On the zero module of rational matrix functions

On the zero module of rational matrix functions

Citation for published version (APA):

Fuhrmann, P. A., & Hautus, M. L. J. (1980). On the zero module of rational matrix functions. (Memorandum

COSOR; Vol. 8015). Technische Hogeschool Eindhoven.

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Published: 01/01/1980

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EINDHOVEN UNIVERSITY OF TECHNOLOGY

Department of Mathematics

PROBABILITY THEORY.. S'l'ATISTICS AND OPERATIONS RESEARCH GROUP

Memorandum COSOR 80-15

On the Zero Module of Rational Matrix Functions

by Paul A. Fuhrmann and M.L.J. Hautus Eindhoven, November 1980 The Netherlands

ON 111E ZERO ~10DlJLE

or

RATIONAL MATRIX FUNCTIONS by

Paul A. Fuhrmann

Department of ~Iathematics

Ben Gurion University of the Negev Beer Sheva, Israel

M. L. J. lIautus

Department of Hathematics Technological University Eindhoven, The Netherlands

1. Introduction

This situation has been remedied to a large extent by the recent paper of Wyman and Sa in (2) which is the main motivation for this short note.

(2.1) (1) If T and U Bre left coprime, then

. pr 2 (p-1_{(F s+P(A])+F}m_[A)) Z(G) "' ' ' -pr 2(Ker P)+Fm[A) (ii) V K T is an (A,B)-invariant subspace in Ker C i f and only i f V

₌

_prlPoKE where P

=

P E

o a a

is a factorization with E

a nonsingular.

2. The Polynomial System Matrix as a ~Umerator

In the next section we state the main results

co~cerning the relation between the zero structure of G and the polynomial system matrix.

Thero are some indications that the pol}~omial

system matrix (1.4) associated ~ith the representation (1.3) of a transfer function G behaves like a numer-ator in a coprime factorization of G. Especially suggestive is a comparison of corollaries 3.12 and 8.6 in [mre and Hautus [5] which give a characterization of the maximal (A,B)-invllriant subspace in Ker C in terms of the representations G

=

T-1U and (1.3). In fact the analysis given in Fuhrmann and Willems [6] can be extended to this case to yield the following. Theorem 2.1: Let P be the pol~lomial system matrix associated with representation (1.3) of a transfer function G,

. TII"l proof follows Fuhrmann and Willems [6] qui te closely and is omitted. The realization referred to in the theorem is the realization associated with the representation (1.3) as outlined in Fuhrmann [7]. Corollary 8.6 in [5] is a direct consequence of Theorem 2.1. pr

l denotes the projection from Fs+p[A] onto

s f

F [A] given by (1) ~ fl' pr

2 is similarly defined.

f

2

(i) If P

=

EaP_a is a factorization of P with

E nonsingular then V

=

prlE K_p is an

(A,B)-a a a

invariant subspace in Ker C.

Our main result is the following.

(ii) If in addition V and T are right coprime

th~n Z(P) ~ Z(G), the isomorphism being the one induced by pr₂.

We note first the following.

Lemma 2.3: Let (~) e F !I+m ((A-I)), then if and only if v e Ker G and u

=

_T-luv.

Theorem 2.2: Let G be a pxm rational matrix func-tIon having the representation (1.3) and let P be the .:ssociated (s+p)x(s+m) polyrlomial system matrix

Proof of Thcnrem 2.3. We break the proof into sever"al steps. (1. 2) (1.31 {1.l) (1.4) Z(G) P

~

c:

The definition of the zero module given by Wyman and 5ain applies just as well to any rational, in par-tieu Iar polynomi.<>l, matrix functi on without any assump-tion of pronerness. Thus for any pxm rational matrix function G, the base field F being arbitrary, we define the zero module by

G-l(FP[A))+Fm[A] Ke!' G+FIll[A)

Of course in analogy with the scalar case one expects the zero information to be inCluded in the numerator Ofcrl)' coprime fa::torization of (.. In fact

i f

Moddle theoretic methods have been introduced into system theory by Kalman (1) and hUH since proved to be central to the theory of linear systems. Their greatest impact has been initially in the analysis and solution of the realization Froblem and the study of feedback in later stages. In particular the state module of a

trall~fer function is detennincd, up to isomorphism, by its pole structure. Surprisingly, as rightly roj~ted

Ollt by Wy111an and Sain [2]. no attel:lpt has been ma(le in analyzing the :ero structure of rational matrix func-tions from a module theoretic point of view. Possibly the closest in spirit, though highly indirect, is the geometric control theory analysis using the quotient of tr.e n:;:xim~: (J'"ll)-invariant subspace in Ker C by the maximal reachability .sUbspace in Ker C using a prop-erly defined state feedback map, where (A,B,C) is any canonical reaJizat iou of the transfer function.

I f instead of representations of the form (1.2) of a transfer function G we consider Rosenbrock [01] type

rcpre~cntationsof tbe [onn

arc rcspecti~elylEft and right coprime factorizations of G then it is easily checked that Z(G)

=

Z~'J) ~ :~~.~). ~71 t;~j.5 ~,O~·~;l~~,"t..~(jI' t~IC \\U1:k. of ?Ugil and

Shelton [3] is also relevant.

with V,T,U and \'I polynomial matrices of which T is assumed nonsingular,.then the polar information, assuming Jeft coprimeness of T ami lJ and right coprimrness of T and V, is determined by T, and a natural question is the representation of the zero module in terms of the data T,U,V and W. rollowing Rosenbrock we define the polynomial system matrix asso-ciated ...,ith 0.3) to be the polynomial matrix

(a) By Lemma Z.3 Ker G

=

pr~Ker P and so ler G+rID[AJ

=

prZCKer p)+rrn[Ai.

(b) Wo show G-IcrP[AJ)C:' przCp-IcrS+P[A)). Indeed if

h₂ E G-I(FP[AJ) then Gh₂

=

Pz e FP[AJ. Define hI by

1 hI

°

hI

=

-T- Uh_Z then P(~ )

= ( )

e FS+P[AJ which

lZ P2

proves the inclusion. In particUlar we obtain the inclusion G-l(FPPJJ+rm[AJC przp-l(rS+p[AJ)+rmp.). (c) Using left coprimeness of T and U we will show

(z.Z) By the left coprimeness of T and U we have

h

tion implies l'(h1) e FS+P[AJ. Also there exist k

2

and a polynomial g e Fm[:\] such that

P(~2+g)

(~)

and 50 by Lemma 2.3 G(h

2+g)

=

O. We choose g2

=

g -1

and will show that gl = hl+T U(h

2+g) is a poly-nomial. By right coprimeness there exists Sand R such that ST-RV = from which T-IU = SU-RVT-IU

hI

SU+RW-RG follows. Since P(h) ~ FS+P[AJ it (ollew!

hI Z

that (S R)P(h) h

l+(SU+RII') h2 is a polynomial.

Fur-2 -1

thermore &1

=

hl+T U(h2+ g) = hl+CSU+RW-RG) (h₂+g) h₁+(SU+RII') h

2+(SU+RW) g is a polynomial using the fact_{h +g} that h

2+g e Ker G. This implies Ch1 1) e Ker P and 2+g2

proves (ii).

and since the inverse inclusion holds always the equali-ty follows which proves Ci).

[2J B. F. Wyman and M. K. Sain, "The zero module and essential inverse systems," to appear

[3J A. C. Pugh and A. K. Shelton, "On a ne',> definition of strict system equivalence," Int. J. Control, vol. 27, pp. 657-672: 1978.

[4J H. H. Rosenbrock, State Space and ~lultivariable

Theory, New York~ J. Wiley

&

Sons, 1970. [5] E. Emre and ~l.L.J. Havtus. "A polynomial

charac-terization of (A,B)-invariant and reachability subspaces," SIAM J. Control and Optimization, vol. 18, pp. 420-436, 1980.

References

[IJ R. E. Kalman, P. L. Fal~ and M. A. Arbib, Topics in Mathematical System The0JCY.., NE.'W York: ~1cGraw­

Hill, 1969.

[6J P. A. Fuhrmann and J. C. \~illems, itA study of (A,B)-invariant subspaces via polynomial models" Int. J. Control, vol. 31, pp. 467-494, 1930.

[7J P. A. Fuhrmann, "On stnct system equivalence and similarity," Int. J. Contra), vol. 25. Pl'. 5-10

1977. a k

(2.4)

Conversely if

there exist hI and poIyno-hl+&l

P( h +g ) =

(~)

or

2 2

"2 ~ prZlker ?)+fm[AJ and so chat is

such that

If _hZE pr2r-l(Fs+P[~]) there exists hI such that

h y

Pl 1)

= (

1) E rS+P[A). By (2.3) there exist polynomial

11 2 yz .

vectors 1'1₁ and 1'1

2 for which T1'11+U1'12

=

Yl' We

obtain the equations Thl+Th

Z

=

Tn +Un_11 2 and

-Vh_{l +l\'h 2} = Y

_r

Since hI = 1'1

1- T U(hz-112) we obtain by substitution that h

Z- 2 € G-I(rP[AJ) or h₂ € G-I(FPp])+Fm[A]. Inclusion (2.Z) implies

pr

2(p-l (Fs+P [AJ) +Fm(A]

C

G- I(rP[AJ) +rm(AJ

mials fl

2+g2 € pr2f.:er.F we have

(d) If _h2 e pr₂CKer P)+rm[:\] then there exist and polynomial g such that P(hk ) (0) or

2+g

°

E Ker F+Fs+ID[:\] whh;h implies

(e) In an analogous way one proves Pr2(p-l(rs+P[:\J))+Fm[:\J

pr

2(p-I(Fs+P[AJ)+F s +m[AJ) (2.6) (f) Equalities (~ and (2.6) and part (i) imply that the induced mal' pr₂: Z(P) -+ Z(G) is surjective.

(g) As a final step we sho,,' that if V and Tare right coprime then the map pr

2 is also inje.:.tive and so an isomorphism. To this elld assume

hI -1 Sf!, .s+m hI

(h ) t: P (r [A])+l' [AJ and l'r

2(h) e pr2(Kn p)

2 h 2

+Fm[:\]. We will show (hI) € Ker P+F s +m[:\]. Our assum-2