Structural Relaxation Approach to H∞ Control With Quadratic Fuzzy Lyapunov Function for Continuous-Time Takagi–Sugeno Fuzzy Systems

This article presents an investigation of the stability analysis and
$H_\infty$
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
membership-quadratic framework
, 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
$H_\infty$
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.