H∞ control for interval type-2 Takagi–Sugeno fuzzy systems via the membership-quadratic framework

This paper aims to explore $H_{\infty }$ control synthesis for interval type-2 Takagi–Sugeno fuzzy systems using novel relaxation techniques. To tackle the conservatism of the previous works with respect to the utilization of membership properties, relaxation techniques are proposed within two distinct aspects. Based on a comprehensive interpretation of the properties of membership functions, an extrema-based algorithm to develop a convex polytope enclosing type-2 fuzzy membership distributions is proposed for membership-dependent stability analysis. To account for real-world conditions, this work utilizes constraints on the derivatives of lower and upper membership functions rather than the embedded membership function, in contrast to recent studies. In such a way as Finsler's lemma and relaxation principle, relaxation techniques that exploit lower, upper, and embedded membership properties with the reasonable constraints are established within a membership-quadratic framework with a matrix shrink lemma to save computational resources. As a result, the less conservative stabilization condition in the shape of the membership-quadratic framework is facilitated by the proposed algorithm and relaxation techniques. Finally, the effectiveness of the proposed methods is demonstrated through numerical examples.