Enhanced Control of Nonlinear Cyber-Physical Systems: A Higher-Order Relaxation Matrix-Based Convex Optimization Algorithm

This paper investigates the static output feedback (SOF) control problem of T-S fuzzy systems based on higher-order Lyapunov functions (HO-LFs). Firstly, to alleviate the conservatism of controller design, an advanced high-order time-varying relaxation matrix (HO-TVRM) approach is proposed. The HO-TVRM approach can completely adapt to the homogeneous polynomial analysis framework and effectively utilize the potential characteristics of fuzzy control to broaden the feasible domain of the reference problem. Secondly, to deal with the non-convex constraints caused by SOF control, a more flexible switching sequence convex optimization (SSCO) algorithm is developed, which adopts a switching-type optimization variable structure and combines variable-weight-based switching mechanism to alternately linearize non-convex terms during the iteration process. As a result, a convex optimization problem that approximates the original problem is obtained. Finally, simulations are presented to testify the effectiveness of the relaxed SOF control strategy. Note to Practitioners—As is well known, SOF control is a tricky bilinear matrix inequality (BMI) problem that involves the product of relaxation variables and controller gains. Existing methods typically introduce additional equality constraints and combine matrix decomposition techniques to decouple controllers, relaxation variables, and Lyapunov variables. In the decoupling process, treating the controller as an indirect optimization parameter inevitably introduces approximation errors that degrade computational precision. Therefore, this paper proposes a new SSCO algorithm that does not rely on any auxiliary constraints and treats the controller as a direct optimization parameter, which improves computational accuracy. On the other hand, this paper focuses on the SOF control problem from the perspective of fuzzy membership functions, which provides insights into designing new switching mechanisms using the potential features of fuzzy control. At the same time, according to the homogeneous matrix polynomial technique, a class of high-order matrices that fully matching the homogeneous polynomial analysis framework is obtained by integrating membership functions into the controllers, relaxation matrices, and Lyapunov variables, which provides the possibility of incorporating the switching mechanism to establish the relaxed performance constraints.