Generalized Nonquadratic Stability of Continuous-Time Takagi–Sugeno Models

Abstract

This paper presents a relaxed approach for stabilization and H ∞ disturbance rejection of continuous-time Takagi-Sugeno models in descriptor form. Based on Finsler's Lemma, the control law can be conveniently decoupled from a non-quadratic Lyapunov function. These developments include and outperform previous results on the same subject while preserving the advantage of being expressed as linear matrix inequalities. Two examples are presented to illustrate the improvements.

Fuzzy control turns 50: 10 years later

“…If feasible, local stabilisation is guaranteed in the modelled region without the need of a posteriori checks. The idea has been developed in [71,67] and embedded into multiple-sum Lyapunov functions in [72,73].…”

LMI conditions for local stability and stabilization of continuous-time T-S fuzzy systems

“…Research directions to reduce the conservatism include approaches searching for general classes of Lyapunov functions, such as piecewise Lyapunov functions [9,10], fuzzy Lyapunov functions (FLFs) [11][12][13][14][15][16], a class of Lyapunov functions using line-integral [17]; polynomial Lyapunov functions [18][19][20][21]; augmented FLFs [22][23][24]; methods using information on MFs' shape [25][26][27], Pólya's theorem [28][29][30][31][32][33]; and local stability approaches [34][35][36][37][38][39][40][41][42][43][44][45]. Amongst the promising research topics, in this paper, we focus on the local stability approaches, which have turned out to be effective in improving the global approaches further [34][35][36][37][38][39][40][41][42]…”

“…Amongst the promising research topics, in this paper, we focus on the local stability approaches, which have turned out to be effective in improving the global approaches further [34][35][36][37][38][39][40][41][42][43][44][45][46][47][48][49][50][51]. Evaluation of the local stability is the problem of determining if there exists a neighborhood of the equilibrium point, called the domain of attraction (DA) [46], such that all trajectories of the system emanating from any initial point in the DA asymptotically converges to the equilibrium point.…”

“…Recently, the problems of assessing the local stability, designing locally stabilizing control laws, and estimating the DA of T-S fuzzy systems have been addressed by many authors [34][35][36][37][38][39][40][41][42][43][44][45].…”

Stability analysis of polynomial fuzzy models via polynomial fuzzy Lyapunov functions

“…This paper is based on two recent works: the first one [Sala09a,Sala09b] provides a systematic way of obtaining exact polynomial fuzzy representations of nonlinear models via a Taylor-series approach, thus generalizing sector nonlinearity approach; the second one [Guerra09,Bernal10] shows how to escape from the quadratic framework by combining local analysis and fuzzy Lyapunov functions for continuous-time TS models. Since local analysis can be easily included via Lagrange multipliers and the Positivstellensatz argumentation in the polynomial framework [Prajna04a,Sala09b], the use of more general Lyapunov functions such as the polynomial fuzzy ones is investigated in this paper as a generalization of those employed in [Guerra09].…”