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Realization of invariant system descriptions from Markov sequences

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Realization of invariant system descriptions from Markov sequences
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Algebra ( jstor )
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Covariance ( jstor )
Factorization ( jstor )
Integers ( jstor )
Linear systems ( jstor )
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Matrices ( jstor )
Stochastic models ( jstor )
Transfer functions ( jstor )
Dissertations, Academic -- Electrical Engineering -- UF
Electrical Engineering thesis Ph. D
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Thesis--University of Florida.
Bibliography:
Bibliography: leaves 114-123.
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Typescript.
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Vita.
Statement of Responsibility:
by James Vincent Candy.

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REALIZATION OF INVARIANT SYSTEM DESCRIPTIONS FROM MARKOV SEQUENCES


By

JAMES VINCENT CANDY


A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL
OF THE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR TH
DEGREE OF DOCTOR OF PHILOSOPHY


UNIVERSITY OF FLORIDA


1976





























To my wife, Patricia,and daughter, Kirstin,for unending faith,

encouragement and understanding. To my mother, Anne, for her constant

support and my mother-in-law, Ruth, for her encouragement. To "big" Ed,

my father-in-law, whose sense of humor often lifted my sometimes low

spirit.


:' .. ~'~~-~P1"*-~qnq8-.s~-rrarrm~mtin*a-s--- --il----~---~i~a~rCI~AI~~-I- -. --~-














ACKNOWLEDGMENTS


I would like to express my sincere appreciation to the members

of my supervisory committee: Dr. Thomas E. Bullock, Chairman, and

Dr. Michael E. Warren, Cochairman, Dr. Donald G. Childers,

Dr. Z.R. Pop-Stojanovic and Dr. V.M. Popov. A special thanks to

Dr. Thomas E. Bullock and Dr. Michael E. Warren for their constant

encouragement, unending patience, and invaluable suggestions in the

course of this research.

I would also like to thank my fellow students and friends,

Zuonhua Luo, Arun Majumdar, Jose DeQueiroz, and Jaime Roman, for

many fruitful discussions and suggestions.


-- iii
















TABLE OF CONTENTS


ACKNOWLEDGMENTS ........................................... iii

LIST OF SYMBOLS .......... .................. ... ....... ... vi

ABSTRACT .............................. ..... ..... ............. vii

CHAPTER 1: INTRODUCTION ................................. ...... 1

1.1 Survey of Previous Work in Canonical Forms
for Linear Systems ............................ 2
1.2 Survey of Previous Work in Realization Theory.. 5
1.3 Purpose and Chapter Outline ................... 10
1.4 Notation ..................................... 11

CHAPTER 2: REALIZATION OF INVARIANT SYSTEM DESCRIPTIONS ...... 12

2.1 Realization Theory ............................ 12
2.2 Invariant System Descriptions ................. 18
2.3 Canonical Realization Theory .................. 33
2.4 Some New Realization Algorithms .............. 45

CHAPTER 3: PARTIAL REALIZATIONS ............................. 54

3.1 Nested Algorithm ............................. 54
3.2 Minimal Extension Sequences ................... 64
3.3 Characteristic Polynomial Determination by
Coordinate Transformation ..................... 74

CHAPTER 4: STOCHASTIC REALIZATION VIA INVARIANT SYSTEM
DESCRIPTIONS ...................................... 78

4.1 Stochastic Realization Theory ................ 81
4.2 Invariant System Description of the
Stochastic Realization ........................ 87
4.3 Stochastic Realizations Via Trial and Error ... 97
4.4 Stochastic Realization Via the Kalman Filter .. 104

CHAPTER 5: CONCLUSIONS ....................................... 111

5.1 Summary ....................................... 111
5.2 Suggestions for Future Research ............... 112


~1_1__1 __~~__ j











TABLE OF CONTENTS (Continued)


REFERENCES ................................................ 114

BIOGRAPHICAL SKETCH ................................. ....... 124


_ ____r___ll___ Cr ~_I_ _~__ 1__1_1 _















LIST OF MATHEMATICAL SYMBOLS


Usage

A aT
A-1
A-T
p(A)
IA| or det A
diag A
xty
x>y
XcY


xeX
X-+Y
X:=
xoy
{.)


dim X


XCA).
/x
max(.)
Z
K


Meaning


First Usage


Symbol
T
-1
-T

P
I I


Transpose of A, a
Inverse of A
Inverse of AT
Rank of A
Determinant of A
Diagonal elements of A
x is not equal to y
x is greater than y
X is contained in or a
subset of Y
x is an element of X
Map (set X into set Y)
x is defined by
Abstract group operation
Sequence or set with
elements
Summation
Infinity
Footnote
End of proof
Group action operator
Dimension of vector
space X
if and only if
Eigenvalues ,of A
Square root of x
Maximum value of
Positive integers
Field


pg.
pg.
pg.
pg.
pg.
pg.
pg.
pg.
pg.


pg.
pg.
pg.
pg.
pg.


pg.
pg.
pg.
pg.
pg.
pg.


pg.
pg.
pg.
pg.
pg.
pg.


13
13
89
13
30,21
103
30
16
18


12
20
13
19
13


13
13
2
34
21
15


14
96
99
23
12
12


iff
X
/


~~1~1_ _I_ I









Abstract of Dissertation Presented to the Graduate Council
of the university of Florida in Partial Fulfillment of
the Requirements for the Degree of Doctor of Philosophy


REALIZATION OF INVARIANT SYSTEM DESCRIPTIONS
FROM MARKOV SEQUENCES

By

James Vincent Candy

March, 1976

Chairman: Dr. Thomas E. Bullock
Cochairman: Dr. Michael E. Warren
Major Department: Electrical Engineering

The realization of infinite and finite Markov sequences for multi-

dimensional systems is considered, and an efficient algorithm to extract

the invariants of the sequence under a change of basis in the state

space is developed. Knowledge of these invariants enables the deter-

mination of the corresponding canonical form, and an invariant system

description under this transformation group. For the partial realization

problem, it is shown that this algorithm possesses some attractive

nesting properties. If the realization is not unique, the class of

all possible solutions is found.

The stochastic version of the realization problem is also examined.

It is shown that the transformation group which must be considered is

richer than the general linear group of the deterministic problem. The

invariants under this group are specified and it is shown that they can

be determined from a realization of the measurement covariance sequence.

Knowledge of these invariants is sufficient to specify an invariant

system description for the stochastic problem. The link between the









realization from the measurement covariance sequence, the white noise

model and the steady state Kalman filter is established.


viii















CHAPTER 1

INTRODUCTION

Special state space representations of linear dynamic systems have

long been the motivation for extensive research. These models are

generally used to simplify a problem, such as pole placement, by

introducing arrays which require the fewest number of parameters while

exhibiting the most pertinent information. In general, system represen-

tations have been studied in literature as the problem of determining

canonical forms; Canonical forms have been used in observer design,

exact model matching methods, feedback system design, and Kalman filtering

techniques. In realization theory, canonical forms for linear multi-

variable systems are important. Since it is only possible to model a

system within an equivalence class, the ability to characterize the class

by a unique element is beneficial.

The problem of determining a canonical form has its roots in

invariant theory. Over the past decade many so-called "canonical" system

representations have evolved in the literature, but unfortunately these

representations were obtained from a particular application or from

computational considerations and not derived from the invariant theory

point of view. Generally, these representations are not even unique and

therefore cannot be called a canonical form. Representations derived

in this manner have generally been a source of confusion as evidenced by

the ambiguity surrounding the word canonical itself. In this dissertation









we follow an algebraic procedure to obtain unique system representations,

i.e., we insure that these representations dre in fact canonical forms.

In simple terms this approach seeks the determination of certain entities

called invariants obtained by applying particular transformation groups

*(e.g., change of basis in the state space) to a well-defined set

representing a system parameterization. The invariants are the basic

structural elements of a system which do not change under this trans-

formation and are used to specify the corresponding canonical form. This

approach insures that the ambiguities prevalent in earlier work are

removed. Initially, we develop a simple solution to the problem of

determining a state space model from the unit pulse response of a given

linear system and then extend these results to the stochastic case where

the system is driven by a random input. The technique developed to

extract the invariants from this (response) sequence not only provides

a simple solution to the realization problem, but also gives more insight

into the system structure.


1.1 Survey of Previous Work in Canonical Forms for Linear Systems

The study of canonical forms for linear dynamic systems evolved

slowly in the sixties. The main impetus of investigation was initiated

by Kalman (1962,1963) when he compared two different methods for describing

linear dynamic systems: (1) internally by the state space representation

denoted by the triple (F,G,H), or (2) externally by the transfer

function--the input/output description. Development over the past

decade in such areas as optimal control, decoupling theory, estimation

and filtering, identification theory, etc., have relied heavily on the

tThis defines (simply) the realization problem.


j









state space representation for analysis and design. In early literature,

however, transfer function representations were used. For highly

complex systems it is much easier to determine external behavior rather

than internal,since the state variables are normally not available for

measurement. As pointed out by Kalman (1963) the language of these

representations may be different, but both describe the same problems and

are related. Many researchers have investigated the relationship between

both representations, but always with one common goal--to obtain a

state space model which specifies the external description directly by

inspection. Kalman (1963) and later Johnson and Wonham (1964),

Silverman (1966) have shown that there exists a canonical form (under

change of basis in the state space) in the scalar case for the triple

(F,g,h) where F is in companion matrix form (see Hoffman and Kunze (1971))
and g is a unit column vector. It was shown that there exists a one to

one correspondence between the non-zero/non-unit elements of the triple

and the transfer function. This representation was used by Bass and Gura

(1965) to solve the pole-positioning problem and recently by Wolovich

(1972b)in solving the exact model matching problem.

The progress in determining a canonical form for the internal

description of multivariable systems came more slowly. The earliest work

appears to be that of Langenhop (1964) in which he develops a representation

to study system stability. Brunovsky (1966,1970) was probably the first

to recognize the invariant properties of the canonical form for the

controllable pair (F,G). Tuel (1966,1967) developed canonical forms for

multivariable systems in his investigation of the quadratic optimization

problem. Subsequently, Luenberger (1967) proposed certain sets of

canonical forms for controllable pairs; however, his development allowed









the possibility of nonuniqueness of these representations. Bucy (1968)

extended the results of Langenhop and Luenberger when he developed a

canonical form for certain subclasses of observable systems, but he

too was unaware of its invariant properties. Proceeding from the

external system description many researchers began to realize the

usefulness in the development of canonical forms. Popov (1969) developed

a canonical form for the transfer function in his investigation of

irreducible system representations. Gilbert (1969) examined the invariant

properties of a system with feedback applied to solve the decoupling

problem. Dickinson et al. (1974a) discuss the construction and appli-

cation of these canonical forms for the transfer function matrix in a

recent survey. The properties of canonical forms were not fully

understood initially. In fact, the basic question of their uniqueness

posed many doubts as to their usefulness. This issue wasn't resolved

until the work of Rosenbrock, Kalman, and Popov in the early seventies.

The properties of the Luenberger forms were clarified by the

results of Rosenbrock (1970) and Kalman (1971a) in their studies of the

minimal column indices (or Kronecker indices) of the matrix pencil

[Iz-F,G], or more commonly, the indices of the pair (F,G). These indices

were shown to be invariants under the following transformations: change

of basis in the state space, input change of basis, and state feedback.

These results precisely resolve the question of what can (or cannot)

be altered by applying feedback to a linear multivariable system. At

the same time Popov (1972) examined the properties of the controllable

pair (F,G) under the same transformations in a very precise, step-by-step,

algebraic procedure todetermine the corresponding invariants. He shows

clearly that obtaining the invariants under a particular transformation









group is the only information required to specify the corresponding

canonical form. Wonham and Morse (1972) obtained the feedback invariants

of the controllable pair from the not as lucid geometric viewpoint.

Their results were identical to those of Brunovsky and-4osenbrock.

Morse (1973) examined the invariants of the triple (F,G,H) under a lars

group of transformations which includes output change of basis. A

complete set of feedback invariants of this triple still remains an

open problem, but some fragmentary results were presented by Wang and

Davison (1972) when they investigated certain sets of restricted triples.

Along these lines Rissanen (1974), Caines and Rissanen (1974), .

Mayne (1972a,b),Weinert and Anton (1972), Tse and Weinert (1973,1975),

Glover and Willems' (1974) examined the identification problem from the

invariant theory viewpoint and obtained some rather interesting results.

Recent results in decoupling theory were obtained by Warren and Eckberg

(1973), Concheiro (1973), and Forney (1975) using the Kronecker invariants.

Probably the most extensive survey of these results has been compiled

by Denham (1974) and we refer the interested reader to this paper.

We temporarily leave this area to consider one specific application of

these results--the realization problem.

1.2 Survey of Previous Work in Realization Theory

The first realization problem proposed for control systems was

the determination of a state space model (internal description) from

a given transfer function (external description). Gilbert (1963) and

Zadeh and Desoer (1963) describe realization procedures based on the

determination of the rank of the residue matrices of the given transfer

function matrix, but unfortunately these procedures only apply to the


I









case of simple poles. Kalman (1963) proposed an algorithm whereby

the given transfer function is realized as a parallel combination of

single input, controllable subsystems in companion form, and then

applied the "canonical structure theory" (Kalman (1962)) to delete the
uncontrollable dynamics. This technique handles simple as well as

multiple transfer function poles. Later Kalman (1965) showed the

equivalence of the realization problem of control theory to the

corresponding network theory formulation.

A significant advance in realization theory was given by Ho and
Kalman (1966). They showed that the state space model could be found

from the impulse response sequence provided the system under investi-

gation is finite dimensional. They also developed an algorithm based

on forming the generalized Hankel array from the given sequence and

then extracted the state space triple from it. Shortly after the pub-

lication of Ho's algorithm, Youla and Tissi (1966) working in network

synthesis and Silverman and Meadows (1966) in control theory developed

similar realization techniques again based on the impulse response sequence.

Ho's algorithm gave new impetus to realization theory. Several

authors have provided alternate or improved realization algorithms based

on the Hankel array formulation. Mayne (1968), Panda and Chen (1969),

Roveda and Schmid (1970), Rosenbrock (1970), Lal et al. (1972) and even

more recently Huang (1974), Rozsa and Sinha (1975) among others,
considered the older transfer function matrix approach, while Rissanen

(1971,1974), Silverman (1971), Ackermannand Bucy (1971), Chen and Mital

(1972), Mital and Chen (1973), and Bonivento et al. (1973) approached

the problem from the Hankel array formulation.








Rissanen (1974), Furata and Paquet (1975), Roman (1975),
Dickinson et al. (1974a,b) have recently considered the problem of

realizing a given infinite impulse response matrix sequence with a
polynomial matrix pair. Such a pair is referred to as a matrix-
fraction description of the system and is becoming well known in

control literature largely due to the ground work established by

Popov (1969), Rosenbrock (1970), Wolovich (1972a,b, 1973a,b) and
others.
Kalman (1971b), Tether (1970), and Godbole (1972) later

considered the more realistic case where only a finite number of

terms of the impulse response sequence are specified. This is
commonly known as the partial realization problem and corresponds
in the scalar case to the classical Pade approximation problem.

Generally most realization altorithms can be used to process partial

data, but usually at a loss of efficiency and even more seriously
the possibility of yielding misleading results. A wealth of new
techniques have recently been published to handle this very special,
yet realistic variant of the realization problem. Rissanen (1972a,b),

Ackermann (1972), Dickinson et al. (1974a), Roman and Bullock (1975a),
Anderson et al. (1975) published some efficient and improved algorithms
to solve this problem.
Also of recent interest is the development of algorithms which

realize a system directly in a canonical form (under a change of basis

in the state space), i.e., algorithms which solve the canonical realiza-

tion problem. The algorithms of Ackermann (1972), Bonivento et al.
(1973), Rissanen (1974), Dickinson et al. (1974a), Rozsa and Sinha

(1975), Luo (1975), and Roman and Bullock (1975a) solve this problem.









One of the main contributions of this dissertation is to use the

results developed from invariant theory to solve the realization and

partial realization problems in the deterministic as well as stochastic

cases. The realization of a system directly in a canonical form actually

reduces to first determining which transformation groups are present,

specifying the corresponding invariants, and then developing a method to

extract these invariants from the given unit pulse response sequence.

This philosophy is basic to any canonical realization scheme and actually

provides an explicit formula which is applied throughout this dissertation.

In the last few years, several interesting extensions have emerged

from the original concept of realization theory. The major motivation

evolved just after the development of the Kalman filter (see Kalman (1961))

in estimation theory because a priori knowledge of the state space

model and noise statistics are required to begin data processing. The

link between the filtering and realization problem was established by

Kalman (1965) just prior to the advent of Ho's algorithm. The work of

Gopinath (1969), Budin (1971,1972), Bonivento et al. (1973), and Audley

and Rugh (1973,1975) were concerned with the more general problem of.

obtaining a state space representation given a general input/output

sequence of the system in both deterministic and stochastic cases. The

stochastic version of the realization problem has not received quite

as much attention as the deterministic case mainly due to its g-eater

complexity and high dependence on the adequacy of covariance estimators.

The realization of stochastic systems was studied by Faurre (1967,1970)

and more recently by Rissanen and Kailath (1972), Gupta and Fairman (1974)

tThe Hankel array formulation is used exclusively in this dissertation;









and Akaike (1974a,b). From the transfer function viewpoint this problem

has been solved using spectral factorization as originally introduced

by Wiener (1955,1959) and studied by others such as Gokhberg and Krein

(1960), Youla (1961), Davis (1963), Motyka and Cadizow (1967), and

Strintzis (1972). The link between the stochastic realization problem

and spectral factorization evolved from the work in stability theory by

Popov (1961,1964), Yakubovich (1963), Kalman (1963), Szeg6 and

Kalman (1963). The equations establishing this link were derived in

the Kalman-Yakubovich-Popov lemma for continuous systems and the Kalman-

Szego-Popov lemma for discrete time systems. Newcomb (1966), Anderson

(1967a,b,1969), and Denham (1975) extended these results and provided

techniques to solve these equations. Defining the invariants of these

problems is still an area of active research as evidenced by the recent

work of Denham (1974), Glover (1973), and Dickinson et al. (1974b).

This is one area developed in this dissertation. It will be shown that

the invariants of the stochastic realization problem not only lends more

insight into the structure of the problem, but also yields some new
results.

Research in realization theory and its applications continues as

evidenced by the recent results of Rissanen (1975) in estimation theory,

Ackermann (1975) in feedback system design,De Jong (1975) in the

numerical aspects of the problem and Roman and Bullock (1975b) in

observer theory. The results presented in this dissertation tie together

some previously well-known results in stochastic realization and filtering

theory as well as provide a technique which can be used to study

other problems.









1.3 Statement of Purpose and Chapter Outline

It is the purpose of this dissertation to provide an extensive

discussion of the realization problem in both the deterministic and

stochastic cases as well as specify the invariants under particular

transformation groups in each case. It is also desired to develop a

simple and efficient algorithm to solve the canonical realization problem.

This algorithm is to be modified to process data sequentially such that

only the pertinent information--the invariants, are extracted from the

given sequence. In the case of a fixed finite unit pulse response

sequence (the partial realization problem), the solution is to be

obtained such that all possible degrees of freedom are specified. The

relationship between the stochastic realization and steady state Kalman

filtering problems are discussed by again examining the corresponding

invariants. In so doing, a considerable amount of knowledge about the

existence and structure of realizations and the steady state filter is

gained.

The basic theoretical essentials of realization and invariant theory

are reviewed in Chapter 2. A "formula" essentially outlined in Popov

(1973) and Kalman (1974) is developed which will be applied to various

realization problems throughout the text. Some new theoretical results

in canonical realization theory are established and used to develop a

new canonical realization algorithm.

In Chapter 3 the algorithm is modified to handle sequentially

the case of partial data and also that of a fixed finite sequence. New

results evolve which completely characterize the class of all minimal
partial realizations and extension sequences as well as determining the

characteristic equation in a simple manner.









The stochastic case of the canonical realization problem is in-

vestigated in Chapter 4. A complete set of independent invariants is

found to characterize the corresponding solution. Equivalent solutions

to this problem as well as to the steady state Kalman filtering problem

are studied and it is shown that the filter parameters can be specified

by solving an analogous realization problem.

The specific contributions of this research and further research

possibilities are outlined in Chapter 5.

Examples are used generously throughout this work to illustrate the

various algorithms discussed and to point out significant details that

are otherwise difficult to see. A comment on notation to be used through-

out this dissertation closesdthis chapter.


1.4 Notation

Uppercase letters denote matrices, and vectors are represented by

underlined lowercase letters. Lowercase letters are used to represent

scalars and integers. All matrices and vectors appearing in this work

are assumed to be real and constant. A = [a..] is an nxm matrix with
m ijmis an nxm matrix with
elements a.i; On is the nxm null matrix with row and column vectors

given by T and 0; In represents the nxn identity matrix, and eT or

stands for its j-th row or j-th column; jem means j=1,2,...,m.
~J














CHAPTER 2

REALIZATION OF INVARIANT SYSTEM DESCRIPTIONS

In this chapter we present a brief review of the major results

in realization theory. We establish a basic "formula" and apply it to

various system representations. It is shown that this approach greatly

simplifies the realization problem. Two new algorithms for realization

are developed which appear to be more efficient than previous techniques

because they extract only the minimal information necessary to specify

a system from the given input/output sequence in an extremely simple

fashion. All of the essential theory is developed and a multivariable

example is presented.

2.1 Realization Theory

A real finite dimensional linear constant dynamic system has

internal description given by the state variable equations in discrete

time as,


x+l = Fx + Guk


Y-= Hk (2.1-1)

where keZ xeKn=X, ueKm=U, eKP=Y and F, G, H are nxn, nxm, pxn matrices

over the field K. X,U,Y are the state, input, and output spaces,

respectively.









The external system description may be given either in rational

form as,


T(z) = H(Iz-F)-IG (z complex) (2.1-2)

or equivalently as an infinite matrix power series

kk
T(z) = E Azk (2.1-3)
k=l

where the sequence {Ak} is the unit pulse response or Markov sequence

of (2.1-1). The Markov parameters are

Ak = HFkG k=1,2,... (2.1-4)


The problem of determining the internal description (F,G,H) from the
external description (T(z) or {Ak}) is the realization problem. Out of

all possible realizations, E:=(F,G,H) having the same Markov parameters,

those of smallest dimension are minimal realizations.

Prior to stating some of the significant results from realization

theory several useful definitions will be given. The j-controllability

and j-observability matrices are the nxmj and pjxn arrays,

W = [G j1G] and = [HT (HF'-1)T]. The pair (F,G)

is completely controllable if p(W )=n and the pair (F,H) is completely

observable if p(V )=n. Throughout this dissertation we will only be

concerned with systems possessing these properties. For a completely

controllable and observable system, the controllability index, u, and

the observability index, v, are the least positive integers such that the

rank of W and V is n.
1I v


I








If two minimal realizations E, 7 are equivalent under a change of
basis in X, then there exists a nonsingular T such that
(Fj,,~,)t = (TFT-1,TG,HT-1). It also follows by direct substitution that
the controllability and observability indices of these realizations are
identical and

W = TW. for j = 1,2,...
j J
Vi = ViT-1 for i = 1,2,...

The generalized NxN' block submatrix of the doubly infinite Hankel
array is given by


rAl

SN,N' :

AN


... AN'



.. AN+N'-1


Implicit in the realization problem is determining when a finite
dimensional realization exists and, if so, its corresponding minimal
dimension. The following proposition by Silverman gives the necessary
and sufficient conditions for {Ak} to have finite dimensional realiza-
tion.


Proposition. (2.1-5)


An infinite sequence {Ak} is realizable iff there
exist positive integers p,v,n such that

p(S ) = (S + ) = (S +j, +1)=n. or j=0,1,...

Further, if {Ak} is realizable, then p,v are the
controllability and observability indices and n is
the dimension of the minimal realization.


tThis notation means F = TFT", G = TG, and H = HT-1









Proof. See Silverman (1971).

Note that the essential point established in Ho and Kalman (1966), which

is used in the proof of the above proposition is that Z is a minimal

realization iff it is completely controllable and observable. Since
S =V W it follows for dim: = n that: p(S ) min[p(V ),p(W )]=n.

This result is essential to construct any realization algorithm. In
(2.1-5) the crucial point of finite dimensionality is carefully woven

into necessary and sufficient conditions for an infinite sequence to be

realizable. What if only partial information about the system is

available in the form of a finite Markov sequence? Is this sequence

realizable? What is the relationship between the minimal realization

and one based only on partial data? These are only a few of the questions

which must be resolved when we are limited to partial data.

Intrinsic in the realization from a finite Markov sequence is the
fact that enough data are contained in S to recover the infinite

sequence, i.e., knowledge of {A1,...,Av,-1} is sufficient to determine

{Ak}, k=1,2,.... But in reality the only way to be sure of this is
knowledge of the actual system dimension (or at least an upper bound). A
minimal partial realization is a realization of smallest dimension
determined from a finite Markov sequence {Ak},keM which realizes the

sequence up to M terms. The order of the partial realization is M and

the realization is denoted by Z(M). The realization induces an extension
k-1
of {Ak}, i.e., Ak=HF k-G for k>M. The following basic result analogous
to (2.1-5) answers the realizability question when only partial data are

given. For a proof, see Kalman (1971).









Proposition. (2.1-6) (Realizability Criterion) The minimal partial

realization problem of order M possesses a

solution, E(M) iff there exist positive integers..

v,v, M = v~-pM, such that

(R) p( 5~) = p(SV+1,) = p(Sv,+1)

where dimE(M) = p(S ) = n.
v,11

In this proposition (R) is designated the rank condition. Also, it

is important to note that when (R) is satisfied the minimal extension

(of S(M)) is unique (see Tether (1970) for proof), but E(M) is not

unique because there exist other minimal partial realizations equiv-

alent to E(M) under a change of basis in X.

We must consider three possible cases when only partial data is

available. In the first case enough data is available such that M>M

for known n; thus, a minimal realization is found. Second, v and p

are available such that (R) is satisfied. In this case a minimal par-

tial realization can be found, but this in no way insures it is also a

minimal realization of the infinite sequence, since the rank of S

may increase as v,u increase. Third, the rank condition does not hold.

How can a realization be found when no more data is available? The

only possibility in this case is to extend the sequence until (R) is

satisfied, but there can exist many extensions satisfying (R) while

giving nonminimal realizations. For this reason define a minimal

extension as any that corresponds to a minimal (partial) realization.

To obtain minimality we must somehow select the right extension among

the many possible.












Prior to summarizing the main results of Kalman (1971) and Tether

(1970), define the incomplete Hankel array associated with a given
partial sequence {Ak}, keM as


A AA
1 A2 .. AM
A2 A3 A.... *
S(M,M) := .

AM *

where the asterisks denote positions where no data is available. The

rank of S(M,M) is the number of linearly independent rows (columns)

determined by comparing only the data specified elements in each row

(column) with the preceding rows (columns) with the cognizance that

upon the availability of more data this number can only remain the same

or increase. Thus, the rank is a lower bound for any extension when the

* are filled in-consistent with the preservation of the Hankel pattern.

Both Kalman and Tether show that there are three pertinent integers
associated with the incomplete Hankel array. They are defined as: n(M),

v(M), p(M) and correspond to the rank of S(M,M), the observability index,

and the controllability index of the given data. The latter two are

lower bounds (separately) for v and p. Knowledge of either v(M), or

y(M) enables us to construct extensions, since they are the least integers
such that (R) holds for all minimal extensions.

It should also be noted that the integers n,v,p,... are actually

non-decreasing functions of the amount of data available, M, and should

be written, n(M), v(M), p(M) etc. to be precise. However, the argument

tIt also follows from this that the p(S(M,M)) is a lower bound for dim.E
(see Kalman (1971)).









M will be understood throughout this dissertation in order to maintain

notational simplicity.

There is one more variant of the partial realization problem that

must be considered. A sequence of minimal partial realizations such

that each lower order realization is contained in one of higher order

will be called a nested realization. Symbolically, this is given by

...S(M)EE(M)I... for M
appear as submatrices of the corresponding matrices in Z(M). The solution

to this problem is most desirable from the computational viewpoint,

since each higher order model can be realized by calculating just a few

new elements in the corresponding realization. Rissanen (1971) has

given an efficient recursive algorithm to determine this solution.

Another related problem of interest is determining a unique member of

equivalent systems under similarity and is discussed in the following

section.

2.2 Invariant System Descriptions

In this section we review some of the fundamental ideas encountered

when examining the invariants of multivariable linear systems. The

framework developed here will be used throughout this dissertation in

formulating and solving various realization problems. Not only does

this formulation enable the determination of unique system representations

under some well-known transformations, but it also provides insight into

the structure of the systems considered. First, we briefly define the

essential terminology and then use it to describe some of the more common

sets of canonical forms employed in many recent applications (e.g., Roman

and Bullock (1975a,b), Tse and Weinert (1975)).









For any two sets X and Y, a subset R = X x Yt is called a binary

relation on X to Y (or, a relation "between" X and Y). Then (x,y)eR

is usually written as xRy and is read: "x stands in the relation R to

y". If for X=Y this relation is reflexive, symmetric, and transitive,
then it is an equivalence relation E on X given by xEy for x,ysX. The

set of all elements z equivalent to x is denoted by E(x).= {zeXfxEz} and

is called the equivalence class or orbit of x for the equivalence relation

E. The set of all such equivalence classes is called the quotient set

or orbit space and is given by X/E. Thus, the relation E of X partitions

the set X into a family of mutually disjoint subsets or orbits by sending

elements which are related into the same equivalence class.

Consider a fixed group Gtt of transformations acting on a set X.

Then the elements xl,x2 of X are equivalent under the action of G iff

there exists a transformation TeG which maps x1 into x2. This is basically

the "formula" we will apply throughout, i.e., we first formulate the set

of elements (generally the internal system description), then define a

transformation group; and finally determine the orbits under the action
of G. To be more precise, let us first define the function f mapping

a set X into Y as an invariant for E if for x1,x2 X, x1Ex2 implies

f(x i)f(x2). In addition if f(xl)=f(x2) implies xlEx2, then f is a


This is the standard Cartesian product, XxY = {(x,y)|xeX, yeY}

tHere we mean "group" in the standard algebraic sense, i.e., (Go)
where G is a closed set of elements each possessing an inverse and
the identify element; o is an associative binary operation. When
o is understood, the group is merely denoted by G.
tNote that an invariant is actually a function, but common usage
refers to its image as the invariant. We will also use this terminology
throughout this dissertation.








complete invariant. In general we will be interested in a complete
system of invariants for E given by the.set of invariants {fi} where
f:t X + Y1xY2x...xY fi is an invariant for E, and fl(x,)= (x2),...'

fn(xl)=fn(x2) imply xEx2. Completeness of this set of invariants
means that the set is sufficient to specify the orbit of x, i.e., there

is a one to one correspondence between the equivalence classes in X
and the image of f. If the set of complete invariants is independent,
then the map f: X-YlX...xYn is surjective. This property means that
corresponding to every set of values of the invariants there always exists
an n-tuple in Y specified by this set. A complete system of independent
invariants will be called an algebraic basis.
Generally, we consider a subset of X (e.g., in system theory a
controllable system). Correspondingly, let fo be a function mapping the
subset X of X into set Y, then fo is a restriction of f if f (x)=f(x)
for each xeXo. We can uniquely characterize an equivalence class E(x)
by means of the set of values of the functions fi(x), ien where the {fi}
constitute a complete set of invariants for E on X. If the corresponding
complete invariant f is restricted such that its image is itself a
subset of X, then we have specified a set of canonical forms C for
E on X. To be more precise, a canonical form C for X under E is a
member of a subset CcX such that: (1) for every xeX there exists one and
only one cEC for which xEc, and since C is the image of a complete
invariant f, then (2) for any xeX and cl, c2EC, xEcl,and xEc2 implies
f(x)=f(cl)=f(c2)=cl=c2 (invariance); (3) for any ceC if f(x1)=c and
f(x2)=c, then x1Ex2 (completeness). Thus, c=f(x) is a unique member of

This notation is actually f=(fl,...,f ):x+Y1...xYn but it is
shortened when the set {fi} is clearly understood.


FR~-"YI i









E(x) for every xeX. With these definitions in mind, our "formula"

becomes

(i) Formulate the set of elements;

(ii) Define the transformation group;

(iii) Determine a set of complete invariants under this

transformation group; and

(iv) Develop the canonical form in terms of the corresponding

invariants. (2.2-1)

We now apply (2.2-1) to various restricted sets related to multivariable

systems. This approach is essentially given in Kalman (1971a),Popov

(1972), Rissanen (1974), or Denham (1974). In this sequel we review the

main results of Popov. First, define the set of matrix pairs (F,G)

as

X = {(F,G)IFeKnxn, GeKnxm; (F,G)controllable}


The general linear group, which corresponds to a change of basis

in the state space, is specified by the set

GL(n):= {TITeKnxn; det TO} (2.2-2)

with the group operation standard matrix multiplication, i.e.,

To T = T T.

In order to determine the orbits of X under the action of GL(n),

it is first necessary to specify the action operator "+"

T + (F,G):= (TFT-1,TG)


In general the problem of determining a canonical form is quite
difficult. However in this dissertation we consider restricted sets
which make the problem much simpler. For a thorough discussion of this
problem see Kalman (1973).


1








or alternately we can say that the action of GL(n) on Xo induces

F + TFT-1
G + TG

The action of GL(n) induces an equivalence relation on Xo. We

indicate (F,G)ET(F,G) if there exists TeGL(n) such that (T,G)=T+(F,G).

Dual results are defined for the observable pair (F,H) and the
analogous set denoted by X .
The third step of (2.2-1) is established in Popov (1972), but
first consider the following definitions. For a controllable pair (F,G)

define the j-th controllability index p ., jEm as the smallest

positive integer such that the vector F 3gj is a linear combination of
its predecessors, where a predecessor of Figj is any vector Frgs where
rm+s
we have assumed p(G) = m. Throughout this dissertation we use the

following definition of predecessor independence: a row or column vec-
tor of a given array is independent if it is not a linear combination of
its regular predecessors. The following results were established by

Popov (1972)

Proposition. (2.2-3) (1) The regular vectors are linearly independent;
(2) The controllability indices satisfy the
m
following relationship, E j = n; (3) Ti.ere exists
n=1
exactly one set of ordered scalars, ajkscK defined

for jem, kej-l, s = O,l,...,min(pj ,k-1) and for jEm,

k = j,...,m,s = 0,l,...,min(pj,pk) 1 such that

tThroughout this dissertation we use the overbar on a set to denote the
dual set.
ttThese indices are also called the Kronecker indices.








j-1 min(sj.,k-l) m min(,ljpk)-1
F jg. Z Z ajks gk + E ajks Fk'
3 k=1 s=O k=j s=O

This proposition follows directly from the controllability of (F,G) and
indicates that the regular vectors forma basis where the a's are the
coefficients of linear dependencies. The set [{Pj},{ajks}], j,kem,
s=0,...,j.-I are defined as the controllability invariants of (F,G),
and v=max(pj). The main result of Popov is:

Proposition. (2.2-4) The controllability invariants are a complete
set of independent invariants for (F,G)eX under
the action of GL(n).

The proof of this proposition is given in Popov (1972) and consists of
verifying the invariance, completeness, and independence of [{1j},{ajks}].
Invariance follows directly from Proposition (2.2-3), since (F,G)ET(F,G),
then T'gk can be replaced by TFSgk in the given recursion and the
controllability invariants remain unchanged. Completeness is shown by
constructing a TeGL(n) such that for two pairs of matrices (F,G),
(F,G)eX with identical controllability invariants, (F,G) = (TFT1, TG)
or (F,G)ET(F,G). Independence of the controllability invariants is
obtained by constructing a canonical form determined only in terms of
these invariants. Thus, by introducing a finite set of indices {pj},
Popov shows that this set along with the {ajks} are invariants under the
action of GL(n). The main reason for specifying a set of complete and in-
dependent invariants is that it enables us to uniquely characterize the
orbit of (F,G). It should also be noted that dual results hold for the observ-
able pair (F,H), and it follows that the observability invariants are the


~~I __L




24



set [{vi},{ ist}], i,sep, t=O,...,v.i- where the {vi}are the observa-

bility indices.

The last step of (2.2-1) is to specify the corresponding canonical

forms under GL(n). These forms are commonly called the Luenberger

forms and are specified by the controllability and observability

invariants. They are defined by the pairs (FC,Gc), (FR,HR) where the

subscripts C,R reference the fact that the regular vectors span either

the columns of W+,1 or the rows of V +l'
p + 1 H+ 1

FC = [t2 l1 2 .. emm] (2.2-5)


GC = el e ql+l1 e l+l

J
qj = s s jem
Ss=1 s


~ T
-2


T
eT
e1 T

T -J
e











i
FR R = (2.2-6)
T T
er P-+2 __r 0+1


T
erp
T


ri = Vs iE2
s=l1


-i---------*-------i--_~-_ --








where aj, i are n column, n row vectors containing {aI {Bi s}

respectively over appropriate indices and zeros in the other places.

Luenberger (1967) shows that the transformation, TC, required to

obtain the pair (FC,Gc) is determined from the columns of W as


TC =[T1 T2 ... Tm (2.2-7)
where Tj =[gj Fg lF1j] jFm

pCG) = m, and gj is the j-th column of G.

Similar results hold for the pair (FR,HR) and is specified by TR

constructed from the rows of V .

Unfortunately Luenberger (1967) in attempting to develop multi-

variable system representations did not determine the invariants under

GL(n). It is essential to use the approach outlined in (2.2-1) in

order to obtain the corresponding canonical forms or else it is possible

to obtain erroneous results. The following example due to Denham (1974),

shows that the Luenberger form, as originally stated is not canonical.

If we are given the pair (F,G) as


0 0 1 1 1 0
----,-------

1 0 2 1 0 0
F = G
0 1 2 1 0 0

0 0 1 1_ 0 1








These matrices are in the form of (2.2-5), but it is easliy verified

by constructing TC that the controllability invariants are in fact

pl=2, P2=2 and 21 = [-1 -1 -1 1], a2 = [-2 0 -2 4T1 The problem

with the Luenberger forms is that the maps 7: X0+ XO/E are not well

defined. Thus, the image of the maps are indeed canonical forms, but

as shown here for (F,G)eXO/E, we need not have rT(F,G)=(F,G), i.e., the

mapping does not leave the canonical forms unchanged. The point to
remember is that the invariants are the necessary entities of interest

which must be determined.

The procedure to construct the transformation matrix TC of (2.2-7)
is called the Luenberger second plan. The first Luenberger plan
consists of examining the columns of ,n' given by


n = Wn = [gl ... Fn-g1 "' gm Fn- m] (2.2-8)


where Y is an nmxnm permutation matrix, for predecessor independence.
Thus, we can define a new set of invariants (under GL(n)) [{f }, {ajks}]
completely analogous to the controllability invariants. The canonical

forms associated with the invariants obtained in this fashion have







tThis procedure amounts to examining the column vectors of Wn for
predecessor independence, i.e., examine g1 ... g Fg1 ... Fg .
Fn-191 ... Fn-lgm.


i I









become known as the Bucy forms which were derived directly from the
results of Langenhop (1964), Luenberger (1967), and Bucy (1968). We
refer the interested reader to these references as well as the recent
survey by Denham (1974). Here we will be satisfied to note that the

procedure of (2.2-1) applies with the set of controllable pairs (F,G)

restricted to the {j} invariants rather than {pj}. Analogous to the
Luenberger forms, we define the row and column Bucy forms as (FBR,HBR),

(FBC,GBC) respectively. The row form is given by

T
L eT
11 2-1
0
L21 L22 He +1
FBR BR (2.2-9)

Lpl Lp L eT + +..+p+1

where

I ..-1 V
L.. = ---- ; L.. = ]
11 ST 13 -T"
L Bii ij

I P
v. > 0 and satisfy E v =n ;
s=l

B, j are v.,vj row vectors containing ist } invariants.

The transformation, TBR,required to obtain the pair (FBR,HBR) is


T T T VlT
BR = [T T F B T (2.2-10)


B i i









The importance of the Bucy form is that the characteristic equation can

be found by inspection of the block diagonal arrays of FBRt. Since FB
FBR
is block lower triangular, the characteristic equation is given as

XFBR(z) = det(Iz-FBR) = (z) ... XL (z) (2.2-11)
FBR 1BR Lpp


where the Lii are the companion matrices of (2.2-9). Similar results

hold for the pair (FBCGBc) and the transformation is specified by TBC

constructed from the columns of Wn.
This completes the discussion of invariants and canonical forms for
controllable or observable pairs. To extend these results to matrix

triples (internal system description), it is more convenient to examine
an alternate characterization of the corresponding equivalence class--
the Markov sequence of (2.1-4). This approach was used by Mayne (1972b)
and Rissanen (1974), in order to determine the orbits of Z under GL(n).

It is obvious that the sequence is invariant under this group action

Aj = (HT-1)(TFT-1)j-1(TG) = HFj-1G (2.2-12)


Consequently every element of Aj can be considered an invariant of Z

with respect to GL(n); therefore, two systems which are equivalent under

GL(n) possess identical Markov sequences. The converse is also true, i.e.,

any two systems with identical Markov sequences are equivalent.
The standard approach to investigate a system characterized by its
Markov sequence is to form the Hankel array, SN,N, where we define S ,


It should be noted that the Bucy form is not a canonical form if the
transformation group includes a change of basis in either input or output
spaces, while the Luenberger form is still a canonical form.








iAN and S., jeN' as the block rows and columns of SN,N, and the
block column and row vectors, a or aT denote the r-th column of S
.r s. ,.
or the s-th row of S,1 for remN', spN. Rissanen (1974) has shown
that by examining the set
X1 = {2 I E controllable and observable with {ii} invariants}
under the action of GL(n) that

Proposition. (2.2-13) The set of controllability invariants and block
column vectors, [{1j},{ajks},{a.i}] for the

appropriate indices constitute an algebraic basis
for any SeX1 under the action of GL(n).

The proof of this proposition is given in Rissanen (1974) and consists of
showing that any two members of X1 with identical Markov sequences
are equivalent under GL(n). Thus, invariance follows by showing that a
dependent column vector of the Hankel array can be uniquely represented
in terms of the set [{CI-},{acjks}]. These parameters remain unchanged
under GL(n); therefore, they are invariants. The block column vectors,
a.t satisfy a recursion analogous to (2.2-3), i.e.,

n-I min(j,Pk-1) m min(lj,Pk)-l

a.j+m. = aksa.j+ms + zE jksa.j+ms
k=l s=O k=j s=O

Thus, all dependent block columns can be generated directly from the set,
{a.t} of regular block column vectors. These vectors are invariants under
GL(n), since every column vector of Aj is an invariant as shown in
(2.2-12). Completeness follows immediately from the above recursion,
since any two members of X1 possessing identical invariants satisfy the
above recursion and therefore have identical Markov sequences.









Independence is shown by constructing the Luenberger form of (2.2-5) and (2.2-14)

below* directly from these invariants.

The dual result yields another basis on X1, [{i},{B.ist},{a ].

The corresponding canonical forms for ZeX1 or X1 are given by the

Luenberger pairs of (2.2-5,2.2-6) and

HC = [a.1 ... a.(l )m+ll.. la.m .. a. mm (2.2-14)

and
T
al

aT
a(l-1)p+l.


GR

a
P*

T
Vpp.


and the canonical triples are denoted by EC and ZR respectively.
Rissanen (1974) also shows that a canonical form for the transfer

function can be constructed from the invariants of (2.2-13). This is

possible because the determination of canonical forms for Z based on the

Markov parameters is independent of the origin of Ak's. Rissanen defines
the (left) matrix fraction description (MFD) as

T(z) := B-1(z)D(z) (2.2-15)

where B(z) = z Bi.z for IB \f 0
i=O '
v-1
D(z) E Diz .
i=0 I









The relationship of the MFD to the Hankel array, S +1,+l ,follows by
writing (2.2-15) as

B(z)T(z) = D(z) (2.2-16)

and equating coefficients of the negative powers of z to obtain the

recursion

BAj + B1Aj+I + + BA =j j=1 ,2...


expanding over j gives the relation over the block Hankel rows as

[B ... B] B S T p ) (2.2-17)

T
v+ ,.

where the pxp(v+l) matrix of Bi's is called the coefficient matrix of

B(z). Similarly equating coefficients of the positive powers of z
gives the recursion

Dk = Bk+A + Bk+A + ... + BA-k k=0,l,...,v-1

or expanding over k gives the relation in terms of the first block Hankel
column as


v-1 Bv
0
D2 B B
Dv-2 v- v S (2.2-18)


DO Bl B2 ... B









The canonical forms for both left and right MFD's are defined by the
polynomial pairs (BR(z),DR(z)) and (BC(z),DC(z)) respectively, where
R and C have the same meaning as in (2.2-5,2.2-6) and the former is
given by


T
b
-1
BR(z) := :

bT
-p


p


Izv


; bT K
ki Kp


(2.2-19)


for


T T
b = OTk


where k=i+pvi and Bkj ar


S {ist

Bkj : 0
1


*T
1 k2 *"'* k(i+pvi) l0-i] i

e given by

j=i,i+p,...,i+p(vi-1)

jfi,i+p,...,i+p(vi-1)

j=i+pvi


and DR(z) is determined from (2.2-18).

Dual results hold for the corresponding column vectors, bj, jem of the
-J
coefficient array of IC,(z) in terms of the controllability invariants.
This completes the discussion of canonical forms for Z or T(z).
Note that analogous forms can easily be determined for the Bucy forms

if X1 is restricted to {vj}. Henceforth, when we refer to an invariant

system description, we will mean any representation completely specified
by an algebraic basis. In the next section we develop the theory
necessary to realize these representations directly from the Markov
sequence.









2.3 Canonical Realization Theory

In this section we develop the theory necessary to solve the

canonical realization problem, i.e., the determination of a minimal

realization from an infinite Markov sequence, directly in a canonical

form for the action of GL(n). Obviously from the previous discussion,

this solution has an advantage over other techniques which do not obtain

Z in any specific form. From the computational viewpoint, the simplest

realization technique would be to extract only the most essential

information from the Markov sequence--the invariants under GL(n). Not

only do the invariants provide the minimal information required to

completely specify the orbit of E, but they simultaneously specify a

unique representation of this orbit--the corresponding canonical form.

Thus, subsequent theory is developed with one goal in mind--to extract

the invariants from the given sequence.

The following lemma provides the theoretical core of the subsequent

algorithms.

Lemma. (2.3-1) Let VN and WN, be any full rank factors of SN,N, = VN WN'

Then each row (column) of SNN,, is dependent iff it is a

dependent row (column) of VN (WN,).

Proof. From the factorization SN,N, = VN WN, it follows if the j-th

row of SN,N, is dependent, then there exists an eKpN, iaj0

such that
T T
SN,N' = OmN'
Since p(WN,)=n, i.e., WN, is of full row rank, it follows that
TSN T =
SS,NIWN' -mN'









or aTVN(WN' N) = OT N'

but det (WNWN' ) 0; thus, a = Om0N, i.e., a dependent row
of SN,N, is a dependent row of VN. Conversely assume that there.
T
exists a nonzero a as before such that
T T
aV = 0 T
VN -mN'
Since p(WN,)=n, it follows that this expression remains unaltered
if post-multiplied by WN,, i.e.,
T T
aVNN, = iN'

and the desired result follows immediately.V

The significance of this lemma is that examining the Hankel rows

(columns) for dependencies is equivalent to examining the rows (columns)

of the observability (controllability) matrix. When these rows (columns)

are examined for predecessor independence, then the corresponding

indices and coefficients of linear dependence have special meaning--

they are the observability (controllability) invariants. Thus, the

obvious corrollary to this lemma is


Corollary. (2.3-2)


If the rows of the Hankel array are examined for

predecessor independence, then the j-th (dependent)

row, where j=i+pvi, iep is given by
i-l min(vi,vs-1) p min(v,v s)-l
T T T
-J = ist-s+pt + s istys+pt
s=1 t=0 s=l t=0

where{Bist an:d{vi} are the observability invariants

and -, kcpN is the k-th row vector of S N,N'









Proof. The proof is immediate from Proposition (2.2-3) and Lemma (2.3-1).V

Note that similar results hold for the columns of the Hankel array when

examined for predecessor independence.

In the solution to some problems knowledge of both controllability

and observability indices are required. Moore and Silverman (1972)

require both indices to design dynamic compensators in order to solve

the exact model matching problem. Similarly the requirement exists in

the design of pole placement compensators and also stable observers as

indicated in Brausch and Pearson (1970) and more recently Roman and

Bullock (1975b). In an on-line application Saridis and Lobbia (1972)

require the controllability invariants as well as the observability

indices to solve the problem of parameter identification and control.

The latter case exemplifies the fact that in some instances it is first

necessary to determine the structural properties of a system from its

external description prior to compensation.

The need for an algorithm which determines both sets of controllability

and observability invariants from an external system description is

apparent. Computationally the simplest and most efficient technique to

determine these invariants would be some type of Gaussian elimination

scheme which utilizes elementary operations (e.g., see Faddeeva (1959)).

If we perform elementary row operations on VN such that the predecessor

dependencies of PVN are identical to those of VN and perform column

operations on WN, so that WNE and WN, have the same dependencies then

examination of SN,N' = PSN,NE is equivalent to the examination of SN,N'

We define SN,N as the structural array of SN,N,. This array is

specified by the indices {vi} and {p.} which are the least integers such

that the row and column vectors of SNN are respectively,
ii **i










rnonzero fa=Q,...,v-1

zero J la=, ..,N- J


Snonzero b=0,...,p -1
a = for 1
S zero b= ..,N-


These results follow since SN,N has identical

as SN,N,, then


SNN'


T
-1


T
-pN


for s=j+mb



predecessor dependencies


where T= 0 if it depends on its predecessors. To find the observability

indices, let a be the index of the last nonzero row of 6i+p t=0,...,N-.
T T
Then if T6 = 0 v = 0 otherwise v. = (a-i)/p+l. Similar results
*
follow when SNN is expressed in terms of the g The following theorem
*
specifies the matrices P and E required to obtain SN,N.

Theorem. (2.3-3) There exist elementary matrices P and E, respectively

lower and upper triangular with unit diagonal elements,

such that SN,N=PSN,N.E has identical predecessor

dependencies as SNN,N

Proof. Let PS N,N=Q where Q is row equivalent to SN,N, and P=[prs].

If the j-th row of SN,N, is dependent on its predecessors, i.e.,

T j-1 T T
T. + Z a = 0
3 k=l jk-k -

then selecting P lower triangular such that


I








0O r Prs = 1 r=s
ajk r>s

gives this relation. From this choice of P it follows that
dependent rows of SN,N, are zero rows of Q. If the j-th row
of SN,N, is regular, then P unit diagonal-lower triangular

insures that the corresponding row of Q is nonzero and regular.

Similar results hold for the columns of SNN with E unit diagonal-
upper triangular.
This choice of P does not alter the column dependencies of SN,N;
for if the i-th column of SN,N, is dependent on its predecessors,
then from Corollary (2.3-2) .i is uniquely represented as a
linear combination of regular vectors in terms of the control-
lability invariants. Since P is unit diagonal-lower triangular,

it is the matrix representation of a nonsingular linear

transformation, Pir=q. where _q is the i-th column vector of Q.

Thus, multiplying on the left every vector Li in (2.3-2) with
this P gives for i=j+mpj
j-1 min(P ,k"-1) m min(j ,pk)-1
=i = Z Z ajksq.k+ms + E Z j ksqk+ms
k=l s=O k=j s=O

Thus, we have shown that selecting P with the given structure
does not alter the predecessor column dependencies of SN,N' or

equivalently Q. Since the column vectors of Q satisfy the

above recursion, SN,N, and Q have identical predecessor column
dependencies, therefore, performing column operations on Q is
analogous to performing them on SN,N, and so we have S =

(PSN,N,)E = QE or the predecessor dependencies of SN,N, and SN,N,
are identical.V









This theorem shows that the indices can be found by performing a sequence

of elementary lower triangular row and upper triangular column operations

in a specified manner on the Hankel array and examining the nonzero rows

and columns of SNN., the structural array of SN,N,. The {ajks} and

({ist} are also easily found by inspection from the proper rows of P and
columns of E as given by

Corollary. (2.3-4) The sets of invariants {8ist',{jiks} or more compactly

the sets of n vectors {. )},{a} are given by the rows

of P and columns of E in (2.3-3) respectively as


i = [qrqr+p "' Pqr+p(v-l)' 1 i+i ir


a. [estes+mt ... es+m(.j. )tT t=mtj+j, j,sm
-JJ

where
1 q=r, r=s
Pqr'est qt


Proof. The proof of this corollary is immediate from Theorem (2.3-3).V

We can also easily extract the set of invariant block row or column

vectors, {a },{a k from the Hankel array and therefore, we have a

solution to the canonical realization problem.

Theorem. (2.3-5) If the generalized Hankel submatrix of rank n is

transformed by elementary row operations to obtain a row

equivalent array, then by proper choice of P the matrix Q

is given by:








TG I TFG ... TF TH,,,
S= --------------------- __ --
nPn-N I pn-N
LmN' L mN'

where (F,G) is a controllable pair and det TO.


Proof. .If x is a minimal realization, then it is well-known that

p(VN)=P(WN,)=n. Since P is an elementary array, then it follows
[PVN]-min[p(P),p(VN)]=n; thus P can be chosen such that

PV -- and det TfO.
N pN-n
Sn
Post multiplication by WN, gives


"VN"''N' pRn N' 1
PVN N' =FG] PSN,N' := Q
n
Multiplication of the arrays gives the desired results.V

Corollary. (2.3-6) If P is selected such that Q is as in (2.3-5) with the

pair (F,G) in Luenberger column form, then the set

of invariants {a.}, jem is given by the columns of
-J
W N wk, kemN' with


-k = k=pjm+j

Proof. If P is selected in Theorem (2.3-5) such that T=T then it
follows that each column of WN, corresponding to the (j+mpj)-th
for each jem contains the {ajks} invariants.V
J ks


The method of selecting P is given in the ensuing algorithm.









Theorem. (2.3-10) Given the infinite realizable Markov sequence
from an unknown system, then EC=(FC,GC,H)n is a
minimal canonical realization of {Ak} with

Fc = [W* I W* 1 ... I W]
C 1 2 Wm
GC is a submatrix of (W+1i)C given by the first m
columns
HC = [a.1 ... al+m(ll) I ... I a .. a m

and W = [wj+ ... w jm], jsm, w is a column
vector of (Wy+1)C.

Proof. Since the sequence is realizable, there exist integers, n,v,p,
satisfying Proposition (2.1-5). If Q is given as in Corollary
(2.3-6), then

(W k)C GC ... FclGC
Q = -------- = -------------------- for k>p+l
0Pv-n pv-n
L mk Omk
Thus, GC is obtained immediately from the first m columns of

(Wk)c. Form two nxn arrays, A and A each constructed by
selecting n regular columns of (Wk)C starting with the first
column for A and the (l+m) column for A The independent
columns of (Wk)C are indexed by the pj and satisfy (2.3-8);
thus, they are unit columns and A is a permutation matrix, i.e.,


A = [w ... w I Wl+m ". 2m I .. I .. +m(p.-) ...], jem
3








Theorem. (2.3-10) Given the infinite realizable Markov sequence
from an unknown system, then EC=(FC,GC,HC)n is a
minimal canonical realization of {Ak) with

FC [ W I W I ... I Wm

GC is a submatrix of (W.+1)C given by the first m
columns
HC = [a.l ... a.l+m(Pll ) I ... a ... am ]

and W = [j+m ... j+m], jm, w is a column
vector of (W +1)C.

Proof. Since the sequence is realizable, there exist integers, n,v,p,
satisfying Proposition (2.1-5). If Q is given as in Corollary
(2.3-6), then

F(Wk)CGC | ... | FC GC
Q = ------- = ----------------- for k>i+l
mk mk
Thus, GC is obtained immediately from the first m columns of

(Wk)C. Form two nxn arrays, A and A each constructed by
selecting n regular columns of (Wk)C starting with the first
column for A and the (l+m) column for A The independent
columns of (Wk)C are indexed by the pj and satisfy (2.3-8);
thus, they are unit columns and A is a permutation matrix, i.e.,


A = [w1 ... w~ l+m 2m ... j+m( 1) jem









where it follows from (2.3-8) that the columns of A form chains
satisfying
m
[w. ...w. j = [eq 1 ... e ] for qj= E P..
3 -34+m1( y1) -qj-1+ll -s=1

Since A is A shifted m columns to the right, each chain of A
is given by [wAj+ ... w J+m ] and again each column is unit
-j- -j+mp.
3 T
except wj+m = aj from Corollary (2.3-6). Thus, FC := A

gives the matrix of (2.2-5). HC is obtained directly from

H G I ... I F G = [a.1 ... a.m I .. a.l+m ... am(k+l)]

since multiplication by the unit columns of (Fc,Gc) select the
n columns of H .V

Analogous results hold for the dual ZR. It should also be noted that
if the Hankel array is transformed to SN,N' and both rows and columns
examined for predecessor independence as before, i.e.,


NN := N,N' U N (2.3-11)

where WN is given in (2.2-8) and T is a permutation array, then all of the
previous theory is applicable. The only exception in this case is that
the Bucy invariants and forms given by IBR and IBC are obtained instead of
the Luenberger forms. These results follow directly from (2.2-1).
In many applications the characteristic polynomial XF(z) is required.
Many efficient classical methods (e.g., see Faddeeva (1959)) exist to
determine XF(z) from the system matrix. Even more recently some
techniques have been developed to extract the characteristic polynomial





43



from the Markov sequence, but in general they are only valid in the cyclic

case (see Candy et al. (1975)). An alternate solution to this problem

is to obtain the Bucy form and use (2.2-11) to find XF(Z) by inspection.

It is possible to realize the system directly in Bucy form as mentioned in

the previous paragraph, but in this dissertation we prefer to take

advantage of the structure of the Luenberger form to construct TBR or

TBC. Superficially, this method does not appear simple because the

transformation matrix and its inverse must be constructed, but the

following lemma shows that TBR can almost entirely be written by inspection

from the observability invariants after the {I.} are known.

Lemma. (2.3-12) The transformation matrix TBR, such that

-1 -1
(FBRGBRHBR) = (TBRFBR' BRG, HTBR

is given by

T = T T
BR B= [T .. B I
1 p

If the given triple is in Luenberger form, ZR' then the

(.ixn) submatrices TB are
T
i-l+l

T
eT ei-+1

S-r+' +2


1FR
T

T > i-V






T-i> R









where


v'.~ are the observability invariants of ER

vi are the invariants associated with EBR and
i
recall ri = E v ro=O.
1 s s o
s=1
Proof. This lemma is proved by direct construction of the T 's.
B i
Since each T satisfies for v.'v.
F 1 1





TB. = F R-1
i h1F'


then analogous to

and therefore
v
hiFR =

R
h.F 'i
h FR


i-l T
property (2.3-8), it follows that h.F =e

i^R RR -r

(hiFRi )FR = eFR = i


v v.-V.-l1 v.-v.-l'
= (hiFR )FR = ~.FR .


In order to construct TBR it is first necessary to find the {ui} from the

rows of [V ]R, but in this case the i 's can generally be found by

inspection while simultaneously building TBR. Also, TBR is generally a

sparse matrix with unit row vectors; therefore, the inverse can easily be

found by solving T RT-B = n directly for the unknown elements of TB .
BR BR n BR'









In the next section we develop some new algorithms which utilize
the theory developed here.

2.4 Some New Realization Algorithms
In this section we present two new algorithms which can be used to
extract both observability and controllability invariants from the given
Markov sequence. Recall from Theorem (2.3-3) that performing row operations
on the Hankel array does not alter the column dependencies, however, it

is possible to obtain the row equivalent array, Q in a form such that
the controllability invariants can easily be found.
The first part of the algorithm consists of performing a restricted
Gaussian elimination (see Faddeeva (1959) for details) procedure on the
Hankel array. This procedure is restricted because there is no row or
column interchange and the leading element or first nonzero element of
each row is not necessarily a one. Define the natural order as 1,2,....

Algorithm. (2.4-1)
(1) Form the augmented array: [IpN I SN,N' I ImN
(2) Perform the following row operations on SN,N. to obtain

EP Q I ImN,:

(i) Set the first row of Q equal to the first Hankel row.
(ii) Search the first column of SN,N, by examining the rows in
their natural order to obtain the first leading element.
This element is qjl.
(iii) Perform row operations (with interchange) to obtain qkl=0,k>j.



tAlternately it is possible to extract the Bucy invariants from Q by
reordering the columns as in (2.2-8) to obtain (=QU and examining
the columns for predecessor dependencies.









(iv) Repeat (ii) and (iii) by searching the columns in their
natural order for leading elements.

(v) Terminate the procedure after all the leading elements have
been determined.

(vi) Check that at least the last p rows of Q are zero. This assures
that the rank condition, (R) is satisfied.

(3) Obtain the observability and controllability indicest as in

Theorem (2.3-3).

(4) Obtain T iep from the appropriate rows of P as in Corollary (2.3-4)
T *
and bi as in (2.2-19) where Bij-pij

(5) Perform the following column operations on Q to obtain [P SN,, E]:

(i) Select the leading element in the first column of Q, qjl.
(ii) Perform column operations (with interchange) to obtain
qjs=0 for s>l.

(iii) Repeat (i) and (ii) until the only nonzero elements in each row
are leading elements.

(6) Obtain cj, jem from the appropriate columns of E as in Corollary

(2.3-4) and -b from the dual of (2.2-19).
-J
(7) From the invariants construct the Luenberger and MFD forms as in

Section (2.2).

If we also require the characteristic polynomial, then we must include:
tt.
(8) Determine the {.}, iep and (simultaneously) construct TBR as in
Lemma (2.3-12).
-1 1
(9) Find TBR by solving for the non unit rows in TT = In
BR BRTBR n



Note that the leading elements have been selected from the rows by examining
the columns in their natural order; therefore, the dependent columns are
not zero as in (2.3-3), but are easily found from this form of Q by
inspection. It should also be noted that the leading elements could have
been selected in the j, (j+m), (j+2m)... columns; therefore, facilitating
the determination of the Bucy invariants and forms.
Alternately the {(3v}, jcm and TBC could be used. These indices can be
found easily from the columns of Q.


I









If we consider the alternate method implied in Corollary (2.3-6), then

the following modifications to the preceding steps are required:

(1)* Start with the following augmented array:

[IpN I SN,N.]

(2)* Obtain [P I Q] as before.

(5)* Perform additional row operations on Q to obtain unit

columns for each column possessing a leading row element, and

perform row interchanges such that (2.3-8) is satisfied

for each jem, i.e., obtain


Q :------n
OpN-n
kmn,

(6)* Obtain the aj, jem, as in (2.3-6).

It should be noted that these algorithms are directly related to

those developed by Ho and Kalman (1966), Silverman (1971), or Rissanen

(1971). As in Ho's algorithm, the basis of the first technique is

performing the special equivalence transformation of Theorem (2.3-3)
*
on SN,N, to obtain SN,N,. The second technique accomplishes the same

objectives by restricting the operations to only the rows of SN,N, which

is analogous to either the Silverman or Rissanen method. The initial

storage requirements in the first method are greater than the second if

mN'>pN, since P and E can be stored in the same locations due to their

lower and upper triangular structure; and (2) P will be altered in the

second method, since row interchanges must be performed in (5)*; whereas,

it remains unaltered in the first method which may be important in some

applications. Consider the following example which is solved using both

techniques.


I









Example. (2.4-2)

Let m=2, p=3, and the Hankel array be given as, S4,4

1 2 2 4 4 8 8 16
1 2 2 4 6 10 13 22

1 0 1 0 3 2 6 6

2 4 4 8 8 16 16 32
2 4 6 10 13 22 28 48

1 0 3 2 6 6 13 16
S = --------------------------
S 4,4
4 8 8 16 16 32 32 64

6 10 13 22 28 48 58 102

3 2 6 6 13 16 27 38
8 16 16 32 32 64 64 128
13 22 28 48 58 102 19 214

6 6 13 16 27 38 56 86

(1) [112 I S4,4 I 8

(2) Performing the row operations as in (2.4-1), obtain [P I Q I 8
where the leading elements are circled,

i] O 2 2 4 4 88 16:
-1 .1 00 00 2 5 6
1 0 -1 -4 0 -5 -7
-2 0 1 0 0 0 0 0 0 0 0
S0 D 1 I 0( 2, 0 1 1
11 1 0 -] 1
-4 0 0 0 0 0 1
-3 0 -1 0 -1 0 0 1
0 1 -2 0 -1 0 D 0 1 07
-8 0 ,0 0 0 0 0 0 0 1
-8 1 -2 0 -2 0 0 0 0 0 1
-1 2 -3 0 -2 0 0 0 0 0 0 1








(3) The indices are obtained by inspection from the independent rows
and columns of Q in accordance with Theorem (2.3-3) as:


v1 : 1

v2 2

v3 =1


P1 = 3

112 = 1


and p(S2,3) = p(S3,3) = p(S2,4) = 4 satisfying (R).


T T
(4) The Ti and bi are determined from the appropriate rows and columns
of P as:


P45 P43] =

P85I P83 =

P65 P633 =


[2 I 0 0]

[3 I 0 1 1]

(-1 -1 1 I 1]


b = 1 P41 P42 P43 P44 I 0 [0 1-2 0 0 1 ]

- = Cp81 P82 P83 P84 P85 P86 P87 P881 01] -3 0-1 0 -1 0 0110]

T = [ IP61 62 P63 P64 P65 P66] = [1 0 -1 1]
-i" 3 i6 p6 p6 1 ]


(5) Performing the column operations, obtain the structural array
SN,N and E as:

[P I S4,4 ) E] where the leading elements are circled,,


= -EP41
= -[P81

= -[p61


P42

P82

I P62










0 000 0 0 0



0 0 0 0 0 0 0 0


0 o0 ( 0 0 0 0 0


7
8


1 -2 -1 1 -4 7

S- 2 4 4

1 -1 T -
1 0 0 0 0

1 -1 -- -3

0 1 0 0

1 0

1


(6) The a. and b. are determined from the appropriate rows and columns

of E as:


e17
e37

e57


e27


el7

"27

e37

e47

e57

e67

e77

0


5
4
1

1
4
0
5
2
0

1

0


5-


2

1
8


' 2


L2


-4

e14

e24

e34

e44


e
14

e34

e54


e24


0



3

-1

1


-1

1

0


3
7


I


1 = N





51




(7) The canonical forms of ZR, BR(z), DR(z) and EC, BC(z), iC(Z) are:


FR
R


T


f3

T

T
"
-3


T
a
1.

a
2.
T
a5
5.

T
a
3.


1 2

1 2

2 4


1 0


z2-2z 0

-3 z2+z

z -z2+z


0

1

z2-zj


; DR() =


z 2z

z+1 2z+2

0 -2z


FC = e2 e3 -1 a 2

GC = [el 4]


HC = [a.1 a.3 a5 1 a 2]
C a1.


z.- [ z2 + 'Z + 4
BC(z) =
i.
8


1
-1
1


_Z3, 2
-z +z


2 4 2

2 6 2
1 3 0



; c(z) =


(8) The {vi} and TBR are determined simultaneously as:


and TBR
BR


T

T

T
eT

-1
a{.


1 0 0 0

0 1 0 0

0 0 1 0

3 0 1 1
3011


BR(z) [


- I-
+ i-

+ 3
IT


z2

z2
1


j2


z2- Iz

z2- 2z
22-z


V1

2=3


I









(9) TB1 is given by solving

1 0
T-1 0 1
-1
BR 0 o0
-3 0


(10) Find FBR and XF(z) as


BR BRR-BR
FBR-TBRFRTBR


2
0
0
2


the equations for the last row as:

0 0
0 0
1 0
-1
-1


0
0
1
2


and


XF(z) = (z-2)(z3-2z2+1) = z4_4z3+4z2+z-2


This

then


completes the first method. If the second method is used instead,

only (5)*, (6)*, and (8)* differ.


(5)* Performing the additional row operations and interchanges to

satisfy (2.3-8) gives:


5 1 1 0 5 7
-- 1 0 0 0 -I 0 -1 ~


5 3 1 1 1
-- 1 0 0 0 0 0 0 1 0 1 1 3
0V
O- o- o o o o -$
0 -4

X 08


are determined


-
i; =w- '


from


the appropriate


1
"-Il

1
0 -4

3


columns of Q as:


(6)* The


a.'s


a =
-1


I





53


This completes the algorithms. In the next chapter the first method is

modified to develop a nested algorithm from finite Markov sequences.















CHAPTER 3

PARTIAL REALIZATIONS

One of the main objectives of this research is to provide an

efficient algorithm to solve the realization problem when only partial

data is given. As new data is made available (e.g., an on-line

application, Mehra (1971)), it must be concatenated with the old

(previous) data and the entire algorithm re-run. What if the rank of

the Hankel array does not change? Effort is wasted, since the previous

solution remains valid. An algorithm which processes only the new data

and augments these results (when required) to the solution is desirable.

Algorithms of this type are nested algorithms.

In this chapter we show how to modify the algorithm of (2.4-1)

to construct a nested algorithm which processes data sequentially.

The more complex case of determining a partial realization from a fixed

number of Markov parameters arises when the rank' condition, abbreviated

(R), is not satisfied. It is shown not only how to determine the minimal

partial realization in this case, but also how to describe the entire

class of partial realizations. In addition, a new recursive technique

is presented to obtain the corresponding class of minimal extensions and

the determination of the characteristic equation is also considered.


3.1 Nested Algorithm

Prior to the work of Rissanen (1971) no earlier recursive methods

appeared in the realization theory literature. Rissanen uses a


I









factorization technique to solve the partial realization problem when

(R) is satisfied. His algorithm not only solves the problem in a
simple manner, but also provides a method for checking (R) simultaneously.

In the scalar case, Rissanen obtains the partial realizations, Z(K),

K=1,2,... imbedded in the nested problem of (2.1), but unfortunately

this is not true in the multivariable case. Also, neither set of

invariants is obtained.

The development of a nested algorithm to solve the partial

realization problem given in this dissertation follows directly from

(2.4-1) with minor modification. There are two cases of interest when

only a finite Markov sequence is available.

Case I. (R) is satisfied assuring that a unique partial

realization exists; or

Case II. (R) is not satisfied and an extension sequence

must be constructed.

The nested algorithm will be given under the assumption that Case I

holds in order to avoid the unnecessary complications introduced in

the second case. The modified algorithm is given below. The corresponding

row or column operations are performed only on the data specified

elements.

Partial Realization Algorithm. (3.1-1)

(1) Same as (1) and (2) of Algorithm (2.4-1) except (iv) is qkjfO

k>j if k is a row whose leading element has been specified.

(2) If (R) is satisfied for some M*=v+p, obtain the invariants as

before in (3), (4) of (2.4-1) and go to (5).If not, continue.









(3) Add the next piece of data, AM+1 and form S(M+1,M+1).
(4) Multiply S(M+1,M+1) by P. Perform row operations (if necessary)

using old leading elements to obtain Q (M+1,M+1). If (R) is

satisfied, continue. If not, go to 3.
(5) Perform column operations as in (5) of (2.4-1) and obtain the
invariants and canonical forms as in (6), (7). Go to 3.

Example (2.4-2) will be processed to demonstrate the modified algorithm

for comparison. Assume that the Markov parameters are sequentially

available at discrete times, i.e., Al is received, then A2, etc., and
the system is to be realized.


Example. (3.1-2) Let the Markov sequence be given

1 2] 2 4 [4 8
A = 1 -2 A = 2 4 A3 6 10 A4
_1 0 1 0 3 2-


by

S8 161 [16 32-
13 22 A5= 28 48
6 6 13 16


and apply the algorithm of (3.1-1). It is found that the rank condition

is first satisfied when A1, A2 are processed, i.e.,


(1) [I6 I S(2,2) 1 14

(2) Performing first row and then
obtain [P I S*(2,2) J E] or

1 0 0
-l 1 O

-2 0 0 1 0 0
-2 0 0 0 1 0 0
0 0 -1 0 0 1 0 0


column operation as in (3.1-1),


1 -2 -1
1 -
0 1


I _








(3) Indices are: v = 1 = 1
V2 =0 2 = 1
V =1

(4) :Invariants are: = -[P41 P43] = [2 ) 0]
__ -1'P 4 = [0 1 0]
S=-[P61 63 = 1

and
-N = [P41 P42 P43 P44 0] [-2 0 0 1 0 0]

T T T
b2 = [0 P21 P22 ] = [0 0 0-1 1 0]

S[P61 P62 P63 P64 P65 P66]= 0 0 -1 0 0 1]

e --1 e4 0
-i l13 114
3 1 e23 e24 -2
22 = ~ e 1 e
S24 2 33 34
-0 O_ e 44 1

(5) Canonical forms are:
1 2
FC I.1 2 GC = [ 1 I e2 HC = [a.1 I 2] = 1 2


z-1 0 1 2
Fc(z) = f C(z) =
-2" z-2 1 0

and
T P T -] -

FR =R IR 1
T 3 .
3 ~ ~_eT2_"3- -








where wTt = -[P 21 P23] =[1 0]


z-2 0 0 1 2
BR(z) = -2 z 0; DR(Z) = 0 0
0 0 z-1- 1 0

The rank condition is next satisfied when A1,...,A5 are processed,
.e, M =5 and we obtain [P (5,5) E as:
i.e., M =5 and we obtain [P | S (5,5) | E] as:


[P I Q(5,5)] =


1

-1

-1

-2

-2

1

-4

-3

0

-8

-8

-1

-16
-24

-8


() 2 2 4 4 8 8 16 16.32

0 0. 0 ( 2 5 61216

0 -1 -4 -1 -6 -2 -10 -3 -16
0 0 0 0 0 0 0 0

0 0 (J) 2 5 6 12 16
0 0 0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0

0 0

0 0


and performing the column operations give [S (5,5) I E~

wT is found easily from HRGR=A1 or solving for the second row of Hp,
wTGR [1 2].
GR


1

0 1

0 0 1

0 0 0 1

0 0 0 0


1

0 1

0 0

0 0

1 -1

0 0

0 -1

1 -2

0 0

1 -2

2 -3

0 0
0 -4

0 -5


1

0 1

0 -1

0 0

0 -1

0 -1

0 0

0 -2

0 -2

0 0
0 0

0 0









) 0 0 0 0 0 0 0 0 0 1 -2 -1 -1 -i- 4 10 -12
4 2
0 0 0 0 @ 0 0 0 0 0 1 i 1 -L' -6 -14
0o( 0 0 0 0 0 0 0 1 -1 -- -3 -- -- 15 20

0 0 0 0 0 0 0 01 0 0 0 0 0 0

0 0 () 0 0 0 0 0 1 -1 -3 -6 -8

0 0 0 0 0 0 0 0 1 0 0 0 0
0 0 0 0 0 0 1 0 0 0
0 0 0 0 0 0. 1 0 0

0 0 0 0 0 0 1 0

0 0 0 0 1

0 0 0 0
0 0 0 0
a 0

a a




The results in this case are identical to those of Example (2.4-2).

Let us examine the nesting properties of this realization algorithm.

Temporarily, we resort to using data dependent notation for this

discussion with the same symbols as defined previously in the previous
sections, e.g., the minimal partial realization of order M is given by

Z(M) :=- (F(M),G(M),H(M)). Thus, Z(M+k) is a (M+k)-order partial

realization. We also assume for this discussion that E(M) is in row

canonical form; therefore, it can be expressed in terms of the set of
invariants, [{vi(M)},{ (M)},{aT(M)}]. If E(M) is an n dimensional,

minimal partial realization specified by these invariants, then there

are n regular vectors, -+pt(M) spanning the rows of S(M,M). Furthermore,
Ss+pt








each dependent row vector, J.(M) is uniquely represented as a linear
combination of regular vectors in terms of the observability invariants

and it can be generated from the recursion of Corollary. (2.3-2). Similarly,

it follows from Proposition (2.2-13) that the dependent block row

vectors, aj (M) satisfy an analogous recursion. The following lemma
j.
describes the nesting properties of minimal partial canonical realizations.

Recall that M is the integer of Proposition (2.1-6) given by M =v+p.

Lemma. (3.1-3) Let there exist an integer M*(M)IM such that the rank

condition is satisfied and let Z(M) be the corresponding
minimal partial canonical realization of {Ar}, reM
specified by the set of invariants [{v.(M)},{ist(M)},

{a (M)}]. Then

vi(M) = vi(M+k)
Bist(M) = ... = ist(M+k)

aT (M) = = aT (M+k)
j. J.

iff p(S(M,M))=p(S(M+1,M+1) = ... = p(S(M+k,M+k))
for the given k.

Proof. If v.(M) = ... = v.(M+k), etc., then the minimal canonical
1 1
partial realizations are identical, E(M)=E(M+1)= ... =E(M+k).

It follows that p(S(M,M))=dimE(M)=p(S(M+1,M+I))=dimE(M+I)=

p(S(M+k,M+k)).
Conversely, P(S(M,M))=P(S(M+1,M+1))= ... =P(S(M+k,M+k)) implies
dimZ(M)=dimE(M+1)=... =dimE(M+k). Since E(M) is a unique minimal

canonical partial realization, so is Z(M*). Furthermore, since

each realization has the same dimension, each realization has









has M*(M)=M*(M+1)= ... = M*(M+k) so that each canonical
realization is equal to Z(M*); therefore, Z(M)=Z(M+1)= ... =Z(M+k).V

Next we examine the case where E(M) and z(M+k) are of different
dimension. The nesting properties are given in the following lemma.

Lemma. (3.1-4) Let there exist integers, M*(M)M, M (M+k)-M+k such
that the rank condition is satisfied (separately) and
E(M), Z(M+k) are minimal partial canonical realizations
of {Ar} whenreM andrsM+k, respectively, for given k.
If p(S(M+k,M+k))>p(S(M,M)), then vi(M+k)-vi(M), iep.
T T
Furthermore, a .(M+k)=a. (M), j=i,i+p,...,i+p(vi(M)-l).
j. j. 1
Proof. Since p(S(M+k,M+k))>p(S(M,M)), M*(M+k)>M*(M) and therefore,

Sv(M),(M) is a submatrix of Sv(M+k),,(M+k). If the j-th row

of Sv(M),p(M) is regular, it follows that the j-th row of

Sv(M+k),i(M+k) is also regular by the nature of the Hankel
pattern, i.e., the rows of Sv(M),p(M) are subrows of

Sv(M+k),p(M+k). The addition of more data (AM+1,...,AM+k) to
S(M,M) makes previously dependent rows become independent rows
but previously independent rows remain independent; thus, the
v.(M) can only increase or remain the same, i.e., v.(M+k) -
vi(M), iep. The set of regular {a (M+k)} are specified by the
vi(M+k)'s; therefore aT (M+k)=a (M), j=i,i+p,...,i+p(v.(M)-1),
( k'. 1
since vi(M+k)Qvi(M), iep.V

The results of these two lemmas are directly related to the nesting
properties of the partial realization algorithm. First, define JM as the
set of indices of regular Hankel row vectors based on M Markov parameters


1









available, i.e., J = {1,1+p,... ,1+p( .(M)-l),...,p,2p,... ,pv (M)}
and similarly denote the row vectors of P1 the elementary row matrix
of the previous chapter, by T(M). From Lemma (3.1-3), it follows
S* T T
that JM J+k and i+p(M)(M) =... i+ (M+k)(M+k) since

the observability invariants are identical. The vi specify the
elements in J and along with the Bist, they specify the elements of
T
Pi+p. (M)(M) (see Corollary (2.3-4)). From Lemma (3.1-4) it is clear

that JMJM+k since vi(M+k)Zvi(M).
Reconsider Example (3.1-2), to see these properties. In this
case we have M=2, k=3, M (2)=2, M*(5)=5, and p(S(5,5))>p(S(2,2)) as
in Lemma (3.1-4); therefore, J*cJ*, since J2 = {1,3} and J* = {1,3,2,5}.

The observability indices are identical except for v2(5)>v2(2); thus,
{a (2),a, (2)}c{aT (5),aT (5),aT (5),aT (5)} since aT (2) = aj (5)
1 3. I 3. 2. 5. J. j.
for j=1,3. We also know from Example (2.4-2) that Z(5) is the solution
to the realization problem and therefore the properties of Lemma (3.1-3)
will hold for {AM}, M>5. Table (3.1-5) summarizes these properties.
The results for the dual case also follow directly. We now proceed to
the case of constructing minimal partial realizations when.(R) is
not satisfied, i.e., the construction of minimal extensions.





63
















Table. (3.1-5) Nesting Properties of Algorithm (3.1-1)


Augment M-4+k


JMJM+k


n(M+k)=n(M)


n(M+k)>n(M)


T
pi+pvi


Vi


Bist


where (R) is satisfied for some k and C means that the

corresponding invariants, vectors, or indices are nested or

contained in a set of higher order.


I









3.2 Minimal Extension Sequences

In this section we discuss the more common and difficult

problem of obtaining a minimal partial realization from a finite

Markov sequence when (R) is not satisfied. Two different approaches

for the solution of this problem have evolved. The first is based

on constructing an extension sequence so that (R) is satisfied

and the second is based on extracting a set of invariants from

the given data. We will show that these methods are equivalent

in the sense that they may both lead to the same solution. In order

to do this the existing algorithm is extended to obtain the more

general results of Roman and Bullock (1975a). Also a new recursive

method for obtaining the entire class of minimal extensions is

presented. It is shown that the existing algorithm does in fact

yield a particular solution to this problem which is valuable in

many modeling applications.

In the first approach, Kalman (1971b),Tether (1970), and

subsequently Godbole (1972) examine the incomplete Hankel array

to determine if (R) is satisfied. If so, the corresponding minimal

partial realization is found. If not, a minimal extension is con-

structed such that (R) holdsand a realization is found as before.

They show that a minimal extension can always be found, but in

general it will be arbitrary. They also show that this extension

must be constructed so that the rank of S(M,M) remains constant

and the existing row or column dependencies are unaltered.

Considerable confusion has resulted from the degrees of freedom









available in the choice of minimal extensions. In fact, initially,

the major motivation for construction an extension was that it

was necessary in order to be able to apply Ho's algorithm. Un-

fortunately, these approaches obscure the possible degrees of

freedom and may lead to the construction of non-minimal extensions

as shown by Godbole (1972).

Roman and Bullock (1975a)developed the second approach to

the solution of this problem. They show that examining the columns

or rows of the Hankel array for predecessor independence yields

a systematic procedure for extracting either set of invariants

imbedded in the data. They also show that some of these would-be

invariants are actually free parameters which can be used to

describe the entire class of minimal partial realizations. These

results precisely specify the number and intrinsic relationship

between these free parameters. Unfortunately Roman and Bullock

did not attempt to connect their results precisely with those in

Kalman (1971b),Tether (1970). It will be shown that this connection

offers further insight into the problem as well as new results

which completely describe the corresponding class of minimal extensions.

Before we state the algorithm to extract all invariants available

in the data, let us first motivate the technique. When operating

on the incomplete Hankel array, only the elements specified by the

data are used. It is assumed that the as yet unspecified elements

will not alter the existing predecessor dependencies when they are

specified by an extension sequence. Since the predecessor dependencies

are found by examining only the data in S(M,M), we must examine

complete submatrices of S(M,M) in order to extract the invariants









associated with a particular chain (see Roman and Bullock (1975a)).

Therefore, it is possible that a dependent vector, say !T of a sub-
-1
matrix of S(M,M) later corresponds to an independent vector in S(M,M).

When representing any other dependent vector in this submatrix

in terms of regular predecessors, T. must be included, since it is
-1I
a regular vector of S(M,M) under the above assumption. In this represen-

tation the coefficient of linear dependence corresponding to Y
-1
is arbitrary. Reconsider Example (3.1-2) for{Ai}, i=1,2,3 where we

only consider the (row) map P.


Example. (3.2-1) For A1, A2, A3 of (3.1-2) we have P: S(3,3)+Q(3,3) or


1 2 2 414 8~ -@2 2 414 8

1 2 2 416 10 0 0 0 0)2

1.0 1 03 2 .0_1-4-1-6

2448 0 0 0 0
P
2 4 6 10 -- 0 0 2

1 0 3 2 0 0 0 0

48 00

6 10 0 0

3 2 _0 0
The indices are {vl,v2,v3} = {1,2,1}. Since v1=1, the fourth row of

S(3,3) (or equivalently Q(3,3) ) is dependent on its regular predecessors

as shown in the corresponding 3x4 submatrix (in dashed lines) of S(3,3)
^T
(or Q(3,3) ). The second row, say 2 in this submatrix is dependent,
yet it is an independent row of S(3,3) (or Q(3,3) ). Now, expand T of

this submatrix, i.e.,








^T =T + T ^T
4 110 1 120 130 3 121'0)
or

[2 4 4 8] = B110 [1 2 2 4] + B120 [1 2 2 4] + B130[1 0 1 0]


The solution is B110 = 2 120' g130=0; thus, the coefficient 8120 is
ah arbitrary parameter. Note that this recursion is essentially the

technique given in Roman and Bullock (1975a).


Clearly, if (R) is satisfied as in the previous section, then there
exists a complete submatrix (data is available for each element) of S(M*,M*)
in which every regular vector of S(M,M) is always a regular vector

of the submatrix corresponding to a particular chain; thus, there

will be no arbitrary or free parameters.

The algorithm for the case when (R) is not satisfied may be
illustrated by considering row operations on S(M,M) to obtain Q(M,M),

since the identical technique can be applied to obtain S*(M,M). The

arbitrary (column) parameters are found by performing additional

column operations to Q(M,M). As in Example (3.2-1), we must find
the largest submatrix of Q(M,M) for each chain, i.e., if we define
k as the index of the block row of S(M,M) containing the: vector ,
- -1i+p vi
then the largest submatrix of data specified elements corresponding

to the i-th chain is given by the first (i+pvi-l) rows and m(M+1-ki)
columns of Q(M,M). Also, we define Ji,iep as the sets of Hankel row
indices corresponding to each dependent (zero) row of the

(i+pu-1l)x (m(M+1-ki) submatrix of Q(M,M) which becomes independent, i.e.,

it contains a leading element. In Example (3.2-1) for i=l, we have

(l+pv-l)3 and k1=2; thus, m(M+l-kl)=4 and the corresponding submatrix is
given by the first 3 rows and columns of Q(3,3), and. of course, J1={2}.


I_ I








Arbitrary Parameter Partial Realization Algorithm. (3.2-2)

(1) Perform (1) of Algorithm (3.1-1) to obtain [P I Q(M,M)]
(2) For each iep, determine the largest (i+pwl)xm(M+1-k.) sub-
1 1
matrix of Q(M,M) of data specified elements and form the set Ji
T T T
(3) For each iep, replace by + b, ba scalar.

(4) Determine the corresponding canonical forms incorporating
these free parameters.

Dual results hold for the columns. The free-parameters are
fund in analogous fashion by examining the zero columns of the
submatrices of S*(M,M).

Example. (3.2-3) The following example is from Tether (1970).
For m=p=2 and


1 1 2 4 3 10 7 22 15]
A 0 0 0 0 1 3 3
(1) [ P r Q(4,4) ] =

1 ( 1 4 3 10 7 22 15
0 1 0 0 0, 0 ( 3
-4 0 1 0o o-6 -5-18-13
0 0 0 1 0 0 06 1 0 0
2 0 -3 0 1 0 01 0
-1 0 0 1 0 1 0 0o 0o
6 0 -7 0 0 0 1 0 0
-3 0 0 0 0 0 0 1 0 0

It should be noted that when (R) is not satisfied, some of the v. may not
be defined, i.e., the last independent row of a.chain is in the last block
Hankel row. In this case all would-be invariants are arbitrary.





69



The indices are: v1=2, v2=3

(2) For i=1, (1+p -1)=4, kl=3, m(M+1-k,)=4; thus, the corresponding
submatrix is constructed from the first 4 rows and columns of Q(4,4)
(small dashes). J1={2}.

For i=2, (2+pv2-1)=7, k2=4, m(M+1-k2)=2; thus, the corresponding
submatrix of Q(4,4) is given by the first 7 rows and 2 columns (large
ra'shes). J2={2,4,6}.

(3) Replacing the fifth and eighth rows of P with p + bPj2 and
T T T T
p + cP2 + dp + ePg where b,c,d,e are real scalars gives
T
S= [ 2 b -3 0 1 0 0 0]

S[-3-e c 0 d+e 0 e 0 1 ]
p8 = [-3-e c 0 d+e 0 e 0 1


The T are:
-1


_-= [ -2


T = [ 3+e


3 -b


0 0]


-c -(d+e) -e ]


(4) The canonical form is

T
-2
T



F T1
FR= eT

T
-4

4


T
e
-1
HR= GR=
T
eL


Corresponding to these realizations is a minimal extension sequence

which can be found by determining the Markov parameters. These parameters


T
a

Ta
3.


a
a4.

T

6,


0 0

0 0

0 0








are cumbersome to obtain due to the general complexity of the expressions

in zR or ZC; therefore, a technique to determine these extensions
without forming the Markov parameters directly (or the realization) was
developed. This method consists of recursively solving simple linear
equations (one unknown) to obtain the minimal extension. Extensions
constructed in this manner not only eliminate the possibility of non-

minimality as expressed in Godbole (1972), but also describe the entire

class of minimal extensions. The method of constructing the minimal
extension sequence evolves easily from the lower triangular-unit diagonal
structure of P. Since a dependent row of Q(M,M) is a zero row, it

follows from Theorem (2.3-3) that

T
+pvi j qi+pi j 0 for jemM (3.2-4)

where recall that p i+i=0 for j>i+pv.. Thus, by inserting the
1ipq,j 1
unknown extension parameters, x..(r) for


rx11(r) *** xm(r)

Ar

x pl(r) *.. pm(r)


into S(M,M) a system of linear equations is established in terms of the
xi (r)'s by (3.2-4). Due to the structure of P, this system of equations
is decoupled and therefore easily solved.

Example. (3.2-5) Reconsider (3.1-2) for Al, A2. Since (R) is satisfied,
the extension A., j>2 is unique. We would like to obtain
3









x11(3) x12(3)

x21(3) x22(3)

x31(3) x32(3)


Since P maps S(2,2) into Q(2,2), we


have


1 2 2 4
1 2 2 4
1 0 1 0


r-" -
2 4 x11(3)
2 4 x21(3)

_1 0 x31(3)


X -
x12(3)
x22(3)
x32(3)


P
-jo


0


2 2 4
0 0 0


0 -1
0 0 0
0 0 0
0 0 0
I01


-4
0
0
0


and in this case,{vl,v2,v3} ={1,0,1}. Thus, using (3.2-4), we


solving 0 = P4T 13 = [-2


0 0 1 o


x11(3)
x21(3)
x31(3)


for x11(3) gives x11(3)=4


Similarily solving:


T
p44 = 0
T

T


for x12(3) gives x12(3) = 8

for x31(3) gives x31(3) = 1

for x32(3) gives x32(3) = 0


In this example, x21(3)=x11(3) and x22(3)=x12(3), since v2=0.


have








Thus, this example shows that the minimal extension sequence can be
found recursively due to the structure of P. Of course, the problem
of real interest is when (R) is not satisfied and (as in Ho's algorithm)
a minimal extension with arbitrary parameters must be constructed.

Minimal Extension Algorithm. (3.2-6)
(1) Perform (1), (2), (3) of Algorithm (3.2-2).
(2) Determine M* = v+-p. (The values of v,p are determined by the partial data)
(3) Recursively construct the minimal extension {Ar}, r = M+1, ... ,M*
where Ar [xij(r)] P by solving the set of equations for xij(r)
given by
+v = 0, j = m(M+l-k)+l, ... ,m(M*+1-ki), for each iep.
p-i +pv 1i)

and recall that ki is the index of the block row of S(M,M) containing
T
the row vector, Y
-i+pvvi.

Example. (3.2-7) Reconsider (3.2-3) for illustrative purposes.

(1) These results are given in Example (3.2-3)
(2) M*=6; thus, find


A5 = Fxll(5) x12(5)1
x21(5) x22(5)

(3) Recursively solve: Pi2vi r = .
i=2, j=3,4,5,6.

p5 5 = 0 gives x11(5);

p T3 = 0 gives x21(5);


= 0 gives

= 0 gives


A6 = xl'(6) x 2(6)-
x21(6) x22(6)

for i=l, j=5,6,7,8 and for


T


= 0 gives x12(5)

= 0 gives x22(5)


= 0 gives

= 0 gives'


xl1(6);

x21(6) ;


x12(6)

x22(6)









and therefore

46-b 31-b 94-6b 63-6b
A5 = A6
12-d 9-d-e 30-c-3d-5e+de 21-c-3d-5e+de-e2

By solving for the x i's in A5, A6 we obtain the extension as


A x11(5) x11(5)-15~ A 6x11(5)-182 6x11(5)-213
x21(5) x22(5) x21(6) x21(6)+(21(5)- (5-3)-9

The number of degrees of freedom is 4,i.e.,{x11(5),x21 (5),x22(5),x2(6)}.


The technique used to solve the partial realization problem when (R)
is not satisfied was to extract the most pertinent information from the

given data in the form of the invariants, which completely described
the class of minimal partial realizations. A recursive method to obtain
the corresponding class of minimal extensions was also presented in (3.2-6).
This method is equivalent to that of Kalman (1971b) or Tether (1970) for

if the minimal extension is recursively constructed and Ho's algorithm
is applied to the resulting Hankel array the corresponding partial real-
ization will belong to the same class. Note that if the extension is not
constructed in this fashion, it is possible that all degrees of freedom

available may not be found (see Roman (1975)). It should be noted that
the integers v and i are determined from the given data,i.e., knowledge
of the invariants enables the construction of a minimal extension such
that v and p can be found. The approach completely resolves the ambiguity
pointed out by Godbole (1972) arising in the Kalman or Tether technique.

The results given above correspond directly to those presented in









Kalman (1971b) and Tether (1970). They have shown, when (R) is satisfied,

there exists no arbitrary parameters in the minimal partial realization

or corresponding extension. Therefore, the existence of arbitrary

parameters can be used as a check to see if the rank condition holds.

Although it is not essential to construct both sets of invariants, it is

necessary to determine M* which requires v and i; thus, the algorithm

presented has definite advantages over others, since these integers are

simultaneously determined.

In practical modeling applications, the prediction of model

performance is normally necessary; therefore, knowledge of a minimal

extension is required. Also in some of the applications the number of

degrees of freedom may not be of interest, if only one partial realization

is required rather than the entire class. In this case such a model is

easily found by setting all free parameters to zero which corresponds to

merely applying the Algorithm (3.1-1) directly to the data and obtaining

the corresponding canonical forms as before.

Describing the class of minimal extensions offers some advantages

over the state space representation in that it is coordinate free and

indicates the number of degrees of freedom available without compensation.


3.3 Characteristic Polynomial Determination by Coordinate Transformation

In this section we obtain the characteristic equation of the entire

class of minimal.partial realizations described by FR or FC of the

previous section. It is easily obtained by transforming the realized

FR or FC into the Bucy form as before. Recall that the advantage of

this representation over the Luenberger form is that it is possible to

find the characteristic polynomial directly by inspection of FBR in (2.2-11).


I









Even though it is possible to realize the system directly in Bucy form
as implied in the discussion of (2.3-12), it has been found that this

method has serious deficiencies when dealing with finite Markov sequences.

If (R) is satisfied, the partial realization is unique. When (R) is not

satisfied, this technique does not yield all degrees of freedom. For

example, reconsider the arbitrary parameter realization of Example (3.2-3).

This realization is given in Ackermann (1972) as



-0 1 0 0 0 0 1 0 0 0

-2 3 -b 0 0 -2 3 0 0 0
FR = 0 0 0 1 0 FAck = 0 0 0 1 0

0 0 0 0 1 0 0 0 0 1

3+e 0 -c -(d+e) -e 3+e 0 -c -(d+e) -e


Note that one degree of freedom (b=0) has been lost. Similarily

Ledwich and Fortmann (1974) have shown by example that this technique

can also lead to non-minimal realizations. These deficiencies arise due

to the procedure used for the determination of the Bucy invariants. This

procedure does not account for the possibility that an independent row

vector of a particular chain may actually be dependent if it is compared

with portions of the same length of vectors in different chains. To cir-

cumvent the problem, the previous technique will be used,i.e., the system

is realized directly in Luenberger form and transformed to Bucy form. Not

only does this assure minimality as well as the determination of all possible

degrees of freedom, but TBR is almost found by inspection as shown in

(2.3-12). Reconsider the example of the previous section.









Example. (3.3-1) Recall that in (3.2-3) m=p=2, n=5, and v1=2, v2=3,


T = [ -2 3 -b 0 0
.k = [ -2 3 -b 0 0 ]


(1) Simultaneously construct TBR

for predecessor independence


I
-1
T
e
-2
T
TBR -
BTFR
-1 R

BTF
-1 R.

(2) Determine T -1 from
BR


-1
BR


1

0

-2/b

0

0


T
j2= [3+e 0 -c -(d+e) -e i


from (3.3-4) while examining the rows


3 -b

7 -3b


-14 15 -7b -3b' -b


TR R= I which gives
BRBR n


1

3/b

-2/b

0


-1/b

3/b

-2/b


0


-1/b

3/b -1/b


(3) Determine FBR:


-1BR BR RBR
FBRTBRFRTBR=


(4) Find the characteristic polynomial by inspection.


0

0

0

0

-3b-2c-ce


1

0

0

0

3c-2d-2e


0

1

0

0

-c+3d+e


0

0

1

0

-d+2e-2


0

0

0

1

-e+3









XFBR(z) = z+(e-3)z4+(d-2e+2)z'+(c-3d-e)z2+(-3c+2d+2e)z+(b+2c+be)


This example points out some very interesting points. When this

technique is combined with the algorithm of (3.2-2), it offers a method

which can be used to obtain the solution to the stable realization

problem developed in Roman and Bullock (1975b). Also, if the system
were realized directly in Bucy form, then b=O and a degree of freedom is

lost; thus, in Ackermann's example 91=1, while ours is Z1=5. It is

critical that all degrees of freedom are obtained as shown in this case,

since the system is observable from a single output.
This section concludes the discussion of the deterministic case of

the realization problem. In the next chapter we examine the stochastic

version of the realization problem.














CHAPTER 4

STOCHASTIC REALIZATION VIA INVARIANT SYSTEMS DESCRIPTIONS


In this chapter the stochastic realization problem is examined

by specifying an invariant system description under suitable trans-

formation groups for the realization. Superficially, this may appear

to be a direct extension of results previously developed, but this is

not the case. It will be shown that the general linear group used in

the deterministic case is not the only group action which must be

considered when examining the Markov sequence for the corresponding

stochastic case.

Analogous to the deterministic realization problem there are

basically two approaches to consider (see Figure 1): (1) realization

from the matrix power spectral density (frequency domain) by performing

the classical spectral factorization; or (2) realization from the

measurement covariance sequence (time domain) and the solution of a set

of algebraic equations. Direct factorization of the power spectral

density (PSD) matrix is inefficient and may not be very accurate.

Recently developed methods of factoring Toeplitz matrices by using fast

algorithms offer some hope, but are quite tedious. Alternately,

realization from the covariance sequence is facilitated by efficient

realization algorithms and solutions of the Kalman-Szeg6-Popov equations.









REALIZATION FROM FACTO T
COVARIANCE SEQUENCE -PSD -- MTOR TIO
AND ALGEBRAIC METHODS METHODS


STOCHASTIC REALIZATION


Figure 1. Techniques of Solution to the Stochastic
Realization Problem.


The problem considered in this chapter is the determination of a

minimal realization from the output sequence of a linear constant

system driven by white noise. The solution to this problem is well known

(e.g.. see Mehra (1971)) as diagrammed below in Figure 2. The output

sequence of an assumed linear system driven by white noise is correlated

and a realization algorithm is applied to obtain a model whose unit

pulse response is the measurement covariance sequence. A set of algebraic

equations is solved in order to determine the remaining parameters of

the white-noise system.

This problem is further complicated by the fact that the covariance

sequence must be estimated from the measurements. From the practical

viewpoint, the realization is highly dependent on the adequacy of the

estimates. Although in realistic situations the covariance-estimation

problem cannot be ignored, it will be assumed throughout this chapter

that perfect estimates are made in order to concentrate on the realization

portion of the problem.t

In this chapter we present a brief review of the major results

necessary to solve the stochastic realization problem. We use the



Majumdar (1976) has shown in the scalar case that even if imperfect
estimates are made realization theory can successfully be applied.










*White Noise Input






Linear Constant
System


Solve Algebraic
Equations


Output Sequence


Measurement Covariance Sequence


Stochastic Realization


Figure 2. A Solution to the Stochastic Realization Problem


Correlation
Techniques


Realization
Algorithm


m


__ ___









algebraic structure of a transformation group acting on a set to obtain

an invariant system description for this problem. A new realization

algorithm is developed to extract this description from the covariance

sequence. Recently published results establishing an alternate approach
to the solution of this problem are also considered.

4.1 Stochastic Realization Theory

Analogous to the deterministic model of (2.1-1) consider a white-
noise (WN) model given by

SFx + w (4.1-1)

Xk =Hx

where x and y are the real, zero mean, n state and p output vectors,
and w is a real, zero mean, white Gaussian noise sequence. The noise

is uncorrelated with the state vector, A., j 5 k and

Cov(w ,w.):=E[(w.-Ew.)(w.-Ewj)] = Q6i

where 6ij is the Kronecker delta. This model is defined by the triple,
WN :=(F,I,H) of compatible dimensions with (F,H) observable and F a

nonsingular,t stability matrix, i.e., the eigenvalues of F have magnitude

less than 1. The transfer function.of (4.1-1) is denoted by TWN(z).




tIn the discussion that follows the WN model parameters will be used to
obtain a solution to the stochastic realization problem. Denham (1975)
has shown that if the spectral factors of the PSD are of least degree,
i.e., they possess no poles at the origin, then F is a nonsingular matrix.








The corresponding measurement process is given by


-k -k + Y-k (4.1-2)

where z is the p measurement vector and v is a zero mean, white
Gaussian noise sequence, uncorrelated with x., j 5 k with

Cov(vi,.j) = R6ij

Cov(wA,.) = S6ij

for R a pxp positive definite, covariance matrix and S a nxp cross
covariance matrix. Thus, a model of this measurement process is
completely specified by the quintuplet, (F,H,Q,R,S).
When a correlation technique is applied to the measurement process,
it is necessary to consider the state covariance defined by

Sk: =Cov(x, )

We assume that the processes are wide sense stationary; therefore,

k = T, a constant here. It is easily shown from (4.1-1) that the
state covariance satisfies the Lyapunov equation (LE)

I = FIFT + Q (4.1-3)

It is well known (e.g. see Faurre (1967)) that since F is a stability
matrix, corresponding to any positive semidefinite covariancee) matrix Q,
there exists a unique, positive semidefinite solution I to the (LE).
The measurement covariance is given (in terms of lag j) by

Cj:= Cov(k+j'k) = Cov(yk+j,yk)+Cov(yk+j'vk)+Cov!(vk+j'yk)+Cov(k+j ',k)

(4.1-4)









and from (4.1-1) it may be shown that


C. = HFj-I(F~HT+S) j > 0 (4.1-5)

C = HnHT + R
0

The PSD matrix of the measurement process is obtained by taking the

bilateral z-transform of the sequence C. defined in (4.1-4) which gives
3

DZ(z) = H(Iz-F) Q(Iz1-FT)l HT+H(Iz-F)-1S+ST(Iz-1-FT) -HT+R
(4.1-6)
It is important to note that this expression is the frequency domain

representation of the measurement process which can alternately be

expressed directly in terms of the measurement covariance sequence as

00 -
4Z(Z) = Z C.z
j=-o, J

Since the measurement process is stationary and z is real, C =CT and
-k k
therefore the PSD can be decomposed as

00 .z
(z) = Z C.z" + C + Z CTz. (4.1-7)
Sj=l J j=l J

Note that {C.} is analogous to the Markov sequence of the deterministic

realization problem. We define the problem of determining a quintuplet,

(F,H,Q,R,S) in (4.1-6) from Z(z) or {C,} as the stochastic realization

problem.

In this chapter we are only concerned with the realization from the

measurement covariance sequence. When a realization algorithm is applied

to the covariance sequence, we define the resulting realization as the

Kalman-Szegi-Popov (KSP) model because of the parameter constraints








(to follow) which evolve from the generalized Kalman-Szegb-Popov lemma

(see Popov (1973)t). Thus, we specify the KSP model as the realization

of {C .}t defined by the quadruple, KSP:=(A,B,C,D) of appropriate

dimension with transfer function, TKSP(z)=C(Iz-A) -B+D. Note that since
the unit pulse response of the KSP model is simply related to the
measurement covariance sequence, then (4.1-7) can be written as the

sum decomposition.

S(z) = TKSP(z)+TKp(z-1) = C(Iz-A)-1B+D+DT+BT(Iz'1-AT)-1CT (4.1-8)


The relationship between the KSP model and the stochastic realization
of the measurement process is shown in the following proposition by

Glover (1973).

Proposition (4.1-9) Let ZKSP=(A,B,C,D) be a minimal realization of {Cj}.

Then the quintuplet (F,H,Q,R,S) is a minimal stochastic

realization of the measurement process specified

by (4.1-1) and (4.1-2), if there exists a positive
definite, symmetric matrix H and TeGL(n) such that
the following KSP equations are satisfied:

n-AlIAT =Q

D+DT-ICT = R

B-AHCT = S
where A=T-1FT and C=HT.

The proof of this proposition is given in Glover (1973) and

This book was published in Romanian in 1966, but the English version
became available in 1973.
tNote that the sequence, {C.}1 is related to the measurement covariance
sequence as Co =Co and C.=C. for j > 0.
o o 33








corresponds directly to the results presented by Anderson (1969) in
the continuous case. The proof follows by comparing the two distinct
representations of 0Z(z) given by (4.1-6) and (4.1-8). Minimality of
(F,H,Q,R,S) is obtained directly from Theorem (3.7-2) of Rosenbrock
(1970). The KSP equations are obtained by equating the sum decomposition
of (4.1-8) to (4.1-6).
This proposition gives an indirect method to check whether a given

EKSP and stochastic realization, (F,H,Q,R,S) correspond to the same
covariance sequence. Attempts to use the KSP equations to construct
all realizations, (F,H,Q,R,S) with identical {C.} from ZKSP and T by
choice of all possible symmetric, positive definite matrices, I will
not work in general because all I's do not correspond to Q,R,S matrices
that have the properties of a covariance matrix, i.e.,

A:= Cov( [w v) Q S (4.1-10)
T '6
Li R

First, it is necessary to question if the stochastic realization problem
always has a solution, or equivalently, when is there a i so that
(4.1-10) holds. Fortunately, the well-known PSD property, on the unit circle (see e.g. Gokhberg and Krein (1960) and Youla (1961))
is sufficient to insure the existence of a solution. This result is
available in the generalized Kalman-Szegb-Popov lemma (see Popov (1973)).

Proposition (4.1-11) If (F,H) is completely observable, then DZ(Z) Z 0
on the unit circle is equivalent to the existence
of a quintuplet, ( ,,R,,) such that








z(z) = [f(Iz-?)-1 I p[Q (Iz-1-FT)-1"T

i" Rj- p
where

STV [VT T >


The proof of this proposition is given in Popov (1973) and essentially
consists of showing there exists a spectral factorization of the given

PSD. Thus, this proposition assures us that there exists at least one
solution to the stochastic realization problem.
Proposition (4.1-9) shows that once ZKSP, T, and n are determined

then a stochastic realization, (F,H,Q,R,S) may be specified; however, it
does not show how to determine I. Recently many researchers (e.g. Glover
(1973), Denham (1974,1975), Tse and Weinert (1975)) have studied this
problem. They were interested in obtaining only those solutions to the
KSP equations of (4.1-9) which correspond to a stochastic realization
such that AO of (4.1-11). Denham (1975) has shown that any solution,
n*, of the KSP equations which corresponds to a factorization as in

(4.1-11) with V=KN, W=N for K=Knxp, NeKPxP, K full rank and N symmetric
positive definite, is in fact a solution of a discrete Riccati equation.
V 'V TT T T
This can readily be seen by substituting, (Q,R,S) =(KNN KNN KNN)
of (4.1-11) into (4.1-9)

I*-AI*AT = KNNTKT (4.1-12)

D+DT-CT*CT = NNT

B-AJ*CT = KNNT for A = T- FT, C=HT, TeGL(n).








Solving the last equation for K and substituting for NNT yields

K = (B-AI*CT )(D+DT-C*CT)-1 (4.1-13)

Now substituting (4.1-13) and NNT in the first equation shows that I*

satisfies

= AI*AT-(B-An*cT)(D+DT-_*cT) (B-AH*CT)T (4.1-14)


a discrete Riccati equation. Thus, in this case the stochastic

realization problem can be solved by (1) obtaining a realization,

EKSP from {C.}; (2) solving (4.1-14) for 1*; (3) determining NNT from
(4.1-12) and K from (4.1-13); and (4) determining Q,R,S from K and NN.

A quintuplet specified by T and I* obtained in this manner is guaranteed

to be a stochastic realization, but at the computational expense of solving

a discrete Riccati equation. Note that solutions of the Riccati equation

are well known and it has been shown that there exists a unique, I*,

which gives a stable, minimum phase, spectral factor (e.g. see Faurre

(1970), Willems (1971), Denham (1975), Tse and Weinert (1975)). We
will examine this approach more closely in a subsequent section, but
first we must find an invariant system description for the stochastic
realization.


4.2 Invariant System Description of the Stochastic Realization

Suppose we obtain two stochastic realizations by different methods
from the same PSD. We would like to know whether or not there is any

way to distinguish between these realizations. To be more precise,

we would like to know whether or not it is possible to uniquely

characterize the class of all realizations possessing the same PSD.
We first approach this problem from a purely algebraic viewpoint.








We define a set of quintuplets more general than the stochastic

realizations, then consider only those transformation groups acting

on this set which leave the PSD or equivalently {C invariant, and

finally specify various invariant system descriptions under these

groups which subsequently prove useful in specifying a stochastic

realization algorithm. The groups employed were first presented by

Popov (1973) in his study of hyperstability. The results we obtain

are analogous to those of Popov as well as those obtained in the

quadratic optimization problem (e.g. see Willems (1971)).

Define the set

X2 = {(F,H,Q,R,S)I FeKnxn,HeKPxn,QeKnxnReKPxPSeKnxp; Q,R symmetric}

and consider the following transformation group specified by the set

GK := {L ILEKnxn; L symmetric}

and the operation of matrix addition. Let the action of GK on X2 be
n b
defined by

L (F,H,Q,R,S) := (F,H,Q-FLFT+L,R-HLHT,S-FLHT) (4.2-1)

This action induces an equivalence relation on X2 written for each pair

(F,H,Q,-R,S), (F,H,Q,R,S)eX2 as (F,H,Q,R,S)EL(F,H,Q,R,S) iff there exists
a LEGKn such that (F,H,Q,-R,S) = L +(F,H,Q,R,S).

This group and GL(n) are essential to this discussion, but we must

consider their composite action. Therefore, we define the transformation

group, GRn which is the cartesian product of GL(n) and GK ,

GRn := GL(n)xGK The following proposition specifies GR .
nn n


i








Proposition. (4.2-2) The closed set GRn and operation o form a group
where
GRn = {(T,L) IT GL(n);LeGK n

and the group operation is given by

(T,)o(T,L) = (TT,L+T- LTT).

Proof. This proof of this proposition follows by verifying the standard
group axioms with respective identity and inverse elements

(In,0O) and (T-1,-TLTT).v

Let the action of GRn on X2 be defined by

(T,L) + (F,H,Q,R,S):=(TFT-1,HT-',T(Q-FLFT+L)TT,R-HLHT,T(S-FLHT)) (4.2-3)

An element (F,HQ,R,<) of the set X2 is said to be equivalent to the
element (F,H,Q,R,S) of X2 if there exists a (T,L)eGR such that
(F,H,Q,R,S)=(T,L)+(F,H,Q,R,S). This relation is reflexive

(F,H,Q,R,S) = (InOn) +(F,H,Q,R,S)

and symmetric

(T-1,TLTT)+(F,H,Q,R,S)=(T-1,-TLTT)+((T,L)+(F,H,Q,R,S)) =
((T-1,TLTT)o(T,L))+(F,H,Q,R,S)=(nn0 )+(F,H,Q,R,S) .

Transitivity follows from (F,H,Q,R,S)=(T,L)+(F,H,Q,R,S) and

(F,H,Q,R,S)= T,T)+(T,H,Qs,,S)=(T,f)+((T,L)+(F,H,Q,R,S))=(f,L)+ (F,H,Q,R,S).

Thus, GRn induces an equivalence relation on X2 which we denote by
ETL and (4.2-3) defines the partitioning of X2 into classes. Note that
our first objective has been satisifed, i.e., two ETL-equivalent quin-
tuplets have the same PSD; for if we let the pair (F,H,Q,R,S),

(F,H,Q,R,S)EX2 then if (T,L)eGRn


_ i __









z(z)=H(Iz-F)-Q(Iz-FT)- +H(IzF) S+S(Iz-- -H+R

=(HT-')T(Iz-F)-1T-1(T(Q-FLFT+LT)TT)TT(Iz -FT )ITT(HT1 )T

+(HT-I)T(Iz-F)-1T-1T(S-FLHT)+(ST-HLFT)TTT-T(Iz-1-FT)1 TT(HT 1)T

+R-HLHT (4.2-4)
or

D (z)=Z (z)+H(Iz-F)I [L-FLFT-FL(Iz-1-FT)-(Iz-F)LFT-(Iz-F)L -FT)(Iz I1-FT)-1HT

which gives QZ(z) = DZ(z). The measurement covariance sequence is also
invariant under the action of GRn on X2 because the PSD is also given by


DZ(z) = Z C.z-J. Thus, we will call any two systems represented by the
j=- j

quintuplet of X2 covariance equivalent, if they are ETL-equivalent.
Clearly, any two covariance equivalent systems have identical PSD's

(or measurement covariance sequences). Conversely, any two systems with
identical PSD's are covariance equivalent (see Popov (1973) for proof).

In order to uniquely characterize the class of covariance equi-

valent quintuplets we must determine an invariant system description
for X2 under the action of GR The number of invariants may be found
by counting the parameters. If we define, M :=dim(F,H,Q,R,S) and
M2:=dim(T,L), then there are M1=n2+np+n(n+l)+p(p+l)+np parameters

specifying this quintuplet and GR acts on M2=n2 +n(n+l) of them; thus,

there exist M1-M2=2np+'p(p+l) invariants. If we consider the transfor-

mation, (TR,L)EGR, to the Luenberger row coordinates, then np of these
invariants specify the canonical pair (FR,HR) of (2.2-6). The action

of (TR,L) on Q,R,S is given by


a_ _~ __ ~I ~








R = TR(Q-(FTR -)(TRLTR )(FTR- ) +L)TR Q-FRLRFR +L (4.2-5)

RR = R-(HTR-1)TRLTRT)(HTR T = R-HRLRHRT (4.2-6)

R = TR(S-(FTR-1)(TRLTRT)(HTR-1)) = SR-FRLRHRT (4.2-7)

where LR = TRLTRT, FR = TRFTR-, HR = HTR-, QR = TRQTRT SR = TRS.
The transformation LR acts on n(n+l) parameters of the total
n(n+l)+'-p(p+l)+np parameters available in QRR,SR as shown above for the
given (FR,HR). Once this action is completed the remaining np+p(p+l)
parameters are invariants. There are only four possible ways that LR
can act on the triple, (Q,R,S):

(i) LR acts only on QR; (4.2-8)
(ii) LR acts first to specify SR with the remaining elements
of LR acting on QR;
(iii) LR acts first to specify R with the remaining elements of

LR free to act on QR or SR or both; and
(iv) LR acts on any combination of elements in Q,R,S.

If we choose to restrict the action of LR to only the kn(n+l)
elements of QR, then for any choice of QR (given (FR,HR) and any QR)'
the transformation LR is uniquely defined. Since FR is a nonsingular
stability matrix, then it is well -known (Gantmacher (1959)) from (4.2-5)
that LR is the unique solution of L-FL F = Q* for Q* = RQ It
is important to note that the elements of QR are completely free, but
once they are selected, LR is fixed by (4.2-5) for any QR and therefore
the np elements of SR and the p(p+l) elements of RR are the invariants.
Thus, for a particular choice of QR we can uniquely specify the equivalence


_I_ ______ ___I I _








class of X2 under the action of GR i.e., (FRHR'R,RR,SR) is a
canonical form for ETL-equivalence on X2.
On the other hand, if we choose to let LR act on the np elements
of SR, then from (4.2-7) only np-p(p-l) elements of LR are uniquely

specified, i.e., since LR is symmetric and

LRH R rl+ l rp-1+1

there are p(p+l) redundant elements in the R.'s. Thus, for any choice
of SR (given (FR,HR) and any SR), np-p(p-l) elements of LR are uniquely
defined by (4.2-7) and the remaining elements of LR are free to act on QR.
In other words np-p(p-l) elements of QR are invariants,t as well as
the elements of RR, since any choice of SR specifies the elements of
LR in (4.2-6).

Similarly restricting the action of LR to act on the elements of R
specifies ;p(p+l) elements of LR from (4.2-6) and we are free to allow
the remaining elements of LR to act exclusively on QR or SR or both.

Clearly, there are many choices available to distribute the action of LR

on QR,R,SR; however, the important point is that once the choice is made,
the invariants are specified.
Any choice of symmetric QR is acceptable, since LR is uniquely
determined from (4.2-5) for given QRFR, but this is not the case when

an SR is selected. First recall that FR is nonsingular (see footnote

p.81). Then if we define SR:=SR-'R it follows from (4.2-7) that

FR-1S LRHRT (4.2-9)

and then

tThis was pointed out by Luo (1975) and Majumdar (1975).


_I




Full Text
91
Qr = ^(Qrn^bCTRLVjtFt^^+LjT^QR-F^F^+L^ (4.2-5)
Rr = (HTr_1)(TrLTrT)(HTR~1)T = R-HrLrHrT (4.2-6)
SR = Tr(S(FTr""^ ) (TrLTrT) (HTr_1 )T) = Sr-FrLrHrT (4.2-7)
where LR = TRLTRT, Fr = TrFTr*1, Hr = HTR-1, QR = TRQTRT, SR = TRS.
The transformation LR acts on %n(n+l) parameters of the total
%n(n+l)+%p(p+l)+np parameters available in Qr,R,Sr as shown above for the
giveh (Fr,Hr). Once this action is completed the remaining np+*sp(p+l)
parameters are invariants. There are only four possible ways that LR
can act on the triple, (Q,R,S):
(i) Lr acts only on QR; (4.2-8)
(ii) Lr acts first to specify SR with the remaining elements
of Lr acting on QR;
(iii) L_ acts first to specify R with the remaining elements of
K
Lr free to act on QR or SR or both; and
(iv) Lr acts on any combination of elements in Q,R,S.
If we choose to restrict the action of LR to only the %n(n+l)
elements of Qr, then for any choice of QR (given (Fr,Hr) and any QR),
the transformation Lr is uniquely defined. Since FR is a nonsingular
stability matrix, then it is well known (Gantmacher (1959)) from (4.2-5)
that Lr is the unique solution of Lr-FrLrFrT = Q* for Q* = Qr-Qr*
is important to note that the elements of QR are completely free, but
once they are selected, LR is fixed by (4.2-5) for any QR and therefore
the np elements of ¥r and the Jsptp+l) elements of RR are the invariants.
Thus, for a particular choice of QR we can uniquely specify the equivalence


116
C. T. Chen and D. P. Mital
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[1963] "Factoring the Spectral Matrix," IEEE Trans, on Auto. Contr.,
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[1.975] Numerical Aspects of Realization Algorithms in Linear
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[1974] "Canonical Forms for the Identification of Multivariable
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[1975] "On the Factorization of Discrete-Time Rational Spectral
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90
z(z)=K(Iz-?)-1^(Iz-1-?r)-1(Iz-?)-^+ST(I2-1-?T)-1K
=(HT"1)T(Iz-F)'1T"1(T(Q-FLFT+L)TT)T"T(Iz"1-FT)"1TT(HT1)T
+ (HT"1)T(Iz-F)"1T"1T($-FLHT)+(ST-HLFT)TVT(Iz'1-FT)1TT(Ht1)T
+R-HLH1 (4.2-4)
or
$z(z)=$z(z)+H(Iz-F)'1[L-FLFT-FL(Iz"1-FT)-(Iz-F)LFT-(Iz-F)L(Iz"1-FT)](Iz1-FT)~1HT
%
which gives $z(z) = $z(z). The measurement covariance sequence is also
invariant under the action of GR on X0 because the PSD is also given by
n 2
GO
$z(z) = 2 C.z"'5. Thus, we will call any two systems represented by the
j=- 3
quintuplet of Xp covariance equivalent, if they are E-^-equi valent.
Clearly, any two covariance equivalent systems have identical PSD's
(or measurement covariance sequences). Conversely, any two systems with
identical PSD's are covariance equivalent (see Popov (1973) for proof).
In order to uniquely characterize the class of covariance equi
valent quintuplets we must determine an invariant system description
for X0 under the action of GR The number of invariants may be found
by counting the parameters. If we define, :=dim(F,H,Q,R,S) and
p
Mz:=dim(T,L), then there are M^=n +np+%n(n+l)+%p(p+l)+np parameters
2
specifying this quintuplet and GRn acts on l^n +J^n(n+1) of them; thus,
there exist M^-Mz=2np+i2p(p+l) invariants. If we consider the transfor
mation, (TD,L)eGRn to the Luenberger row coordinates, then np of these
k n
invariants specify the canonical pair (FR,H^) of (2.2-6). The action
of (TR,L) on Q,R,S is given by


11
The stochastic case of the canonical realization problem is in
vestigated in Chapter 4. A complete set of independent invariants is
found to characterize the corresponding solution. Equivalent solutions
to this problem as well as to the steady state Kalman filtering problem
are studied and it is shown that the filter parameters can be specified
by solving an analogous realization problem.
The specific contributions of this research and further research
possibilities are outlined in Chapter 5.
Examples are used generously throughout this work to illustrate the
various algorithms discussed and to point out significant details that
are otherwise difficult to see. A comment on notation to be used through
out this dissertation cl oses0this chapter.
1.4 Notation
Uppercase letters denote matrices, and vectors are represented by
underlined lowercase letters. Lowercase letters are used to represent
scalars and integers. All matrices and vectors appearing in this work
are assumed to be real and constant. An = [a. is an nxm matrix with
m L lj m
elements a.. .; 0^ is the nxm null matrix with row and column vectors
T T
given by 0^ and 0^; In represents the nxn identity matrix, and e. or
e. stands for its j-th row or j-th column; jqn means j=T,2,...,m.
vi
\


4
the possibility of nonuniqueness of these representations. Buey (1968)
extended the results of Langenhop and Luenberger when he developed a
canonical form for certain subclasses of observable systems, but he
too was unaware of its invariant properties. Proceeding from the
external system description many researchers began to realize the
usefulness in the development of canonical forms. Popov (1969) developed
a canonical form for the transfer function in his investigation of
irreducible system representations. Gilbert (1969) examined the invariant
properties of a system with feedback applied to solve the decoupling
problem. Dickinson et al. (1974a) discuss the construction and appli
cation of these canonical forms for the transfer function matrix in a
recent survey. The properties of canonical forms were not fully
understood initially. In fact, the basic question of their uniqueness
posed many doubts as .to their usefulness. This issue wasn't resolved
until the work of Rosenbrock, Kalman, and Popov in the early seventies.
The properties of the Luenberger forms were clarified by the
results of Rosenbrock (1970) and Kalman (1971a) in their studies of the
minimal column indices (or Kronecker indices) of the matrix pencil
[Iz-F,G], or more commonly, the indices of the pair (F,G). These indices
were shown to be invariants under the following transformations: change
of basis in the state space, input change of basis, and state feedback.
These results precisely resolve the question of what can (or cannot)
be altered by applying feedback to a linear multivariable system. At
the same time Popov (1972) examined the properties of the controllable
pair (F,G) under the same transformations in a very precise, step-by-step,
algebraic procedure to.determine the corresponding invariants. He shows
clearly that obtaining the invariants under a particular transformation


48
Example. (2.4-2)
Let m=2, p=3, and the Hankel array be given as, ^
-
1
2
2
4
4
8
8
16
1
2
2
4
6
10
13
22
1
0
1
0
3
2
6
6
2
4
4
8
8
16
16
32
2
4
6
10
13
22
28
48
1
0
3
2
6
6
13
16
4
8
8
16
16
32
32
64
6
10
13
22
28
48
58
102
3
2
6
6
13
16
27
38
8
16
16
32
32
64
64
128
13
22
28
48
58
102
119
214
6
6
13
16
27
38
56
86
^ ^12 I S4,4 I V
. Y
(2) Performing the row operations as in (2.4-1), obtain [P | Q I I0],
O
where the leading elements are circled,
]
2 2 4 4 8
8
16:
-1
1
0
o
o
o

ro
5
6
Jl
2
i
T
1
0
@-1 -4 0 -5
_i_
2
-7
-2
b
0
1
0
0
0 0 0 0 0
0
0
1
2
_ 5 .
2
0
0
1
0
0 0 201
1
2
1
1
1
-1
0
-]
1
-4
0
b
0
0
0
1
r8
-3
0
-i

-1
0
0
1
0
1
-2
0
-1
0
0
0
1
o
00 -^1
-8
0
.0
0
. 0
0
0
0
0
1
-8
1
-2
0
-2
0
0
0
0
0 1
-1
2
-3
0
-2
0
0
0
0
0 0
1


25
where g^., 3.j are n column, n row vectors containing {cu^}, {3..^}
respectively over appropriate indices and zeros in the other places.
Luenberger (1967) shows that the transformation, T^, required to
obtain the pair (F^.G^) is determined from the columns of Wn, as
where
TC*T1 T2 <2-2-7>
Tj =t9j Fgj F J_1gj] jera
p(G) = m, and g. is the j-th column of G.
vl
Similar results hold for the pair and is specified by
constructed from the rows of V .
n
Unfortunately Luenberger (1967) in attempting to develop multi-
variable system representations did not determine the invariants under
GL(n). It is essential to use the approach outlined in (2.2-1) in
order to obtain the corresponding canonical forms or else it is possible
to obtain erroneous results. The following example due to Denham (1974),
shows that the Luenberger form, as originally stated is not canonical.
If we are given the pair (F,G) as
"o
0
1
1 1
¡ 1
~i
o"
1
1
0
2
1
! 1
0
0
F =
G =
0
1
2
S 1
1
l
0
0
_0
0
1
1
l
! 1j
_0
1_


118
R. D. Gupta and F. W. Fairman
[1974] "Parameter Estimation for Multivariable Systems,"
IEEE Trans, on Auto. Contr., Vol.'AC-19, pp. 546-549.
B. L. Ho and R. E. Kalman
[1966] "Contruction of Linear State Variable Models from Input/
Output Functions," Regelungstechnik, VoT. 14, pp. 545-548.
K. Hoffman and R. Kunze
f1971] Linear Algebra, Prentice-Hall Pubs., Second Edition, New Jersey
H. L. Huang
[1974] "A Generalized-Jordan-Form-Approach One-Step Irreducible
Realization of Matrices," IEEE Trans, on Auto. Contr.,
Vol. AC-19, pp. 271-272.
C. D. Johnson and W. M. Wonham
[1964] "A Note on the Transformation to Canonical (Phase-Variable)
Form," IEEE Trans, on Auto. Contr., Vol. AC-9,
pp. 312-313.
R. E. Kalman
[1960] "Control System Analysis and Design Via the Second
Method of Lyapunov," J. of Basic Engr., Vol. 82D,
pp. 394-499.
[1961] "A New Approach to Linear Filtering and Prediction
Problems," J. of Basic Engr., Vol. 82D, pp. 35-43.
[1962] "Canonical Structure of Linear Dynamical Systems,"
Proc. Nat. Acad, of Sci. (USA), Vol. 48, pp. 596-600.
[1963] "Mathematical Description of Linear Dynamical Systems,"
SIAM J. on Contr., Vol. 1, pp. 152-192.
[1964] "Lyapunov Functions for the Problem of Lure in Automatic
Control," Proc. Nat. Acad. Sci. (USA), Vol. 49, pp. 201-205.
[1965] "Linear Stochastic Filtering Theory-Reappraisal and
Outlook," Proc. of Symposium of System Theory, Brooklyn,
N. Y., pp. 197-205.
[1971a] "Kronecker Invariants and Feedback," Proc. 1971 Conf.
on Ord. Diff. Eg., National Research Laboratory
Mathematics Research Center, Washington, D. C.
[1971b] "On Minimal Partial Realizations of a Linear Input/Output
Map," in Aspects of Network and System Theory, R. E.
Kalman and N. De Claris (eds.), pp. 385-407, Holt,
Rinehart and Winston., N.Y.
[1973] "Global Structure Classes of Linear Dynamical Systems,"
presented at the NATO ASI Geometric and Algebraic
Methods for Nonlinear Systems, London, England.
[1974] Class Notes on System TheoryCourse by R. E. Kalman
at the University of Florida, Gainesville, Fla.
*


92
class of under the action of GRn, i.e., (FR,HR,QR,RR,SR) is a
canonical form for Ej^-equivalence on X2.
On the other hand, if we choose to let LD act on the np elements
K
of SR, then from (4.2-7) only np-%p(p-l) elements of LR are uniquely
specified, i.e., since LR is symmetric
and
LRHR f-l 4^+1
* +i ]
Vi 1
there are *sp(p+l) redundant elements in the £.'s. Thus, for any choice
J
of (given (FR,HR) and any SR), np-Jgp(p-l) elements of LR are uniquely
defined by (4.2-7) and the remaining elements of LR are free to act on QR
In other words np-^p(p-l) elements of QR are invariants, as well as
the elements of RR, since any choice of ifR specifies the elements of
Lr in (4.2-6).
Similarly restricting the action of LR to act on the elements of R
specifies %p(p+l) elements of LR from (4-2-6) and we are free to allow
the remaining elements of LR to act exclusively on QR or SR or both.
Clearly, there are many choices available to distribute the action of LR
on Qr,R,Sr; however, the important point is that once the choice is made,
the invariants are specified.
Any choice of symmetric (IR is acceptable, since LR is uniquely
determined from (4.2-5) for given QR,FR, but this is not the case when
an SR is selected. First recall that FR is nonsingular (see footnote
p.81). Then if we define SR: =SR-S"R it follows from (4.2-7) that
FR 1sR = LRHR (4.2-9)
and then
^This was pointed out by Luo (1975) and Majumdar (1975).


13
The external system description may be given either in rational
form as,
T(z) = H(Iz-F)~^G (z complex) (2.1-2)
or equivalently as an infinite matrix power series
T(z) = Z A zk (2.1-3)
k=l K
where the sequence {A^} is the unit pulse response or Markov sequence
of (2.1-1). The Markov parameters are
Ar> HF^G k=l ,2,... (2.1-4)
The problem of determining the internal description (F,G,H) from the
external description (T(z) or {A^>) is the realization problem. Out of
all possible realizations, Z:=(F,G,H) having the same Markov parameters,
those of smallest dimension are minimal realizations.
Prior to stating some of the significant results from realization
theory several useful definitions will be given. The j-control!ability
and j-observability matrices are the nxmj and pjxn arrays,
W, [G | ... | Fj_1G] and v! = [HT | ... | (HFj"1)T]. The pair (F,G)
J J
is completely controllable if p(Wn)=n and the pair (F,H) is completely
observable if p(Vn)=n. Throughout this dissertation we will only be
concerned with systems possessing these properties. For a completely
controllable and observable system, the controllability index, y, and
the observability index, v, are the least positive integers such that the
rank of W and V is n.
y v


REFERENCES
J. E. Ackermann
[1972] "On Partial Realizations," IEEE Trans, on Auto. Contr.,
Vol. AC-17, pg. 381.
[1975] "On the Synthesis of Linear Control Systems with
Specified Characteristics," Proc. 1975 IFAC Congress,
Boston, Mass., pp. 88-92.
J. E. Ackermann and R. S. Buey
[1971] "Canonical Minimal Realization of a Matrix of Impulse
Response Sequences," Inf, and Contr., Vol. 19, pp. 224-231.
H. Akaike
[1974a] "Stochastic Theory of Minimal Realization," IEEE Trans,
on Auto. Contr., Vol. AC-19, pp. 667-674.
[1974b] "A New Look at the Statistical Model Identification,"
IEEE Trans, on Auto. Contr., Vol. AC-19, pp. 716-723.
[1975] "Markovian Representation of Stochastic Processes by
Canonical Variables," SIAM J. on Contr., Vol. 13, pp. 162-173.
>
B. D. 0. Anderson
[1967a] "A System Theory Criterion for Positive Real Matrices,"
SIAM J. on Contr., Vol. 5, pp. 171-182.
[1967b] "An Algebraic Solution to the Spectral Factorization
Problem," IEEE Trans, on Auto. Contr., Vol. AC-12, pp. 410-414.
[1969] "The Inverse Problem of Stationary Covariance Generation,"
J. of Statistical Physics, Vol. 1, pp. 133-147.
B. D. 0. Anderson and S. Vongpanitlerd
[1973] Network Analysis and Synthesis, Prentice-Hall Inc., New Jersey.
B. M. Anderson, F. M. Brasch, and P. V. Lopresti
[1975] "The Sequential Construction of Minimal Partial
Realizations from Finite Input Output Data," SIAM J. on
Contr., Vol. 13, pp. 552-570.
D. R. Audley and W.J. Rugh
[1973] "On the H-Matrix System Representation," IEEE Trans, on
Auto. Contr., Vol. AC-18, pp. 235-242.
D. R. Audley, S. L. Baumgartner and W. J. Rugh
[1975] "Linear System Realization Based on Data Set Representations,"
IEEE Trans, on Auto Contr., Vol. AC-20, pg. 432.
134


76
Example. (3.3-1) Recall that in (3.2-3) m=p=2, n=5, and Vj=2, V2=3,
= [ -2 3-bOO] §J2 = [3+e 0 -c -(d+e) -e ]
(1)Simultaneously construct TBR from (3.3-4) while examining the rows
for predecessor independence
-1
1
0
0
0
o"
4
0
1
0
0
0
4
s
-2
3
-b
0
0
4fr
-6
7
-3b
-b
0
/.
-14
15
-7b
-3 b'
-b
1 1
(2)Determine TBR from TBR TBR = In which gives
f1
'BR
1
0
-2/b
0
0
(3)Determine Fbr:
F =T F T"^ =
rBR BRrR BR
0
0
0
0
1
3/b
2/b
0
-1/b
3/b
-2/b
0
-1/b
3/b -1/b
1
0
0
0
-3b-2c-ce 3c-2d-2e
0
1
0
0
-c+3d+e
0
0
1
0
-d+2e-2
0
0
0
1
-e+3
(4)Find the characteristic polynomial by inspection.


24
set [{v.-LB.j^}], i,se£,. where the {v..} are the observa-
bi1ity indices.
The last step of (2.2-1) is to specify the corresponding canonical
forms under GL(n). These forms are commonly called the Luenberger
forms and are specified by the controllability and observability
invariants. They are defined by the pairs (Fq.Gq), (FrjHr) where the
subscripts C,R reference the fact that the regular vectors span either
the columns of W
P+1
or the rows of V
v+1
(2.2-5)
3
l P- Jem
s=l s
F
R
J
H
R
(2.2-6)
P
T
r. = Z v ie£
1 s=l s


10
1.3 Statement of Purpose and Chapter Outline
It is the purpose of this dissertation to provide an extensive
discussion of the realization problem in both the deterministic and
stochastic cases as well as specify the invariants under particular
transformation groups in each case. It is also desired to develop a
simple and efficient algorithm to solve the canonical realization problem
This algorithm is to be modified to process data sequentially such that
only the pertinent informationr-the invariants, are extracted from the
given sequence. In the case of a fixed finite unit pulse response
sequence (the partial realization problem), the solution is to be
obtained such that all possible degrees of freedom are specified. The
relationship between the stochastic realization and steady state Kalman
filtering problems are discussed by again examining the corresponding
invariants. In so doing, a considerable amount of knowledge about the
existence and structure of realizations and the steady state filter is
gained.
The basic theoretical essentials of realization and invariant theory
are reviewed in Chapter 2. A "formula" essentially outlined in Popov
(1973) and Kalman (1974) is developed which will be applied to various
realization problems throughout the text. Some new theoretical results
in canonical realization theory are established and used to develop a
new canonical realization algorithm.
In Chapter 3 the algorithm is modified to handle sequentially
the case of partial data and also that of a fixed finite sequence. New
results evolve which completely characterize the class of all minimal
partial realizations and extension sequences as well as determining the
characteristic equation in a simple manner.


REALIZATION OF INVARIANT SYSTEM DESCRIPTIONS FROM MARKOV SEQUENCES
By
JAMES VINCENT CANDY
A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL
OF THE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1976

To my wife, Patricia, and daughter, Kirs tin,, for unending faith,
encouragement and understanding. To my mother, Anne, for her constant
support and my mother-in-law, Ruth, for her encouragement. To "big" Ed
my father-in-law, whose sense of humor often lifted my sometimes low

ACKNOWLEDGMENTS
I would like to express my sincere appreciation to the members
of my supervisory committee: Dr. Thomas E. Bullock, Chairman, and
Dr. Michael E. Warren, Cochairman, Dr. Donald G. Childers,
Dr. Z.R. Pop-Stojanovic and Dr. V.M. Popov. A special thanks to
Dr. Thomas E. Bullock and Dr. Michael E. Warren for their constant
encouragement, unending patience, and invaluable suggestions in the
course of this research.
I would also like to thank my fellow students and friends,
Zuonhua Luo, Arun Majumdar, Jos DeQueiroz, and Jaime Roman, for
many fruitful discussions and suggestions.

TABLE OF CONTENTS
ACKNOWLEDGMENTS .. iii
LIST OF SYMBOLS vi
ABSTRACT vii
CHAPTER 1: INTRODUCTION 1
1.1 Survey of Previous Work in Canonical Forms
for Linear Systems .. ... 2
1.2 Survey of Previous Work in Realization Theory.. 5
1.3 Purpose and Chapter Outline 10
1.4 Notation 11
CHAPTER 2: REALIZATION OF INVARIANT SYSTEM DESCRIPTIONS 12
2.1 Realization Theory ........ 12
2.2 Invariant System Descriptions .18
2.3 Canonical Realization Theory 33
2.4 Some New Realization Algorithms 45
CHAPTER 3: PARTIAL REALIZATIONS 54
3.1 Nested Algorithm ..... ........ 54
3.2 Minimal Extension Sequences ... 64
3.3 Characteristic Polynomial Determination by
Coordinate Transformation ........ ..... 74
CHAPTER 4: STOCHASTIC REALIZATION VIA INVARIANT SYSTEM
DESCRIPTIONS 78
4.1 Stochastic Realization Theory 81
4.2 Invariant System Description of the
Stochastic Realization ......... ........... 87
4.3 Stochastic Realizations Via Trial and Error ... 97
4.4 Stochastic Realization Via the Kalman Filter .. 104
CHAPTER 5: CONCLUSIONS Ill
5.1 Summary Ill
5.2 Suggestions for Future Research 112
iv

TABLE OF CONTENTS (Continued)
REFERENCES .....114
BIOGRAPHICAL SKETCH .....124
V
\
V

LIST OF MATHEMATICAL SYMBOLS
Symbol
Usage
Meaning
First Usage
T
AT, aT
Transpose of A, ai
pg. 13
-1
A1
Inverse of A
pg. 13
-T
at
Inverse of AT
pg. 89
P
p(A)
Rank of A
pg. 13
|.|
|A| or det A
Determinant of A
pg. 30,21
diag A
Diagonal elements of A
pg. 103
\
x^y
x is not equal to y
pg. 30
>
x>y
x is greater than y
pg. 16
c
XcY
X is contained in or a
subset of Y
pg. 18
e
xeX
x is an element of X
pg. 12
X->Y
Map (set X into set Y)
pg. 20
: =
x: =
x is defined by
pg. 13
0
xoy
Abstract group operation
pg. 19
{ }
{.>
Sequence or set with
elements
pg. 13
Z
Summation
pg. 13
00
Infinity
pg. 13
t
Footnote
pg. 2
V
End of proof
pg. 34
4
Group action operator
pg. 2i
dim X
Dimension of vector
space X
. pg. 15
iff
if and only if
pg. 14
X
X(A). ,
Eigenval u'es of A
pg- 9e
/
/x
Square root of x
pg. 99
max()
Maximum value of
pg. 23
Z+
Positive integers
pg. 12'
K
Field
pg. 12
vi

Abstract of Dissertation Presented to the Graduate Council
of the University of Florida in Partial Fulfillment of
the Requirements for the Degree of Doctor of Philosophy
REALIZATION OF INVARIANT SYSTEM DESCRIPTIONS
FROM MARKOV SEQUENCES
By
James Vincent Candy
March, 1976
Chairman: Dr. Thomas E. Bullock
Cochairman: Dr. Michael E. Warren
Major Department: Electrical Engineering
The realization of infinite and finite Markov sequences for multi
dimensional systems is considered, and an efficient algorithm to extract
the invariants of the sequence under a change of basis in the state
space is developed. Knowledge of these invariants enables the deter
mination of the corresponding canonical form, and an invariant system
description under this transformation group. For the partial realization
problem, it is shown that this algorithm possesses some attractive
nesting properties. If the realization is not unique, the class of
all possible solutions is found.
The stochastic version of the realization problem is also examined.
It is shown that the transformation group which must be considered is
richer than the general linear group of the deterministic problem. The
invariants under this group are specified and it is shown that they can
be determined from a realization of the measurement covariance sequence.
Knowledge of these invariants is sufficient to specify an invariant
system description for the stochastic problem. The link between the
vii

realization from the measurement covariance sequence, the white noise
model and the steady state Kalman filter is established.
vm

CHAPTER 1
INTRODUCTION
Special state space representations of linear dynamic systems have
long been the motivation for extensive research. These models are
generally used to simplify a problem, such as pole placement, by
introducing arrays which require the fewest number of parameters while
exhibiting the most pertinent information. In general, system represen-
. \ '
tations have been studied in literature as the problem of determining
canonical forms; Canonical forms have been used in observer design,
exact model matching methods, feedback system design, and Kalman filtering
techniques. In realization theory, canonical forms for linear multi-
variable systems are important. Since it is only possible to model a
system within an equivalence class, the ability to characterize the class
by a unique element is beneficial.
The problem of determining a canonical form has its roots in
invariant theory. Over the past decade many so-called "canonical" system
representations have evolved in the literature, but unfortunately these
representations were obtained from a particular application or from
computational considerations and not derived from the invariant theory
point of view. Generally, these representations are not even unique and
therefore cannot be called a canonical form. Representations derived
in this manner have generally been a source of confusion as evidenced by
the ambiguity surrounding the word canonical itself. In this dissertation
1

2
we follow an algebraic procedure to obtain unique system representations,
i.e., we insure that these representations re in fact canonical forms.
In simple terms this approach seeks the determination of certain entities
called invariants obtained by applying particular transformation groups
(e.g., change of basis in the state space) to a well-defined set
representing a system parameterization. The invariants are the basic
structural elements of a system which do not change under this trans
formation and are used to specify the corresponding canonical form. This
approach insures that the ambiguities prevalent in earlier work are
removed. Initially, we develop a simple solution to the problem of
determining a state space model from the unit pulse response of a given
linear systerr and then extend these results to the stochastic case where
the system is driven by a random input. The technique developed to
extract the invariants from this (response) sequence not only provides
a simple solution to the realization problem, but also gives more insight
into the system structure.
1.1 Survey of Previous Work in Canonical Forms for Linear Systems
The study of canonical forms for linear dynamic systems evolved
slowly in the Sixties. The main impetus of investigation was initiated
by Kalman (1962,1963) when he compared two different methods for describing
linear dynamic systems: (1) internally by the state space representation
denoted by the triple (F,G,H), or (2) externally by the transfer
function--the input/output description. Development over the past
decade in such areas as optimal control, decoupling theory, estimation
and filtering, identification theory, etc., have relied heavily on the
_
This defines (simply) the realization problem.

3
state space representation for analysis and design. In early literature,
however, transfer function representations were used. For highly
complex systems it is much easier to determine external behavior rather
than internal, since the state variables are normally not available for
measurement. As pointed out by Kalman (1963) the language of these
representations may be different, but both describe the same problems and
are related. Many researchers have investigated the relationship between
both representations, but always with one common goalto obtain a
state space model which specifies the external description directly by
inspection. Kalman (1963) and later Johnson and Wonham (1964),
Silverman (1966) have shown that there exists a canonical form (under
change of basis in the state space) in the scalar case for the triple
(F,g,h) where F is in companion matrix form (see Hoffman and Kunze (1971))
and g is a unit column vector. It was shown that there exists a one to
one correspondence between the non-zero/non-unit elements of the triple
and the transfer function. This representation was used by Bass and Gura
(1965) to solve the pole-positioning problem and recently by Wolovich
(1972b)in solving the exact model matching problem. ,
The progress in determining a canonical form for the internal
description of multivariable systems came more slowly. The earliest work
appears to be that of Langenhop (1964) in which he develops a representation
to study system stability. Brunovsky (1966,1970) was probably the first
to recognize the invariant properties of the canonical form for the
controllable pair (F,G). Tuel (1966,1967) developed canonical forms for
multivariable systems in his investigation of the quadratic optimization
problem. Subsequently, Luenberger (1967) proposed certain sets of
canonical forms for controllable pairs; however, his development allowed

4
the possibility of nonuniqueness of these representations. Buey (1968)
extended the results of Langenhop and Luenberger when he developed a
canonical form for certain subclasses of observable systems, but he
too was unaware of its invariant properties. Proceeding from the
external system description many researchers began to realize the
usefulness in the development of canonical forms. Popov (1969) developed
a canonical form for the transfer function in his investigation of
irreducible system representations. Gilbert (1969) examined the invariant
properties of a system with feedback applied to solve the decoupling
problem. Dickinson et al. (1974a) discuss the construction and appli
cation of these canonical forms for the transfer function matrix in a
recent survey. The properties of canonical forms were not fully
understood initially. In fact, the basic question of their uniqueness
posed many doubts as .to their usefulness. This issue wasn't resolved
until the work of Rosenbrock, Kalman, and Popov in the early seventies.
The properties of the Luenberger forms were clarified by the
results of Rosenbrock (1970) and Kalman (1971a) in their studies of the
minimal column indices (or Kronecker indices) of the matrix pencil
[Iz-F,G], or more commonly, the indices of the pair (F,G). These indices
were shown to be invariants under the following transformations: change
of basis in the state space, input change of basis, and state feedback.
These results precisely resolve the question of what can (or cannot)
be altered by applying feedback to a linear multivariable system. At
the same time Popov (1972) examined the properties of the controllable
pair (F,G) under the same transformations in a very precise, step-by-step,
algebraic procedure to.determine the corresponding invariants. He shows
clearly that obtaining the invariants under a particular transformation

5
group is the only information required to specify the corresponding
canonical form. Wonham and Morse (1972) obtained the feedback invariants
of the controllable pair from the not as lucid geometric viewpoint.
Their results were identical to those of Brunovsky and ftosenbrock. :
Morse (1973) examined the invariants of the triple (F,G,H) under a larg
group of transformations which includes output change of basis. A
complete set of feedback invariants of this triple still remains an
open problem, but some fragmentary results were presented by Wang and
Davison (1972) when they investigated certain sets of restricted triples.
Along these lines Rissanen (1974), Caines and Rissanen (1974), .
Mayne (1972a,b),Weinert and Anton (1972), Tse and Weinert (1973,1975),
Glover and Willems' (1974) examined the identification problem from the
invariant theory viewpoint and obtained some rather interesting results.
Recent results in decoupling theory were obtained by Warren and Eckberg
(1973), Concheiro (1973), and Forney (1975) using the Kronecker invariants
Probably the most extensive survey of these results has been compiled
by Denham (1974) and we refer the interested reader to this paper.
We temporarily leave this area to consider one specific application of
these resultsthe realization problem.
1.2 Survey of Previous Work in Realization Theory
The first realization problem proposed for control systems was
the determination of a state space model (internal description) from
a given transfer function (external description). Gilbert (1963) and
Zadeh and Desoer (1963) describe realization procedures based on the
determination of the rank of the residue matrices of the given transfer
function matrix, but unfortunately these procedures only apply to the

6
case of simple poles. Kalman (1963) proposed an algorithm whereby
the given transfer function is realized as a parallel combination of
single input, controllable subsystems in companion form, and then
applied the "canonical structure theory" (Kalman (1962)) to delete the
uncontrollable dynamics. This technique handles simple as well as
multiple transfer function poles. Later Kalman (1965) showed the
equivalence of the realization problem of control theory to the
corresponding network theory formulation.
A significant advance in realization theory was given by Ho and
Kalman (1966). They showed that the state space model could be found
from the impulse response sequence provided the system under investi
gation is finite dimensional. They also developed an algorithm based
on forming the generalized Hankel array from the given sequence and
then extracted the state space triple from it. Shortly after the pub
lication of Ho's algorithm, Youla and Tissi (1966) working in network
synthesis and Silverman and Meadows (1966) in control theory developed
similar realization techniques again based on the impulse response sequence.
Ho's algorithm gave new impetus to realization theory. Several
authors have provided alternate or improved realization algorithms based
on the Hankel array formulation. Mayne (1968), Panda and Chen (1969),
Roveda and Schmid (1970), Rosenbrock (1970), Lai et al. (1972) and even
more recently Huang (1974), Rozsa and Sinha (1975) among others,
considered the older transfer function matrix approach, while Rissanen
(1971,1974), Silverman (1971), Ackermannand Buey (1971), Chen and Mita!
(1972), Mita! and Chen (1973), and Bonivento et al. (1973) approached
the problem from the Hankel array formulation.

7
Rissanen (1974), Furata and Paquet (1975), Roman (1975),
Dickinson et al. (1974a,b) have recently considered the problem of
realizing a given infinite impulse response matrix sequence with a
polynomial matrix pair. Such a pair is referred to as a matrix-
fraction description of the system and is becoming well known in
control literature largely due to the ground work established by
Popov (1969), Rosenbrock (1970), Wolovich (1972a,b, 1973a,b) and
others.
Kalman (1971b), Tether (1970), and Godbole (1972) later
considered the more realistic case where only a finite number of
terms of the impulse response sequence are specified. This is
commonly known as the partial realization problem and corresponds
in the scalar case to the classical Pad approximation problem.
Generally most realization altorithms can be used to process partial
data, but usually at a loss of efficiency and even more seriously
the possibility of yielding misleading results. A wealth of new
techniques have recently been published to handle this very special,
yet realistic variant of the realization problem. Rissanen (1972a,b),
Ackermann (1972), Dickinson et al. (1974a), Roman and Bullock (1975a),
Anderson et al. (1975) published some efficient and improved algorithms
to solve this problem.
Also of recent interest is the development of algorithms which
realize a system directly in a canonical form (under a change of basis
in the state space), i.e., algorithms which solve the canonical realiza
ti on problem. The algorithms of Ackermann (1972), Bonivento et al.
(1973), Rissanen (1974), Dickinson et al. (1974a), Rozsa and Sinha
(1975), Luo (1975), and Roman and .Bullock (1975a) solve this problem.

8
One of the main contributions of this dissertation is to use the
results developed from invariant theory to solve the realization and
partial realization problems in the deterministic as well as stochastic
cases. The realization of a system directly in a canonical form actually
reduces to first determining which transformation groups are present,
specifying the corresponding invariants, and then developing a method to
4*
extract these invariants from the given unit pulse response sequence.
This philosophy is basic to any canonical realization scheme and actually
provides an explicit formula which is applied throughout this dissertation.
In the last few years, several interesting extensions have emerged
from the original concept of realization theory. The major motivation
evolved just after the development of the Kalman filter (see Kalman (1961))
in estimation theory because a priori knowledge of the state space
model and noise statistics are required to begin data processing. The
link between the filtering and realization problem was established by
Kalman (1965) just prior to the advent of Ho's algorithm. The work of
Gopinath (1969), Budin (1971,1972), Bonivento et al. (1973), and Audley
and Rugh (1973,1975) were concerned with the more general problem of,
obtaining a state space representation given a general input/output
sequence of the system in both deterministic and stochastic cases. The
stochastic version of the realization problem has not received quite
as much attention as the deterministic case mainly due to its greater
complexity and high dependence on the adequacy of covariance estimators.
The realization of stochastic systems was studied by Faurre (1967,1970)
and more recently by Rissanen and Kailath (1972), Gupta and Fairman (1974)
4* *
The Hankel array formulation is used exclusively in this dissertation.-

9
and Akaike (1974a,b). From the transfer function viewpoint this problem
has been solved using spectral factorization as originally introduced
by Wiener (1955,1959) and studied by others such as Gokhberg and Krein
(1960), Youla (1961), Davis (1963), Motyka and Cadzow (1967), and
Strintzis (1972). The link between the stochastic realization problem
and spectral factorization evolved from the work in stability theory by
Popov (1961,1964), Yakubovich (1963), Kalman (1963), Szego and
Kalman (1963). The equations establishing this link were derived in
the Kalman-Yakubovich-Popov lemma for continuous systems and the Kalman-
Szego-Popov lemma for discrete time systems. Newcomb (1966), Anderson
(1967a,b,1969), and Denham (1975) extended these results and provided
techniques to solve these equations. Defining the invariants of these
problems is still an area of active research as evidenced by the recent
work of Denham (1974), Glover (1973), and Dickinson et al. (1974b).
This is one area developed in this dissertation. It will be shown that
the invariants of the stochastic realization problem not only lends more
insight into the structure of the problem, but also yields some new
results.
Research in realization theory and its applications continues as
evidenced by the recent results of Rissanen (1975) in estimation theory,
Ackermann (1975) in feedback system design,De Jong (1975) in the
numerical aspects of the problem and Roman and Bullock (1975b) in
observer theory. The results presented in this dissertation tie together
some previously well-known results in stochastic realization and filtering
theory as well as provide a technique which can be used to study
other problems.

10
1.3 Statement of Purpose and Chapter Outline
It is the purpose of this dissertation to provide an extensive
discussion of the realization problem in both the deterministic and
stochastic cases as well as specify the invariants under particular
transformation groups in each case. It is also desired to develop a
simple and efficient algorithm to solve the canonical realization problem
This algorithm is to be modified to process data sequentially such that
only the pertinent informationr-the invariants, are extracted from the
given sequence. In the case of a fixed finite unit pulse response
sequence (the partial realization problem), the solution is to be
obtained such that all possible degrees of freedom are specified. The
relationship between the stochastic realization and steady state Kalman
filtering problems are discussed by again examining the corresponding
invariants. In so doing, a considerable amount of knowledge about the
existence and structure of realizations and the steady state filter is
gained.
The basic theoretical essentials of realization and invariant theory
are reviewed in Chapter 2. A "formula" essentially outlined in Popov
(1973) and Kalman (1974) is developed which will be applied to various
realization problems throughout the text. Some new theoretical results
in canonical realization theory are established and used to develop a
new canonical realization algorithm.
In Chapter 3 the algorithm is modified to handle sequentially
the case of partial data and also that of a fixed finite sequence. New
results evolve which completely characterize the class of all minimal
partial realizations and extension sequences as well as determining the
characteristic equation in a simple manner.

11
The stochastic case of the canonical realization problem is in
vestigated in Chapter 4. A complete set of independent invariants is
found to characterize the corresponding solution. Equivalent solutions
to this problem as well as to the steady state Kalman filtering problem
are studied and it is shown that the filter parameters can be specified
by solving an analogous realization problem.
The specific contributions of this research and further research
possibilities are outlined in Chapter 5.
Examples are used generously throughout this work to illustrate the
various algorithms discussed and to point out significant details that
are otherwise difficult to see. A comment on notation to be used through
out this dissertation cl oses0this chapter.
1.4 Notation
Uppercase letters denote matrices, and vectors are represented by
underlined lowercase letters. Lowercase letters are used to represent
scalars and integers. All matrices and vectors appearing in this work
are assumed to be real and constant. An = [a. is an nxm matrix with
m L lj m
elements a.. .; 0^ is the nxm null matrix with row and column vectors
T T
given by 0^ and 0^; In represents the nxn identity matrix, and e. or
e. stands for its j-th row or j-th column; jqn means j=T,2,...,m.
vi
\

CHAPTER 2
REALIZATION OF INVARIANT SYSTEM DESCRIPTIONS
In this chapter we present a brief review of the major results
in realization theory. We establish a basic "formula" and apply it to
various system representations. It is shown that this approach greatly
simplifies the realization problem. Two new algorithms for realization
are developed which appear to be more efficient than previous techniques
because they extract only the minimal information necessary to specify
a system from the given input/output sequence in an extremely simple
fashion. All of the essential theory is developed and a multivariable
example is presented.
2.1 Realization Theory
A real finite dimensional linear constant dynamic system has
internal description given by the state variable equations in discrete
time as,
^<+1
+ GJk
(2.1-1)
where keZ+, xeKn=X, u£Km=U, yeK^Y and F, G, H are nxn, nxm, pxn matrices
over the field K. X,U,Y are the state, input, and output spaces,
respectively.
12

13
The external system description may be given either in rational
form as,
T(z) = H(Iz-F)~^G (z complex) (2.1-2)
or equivalently as an infinite matrix power series
T(z) = Z A zk (2.1-3)
k=l K
where the sequence {A^} is the unit pulse response or Markov sequence
of (2.1-1). The Markov parameters are
Ar> HF^G k=l ,2,... (2.1-4)
The problem of determining the internal description (F,G,H) from the
external description (T(z) or {A^>) is the realization problem. Out of
all possible realizations, Z:=(F,G,H) having the same Markov parameters,
those of smallest dimension are minimal realizations.
Prior to stating some of the significant results from realization
theory several useful definitions will be given. The j-control!ability
and j-observability matrices are the nxmj and pjxn arrays,
W, [G | ... | Fj_1G] and v! = [HT | ... | (HFj"1)T]. The pair (F,G)
J J
is completely controllable if p(Wn)=n and the pair (F,H) is completely
observable if p(Vn)=n. Throughout this dissertation we will only be
concerned with systems possessing these properties. For a completely
controllable and observable system, the controllability index, y, and
the observability index, v, are the least positive integers such that the
rank of W and V is n.
y v

14
If two minimal realizations Z, t are equivalent under a change of
basis in X, then there exists a nonsingular T such that
(F,G,H)^ = (TFT\tG,HT_1 ). It also follows by direct substitution that
the controllability and observability indices of these realizations are
identical and
W. = TW. for j = 1,2,...
J J
V. = V.T"1 for i = 1,2,...
The generalized NxN' block submatrix of the doubly infinite Hankel
array is given by
SN,N' =
Implicit in the realization problem is determining when a finite
dimensional realization exists and, if so, its corresponding minimal
dimension. The following proposition by Silverman gives the necessary
and sufficient conditions for {A^} to have finite dimensional realiza
tion.
Proposition. (2.1-5) An infinite sequence {A^} is realizable iff there
exist positive integers y,v,n such that
otVuu+j) =n 'or j'01
Further, if {A^} is realizable, then p,v are the
controllability and observability indices and n is
the dimension of the minimal realization.
^This notation means F = TFT"\ G = TG, and H = HT ^.
AN'
W-l

15
Proof. See Silverman (1971).
Note that the essential point established in Ho and Kalman (1966), which
is used in the proof of the above proposition is that Z is a minimal
realization iff it is completely controllable and observable. Since
$ *V W it follows for dims n that: p(S ) min[p(V ),p(W )]=n.
v,p v y v,y v y
This result is essential to construct any realization algorithm. In
(2.1-5) the crucial point of finite dimensionality is carefully woven
into necessary and sufficient conditions for an infinite sequence to be
realizable. What if only partial information about the system is
available in the form of a finite Markov sequence? Is this sequence
realizable? What is the relationship between the minimal realization
and one based only on partial data? These are only a few of the questions
which must be resolved when we are limited to partial data.
Intrinsic in the realization from a finite Markov sequence is the
fact that enough data are contained in S to recover the infinite
a v,y
sequence, i.e., knowledge of (A^,...,A is sufficient to determine
{Ak>, k-1,2,... But in reality the only way to be sure of this is
knowledge of the actual system dimension (or at least an upper bound). A
minimal partial realization is a realization of smallest dimension
determined from a finite Markov sequence {A^},keM_ which realizes the
sequence up to M terms. The order of the partial realization is M and
the realization is denoted by Z(M). The realization induces an extension
k-1
of {Ak>, i.e., Ak=HF G for k>M. The following basic result analogous
to (2.1-5) answers the realizability question when only partial data are
given. For a proof, see Kalman (1971).

16
Proposition. (2.1-6) (Realizability Criterion) The minimal partial
realization problem of order M possesses a
solution, £(M) iff there exist positive integers
*
v,y, M = \H-y
where dimE(M) = p(S^ ) = n.
In this proposition (R) is designated the rank condition. Also, it
is important to note that when (R) is satisfied the minimal extension
(of 2(M)) is unique (see Tether (1970) for proof), but S(M) is not
unique because there exist other minimal partial realizations equiv
alent to S(M) under a change of basis in X.
We must consider three possible cases when only partial data is
*
available. In the first case enough data is available such that M>M
for known n; thus, a minimal realization is found. Second, v and y
are available such that (R) is satisfied. In this case a minimal par
tial realization can be found, but this in no way insures it is also a
minimal realization of the infinite sequence, since the rank of S
v
may increase as v,y increase. Third, the rank condition does not hold
How can a realization be found when no more data is available? The
only possibility in this case is to extend the sequence until (R) is
satisfied, but there can exist many extensions satisfying (R) while
giving nonminimal realizations. For this reason define a minimal
extension as any that corresponds to a minimal (partial) realization.
To obtain minimality we must somehow select the right extension among
the many possible.

17
Prior to summarizing the main results of Kalman (1971) and Tether
(1970), define the incomplete Hankel array associated with a given
partial sequence {A^}, keM. as
where the asterisks denote positions where no data is available. The
rank of S(M,M) is the number of linearly independent rows (columns)
determined by comparing only the data specified elements in each row
(column) with the preceding rows (columns) with the cognizance that
upon the availability of more data this number can only remain the same
4*
or increase. Thus, the rank is a lower bound for any extension when the
* are filled in-consistent with the preservation of the Hankel pattern.
Both Kalman and Tether show that there are three pertinent integers
associated with the incomplete Hankel array. They are defined as: n(M),
v(M), y(M) and correspond to the rank of S(M,M), the observability index,
and the controllability index of the given data. The latter two are
lower bounds (separately) for v and y. Knowledge of either v(M), or
y(M) enables us to construct extensions, since they are the least integers
such that (R) holds for all minimal extensions.
It should also be noted that the integers n,v,y,... are actually
non-decreasing functions of the amount of data available, M, and should
be written, n(M), v(M), y(M) etc. to be precise. However, the argument
^It also follows from this that the p(S(M,M)) is a lower bound for dim 2
(see Kalman (1971)).

18
M will be understood throughout this dissertation in order to maintain
notational simplicity.
There is one more variant of the partial realization problem that
must be considered. A sequence of minimal partial realizations such
that each lower order realization is contained in one of higher order
will be called a nested realization. Symbolically, this is given by
...-E(M)-S(M)-... for M appear as submatrices of the corresponding matrices in £(M). The solution
to this problem is most desirable from the computational viewpoint,
since each higher order model can be realized by calculating just a few
new elements in the corresponding realization. Rissanen (1971) has
given an efficient recursive algorithm to determine this solution.
Another related problem of interest is determining a unique member of
equivalent systems under similarity and is discussed in the following
section.
2.2 Invariant System Descriptions
In this section we review some of the fundamental ideas encountered
when examining the invariants of multivariable 1 inear systems. The
framework developed here will be used throughout this dissertation in
formulating and solving various realization problems. Not only does
this formulation enable the determination of unique system representations
under some well-known transformations, but it also provides insight into
the structure of the systems considered. First, we briefly define the
essential terminology and then use it to describe some of the more common
sets of canonical forms employed in many recent applications (e.g., Roman
and Bullock (1975a,b), Tse and Weinert (1975)).

4
For any two sets X and Y, a subset R c X x Y is called a binary
relation on X to Y (or, a relation "between" X and Y). Then (x,y)eR
is usually written as xRy and is read: "x stands in the relation R to
y". If for X=Y this relation is reflexive, symmetric, and transitive,
then it is an equivalence relation E on X given by xEy for x,yeX. The
set of all elements z equivalent to x is denoted by E(x),= {zeXjxEz} and
is called the equivalence class or orbit of x for the equivalence relation
E. The set of all such equivalence classes is called the quotient set
or orbit space and is given by X/E. Thus, the relation E of X partitions
the set X into a family of mutually disjoint subsets or orbits by sending
elements which are related into the same equivalence class.
ff
Consider a fixed group G of transformations acting on a set X.
Then the elements Xj,Xg of X are equivalent under the action of G iff
there exists a transformation TeG which maps x-j into.Xg.. This is basically
the "formula" we will apply throughout, i.e., we first formulate the set
of elements (generally the internal system description), then define a
transformation group; and finally determine the orbits under the action
of G. To be more precise, let us first define the function f mapping
a set X into Y as an invariant71^for E if for x-j^eX, x^Ex^ implies
f(Xi)=f(x2). In addition if f.(xi)asf(Xg) implies x-jExg, then f is a
X ^ ~
This is the standard Cartesian product, XxY = {(x,y)|xeX, yeY} .
.Here we mean "group" in the standard algebraic sense, i.e., (G)
where G is a closed set of elements each possessing an inverse and
the identify element; 0 is an associative binary operation. When
o is understood, the group is merely denoted by G.
4.4-4.
Note that an invariant is actually a function, but common usage
refers to its image as the invariant. We will also use this terminology
throughout this dissertation.

20
complete invariant. In general we will be interested in a complete
system of invariants for E given by the,set of invariants (f^} where
4*
f : X + Y1xY2X. xYn> f. is an invariant for E, and f-j (x^i (xg) * >
ffi(X1)~fn(x2) imPly x]Ex2* Completeness of this set of invariants
means that the set is sufficient to specify the orbit of x, i.e., there
is a one to one correspondence between the equivalence classes in X
and the image of f. If the set of complete invariants is independent,
then the map f: X+Y-jX.. ,xYn is surjective. This property means that
corresponding to every set of values of the invariants there always exists
an n-tuple in Y specified by this set. A complete system of independent
invariants will be called an algebraic basis.
Generally, we consider a subset of X (e.g., in system theory a
controllable system). Correspondingly, let f be a function mapping the
subset XQ of X into set Y, then f is a restriction of f if fQ(x)=f(x)
for each xeXQ. We can uniquely characterize an equivalence class E(x)
by means of the set of values of the functions f.(x), ien. where the {f..}
constitute a complete set of invariants for E on X. If the corresponding
complete invariant f is restricted such that its image is itself a
subset of X, then we have specified a set of canonical forms C for
E on X. To be more precise, a canonical form C for X under E is a
member of a subset C<=X such that: (1) for every xeX there exists one and
only one ceC for which xEc, and since C is the image of a complete
invariant f, then (2) for any xeX and c-j, C2eC, xEc^, and xEc2 implies
f(x)=f(c-|)=f(c2)=c-j=C2 (invariance); (3) for any ceC if f(x-|)=c and
f(x2)=c, then x-|Ex2 (completeness). Thus, c=f(x) is a unique member of
^This notation is actually f=(f-|,... ,fn) :x-^Y^x.. .xYn, but it is
shortened when the set {f.} is clearly understood.

21
E(x) for every xeX. With these definitions in mind, our "formula"
becomes
(i)Formulate the set of elements;
(ii)Define the transformation group;
(iii)Determine a set of complete invariants under this
transformation group; and
(iv)Develop the canonical form in terms of the corresponding
invariants. (2.2-1)
We now apply (2.2-1) to various restricted sets related to multivariable
systems. This approach is essentially given in Kalman (1971a), Popov
(1972), Rissanen (1974), or Denham (1974). In this sequel we review the
main results of Popov. First, define the set of matrix pairs (F,G)
as
XQ = i(F,G)|FeKnxn, GeKnxm; (F,G)controllable}
The general linear group, which corresponds to a change of basis
in the state space, is specified by the set
GL(n):= {T|TeKnxn; det T?0} (2.2-2)
with the group operation standard matrix multiplication, i.e.,
T o T = T T.
In order to determine the orbits of XQ under the action of GL(n),
it is first necessary to specify the action operator
T + (F,G):= (TFT-1,TG)
4*
In general the problem of determining a canonical form is quite
difficult. However in this dissertation we consider restricted sets
which make the problem much simpler. For a thorough discussion of this
problem see Kalman (1973).

22
or alternately we can say that the action of GL(n) on XQ induces
F TFT"1
G + TG
The action of GL(n) induces an equivalence relation on XQ. We
indicate (F,G)Ej(F,G) if there exists TcGL(n) such that (F,G)=Tt(F,G).
Dual results are defined for the observable pair (F,H) and the
A.
analogous set denoted by XQ.
The third step of (2.2-1) is established in Popov (1972), but
first consider the following definitions. For a controllable pair (F,G)
4*J*
define the j-th controllability index y., jem as the smallest
positive integer such that the vector F Jg. is a linear combination of
J
4 y
its predecessors, where a predecessor of F g. is any vector F g^ where
J *
rm+s J J
we have assumed p(G) = m. Throughout this dissertation we use the
following definition of predecessor independence: a row or column vec
tor of- a given array is independent if it is not a linear combination of
its regular predecessors. The following results were established by
Popov (1972)
Proposition. (2.2-3) (1) The regular vectors are linearly independent;
(2) The controllability indices satisfy the
m
following relationship, E y. = n; (3) There exists
n=l J
exactly one set of ordered scalars, a^cK defined
for jem, kej-1, s = 0,1,...,min(pj,pk-T) and for jem,
k = j,...,m,s = 0,l,...,min(y.,y(c) 1 such that
+
Throughout this dissertation we use the overbar on a set to denote the
dual set.
^These indices are also called the Kronecker indices.

23
yi
F Jg. *= E
J k=l
J-l nnn(y ,y.-1)
" 1 % + z
s=0 JKS K k=j
m min(y.,yk)-l
JE
s=0
ajksF gi
This proposition follows directly from the controllability of (F,G) and
indicates that the regular vectors form a basis where the a's are the
coefficients of linear dependencies. The set [{yj},{ctjks}], j,kem,
s*0,...,y.-1 are defined as the controllability invariants of (F,G),
J
and y=max(y.). The main result of Popov is:
J
Proposition. (2.2-4) The controllability invariants are a complete
set of independent invariants for (F,G)eX under
the action of GL(n).
The proof of this proposition is given in Popov (1972) and consists of
verifying the invariance, completeness, and independence of [(y^},-Cctjj Invariance follows directly from Proposition (2.2-3), since (F,G)Ej(F,G),
then can be replaced by TFsgk in the given recursion and the
controllability invariants remain unchanged. Completeness is shown by
constructing a TeGL(n) such that for two pairs of matrices (F,G),
(F,G)eX0 with identical controllability invariants, (F,G) = (TFT""^, TG)
or (F,G)Ej(F,G). Independence of the controllability invariants is
obtained by constructing a canonical form determined only in terms of
these invariants. Thus, by introducing a finite set of indices (y^},
Popov shows that this set along with the {ajks} are invariants under the
action of GL(n). The main reason for specifying a set of complete and in
dependent invariants is that it enables us to uniquely characterize the
orbit of (F,G). It should also be noted that dual results hold for the observ
able pair (F,H), and it follows that the observabi1 ity invariants are the

24
set [{v.-LB.j^}], i,se£,. where the {v..} are the observa-
bi1ity indices.
The last step of (2.2-1) is to specify the corresponding canonical
forms under GL(n). These forms are commonly called the Luenberger
forms and are specified by the controllability and observability
invariants. They are defined by the pairs (Fq.Gq), (FrjHr) where the
subscripts C,R reference the fact that the regular vectors span either
the columns of W
P+1
or the rows of V
v+1
(2.2-5)
3
l P- Jem
s=l s
F
R
J
H
R
(2.2-6)
P
T
r. = Z v ie£
1 s=l s

25
where g^., 3.j are n column, n row vectors containing {cu^}, {3..^}
respectively over appropriate indices and zeros in the other places.
Luenberger (1967) shows that the transformation, T^, required to
obtain the pair (F^.G^) is determined from the columns of Wn, as
where
TC*T1 T2 <2-2-7>
Tj =t9j Fgj F J_1gj] jera
p(G) = m, and g. is the j-th column of G.
vl
Similar results hold for the pair and is specified by
constructed from the rows of V .
n
Unfortunately Luenberger (1967) in attempting to develop multi-
variable system representations did not determine the invariants under
GL(n). It is essential to use the approach outlined in (2.2-1) in
order to obtain the corresponding canonical forms or else it is possible
to obtain erroneous results. The following example due to Denham (1974),
shows that the Luenberger form, as originally stated is not canonical.
If we are given the pair (F,G) as
"o
0
1
1 1
¡ 1
~i
o"
1
1
0
2
1
! 1
0
0
F =
G =
0
1
2
S 1
1
l
0
0
_0
0
1
1
l
! 1j
_0
1_

26
These matrices are in the form of (2.2-5), but it is easliy verified
by constructing that the controllability invariants are in fact
Pl=2, ^=2 and a-j = [-1-1 -1 1]I ol, = [-2 0 -2 4]I The problem
with the Luenberger forms is that the maps it: Xq-* Xq/E are not well
defined. Thus, the image of the maps are indeed canonical forms, but
as shown here for (F,G)eXq/E, we need not have tt(F,G)=(F,G), i.e., the
mapping does not leave the canonical forms unchanged. The point to
remember is that the invariants are the necessary entities of interest
which must be determined.
4*
The procedure to construct the transformation matrix Tq of (2.2-7)
is called the Luenberger second plan. The first Luenberger plan
consists of examining the columns of i^n, given by
V ... F--'g,
F"'\l '2-2-8>
where is an nmxnm permutation matrix, for predecessor independence.
Thus, we can define a new set of invariants (under GL(n)) [{f^.}, ^jks^
completely analogous to the controllability invariants. The canonical
forms associated with the invariants obtained in this fashion have
4*
This procedure amounts to examining the column vectors of Wft for
predecessor independence, i.e., examine g-j ... gm Fg^ ... Fgm . .

27
become known as the Buey forms which were derived directly from the
results of Langenhop (1964), Luenberger (1967), and Buey (1968). We
refer the interested reader to these references as well as the recent
survey by Denham (1974). Here we will be satisfied to note that the
procedure of (2.2-1) applies with the set of controllable pairs (F,G)
restricted to the {y.} invariants rather than {y.}. Analogous to the
J J
Luenberger forms, we define the row and column Buey forms as (^dr^br)*
(Fbc>GBc) respectively. The row form is given by
L11
fbr=
L21
L22



0
>
HBR =
T
v +1
V] + l



Lpl
Lp2 ...
L
PP
+ +% +i
VI -^Vp.f
(2.2-9)
where
L..
n
I V
T.
U
'Xi
'o>r
'vr
8..
v. > 0 and satisfy E v =n ;
1 s=l 5
a. ^
KI.. are v.,v-, row vectors containing (3. invariants.
1 J! 1 SC
n
ij
The transformation, TRB,required to obtain the pair (FgR>HBR) is
' BR
'BR
[T
1
(2.2-10)
where tI = [hT(h_.F)T
D.j I I
Vl T
(h,F 1 ) ], ie£

28
The importance of the Buey form is that the characteristic equation can
JL
be found by inspection of the block diagonal arrays of FBR Since FBR
is block lower triangular, the characteristic equation is given as
Xp (z) = det(Iz-FpR) = Xj (z)...x, (z) (2.2-11)
hBR bK L11 Lpp
where the L.. are the companion matrices of (2.2-9). Similar results
hold for the pair (Fg^.Gg^,) and the transformation is specified by TB(,
constructed from the columns of W .
n
This completes the discussion of invariants and canonical forms for
controllable or observable pairs. To extend these results to matrix
triples (internal system description), it is more convenient to examine
ft
an alternate characterization of the corresponding equivalence class
the Markov sequence of (2.1-4). This approach was used by Mayne (1972b)
and Rissanen (1974), in order to determine the orbits of Z under GL(n).
It is obvious that the sequence is invariant under this group action
A. = (HT^MTFT-VVtG) = HF'3'1G (2.2-12)

Consequently every element of A. can be considered an invariant of Z
J
with respect to GL(n); therefore, two systems which are equivalent under
GL(n) possess identical Markov sequences. The converse is also true, i.e.,
any two systems with identical Markov sequences are equivalent.
The standard approach to investigate a system characterized by its
Markov sequence is to form the Hankel array, N, where we define sT ,
+
It should be noted that the Buey form is not a canonical form if the
transformation group includes a change of basis in either input or output
spaces, while the Luenberger form is still a canonical form.

29
ieN and S jeN1 as the block rows and columns of SM and the
J IN5I1
block column and row vectors, a or a^ denote the r-th column of si
or the s-th row of S for remN1, sepN. Rissanen (1974) has shown
9
that by examining the set
X, = {£ I I controllable and observable with {y.} invariants}
I J
under the action of GL(n) that
Proposition. (2.2-13) The set of controllability invariants and block
column vectors, [{yj>,-Caj|9-[a }.] for the
appropriate indices constitute an algebraic basis

for any ZeX^ under the action of GL(n).
The proof of this proposition is given in Rissanen (1974) and consists of
showing that any two members of X^ with identical Markov sequences
are equivalent under GL(n). Thus, invariance follows by showing that a
dependent column vector of the Hankel array can be uniquely represented
in terms of the set [{y.},{a.. _}]. These parameters remain unchanged
J JKS
under GL(n); therefore, they are invariants. The block column vectors,
a t satisfy a recursion analogous to (2.2-3), i.e.,
n-1 min(yJ.,y[<-l) m min(y.,uk)-l
z
k=l
Z o .1 a , + Z
jks .j+ms
s=0 k=j
ajksa.j+ms
s=0
Thus, all dependent block columns can be generated directly from the set,
{a of regular block column vectors. These vectors are invariants under
GL(n), since every column vector of A. is an invariant as shown in
J
(2.2-12). Completeness follows immediately from the above recursion,
since any two members of X^ possessing identical invariants satisfy the
above recursion and therefore have identical Markov sequences.

30
Independence is shown by constructing the Luenberger form of (2.2-5) and (2.2-14)
below^ directly from these invariants.
The dual result yields another basis on X,, [{v.},{g. .},{aT }].
I I I S L J
The corresponding canonical forms for EeX-j or are given by the
Luenberger pairs of (2.2-5,2.2-6) and
and
a' (ia]-1 )m+ l I
(v^ljp+l.
]
(2.2-14)
and the canonical triples are denoted by and respectively.
Rissanen (1974) also shows that a canonical form for the transfer
function can be constructed from the invariants of (2.2-13). This is
possible because the determination of canonical forms for £ based on the
Markov parameters is independent of the origin of A^'s. Rissanen defines
the (left) matrix fraction description (MFD) as
T(z) := B"1(z)D(z) (2.2-15)
v ,
where B(z) = z B.z for |B \f 0
i=0 1 v
V-l 4
D(z) = I D.z1 .
i=0 1

31
The relationship of the MFD to the Hankel array, Sv+^ ^follows by
writing (2.2-15) as
B(z)T(z) = D(z) (2.2-16)
and equating coefficients of the negative powers of z to obtain the
recursion
BoAj + BlVl 'f + BvAj+v = mj j=1>2
expanding over j gives the relation over the block Hankel rows as
[B0 B, ... Bu]
where the pxp(v+l) matrix of B.'s is called the coefficient matrix of
B(z). Similarly equating coefficients of the positive powers of z
gives the recursion \
1
1 5 *
t
Vi,.
= 0.
m(y+l)
(2.2-17)
D, = B, ,A, + Bli0A0 + ... + B A .
k k+1 1 k+2 2 v v-k
k=0,l,...,v-l
or expanding over k gives the relation in terms of the first block Hankel
column as
Dv-1
Bv
O
1
Dv-2
II
Bv-1
Bv
*
o
1
_B1
Bn ... B
2 v
(2.2-18)

32
The canonical forms for both left and right MFD's are defined by the
polynomial pairs (BR(z),DR(z)) and ("Bc(z) ,Dc(z)) respectively, where
R and C have the same meaning as in (2.2-5,2.2-6) and the former is
given by
11
CL
Izv
; bT e K^v
(2.2-19)
for
4 = [4(V-V,.)
* *
6ki ek2 ...
ek(i+pv.) I
*
where
k=i+pvi and Bkj
are given by
{6ist}
j=i,i+p,..
.i+P.(V|-l)
*
Bkjs<
0
j^i,i+p,..
.*i+p(v.-T)
. 1
j=i+pvi
and DR(z) is determined from (2.2-18).
Dual results hold for the corresponding column vectors, b., jem of the
J
coefficient array of !q(z) in terms of the controllability invariants.
This completes the discussion of canonical forms for £ or T(z).
Note that analogous forms can easily be determined for the Buey forms
if X, is restricted to {v.}. Henceforth, when we refer to an invariant
. J
system description, we will mean any representation completely specified
by an algebraic basis. In the next section we develop the theory
necessary to realize these representations directly from the Markov
sequence

33
2.3 Canonical Realization Theory
. *
In this section we develop the theory necessary to solve the
canonical realization problem, i.e., the determination of a minimal
realization from an infinite Markov sequence, directly in a canonical
form for the action of GL(n). Obviously from the previous discussion,
this solution has an advantage over other techniques which do not obtain
E in any specific form. From the computational viewpoint, the simplest
realization technique would be to extract only the most essential
information from the Markov sequence--the invariants under GL(n). Not
only do the invariants provide the minimal information required to
completely specify the orbit of Z, but they simultaneously specify a
unique representation of this orbitthe corresponding canonical form.
Thus, subsequent theory is developed with one goal in mind--to extract
the invariants from the given sequence.
The following lemma provides the theoretical core of the subsequent
algorithms.
Lemma. (2.3-1) Let and W^, be any full rank factors of = V^,
Then each row (column) of is dependent iff it is a
dependent row (column) of (W^,).
Proof. From the factorization = V^, it follows if the j-th
row of is dependent, then there exists an aTe:KpN, a^O
such that
T_ nT
~ N,N' -W
Since p(WN,)=n, i.e., W^, is of full row rank, it follows that
-TsN N' WN' =

34
or .V^-r
T T T
but det (W^, WN,} t 0; thus, a = 0^, i.e., a dependent row
of is a dependent row of V^. Conversely assume that there .
exists a nonzero aT as before such that
v nT
VN ^tnN'
Since p(W^,)=n, it follows that this expression remains unaltered
if post-multiplied by W^,, i.e.,
A/V 4-
and the desired result follows immediately.V
The significance of this lemma is that examining the Hankel rows
(columns) for dependencies is equivalent to examining the rows (columns)
of the observability (controllability) matrix. When these rows (columns)
are examined for predecessor independence, then the corresponding
indices and coefficients of linear dependence have special meaning--
they are the observability (controllability) invariants. Thus, the
obvious corro!ary to this lemma is \
Corollary. (2.3-2) If the rows of the Hankel array are examined for
predecessor independence, then the j-th (dependent)
row, where j=i+pv., ie£ is given by
P
+ Z
i-l min(v.,v -1)
T 1 5 T
Ij = 2 z
s-1 t=0
p min(vi,vs)-l
isre+pt
s=l
Z
t=0
^ist-s+pt
whereig^^} an.d{v/} are the observability invariants
and kepN is the k-th row vector of .

35
Proof. The proof is immediate from Proposition (2.2-3) and Lemma (2.3-1).V
Note that similar results hold for the columns of the Hankel array when
examined for predecessor independence.
In the solution to some problems knowledge of both controllability
and observability indices are required. Moore and Silverman (1972)
require both indices to design dynamic compensators in order to solve
the exact model matching problem. Similarly the requirement exists in
the design of pole placement compensators and also stable observers as
indicated in Brausch and Pearson (1970) and more recently Roman and
Bullock (1975b). In an on-line application Saridis and Lobbia (1972)
require the controllability invariants as well as the observability
indices to solve the problem of parameter identification and control.
The latter case exemplifies the fact that in some instances it is first
necessary to determine the structural properties of a system from its
external description prior to compensation.
The need for an algorithm which determines both sets of controllability
and observability invariants from an external system description is
apparent. Computationally the simplest and most efficient technique to
determine these invariants would be some type of Gaussian elimination
scheme which utilizes elementary operations (e.g., see Faddeeva (1959)).
If we perform elementary row operations on such that the predecessor
dependencies of PV^ are identical to those of and perform column
operations on W^, so that W^,E and W^, have the same dependencies then
a
examination of = PS^ niE is equivalent to the examination of .
*
We define ^ as the structural array of ^,. This array is
specified by the indices {v^} and {y^} which are the least integers such
that the row and column vectors of ^ are respectively,

36
nonzero i
zero
for
ra=0;,...
.a=Vj..N-1
for k=i+pa
'nonzero'
for 4
b-0,... ,y -1
_ zero
[b=y .,... ,N-1
J
for s=j+mb
These results follow since ^ has identical predecessor dependencies
as SNjN,, then
N.N'
%>N
IT
where ^. = 0 if it depends on its predecessors. To find the observability
indices, let a be the index of the last nonzero row of ¡+pt t=0,l,...,N-1.
T T
Then if 6_. = jD v- = 0 otherwise = (a-i)/p+l. Similar results

follow when is expressed in terms of the c^. The following theorem
*
specifies the matrices P and E required to obtain
Theorem. (2.3-3) There exist elementary matrices P and E, respectively
lower and upper triangular with unit diagonal elements,
*
such that N=PS^ ^(E has identical predecessor
dependencies as
Proof. Let PS^ M,=Q where Q is row equivalent to ¡^i and P=[pr$].
If the j-th row of ^, is dependent on its predecessors, i.e.,
T T T
V yi ;*
then selecting P lower triangular such that

37
P
rs
gives this relation. From this choice of P it follows that
dependent rows of are zero rows of Q. If the j-th row
of SN is regular, then P unit diagonal-lower triangular
insures that the corresponding row of Q is nonzero and regular.
Similar results hold for the columns of ^, with E unit diagonal
upper triangular.
This choice of P does not alter the column dependencies of
for if the i-th column of is dependent on its predecessors,
then from Corollary (2.3-2) £. is uniquely represented as a
linear combination of regular vectors in terms of the control- .
lability invariants. Since P is unit diagonal-lower triangular,
it is the matrix representation of a nonsingular linear
transformation, Pr^q^. where q. is the i-th column vector of Q.
Thus, multiplying on the left every vector £. in (2.3-2) with
this P gives for i-j+mp.
J
3-1 minivyy^l)
q = Z Z
k=l s=0
m min(u.,u.)-l
Thus, we have shown that selecting P with the given structure
does not alter the predecessor column dependencies of S^ or
equivalently Q. Since the column vectors of Q satisfy the
above recursion, Sf^ and Q have identical predecessor column
dependencies, therefore, performing column operations on Q is
*
analogous to performing them on S^ and so we have SfJ
. *
(PS^ n,)E = QE or the predecessor dependencies of S^ N, and S^ M
are identical.V

38
This theorem shows that the indices can be found by performing a sequence
of elementary lower triangular row and upper triangular column operations
in a specified manner on the Hankel array and examining the nonzero rows

and columns of S^,, the structural array of The {cu^} and
{BTjsare also easily found by inspection from the proper rows of P and
columns of E as given by
Corollary. (2.3-4) The sets of invariants or more compactly
j
the sets of n vectors {8.},{a.} are given by the rows
of P and columns of E in (2.3-3) respectively as
4 CVV+P Pqr+P(vrl)] q'pV1 1>re£
a. [e .e .
-j st s+mt
es+m(prl)t] j-5£a
where
Pqrest
:
q=r, r=s
qt
Proof. The proof of this corollary is immediate from Theorem (2.3-3).V
We can also easily extract the set of invariant block row or column
vectors, {a! },{a } from the Hankel array and therefore, we have a
J - >
solution to the canonical realization problem.
Theorem. (2.3-5) If the generalized Hankel submatrix of rank n is
transformed by elementary row operations to obtain a row
equivalent array, then by proper choice of P the matrix Q
is given by:

39
TG | TFG
... j TFN'_1G
'V
pn-M
mN
0Pn-N
_umN _
Q =
where (F,G) is a controllable pair and det TVO.
Proof. .If z is a minimal realization, then it is well-known that
p(VN)=p(W^,)=n. Since P is an elementary array, then it follows
[PV^] PV =
r N
-I..
p=
and det T^O.
Post multiplication by W^, gives
[G | FG | ... |
PVV =
_T__
pN-
FN'"1G]
= PS
N,N1
Multiplication of the arrays gives the desired results.V
Corollary. (2.3-6) If P*is selected such that Q is as in (2.3-5) with the
pair (F,G) in Luenberger column form, then the set
of invariants {a-}, jem is given by the columns of
J
Wl\|> > w^ kemN1 with
ak = k=pjm+j
Proof. If P is selected in Theorem (2.3-5) such that T=T^, then it
follows that each column of W.,, corresponding to the (j+mp.)-th
'* u
for each jem contains the {a^} invariants.V
t.
The method of selecting P is given in the ensuing algorithm.

41
Theorem. (2.3-10) Given the infinite realizable Markov sequence
from an unknown system, then SQ=(,rQ>GQ>H(.)n is a
minimal canonical realization of {Ar} with
7C X
Fc = [W, | H2
*_
w ]
nr
Gc is a submatrix of (Wu+i)r given by the first m
u columns y
HC ta.l
a.l+m(y^-l)
a ... a_ 1
. m mu '
Km
and j fcj+m j+m
vector of
Vt
], jqn, Wr is a col
umn
Proof. Since the sequence is realizable, there exist, integers, n,v,y,
satisfying Proposition (2.1-5). If Q is given as in Corollary
(2.3-6), then
Q =
"Wk>c.r
0pv-"
L mk _
i1""'
for k>y+l
Thus, Gc is obtained immediately from the first m columns of
*
(Wr)c. Form two nxn arrays, A and A each constructed by
selecting n regular columns of (Wr)q starting with the first
*
column for A and the (1+m) column for A The independent
columns of (Wr)q are indexed by the y. and satisfy (2.3-8);
thus, they are unit columns and A is a permutation matrix, i.e,,
A = [w, ... | w
-1+m
w
2m
*' -j+miyj-l)
], jem

41
Theorem. (2.3-10) Given the infinite realizable Markov sequence
from an unknown system, then SQ=([:Q>GcH(,)n is a
minimal canonical realization of A^} with
Fc [, | W2
WJ
nr
Gc is a submatrix of (W +1)c given by the first
. columns .
m
HC = t\i
1+m(u1-1)
,m
am ]
^m
umn
and Uj = C%+m Sj+rapj]- k fs a 1
vector of (W^+^)c.
Proof. Since the sequence is realizable, there exist integers, n,v,y,
satisfying Proposition (2.1-5). If Q is given as in Corollary
(2.3-6), then
Q =
Gc 1 1 Fc Gc
jr\
I!
/"c"
for k>y+l
Thus, Gc is obtained immediately from the first m columns of

(Wk)c* ^orm two nxn arrays ^ and A each constructed by
selecting n regular columns of (W^)c starting with the first

column for A and the (1+m) column for A The independent
columns of (\)c are indexed by the y. and satisfy (2.3-8);
thus, they are unit columns and A is a permutation matrix, i.e,,
A = [w-, .
I 1+m
2m
w.
j+m(yj-l) *

42
where it follows from (2.3-8) that the columns of A form chains
satisfying
-j+miUj-l)-* ~
e ]
m
for q .= Eu. .
3 s=l 3
* *
Since A is A shifted m columns to the right, each chain of A
is given by [w.j+m ... Wj+m^ ] and again each column is unit
J T
except Wj+miJ = aj from Corollary (2.3-6). Thus, := A A
gives the matrix of (2.2-5). is obtained directly from
HcCGe
k-1
Fq Gq] = [a | ... a
m
a.l+m a.m(k+l)
L
since multiplication by the unit columns of (F^.G^J select the
n columns of H^.V
Analogous results hold for the dual ER. It should also be noted that
if the Hankel array is transformed to and both rows and columns
examined for predecessor independence as before, i.e.,
?S
N,N'
%
U =
'b V
vnV
(2.3-11)
Of
where is given in (2.2-8) and T is a permutation array, then all of the
previous theory is applicable. The only exception in this case is that
the Buey invariants and forms given by 3igR and liBC are obtained instead of
the Luenberger forms. These results follow directly from (2.2-1).
In many applications the characteristic polynomial xR(z) is required.
Many efficient classical methods (e.g., see Faddeeva (1959)) exist to
determine XR(z) from the system matrix. Even more recently some
techniques have been developed to extract the characteristic polynomial

43
from the Markov sequence, but in general they are only valid in the cyclic
case (see Candy et al. (1975)). An alternate solution to this problem
is to obtain the Buey form and use (2.2-11) to find xp(z) by inspection.
It is possible to realize the system directly in Buey form as mentioned in
the previous paragraph, but in this dissertation we prefer to take
advantage of the structure of the Luenberger form to construct Tg^ or
Tgg. Superficially, this method does not appear simple because the
transformation matrix and its inverse must be constructed, but the
following lemma shows that Tgg can almost entirely be written by inspection
from the observability invariants after the {v.} are known.
is given by
\
If the given triple is in Luenberger form, ZD, then the
(v^xn) submatrices Tg are
'V > w
v.-v. or T.
v.>v.
B
V .-VI

44
where
v.,jg are the observability invariants of ZR
i\,
v.¡ are the invariants associated with ZRR and
recall r. = Z v, ro=0>
Proof. This lemma is proved by direct construction of the TD 's,
Since each T0 satisfies for v.*v.
i i
hi
Bi*
h.F
v.-l
i' R
hiFR
'V ,
Vi-1
v.-l T
then analogous to property (2.3-8), it follows that h.FD =e
1 K ¥
1
and therefore
v.-l
V- V.-l j TP
hiFR -Fr = 4.FR = 4
V1 vi Vv1
Â¥r ' btfVvv
iFR
In order to construct TRR it is first necessary to find the {v..} from the
rows of [Vn]R, but in this case the v.'s can generally be found by
inspection while simultaneously building TRR. Also, TRR is generally a
sparse matrix with unit row vectors; therefore, the inverse can easily be
-1 1
found by solving M >n directly for the unknown elements of TRR.

45
jr\
In the next section we develop some new algorithms which utilize
the theory developed here.
2.4 Some New Realization Algorithms
In this section we present two new algorithms which can be used to
extract both observability and controllability invariants from the given
Markov sequence. Recall from Theorem (2.3-3) that performing row operations
on the Hankel array does not alter the column dependencies, however, it
is possible to obtain the row equivalent array, Q in a form such that
the controllability invariants can easily be found.
The first part of the algorithm consists of performing a restricted
Gaussian elimination (see Faddeeva (1959) for details) procedure on the
Hankel array. This procedure is restricted because there is no row or
column interchange and the leading element or first nonzero element of
each row is not necessarily a one. Define the natural order as 1,2,... .
Algorithm. (2.4-1)
(1) Form the augmented array: [IpN | S^, | ImN)] .
(2) Perform the following row operations on N, to obtain
Cp I Q I ImN'3:
(i) Set the first row of Q equal to the first Hankel row.
(ii) Search the first column of S^ ^, by examining the rows in
their natural order to obtain the first leading element.
This element is q^.
(iii) Perform row operations (with interchange) to obtain q^-j =0,k>j.
4*
Alternately it is possible to extract the Buey invariants from Q by
reordering the columns as in (2.2-8) to obtain ()=QU and examining
the columns for predecessor dependencies.

46
(iv) Repeat (ii) and (i i i) by searching the columns in their
natural order for leading elements.
(v) Terminate the procedure after all the leading elements have
been determined.
(vi) Check that at least the last p rows of Q are zero. This assures
that the rank condition, (R) is satisfied.
(3) Obtain the observability and controllability indices^ as in
Theorem (2.3-3).
T
(4) Obtain £. iejj from the appropriate rows of P as in Corollary (2.3-4)
T *
and jb. as in (2.2-19) where $...=p...
* J J
(5) Perform the following column operations on Q to obtain [P | S* Nl | E]:
(i) Select the leading element in the first column of Q, .
(ii) Perform column operations (with interchange) to obtain
qjs=0 for s>l.
(iii) Repeat (i) and (if) until the only nonzero elements in each row
are leading elements.
(6) Obtain a., jem from the appropriate columns of E as in Corollary
J
(2.3-4) and ¥. from the dual of (2.2-19).
(7) From the invariants construct the Luenberger and MFD forms as in
Section (2.2).
If we also require the characteristic polynomial, then we must include:
4*4*
(8) Determine the v.}, cjd and (simultaneously) construct TgR as in
Lemma (2.3-12).
(9) Find Til by solving for the non unit rows in TnnT¡i I .
BR BR BR n
^Note that the leading elements have been selected from the rows by examining
the columns in their natural order; therefore, the dependent columns are
not zero as in (2.3-3), but are easily found from this form of Q by
inspection. It should also be noted that the leading elements could have
been selected in the j, (j+m), (j+2m)... columns; therefore, facilitating
the determination of the Buey invariants and forms.
4.4* r\j
Alternately the {vj}, jem and Tjjc could be used. These indices can be
found easily from the columns of Q.

47
If we consider the alternate method implied in Corollary (2.3-6), then
the following modifications to the preceding steps are required:
(1)* Start with the following augmented array:
^pN I SN,N'l
(2)* Obtain [P | Q] as before.
(5)* Perform additional row operations on Q to obtain unit
columns for each column possessing a leading row element, and
perform row interchanges such that (2.3-8) is satisfied
for each jem, i.e., obtain
(6)* Obtain the a., jem, as in (2.3-6).
J
It should be noted that these algorithms are directly related to
those developed by Ho and Kalman (1966), Silverman (1971), or Rissanen
(1971). As in Ho's algorithm, the basis of the first technique is
performing the special' equivalence transformation of Theorem (2.3-3)
rk
on S^ to obtain S^ The second technique accomplishes the same
objectives by restricting the operations to only the rows of S^ which
is analogous to either the Silverman or Rissanen method. The initial
storage requirements in the first method are greater than the second if
mN'>pN, since P and E can be stored in the same locations due to their
lower and upper triangular structure; and (2) P will be altered in the
second method, since row interchanges must be performed in (5)*; whereas,
it remains unaltered in the first method which may be important in some
applications. Consider the following example which is solved using both
techniques.

48
Example. (2.4-2)
Let m=2, p=3, and the Hankel array be given as, ^
-
1
2
2
4
4
8
8
16
1
2
2
4
6
10
13
22
1
0
1
0
3
2
6
6
2
4
4
8
8
16
16
32
2
4
6
10
13
22
28
48
1
0
3
2
6
6
13
16
4
8
8
16
16
32
32
64
6
10
13
22
28
48
58
102
3
2
6
6
13
16
27
38
8
16
16
32
32
64
64
128
13
22
28
48
58
102
119
214
6
6
13
16
27
38
56
86
^ ^12 I S4,4 I V
. Y
(2) Performing the row operations as in (2.4-1), obtain [P | Q I I0],
O
where the leading elements are circled,
]
2 2 4 4 8
8
16:
-1
1
0
o
o
o

ro
5
6
Jl
2
i
T
1
0
@-1 -4 0 -5
_i_
2
-7
-2
b
0
1
0
0
0 0 0 0 0
0
0
1
2
_ 5 .
2
0
0
1
0
0 0 201
1
2
1
1
1
-1
0
-]
1
-4
0
b
0
0
0
1
r8
-3
0
-i

-1
0
0
1
0
1
-2
0
-1
0
0
0
1
o
00 -^1
-8
0
.0
0
. 0
0
0
0
0
1
-8
1
-2
0
-2
0
0
0
0
0 1
-1
2
-3
0
-2
0
0
0
0
0 0
1

49
(3)The indices are obtained by inspection from the independent rows
and columns of Q in accordance with Theorem (2.3-3) as:
v1 1 u1 = 3
^2 2 u2 ^
and p(S2s3) = p(S3 3) p(S2 = 4 satisfying (R).
T T
(4)The jfj and bjj are determined from the appropriate rows and columns
of P as:
1 ~ -CP41 I P42 P45 I P433 = [2 I 0 0 I 0]
-2 = "^p81 I p82 p85 I p83-* = I 0 V'M3
-3 = "^p61 I p62 p65 I p63^ = C"1 I 1 Ml
-1 = tO-3 I P4i P42 p43 p44 I 5^3 = [O3 I -2 0 0 1 I ^
2 = ^-p81 P82 P83 P84 P85 P86 P87 p88 I 3=E"3 O*1 0-1 0 011 3
-3 = % IP61 P62 p63 P64 P65 P66"* = % ¡ 1 1 _1 0 _1 13
(5)Performing the column operations, obtain the structural array
k
N, and E as:
... Jc
[P I 4 I E] where the leading elements are circled,.

50
0 0 0 0 0 0 0
0 0 0 00 o o
000 0 0 0 0 0
0000 0 000
0 000.0.0 o o
i _jl _jl n JL i. 13
2 2 U 4 8 4
10 o o o
1 -1 -4 -3
0 10 0
1 o
The a. and b. are determined from the appropriate rows and columns
J J
of E as:
e17
r 5-i
-T
e14
~ -1
e37
, 1
4
e34
1
e57
.5
2
ao ~ "*
e54
0
e27
1
8
e24
3
T
el 7
e27
- _5 -
4
1
8
^4
S4
e37
1
4
e14
1
e47
=
0
I -2 =
e24
s
3
" ~T
e57
5
2
e34
-1
e67
0
e44
1
e77
1
if
0
0

51
(7) The canonical forms of zR, BR(z), DR(z) and Eg, Bg(z),. DG(z) are
z2-2z 0
0
z
2z ~
Br(z) =
-3 z2+z
2
z -z +z
1
z2-z_
' VZ> =
z+1
0
2z+2
-2z
Fq = [eg £3 ] I 2-1
GC = [^1 ^4]
HC = *-a.l a.3 a.5
a>2] *
1
1
LI
2 4 I 2
2 6,2
1 3 0 _
Bc(z) =
TZ + fz +
i.
8
3. _2
-z +z
3 3 2
z fz
; Dc(z)
z fz £
z -z + £
z2- fz + f
(8) The {v.} and TgR
are determined simultaneously as:
1¡
1 1
1
1
0
0
0
and tbr
4
0
1
0
0
4
0
0
1
0
1
1
3
0
1
1

52
(9) T"1 is given by solving the equations for the last row
as:
r1
1BR
1
0
0
3
0
1
.0
0
0 0
0 0
1 0
-1 1
(10) Find FBR and Xp(z) as
F =T F
BR VBRVbR
2
0
0
2
0 0
0 1
0 0
1 0
0
0
1
2
and
XF(z) = (z-2)(z3-2z2+l) = z4-4z3+4z2+z-2
This completes the first method. If the second method is used instead,
then only (5)*, (6)*, and (8)* differ.
(5)* Performing the additional row operations and interchanges to
satisfy (2.3-8) gives:
5 1
T
1
-T
1
l
T
* -T 0
0
1
T
l
T
3
8
0
0 0
0 -$-
(D 0 0-1 0 --z- -f T
o 0 Q 1 o i *
0 0 0 0 1 £ 3
0 o I- 0
9 1
T -T
1 3
0
(6)* The a.'s are determined from the appropriate columns of Q as:
J
-T
"-1
-i
1
-1 =
5
T
= W7 2 =
0
I
L_ T _
- -
= W4

53
This completes the algorithms. In the next chapter the first method is
modified to develop a nested algorithm from finite Markov sequences.

CHAPTER 3
PARTIAL REALIZATIONS
One of the main objectives of this research is to provide an
efficient algorithm to solve the realization problem when only partial
data is given. As new data is made available (e.g., an on-line
application, Mehra (1971)), it must be concatenated with the old
(previous) data and the entire algorithm re-run. What if the rank of
the Hankel array does not change? Effort is wasted, since the previous
solution remains valid. An algorithm which processes only the new data
and augments these results (when required) to the solution is desirable.
Algorithms of this type are nested algorithms.
In this chapter we show how to modify the algorithm of (2.4-1)
to construct a nested algorithm which processes data sequentially.
The more complex case of determining a partial realization from a fixed
number of Markov parameters arises when the rank condition, abbreviated
(R), is not satisfied. It is shown not only how to determine the minimal
partial realization in this case, but also how to describe the entire
class of partial realizations. In addition, a new recursive technique
is presented to obtain the corresponding class of minimal extensions and
the determination of the characteristic equation is also considered.
3.1 Nested Alqorithm
Prior to the work of Rissanen (1971) no earlier recursive methods
appeared in the realization theory literature. Rissanen uses a
54

55
factorization technique to solve the partial realization problem when
(R) is satisfied. His algorithm not only solves the problem in a
simple manner, but also provides a method for checking (R) simultaneously.
In the scalar case, Rissanen obtains the partial realizations, Z(K),
K=l,2,... imbedded in the nested problem of (2.1), but unfortunately
this is not true in the multivariable case. Also, neither set of
invariants is obtained.
The development of a nested algorithm to solve the partial
realization problem given in this dissertation follows directly from
(2.4-1) with minor modification. There are two cases of interest when
only a finite Markov sequence is available.
Case I. (R) is satisfied assuring that a unique partial
realization exists; or
Case II. (R) is not satisfied and an extension sequence
must be constructed.
The nested algorithm will be given under the assumption that Case I
holds in order to avoid the unnecessary complications introduced in
the second case. The modified algorithm is given below. The corresponding
row or column operations are performed only on the data specified
elements.
Partial Realization Algorithm. (3.1-1)
(1) Same as (1) and (2) of Algorithm (2.4-1) except (iv) is q^O
k>j if k is a row whose leading element has been specified.
(2) If (R) is satisfied for some M*=v+y, obtain the invariants as
before in (3), (4) of (2.4-1) and go to (5). If not, continue.

56
(3) Add the next piece of data, Am+-j and form S(M+1,M+1).
(4) Multiply S(M+1,M+1) by P. Perform row operations (if necessary)
using old leading elements to obtain Q (M+1,M+1). If (R) is
satisfied, continue. If not, go to 3.
(5) Perform column operations as in (5) of (2.4-1) and obtain the
invariants and canonical forms as in (6), (7). Go to 3.
Example (2.4-2) will be processed to demonstrate the modified algorithm
for comparison. Assume that the Markov parameters are sequentially
available at discrete times, i.e., A^ is received, then Ag, etc., and
the system is to be realized.
Example. (3.1-2) Let the Markov sequence be given by
"l 2'
C\i
1
4
"4 8
8 16
'16 32~
A1 *
1 -2
J _
~
,2
_1
4
0_
, a3 =
6 10
_3 2_
ii
13 22
6 6
V
28 48
J3 16_
and apply the algorithm of (3.1-1). It is found that the rank condition
is first satisfied when A^, Ag are processed, i.e., v
(1) [I6 | S(2,2) | I4]
(2) Performing first row and then column operation as in (3.1-1),
obtain [P ) S*(2,2) ¡ E] or
l

0
0
0
1 -2 -1
0
-1
1
0
0
0
0
0
1 -i
-2
-1
0
1
0

0
0
0 1
0
-2
0
0
1
0
0
1
-2
0
0
0 1
0
0
__0
0
-1
0 0 1
0
0

57
(3) Indices are: v-j = 1
v2 = 0
h = 1
y2 = 1
oo
n
(4) ^Invariants are: Is-
tP41
1 P43] [2
1 0]
4--
CP61
O
| 1
II
1 1
OO
*£>
a.
1 1]
and
-1 = ^P41 P42
P
K43
P44 1 $
[-2 0
0 10 0]
2 = *--3 1 P21
P
2
1 o{] = [0
0 0-1
i 0]
-3 = ^P61 P62
P
63
P P P
64 K65
0-1001]
el 3
-1 '
e14
*0 "
eT3
1
a
2 '
e14
_0
e23
1
~T
e24
-2
T *."
e23
v
1
T
e24
2
. b, =
e33
1
5 bo~
e34
0
tm-
_0
_ 0-
-e44-
_1 _

58
where wT+ = -[P2] | P^] = [1 | o]
z-2
o
o
J
~i
2
-Z
2 0
; dr(z) =
0
0
_ 0
0 2-l_
_ i
0_
The rank condition is next satisfied when A1,Ag.are processed,
i.e., M =5 and we obtain [P | S (5,5) | E] as:
[P I Q(5,5)] =
1

2
2
4
4
8
8
16
16
.32
-1
1

0
0.
0
0

2
5
6
12
16
^1
0
1
0(
-1
-4
-1
-6
-2
-10
-3
-16
-2
0
0
1
0
0
0
0
0
0
0
0
-2
0
0
0
1
0
0

2
5
6
12
16
1
1
-1
0
-1
1
0
0
0
0
0
0
0
0
-4
0
0
0
0
0
1
0
0
0
0
0
0
-3
0
-1
0
-1
0
0
1
0
0
0
0
0
0
0
1
-2
0
-1
0
0
0
1
0
0
0
0
0
0
-8
0
0
0
0
0
0
0
0
1
0
0
0
0
-8
1
-2
0
-2
0
0
0
0
0
1
0
0
0
0
-1
2
-3
0
-2
0
0
0
0
0
0
1
0
0
0
0
-16
0
0
0
0
0
0
0
0
0
0
0
1
0
0
-24
0
-4
0
0
0
0
0
0
0
0
0
0 1
0
0
-8
0
-5
0
0
0
0
0
0
0
0
0
0 0 1
0
0
and performing the column operations give [S (5,5) J E]
+W is found easily from HrGr=A-j or solving for the second row of HR,
wTGr [1 2].

59
XD
0
0
0
0
0
0
0
0
1 -2 -1
-1
- T
4
_5
4
7
2
-10
-12"-
0
0
0
0

0
0
0
0
0
1 4
*
-1
8
1.3
4
-6
-14
0

0
0
0
0
0
0
0
0
1
-1
__5.
2
-3
a.
4
2
15
20
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0

0
0
0
0
0
1
-1
5
~T
-3
-6
-8
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0.
1
0
0
0
0
0
0
0
0
1
0
0
0
0
0
1
0
0
0
0
a
0
0
0
a a
a a
_o: a
The results in this case are identical to those of Example (2.4-2).
Let us examine the nesting properties of this realization algorithm.
Temporarily, we resort to using data dependent notation for this
discussion with the same symbols as defined previously in the previous
sections, e.g., the minimal partial realization of order M is given by
S(M) (F(M),G(M),H(M)). Thus, I(M+k) is a (M+k)-order partial
realization. We also assume for this discussion that £(M) is in row
canonical form; therefore, it can be expressed in terms of the set of
invariants, [{v.j(M)},{B.¡st(M)},{aT(M)}]. If S(M) is an n dimensional,
minimal partial realization specified by these invariants, then there
T
arc n regular vectors, ¥g+pt(M) spanning the rows of S(M,M). Furthermore,

60
T
each dependent row vector, f.(M) is uniquely represented as a linear
J
combination of regular vectors in terms of the observability invariants
and it can be generated from the recursion of Corollary (2)3-2); Similarly
it follows from Proposition (2.2-13) that the dependent block row
T
vectors, a. (M) satisfy an analogous recursion. The following lemma
J *
describes the nesting properties of minimal partial canonical realizations
Recall that M is the integer of Proposition (2.1-6) given by M =v+y.
Lemma. (3.1-3) Let there exist an integer M (M)-M such that the rank
condition is satisfied and let £(M) be the corresponding
minimal partial canonical realization of {Ar>, rcM
specified by the set of invariants [v.(M)},{fi. .(M)},
I l U
ia^(M)}]. Then
J '
v.(M) = ... = v.(M+k)
6-st(M) = ... = 3ist(M+k)
al (M) = .... = aT (M+k)
J J
' . ' A. ,
iff p(S(M,M))=p(S(M+l,M+l) = ... = p(S(M+k,M+k))
for the given k.
Proof. If v. (M) = ... = v. (M+k), etc., then the minimal canonical
partial realizations are identical, £(M) = £(M+1)= ... = £(M+k).
It follows that p(S(M,M))=dim£(M)=p(S(M+l,M+l))-dim£(M+l) =
p(S(M+k,M+k)).
Conversely, P(S(M,M))=P(S(M+1,M+1))= ... =P(S(M+k,M+k)) implies
dim£(M)=dim£(M+l)=... =dim£(M+k). Since £(M) is a unique minimal
canonical partial realization, so is £(M ). Furthermore, since
each realization has the same dimension, each realization has

61
has M (M)=M (M+l)= ... = M (M+k) so that each canonical
realization is equal to Z(M*); therefore, £(M)=£(M+1)= ... =Z(M+k).V
Next we examine the case where £(M) and z(M+k) are of different
dimension. The nesting properties are given in the following lemma.
^ "At
Lemma. (3.1-4) Let there exist integers, M (M)^M, M (M+k)^M+k such
that the rank condition is satisfied (separately) and
£(M), Z(M+k) are minimal partial canonical realizations
of (Ar> when reM and reM+k, respectively, for given k.
If p(S(M+k,M+k))>p(S(M,M)), then v.(M+k)*v.(M), ie£.
Furthermore, a. (M+k)=a. (M), j=i,i+p,...,i+p(v.(M)-l).
J J
Proof. Since p(S(M+k,M+k))>p(S(M,M)), M*(M+k)>M*(M) and therefore,
Sv(M),y(M) is a submatHx of Sv(M+k).uCM+k)* If the row
of is regular, it follows that the j-th row of
Sv(M+k) y(M+k) ls also re9u1ar by the nature of the Hankel
pattern, i.e., the rows of Sv^ are subrows of
Sv(M+k) ,y(M+k) The addition of more data (AM+],... ,AM+|<) to
S(M,M) makes previously dependent rows become independent rows
but previously independent rows remain independent; thus, the
v..(M) can only increase or remain the same, i.e., v..(M+k) ^
T
v.(M), ic£. The set of regular {a.(M+k)} are specified by the
' J
v.(M+k)'s; therefore ai (M+k)=aT (M), j=i,i+p,.. .,i+p(v.(M)-l),
1 J J 1
since vi(M+k)-vi(M), ie£.V
The results of these two lemmas are directly related to the nesting
4
properties of the partial realization algorithm. First, define JM as the
set of indices of regular Hankel row vectors based on M Markov parameters

62
available, i.e., {1,1+p,...,l+p(v1(M)-l),...,p,2p,...,pvi(M)}
and similarly denote the row vectors of the elementary row matrix
of the previous chapter, by (M). From Lemma (3.1-3), it follows
that jJ J*+k and 4+pv.(H)(M) £.T+pv1 (M+k)(M+k) si"ce
the observability invariants are identical. The specify the
elements in and along with the 3ist they specify the elements of
pi+pv.(M)(M) (see Coro11ary (2-3-4)). From Lemma (3.1-4) it is clear
that J¡¡jcjj¡¡+k since v. (M+k^v^M).
Reconsider Example (3.1-2), to see these properties. In this
case we have M=2, k=3, M*(2>2, M*(5)=5, and p(S(5,5))>p(S(2,2)) as
in Lemma (3.1-4); therefore, since 3-? = and J5 = d*3,2,5}.
The observability indices are identical except for v2(5)>v2(2); thus,
iaj (2) ,a2> (2)}<={aj^ (5) ,a^ (5) ,a2>(5) ,a^ (5)} since aT (2) = a! (5)
for j=l,3. We also know from Example (2.4-2) that £(5) is the solution
to the realization problem and therefore the properties of Lemma (3.1-3)
will hold for {A|yj}, M>5. Table (3.1-5) summarizes these properties.
The results for the dual case also follow directly. We now proceed to
the case of constructing minimal partial realizations when (R) is
not satisfied, i.e., the construction of minimal extensions.

63
Table. (3.1-5) Nesting Properties of Algorithm (3.1-1)
Augment M^M+k n(M+k)=n(M) n(M+k)>n(M)
JM"^M+k "
fi-i+pv.
>
c
where (R) is satisfied for some k and means that the
corresponding invariants, vectors, or indices are nested or
contained in a set of higher order.
Vj
5ist
J.

64
3.2 Minimal Extension Sequences
In this section we discuss the more common and difficult
problem of obtaining a minimal partial realization from a finite
Markov sequence when (R) is not satisfied. Two different approaches
for the solution of this problem have evolved. The first is based
on constructing an extension sequence so that (R) is satisfied
and the second is based on extracting a set of invariants from
the given data. We will show that these methods are equivalent
in the sense that they may both lead to the same solution. In order
to do this the existing algorithm is extended to obtain the more
general results of Roman and Bullock (1975a). Also a new recursive
method for obtaining the entire class of minimal extensions is
presented. It is shown that the existing algorithm does in fact
yield a particular solution to this problem which is valuable in
many modeling applications.
In the first approach, Kalman (1971b), Tether (1970) and
subsequently Godbole (1972) examine the incomplete Hankel array
to determine if (R) is satisfied. If so, the corresponding minimal
partial realization is found. If not, a minimal extension is con
structed such that (R) holdsand a realization is found as before.
They show that a minimal extension can always be found, but in
general it will be arbitrary. They also show that this extension
must be constructed so that the rank of S(M,M) remains constant
and the existing row or column dependencies are unaltered.
Considerable confusion has resulted from the degrees of freedom

65
available in the choice of minimal extensions. In fact, initially,
the major motivation for constructing an extension was that it
was necessary in order to be able to apply Ho's algorithm. Un
fortunately, these approaches obscure the possible degrees of
freedom and may lead to the construction of non-minimal extensions
as shown by Godbole (1972).
Roman and Bullock (1975a)developed the second approach to
the solution of this problem. They show that examining the columns
or rows of the Hankel array for predecessor independence yields
a systematic procedure for extracting either set of invariants
imbedded in the data. They also show that some of these would-be
invariants are actually free parameters which can be used to
describe the entire class of minimal partial realizations. These
results precisely specify the number and intrinsic relationship
between these free parameters. Unfortunately Roman and Bullock
did not attempt to connect their results precisely with those in
Kalman (1971b),Tether (1970). It will be shown that this connection
offers further insight into the problem as well as new results
which completely describe the corresponding class of minimal extensions.
Before we state the algorithm to extract all invariants available
in the data, let us first motivate the technique. When operating
on the incomplete Hankel array, only the elements specified by the
data are used. It is assumed that the as yet unspecified elements
will not alter the existing predecessor dependencies when they are
specified by an extension sequence. Since the predecessor dependencies
are found by examining only the data in S(M,M), we must examine
complete submatrices of S(M,M) in order to extract the invariants

66
associated with a particular chain (see Roman and Bullock (1975a)).
Therefore, it is possible that a dependent vector, say ^ of a sub-
matrix of S(M,M) later corresponds to an independent vector in S(M,M).
When representing any other dependent vector in this submatrix
m terms of regular predcessors, ¥. must be included, since it is
a regular vector of S(M,M) under the above assumption. In this represen-
tation the coefficient of linear dependence corresponding to
is arbitrary. Reconsider Example (3.1-2) for{A..}, i =1,2,3 where we
only consider the (row) map P.
Example. (3.2-1) For A^, Ag, A3 of (3.1-2) we have P: S(3,3)-K)(3,3) or
" 1 2 2 414 8
2 2 414 8*
1 2 2 416 10
1
000 0(2)2
IJ_J3 2
0^|)-1-4jl-6
2 4 4 8
0 0 0 0
2 4 6 10
P
0 0(2)2
10 3 2
0 0 0 0
4 8
0 0
6 TO
0 0
.3 2
_ 0 0
The indices are = {1,2,1}. Since v^l, the fourth row of
S(3,3) (or equivalently Q(3,3) ) is dependent on its regular predecessors
as shown in the corresponding 3x4 submatrix (in dashed lines) of S(3,3)
(or Q(3,3) ). The second row, say ^ > in this submatrix is dependent,
yet it is an independent row of S(3,3) (or Q(3,3) ). Now, expand of
this submatrix, i.e.,

67
= 3110 + e!20 -2 +f3130 -3 {312r0)
or
[2 4 4 8] = 3110 [1 2 2 4]+ g]20 [1 2 2 4]+ g^H 01 0] .
The solution is = 2 g^o ^130= t*1us* t*ie coefficient B-^O 1s
ah arbitrary parameter. Note that this recursion is essentially the
technique given in Roman and Bullock (1975a).
Clearly, if (R) is satisfied as in the previous section, then there
exists a complete submatrix (data is available for each element) of S(M*,M*)
in which every regular vector of S(M,M) is always a regular vector
of the submatrix corresponding to a particular chain; thus, there
will be no arbitrary or free parameters.
The algorithm for the case when (R) is not satisfied may be
illustrated by considering row operations on S(M,M) to obtain Q(M,M),
since the identical technique can be applied to obtain S*(M,M). The
arbitrary (column) parameters are found by performing additional
column operations to Q(M,M). As in Example (3.2-1), we must find
the largest submatrix of Q(M,M) for each chain, i.e., if we define
k_. as the index of the block row of S(M,M) containlirig the vector. ,
then the largest submatrix of data specified elements corresponding
to the i-th chain is given by the first (i+pv^-1) rows and m(M+l-k.)
columns of Q(M,M). Also, we define J|ie£ as the sets of Hankel row
indices corresppnding to each dependent (zero) row of the
(i+pMj-l)x (m(M+l-kj) submatrix of Q(M,M) which becomes independent, i.e.,
it contains a leading element. In Example (3.2-1) for i=l, we have
(1 +pv1-1)=3 and k-j =2; thus, nKM+l-k-^A and the corresponding submatrix is
given by the first 3 rows and 4 columns of Q(3,3), and.of course, J^={2}.

68
Arbitrary Parameter Partial Realization Algorithm. (3.2-2)
(1) Perform (1) of Algorithm (3.1-1) to obtain [P | Q(M,M)]t.
(2) For each ie£, determine the largest (i+pYl)xm(M+l-k..) sub
matrix of Q(M,M) of data specified elements and form the set J..
(3) For each ie£, replace pi by J + z bJ, b a scalar.
H i H i seJ^
(4) Determine the corresponding canonical forms incorporating
these free parameters.
Dual results hold for the columns. The fre-parameters are
fpund in analogous fashion by examining the zero columns of the
submatrices of S*(M,M).
Example. (3.2-3) The following example is from Tether (1970).
For m=p=2 and
"1 f
4 3
10 1
22 15
Ar
_0 0_
_0 0_
,A3-
_ 1 1_
4*
II
3 3 _
(1) [ P f Q(4,4) ] =
_1

1
4
3 j
10
7 22 15
0
1
0
0
0
!

1 3 3
-4
0
1
0
-6
5i
18-
13
0
0
0
1
0
0
0
!
Jj
0
0
2
0
-3
0
1
0
0
0
0
-1
0
0
1
0
1
0
0
0

6
0
-7
0
0
0 1
0
0
-3
0
0
0
0
0 0 1
0
0
4*
It should be noted that when (R) is not satisfied, some of thev. may not
be defined, i.e., the last independent row of a.chain is in the last block
Hankel row. In this case all_ would-be invariants are arbitrary.

69
The indices are: v-|=2, V2=3
(2) For 1=1, (1+p ^-1)=4, k.j=3, m'(M+T-k1 )=4;' thus, the corresponding
submatrix is constructed from the first 4 rows and columns of Q(4,4)
(small, dashes). J-j={2}.
For i=2, (2+pv2-l)=7, k2=4, m(M+l-k2)=2; thus, the corresponding
submatrix of Q(4,4) is given by the first 7 rows and 2 columns (large
r'a'shes). J2={2,4,6}.
I T
(3) Replacing the fifth and eighth rows of P with jDj. + b£2 and
£¡ + c4 +
T T
+ ej^. where b,c,d,e are real scalars gives
= [2 b -3 0 1 0 0 0 ]
= [-3-e c 0 d+e 0 e 0 1 ]
The §1 are:
g_{ = [ -2 3 -b 0 0]
g_2 = [ 3+e 0 -c -(d+e) -e ]
(4) The canonical form is
Corresponding to ttiese realizations is a minimal extension sequence
which can be found by determining the Markov parameters. These parameters
4

70
are cumbersome to obtain due to the general complexity of the expressions
in Er or therefore, a technique to determine these extensions
without forming the Markov parameters directly (or the realization) was
developed. This method consists of recursively solving simple linear
equations (one unknown) to obtain the minimal extension. Extensions
constructed in this manner not only eliminate the possibility of non
minimality as expressed in Godbole (1972), but also describe the entire
class of minimal extensions. The method of constructing the minimal
extension sequence evolves easily from the lower triangular-unit diagonal
structure of P. Since a dependent row of Q(M,M) is a zero row, it
follows from Theorem (2.3-3) that
for jemM
(3.2-4)
where recall that p.^ .=0 for j>i+pv..
i+pv^,j r 1
unkndwn extension parameters, x.y(r) fr
Thus, by inserting the
'lm
(r)
into S(M,M) a system of linear equations is established in terms of the
x..(r)s by (3.2-4). Due to the structure of P, this system of equations
i J
is decoupled and therefore easily solved.
Example. (3.2-5) Reconsider (3.1-2) for Since (R) is satisfied,
the extension A., j>2 is unique. We would like to obtain
0

71
A-
*11(3)
x21(3)
x31(3)
Since P maps S(2,2) into 0(2,2), we
x12(3)
x22(3)
x32(3)
have
2 2 4
2 2 4~
12 2 4
0 0 0 0
1 0 1 0
P
0 -1 -4
0 0 ^0 0
2 A J x-ji (3) x]2(3)
2 4 j x2i(3) x22(3)
0 0 j 0 0
1 0 | x3i (3) x32(3)_
0 0 | 0 0
and in this case,{v.j ,v2,v3> ={1,0,1}. Thus, using (3.2-4), we have
solving 0 = £4 £3 = C-2 0 0
1 0 Q]
2
2
1
for x11(3) gives x11(3)
X1 -j (3)
x2i(3)
X31(3)
Similarily solving:
¡^ = 0 for x-jg(3) gives x12(3) = 8
£3 = 0 for x3i(3) gives x^(3) =1
£^1^ = 0 for x32(3) gives x32(3) = 0
In this example, x2-j(3)=x^(3) and x22(3)=x-|2(3), since v2=0.

72
Thus, this example shows that the minimal extension sequence can be
found recursively due to the structure of P. Of course, the problem
of real interest is when (R) is not satisfied and (as in Ho's algorithm)
a minimal extension with arbitrary parameters must be constructed.
Minimal Extension Algorithm. (3.2-6)
(1) Perform (1), (2), (3) of Algorithm (3.2-2).
(2) Determine M* = v+y. (The values of v,y are determined by the partial data)
(3) Recursively construct the minimal extension {A^,}, r = M+l, ... ,M*
where Ar = [x^.(r)] by solving the set of equations for j(r)
given by
j+pv. Lj = 0 j m(M+l-k.)+l, ... ,m(M*+l-k.), for each iej>.
and recall that k. is the index of the block row of S(M,M) containing
the row vector,
i+pv-.
Example. (3.2-7) Reconsider (3.2-3) for illustrative purposes.
(1) These results are given in Example (3.2-3)
(2) M*=6; thus, find
A5 =
x-j 1 (5) x-j 2(5)
A6
x-j 1 (6) x-j2(6)
x21(5) x22(5)
X21(6) x22(6)
(3) Recursively solve: pT+2v £j 0 for i=1> j=5,6,7,8 and for
i=2, j=3,4,5,6.
£5 I5 = 0 gives x^iB); 1^ = 0 gives x12(5)
£^ r3 =0 gives x21(5); ^ = 0 gives x22(5)
£5 I7 = 0 gives xn(6); ^ = 0 gives Xj2(6)
£g Z5 = 0 gives x21(6); 1^ = 0 gives x22(6)

73
and therefore
A5"
46-b
31 -b
94-6b
63-6b
12-d
9-d-e
30-c-3d-5e+de
21-c-3d-5e+de-e2_
By solving for the x^.'s in Ag, Ag we obtain the extension as
x^B)
Xll(5)-15"
A, =
~6x11(5)-182 6x11 (5)-213
x2i(5)
x22 ^
6
X21 X21 (6)+(x21 (5)-X22(5)-3)2-9
The number of degrees of freedom is 4,i .e. .{x^(5),x21 (5),x22(5),x2-| (6)}.
The technique used to solve the parcial realization problem when (R)
is not satisfied was to extract the most pertinent information from the
given data in the form of the invariants, which completely described
the class of minimal partial realizations. A recursive method to obtain
the corresponding class of minimal extensions was also presented in {3.2-6)
This method is equivalent to that of Kalman (1971b) or Tether (1970) for
if the minimal extension is recursively constructed and Ho's algorithm
is applied to the resulting Hankel array the corresponding partial real
ization will belong to the same class. Note that if the extension is not
constructed in this fashion, it is possible that all degrees of freedom
available may not be found (see Roman (1975)). It should be noted that
the integers v and y are determined from the given data,i.e., knowledge
of the invariants enables the construction of a minimal extension such
that v and y can be found. The approach completely resolves the ambiguity
pointed out by Godbole (1972) arising in the Kalman or Tether technique.
The results given above correspond directly to those presented in

74
Kalman (1971b) and Tether (1970). They have shown, when (R) is satisfied,
there exists no arbitrary parameters in the minimal partial realization
or corresponding extension. Therefore, the existence of arbitrary
parameters can be used as a check to see if the rank condition holds.
Although it is not essential to construct both sets of invariants, it is
necessary to determine M* which requires v and y; thus, the algorithm
presented has definite advantages over others, since these integers are
simultaneously determined.
In practical modeling applications, the prediction of model
performance is normally necessary; therefore, knowledge of a minimal
extension is required. Also in some of the applications the number of
degrees of freedom may not be of interest, if only one partial realization
is required rather than the entire class. In this case such a model is
easily found by setting all free parameters to zero which corresponds to
merely applying the Algorithm (3.1-1) directly to the data and obtaining
the corresponding canonical forms as before.
Describing the class of minimal extensions offers some advantages
over the state space representation in that it is coordinate free and
indicates the number of degrees of freedom available without compensation.
3.3 Characteristic Polynomial Determination by Coordinate Transformation
In this section we obtain the characteristic equation of the entire
class of minimal partial realizations described by Fr or F^, of the
previous section. It is easily obtained by transforming the realized
Fr or Fc into the Buey form as before. Recall that the advantage of
this representation over the Luenberger form is that it is possible to
find the characteristic polynomial directly by inspection of FgR in (2.2-11).

75
Even though it is possible to realize the system directly in Buey form
as implied in the discussion of (2.3-12), it has been found that this
method has serious deficiencies when dealing with finite Markov sequences.
If (R) is satisfied, the partial realization is unique. When (R) is not
satisfied, this technique does not yield all degrees of freedom. For
example, reconsider the arbitrary parameter realization of Example (3.2-3).
This realization is given in Ackermann (1972) as
Q
1
0
0
O'
"0
1
0
0
0
-2
3
-b
0
0
-2
3
0
0
0
0
0
0
1
0
n
. 11
0
0
0
1
0
0
0
0
0
1
0
0
0
0
1
3+e
0
-c
-(d+e)
-e_
_3+e
0
-c
-(d+e)
-e_
Note that one degree of freedom (b=0) has been lost. Similarity
Ledwich and Fortmann (1974) have shown by example that this technique
can also lead to non-minima! realizations. These deficiencies arise due
to the procedure used for the determination of the Buey invariants. This
procedure does not account for the possibility that an independent row
vector of a particular chain may actually be dependent if it is compared
with portions of the same length of vectors in different chains. To cir
cumvent the problem, the previous technique will be used,i .e., the system
is realized directly in Luenberger form and transformed to Buey form. Not
only does this assure minimality as well as the determination of all possible
degrees of freedom, but Tg^ is almost found by inspection as shown in
(2.3-12). Reconsider the example of the previous section.

76
Example. (3.3-1) Recall that in (3.2-3) m=p=2, n=5, and Vj=2, V2=3,
= [ -2 3-bOO] §J2 = [3+e 0 -c -(d+e) -e ]
(1)Simultaneously construct TBR from (3.3-4) while examining the rows
for predecessor independence
-1
1
0
0
0
o"
4
0
1
0
0
0
4
s
-2
3
-b
0
0
4fr
-6
7
-3b
-b
0
/.
-14
15
-7b
-3 b'
-b
1 1
(2)Determine TBR from TBR TBR = In which gives
f1
'BR
1
0
-2/b
0
0
(3)Determine Fbr:
F =T F T"^ =
rBR BRrR BR
0
0
0
0
1
3/b
2/b
0
-1/b
3/b
-2/b
0
-1/b
3/b -1/b
1
0
0
0
-3b-2c-ce 3c-2d-2e
0
1
0
0
-c+3d+e
0
0
1
0
-d+2e-2
0
0
0
1
-e+3
(4)Find the characteristic polynomial by inspection.

77
Xr (z) = z5+(e-3)z4+(d-2e+2)z3+(c-3d-e)z2+(-3c+2d+2e)z+(b+2c+be)
rBR
This example points out some very interesting points. When this
technique is combined with the algorithm of (3.2-2), it offers a method
which can be used to obtain the solution to the stable realization
problem developed in Roman and Bullock (1975b). Also, if the system
were realized directly in Buey form, then b=0 and a degree of freedom is
lost; thus, in Ackermann's example Vj=l, while ours is v^=5. It is
critical that al_]_ degrees of freedom are obtained as shown in this case,
since the system is observable from a single output.
This section concludes the discussion of the deterministic case of
the realization problem. In the next chapter we examine the stochastic
version of the realization problem.

CHAPTER 4
STOCHASTIC REALIZATION VIA INVARIANT SYSTEMS DESCRIPTIONS,
In this chapter the stochastic realization problem is examined
by specifying an invariant system description under suitable trans
formation groups for the realization. Superficially, this may appear
to be a direct extension of results previously developed, but this is
not the case. It will be shown that the general linear group used in
the deterministic case is not the only group action which must be
considered when examining the Markov sequence for the corresponding
stochastic case. .
Analogous to the deterministic realization problem there are
basically two approaches to consider (see Figure 1): (1) realization
from the matrix power spectral density (frequency domain) by performing
the classical spectral factorization; or (2) realization from the
measurement covariance sequence (time domain) and the solution of a set
of algebraic equations. Direct factorization of the power spectral
density (PSD) matrix is inefficient and may not be very accurate.
Recently developed methods of factoring Toeplitz matrices by using fast
algorithms offer some hope, but are quite tedious. Alternately,
realization from the covariance sequence is facilitated by efficient
realization algorithms and solutions of the Kalman-Szego-Popov equations.
78

79
REALIZATION FROM
COVARIANCE SEQUENCE ^ PSD
AND ALGEBRAIC METHODS
STOCHASTIC REALIZATION
Figure 1. Techniques of Solution to the Stochastic
Realization Problem.
The problem considered in this chapter is the determination of a
minimal realization from the output sequence of a linear constant
system driven by white noise. The solution to this problem is well known
(e.g.. see Mehra (1971)) as diagrammed below in Figure 2. The output
sequence of an assumed linear system driven by white noise is correlated
and a realization algorithm is applied to obtain a model whose unit
pulse response is the measurement covariance sequence. A set of algebraic
equations is solved in order to determine the remaining parameters of
the white-noise system..
This problem is further complicated by the fact that the covariance
sequence must be estimated from the measurements. From the practical
viewpoint, the realization is highly dependent on the adequacy of the
estimates. Although in realistic situations the covariance-estimation
problem cannot be ignored, it will be assumed throughout this chapter
that perfect estimates are made in order to concentrate on the realization
-J*
portion of the problem.
In this chapter we present a brief review of the major results
necessary to solve the stochastic realization problem. We use the
4-'
Majumdar (1976) has shown in the scalar case that even if imperfect
estimates are made realization theory can successfully be applied.
FACTORIZATION
METHODS

80
White Noise Input
Stochastic Realization
Figure 2. A Solution to the Stochastic Realization Problem

81
algebraic structure of a transformation group acting on a set to obtain
an invariant system description for this problem. A new realization
algorithm is developed to extract this description from the covariance
sequence. Recently published results establishing an alternate approach
to the solution of this problem are also considered.
4.1 Stochastic Realization Theory
Analogous to the deterministic model of (2.1-1
) consider a white-
noise (WN) model given by
Vi = F*k + *k
(4.1
4 =H4
where and ^ are the real, zero mean, n state and p output vectors,
and Wj, is a real, zero mean, white Gaussian noise sequence. The noise
is uncorrelated with the state vector, X., j k and
J
Cov(wi,wj.):=E[(wi-Ewi)(Wj-Ew[.)T] = x.
where 6.. is the Kronecker delta. This model is defined by the triple,
ij
ZWN:=(F,In*H) compatible dimensions with (F,H) observable and F a
nonsingular,^ stability matrix, i.e., the eigenvalues of F have magnitude
less than 1. The transfer function of (4.1-1) is denoted by TWN(z).
+In the discussion that follows the WN model parameters will be used to
obtain a solution to the stochastic realization problem. Denham (1975)
has shown that if the spectral factors of the PSD are of least degree,
i.e., they possess no poles at the origin, then F is a nonsingular matrix

82
The corresponding measurement process is given by
h = h + \
(4.1-2)
where is the p measurement vector and v^ is a zero mean, white
Gaussian noise sequence, uncorrelated with x.., j k with
J
Covtvj.Vj) =
Covfw^Vj) = S5, j
for R a pxp positive definite, covariance matrix and S a nxp cross
covariance matrix. Thus, a model of this measurement process is
completely specified by the quintuplet, (F,H,Q,R,S).
When a correlation technique is applied to the measurement process,
it is necessary to consider the state covariance defined by
n^Covtx^,)^)
We assume that the processes are wide sense stationary; therefore,
\
nk = n, a constant here. It is easily shown from (4.1-1) that the
state covariance satisfies the Lyapunov equation (LE)
n = FIIFT + Q (4.1-3)
It is well known (e.g. see Faurre (1967)) that since F is a stability
matrix, corresponding to any positive semidefinite (covariance): matrix Q,
there exists a unique, positive semidefinite solution n to the (LE).
The measurement covariance is given (in terms of lag j) by
cj:= Gov(%j-4) = Cov(^+j4,+Cov(Vj^)+Co'' (4.1-4)

83
and from (4.1-1) it may be shown that
C. = HFJ_1(FnHT+S) j > 0 (4.1-5)
J
Co = HnHT + R
The PSD matrix of the measurement process is obtained by taking the
bilateral z-transform of the sequence C. defined in (4,1-4) which gives
3
$z(z) = H(Iz-F)'1Q(Iz"1-FT)"1HT+H(Iz-F)1S+ST(Iz"1-FT)'1HT+R
(4.1-6)
It is important to note that this expression is the frequency domain
representation of the measurement process which can alternately be
expressed directly in terms of the measurement covariance sequence as
00
$z(z) = E C.Z_J
j=-3
T
Since the measurement process is stationary and z is real, C and
therefore the PSD can be decomposed as
00 00 .
Mz) = 2 c Z*J + C + E C z3 (4.1-7)
L j=1 J 0 j=1 J
Note that {C.> is analogous to the Markov sequence of the deterministic
J
realization problem. We define the problem of determining a quintuplet,
(F,H,Q,R,S) in (4.1-6) from ^(z) or {Cj> as the stochastic realization
problem.
In this chapter we are only concerned with the realization from the
measurement covariance sequence. When a realization algorithm is applied
to the covariance sequence, we define the resulting realization as the
Kalman-Szegb-Popov (KSP) model because of the parameter constraints

84
(to follow) which evolve from the generalized Kalman-Szego-Popov lemma
a.
(see Popov (1973) ). Thus, we specify the KSf3 model as the realization
of {C.l defined by the quadruple, E^cd:=(A,B,C,D) of appropriate
J 0 Iw r
dimension with transfer function, TK<.p(z)=C(Iz-A)"^B+D. Note that since
the unit pulse response of the KSP model is simply related to the
measurement covariance sequence, then (4.1-7) can be written as the
sum decomposition.
*z(z) = T^pUJ+T^pU"1) = C(Iz-A)1B+D+DT+BT(Iz"1-AT)_1CT (4.1-8)
The relationship between the KSP model and the stochastic realization
of the measurement process is shown in the following proposition by
Glover (1973).
Proposition (4.1-9) Let zKSp=(A,B,C,D) be a minimal realization of {Cj}.
Then the quintuplet (F,H,Q,R,S) is a minimal stochasti
realization of the measurement process specified
by (4.1-1) and (4.1-2), if there exists a positive
definite, symmetric matrix n and TeGL(n) such that
the following KSP equations are satisfied:
n-AnAT * Q
D+DT-CnCT = R
B-AHC1" s
where A=T-1FT and C=HT.
The proof of this proposition is given in Glover (1973) and > ;
-J-
This book was published in Romanian ini966, but the English version
became available in 1973.
4j* oo
Note that the sequence, -CC^>Q is related to the measurement covariance
sequence as Cn-hC and C-=C. for j > 0.
0 0 J J

85
corresponds directly to the results presented by Anderson (1969) in
the continuous case. The proof follows by comparing the two distinct
representations of $z(z) given by (4.1-6) and (4.1-8). Minimality of
(F,H,Q,R,S) is obtained directly from Theorem (3.7-2) of Rosenbrock
(1970). The KSP equations are obtained by equating the sum decomposition
of (4.1-8) to (4.1-6).
This proposition gives an indirect method to check whether a given
KSP anc* stoc^ast'lc realization, (F,H,Q,R,S) correspond to the same
covariance sequence. Attempts to use the KSP equations to construct
all realizations, (F,H,Q,R,S) with identical {CL} from and T by
choice of all possible symmetric, positive definite matrices, H will
not work in general because all n's do not correspond to Q,R,S matrices
that have the properties of a covariance matrix, i.e.,
A:= Cov( pw.
V
[wj v¡]) .
ST R.
6,. 0
(4.1-10)
First, it is necessary to question if the stochastic realization problem
always has a solution, or equivalently, when is there a n so that
(4.1-10) holds. Fortunately, the well-known PSD property, 4>z(z) 0
on the unit circle (see e.g. Gokhberg and Krein (I960) and Youla (1961))
is sufficient to insure the existence of a solution. This result is
available in the generalized Kalman-Szego-Popov lemma (see Popov (1973)).
Proposition (4.1-11) If (F,H) is completely observable, then $z(z) 0
on the unit circle is equivalent to the existence
of a quintuplet, (?,fr,$,$,^¡) such that

86
$z(z)
[F(iz-f')'1 ip]
'V
Q
S
'(i2-1-f'T)-19r
where
a,
Oj-

Q
s
V
O.T
-W_
Ls
RJ
yj a.
S R
[VT WT]
^ 0
The proof of this proposition is given in Popov (1973) and essentially
consists of showing there exists a spectral factorization of the given
PSD. Thus, this proposition assures us that there exists at least one
solution to the stochastic realization problem.
Proposition (4.1-9) shows that once T, and n are determined
then a stochastic realization, (F,H,Q,R,S) may be specified; however, it
does not show how to determine n. Recently many researchers (e.g. Glover
(1973), Denham (1974,1975), Tse and Weinert (1975)) have studied this
problem. They were interested in obtaining only those solutions to the
KSP equations of (4.1-9) which correspond to a stochastic realization
such that A^O of (4.1-11). Denham (1975) has shown that any solution,
n*, of the KSP equations which corresponds to a factorization as in
(4.1-11) with V=KN, W=N for K=Knxp, NeKpxp, K full rank and N symmetric
positive definite, is in fact a solution of a discrete Riccati equation.
This can readily be seen by substituting, (Q,R,S) = (KNNTKT,NNT,KNNT)
of (4.1-11) into (4.1-9)
n*-An*AT = knnV
d+dt-cji*ct = nnt
(4.1-12)
B-An*CT = KNNT for A = T~]FT, C=HT, TeGL(n)

87
T
Solving the last equation for K and substituting for NN yields
K = (B-An*CT) (D+DT-CII*CT)"1 (4.1-13)
Now substituting (4.1-13) and NN^ in the first equation shows that n*
satisfies
n* = An*AT-(B-An*cT)(D+DT-cn*cT)"1(B-An*cT)T (4.1-14)
a discrete Riccati equation. Thus, in this case the stochastic
realization problem can be solved by (1) obtaining a realization,
^KSP ^rom'{Cj}.; (2) solving (4.1-14) for n*; (3) determining NN^ from
(4.1-12) and K from (4.1-13); and (4) determining Q,R,S from K and NN1.
A quintuplet specified by T and n* obtained in this' manner is guaranteed
to be a stochastic realization, but at the computational expense of solving
a discrete Riccati equation. Note that solutions of the Riccati equation
are well known and it has been shown thpt there exists a unique, n*,
which gives a stable, minimum phase, spectral factor (e.g. see Faurre
(1970), Willems (1971), Denham (1975), Tse and Weinert (1975)). We
will examine this approach more closely in a subsequent section, but
first we must find an invariant system description for the stochastic
realization.
4.2 Invariant 'System Description of the Stochastic Realization
Suppose we obtain two stochastic realizations by different methods
from the same PSD. We would like to know whether or not there is any
way to distinguish between these realizations. To be more precise,
we would like to know whether or not it is possible to uniquely
characterize the class of all realizations possessing the same PSD.
We first approach this problem from a purely algebraic viewpoint.

88
We define a set of quintuplets more general than the stochastic
realizations, then consider only those transformation groups acting
on this set which leave the PSD or equivalently (C .} invariant, and
*3
finally specify various invariant system descriptions under these
groups which subsequently prove useful in specifying a stochastic
realization algorithm. The groups employed were first presented by
Popov (1973) in his study of hyperstability. The results we obtain
are analogous to those of Popov as well as those obtained in the
quadratic optimization problem (e.g. see Willems (1971)).
Define the set
X2 = ((F,H,Q,R,S)| FeKnxn,HeKpxn,QeKnxri,ReKpxp,SeKnxp; Q,R symmetric}
and consider the following transformation group specified by the set
GKn := {L | LxKnxn; L symmetric}
and the operation of matrix addition. Let the action of GK^ on X2 be
defined by
L t (F,H,Q,R,S) := (F,H,Q-FLFT+L,R-HLHT,S-FLHT) V (4.2-1)
This action induces an equivalence relation on X2 written for each pair
(F,H,Q,R,S), (F,H,Q,R,S)eX2 as (F,H,Q,R,S)EL(F,H,Q,R,S) iff there exists
a LeGKn such that (F,H,Q,R,S) = L T(F,H,Q,R,S).
This group and GL(n) are essential to this discussion, but we must
consider their composite action. Therefore, we define the transformation
group, GRn which is the cartesian product of GL(n) and GKn,
GRn := GL(n)xGKn. The following proposition specifies GRn*

89
Proposition. (4.2-2) The closed set GRn and operation form a group
where
GRn = {(T,L) | TeGL(n);LeGKn}
and the group operation is given by
(T,L)o(T,L) = (TTfL+T"1LT"T).
Proof. This proof of this proposition follows by verifying the standard
group axioms with respective identity and inverse elements
(In.0n) and (T_1,-TLTT).V
Let the action of GR on X0 be defined by
(T,L) 4- (F,H,Q,R,S) : = (TFT_1 ,HT~\t(Q-FLFT+L)TT,R-HLHT,T(S-.FLHT) ) (4.2-3)
An element (F,h7q7R,1>) of the set is said to be equivalent to the
element (F,H,Q,R,S) of X2 if there exists a (T,L)eGRn such that
(F,H>'Q,R,S) = (T,L)4'(F,H,Q,R>S). This relation is reflexive
(F,H,Q,R,S) = (InOn) + (F.H,Q,R,S)
\
and symmetric
(r1iTLTT)T(F,H,Q,RiS) = (r1>-TLTT)f((T,L)T(FsHsQ>R>S)) =
((T"1JLTT)o(T,L))+(F,H,Q,R,S)=(In,On)T(F,H,Q,R,S) .
Transitivity follows from (F,H,Q,R,S)=(T,L)4'(F,HsQ,R,S) and
(F,H,Q,R,S)=(T,Lj 4'(Tr,lT,^,R,S^) = (T,r)T((T ,L)+(F,H,Q,R,S.)) = (f ,L)4-(F,H,Q,R,S).
Thus, GRn induces an equivalence relation on X2 which we denote by
ETL and (4.2-3) defines the partitioning of X2 into classes. Note that
our first objective has been satisifed, i.e., two EyL-equivalent quin
tuplets have the same PSD; for if we let the pair (F,H,QRS),
(F,H,Q,R,S)eX2 then if (T,L)eGRn

90
z(z)=K(Iz-?)-1^(Iz-1-?r)-1(Iz-?)-^+ST(I2-1-?T)-1K
=(HT"1)T(Iz-F)'1T"1(T(Q-FLFT+L)TT)T"T(Iz"1-FT)"1TT(HT1)T
+ (HT"1)T(Iz-F)"1T"1T($-FLHT)+(ST-HLFT)TVT(Iz'1-FT)1TT(Ht1)T
+R-HLH1 (4.2-4)
or
$z(z)=$z(z)+H(Iz-F)'1[L-FLFT-FL(Iz"1-FT)-(Iz-F)LFT-(Iz-F)L(Iz"1-FT)](Iz1-FT)~1HT
%
which gives $z(z) = $z(z). The measurement covariance sequence is also
invariant under the action of GR on X0 because the PSD is also given by
n 2
GO
$z(z) = 2 C.z"'5. Thus, we will call any two systems represented by the
j=- 3
quintuplet of Xp covariance equivalent, if they are E-^-equi valent.
Clearly, any two covariance equivalent systems have identical PSD's
(or measurement covariance sequences). Conversely, any two systems with
identical PSD's are covariance equivalent (see Popov (1973) for proof).
In order to uniquely characterize the class of covariance equi
valent quintuplets we must determine an invariant system description
for X0 under the action of GR The number of invariants may be found
by counting the parameters. If we define, :=dim(F,H,Q,R,S) and
p
Mz:=dim(T,L), then there are M^=n +np+%n(n+l)+%p(p+l)+np parameters
2
specifying this quintuplet and GRn acts on l^n +J^n(n+1) of them; thus,
there exist M^-Mz=2np+i2p(p+l) invariants. If we consider the transfor
mation, (TD,L)eGRn to the Luenberger row coordinates, then np of these
k n
invariants specify the canonical pair (FR,H^) of (2.2-6). The action
of (TR,L) on Q,R,S is given by

91
Qr = ^(Qrn^bCTRLVjtFt^^+LjT^QR-F^F^+L^ (4.2-5)
Rr = (HTr_1)(TrLTrT)(HTR~1)T = R-HrLrHrT (4.2-6)
SR = Tr(S(FTr""^ ) (TrLTrT) (HTr_1 )T) = Sr-FrLrHrT (4.2-7)
where LR = TRLTRT, Fr = TrFTr*1, Hr = HTR-1, QR = TRQTRT, SR = TRS.
The transformation LR acts on %n(n+l) parameters of the total
%n(n+l)+%p(p+l)+np parameters available in Qr,R,Sr as shown above for the
giveh (Fr,Hr). Once this action is completed the remaining np+*sp(p+l)
parameters are invariants. There are only four possible ways that LR
can act on the triple, (Q,R,S):
(i) Lr acts only on QR; (4.2-8)
(ii) Lr acts first to specify SR with the remaining elements
of Lr acting on QR;
(iii) L_ acts first to specify R with the remaining elements of
K
Lr free to act on QR or SR or both; and
(iv) Lr acts on any combination of elements in Q,R,S.
If we choose to restrict the action of LR to only the %n(n+l)
elements of Qr, then for any choice of QR (given (Fr,Hr) and any QR),
the transformation Lr is uniquely defined. Since FR is a nonsingular
stability matrix, then it is well known (Gantmacher (1959)) from (4.2-5)
that Lr is the unique solution of Lr-FrLrFrT = Q* for Q* = Qr-Qr*
is important to note that the elements of QR are completely free, but
once they are selected, LR is fixed by (4.2-5) for any QR and therefore
the np elements of ¥r and the Jsptp+l) elements of RR are the invariants.
Thus, for a particular choice of QR we can uniquely specify the equivalence

92
class of under the action of GRn, i.e., (FR,HR,QR,RR,SR) is a
canonical form for Ej^-equivalence on X2.
On the other hand, if we choose to let LD act on the np elements
K
of SR, then from (4.2-7) only np-%p(p-l) elements of LR are uniquely
specified, i.e., since LR is symmetric
and
LRHR f-l 4^+1
* +i ]
Vi 1
there are *sp(p+l) redundant elements in the £.'s. Thus, for any choice
J
of (given (FR,HR) and any SR), np-Jgp(p-l) elements of LR are uniquely
defined by (4.2-7) and the remaining elements of LR are free to act on QR
In other words np-^p(p-l) elements of QR are invariants, as well as
the elements of RR, since any choice of ifR specifies the elements of
Lr in (4.2-6).
Similarly restricting the action of LR to act on the elements of R
specifies %p(p+l) elements of LR from (4-2-6) and we are free to allow
the remaining elements of LR to act exclusively on QR or SR or both.
Clearly, there are many choices available to distribute the action of LR
on Qr,R,Sr; however, the important point is that once the choice is made,
the invariants are specified.
Any choice of symmetric (IR is acceptable, since LR is uniquely
determined from (4.2-5) for given QR,FR, but this is not the case when
an SR is selected. First recall that FR is nonsingular (see footnote
p.81). Then if we define SR: =SR-S"R it follows from (4.2-7) that
FR 1sR = LRHR (4.2-9)
and then
^This was pointed out by Luo (1975) and Majumdar (1975).

93
HRFR_1S* = HRLRHRT (4.2-10)
-1 *
In general, HRFR SR is not symmetric; therefore, the set of acceptable
S is restricted by (4.2-10). Since any square matrix can be decomposed
as the sum of a symmetric and skew-symmetric matrix, i.e.,
hrfr 1sr = (hrfr 1sr^sym
+ (hrfr sr)skw
A
then from (4.2-10) ^sp(p-1) elements of SR are constrained to satisfy
(for given (FR,HR))
(Hrfr Sr)$kw = 0p (4.2-11)
We limit our discussion to only cases (i) and (ii) of (4.2-8)
because the techniques employed to obtain the invariant system description
will be used in the next section to determine a solution to the
stochastic realization problem. Thus, we have satisfied our second
objective, i.e., we have specified a unique characterization of covariance
equivalent systemsan invariant system description for X2 under the
action of GR \
n
It is important to note that when QR is selected corresponding to
case (i) of (4.2-8), then SR and R are uniquely specified, but when S"R
is selected as in case (ii), R is again uniquely specified; however, this
is not true for Q. There is a family of Q 1s which correspond to this
S"R and R because only np-%p(p-l) elements of QR are fixed. Consider the
following example which not only illuminates this point, but also
shows how to uniquely characterize the class of covariance equivalent
systems by determining an invariant system description corresponding
to both (i) and (ii) of (4.2-8).

94
Example. (4.2-12) Suppose we are given the stochastic realization,
(F,H,Q,R,S) as
F =
0
1
o'
ro
i
01
rjin
5
-7
5"
0
0
1
, H =
V
JO)
il
-7
1
-1
_
24
3
8
1
2
Lo
i
1J
_ 5
-1
2_
R =
'2
i"
" 1 1"
J
4.
* S =
0 0
_-l _1_
and we would like to obtain (FR,HR,QR,RR,SR) corresponding to (i) of
(4.2-8)
(1)Use the transformation, (TR,I) to obtain, (FR,HR,QR,R,SR) as
-1
1
0
"l
0
0
"1
0
0
o
o"
0
1
0
7
1
1
hr
0
1
0_
* qr =
0
o
1
0
0
3 6 6 7
SR
-1
-1
1
L 4
24
2880-J
24
24
V
(2)First, select a QR as
0 0 J-i
u u 24
0 1
12
13 1 3051
2 4 12 ,144 0
then solving (4.2-5), we obtain and therefore
"l
f
' /V
" 1
-T
J
3_
li
-1
-i
2 4
3
(3)If we choose to select an SR instead, corresponding to (ii) of
(4.2-8) we must first determine the constraint imposed in (4.2-1).
The choice 5^2 trivially satisfies this constraint; thus,we solve
(4.2-5) for the (np-%p(p-l)) elements of LR

95
L
R
1 1 -1
1 1 -1
-1 -1
33
and therefore
2
-1
0
RR 0 3 Or 1 ^2_£33^ (-It^33)
This example illustrates two methods of specifying an invariant
system description of the given stochastic realization. It also
points out that selecting Q in (2) uniquely specifies R and S; however,
selecting S in (3) uniquely fixes R, but not Q. Thus, there is an
entire family of Q's which have the same R and S and each particular Q
specifies a canonical form for (F,H,Q,R,S) on X_ under the action of GR .
c n
We must place these results into the proper perspective, since we
are primarily concerned with the stochastic realization problem. Suppose
Clearly, we are free to choose any coordinate system, (T,I_)eGR Once
the coordinates are selected F and H are fixed from (4.1-9), since
-1-1
F=TAT H=CT but the major problem of finding an not only such that
the KSP equations are satisfied, but also so that A-0 still remains.
The above methods of specifying an invariant system description partially
resolve this problem. The first method shows that for given (F,H), the
matrices R and S are fixed once a Q is specified; therefore, this quintuplet
is an invariant system description, but whether or not it is a stochastic
realization corresponding to the same PSD or {Cj} as can only be
resolved by first determining if there exists a T and n such that the KSP

96
equations are satisfied. Clearly any choice of Q uniquely specifies
a n for given F; for if, there exist two solutions and corresponding
to identical Q,F, then n^=FH^F^+Q and n^FH^F^+Q. Subtract these equations
to obtain n*-FII*F"I"=On for -n^. It is well known (e.g. see
Gantmacher (1959)) that n*=0 is a unique solution of FH*-H*F*"^=0 ,
n n
sinc^ A(F)^X(-F~T) in this case. Therefore, selecting a Q uniquely
specifies a n and of course fixes R and S which can now be obtained from
the remaining KSP equations. Practical considerations in selecting a
Q which yields a positive definite II will be discussed in the next
section. Here the point is for given F and H (modulo TcGKn))
are obtained from (4.1-9); moreover, selecting Q uniquely specifies a
n which fixes R and S such that the KSP equations are satisfied. The
resulting model, (F,H,Q,R,S) has the same PSD or equivalently {C.} as
SKSP but ^ stl"^ may not satisfy A^O. Obtaining stochastic realizations
such that the latter condition is satisfied is the subject of the next
section.
Similar results can be obtained by using the second method of
specifying an invariant system description; however, recall in this
case that only np-%p(p-l) elements of Q are uniquely specified. Since
n is linearly related to Q for given F through the (LE), then the same
number of elements are uniquely specified in n. Thus, when we select
T=TR, the observability invariants of (2.2-4) uniquely specify the pair
(FR,HR) and for any choice of QR (or alternately SR) we specify an
invariant system description for the stochastic realization by
(F^Hj^Qj^RpjS^). In the next section we develop an algorithm based
on these techniques to obtain a stochastic realization.

97
4.3 Stochastic Realization Via Trial and Error
In this section, we develop an algorithm-to obtain a stochastic
realization from the measurement covariance sequence. We would like
to find this realization directly in a form which uniquely characterizes
the class of covariance equivalent quintuplets, i.e., quintuplets which
have the same PSD or equivalently {C.}. From Section (4.1) we already
know that one way to obtain a stochastic realization is to solve a
discrete Riccati equation; however, this technique can become computa
tionally burdensome when system order is large. Therefore, we would
like to develop an algorithm to directly extract an invariant system
description (under GR^) of the stochastic realization from the measurement
covariance sequence which does not require a solution of the Riccati
equation.
We briefly recall the results of earlier chapters to obtain a
canonical realization of the KSP model.. We show how constraints which
evolve from the stochastic nature of the problem can be used to obtain
a stochastic realization.
The canonical realization of from {Cj> follows by recalling
that the Hankel arrayadmits the factorization.
SN,N' = VNWN*
where Vj^jW^, are the corresponding observability and controllability
matrices. Thus, by applying the canonical realization algorithm of
(2.4-1) to {C.} we obtain the set [{v^}{6.-c+.}{a. }] which uniquely
j 1 1ST J .
specifies, (A,C)=(FR,HR) of (2.2-6) and B=GR of (2.2-14).
^Note that in realizing ien from (C.} we start with C, and not Cn;
l\b r j y I U
therefore, R is uniquely determined once ITH is found.

98
From Proposition (4.1-9), the observable pair.(A,C) of the KSP
model and (F,H) of the WN model are E^-equivalent; therefore, the
invariants are identical. The link between the canonical realization
of ZKSp and the stochastic realization is provided by the KSP equations
of (4.1-9), i.e., the (LE) and
(A.B.C) = (FR,FRnRHRT+sR,HR) (4.3-1)
D+DT = HRnRHRT+R
Recall that under the action of GR^ on X^, there are 2np+^p(p+l)
invariants--np specifying (FR,HR) and np+%p(p+l) specifying QR,R,SR.
It is possible to extract these np+%p(p+l) invariants from the
measurement covariance sequence using the KSP model realization,
(A,B,C,D). As before, if we assume the action of LR is restricted to
only the elements of QR, then for any choice of QR a unique nR is
specified by the (LE) and therefore R and SR are uniquely obtained from
(4.3-2)
T
R = W-DT-HRnRHRT
SR B-FRnRHR
On the other hand, suppose the action of LR is restricted to SR
and Qr, then for any choice of SR, R and np-%p(p-l) elements of QR are
fixed. Since nR is linearly related to QR through the (LE), the same
number of elements are uniquely specified in nR. We are free to select
the remaining elements in QR and n^. The realization invariants,
}, of the previous chapters allow us to uniquely specify
the invariants of (F,H,Q,R,S) from the KSP equations. Therefore, using
EKSp and (4.1-9) we are able to extract the 2np+%p(p+l) invariants of
(F,H,Q,R,S) from the measurement covariance sequence.

99
In many practical situations, it is known a priori that the
system and measurement noise sequences are uncorrelated. This case
has been considered by many researchers (e.g. Faurre (1967), Anderson
(1969), Mehra (1971), Rissanen and Kailath (1972), etc.) and it
corresponds to setting S.-o in the WN model of (4.1-1). It is crucial
to note that with this choice of S, it appears that the only trans
formation group which leaves the PSD invariant is GL(n). From (4.2-4)
it is clear that GRn not just GL(n) must be considered; therefore,
there are %n(n+l) fewer invariants when GRn rather than GL(n) acts on X2.
Recall the first technique outlined in Section (4.2) to obtain
(F,H,Q,R,S) from {Cj}: realize S^p, select a Q, specify a n from the
(LE), and then find R and 5 from the KSP equations. The selection of a
proper Q is essential to obtain a quintuplet of X2 that is a stochastic
realization. Therefore, it is useful to consider constraints which
evolve from the fact that Q and II,R,S are stationary covariance matrices.
For given (FR,HR)+, each choice of QR uniquely specifies a nR and hence
R,SR as in (4.3-2), i.e., (FR,HR,QR,R,SR) is a canonical form on X^ for
E-^-equivalence. Since F is a stability matrix, it follows from stability
theory that if (F,/Q)^ is completely controllable, corresponding to
each Q^O there exists a unique positive definite solution n to the (LE).
Therefore, restricting the choice of QR to be non-negative definite
simultaneously satisfies this "stability constraint" as well as the fact
that Qr must be a covariance matrix.
The results of the generalized Kalman-Szego-Popov lemma of (4.1-11)
assures us that there exists at least one realization such that A is a
+Here we assume the action of Gl(n) is completed with T=TR.
++/q is any full rank factor of Q, i.e., QVQ/Q"*"

100
covariance matrix; thus, the condition A-0 reduces to
det(Q-SR_1ST) ^ 0 (4.3-3)
since R is a positive definite covariance matrix. On the other hand,
if we consider the special case S=0p, then this constraint reduces the
condition A-0 to
det(Q) 0 for R > 0 (4.3-4)
Thus, the choice of admissible Q,R,S must be restricted such that
these constraints are satisfied. Recall that one possible choice is
(Q,R,S) = (KNnV,NN^,KNNT) where K and NN^"'are specified by IT*, the
unique solution of the discrete Riccati equation. Of course, if a
canonical realization algorithm is applied to {C^}, then is found
with T=TR, the Luenberger row coordinates, and (A,B,C)=(FR,GR,HR).
If a positive semidefinite is selected, then a nR>0 is uniquely
specified and therefore R and SR are found from the KSP equations. The
quintuplet, (FR,HR,QR,R,'R) is an invariant system description (under
GR ) and also a stochastic realization, if the above constraints are
n
satisfied. Note that the Riccati equation need not be solved. The
following algorithm summarizes this technique as well as the alternate method
discussed in Section (4.2).
Stochastic Realization Algorithm (4.3-5)
Step 1. Obtain from {Cj} as in (2.4-1).
Step 2. Select a positive semidefinite Qp and solve the (LE)
for nR.
Step 3. Solve (4.3-2) for R and SR.

TOT
Step 4. Check that (4.3-3) is satisfied. If so, stop. If not,
choose another QR-0 and go to 2. If numerous choices
of simple Q^O do not yield a stochastic realization, solve
the discrete Riccati equation of (4.1-13) and go to 3.
Or
Step 2* Select an SR satisfying (4.2-10).
Step 3* Solve (4.3-2) for R and Check that det (R)>0.
If so, continue. If not, go to 2*.
Step 4* Determine QR from the (LE) and select its free elements
to satisfy (4.3-4) if ^=0p or (4.3-3). If so, stop.
If not possible select another simple SR, i.e., go to 2*,
or try the first procedure, i.e., go to 2.
Consider the following example which illustrates this algorithm.
Example (4.3-6)
For m=3, p=2 the measurement covariance sequence is
20 59
iT5T
3
13 0 9
" 1,5 0
1
9 13
150
4 1
TT
22 6 1
- TO T "
79 1
6W
tl
o
o
3
7
w
O
II
66
25
,1 6
~ TT
S*
O
ro
H
13 9 1
6 0 0 "
1 9 3
6 0 0
c3 5
115 55
720 0
3 3 3 7
720 0
u.
U
1 5577
6155
~ 275717
561964
720 0
7200
C5
1036800
1036800
1 043 1
285588
21434293
4057079
_1 728 0
- 311040
12441600
" 124416 00.
Applying the algorithm of (2.4-1) we obtain
(1) The observability invariants are: v-|=l, V2=2 and
4 = t -i i i o]
J r i i __7_ Li
2 L 4 j 2 4 12*^

102
T T T T
(2)The {a. } invariants are {a^ ,a2 ,a4 }; therefore, the KSP model is
A=Fr
4
13 0 9
150
1
4
, B =
66
_ ~ _
1 6
T5
4
1391
1 9 3
' .
600

6 0 0
, C=Hn
-1
J
(3)Using the first approach of selecting a QR-0:
(i) 0^=1^ and solving the (LE) gives
n
R
67787
1261
82 9 9'
6 6 0 0
6 6 0
3 3 0 0
126 1
9 3 1
997
660
3 3 0
1650
8299
997
6 0 1
3 30 0
16 50
3 3 0
(it) Solving (4.2-14) for R and SR gives
' *
*
76 0 3
719
2419
59
6 6 0 0
66 0
22 0 0
660
and SR =
41 3
59
719
1379
~ 3 3 0 0
_ 1 6 50
660
3 3 0 .
59
1 77
nsa :
-
6 0 0
66 0 0
(4) det(I3-SRR"1SR^)>0
or using the second approach
(5) Let SR=02> then B=FRnRHRT and
- 1 6 09
150
= Cu-, n2] =
2 3
6 6 16
~25 ~ *25
and

103
where
7
-2
-24"
T 1609
"TFT
2
66
- nr
1
-6
-2
-24
and therefore nR =
2
3
16
- TF
1
0
6 8
1 6
0
-TT
~TT
n22_
(i) From (4.2-2) R is
R = C0-H^irH-r
3 1
1 4
and det R >0
(ii) Using the (LE) and (i) we have
0 0
i
3-n
R
1
0
22
T2H22-
0 12n22" 6
143
il
1
T
2053
144 22 2880
(iii) Choose q22=1 so that (4.3-4) is satisfied, then n22~2 and
Qr = diag(1 ,1 28 8 o)
This example points out some very important facts. First, it
follows that the measurement covariance sequence contains all of the
essential information necessary to specify a stochastic realization.
By choosing QR, nR is determined from the (LE) and the matrices R,SR
are uniquely found from (4.3-2) for given (FR,HR). Alternately,
selecting SD=0n, which satisfies (4.2-10), R is uniquely specified
k p
and np-%p(p-l) elements of QR and nR are invariants. The remaining
elements of QR and nR are free. This example also shows that there
may exist a measurement process with uncorrelated system and measurement
noise (S=0p) equivalent (under GRn) to a model with correlated noise
(S/0p), i.e., they both have identical PSD's or {C-}.
r w

104
Thus, the Riccati equation solution has essentially been
circumvented by this algorithm. However, if one not only desires
a stochastic realization, but also a stable minimum phase, spectral
factor, then the Riccati equation solution should be investigated.
This the subject of our next section.
4.4 Stochastic Realization Via the Kalman Filter
In this section we present a special case of the Riccati equation
approach to solving the stochastic realization problem. This approach
is a special case of the factorization (Denham (1975)) discussed in
Section (4.1) because we require the unique, steady state solution to
the discrete Riccati equation. It is well known (e.g. see Tse and
Weinert (1975)) that the steady state solution uniquely specifies the
optimal or Kalman gain. The significance of obtaining a stochastic
realization via Kalman gain is twofold.,, First, since the Kalman gain
is unique (modulo GL(n)), so is the corresponding stochastic realization.
Second and even more important, knowledge of this gain specifies a
stable, minimum phase spectral factor (e.g. see Faurre (1970) or
Willems (1971)). The importance of this approach compared to that of the
last section is that once the gain is specified, a stochastic realization
is guaranteed immediately, while this is not true using the trial and
error technique. However, the price paid for so elegant a solution to
the stochastic realization problem is the computational burden of solving
the Riccati equation.
We use the innovations representation of the optimal filter and
briefly develop it in the standard manner--from the estimation theory
viewpoint. We then examine the realization of this model from the

105
measurement covariance sequence in the steady state, stationary case
and show how this realization can be used to Represent the measurement
process of (4.1-2). Care is taken to formulate this realization problem
in precisely the same manner as the WN model of Section (4.1) in order
to emphasize the striking similarity between these two distinctly
different models of the same measurement process. Finally, we present
the algorithm to solve the stochastic realization problem using the
innovations representation. It should be emphasized that this technique
was presented in Mehra (1971) and improved in Carew and Belanger (1973)
and Tse and Weinert (1975).
The basic filtering problem is to find the best minimum error
covariance estimate of the state vector of the WN model in'terms of the
currently available measurement sequence, z^. A convenient model used
in the Kalman theory is the innovations (INV) representation given by
^k+l|k = F4|k-1 + Kk% (4.4-1)
. /y-
4|k-l = H-k| k-1 \
-k = ^k | k-1 + ~k
where _x, z, e_ are n state estimate vector, p measurement vector, and
p innovation (of z) vector and is the optimal estimate of
given z^.z^ ,... ,z^ .
The innovations sequence, {e^} is a zero mean, white Gaussian
process which is related to the WN model by
% ~ H^kIk-1 + -k
%
where x^.j jjC_i is the error in the estimate of x^, given zQ,z^,... ,z^^
defined by

106
k| k-1 = h 4| k-1
and is the measurement noise. Note that the'{z^} of (4.4-1) is
precisely the measurement process of (4.1-2); for if, we substitute
the above expressions of ancl 2ik||<-l int0 (4*4-1) we obtain
(4.1-2).
The respective covariances of _x, x_, £ are denoted by the nxn
%
matrices, n, E, n and (R )^ is the innovation covariance which satisfies
(4.4-2)
It is well known that the Kalman gain, K^, satisfies
Kk= (FV|1+s)(Re>"1k (4*4-3)
O/ '
where satisfies a discrete Riccati equation
*k fVifT +1 Kk-i(Vk-iKk-i (4-4-4)
The standard solution to the estimation problem is to solve (4.4-4)
and (4.4-2) for nk and then to calculate the corresponding Kalman gain,
K^ from (4.4-3). Since the observable pair (F,H) is known, the state
estimate is updated using the INV model. If we consider the stationary,
steady state case, then K^ = K^ = ... = K, 11^ = n^-j = ... = H, and
therefore, (R ). = (R ). = ... = R The stationary, steady state,
S K £ K** I £
INV model is given by the quadruple, E^y = (F,K,H,Ip) and R£ with K
the steady state Kalman gain and R£ the innovation covariance. The
transfer function is given by T^U) = H(Iz-F)\ +. I The Kalman
filter accepts as inputs the current measurement sequence of the WN
model and has as its.state the best minimum error covariance estimate of
the corresponding state vector.

107
The stochastic realization problem can be reformulated in terms
of the INV model in precisely the same manner as the WN model of
(4.1-1) and (4.1-2). Thus, expressions analogous to (4.1-4) and
(4.1-5) can be derived and therefore the measurement covariance sequence
is alternately given (in terms of lag j) by
C. = HFJ_1(FnHT+KR ) j>0
J £
Cn = HilHT+R
0 e
where the state estimate covariance matrix n satisfies
/s ^ T T .
n-FnF = kr.k'
e
analogous to the (LE) of (4.1-3) in n.
The PSD matrix of the measurement process in terms of the INV
model is obtained in precisely the same way as (4.1-6); therefore, we
have ,
$7(z)=H(Iz-F)_1KR KT(Iz1-FT)1HT+H(Iz-F)"1KR +R KT(Iz"1-FT)_1HT+R
L e e e e
\ (4.4-7)
It is also possible to express the PSD in factored form; thus, by
simple manipulation (4.4-7) can be written as
$z(z)=CH(Iz-F)'1K+Ip]Re[Ip+KT(Iz*1-FT)"1HT>TINV(z)ie(z)T]NV(z"1) (4.4-8)
Since R >0, the PSD is positive definite and (4.1-11) is always satisfied;
£
therefore, we can specify a stochastic realization immediately, once
F,H,K,R are determined. This stochastic realization is defined by £
e e
Ze := (FH>QINVsRINV>SINV) = (F>H*KReKTRe>KRe) (4*4-9)
(4.4-5)
(4.4-6)
where

108
Note that this realization is just a special case of the factorization
of Denham (1975) discussed in Section (4.1) with n*=n in (4.1-13),
K the Kalman gain in (4.1-12) and R£=NN^.
Clearly, the relationship between the canonical realization of
E^p and E£ of (4.4-9) is provided by the KSP equations, i.e., (4.4-6)
and
(A,B,C) (FR,FRHRHR +(Sinv)rHr)
(4.4-10)
D+D = HRnRHR+RINV
Note that since K and R are unique, then E is unique (e.g. see Tse
and Weinert (1975) or Denham (1975) for proof). Therefore, it is futile
to attempt to determine E£ from the trial and error algorithm of (4.3-5)
because this quintuplet is a unique stochastic realization (modulo GL(n)).
Recall from Section (4.2), if we let T=TR, then np of the total
2np+%p(p+l) invariants specify the pair (FR,HR) and it follows that np
specify Kr and %p(p+l) specify R£. Thus, the canonical realization of
'Xj 'X/ ^
the INV model is analogous to the WN model; however, unlike the QR,RR,SR
obtained in the WN case by trial and error, (QINV)R> RjNV> ^Sinv^R are
uniquely specified by KD and R in (4.4-9). The following diagram
summarizes the relationship between these two distinct approaches to
obtaining a stochastic realization.
REALIZATION FROM {C^
J
PSD
KSP
(Trial and Error)
FACTORIZATION METHODS
E
INV
(Riccati Equation)
(F,H,Q,R,S)
Figure 3, Solutions of the Stochastic Realization Problem

109
As a matter of completeness, we would like to briefly present an
algorithm to obtain the stochastic realization using the Riccati
equation approach. Mehra (1970,1971), Carew and Belanger (1973)
and even more recently Tse and Weinert (1975) have proposed iterative
schemes to obtain n,K,R£, but the theoretical connection to the KSP
equations and the stochastic realization was never established. Their
results are summarized below and we refer the interested reader to these
references for a detailed discussion of convergence properties and
simulation results.
Iterative Solution to the Riccati Equation (4.4-11)
Step 1. Set nQ = 0
Step 2. (R£).-. = D+D^-Hn.HT, where i is the i-th iteration step.
Step 3. K. = (B-Fn.HT)(R )T]
1 1 £ 1
Step 4. n.,, = Fn-FT+K.(R ).kI
K i+l i i e i i
Once K and R are found in this manner, then the stochastic realization
e
follows from (4.4-9). The following algorithm: summarizes the realization
technique using the INV model. <
Stochastic Realization Algorithm via INV Model (4.4-12)
Step 1. Obtain ZKSp from {C^} as in (2.4-1).
Step 2. Use the iterative technique of (4.4-11) to obtain K, R£.
Step 3. Determine QINV,RjNy,SINV from K and R£ as in (4.4-10).
Thus, we have two algorithms to obtain an invariant system description
for the stochastic realization using either the lsiN model or the INV model.
The following figure summarizes these techniques.
*

no
Figure 4. Stochastic Realization Algorithms,

CHAPTER 5
CONCLUSIONS
5.1 Summary
This dissertation has contributed results in realization theory
for both deterministic and stochastic cases. It was shown that by
carefully specifying the invariants of the realization problem under
a change of basis in the state space that a simple and efficient
algorithm to extract these entities from the Markov sequence could be
developed. This technique provides a solution to the realization problem
directly in a canonical form, and an invariant system description under
this transformation group is.'specified. The partial realization problem
was solved by modifying this technique to develop a nested algorithm.
It was shown that this method specifies the class of minimal partial
canonical realizations. A new recursive technique to determine the
corresponding class of minimal extensions while conserving all degrees
of freedom available was developed. These results bridge the gap between
the more classical approach of constructing a minimal extension and
that of extracting the realization invariants. The characteristic
equation is determined from the transition matrix in a convenient
coordinate system by inspection. These coordinates were easily obtained
from the given solution to the partial realization problem.
In the stochastic realization problem it was shown that the
transformation group which must be considered is richer than the general
111

112
linear group of the deterministic problem. The equivalence class under
this group was specified and it was shown how the additional constraints
imposed by the stochastic realization further restrict the selection of
free parameters available in the corresponding noise covariance matrices.
Specifying the invariants under this transformation group enabled the
development of a trial and error algorithm to obtain a stochastic
realization without requiring a Riccati equation solution.
The link between the KSP, WN and steady state Kalman filter was
presented. It was shown that realization of the KSP model allowed both
representations to be determined. It was shown that determination of
the filter parameters uniquely specifies a stochastic realization. An
algorithm requiring the solution of a Riccati equation was also presented.
5.2 Suggestions for Future Research
The results given in this dissertation open several interesting
*
possibilities for future research. Applying the algebraic framework
of a transformation group acting on a set offers definite advantages
over unstructured approaches. Simple equivalent solutions which confirm
physical intuition may evolve. It may be possible to specify a set of
invariants under the action of this transformation group which yields
considerable insight into the problem structure. If the problem possesses
additional constraints, it may be possible to utilize this information
to influence the choice of free parameters available. Many problems
of current interest can be examined in this framework (e.g. identification,
exact model matching, and stable observer design problems).
Efficient covariance estimators should be examined in order to
facilitate the development of realization algorithms which yield useful

113
results with realistic noisy data. Along these lines the use of
maximum likelihood estimators by Caines and Rissanen (1974) and the
least squares estimates in the technique of Majumdar (1976) should
be investigated further.
The use of Markov sequences to design controllers to solve the
model following problem (e.g. see Moore and Silverman (1972)) should
be examined by first defining the problem invariants and then inves
tigating the possibility of using the realization algorithms of
Chapters 2 or 3 to extract them. The use of the class (under GL(n))
of minimal extension sequences developed directly from a given finite
sequence may prove instrumental in this technique and should be studied.
An efficient technique to factor Toeplitz matrices (see Rissanen
and Kailath (1972)) should be developed by extracting the invariants
of the stochastic realization specified in Chapter 4. Analogous techniques
for the equivalent frequency domain representation of this problem should
also be investigated.

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BIOGRAPHICAL SKETCH
James Vincent Candy was born in Astoria, New York on January 21, 1944
He graduated from Holy Cross High School, Flushing, New York in June, 1961
He received the degree of Bachelor of Science in Electrical Engineering
in June, 1966 from the University of Cincinnati, Cincinnati, Ohio.
Upon graduating he worked with the General Electric Company for 9 months.
Then he enlisted in the Air Force of the United States in April, 1967.
He received a commission as a Second Lieutenant in June, 1967 after
completion of Officers Training School at Lackland AFB, Texas. He
spent the majority of his four years' active duty at Eglin AFB, Fla. as
a Threat Systems Engineer and Test Director until separated in June, 1971
as a Captain. In January 1968, he began study at the University of
Florida Extension School (GENEYSIS) for a Master of Science Degree in
Electrical Engineering. He completed his residency requirements in
March, 1972 and received the M.S.E. from the University of Florida.
From March, 1972, until the present time he has done work toward the
degree of Doctor of Philosophy.
James Vincent Candy is married to the former Patricia Meyers and
they have one lovely daughter, Kirstin Patrice. He is a member of
Phi Kappa Theta, Phi Kappa Phi, Eta Kappa Nu and the Institute of
Electrical and Electronics Engineers.
124

I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and is fully
adequate, in scope and quality, as a dissertation for the degree of
Doctor of Philosophy.
Thomas E. Bullock, Chairman
Professor of Electrical Engineering
I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and is fully
adequate, in scope and quality, as/a dissertation for the-^degree of
Doctor of Philosophy.
MTchael E. Warrenchairman
Assistant Professor of
Electrical Engineering
I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and is fully
adequate, in scope and quality, as a dissertation for the degree of
Doctor of Philosophy.
Donald G. Childers
Professor of Electrical Engineering
I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and is fully
adequate, in scope and quality, as a dissertation for the degree of
Doctor of Philosophy.
Zoran R. Pop/Stojanovic
Professor of Mathematics
I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and is fully
adequate, in scope and quality, as a dissertation for the degree of
Doctor of Philosophy.
'i ufa
opov
Vasile M. Popov
Professor of Mathematics

This dissertation was submitted to the Graduate Faculty of the College
of Engineering and to the Graduate Council, and was accepted as partial
fulfillment of the requirements for the degree of Doctor of Philosophy.
Dean, Graduate School

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103
where
7
-2
-24"
T 1609
"TFT
2
66
- nr
1
-6
-2
-24
and therefore nR =
2
3
16
- TF
1
0
6 8
1 6
0
-TT
~TT
n22_
(i) From (4.2-2) R is
R = C0-H^irH-r
3 1
1 4
and det R >0
(ii) Using the (LE) and (i) we have
0 0
i
3-n
R
1
0
22
T2H22-
0 12n22" 6
143
il
1
T
2053
144 22 2880
(iii) Choose q22=1 so that (4.3-4) is satisfied, then n22~2 and
Qr = diag(1 ,1 28 8 o)
This example points out some very important facts. First, it
follows that the measurement covariance sequence contains all of the
essential information necessary to specify a stochastic realization.
By choosing QR, nR is determined from the (LE) and the matrices R,SR
are uniquely found from (4.3-2) for given (FR,HR). Alternately,
selecting SD=0n, which satisfies (4.2-10), R is uniquely specified
k p
and np-%p(p-l) elements of QR and nR are invariants. The remaining
elements of QR and nR are free. This example also shows that there
may exist a measurement process with uncorrelated system and measurement
noise (S=0p) equivalent (under GRn) to a model with correlated noise
(S/0p), i.e., they both have identical PSD's or {C-}.
r w


LIST OF MATHEMATICAL SYMBOLS
Symbol
Usage
Meaning
First Usage
T
AT, aT
Transpose of A, ai
pg. 13
-1
A1
Inverse of A
pg. 13
-T
at
Inverse of AT
pg. 89
P
p(A)
Rank of A
pg. 13
|.|
|A| or det A
Determinant of A
pg. 30,21
diag A
Diagonal elements of A
pg. 103
\
x^y
x is not equal to y
pg. 30
>
x>y
x is greater than y
pg. 16
c
XcY
X is contained in or a
subset of Y
pg. 18
e
xeX
x is an element of X
pg. 12
X->Y
Map (set X into set Y)
pg. 20
: =
x: =
x is defined by
pg. 13
0
xoy
Abstract group operation
pg. 19
{ }
{.>
Sequence or set with
elements
pg. 13
Z
Summation
pg. 13
00
Infinity
pg. 13
t
Footnote
pg. 2
V
End of proof
pg. 34
4
Group action operator
pg. 2i
dim X
Dimension of vector
space X
. pg. 15
iff
if and only if
pg. 14
X
X(A). ,
Eigenval u'es of A
pg- 9e
/
/x
Square root of x
pg. 99
max()
Maximum value of
pg. 23
Z+
Positive integers
pg. 12'
K
Field
pg. 12
vi


94
Example. (4.2-12) Suppose we are given the stochastic realization,
(F,H,Q,R,S) as
F =
0
1
o'
ro
i
01
rjin
5
-7
5"
0
0
1
, H =
V
JO)
il
-7
1
-1
_
24
3
8
1
2
Lo
i
1J
_ 5
-1
2_
R =
'2
i"
" 1 1"
J
4.
* S =
0 0
_-l _1_
and we would like to obtain (FR,HR,QR,RR,SR) corresponding to (i) of
(4.2-8)
(1)Use the transformation, (TR,I) to obtain, (FR,HR,QR,R,SR) as
-1
1
0
"l
0
0
"1
0
0
o
o"
0
1
0
7
1
1
hr
0
1
0_
* qr =
0
o
1
0
0
3 6 6 7
SR
-1
-1
1
L 4
24
2880-J
24
24
V
(2)First, select a QR as
0 0 J-i
u u 24
0 1
12
13 1 3051
2 4 12 ,144 0
then solving (4.2-5), we obtain and therefore
"l
f
' /V
" 1
-T
J
3_
li
-1
-i
2 4
3
(3)If we choose to select an SR instead, corresponding to (ii) of
(4.2-8) we must first determine the constraint imposed in (4.2-1).
The choice 5^2 trivially satisfies this constraint; thus,we solve
(4.2-5) for the (np-%p(p-l)) elements of LR


42
where it follows from (2.3-8) that the columns of A form chains
satisfying
-j+miUj-l)-* ~
e ]
m
for q .= Eu. .
3 s=l 3
* *
Since A is A shifted m columns to the right, each chain of A
is given by [w.j+m ... Wj+m^ ] and again each column is unit
J T
except Wj+miJ = aj from Corollary (2.3-6). Thus, := A A
gives the matrix of (2.2-5). is obtained directly from
HcCGe
k-1
Fq Gq] = [a | ... a
m
a.l+m a.m(k+l)
L
since multiplication by the unit columns of (F^.G^J select the
n columns of H^.V
Analogous results hold for the dual ER. It should also be noted that
if the Hankel array is transformed to and both rows and columns
examined for predecessor independence as before, i.e.,
?S
N,N'
%
U =
'b V
vnV
(2.3-11)
Of
where is given in (2.2-8) and T is a permutation array, then all of the
previous theory is applicable. The only exception in this case is that
the Buey invariants and forms given by 3igR and liBC are obtained instead of
the Luenberger forms. These results follow directly from (2.2-1).
In many applications the characteristic polynomial xR(z) is required.
Many efficient classical methods (e.g., see Faddeeva (1959)) exist to
determine XR(z) from the system matrix. Even more recently some
techniques have been developed to extract the characteristic polynomial


4
For any two sets X and Y, a subset R c X x Y is called a binary
relation on X to Y (or, a relation "between" X and Y). Then (x,y)eR
is usually written as xRy and is read: "x stands in the relation R to
y". If for X=Y this relation is reflexive, symmetric, and transitive,
then it is an equivalence relation E on X given by xEy for x,yeX. The
set of all elements z equivalent to x is denoted by E(x),= {zeXjxEz} and
is called the equivalence class or orbit of x for the equivalence relation
E. The set of all such equivalence classes is called the quotient set
or orbit space and is given by X/E. Thus, the relation E of X partitions
the set X into a family of mutually disjoint subsets or orbits by sending
elements which are related into the same equivalence class.
ff
Consider a fixed group G of transformations acting on a set X.
Then the elements Xj,Xg of X are equivalent under the action of G iff
there exists a transformation TeG which maps x-j into.Xg.. This is basically
the "formula" we will apply throughout, i.e., we first formulate the set
of elements (generally the internal system description), then define a
transformation group; and finally determine the orbits under the action
of G. To be more precise, let us first define the function f mapping
a set X into Y as an invariant71^for E if for x-j^eX, x^Ex^ implies
f(Xi)=f(x2). In addition if f.(xi)asf(Xg) implies x-jExg, then f is a
X ^ ~
This is the standard Cartesian product, XxY = {(x,y)|xeX, yeY} .
.Here we mean "group" in the standard algebraic sense, i.e., (G)
where G is a closed set of elements each possessing an inverse and
the identify element; 0 is an associative binary operation. When
o is understood, the group is merely denoted by G.
4.4-4.
Note that an invariant is actually a function, but common usage
refers to its image as the invariant. We will also use this terminology
throughout this dissertation.


64
3.2 Minimal Extension Sequences
In this section we discuss the more common and difficult
problem of obtaining a minimal partial realization from a finite
Markov sequence when (R) is not satisfied. Two different approaches
for the solution of this problem have evolved. The first is based
on constructing an extension sequence so that (R) is satisfied
and the second is based on extracting a set of invariants from
the given data. We will show that these methods are equivalent
in the sense that they may both lead to the same solution. In order
to do this the existing algorithm is extended to obtain the more
general results of Roman and Bullock (1975a). Also a new recursive
method for obtaining the entire class of minimal extensions is
presented. It is shown that the existing algorithm does in fact
yield a particular solution to this problem which is valuable in
many modeling applications.
In the first approach, Kalman (1971b), Tether (1970) and
subsequently Godbole (1972) examine the incomplete Hankel array
to determine if (R) is satisfied. If so, the corresponding minimal
partial realization is found. If not, a minimal extension is con
structed such that (R) holdsand a realization is found as before.
They show that a minimal extension can always be found, but in
general it will be arbitrary. They also show that this extension
must be constructed so that the rank of S(M,M) remains constant
and the existing row or column dependencies are unaltered.
Considerable confusion has resulted from the degrees of freedom


106
k| k-1 = h 4| k-1
and is the measurement noise. Note that the'{z^} of (4.4-1) is
precisely the measurement process of (4.1-2); for if, we substitute
the above expressions of ancl 2ik||<-l int0 (4*4-1) we obtain
(4.1-2).
The respective covariances of _x, x_, £ are denoted by the nxn
%
matrices, n, E, n and (R )^ is the innovation covariance which satisfies
(4.4-2)
It is well known that the Kalman gain, K^, satisfies
Kk= (FV|1+s)(Re>"1k (4*4-3)
O/ '
where satisfies a discrete Riccati equation
*k fVifT +1 Kk-i(Vk-iKk-i (4-4-4)
The standard solution to the estimation problem is to solve (4.4-4)
and (4.4-2) for nk and then to calculate the corresponding Kalman gain,
K^ from (4.4-3). Since the observable pair (F,H) is known, the state
estimate is updated using the INV model. If we consider the stationary,
steady state case, then K^ = K^ = ... = K, 11^ = n^-j = ... = H, and
therefore, (R ). = (R ). = ... = R The stationary, steady state,
S K £ K** I £
INV model is given by the quadruple, E^y = (F,K,H,Ip) and R£ with K
the steady state Kalman gain and R£ the innovation covariance. The
transfer function is given by T^U) = H(Iz-F)\ +. I The Kalman
filter accepts as inputs the current measurement sequence of the WN
model and has as its.state the best minimum error covariance estimate of
the corresponding state vector.


86
$z(z)
[F(iz-f')'1 ip]
'V
Q
S
'(i2-1-f'T)-19r
where
a,
Oj-

Q
s
V
O.T
-W_
Ls
RJ
yj a.
S R
[VT WT]
^ 0
The proof of this proposition is given in Popov (1973) and essentially
consists of showing there exists a spectral factorization of the given
PSD. Thus, this proposition assures us that there exists at least one
solution to the stochastic realization problem.
Proposition (4.1-9) shows that once T, and n are determined
then a stochastic realization, (F,H,Q,R,S) may be specified; however, it
does not show how to determine n. Recently many researchers (e.g. Glover
(1973), Denham (1974,1975), Tse and Weinert (1975)) have studied this
problem. They were interested in obtaining only those solutions to the
KSP equations of (4.1-9) which correspond to a stochastic realization
such that A^O of (4.1-11). Denham (1975) has shown that any solution,
n*, of the KSP equations which corresponds to a factorization as in
(4.1-11) with V=KN, W=N for K=Knxp, NeKpxp, K full rank and N symmetric
positive definite, is in fact a solution of a discrete Riccati equation.
This can readily be seen by substituting, (Q,R,S) = (KNNTKT,NNT,KNNT)
of (4.1-11) into (4.1-9)
n*-An*AT = knnV
d+dt-cji*ct = nnt
(4.1-12)
B-An*CT = KNNT for A = T~]FT, C=HT, TeGL(n)


This dissertation was submitted to the Graduate Faculty of the College
of Engineering and to the Graduate Council, and was accepted as partial
fulfillment of the requirements for the degree of Doctor of Philosophy.
Dean, Graduate School


56
(3) Add the next piece of data, Am+-j and form S(M+1,M+1).
(4) Multiply S(M+1,M+1) by P. Perform row operations (if necessary)
using old leading elements to obtain Q (M+1,M+1). If (R) is
satisfied, continue. If not, go to 3.
(5) Perform column operations as in (5) of (2.4-1) and obtain the
invariants and canonical forms as in (6), (7). Go to 3.
Example (2.4-2) will be processed to demonstrate the modified algorithm
for comparison. Assume that the Markov parameters are sequentially
available at discrete times, i.e., A^ is received, then Ag, etc., and
the system is to be realized.
Example. (3.1-2) Let the Markov sequence be given by
"l 2'
C\i
1
4
"4 8
8 16
'16 32~
A1 *
1 -2
J _
~
,2
_1
4
0_
, a3 =
6 10
_3 2_
ii
13 22
6 6
V
28 48
J3 16_
and apply the algorithm of (3.1-1). It is found that the rank condition
is first satisfied when A^, Ag are processed, i.e., v
(1) [I6 | S(2,2) | I4]
(2) Performing first row and then column operation as in (3.1-1),
obtain [P ) S*(2,2) ¡ E] or
l

0
0
0
1 -2 -1
0
-1
1
0
0
0
0
0
1 -i
-2
-1
0
1
0

0
0
0 1
0
-2
0
0
1
0
0
1
-2
0
0
0 1
0
0
__0
0
-1
0 0 1
0
0


35
Proof. The proof is immediate from Proposition (2.2-3) and Lemma (2.3-1).V
Note that similar results hold for the columns of the Hankel array when
examined for predecessor independence.
In the solution to some problems knowledge of both controllability
and observability indices are required. Moore and Silverman (1972)
require both indices to design dynamic compensators in order to solve
the exact model matching problem. Similarly the requirement exists in
the design of pole placement compensators and also stable observers as
indicated in Brausch and Pearson (1970) and more recently Roman and
Bullock (1975b). In an on-line application Saridis and Lobbia (1972)
require the controllability invariants as well as the observability
indices to solve the problem of parameter identification and control.
The latter case exemplifies the fact that in some instances it is first
necessary to determine the structural properties of a system from its
external description prior to compensation.
The need for an algorithm which determines both sets of controllability
and observability invariants from an external system description is
apparent. Computationally the simplest and most efficient technique to
determine these invariants would be some type of Gaussian elimination
scheme which utilizes elementary operations (e.g., see Faddeeva (1959)).
If we perform elementary row operations on such that the predecessor
dependencies of PV^ are identical to those of and perform column
operations on W^, so that W^,E and W^, have the same dependencies then
a
examination of = PS^ niE is equivalent to the examination of .
*
We define ^ as the structural array of ^,. This array is
specified by the indices {v^} and {y^} which are the least integers such
that the row and column vectors of ^ are respectively,


5
group is the only information required to specify the corresponding
canonical form. Wonham and Morse (1972) obtained the feedback invariants
of the controllable pair from the not as lucid geometric viewpoint.
Their results were identical to those of Brunovsky and ftosenbrock. :
Morse (1973) examined the invariants of the triple (F,G,H) under a larg
group of transformations which includes output change of basis. A
complete set of feedback invariants of this triple still remains an
open problem, but some fragmentary results were presented by Wang and
Davison (1972) when they investigated certain sets of restricted triples.
Along these lines Rissanen (1974), Caines and Rissanen (1974), .
Mayne (1972a,b),Weinert and Anton (1972), Tse and Weinert (1973,1975),
Glover and Willems' (1974) examined the identification problem from the
invariant theory viewpoint and obtained some rather interesting results.
Recent results in decoupling theory were obtained by Warren and Eckberg
(1973), Concheiro (1973), and Forney (1975) using the Kronecker invariants
Probably the most extensive survey of these results has been compiled
by Denham (1974) and we refer the interested reader to this paper.
We temporarily leave this area to consider one specific application of
these resultsthe realization problem.
1.2 Survey of Previous Work in Realization Theory
The first realization problem proposed for control systems was
the determination of a state space model (internal description) from
a given transfer function (external description). Gilbert (1963) and
Zadeh and Desoer (1963) describe realization procedures based on the
determination of the rank of the residue matrices of the given transfer
function matrix, but unfortunately these procedures only apply to the


6
case of simple poles. Kalman (1963) proposed an algorithm whereby
the given transfer function is realized as a parallel combination of
single input, controllable subsystems in companion form, and then
applied the "canonical structure theory" (Kalman (1962)) to delete the
uncontrollable dynamics. This technique handles simple as well as
multiple transfer function poles. Later Kalman (1965) showed the
equivalence of the realization problem of control theory to the
corresponding network theory formulation.
A significant advance in realization theory was given by Ho and
Kalman (1966). They showed that the state space model could be found
from the impulse response sequence provided the system under investi
gation is finite dimensional. They also developed an algorithm based
on forming the generalized Hankel array from the given sequence and
then extracted the state space triple from it. Shortly after the pub
lication of Ho's algorithm, Youla and Tissi (1966) working in network
synthesis and Silverman and Meadows (1966) in control theory developed
similar realization techniques again based on the impulse response sequence.
Ho's algorithm gave new impetus to realization theory. Several
authors have provided alternate or improved realization algorithms based
on the Hankel array formulation. Mayne (1968), Panda and Chen (1969),
Roveda and Schmid (1970), Rosenbrock (1970), Lai et al. (1972) and even
more recently Huang (1974), Rozsa and Sinha (1975) among others,
considered the older transfer function matrix approach, while Rissanen
(1971,1974), Silverman (1971), Ackermannand Buey (1971), Chen and Mita!
(1972), Mita! and Chen (1973), and Bonivento et al. (1973) approached
the problem from the Hankel array formulation.


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AUTHOR: Candy, James
TITLE: Realization of Invariant system descriptions from markov (record
number: 180838)
PUBLICATION DATE: 1976
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80
White Noise Input
Stochastic Realization
Figure 2. A Solution to the Stochastic Realization Problem


68
Arbitrary Parameter Partial Realization Algorithm. (3.2-2)
(1) Perform (1) of Algorithm (3.1-1) to obtain [P | Q(M,M)]t.
(2) For each ie£, determine the largest (i+pYl)xm(M+l-k..) sub
matrix of Q(M,M) of data specified elements and form the set J..
(3) For each ie£, replace pi by J + z bJ, b a scalar.
H i H i seJ^
(4) Determine the corresponding canonical forms incorporating
these free parameters.
Dual results hold for the columns. The fre-parameters are
fpund in analogous fashion by examining the zero columns of the
submatrices of S*(M,M).
Example. (3.2-3) The following example is from Tether (1970).
For m=p=2 and
"1 f
4 3
10 1
22 15
Ar
_0 0_
_0 0_
,A3-
_ 1 1_
4*
II
3 3 _
(1) [ P f Q(4,4) ] =
_1

1
4
3 j
10
7 22 15
0
1
0
0
0
!

1 3 3
-4
0
1
0
-6
5i
18-
13
0
0
0
1
0
0
0
!
Jj
0
0
2
0
-3
0
1
0
0
0
0
-1
0
0
1
0
1
0
0
0

6
0
-7
0
0
0 1
0
0
-3
0
0
0
0
0 0 1
0
0
4*
It should be noted that when (R) is not satisfied, some of thev. may not
be defined, i.e., the last independent row of a.chain is in the last block
Hankel row. In this case all_ would-be invariants are arbitrary.


120
R. K. Mehra
[1970] "On the Identification of Variances and Adaptive
Kalman Filtering," IEEE Trans. Auto. Contr., Vol.
AC-15, pp. 175-184.
[1971] "On-Line Identification of Linear Dynamic Systems with
Applications to Kalman Filtering," IEEE Trans, on Auto.
Contr., Vol. AC-16, pp. 12-20.
D. P. Mi tal and C. T. Chen
[1973] "Irreducible Canonical Form Realization of a Rational
Matrix," Int. J. Contr., Vol. 18, pp. 881-887.
B. C. Moore and L. M. Silverman
[1972] "Model Matching by State Feedback and Dynamic Compen
sation," IEEE Trans on Auto. Contr,,Vol. AC-17, pp. 491.
A. S. Morse
[1973] "Structural Invariants of Linear Multivariable Systems,"
SIAM J. on Contr., Vol. IT, pp 446-465.
P. R. Motyka and J. A. Cadzow
[1967] "The Factorization of Discrete-Process Spectral Matrices,"
IEEE Trans, on Auto. Contr., Vol. AC-12, pp 698-706.
R. W. Newcomb
[1966] Linear Multiport Synthesis, Me Graw-Hill, New York. '
S. P. Panda and C. T. Chen -
[1969] "Irreducible Jordan Form Realization of a Rational Matrix,"
IEEE Trans, on Auto. Contr., Vol.AC-14, pp. 66-69.
V. M. Popov
[1961] "Absolute Stability of Nonlinear Systems of Automatic
Control," Avt. i Telemekh., Vol. 22, pp. 961-979.
[1964] "Hyperstability and Optimality of Automatic Systems
With Several Control Functions," Rev. Roum. Sci. Tech.
Electr. Enerqetique, Vol. 10, pp. 629-690.
[1969] "Some Properties of the Control Systems with Irreducible
Matrix Transfer Functions," in Seminar on Pifferentia!
Equations and Dynamical Systems-II. Springer,; Berlin,
pp. 169-180.
[1972] "Invariant Description of Linear, Time-Invariant
Controllable Systems," SIAM J. On Contr., Vol. 10, pp. 252-264
[1973] Hyperstability of Control Systems, Springer-Verlag, N.Y.
J. Rissanen
[1971] "Recursive Identification of Linear Systems," SIAM J.
on Contr., Vol. 9, pp 420-430.
[1972a] "Realization of Matrix Sequences," IBM Research Report, RJ
1032, San Jose, Calif.
[1972b] "Recursive Evaluation of Pade Approximants for Matrix
Sequences," IBM J. df Res. and Dev., Vol. 16, pp. 401-406.


83
and from (4.1-1) it may be shown that
C. = HFJ_1(FnHT+S) j > 0 (4.1-5)
J
Co = HnHT + R
The PSD matrix of the measurement process is obtained by taking the
bilateral z-transform of the sequence C. defined in (4,1-4) which gives
3
$z(z) = H(Iz-F)'1Q(Iz"1-FT)"1HT+H(Iz-F)1S+ST(Iz"1-FT)'1HT+R
(4.1-6)
It is important to note that this expression is the frequency domain
representation of the measurement process which can alternately be
expressed directly in terms of the measurement covariance sequence as
00
$z(z) = E C.Z_J
j=-3
T
Since the measurement process is stationary and z is real, C and
therefore the PSD can be decomposed as
00 00 .
Mz) = 2 c Z*J + C + E C z3 (4.1-7)
L j=1 J 0 j=1 J
Note that {C.> is analogous to the Markov sequence of the deterministic
J
realization problem. We define the problem of determining a quintuplet,
(F,H,Q,R,S) in (4.1-6) from ^(z) or {Cj> as the stochastic realization
problem.
In this chapter we are only concerned with the realization from the
measurement covariance sequence. When a realization algorithm is applied
to the covariance sequence, we define the resulting realization as the
Kalman-Szegb-Popov (KSP) model because of the parameter constraints


41
Theorem. (2.3-10) Given the infinite realizable Markov sequence
from an unknown system, then SQ=(,rQ>GQ>H(.)n is a
minimal canonical realization of {Ar} with
7C X
Fc = [W, | H2
*_
w ]
nr
Gc is a submatrix of (Wu+i)r given by the first m
u columns y
HC ta.l
a.l+m(y^-l)
a ... a_ 1
. m mu '
Km
and j fcj+m j+m
vector of
Vt
], jqn, Wr is a col
umn
Proof. Since the sequence is realizable, there exist, integers, n,v,y,
satisfying Proposition (2.1-5). If Q is given as in Corollary
(2.3-6), then
Q =
"Wk>c.r
0pv-"
L mk _
i1""'
for k>y+l
Thus, Gc is obtained immediately from the first m columns of
*
(Wr)c. Form two nxn arrays, A and A each constructed by
selecting n regular columns of (Wr)q starting with the first
*
column for A and the (1+m) column for A The independent
columns of (Wr)q are indexed by the y. and satisfy (2.3-8);
thus, they are unit columns and A is a permutation matrix, i.e,,
A = [w, ... | w
-1+m
w
2m
*' -j+miyj-l)
], jem


61
has M (M)=M (M+l)= ... = M (M+k) so that each canonical
realization is equal to Z(M*); therefore, £(M)=£(M+1)= ... =Z(M+k).V
Next we examine the case where £(M) and z(M+k) are of different
dimension. The nesting properties are given in the following lemma.
^ "At
Lemma. (3.1-4) Let there exist integers, M (M)^M, M (M+k)^M+k such
that the rank condition is satisfied (separately) and
£(M), Z(M+k) are minimal partial canonical realizations
of (Ar> when reM and reM+k, respectively, for given k.
If p(S(M+k,M+k))>p(S(M,M)), then v.(M+k)*v.(M), ie£.
Furthermore, a. (M+k)=a. (M), j=i,i+p,...,i+p(v.(M)-l).
J J
Proof. Since p(S(M+k,M+k))>p(S(M,M)), M*(M+k)>M*(M) and therefore,
Sv(M),y(M) is a submatHx of Sv(M+k).uCM+k)* If the row
of is regular, it follows that the j-th row of
Sv(M+k) y(M+k) ls also re9u1ar by the nature of the Hankel
pattern, i.e., the rows of Sv^ are subrows of
Sv(M+k) ,y(M+k) The addition of more data (AM+],... ,AM+|<) to
S(M,M) makes previously dependent rows become independent rows
but previously independent rows remain independent; thus, the
v..(M) can only increase or remain the same, i.e., v..(M+k) ^
T
v.(M), ic£. The set of regular {a.(M+k)} are specified by the
' J
v.(M+k)'s; therefore ai (M+k)=aT (M), j=i,i+p,.. .,i+p(v.(M)-l),
1 J J 1
since vi(M+k)-vi(M), ie£.V
The results of these two lemmas are directly related to the nesting
4
properties of the partial realization algorithm. First, define JM as the
set of indices of regular Hankel row vectors based on M Markov parameters


105
measurement covariance sequence in the steady state, stationary case
and show how this realization can be used to Represent the measurement
process of (4.1-2). Care is taken to formulate this realization problem
in precisely the same manner as the WN model of Section (4.1) in order
to emphasize the striking similarity between these two distinctly
different models of the same measurement process. Finally, we present
the algorithm to solve the stochastic realization problem using the
innovations representation. It should be emphasized that this technique
was presented in Mehra (1971) and improved in Carew and Belanger (1973)
and Tse and Weinert (1975).
The basic filtering problem is to find the best minimum error
covariance estimate of the state vector of the WN model in'terms of the
currently available measurement sequence, z^. A convenient model used
in the Kalman theory is the innovations (INV) representation given by
^k+l|k = F4|k-1 + Kk% (4.4-1)
. /y-
4|k-l = H-k| k-1 \
-k = ^k | k-1 + ~k
where _x, z, e_ are n state estimate vector, p measurement vector, and
p innovation (of z) vector and is the optimal estimate of
given z^.z^ ,... ,z^ .
The innovations sequence, {e^} is a zero mean, white Gaussian
process which is related to the WN model by
% ~ H^kIk-1 + -k
%
where x^.j jjC_i is the error in the estimate of x^, given zQ,z^,... ,z^^
defined by


97
4.3 Stochastic Realization Via Trial and Error
In this section, we develop an algorithm-to obtain a stochastic
realization from the measurement covariance sequence. We would like
to find this realization directly in a form which uniquely characterizes
the class of covariance equivalent quintuplets, i.e., quintuplets which
have the same PSD or equivalently {C.}. From Section (4.1) we already
know that one way to obtain a stochastic realization is to solve a
discrete Riccati equation; however, this technique can become computa
tionally burdensome when system order is large. Therefore, we would
like to develop an algorithm to directly extract an invariant system
description (under GR^) of the stochastic realization from the measurement
covariance sequence which does not require a solution of the Riccati
equation.
We briefly recall the results of earlier chapters to obtain a
canonical realization of the KSP model.. We show how constraints which
evolve from the stochastic nature of the problem can be used to obtain
a stochastic realization.
The canonical realization of from {Cj> follows by recalling
that the Hankel arrayadmits the factorization.
SN,N' = VNWN*
where Vj^jW^, are the corresponding observability and controllability
matrices. Thus, by applying the canonical realization algorithm of
(2.4-1) to {C.} we obtain the set [{v^}{6.-c+.}{a. }] which uniquely
j 1 1ST J .
specifies, (A,C)=(FR,HR) of (2.2-6) and B=GR of (2.2-14).
^Note that in realizing ien from (C.} we start with C, and not Cn;
l\b r j y I U
therefore, R is uniquely determined once ITH is found.


37
P
rs
gives this relation. From this choice of P it follows that
dependent rows of are zero rows of Q. If the j-th row
of SN is regular, then P unit diagonal-lower triangular
insures that the corresponding row of Q is nonzero and regular.
Similar results hold for the columns of ^, with E unit diagonal
upper triangular.
This choice of P does not alter the column dependencies of
for if the i-th column of is dependent on its predecessors,
then from Corollary (2.3-2) £. is uniquely represented as a
linear combination of regular vectors in terms of the control- .
lability invariants. Since P is unit diagonal-lower triangular,
it is the matrix representation of a nonsingular linear
transformation, Pr^q^. where q. is the i-th column vector of Q.
Thus, multiplying on the left every vector £. in (2.3-2) with
this P gives for i-j+mp.
J
3-1 minivyy^l)
q = Z Z
k=l s=0
m min(u.,u.)-l
Thus, we have shown that selecting P with the given structure
does not alter the predecessor column dependencies of S^ or
equivalently Q. Since the column vectors of Q satisfy the
above recursion, Sf^ and Q have identical predecessor column
dependencies, therefore, performing column operations on Q is
*
analogous to performing them on S^ and so we have SfJ
. *
(PS^ n,)E = QE or the predecessor dependencies of S^ N, and S^ M
are identical.V


15
Proof. See Silverman (1971).
Note that the essential point established in Ho and Kalman (1966), which
is used in the proof of the above proposition is that Z is a minimal
realization iff it is completely controllable and observable. Since
$ *V W it follows for dims n that: p(S ) min[p(V ),p(W )]=n.
v,p v y v,y v y
This result is essential to construct any realization algorithm. In
(2.1-5) the crucial point of finite dimensionality is carefully woven
into necessary and sufficient conditions for an infinite sequence to be
realizable. What if only partial information about the system is
available in the form of a finite Markov sequence? Is this sequence
realizable? What is the relationship between the minimal realization
and one based only on partial data? These are only a few of the questions
which must be resolved when we are limited to partial data.
Intrinsic in the realization from a finite Markov sequence is the
fact that enough data are contained in S to recover the infinite
a v,y
sequence, i.e., knowledge of (A^,...,A is sufficient to determine
{Ak>, k-1,2,... But in reality the only way to be sure of this is
knowledge of the actual system dimension (or at least an upper bound). A
minimal partial realization is a realization of smallest dimension
determined from a finite Markov sequence {A^},keM_ which realizes the
sequence up to M terms. The order of the partial realization is M and
the realization is denoted by Z(M). The realization induces an extension
k-1
of {Ak>, i.e., Ak=HF G for k>M. The following basic result analogous
to (2.1-5) answers the realizability question when only partial data are
given. For a proof, see Kalman (1971).


21
E(x) for every xeX. With these definitions in mind, our "formula"
becomes
(i)Formulate the set of elements;
(ii)Define the transformation group;
(iii)Determine a set of complete invariants under this
transformation group; and
(iv)Develop the canonical form in terms of the corresponding
invariants. (2.2-1)
We now apply (2.2-1) to various restricted sets related to multivariable
systems. This approach is essentially given in Kalman (1971a), Popov
(1972), Rissanen (1974), or Denham (1974). In this sequel we review the
main results of Popov. First, define the set of matrix pairs (F,G)
as
XQ = i(F,G)|FeKnxn, GeKnxm; (F,G)controllable}
The general linear group, which corresponds to a change of basis
in the state space, is specified by the set
GL(n):= {T|TeKnxn; det T?0} (2.2-2)
with the group operation standard matrix multiplication, i.e.,
T o T = T T.
In order to determine the orbits of XQ under the action of GL(n),
it is first necessary to specify the action operator
T + (F,G):= (TFT-1,TG)
4*
In general the problem of determining a canonical form is quite
difficult. However in this dissertation we consider restricted sets
which make the problem much simpler. For a thorough discussion of this
problem see Kalman (1973).


7
Rissanen (1974), Furata and Paquet (1975), Roman (1975),
Dickinson et al. (1974a,b) have recently considered the problem of
realizing a given infinite impulse response matrix sequence with a
polynomial matrix pair. Such a pair is referred to as a matrix-
fraction description of the system and is becoming well known in
control literature largely due to the ground work established by
Popov (1969), Rosenbrock (1970), Wolovich (1972a,b, 1973a,b) and
others.
Kalman (1971b), Tether (1970), and Godbole (1972) later
considered the more realistic case where only a finite number of
terms of the impulse response sequence are specified. This is
commonly known as the partial realization problem and corresponds
in the scalar case to the classical Pad approximation problem.
Generally most realization altorithms can be used to process partial
data, but usually at a loss of efficiency and even more seriously
the possibility of yielding misleading results. A wealth of new
techniques have recently been published to handle this very special,
yet realistic variant of the realization problem. Rissanen (1972a,b),
Ackermann (1972), Dickinson et al. (1974a), Roman and Bullock (1975a),
Anderson et al. (1975) published some efficient and improved algorithms
to solve this problem.
Also of recent interest is the development of algorithms which
realize a system directly in a canonical form (under a change of basis
in the state space), i.e., algorithms which solve the canonical realiza
ti on problem. The algorithms of Ackermann (1972), Bonivento et al.
(1973), Rissanen (1974), Dickinson et al. (1974a), Rozsa and Sinha
(1975), Luo (1975), and Roman and .Bullock (1975a) solve this problem.


G. D. Forney
[1975] "Minimal Bases of Rational Vector Spaces with Applications
to Multivariable Linear Systems," SIAM J. on Contr., Vol.
17, pp. 192-212.
K. Furuta and J. G. Paquet
[1975] "Determination of Matrix Transfer Function in the Form
of Matrix Fraction from Input-Output Observations,"
IEEE Trans, on Auto. Contr., Vol. AC-20, pp 392-396.
F. R. Gantmacher
[1959] The Theory of Matrices, Vols.T and 2, Chelsea Publishing Co.,N.Y
M. R. Gevers and T. Kailath
[1973] "An Innovations Approach to Least Squares Estimation-
Part VI: Discrete-Time Innovations Representation and
Recursive Estimation," IEEE Trans, on Auto. Contr., Vol.
AC-18, pp. 588-600.
E. G. Gilbert
[1963] "Controllability and Observability in Multivariable
Control Systems," SIAM J. on Contr., Vol. 1, pp. 128-151.
[1969] "The Decoupling of Multivariable Systems by State
Feedback," SIAM J. on Contr., Vol.7, pp. 51-63.
K. Glover
[1973] "Structural Aspects of System Identification," Rep.
ESL-R-516, Electronic Systems Laboratory, M.I.T.,
Cambridge, Mass.
K. Glover and J. Willems
[1974] "Parameterizations of Linear Dynamical Systems: Canonical
Forms and Identifability," IEEE Trans, on Auto. Contr.,
Vol. AC-19, pp. 640-646.
S. S. Godbole
[1972] "Comments on 'Contruction of Minimal Linear State-
Variable Models from Finite Input-Output Data,'"
IEEE Trans, on Auto. Contr., Vol. AC-17, pp. 173-175.
I. Gokhberg and M. G. Krein
[1960] "Systems of Integral Equations on a Half Line with Kernels
Depending on the Difference of Arguments," Uspekh: Mat.
Naut. 13, pp. 217-287.
B. Gopinath
[1969] "On the Identifcation of Linear Time Invariant Systems
from Input-Output Data," Bell Syst. Tech. J., Vol. 48,
pp. 1101-1113.


102
T T T T
(2)The {a. } invariants are {a^ ,a2 ,a4 }; therefore, the KSP model is
A=Fr
4
13 0 9
150
1
4
, B =
66
_ ~ _
1 6
T5
4
1391
1 9 3
' .
600

6 0 0
, C=Hn
-1
J
(3)Using the first approach of selecting a QR-0:
(i) 0^=1^ and solving the (LE) gives
n
R
67787
1261
82 9 9'
6 6 0 0
6 6 0
3 3 0 0
126 1
9 3 1
997
660
3 3 0
1650
8299
997
6 0 1
3 30 0
16 50
3 3 0
(it) Solving (4.2-14) for R and SR gives
' *
*
76 0 3
719
2419
59
6 6 0 0
66 0
22 0 0
660
and SR =
41 3
59
719
1379
~ 3 3 0 0
_ 1 6 50
660
3 3 0 .
59
1 77
nsa :
-
6 0 0
66 0 0
(4) det(I3-SRR"1SR^)>0
or using the second approach
(5) Let SR=02> then B=FRnRHRT and
- 1 6 09
150
= Cu-, n2] =
2 3
6 6 16
~25 ~ *25
and


107
The stochastic realization problem can be reformulated in terms
of the INV model in precisely the same manner as the WN model of
(4.1-1) and (4.1-2). Thus, expressions analogous to (4.1-4) and
(4.1-5) can be derived and therefore the measurement covariance sequence
is alternately given (in terms of lag j) by
C. = HFJ_1(FnHT+KR ) j>0
J £
Cn = HilHT+R
0 e
where the state estimate covariance matrix n satisfies
/s ^ T T .
n-FnF = kr.k'
e
analogous to the (LE) of (4.1-3) in n.
The PSD matrix of the measurement process in terms of the INV
model is obtained in precisely the same way as (4.1-6); therefore, we
have ,
$7(z)=H(Iz-F)_1KR KT(Iz1-FT)1HT+H(Iz-F)"1KR +R KT(Iz"1-FT)_1HT+R
L e e e e
\ (4.4-7)
It is also possible to express the PSD in factored form; thus, by
simple manipulation (4.4-7) can be written as
$z(z)=CH(Iz-F)'1K+Ip]Re[Ip+KT(Iz*1-FT)"1HT>TINV(z)ie(z)T]NV(z"1) (4.4-8)
Since R >0, the PSD is positive definite and (4.1-11) is always satisfied;
£
therefore, we can specify a stochastic realization immediately, once
F,H,K,R are determined. This stochastic realization is defined by £
e e
Ze := (FH>QINVsRINV>SINV) = (F>H*KReKTRe>KRe) (4*4-9)
(4.4-5)
(4.4-6)
where


73
and therefore
A5"
46-b
31 -b
94-6b
63-6b
12-d
9-d-e
30-c-3d-5e+de
21-c-3d-5e+de-e2_
By solving for the x^.'s in Ag, Ag we obtain the extension as
x^B)
Xll(5)-15"
A, =
~6x11(5)-182 6x11 (5)-213
x2i(5)
x22 ^
6
X21 X21 (6)+(x21 (5)-X22(5)-3)2-9
The number of degrees of freedom is 4,i .e. .{x^(5),x21 (5),x22(5),x2-| (6)}.
The technique used to solve the parcial realization problem when (R)
is not satisfied was to extract the most pertinent information from the
given data in the form of the invariants, which completely described
the class of minimal partial realizations. A recursive method to obtain
the corresponding class of minimal extensions was also presented in {3.2-6)
This method is equivalent to that of Kalman (1971b) or Tether (1970) for
if the minimal extension is recursively constructed and Ho's algorithm
is applied to the resulting Hankel array the corresponding partial real
ization will belong to the same class. Note that if the extension is not
constructed in this fashion, it is possible that all degrees of freedom
available may not be found (see Roman (1975)). It should be noted that
the integers v and y are determined from the given data,i.e., knowledge
of the invariants enables the construction of a minimal extension such
that v and y can be found. The approach completely resolves the ambiguity
pointed out by Godbole (1972) arising in the Kalman or Tether technique.
The results given above correspond directly to those presented in


REALIZATION OF INVARIANT SYSTEM DESCRIPTIONS FROM MARKOV SEQUENCES
By
JAMES VINCENT CANDY
A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL
OF THE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1976


58
where wT+ = -[P2] | P^] = [1 | o]
z-2
o
o
J
~i
2
-Z
2 0
; dr(z) =
0
0
_ 0
0 2-l_
_ i
0_
The rank condition is next satisfied when A1,Ag.are processed,
i.e., M =5 and we obtain [P | S (5,5) | E] as:
[P I Q(5,5)] =
1

2
2
4
4
8
8
16
16
.32
-1
1

0
0.
0
0

2
5
6
12
16
^1
0
1
0(
-1
-4
-1
-6
-2
-10
-3
-16
-2
0
0
1
0
0
0
0
0
0
0
0
-2
0
0
0
1
0
0

2
5
6
12
16
1
1
-1
0
-1
1
0
0
0
0
0
0
0
0
-4
0
0
0
0
0
1
0
0
0
0
0
0
-3
0
-1
0
-1
0
0
1
0
0
0
0
0
0
0
1
-2
0
-1
0
0
0
1
0
0
0
0
0
0
-8
0
0
0
0
0
0
0
0
1
0
0
0
0
-8
1
-2
0
-2
0
0
0
0
0
1
0
0
0
0
-1
2
-3
0
-2
0
0
0
0
0
0
1
0
0
0
0
-16
0
0
0
0
0
0
0
0
0
0
0
1
0
0
-24
0
-4
0
0
0
0
0
0
0
0
0
0 1
0
0
-8
0
-5
0
0
0
0
0
0
0
0
0
0 0 1
0
0
and performing the column operations give [S (5,5) J E]
+W is found easily from HrGr=A-j or solving for the second row of HR,
wTGr [1 2].


ACKNOWLEDGMENTS
I would like to express my sincere appreciation to the members
of my supervisory committee: Dr. Thomas E. Bullock, Chairman, and
Dr. Michael E. Warren, Cochairman, Dr. Donald G. Childers,
Dr. Z.R. Pop-Stojanovic and Dr. V.M. Popov. A special thanks to
Dr. Thomas E. Bullock and Dr. Michael E. Warren for their constant
encouragement, unending patience, and invaluable suggestions in the
course of this research.
I would also like to thank my fellow students and friends,
Zuonhua Luo, Arun Majumdar, Jos DeQueiroz, and Jaime Roman, for
many fruitful discussions and suggestions.


9
and Akaike (1974a,b). From the transfer function viewpoint this problem
has been solved using spectral factorization as originally introduced
by Wiener (1955,1959) and studied by others such as Gokhberg and Krein
(1960), Youla (1961), Davis (1963), Motyka and Cadzow (1967), and
Strintzis (1972). The link between the stochastic realization problem
and spectral factorization evolved from the work in stability theory by
Popov (1961,1964), Yakubovich (1963), Kalman (1963), Szego and
Kalman (1963). The equations establishing this link were derived in
the Kalman-Yakubovich-Popov lemma for continuous systems and the Kalman-
Szego-Popov lemma for discrete time systems. Newcomb (1966), Anderson
(1967a,b,1969), and Denham (1975) extended these results and provided
techniques to solve these equations. Defining the invariants of these
problems is still an area of active research as evidenced by the recent
work of Denham (1974), Glover (1973), and Dickinson et al. (1974b).
This is one area developed in this dissertation. It will be shown that
the invariants of the stochastic realization problem not only lends more
insight into the structure of the problem, but also yields some new
results.
Research in realization theory and its applications continues as
evidenced by the recent results of Rissanen (1975) in estimation theory,
Ackermann (1975) in feedback system design,De Jong (1975) in the
numerical aspects of the problem and Roman and Bullock (1975b) in
observer theory. The results presented in this dissertation tie together
some previously well-known results in stochastic realization and filtering
theory as well as provide a technique which can be used to study
other problems.


W. A. Wolovich
[1972a] "On the Synthesis of Multivariable Systems," 1972 JACC Pre
prints, Stanford, Calif, pp. 158-165.
[1972b] "The Use of State Feedback for Exact Model Matching,"
SIAM J. on Contr., Vol. 10, pp. 512-523.
[1973a] "Frequency Domain State Feedback and Estimation,"
Int. J. Contr., Vol. 17, pp. 447-428.
[1973b] "Multivariable System Synthesis with Step Disturbance
Rejection," Proc. 1973 IEEE CDC, San Diego, Calif., pp 320-
325.
W. M. Wonham and A. S. Morse
[1972] "Feedback Invariants of Linear Multivariable Systems,"
Automtica, Vol. 8, pp. 93-100.
V. A. Yakubovich
[1963] "Absolute-Stability of Nonlinear Control Systems in
Critical Cases," Avt. i Telem., Vol. 24, pp. 293-303.
D. C. Youla
[1961] "On the Factorization of Rational Matrices," IRE Trans,
Info. Thy., Vol. IT-7, pp 172-189.
D. C. Youla and P. Tissi
[1966] "N-port Synthesis via Reactance Extraction," IEEE Int. Conv.
Rec., Vol. 14, pp. 183-205.
L. A. Zadeh and C. A. Desoer
F19631 Linear System Theory: The State Space Approach., McGraw-


33
2.3 Canonical Realization Theory
. *
In this section we develop the theory necessary to solve the
canonical realization problem, i.e., the determination of a minimal
realization from an infinite Markov sequence, directly in a canonical
form for the action of GL(n). Obviously from the previous discussion,
this solution has an advantage over other techniques which do not obtain
E in any specific form. From the computational viewpoint, the simplest
realization technique would be to extract only the most essential
information from the Markov sequence--the invariants under GL(n). Not
only do the invariants provide the minimal information required to
completely specify the orbit of Z, but they simultaneously specify a
unique representation of this orbitthe corresponding canonical form.
Thus, subsequent theory is developed with one goal in mind--to extract
the invariants from the given sequence.
The following lemma provides the theoretical core of the subsequent
algorithms.
Lemma. (2.3-1) Let and W^, be any full rank factors of = V^,
Then each row (column) of is dependent iff it is a
dependent row (column) of (W^,).
Proof. From the factorization = V^, it follows if the j-th
row of is dependent, then there exists an aTe:KpN, a^O
such that
T_ nT
~ N,N' -W
Since p(WN,)=n, i.e., W^, is of full row rank, it follows that
-TsN N' WN' =


32
The canonical forms for both left and right MFD's are defined by the
polynomial pairs (BR(z),DR(z)) and ("Bc(z) ,Dc(z)) respectively, where
R and C have the same meaning as in (2.2-5,2.2-6) and the former is
given by
11
CL
Izv
; bT e K^v
(2.2-19)
for
4 = [4(V-V,.)
* *
6ki ek2 ...
ek(i+pv.) I
*
where
k=i+pvi and Bkj
are given by
{6ist}
j=i,i+p,..
.i+P.(V|-l)
*
Bkjs<
0
j^i,i+p,..
.*i+p(v.-T)
. 1
j=i+pvi
and DR(z) is determined from (2.2-18).
Dual results hold for the corresponding column vectors, b., jem of the
J
coefficient array of !q(z) in terms of the controllability invariants.
This completes the discussion of canonical forms for £ or T(z).
Note that analogous forms can easily be determined for the Buey forms
if X, is restricted to {v.}. Henceforth, when we refer to an invariant
. J
system description, we will mean any representation completely specified
by an algebraic basis. In the next section we develop the theory
necessary to realize these representations directly from the Markov
sequence


43
from the Markov sequence, but in general they are only valid in the cyclic
case (see Candy et al. (1975)). An alternate solution to this problem
is to obtain the Buey form and use (2.2-11) to find xp(z) by inspection.
It is possible to realize the system directly in Buey form as mentioned in
the previous paragraph, but in this dissertation we prefer to take
advantage of the structure of the Luenberger form to construct Tg^ or
Tgg. Superficially, this method does not appear simple because the
transformation matrix and its inverse must be constructed, but the
following lemma shows that Tgg can almost entirely be written by inspection
from the observability invariants after the {v.} are known.
is given by
\
If the given triple is in Luenberger form, ZD, then the
(v^xn) submatrices Tg are
'V > w
v.-v. or T.
v.>v.
B
V .-VI


82
The corresponding measurement process is given by
h = h + \
(4.1-2)
where is the p measurement vector and v^ is a zero mean, white
Gaussian noise sequence, uncorrelated with x.., j k with
J
Covtvj.Vj) =
Covfw^Vj) = S5, j
for R a pxp positive definite, covariance matrix and S a nxp cross
covariance matrix. Thus, a model of this measurement process is
completely specified by the quintuplet, (F,H,Q,R,S).
When a correlation technique is applied to the measurement process,
it is necessary to consider the state covariance defined by
n^Covtx^,)^)
We assume that the processes are wide sense stationary; therefore,
\
nk = n, a constant here. It is easily shown from (4.1-1) that the
state covariance satisfies the Lyapunov equation (LE)
n = FIIFT + Q (4.1-3)
It is well known (e.g. see Faurre (1967)) that since F is a stability
matrix, corresponding to any positive semidefinite (covariance): matrix Q,
there exists a unique, positive semidefinite solution n to the (LE).
The measurement covariance is given (in terms of lag j) by
cj:= Gov(%j-4) = Cov(^+j4,+Cov(Vj^)+Co'' (4.1-4)


70
are cumbersome to obtain due to the general complexity of the expressions
in Er or therefore, a technique to determine these extensions
without forming the Markov parameters directly (or the realization) was
developed. This method consists of recursively solving simple linear
equations (one unknown) to obtain the minimal extension. Extensions
constructed in this manner not only eliminate the possibility of non
minimality as expressed in Godbole (1972), but also describe the entire
class of minimal extensions. The method of constructing the minimal
extension sequence evolves easily from the lower triangular-unit diagonal
structure of P. Since a dependent row of Q(M,M) is a zero row, it
follows from Theorem (2.3-3) that
for jemM
(3.2-4)
where recall that p.^ .=0 for j>i+pv..
i+pv^,j r 1
unkndwn extension parameters, x.y(r) fr
Thus, by inserting the
'lm
(r)
into S(M,M) a system of linear equations is established in terms of the
x..(r)s by (3.2-4). Due to the structure of P, this system of equations
i J
is decoupled and therefore easily solved.
Example. (3.2-5) Reconsider (3.1-2) for Since (R) is satisfied,
the extension A., j>2 is unique. We would like to obtain
0


74
Kalman (1971b) and Tether (1970). They have shown, when (R) is satisfied,
there exists no arbitrary parameters in the minimal partial realization
or corresponding extension. Therefore, the existence of arbitrary
parameters can be used as a check to see if the rank condition holds.
Although it is not essential to construct both sets of invariants, it is
necessary to determine M* which requires v and y; thus, the algorithm
presented has definite advantages over others, since these integers are
simultaneously determined.
In practical modeling applications, the prediction of model
performance is normally necessary; therefore, knowledge of a minimal
extension is required. Also in some of the applications the number of
degrees of freedom may not be of interest, if only one partial realization
is required rather than the entire class. In this case such a model is
easily found by setting all free parameters to zero which corresponds to
merely applying the Algorithm (3.1-1) directly to the data and obtaining
the corresponding canonical forms as before.
Describing the class of minimal extensions offers some advantages
over the state space representation in that it is coordinate free and
indicates the number of degrees of freedom available without compensation.
3.3 Characteristic Polynomial Determination by Coordinate Transformation
In this section we obtain the characteristic equation of the entire
class of minimal partial realizations described by Fr or F^, of the
previous section. It is easily obtained by transforming the realized
Fr or Fc into the Buey form as before. Recall that the advantage of
this representation over the Luenberger form is that it is possible to
find the characteristic polynomial directly by inspection of FgR in (2.2-11).


26
These matrices are in the form of (2.2-5), but it is easliy verified
by constructing that the controllability invariants are in fact
Pl=2, ^=2 and a-j = [-1-1 -1 1]I ol, = [-2 0 -2 4]I The problem
with the Luenberger forms is that the maps it: Xq-* Xq/E are not well
defined. Thus, the image of the maps are indeed canonical forms, but
as shown here for (F,G)eXq/E, we need not have tt(F,G)=(F,G), i.e., the
mapping does not leave the canonical forms unchanged. The point to
remember is that the invariants are the necessary entities of interest
which must be determined.
4*
The procedure to construct the transformation matrix Tq of (2.2-7)
is called the Luenberger second plan. The first Luenberger plan
consists of examining the columns of i^n, given by
V ... F--'g,
F"'\l '2-2-8>
where is an nmxnm permutation matrix, for predecessor independence.
Thus, we can define a new set of invariants (under GL(n)) [{f^.}, ^jks^
completely analogous to the controllability invariants. The canonical
forms associated with the invariants obtained in this fashion have
4*
This procedure amounts to examining the column vectors of Wft for
predecessor independence, i.e., examine g-j ... gm Fg^ ... Fgm . .


104
Thus, the Riccati equation solution has essentially been
circumvented by this algorithm. However, if one not only desires
a stochastic realization, but also a stable minimum phase, spectral
factor, then the Riccati equation solution should be investigated.
This the subject of our next section.
4.4 Stochastic Realization Via the Kalman Filter
In this section we present a special case of the Riccati equation
approach to solving the stochastic realization problem. This approach
is a special case of the factorization (Denham (1975)) discussed in
Section (4.1) because we require the unique, steady state solution to
the discrete Riccati equation. It is well known (e.g. see Tse and
Weinert (1975)) that the steady state solution uniquely specifies the
optimal or Kalman gain. The significance of obtaining a stochastic
realization via Kalman gain is twofold.,, First, since the Kalman gain
is unique (modulo GL(n)), so is the corresponding stochastic realization.
Second and even more important, knowledge of this gain specifies a
stable, minimum phase spectral factor (e.g. see Faurre (1970) or
Willems (1971)). The importance of this approach compared to that of the
last section is that once the gain is specified, a stochastic realization
is guaranteed immediately, while this is not true using the trial and
error technique. However, the price paid for so elegant a solution to
the stochastic realization problem is the computational burden of solving
the Riccati equation.
We use the innovations representation of the optimal filter and
briefly develop it in the standard manner--from the estimation theory
viewpoint. We then examine the realization of this model from the


27
become known as the Buey forms which were derived directly from the
results of Langenhop (1964), Luenberger (1967), and Buey (1968). We
refer the interested reader to these references as well as the recent
survey by Denham (1974). Here we will be satisfied to note that the
procedure of (2.2-1) applies with the set of controllable pairs (F,G)
restricted to the {y.} invariants rather than {y.}. Analogous to the
J J
Luenberger forms, we define the row and column Buey forms as (^dr^br)*
(Fbc>GBc) respectively. The row form is given by
L11
fbr=
L21
L22



0
>
HBR =
T
v +1
V] + l



Lpl
Lp2 ...
L
PP
+ +% +i
VI -^Vp.f
(2.2-9)
where
L..
n
I V
T.
U
'Xi
'o>r
'vr
8..
v. > 0 and satisfy E v =n ;
1 s=l 5
a. ^
KI.. are v.,v-, row vectors containing (3. invariants.
1 J! 1 SC
n
ij
The transformation, TRB,required to obtain the pair (FgR>HBR) is
' BR
'BR
[T
1
(2.2-10)
where tI = [hT(h_.F)T
D.j I I
Vl T
(h,F 1 ) ], ie£


95
L
R
1 1 -1
1 1 -1
-1 -1
33
and therefore
2
-1
0
RR 0 3 Or 1 ^2_£33^ (-It^33)
This example illustrates two methods of specifying an invariant
system description of the given stochastic realization. It also
points out that selecting Q in (2) uniquely specifies R and S; however,
selecting S in (3) uniquely fixes R, but not Q. Thus, there is an
entire family of Q's which have the same R and S and each particular Q
specifies a canonical form for (F,H,Q,R,S) on X_ under the action of GR .
c n
We must place these results into the proper perspective, since we
are primarily concerned with the stochastic realization problem. Suppose
Clearly, we are free to choose any coordinate system, (T,I_)eGR Once
the coordinates are selected F and H are fixed from (4.1-9), since
-1-1
F=TAT H=CT but the major problem of finding an not only such that
the KSP equations are satisfied, but also so that A-0 still remains.
The above methods of specifying an invariant system description partially
resolve this problem. The first method shows that for given (F,H), the
matrices R and S are fixed once a Q is specified; therefore, this quintuplet
is an invariant system description, but whether or not it is a stochastic
realization corresponding to the same PSD or {Cj} as can only be
resolved by first determining if there exists a T and n such that the KSP


100
covariance matrix; thus, the condition A-0 reduces to
det(Q-SR_1ST) ^ 0 (4.3-3)
since R is a positive definite covariance matrix. On the other hand,
if we consider the special case S=0p, then this constraint reduces the
condition A-0 to
det(Q) 0 for R > 0 (4.3-4)
Thus, the choice of admissible Q,R,S must be restricted such that
these constraints are satisfied. Recall that one possible choice is
(Q,R,S) = (KNnV,NN^,KNNT) where K and NN^"'are specified by IT*, the
unique solution of the discrete Riccati equation. Of course, if a
canonical realization algorithm is applied to {C^}, then is found
with T=TR, the Luenberger row coordinates, and (A,B,C)=(FR,GR,HR).
If a positive semidefinite is selected, then a nR>0 is uniquely
specified and therefore R and SR are found from the KSP equations. The
quintuplet, (FR,HR,QR,R,'R) is an invariant system description (under
GR ) and also a stochastic realization, if the above constraints are
n
satisfied. Note that the Riccati equation need not be solved. The
following algorithm summarizes this technique as well as the alternate method
discussed in Section (4.2).
Stochastic Realization Algorithm (4.3-5)
Step 1. Obtain from {Cj} as in (2.4-1).
Step 2. Select a positive semidefinite Qp and solve the (LE)
for nR.
Step 3. Solve (4.3-2) for R and SR.


77
Xr (z) = z5+(e-3)z4+(d-2e+2)z3+(c-3d-e)z2+(-3c+2d+2e)z+(b+2c+be)
rBR
This example points out some very interesting points. When this
technique is combined with the algorithm of (3.2-2), it offers a method
which can be used to obtain the solution to the stable realization
problem developed in Roman and Bullock (1975b). Also, if the system
were realized directly in Buey form, then b=0 and a degree of freedom is
lost; thus, in Ackermann's example Vj=l, while ours is v^=5. It is
critical that al_]_ degrees of freedom are obtained as shown in this case,
since the system is observable from a single output.
This section concludes the discussion of the deterministic case of
the realization problem. In the next chapter we examine the stochastic
version of the realization problem.


Abstract of Dissertation Presented to the Graduate Council
of the University of Florida in Partial Fulfillment of
the Requirements for the Degree of Doctor of Philosophy
REALIZATION OF INVARIANT SYSTEM DESCRIPTIONS
FROM MARKOV SEQUENCES
By
James Vincent Candy
March, 1976
Chairman: Dr. Thomas E. Bullock
Cochairman: Dr. Michael E. Warren
Major Department: Electrical Engineering
The realization of infinite and finite Markov sequences for multi
dimensional systems is considered, and an efficient algorithm to extract
the invariants of the sequence under a change of basis in the state
space is developed. Knowledge of these invariants enables the deter
mination of the corresponding canonical form, and an invariant system
description under this transformation group. For the partial realization
problem, it is shown that this algorithm possesses some attractive
nesting properties. If the realization is not unique, the class of
all possible solutions is found.
The stochastic version of the realization problem is also examined.
It is shown that the transformation group which must be considered is
richer than the general linear group of the deterministic problem. The
invariants under this group are specified and it is shown that they can
be determined from a realization of the measurement covariance sequence.
Knowledge of these invariants is sufficient to specify an invariant
system description for the stochastic problem. The link between the
vii


47
If we consider the alternate method implied in Corollary (2.3-6), then
the following modifications to the preceding steps are required:
(1)* Start with the following augmented array:
^pN I SN,N'l
(2)* Obtain [P | Q] as before.
(5)* Perform additional row operations on Q to obtain unit
columns for each column possessing a leading row element, and
perform row interchanges such that (2.3-8) is satisfied
for each jem, i.e., obtain
(6)* Obtain the a., jem, as in (2.3-6).
J
It should be noted that these algorithms are directly related to
those developed by Ho and Kalman (1966), Silverman (1971), or Rissanen
(1971). As in Ho's algorithm, the basis of the first technique is
performing the special' equivalence transformation of Theorem (2.3-3)
rk
on S^ to obtain S^ The second technique accomplishes the same
objectives by restricting the operations to only the rows of S^ which
is analogous to either the Silverman or Rissanen method. The initial
storage requirements in the first method are greater than the second if
mN'>pN, since P and E can be stored in the same locations due to their
lower and upper triangular structure; and (2) P will be altered in the
second method, since row interchanges must be performed in (5)*; whereas,
it remains unaltered in the first method which may be important in some
applications. Consider the following example which is solved using both
techniques.


36
nonzero i
zero
for
ra=0;,...
.a=Vj..N-1
for k=i+pa
'nonzero'
for 4
b-0,... ,y -1
_ zero
[b=y .,... ,N-1
J
for s=j+mb
These results follow since ^ has identical predecessor dependencies
as SNjN,, then
N.N'
%>N
IT
where ^. = 0 if it depends on its predecessors. To find the observability
indices, let a be the index of the last nonzero row of ¡+pt t=0,l,...,N-1.
T T
Then if 6_. = jD v- = 0 otherwise = (a-i)/p+l. Similar results

follow when is expressed in terms of the c^. The following theorem
*
specifies the matrices P and E required to obtain
Theorem. (2.3-3) There exist elementary matrices P and E, respectively
lower and upper triangular with unit diagonal elements,
*
such that N=PS^ ^(E has identical predecessor
dependencies as
Proof. Let PS^ M,=Q where Q is row equivalent to ¡^i and P=[pr$].
If the j-th row of ^, is dependent on its predecessors, i.e.,
T T T
V yi ;*
then selecting P lower triangular such that


TABLE OF CONTENTS
ACKNOWLEDGMENTS .. iii
LIST OF SYMBOLS vi
ABSTRACT vii
CHAPTER 1: INTRODUCTION 1
1.1 Survey of Previous Work in Canonical Forms
for Linear Systems .. ... 2
1.2 Survey of Previous Work in Realization Theory.. 5
1.3 Purpose and Chapter Outline 10
1.4 Notation 11
CHAPTER 2: REALIZATION OF INVARIANT SYSTEM DESCRIPTIONS 12
2.1 Realization Theory ........ 12
2.2 Invariant System Descriptions .18
2.3 Canonical Realization Theory 33
2.4 Some New Realization Algorithms 45
CHAPTER 3: PARTIAL REALIZATIONS 54
3.1 Nested Algorithm ..... ........ 54
3.2 Minimal Extension Sequences ... 64
3.3 Characteristic Polynomial Determination by
Coordinate Transformation ........ ..... 74
CHAPTER 4: STOCHASTIC REALIZATION VIA INVARIANT SYSTEM
DESCRIPTIONS 78
4.1 Stochastic Realization Theory 81
4.2 Invariant System Description of the
Stochastic Realization ......... ........... 87
4.3 Stochastic Realizations Via Trial and Error ... 97
4.4 Stochastic Realization Via the Kalman Filter .. 104
CHAPTER 5: CONCLUSIONS Ill
5.1 Summary Ill
5.2 Suggestions for Future Research 112
iv


CHAPTER 4
STOCHASTIC REALIZATION VIA INVARIANT SYSTEMS DESCRIPTIONS,
In this chapter the stochastic realization problem is examined
by specifying an invariant system description under suitable trans
formation groups for the realization. Superficially, this may appear
to be a direct extension of results previously developed, but this is
not the case. It will be shown that the general linear group used in
the deterministic case is not the only group action which must be
considered when examining the Markov sequence for the corresponding
stochastic case. .
Analogous to the deterministic realization problem there are
basically two approaches to consider (see Figure 1): (1) realization
from the matrix power spectral density (frequency domain) by performing
the classical spectral factorization; or (2) realization from the
measurement covariance sequence (time domain) and the solution of a set
of algebraic equations. Direct factorization of the power spectral
density (PSD) matrix is inefficient and may not be very accurate.
Recently developed methods of factoring Toeplitz matrices by using fast
algorithms offer some hope, but are quite tedious. Alternately,
realization from the covariance sequence is facilitated by efficient
realization algorithms and solutions of the Kalman-Szego-Popov equations.
78


45
jr\
In the next section we develop some new algorithms which utilize
the theory developed here.
2.4 Some New Realization Algorithms
In this section we present two new algorithms which can be used to
extract both observability and controllability invariants from the given
Markov sequence. Recall from Theorem (2.3-3) that performing row operations
on the Hankel array does not alter the column dependencies, however, it
is possible to obtain the row equivalent array, Q in a form such that
the controllability invariants can easily be found.
The first part of the algorithm consists of performing a restricted
Gaussian elimination (see Faddeeva (1959) for details) procedure on the
Hankel array. This procedure is restricted because there is no row or
column interchange and the leading element or first nonzero element of
each row is not necessarily a one. Define the natural order as 1,2,... .
Algorithm. (2.4-1)
(1) Form the augmented array: [IpN | S^, | ImN)] .
(2) Perform the following row operations on N, to obtain
Cp I Q I ImN'3:
(i) Set the first row of Q equal to the first Hankel row.
(ii) Search the first column of S^ ^, by examining the rows in
their natural order to obtain the first leading element.
This element is q^.
(iii) Perform row operations (with interchange) to obtain q^-j =0,k>j.
4*
Alternately it is possible to extract the Buey invariants from Q by
reordering the columns as in (2.2-8) to obtain ()=QU and examining
the columns for predecessor dependencies.


93
HRFR_1S* = HRLRHRT (4.2-10)
-1 *
In general, HRFR SR is not symmetric; therefore, the set of acceptable
S is restricted by (4.2-10). Since any square matrix can be decomposed
as the sum of a symmetric and skew-symmetric matrix, i.e.,
hrfr 1sr = (hrfr 1sr^sym
+ (hrfr sr)skw
A
then from (4.2-10) ^sp(p-1) elements of SR are constrained to satisfy
(for given (FR,HR))
(Hrfr Sr)$kw = 0p (4.2-11)
We limit our discussion to only cases (i) and (ii) of (4.2-8)
because the techniques employed to obtain the invariant system description
will be used in the next section to determine a solution to the
stochastic realization problem. Thus, we have satisfied our second
objective, i.e., we have specified a unique characterization of covariance
equivalent systemsan invariant system description for X2 under the
action of GR \
n
It is important to note that when QR is selected corresponding to
case (i) of (4.2-8), then SR and R are uniquely specified, but when S"R
is selected as in case (ii), R is again uniquely specified; however, this
is not true for Q. There is a family of Q 1s which correspond to this
S"R and R because only np-%p(p-l) elements of QR are fixed. Consider the
following example which not only illuminates this point, but also
shows how to uniquely characterize the class of covariance equivalent
systems by determining an invariant system description corresponding
to both (i) and (ii) of (4.2-8).


115
R. W. Bass and I. Gura
[1965] "High Order System Design via S.tate-Space Considerations,"
Proc. Joint Auto. Contr. Conf., Rensselaer, N. Y.,
pp. 311-318.
C. Bonivento, R. Guidorzi, and G. Marro
[1973] "Irreducible Canonical Realizations from External Data
Sequences," Int. J. Contr., Vol. 17, pp. 553-563.
F. M. Brausch and J. B. Pearson
[1970] "Pole Placement Using Dynamic Compensators,"IEEE Trans,
on Auto. Contr., Vol. AC-15, pp. 34-43.
P. Brunovsky
[1966] "On Stabilization of Linear Systems Under a Certain
Class of Persistent Perturbations," Differential
Equations, Vol. 2, pp. 401-406.
[1970] "A Classification of Linear Controllable Systems,"
Kibern., Vol. 3, pp. 173-187.
R. S. Buey
[1968] "Canonical Forms for Mutivariable Systems," IEEE Trans,
on Auto. Contr., Vol. AC-13, pp. 567-569.
M. A. Budin
[1971] "Minimal Realization of Discrete Linear Systems from
Input-Output Observations," IEEE Trans, on Auto. Contr.,
Vol. AC-16, pp. 305-401.
[1972] "Minimal Realization of Cohtinuous Linear Systems from
Input-Output Observations," IEEE Trans, on Auto. Contr.,
Vol.. AC-17, pp. 252-253.
T. E. Bullock and J. V. Candy
[1974] "Modeling of Wind Tunnel Noise Using Spectral Factorization
and Realization Theory," Proc. IEEE Southeastcon,
Orlando, Florida.
P. E. Caines and J. Rissanen
[1974] "Maximum Likelihood Estimation of Parameters in Multi-
variable Gaussian Stochastic Processes," IEEE Trans.
Inform. Theory, Vol. IT-20, pp. 102-104.
J. V. Candy, M. E. Warren and T. E. Bullock
[1975] "An Algorithm for the Determination of System Invariants
and Canonical Forms," Proc. 1975 Southeastcon, Auburn,
Alabama.
B. Carew and P. R. Belanger
[19731 "identification of Optimum Filter Steady-State Gain for
Systems with Unknown Noise Covariances," IEEE Trans, on
Auto. Contr., Vol. AC-18, pp. 582-587.


22
or alternately we can say that the action of GL(n) on XQ induces
F TFT"1
G + TG
The action of GL(n) induces an equivalence relation on XQ. We
indicate (F,G)Ej(F,G) if there exists TcGL(n) such that (F,G)=Tt(F,G).
Dual results are defined for the observable pair (F,H) and the
A.
analogous set denoted by XQ.
The third step of (2.2-1) is established in Popov (1972), but
first consider the following definitions. For a controllable pair (F,G)
4*J*
define the j-th controllability index y., jem as the smallest
positive integer such that the vector F Jg. is a linear combination of
J
4 y
its predecessors, where a predecessor of F g. is any vector F g^ where
J *
rm+s J J
we have assumed p(G) = m. Throughout this dissertation we use the
following definition of predecessor independence: a row or column vec
tor of- a given array is independent if it is not a linear combination of
its regular predecessors. The following results were established by
Popov (1972)
Proposition. (2.2-3) (1) The regular vectors are linearly independent;
(2) The controllability indices satisfy the
m
following relationship, E y. = n; (3) There exists
n=l J
exactly one set of ordered scalars, a^cK defined
for jem, kej-1, s = 0,1,...,min(pj,pk-T) and for jem,
k = j,...,m,s = 0,l,...,min(y.,y(c) 1 such that
+
Throughout this dissertation we use the overbar on a set to denote the
dual set.
^These indices are also called the Kronecker indices.


65
available in the choice of minimal extensions. In fact, initially,
the major motivation for constructing an extension was that it
was necessary in order to be able to apply Ho's algorithm. Un
fortunately, these approaches obscure the possible degrees of
freedom and may lead to the construction of non-minimal extensions
as shown by Godbole (1972).
Roman and Bullock (1975a)developed the second approach to
the solution of this problem. They show that examining the columns
or rows of the Hankel array for predecessor independence yields
a systematic procedure for extracting either set of invariants
imbedded in the data. They also show that some of these would-be
invariants are actually free parameters which can be used to
describe the entire class of minimal partial realizations. These
results precisely specify the number and intrinsic relationship
between these free parameters. Unfortunately Roman and Bullock
did not attempt to connect their results precisely with those in
Kalman (1971b),Tether (1970). It will be shown that this connection
offers further insight into the problem as well as new results
which completely describe the corresponding class of minimal extensions.
Before we state the algorithm to extract all invariants available
in the data, let us first motivate the technique. When operating
on the incomplete Hankel array, only the elements specified by the
data are used. It is assumed that the as yet unspecified elements
will not alter the existing predecessor dependencies when they are
specified by an extension sequence. Since the predecessor dependencies
are found by examining only the data in S(M,M), we must examine
complete submatrices of S(M,M) in order to extract the invariants


28
The importance of the Buey form is that the characteristic equation can
JL
be found by inspection of the block diagonal arrays of FBR Since FBR
is block lower triangular, the characteristic equation is given as
Xp (z) = det(Iz-FpR) = Xj (z)...x, (z) (2.2-11)
hBR bK L11 Lpp
where the L.. are the companion matrices of (2.2-9). Similar results
hold for the pair (Fg^.Gg^,) and the transformation is specified by TB(,
constructed from the columns of W .
n
This completes the discussion of invariants and canonical forms for
controllable or observable pairs. To extend these results to matrix
triples (internal system description), it is more convenient to examine
ft
an alternate characterization of the corresponding equivalence class
the Markov sequence of (2.1-4). This approach was used by Mayne (1972b)
and Rissanen (1974), in order to determine the orbits of Z under GL(n).
It is obvious that the sequence is invariant under this group action
A. = (HT^MTFT-VVtG) = HF'3'1G (2.2-12)

Consequently every element of A. can be considered an invariant of Z
J
with respect to GL(n); therefore, two systems which are equivalent under
GL(n) possess identical Markov sequences. The converse is also true, i.e.,
any two systems with identical Markov sequences are equivalent.
The standard approach to investigate a system characterized by its
Markov sequence is to form the Hankel array, N, where we define sT ,
+
It should be noted that the Buey form is not a canonical form if the
transformation group includes a change of basis in either input or output
spaces, while the Luenberger form is still a canonical form.


51
(7) The canonical forms of zR, BR(z), DR(z) and Eg, Bg(z),. DG(z) are
z2-2z 0
0
z
2z ~
Br(z) =
-3 z2+z
2
z -z +z
1
z2-z_
' VZ> =
z+1
0
2z+2
-2z
Fq = [eg £3 ] I 2-1
GC = [^1 ^4]
HC = *-a.l a.3 a.5
a>2] *
1
1
LI
2 4 I 2
2 6,2
1 3 0 _
Bc(z) =
TZ + fz +
i.
8
3. _2
-z +z
3 3 2
z fz
; Dc(z)
z fz £
z -z + £
z2- fz + f
(8) The {v.} and TgR
are determined simultaneously as:
1¡
1 1
1
1
0
0
0
and tbr
4
0
1
0
0
4
0
0
1
0
1
1
3
0
1
1


TABLE OF CONTENTS (Continued)
REFERENCES .....114
BIOGRAPHICAL SKETCH .....124
V
\
V


55
factorization technique to solve the partial realization problem when
(R) is satisfied. His algorithm not only solves the problem in a
simple manner, but also provides a method for checking (R) simultaneously.
In the scalar case, Rissanen obtains the partial realizations, Z(K),
K=l,2,... imbedded in the nested problem of (2.1), but unfortunately
this is not true in the multivariable case. Also, neither set of
invariants is obtained.
The development of a nested algorithm to solve the partial
realization problem given in this dissertation follows directly from
(2.4-1) with minor modification. There are two cases of interest when
only a finite Markov sequence is available.
Case I. (R) is satisfied assuring that a unique partial
realization exists; or
Case II. (R) is not satisfied and an extension sequence
must be constructed.
The nested algorithm will be given under the assumption that Case I
holds in order to avoid the unnecessary complications introduced in
the second case. The modified algorithm is given below. The corresponding
row or column operations are performed only on the data specified
elements.
Partial Realization Algorithm. (3.1-1)
(1) Same as (1) and (2) of Algorithm (2.4-1) except (iv) is q^O
k>j if k is a row whose leading element has been specified.
(2) If (R) is satisfied for some M*=v+y, obtain the invariants as
before in (3), (4) of (2.4-1) and go to (5). If not, continue.


127
[1974] "Basis of Invariants and Canonical Forms for Linear
Dynamic Systems," Automtica, Vol. 10, pp. 175-182.
[1975] "Canonical Markovian Representations and Linear
Prediction," Proc. 1975 IFAC Congress, Boston, Mass.
J. Rissanen and T. Kailath
[1972] "Partial Realizations of Random Systems," Automtica,
Vol. 8, pp. 389-396.
J. Roman
[1975] Low Order Observer Design Via Realization Theory,:
Ph.D. Dissertation, Univ. of Florida, Gainesville,
Florida.
J. Roman and T. E. Bullock
[1975a] "Minimal Parital Realizations in a Canonical Form,"
IEEE Trans, on Auto. Contr., Vol. AC-20, pp. 529-533.
[1975b] "Design of Minimal Order Stable Observers to Estimate
Linear Functions of the State via Realization Theory,"
IEEE Trans, on Auto. Contr., Vol. AC-20, pp. 613-623.
H. H. Rosenbrock
[1970] "State-Space and Multivariable Theory, John Wiley and
Sons, Inc. N.Y.
C. A. Roveda and R. M. Schmid
[1970] "Upper Bound on the Dimension of Minimal Realizations of
Linear Time Invariant Dynamical Systems," IEEE Trans,
on Auto. Contr., Vol. AC-15, pp. 639-644.
P. Rozsa and N. Sinha
[1975] "Minimal Realization of a Transfer Function Matrix in
Canonical Forms," Int. J. Contr., Vol. 21, pp. 273-284.
G. N. Saridis and R. Lobbia
[1972] "Parameter Identification and Control of Linear Discrete
Time Systems," IEEE Trans, on Auto. Contr., Vol. AC-17,
pg. 491.
L. M. Silverman
[1966] "Transformation of Time Variable Systems to Canonical
Form," IEEE Trans, on Auto. Contr.., Vol. AC-11, pp. 300-
303.
[197T] "Realization of Linear Dynamical Systems," IEEE Trans,
on Auto. Contr,, Vol. AC-16, pp. 554-567.
L. M. Silverman and H. E. Meadows
[1966] "Equivalence and Synthesis of Time Variable Linear
Systems," Proc 4-th Allerton Confr. Circuit and
System Theory, pp. 776-784.


20
complete invariant. In general we will be interested in a complete
system of invariants for E given by the,set of invariants (f^} where
4*
f : X + Y1xY2X. xYn> f. is an invariant for E, and f-j (x^i (xg) * >
ffi(X1)~fn(x2) imPly x]Ex2* Completeness of this set of invariants
means that the set is sufficient to specify the orbit of x, i.e., there
is a one to one correspondence between the equivalence classes in X
and the image of f. If the set of complete invariants is independent,
then the map f: X+Y-jX.. ,xYn is surjective. This property means that
corresponding to every set of values of the invariants there always exists
an n-tuple in Y specified by this set. A complete system of independent
invariants will be called an algebraic basis.
Generally, we consider a subset of X (e.g., in system theory a
controllable system). Correspondingly, let f be a function mapping the
subset XQ of X into set Y, then f is a restriction of f if fQ(x)=f(x)
for each xeXQ. We can uniquely characterize an equivalence class E(x)
by means of the set of values of the functions f.(x), ien. where the {f..}
constitute a complete set of invariants for E on X. If the corresponding
complete invariant f is restricted such that its image is itself a
subset of X, then we have specified a set of canonical forms C for
E on X. To be more precise, a canonical form C for X under E is a
member of a subset C<=X such that: (1) for every xeX there exists one and
only one ceC for which xEc, and since C is the image of a complete
invariant f, then (2) for any xeX and c-j, C2eC, xEc^, and xEc2 implies
f(x)=f(c-|)=f(c2)=c-j=C2 (invariance); (3) for any ceC if f(x-|)=c and
f(x2)=c, then x-|Ex2 (completeness). Thus, c=f(x) is a unique member of
^This notation is actually f=(f-|,... ,fn) :x-^Y^x.. .xYn, but it is
shortened when the set {f.} is clearly understood.


79
REALIZATION FROM
COVARIANCE SEQUENCE ^ PSD
AND ALGEBRAIC METHODS
STOCHASTIC REALIZATION
Figure 1. Techniques of Solution to the Stochastic
Realization Problem.
The problem considered in this chapter is the determination of a
minimal realization from the output sequence of a linear constant
system driven by white noise. The solution to this problem is well known
(e.g.. see Mehra (1971)) as diagrammed below in Figure 2. The output
sequence of an assumed linear system driven by white noise is correlated
and a realization algorithm is applied to obtain a model whose unit
pulse response is the measurement covariance sequence. A set of algebraic
equations is solved in order to determine the remaining parameters of
the white-noise system..
This problem is further complicated by the fact that the covariance
sequence must be estimated from the measurements. From the practical
viewpoint, the realization is highly dependent on the adequacy of the
estimates. Although in realistic situations the covariance-estimation
problem cannot be ignored, it will be assumed throughout this chapter
that perfect estimates are made in order to concentrate on the realization
-J*
portion of the problem.
In this chapter we present a brief review of the major results
necessary to solve the stochastic realization problem. We use the
4-'
Majumdar (1976) has shown in the scalar case that even if imperfect
estimates are made realization theory can successfully be applied.
FACTORIZATION
METHODS


52
(9) T"1 is given by solving the equations for the last row
as:
r1
1BR
1
0
0
3
0
1
.0
0
0 0
0 0
1 0
-1 1
(10) Find FBR and Xp(z) as
F =T F
BR VBRVbR
2
0
0
2
0 0
0 1
0 0
1 0
0
0
1
2
and
XF(z) = (z-2)(z3-2z2+l) = z4-4z3+4z2+z-2
This completes the first method. If the second method is used instead,
then only (5)*, (6)*, and (8)* differ.
(5)* Performing the additional row operations and interchanges to
satisfy (2.3-8) gives:
5 1
T
1
-T
1
l
T
* -T 0
0
1
T
l
T
3
8
0
0 0
0 -$-
(D 0 0-1 0 --z- -f T
o 0 Q 1 o i *
0 0 0 0 1 £ 3
0 o I- 0
9 1
T -T
1 3
0
(6)* The a.'s are determined from the appropriate columns of Q as:
J
-T
"-1
-i
1
-1 =
5
T
= W7 2 =
0
I
L_ T _
- -
= W4


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34
or .V^-r
T T T
but det (W^, WN,} t 0; thus, a = 0^, i.e., a dependent row
of is a dependent row of V^. Conversely assume that there .
exists a nonzero aT as before such that
v nT
VN ^tnN'
Since p(W^,)=n, it follows that this expression remains unaltered
if post-multiplied by W^,, i.e.,
A/V 4-
and the desired result follows immediately.V
The significance of this lemma is that examining the Hankel rows
(columns) for dependencies is equivalent to examining the rows (columns)
of the observability (controllability) matrix. When these rows (columns)
are examined for predecessor independence, then the corresponding
indices and coefficients of linear dependence have special meaning--
they are the observability (controllability) invariants. Thus, the
obvious corro!ary to this lemma is \
Corollary. (2.3-2) If the rows of the Hankel array are examined for
predecessor independence, then the j-th (dependent)
row, where j=i+pv., ie£ is given by
P
+ Z
i-l min(v.,v -1)
T 1 5 T
Ij = 2 z
s-1 t=0
p min(vi,vs)-l
isre+pt
s=l
Z
t=0
^ist-s+pt
whereig^^} an.d{v/} are the observability invariants
and kepN is the k-th row vector of .


69
The indices are: v-|=2, V2=3
(2) For 1=1, (1+p ^-1)=4, k.j=3, m'(M+T-k1 )=4;' thus, the corresponding
submatrix is constructed from the first 4 rows and columns of Q(4,4)
(small, dashes). J-j={2}.
For i=2, (2+pv2-l)=7, k2=4, m(M+l-k2)=2; thus, the corresponding
submatrix of Q(4,4) is given by the first 7 rows and 2 columns (large
r'a'shes). J2={2,4,6}.
I T
(3) Replacing the fifth and eighth rows of P with jDj. + b£2 and
£¡ + c4 +
T T
+ ej^. where b,c,d,e are real scalars gives
= [2 b -3 0 1 0 0 0 ]
= [-3-e c 0 d+e 0 e 0 1 ]
The §1 are:
g_{ = [ -2 3 -b 0 0]
g_2 = [ 3+e 0 -c -(d+e) -e ]
(4) The canonical form is
Corresponding to ttiese realizations is a minimal extension sequence
which can be found by determining the Markov parameters. These parameters
4


30
Independence is shown by constructing the Luenberger form of (2.2-5) and (2.2-14)
below^ directly from these invariants.
The dual result yields another basis on X,, [{v.},{g. .},{aT }].
I I I S L J
The corresponding canonical forms for EeX-j or are given by the
Luenberger pairs of (2.2-5,2.2-6) and
and
a' (ia]-1 )m+ l I
(v^ljp+l.
]
(2.2-14)
and the canonical triples are denoted by and respectively.
Rissanen (1974) also shows that a canonical form for the transfer
function can be constructed from the invariants of (2.2-13). This is
possible because the determination of canonical forms for £ based on the
Markov parameters is independent of the origin of A^'s. Rissanen defines
the (left) matrix fraction description (MFD) as
T(z) := B"1(z)D(z) (2.2-15)
v ,
where B(z) = z B.z for |B \f 0
i=0 1 v
V-l 4
D(z) = I D.z1 .
i=0 1


3
state space representation for analysis and design. In early literature,
however, transfer function representations were used. For highly
complex systems it is much easier to determine external behavior rather
than internal, since the state variables are normally not available for
measurement. As pointed out by Kalman (1963) the language of these
representations may be different, but both describe the same problems and
are related. Many researchers have investigated the relationship between
both representations, but always with one common goalto obtain a
state space model which specifies the external description directly by
inspection. Kalman (1963) and later Johnson and Wonham (1964),
Silverman (1966) have shown that there exists a canonical form (under
change of basis in the state space) in the scalar case for the triple
(F,g,h) where F is in companion matrix form (see Hoffman and Kunze (1971))
and g is a unit column vector. It was shown that there exists a one to
one correspondence between the non-zero/non-unit elements of the triple
and the transfer function. This representation was used by Bass and Gura
(1965) to solve the pole-positioning problem and recently by Wolovich
(1972b)in solving the exact model matching problem. ,
The progress in determining a canonical form for the internal
description of multivariable systems came more slowly. The earliest work
appears to be that of Langenhop (1964) in which he develops a representation
to study system stability. Brunovsky (1966,1970) was probably the first
to recognize the invariant properties of the canonical form for the
controllable pair (F,G). Tuel (1966,1967) developed canonical forms for
multivariable systems in his investigation of the quadratic optimization
problem. Subsequently, Luenberger (1967) proposed certain sets of
canonical forms for controllable pairs; however, his development allowed


realization from the measurement covariance sequence, the white noise
model and the steady state Kalman filter is established.
vm


62
available, i.e., {1,1+p,...,l+p(v1(M)-l),...,p,2p,...,pvi(M)}
and similarly denote the row vectors of the elementary row matrix
of the previous chapter, by (M). From Lemma (3.1-3), it follows
that jJ J*+k and 4+pv.(H)(M) £.T+pv1 (M+k)(M+k) si"ce
the observability invariants are identical. The specify the
elements in and along with the 3ist they specify the elements of
pi+pv.(M)(M) (see Coro11ary (2-3-4)). From Lemma (3.1-4) it is clear
that J¡¡jcjj¡¡+k since v. (M+k^v^M).
Reconsider Example (3.1-2), to see these properties. In this
case we have M=2, k=3, M*(2>2, M*(5)=5, and p(S(5,5))>p(S(2,2)) as
in Lemma (3.1-4); therefore, since 3-? = and J5 = d*3,2,5}.
The observability indices are identical except for v2(5)>v2(2); thus,
iaj (2) ,a2> (2)}<={aj^ (5) ,a^ (5) ,a2>(5) ,a^ (5)} since aT (2) = a! (5)
for j=l,3. We also know from Example (2.4-2) that £(5) is the solution
to the realization problem and therefore the properties of Lemma (3.1-3)
will hold for {A|yj}, M>5. Table (3.1-5) summarizes these properties.
The results for the dual case also follow directly. We now proceed to
the case of constructing minimal partial realizations when (R) is
not satisfied, i.e., the construction of minimal extensions.


29
ieN and S jeN1 as the block rows and columns of SM and the
J IN5I1
block column and row vectors, a or a^ denote the r-th column of si
or the s-th row of S for remN1, sepN. Rissanen (1974) has shown
9
that by examining the set
X, = {£ I I controllable and observable with {y.} invariants}
I J
under the action of GL(n) that
Proposition. (2.2-13) The set of controllability invariants and block
column vectors, [{yj>,-Caj|9-[a }.] for the
appropriate indices constitute an algebraic basis

for any ZeX^ under the action of GL(n).
The proof of this proposition is given in Rissanen (1974) and consists of
showing that any two members of X^ with identical Markov sequences
are equivalent under GL(n). Thus, invariance follows by showing that a
dependent column vector of the Hankel array can be uniquely represented
in terms of the set [{y.},{a.. _}]. These parameters remain unchanged
J JKS
under GL(n); therefore, they are invariants. The block column vectors,
a t satisfy a recursion analogous to (2.2-3), i.e.,
n-1 min(yJ.,y[<-l) m min(y.,uk)-l
z
k=l
Z o .1 a , + Z
jks .j+ms
s=0 k=j
ajksa.j+ms
s=0
Thus, all dependent block columns can be generated directly from the set,
{a of regular block column vectors. These vectors are invariants under
GL(n), since every column vector of A. is an invariant as shown in
J
(2.2-12). Completeness follows immediately from the above recursion,
since any two members of X^ possessing identical invariants satisfy the
above recursion and therefore have identical Markov sequences.


119
R. E. Kalman and R. S. Buey
[1961] "New Results in Linear Filtering and Prediction Theory,"
J. of Basic Engr. Vol. 83, pp. 95-108.
R. E. Kalman, P. L. Falb, and M. A. Arbib
[1969] Topics in Mathematical System Theory, McGraw-Hill, Inc. N.Y
M. Lai, S. C. Puri, and H. Singh
[1972] "On the Realization of Linear Time-Invariant Dynamical
Systems," IEEE Trans, on Auto. Contr., Vol. AC-17,
pp. 251-252:
C. E. Langenhop
[1964] "On the Stabilization of Linear Systems," Proc. Amer.
Math. Soc., Vol. 15, pp. 735-742.
G. Ledwich and T.E. Fortmann
[1974] "Comments on 'On Partial Realizations'," IEEE Trans, on
Auto. Contr., Vol. AC-19, pp. 625-626.
D. G. Luenberger
[1966] "Observers for Multivariable Systems," IEEE Trans, on
Auto. Contr., Vol. AC-11, pp. 190-197.
[1967] "Canonical Forms for Linear Multivariable Systems," IEEE
Trans, on Auto. Contr., Vol. AC-12, pp. 290-293.
Z. Luo
[1975] Discrete Kalman Filtering and Stochastic Identification
Using a Generalized Companion Form, Ph.D. Dissertation,
Univ. of Florida, Gainesville, Florida.
Z. Luo and T.E. Bullock
[1975] "Discrete Kalman Filtering Using a Generalized Companion
Form," IEEE Trans. Auto. Contr., Vol. AC-20, pp. 227-230.
A. Majumdar
1975] Private communication.
1976] Modeling and Identification of the Nerve Excitation
Phenomena, Ph.D. Dissertation, Univ. of Florida,
Gainesville, Fla.
D. Q. Mayne
[1968] "Computational Procedure for the Minimal Realization of
Transfer-Function Matrices," Proc. IEEE, Vol. 115,
pp. 1363-1368.
[1972a] "Parameterization and Identification of Linear Multi-
variable Systems," Lecture Notes in Mathematics,
Springer, Berlin, Vol. 294, pp. 56-61.
[1972b] "A Canonical Model for the Identification of Multi-
variable Linear Systems," IEEE Trans, on Auto. Contr.,
Vol. AC-17, pp. 728-729.


CHAPTER 5
CONCLUSIONS
5.1 Summary
This dissertation has contributed results in realization theory
for both deterministic and stochastic cases. It was shown that by
carefully specifying the invariants of the realization problem under
a change of basis in the state space that a simple and efficient
algorithm to extract these entities from the Markov sequence could be
developed. This technique provides a solution to the realization problem
directly in a canonical form, and an invariant system description under
this transformation group is.'specified. The partial realization problem
was solved by modifying this technique to develop a nested algorithm.
It was shown that this method specifies the class of minimal partial
canonical realizations. A new recursive technique to determine the
corresponding class of minimal extensions while conserving all degrees
of freedom available was developed. These results bridge the gap between
the more classical approach of constructing a minimal extension and
that of extracting the realization invariants. The characteristic
equation is determined from the transition matrix in a convenient
coordinate system by inspection. These coordinates were easily obtained
from the given solution to the partial realization problem.
In the stochastic realization problem it was shown that the
transformation group which must be considered is richer than the general
111


99
In many practical situations, it is known a priori that the
system and measurement noise sequences are uncorrelated. This case
has been considered by many researchers (e.g. Faurre (1967), Anderson
(1969), Mehra (1971), Rissanen and Kailath (1972), etc.) and it
corresponds to setting S.-o in the WN model of (4.1-1). It is crucial
to note that with this choice of S, it appears that the only trans
formation group which leaves the PSD invariant is GL(n). From (4.2-4)
it is clear that GRn not just GL(n) must be considered; therefore,
there are %n(n+l) fewer invariants when GRn rather than GL(n) acts on X2.
Recall the first technique outlined in Section (4.2) to obtain
(F,H,Q,R,S) from {Cj}: realize S^p, select a Q, specify a n from the
(LE), and then find R and 5 from the KSP equations. The selection of a
proper Q is essential to obtain a quintuplet of X2 that is a stochastic
realization. Therefore, it is useful to consider constraints which
evolve from the fact that Q and II,R,S are stationary covariance matrices.
For given (FR,HR)+, each choice of QR uniquely specifies a nR and hence
R,SR as in (4.3-2), i.e., (FR,HR,QR,R,SR) is a canonical form on X^ for
E-^-equivalence. Since F is a stability matrix, it follows from stability
theory that if (F,/Q)^ is completely controllable, corresponding to
each Q^O there exists a unique positive definite solution n to the (LE).
Therefore, restricting the choice of QR to be non-negative definite
simultaneously satisfies this "stability constraint" as well as the fact
that Qr must be a covariance matrix.
The results of the generalized Kalman-Szego-Popov lemma of (4.1-11)
assures us that there exists at least one realization such that A is a
+Here we assume the action of Gl(n) is completed with T=TR.
++/q is any full rank factor of Q, i.e., QVQ/Q"*"


108
Note that this realization is just a special case of the factorization
of Denham (1975) discussed in Section (4.1) with n*=n in (4.1-13),
K the Kalman gain in (4.1-12) and R£=NN^.
Clearly, the relationship between the canonical realization of
E^p and E£ of (4.4-9) is provided by the KSP equations, i.e., (4.4-6)
and
(A,B,C) (FR,FRHRHR +(Sinv)rHr)
(4.4-10)
D+D = HRnRHR+RINV
Note that since K and R are unique, then E is unique (e.g. see Tse
and Weinert (1975) or Denham (1975) for proof). Therefore, it is futile
to attempt to determine E£ from the trial and error algorithm of (4.3-5)
because this quintuplet is a unique stochastic realization (modulo GL(n)).
Recall from Section (4.2), if we let T=TR, then np of the total
2np+%p(p+l) invariants specify the pair (FR,HR) and it follows that np
specify Kr and %p(p+l) specify R£. Thus, the canonical realization of
'Xj 'X/ ^
the INV model is analogous to the WN model; however, unlike the QR,RR,SR
obtained in the WN case by trial and error, (QINV)R> RjNV> ^Sinv^R are
uniquely specified by KD and R in (4.4-9). The following diagram
summarizes the relationship between these two distinct approaches to
obtaining a stochastic realization.
REALIZATION FROM {C^
J
PSD
KSP
(Trial and Error)
FACTORIZATION METHODS
E
INV
(Riccati Equation)
(F,H,Q,R,S)
Figure 3, Solutions of the Stochastic Realization Problem


67
= 3110 + e!20 -2 +f3130 -3 {312r0)
or
[2 4 4 8] = 3110 [1 2 2 4]+ g]20 [1 2 2 4]+ g^H 01 0] .
The solution is = 2 g^o ^130= t*1us* t*ie coefficient B-^O 1s
ah arbitrary parameter. Note that this recursion is essentially the
technique given in Roman and Bullock (1975a).
Clearly, if (R) is satisfied as in the previous section, then there
exists a complete submatrix (data is available for each element) of S(M*,M*)
in which every regular vector of S(M,M) is always a regular vector
of the submatrix corresponding to a particular chain; thus, there
will be no arbitrary or free parameters.
The algorithm for the case when (R) is not satisfied may be
illustrated by considering row operations on S(M,M) to obtain Q(M,M),
since the identical technique can be applied to obtain S*(M,M). The
arbitrary (column) parameters are found by performing additional
column operations to Q(M,M). As in Example (3.2-1), we must find
the largest submatrix of Q(M,M) for each chain, i.e., if we define
k_. as the index of the block row of S(M,M) containlirig the vector. ,
then the largest submatrix of data specified elements corresponding
to the i-th chain is given by the first (i+pv^-1) rows and m(M+l-k.)
columns of Q(M,M). Also, we define J|ie£ as the sets of Hankel row
indices corresppnding to each dependent (zero) row of the
(i+pMj-l)x (m(M+l-kj) submatrix of Q(M,M) which becomes independent, i.e.,
it contains a leading element. In Example (3.2-1) for i=l, we have
(1 +pv1-1)=3 and k-j =2; thus, nKM+l-k-^A and the corresponding submatrix is
given by the first 3 rows and 4 columns of Q(3,3), and.of course, J^={2}.


CHAPTER 1
INTRODUCTION
Special state space representations of linear dynamic systems have
long been the motivation for extensive research. These models are
generally used to simplify a problem, such as pole placement, by
introducing arrays which require the fewest number of parameters while
exhibiting the most pertinent information. In general, system represen-
. \ '
tations have been studied in literature as the problem of determining
canonical forms; Canonical forms have been used in observer design,
exact model matching methods, feedback system design, and Kalman filtering
techniques. In realization theory, canonical forms for linear multi-
variable systems are important. Since it is only possible to model a
system within an equivalence class, the ability to characterize the class
by a unique element is beneficial.
The problem of determining a canonical form has its roots in
invariant theory. Over the past decade many so-called "canonical" system
representations have evolved in the literature, but unfortunately these
representations were obtained from a particular application or from
computational considerations and not derived from the invariant theory
point of view. Generally, these representations are not even unique and
therefore cannot be called a canonical form. Representations derived
in this manner have generally been a source of confusion as evidenced by
the ambiguity surrounding the word canonical itself. In this dissertation
1


CHAPTER 3
PARTIAL REALIZATIONS
One of the main objectives of this research is to provide an
efficient algorithm to solve the realization problem when only partial
data is given. As new data is made available (e.g., an on-line
application, Mehra (1971)), it must be concatenated with the old
(previous) data and the entire algorithm re-run. What if the rank of
the Hankel array does not change? Effort is wasted, since the previous
solution remains valid. An algorithm which processes only the new data
and augments these results (when required) to the solution is desirable.
Algorithms of this type are nested algorithms.
In this chapter we show how to modify the algorithm of (2.4-1)
to construct a nested algorithm which processes data sequentially.
The more complex case of determining a partial realization from a fixed
number of Markov parameters arises when the rank condition, abbreviated
(R), is not satisfied. It is shown not only how to determine the minimal
partial realization in this case, but also how to describe the entire
class of partial realizations. In addition, a new recursive technique
is presented to obtain the corresponding class of minimal extensions and
the determination of the characteristic equation is also considered.
3.1 Nested Alqorithm
Prior to the work of Rissanen (1971) no earlier recursive methods
appeared in the realization theory literature. Rissanen uses a
54


122'
M. G. Strintzis
[1972] "A Solution to the Matrix Factorization Problem,"
IEEE Trans, on Info. Th,y., Vol. IT-18, pp. 225-232.
G. Szeg'd and R. E. Kalman
[1963] "Sur la Stabilite Absolve d'un Systeme D'Equations aux
Differences Finies," Compte Rendus a L'Academie des
Sciences, pp. 388-390.
A. J. Tether
[1970] "Construction of Minimal Linear State-Variable Models
from Finite Input-Output Data," IEEE Trans, on Auto.
Contr., Vol. AC-15, pp. 427-436.
E. Tse and H. L. Weinert
[1973] "Extension of 'On the Identifiability of Parameters',"
IEEE Trans, on Auto. Contr., Vol. AC-18, pp. 687-688.
[1975] "Structure Determination and Parameter Identification
for Multivariable Stochastic Linear Systems," IEEE Trans,
on Auto. Contr., Vol. AC-20, pp. 596-603.
W. Tuel
[1966] "Canonical Forms for Linear Systems-Pt. 1," IBM Res.
Lab., RJ 375, San Jose, Calif.
[1967] "An Improved Algorithm for the solution of Discrete
Regulation Problems," IEEE Trans, on Auto. Contr.,
Vol. AC-12, pp. 522-528.
S. H. Wang and E. J. Davison
[1972] "Canonical Forms of Linear Multivariable Systems,"
Univ. Toronto, Toronto, Canada, Control Syst. Rept. 7203.
M. E. Warren arid A. E. Eckberg
[1973] "On the Dimensions of Controllability Subspaces: A
Characterization via Polynomial Matrices and Kronecker
Invariants," 1974 JACC Preprints, Austin, Texas, pp. 157-
163.
H. Weinert and J. Anton
[1972] "Canonical Forms for Multivariable System Identification,"
Proc. Conf. Decision and Contr., New Orleans, La.
N. Wiener
[1955] "On the Factorization of Matrices," Comment. Math. Helv.,
Vol. 29, pp. 97-111.
[1959] "The Prediction Theory of Multivariate Stochastic
Processes: I-The Regularity Condition," Acta. Math.,
Vol. 98, pp. 111-150.
J. C. Willems
[1971] "Least Squares Stationary Optimal Control and the Algebraic
Ricatti Equation," IEEE Trans, on Auto. Contr., Vol. AC-16,
pp. 621-634.


14
If two minimal realizations Z, t are equivalent under a change of
basis in X, then there exists a nonsingular T such that
(F,G,H)^ = (TFT\tG,HT_1 ). It also follows by direct substitution that
the controllability and observability indices of these realizations are
identical and
W. = TW. for j = 1,2,...
J J
V. = V.T"1 for i = 1,2,...
The generalized NxN' block submatrix of the doubly infinite Hankel
array is given by
SN,N' =
Implicit in the realization problem is determining when a finite
dimensional realization exists and, if so, its corresponding minimal
dimension. The following proposition by Silverman gives the necessary
and sufficient conditions for {A^} to have finite dimensional realiza
tion.
Proposition. (2.1-5) An infinite sequence {A^} is realizable iff there
exist positive integers y,v,n such that
otVuu+j) =n 'or j'01
Further, if {A^} is realizable, then p,v are the
controllability and observability indices and n is
the dimension of the minimal realization.
^This notation means F = TFT"\ G = TG, and H = HT ^.
AN'
W-l


I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and is fully
adequate, in scope and quality, as a dissertation for the degree of
Doctor of Philosophy.
Thomas E. Bullock, Chairman
Professor of Electrical Engineering
I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and is fully
adequate, in scope and quality, as/a dissertation for the-^degree of
Doctor of Philosophy.
MTchael E. Warrenchairman
Assistant Professor of
Electrical Engineering
I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and is fully
adequate, in scope and quality, as a dissertation for the degree of
Doctor of Philosophy.
Donald G. Childers
Professor of Electrical Engineering
I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and is fully
adequate, in scope and quality, as a dissertation for the degree of
Doctor of Philosophy.
Zoran R. Pop/Stojanovic
Professor of Mathematics
I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and is fully
adequate, in scope and quality, as a dissertation for the degree of
Doctor of Philosophy.
'i ufa
opov
Vasile M. Popov
Professor of Mathematics


84
(to follow) which evolve from the generalized Kalman-Szego-Popov lemma
a.
(see Popov (1973) ). Thus, we specify the KSf3 model as the realization
of {C.l defined by the quadruple, E^cd:=(A,B,C,D) of appropriate
J 0 Iw r
dimension with transfer function, TK<.p(z)=C(Iz-A)"^B+D. Note that since
the unit pulse response of the KSP model is simply related to the
measurement covariance sequence, then (4.1-7) can be written as the
sum decomposition.
*z(z) = T^pUJ+T^pU"1) = C(Iz-A)1B+D+DT+BT(Iz"1-AT)_1CT (4.1-8)
The relationship between the KSP model and the stochastic realization
of the measurement process is shown in the following proposition by
Glover (1973).
Proposition (4.1-9) Let zKSp=(A,B,C,D) be a minimal realization of {Cj}.
Then the quintuplet (F,H,Q,R,S) is a minimal stochasti
realization of the measurement process specified
by (4.1-1) and (4.1-2), if there exists a positive
definite, symmetric matrix n and TeGL(n) such that
the following KSP equations are satisfied:
n-AnAT * Q
D+DT-CnCT = R
B-AHC1" s
where A=T-1FT and C=HT.
The proof of this proposition is given in Glover (1973) and > ;
-J-
This book was published in Romanian ini966, but the English version
became available in 1973.
4j* oo
Note that the sequence, -CC^>Q is related to the measurement covariance
sequence as Cn-hC and C-=C. for j > 0.
0 0 J J


57
(3) Indices are: v-j = 1
v2 = 0
h = 1
y2 = 1
oo
n
(4) ^Invariants are: Is-
tP41
1 P43] [2
1 0]
4--
CP61
O
| 1
II
1 1
OO
*£>
a.
1 1]
and
-1 = ^P41 P42
P
K43
P44 1 $
[-2 0
0 10 0]
2 = *--3 1 P21
P
2
1 o{] = [0
0 0-1
i 0]
-3 = ^P61 P62
P
63
P P P
64 K65
0-1001]
el 3
-1 '
e14
*0 "
eT3
1
a
2 '
e14
_0
e23
1
~T
e24
-2
T *."
e23
v
1
T
e24
2
. b, =
e33
1
5 bo~
e34
0
tm-
_0
_ 0-
-e44-
_1 _


To my wife, Patricia, and daughter, Kirs tin,, for unending faith,
encouragement and understanding. To my mother, Anne, for her constant
support and my mother-in-law, Ruth, for her encouragement. To "big" Ed
my father-in-law, whose sense of humor often lifted my sometimes low


38
This theorem shows that the indices can be found by performing a sequence
of elementary lower triangular row and upper triangular column operations
in a specified manner on the Hankel array and examining the nonzero rows

and columns of S^,, the structural array of The {cu^} and
{BTjsare also easily found by inspection from the proper rows of P and
columns of E as given by
Corollary. (2.3-4) The sets of invariants or more compactly
j
the sets of n vectors {8.},{a.} are given by the rows
of P and columns of E in (2.3-3) respectively as
4 CVV+P Pqr+P(vrl)] q'pV1 1>re£
a. [e .e .
-j st s+mt
es+m(prl)t] j-5£a
where
Pqrest
:
q=r, r=s
qt
Proof. The proof of this corollary is immediate from Theorem (2.3-3).V
We can also easily extract the set of invariant block row or column
vectors, {a! },{a } from the Hankel array and therefore, we have a
J - >
solution to the canonical realization problem.
Theorem. (2.3-5) If the generalized Hankel submatrix of rank n is
transformed by elementary row operations to obtain a row
equivalent array, then by proper choice of P the matrix Q
is given by:


39
TG | TFG
... j TFN'_1G
'V
pn-M
mN
0Pn-N
_umN _
Q =
where (F,G) is a controllable pair and det TVO.
Proof. .If z is a minimal realization, then it is well-known that
p(VN)=p(W^,)=n. Since P is an elementary array, then it follows
[PV^] PV =
r N
-I..
p=
and det T^O.
Post multiplication by W^, gives
[G | FG | ... |
PVV =
_T__
pN-
FN'"1G]
= PS
N,N1
Multiplication of the arrays gives the desired results.V
Corollary. (2.3-6) If P*is selected such that Q is as in (2.3-5) with the
pair (F,G) in Luenberger column form, then the set
of invariants {a-}, jem is given by the columns of
J
Wl\|> > w^ kemN1 with
ak = k=pjm+j
Proof. If P is selected in Theorem (2.3-5) such that T=T^, then it
follows that each column of W.,, corresponding to the (j+mp.)-th
'* u
for each jem contains the {a^} invariants.V
t.
The method of selecting P is given in the ensuing algorithm.


113
results with realistic noisy data. Along these lines the use of
maximum likelihood estimators by Caines and Rissanen (1974) and the
least squares estimates in the technique of Majumdar (1976) should
be investigated further.
The use of Markov sequences to design controllers to solve the
model following problem (e.g. see Moore and Silverman (1972)) should
be examined by first defining the problem invariants and then inves
tigating the possibility of using the realization algorithms of
Chapters 2 or 3 to extract them. The use of the class (under GL(n))
of minimal extension sequences developed directly from a given finite
sequence may prove instrumental in this technique and should be studied.
An efficient technique to factor Toeplitz matrices (see Rissanen
and Kailath (1972)) should be developed by extracting the invariants
of the stochastic realization specified in Chapter 4. Analogous techniques
for the equivalent frequency domain representation of this problem should
also be investigated.


49
(3)The indices are obtained by inspection from the independent rows
and columns of Q in accordance with Theorem (2.3-3) as:
v1 1 u1 = 3
^2 2 u2 ^
and p(S2s3) = p(S3 3) p(S2 = 4 satisfying (R).
T T
(4)The jfj and bjj are determined from the appropriate rows and columns
of P as:
1 ~ -CP41 I P42 P45 I P433 = [2 I 0 0 I 0]
-2 = "^p81 I p82 p85 I p83-* = I 0 V'M3
-3 = "^p61 I p62 p65 I p63^ = C"1 I 1 Ml
-1 = tO-3 I P4i P42 p43 p44 I 5^3 = [O3 I -2 0 0 1 I ^
2 = ^-p81 P82 P83 P84 P85 P86 P87 p88 I 3=E"3 O*1 0-1 0 011 3
-3 = % IP61 P62 p63 P64 P65 P66"* = % ¡ 1 1 _1 0 _1 13
(5)Performing the column operations, obtain the structural array
k
N, and E as:
... Jc
[P I 4 I E] where the leading elements are circled,.


81
algebraic structure of a transformation group acting on a set to obtain
an invariant system description for this problem. A new realization
algorithm is developed to extract this description from the covariance
sequence. Recently published results establishing an alternate approach
to the solution of this problem are also considered.
4.1 Stochastic Realization Theory
Analogous to the deterministic model of (2.1-1
) consider a white-
noise (WN) model given by
Vi = F*k + *k
(4.1
4 =H4
where and ^ are the real, zero mean, n state and p output vectors,
and Wj, is a real, zero mean, white Gaussian noise sequence. The noise
is uncorrelated with the state vector, X., j k and
J
Cov(wi,wj.):=E[(wi-Ewi)(Wj-Ew[.)T] = x.
where 6.. is the Kronecker delta. This model is defined by the triple,
ij
ZWN:=(F,In*H) compatible dimensions with (F,H) observable and F a
nonsingular,^ stability matrix, i.e., the eigenvalues of F have magnitude
less than 1. The transfer function of (4.1-1) is denoted by TWN(z).
+In the discussion that follows the WN model parameters will be used to
obtain a solution to the stochastic realization problem. Denham (1975)
has shown that if the spectral factors of the PSD are of least degree,
i.e., they possess no poles at the origin, then F is a nonsingular matrix


46
(iv) Repeat (ii) and (i i i) by searching the columns in their
natural order for leading elements.
(v) Terminate the procedure after all the leading elements have
been determined.
(vi) Check that at least the last p rows of Q are zero. This assures
that the rank condition, (R) is satisfied.
(3) Obtain the observability and controllability indices^ as in
Theorem (2.3-3).
T
(4) Obtain £. iejj from the appropriate rows of P as in Corollary (2.3-4)
T *
and jb. as in (2.2-19) where $...=p...
* J J
(5) Perform the following column operations on Q to obtain [P | S* Nl | E]:
(i) Select the leading element in the first column of Q, .
(ii) Perform column operations (with interchange) to obtain
qjs=0 for s>l.
(iii) Repeat (i) and (if) until the only nonzero elements in each row
are leading elements.
(6) Obtain a., jem from the appropriate columns of E as in Corollary
J
(2.3-4) and ¥. from the dual of (2.2-19).
(7) From the invariants construct the Luenberger and MFD forms as in
Section (2.2).
If we also require the characteristic polynomial, then we must include:
4*4*
(8) Determine the v.}, cjd and (simultaneously) construct TgR as in
Lemma (2.3-12).
(9) Find Til by solving for the non unit rows in TnnT¡i I .
BR BR BR n
^Note that the leading elements have been selected from the rows by examining
the columns in their natural order; therefore, the dependent columns are
not zero as in (2.3-3), but are easily found from this form of Q by
inspection. It should also be noted that the leading elements could have
been selected in the j, (j+m), (j+2m)... columns; therefore, facilitating
the determination of the Buey invariants and forms.
4.4* r\j
Alternately the {vj}, jem and Tjjc could be used. These indices can be
found easily from the columns of Q.


8
One of the main contributions of this dissertation is to use the
results developed from invariant theory to solve the realization and
partial realization problems in the deterministic as well as stochastic
cases. The realization of a system directly in a canonical form actually
reduces to first determining which transformation groups are present,
specifying the corresponding invariants, and then developing a method to
4*
extract these invariants from the given unit pulse response sequence.
This philosophy is basic to any canonical realization scheme and actually
provides an explicit formula which is applied throughout this dissertation.
In the last few years, several interesting extensions have emerged
from the original concept of realization theory. The major motivation
evolved just after the development of the Kalman filter (see Kalman (1961))
in estimation theory because a priori knowledge of the state space
model and noise statistics are required to begin data processing. The
link between the filtering and realization problem was established by
Kalman (1965) just prior to the advent of Ho's algorithm. The work of
Gopinath (1969), Budin (1971,1972), Bonivento et al. (1973), and Audley
and Rugh (1973,1975) were concerned with the more general problem of,
obtaining a state space representation given a general input/output
sequence of the system in both deterministic and stochastic cases. The
stochastic version of the realization problem has not received quite
as much attention as the deterministic case mainly due to its greater
complexity and high dependence on the adequacy of covariance estimators.
The realization of stochastic systems was studied by Faurre (1967,1970)
and more recently by Rissanen and Kailath (1972), Gupta and Fairman (1974)
4* *
The Hankel array formulation is used exclusively in this dissertation.-


87
T
Solving the last equation for K and substituting for NN yields
K = (B-An*CT) (D+DT-CII*CT)"1 (4.1-13)
Now substituting (4.1-13) and NN^ in the first equation shows that n*
satisfies
n* = An*AT-(B-An*cT)(D+DT-cn*cT)"1(B-An*cT)T (4.1-14)
a discrete Riccati equation. Thus, in this case the stochastic
realization problem can be solved by (1) obtaining a realization,
^KSP ^rom'{Cj}.; (2) solving (4.1-14) for n*; (3) determining NN^ from
(4.1-12) and K from (4.1-13); and (4) determining Q,R,S from K and NN1.
A quintuplet specified by T and n* obtained in this' manner is guaranteed
to be a stochastic realization, but at the computational expense of solving
a discrete Riccati equation. Note that solutions of the Riccati equation
are well known and it has been shown thpt there exists a unique, n*,
which gives a stable, minimum phase, spectral factor (e.g. see Faurre
(1970), Willems (1971), Denham (1975), Tse and Weinert (1975)). We
will examine this approach more closely in a subsequent section, but
first we must find an invariant system description for the stochastic
realization.
4.2 Invariant 'System Description of the Stochastic Realization
Suppose we obtain two stochastic realizations by different methods
from the same PSD. We would like to know whether or not there is any
way to distinguish between these realizations. To be more precise,
we would like to know whether or not it is possible to uniquely
characterize the class of all realizations possessing the same PSD.
We first approach this problem from a purely algebraic viewpoint.


63
Table. (3.1-5) Nesting Properties of Algorithm (3.1-1)
Augment M^M+k n(M+k)=n(M) n(M+k)>n(M)
JM"^M+k "
fi-i+pv.
>
c
where (R) is satisfied for some k and means that the
corresponding invariants, vectors, or indices are nested or
contained in a set of higher order.
Vj
5ist
J.


85
corresponds directly to the results presented by Anderson (1969) in
the continuous case. The proof follows by comparing the two distinct
representations of $z(z) given by (4.1-6) and (4.1-8). Minimality of
(F,H,Q,R,S) is obtained directly from Theorem (3.7-2) of Rosenbrock
(1970). The KSP equations are obtained by equating the sum decomposition
of (4.1-8) to (4.1-6).
This proposition gives an indirect method to check whether a given
KSP anc* stoc^ast'lc realization, (F,H,Q,R,S) correspond to the same
covariance sequence. Attempts to use the KSP equations to construct
all realizations, (F,H,Q,R,S) with identical {CL} from and T by
choice of all possible symmetric, positive definite matrices, H will
not work in general because all n's do not correspond to Q,R,S matrices
that have the properties of a covariance matrix, i.e.,
A:= Cov( pw.
V
[wj v¡]) .
ST R.
6,. 0
(4.1-10)
First, it is necessary to question if the stochastic realization problem
always has a solution, or equivalently, when is there a n so that
(4.1-10) holds. Fortunately, the well-known PSD property, 4>z(z) 0
on the unit circle (see e.g. Gokhberg and Krein (I960) and Youla (1961))
is sufficient to insure the existence of a solution. This result is
available in the generalized Kalman-Szego-Popov lemma (see Popov (1973)).
Proposition (4.1-11) If (F,H) is completely observable, then $z(z) 0
on the unit circle is equivalent to the existence
of a quintuplet, (?,fr,$,$,^¡) such that


60
T
each dependent row vector, f.(M) is uniquely represented as a linear
J
combination of regular vectors in terms of the observability invariants
and it can be generated from the recursion of Corollary (2)3-2); Similarly
it follows from Proposition (2.2-13) that the dependent block row
T
vectors, a. (M) satisfy an analogous recursion. The following lemma
J *
describes the nesting properties of minimal partial canonical realizations
Recall that M is the integer of Proposition (2.1-6) given by M =v+y.
Lemma. (3.1-3) Let there exist an integer M (M)-M such that the rank
condition is satisfied and let £(M) be the corresponding
minimal partial canonical realization of {Ar>, rcM
specified by the set of invariants [v.(M)},{fi. .(M)},
I l U
ia^(M)}]. Then
J '
v.(M) = ... = v.(M+k)
6-st(M) = ... = 3ist(M+k)
al (M) = .... = aT (M+k)
J J
' . ' A. ,
iff p(S(M,M))=p(S(M+l,M+l) = ... = p(S(M+k,M+k))
for the given k.
Proof. If v. (M) = ... = v. (M+k), etc., then the minimal canonical
partial realizations are identical, £(M) = £(M+1)= ... = £(M+k).
It follows that p(S(M,M))=dim£(M)=p(S(M+l,M+l))-dim£(M+l) =
p(S(M+k,M+k)).
Conversely, P(S(M,M))=P(S(M+1,M+1))= ... =P(S(M+k,M+k)) implies
dim£(M)=dim£(M+l)=... =dim£(M+k). Since £(M) is a unique minimal
canonical partial realization, so is £(M ). Furthermore, since
each realization has the same dimension, each realization has


2
we follow an algebraic procedure to obtain unique system representations,
i.e., we insure that these representations re in fact canonical forms.
In simple terms this approach seeks the determination of certain entities
called invariants obtained by applying particular transformation groups
(e.g., change of basis in the state space) to a well-defined set
representing a system parameterization. The invariants are the basic
structural elements of a system which do not change under this trans
formation and are used to specify the corresponding canonical form. This
approach insures that the ambiguities prevalent in earlier work are
removed. Initially, we develop a simple solution to the problem of
determining a state space model from the unit pulse response of a given
linear systerr and then extend these results to the stochastic case where
the system is driven by a random input. The technique developed to
extract the invariants from this (response) sequence not only provides
a simple solution to the realization problem, but also gives more insight
into the system structure.
1.1 Survey of Previous Work in Canonical Forms for Linear Systems
The study of canonical forms for linear dynamic systems evolved
slowly in the Sixties. The main impetus of investigation was initiated
by Kalman (1962,1963) when he compared two different methods for describing
linear dynamic systems: (1) internally by the state space representation
denoted by the triple (F,G,H), or (2) externally by the transfer
function--the input/output description. Development over the past
decade in such areas as optimal control, decoupling theory, estimation
and filtering, identification theory, etc., have relied heavily on the
_
This defines (simply) the realization problem.


96
equations are satisfied. Clearly any choice of Q uniquely specifies
a n for given F; for if, there exist two solutions and corresponding
to identical Q,F, then n^=FH^F^+Q and n^FH^F^+Q. Subtract these equations
to obtain n*-FII*F"I"=On for -n^. It is well known (e.g. see
Gantmacher (1959)) that n*=0 is a unique solution of FH*-H*F*"^=0 ,
n n
sinc^ A(F)^X(-F~T) in this case. Therefore, selecting a Q uniquely
specifies a n and of course fixes R and S which can now be obtained from
the remaining KSP equations. Practical considerations in selecting a
Q which yields a positive definite II will be discussed in the next
section. Here the point is for given F and H (modulo TcGKn))
are obtained from (4.1-9); moreover, selecting Q uniquely specifies a
n which fixes R and S such that the KSP equations are satisfied. The
resulting model, (F,H,Q,R,S) has the same PSD or equivalently {C.} as
SKSP but ^ stl"^ may not satisfy A^O. Obtaining stochastic realizations
such that the latter condition is satisfied is the subject of the next
section.
Similar results can be obtained by using the second method of
specifying an invariant system description; however, recall in this
case that only np-%p(p-l) elements of Q are uniquely specified. Since
n is linearly related to Q for given F through the (LE), then the same
number of elements are uniquely specified in n. Thus, when we select
T=TR, the observability invariants of (2.2-4) uniquely specify the pair
(FR,HR) and for any choice of QR (or alternately SR) we specify an
invariant system description for the stochastic realization by
(F^Hj^Qj^RpjS^). In the next section we develop an algorithm based
on these techniques to obtain a stochastic realization.


TOT
Step 4. Check that (4.3-3) is satisfied. If so, stop. If not,
choose another QR-0 and go to 2. If numerous choices
of simple Q^O do not yield a stochastic realization, solve
the discrete Riccati equation of (4.1-13) and go to 3.
Or
Step 2* Select an SR satisfying (4.2-10).
Step 3* Solve (4.3-2) for R and Check that det (R)>0.
If so, continue. If not, go to 2*.
Step 4* Determine QR from the (LE) and select its free elements
to satisfy (4.3-4) if ^=0p or (4.3-3). If so, stop.
If not possible select another simple SR, i.e., go to 2*,
or try the first procedure, i.e., go to 2.
Consider the following example which illustrates this algorithm.
Example (4.3-6)
For m=3, p=2 the measurement covariance sequence is
20 59
iT5T
3
13 0 9
" 1,5 0
1
9 13
150
4 1
TT
22 6 1
- TO T "
79 1
6W
tl
o
o
3
7
w
O
II
66
25
,1 6
~ TT
S*
O
ro
H
13 9 1
6 0 0 "
1 9 3
6 0 0
c3 5
115 55
720 0
3 3 3 7
720 0
u.
U
1 5577
6155
~ 275717
561964
720 0
7200
C5
1036800
1036800
1 043 1
285588
21434293
4057079
_1 728 0
- 311040
12441600
" 124416 00.
Applying the algorithm of (2.4-1) we obtain
(1) The observability invariants are: v-|=l, V2=2 and
4 = t -i i i o]
J r i i __7_ Li
2 L 4 j 2 4 12*^


31
The relationship of the MFD to the Hankel array, Sv+^ ^follows by
writing (2.2-15) as
B(z)T(z) = D(z) (2.2-16)
and equating coefficients of the negative powers of z to obtain the
recursion
BoAj + BlVl 'f + BvAj+v = mj j=1>2
expanding over j gives the relation over the block Hankel rows as
[B0 B, ... Bu]
where the pxp(v+l) matrix of B.'s is called the coefficient matrix of
B(z). Similarly equating coefficients of the positive powers of z
gives the recursion \
1
1 5 *
t
Vi,.
= 0.
m(y+l)
(2.2-17)
D, = B, ,A, + Bli0A0 + ... + B A .
k k+1 1 k+2 2 v v-k
k=0,l,...,v-l
or expanding over k gives the relation in terms of the first block Hankel
column as
Dv-1
Bv
O
1
Dv-2
II
Bv-1
Bv
*
o
1
_B1
Bn ... B
2 v
(2.2-18)


53
This completes the algorithms. In the next chapter the first method is
modified to develop a nested algorithm from finite Markov sequences.


44
where
v.,jg are the observability invariants of ZR
i\,
v.¡ are the invariants associated with ZRR and
recall r. = Z v, ro=0>
Proof. This lemma is proved by direct construction of the TD 's,
Since each T0 satisfies for v.*v.
i i
hi
Bi*
h.F
v.-l
i' R
hiFR
'V ,
Vi-1
v.-l T
then analogous to property (2.3-8), it follows that h.FD =e
1 K ¥
1
and therefore
v.-l
V- V.-l j TP
hiFR -Fr = 4.FR = 4
V1 vi Vv1
Â¥r ' btfVvv
iFR
In order to construct TRR it is first necessary to find the {v..} from the
rows of [Vn]R, but in this case the v.'s can generally be found by
inspection while simultaneously building TRR. Also, TRR is generally a
sparse matrix with unit row vectors; therefore, the inverse can easily be
-1 1
found by solving M >n directly for the unknown elements of TRR.


98
From Proposition (4.1-9), the observable pair.(A,C) of the KSP
model and (F,H) of the WN model are E^-equivalent; therefore, the
invariants are identical. The link between the canonical realization
of ZKSp and the stochastic realization is provided by the KSP equations
of (4.1-9), i.e., the (LE) and
(A.B.C) = (FR,FRnRHRT+sR,HR) (4.3-1)
D+DT = HRnRHRT+R
Recall that under the action of GR^ on X^, there are 2np+^p(p+l)
invariants--np specifying (FR,HR) and np+%p(p+l) specifying QR,R,SR.
It is possible to extract these np+%p(p+l) invariants from the
measurement covariance sequence using the KSP model realization,
(A,B,C,D). As before, if we assume the action of LR is restricted to
only the elements of QR, then for any choice of QR a unique nR is
specified by the (LE) and therefore R and SR are uniquely obtained from
(4.3-2)
T
R = W-DT-HRnRHRT
SR B-FRnRHR
On the other hand, suppose the action of LR is restricted to SR
and Qr, then for any choice of SR, R and np-%p(p-l) elements of QR are
fixed. Since nR is linearly related to QR through the (LE), the same
number of elements are uniquely specified in nR. We are free to select
the remaining elements in QR and n^. The realization invariants,
}, of the previous chapters allow us to uniquely specify
the invariants of (F,H,Q,R,S) from the KSP equations. Therefore, using
EKSp and (4.1-9) we are able to extract the 2np+%p(p+l) invariants of
(F,H,Q,R,S) from the measurement covariance sequence.


88
We define a set of quintuplets more general than the stochastic
realizations, then consider only those transformation groups acting
on this set which leave the PSD or equivalently (C .} invariant, and
*3
finally specify various invariant system descriptions under these
groups which subsequently prove useful in specifying a stochastic
realization algorithm. The groups employed were first presented by
Popov (1973) in his study of hyperstability. The results we obtain
are analogous to those of Popov as well as those obtained in the
quadratic optimization problem (e.g. see Willems (1971)).
Define the set
X2 = ((F,H,Q,R,S)| FeKnxn,HeKpxn,QeKnxri,ReKpxp,SeKnxp; Q,R symmetric}
and consider the following transformation group specified by the set
GKn := {L | LxKnxn; L symmetric}
and the operation of matrix addition. Let the action of GK^ on X2 be
defined by
L t (F,H,Q,R,S) := (F,H,Q-FLFT+L,R-HLHT,S-FLHT) V (4.2-1)
This action induces an equivalence relation on X2 written for each pair
(F,H,Q,R,S), (F,H,Q,R,S)eX2 as (F,H,Q,R,S)EL(F,H,Q,R,S) iff there exists
a LeGKn such that (F,H,Q,R,S) = L T(F,H,Q,R,S).
This group and GL(n) are essential to this discussion, but we must
consider their composite action. Therefore, we define the transformation
group, GRn which is the cartesian product of GL(n) and GKn,
GRn := GL(n)xGKn. The following proposition specifies GRn*


59
XD
0
0
0
0
0
0
0
0
1 -2 -1
-1
- T
4
_5
4
7
2
-10
-12"-
0
0
0
0

0
0
0
0
0
1 4
*
-1
8
1.3
4
-6
-14
0

0
0
0
0
0
0
0
0
1
-1
__5.
2
-3
a.
4
2
15
20
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0

0
0
0
0
0
1
-1
5
~T
-3
-6
-8
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0.
1
0
0
0
0
0
0
0
0
1
0
0
0
0
0
1
0
0
0
0
a
0
0
0
a a
a a
_o: a
The results in this case are identical to those of Example (2.4-2).
Let us examine the nesting properties of this realization algorithm.
Temporarily, we resort to using data dependent notation for this
discussion with the same symbols as defined previously in the previous
sections, e.g., the minimal partial realization of order M is given by
S(M) (F(M),G(M),H(M)). Thus, I(M+k) is a (M+k)-order partial
realization. We also assume for this discussion that £(M) is in row
canonical form; therefore, it can be expressed in terms of the set of
invariants, [{v.j(M)},{B.¡st(M)},{aT(M)}]. If S(M) is an n dimensional,
minimal partial realization specified by these invariants, then there
T
arc n regular vectors, ¥g+pt(M) spanning the rows of S(M,M). Furthermore,


16
Proposition. (2.1-6) (Realizability Criterion) The minimal partial
realization problem of order M possesses a
solution, £(M) iff there exist positive integers
*
v,y, M = \H-y
where dimE(M) = p(S^ ) = n.
In this proposition (R) is designated the rank condition. Also, it
is important to note that when (R) is satisfied the minimal extension
(of 2(M)) is unique (see Tether (1970) for proof), but S(M) is not
unique because there exist other minimal partial realizations equiv
alent to S(M) under a change of basis in X.
We must consider three possible cases when only partial data is
*
available. In the first case enough data is available such that M>M
for known n; thus, a minimal realization is found. Second, v and y
are available such that (R) is satisfied. In this case a minimal par
tial realization can be found, but this in no way insures it is also a
minimal realization of the infinite sequence, since the rank of S
v
may increase as v,y increase. Third, the rank condition does not hold
How can a realization be found when no more data is available? The
only possibility in this case is to extend the sequence until (R) is
satisfied, but there can exist many extensions satisfying (R) while
giving nonminimal realizations. For this reason define a minimal
extension as any that corresponds to a minimal (partial) realization.
To obtain minimality we must somehow select the right extension among
the many possible.


CHAPTER 2
REALIZATION OF INVARIANT SYSTEM DESCRIPTIONS
In this chapter we present a brief review of the major results
in realization theory. We establish a basic "formula" and apply it to
various system representations. It is shown that this approach greatly
simplifies the realization problem. Two new algorithms for realization
are developed which appear to be more efficient than previous techniques
because they extract only the minimal information necessary to specify
a system from the given input/output sequence in an extremely simple
fashion. All of the essential theory is developed and a multivariable
example is presented.
2.1 Realization Theory
A real finite dimensional linear constant dynamic system has
internal description given by the state variable equations in discrete
time as,
^<+1
+ GJk
(2.1-1)
where keZ+, xeKn=X, u£Km=U, yeK^Y and F, G, H are nxn, nxm, pxn matrices
over the field K. X,U,Y are the state, input, and output spaces,
respectively.
12


50
0 0 0 0 0 0 0
0 0 0 00 o o
000 0 0 0 0 0
0000 0 000
0 000.0.0 o o
i _jl _jl n JL i. 13
2 2 U 4 8 4
10 o o o
1 -1 -4 -3
0 10 0
1 o
The a. and b. are determined from the appropriate rows and columns
J J
of E as:
e17
r 5-i
-T
e14
~ -1
e37
, 1
4
e34
1
e57
.5
2
ao ~ "*
e54
0
e27
1
8
e24
3
T
el 7
e27
- _5 -
4
1
8
^4
S4
e37
1
4
e14
1
e47
=
0
I -2 =
e24
s
3
" ~T
e57
5
2
e34
-1
e67
0
e44
1
e77
1
if
0
0


23
yi
F Jg. *= E
J k=l
J-l nnn(y ,y.-1)
" 1 % + z
s=0 JKS K k=j
m min(y.,yk)-l
JE
s=0
ajksF gi
This proposition follows directly from the controllability of (F,G) and
indicates that the regular vectors form a basis where the a's are the
coefficients of linear dependencies. The set [{yj},{ctjks}], j,kem,
s*0,...,y.-1 are defined as the controllability invariants of (F,G),
J
and y=max(y.). The main result of Popov is:
J
Proposition. (2.2-4) The controllability invariants are a complete
set of independent invariants for (F,G)eX under
the action of GL(n).
The proof of this proposition is given in Popov (1972) and consists of
verifying the invariance, completeness, and independence of [(y^},-Cctjj Invariance follows directly from Proposition (2.2-3), since (F,G)Ej(F,G),
then can be replaced by TFsgk in the given recursion and the
controllability invariants remain unchanged. Completeness is shown by
constructing a TeGL(n) such that for two pairs of matrices (F,G),
(F,G)eX0 with identical controllability invariants, (F,G) = (TFT""^, TG)
or (F,G)Ej(F,G). Independence of the controllability invariants is
obtained by constructing a canonical form determined only in terms of
these invariants. Thus, by introducing a finite set of indices (y^},
Popov shows that this set along with the {ajks} are invariants under the
action of GL(n). The main reason for specifying a set of complete and in
dependent invariants is that it enables us to uniquely characterize the
orbit of (F,G). It should also be noted that dual results hold for the observ
able pair (F,H), and it follows that the observabi1 ity invariants are the


109
As a matter of completeness, we would like to briefly present an
algorithm to obtain the stochastic realization using the Riccati
equation approach. Mehra (1970,1971), Carew and Belanger (1973)
and even more recently Tse and Weinert (1975) have proposed iterative
schemes to obtain n,K,R£, but the theoretical connection to the KSP
equations and the stochastic realization was never established. Their
results are summarized below and we refer the interested reader to these
references for a detailed discussion of convergence properties and
simulation results.
Iterative Solution to the Riccati Equation (4.4-11)
Step 1. Set nQ = 0
Step 2. (R£).-. = D+D^-Hn.HT, where i is the i-th iteration step.
Step 3. K. = (B-Fn.HT)(R )T]
1 1 £ 1
Step 4. n.,, = Fn-FT+K.(R ).kI
K i+l i i e i i
Once K and R are found in this manner, then the stochastic realization
e
follows from (4.4-9). The following algorithm: summarizes the realization
technique using the INV model. <
Stochastic Realization Algorithm via INV Model (4.4-12)
Step 1. Obtain ZKSp from {C^} as in (2.4-1).
Step 2. Use the iterative technique of (4.4-11) to obtain K, R£.
Step 3. Determine QINV,RjNy,SINV from K and R£ as in (4.4-10).
Thus, we have two algorithms to obtain an invariant system description
for the stochastic realization using either the lsiN model or the INV model.
The following figure summarizes these techniques.
*


17
Prior to summarizing the main results of Kalman (1971) and Tether
(1970), define the incomplete Hankel array associated with a given
partial sequence {A^}, keM. as
where the asterisks denote positions where no data is available. The
rank of S(M,M) is the number of linearly independent rows (columns)
determined by comparing only the data specified elements in each row
(column) with the preceding rows (columns) with the cognizance that
upon the availability of more data this number can only remain the same
4*
or increase. Thus, the rank is a lower bound for any extension when the
* are filled in-consistent with the preservation of the Hankel pattern.
Both Kalman and Tether show that there are three pertinent integers
associated with the incomplete Hankel array. They are defined as: n(M),
v(M), y(M) and correspond to the rank of S(M,M), the observability index,
and the controllability index of the given data. The latter two are
lower bounds (separately) for v and y. Knowledge of either v(M), or
y(M) enables us to construct extensions, since they are the least integers
such that (R) holds for all minimal extensions.
It should also be noted that the integers n,v,y,... are actually
non-decreasing functions of the amount of data available, M, and should
be written, n(M), v(M), y(M) etc. to be precise. However, the argument
^It also follows from this that the p(S(M,M)) is a lower bound for dim 2
(see Kalman (1971)).


66
associated with a particular chain (see Roman and Bullock (1975a)).
Therefore, it is possible that a dependent vector, say ^ of a sub-
matrix of S(M,M) later corresponds to an independent vector in S(M,M).
When representing any other dependent vector in this submatrix
m terms of regular predcessors, ¥. must be included, since it is
a regular vector of S(M,M) under the above assumption. In this represen-
tation the coefficient of linear dependence corresponding to
is arbitrary. Reconsider Example (3.1-2) for{A..}, i =1,2,3 where we
only consider the (row) map P.
Example. (3.2-1) For A^, Ag, A3 of (3.1-2) we have P: S(3,3)-K)(3,3) or
" 1 2 2 414 8
2 2 414 8*
1 2 2 416 10
1
000 0(2)2
IJ_J3 2
0^|)-1-4jl-6
2 4 4 8
0 0 0 0
2 4 6 10
P
0 0(2)2
10 3 2
0 0 0 0
4 8
0 0
6 TO
0 0
.3 2
_ 0 0
The indices are = {1,2,1}. Since v^l, the fourth row of
S(3,3) (or equivalently Q(3,3) ) is dependent on its regular predecessors
as shown in the corresponding 3x4 submatrix (in dashed lines) of S(3,3)
(or Q(3,3) ). The second row, say ^ > in this submatrix is dependent,
yet it is an independent row of S(3,3) (or Q(3,3) ). Now, expand of
this submatrix, i.e.,


41
Theorem. (2.3-10) Given the infinite realizable Markov sequence
from an unknown system, then SQ=([:Q>GcH(,)n is a
minimal canonical realization of A^} with
Fc [, | W2
WJ
nr
Gc is a submatrix of (W +1)c given by the first
. columns .
m
HC = t\i
1+m(u1-1)
,m
am ]
^m
umn
and Uj = C%+m Sj+rapj]- k fs a 1
vector of (W^+^)c.
Proof. Since the sequence is realizable, there exist integers, n,v,y,
satisfying Proposition (2.1-5). If Q is given as in Corollary
(2.3-6), then
Q =
Gc 1 1 Fc Gc
jr\
I!
/"c"
for k>y+l
Thus, Gc is obtained immediately from the first m columns of

(Wk)c* ^orm two nxn arrays ^ and A each constructed by
selecting n regular columns of (W^)c starting with the first

column for A and the (1+m) column for A The independent
columns of (\)c are indexed by the y. and satisfy (2.3-8);
thus, they are unit columns and A is a permutation matrix, i.e,,
A = [w-, .
I 1+m
2m
w.
j+m(yj-l) *


72
Thus, this example shows that the minimal extension sequence can be
found recursively due to the structure of P. Of course, the problem
of real interest is when (R) is not satisfied and (as in Ho's algorithm)
a minimal extension with arbitrary parameters must be constructed.
Minimal Extension Algorithm. (3.2-6)
(1) Perform (1), (2), (3) of Algorithm (3.2-2).
(2) Determine M* = v+y. (The values of v,y are determined by the partial data)
(3) Recursively construct the minimal extension {A^,}, r = M+l, ... ,M*
where Ar = [x^.(r)] by solving the set of equations for j(r)
given by
j+pv. Lj = 0 j m(M+l-k.)+l, ... ,m(M*+l-k.), for each iej>.
and recall that k. is the index of the block row of S(M,M) containing
the row vector,
i+pv-.
Example. (3.2-7) Reconsider (3.2-3) for illustrative purposes.
(1) These results are given in Example (3.2-3)
(2) M*=6; thus, find
A5 =
x-j 1 (5) x-j 2(5)
A6
x-j 1 (6) x-j2(6)
x21(5) x22(5)
X21(6) x22(6)
(3) Recursively solve: pT+2v £j 0 for i=1> j=5,6,7,8 and for
i=2, j=3,4,5,6.
£5 I5 = 0 gives x^iB); 1^ = 0 gives x12(5)
£^ r3 =0 gives x21(5); ^ = 0 gives x22(5)
£5 I7 = 0 gives xn(6); ^ = 0 gives Xj2(6)
£g Z5 = 0 gives x21(6); 1^ = 0 gives x22(6)


no
Figure 4. Stochastic Realization Algorithms,


BIOGRAPHICAL SKETCH
James Vincent Candy was born in Astoria, New York on January 21, 1944
He graduated from Holy Cross High School, Flushing, New York in June, 1961
He received the degree of Bachelor of Science in Electrical Engineering
in June, 1966 from the University of Cincinnati, Cincinnati, Ohio.
Upon graduating he worked with the General Electric Company for 9 months.
Then he enlisted in the Air Force of the United States in April, 1967.
He received a commission as a Second Lieutenant in June, 1967 after
completion of Officers Training School at Lackland AFB, Texas. He
spent the majority of his four years' active duty at Eglin AFB, Fla. as
a Threat Systems Engineer and Test Director until separated in June, 1971
as a Captain. In January 1968, he began study at the University of
Florida Extension School (GENEYSIS) for a Master of Science Degree in
Electrical Engineering. He completed his residency requirements in
March, 1972 and received the M.S.E. from the University of Florida.
From March, 1972, until the present time he has done work toward the
degree of Doctor of Philosophy.
James Vincent Candy is married to the former Patricia Meyers and
they have one lovely daughter, Kirstin Patrice. He is a member of
Phi Kappa Theta, Phi Kappa Phi, Eta Kappa Nu and the Institute of
Electrical and Electronics Engineers.
124


112
linear group of the deterministic problem. The equivalence class under
this group was specified and it was shown how the additional constraints
imposed by the stochastic realization further restrict the selection of
free parameters available in the corresponding noise covariance matrices.
Specifying the invariants under this transformation group enabled the
development of a trial and error algorithm to obtain a stochastic
realization without requiring a Riccati equation solution.
The link between the KSP, WN and steady state Kalman filter was
presented. It was shown that realization of the KSP model allowed both
representations to be determined. It was shown that determination of
the filter parameters uniquely specifies a stochastic realization. An
algorithm requiring the solution of a Riccati equation was also presented.
5.2 Suggestions for Future Research
The results given in this dissertation open several interesting
*
possibilities for future research. Applying the algebraic framework
of a transformation group acting on a set offers definite advantages
over unstructured approaches. Simple equivalent solutions which confirm
physical intuition may evolve. It may be possible to specify a set of
invariants under the action of this transformation group which yields
considerable insight into the problem structure. If the problem possesses
additional constraints, it may be possible to utilize this information
to influence the choice of free parameters available. Many problems
of current interest can be examined in this framework (e.g. identification,
exact model matching, and stable observer design problems).
Efficient covariance estimators should be examined in order to
facilitate the development of realization algorithms which yield useful


89
Proposition. (4.2-2) The closed set GRn and operation form a group
where
GRn = {(T,L) | TeGL(n);LeGKn}
and the group operation is given by
(T,L)o(T,L) = (TTfL+T"1LT"T).
Proof. This proof of this proposition follows by verifying the standard
group axioms with respective identity and inverse elements
(In.0n) and (T_1,-TLTT).V
Let the action of GR on X0 be defined by
(T,L) 4- (F,H,Q,R,S) : = (TFT_1 ,HT~\t(Q-FLFT+L)TT,R-HLHT,T(S-.FLHT) ) (4.2-3)
An element (F,h7q7R,1>) of the set is said to be equivalent to the
element (F,H,Q,R,S) of X2 if there exists a (T,L)eGRn such that
(F,H>'Q,R,S) = (T,L)4'(F,H,Q,R>S). This relation is reflexive
(F,H,Q,R,S) = (InOn) + (F.H,Q,R,S)
\
and symmetric
(r1iTLTT)T(F,H,Q,RiS) = (r1>-TLTT)f((T,L)T(FsHsQ>R>S)) =
((T"1JLTT)o(T,L))+(F,H,Q,R,S)=(In,On)T(F,H,Q,R,S) .
Transitivity follows from (F,H,Q,R,S)=(T,L)4'(F,HsQ,R,S) and
(F,H,Q,R,S)=(T,Lj 4'(Tr,lT,^,R,S^) = (T,r)T((T ,L)+(F,H,Q,R,S.)) = (f ,L)4-(F,H,Q,R,S).
Thus, GRn induces an equivalence relation on X2 which we denote by
ETL and (4.2-3) defines the partitioning of X2 into classes. Note that
our first objective has been satisifed, i.e., two EyL-equivalent quin
tuplets have the same PSD; for if we let the pair (F,H,QRS),
(F,H,Q,R,S)eX2 then if (T,L)eGRn


75
Even though it is possible to realize the system directly in Buey form
as implied in the discussion of (2.3-12), it has been found that this
method has serious deficiencies when dealing with finite Markov sequences.
If (R) is satisfied, the partial realization is unique. When (R) is not
satisfied, this technique does not yield all degrees of freedom. For
example, reconsider the arbitrary parameter realization of Example (3.2-3).
This realization is given in Ackermann (1972) as
Q
1
0
0
O'
"0
1
0
0
0
-2
3
-b
0
0
-2
3
0
0
0
0
0
0
1
0
n
. 11
0
0
0
1
0
0
0
0
0
1
0
0
0
0
1
3+e
0
-c
-(d+e)
-e_
_3+e
0
-c
-(d+e)
-e_
Note that one degree of freedom (b=0) has been lost. Similarity
Ledwich and Fortmann (1974) have shown by example that this technique
can also lead to non-minima! realizations. These deficiencies arise due
to the procedure used for the determination of the Buey invariants. This
procedure does not account for the possibility that an independent row
vector of a particular chain may actually be dependent if it is compared
with portions of the same length of vectors in different chains. To cir
cumvent the problem, the previous technique will be used,i .e., the system
is realized directly in Luenberger form and transformed to Buey form. Not
only does this assure minimality as well as the determination of all possible
degrees of freedom, but Tg^ is almost found by inspection as shown in
(2.3-12). Reconsider the example of the previous section.


71
A-
*11(3)
x21(3)
x31(3)
Since P maps S(2,2) into 0(2,2), we
x12(3)
x22(3)
x32(3)
have
2 2 4
2 2 4~
12 2 4
0 0 0 0
1 0 1 0
P
0 -1 -4
0 0 ^0 0
2 A J x-ji (3) x]2(3)
2 4 j x2i(3) x22(3)
0 0 j 0 0
1 0 | x3i (3) x32(3)_
0 0 | 0 0
and in this case,{v.j ,v2,v3> ={1,0,1}. Thus, using (3.2-4), we have
solving 0 = £4 £3 = C-2 0 0
1 0 Q]
2
2
1
for x11(3) gives x11(3)
X1 -j (3)
x2i(3)
X31(3)
Similarily solving:
¡^ = 0 for x-jg(3) gives x12(3) = 8
£3 = 0 for x3i(3) gives x^(3) =1
£^1^ = 0 for x32(3) gives x32(3) = 0
In this example, x2-j(3)=x^(3) and x22(3)=x-|2(3), since v2=0.


18
M will be understood throughout this dissertation in order to maintain
notational simplicity.
There is one more variant of the partial realization problem that
must be considered. A sequence of minimal partial realizations such
that each lower order realization is contained in one of higher order
will be called a nested realization. Symbolically, this is given by
...-E(M)-S(M)-... for M appear as submatrices of the corresponding matrices in £(M). The solution
to this problem is most desirable from the computational viewpoint,
since each higher order model can be realized by calculating just a few
new elements in the corresponding realization. Rissanen (1971) has
given an efficient recursive algorithm to determine this solution.
Another related problem of interest is determining a unique member of
equivalent systems under similarity and is discussed in the following
section.
2.2 Invariant System Descriptions
In this section we review some of the fundamental ideas encountered
when examining the invariants of multivariable 1 inear systems. The
framework developed here will be used throughout this dissertation in
formulating and solving various realization problems. Not only does
this formulation enable the determination of unique system representations
under some well-known transformations, but it also provides insight into
the structure of the systems considered. First, we briefly define the
essential terminology and then use it to describe some of the more common
sets of canonical forms employed in many recent applications (e.g., Roman
and Bullock (1975a,b), Tse and Weinert (1975)).