., (9*6)) is satisfied. (9.I3) REMARK. Assume that the initial state is known and that rank G = m (i.e., dim im G = in). It can be shown that the input sequence correspond ing to an initially modified output sequence (v(0), y(0), ..., y(r)) is unique iff (9*5) (or (9*6)) is true. 10. Unknown Input Initial State Observability. Consider the system (F, G, H) given by (8.l) and (8.2). As before r = 1, 2, ... . We wish to find x(0) based on the measurement of the output sequence y(0), ..., y(r). Suppose that we are allowed to make an extra measurement at the final time r. (Suppose for example, that the "motion" of the system stops at the end (at the time r) and that we can examine the system more carefully to get an extra information at the time.) We will denote this extra information by (10.1) vg(r) = Jgx(r), where ve(r) Â£ ks and Jg: X >kS is a k-homomorphism (for some integer s > 0). Then (10.2) v (r) J ev r e y(r)_ H_ (13.4) COROLLARY. Let x Â£ X. Then there exists a feedback K : X ->U such that y(t) = 0, for all t = r, r + 1, ... where y(t) is the output of the system (F + GK^, 0, H) for x(o) = x# iff (13-5) x* G _ contr / % 1 \aX (13.6) COROLLARY. The output zeroing problem by state feedback has a solution iff (13-7) contr \ (kerH),X max' (n) = X. PROOF. Immediate from Theorem (13.2). 14. Disturbance Decoupling Consider a finite-dimensional, constant coefficient, discrete time, linear dynamical system with disturbance given by (14.1) x(t + l) = Fx(t) + Gu(t) + Dv(t), t = 0, 1, ... (14.2) y(t) = Hx(t), t = 0, 1, ... where -X;= kn, U km, Y := kP, V : = kS (v(t) V, t = 0, 1, .. ) and k is a field. We may denote the system (l4.l), (l4.2) as (F, G, H, D). As before r denotes a positive integer. (14.3) PROBLEM. (r-step Disturbance Decoupling Problem) Given (F, G, H, D) with x(0) = 0, find (if possible) a feedback K : X U such that g y(t) = 0, t = 0, ..., r for any v(t) G k t = 0, 1, ..., where y(t) is the output of the system (F + GK^ 0, H, D) with x(0) = 0. i.e., (9.6)rank HG = rank G. PROOF. By Theorem (8.32) and Theorem (2.5) (F, G, H) is unknown input observable at the final time r iff (9.7) X (r) = 0. w o,kerHv ' Sinoe Xo^kerH(i) C ^kerH^ + l), i = > ^ follows from Lemma (2.6) that (9*7) holds iff (9.8) 0 = X^kerH(l) = im GHker H. It remains to show that (9-8) holds iff (9-5) holds. Suppose im G O ker H = 0. Let m' := rank G = dim im G. Denote by [g g0, g ,) a basis of im G. We claim that Hg.., Hg, ..., Hg , x m m" x m are linearly independent. In fact, let ,Z,a.Hg. =0 for some a. Â£ k, j = TT1 0JO J 2,,..., m, not all zero. Then H .Z,Qhg. = 0, which implies Ill J X J J g.. = 0 since im G O ker H = 0. Now since g^, ..., g^, are linearly independent, we get a. = 0 for all j = 1, ..., m". Contradiction. Hence, J dim H im G = m'. 1, Conversely, suppose im G Oker H 4 (0j. Let 0 / e^ Â£ im G ^ ker H. Trivially e^ is a basis of Span e^ C im G O ker H. Extend e^ to form a basis (e^, e^, ..., e^,) of im G. Then (9-9) im HG = Span {He ..., He^,} = Span {He, ..., He , 2 m Therefore, dim im HG < m'. (9.10) COROLLARY. Let m = p = 1 and define polynomials Jl(z) and X(z) by il(z)/x(z) = H(zl F) ~*"G (5/ 0). Suppose the initial state is known. Then the system (F, G, H) is unknown-input observable at the final ON SOME STRUCTURAL PROPERTIES OF LINEAR CONTROL SYSTEMS REACHABILITY FROM W IN N AND CONTROLLABILITY TO W IN N By FUMIO HAMANO A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1979 10 and reach x W) o N (J + 1). Vie now study properties of the sequence W, X^e^ch(l), X^e^|ch(2), ... hy looking at some properties of the sequence X^. ^(i), i = 0, 1, ... . The sequence 7L. N(i), i = 0, 1, ... (equivalently, W, X^^ch(l), N (2), ...) has conditional monotone properties. (2.6) IEMMA. (i) If X^ N(i) C XW w(i + 1) for some integer i > 0, then X^ N(j) C X^. N(j + l) for all integer j > i. (ii) If X^ N(i) D X^j. ^(i + 1) for some integer i > 0, then Xw^N(j) D Xtf N(d + 1) for all integer > i. (iii) If Xw (i) = X^j N(i + 1) for some integer i > 0, then Xw>N(i) = X^N(i + ) for all = 0, 1, ... . PROOF, (i) Let X^N(i) CX^N(i + 1) for some integer Assume that X^^ N(s) C X^^ N(s + l) holds for some integer s > i. i > 0. Then X^N(s + 2) = (fx^jH(b + 1) + im G) On D (FXw>n(s) + im G)PlN = + 1). Therefore, hy induction V(3)cxw,n<3 + 1) for all integer > i. (ii) Similar to the proof of (i). (iii) Suppose X^ N(i) = X^ N(i + 1) for some integer i If we assume that X^ ^(s) = X^ ^(s + l) for some integer s > i, we have > 0. then CHAPTER VI. CONCLUSION We have introduced the r-etep reachable subspace X^e^C^(r) from W in N and the r-step controllable subspace X^.^r(r) to W in N of the finite-dimensional, constant, discrete-time, linear dynamical system (F, G, H) over a field k. We have characterized these subspaces and discovered several interesting properties pertaining to them. The notions X^e^ch(r). and X^^r(r) are natural generalizations of reachable and controllable subspaces when we are interested in structural properties of state trajectories contained in the subspace N and when initial and final states of the trajectories may not be zero. The significance of X^e^|ch(r) and 3^^tr(r) in control problems has been demonstrated in Chapters IV and V. Among the applications pre sented in these chapters, the unknown-input observability at the final time r is the most important in the sense that it has motivated the author to study the subjects treated here. As has been mentioned in the concluding remarks of each chapter, there are several other interesting topics concerning X^e^C^(r) which are still open to further research. The relation between the transfer function M^(zl F) and the subspaces X^e^ch(r), X^^r(r) should also be studied. 52 2 In this chapter we shall sometimes refer to a pair (F, G) with the continuous time interpretation for comparision. This is defined as follows: We choose the continuous time and set k to he either the set of real or complex numbers. Then we replace (0.1) by ( 0.1)' x(t) = Fx(t) + Gu(t), t > 0, in the definitions of the (discrete time) system (F, G, H) and the (discrete time) pair (F, G). Since continuous time systems are not what , we are interested in, unless otherwise specified the system (F, G, H) and the pair (F, G) will always be in discrete time. Our main concerns are placed in the structural properties of the trajectories governed by (O.l). Due to the algebraic nature of difference equations, these properties can be studied in a purely algebraic way without losing the intuition of the original dynamical nature of (O.l). At this stage it seems appropriate to give a quick review of related concepts which have been treated in the literature: (i) The reachable subspace (of the pair (F, G)), is defined to be the set of states which can be reached from the zero state (via some trajectories) in a finite number of steps. It is known that X**60,011 is equal to (0.3) im G + F im G + ... + Fn_1 im G. The pair (F, G) is said to be reachable if and only if xreac^ x. (ii) The controllable subspace (of the pair (F, G)), xcorrtr, is the set of states from which the zero state can be reached (via some trajectories) in a finite number of steps. It is characterized by (0.4) XCOntr = (Fn)_1 (im G + F im G + ... + Fn_1 im G) where (Fn) := {x X: Fn x Xg} for a subspace Xg of X. Note that we have a60,0*1 c XCOn^r in general. The pair (F, G) is called k shall denote it by V (N). J max' ' (v) Closely related to V (N) is the subspace called the maximal reachability subspace contained in N which is denoted by X~ (N). It is defined as follows: Let K: X -+U be a k-homomorphism such that (F + GK)V (n) CV (N) and let L: U -U be another k-homomorphism max max , satisfying im GL = im GOV (N). xXeacn(N), then, is defined to be max max the reachable subspace of the pair (F + GK, GL). It is evident that (F + GK)Xreach(N) c XreaC^1(N) if one uses (0.3) and the Cayley-Hamilton max max theorem. Consequently, X^fty (T) is an F mod G invariant subspace contained in N (or equivalently in V^y(lQ). Also, since = V (V (N)) trivially, it follows that X^^fa) = xrach(v (U)). max' max' 7' max ' max max' The definition of ]Coh(N) does not depend on whether the pair (F, G) represents a discrete time system or a continuous time system. However, there is an important difference in interpretation of X^ach(lQ. In the continuous time case can be interpreted as the set of states that can be reached from the zero state in a finite time via trajectories contained (at each time) in N. On the contrary this inter pretation fails in the discrete time case. This point has not been clarified in the literature, not to mention its importance. The above observation then raises the following questions: What is the set of states reach reachable from the zero state via trajectories contained in N, XI , in the (discrete time) system (F, G, H)? How is it characterized? Are there any interesting properties? A natural question which comes next is: What is then the set of states controllable to the zero state via trajectories in N? What are its properties? In Chapters II and III we shall attempt to answer the above questions in more general contexts. We shall introduce the following new notion in Chapter II. Let r be a positive integer, and let W and N be subspaces of X satisfying W C N. The r-step reachable subspace from W in N, denoted by resell ^ (r), is the set of states which can be reached from some states in W in r steps via trajectories contained (at each instant of time) 30 t = 0, r 1, and an initial state x G ker J such that the resulting x(t) G ker H, t = 1, . ., r and x(r) = x^. The set of states unknown-input indistinguishable from 0 at the final time r is a subspace of X. The unknown-innut unobservable subsoace at the final U0 (8.8) DEFINITION. time r, denoted as X^rj(r), is the set of states that are unknown- input indistinguishable from 0 at the final time r *1 x0 G X are unknown-input indistin- (8.9) DEFINITION. Two states x guishable at the final time r iff there exist pairs ((u+1)^_^, x^n) ((ut2 ^t-o* ^o2/ '~'J~ / '(^0 j = 1, 2 and initial ataees x(0) = x ({u+0)J~^, x 0) of input sequences u(t) = u, t = 0, ..., r 1; tl t=o ol 9 oj- 1, 2, such that the following conditions hold: (i) x^ is the final state (at the time r) corresponding to the Palr (fVw V5 where j =1? 2, (ii) the initially modified output sequences corresponding to the pairs coincide. The significance of ^grj(r) (8,10) PROPOSITION. Two states is clarified by xn, xv G X are unknown-input indistinguishable at the final time r iff (8.11) x^ x^g G ^erj(r) PROOF] [Necessity.] Let {v., y ., J -*-J ' yr.i! be the r-1 initially modified output sequence corresponding to the pair ((u , CJ oO x .) where 03 j = 1, 2. Suppose v1 = v2 and ytl = y t2J 1, r. Let the new initial state of (F, G, H) be x(0) x T x O* ol o2 Since vl v. we have 15 where ?r(B, A, C) is the matrix obtained by eliminating the first block row in fr(B, A, C) (hence, fr(B, A, C) consists of r block rows). From Lemma (2.6) it follows that X^. ^(i) C! X^ ^(i + l), i = 0, 1, ..., iff (4.2) wcx^N(i) and that X^N(i) d ^ jjC1 + 1)* 1 = > 1> > iff (4.3) WDX^n(1). The natural question is then what is the significance of the conditions (4.2) and (4.3). The significance of (4.2) will be partially answered in Chapter III, Section 5. However, the implication of (4.3) is yet to be clarified. There are other questions to ask: when does the sequence X^ ^ (i), 1, 2, ... "oscillate", i.e., X^^ch(j) 4 X^JJCh(j + l) fr every j > 0?; when does the sequence "oscillate" at the outset and stop its oscillation some time later?; etc. These are open problems. * 29 and y(t) Â£ Y, t = 0, 1, respectively. As a result of (8.5) we also have v(x ) = Jx_ which we call v . o o o Considering the initial state, we have (8.10 Let *(x(0)) 7n J y(x(o)) H x(0), V(x(0)) T . J y(x(o)) > 0 H_ v(O) :: For each r = 1, 2, ... the sequence {v(0), y(l), ..., y(r)) is called an initially modified output sequence (till time r). When we discuss two initially modified output sequences we prefer to denote them by (v. y ,} and {v oi> JU) *riJ 1 02> J,12 * Jr2 (8.5)PROBLEM. Given an initially modified output sequence (v(o), y(l), ..., y(r)) of (F, G, H), find the corresponding final state x(r) Â£ X for r = 1, 2, ... . Note that the definition of v(0) and J and (8.4) clearly implies (8.6) ker J C ker H. (See Section 11 for the remark on the condition (8.6).) We say that the system (F, G, H) is unknown-input observable at the final time r iff Problem (8.5) has a unique solution. It is not always the case that this problem has a unique solution. In fact, it will be seen that rank HG = rank G is necessary for the final state x(r) to be uniquely determinable. (See Theorem (9-4).) So the important question is "to what extent can we recover x(r)?" (8.7) DEFINITION. x# Â£ X is unknown input indistinguishable from 0 at the final time r iff there exist an input sequence u(t) Â£ U, 43 (10.14) fr(ln, Je, H)(u^) = fr(ln, Je, H)(r + ) (for some u C ker JqI H)) f (I J H)(u ). r' n e' r The linearity and the uniqueness can he shown similarly to Lemma (8.24). By Proposition (10.8) (or the remark after the proposition) and Theorem (10.12), we get the necessary and sufficient condition for (F, G, H) to he unknown input initial state observable at the time r. (10.15) THEOREM. The system (F, G, H) is unknown input initial state observable at the time r iff <*> D (10.17) REMARK. If r > n and if Jg = H, then (10.16) is equivalent to (10.18) V (ker H) = 0, ' max' * where vmax(ker H) the raaxiraal F mod G invariant subspace in ker H. 11. Concluding Remarks If one does not go through the arguments at the beginnings of Sections 8 and 10, i.e., if one does not want such conditions as (8.6) and (10.4), one can easily accomodate the discussion in this chapter to the new situations. The modified versions of X^e^ch(r) and X^^r(r) (without the assumption of WCl) should be used accordingly. (See Section 4 of Chapter II and Section 7 of Chapter III.) Techniques similar to those used in this chapter can be applied to study unknown input observability problems in non-constant dynamical systems (F(t), G(t), H(t)). (The problem statements (8.5) and (IO.5) should be modified in the obvious ways.) The unknown input unobservable uo subspace X (s, s + r) at the final time based on the observation over the 32 is a linear space over k where addition and scalar multi plication are defined in the obvious ways. X? T(r) is characterized as follows: kerJv ' (8.12) THEOREM. (8.) *SrJM - where xJerifkerH^ is defined by (2.1) and-(2.2). PROOF. Immediate by using Definitions (2.1) and (2.2) and Definitions (8.7) and (8.8). Proposition (8.10) tells us that to recover x(r) we cannot do IK) better than identifying the equivalence class x(r) + ^erj(r)* Then the question is, "can we really find the equivalence class containing x(r)?" (8.14) THEOREM. JJO If x(r) is the final state at the time r, then x(r) + Xjcerj(r) can be Uniquely determined based on the knowledge of the corresponding initially modified output sequence (v(0), y(l), ..., y(r)}. PROOF. Given the initially modified output sequence (v(0), y(l), ..., y(r)} of (F, G, H). Choose an xq X satisfying (8.15) v(0) = Jxq, e.g., on a fixed coordinate basis, xq may be taken to be (8.16) Xo := J+v(0) where J* is the pseudo-inverse of J. (See Appendix for the definition of the pseudo-inverse.) Then the initial state x(0) is written as (8.17) x(0) = xq + x^ 56 - PSr(Bkerj) (uri) + pgr^BkerJ^ ^Ur2^ = ^rJ(yrl)+ cer J\ r r r r r (yr) To show the uniqueness of the map suppose $^erJ: r^ker J) - x/gr(Bkerj) ker fr(B^erj^ H) is another map for which the diagram commutes. Then for each u Â£ (ker j) r r $cerJf (B H h) (u ) = pg (B. _) (u ) rr, r' kerJ v r r kerJ r ,kerJ = erJfr(W H (8.25) follows from Theorems (3.1) and (8.12). (8.29) REMARK. ^erJ: ^(ker J) ->X/XÂ¡^rJ(r) does not depend on the choice of B^erj. (8.30) REMARK. For a fixed coordinate basis ^erJ; Tr(ker J) -> X/X^rj(r) &y defined by (8.31) iff" := P^OW,) H> K> j* where **r(Bkerj> H> H) is the pseudo-inverse of fr(Bkerj> H). (See Appendix.) As an immediate consequence of Proposition (8.10) and Theorems (8.12) and (8.lU) we have (8.32)THEOREM. A system (F, G, H) is unknown-input observable at the final time r iff (8-35) rH=0- the initial state x(0). (13.2) THEOREM. Let x^ G X. Then there exists a state feedback K: X -U for which there is an integer i > 0 such that y(t) = 0, t > i for x(0) = x* where y(t) is the output of the system (F + GK, 0, H) due to the initial state x^ iff (13.3) rcontr (n) where V (ker H) is defined by (5.14). max PROOF. ["if"] Since V (ker H) is an F mod G invariant sub max space in N, by Theorem (5.19) there is a feedback K^: X -4 U such that (F + x(n) C Vmax(ker H). This implies that, if x(0) = x#, then x(l) = (F + GK)x# x(2) = (F + GK)x(l) = (F + GK)2x^ x(n) = (F + GK)x(n l) = ... = (F + GK)nx G V (ker H). ^ max If K is so chosen as to satisfy (F + GK) V (ker H) C V (ker H) max (see the proof of Theorem (5.19))* we have x(t) = (F + GK) nx(n) G V (ker H) max for all t > n. Thus, y(t) = Hx(t) = 0 for all t > n. ["only if"] Let x(t) 6 X, t = 0, 1, ..., be the trajectory of the system (F + GK, 0, H) with the initial state x(0) = x#. Then by assumption we have for some integer i > 0 x(i + j) G ker H, j = 0, ..., n. Therefore, by Corollary (5.I5) we know x(i) G V (ker H), which implies max that x(0) = x^ G v^(vmax(ker x) "by Definition (5.2) and (5.3). Appealing to Proposition (5.17) we conclude 3 (in the continuous time case control1able if and only if xCon^r = X. and Xcontr can he defined similarly. The characterization of yContr , however, is different from the one in the discrete time case, corrtr namely, we have for the continuous time case X = im G + F im G + + F^1 im G = Xreach.) unob (iii) The unobservahle subspace, xww, is the set of initial states which can not be distinguished from the zero state by any input/output experiment. The subspace X^0^ is given by / vunob . (0.5) X = ker H HF The system (F, G, H) (or simply the pair (H, F)) is said to be observable if and only if X^0*5 = 0. The condition X^0*5 = 0 is necessary and sufficient for the initial state x(0) to be uniquely determined based on a sufficiently long interval of input/output measurement. For more details about reachability, controllability and observability the reader should refer to KAIMAN [1968], FURUTA [1973, Chapter 2, Section 7 through 10], MARRO [1975, Chapter 6] and WONHAM [1974, Chapters 1, 2 and 3]. (iv) A subspace V is called an F mod G invariant subspace if and only if (0.6) FVCV + imG. It is well-known that (0.6) holds if and only if there exists a k-homomorphism ("state feedback") K: X -U satisfying (0.7) (F + gk)v C V. The set of F mod G invariant subspaces is closed under subspace addition. Therefore, for a given subspace N of X there is an F mod G invariant subspace which contains any other F mod G invariant subspace. This is called the maximal F mod G invariant subspace contained in R, and we 24 (502) (F + GK )Je <2 N, J = 0, I Sj* hold (533) REMARK. The choice of K^: X ->U (r > 0) is by no means unique. This (limited) freedom in choosing K^ (r >0) is rather useful in application. (See Chapter V.) r 1; = 1, (5.34)REMARK. The inputs or r) (but not necessarily uniquely). See Remark (6.3). j = 0, u. s (s. = 1. 0sj I based on which K^ is defined, can be determined explicitly Recall the definition of X^e^C^(r). Given x GW, o there may not , r be any input sequence producing x(0) = x x(t) Â£ I, t = 1, reach ^ and x(r) Â£ ^ ^ (r), r > 0. The condition that x^ must satisfy to have such an input sequence is as follows. Let xq Â£ W. Then there is an input sequence u(t) Â£ U, t = 0, ..., r 1 such that the state xq, x(t) G N, t = 1, .. iff trajectory satisfies x(0) r and x(r) G 1, N , 0 ,rreach/ \ Yhere X := (r). We now turn to the question of what is the implication of (4.2) which has been posed at the end of Chapter II. co nt Y* / (5.35)LEMMA. Let W(i) (i), i > 0. Then for each integer i > 0, (4.2) holds iff (5.36)W vreach ,. N PROOF. Suppose (b.2) holds. Trivially, W !) ^(i). To show the converse inclusion let w G W. Then by (4.2) there exist x^ G W, ux G U such that w = Fx^ + Gu^ G N. Since w G W we.obtain w = Fx^ + Gu^ G W. Repeating the above argument i times, we see that there also and "feedback properties of the above notions are first given, and then applications to the unknown-input observability and other problems in control systems are given to show the significance of the results. vi 19 happens for some i, j = 0, 1, ..., the sequence (i), Â£=0,1, ... will stop increasing or decreasing in a finite number of steps since X is finite dimensional. If it is the case, let |i be the least integer i > 0 such that X^ i.) = X^. ^(i + l). (5.9) LEMMA, (i) Lf WCX^jj(l), then p < dim N dim W < n. (ii) _If Â¥DX^^(l), then M < dim W < n. PROOF. Immediate from Lemma (5-7) and the above comment. (5*10) LEMMA. Let p be as in the paragraph preceeding Lemma (7.9)* Then (5-H) c y N(n) + im G. (5.12) yH(p)cN. So X^fr(p) is an F mod G invariant subspace in N. PROOF. By the definition of p, X^. ^(p) = ^(p + l). Therefore + = F_1(XW (M) + im G)ON. Hence, Ry(n) + im G, yN(p) Cl. n (5.13) REMARK. Lemma (5.IO) guarantees the existence of a feedback K: X -U such that (F + GK) X^JJtr(p) CX^JJtr(p). (See Lemma (1.4).) The set of F mod G invariant subspaces is closed under subspace addition. (See BASILE and MARRO [1968a, Section 2, Assertion 1] and WONHAM [1974, Lemma (5-3)]*) Therefore, the following is well defined: (5.14) V (N) : = max[V C N: FV C V + im G]. max 56 M. K. SAIN and J. L. MASSEY [1969] "invertibility of linear time-invariant dynamical systems", TWER Trans. Auto. Control, AC-14: 141-149. L. M. SILVERMAN [1976] "Discrete Reccati equations: alternative algorithms, asymptotic properties and system theoretic interpretations", in Control and Dynamical Systems: Advances in Theory and Applications, Vol. 12 (edited by C. T. Leondes), Academic Press, pages 513-586. L. M. SILVERMAN and H. J. PAYNE [1971] "Input-output structure of linear systems with application to the decoupling problem", SIAM J. Control, 9: 199-233. E. D. SONTAG [1979] "On the observability of polynomial systems, I: finite-time problems", SIAM J. Control and Optimization, 17: 139-151. L. WEISS and R. E. KAIMAN [1965] "Contributions to linear system theory", Int. J. Engineering Science, J: 141-171. W. M. WONHAM [1974] Linear Multivariable Control: A.Geometric Approach,. Springer, New York. .. - W. M. WONHAM and A. S. MORSE [1970] "Decoupling and pole assignment in linear multivariable systems: A geometric approach", AM J. Control,"8f1-18. and fr(*, , Chapter III. ) is as in Section 3 of Chapter II Define ft (I ) := kn+rni and p (ker r n r and Section 6 of J ) := im f (I J H). e r n e (lO.ll) THEOREM. There is a unique homomorphism ithat the following diagram commutes: >We(r) ft (I ) r n f (I J H) rv n e T (ker J ) rx e P r X cr I *r -ku erJe (r) where I is the n x n identity matrix (or the identity map kn ->kn), ft IU f (, , ) and P are as in Sections 3 and 5, cr: X X/X. (r) r r kerJe is the canonical projection. PROOF. By Theorems (6.1) and (10.6) we have i10-11 Cje(r) Pr ker fr(l Je H>' Let y G T (ker J ). There is an u G ft (I ) such that r rk e r rs n (10.12) yr = fr(ln, Je, H)(ur). Define ^(ker Jg) ->X/X^rJe(r) = X/Pp ker fr(ln, J,, H) by (10.13) *r(yr) = aPrr. We claim that is well-defined. In fact, let G ^r^n^ be anot':ier vector satisfying (10.12). Then ' G ker f (I J H). r r r n e Therefore, 55 K. FURUTA f1975] Senkei Shistem Seigyo Riron (Theory on Linear Systems and Control), Shohkodoh, Tokyo. F. HAMANO and K. FTJRUTA [1975] "Localization of disturbance and output decomposition in decentralized linear multi-variable systems", Int. J. Control, 22: 551-562* B. HARTLEY and T. 0. HAWKES [1970] Rings, Modules and Linear Algebra, Chapman and Hall, London. D. G. LUENBERGER [1966] "Observers for multivariable systems", IEEE Trans. Aut. Control, AC-11: 190-197* R. E. KALMAN [1963] "Mathematical description of linear dynamical systems", SIAM J. Control, 1: 152-192. [1968] "Lectures on controllability and observability", Proc. C.I.M.E. Summer School, Edlzioni, Cremonese, Roma, 1-149* R. E. KALMAN, P. L. FALB and M. A. ARBIB [1969] Topics in Mathematical System Theory, McGraw-Hill, New York. R. LASCHI and G. MARRO [1969) "Alcune considerazioni sull'osservabilita del sistemi dinamici con ingressi inaccesibility", Rendiconti Della LXX Riunione Annuale AEI. G. MARRO [1975] Fondamenti di Theoria dei System!, Patron, Bologna. B. C. MOORE and A. LAtJB [1978] "Computation of supremal (A, B)-invariant and controllability subspaces", IEEE Trans. Auto. Control, AC-23: 783-792. (l4.4) THEOREM. The r-step disturbance decoupling problem has a solu tion iff (111.5) m D C \erH>kerH(r !) PROOF. [Sufficiency] By assumption x(0) = 0. Let K X -i be as in Theorem (5.19 ii) where we assume W = N = ker H. Replacing u(t) in (l4.l) by K ^(t), we get (14.6) x(t + 1) = (F + GKr_1)x(t) + Dv(t). g Consider the system (14.6), (l4.2). Let v C k j = 0, 1, ... For ti each j = 0, ..., r 1 suppose the disturbance v(t), t = 0, 1, ..., be such that v(j) = v.(^ 0 possibly) and that v(t) =0 if t / j. Then we have *t = Oj y J } x(t) = (F + GK )t"1Dv., t = j + 1, j + 2, ... . Therefore, by Theorem (5*19 ii) with W and N both replaced by ker H, y_j(t) := Hx(t) =0, t = 0, ..., r + j. Now we superpose all the disturbance used above, i.e., we use v(t) such that v(t) = for t = 0, .... r 1. Then the output y(t), t = 0, 1, ..., satisfies r-1 y(t) = X0 y^(t) = o, t = o, ..., r. [Necessity] Suppose that the problem has a solution but that im D ^kgyHrker{j(r i)* Since x(0) = 0, we have x(l) Dv(0). There must be v kS such that Dv X^n^.r. (r l). Then by o o K6rii) Kern Definitions (2.2), (2.3) there does not exist an input sequence u(t) f U, t = 1, 2, ..., such that the corresponding state trajectory 37 9. Unknown Input Final State ObservabilityPart 2 (Special Cases). Let r = 1, 2, ... Recall Theorem (8.12). The unknown input un- Tin observable subspace X^rJ(r) at the final time r depends on the time r in general. However, in some cases the time dependence disappears in a finite time. (See Lemma (2.6) and the paragraph prior to Lemma (2.8).) And if (9.1) ker J C (F ker J + im G) Oker H holds, more can be said, namely, the sequence X^rJ(i), i 1, 2, ..., is monotonically nondecreasing (Lemma (2.6 i)) as well as it stops increasing in at most n steps, i.e., (9'2) O1') X2rJ(n + 1) = (See Lemma (2.8 i).) Also, if (9.3) ker J 3 (F ker J + im G) Oker H uo is true, then the sequence xkerj(i) i = 1, 2, ..., is monotonically nonincreasing (Lemma (2.6 ii)) and (9-2) holds. (See Lemma (2.8 ii).) Note that if ker J = 0, i.e., the initial state is known a priori, then (9.1) holds, and that if ker J = ker H, i.e., if the initial state is not known a priori, then (9.3) holds. Therefore, for these cases the above statements are true. Let us consider the first situation where ker J = 0 (i.e., the initial state is a priori known). The condition under which Problem (8.5) has a unique solution becomes particularly..simple. (9-4) THEOREM. If ker J 0, then the system (F, G, H) is unknown input observable at the final time r iff (9*5) dim im HG = dim im G, 40 Denote v (r)~ e , J : J e y(r) e H How we shall call the sequence (y(0), y(r l), v (r)) r-modified e output sequence. Consequently, we have (10.3) PROBLEM. Given an r-modified output sequence (y(0), ..., y(r l), vg(r)), find the corresponding initial state x(0). By (10.2), clearly, (10.4) ker J C ker H. e (See the remark on this condition in Section 11.) We say that the system (F, G, H) is unknown-input initial state observable at the time r iff Problem (IO.3) has a unique solution. A state x# 6 ker H is said to be unknown-input initial state indis tinguishable from ,0 at the time r iff there is an input sequence u(t) U, t = 0, 1, ..., r 1 such that the corresponding state sequence satisfies x(0). x#, x(t) ker H, t = 0, ..., r 1 and x(r) ker Je. The set of states that are unknown-input initial state indistinguishable from 0 at the time r is a subspace of X. (10.5)DEFINITION. The unknown-input initial state unobservable subsuace at the time r, denoted by ^erje(r) j is the set of states that are unknown jnput initial state indistinguishable' from 0 at the time r. i From Definitions (10.5) and (5.3) we obtain 0.0.6) IHEOEM. CeAerHM- Recall the results given in Chapter HI. We know the characterizations and the properties of ^^)!terH(r). The following result given by SILVERMAN [1976, Section III, A, the Jx(0) = Jx i.e., x(0) Â£ ker J. 01 JXo2 T1 V2 = 0 Apply the input u(t) = u u t2 0, (F, G, H). Then it is straightforward to show 1 to the system x(r) x*p x*2 y(-t) = ytl yt2 = o, t = o, ..., r. Therefore, x^ x^2 Â£ X^rJ(r). [Sufficiency.] Suppose (8.11) holds. Write x^ := x1 x0. Then clearly x^ Â£ j(r). Therefore, there exist an input sequence (u,), and an initial state x Â£ ker J of (F, G, H) such that the t t=o o 7 corresponding trajectory satisfies x(t) Â£ ker H, t = 0, r and x(r) = x#. Define a new input sequence and a new initial state of (F, G, H) by u(t) := ut2 + u t = 0, ..., r 1 and x(0) := Xq2 + xq where I* 1 xq2 and ^ut2^t-o are state and input sequence, respectively, giving rise to the final state x*2. It is easy to show that v(0) = Jx(0) = Jxq2, y(t) = yt2> t = 0, ..., r, x(r) = x^2 + x* = x#1, where fy^^t-o 0U^PU^ sequence produced by {u^2}^_q and xq2. It follows from Proposition (8.10) that (8.1l) defines equivalence classes, each of which consists of the states unknown-input indistinguishable at the final time r and the collections of these classes are denoted by x/x UO kerJ (r). (in which is replaced by BkgrJ and by H). Note that y(0) = 0 since x(0) G ker J C ker H. The corresponding final state x(r) G X is obtained by (8.25) x(r) = gr(\erJ) u(0) u(r 1) where r) is defined in Section 3 (Chapter II). ,U0 M^erJ' Our problem is to determine x(r) 4 X^rj(r) from the knowledge of y(l), ..., y(r). (v(0) = Jx(0) = 0 since x(0) G ker J. Also y(0) 0.) Define i i u u r-1 _ n 4rm t n m A ,, n > G k : | G k u. fe k j = 0, ...,,r lj 0 where n' := dim ker J and B, is a basis matrix of ker J. Write kerJ T (ker J) := im f (B, t, H, H). r' rv kerJ* 1 (8.24) LEMMA. [There is ker VW a unique homomorphism ^er^: r^ker J) -> H, H) such that the following diagram commutes: (B, t) rv kerJ kerJ* H, H) X P ry(ker J) ! /erj ker VW H, H) where ?T('> '> *) and g^() are as in Section 3 and p: X - X/gr(Bkerj) ker fr(BkerJ> H, H) is the canonical projection. which means that the output y(t), t = 0, 1, ... of the system (F + GK, 0, D, H) with x(o) =0 is zero for all t = 0, 1, ... . It remains to show that there is an integer i > 0 satisfying (F + GK)1 = 0. If (4.10) holds, then X = X^Jtr(n) C X^Jtr(n) for any subspace W of X. So X = XTS^r(n) where W := X^.n^rTT(n). , W,X o.kerH' Noticing that 0 and X w(n) are F mod G invariant subspaces and O j K6!Tri repeatedly using Theorem (5*19 i)> we see that there is a state feedback K: X ~>U satisfying (l4.ll), (14.12) and CL.,.11,) (f + gk)"xcxÂ£>). Hence, (14.15) (F + GK)2nX = 0. [Necessity] Let K: X -> U be a solution of the problem. Then clearly (F + GK)n = 0. Hence XC^r(n) = X. Now define O ) . *1 V := im D + (F + GK) im D + ... + (F + GK) im D. Then clearly im D C V, (F + GK)V C V and V C ker H. Since (F + GK)n we have (F + GK)^ = 0. Therefore, V C xcon"^r . O KG^Tii rcontr = 0, Thus im D C V C x;;kÂ¡;H(n)- D (l4.l6) REMARK. Recall (0.4). We have X^Jtr(n) = xCOntr. 15 Concluding Remarks There is an important difference between Xre^C^(n) and Xreac'1(N). o,N v max v By choosing K: X ->U the spectrum of (F + GK)|xreac (n) can be assigned T13.X !TG3C]l arbitrarily, while to discuss the spectrum of (F + GK)Â¡X^ w (n) may not make sense since (F + GK) Xq (n) C XQ (n) may not hold (unless ) y X*G9/0l^l N is an F mod G invariant subspace). However, though X. (r) loses contr w,N the arbitrary pole assignability, X^ (r) can treat stabilizability by state feedback (Theorem (l2.l)). The usefulness of Theorem (12.1) is yet to be clarified. Fx* + Gu V. (iii) There exists a k-homomorphism K: X -+U ("state feedback") such that (1.5) (f + gk)vcv. PROOF. The proof for the equivalence between (i) and (iii) can be found in BASILE and MARRO [1968a, Theorem 3], WONHAM and MORSE [1970. Lemma (3.2)] and WONHAM [1974, Lemma (4.2)]. For convenience we shall give the proof here. Suppose that (1.5) holds. Let x* V. Then (F + GK)x# = Fx* + GKx* V. So Fx* = v + GKx* for some v V, which implies (1.3). Conversely, assume that (1.3) holds. Let {v., ..., v ) be x Si a basis of V where q := dim V. Then (1.3) implies that for each i 1, ..., q there exist w^ V and u^ Â£ U satisfying Fv^ = w^ + Gu^. Now define a k-homomorphism K: X -U by Kv^ = -u^, i = 1, ..., q. Then we have (F + GK)v^ = w^ Â£ V, i = 1, ..., q, which implies (1.5). The equivalence of (i) and (ii) is easily proved and omitted here. 2. Reachability from W in N We consider a finite dimensional, constant, discrete time, linear dynamical system (F, G) given by (l.l). From here on, W and N will be subspaces of X satisfying W C N C X, and r will be a positive integer. (2.1) DEFINITION. A state x# N is r-step reachable from W in N iff there exist another state xq^ W and an input sequence u(t) U, t = 0, ..., r 1 such that x(0) = xq^, x(t) N, t = 0, ..., r and x(r) = x#. The set of states x^'s satisfying the above requirement form a (linear) subspace. |