Fuzzy Covariance Control for Interval Type-2 Fuzzy Stochastic Systems with Individual State Variance Constraints

Abstract

A fuzzy covariance control strategy is proposed for a class of uncertain nonlinear stochastic systems by means of the Interval Type-2 (IT2) Takagi-Sugeno Fuzzy Stochastic Model (TSFSM). By leveraging the IT2 TSFSM, the dynamics of nonlinear systems subject to uncertainties and stochastic behaviors can be characterized in a more comprehensive manner. In line with the IT2 membership function design of the TSFSM, the fuzzy controller is developed using the state feedback technique. Subsequently, the state covariance matrices and the variance constraints are constructed for each subsystem of the IT2 TSFSM. By reformulating the closed-loop model into parallel and cross-terms, the number of fuzzy rules required for the analysis can be effectively reduced. This approach also allows for the integration of covariance control theory into the IT2 fuzzy controller design. Using Lyapunov theory, stability criteria involving upper and lower-bound parameters and state covariance matrices are derived. Unlike typical Type-1 (T1) fuzzy control approaches, the upper and lower-bound membership functions complicate the integration of covariance control theory, which is also overcome in this research. Additionally, the stability and variance constraints are further achieved by applying an upper-bound matrix. The stability conditions are also converted into a Linear Matrix Inequality (LMI) problem. Eventually, a comparative simulation is conducted on the nonlinear ship steering system to demonstrate the validity and applicability of the proposed IT2 fuzzy covariance control strategy.