<inline-formula><tex-math>$H_{\infty} $</tex-math></inline-formula> Fuzzy Control Synthesis for a Large-Scale System With a Reduced Number of LMIs

Abstract

This paper introduces an H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> fuzzy control synthesis method for a nonlinear large-scale system with a reduced number of linear matrix inequalities (LMIs). It is well known that a nonlinear large-scale system can be transformed to a Takagi-Sugeno (T-S) fuzzy system by using “sector nonlinearity” or “local approximation in fuzzy partition spaces” methods. Next, in order to achieve the fuzzy control design for this T-S fuzzy system, we solve the stabilization conditions that are represented by the LMIs. However, if the number of LMIs is large, the control design process may become very complicated. In this study, based on the Lyapunov method and S-procedure, several theorems are proposed for the synthesis of parallel distributed compensation (PDC)-type fuzzy control such that the nonlinear large-scale system achieves H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> control performance, and the number of LMIs to be solved is reduced explicitly. As a result, the control design process will become much easier. Furthermore, if the modeling error between the nonlinear system and T-S fuzzy system exists, the robust H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> control performance and the number reduction of LMIs are also achieved by the proposed theorem. Several examples are presented in this paper to show the number reduction effect of LMIs and the effectiveness of the proposed controller synthesis.