Globally fuzzy model based adaptive variable structure control for a class of nonlinear time-varying systems

In this paper, a nonlinear time-varying dynamic system is first approximated by N fuzzy-based linear state-space subsystems. To track a trajectory dominant by a specific frequency, the reference models with desired amplitude and phase features are established by the same fuzzy sets of the system rule. It is known that linear state feedback control for each fuzzy subsystem is inferior to that using nonlinear feedback control. It is also known that most of the fuzzy adaptive controls must be in a specific domain for the function approximation. To overcome the above shortcomings, we propose a globally fuzzy model based adaptive variable structure control with a switching function to determine when the learning law should be used. As the norm of the switching surface is inside of a defined set, the learning law starts; simultaneously, as it is outside of the other set which is larger than the previous defined set, the learning law stops. In this situation, the proposed control is verified to converge into a convex set, which is smaller than the set for the function approximation. For the purpose of smoothing the discontinuity of control input, a transition between outside and inside of approximated set is also assigned. Under these circumstances, the proposed control can automatically tune as a control without or with the learning compensation of uncertainties. Finally, the stability of the overall system is verified by Lyapunov stability theory.