Stability and Stabilization of 2D Singular Systems: A Strict LMI Approach

Abstract

This paper deals with the stability and stabilization of 2D singular systems described by a Roesser model. The proposed results are presented in terms of a strict Linear Matrix Inequality (LMI), which makes possible to elaborate a new sufficient admissibility condition. Based on this condition the design of a state feedback controller is also treated, deriving a sufficient condition for the admissibility of the closed loop system. A numerical example is given at the end of the paper to illustrate the effectiveness of the proposed methodology.

Admissibility and robust H∞ controller design for uncertain 2D singular discrete systems with interval time‐varying delays

“…Then, the convex optimization problem can be easily and directly solved by the function mincx of Matlab LMI Toolbox. Remark 6. when the delays in 2-D uncertain singular system (13) are constants, that is, h 1 =h 2 =h and d 1 =d 2 =d, Theorem 3 does not directly provide a feasible solution. However, in that case, by modifying the Lyapunov Krasovskii's functional given in (18) and 𝜉(𝚤, 𝚥) as…”

“…Corollary 1. For a given matrix K c and constant delays h, d the 2-D singular closed-loop delayed system (13) with out parameter uncertainties is admissible with a prescribed H ∞ performance level 𝛾 > 0 if there exist symmetric positive-definite matrices P h , P v , Q h , Q v , Z h , Z v , N h , N v , appropriately dimensioned matrices M i , S, Y, W, and any symmetric matrices R h , R v , (i = 1, 2, 3), such that the following LMIs holds…”

“…Two-dimensional (2-D) systems get an appreciable attentions in the past few decades [1][2][3][4][5] due to their wide applications in many practical areas such as digital data filtering, image processing, [6][7][8] thermal engineering, 9 electrical circuits networks 10 and in the modeling of partial differential equations by Marszalek,11 such as the Darboux equation used in modeling air drying, water stream heating, gas absorption, and so forth. Many control and estimation problems for 2-D systems have been investigated in the literature, for examples stability analysis, stabilization and filter design, for discrete and continuous case has been studied in the work of (Yang et al; 12 Elloumi et al; 13 Badie et al; 14 Benzaouia et al; 15 Hmamed et al; 16 El-Kasri et al 17 ). The phenomena of time delays are considered here, as they are frequently encountered in many control systems such as communication, process control systems, switched control systems and networked control systems.…”

H∞ Control for 2D Singular Continuous Systems

“…Furthermore, 2D singular systems have received great interest from the academic community due to their wide applications to describe physical systems in several practical areas [27,28]. However, a great number of fundamental results on 1D singular systems have been extended to 2D singular systems, but the majority of existing results have been devoted to the discrete case [27,[29][30][31][32][33][34][35][36]. It should be pointed out that in [36], we established necessary and sufficient conditions for the admissibility and the admissibilization using strict LMIs for 2D singular systems described by the Roesser model in the discrete form.…”

“…However, a great number of fundamental results on 1D singular systems have been extended to 2D singular systems, but the majority of existing results have been devoted to the discrete case [27,[29][30][31][32][33][34][35][36]. It should be pointed out that in [36], we established necessary and sufficient conditions for the admissibility and the admissibilization using strict LMIs for 2D singular systems described by the Roesser model in the discrete form. Recently, based on this result, we proposed sufficient conditions for the regularity, causality, and stability for 2D discrete singular systems in the stochastic case in [37].…”