Stabilization of Two-Dimensional T–S Fuzzy Systems

Abstract

In the last two decades, the two-dimensional (2D) system theory has been paid considerable attention by many researchers. The 2D linear models were introduced in the 70s [1,2] [6]). On the other hand, the stabilization problem has not been fully investigated and is not completely solved. The stability of 2D discrete linear systems can be reduced to checking the stability of 2D characteristic polynomial [7,8]. This appears to be a difficult task for the control synthesis problem. In the literature, various types of easily checkable, but only sufficient conditions for asymptotic stability and stabilization problems for 2D discrete linear systems have been proposed [9][10][11][12][13].This chapter is interested to nonlinear 2D model Roesser systems described with T-S models. The obtained fuzzy system is then a set of r linear 2D systems linked between themselves by membership functions. The objective of this work is to design stabilizing controllers for this class of systems. To the best of our knowledge, no works have been directly considered fuzzy 2D systems before, except the work of the authors [14]. To this end, common Lyapunov quadratic and multiple Lyapunov functions are used. In this context, sufficient conditions of stabilizability are presented. Furthermore, these conditions are presented in the form of a set of LMIs for the state feedback control case. The results of this chapter were initially published in [14][15][16].

H∞ tracking control of two‐dimensional fuzzy networked systems

“…Specifically, due to the inevitable nonlinearity, the Takagi‐Sugeno (T‐S) fuzzy model has been widely employed in modeling and control problems of 2D nonlinear systems. The stability and stabilization of 2D T‐S fuzzy systems have been studied in References 4‐12. And there has been a flurry of research on H ∞ control of 2D T‐S fuzzy systems, 10,13‐15 including 2D fuzzy systems with stochastic perturbation 14 and that with time‐delays and missing measurements 15 …”

Output feedback stabilization of two-dimensional fuzzy systems

“…So far, some efforts have been made to utilize fuzzy-model-based control for 2D nonlinear and other complex systems. Sufficient conditions of stabilizability are presented for 2D fuzzy systems described with Roesser and Fornasini-Marchesini models (Benzaouia and El Hajjaji, 2014;Li and Wang, 2012;Chen et al, 2013;Hien and Trinh, 2017). However, the above controllers considered are state feedback controllers.…”