Structural Stability and Equivalence of Linear 2D Discrete Systems

Transcription

1 Proceedings of the 6th IFAC Symposium on System Structure and Control, Istanbul, Turkey, June -4, 016 FrM11. Structural Stability and Equivalence of Linear D Discrete Systems Olivier Bachelier, Ronan David, Nima Yeganefar Thomas Cluzeau University of Poitiers ; LIAS-ENSIP, Bâtiment B5, rue Pierre Brousse, TSA 41105, Poitiers cede, France {olivier.bachelier,ronan.david,nima.yeganefar}@univ-poitiers.fr. University of Limoges ; CNRS ; XLIM UMR 75, 13 avenue Albert Thomas, Limoges cede, France thomas.cluzeau@unilim.fr. Abstract: We study stability issues for linear two-dimensional D discrete systems by means of the constructive algebraic analysis approach to linear systems theory. We provide a general definition of structural stability for linear D discrete systems which coincides with the eisting definitions in the particular cases of the classical Roesser and Fornasini-Marchesini models. We then study the preservation of this structural stability by equivalence transformations. Finally, using the same framework, we consider the stabilization problem for equivalent linear systems. Keywords: System theory, algebraic approaches, multidimensional systems, discrete systems, structural stability, stabilization methods 1. INTRODUCTION The algebraic analysis or behavioral approach to linear systems theory is a unified mathematical framework to study multidimensional systems of linear functional equations appearing in control theory, engineering sciences, mathematical physics,.... See for instance [0, 5, 1, 6, 9, 3, 7, 14] and the references therein. A linear system can always be written as R η = 0, where R D q p is a q p matri with entries in a noncommutative polynomial ring D of functional operators and η is a vector of p unknown functions which belongs to a functional space. We then introduce the finitely presented left D-module M := D 1 p /D 1 q R. If F is a functional space having a left D-module structure, we consider the linear system or behavior ker F R. := {η F p R η = 0} and we have Malgrange s isomorphism ker F R. = hom D M, F see [0] which shows that system properties of ker F R. can be studied by means of module properties of M and F. Moreover, we nowadays have constructive algebraic techniques e.g., constructive homological algebra using noncommutative Gröbner basis computations and their implementations in several computer algebra systems at our disposal to study/check module properties of M. See for instance [6, 3] and the references therein. The contribution of this paper consists in using the framework of the constructive algebraic analysis approach to linear systems theory to investigate stability and stabilization issues for linear D discrete systems. Note that the algebraic analysis approach has already been used to study stability and stabilization problems for linear multidimensional systems: see for instance [] or more recently [6, 4] and the references therein. In the present work, we provide This work was supported by the ANR-13-BS MSDOS. general definitions of structural stability and stabilization for linear D discrete systems which are coherent with the eisting definitions in the particular cases of the classical Roesser and Fornasini-Marchesini models. Moreover, we take advantage of the results in [11, 1, 8] which constructively tackle the equivalence problem for linear systems in order to focus on how structural stability and stabilization properties are transmitted from a given linear system to an equivalent one. In particular, we study the impact of applying a state feedback control law to stabilize a system on an equivalent system. Finally, all our results are applied to the particular case of a generalized Fornasini-Marchesini model and its equivalent Roesser model see [8]. The paper is organized as follows. In Section, we recall some useful facts concerning Roesser and generalized Fornasini-Marchesini models and the equivalence of linear systems within the constructive algebraic analysis approach to linear systems theory. In Section 3, we introduce a general definition of structural stability for linear D discrete systems and we study its preservation via equivalence transformations. In Section 4, we consider stabilization issues in the same framework. Finally, in Section 5, we apply the previous results to the case of a generalized Fornasini-Marchesini model and its equivalent Roesser model. Notation: We note Q resp. C the field of rational resp. comple numbers. In the whole paper, R D d1 d means that R is a matri with d 1 rows and d columns whose entries are in a ring D and I n denotes the identity matri of dimension n. If C = C { } denotes the etension of C in Aleandrov s sense, then we introduce the following two subsets of C : S := {z 1, z C i = 1,, z i 1}, and D := {z 1, z C i = 1,, z i 1}. Copyright 016 IFAC 137

2 . PRELIMINARIES.1 Classical linear D discrete systems In the present paper, we shall focus on linear D discrete systems, i.e., systems of linear equations whose dependent variables are discrete functions sequences of two independent variables denoted by i and j. In particular, we shall consider two classical eplicit models of such systems which are the Roesser model [4] and the generalized Fornasini-Marchesini model [15]. Let us recall the particular form of these two models and how structural stability has been defined in both cases. Roesser models: Roesser models have been introduced in [4]. They correspond to linear D discrete systems for which the equations are written under the eplicit particular form: h i + 1, j A11 A v = 1 h i, j B1 i, j + 1 A 1 A v + ui, j, i, j B 1 where h resp. v is the horizontal resp. vertical state vector of dimension d h resp. d v, u is the input vector of dimension d u, A 11 Q d h d h, A 1 Q d h d v, A 1 Q dv d h, A Q dv dv, B 1 Q d h d u, B Q dv du. A notion of structural stability has been introduced for such particular models. See [19, 1, 3] and the references therein. Definition 1. A Roesser model 1 is said to be structurally stable if λ 1, λ S, det λ1 I dh A 11 A 1 0. A 1 λ I dv A If 1 is not structurally stable, then applying the state feedback control law ui, j = K 1 K h i, j v, 3 i, j where K 1 Q du d h and K Q du dv to 1, we obtain the new Roesser model h i + 1, j A11 + B v = 1 K 1 A 1 + B 1 K h i, j i, j + 1 A 1 + B K 1 A + B K v. i, j If the latter model is structurally stable, then we say that the state feedback control law 3 stabilizes 1. Fornasini models: These models take their origin in the work by Fornasini and Marchesini. See for instance [15] and the references therein. They were generalized by Kurek [18] and Kaczorek [16, 17]. In the present paper, we call Fornasini model, a linear D discrete system for which the equations are written under the eplicit particular form: i + 1, j + 1 = F 1 i + 1, j + F i, j F 3 i, j + G 1 ui + 1, j + G ui, j G 3 ui, j, 4 where is the state vector of dimension d, u is the input vector of dimension d u, F 1 Q d d, F Q d d, F 3 Q d d, G 1 Q d du, G Q d du, G 3 Q d du. To our knowledge, a notion of structural stability for Fornasini models 4 has been defined only in the particular case F 3 = G 3 = 0. See [15, 19, 3] and the references therein. Definition. A Fornasini model 4 with F 3 = G 3 = 0 is said to be structurally stable if λ 1, λ D, deti d λ 1 F 1 λ F 0. 5 If a Fornasini model 4 with F 3 = G 3 = 0 is not structurally stable, then applying the state feedback control law ui, j = K i, j, 6 where K Q du d to 4, we obtain the new Fornasini model i+1, j+1 = F 1 +G 1 K i+1, j+f +G K i, j+1. If the latter model is structurally stable, then we say that the state feedback control law 6 stabilizes 4. We have the following straightforward lemma: Lemma 3. With the above notation, the condition 5 is equivalent to: λ 1, λ S, detλ 1 λ I d λ 1 F 1 λ F 0. 7 Remark 4. Note that the conditions, 5, and 7 of structural stability can be effectively checked using the efficient algorithm recently developed in [5] and implemented in the computer algebra system Maple. Let D = Q σ i, σ j denote the commutative ring of partial forward shift operators with constant rational coefficients, i.e., for a bivariate sequence fi, j, we have σ i fi, j = fi + 1, j, σ j fi, j = fi, j + 1, and we further have σ i σ j = σ j σ i, where σ i σ j stands for the composition of operators σ i σ j. An operator P D can be written as P = m,l p ml σi m σj l, where p ml Q, the sum is finite, and, for a bivariate sequence fi, j, we thus have P fi, j = m,l p ml fi + m, j + l. Within the algebraic analysis approach to linear systems theory: 1 The Roesser model 1 is written as R η = 0, where R D d h+d v d h +d v+d u and η are defined by Idh σ R = i A 11 A 1 B 1, η = h A 1 I dv σ j A B v. u It is then studied by means of the factor D-module M = D 1 d h+d v+d u /D 1 d h+d v R, The Fornasini model 4 is written as R η = 0, where R D d d+du is defined by R = I d σ i σ j F 1 σ i F σ j F 3 G 1 σ i G σ j G 3, and η = T u T T. It is then studied by means of the D-module M = D 1 d+du /D 1 d R.. Equivalence in the framework of algebraic analysis Using the framework of the algebraic analysis approach to linear systems theory recalled in the introduction, equivalent linear systems correspond to isomorphic left D-modules and the equivalence problem has been constructively studied in the recent works [11, 1, 8]. Let us summarize part of the results obtained in the latter works: Lemma 5. Let R D q p, R D q p and consider the associated left D-modules M = D 1 p /D 1 q R and M = D 1 p /D 1 q R. 1 The eistence of a homomorphism f hom D M, M is equivalent to the eistence of P D p p and Q D q q satisfying the identity R P = Q R

3 Then, the homomorphism f hom D M, M is defined by fπλ = π λ P for all λ D 1 p, where π : D 1 p M and π : D 1 p M denote the canonical projections onto M and M. With the previous notation, f is an isomorphism meaning that M is isomorphic to M which is denoted by M = M iff there eist P D p p, Q D q q, Z D p q, and Z D p q satisfying R P = Q R, P P + Z R = I p, P P + Z R = I p. 9 Algorithms for computing homomorphisms of finitely presented left D-modules are given in [9] and have been implemented both in the Maple package OreMorphisms [10] based on OreModules [7] and in the Mathematica package OreAlgebraicAnalysis [13]. When D is commutative, which is the case of the ring Q σ i, σ j considered in the following, we can compute a representation of all D-homomorphisms between finitely presented D-modules. For more details, see [9]. Moreover algorithms for deciding whether a given homomorphism of finitely presented left D-modules is an isomorphism and if so, compute the matrices appearing in Lemma 5. are implemented in the Maple package OreMorphisms [10]. Corollary 6. Let R D q p, R D q p and consider the associated left D-modules M = D 1 p /D 1 q R and M = D 1 p /D 1 q R. Let f hom D M, M be an isomorphism given by a matri P D p p such that there eists Q D q q satisfying 8. Then, with the notation of Lemma 5, if F is a left D-module, we have the following isomorphism of linear systems: P. : ker F R. ker F R., η η := P η, whose inverse is given by P. : ker F R. ker F R., η η := P η. In other words, the invertible changes of variables η = P η and η = P η provide a 1-1 correspondence between F- solutions of R η = 0 and F-solutions of R η = 0, i.e., we have: R η = 0 R η = 0. In [8], the above techniques are applied to study the equivalence problem between Roesser models 1 and Fornasini models 4. In particular, it is proved that 4 is always equivalent to a Roesser model. Let us eplicitly recall this equivalence which will be useful in Section 5. Lemma 7. Let R D d d+du be the matri associated to the Fornasini model 4 see Subsection.1 and let M = D 1 d+du /D 1 d R be the associated D-module. If we define the Roesser model h i + 1, j v = F F F 1 + F 3 F G 1 + G 3 I i, j + 1 d F 1 G 1 h i, j v i, j + G 0 u i, j, I du where v i, j = v 1i, j T 10 v i, j T T, the associated matri R D d+du d+ du given by Id σ i F F F 1 + F 3 F G 1 + G 3 G R = I d I d σ j F 1 G 1 0, 0 0 I du σ j I du and M = D 1 d+ du /D 1 d+du R the associated D-module, then we have the following results: 1 The homomorphism f hom D M, M defined by the matri 0 Id 0 0 P = D d+du d+ du, 0 0 I du 0 is an isomorphism so that M = M. The identities 8 and 9 of Lemma 5 are then satisfied by the following matrices: Q = I d I d σ i F G D d d+du, I d σ j F 1 G 1 P I = d 0 0 I D d+ du d+du, du 0 I du σ j Id Q = 0 D 0 d+du d, Z = D 0 d+du d, 0 0 I d 0 Z = I du D d+ du d+du. Corollary 6 implies that, if F is a Q σ i, σ j -module, then ker F R. = ker F R., i.e., there is a 1-1 correspondence between F-solutions of 4 and F-solutions of 10. More precisely, if we denote ηi, j = i, j T ui, j T T, and η i, j := h i, j T v 1i, j T v i, j T u i, j T T, then if ηi, j is a solution of 4, then i, j + 1 F 1 i, j G 1 ui, j η i, j = P i, j ηi, j = ui, j, ui, j + 1 is solution of 10. Conversely, if η i, j is a solution of 10, then ηi, j = P η v i, j = 1 i, j v, i, j is solution of STRUCTURAL STABILITY From now on D = Q σ i, σ j denotes the commutative polynomial ring defined in Subsection.1. For a matri R with entries in D, let us denote by R the matri obtained from R by replacing the shift operator σ i resp. σ j by a new comple variable z 1 resp. z. The algebraic analysis approach to linear systems theory recalled in the introduction makes no distinction between the different variables of a linear system R η = 0, i.e., all the components of the vector η are treated in the same way. Although, as soon as one is concerned with stability and stabilization issues, the state variables and the input variables of a linear system do not play the same role. Consequently, in the sequel, we still consider linear systems written as R η = 0, where R D q p but we split the vector η of p unknown sequences into a subvector of state variables of dimension d and a subvector of input variables u of dimension d u so that 139

4 p = d + d u. Splitting the matri R accordingly, i.e., R = R 1 R, with R 1 D q d, R D q du, we have: R η = 0 R 1 R = 0 R u 1 + R u = 0. The linear system R 1 = 0 is then the autonomous linear system associated to R η = 0. Definition 8. A linear system R 1 + R u = 0, where R 1 D q d and R D q du is said to be structurally stable if λ 1, λ S, R1 λ 1, λ y = 0 y = Note that in 11, R 1 λ 1, λ C q d stands for the matri R 1 z 1, z evaluated at the point λ 1, λ S. The condition 11 is then equivalent to the fact that, for all λ 1, λ S, the matri R 1 λ 1, λ admits a left inverse, i.e., for all λ 1, λ S, there eists L λ1,λ C d q such that L λ1,λ R 1 λ 1, λ = I d. Moreover, if d = q, then R 1 is a square matri and 11 is also equivalent to: λ 1, λ S, det R 1 λ 1, λ 0. We thus have the following straightforward lemma: Lemma 9. Definition 8 applied to the particular case of Roesser models 1 resp. Fornasini models 4 with F 3 = G 3 = 0 is equivalent to Definition 1 resp. Definition. We shall now consider the problem of the preservation of structural stability introduced in Definition 8 by equivalence transformations. Theorem 10. Let us consider the following two linear D discrete systems: R 1 + R u = 0, R 1 D q d, R D q du, 1 R 1 + R u = 0, R 1 D q d, R D q d u. 13 If the autonomous linear systems R 1 = 0 and R 1 = 0 are equivalent, i.e., M 1 := D 1 d /D 1 q R 1 = M 1 := D 1 d /D 1 q R 1, then 1 is structurally stable iff 13 is structurally stable. Proof. From Corollary 6, if M 1 = M 1, then there is a 1-1 correspondence between solutions of R 1 η = 0 and solutions of R 1 η = 0 via the eplicit changes of variables η = P η and η = P η for two matrices P D d d and P D d d, i.e., we have: η =P η = R 1 η = 0 R = 1 η = 0. η=p η As D is a commutative ring, we can replace the shift operators σ i and σ j by the comple variables z 1 and z without affecting the equality in the identities 8 and 9 of Lemma 5 applied to R 1 associated to M 1 and R 1 associated to M 1. We can then evaluate the obtained identities at λ 1, λ C so that we get: λ 1, λ C, y =P λ 1,λ y = R 1 λ 1, λ y = 0 = R 1 λ 1, λ y = 0. y=p λ 1,λ y Now let us assume w.l.o.g. that 1 is structurally stable. If 13 is not structurally stable, then, there eists λ 1, λ S and y 0 such that R 1 λ 1, λ y = 0. This would imply that we have R 1 λ 1, λ P λ 1, λ y = 0 for P λ 1, λ y 0 indeed from P λ 1, λ P λ 1, λ + Z λ 1, λ R 1 λ 1, λ = I d, we get that P λ 1, λ y = 0 implies y = 0 which contradicts the fact that 1 is structurally stable. Remark 11. Theorem 10 claims that if the autonomous parts R 1 = 0 and R 1 = 0 of 1 and 13 are equivalent, then 1 is structurally stable iff 13 is structurally stable. Note however that the equivalence of the whole linear systems 1 and 13 does not necessarily imply that 1 is structurally stable iff 13 is structurally stable because it does not necessarily imply the equivalence of the corresponding autonomous parts. This is due to the fact that the change of variables associated to an equivalence transformation may mi the state variables and the input variables. For eample, the linear systems i+1, j+ui+ 1, j ui, j = 0 and i+1, j i, j+u i+1, j = 0, are equivalent in the sense of algebraic analysis e.g., take the equivalence transformation sending the state variable onto the input variable u and the input variable u onto the state variable, the first one is structurally stable but the second one is not structurally stable. Indeed the autonomous linear systems σ i = 0 and σ i 1 = 0 are not equivalent in the sense of algebraic analysis. 4. STABILIZATION Definition 1. A linear system R 1 + R u = 0, with R 1 D q d and R D q du is stabilized by the state feedback control law u = K with K Q du d if the linear autonomous system R1 R R s s = 0, R s := D K I q+du d+du, 14 du is structurally stable in the sense of Definition 8, i.e., λ 1, λ S, Rs λ 1, λ y = 0 y = Remark 13. In Definition 1, the notation s for the whole variable in 14 aims at highlighting the fact that the closed-loop model is autonomous. Indeed, s is now the closed-loop state vector with no input subvector. Hence the structural stability of such a model should be tested owing to the whole matri R s. As for Condition 11 in Definition 8, the characterization 15 above can also be epressed in terms of the eistence of a left inverse for R s λ 1, λ C q+du d+du, and, in the particular case q = d, in terms of the non cancellation of detr s λ 1, λ. Lemma 14. Definition 1 applied to the particular case of Roesser models 1 resp. Fornasini models 4 with F 3 = G 3 = 0 is equivalent to the corresponding notions recalled in Subsection.1. In the sequel, we shall need to consider control laws that are not state feedbacks of the form u = K with K Q du d. We thus generalize Definition 15 to every control law of the form T + T u u = 0 with T D du d and T u D du du. Note that T and T u can involve shift operators so that such a control law may not be causal. Definition 15. A linear system R 1 + R u = 0, with R 1 D q d and R D q du is stabilized by the control law T + T u u = 0 with T D du d and T u D du du if the linear autonomous system 140

5 R1 R R s s = 0, R s := D T T q+du d+du u is structurally stable in the sense of Definition 8, i.e., λ 1, λ S, Rs λ 1, λ y = 0 y = Let us now study the impact of applying a state feedback control law to a linear system on an equivalent one. Proposition 16. Let us consider the two linear D discrete systems 1 and 13. Let us assume that 1 and 13 are equivalent, and, using the notation of Lemma 5, let P11 P P = 1 D P 1 P d+du d +d u, where P 11 D d d, P 1 D d d u, P 1 D du d, P D du d u, denote the matri defining the isomorphism between the D-modules respectively associated to 1 and 13, and P P = 11 P 1 P 1 P D d +d u d+du, where P 11 D d d, P 1 D d du, P 1 D d u d, P D d u du, denote the matri defining the inverse morphism. Then, we have the following results: 1 Applying the state feedback control law u = K with K Q du d to 1 is equivalent to applying the control law K P 11 +P 1 + K P 1 +P u = 0 to 13. Applying the state feedback control law u = K with K Q d u d to 13 is equivalent to applying the control law K P 11 + P 1 + K P 1 + P u = 0 to 1. Proof. Applying the state feedback control law u = K, with K Q du d to 1 amounts to adding the new equations K I du = 0 to 1. The result is then u straightforward since the change of variables associated to the equivalence transformation is given by the relation = P u u, so that the new equivalent equations on the variables and u to be added to 13 are then given by K I du P u = 0, which is equivalent to K P 11 + P 1 + K P 1 + P u = 0. The second assertion can be proved similarly. We have the following consequence of Proposition 16: Corollary 17. With the notation and assumptions of Proposition 16, we have the following results: 1 The linear autonomous systems R1 R R s s = 0, R s =, K I du and R s s = 0, R s R = 1 R, K P 11 + P 1 K P 1 + P are equivalent in the sense of algebraic analysis. The linear autonomous systems R s s = 0, R s R = 1 R K, I du and R R s s = 0, R s = 1 R K P 11 + P 1 K P 1 + P, are equivalent in the sense of algebraic analysis. Proof. Let us eplicitly give the equivalence announced in the corollary. From Lemma 5, the equivalence of 1 and 13 implies the eistence of matrices P D d+du d +d u, Q D q q, P D d +d u d+du, Q D q q, Z = Z1 T Z T T D d+du q and Z = Z 1 T Z T T D d +d u q such that 8 and 9 are satisfied with the matrices R = R 1 R D q d+du and R = R 1 R D q d +d u. Then one can check that we have the following identities: Q 0 R s P = R 0 I du s, R s P Q = 0 R K Z 1 Z I s, du P P Z1 0 + R Z 0 s = I d+d u, P Z P Z R 0 s = I d +d u, which, from Lemma 5, proves the first assertion of the corollary. The second assertion can be proved similarly. Finally we have the following result: Corollary 18. With the notation and assumptions of Proposition 16, we get that: 1 The state feedback control law u = K with K Q du d stabilizes 1 iff the control law K P 11 + P 1 + K P 1 + P u = 0 stabilizes 13. The state feedback control law u = K with K Q d u d stabilizes 13 iff the control law K P 11 + P 1 + K P 1 + P u = 0 stabilizes 1. Proof. This can be straightforwardly deduced from Corollary 17, Theorem 10 applied to the equivalent autonomous linear systems R s s = 0 and R s s = 0 and Definitions 1 and 15 of stabilization. 5. FORNASINI AND ROESSER MODELS Lemma 7 shows that a Fornasini model 4 is equivalent to the Roesser model 10. Using the results obtained above, we then get the following two theorems: Theorem 19. The Fornasini model 4 is structurally stable iff the equivalent Roesser model 10 is structurally stable. Proof. From Definition 8, 4 is structurally stable iff λ 1, λ S, detr 1 λ 1, λ 0, where R 1 λ 1, λ = I d λ 1 λ F 1 λ 1 F λ F 3. On the other hand, 10 is structurally stable iff λ 1, λ S, detr 1 λ 1, λ 0, where Id λ 1 F F F 1 + F 3 F G 1 + G 3 R 1 λ 1, λ = Now, where I d I d λ F 1 G I du λ detr 1 λ 1, λ = detuλ 1, λ deti du λ, Uλ 1, λ := Id λ 1 F F F 1 + F 3, I d I d λ F

6 and detuλ 1, λ is equal to det I d λ 1 F I d λ F 1 I d F F 1 + F 3, since all the matrices are square and I d commutes with every square matri of size d. We then get detr 1 λ 1, λ = λ du detr 1 λ 1, λ, so that, for all λ 1, λ S, detr 1 λ 1, λ 0 is equivalent to detr 1 λ 1, λ 0 because, for all λ 1, λ S, λ 0. Theorem 0. Let us consider a Fornasini model 4 and the equivalent Roesser model 10. The state feedback control law u = K h v, 17 with K = K 1 K, K 1 Q du d, K Q du d+du and K = K 1 K with K 1 Q du d, K Q du du, stabilizes the Roesser model 10 iff the control law K 1 I d σ j F 1 K 1 +K 1 G 1 K + I du σ j u = 0, stabilizes the Fornasini model 4. Proof. From Corollary 18., the state feedback control law 17 stabilizes 10 iff the control law K P 11 + P 1 + K P 1 + P u = 0 stabilizes 4. Now using the formula for P D d+ du d+du provided in Lemma 7., we get the desired result. Putting together Theorem 0 and the recent results obtained in [1] allows us to develop a new method for stabilizing linear D discrete Fornasini models. See []. REFERENCES [1] O. Bachelier, W. Paszke, N. Yeganefar, D. Mehdi and A. Cherifi. LMI necessary and sufficient stability conditions for D Roesser models. IEEE Trans. on Automatic Control, 613, , 016. [] O. Bachelier, T. Cluzeau, R. David, and N. Yeganefar. Structural stabilization of linear D discrete systems using equivalence transformations. Submitted, 016. [3] O. Bachelier, N. Yeganefar, D. Mehdi, and W. Paszke. On structural stabilization of state-space D models. Submitted, 015. [4] H. Bourlès, B. Marinescu, and U. Oberst. Weak eponential stability of linear time-varying differential behaviors. Linear Algebra Appl., 486, , 015. [5] Y. Bouzidi, A. Quadrat, and F. Rouillier. Computer algebra methods for testing the structural stability of multidimensional systems. Proceedings of nds 15, Vila Real Portugal, 07-09/09/015. [6] F. Chyzak, A. Quadrat, and D. Robertz. Effective algorithms for parametrizing linear control systems over Ore algebras. Appl. Algebra Engrg. Comm. Comput., 16, , 005. [7] F. Chyzak, A. Quadrat, and D. Robertz. OreModules: A symbolic package for the study of multidimensional linear systems. Springer LNCIS, 35, 33-64, OreModules.html [8] T. Cluzeau. A constructive algebraic analysis approach to the equivalence of multidimensional linear systems. Proceedings of nds 15, Vila Real Portugal, 07-09/09/015. [9] T. Cluzeau, A. Quadrat. Factoring and decomposing a class of linear functional systems. Linear Algebra Appl., 48, , 008. [10] T. Cluzeau, A. Quadrat. OreMorphisms: A homological algebraic package for factoring, reducing and decomposing linear functional systems. Springer LNCIS, 388, , OreMorphisms.html [11] T. Cluzeau, A. Quadrat. A constructive version of Fitting s theorem on isomorphisms and equivalences of linear systems. Proceedings of nds 11, Poitiers France, 05-07/09/011. [1] T. Cluzeau, A. Quadrat. Isomorphisms and Serre s reduction of linear systems. Proceedings of nds 13, Erlangen Germany, 09-11/09/013. [13] T. Cluzeau, A. Quadrat, and M. Tõnso. OreAlgebraicAnalysis: A Mathematica package for the algorithmic study of linear functional systems, OreAlgebraicAnalysis.html [14] P. Dabkowski, K. Galkowski, E. Rogers, and A. Kummert. Strong practical stability and stabilization of discrete linear repetitive processes. Multidimensional Systems and Signal Processing, 04: , 009. [15] E. Fornasini, G. Marchesini. Doubly indeed dynamical systems: state space models and structural properties. Math. Systems Theory, 1, 59-7, [16] T. Kaczorek. The singular general model of -D systems and its solution. IEEE Trans. on Automatic Control, 3311, , [17] T. Kaczorek. Two-Dimensional Linear Systems. Springer Verlag, [18] J. E. Kurek. The general state-space model for a two-dimensional linear digital system. IEEE Trans. on Automatic Control, 30, , [19] L. Li, L. Xu, and Z. Lin. Stability and stabilization of linear multidimensional discrete systems in the frequency domain. International Journal of Control, 8611: , 013. [0] B. Malgrange. Systèmes différentiels à coefficients constants. Séminaire Bourbaki 196/63, 1-11, 196. [1] U. Oberst. Multidimensional constant linear systems. Acta Appl. Math., 0, 1-175, [] A. Quadrat. Every internally stabilizable multidimensional system admits a doubly coprime factorization. Proceedings of MTNS 04, Leuven Belgium, 05-09/07/004. [3] A. Quadrat. An introduction to constructive algebraic analysis and its applications. Les cours du CIRM, JNCF 010, 1, , 010. [4] R. P. Roesser. A discrete state-space model for linear image processing. IEEE Trans. on Automatic Control, 01, 1-10,1975. [5] H. H. Rosenbrock. State Space and Multivariable Theory. Nelson-Wiley, London, UK,1970. [6] M. Scheicher, U. Oberst. Multidimensional discrete stability by Serre categories and the construction and parametrization of observers via Gabriel localizations. SIAM J. Control Optimization 51013, [7] J. Wood, E. Rogers, and D. H. Owens. Behaviours, modules, and duality. Multidimensionl Signals, Circuits and Systems, Taylor&Francis, ch. 3, 45-56,