A fuzzy Lyapunov function approach to estimating the domain of attraction for continuous-time Takagi–Sugeno fuzzy systems

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].…”

“…The solution processes are formulated as single-parameter minimization problems subject to LMIs, which can be efficiently solved via a sequence of LMI optimizations such as a bisection algorithm or a line search process, or can be treated as eigenvalue problems (EVPs) [2] directly tractable via LMI solvers [3][4][5]. In comparison with existing results, the distinguished feature of the proposed methods is that while existing approaches [34][35][36][37][38][39]41] employ bounds on each scalar MF, a quadratic bound on the vector containing MFs is considered in this paper. Note that this quadratic bounding technique was also used in [40].…”

Design of polynomial fuzzy observer–controller with membership functions using unmeasurable premise variables for nonlinear systems

“…For the source of double fuzzy summation, Pólya's theory [37,27] is exploited to offer progressively necessary and sufficient conditions which generalizes some earlier 25 works [26,10]. For the source of quadratic Lyapunov function, more general types of Lyapunov function candidates such as fuzzy Lyapunov function [29,5,24,18], piecewise linear Lyapunov function [11,12], switching Lyapunov function [32,21] and polynomial Lyapunov function [4,21] function-independent stability conditions, the membership-function-dependent approach is applied to make the stability conditions depend on membership functions such as using approximated membership functions [30,17], polynomial constraints [38], symbolic variables [39,22,23] and other techniques [3,20,16,18,7]. 35 Another task of the development of FMB control strategy is to extend it to solve control problems [40,33,42,55,43,8,15,25,54].…”