Robust stabilization of the Takagi-Sugeno fuzzy model via bilinear matrix inequalities

Quadratic stability has enabled, mainly via the linear matrix inequality framework, the analysis and design of a nonlinear control system from the local matrices of the system’s Takagi–Sugeno (T–S) fuzzy model. It is well known, however, that there exist stable differential inclusions, hence T–S fuzzy models whose stability is unprovable by a globally quadratic Lyapunov function. At present, literature in the broader area of stability analysis suggests piecewise-quadratic stability as a means to avoid such conservatism. This paper generalizes the idea and proposes a framework that supports less conservative sufficient conditions for the stability of the T–S model by using piecewise-quadratic generalized Lyapunov functions. The advocated approach results in the formulation of the controller synthesis, which, herein, aims for robust stabilization, as a problem of bilinear rather than linear matrix inequalities. Simulation studies, which include an algorithm for solution of bilinear matrix inequalities, demonstrate the proposed method.