Structural Stability and Equivalence of Linear 2D Discrete Systems1

6th IFAC Symposium on System Structure and Control 6th IFAC Symposium on System and Control June 22-24, 2016. Istanbul, TurkeyStructure 6th IFAC Symposium on Structure 6th IFAC Symposium on System System Structure and and Control Control June 22-24, 2016. Istanbul, Turkey Available online at www.sciencedirect.com June 22-24, 2016. Istanbul, Turkey June 22-24, 2016. Istanbul, Turkey

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Structural Stability and Structural Stability and Structural Stability and Structural Stability and Linear 2D Discrete Linear 2D Discrete Linear 2D Discrete Linear 2D Discrete

Equivalence Equivalence Equivalence  Equivalence Systems  Systems Systems Systems 

of of of of

Bachelier, Ronan David, Nima Yeganefar ∗∗ Bachelier, Ronan David, Nima Yeganefar ∗∗ ∗∗ Bachelier, Ronan David, Nima Thomas Bachelier, RonanCluzeau David, ∗∗ Nima Yeganefar Yeganefar ∗∗ Thomas Cluzeau Thomas Thomas Cluzeau Cluzeau ∗∗ ∗ atiment B25, 2 rue Pierre ∗ University of Poitiers ; LIAS-ENSIP, Bˆ ∗ University of Poitiers ; LIAS-ENSIP, Bˆ a timentFrance B25, 2(e-mail: rue Pierre ∗ University of Poitiers ; LIAS-ENSIP, Bˆ a B25, rue Brousse, TSA 41105, 86073 Poitiers cedex, University ofTSA Poitiers ; LIAS-ENSIP, Bˆ atiment timentFrance B25, 2 2(e-mail: rue Pierre Pierre Brousse, 41105, 86073 Poitiers cedex, Brousse, TSA 41105, 86073 Poitiers cedex, France (e-mail: {olivier.bachelier,ronan.david,nima.yeganefar}@univ-poitiers.fr). Brousse, TSA 41105, 86073 Poitiers cedex, France (e-mail: {olivier.bachelier,ronan.david,nima.yeganefar}@univ-poitiers.fr). ∗∗ {olivier.bachelier,ronan.david,nima.yeganefar}@univ-poitiers.fr). University of Limoges ; CNRS ; XLIM UMR 7252, 123 avenue {olivier.bachelier,ronan.david,nima.yeganefar}@univ-poitiers.fr). ∗∗ ∗∗ University of Limoges ; CNRS ; XLIM UMR 7252, 123 avenue ∗∗ University of ;; CNRS ;; XLIM UMR 7252, 123 Albert Thomas, 87060 Limoges cedex, France (e-mail: University of Limoges Limoges CNRS XLIM UMR 7252, 123 avenue avenue Albert Thomas, 87060 Limoges cedex, France (e-mail: Albert Thomas, 87060 Limoges cedex, France (e-mail: [email protected]). Albert Thomas, 87060 Limoges cedex, France (e-mail: [email protected]). [email protected]). [email protected]). Abstract: We study stability issues for linear two-dimensional (2D) discrete systems by means Abstract: We study stability issues for linear two-dimensional (2D) discrete means Abstract: We stability issues linear (2D) discrete systems by means of the constructive algebraic analysis approach to linear systems theory. We systems provide by a general Abstract: We study study stabilityanalysis issues for for linear two-dimensional two-dimensional (2D) discrete systems by means of the constructive algebraic approach to linear systems theory. We provide aa existing general of the constructive algebraic analysis approach to linear systems theory. We provide general definition of structural stability for linear 2D discrete systems which coincides with the of the constructive algebraic analysis approach to linear systems theory. We provide a general definition stability 2D discrete systems which coincides with the existing definition of structural stability for linear 2D systems which existing definitionsof instructural the particular casesfor oflinear the classical Roesser and Fornasini-Marchesini models. We definition ofin structural stability forof linear 2D discrete discrete systems which coincides coincides with with the the existing definitions the particular cases the classical Roesser and Fornasini-Marchesini models. We 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, definitions in the particular cases ofstructural the classical Roesser and Fornasini-Marchesini models. We then study the preservation of this stability by equivalence transformations. Finally, then study the preservation of this structural stability by equivalence transformations. Finally, usingstudy the same framework, we consider the stabilization problem for equivalent linear systems. then the preservation of this structural stability by equivalence transformations. Finally, using the same framework, we consider the stabilization problem for equivalent linear systems. using the same framework, we the problem for equivalent linear systems. using same framework,Federation we consider consider the stabilization stabilization problem equivalent © 2016,the IFAC (International of Automatic Control) Hosting by for Elsevier Ltd. Alllinear rights systems. reserved. Keywords: System theory, algebraic approaches, multidimensional systems, discrete systems, Keywords: System theory, algebraic approaches, multidimensional systems, discrete systems, Keywords: System theory, theory, algebraic approaches, multidimensional multidimensional systems, systems, discrete discrete systems, systems, structural stability, stabilization methods Keywords: System algebraic approaches, structural stability, stabilization methods structural stability, stabilization methods structural stability, stabilization methods 1. INTRODUCTION general definitions of structural stability and stabilization 1. INTRODUCTION INTRODUCTION general of structural stability and stabilization 1. general definitions of structural stability and for lineardefinitions 2D discrete which are coherent with the 1. INTRODUCTION general definitions of systems structural stability and stabilization stabilization for linear 2D discrete systems which are coherent with the for linear 2D discrete systems which are coherent the existing definitions in the particular cases of the classical The algebraic analysis or behavioral approach to linear for linear 2D discrete systems which are coherent with with the existing definitions in the particular cases of the classical The algebraic analysis or behavioral approach to linear existing definitions in the particular cases of the classical Roesser and Fornasini-Marchesini models. Moreover, we The algebraic analysis or behavioral approach to linear systems theoryanalysis is a unified mathematical framework to existing definitions in the particular cases of the classical The algebraic or behavioral approach to linear and Fornasini-Marchesini models. Moreover, we systems theory is is aa unified unified mathematical framework to Roesser Roesser and Fornasini-Marchesini models. Moreover, we take advantage of the results in [11, 12, 8] which construcsystems theory mathematical framework to study multidimensional systems of linear functional equaand Fornasini-Marchesini models. Moreover, we systems theory is a unified mathematical framework to Roesser take advantage of the results in [11, 12, 8] which construcstudy multidimensional systems of linear functional equatake advantage of the results in [11, 12, 8] which constructively tackle the equivalence problem for linear systems in study multidimensional systems of linear functional equations appearing in control theory, engineering sciences, take advantage of the results in [11, 12, 8] which construcstudy multidimensional systems of linear functional equatively tackle the equivalence problem for linear systems in tions appearing in theory, engineering tively tackle problem for linear systems in to focusthe on equivalence how structural stability and stabilization tions appearing in control control theory, engineering sciences, mathematical physics, . . . . See for instance [20, 25,sciences, 21, 6, 9, order tively tackle the equivalence problem for linear systems in tions appearing in control theory, engineering sciences, order to focus on how structural stability and stabilization mathematical physics, . . . . See for instance [20, 25, 21, 6, 9, order to focus on how structural stability and stabilization properties are transmitted from a given linear system to mathematical physics, . . . . See for instance [20, 25, 21, 6, 9, 23, 27, 14] and the references therein. A linear system can order to focus on how structural stability and stabilization mathematical physics, . . . . See for instance [20, 25, 21, 6, 9, properties are transmitted from given linear 23, 27, 14] 14] and the the as references therein. A ∈linear linear system can properties are one. transmitted from aaawe given linear system to an equivalent In particular, study the system impact to of 23, 27, and references system can is a q × p properties always be written R η = 0,therein. where RA Dq×p are transmitted from given linear system to 23, 27, 14] and the as references therein. A ∈linear can q×psystem an equivalent one. In particular, we study the impact of q×p is a q × p always be written R η = 0, where R D an equivalent one. In particular, we study the impact of applying a state feedback control law to stabilize a system q×ppolynomial is a q × p always be written as R η = 0, where R ∈ D matrix with entries in a (noncommutative) an equivalent one. In particular, we study the impact of is a q × p always be written as R η = 0, where R ∈ D applying a state feedback control law to stabilize a system matrix with entries in a (noncommutative) polynomial applying a state feedback control law to stabilize a system on an equivalent system. Finally, all our results are applied matrix with entries in a (noncommutative) polynomial ring D with of functional operators and η is a vector of p applying a state feedback control law to stabilize a system matrix entries in a (noncommutative) polynomial an equivalent system. all our results are applied ring D offunctions functional operators is a vector of p on on an system. Finally, all results the particular case of aFinally, generalized Fornasini-Marchesini ring D functional operators and is of unknown which belongsand to ηηηa functional on an equivalent equivalent system. Finally, all our our results are are applied applied ring D of offunctions functional operators and is a a vector vectorspace. of p p to to the particular case of a generalized Fornasini-Marchesini unknown which belongs to a functional space. to the particular case of a generalized Fornasini-Marchesini model and its equivalent Roesser model (see [8]). unknown functions which belongs to a functional space. We then introduce the finitely presented left D-module to the particular case of a generalized Fornasini-Marchesini unknown functions which belongs to a functional space. model and its equivalent Roesser model (see [8]). We then introduce the finitely D-module and equivalent Roesser model (see We then introduce the presented left D-module M D1×p /(D1×q F is apresented functionalleft space having model model and its its Roesser model (see [8]). [8]).2, we reWe := then introduce the Iffinitely finitely presented left D-module 1×p 1×q R). The paper is equivalent organized as follows. In Section 1×p /(D 1×q R). If F is a functional space having M := D 1×p 1×q M := D /(D R). If F is a functional space having a left D-module structure, weis consider the linear system The paper is organized as follows. In Section we M := D /(D R). If F a functional space having The some paper useful is organized organized as follows. follows. Roesser In Section Section 2,generalwe rerecall facts concerning and2, a left D-module system paper is as In 2, we rea left D-module structure, we consider the linear system behavior kerF structure, (R.) := {ηwe ∈consider F pp | Rthe η =linear 0} and we The call some useful facts concerning Roesser and generalaor left D-module structure, we consider the linear system call some useful facts concerning Roesser and generalized some Fornasini-Marchesini models and the equivalence of p | R η∼= 0} and we or behavior ker (R.) := {η ∈ F call useful facts concerning Roesser and generalF or ker (R.) := F || R = 0} we have Malgrange’s isomorphism (M, F) Fornasini-Marchesini and the equivalence of or behavior behavior kerF := {η {η ∈ ∈ ker F pF (R.) R ηη= =hom 0}Dand and we ized ized Fornasini-Marchesini models and the equivalence of ∼ F (R.) linear systems within themodels constructive algebraic analyhave Malgrange’s isomorphism ker (R.) hom (M, F) = ized Fornasini-Marchesini models and the equivalence of ∼ F D have[20]) Malgrange’s isomorphism kerFproperties (R.) ∼ hom (M, F) = (see which shows that system of ker (R.) linear systems within the constructive algebraic analyD F have Malgrange’s isomorphism ker (R.) hom (M, F) = of ker linear systems within the constructive algebraic analyF D sis approach to linear systems theory. In Section 3, we (see [20]) which shows that system properties (R.) linear systems within the constructive algebraic analyF (see [20]) which shows that system properties of ker (R.) can be studied means of system moduleproperties propertiesofofker MF (and sis approach to linear linear systems oftheory. theory. In Section Section 3, for we (see [20]) whichby shows that (R.) sis approach to systems In 3, we F(and introduce a general definition structural stability can be studied by of module properties of M sis approach to linear systems oftheory. In Section 3, for we can be studied by means of module properties of M (and F). Moreover, we means nowadays have constructive algebraic introduce aa general definition structural stability can be studied by means of module properties of M (and introduce general definition of structural stability for linear 2D discrete systems and we study its preservation F). Moreover, we nowadays have constructive algebraic introduce a general definition of structural stability for F). Moreover, nowadays constructive algebraic techniques (e.g.,we homological algebra using linear 2D discrete systems and we preservation F). Moreover, weconstructive nowadays have have constructive algebraic linear 2D systems study its preservation via equivalence transformations. In study Sectionits we consider techniques homological algebra 2D discrete discrete systems and and we we study its4, preservation techniques (e.g., (e.g., constructive constructive homological algebra using (noncommutative) Gr¨ obner basis computations) and using their linear via equivalence transformations. In Section 4, we consider techniques (e.g., constructive homological algebra using via equivalence transformations. In Section 4, consider stabilization issues in the same framework. Finally, in (noncommutative) Gr¨ o bner basis computations) and their equivalence transformations. Inframework. Section 4, we we consider (noncommutative) o bner basis computations) and their implementations inGr¨ several computer algebra systems at via stabilization issues in the same Finally, ina (noncommutative) Gr¨ o bner basis computations) and their stabilization issues in the same framework. Finally, Section 5, weissues apply in thethe previous results to theFinally, case of in implementations in several computer algebra systems at stabilization same framework. ina implementations in several computer algebra systems at our disposal to study/check module properties of M . See Section 5, we apply the previous results to the case of implementations in several computer algebra systems at Section 5, we apply the previous results to the case of a generalized Fornasini-Marchesini model and its equivalent our disposal to study/check module properties of M . See Section 5, we apply the previous results to the case of a our disposal to study/check module properties M .. See for instance [6, 23] and the references therein. of generalized Fornasini-Marchesini model and its equivalent our disposal to study/check module properties of M See generalized Fornasini-Marchesini model and its equivalent Roesser model. for instance [6, 23] and the references therein. generalized Fornasini-Marchesini model and its equivalent for instance instance [6, [6, 23] 23] and and the references references therein. Roesser model. for model. The contribution of thisthe paper consiststherein. in using the frame- Roesser Roesser Notation:model. We note Q (resp. C) the field of rational (resp. The contribution of this paper consists in using the frameThe contribution of this paper consists in using the framework of the constructive algebraic analysis approach to Notation: We note Q C) the field of 2 (resp. The contribution of this paper consists in using the frameNotation: We (resp. C) field of rational (resp. complex) numbers. In (resp. the whole paper, ∈ rational Ddd11 ×d work ofsystems the constructive constructive algebraic analysis analysis approach to complex) Notation: numbers. We note note Q Q (resp. C) the the fieldR of rational (resp. ×d means work the algebraic to d1 ×d22 means linear of theory to investigate stabilityapproach and stabiIn the whole paper, R ∈ D work of the constructive algebraic analysis approach to d1 ×d2 means complex) numbers. In the whole paper, R ∈ D that R is a matrix with d rows and d columns whose 1 2 linear systems theory to investigate stability and stabicomplex) numbers. Inwith the d whole paper, R ∈columns D means linear systems theory to investigate stability and stabilization issues for linear 2D discrete systems. Note that the that R is a matrix rows and d whose 1 2 identity matrix linear systems theory to investigate stability and stabithat R is matrix d and d whose entries (a ring)with D and n denotes lization issues for linear 2D discrete systems. that the that R are is aain matrix with d11 IIrows rows and the d22 columns columns whose lization issues for linear 2D discrete systems. Note that the algebraic analysis approach has already been Note used to study entries are in (a ring) D and the identity matrix n denotes lization issues for linear 2D discrete systems. Note that the entries are in (a ring) D and I denotes the identity matrix n of dimension n. If C = C ∪ {∞} denotes the extension of C algebraic analysis approach has already been used to study entries are in (a ring) D and I denotes the identity matrix n algebraic analysis approach has been to stability and stabilization for linear multidimendimension n. If C= = then C∪ ∪ {∞} {∞} denotes the extension of C algebraic analysis approachproblems has already already been used used to study study of C C denotes the extension of C of dimension n. If in Alexandrov’s sense, we introduce the following two stability and stabilization problems for linear multidimenof dimension n. If C = C ∪ {∞} denotes the extension of C stability and stabilization problems for linear multidimensional systems: see for instance [22] or more recently [26, 4] in in Alexandrov’s sense, then we introduce the following two 2 2 2 stability and stabilization problems for linear multidimenAlexandrov’s sense, then we introduce the following two sional systems: see for instance [22] or more recently [26, 4] in Alexandrov’s then we introduce the following two 2 : Ssense, 2 := {(z 2 | ∀i = subsets of C , z ) ∈ C 1, 2, |z | ≥ 1}, sional systems: see for instance [22] or more recently [26, 4] 1 2 i and the references therein. In the present work, we provide sional systems: see therein. for instance [22] or more recently [26, 4] subsets2 of C22 : S22 := {(z C22 || ∀i = 1, 2, |z i | ≥ 1}, 2 1 , z2 ) and the references In the present work, we provide subsets of C :: ,S S := )) ∈ ∈ = 1, and of {(z C {(z ∈ C | ∀i ∀i = 1}. 1, 2, 2, |z |zii || ≥ ≥ 1}, 1}, ∈ {(z C22 11| ,,∀izz22= 1, C 2, |z and D22 := and the the references references therein. therein. In In the the present present work, work, we we provide provide subsets 1 z2 ):= i| ≤  := {(z , z ) ∈ C | ∀i = 1, 2, |z | ≤ 1}. and D This work was supported by the ANR-13-BS03-0005 (MSDOS). 2 2 1 2 i D := {(z , z ) ∈ C | ∀i = 1, 2, |z | ≤ 1}. and  1 2 i This work was supported by the ANR-13-BS03-0005 (MSDOS). and D := {(z1 , z2 ) ∈ C | ∀i = 1, 2, |zi | ≤ 1}.   This work was supported by the ANR-13-BS03-0005 (MSDOS). Olivier Olivier Olivier Olivier

This work was supported by the ANR-13-BS03-0005 (MSDOS). Copyright © 2016, 2016 IFAC 136Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © IFAC (International Federation of Automatic Control) Copyright © 2016 IFAC 136 Copyright © 2016 IFAC 136 Peer review under responsibility of International Federation of Automatic Copyright © 2016 IFAC 136Control. 10.1016/j.ifacol.2016.07.514

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2

2. PRELIMINARIES

∀(λ1 , λ2 ) ∈ D , det(Idx − λ1 F1 − λ2 F2 ) = 0.

2.1 Classical linear 2D discrete systems In the present paper, we shall focus on linear 2D 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 explicit models of such systems which are the Roesser model [24] 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 [24]. They correspond to linear 2D discrete systems for which the equations are written under the (explicit) particular form:  h  h      A11 A12 B1 x (i, j) x (i + 1, j) u(i, j), = + A21 A22 B2 xv (i, j + 1) xv (i, j) (1) where xh (resp. xv ) is the horizontal (resp. vertical) state vector of dimension dh (resp. dv ), u is the input vector of dimension du , A11 ∈ Qdh ×dh , A12 ∈ Qdh ×dv , A21 ∈ Qdv ×dh , A22 ∈ Qdv ×dv , B1 ∈ Qdh ×du , B2 ∈ Qdv ×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   2 −A12 λ1 Idh − A11 = 0. ∀(λ1 , λ2 ) ∈ S , det −A21 λ2 Idv − A22 (2) If (1) is not structurally stable, then applying the state feedback control law   h x (i, j) , (3) u(i, j) = (K1 K2 ) xv (i, j)

where K1 ∈ Qdu ×dh and K2 ∈ Qdu ×dv to (1), we obtain the  model    h  hnew Roesser A11 + B1 K1 A12 + B1 K2 x (i, j) x (i + 1, j) = . A21 + B2 K1 A22 + B2 K2 xv (i, j + 1) xv (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 2D discrete system for which the equations are written under the (explicit) particular form: x(i + 1, j + 1) = F1 x(i + 1, j) + F2 x(i, j + 1) + F3 x(i, j) + G1 u(i + 1, j) + G2 u(i, j + 1) + G3 u(i, j), (4) where x is the state vector of dimension dx , u is the input vector of dimension du , F1 ∈ Qdx ×dx , F2 ∈ Qdx ×dx , F3 ∈ Qdx ×dx , G1 ∈ Qdx ×du , G2 ∈ Qdx ×du , G3 ∈ Qdx ×du . To our knowledge, a notion of structural stability for Fornasini models (4) has been defined only in the particular case F3 = G3 = 0. See [15, 19, 3] and the references therein. Definition 2. A Fornasini model (4) with F3 = G3 = 0 is said to be structurally stable if 137

137

(5)

If a Fornasini model (4) with F3 = G3 = 0 is not structurally stable, then applying the state feedback control law u(i, j) = K x(i, j), (6) du ×dx where K ∈ Q to (4), we obtain the new Fornasini model x(i+1, j+1) = (F1 +G1 K) x(i+1, j)+(F2 +G2 K) x(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: 2

∀(λ1 , λ2 ) ∈ S , det(λ1 λ2 Idx − λ1 F1 − λ2 F2 ) = 0. (7) Remark 4. Note that the conditions (2), (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 = Qσi , σj  denote the commutative ring of partial (forward) shift operators with constant rational coefficients, i.e., for a bivariate sequence f (i, j), we have σi f (i, j) = f (i + 1, j), σj f (i, j) = f (i, j + 1), and we further have σi σj = σj σi , where σi σj stands for the composition of operators σi ◦σj . An operator P ∈ D can be  written as P = m,l pml σim σjl , where pml ∈ Q, the sum is finite, and, for a bivariate sequence f (i, j), we thus have P f (i, j) = m,l pml f (i + 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(dh +dv )×(dh +dv +du ) and η are defined by     xh −A12 −B1 I σ − A11 , η =  xv  . R = dh i −A21 Idv σj − A22 −B2 u It is then studied by means of the factor D-module M = D1×(dh +dv +du ) /(D1×(dh +dv ) R), (2) The Fornasini model (4) is written as R η = 0, where R ∈ Ddx ×(dx +du ) is defined by R = (Idx σi σj − F1 σi − F2 σj − F3 −G1 σi − G2 σj − G3 ) , T  and η = xT uT . It is then studied by means of the D-module M = D1×(dx +du ) /(D1×dx R).

2.2 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, 12, 8]. Let us summarize part of the results obtained in the latter works:   Lemma 5. Let R ∈ Dq×p , R ∈ Dq ×p and consider 1×p the associated left D-modules M = D /(D1×q R) and  1×p 1×q   M =D /(D R ). (1) The existence of a homomorphism f ∈ homD (M, M  )  is equivalent to the existence of P ∈ Dp×p and  Q ∈ Dq×q satisfying the identity R P = Q R . (8)

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Then, the homomorphism f ∈ homD (M, M  ) is defined by f (π(λ)) = π  (λ P ) for all λ ∈ D1×p , where  π : D1×p → M and π  : D1×p → M  denote the canonical projections onto M and M  . (2) With the previous notation, f is an isomorphism meaning that M is isomorphic to M  which is denoted   by M ∼ = M  iff there exist P  ∈ Dp ×p , Q ∈ Dq ×q ,   Z ∈ Dp×q , and Z  ∈ Dp ×q satisfying   R P = Q R, P P  + Z R = Ip , P  P + Z  R = Ip . (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 Qσi , σj  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.(2)) are implemented in the Maple package OreMorphisms [10]. 



and M  = D1×(2 dx +2 du ) /(D1×(2 dx +du ) R ) the associated D-module, then we have the following results: (1) The homomorphism f ∈ homD (M, M  ) defined by the matrix   0 Idx 0 0 P = ∈ D(dx +du )×(2 dx +2 du ) , 0 0 Id u 0 is an isomorphism so that M ∼ = M . (2) The identities (8) and (9) of Lemma 5 are then satisfied by the following matrices: Q = (Idx Idx σi − F2 − G2 ) ∈ Ddx ×(2 dx +du ) ,   Idx σj − F1 −G1 I dx 0   P =  ∈ D(2 dx +2 du )×(dx +du ) , 0 I du  0 Id u σ j     Idx 0  (2 dx +du )×dx Q = 0 ∈D , Z= ∈ D(dx +du )×dx , 0 0   0 −Idx 0 0  0 0 Z =  ∈ D(2 dx +2 du )×(2 dx +du ) . 0 0 0  0 0 −Idu

Corollary 6. Let R ∈ Dq×p , R ∈ Dq ×p and consider the associated left D-modules M = D1×p /(D1×q R) and   M  = D1×p /(D1×q R ). Let f ∈ homD (M, M  ) be an Corollary 6 implies that, if F is a Qσi , σj -module, then  = kerF (R .), i.e., there is a 1-1 correspondence isomorphism given by a matrix P ∈ Dp×p such that there kerF (R.) ∼  of (10). More exists Q ∈ Dq×q satisfying (8). Then, with the notation between F-solutions of (4) and F-solutions T  of Lemma 5, if F is a left D-module, we have the following precisely, if we denote η(i, j) = x(i, j)T u(i, j)T , and T  isomorphism of linear systems: η  (i, j) := xh (i, j)T xv1 (i, j)T xv2 (i, j)T u (i, j)T , then P. : kerF (R .) −→ kerF (R.), η  −→ η := P η  , if η(i, j) is a solution of (4), then   whose inverse is given by x(i, j + 1) − F1 x(i, j) − G1 u(i, j) x(i, j)   P  . : kerF (R.) −→ kerF (R .), η −→ η  := P  η. η  (i, j) = P  η(i, j) =  , u(i, j) In other words, the invertible changes of variables η = P η  u(i, j + 1) 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 is solution of (10). Conversely, if η  (i, j) is a solution of have: R η = 0 ⇐⇒ R η  = 0. (10), then   v x1 (i, j) In [8], the above techniques are applied to study the η(i, j) = P η  (i, j) = , xv2 (i, j) equivalence problem between Roesser models (1) and Fornasini models (4). In particular, it is proved that (4) is solution of (4). is always equivalent to a Roesser model. Let us explicitly recall this equivalence which will be useful in Section 5. 3. STRUCTURAL STABILITY Lemma 7. Let R ∈ Ddx ×(dx +du ) be the matrix associated to the Fornasini model (4) (see Subsection 2.1) and let M = D1×(dx +du ) /(D1×dx R) be the associated D-module. From now on D = Qσi , σj  denotes the commutative polynomial ring defined in Subsection 2.1. For a matrix R If we define the  Roesser model  with entries in D, let us denote by R the matrix obtained  h    F 2 F 2 F1 + F 3 F 2 G 1 + G 3 from R by replacing the shift operator σi (resp. σj ) by a x (i + 1, j) xh (i, j)   F1 G1 = Idx new complex variable z1 (resp. z2 ). The algebraic analysis xv (i, j + 1) xv (i, j) 0 0 0 approach to linear systems theory recalled in the introduc  tion makes no distinction between the different variables G2 of a linear system R η = 0, i.e., all the components of the +  0  u (i, j), vector η are treated in the same way. Although, as soon as Id u (10) one is concerned with stability and stabilization issues, the T  state variables and the input variables of a linear system where xv (i, j) = xv1 (i, j)T xv2 (i, j)T , the associated do not play the same role. Consequently, in the sequel, matrix R ∈ D(2 dx +du )×(2 dx +2 du ) given by we still consider linear systems written as R η = 0, where   R ∈ Dq×p but we split the vector η of p unknown sequences Idx σi − F2 −(F2 F1 + F3 ) −(F2 G1 + G3 ) −G2  −Idx I d x σ j − F1 −G1 0 , into a subvector of state variables x of dimension dx and a R = subvector of input variables u of dimension du (so that 0 0 Id u σ j −Idu 138

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p = dx + du ). Splitting the matrix R accordingly, i.e., R = (R1 R2 ), with R1 ∈ Dq×dx , R2 ∈ Dq×du , we have:   x R η = 0 ⇐⇒ (R1 R2 ) = 0 ⇐⇒ R1 x + R2 u = 0. u The linear system R1 x = 0 is then the autonomous linear system associated to R η = 0. Definition 8. A linear system R1 x + R2 u = 0, where R1 ∈ Dq×dx and R2 ∈ Dq×du is said to be structurally stable if   2 ∀(λ1 , λ2 ) ∈ S , R1 (λ1 , λ2 ) y = 0 ⇒ y = 0 . (11) q×dx

Note that in (11), R1 (λ1 , λ2 ) ∈ C stands for the 2 matrix R1 (z1 , z2 ) evaluated at the point (λ1 , λ2 ) ∈ S . The condition (11) is then equivalent to the fact that, for all 2 (λ1 , λ2 ) ∈ S , the matrix R1 (λ1 , λ2 ) admits a left inverse, 2 dx ×q such i.e., for all (λ1 , λ2 ) ∈ S , there exists Lλ1 ,λ2 ∈ C that Lλ1 ,λ2 R1 (λ1 , λ2 ) = Idx . Moreover, if dx = q, then R1 is a square matrix and (11) is also equivalent to:   2 ∀(λ1 , λ2 ) ∈ S , det R1 (λ1 , λ2 ) = 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 F3 = G3 = 0) is equivalent to Definition 1 (resp. Definition 2). 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 2D discrete systems: R1 x + R2 u = 0, R1 ∈ Dq×dx , R2 ∈ Dq×du , (12) 



R1 x + R2 u = 0, R1 ∈ Dq ×dx , R2 ∈ Dq ×du . (13) If the autonomous linear systems R1 x = 0 and R1 x = 0 are equivalent, i.e., M1 := D1×dx /(D1×q R1 ) ∼ = M1 := 1×dx 1×q   D /(D R1 ), then (12) is structurally stable iff (13) is structurally stable.

∼ M  , then there is a Proof. From Corollary 6, if M1 = 1 1-1 correspondence between solutions of R1 η = 0 and solutions of R1 η  = 0 via the explicit changes of variables η = P η  and η  = P  η for two matrices P ∈ Ddx ×dx and P  ∈ Ddx ×dx , i.e., we have: η  =P  η

R1 η = 0

=⇒ R η  = 0. ⇐=  1

η=P η

As D is a commutative ring, we can replace the shift operators σi and σj by the complex variables z1 and z2 without affecting the equality in the identities (8) and (9) of Lemma 5 applied to R1 (associated to M1 ) and R1 (associated to M1 ). We can then evaluate the obtained 2 2 identities at (λ1 , λ2 ) ∈ C so that we get: ∀(λ1 , λ2 ) ∈ C , y  =P  (λ1 ,λ2 ) y

R1 (λ1 , λ2 ) y = 0

=⇒ ⇐=

R1 (λ1 , λ2 ) y  = 0.

y=P (λ1 ,λ2 ) y 

Now let us assume w.l.o.g. that (12) is structurally stable. If (13) is not structurally stable, then, there exists 2 (λ1 , λ2 ) ∈ S and y  = 0 such that R1 (λ1 , λ2 ) y  = 0. This would imply that we have R1 (λ1 , λ2 ) (P (λ1 , λ2 ) y  ) = 0 139

139

for P (λ1 , λ2 ) y  = 0 (indeed from P  (λ1 , λ2 ) P (λ1 , λ2 ) + Z  (λ1 , λ2 ) R1 (λ1 , λ2 ) = Idx , we get that P (λ1 , λ2 ) y  = 0 implies y  = 0) which contradicts the fact that (12) is structurally stable. Remark 11. Theorem 10 claims that if the autonomous parts R1 x = 0 and R1 x = 0 of (12) and (13) are equivalent, then (12) is structurally stable iff (13) is structurally stable. Note however that the equivalence of the whole linear systems (12) and (13) does not necessarily imply that (12) 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 mix the state variables and the input variables. For example, the linear systems x(i+1, j)+u(i+ 1, j) − u(i, j) = 0 and x (i + 1, j) − x (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 x onto the input variable u and the input variable u onto the state variable x ), the first one is structurally stable but the second one is not structurally stable. Indeed the autonomous linear systems σi x = 0 and (σi − 1) x = 0 are not equivalent in the sense of algebraic analysis. 4. STABILIZATION Definition 12. A linear system R1 x + R2 u = 0, with R1 ∈ Dq×dx and R2 ∈ Dq×du is stabilized by the state feedback control law u = K x with K ∈ Qdu ×dx if the linear autonomous system   R1 R 2 ∈ D(q+du )×(dx +du ) , (14) Rs xs = 0, Rs := −K Idu

is structurally stable in the sense of Definition 8, i.e.,   2 ∀(λ1 , λ2 ) ∈ S , Rs (λ1 , λ2 ) y = 0 ⇒ y = 0 . (15) Remark 13. In Definition 12, the notation xs for the whole variable in (14) aims at highlighting the fact that the closed-loop model is autonomous. Indeed, xs 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 matrix Rs . As for Condition (11) in Definition 8, the characterization (15) above can also be expressed in terms of the existence (q+du )×(dx +du )

, and, in of a left inverse for Rs (λ1 , λ2 ) ∈ C the particular case q = dx , in terms of the non cancellation of det(Rs (λ1 , λ2 )). Lemma 14. Definition 12 applied to the particular case of Roesser models (1) (resp. Fornasini models (4) with F3 = G3 = 0) is equivalent to the corresponding notions recalled in Subsection 2.1. In the sequel, we shall need to consider control laws that are not state feedbacks of the form u = K x with K ∈ Qdu ×dx . We thus generalize Definition 15 to every control law of the form Tx x + Tu u = 0 with Tx ∈ Ddu ×dx and Tu ∈ Ddu ×du . Note that Tx and Tu can involve shift operators so that such a control law may not be causal. Definition 15. A linear system R1 x + R2 u = 0, with R1 ∈ Dq×dx and R2 ∈ Dq×du is stabilized by the control law Tx x + Tu u = 0 with Tx ∈ Ddu ×dx and Tu ∈ Ddu ×du if the linear autonomous system

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Rs xs = 0,

Rs :=

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  R 1 R2 ∈ D(q+du )×(dx +du ) T x Tu

and Rs xs = 0, Rs =

is structurally stable in the sense of Definition 8, i.e.,   2 Rs (λ1 , λ2 ) y = 0 ⇒ y = 0 . (16) ∀(λ1 , λ2 ) ∈ S ,

are equivalent in the sense of algebraic analysis.

Proof. Let us explicitly give the equivalence announced in the corollary. From Lemma 5, the equivalence of (12) and (13) implies the existence of matrices P ∈  D(dx +du )×(dx +du ) , Q ∈ Dq×q , P  ∈ D(dx +du )×(dx +du ) ,  Q ∈ Dq ×q , Z = (Z1T Z2T )T ∈ D(dx +du )×q and Z  =  (Z1T Z2T )T ∈ D(dx +du )×q such that (8) and (9) are satisfied with the matrices R = (R1 R2 ) ∈ Dq×(dx +du )  and R = (R1 R2 ) ∈ Dq ×(dx +du ) . Then one can check that we have the following identities:     Q 0 0 Q Rs , Rs , Rs P  = Rs P = 0 Idu K Z 1 − Z 2 I du      Z 0 Z1 0   PP + Rs = Idx +du , Rs = Idx +du , P P + 1 Z2 0 Z2 0 which, from Lemma 5, proves the first assertion of the corollary. The second assertion can be proved similarly.

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 2D discrete systems (12) and (13). Let us assume that (12) and (13) are equivalent, and, using the notation of Lemma 5, let   P11 P12 ∈ D(dx +du )×(dx +du ) , P = P21 P22

where P11 ∈ Ddx ×dx , P12 ∈ Ddx ×du , P21 ∈ Ddu ×dx , P22 ∈ Ddu ×du , denote the matrix defining the isomorphism between the D-modules respectively associated to (12) and (13), and     P11 P12  ∈ D(dx +du )×(dx +du ) , P =   P21 P22

   ∈ Ddu ×dx , ∈ Ddx ×du , P21 ∈ Ddx ×dx , P12 where P11 du ×du  , denote the matrix defining the inverse P22 ∈ D morphism. Then, we have the following results:

Finally we have the following result: Corollary 18. With the notation and assumptions of Proposition 16, we get that:

(1) Applying the state feedback control law u = K x with K ∈ Qdu ×dx to (12) is equivalent to applying the control law (−K P11 +P21 ) x +(−K P12 +P22 ) u = 0 to (13). (2) Applying the state feedback control law u = K  x with K  ∈ Qdu ×dx to (13) is equivalent to apply   ing the control law (−K  P11 + P21 ) x + (−K  P12 +  P22 ) u = 0 to (12).

(1) The state feedback control law u = K x with K ∈ Qdu ×dx stabilizes (12) iff the control law (−K P11 + P21 ) x + (−K P12 + P22 ) u = 0 stabilizes (13). (2) The state feedback control law u = K  x with K  ∈  Qdu ×dx stabilizes (13) iff the control law (−K  P11 +     P21 ) x + (−K P12 + P22 ) u = 0 stabilizes (12).

Proof. Applying the state feedback control law u = K x, with K ∈ Qdu ×dx to(12)  amounts to adding the new x equations (−K Idu ) = 0 to (12). The result is then u straightforward since the change of variables associated to  the  equivalence    transformation is given by the relation x x , so that the new equivalent equations = P u u on the variables x andu to be added to (13) are then x = 0, which is equivalent to given by (−K Idu ) P u  (−K P11 + P21 ) x + (−K P12 + P22 ) 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 R2 , Rs xs = 0, Rs = −K Idu and

Rs xs = 0, Rs =



 R2 R1 , −K P11 + P21 −K P12 + P22

 R1 R2 ,     −K  P11 + P21 −K  P12 + P22



Proof. This can be straightforwardly deduced from Corollary 17, Theorem 10 applied to the equivalent autonomous linear systems Rs xs = 0 and Rs xs = 0 and Definitions 12 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 2 ∀(λ1 , λ2 ) ∈ S , det(R1 (λ1 , λ2 )) = 0, where R1 (λ1 , λ2 ) = (Idx λ1 λ2 − F1 λ1 − F2 λ2 − F3 ). On the other hand, (10) is 2 structurally stable iff ∀(λ1 , λ2 ) ∈ S , det(R1 (λ1 , λ2 )) = 0, where   Idx λ1 − F2 −(F2 F1 + F3 ) −(F2 G1 + G3 )  −Idx I d x λ 2 − F1 −G1 R1 (λ1 , λ2 ) = . 0 0 I d u λ2 Now, det(R1 (λ1 , λ2 )) = det(U (λ1 , λ2 )) det(Idu λ2 ),

are equivalent in the sense of algebraic analysis. (2) The linear autonomous systems    R1 R2    , Rs xs = 0, Rs = −K  Idu

where U (λ1 , λ2 ) := 140



 Idx λ1 − F2 −(F2 F1 + F3 ) , −Idx I d x λ 2 − F1

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and det(U (λ1 , λ2 )) is equal to det ((Idx λ1 − F2 ) (Idx λ2 − F1 ) − Idx (F2 F1 + F3 )) , since all the matrices are square and Idx commutes with every square matrix of size dx . We then get det(R1 (λ1 , λ2 )) = λd2u det(R1 (λ1 , λ2 )),

[9] [10]

2

so that, for all (λ1 , λ2 ) ∈ S , det(R1 (λ1 , λ2 )) = 0 is equiva2 lent to det(R1 (λ1 , λ2 )) = 0 (because, for all (λ1 , λ2 ) ∈ S , λ2 = 0). Theorem 20. Let us consider a Fornasini model (4) and the equivalent Roesser model (10). The state feedback control law  h x   u =K , (17) xv

with K  = (K1 K2 ), K1 ∈ Qdu ×dx , K2 ∈ Qdu ×(dx +du )     and K2 = (K21 K22 ) with K21 ∈ Qdu ×dx , K22 ∈ Qdu ×du , stabilizes the Roesser model (10) iff the control law   (−K1 (Idx σj − F1 ) − K21 ) x+(K1 G1 − K22 + Idu σj ) u = 0, stabilizes the Fornasini model (4). Proof. From Corollary 18.(2), the state feedback control  law (17) stabilizes (10) iff the control law (−K  P11 +    P21 ) x + (−K  P12 + P22 ) u = 0 stabilizes (4). Now using the formula for P  ∈ D(2 dx +2 du )×(dx +du ) provided in Lemma 7.(2), we get the desired result.

[11]

[12] [13]

[14]

[15]

Putting together Theorem 20 and the recent results obtained in [1] allows us to develop a new method for stabilizing linear 2D discrete Fornasini models. See [2].

[16]

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