Structural Relaxation Approach to <i>H</i> <sub>∞</sub> Control With Quadratic Fuzzy Lyapunov Function for Continuous-Time Takagi–Sugeno Fuzzy Systems
Kyung Soo Kim, Jun Hui Lee, PooGyeon Park
Abstract
This article presents an investigation of the stability analysis and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$H_\infty$</tex-math></inline-formula> control synthesis for Takagi–Sugeno fuzzy systems based on a quadratic fuzzy Lyapunov function, which incorporates second-degree information of membership functions. Instead of a multiple summation expression, the Lyapunov function and controller are designed within the structure of the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">membership-quadratic framework</i> , which is a quadratic form with membership-dependent outfactors. A structural relaxation lemma is established based on zero-equality conditions through orthogonal complements and characteristics of the matrix of the quadratic form. On the basis of the proposed structural relaxation approach, the stability conditions for the analysis and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$H_\infty$</tex-math></inline-formula> synthesis are achieved through linear matrix inequalities with elaborate matrix manipulation techniques. Conceptual foundations employing high-degree membership functions are provided to generalize the structural relaxation approach. Comprehensive numerical examples that demonstrate the efficacy of the proposed approach are detailed.