Stability analysis of polynomial fuzzy models via polynomial fuzzy Lyapunov functions

Abstract

ElsevierBernal Reza, MÁ.; Sala, A.; Jaadari, A.; Guerra, T. (2011) AbstractIn this paper, stability of continuous-time polynomial fuzzy models by means of a polynomial generalization of fuzzy Lyapunov functions is studied. Fuzzy Lyapunov functions have been fruitfully used in literature for local analysis of Takagi-Sugeno models, a particular class of the polynomial fuzzy ones.Based on a recent Taylor-series approach which allows a polynomial fuzzy model to exactly represent a nonlinear model in a compact set of the state space, it is shown that a refinement of the polynomial Lyapunov function so as to make it share the fuzzy structure of the model proves advantageous. Conditions thus obtained are tested via SOS software.

Fuzzy control turns 50: 10 years later

“…The SOS approach has given rise to later developments in control [92,93], observers [94], fuzzy polynomial Lyapunov functions [95], both in continuous-time and in discrete-time [96]. Using the above mentioned multipliers, conservatism in local stability results can be reduced [93,97].…”

“…Also, in most modelling problems in which nonlinearities are separately modelled by interpolation between two bounds, the resulting membership functions have tensor-product structure; exploiting this allows reduced conservatism [103,104]. Shapedependent analysis with TS fuzzy models of the partial derivatives of membership functions appears in [95].…”

Control of nonlinear systems by fuzzy observer-controller with unmeasurable premise variables

“…jk andΞ (21) ijk are defined in (15) and (16), respectively. Remark 3: The stability contritions (28) are non-convex, which cannot be solved by current convex programming toolboxes.…”

“…The first method of reducing the conservativeness is considering the permutations of membership functions in the fuzzy summations [9], [10], which can be handled generally by Pólya's theory in [11]. The second approach is exploiting different Lyapunov function candidates such as quadratic Lyapunov function [5], piecewise linear Lyapunov function [12], switching Lyapunov function [13], [14], fuzzy Lyapunov function [15], [16] and polynomial Lyapunov function [14], [17]. The third method is obtaining membershipfunction-dependent stability conditions.…”

Continuous quasi-LPV Systems: how to leave the quadratic Framework?

“…In order to deal with these problems, several works based on quadratic Lyapunov functions were developed [Hong & Langari, 2000], [Lee & al, 2001], ] and nonquadratic Lyapunov functions [Bernal & al 2011b]. Motivated by the new improvements obtained in recent works in non-quadratic approaches for stability analysis and controller design previously cited.…”