Adaptive Fuzzy Tracking Control for Unknown Nonaffine Nonlinear Systems With Delay

Tracking control for a class of unknown nonaffine nonlinear systems with delay is studied by using adaptive fuzzy approximation approach. The real controller is recursively obtained through a Backstepping design procedure. In each step, an adaptive fuzzy logic system is used to approximate the unknown nonlinear functions. To reduce the dependence on a priori knowledge and release the limitation that the fuzzy basis functions must be completely known beforehand, some nonlinearly parameterized fuzzy logic systems are used as approximators. As the relationships of the unknown parameters in the approximators are nonlinear, the adaptive laws cannot be directly obtained to estimate these parameters on-line. To realize the on-line adjusting for corresponding parameters, Taylor expansion is first used to separate the nonlinear parameters, by which the relationships of the unknown parameters become linear.Then, the adaptive laws for estimating the unknown parameters can be designed on the basis of Lyapunov second method. Finally, a simulation example is presented to demonstrate the effectiveness of the approach.Resultsshow that the proposed adaptive fuzzy controller can compensate for the influence of delay and guarantee all closed-loop signals bounded. Besides, the output tracking error converges to an arbitrarily small neighborhood of the origin.