A Fuzzy-Neural Variable Structure Control for Nonlinear Time-Varying Delay Systems

In this paper, a partially known nonlinear dynamic system with input and state time-varying delay was approximated by N fuzzy-based linear subsystems described by state-space model with average-delay. For tracking the trajectory with a primary frequency, the fuzzy reference models with desired amplitude and phase features were established. Similarly, the same fuzzy sets of the system rule were employed to design a fuzzy-neural variable structure control (FNVSC). The proposed control contained a radial basis neural network to learn the uncertainties caused by the fuzzy-model error and the interactions resulting from the other subsystems. As the norm of the switching surface was inside of a defined set (e.g., ||sigma(t)|| < n_sigma2) the learning law started; the proposed method was an adaptive control possessing a compensation of uncertainties. As it was outside of the other set (e.g., ||sigma(t)|| > n_sigma1 , where n_sigma1 > n_sigma2) the learning law stopped; the proposed method became a robust control. A transition between robust control and adaptive control was also assigned to smooth the possible discontinuity of control input. In addition, no assumption about the upper bound of the time-varying delay for the state and the input is required; however, a time-average delay is needed for the controller design. The stability of the overall system was verified by Lyapunov stability theory