Fuzzy Static Output Control of T–S Fuzzy Stochastic Systems via Line Integral Lyapunov Function

Abstract

Considering some unmeasurable states, a fuzzy static output control problem of nonlinear stochastic systems is discussed in this paper. Based on a modelling approach, a Takagi–Sugeno (T–S) fuzzy system, constructed by a family of stochastic differential equations and membership functions, is applied to represent nonlinear stochastic systems. Parallel distributed compensation (PDC) technology is used to construct the static output controller. A line-integral Lyapunov function (LILF) is used to derive some sufficient conditions for guaranteeing the asymptotical stability in the mean square. From the LILF, a potential conservatism produced by the derivative of the membership function is eliminated to increase the relaxation of sufficient conditions. Furthermore, those conditions are transferred into linear matrix inequality (LMI) form via projection lemma. According to the convex optimization algorithm, the feasible solutions are directly obtained to establish the static output fuzzy controller. Finally, a numerical example is applied to demonstrate the effectiveness and usefulness of the proposed design method.

Passive Stabilization of Static Output Feedback of Disturbed Nonlinear Stochastic System

“…Remark 1. In [18], the line-integral Lyapunov function was applied to develop the stability criterion for the stability in the mean square. However, the disturbance effect on the systems was not considered by [18].…”

“…In [18], the line-integral Lyapunov function was applied to develop the stability criterion for the stability in the mean square. However, the disturbance effect on the systems was not considered by [18]. To extend the result, passivity is used in this paper to guarantee the stability and attenuation of the nonlinear stochastic systems in the mean square.…”

“…To avoid the conservatism, a Lyapunov function constructed using the line-integral concept was developed by [15] for eliminating the derivative membership function during the stability analysis. Referring to [16][17][18][19], some diagonal matrices and an off-diagonal matrix are required for forming the positive definite matrices in the line-integral Lyapunov function owing to the path-independent property. Because of the unique structure of positive definite matrices, the difficulty of obtaining LMI conditions is increased.…”

Synthesis of Polynomial Fuzzy Model-Based Designs with Synchronization and Secure Communications for Chaos Systems with H∞ Performance

“…Second, the LMI-based stability conditions will be very tentative if the quadratic Lyapunov function is employed. Thus, some literature has taken relaxed stability conditions into account [13][14][15]. In [13], the conservativeness of stabilization criterion for the T-S fuzzy system was significantly relaxed using the constraints condition of the controller membership functions.…”

“…In [13], the conservativeness of stabilization criterion for the T-S fuzzy system was significantly relaxed using the constraints condition of the controller membership functions. From a line-integral Lyapunov function, a possible conservatism generated by the derivation of the membership function was removed to raise the relaxation of the sufficient conditions [14]. In [15], the piecewise fuzzy Lyapunov function was introduced and less conservatism was attained.…”