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2 Systems & Control Letters 60 (0) 9 7 Contents lists available at ScienceDirect Systems & Control Letters journal homepage: Equivalence of rational representations of behaviors Sasanka. Gottimukkala, Shaik Fiaz, Harry L. Trentelman Johann Bernoulli Institute for Mathematics an Computer Science, University of Groningen, P. O Box 800, 9700 A Groningen, The Netherlans article info abstract Article history: Receive 6 June 00 Receive in revise form October 00 Accepte November 00 Available online 4 December 00 Keywors: Behaviors Rational kernel an image representations Equivalence Rational annihilators R[ξ]-moules This article eals with the equivalence of representations of behaviors of linear ifferential systems. In general, the behavior of a given linear ifferential system has many ifferent representations. In this paper we restrict ourselves to kernel an image representations. Two kernel representations are calle equivalent if they represent one an the same behavior. For kernel representations efine by polynomial matrices, necessary an sufficient conitions for equivalence are well known. In this paper, we eal with the equivalence of rational representations, i. e. kernel an image representations that are efine in terms of rational matrices. As the first main result of this paper, we will erive a new conition for the equivalence of rational kernel representations of possibly noncontrollable behaviors. Seconly we will erive conitions for the equivalence of rational representations of a given behavior in terms of the polynomial moules generate by the rows of the rational matrices. We will also establish conitions for the equivalence of rational image representations. Finally, we will erive conitions uner which a given rational kernel representation is equivalent to a given rational image representation. 00 Elsevier B.. All rights reserve.. Introuction In this article, we eal with the issue of equivalence of representations of a given behavior with the emphasis on rational representations. In the behavioral approach, a mathematical moel of a phenomenon is viewe as a restricte subset of all possible outcomes. More precisely, a mathematical moel is efine as a pair (U, B), with U the universum, with outcomes as its elements, an B the behavior. A ynamical system is viewe as a mathematical moel in which the objects of interest are functions of time: the universum U is a function space. The behavior B of the ynamical system is the set of all time trajectories in U that are compatible with the laws of the system. More precisely, a ynamical system Σ is efine as a triple Σ (T, W, B), with T a subset of R, calle the time axis, W a set calle the signal space, an B a subset of W T (the collection of all maps from T to W) calle the behavior (see []). In the context of linear, finite-imensional, time-invariant systems this leas to the concept of linear ifferential system. A linear ifferential system is efine to be a system whose behavior is equal to the set of solutions of a finite number of higher orer, linear, constant coefficient ifferential equations. This set of ifferential equations is then calle a representation of the behavior, often calle a kernel representation. One an the same behavior amits many ifferent kernel representations. In aition to kernel representations, Corresponing author. Tel.: ; fax: aresses: s.v.gottimukkala@rug.nl (S.. Gottimukkala), s.fiaz@math.rug.nl (S. Fiaz), h.l.trentelman@math.rug.nl (H.L. Trentelman). controllable linear ifferential systems can be represente in many ways as the image of a ifferential operator. Traitionally, kernel an image representations of linear ifferential systems involve polynomial matrices. Recently, in [], the concept of rational representation was efine an elaborate, extening the class of representations to kernel, latent variable, an image representations involving rational matrices (see Sections 3, 5 an 6 of [], respectively). The motivation for this comes from the fact that in systems an control, representations of ynamical systems often involve (rational) transfer matrices. In orer to be able to fit such representations into the behavioral framework in a natural way, the notion of rational representations of behaviors neee to be formalize. Relate material on rational representations of behaviors can be foun in [3 5] an, in an input output framework, in [6 8]. As note above, a given linear ifferential system amits many ifferent representations. Two representations are calle equivalent if they represent one an the same behavior. The issue of equivalence of representations of behaviors has been stuie before, in an input output framework in [9 4], an in a behavioral framework in [,5 7]. In the present paper, we will stuy the equivalence of kernel representations an image representation in terms of rational matrices. In particular, we consier the question how the rational matrices appearing in equivalent rational kernel representations an rational image representations are relate. The outline of this article is as follows. In the remainer of this section we will introuce the notation, an review some basic material on polynomial an rational matrices. In Section we will review linear ifferential systems an their polynomial an rational kernel an image representations. In Section 3 we formally /$ see front matter 00 Elsevier B.. All rights reserve. oi:0.06/j.sysconle

3 0 S.. Gottimukkala et al. / Systems & Control Letters 60 (0) 9 7 state the main problems aresse in this paper. In Section 4 we review the problem of equivalence of polynomial kernel representations. We establish new results here, an obtain, for two given polynomial kernel representations, separate conitions uner which their controllable parts are equal, an their sets of autonomous parts are equal. Combining these conitions, we reobtain the well known classical result on the equivalence of polynomial kernel representations. In Section 5 we will apply these results to obtain up to now unknown conitions uner which rational representations of possibly uncontrollable behaviors are equivalent. In Section 6 we eal with the moule characterization of equivalence of rational kernel representations of a given behavior. In Section 7 we consier the equivalence of image representations. Finally in Section 8 we eal with the question of uner what conitions kernel representations are equivalent to image representations. As announce, first a few wors about the notation an nomenclature use. We use the stanar symbols for the fiels of real an complex numbers R an C. We use R n, R n m, etc. for the real linear spaces of vectors an matrices with components in R. C (R, R w ) enotes the set of infinitely often ifferentiable functions from R to R w. R(ξ) will enote the fiel of real rational functions in the ineterminate ξ. R[ξ] will enote the ring of polynomials in the ineterminate ξ with real coefficients. We will use R(ξ) n, R(ξ) n m, R[ξ] n, R[ξ] n m, etc. for the spaces of vectors an matrices with components in R(ξ), an R[ξ] respectively. If one, or both, imensions are unspecifie, we will use the notation R(ξ) m, R(ξ) n, R[ξ] or R(ξ), etc. Elements of R(ξ) n m are calle real rational matrices, elements of R[ξ] n m are calle real polynomial matrices. A square non-singular polynomial matrix U is calle unimoular if the eterminant of U is a non-zero constant. To conclue this section we state the following well known facts that are use ubiquitously in the analysis in the rest of this paper (see Theorem an Section 6.5. from [8]). Proposition.. Let R R[ξ] p q be a full row rank polynomial matrix. Then there exist unimoular polynomial matrices U an such that R U D 0, where D iag(z, z,...,z p ), z, z,...,z p are monic polynomials obeying the ivision property z i z i+,i,,...,p. The polynomial matrix D 0 is calle the Smith form of R. Proposition.. Let G R(ξ) p q be a full row rank rational matrix. Then there exist unimoular polynomial matrices U an such that G UΠ D 0, where D iag(z, z,...,z p ) an Π iag(π,π,...,π p ). Here, z, z,...,z p are monic polynomials obeying the ivision property z i z i+, i,,...,p an π,π,...,π p are monic polynomials obeying the ivision property π i+ π i, i,,...,p. Also z i an π i are coprime for i,,...,p. The rational matrix Π D 0 is calle the Smith McMillan form of G.. Linear ifferential systems In this section we will review the basic material on linear ifferential systems an their polynomial an rational representations. In the behavioral approach to linear systems, a ynamical system is given by a triple Σ (R, R w, B), where R is the time axis, R w is the signal space, an the behavior B is a linear subspace of C (R, R w ) consisting of all solutions of a set of higher orer, linear, constant coefficient ifferential equations. For any such system Σ (R, R w, B) there exists a real polynomial matrix R with w columns, i.e. R R[ξ] w, such that B w C (R, R w ) R t w 0. () Such a system is calle a linear ifferential system. The set of all linear ifferential systems with w variables is enote by L w. The representation () of the behavior B is calle a polynomial kernel representation of B, an often we write B t. If R has p rows, then the polynomial kernel representation is sai to be minimal if every polynomial kernel representation of B has at least p rows. A given polynomial kernel representation, B t, is minimal if an only if the polynomial matrix R has full row rank (see [], Theorem 3.6.4). The number of rows in any minimal polynomial kernel representation of B, enote by p(b), is calle the output carinality of B. This number correspons to the number of outputs in any input/output representation of B. For a etaile exposition of polynomial representations of behaviors, we refer to []. Recently, in [], a meaning was given to the equation R t w 0, where R(ξ) is a given real rational matrix. In orer to o this, we nee the concept of left coprime factorization over R[ξ]. Definition.. Let G be a real rational matrix. The pair of real polynomial matrices (P, Q ) is calle a left coprime factorization of G over R[ξ] if. et(p) 0,. G P Q, 3. the matrix P(λ) Q (λ) has full row rank for all λ C. A meaning to the equation G w 0, () t with R(ξ) a real rational matrix is then given as follows: Let (P, Q ) be a left coprime factorization of R over R[ξ]. Then we efine: Definition.. Let w C (R, R w ). Then we efine wto be a solution of () if it satisfies the ifferential equation Q t w 0. This space of solutions is inepenent of the particular left coprime factorization. Inee, if R P Q is a secon left coprime factorization then by [8], Theorem 6.5-4, there exists a unimoular U such that P UP an Q UQ. Hence from Theorem 3.6. in [], t t. Thus, () represents the uniquely etermine linear ifferential system Σ R, R w, t L w. Since the behavior B of the system Σ is the central item, often we will speak about the system B L w (instea of Σ L w ). If a behavior B is represente by G t w 0 or : B t, with G(ξ) a real rational matrix, then we call this a rational kernel representation of B. If G has p rows, then the rational kernel representation is calle minimal if every rational kernel representation of B has at least p rows. It can be shown that a given rational kernel representation B t is minimal if an only if the rational matrix G has full row rank. As in the polynomial case, every B L w amits a minimal rational kernel representation. It follows immeiately from Definition. that the number of rows in any minimal rational kernel representation of B is equal to the number of rows in any minimal polynomial kernel representation of B, an therefore equal to p(b), the output carinality of B. In general, if B t is a rational kernel representation, then p(b) rank(g). This follows immeiately from the corresponing result for polynomial kernel representations (see []). Before proceeing, we recall the concepts of autonomous behavior an controllable behavior. We state the following efinitions from []: Definition.3. A behavior B L w is calle autonomous if for all w,w B,w (t) w (t) for t 0 implies w (t) w (t) for all t.

4 S.. Gottimukkala et al. / Systems & Control Letters 60 (0) 9 7 Definition.4. Let B L w. It is calle controllable if for any two trajectories w,w B, there exists a t 0 an a trajectory w B with the property that w(t) w (t) for t 0, an w(t) w (t t ) for t t. We enote the set of all autonomous linear ifferential systems with w variables by L w aut an the set of all controllable linear ifferential systems with w variables by L w contr. It is well known that a behavior B L w is controllable if an only if there exists a positive integer l an a real polynomial matrix M R[ξ] w l such that B w C (R, R w ) C (R, R l ) s.t. w M. (3) t The representation (3) is calle a polynomial image representation of B because B is written as the image of the ifferential operator M t. In this case we will write B im M t. It can be shown that the polynomial matrix M can be chosen of full column rank. Also, M has full column rank if an only if the number of columns is equivalent to m(b), the input carinality of B. This number is equal to w p(b), an equals the number of inputs in any input output representation of B. Even more, M can be chosen to be right prime over R[ξ], equivalently, M(λ) has full column rank for all λ C. In that case, in (3) the latent variable is uniquely etermine by the manifest variable w, an the image representation is calle observable. In [], also the concept of rational image representation was introuce. We will give a brief review here. Let H(ξ) be a real rational matrix. We will first give a meaning to the equation w H. (4) t Of course (4) shoul be interprete as w I H 0, t in the context of (). If H D N is a left coprime factorization over R[ξ] then D (D N) is a left coprime factorization of (I H) an therefore (w, ) satisfies (4) if an only if D t w N t. For a given B L w, the representation B w C (R, R w ) C (R, R ) s.t. w H,(5) t with H R(ξ) w, is calle a rational image representation. In that case, we write B im H t. It was shown in [] that B L w amits a rational image representation if an only if it is controllable. Like for polynomial image representations, the rational matrix H can then be chosen of full column rank, an it has full column rank if an only if the number of columns is equal to the input carinality m(b). 3. Problem formulation In this section, we shall state the main problems aresse in this paper. Problem. Let B, B L w. Let G, G R(ξ) w. Let B t an B t be minimal rational kernel representations. Fin necessary an sufficient conitions on G an G so that B B. Problem. Let B, B L w contr. Let H, H R(ξ) w have full column rank. Let B im H t an B im H t be rational image representations. Fin necessary an sufficient conitions on H an H so that B B. Problem 3. Let B, B L w contr. Let G R(ξ) w have full row rank an H R(ξ) w have full column rank. Let B t an B im H t be a rational kernel an image representation respectively. Fin necessary an sufficient conitions on G an H so that B B. 4. Equivalence of polynomial kernel representations In this section, we iscuss the equivalence of polynomial kernel representations from a slightly ifferent perspective compare to that iscusse in [], an arrive at conitions which we shall use in aressing the issue of equivalence of rational kernel representations. Given a behavior B L w, it can be ecompose into the irect sum of its controllable part B contr, an an autonomous part B aut, i.e. B B aut B contr. This is ealt with in etail in []. It is prove in [] that for a given behavior, the controllable part is unique. It is also shown in [] that for a given behavior, an autonomous part is not unique. Let A(B) {P L w aut P B contr B} (6) enote the set of all autonomous irect summans of B contr in B. The following lemma expresses the equality of behaviors in terms of equality of the controllable parts an equality of the sets of autonomous parts. Lemma 4.. Let B, B L w. Then B B if an only if. B,contr B,contr an. A(B ) A(B ). Proof (Only If). : This part of the proof is obvious. (If): Let P A(B ). Then we have B P + B,contr P + B,contr. Since also P A(B ) the latter equals B. Kernel representations of the behaviors in A(B) are iscusse in []. For the sake of completeness, we shall state the following lemma, which escribes kernel representations of the controllable as well as the autonomous parts of a given behavior. Lemma 4.. Let B L w. Let B t be a minimal polynomial kernel representation, an let U an be unimoular polynomial matrices such that R U D 0, where D 0 is the Smith form of R. Then we have:. B contr ker I 0 t, an. P A(B) if an only if P ker D t 0 0 I W t, t for some unimoular polynomial matrix W, satisfying D 0 D 0 W. This characterization of the autonomous parts of a given behavior is also ealt with in [], in Excercise 5.6. It can be verifie that W mentione above takes the form W I 0, where W W 3 W 3 is 4 any polynomial matrix of appropriate imensions, an W 4 is any unimoular polynomial matrix. Remark 4.3. Let B L w. Let B t be a minimal polynomial kernel representation, an let U an be unimoular polynomial matrices such that R U D 0, where D 0 is the Smith form of R. From the above lemma an the structure of W, a unimoular polynomial matrix, it is clear that P A(B) if an only if it amits a minimal polynomial kernel representation D 0 ker t t, (7) F S t t

5 S.. Gottimukkala et al. / Systems & Control Letters 60 (0) 9 7 where F is an arbitrary polynomial matrix of appropriate imensions, an S is an arbitrary unimoular polynomial matrix. Equivalence of polynomial kernel representations has been ealt with before in []. We recall the following well known result given as Theorem 3.6. in []: Proposition 4.4. Let B, B L w. Let R, R R[ξ] w be such that R t w 0 an R t w 0 are minimal polynomial kernel representations of B an B respectively. Then B B if an only if there exists a unimoular polynomial matrix U such that R UR. In orer to procee, we have the following lemma: Lemma 4.5. Let B L w. Let B t be a minimal polynomial kernel representation. Let R FR be any factorization of R such that F R[ξ] p(b) p(b) is square an non-singular, an R R[ξ] p(b) w such that R (λ) has full row rank for all λ C. Then B contr t. Proof. Let U an be unimoular polynomial matrices such that R U D 0, where D 0 is the Smith form of R. It is clear that B contr ker I 0 t (from Lemma 4.). Now let R FR be any factorization of R such that R (λ) has full row rank for all λ C an F is square an non-singular. We have following ientities: UD 0 FR F R R, FR FR, where R R : R. This implies FR 0, an hence R 0. Since R (λ) (λ) has full row rank for all λ C an R is square, we must have that R is a unimoular polynomial matrix. Therefore R U I 0 an t ker U t I 0 t Bcontr (from Proposition 4.4). The following theorem is the main result of this section. It expresses equality of the controllable parts of two behaviors in terms of their polynomial kernel representations, an it gives aitional conitions uner which the sets of autonomous parts are also equal. Theorem 4.6. Let B, B L w. Let B t an B t be minimal polynomial kernel representations. Then (a) B,contr B,contr if an only if there exist square nonsingular polynomial matrices M an N such that MR NR. (8) (b) Assume that B,contr B,contr. Then for any pair of square nonsingular polynomial matrices M, N such that (8) hols, we have A(B ) A(B ) if an only if M N is a unimoular polynomial matrix. Proof. Let U i an i be unimoular polynomial matrices such that R i U i Di 0 i, where D i 0 is the Smith form of R i, for i,. From Lemma 4., we have B,contr ker I 0 t, an B,contr ker I 0 t. (a) (Only if): Since B,contr B,contr, by Proposition 4.4, there exists a unimoular polynomial matrix U such that I 0 U I 0 hols. Consequently, D I 0 D UD D I 0. It is easy to seethat there exist square nonsingular polynomial matrices M an Ñ, such that M Ñ D UD. Therefore we have MD I 0 ÑD I 0. Define M : MU an N : ÑU. Then we have MR NR. (If): Let G MR NR. Then we have G MU D I 0 NU D I 0. (9) Further, from Lemma 4.5, it is evient that t contr ker I 0 ker I 0 t t. Therefore B,contr B,contr. (b) (Only if): Assume A(B ) A(B ). As B,contr B,contr, from Lemma 4. it is clear that B B. Therefore from Proposition 4.4, we have R UR, (0) where U is a unimoular polynomial matrix. Further we have R M NR. () Since R an R are minimal kernel representations, R an R have full row rank. Therefore from (0) an () it is clear that U M N. (If): As B,contr B,contr, from Proposition 4.4, we have I 0 Ũ I 0, where Ũ is a unimoular polynomial matrix, an it can be checke that takes the Ṽ form 0, where Ṽ, Ṽ are unimoular Ṽ Ṽ polynomial matrices. Further, we have MU D 0 NU D 0. Define M MU an N NU. Then we have D 0 M N D 0. Now, consier any P A(B ), then from Remark 4.3, we know that there exists a square nonsingular polynomial matrix F, an a unimoular polynomial matrix S, such that P ker D 0 t t. We have F t S t D 0 M N D 0 F S F S M N D 0 F S Ṽ M N 0 D 0 0 I. F S Ṽ M It is easy to see that N 0 an S 0 I Ṽ are unimoular polynomial matrices. From Proposition 4.4, we have P ker D 0 t t. From Remark 4.3 F t S t Ṽ t it is clear that P A(B ). The reverse inclusion is obvious. Eviently, from the above theorem we have the following corollary: Corollary 4.7. Let B, B L w. Let B t an B t be minimal polynomial kernel representations. Then B B if an only if there exist square an nonsingular polynomial matrices M, N such that MR NR an M N is a unimoular polynomial matrix. Obviously, Corollary 4.7 is a restatement of Proposition 4.4. However, in combination with Theorem 4.6 it shows the origin of the unimoular matrix U. The corollary has been erive in two stages. Firstly, it has been shown that equality of the controllable parts of a given behavior is equivalent to the existence of square

6 S.. Gottimukkala et al. / Systems & Control Letters 60 (0) an non-singular matrices M an N. Seconly, unimoularity of M N has been shown to be equivalent to equality of the sets of autonomous parts of the behavior. 5. Equivalence of rational kernel representations In this section we aress the question of equivalence of minimal rational kernel representations. We will first recall the concepts of polynomial an rational annihilators of a given behavior from [], Section 7. Definition 5.. Let B L w.. n R[ξ] w is calle a polynomial annihilator of B if n t w 0 for all w B.. n R(ξ) w is calle a rational annihilator of B if n t w 0 for all w B. We enote the set of polynomial annihilators of B L w by B R[ξ] an the set of rational annihilators of B by B R(ξ). It is a well known result that for B L w, B R[ξ] is a finitely generate submoule of the R[ξ]-moule R[ξ] w. Moreover, if B t is a polynomial kernel representation, then this submoule is generate by the rows of R. In the context of rational representations one nees to impose controllability: Theorem 5.. Let B L w. Then B R(ξ) is a subspace of the R(ξ)-linear vector space R(ξ) w if an only if B is controllable. If G t w 0 is a minimal rational kernel representation of B, then the rows of G form a basis of (B contr ) R(ξ), the rational annihilators of the controllable part of B. Proof. The first statement is the content of Statement of Theorem in []. Let G P Q be a left coprime factorization over R[ξ] of G. Then B t is a minimal polynomial kernel representation. Let Q UD I 0 be the Smith form of Q. Then from Lemma 4. we have B contr ker I 0 t. Let n (B contr ) R(ξ). Let n u v be a left coprime factorization of n over R[ξ]. Then by efinition we have v t w 0 for all w B contr. Thus, by Definition 5., v (B contr ) R[ξ]. Consequently, there exists a l R[ξ] such that v l I 0. Hence n u v u l I 0 (u ld U P)(P UD I 0 ) (u ld U P)(P Q ) (u ld U P)G. Define m : u ld U P. Then we have n mg. Thus, n is a R(ξ)-linear combination of the rows of G. Since n was arbitrary, the rows of G span the subspace (B contr ) R(ξ) of the R(ξ)-linear vector space R(ξ) w. Finally, as B t is a minimal rational kernel representation, the rows of G are linearly inepenent over R(ξ). We conclue then that these rows form a basis of (B contr ) R(ξ). Remark 5.3. It follows immeiately from the previous theorem for any behavior B L w contr we have im(b R(ξ)) p(b), i.e. the imension of the linear space of rational annihilators of a controllable behavior is equal to the output carinality of B. The following theorem is an immeiate consequence of Theorem 5.. It gives necessary an sufficient conitions for the controllable parts of two behaviors to be equal in terms of the rational kernel representations. Theorem 5.4. Let B, B L w. Let B t an B t be minimal rational kernel representations. Then the following statements are equivalent: (a) B,contr B,contr. (b) There exists a nonsingular rational matrix W such that G WG. (c) There exist nonsingular polynomial matrices M an N such that MG NG. Proof. The equivalence of (b) an (c) is obvious. We first prove the implication (a) (b). As B,contr B,contr we have (B,contr ) R(ξ) (B,contr ) R(ξ) : T. From Theorem 5., the rows of G an G both form a basis for the subspace T of R(ξ) w. Then, from basic linear algebra, there exists a square, nonsingular rational matrix W such that G WG. Conversely, assume G WG. Let G P Q an G P Q be left coprime factorizations over R[ξ] of G an G. Let W M N be a left coprime factorization over R[ξ] of W. Then both M an N are nonsingular. By efinition we have B t an B t. Then, G WG P Q M NP Q Q P M NP Q. Now factorize P M NP M Ñ. Then we have MQ ÑQ. From Theorem 4.6,(a) follows. Eviently, the above theorem only gives a necessary conition on G an G for the associate behaviors to be equal. Again however, we woul like to obtain conitions that are necessary an sufficient. As shown in Corollary 4.7, in case of polynomial kernel representations, Statement 3 of Theorem 5.4 together with unimoularity of M N serves the purpose. Hence, a first guess is to check whether this also hols true for rational representations. However, the following simple counterexample shows that this is not the case. Example 5.5. G (ξ) an G (ξ). These are equivalent ξ representations since they both represent the {0}-behavior. For all M, N such that MG NG, we have M N, which is not even ξ a polynomial. In orer to procee we nee following efinition: Definition 5.6. A greatest common left ivisor (gcl) of two polynomial matrices P, Q R[ξ] m is any square polynomial matrix D such that P DP an Q DQ, an such that for all square polynomial matrices D satisfying P D P an Q D Q there exists a polynomial matrix F such that D D F. For given polynomial matrices P an Q, we enote by gcl(p, Q ) any greatest common left ivisor (gcl) of P an Q. If P Q has full row rank, then any gcl must be a non-singular polynomial matrix. In that case any two gcl s are relate by post-multiplication with a unimoular polynomial matrix. Now, the following theorem is the first main result of this paper. The theorem states that the aitional conitions on M an N so that the sets of autonomous parts of t an ker G t are also equal involve the greatest common left ivisor matrices gcl(m, MG ) an gcl(n, NG ). More precisely: Theorem 5.7. Let B, B L w. Let B t an B t be minimal rational kernel representations. Assume B,contr B,contr. Then we have A(B ) A(B ) if an only if there exist square nonsingular polynomial matrices M an N such that MG NG, MG NG is a polynomial matrix, an gcl(m, MG ) gcl(n, NG ) is a unimoular polynomial matrix.

7 4 S.. Gottimukkala et al. / Systems & Control Letters 60 (0) 9 7 Proof (Only if). Let U i an i be unimoular polynomial matrices such that G i U i Π Di 0 i i, where Π Di 0 i is the Smith McMillan form of G i, for i,. Assume A(B ) A(B ). Then from Remark 4.3, P A(B ) amits a polynomial kernel representation D 0 ker t t, F S t t similarly it also amits apolynomial kernel representation D 0 ker t t, F S t t where F, F are arbitrary polynomial matrices of appropriate imensions an S, S are unimoular polynomial matrices. From Proposition 4.4, there exists a U, a unimoular polynomial matrix, such that D 0 D 0 S U S. F F Using the assumption that B,contr B,contr, it can be verifie that U must be of the form U U 0, where U U U an U are unimoular polynomial matrices. Therefore we have D 0 U 0 D 0 F S U U F S, which implies Π U U Π D 0 F S U 0 Π U U Π D 0. U F S U Define M : Π U an N : U Π U. Then we have M 0 G 0 I F S N 0 G U Π U. U F S It is evient from the above equation that MG an NG are polynomial matrices an that MG NG. Define L : MG NG. Then we have R : gcl(m, L) I, an similarly R : gcl(n, L) U. Hence, it is evient that R R U is a unimoular polynomial matrix. (If): Assume that L : MG NG is a polynomial matrix. Let gcl(m, L) : R an gcl(n, L) : R. Let G P Q an G P Q be left coprime factorizations of G an G over R[ξ]. Obviously we have P Q M L an P Q N L. Hence, from [8], Lemma 6.5-5, there exist square nonsingular polynomial matrices R, R such that R P Q M L an R P Q N L. Further, using the left primeness of Pi Q i, it can be verifie that R an R are gcl s of M L an N L respectively. Also, since M an N are square nonsingular polynomial matrices, M L an N L have full row rank. Consequently, we have that R an R are nonsingular. Hence, there exist unimoular polynomial matrices U an U such that R R U an R R U. Define M : R U, Ñ : R U. Then we have MQ ÑQ an M Ñ U, which is a unimoular polynomial matrix. Therefore, from Theorem 4.6, we have A(B ) A(B ). The following corollary is the secon main result of this paper. It gives necessary an sufficient conitions on the rational matrices G an G for t an ker G t to be equal. In fact, by combining Theorems 5.4 an 5.7 we immeiately obtain: Corollary 5.8. Let B, B L w. Let B t an B t be minimal rational kernel representations. Then B B if an only if there exist square an nonsingular polynomial matrices M, N such that (a) MG NG, (b) MG NG is a polynomial matrix an (c) gcl(m, MG ) gcl(n, NG ) is a unimoular polynomial matrix. Corollary 5.8 is illustrate below in the following examples. Example 5.9. G (ξ), G (ξ) represent the same behavior: ξ. MG NG with N(ξ) ξ,m(ξ) nonsingular polynomial,. MG NG is polynomial an gcl(n, NG ) gc(ξ, ), gcl(m, MG ) gc(, ). Example 5.0. G (ξ) (ξ ξ), G (ξ) the same behavior: ξ o not represent ξ. their controllable parts are the same: MG NG with N(ξ) ξ, M(ξ) nonsingular polynomial,. for any M, N such that MG NG we must have N(ξ) ξ M(ξ). Hence gcl(m, MG ) gc(m,ξm,ξm) M, while gcl(n, NG ) gc(ξ M,ξM,ξM) ξ M. Remark 5.. We note that, in the case that G an G are polynomial matrices, Corollary 5.8 immeiately yiels Corollary 4.7. Inee, in that case gcl(m, MG ) M an gcl(n, NG ) N so conition (b) becomes: M N is a unimoular polynomial matrix. Accoring to Corollary 5.8, in orer to check the equivalence of rational representations, we nee to check for the existence of square an nonsingular polynomial matrices M an N that satisfy (a), (b) an (c) in Corollary 5.8. The algorithm below achieves this objective. Algorithm. Let G, G R(ξ) k w an let B t an B t be minimal kernel representations. Then,. Solve G WG for W R(ξ) k k. If there exists no solution, eclare B B, else continue further.. Fin a left coprime factorization of W over R[ξ]. Let it be W M N, where M, N are square an nonsingular polynomial matrices. 3. Fin a left coprime factorization of MG over R[ξ]. Let it be MG P Q where P, Q R[ξ]. Then PMG PNG Q is a polynomial matrix. 4. Fin L : gcl(pm, Q ) gcl(pn, Q ). If L is a unimoular polynomial matrix, eclare B B, else eclare B B. Before elaborating on the above algorithm, we state an alternative algorithm to check the equivalence of rational representations of a given behavior. Algorithm. Let G, G R(ξ) k w an let B t an B t be minimal kernel representations. Then,. Fin a left coprime factorization of G over R[ξ]. Let it be G P Q, where P, Q R[ξ].

8 S.. Gottimukkala et al. / Systems & Control Letters 60 (0) Fin a left coprime factorization of G over R[ξ]. Let it be G P Q, where P, Q R[ξ]. 3. Solve Q UQ for U, where U is a unimoular polynomial matrix. If a solution exists, eclare B B, else eclare B B. The first algorithm has two avantages. Firstly, in case the behaviors B, B are not equal, it is alreay eclare in Step-, without actually proceeing to left coprime factorizations. Seconly, it fins in Step- whether the controllable parts of the behavior are equal for the given kernel representations. 6. A moule characterization of equivalence of rational representations In this section, we will give conitions for the equivalence of rational representations of a given behavior in terms of the polynomial moules generate by the rows of the rational matrices. In Section 5, the polynomial an rational annihilators of a given behavior B L w have been introuce an iscusse. For a given behavior B L w, with rational representation B t, we will now first establish the relation between the R[ξ]-moule generate by the rows of the rational matrix G, an the moule of polynomial annihilators of B. In case of polynomial kernel representations, the following proposition is well known. Proposition 6.. Let B L w. Let B t be a minimal polynomial kernel representation. Let B R[ξ] enote the moule of polynomial annihilators of B. Then the rows of R form a basis for B R[ξ]. In the following, for a given rational matrix G R(ξ) w, we will enote by G R[ξ] the set of all linear combinations of the rows of G using coefficients from the polynomial ring R[ξ]. Clearly this set is a R[ξ]-moule. The intersection G R[ξ] R[ξ] w consists of all linear combinations of the rows of G using coefficients from R[ξ] that are polynomial vectors. Clearly, this intersection is a R[ξ]-submoule of R[ξ] w. The following theorem states that this intersection moule is in fact equal to the moule of polynomial annihilators of B. Theorem 6.. Let B L w. Let G R(ξ) w. Let B t be a minimal rational kernel representation. Then G R[ξ] R[ξ] w B R[ξ]. Proof. Let G P Q be a left coprime factorization of G over R[ξ]. We first prove the following inclusion: B R[ξ] G R[ξ] R[ξ] w. By Definition., B t. From Proposition 6., we know that the rows of Q form a basis for B R[ξ]. Let n B R[ξ]. Then there exists a polynomial row vector l such that n lq. Hence n lq lpp Q lpg mg, where m lp is a polynomial row vector. Therefore n G R[ξ] R[ξ] w. We now prove the converse inclusion, G R[ξ] R[ξ] w B R[ξ]. Let U an be unimoular polynomial matrices such that G UΠ D 0, where Π D 0 is the Smith McMillan form of G i. Define P : ΠU an Q : D 0. Then G P Q is a left coprime factorization an B t. Let l G R[ξ] R[ξ] w. Then we have l ng for some polynomial row vector n. Define l : l. Then l nuπ D 0, where l : l ñπ D 0, where ñ : nu z ñ iag,..., z k 0. () π π k Write l l l l w an ñ ñ ñ ñ k. From Eq. () we have l z i ñ i i, for i π i,,...,k an, l i 0 for i > k. As (z i,π i ) are coprime, there exists m i R[ξ] such that ñ i m i π i, for k,,...,k. Let m m m m k. Then n ñu mπu mp so l ng mpg mq. Therefore l B R[ξ]. By combining Theorems 6. an 5.7 we finally get the following complete characterization of the equivalence of rational kernel representations: Theorem 6.3. Let B, B L w. Let B t an B t be minimal rational kernel representations. Then following statements are equivalent:. B B.. B R[ξ] B R[ξ]. 3. There exists square non-singular polynomial matrices M an N such that (a) MG NG, (b) MG NG is a polynomial matrix an (c) gcl(m, MG ) gcl(n, NG ) is a unimoular polynomial matrix. 4. G R[ξ] R[ξ] w G R[ξ] R[ξ] w. 7. Equivalence of rational image representations In this section we will aress the issue of equivalence of rational image representations. In particular, we will establish a solution to Problem as state in Section 3. We first recall the following fact on polynomial an rational image representations of behaviors (see Theorem 9 in []). Theorem 7.. Let B L w. Then the following statements are equivalent:. B is controllable,. B amits a polynomial image representation B im M t, with M R[ξ] w of full column rank. 3. B amits a polynomial image representation B im M t with M R[ξ] w right prime over R[ξ], 4. B amits a rational image representation with B im H t with H R(ξ) w of full column rank. In the sequel, the following result will be useful. The result states that right coprime factorization of a rational image representation leas to a polynomial image representation. Lemma 7.. Let B L w contr. Let H R(ξ)w be such that B im H t. Let H MP be a right coprime factorization over R[ξ]. Then B im M t. Proof. Let H D N be a left coprime factorization over R[ξ]. Then we have M ker D N im t t t. P t

9 6 S.. Gottimukkala et al. / Systems & Control Letters 60 (0) 9 7 Thuswe obtain B w C (R, R w ) s.t. D t w N t M w C (R, R w ), w s.t. t P t w C (R, R w ) s.t. w M. t In orer to procee, we will now first stuy the question uner which conitions two polynomial image representations are equivalent, i.e. represent the same behavior. Theorem Let B, B L w contr. Let M, M R[ξ] w have full column rank, an let B im M t an B im M t. Then B B if an only if there exists a square nonsingular rational matrix R such that M M R.. Let B, B L w contr. Let M, M R[ξ] w be right prime over R[ξ], an let B im M t an B im M t. Then B B if an only if there exists a unimoular polynomial matrix U such that M M U. Proof. We first prove the only if part of Statement. By right primeness, both M (λ) an M (λ) have full column rank for all λ C, so correspon to observable image representations. From B B it follows that also the orthogonal complements coincie, i.e. B B (see [9]). By observability we have B i ker M i t, where M i (ξ) : M i ( ξ) (i, ). By Proposition 4.4 there exists a unimoular polynomial matrix such that M M. This implies M M U, with U : again unimoular. Next, we prove the only if part of Statement. Both M an M have full column rank. Hence, we can factorize M i M i R i, with M i right prime over R[ξ] an R i a nonsingular polynomial matrix (i, ). By nonsingularity, R i t is surjective, an therefore im M i t im Mi t (i, ). Consequently, B B implies im M t im M t. Then, by the only if part of Statement, there exists a unimoular polynomial matrix U such that M M U. This implies M M R, with R : R UR. Finally we prove the if part of Statement. Assume that M M R with R a nonsingular rational matrix. Let R KL be a right coprime factorization of R over R[ξ]. Then we have M L M K, with K an L nonsingular polynomial matrices. Again by the surjectivity of L t an K t, we obtain B im M t im M t K t im M t L t im M t B. This also proves the if part of Statement. Next, we consier controllable behaviors represente by rational image representations. Theorem 7.4. Let B, B L w contr. Let H, H R(ξ) w have full column rank an let B im H t an B im H t. Then B B if an only if there exists a square nonsingular rational matrix R such that H H R. Proof. Let H i M i P i be a right coprime factorization over R[ξ]. Then by Lemma 7., B i im M i t (i, ). By Theorem 7.3, B B implies that there exists a nonsingular rational matrix R such that M M R. Thus H H R, with R : P RP nonsingular. Conversely, if H H R then M M P RP. Then, by Theorem 7.3, im M t im M t so B B. 8. Equivalence of rational kernel an image representations So far, we have erive necessary an sufficient conitions uner which the kernels of two rational ifferential operators represent one an the same behavior an conitions uner which the images of two rational ifferential operators represent the same behavior. In the present section, we will erive necessary an sufficient conitions for behaviors given as the kernel of a rational ifferential operator an the image of a rational ifferential operator to be equal. Theorem 8.. Let B, B L w contr. Let B t an B im H t be a minimal rational kernel representation an a rational image representation of B an B, respectively, where G R(ξ) p w has full row rank an H R[ξ] w m has full column rank. Then B B if an only if GH 0 an p + m w. Proof. Let G P Q an H ND be left an right coprime factorizations of G an H, respectively over R[ξ]. Then from Definition. an Lemma 7., we have B t an B im N t. As B, B L w contr, we recall that B R(ξ) an B R(ξ) are subspaces of the R(ξ)-linear vector space R(ξ) w (see Theorem 5.) with im(b R(ξ) ) p an im(b R(ξ) ) w m. (Only if): As B B we have p w m. Further, since t im N t, we have Q t N t 0 for all C (R, R w m ). Consequently, we have QN 0 so GH 0. (If) We have GH 0 if an only if P QND 0 if an only if QN 0. In orer to procee we first prove B B. Let w B, then there exists an C (R, R w m ) such that w N t. Then Q t w Q t N t 0 (since Q t N t 0). Therefore w B, so B B. Finally B B implies that B R(ξ) B R(ξ). By using the assumption that p w m, the imensions of these two subspaces are equal, so we must have B R(ξ) B R(ξ). Therefore we conclue that B B. 9. Conclusions In this paper we have ealt with the equivalence of representations of a given behavior with emphasis on rational representations. We have obtaine new necessary an sufficient conitions for the equivalence of polynomial kernel representations, illustrating the origin of the unimoular matrix appearing in equivalent polynomial kernel representations. As the first major contribution of this paper, we have obtaine necessary an sufficient conitions for the equivalence of rational kernel representations of controllable as well as uncontrollable behaviors. As the secon major contribution of this paper, we also have erive conitions for the equivalence of rational representations of a given behavior in terms of the polynomial moules generate by the rows of the rational matrices. Further, we have obtaine necessary an sufficient conitions for the equivalence of image representations in the context of both polynomial an rational representations. Finally, we have obtaine necessary an sufficient conitions for the equality of behaviors efine as the kernel of a rational ifferential operator an the image of a rational ifferential operator. Acknowlegements The authors woul like to thank Prof. Jan C. Willems an Prof. Kiyotsugu Takaba for their valuable suggestions on certain sections of this paper.

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