H∞ Control of T-S Fuzzy Singularly Perturbed Systems Using Multiple Lyapunov Functions

In this paper, H∞ controller synthesis of T-S fuzzy singularly perturbed systems based on fuzzy and non-fuzzy multiple Lyapunov functions is discussed. By assuming some lower bounds for the grades of fuzzy membership functions and using the elimination lemma, the design conditions are presented in the form of linear matrix inequalities (LMIs). Considering ε as the singular perturbation parameter, it is shown that the ε-dependent controller in the absence of disturbances, results in an asymptotically stable closed-loop system, and in the presence of disturbances, satisfies the H∞-norm condition for all ε∈(0,ε∗]. The resulting LMIs are feasible for larger values of ε∗ compared to those of the previous methods. Moreover, for the case that the value of ε is not available for feedback, Finsler’s lemma is used to separate the controller gains and the ε-dependent Lyapunov matrix, and to achieve an ε-independent control. An example is presented to illustrate the validity of the design techniques.