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Journal of Control Science and Engineering
Volume 2012 (2012), Article ID 609276, 6 pages
http://dx.doi.org/10.1155/2012/609276
Research Article

Strict System Equivalence of 2D Linear Discrete State Space Models

Department of Mathematics and Statistics, Sultan Qaboos University, P.O. Box 36, Al-Khodh, 123 Muscat, Oman

Received 10 October 2011; Revised 30 January 2012; Accepted 6 February 2012

Academic Editor: L. Z. Yu

Copyright © 2012 Mohamed S. Boudellioua. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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=[𝑧1,𝑧2] denote the ring of polynomials in the indeterminates 𝑧1 and 𝑧2 with coefficients in a specified field of real numbers . 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 𝑥𝑝 is the state vector, 𝑢𝑛 is the input vector, and 𝑦𝑚 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=𝑇𝑈𝑉𝑊.(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 𝑃1𝑆1𝑃𝑇1𝑆𝑇1𝑇(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 𝑃𝐷𝑝×𝑞, the 𝑖th-order determinantal ideal 𝑖 of 𝑃 be the polynomial ideal generated by the 𝑖×𝑖 minors of 𝑃.

The determinantal ideals 𝑖 of 𝑃 satisfy the following:012𝑡,(5) where 𝑡 is the normal rank of 𝑃 and 0=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(𝑚,𝑛) are related by zero coprime equivalence, and let [𝑃]𝑗 for 𝑗=1,,=min(𝑟[𝑃]+𝑚,𝑟[𝑃]+𝑛) denote the ideal generated by the 𝑗×𝑗 minors of 𝑃, and let [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 [𝑃]𝑖=[𝑄]𝑘𝑖,𝑖=0,,,(7) where =min(1,𝑘1) and 𝑖>, [𝑃]𝑖=1,(8) or [𝑄]𝑘𝑖=1(9) in case 𝑖< or 𝑖<𝑘.

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

Definition 5. Let (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(𝑚,𝑛) are said to be F-SSE if they are related by the following: 𝑀0𝑋𝐼𝑚𝑆1𝑇1𝑈1𝑉1𝑊1𝑃2=𝑇2𝑈2𝑉2𝑊2𝑃1𝑁𝑌0𝐼𝑛𝑆2(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,𝑇𝑉.(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)=𝐴11𝐴12𝐴21𝐴22𝑥1(𝑖,𝑗)𝑥2(𝑖,𝑗)+𝐵1𝐵2𝑢(𝑖,𝑗),𝑦(𝑖,𝑗)=𝐶1𝐶2𝑥1(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), 𝐴𝑖𝑗, 𝐵𝑖, 𝐶𝑖, 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:𝑃R=𝑧1𝐼𝐴11𝐴12𝐵1𝐴21𝑧2𝐼𝐴22𝐵2𝐶1𝐶2𝐷.(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 is𝑃𝐴=𝑧1𝑧2𝐼𝑧1𝐴2𝑧2𝐴1+𝐴1𝐴2𝐵𝐶0.(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 is𝑃FM1=𝑧1𝑧2𝐼𝑧1𝐴2𝑧2𝐴1𝐴0𝐵𝐶0.(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:𝑃FM2=𝑧1𝑧2𝐼𝑧1𝐴2𝑧2𝐴1𝑧1𝐵2+𝑧2𝐵1𝐶0.(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=0𝐵2𝐶=𝐶1𝐶2.(20) With these assumptions, the system matrix in (19) become:PFM2s=𝑧2𝑧1𝐼𝐴11𝑧2𝐴12𝑧2𝐵1𝑧1𝐴21𝑧1𝑧2𝐼𝐴22𝑧1𝐵2𝐶1𝐶20.(21) The PMD in (21) is clearly not F-SSE to the PMD in (13). However if a weaker transformation over the rational ring (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 𝑃FM1 in the form (17), then 𝑃FM1 is related to the Roesser matrix description 𝑃𝑅 in the form (13) by the following F-SSE transformation: 𝑆1𝑃FM1=𝑃𝑅𝑆2,(22) where 𝑆1=𝐼0000𝐼,𝑆2=𝑧2𝐼𝐴20𝐼00𝐼,(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 𝑧1𝑧2𝐼𝑧1𝐴2𝑧2𝐴1𝐴0𝐵00𝐶0.(25) The zero left coprimeness of 𝑃𝑅 and 𝑆1 follows from the fact that the compound matrix 𝑃𝑅𝑆1𝑧1𝐼𝐴1𝐴1𝐴2𝐴0𝐵𝐼0𝐼𝑧2𝐼𝐴20000𝐶00𝐼(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 𝑃FM1 and 𝑆2 follows from the fact that the compound matrix 𝑃FM1𝑆2𝑧1𝑧2𝐼𝑧1𝐴2𝑧2𝐴1𝐴0𝐵𝐶0𝑧2𝐼𝐴20𝐼00𝐼(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, 𝐴𝑖, 𝐵𝑖, 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:𝑃GM=𝑧1𝑧2𝐸𝑧1𝐴2𝑧2𝐴1𝐴0𝑧1𝐵2+𝑧2𝐵1+𝐵0𝐶𝐷.(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 description𝑃SFM1=𝑧1𝑧2𝐸𝑧1𝐴2𝑧2𝐴1𝐴0𝐵0𝐶𝐷(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),𝑃SFM2=𝑧1𝑧2𝐸𝑧1𝐴2𝑧2𝐴1𝐴0𝑧1𝐵2+𝑧2𝐵1𝐶𝐷.(31) The matrix description associated with the singular Attasi model (SA) is obtained from (30) by setting 𝐴0=𝐴1𝐴2=𝐴2𝐴1, that is,𝑃SA𝑧1,𝑧2=𝑧1𝑧2𝐸𝑧1𝐴2𝑧2𝐴1+𝐴1𝐴2𝑧1𝐵2+𝑧2𝐵1+𝐵0𝐶𝐷.(32) A different type of singular model is the singular Roesser-type model (SR):𝐸𝑥1(𝑖+1,𝑗)𝑥2(𝑖,𝑗+1)=𝐴11𝐴12𝐴21𝐴22𝑥1(𝑖,𝑗)𝑥2(𝑖,𝑗)+𝐵1𝐵2𝑢(𝑖,𝑗)𝑦(𝑖,𝑗)=𝐶1𝐶2𝑥1(𝑖,𝑗)𝑥2(𝑖,𝑗)+𝐷𝑢(𝑖,𝑗),(33) where the matrix 𝐸 is singular. The matrix description arising from (33) is of the form𝑃SR=𝑧1𝐸2+𝑧2𝐸1𝐴𝐵𝐶𝐷.(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 𝑃GM in the form (29), then 𝑃GM is F-SSE to a singular Roesser matrix description 𝑃SR in the form (34). That is, 𝑆1𝑃GM=𝑃SR𝑆2,(35) where 𝑃SR=𝐼𝑧1𝐼000𝑧2𝐸𝐴1𝑧2𝐴2𝐴0𝑧1𝐵2+𝑧2𝐵1+𝐵0000𝐶𝐷𝐼𝑚000𝐼𝑛0𝐼𝑛000𝐼𝑚0.(36) The matrices 𝐸1, 𝐸2, 𝐴, 𝐵, 𝐶, 𝐷 corresponding to (34) are obtained from the matrices 𝐸, 𝐴𝑖, 𝐵𝑖, 𝐶, 𝐷 given in (29). 𝑆1=00𝐼000000𝐼𝑚,𝑆2=𝑧1𝐼0𝐼00𝐼𝑛𝐶𝐷0𝐼𝑛.(37)

Proof. Consider the following system transformation: 𝑆1𝑃GM=𝑃SR𝑆2,(38) where 𝑆1=00𝐼𝑟000000𝐼𝑚,𝑆2=𝑧1𝐼𝑟0𝐼𝑟00𝐼𝑛𝐶𝐷0𝐼𝑛,(39) and the resulting singular Roesser-type system matrix 𝑃SR is given by 𝐼𝑟𝑧1𝐼𝑟000𝑧2𝐸𝐴1𝑧2𝐴2𝐴0𝑧1𝐵2+𝑧2𝐵1+𝐵0000𝐶𝐷𝐼𝑚000𝐼𝑛0𝐼𝑛000𝐼𝑚0.(40) The matrices 𝐸1, 𝐸2, 𝐴, 𝐵, 𝐶, and 𝐷 corresponding to (34) are 𝐸1=0000𝐸𝐴2𝐵1000000000,𝐸2=0𝐼𝑟0000𝐵2000000000,𝐴=𝐼𝑟000𝐴1𝐴0𝐵000𝐶𝐷𝐼𝑚00𝐼𝑛0,𝐵=000𝐼𝑛𝑇,𝐶=000𝐼𝑚,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 00𝑧1𝑧2𝐸𝑧1𝐴2𝑧2𝐴1𝐴0𝑧1𝐵2+𝑧2𝐵1+𝐵00000𝐶𝐷.(42) The zero left coprimeness of 𝑃SR and 𝑆1 follows from the fact that the matrix 𝑃SR𝑆1(43) has the highest-order minor |||||||||||||||𝐼𝑟0000𝑧2𝐸𝐴100𝐼𝑟00𝐼𝑚00000𝐼𝑛000𝐼𝑚00𝐼𝑚|||||||||||||||=1(44) obtained by deleting the second and third block columns. Similarly the zero right coprimeness of 𝑃GM and 𝑆2 follows from the fact that the matrix 𝑃GM𝑆2(45) has the highest-order minor ||||||𝐼𝑟00𝐼𝑛||||||=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.

References

  1. H. H. Rosenbrock, State Space and Multivariable Theory, Nelson-Wiley, London, UK, 1970.
  2. P. A. Fuhrmann, “On strict system equivalence and similarity,” International Journal of Control, vol. 25, no. 1, pp. 5–10, 1977. View at Scopus
  3. D.S. Johnson, Coprimeness in multidimensional system theory and symbolic computation, Ph.D. thesis, Loughborough University of Technology, Leicestershire, UK, 1993.
  4. A. C. Pugh, S. J. McInerney, M. Hou, and G. E. Hayton, “A transformation for 2-D systems and its invariants,” in Proceedings of the 35th IEEE Conference on Decision and Control, pp. 2157–2158, Kobe, Japan, December 1996. View at Scopus
  5. A. C. Pugh, S. J. McInerney, M. S. Boudellioua, D. S. Johnson, and G. E. Hayton, “A transformation for 2-D linear systems and a generalization of a theorem of Rosenbrock,” International Journal of Control, vol. 71, no. 3, pp. 491–503, 1998. View at Scopus
  6. E. Zerz, “On strict system equivalence for multidimensional systems,” International Journal of Control, vol. 73, no. 6, pp. 495–504, 2000. View at Scopus
  7. A. C. Pugh, S. J. McInerney, M. S. Boudellioua, and G. E. Hayton, “Matrix pencil equivalents of a general 2-D polynomial matrix,” International Journal of Control, vol. 71, no. 6, pp. 1027–1050, 1998. View at Scopus
  8. A. C. Pugh, S. J. McInerney, and E. M. O. El-Nabrawy, “Equivalence and reduction of 2-D systems,” IEEE Transactions on Circuits and Systems II, vol. 52, no. 5, pp. 271–275, 2005. View at Publisher · View at Google Scholar · View at Scopus
  9. M. S. Boudellioua, “An equivalent matrix pencil for bivariate polynomial matrices,” International Journal of Applied Mathematics and Computer Science, vol. 16, no. 2, pp. 175–181, 2006. View at Scopus
  10. M. S. Boudellioua, “On the simplification of systems of linear multidimensional equations,” in The Sage Days 24 Workshop on Symbolic Computation in Differential Algebra and Special Functions, Hagenberg, Austria, 2010.
  11. R. P. Roesser, “A discrete state-space model for linear image processing,” IEEE Transactions on Automatic Control, vol. 20, no. 1, pp. 1–10, 1975. View at Scopus
  12. S. Attasi, “Systemes lineaires a deux indices,” Tech. Rep. 31, INRIA, Le Chesnay, France, 1973.
  13. E. Fornasini and G. Marchesini, “State space realization theory of twodimensional filters,” IEEE Transactions on Automatic Control, vol. 21, no. 4, pp. 484–492, 1976. View at Scopus
  14. T. Kaczorek, “Singular models of 2-D systems,” in Proceedings of the 12th World Congress on Scientific Computation, Paris, France, 1988.
  15. E. Zerz, “Primeness of multivariate polynomial matrices,” Systems and Control Letters, vol. 29, no. 3, pp. 139–145, 1996. View at Publisher · View at Google Scholar · View at Scopus
  16. E. Zerz, Topics in Multidimensional Linear Systems Theory, Springer, London, UK, 2000.
  17. A. C. Pugh, S. J. McInerney, and E. M. O. El-Nabrawy, “Zero structures of n-D systems,” International Journal of Control, vol. 78, no. 4, pp. 277–285, 2005. View at Publisher · View at Google Scholar · View at Scopus
  18. E. Fornasini and G. Marchesini, “Doubly-indexed dynamical systems: State-space models and structural properties,” Mathematical Systems Theory, vol. 12, no. 1, pp. 59–72, 1978. View at Publisher · View at Google Scholar · View at Scopus
  19. M. Morf, B. C. Levy, and S. Y. Kung, “New results in 2-D systems theory: part II: state-space models—realization and the notions of controllability, observability, and minimality,” Proceedings of the IEEE, vol. 65, no. 6, pp. 945–961, 1977. View at Scopus
  20. T. Kaczorek, Two-Dimensional Linear Systems, Springer, London, UK, 1985.
  21. T. Kaczorek, “Equivalence of singular 2-D linear models,” Bulletin of the Polish Academy of Sciences-Technical Sciences, vol. 37, no. 5-6, pp. 263–267, 1989.