Lecture Notes in Control and Information Sciences Edited by A.V. Balakrishnan and M.Thoma
58
Mathematical Theory of Networks and Systems Proceedings of the MTNS-83 International Symposium Beer Sheva, Israel, June 20-24, 1983
Edited by RA. Fuhrmann
Series Editors A.V. Balakrishnan • M. Thoma
Advisory Board L. D. Davisson • A. G. J. MacFarlane • H. Kwakernaak J. L. Massey • Ya. Z. Tsypkin • A. J. Viterbi
Editor Paul A. Fuhrmann Dept. of Mathematics and Computer Science Ben-Gurion University of the Negev Beer Sheva 84120 Israel
AMS Subject Classifications (1980): 9306, 9406
ISBN 3-540-13168-X Springer-Verlag Berlin Heidelberg New York Tokyo ISBN 0-387-13168-X Springer-Verlag New York Heidelberg Berlin Tokyo
This work is subject to copyright. All rights are reserved, whether the whole or p~.rt of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to "Verwertungsgeseltschaft Wort", Munich.
© by Springer-Verlag Berlin Heidelberg 1984 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/13ergstr. 2061/3020-543210
This volume is dedicated to M.S. Livsic -
a man of great courage, intellect and vision.
SYMPOS I~UM CHAIRMAN:
~NS ORGANIZING COMMITTEE:
P.A. Fuhrmann, J.S. Baras, R.W. Brockett, P. Dewilde, H. Dym, A. Peintuch, I.C. Go,berg, D. Hinrichsen, R. Saeks
~NS PROGRAM COMMITrEE:
C. Byrnes, M.L.J. Hautus, M. Hazewinkel, J.W. Helton, H. Heymann, A. Isidori~ T. Kailath, R.E. Kalman, E.W. Kamen, P. Krishnaprasad, M.S. Livsic, G. Marchesini, S.K. Mitter, A.S. Morse, H.Fo Munzner, W. Porter, L.M. Silve~man, E.D. Sontag, H.J. Sussmann, D. Tabak, A. Tannenbaum, J.C. Willems, H.S. Witsenhausen
MTNS STEERING CO~ITrEE:
W.N. Anderson, R. deSantis, C.A. Desoer, P. Dewilde [Chairman], A. Peintuch, I.C. Gohberg, J.W. Helton, N. Levan, R.W. Newcomb, W.A. PorteT, R. Sacks, G.E. TTapp, A. Zemanian.
PREFACE
This volume is based on the lectures presented at the
International Symposium of Networks and Systems (MTNS-83)
held at the campus of Ben Gurion University of the Negev,
June 20-24, 1983.
The symposium was supported by Ben Gurion University of
the Negev and The Israeli Academy of Sciences, as well as
indirectly by the United States NSF which provided a signifi-
cant block travel grant for American participants. All this
support is greatly appreciated.
the conference and the setting up of the special sessions.
Especially I would like to mention A. Bultheel, C.I. Byrnes,
P. Dewilde, I.C. Gohberg, D. Hinrichsen, E. Jonckheere,
A. Lindquist, E.D. Sontag and R. Saeks. Y. Magen the
Academic Secretary of Ben Gurion University was particularly
helpful. To all of them I am deeply grateful.
It is with growing satisfaction that we note that the
MINS conference has become a regular meeting ground for Net-
work and System Theorists and Mathematicians. That the
attendance and high standards are keeping up attests to the
continuing need for such a meeting and to its importance to
all participants.
Beer Sheva
October 1983
Z. BARTOSIEWICZ
Y. BISTRITZ
O.H. BOSGRA
On the Design Problem for Linear Systems
Linear Fractional Par~meterizations of Matrix Function Spaces and a New Proof of the Youla- Jabr-Bongiorno Parametrization for Stabilizing Compensators
Minimal Order Representation, Estimation and Feedback of Continuous-Time Stochastic Linear Systems
Wiener-Hopf Factorization and Realization
Closedness of an Attainable S e t of a Delay System
A New Unit Circle Stability Criterion
On t h e Structure and P a r a m e t r i z a t i o n of Non- Minimal Partial Realizations
Uniqueness of Circuits and Systems Containing One Nonlinearity
Robotic Manipulators and the Product of Exponential Formula
Applications of Pade Approximants and Continued Fractions
Toward a Global Theory of (f,g)-Invariant Distribution with Singularities
An Algebraic Notion of Zeros for Systems Over Rings
On Semigroup Formulations of Unbounded Obse rv - a t i o n s and Control Action for Distributed Systems
Spectral Properties of Finite Toeplitz Matrices
16
24
42
63
69
88
101
120
130
149
166
183
194
S t o c h a s t i c Model Reduc t ion and Reduced- Orde r Kalman-Bucy F i l t e r i n g 214
IX
D. GOL LMANN
P. HARSHAVARDIIANA, E.A. JONCKHEERE, L.M. SI LVERMAN
M.L,J. HAUTUS
M. HAZEWINKEL
Spectral Approximation and Estimation with Scattering Functions
O r t h o g o n a l F i l t e r s : A Numer ica l Approach to F i l t e r i n g Theory
Generalized Solutions of Semistate Equations
Formal Orthogonal Polynomials and Fade Approximants in a Non-Commutative Algebra
U n i t a r y I n t e r p o l a t i o n , F a c t o r i z a t i o n I n d i c e s and Block Hankel M a t r i c e s
A Continuation-Type Method for Solving Equations
The Distance Between a System and the Set of Uncontrollable Systems
The Resolution Topology and Internal St~ilization
On the Inversion of Nonlinear Multivariable Systems
On Some Connections Between 2D Systems Theory and the Theory of Systems Over Rings
H ~ - Optimal Feedback Controllers for Linear Multivariable Systems
A Sign Characteristic for Selfadjoint Rational Matrix Functions
Partial Realization by Discrete-Time Inter- nally Bilinear Systems: An Algorithm
Approximation and Control of Symmetric Systems on the Circle
A Canonical Form for the Algebraic Riccati Equation
Stochastic Balancing and Approximation- Stability and Minimality
Disturbance Rejection for Systems over Rings
The Linear Systems Lie Algebra, the Segal- Shale-Weil Representation and all Kalman- Bucy Filters
234
253
268
278
293
299
303
315
323
331
347
363
370
376
389
406
427
433
A. ISIDORI
E.A. JONCKHEERE
D. KANEVSKY
N. KRAVITSKY
P.S. KRISHNAPRASAD
R.P. KURSHAN
Specia l S t ruc tu re , D e c e n t r a l i z a t i o n and Symmetry fo r Linear Systems
A Canonical Form for Static Linear Output Feedback
Nonlinear Control of Aircraft
Principal Component Analysis of Flexible Systems - Open-Loop Case
Reduced Compensator Design via Log-Balancing - a Case Study
A LocalTheory of Linear Systems with Non- Commensurate Time Delays
An Approach to the Description of Some Algebraic Varieties Arising in System Theory
A Fast Graph Theoretic Algorithm for the Feedback Decoupling Problem of Nonlinear Systems
On the Regulator Problem with Internal Stability
Sta te Space Theory - A Banach Space Approach
On Determinantal Representa t ion o f Algebraic Curves
On Certain Infinite-Dimensional Lie Algebras and Related System-Theoretic Problems
Coordination of Concurrent Probabilistic Processes
437
441
463
473
494
513
521
541
550
563
574
584
591
605
E.B, LEE, S.H. ZAK
A. LEPSCHY, U. VIARO
A Volterra Series Interpretation of Some Higher Order Conditions in Optimal Control
Remarks on Minimal Realizations of 2-D Systems
Some Problems in Model Order Reduction Using Frequency-Domain Methods
615
628
639
Xl
System Theory and Wave Dispersion
On Exact Controllability in Hilbert Spaces
Forward and Backward Semimartingale Represen- tations for Stationary Increments Processes
Nonlinear Control Theory and Symbolic Algebraic Manipulations
A Sunmlary of Recent Results on Deadbeat Control Using Periodic Feedback
Stochastic Control with Average Constraint
Analytic Theory of Random Fields Estimation and Filtering
A Strategy for Decentralized Control of Stably Connected Systems
The Concept of Power Dominant Systems
A Lattice Theoretical Characterization of Network Systems
A Complete Phase Portrait for the Matrix Riccati Equation
An Approximation Theorem in Nonlinear Sampling
Some Recent Results on Pathwise Nonlinear Filtering
A Lie-Volterra Expansion for Nonlinear Systems
657
663
676
686
725
741
753
764
774
787
796
804
806
813
822
T.J. TARN, J.W. CLARK, G.M. HUANG
On Weak Po le P lacement o f L inea r Systems Depending on Pa ramete r s
Analytic Controllability of Quantum-Mechanical Systems
829
840
E.A. TRACHTENBERG Systems ovew Finite Groups as Suboptimal Wiener Filters: A Comparative Study 856
Xll
P. VAN DOOREN A Unitary Method for Deadbeat Control 864
J.H. van SCHUPPEN Dissipative Stochastic Control Systems 881
V. VINNIKOV Determinantal Representa t ions of Real Cubics and Canonical Forms of Corresponding Triples of Matrices 882
J.C. WILLEMS Modelling a Time Series by a Linear Time- Invariant System 899
E. ZEHEB Output Feedback Stabilization of Delay Linear Systems with Uncertain Parameters 903
ON THE DESIGN PROBLEM FOR LINEAR SYSTEMS
A. C. Antoulas
Houston, Texas 77251, U.S.A.
CH-8092 Z~rich, Switzerland
ABSTRACT. A unifying theory for the design of linear multi-
variable systems is presented. A non-linear rational equation
which is thereby involved, is solved using closed formulae. More-
over, all solutions satisfying appropriate properness, stability,
and minimality conditions are parametrized. The main ingredient of this approach is the concept of partial realizations.
i. INTRODUCTION.
Given the r x q, r × m, p × q proper rational matrices Zll, Z12, Z21 and
the p × m strictly proper rational matrix Z22 we are looking for a parametriza-
tion of the set of all rational matrices C, Z which satisfy the non-linear Y
equation:
Y
Many important synthesis problems in linear systems involve the solution of the
above equation. For example:
I. The observer problem. ZI2 = I, Z22 = 0. (See EMRE and HAUTUS [1980],
ANTOULAS [1983a].)
and MARRO [1972], EMRE and HAUTUS [1980].)
= I, Z22 = 0, Z = 0. (See BASILE Y
III. The exact model matching or model following problem. Z21 = I, Z22 = 0,
Z = 0. CSee WOLOVICH [1972], MORSE [1973], EMRE and HAUTUS [1980], ANTOULAS [1983a].) Y
IV. Disturbance rejection by state feedback. ZII = HZ21 , ZI2 = HZ22 ,
constant, C is constant, and Z = 0. (See WONHAM [1979].) Y
where H is
V. Approximate model matching with error minimization, z12 = I, z22 = 0,
11Zyll = minimal, where II.ll is an appropriate noxm.
VI. Unity feedback compensation. Z21 = I, Zll = z22, ZII = 0, Z = stable. ............. y
(See ANTSAKLIS and SAIN [1981], DESOER and CHEN [1981].)
VII. Disturbance rejection by measurement dynamic feedback. Z = 0. (See Y
SCHUMACHER [1980], [1982], WILLEMS and COMMAULT [1981], IMAI and AKASHI [1981].)
VIII. The tracking and/or the regulation problem. (i) Special case: the regulated
and the measured variables are the same: ZII = Z21, ZI2 = Z22, Zy = stable.
(ii) General case: the regulated and the measured variables are different:
Z = stable. (See WONHAM [1979], WONHAM and PEARSON [1974], YOUNG and WILLEMS [1972], Y
BENGTSSON [1977], CHENG and PEARSON [1978], ANTSAKLIS and PEARSON [1978], WOLOVICH
and FERREIRA [1979], PERNEBO [1981], CHENG and PEARSON [1981], KHARGONEKAR and
OZGULER [1982], HAUTUS [1982].)
Z = stable, Y
X. Tracking and regulation with sensitivity minimization. Same as IX. (See
YOULA, BONGIORNO, and JABR [1976], YOULA, JABR, and BONGIORNO [1976], ZAMES and
FRANCIS [1981], FRANCIS and ZAMES [1982], CHANG and PEARSON [1982], WILLEMS [1981],
[1982a], 1982b].)
In all of the above problems, the first requirement on the solutions C, Z is Y
that they should be physically realizable, i.e. causal:
(1.2) C, Z : proper rational. Y
In addition, they have to satisfy the stability requirements
(1.3) C : internally stabilizing; Z : stable. Y
Finally, in most of the problems we just mentioned, C is the transfer function of
the compensator and Z , the resulting closed-loop transfer function. Thus, the Y
restriction
where 6(.) denotes the MacMillan degree, is of interest.
In the sequel we will give a detailed outline of the complete solution of equa-
tion (i.I) subject to conditions (1.2), (1.3), (1.4). (See section three and in
particular Main Theorem (3.14) and Remarks (3,15).)
The parametrization of all solutions which satisfy (1.2-4) is given in terms of
closed formulae. The concept which allows this effective parametrization, as al-
ready shown in ANTOULAS [1983a,b], is that of partial realizations of a given finite
sequence of matrices. As it turns out, in this context, partial realization and
causality are equivalent.
2. PRELIMINARIES.
Let k be the field of real numbers and L a k-linear space. L(iz-1)) is the
k-linear space of all rational (formal power) series £ = Z A z -t, where A 6 L, t>T t t
T is an integer, and there exists p 6 k[z], p ~ 0, such that pA @ L[z] ;
L[[z-1]] denotes the k-subspace of Li(z-1)), obtained for T = 0, which is
called the set of proper rational series. We will also use z-iL[[z-l]] to denote
the strictly proper rational series iT = i]. Recall that, by realization theory,
A 6 z-iL[[z-l]] if, and only if, there exist constant matrices iF, G, H) such that
A = H(zI - F)-IG. (For details on the polynomial set-up see FUHRMANN [1976].)
Let A be a subset of the complex plane, symmetric with respect to the real
axis; A will be referred to as the stability region. Depending on the problem,
A might be the open left half of the complex plane, the open unit disk, or subsets
thereof.
Every Z E L((z-l)) can be uniquely decomposed as a sum
(2.1) Z = Z + Z , +
where all the poles of Z are unstable (i.e. belong to the complement of A) while +
the poles of Z are stable (i.e. belong to A). A square nonsingular polynomial
matrix M can always be factored as a product:
(2.2) M = M+M_ = <~+,
where the roots of det M+, det ~+ are unstable, and the roots of det M_, det M_
are stable. The factors M+(M_), M_ (<) are unique up to right, left multiplica-
tion by a unit, respectively.
The m × m non-singular polynomial matrix D is column reduced if, and only if,
deg det D = deg d I + ... + deg dm, where deg dt denotes the degree of the t-th
column d of D. Any non-singular D can be transformed to column reduced form, t
by right multiplication with a unimodular matrix. Similarly, the p × p non-singular
polynomial matrix Q is row reduced if, and only if, its transpose is column re-
duced.
(2.4) r := k[[z-l]].
(2.5) PROPOSITION. Le___t D be an m × m non-singular column-reduced polynomial
matrix with column indices Kt, t 6 m. Let Q be a p × p nQD-singular
row-reduced polynomial matri~ w~th ro~ ~ndices vt, t 6 ~. It follows
D-IRQ-I -i -i -- = Ddiag ~Qdiag'
where Ddiag := diag(zKl' "''' zKm) ' Qdiag := diag(z , ..., zVP).
The proof of the above result, can be found in ANTOULAS [1983b].
Let S be a sequence of m × p constant matrices
S = (A I, .... A ),
which contain some undetermined elements. In particular, let (At) il, the (i,j)-th
element of At, be determined for t 6 ~13~''' where 0 ~< ~ji ~ < ~' i 6 _m, j 6 ~,
and undetermined (free) otherwise. The canonical triple (F, G, H) is a partial
realization of S iff
, t-i (At) ij = hiF gj, t ~ --13~''' i E _m, j E p,
where h!l denotes the i-th row of H and gj denotes the j-th column of G. The
formal power series A = t~oAt z-t 6 R, is a partial power series realization of S
iff (At)ij = (At)ij, t 6 ~ij, i C 2, j 6 £. We will denote the set of all
partial power series realizations of S by
(2.6) ~ := {A = t>oZ At z-I C R: (At)ij = (At)ij' t C --13P'., i 6 _m, j E _P}-
The subset of stable elements of R is --S
(2.7) (Rs) - := {A C R S: poles of A in A},
and the subset of minimal-MacMillan-degree elements of (R_s) is
R )min := {A E (R-S) : ~(A) = minimal}. (2.8) (--S -
As before, 6(-) denotes the MacMillan degree. For an account on the partial reali-
zation problem, see ANTOULAS [1983b, c].
We conclude this preparatory section with a result which is used to determine
whether it is possible to satisfy condition (1.3) or not.
(2.9) LEMMA. Given are the r × q, r x m, p x q rational matrices Z I, Z 2, Z 3.
Let Z 1 = (Zl) + + (ZI) - according to (2.1). There exist m × p, r x q stable ra-
tional matrices Z , Z such that Z 1 = Z2ZxZ 3 + Zy, if, and only if there exist x y
m x p, r × q polynomial matrices x, Y such that (Zl) + = Z2XZ 3 + Y.
The proof of this lemma can be found in ANTOULAS [1983b]. We will only show here
how to check the above condition, and how to compute the polynomial matrices X, Y
if they exist. Let [M] c°l denote the column matrix which is obtained from the
(rectangular) matrix M, by consecutively transposing each one of its rows and placing
them each on top of the other. Let also A~B denote the Kronecker product of the
matrices A and B.
{[( +] z20z ~} = (x N) w
where w is a scalar (unstable) nonzero polynomial, x, y
-i i vectors, and Z2 ~Z 3 = ND It is readily checked that
W(Zl) + = Z2XZ 3 + Y,
where X, Y are the unique polynomial matrices such that:
The existence of stable solutions is equivalent to the condition
are polynomial comlumn
W = constant.
3. THE RESULTS.
In the present section a detailed outline of the complete solution of equation
(i.i) subject to conditions (1.2), (1.3), (1.4) will be described. For the missing
proofs the reader is referred to ANTOULAS [1983b].
The first step is to reduce the solution of equation (i.i) with data ZII, z12,
z21, Z22 , where Z22 ~ 0, to the solution of a similar equation with data Z1, Z2,
Z 3, 0. The resulting equation turns out to be linear and at the same, condition:
C = internally stabilizing, is transformed to condition: Z = stable, where Z is x x
a new parameter. Let
-i = T-Iu, Z22 = LM
be polynomial coprime representations. There exist polynomial matrices A, B, of
appropriate size such that
TA + UB = I,
where I is the identity matrix. The new rational parameter Z which replaces C x
is defined as follows:
-i (3.1) C(I + Z22C) = (B + MZx)T,
with det (LZ - A) ~ 0. (It should be noted, that if C is proper rational, i.e. x
(1.2) is satisfied, this condition is automatically satisfied.) This implies
(3.2) C = (MZ + B)(LZ - A) x x
-i
Using the above transformation, equation (1.1) becomes
(3.3) Z 1 = Z2ZxZ 3 + Z , Y
where ZI, Z2, Z 3 are the rational matrices defined by
(3.4) z I = Zll - ZI2BTZ21, Z 2 = ZI2M, Z 3 = TZ21.
Following YOULA, JABR, and BONGIORNO [1976, Lemma 3], we conclude that C is inter-
nally stabilizing if and only if Z is stable rational. Thus, requirement (1.3) is x
equivalent to
(3.5) Z , Z : stable. x y
The problem reduces therefore, to the solution of equation (3.3) subject to (1.2),
(3.5), (1.4).
It is assumed that equation (3.3) admits stable solutions; in other words accord-
ing to Lemma (2.9), there exist polynomial matrices X, Y such that (ZI) + = Z2XZ 3 +
Y. I t follows that
(3.6) Z 1 = Z2XZ 3 + Z,
where Z is some stable rational matrix. Let
-i -i Z 2 = ND , Z 3 = Q P,
be polynomial coprime representations. We factorize D, Q according to (2.2):
D = D+D_, Q = Q Q+.
Without loss of generality, D+ and Q+ can be chosen so that
(3.7) MD+ is column reduced, with column indices .K i, ± ~ m;_
Q+T is row reduced, with row indices ~, 3 ~ ~- 3
Furthermore, in (3.6), X, Z can be chosen (without loss of generality) so that
z-i (3.8) @ := D-I(M-1B+ + X)Q+ I- e _R,
in other words, the quantity @ can be chosen to be strictly proper rational. (3.7)
and (3.8) are technical assumptions which will be used in the sequel.
we have thus replaced the rational data Z1, Z2, Z 3 of the problem with the
polynomial data
and the stable rational matrix
(3.9b) Z.
The easy problem of solving (3.3) over the field of rational functions (i.e. with-
out reference to conditions (1.2-4)) will be attacked first. Using the data (3.9)
we can write down a parametrization of all rational solutions of (3.3) by inspection;
the rational matrices Z x, Z satisfy equation (3.3) if and only if Y
(3.10a) Z = + X, x D+gQ+
(3.10b) Z = - ND-I~Q-IP + Z. y - _
A is an arbitrary rational matrix.
This parametrization of all rational solutions of (3.3) forms the basis for the pa-
rametrization of all rational solutions which satisfy (1.2-4), which will be developed
below. In fact, all we need to do is to restict the parameter g to lie in appropri-
ate subsets of the set of m x p rational matrices.
With respect to requirement (1.2), we notice that due to our properness assumptions
on the original data Z, the properness of C implies the properness of Z . From • 3 Y
(3.1) and (3.10) follows that C is proper rational if and only if the expression
BT + M(D+AQ+ + X)T is proper rational. In other words, the following linear equation
has to be solved for 4:
(3.11) BT + M(D+AQ+ + X)T = 0 mod R.
Recall (3.8). A satisfies the above equation iff
of (3.7) , Proposition (2.5) implies
g 0 + (MD+)-IR(Q+T) -I. Because
-i -i A E 0 + (MD+)diagR(Q+T)diag,
where (MD+)diag := diag (z <I ..... z<m), and (Q+T)diag := diag (z Vl, ..., z~P). Let
~ij := Ki + ~j - i, i E in, j E p.
The above relationship yields
(A).. e (8).. + z-~ij-l[, i e m, j e p, z3 ~3
where (M).. 13
expansion
denotes the (i, j)-th element of the matrix M, and r is defined by
8 is strictly proper rational, we can write the formal power series
z-i -2 k mxp, t > 0. 0 = A 1 + A2z + ... , A t
We conclude
(8)l 3 + z-Dij-lr = t>oZ (At)ijz-t + z-~ij-lr = Z (A) z -t + z-~ij-lr, ' -- -- tc~i j t ij --
for i £ ~, j £ p; the last equality follows from the definition (2.4) of ~. We
conclude therefore, that ~ satisfies equation (3.11) if and only if its elements
(4).. can be expressed as follows: m3
(Al)ijz-i -~ij + z-~ij-16, i E m, j e £, (A)ij = + ... + (A~ij)ijz
for some 6 E ~. Since (At)ij are fixed and ~ is arbitrary, the above relation-
ship shows that (A).. is a partial (power series) realization of the scalar sequence ~3
Sij := ((Al)ij ..... (A )..), i e m, j ~ p. ~ij 13 -- --
We define a sequence of m × p matrices S, made out of the scalar sequences S.. 13
10
in the following way:
(3.12) S := (A 1 ..... A~), (A t )ij := (At)ij, t £ -~13~'~' i £ _m, j £ _P,
where ~ := max {~..: i E m, j e p}. Our definition of S implies that the ele- 13
ments (At)ij , for t = ~ij + 1 ..... U, i c m, j e ~, remain undetermined (free).
Recall definition (2.6). The above discussion can be summarized as follows.
(3.13) PROPOSITION. A is a solution of equation (3.11) if and only if A e R . --S
This result together with formulae (3.10), implies that C, z are proper ratio- y
nal solutions of equation (i.i) if and only if the parameter A lies in the set R ° --S
We now turn our attention to the parametrization of all stable solutions Z , Z . x y
From formulae (3.10) it follows that if A is a stable rational matrix, the result-
ing Z , Z are also stable. The converse however, need not necessarily hold; i.e. x y
there might exist unstable A which give rise to stable Z , Z . This arises becau- x y
se of possible unstable cross cancellations between Z 2 and Z 3. In order to descri-
be this phenomenon quantitatively, we have to use Kronecker products of matrices; in-
tuitively however, this means that Z 2 and Z 3 might have 'common' unstable poles
and zeros. This complication which arises in the parametrization of all stable solu-
tions, is of technical nature (i.e. not fundamental). Because of space limitations
it will not be further discussed here. The reader is referred to ANTOULAS [1983b].
Thus, due to unstable cross cancellations, the set of parameters ~ which give
rise to stable Z , Z , in general contains properly the subset of stable 4. Ge- x y
nerically however, the two sets are equal. The main result which follows is stated
for the case where the data do not exhibit unstable cross cancellations. Recall no-
tation (2.6), (2.7), (2.8).
(3.14) MAIN THEOREM. Let ZII, ZI2, Z21 be given r × q, r × m, p x q, proper
rational matrices, and Z22 a given p × m strictly proper rational matrix. We
define the rational matrices ZI, Z2, Z 3 b~ (3.4). It is assumed that: (i) Zl, Z2,
Z 3 satisfy the condition of Lemma (2.9), i.e. condition (1.3) can be fulfilled;
(ii) there are no unstable cross cancellations occurin@ between Z2, Z 3. Then C,
Z are rational matrices which satisfy e~uatign (i.i) if, and only if, C is given Y
by formulae (3.2), (3.10a) and Z is 9iven by formula (3.10b), where A is an Y
arbitrary m x p rational matrix. Moreover
(a) C and Z are proper rational solutions if, and only if, ~ e R . y --S
(b) C is proper rational and internally stabilizing, while z is proper Y
rational and stable if, and only if, A £ (Rs) .
min (c) in addition, 6(C) is minimal if, and only if, ~ E (Rs)_ .
11
(3.15) REMARKS. (a) The main features of the comolete solution of (i.i} subject
to (1.2),(1.3), (1.4) just given are the following. By introducing the new para-
meter Z to replace C, we linearize the original non-linear equation. At the x
same time, the internal stabilizability property of C becomes equivalent to the
stability of the new parameter. The resulting linear equation is solved in a
straightforward way, using closed formulae which are simple functions of the data
of the problem. The rational solutions are parametrized at first, in terms of a
matrix parameter, A . Having appropriately arranged our formulae, the requirements
(1.2), (1.3), (1.4) are translated as restrictions on the range of the parameter A.
Properness results in the restriction of A to the set of partial realizations of
the sequence S defined by (3.12). The stability requirements further restrict A
to be a stable partial realization of S. Finally, minimality of the MacMillan
degree of C restricts A to the subset of minimal partial stable realizations
of S.
The resulting approach gives a clear and complete insight into the synthesis
(design) problem for linear multivariable systems. Most important, it is flexible
and leaves room for the fulfillment of additional or different sets of requirements.
(b) The existence of solutions to equation (1.1) which satisfy
requirements (1.3) is settled by Lemma (2.9). This is an alternative new necessary
and sufficient condition for the solvability of the so-called RPIS (Regulator problem
with internal stability) which was first formulated by WONHAM and PEARSON [1974]
(see also WONHAM [1979, Chapter 6]). Recent investigations of RPIS are given by
CHENG and PEARSON [1981], and KHARGONEKAR and OZGULER [1982]. The latter paper
answers the existence question in terms of the skew coprimeness of appropriately de-
fined polynomial matrices. Our formulation of the existence question shows that
there is an abstract equivalence between the skew coprimeness of polynomial matrices
and the existence of polynomial solutions to an associated rational equation.
(c) It is well known (see ANDERSON, BOSE, ~nd JURY [1975]) that
the computation of minimal partial stable realizations of a given sequence S in-
volves the application of the Routh-Hurwitz stability test, which in turn, involves
the solution of a system of polynomial inequalities in several variables. From (e)
of the Main Theorem, it follows therefore that the complexity of computing proper,
minimal, stabilizing and regulating compensators (i.e. solutions to (i.i) subject
to (1.2-4)) is the same as the complex it[ of solving a system of polynomial inequali-
ties in several variables.
(d) Proposition (3.13) shows that the properness (causality) of
C is expressed through the restriction of the range of A from arbitrary rational
to the set of partial realizations of the sequence S (defined by (3.12)). This
fact is general. It says that a system is causal if, and only if, in an appropriate
parametrization, the parameter A is a partial realization of a given sequence; in
other words, the first few coefficients of the formal power series expansion of A
must be fixed, while the rest are free. We conclude that causality is equivalent to
partial realization.
This result of fundamental significance establishes, at the same time, a connec-
tion between causality and formal power series. The latter topic has attracted the
attention and efforts of many researchers. It has proved a powerful tool for dea-
ling not only with linear but most importantly, with non-linear systems as shown
originally by FLIESS [1975]. It is conjectured at this point that the equivalence
of causality with partial realization is valid not only for linear, but for non-
linear systems as well.
(e) The theory developed in this section allows the determination
of the family of all compensators C of a given MacMillan degree which satisfy
(1.2-4). This is not possible with the existing methods.
In particular, C can be chosen to be constant, i.e. the problem is solvable
with constant output feedback, if and only if the MacMillan degree of the minimal
partial stable realizations of S is equal to the MacMillan degree of
Z22 : 6(A) = 6(Z22 ), A C (~)min (for notation see Main Theorem).
13
REFERENCES.
B. D. O. ANDERSON, N. K. BOSE, and E. I. JURY
[1975] "Output feedback stabilization and related problems - Solution via decision methods', IEEE Transactions on Automatic Control, AC-20: 53-66.
A. C. ANTOULAS
[1983a] "New results on the algebraic theory of linear systems: The solution of the cover problems", Linear Algebra and Applications, Special issue on
Linear Control Systems.
[1983b] "A new approach to synthesis problems in linear system theory", Report
8302, Dept. of Elec. Eng., Rice University.
[1983c] "On the partial realization problem", (in preparation).
P. J. ANTSAKLIS and J. B. PEARSON
[1978] "Stabilization and regulation in linear multivariable systems", IEEE
Transactions on Automatic Control, AC-23: 928-930.
P. J. ANTSAKLIS and M. K. SAIN
[1981] "Unity feedback compensation of unstable plants", Proceedings of 20th Annual IEEE Conference on Decision and Control, San Diego.
G. BASILE and G. MARRO
[1972] "A new characterization of some structural properties of linear systems:
Unknown-input observability, invertibility, and functional controllabi-
lity", Int. J. Control, 17: 931-943.
G. BENGTSSON
[1977] "Output regulation and internal models - A frequency domain approach",
Automatica, 13: 333-345.
[1982] "optimal disturbance reduction in linear multivariable systems", Techni-
cal report no. 8214, Department of Electrical Engineering, Rice Uni-
versity.
[1978] "Frequency domain synthesis of linear multivariable regulators", IEEE
Transactions on Automatic Control, AC-23: 3-15.
L. CHENG and J. B. PEARSON
[1981] "Synthesis of linear multivariable regulators", IEEE Transactions on Automatic Control, AC-26: 194-202.
C. A. DESOER and M. J. CHEN
[1981] "Design of multivariable feedback systems with stable plants", IEEE
Transactions on Automatic Control, AC-26: 408-415.
E. EMRE and M. L. J. HAUTUS
[1980] "A polynomial characterization of (A,B)-invariant and reachability sub-
spaces", SIAM J. Control Opt., 18: 420-436.
14
M. FLIESS
[1975] "Un outil alg~brique: Les s~ries formelles noncon~nutatives", Proceedings of Symposium on Algebraic System Theory, editors G. Manchesini and
S. Mitter, Springer, p. 131.
B. A. FRANCIS and G. ZAMES
[1982] "On optimal sensitivity theory for SISO feedback systems", Technical re-
port, Electrical Engineering Department, University of Waterloo.
P. A. FUHRMANN
M. L. J. HAUTUS
[1982] "Linear matrix equations with applications to the regulator problem",
Technical report, Department of Mathematics, Eindhoven University of
Technology.
[1981] "Disturbance localization and pole shifting by dynamic compensation", IEEE Transactions on Automatic Control, AC-26: 226-235.
P. P. KHARGONEKAR and A. B. OZGULER
[1982] "Regulator problem with internal stability", Technical report, Electri-
cal Engineering Department, University of Florida.
A. S. MORSE
[1973] "Structure and design of linear model following systems", IEEE Transac-
tions on Automatic Control, AC-18: 346-354.
L. PERNEBO
[1981] "An algebraic theory for the design of controllers for linear multivari-
able systems; Part I: Structure matrices and feedforward design; Part II:
Feedback realizations and feedback design", IEEE Transactions on Automatic
Control, AC-26: 171-194.
J. M. SCHUMACHER
matic Control, AC-25~ 1133-1138.
matic Control, AC-27: 1211-1221.
[1981] "Disturbance decoupling by measurement feedback with stability or pole
placement", SIAM J. Control Opt., 19: 490-504.
J. C. WILLEM~
[1981] "Almost invariant subspaces: an approach to high gain feedback design -
Part I: Almost controlled invariant subspaces", IEEE Transactions on
Automatic Control, AC-26: 235-252.
J. C. WILLEM~q
[1982] "Almost invariant subspaees: an approach to high gain feedback design -
Part II: Almost conditionally invariant subspaces", IEEE Transactions
on Automatic Control, AC-27: i071-1085.
[1982] "Approximate disturbance decoupling by measurement feedback", Proceedings
of International Conference on Linear and Nonlinear Systems, Bielefeld.
W. A. WOLOVICH
[1972] "The use of state feedback for exact model matching n, SIAM J. Control
Opt., i0: 512-523.
[1979] "Output regulation and tracking in linear multivariable systems", IEEE Transactions on Automatic Control, AC-24: 460-465.
W. M. WONHAM and J. B. PEARSON
[1974] "Regulation and internal stabilization in linear multivariable systems", SIAM J. Control and Opt., 12: 5-18.
w. M. WONHAM
D. C. YOULA, J. J. BONGIORNO, and H. A. JABR
[1976] "Modern Wiener-Hopf design of optimal controllers, Part I: the single
input-output case", IEEE Transactions on Automatic Control, AC-21: 3-13.
D. C. YOULA, H. A. JABR, and J. J. BONGIORNO
[1976] "Modern Wiener-Hopf design of optimal controllers, Part If: the multi-
variable case", IEEE Transactions on Automatic Control, AC-21: 318-338.
P. C. YOUNG and J. C. WILLEMS
[1972] "An approach to the general multivariable servomechanism problem", Int.
J. Control, 15: 961-979.
G. ZAMES and B. A. FRANCIS
[1981] "A new approach to classical frequency methods: Feedback and minimax
sensitivity", Proceedings of the IEEE Conference of Decision and
Control, pp. 867-874.
NEW PROOF OF THE YOULA-JABRoBONGIORNO PARAMETERIZATION
FOR STABILIZING COMPENSATORS*
Blacksburg, Virginia 24061
La Jolla, California 92093
This talk has two purposes. One is to call attention to a recent campaign to systematically study linear
fractional parameterizations of function spaces. The second purpose is to give a new proof of the Youla-Jabr-
Bongiorno parameterization of all compensators which stabilize a given plant. This is a linear fractional parame-
terization and is one specialized example of our general study of linear fractional map~. Further examples of
linear fractional parameterizations are those of Nevanlinna type (scalar case), Adamajan-Arov-Krein [A-A-K]
and Arsene-Ceausescu-Foias [A-C-F] type (vector case) which are obtained in designing certain digital filters,
and in many gain equalization studies [B-HI] and [H1,2].
The first section of this note gives an outline of our general Lie group approach to the subject and the
second focuses on the [Y-J-B] parameterization in a way as analogous to classical circuit theory as possible.
Much of classic~ circuit theory (e.g. Darlington's theorem) amounts to a study of rational functions with values
in the matrix Lie group U(n,n); the analogous results for the Lie group SL(n,~D give the [Y-J-B] parameteri-
zation. We focus on the [Y-J-B] results rather than other aspects of our study, not because they are more
representative, but because they are more unexpected. Also this feature received little attention in the
(mathematics) papers [B-H2-6] which have already been written.
§1. A survey of results on linear fractional parameterizations
The linear fractional maps of interest are of the form
Gz(s)(io,) = [~(i~)s(j~,) +/~(i~)][,~(i~)s(io~)+:¢(i~,)] -t
[~ ~} is a (re+n) x (m+n) matrix-valued rational function. Its values are invertible except at a where g -
finite number of points. The map Gg acts on m x n matrix-valued rational functions. Our notation is
M,~n = m x n matrices, BMmn denotes matrices of norm ~< 1, and the prefix R means rational functions.
For example, RMm means the Mran matrices with rational entries. Thus Gg: RMrrm "* RMmn.
The articles [B-H2-6] constitute a study of such maps where g meets the restriction that g(jo) belongs
to a fixed classical Lie group I' of matrices. There are ten types of classical simple Lie groups which we now
list according to which article treats them:
*Research was supported in part by the National Science Foundation.
17
[B-H4] GL(n,(L') [B-Hr] SL(n,O, GL(n,R), O(n), a solvable group
At the present time the following groups are known to occur in engineering:
U ( n , n ) - Iossless circuits
Sp(n ,~ - reciprocal circuits
Sp(n,R) - Iossless reciprocal
0* (2n) - anti-reciprocal
The results obtained are of several types
(1) Description of Gz(BH~(Mmn)) or Gg(H~(Mn)), for example as the set of all solutions to a fixed
interpolation problem with symmetries determined by F. We give an explicit solution to such prob-
lems.
(2) Spectral factorizations of matrix functions A satisfying various symmetries in addition to or instead
of A(jo~) = AOoJ)*.
(3) Description of orbits {Gg(S): g E RF} =a Osr of some key functions S under each of the classical
Lie groups r. The case F = U(I,I) and S ~ 0 is the original Darlington theorem.
(4) Description of all H~(Mmn) functions meeting certain symmetries and lying in a given 'disk' in
matrix function space.
(5) Various "F-inner-outer" factorizations which extend the classical one for r = U(n).
There is one basic type of theorem used to obtain all of these results. It is motivated by the observation that
most serious studies of ordinary linear fractional maps don't work directly with G s but with the coefficient
matrix g. The action of G s on s corresponds to the action of g on (a Grassmannian of) subspaces. If this
is the basic wisdom for constant coefficient maps, what is the appropriate generalization to g whose entries are
functions and to representations as in (1) where H °° plays a special role?
The answer turns out to be surprisingly elegant. To describe it, we transfer our attention from the jco
axis to the circle, e ~°,
Naturally we shall be interested in the action of g E RF C R G L ( m + n , ~ on L2(~Y'+n). The 'Grassman-
nian' of interesting subspaces consists of the closed subspaces M C L2(~ #+n) which are invariant under Mei0, multiplication by e i°. That is
f E M ~ ei°f(ei°) E M .
Finally the main theorem turns out to be a representation for such spaces as
M = g H 2 ( ~ ÷n) . (*)
A major point is that one space M can be represented as in (*) with several different g's. So one is free to put
added restrictions on g. For example, the classical result of this type is the Bcurling-Lax theorem which says
that if M is full range simply invariant, then one can take g(e i°) in the Lie group U ( m + n ) .
18
In [B-H2] we showed that for most such M one could take g(e i°) in U(m,n). That started the Lie
groups business. The central theorems in [B-H2-6] say precisely which M have a representation (*) with g(e i°)
in a particular Lie group. For lower rank invariant subspaces M, in many cases we have also determined which
IOm] H2(~ra+n) with g(ei°) in a particular group. This type of representation ones are of the form M ~ g in
theorem is crucial for the orbit description problem ((3) above). Fortunately, these representations are easy to
prove, at least intuitively. Obtaining the applications (1), (2), (3), (4) and (5) of these theorems is what takes
most of the work, primarily because ten different groups and five different applications per group leads to lots
of special cases.
Consider the system
r a 4 ~ a l - a2 p i a 3
T Figure 1.
with the plant P a given function in RMn.
Our approach to the parameterization is no shorter than the improved versions in [D-L-M-S] and [F-V] or
than the Grassmannian approach we used in [B-H6]. However, it does have a conceptual simplicity especially
for those who know Darlington style circuit theory. We try to prove the theorem entirely with pictures; we
succeed except for one gap where some algebraic computations are required to justify the equivalence of two of
the pictures.
The first step is to redraw Figure 1 in a more symmetric form.
a4 y a3
19
Corollary 2.3 from [B-H6] is a Darlingtontheorem for SL(n,R) as opposed to the usual one for U(n,n) (see
[13-H1,§3], and [B-H3]). It says that one can construct a g in RSL(2n,6) (to be more precise go and
[ . . . . . . . . . . |
- p
Figure 3.
One can construct such a go from a right coprime factorization P = ND -t OW,D E H'~(Mn)) for P. This
induces us to consider the system
a21' I go LL a 5 a 6
a 8
Figure 4.
with Q = 0. Stability of the system in Figures 1 or 2 means that the aj in Figure 4 are in H2(~) provided d'
and r' are zero and d and r are in H2(gY). Then all of the aj are in H2(g"), The fact that both go and
gffl are in H~(Mza) now comes into play. Intuitively, the consequence of this is that exciting the system with
d,r is equivalent to exciting it with d',r'. As far as stability is concerned signals on one side are not destroyed
by go in passing to the other side (go has no transmission zeroes). This is the content of Lemma 1 given
below. The actual proof involves some algebra which we postpone to the end. Given this, we have that the
20
stability of the system in Figures 1 or 2 is equivalent to the system in Figure 4 with d,r = 0 being stable
under excitations d',r' in H2(~). That is, all signals in the system given by
' ~ " go L " < a3 I" a8 r"
Figure 5.
are in H2(~9) provided d',r' are in H2(~n). Now it is trivial to check that the simple system
J ~d-
Figure 6.
is stable if and only if H is in H~(Mn). Combine Figures 5 and 6 to see that the C which make Figure 5
stable are those for which the picture equation
I !
Figure 7.
yields a function H in H'(Mn). If we solve for C, we see that the stabilizing compensators are those of the
form
21
It remains only to give Lemma I and to prove it. The key definitions are based on Figure 4; for the pur-
poses of the lemma itself, we need not assume that Q ffi 0. Associate two subsystems Z and Z' to Figure 4.
For the system Z' demand that d',r ' are always 0; the inputs to Z are d,r while its outputs are taken to be
the resulting al,a2,a3,a4. Similarly, Z has r,d = 0, inputs d',r ' and outputs as,a6,a7,as, As usual we say that
(or Z') is stable provided that H2(~) inputs lead to H2(~) outputs.
LEMMA 1. Suppose go and go-1 are in H~(M2n). Then stability of the system Z is equivalent to stability of the
system Z'.
To prove this we introduce systems F-o and Z0 which are slightly larger than Z and Z'. Define Zo
from Figure 4 by setting d,r to be zero; its inputs are d',r ' and outputs are all of the aj. Similarly Z0 has
d',r' zero, inputs d,r and outputs equal to all the aj's. If go and g6 "1 are in H**(M2n), then X is stable if
and only if Zo is stable, because
implies as
las}--Ia2} go as a3
E H2(~) . Similarly Z ' is stable if and only if 7.~ is stable. Thus
LEMMA 2. Stability of the system F..o is equivalent to stability of the system Z~, if and only if go and gff i are in
H~(M~) .
[a l H " 1°'1 u,- H '- Proof. Set u~ a3 ' 112 =" as ' u3 a6 ' a7 '
We set about to eliminate u 3 -
In this notation the defining equations of the system in Figure 4 are
Ul ~ [S[]a4 + A HI ~ goU2
CO U2 = [_~Qla6 + A' u4 ~ 10 (~]u3 •
[a41 and fmd that a mapping fl defined bY a6
f t - 'u3 ffi [1Cla4 + gol_..~la6
is the key. We obtain
22
U3 = M3[ I --go][~'] where M3 ~ 11
u , = [ff31" [I -go'[~,] ~ M4[I -go][~,] •
Stability of ,Y0 means MjA E H2(~ n) for all j provided A E H2(~n), that is, each Mj E .H~(M2n). Stability
of E~ means MjgoA' E H2(~ a) for all j provided A' E H2(~"), that is, each M3g0 E H~(M2,). These two
sets of conditions are equivalent (for all choices of the Mj's) if and only if both go and gffl are in H~(M2a).
REFERENCES
[A-A-K]
[A-C-FI
[al
[B-Hll
[B-H2I
IB-H3]
[B-H41
[B-H5]
[B-H61
Adamajan, V.M., Arov, D.Z. and Krein, M.G.: "Infinite Hankel block matrices and related exten- sion problems, Amer. Math. Soc. Transl, 111 (t978), 133-156.
Arsene, Gr., Ceausescu, Z., and Foias, C.: "On intertwining dilations, VIII," Z Operator Theory. 4 (1980), 55-91.
Ball, J,A.: "A non-Euclidean Lax-Beurling theorem with applications to matricial Nevanlinna-Pick interpolation," in Proceedings of Toeplitz Memorial Conference, BirkMuser (1981), 67-84.
Ball, J.A. and HeRon, J.W,: "Lie groups over the field of rational functions, signed spectral factoriza- tion, signed interpolation, and amplifier design," Z Operator Theory, 8 (1982), 19-64.
Ball, J.A., and Helton, J.W.: "A Beurling-Lax theorem for the Lie group U(m,n) which contains most classical interpolation theory," J. Operator Theory, 9 (1983), 107-142.
Ball, J.A., and Helton, LW.: "Factorization results related to a shift in an indefinite metric," Integral Equations and Operator Theory, 5 (1982), 632-658.
Ball, J.A. and Helton, J.W.: "Beurling-Lax representations using classical Lie groups with many appli- cations II: GL(n,~)," Integral Equations and Operator Theory, to appear.
Ball, J.A. and Helton, J.W.: "Beurling-Lax representations using classical Lie groups with many appli- cations III: groups preserving forms," preprint.
Ball, J.A. and HeRon, J.W.: "Baurling-Lax representations using classical Lie groups with many appli- cations IV: GL(n,R), U*(2n), SL(n,~) and a solvable group," preprint.
[D-L-M-S] Desoer, C., Liu, R.W., Murray, J. and Sacks, R.: "Feedback system design: the fractional representa- tion approach to analysis and synthesis," IEEE Trans. Automatic Control, AC-25 (1980), 399-412.
IF-V] Francis, B.A. and Vidyasager, M.: "Algebraic and topological aspects of the regulator problem for lumped linear systems," to appear.
[H1] Helton, J.W.: "Operator theory techniques for distributed systems", Proceedings Eleventh Annual Aller- ton Conference on Circuits and Systems Theory, 1976.,
23
[H2] Helton, J.W.: "Non-Euclidean functional analysis and electronics," Bull. AMS. 7 (July 1982), 1-64.
[Y-J-B] Youla, D.C., Jabr, H.A. and Bongiorno, J.J.: "Modem Wiener-Hopf design of optimal controllers: I and II," IEEE Trans. Aut. Control, AC-21 (Feb. 1977), 3-13; (June 1977), 319-338.
MINIMAL ORDER REPRESENTATION, ESTIMATION AND FEEDBACK OF CONTINUOUS-TIME STOCHASTIC LINEAR S-Y-~TEMS.
Yoram Baram Department of Electronic Systems
Faculty of Engineering, Tel-Aviv University, Israel.
Abstract
The transition from a given state space representation of a continu-
ous-time, stochastic linear system to its minimal state, minimal es-
timator and minimal output representations is investigated. The an-
alysis is centered about the minimal state predictor, which is shown
to be the connecting link between the different representations.
The role of minimal representation in feedback system design is also
examined.
The subject of minimal order representation of linear systems often
arises in control and estimation problems. It has been treated in
the system theoretic literature mainly in two different contexts.
One is the minimal description of a given system as an input-output
relationship and the other is the minimal representation of a single
process as the output of a linear system. The input-output repre-
sentation problem may be treated entirely in the framework of deter-
ministic systems ([I]-[3]). The state-space representation of an
input-output relationship is non-unique and it is minimal if and
only if it is both controllable and observable. The representation
of a single observed process as the output of a linear system is
known as stochastic realization. Minimal stochastic realizations
have been characterized with respect to the second order statistics
25
([4],[5]) and with respect to the probabilistic structure ([6]-[11])
of the given process. Neither the input, nor the state of the state
space representation of a process are unique.
In control and estimation applications, the state variables of a
given state space model are often physically meaningful and the
model is regarded as a representation of the state process~ not
merely of the input-output relationship or the output process alone.
The question then arises as to whether the given state representa-
tion is of minimal order or whether a lower order representation can
be found from which the original state of the system can be obta-
ined. It is well known that the order of a minimal state observer
of a deterministic system is equal to the difference between its
state and output dimensions ([12]). This is not true in the sto-
chastic case, where the Kalman-Buey filter, whose order equals that
of the given representation, is the state "observer". It has been
shown ([13],[14]) that a reduced order version of the Kalman-Bucy
filter may be derived when the observations covariance matrix is
singular, using output derivatives as observations. As pointed out
in this paper, a given state-space representation of a stochastic
system may not be a minimal order representation of its state and
the corresponding Kalman-Bucy filter may not be the minimal order
state estimator, even when the observation covariance matrix is
non-singular.
The main purpose of this paper is to present a logical transition
and simple mathematical transformations from a given system in state
space form to its minimal order state and output representations,
establishing a certain linkage between control and estimation theo-
ry, on the one hand, and stochastic realization theory, on the
26
other. In the stationary case, the transformation from the given
system to the minimal output representation may be obtained from the
spectral factor representation or the Hankel correlation matrix of
the output. This approach is, however~ rather indirect. In the
present paper we take a direct approach and obtain minimal order
state and output representations by applying transformations to the
state of the given system. This results in linear transformations
on the system's matrices. The approach exploits the information
structure of a linear system and, in this sense, resembles the ap-
proach taken in stochastic realization ([6]-[11]). However, while
the stochastic realization approach would be centered in the present
context about the output predictor [6], our approach, which also ad-
dresses the other (state) variables of the system is centered about
the state predictor. After deflnining the minimal state of a system
and obtaining a minimal state representation, we define the system's
minimal predictor, which is the connecting link between the given
representation, the minimal innovations representation (or the Kal-
man-Bucy filter) and the minimal output representation. The trans-
formation from the given representation to the minimal output repre-
sentation and the minimal output estimator are then specified.
Finally, the role of minimal state and output estimators in feedback
system design and representation is discussed.
2. The Minimal State of a Stochastic Linear System
The state of a system is a fundamental concept in systems theory.
The significance of the role played by the state in the description
of a physical system can be seen from the following informal, yet,
complete characterization ([15], p. 11): "The state of a system is
x if by knowing x one knows all that is needed to know about the
27
system in order to determine its future behaviour, assuming that the
future stimuli can be observed". We note that the "future behavi-
our" means the future values of all the variables of the system.
This intuitive definition of state implies that its mathematical re-
presentation should be minimal in some sense. Since the state of a
system would be normally specified by a vector, this means that the
dimension of this vector should be as small as possible.
Consider the system
x(t) : Ax(t) + Bw(t) x(t)c R p (2.1a)
y(t) : Cx(t) + Dw(t) y(t)e R q (2.1b)
where x(O) is zero mean with covariance H and w(t) is a white o
noise process, uncorrelated with x(O), with covariance Q(t). (It is
well recognized that (2. la) is a formal representation of the
:mathematically more precise (Ito) representation dx(t) : Ax(t)dt +
Bdu(t) , where u(t) is a zero mean process of uncorrelated increments
with E{d~(t)d~(t) T} = Q(t)d~. Since the form (2.1a) is commonly
used in control and estimation applications, we shall also use it in
this paper, with w(t)dt interpreted as dU(t) for mathematical pre-
ciseness). The future space of the system at time t is given by
F+(t) : {xi(s) , yj(s), i=1,...,p, j=1,...,q, s _> t}. Let us denote
by X+(t) = {xi(s) , i=1,...,p, s > t} the space of the future states
and by Y+(t) = {yi(s), i:1,...,q, s > t} the space of the future
outputs. We have 8
t
y(s)=Cx(s) + Dw(s) (2.2b)
It can be seen that, given the future inputs, X+(t), Y+(t) and F+(t)
are all spanned by x(t), in the sense that each element of these
spaces can be expressed as a linear combination of the elements of
x(t). Since x(t) is commonly recognized as the state of the system
28
(2.1), and since the intuitive characterization of state suggests
that it should be of minimal dimension, a question of interest is
whether x(t) is minimal, i.e., whether there does not exist a vector
of lower dimension that spans (modulo future inputs) F+(t).
Clearly, any vector that spans X+(t) also spans F÷(t) and Y÷(t). We
therefore confine our attention presently to vectors that span
x+(t).
A base of X+(t) is defined as a vector with mutually orthonormal
components and minimal dimension which spans X+(t). (Recall that x i
and xj are said to be orthonormal if E{x i xj} = 6i,j' where 6i,j is
the Kronecker delta). Clearly, any base of X+(t) is a minimal state
of the system. Let P(t) denote the covariance of x(t) and let U(t)
denote the matrix whose columns are the normalized eigenvectors of
P(t), corresponding to the non-zero eigenvalues (the nomalization is
done by dividing the eigenvectors by the square roots of the corres-
ponding eigenvalue). Let us define
u(t) : U(t)Tx(t) (2.3)
Then, by theorem A.I, given in the appendix, u(t) has orthonormal
components, dim u(t) = rank P and
x(t) : u(t)u(t) (2.4)
It follows that X+(t) is spanned by u(t). In order to see that u(t)
is a base of X+(t), suppose that there exists a vector G(t), whose
dimension is smaller than that of u(t), that spans X+(t). Then
there exists a transformation U(t) such that x(t)=[~(t)u(t) (since
x(t) eX+(t)). But then P(t):U(t)E{u(t)u(t)T}u T, yielding rank
P(t) < dim u(t) < dim u(t), a contradiction. It follows that u(t)
is a base for X+(t). Substituting (2.3) and (2.4) into (2.1), we
get
29
y(t) : CU(t)u(t) + Dw(t) (2.5b)
Since u(t) is a minimal state of the system (2.1), we call (2.5) a
minimal state representation of (2.1).
It follows from the above analysis that x(t) is the minimal state of
the system (2.1) if and only if P(t) is of full rank. P(t) satis-
fies the Lyapunov equation
P(t) = AP(t) + P(t)A T + BQ(t)B T (2.6)
Suppose that Q(t)=Q, a constant matrix. It is well known that P(t)
has a limit value P if and only if A has no eigenvalues in the right
half plane. In the stationary case we have U(t)=O and the minimal
state representation is given by
6(t) = uT(t)AU(t)u(t) + uT(t)Bw(t)
y(t) = CU(t)u(t) + Dw(t)
It is well known that P is full rank if and only if the pair
(A,BQ I/2) is controllable. It follows that a given stationary sys-
tem is a minimal representation of its state if and only if it is
controllable.
3. The Miminal Predictor of a Stochastic Linear System.
The assumption that the inputs of a system can be observed, embedded
in the intuitive characterization of the state of the system, holds
for (indeed defines) deterministic systems. It is, however, hypoth-
etical (yet, formally legitmate for the purpose of definition) for
stochastic systems, whose inputs, at least partially, cannot be ob-
served. Furthermore, while the state of a deterministic system may
be observed from the output, given a structural (observability) con-
dition, the state of a stochastic system is, normally, not accessi-
ble. The informative role played by the state of a deterministic
3O
system is played in the stochastic case by the systems's predictor,
defined below.
Let Y-(t):{y(~),~ <t} denote the set of past observations on a given
system.We shall use the term "prediction" to mean projection in some
sense on Y-(t). Consider the following definition: x is a predic- A
tot of a system if by knowing x one knows all that is needed to
know, in addition to the predictable inputs, in order to predict the
system's future behaviour. We note that the "predictable inputs"
are the projections of the inputs on Y-(t). It can be seen that the
predictor plays a conceptual role similar to that of the state, in
that it provides an information linkage between the past and the fu-
ture of the system. However, while the state of a given stochastic
system is normally unknown, its predictor is known.
Consider the continuous time linear system (2.1). Let us denote by
X(t)=X+(t)IY-(t)={x(s It), s>t} the space of state predictions and by
Y(t)=Y+(t) I Y-('t):{y(s It), s_>t} the space of output predictions,
where x(s It)=x(s) IY-(t) and y(slt):y(s) IY-(t) are the respective
mean-square projections. Since the inputs are unpredictable, i.e.,
w(s)IY-(t)=0 for s>t, we have
x(slt) : eA(s-t)x(tlt) (3.1)
y(slt) = Cx(slt) (3.2)
It follows that both X(t) and Y(t) are spanned by x(tlt) , which
qualifies, then, as a predictor of the system. It is well known
that x(t]t) is produced by the Kalman filter
x(tlt) = Ax(tlt) + K(t)~(t) (3.3a)
where [ (t) satisfies the Riccatl equation
X(t)=A [(t)+ ~. (t)AT+BQ(t)BT-[[ (t)cT+BQ(t)DT]R-I(t)[C [ (t)÷nTQ(t)S]
with
We have
y(t) = Cx(tlt) + u(t) (3.3b)
It is assumed that the Kalman filter (3.3a) exists, which is the
case if and only if the observations eovarianee matrix R(t) is
non-singular. The system (3.3) is called the innovations represen-
tation of (2. I). We note that the predictor of a linear system is
the state of its innovations representation. The state of the inno-
vations representation corresponds directly to that of the original
system. Yet, only the output of (3.3) is the same as that of the
original system. We therefore call (3.3) an output representation
of (2.1).
Let us denote by E(t) the covariance matrix of x(tlt). Then H(t)
satisfies the Lyapunov equation
~(t) = AH(t) + II(t)A T + K(t)R(t)K(t) T H(O) = ~o (3.4)
Let T(t) denote the matrix whose columns are the eigenvectors of
E(t) corresponding to non-zero eigenvalues. Clearly, the vector
z(t) = T(t)Tx(t It) (3.5)
spans X(t), i.e., it is a predictor of (2.1). Furthermore, z(t) is
orthonormal and dim z(t)=rank ~(t). In order to see that z(t) is in
fact a minimal order predictor of the system, suppose that ~(t) is
another predictor and that dim ~(t) < dim z(t). Then there exists a
transformation T(t) such that x(t It)=T(t)z(t) (as x(t It) is a member
of X(t)). It follows that ~(t):T(t)E[z(t)z(t)T}T(t) T, implying rank
32
t)<dim ~(t)<dim z(t), a contradiction. The vector z(t) is then a
base of X(t), hence, a minimal predictor of (2.1). By Theorem A.I
we have
~(t) : [TT(t)AT(t) + TT(t)T(t)]z(t) + TT(t)K(t)v(t) (3.7a)
and
y(t) : cTT(t) z(t) + v(t) (3.7b)
The system (3.7) is a minimal predictor representation of the given
system and (3.7a) is a minimal order state estimator of the system.
Suppose that Q(t)=Q, then ~(t) has a limit value, ~ , if and only if
A has no eigenvalues in the right half plane and ~ is of full rank
if and only if the pair (A,KR I/2) is controllable, where K is the
limit value of K(t). In the stationary case the minimal predictor
representation is
(3.8a)
(3.8b)
In this case we have dim
z(t)<dim x(t) with equality if and only if (A,KR I/2) is controll-
able. The minimal order state estimator for the system (2.1) is
then given by (3.8a) with the transformation (3.6) back to the ori-
ginal state estimate. It is not difficult to see that X(t) is also
spanned by u(tlt)=u(t)IY-(t) , where u(t) is the minimal state of the
system (2.1), defined by (2.3). However, u(t It) is not necessarily
a minimal predictor of (2.1), as controllability of (A,BQ I/2) does
not imply controllability of (A,KRI/2).
4. Minimal Output Representation
So far we have considered a given state space model mainly as a
state representation and have presented transformations to minimal
3S
order representations and estimators of the state. In certain ap-
plications, only the output of a system is of interest. We now con-
sider the transformation from a given system to its' minimal order
output representation and to its minimal order output estimator.
Consider the system (2. I).
erated by
scalars al(s-t) , i=O,...,p-1, such that
eA(S_t ) p-i : ~ ai(s-t)Ai
and
- ] C
CA
1~ = .
s>t (4.1)
(4.2)
v'(t) =~x(tlt) :~T(t) z(t) (4.3)
A base for Y(t) may be obtained by applying an orthonormalization
transformation to the rows of ~T(t). (First, ~ may be replaced by,
say, its first independent rows or by its eigenrows corresponding to
non-zero eigenvalues). Denote the resulting matrix by V(t) and let
v(t) = V(t)z(t) (4.4)
34
It should be noted that when H(t) is of full rank, dim v(t) : rank
. To see that v(t) is a base for Y(t), suppose that there exists
a vector of lower dimension ~(t) that spans Y(t). Then there exists
a transformation V(t) such that v(t)=V(t) ~(t) , implying rank
cov(v(t))<dim v(t)<dim v(t). But, by (4.41, cov(v(t)) is of full
rank, a contradiction. Hence v(t) is a base for Y(t). Writing
v(t) : v(t) I v(t) + n(t) (4.6a)
we would like to show that w(t) is an unpredictable process, i.e.,
that n(t) IY-(t):0. We have n(t)IY-(t) : ~(t)IY-(t)-v(t)Iv(t)IY-(t).
First note that dv(t)=v(t+dt)-v(t) where v(t+dt) is a base of
Y(t+dt) . Projecting the latter on v(t) , we get
Y+(t+dt) IY-(t+dt) IY+(t) IY-(t):Y+(t+dt)IY+(t)IY-(t):Y+(t+dt)IY-(t),
where the first equality follows from the fact that
Y-(t+dt)~ Y+(t)l Y-(t) and the second from the fact that
Y+(t+dt)CY+(t). It follows that v(t) Iv(t) IY-(t):v(t) IY-(t), and,
consequently, n(t)IY-(t)=0. Hence, n(t) is an unpredictable pro-
cess.
y(t) = y(t)Iv(t) + ~(t).
The output representation (4.6) may be written in the form
~(t) = F(t) v(t) + n(t)
F(t) = E{~(t)vT(t)}
H(t) = E{y(t)vT(t)}
In the stationary case, when K, ~ and, consequently, T and V are
constant, we get the minimal output representation of (2.11 as
v(t) = F v(t) + n(t)
y(t) = H v(t) + v(t)
F = vTTATv T
H = CTV T
and
cov{~(t)} -- R
It should be noted that, by (4.5), when ~(t) is of full rank (hence
T(t) is of full rank), the order of the minimal output representa-
tion is equal to the rank of the observability matrix. In the sta-
tionary case, a necessary and sufficient condition that a given
state-space model is a minimal output representation is that
(A,KRI/2,C) is controllable and observable.
In many applications the output of a linear system of the form (2,1)
is viewed as being composed of a "useful signal" Cx(t) and additive
"measurement noise" v(t). Output filtering means obtaining an esti-
mate of the useful signal i.e., the value of Cx(tl t)=y(tlt). A min-
imal order output filter may then be obtained from the Kalman-Bucy
filter corresponding to (4.6)
(as v(t It)cv(t)) with
y(tlt) : H(t) v(t). (4.8b)
We have seen that the state of the minimal output representation of
a system is the minimal predictor of the output process. This is
consistent with the characterization of the minimal state of a sta-
tionary process suggested by Akaike [6],[7], who provided dual re-
presentations for the projections of the future of the process on
its past and vice versa. These concepts were generalized by Picci
[9] and by Lindquist and Picci [11]. In the context of estimation
36
and control of dynamical systems, the predictor version of this in-
terpretation of state, which was shown in [16] to apply to nonsta-
tionary processes generated by time invariant linear systems, is of
particular interest. While the stochastic realization approach
starts with the correlation function or with the spectral density
function of the given process, we have obtained the minimal output
predictor by a transformation on the minimal state predictor (or,
simply, the minimal predictor) of the given system.
5. Minimal Order State and Output Feedback
Consider the system
y(t) = Cx(t) + Dw(t)
where w(t) is a zero-mean white noise process and r(t) is a control
process, chosen so as to satisfy some optimality criterion. Certain
common criteria lead to a choice of r(t) in the form (e.g., [18])
r(t) : -Sx(t) x(tl t) (5.2)
where S x is a gain matrix. The predictor x(t I t) is produced by the
associated Kalman filter
x(t It) = [A-GSx(t)]x(tl t) + K'(t) v(t) (5.3)
where K'(t) and v(t) are the corresponding Kalman gain and innova-
tions process. A minimal order predictor is given by
z(t) = T'(t)Tx(tl t) (5.4)
where T'(t) is the matrix of eigenvectors of cov(x(t it)) correspond-
ing to non-zero eigenvalues. By Theorem A. I, we have
x(t It) = T'(t) z(t) (5.5)
Substituting (5.4) and (5.5) into (5.3) we get
z(t)=T' (t) T[A-GSx (t) IT' (t) z(t) +T' ( t)TK(t)v (t) (5.6)
and the feedback system is given by
~(t) -- Ax(t) + Bw(t)
T(t)T[A-GSx (t) ]K(t)
T(t)T[A-GSx(t)]T(t)T+T(t)TK(t)
The order of the resulting system is dim x(t) = dim x(t)+dim z(t).
However, it can be seen that the prediction space of this system,
which is clearly spanned by x(t t), is also spanned by the minimal
predictor z(t). It follows that the order of the minimal output re-
presentation of the system (5.7) is equal to dim z(t). This gives a
minimal representation interpretation to optimal control. While the
orders of the minimal predictor and the minimal output representa-
tions of an arbitrary feedback system may be greater than those of
the controlled plant, the orders of the minimal predictor and output
representations of an optimal feedback system are not. In the sta-
tionary case the equality dim z(t)=dim x(t) holds if and only if the
pair (T'T[A-GSx]T ', T'K(t)R I/2) is controllable.
Next suppose that, instead of state feedback, output feedback is de-
sired. Then the feedback process has the form
r(t) = -Sy y(t It)
: -SyCx(tlt) = -SyCT'(t) z(t)
Replacing in the state feedback system S x by SyC, it can be seen
38
that the order of the minimal ouput representation of this system
is, as in the state feedback case, equal to dim Zn, which is, in
turn, smaller than or equal to the plant order.
6. Conclusion
This paper has investigated the transformation from a given system
in state space form to its minimal state and output representations
and estimators. We have seen that a given state space representa-
tion is a minimal output representation if and only if the state
predictor is of full rank and the system is observable. The innova-
tions representation, based on the state predictor, links the given
representation and the minimal output representation. In this paper
we were mainly concerned with continuous time, statistically nonsta-
tionary systems. A more specific discussion of the stationary case,
which lends itself to a detailed structural analysis, and a discus-
sion of the discrete-time case are deferred to later publication.
Theorem A.I
Appendix
Let x be a random vector with zero mean and covariance P. Suppose
that rank P=q. Let T denote the matrix of the normalized eigenvec-
tors of P, corresponding to non-zero eigenvalues. (The normaliza-
tion is done by dividing each eigenvector by the square root of the
corresponding eigenvalue). Define
and
39
Proof
Let us denote by T the matrix of eigenvectors of P corresponding
zero eigenvalues. Defining
E{y I y~} : Iq
and
Denoting
(A.3)
CA.4)
Y2 = 0
X =
z x -- [T T] : Tz
0
I ,
2. R.E. Kalman, "Mathematical Description of Linear Dynamical Systems", SIAM J. Control, Vol. I, No. 2, pp. 152-192, 1963.
. L.M. Silverman, "Realization of Linear Dynamical Systems", IEEE Trans. on Automat. Contr. Vol. AC-16, No. 6, 1971.
4. N~ Wiener, Extrapolation, Interpolation and Smoothing of Stationary Time Series, The M.I.T. Press, 1949.
5. B.D.O. Anderson, "The Inverse Problem of Stationary Covariance Generation", J. Statist. Phys. Vol. I, pp. 133-147, 1969.
6. H. Akaike, "Markovian Representation of Stochastic Processes by Canonical Variables", SIAM J. Control, Vol. 13, No. 1, 1975.
7. H. Akaike, "Stochastic Theory of Minimal Realization", IEEE Trans. on Automatic Control, Vol. AC-19, pp. 716-723, 1974.
8. P.L. Faurre, "Stochastic Realization Algorithms", in System Identification, R.K. Mehra and D.G. Lainiotis, Eds., New York, Academic Press, 1976.
9. G. Picci, "Stochastic Realization of Gaussian Processes", Proe. IEEE, Vol. 64, Jan. 1976.
10. A. Lindquist and G. Picci, "On the Stochastic Realization Problem", SIAM J. Contr. Optimiz., Vol. 17, May 1979.
II. A. Lindquist and G. Picci, "On the Structure of Minimal Splitting Subspaces in Stochastic Realization Theory", Proe. CDC, New Orleans, Dec. 1977.
12. D.G. Luenberger, "An Introduction to Observers", IEEE Trans. on Automat. Contr. Vol. AC-16, No. 6, 1971.
13. A.E. Bryson and D.E. Johansen, "Linear Filtering for Time Varying Systems Using Measurements Containing Colored Noise", IEEE Trans. on Automat. Contr. Vol. AC-IO, No. I, 1965.
14. E. Tse and M. Athans, ,Optimal Minimal Order Observer - Estimators for Discrete Linear Time- Varying Systems", IEEE Trans. on Automat. Contr. Vol. AC-15, No. 4, 1970.
15. R.W. Brockett, Finite Dimensional Linear Systems, Wiley, 1970.
41
16. Y. Baram, Realization and Reduction of Markovian Models from Nonstationary Data", IEEE Trans. on Automat. Contr. Vol. AC-26, No. 6, 1981.
17. T. Kailath, Linear Systems, Prentlce-Hall, 1980.
18. A.E. Bryson, Jr. and Y-C Ho, Applied Optimal Control, Ginn and Company, 1969.
WIENER-HOPF FACTORIZATION AND REALIZATION
subfaculteit Wiskunde, Vrije Universiteit, Amsterdam
Department of Mathematics, Tel-Aviv University, Tel-Aviv
ABSTRACT
Explicit formulas for Wiener-Hopf factorization of rational matrix and analytic ope-
rator functions relative to a closed contour are constructed. The formulas are given
in terms of a realization for the functions. Also formulas for the factorization in-
dices are presented.
Systems of singular integral equations, vector-valued Wiener-Hopf integral equa-
tions, equations involving block Toeplitz matrices etc. can be solved when a Wiener-
Hopf factorization of the symbol of the equation is known (see, e.g., [GK,GF,GKr, K]),
and in general the solutions of the equations can be obtained as explicit as the fac-
tors in the factorization are known. Recall that a (right) Wien~P-Hopffaotor~zation
relative to a contour r of a continuous m x m matrix function W on r is a representation
of W in the form:
(0.i) w(X) = w_(X)
w+(1), I E F.
J Here F is the positively oriented boundary of an open set with a finite number of
components in the Riemann sphere ~ and F consists of a finite number of non-inter- +
secting closed rectifiable Jordan curves. The point E l lies in the inner domain ~F
of F and C 2 is in the outer domain ~ of F. The m × m matrix function W (resp. ~+)
has invertible values, is continuous on ~r U F (resp. ~ u F) and analytic on ~F (resp-
~ ). The integers KI,...,Km, which are assumed to be arranged in increasing order, are
called the (r~ght) factor~zation ~nd~ce~ of w with respect to F. ~he factorization in-
dices are uniquely determined by the original function W and do not depend on the
particular form of the factors W_ and }~. The Wiener-Hopf factorization is called
c~on{dal if all factorization indices are equal to zero, i.e., if
43
(0.2) W(1) = W_(1)w+(1), I e F,
with W and W+ as above. By interchanging the roles of W_ and W+ in (0.I) and (0.2)
one obtains a left Wiener-Hopf factorization and a canonical left Wiener-Hopf factori-
zation, respectively. In this paper we shall only deal with right factorizations.
An arbitrary continuous m x m matrix function need not have a Wiener-Hopf factori-
zation, but necessary and sufficient conditions for its existence are known (see, e.g.,
[CG]). For the scalar case m = i explicit formulas for the factors in the Wiener-Hopf
factorization are available (cf. [K]), but in the matrix case analogous formulas do not
exist. For rational matrix functions there is an algorithm which yields the Wiener-
Hopf factorization in a finite number of steps, but again this algorithm does not pro-
vide explicit expressions for the factors.
The main goal of this paper is to produce explicit formulas for Wiener-Hopf fact-
orization of a non-slngular rational m x m matrix function W. Without loss of generality
one may assume that W(~) = Im, where i m denotes the m× m identity matrix. (Apply a
suitable MSbius transformation and normalize to I m at ~.) But then is is known from
system theory that W admits a representation of the form
(0.3) W(~) = I m + C(IIn-A)-IB,
where A, B and C are matrices of appropriate sizes. Our analysis will be based on (0.3).
The identity (0.3) is called a realization of W (cf. e.g., [Ba,KFA,BGKI]).
The case of canonical Wiener-Hopf factorization of rational matrix functions of
the form (0.3) has been treated already in [BGKI], Section 4.4. There a necessary and
sufficient condition for canonical Wiener-Hopf factorization and explicit formulas for
the factors were obtained on the basis of a geometric factorization principle which
gives the factors in terms of invariant subspaces of A and A - BC.
The present paper is a continuation of [BGKI]. The same geometric factorization
principle is used, but the construction of the factors is much more involved. It turns
out that in general one cannot apply the factorization principle directly to the ma-
trices A, B and C appearing in (0.3). Instead, starting from (0.3), one has to con-
struct in a special way a new realization W(I) = I m+C(l-~)-l~, which permits to ob-
tain the factors in terms of invariant subspaces of A and A - BC. The final formulas
are again in terms of the original matrices A, B, C and oertain auxiliary operators
introduced in the construction of the new realization.
The construction of the Wiener-Hopf factorization referred to above is carried
out in Sections 2-6 of this paper. Much of what is done there extends to the infinite
dimensional case involving analytic operator functions instead of rational matrices.
This is the topic of Sections 7 and 8.
We only present results; no proofs are given. For more information, the reader
is referred to the detailed report [BGK2]. Notation and terminology are taken from
[BGKV]. Often matrices will be viewed as operators acting between finite dimensional
44
1. CANONICAL WIENER-HOPF FACTORIZATION
A system (or node) is a quintet 8 = (A,B,C;X,Y) consisting of two complex Banach
spaces X,Y and three (bounded linear) operators A : X ÷ X, B : Y ÷ X, C : X -~ Y. The
function We(1) = Iy + C(IIx-A)-IB is called the transfer function of @. The space X
is referred to as the state space of @.
In Sections 1- 6 the emphasis will be on systems @ = (A,B,C;X,Y) with finite
dimensional X and Y. Such systems are called finite dimensional. A common way to give
a finite dimensional system is to specify three matrices A, B and C of appropriate
sizes. If @ = (A,B,C;X,Y) is a finite dimensional system, then its transfer function
W@ can be identified with a rational m x m matrix having the value I m at ~. Here m =
dim Y. Conversely, such a rational matrix can always be written as the transfer func-
tion of a finite dimensional system.
As in the introduction, let F be a contour on the Riemann sphere C . We say that
the system @ = (A,B,C;X,Y) has no spectrum on P if neither A nor A x = A-BC has spectru~
on F. The transfer function of a system @ without spectrum on F is defined and takes
invertible values on F. In fact We(1) -I = Iy -C(IIx-AX)-IB. Note that A x depends not
only on A but also on B and C.
If T is an operator on a Banach space Z, we often write I-T instead of II Z-T.
Also the identity operator I Z on Z is sometimes simply denoted by I. The spectrum of
T w i l l be d e n o t e d by O'(T). R e c a l l t h a t ~F and f~ s t a n d f o r t h e i n n e r domain and o u t e r
domain of F, respectively.
THEOREM i.i. Let W be the transfer function of a finite dimensional system 8 =
(A,B,C;X,Y) without spectrum on F. Let M be the spectral subspace of A corresponding
+ and let M × be the spectral subspace of A × corresponding to to the part of O(A) in ~F'
the part of O(A×)in ~F" Then W admits a canonical Wiener-Hopf factorization with res-
pect to F if and only if
(i.l) X = M • M ×.
If (1.1) is satisfied and K is the projection of x along M onto M ×, then a canonical
Wiene~Hopf factorization of w with respect to F is given by W(1) = W_ (/)w+(1), where
W_(1) = I + C(I-A)-I(I-~)B,
W+(I) -I = I - C (I - Ax)-]HB.
The space M introduced in Theorem I. I is the (direct) sum of the generalized +
e i g e n s p a c e s c o r r e s p o n d i n g t o t h e e i g e n v a l u e s o f A l y i n g i n ~F" S i m i l a r l y , M x i s t h e
45
(direct) sum of the generalized eigenspaces corresponding to the eigenvalues of A x
located in ~. If ~ is a bounded open set in ~ (and hence F lies in the finite com-
plex plane) , then also
F
r
We call M,M × the pair of spectral subspaces associated with 8 and F.
The sufficiency of the condition (1.1) and the formulas for W and W+ are covered
by Theorem 1.5 in [BGKI]. Theorem 4.9 in [BGKI] yields the necessity of condition (i.i)
for the case when 8 is a minimal finite dimensional system. (The fact that in [BGKI ] +
the set ~F is assumed to be bounded is not important and the proofs of Theorems 1.5
and 4.9 in [BGKI] go through for the more general curves considered here.) The nec-
essity of the condition (I. i) for possibly non-minimal systems will appear in this
paper as a corollary of our main factorization theorem (Theorem 6.1).
2. DISCUSSION OF THE PROBLEM AND SYSTEMS WITH CENTRALIZED SINGULARITIES
~'or the convenience of the reader we recall the geometric factorization principle
mentioned in the introduction. Let 8 = (A,B,C;X,Y) be a system, let N be an invariant
8ubspace for A and let N x be an invariant 8ubspace for A × = A- BC. S~-ppose x = N • NX.
Then the transfer function W of 8 admits the factorization W(h) = WI(~)W2(h), where
W1(1) = I + C(h-A)-I(I-Q)B,
W2(1) = I + CQ(I-A)-IB.
Here Q is the projection of X onto N x along N.
Now let us return to the situation considered in Theorem 1.1. Clearly M is an in-
variant subspace for A and M × is an invariant subspace for A x. In fact M is the largest
+ M × A-invariant subspace such that O(AIM) c ~P and is the largest A×-invariant subspace
for which o (Ax 1 MX ) c ~ . If X = M ~ M ×, the geometric factorization principle yields a
factorization of W. The fact that M and M x are special invariant subspaces for A and
A x, respectively, implies that this factorization is a canonical Wiener-Hopf factori-
zation.
In order to construct a non-canonical Wiener-Hopf factorization for W via the
factorization principle one needs a realization of W with special properties. To see
this suppose such a factorization is available. To be more specific, let
(2.1) W(~) = W_(~)D(1)W+(~), ~ ~ F
be a Wiener-Hopf factorization of W relative to F. Here D(~) has the diagonal form
of the middle term in formula (0.1). Now write the factors in (2.1) as transfer func-
tions. So let @_ = (A ,B_,C ;X_,Y) be a system with transfer function W_, and let
46
@+ = (A+,B+,C+;X+,Y) be a system with transfer function W+. In view of the special
properties of W_ and W+ it is natural to assume that the spectra of A and A x _ are in
+ and × are in ~F. Further, let @D (AD,BD,CD;XD,Y) be a ~F and the spectra of A+ A+ =
minimal system with transfer function D, and consider @ = @_@D@+, where the product
of the systems is taken in the sense of [BGKI] and [BGKV]. Because of (2.1) the trans-
fer function of 0 coincides with W- A detailed inspection of the properties of @ ~howg
that @ = (A,B,C;X,Y) is a system with cent~l~z~ ~ngu~a1~t~e8 relative to the con-
tour F (and the points el,e2). By this we mean that there exists a decomposition of
the state space X into four subspaces, X = X 1 • X 2 @ X 3 • X4, such that with respect
to this decomposition the operators A, A ×, B and C can be written in the form
A 1 . * , A 1
x 0 A 3 0 A 3
0 0 A 4 * * A
B = I B I
and the following properties hold:
x + (i) the spectra of AI,A 1 are in ~F;
(ii) the spectra of A4,A 4 are in ~;
(iii) the operators A 2 - £. and A~ - e 2 have the same nilpotent Jordan normal form in I~ 7 ~' t
(finite) bases {djk}kJl,gtl and {ejk}k__31,j=1 of X 2, respectively, i.e.,
(A 2- £1)djk = dj,k+l, k = 1 .... ,ej,
(A 2 - 62) ejk = ej ,k+l ' k = 1,... ,e 9 ,
where dj,~j+l = ej,~j+1 = 0 and j = l,...,t, and the two bases are related by
ejk = ~ <k~l>(£1-E2)~dj,k-~'
k l (k dJ k = 9~0 \ ~ / 2- £1)~ej,k-V'
where k = l,...,~j and j = I ..... t;
(iv) rank B 2 = rank C 2 = t with t as in (iii);
x e I and A 3 - £2 have the same nilpotent Jordan normal form in (v) the operator A 3 -
47
s ~" S (finite) bases {fjk}k~l,j=1 and {gjk}k21,j=l of x3, respectively, i.e.,
(At- El)fjk = fj,k+l' k = 1 ..... ~j,
(A3- £2)gjk = gj,k+l' k = I,...,~ 9,
where fj,~j+l = gJ,~j+l = 0 and j = i ..... s, and the two bases are related by
k-i {~+~-k~(~ gjk = ~0 \ ~ / 2- £1)Wfj,k-~ '
k-I (~+~j-kh(~ ~2)~gj fjk = 9~0 \ D / I - ,k-~"
where k = l,...,~j and j = I .... ,s;
(vi) rank B 3 = rank C 3 = s with s as in (v].
We shall suppose that the numbers ~. and ~. appearing in (iii) and (v) are ordered in ] ] the following way:
~i -> 62 >- " "" -> ~t' ~I -< ~2 -< ''' -< ~s"
Clearly a system with centralized singularities has no spectrum on F. It turns out
that having centralized singularities is exactly the special property we are looking
for. This is clear from the remarks just made and the following theorem.
THEOREM 2. i. Let W be the transfer function of a finite dimensional system 8 =