Symbolic computation techniques are used to obtain a canonical form for polynomial matrices arising from discrete 2D linear state-space systems. The canonical form can be regarded as an extension of the companion form often encountered in the theory of 1D linear systems. Using previous results obtained by Boudellioua and Quadrat (2010) on the reduction by equivalence to Smith form, the exact connection between the original polynomial matrix and the reduced canonical form is set out. An example is given to illustrate the computational aspects involved. 1. Introduction Canonical forms play an important role in the modern theory of linear systems. In particular, the so-called companion matrix has been used by many authors in the analysis and synthesis of 1D linear control systems. For instance, Barnett [1] showed that many of the concepts encountered in 1D linear systems theory such as controllability, observability, stability, and pole assignment can be nicely linked via the companion matrix. Boudellioua [2] suggested a matrix form which can be regarded as a 2D companion form for a class of bivariate polynomials. These polynomials arise in the study of 2D linear discrete state-space systems describing, for example, 2D image processing systems, as suggested by Roesser [3]. However in that paper, the author did not establish the exact connection between the original matrix and the reduced canonical form. In this paper, using symbolic computation based on the OreModules [4] Maple package the connection between the original polynomial matrix and the canonical form is established. 2. Polynomial Matrices Arising from Linear 2D Systems A 2D system is a system in which information propagates in two independent directions. These systems arise from applications such as image processing and iterative circuits. Several authors (Attasi [5], Fornasini and Marchesini [6], and Roesser [3]) have proposed different state-space models for 2D discrete linear systems. However, it has been shown that Roesser’s model is the most satisfactory and the most general model since the other models can be embedded in it. The model of Roesser is one in which the local state is divided into horizontal and vertical states which are propagated, respectively, horizontally and vertically by first order difference equations. The model has the form: where is the horizontal state vector, is the vertical state vector, is the input vector, and , , , , , and are real constant matrices of appropriate dimensions. System (1) can be written in the polynomial form: where represents an advance operator
References
[1]
S. Barnett, “A matrix circle in linear control theory,” Bulletin of the Institute of Mathematics and Its Applications, vol. 12, no. 6, pp. 173–176, 1976.
[2]
M. S. Boudellioua, “A canonical matrix representation of 2-D linear discrete systems,” Journal for Engineering Sciences, vol. 11, no. 2, pp. 81–90, 1999.
[3]
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.
[4]
F. Chyzak, A. Quadrat, and D. Robertz, “OreModules: a symbolic package for the study of multidimensional linear systems,” in Applications of Time-Delay Systems, J. Chiasson and and J.-J. Loiseau, Eds., vol. 352 of Lecture Notes in Control and Information Sciences, pp. 233–264, Springer, 2007, /http://wwwb.math.rwth-aachen.de/OreModules.
[5]
S. Attasi, “Systemes lineaires a deux indices,” Tech. Rep. 31, IRIA, Paris, France, 1973.
[6]
E. Fornasini and G. Marchesini, “State-space realization theory of two-dimensional filters,” IEEE Transactions on Automatic Control, vol. 21, no. 4, pp. 484–492, 1976.
[7]
H. H. Rosenbrock, State-Space and Multivariable Theory, John Wiley & Sons, London, UK, 1970.
[8]
M. G. Frost and C. Storey, “Equivalence of a matrix over [s; z] with its Smith form,” International Journal of Control, vol. 28, no. 5, pp. 665–671, 1978.
[9]
M. G. Frost and M. S. Boudellioua, “Some further results concerning matrices with elements in a polynomial ring,” International Journal of Control, vol. 43, no. 5, pp. 1543–1555, 1986.
[10]
E. B. Lee and S. H. Zak, “Smith forms over ,” IEEE Transactions on Automatic Control, vol. 28, no. 1, pp. 115–118, 1983.
[11]
Z. Lin, M. S. Boudellioua, and L. Xu, “On the Equivalence and Factorization of Multivariate Polynomial Matrices,” in Proceedings of the IEEE International Symposium on Circuits and Systems, pp. 4911–4914, Island of Kos, Greece, May 2006.
[12]
M. S. Boudellioua and A. Quadrat, “Serre's reduction of linear functional systems,” Mathematics in Computer Science, vol. 4, no. 2, pp. 289–312, 2010.
[13]
A. Fabianska and A. Quadrat, “Applications of the Quillen-Suslin theorem in multidimensional systems theory,” in Grobner Bases in Control Theory and Signal Processing, H. Park and G. Regensburger, Eds., vol. 3 of Radon Series on Computation and Applied Mathematics, pp. 23–106, de Gruyter publisher, 2007.
[14]
D. Quillen, “Projective modules over polynomial rings,” Inventiones Mathematicae, vol. 36, pp. 167–171, 1976.
[15]
A. A. Suslin, “Projective modules over polynomial rings are free,” Soviet Mathematics—Doklady, vol. 17, no. 4, pp. 1160–1164, 1976.
[16]
E. Zerz, Topics in Multidimensional Linear Systems Theory, Springer, London, UK, 2000.
