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. (MemorandumCOSOR; Vol. 8015). Technische Hogeschool Eindhoven.

Document status and date: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. FuhrmannDepartment of ~Iathematics

Ben Gurion University of the NegevBeer Sheva, Israel

M. L. J. lIautusDepartment of HathematicsTechnological UniversityEindhoven, The Netherlands

1. Introduction

This situation has been remedied to a large extentby the recent paper of Wyman and Sain (2) which is themain motivation for this short note.

(2.1)

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

. pr2

(p- 1 (F s+P(A])+Fm[A))Z(G) "' --'-'-------­

pr2(Ker P)+Fm[A)

(ii) V KT is an (A,B)-invariant subspace in

Ker C if and only if V = prlPoKE where P = P Eo aais a factorization with Ea nonsingular.

2. The Polynomial System Matrix as a ~Umerator

In the next section we state the main resultsco~cerning the relation between the zero structure ofG 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 numer­ator in a coprime factorization of G. Especiallysuggestive is a comparison of corollaries 3.12 and 8.6in [mre and Hautus [5] which give a characterization ofthe maximal (A,B)-invllriant subspace in Ker C in

terms of the representations G = T- 1U and (1.3). Infact the analysis given in Fuhrmann and Willems [6] canbe extended to this case to yield the following.

Theorem 2.1: Let P be the pol~lomial system matrixassociated with representation (1.3) of a transferfunction G,

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

2.1. prl denotes the projection from Fs+p[A] ontos fF [A] given by (1) ~ fl' pr2 is similarly defined.

f2

(i) If P = EaPa is a factorization of P with

E nonsingular then V = prlE Kp 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 coprimeth~n Z(P) ~ Z(G), the isomorphism being the oneinduced 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 func­tIon 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 intosever"al steps.

(1. 2)

(1.31

{1.l)

(1.4)

Z(G)

P ~ c:

The definition of the zero module given by Wymanand 5ain applies just as well to any rational, in par­tieu Iar polynomi.<>l, matrix functi on without any assump­tion of pronerness. Thus for any pxm rational matrixfunction G, the base field F being arbitrary, wedefine the zero module by

G-l(FP[A))+Fm[A]

Ke!' G+F Ill [A)

Of course in analogy with the scalar case oneexpects the zero information to be inCluded in thenumerator Ofcrl)' coprime fa::torization of (.. In factif

Moddle theoretic methods have been introduced intosystem theory by Kalman (1) and hUH since proved to becentral to the theory of linear systems. Their greatestimpact has been initially in the analysis and solutionof the realization Froblem and the study of feedback inlater stages. In particular the state module of atrall~fer function is detennincd, up to isomorphism, byits pole structure. Surprisingly, as rightly roj~ted

Ollt by Wy111an and Sain [2]. no attel:lpt has been ma(le inanalyzing the :ero structure of rational matrix func­tions from a module theoretic point of view. Possiblythe closest in spirit, though highly indirect, is thegeometric control theory analysis using the quotient oftr.e n:;:xim~: (J'"ll)-invariant subspace in Ker C by themaximal reachability .sUbspace in Ker C using a prop­erly defined state feedback map, where (A,B,C) is anycanonical reaJizat iou of the transfer function.

If instead of representations of the form (1.2) ofa transfer function G we consider Rosenbrock [01] typercpre~cntations of tbe [onn

arc rcspecti~ely lEft and right coprime factorizationsof 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 andShelton [3] is also relevant.

with V,T,U and \'I polynomial matrices of which Tis assumed nonsingular,.then the polar information,assuming Jeft coprimeness of T ami lJ and rightcoprimrness of T and V, is determined by T, and anatural question is the representation of the zeromodule in terms of the data T,U,V and W. rollowingRosenbrock we define the polynomial system matrix asso­ciated ...,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- UhZ then P(~ ) = ( ) e FS+P[AJ whichlZ 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

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

2and a polynomial g e F

m[:\] such that P(~2+g) (~)

and 50 by Lemma 2.3 G(h2+g) = O. We choose g2 = g-1and 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-IUhI

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

that (S R)P(h) hl +(SU+RII') h2 is a polynomial. Fur-2 -1

thermore &1 = hl+T U(h2+g) = hl+CSU+RW-RG) (h 2+g)

h1+(SU+RII') h2+(SU+RW) g is a polynomial using the facth +g

that h2+g e Ker G. This implies C 1 1) e Ker P andh

2+g

2proves (ii).

and since the inverse inclusion holds always the equali­ty follows which proves Ci).

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

[3J A. C. Pugh and A. K. Shelton, "On a ne',> definitionof 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 charac­terization of (A,B)-invariant and reachabilitysubspaces," SIAM J. Control and Optimization,vol. 18, pp. 420-436, 1980.

References

[IJ R. E. Kalman, P. L. Fal~ and M. A. Arbib, Topicsin 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 andsimilarity," Int. J. Contra), vol. 25. Pl'. 5-101977.

a k

(2.4)

Conversely if

there exist hI and poIyno­hl+&l

P( h +g ) = (~) or2 2

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

such that

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

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

112 yz .

vectors 1'1 1 and 1'1 2 for which T1'11+U1'1

2= Yl' We

obtain the equations Thl+ThZ = Tn +Un2

and_11

-Vh l +l\'h 2 = Yr Since hI = 1'11

- T U(hz-112) we obtain

by substitution that hZ

- 2 € G-I(rP[AJ) or

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

pr2(p-l (Fs+P [AJ) +Fm(A] C G- I (rP [AJ) +rm(AJ

mials

fl2

+g2

€ pr2

f.:er.F

we have

(d) If h2 e pr2 CKer P)+rm[:\] then there exist

and polynomial g such that P(hk ) (0) or2+g °

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

(e) In an analogous way one proves

Pr2(p-l(rs+P[:\J))+Fm[:\J

pr2(p-I(Fs+P[AJ)+Fs+m[AJ) (2.6)

(f) Equalities (~ and (2.6) and part (i) imply thatthe induced mal' pr2 : Z(P) -+ Z(G) is surjective.

(g) As a final step we sho,,' that if V and Tareright coprime then the map pr2 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'r2(h) e pr2(Kn p)

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

2