Fuzzy adaptive control for pure-feedback system via time scale separation

A fuzzy adaptive control method is proposed for a class of completely non-affine purefeedback nonlinear systems. By the combination of back-stepping control method and time scale separation, the virtual/actual control inputs are derived from the solutions of a series of fast dynamical equations. This strategy avoids the drawback of "explosion of complexity" inherently existing in the conventional back-stepping design for the pure-feedback system as the dynamic surface control (DSC) method does for the strict-feedback nonlinear system. By using mean value theorem, error system dynamic is obtained for each subsystem. Thus, Lyapunov theory can be employed for the stability analysis. It shows that the developed fuzzy adaptive control scheme achieves semi-global uniform ultimate boundedness of all the signals in the closed loop. Simulation results are presented to show the effectiveness of the approach.