Stability and the matrix Lyapunov equation for discrete 2-dimensional systems

Abstract

The stability of two-dimensional, linear, discrete systems is examined using the 2-D matrix Lyapunov equation. While the existence of a positive definite solution pair to the 2-D Lyapunov equation is sufficient for stability, the paper proves that such existence is not necessary for stability, disproving a long-standing conjecture.

On Hermitian and Skew-Hermitian Splitting Ietration Methods for the Continuous Sylvester Equations

“…BAI The continuous Sylvester equation (1.1) has numerous applications in control and system theory [30,36,39], stability of linear systems [22], analysis of bilinear systems [32], power systems [25], linear algebra [16], signal processing [1], image restoration [11], filtering [21,23], model order reduction [35], numerical methods for differential equations [8,9], iterative methods for algebraic Riccati equations [7,[18][19][20]30], matrix nearness problem [34,41], finite element model updating [13,26], block-diagonalization of matrices [16,31] and so on. Many of these applications lead to stable Sylvester equations, i.e., Assumption (A 3 ) made in the above is satisfied.…”

New algorithms for the derivation of the transfer-function matrices of 2-D state-space discrete systems

“…Example 1 is a 2-D discrete system of order (2,6), which was used in [6] for stability analysis of 2-D systems. The system is represented by the Roesser state-space model with the matrices: Algorithms 1 and 2 proposed and the algorithms in [19] and [22] led to the transfer function where The amounts of computation required by the various algorithms are listed in Table I. Example 2 is a two-input two-output system represented by the Roesser model of order (2, 2), which was used to illustrate the algorithm in [19].…”

“…These include methods for stability analysis [2]- [6], analysis of finite-wordlength effects [7], [8], design [9], [10], model reduction [11]- [13], and relevant computation issues [14], [15]. Since many of the available analysis and design methods are applicable only to the 2-D transfer-function matrix, it is often necessary to derive the transfer-function matrix from a state-space description of the system.…”

“…To date, no efficient algorithms for the derivation of the 2-D transferManuscript received October 11, 1994; revised May 10, 1996 (2) 1057-7122/97$10.00 © 1997 IEEE in (2) can be written as (3) where is an integer, and are polynomials in of order not greater than , and It follows that (4) where denotes the characteristic polynomial of in variable . Further, from (2) and the formula of matrix inversion [25], transfer function can be expressed as (5) where Note that is a common term in , , , and ; hence, the above equations can be expressed as (6) By using a well-known formula for the transfer function of a 1-D SISO state-space model (see [25,Appendix A.13]), (5) can be rewritten as (7) where and are the characteristic polynomials of and , respectively. Note that the denominator in (7) is a monic polynomial in but the denominator in (3) is a polynomial in with as the coefficient of .…”

Two Dimensional Sliding Mode Control

“…2-D systems are often applied to theoretical aspects like filter design, image processing, and recently, Iterative Learning Control methods (see for example Roesser, 1975;Hinamoto, 1993;Whalley, 1990;Al-Towaim, 2004;Hladowski et al, 2008). Over the past two decades, the stability of multidimensional systems in various models has been a point of high interest among researchers (Anderson et al, 1986;Kar, 2008;Singh, 2008;Bose, 1994;Kar & Singh, 1997;Lu, 1994). Some new results on the stability of 2-D systems have been presented -specifically with regard to the Lyapunov stability condition which has been developed for RM (Lu, 1994).…”