Strict System Equivalence of 2D Linear Discrete State Space Models

Abstract

The connection between the polynomial matrix descriptions (PMDs) of the well-known regular and singular 2D linear discrete state space models is considered. It is shown that the transformation of strict system equivalence in the sense of Fuhrmann provides the basis for this connection. The exact form of the transformation is established for both the regular and singular cases.

1. Introduction

Multidimensional linear systems theory has attracted many researchers in recent years. The reason is the wide application in areas such as signal processing, linear multipass processes, iterative learning control systems, and delay-differential systems. An area of interest has been the equivalence of systems. There are various notions of equivalence proposed in the literature, and the most basic concept of equivalence would require the preservation of the transfer function matrix of the system. For 1D systems, Rosenbrock [1] introduced the well-known definition of strict system equivalence, using a polynomial matrix description (PMD) setting. Subsequently, Rosenbrock’s strict system equivalence was generalized by Fuhrmann [2] for PMDs not necessarily having the same size. A 2D version of Fuhrmann’s definition of strict system equivalence has been proposed by Johnson [3] and Pugh et al. [4]. This transformation often referred to as zero coprime system equivalence or Fuhrmann’s strict system equivalence (F-SSE) displays some interesting properties and has been shown by Pugh et al. [5] and Zerz [6] to provide the basis of a 𝑛-D generalization of Rosenbrock’s least-order characterization. Recently Pugh et al. [7, 8] and Boudellioua [9] have shown that F-SSE forms the basis of the reduction of an arbitrary 2D PMD to various singular state space forms. Furthermore this transformation was shown to form the basis of a simplification method in linear multidimensional systems [10]. In this paper, it is shown that F-SSE provides the connection between the various 2Dstate space linear models proposed by Roesser [11], Attasi [12], Fornasini-Marchesini [13], and Kaczorek [14].

2. Equivalence of 2D PMDs

Let 𝐷=ℝ[𝑧₁,𝑧₂] denote the ring of polynomials in the indeterminates 𝑧₁ and 𝑧₂ with coefficients in a specified field of real numbers ℝ. The notion of polynomial matrix description was first introduced by Rosenbrock [1]. It arises from the polynomial system description𝑇𝑥=𝑈𝑢,𝑦=𝑉𝑥+𝑊𝑢,(1) where 𝑥∈ℝ^𝑝 is the state vector, 𝑢∈ℝ^𝑛 is the input vector, and 𝑦∈ℝ^𝑚 is the output vector. 𝑇, 𝑈, 𝑉, and 𝑊 are polynomial matrices with elements in 𝐷 of dimensions 𝑟×𝑝, 𝑟×𝑛, 𝑚×𝑝, and 𝑚×𝑛, respectively. The system in (1) gives rise to the PMD𝑃=⎛⎜⎜⎝𝑇𝑈−𝑉𝑊⎞⎟⎟⎠.(2) A matrix description (2) in which 𝑇 has full row rank and 𝑉=𝐻𝑇 for some rational matrix 𝐻 is called an admissible matrix description, and its transfer function is𝐺=𝐻𝑈+𝑊.(3) The following definitions are needed for the results of the paper.

Definition 1. Two polynomial matrices 𝑃₁ and 𝑆₁ of appropriate dimensions and with elements in 𝐷 are said to be zero left (right) coprime if the matrix 𝑃₁𝑆₁𝑃^𝑇₁𝑆^𝑇₁^𝑇(4) admits a right inverse over 𝐷.

As in the 1D case, the zero structure of a multivariate polynomial matrix is a crucial indicator of the system behavior. Zerz [15] has shown that the controllability and observability of a system in the behavioral setting is connected to the zero structure of the associated polynomial matrix. The zero structure of a multivariate polynomial matrix is completely captured by the determinantal ideals as given by Zerz [16].

Definition 2. Let 𝑃∈𝐷^𝑝×𝑞, the 𝑖th-order determinantal ideal ℐ_𝑖 of 𝑃 be the polynomial ideal generated by the 𝑖×𝑖 minors of 𝑃.

The determinantal ideals ℐ_𝑖 of 𝑃 satisfy the following:ℐ₀⊇ℐ₁⊇ℐ₂⊇⋯ℐ_𝑡,(5) where 𝑡 is the normal rank of 𝑃 and ℐ₀=𝐷.

Prior to introducing the concept of Fuhrmann’s strict system equivalence in the 2Dsetting, the following transformation relating two polynomial matrices is noted.

Definition 3. Two polynomial matrices 𝑃₁ and 𝑃₂ with elements in 𝐷 are said to be zero coprime equivalent if they are related by 𝑆₂𝑃₁=𝑃₂𝑆₁,(6) where 𝑃₁, 𝑆₁ are zero left coprime and 𝑃₂, 𝑆₂ are zero right coprime.

The transformation of zero coprime equivalence can be regarded as the classical unimodular equivalence coupled with a trivial expansion or deflation of the matrices. Despite its apparently restrictive form, this transformation displays some interesting properties. Johnson [3] and Pugh et al. [4, 17] have shown that zero coprime equivalence exhibits fundamental algebraic properties amongst its invariants as illustrated by the following.

Lemma 4 (Pugh et al. [17]). Suppose that two polynomial matrices 𝑃 and 𝑄∈ℙ(𝑚,𝑛) are related by zero coprime equivalence, and let ℐ^[𝑃]_𝑗 for 𝑗=1,…,ℎ=min(𝑟^[𝑃]+𝑚,𝑟^[𝑃]+𝑛) denote the ideal generated by the 𝑗×𝑗 minors of 𝑃, and let ℐ^[𝑄]_𝑖 for 𝑖=1,…,𝑘=min(𝑟^[𝑄]+𝑚,𝑟^[𝑄]+𝑛) denote the ideal generated by the 𝑖×𝑖 minors of 𝑄. Then ℐ^[𝑃]_ℎ−𝑖=ℐ^[𝑄]_𝑘−𝑖,𝑖=0,…,ℎ,(7) where ℎ=min(ℎ−1,𝑘−1) and ∀𝑖>ℎ, ℐ^[𝑃]_ℎ−𝑖=⟨1⟩,(8) or ℐ^[𝑄]_𝑘−𝑖=⟨1⟩(9) in case 𝑖<ℎ or 𝑖<𝑘.

In the context of PMDs a special case of the transformation given in (6) is used.

Definition 5. Let ℙ(𝑚,𝑛) denote the class of (𝑟+𝑚)×(𝑝+𝑛) admissible PMDs where 𝑚, 𝑛, and 𝑟+𝑝 are fixed positive integers. Two PMDs 𝑃₁ and 𝑃₂∈ℙ(𝑚,𝑛) are said to be F-SSE if they are related by the following: ⎛⎜⎜⎝𝑀0𝑋𝐼_𝑚⎞⎟⎟_{𝑆1}⎠⎛⎜⎜⎝𝑇₁𝑈₁−𝑉₁𝑊₁⎞⎟⎟_{𝑃2}⎠=⎛⎜⎜⎝𝑇₂𝑈₂−𝑉₂𝑊₂⎞⎟⎟_{𝑃1}⎠⎛⎜⎜⎝𝑁𝑌0𝐼_𝑛⎞⎟⎟_{𝑆2}⎠(10) where 𝑃₁, 𝑆₁ are zero left coprime, 𝑃₂, 𝑆₂ are zero right coprime, and 𝑀, 𝑁, 𝑋, and 𝑌 are polynomial matrices over 𝐷 of appropriate dimensions.

Lemma 6 (Pugh et al. [17]). The transformation of F-SSE preserves the transfer function of 𝑃 and the determinantal ideals in the sense of Lemma 4 of the matrices 𝑇,𝑃,𝑇𝑈,⎛⎜⎜⎝𝑇−𝑉⎞⎟⎟⎠.(11)

3. Equivalence of Regular 2D Linear Models

State space models play an important role in the theory of 1D finite-dimensional linear systems. During the last decades extensions have been made of the state space representation to the more general 2D systems. One such extension is the discrete linear state space models proposed by Roesser [11], Attasi [12] and Fornasini-Marchesini [13]. It is well known that these models are not independent of each other and that the standard Roesser model is the most general in which the two other models can be embedded. In the following the exact polynomial connection between the various models is established.

The first 2D standard discrete linear model was introduced by Roesser [11] and can be described by the following equations:⎛⎜⎜⎝𝑥₁(𝑖+1,𝑗)𝑥₂(𝑖,𝑗+1)⎞⎟⎟⎠=⎛⎜⎜⎝𝐴₁₁𝐴₁₂𝐴₂₁𝐴₂₂⎞⎟⎟⎠⎛⎜⎜⎝𝑥₁(𝑖,𝑗)𝑥₂(𝑖,𝑗)⎞⎟⎟⎠+⎛⎜⎜⎝𝐵₁𝐵₂⎞⎟⎟⎠𝑢(𝑖,𝑗),𝑦(𝑖,𝑗)=𝐶₁𝐶₂⎛⎜⎜⎝𝑥₁(𝑖,𝑗)𝑥₂(𝑖,𝑗)⎞⎟⎟⎠+𝐷𝑢(𝑖,𝑗),(12) where 𝑥₁(𝑖,𝑗), 𝑥₂(𝑖,𝑗) are, respectively, the horizontal and vertical state vectors, 𝑢(𝑖,𝑗) is the input vector, 𝑦(𝑖,𝑗) is the output vector at cell (𝑖,𝑗), 𝐴_𝑖𝑗, 𝐵_𝑖, 𝐶_𝑖, and 𝐷 are constant real matrices of appropriate dimensions. Then, taking the 2D 𝑧-transform of (12) and assuming zero boundary conditions yield the PMD:𝑃_R=⎛⎜⎜⎜⎜⎝𝑧₁𝐼−𝐴₁₁−𝐴₁₂𝐵₁−𝐴₂₁𝑧₂𝐼−𝐴₂₂𝐵₂−𝐶₁−𝐶₂𝐷⎞⎟⎟⎟⎟⎠.(13) The second 2D discrete linear model was introduced by Attasi [12]:𝑥(𝑖+1,𝑗+1)=𝐴₂𝑥(𝑖+1,𝑗)+𝐴₁𝑥(𝑖,𝑗+1)−𝐴₁𝐴₂𝑥(𝑖,𝑗)+𝐵𝑢(𝑖,𝑗),𝑦(𝑖,𝑗)=𝐶𝑥(𝑖,𝑗),(14) in which 𝐴₁ and 𝐴₂ are assumed to commute. The corresponding matrix description for this model is𝑃_𝐴=⎛⎜⎜⎝𝑧₁𝑧₂𝐼−𝑧₁𝐴₂−𝑧₂𝐴₁+𝐴₁𝐴₂𝐵−𝐶0⎞⎟⎟⎠.(15) An alternative model was proposed by Fornasini and Marchesini [13] which is given by𝑥(𝑖+1,𝑗+1)=𝐴₂𝑥(𝑖+1,𝑗)+𝐴₁𝑥(𝑖,𝑗+1)+𝐴₀𝑥(𝑖,𝑗)+𝐵𝑢(𝑖,𝑗),𝑦(𝑖,𝑗)=𝐶𝑥(𝑖,𝑗).(16) This model is referred to as the first Fornasini-Marchesini model, and the corresponding matrix description is𝑃_FM1=⎛⎜⎜⎝𝑧₁𝑧₂𝐼−𝑧₁𝐴₂−𝑧₂𝐴₁−𝐴₀𝐵−𝐶0⎞⎟⎟⎠.(17) Another model given by Fornasini and Marchesini [18] is the so-called second Fornasini-Marchesini model:𝑥(𝑖+1,𝑗+1)=𝐴₂𝑥(𝑖+1,𝑗)+𝐴₁𝑥(𝑖,𝑗+1)+𝐵₁𝑢(𝑖,𝑗+1)+𝐵₂𝑢(𝑖+1,𝑗),𝑦(𝑖,𝑗)=𝐶𝑥(𝑖,𝑗).(18) The corresponding system matrix is given by:𝑃_FM2=⎛⎜⎜⎝𝑧₁𝑧₂𝐼−𝑧₁𝐴₂−𝑧₂𝐴₁𝑧₁𝐵₂+𝑧₂𝐵₁−𝐶0⎞⎟⎟⎠.(19)

Because of the assumption on commutativity on 𝐴₁ and 𝐴₂, it has been shown by Morf et al. [19] that Attasi’s model (14) is a special case of that of the first Fornasini-Marchesini model (16) and that the latter can be written in the form of Roesser’s model (12) which itself is a special case of Fornasini-Marchesini second model (18) upon assuming the following in (18):𝐴₁=⎛⎜⎜⎝𝐴₁₁𝐴₁₂00⎞⎟⎟⎠,𝐴₂=⎛⎜⎜⎝00𝐴₂₁𝐴₂₂⎞⎟⎟⎠,𝐵₁=⎛⎜⎜⎝𝐵₁0⎞⎟⎟⎠,𝐵₂=⎛⎜⎜⎝0𝐵₂⎞⎟⎟⎠𝐶=𝐶₁𝐶₂.(20) With these assumptions, the system matrix in (19) become:PFM2s=⎛⎜⎜⎜⎜⎝𝑧₂𝑧₁𝐼−𝐴₁₁−𝑧₂𝐴₁₂𝑧₂𝐵₁−𝑧₁𝐴₂₁𝑧₁𝑧₂𝐼−𝐴₂₂𝑧₁𝐵₂𝐶₁𝐶₂0⎞⎟⎟⎟⎟⎠.(21) The PMD in (21) is clearly not F-SSE to the PMD in (13). However if a weaker transformation over the rational ring ℝ(𝑧₁,𝑧₂) is used, the second Fornasini-Marchesini system matrix (21) can be clearly reduced to a Roesser’s model (12) by simply cancelling the common factors 𝑧₂ and 𝑧₁ in the first and second block rows of PFM2s, respectively. In the following, the exact form of the connection between the PMDs (13) and (17) is established and shown to be F-SSE.

Theorem 7. Given an arbitrary Fornasini-Marchesini matrix description 𝑃_FM1 in the form (17), then 𝑃_FM1 is related to the Roesser matrix description 𝑃_𝑅 in the form (13) by the following F-SSE transformation: 𝑆₁𝑃_FM1=𝑃_𝑅𝑆₂,(22) where 𝑆₁=⎛⎜⎜⎜⎜⎜⎝𝐼0000𝐼⎞⎟⎟⎟⎟⎟⎠,𝑆₂=⎛⎜⎜⎜⎜⎜⎝𝑧₂𝐼−𝐴₂0𝐼00𝐼⎞⎟⎟⎟⎟⎟⎠,(23)𝐴₁₁=𝐴₁,𝐴₁₂=𝐴₁𝐴₂+𝐴₀,𝐴₂₁=𝐼,𝐴₂₂=𝐴₂,𝐵₁=𝐵,𝐵₂=0,𝐶₁=0,𝐶₂=𝐶.(24)

Proof. Clearly the transformation in (22) is in the required form (10), so it remains to prove the equality and the zero coprimeness of the matrices. In fact, it can be easily verified that the LHS and RHS of (22) both yield the matrix ⎛⎜⎜⎜⎜⎝𝑧₁𝑧₂𝐼−𝑧₁𝐴₂−𝑧₂𝐴₁−𝐴₀𝐵00−𝐶0⎞⎟⎟⎟⎟⎠.(25) The zero left coprimeness of 𝑃_𝑅 and 𝑆₁ follows from the fact that the compound matrix 𝑃_𝑅𝑆₁≡⎛⎜⎜⎜⎜⎝𝑧₁𝐼−𝐴₁−𝐴₁𝐴₂−𝐴₀𝐵𝐼0−𝐼𝑧₂𝐼−𝐴₂0000−𝐶00𝐼⎞⎟⎟⎟⎟⎠(26) has the highest-order minor which is equal to 1, obtained by deleting the block columns 2 and 3. Similarly the zero right coprimeness of 𝑃_FM1 and 𝑆₂ follows from the fact that the compound matrix ⎛⎜⎜⎝𝑃_FM1𝑆₂⎞⎟⎟⎠≡⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝𝑧₁𝑧₂𝐼−𝑧₁𝐴₂−𝑧₂𝐴₁−𝐴₀𝐵−𝐶0𝑧₂𝐼−𝐴₂0𝐼00𝐼⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠(27) has the highest-order minor which is equal to 1, obtained from the last two block rows.

4. Equivalence of Singular 2D Linear Models

One of the limitations of the regular 2D models is they can be used only to describe 2D proper transfer functions. In other words, they are suitable only for the representation of northeast quarter plane 2D systems. To overcome this problem of causality, several versions of these models have been proposed by Kaczorek [14, 20] and have been shown in [21] to be equivalent in the sense that they can all be embedded in the 2D singular general model. In the following, the PMDs associated with the various singular models are considered and the exact form of the transformation linking them is again established.

The singular 2D general model (GM) is given by Kaczorek [14]:𝐸𝑥(𝑖+1,𝑗+1)=𝐴₂𝑥(𝑖+1,𝑗)+𝐴₁𝑥(𝑖,𝑗+1)+𝐴₀𝑥(𝑖,𝑗)+𝐵₂𝑢(𝑖+1,𝑗)+𝐵₁𝑢(𝑖,𝑗+1)+𝐵₀𝑢(𝑖,𝑗)𝑦(𝑖,𝑗)=𝐶𝑥(𝑖,𝑗)+𝐷𝑢(𝑖,𝑗),(28) where 𝐸, 𝐴_𝑖, 𝐵_𝑖, 𝐶, 𝐷 are real matrices of appropriate dimensions and 𝐸 is singular. Then taking the 2D 𝑧-transform of (28) and assuming zero boundary conditions yield the PMD:𝑃_GM=⎛⎜⎜⎝𝑧₁𝑧₂𝐸−𝑧₁𝐴₂−𝑧₂𝐴₁−𝐴₀𝑧₁𝐵₂+𝑧₂𝐵₁+𝐵₀−𝐶𝐷⎞⎟⎟⎠.(29) The matrix description in (29) describes a number of singular 2D linear models as special cases. In particular setting 𝐵₁=𝐵₂=0 in (29) gives rise to the matrix description𝑃_SFM1=⎛⎜⎜⎝𝑧₁𝑧₂𝐸−𝑧₁𝐴₂−𝑧₂𝐴₁−𝐴₀𝐵₀−𝐶𝐷⎞⎟⎟⎠(30) which is associated with the first singular Fornasini-Marchesini model (SFM1). Alternatively when 𝐵₀=0 in (29), the resulting matrix description is that of the second singular Fornasini-Marchesini model (SFM2),𝑃_SFM2=⎛⎜⎜⎝𝑧₁𝑧₂𝐸−𝑧₁𝐴₂−𝑧₂𝐴₁−𝐴₀𝑧₁𝐵₂+𝑧₂𝐵₁−𝐶𝐷⎞⎟⎟⎠.(31) The matrix description associated with the singular Attasi model (SA) is obtained from (30) by setting 𝐴₀=−𝐴₁𝐴₂=−𝐴₂𝐴₁, that is,𝑃_SA𝑧₁,𝑧₂=⎛⎜⎜⎝𝑧₁𝑧₂𝐸−𝑧₁𝐴₂−𝑧₂𝐴₁+𝐴₁𝐴₂𝑧₁𝐵₂+𝑧₂𝐵₁+𝐵₀−𝐶𝐷⎞⎟⎟⎠.(32) A different type of singular model is the singular Roesser-type model (SR):𝐸⎛⎜⎜⎝𝑥₁(𝑖+1,𝑗)𝑥₂(𝑖,𝑗+1)⎞⎟⎟⎠=⎛⎜⎜⎝𝐴₁₁𝐴₁₂𝐴₂₁𝐴₂₂⎞⎟⎟⎠⎛⎜⎜⎝𝑥₁(𝑖,𝑗)𝑥₂(𝑖,𝑗)⎞⎟⎟⎠+⎛⎜⎜⎝𝐵₁𝐵₂⎞⎟⎟⎠𝑢(𝑖,𝑗)𝑦(𝑖,𝑗)=𝐶₁𝐶₂⎛⎜⎜⎝𝑥₁(𝑖,𝑗)𝑥₂(𝑖,𝑗)⎞⎟⎟⎠+𝐷𝑢(𝑖,𝑗),(33) where the matrix 𝐸 is singular. The matrix description arising from (33) is of the form𝑃_SR=⎛⎜⎜⎝𝑧₁𝐸₂+𝑧₂𝐸₁−𝐴𝐵−𝐶𝐷⎞⎟⎟⎠.(34) In the following we establish the exact form of the connection between the PMD in the singular general form (29) and the corresponding singular Roesser form in (34). We will show that the transformation involved is that of F-SSE. The connections between all other singular models and the singular Roesser model are deduced as special cases.

Theorem 8. Given an arbitrary (𝑟+𝑚)×(𝑟+𝑛) singular 2D general matrix description 𝑃_GM in the form (29), then 𝑃_GM is F-SSE to a singular Roesser matrix description 𝑃_SR in the form (34). That is, 𝑆₁𝑃_GM=𝑃_SR𝑆₂,(35) where 𝑃_SR=⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝𝐼−𝑧₁𝐼000𝑧₂𝐸−𝐴₁−𝑧₂𝐴₂−𝐴₀𝑧₁𝐵₂+𝑧₂𝐵₁+𝐵₀000−𝐶𝐷𝐼_𝑚000−𝐼_𝑛0𝐼_𝑛000−𝐼_𝑚0⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠.(36) The matrices 𝐸₁, 𝐸₂, 𝐴, 𝐵, 𝐶, 𝐷 corresponding to (34) are obtained from the matrices 𝐸, 𝐴_𝑖, 𝐵_𝑖, 𝐶, 𝐷 given in (29). 𝑆₁=⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝00𝐼000000𝐼_𝑚⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠,𝑆₂=⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝𝑧₁𝐼0𝐼00𝐼_𝑛𝐶−𝐷0𝐼_𝑛⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠.(37)

Proof. Consider the following system transformation: 𝑆₁𝑃_GM=𝑃_SR𝑆₂,(38) where 𝑆₁=⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝00𝐼_𝑟000000𝐼_𝑚⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠,𝑆₂=⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝𝑧₁𝐼_𝑟0𝐼_𝑟00𝐼_𝑛𝐶−𝐷0𝐼_𝑛⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠,(39) and the resulting singular Roesser-type system matrix 𝑃_SR is given by ⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝𝐼_𝑟−𝑧₁𝐼_𝑟000𝑧₂𝐸−𝐴₁−𝑧₂𝐴₂−𝐴₀𝑧₁𝐵₂+𝑧₂𝐵₁+𝐵₀000−𝐶𝐷𝐼_𝑚000−𝐼_𝑛0𝐼_𝑛000−𝐼_𝑚0⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠.(40) The matrices 𝐸₁, 𝐸₂, 𝐴, 𝐵, 𝐶, and 𝐷 corresponding to (34) are 𝐸₁=⎛⎜⎜⎜⎜⎜⎜⎝0000𝐸−𝐴₂𝐵₁000000000⎞⎟⎟⎟⎟⎟⎟⎠,𝐸₂=⎛⎜⎜⎜⎜⎜⎜⎝0−𝐼_𝑟0000𝐵₂000000000⎞⎟⎟⎟⎟⎟⎟⎠,𝐴=⎛⎜⎜⎜⎜⎜⎜⎝−𝐼_𝑟000𝐴₁𝐴₀−𝐵₀00𝐶−𝐷−𝐼_𝑚00𝐼_𝑛0⎞⎟⎟⎟⎟⎟⎟⎠,𝐵=000𝐼_𝑛^𝑇,𝐶=000𝐼_𝑚,𝐷=0.(41) Clearly the transformation in (38) is in the required form (10), so it remains to prove the equality and the zero coprimeness of the matrices. In fact, it can be easily verified that the LHS and RHS of (38) both yield the matrix ⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝00𝑧₁𝑧₂𝐸−𝑧₁𝐴₂−𝑧₂𝐴₁−𝐴₀𝑧₁𝐵₂+𝑧₂𝐵₁+𝐵₀0000𝐶−𝐷⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠.(42) The zero left coprimeness of 𝑃_SR and 𝑆₁ follows from the fact that the matrix 𝑃_SR𝑆₁(43) has the highest-order minor |||||||||||||||𝐼_𝑟0000𝑧₂𝐸−𝐴₁00𝐼_𝑟00𝐼_𝑚00000𝐼_𝑛000−𝐼_𝑚00𝐼_𝑚|||||||||||||||=1(44) obtained by deleting the second and third block columns. Similarly the zero right coprimeness of 𝑃_GM and 𝑆₂ follows from the fact that the matrix ⎛⎜⎜⎝𝑃_GM𝑆₂⎞⎟⎟⎠(45) has the highest-order minor ||||||𝐼_𝑟00𝐼_𝑛||||||=1(46) obtained by deleting all the block rows except the 4th and 5th.

5. Conclusions

The paper established the connection between the PMDs associated with the various regular and singular 2D linear discrete models. The transformation of strict system equivalence in the sense of Fuhrmann turned out to provide such connection. The results show the relevance of this transformation in another area of 2D polynomial systems theory.

Acknowledgment

The author wishes to express his thanks to the Sultan Qaboos University (Oman) for their support in carrying out this research work.