Direct Adaptive Fuzzy Control Scheme With Guaranteed Tracking Performances For Uncertain Canonical Nonlinear Systems

In our recent work [22], we propose an indirect adaptive fuzzy control scheme for uncertain unparametrizable nonlinear systems, which ensures that the number of adaptive law does not increase with the number of fuzzy rules, and the time derivative of the chosen Lyapunov function is negative semidefinite. However, the scheme involves a class of high-order smooth functions, and their derivatives in time, which can make the controller structure become sophisticated especially when the relative degree of system is quite large. To overcome this problem, in the paper, we propose a new direct adaptive fuzzy control scheme based on a class of reduced-order smooth functions. With the scheme, no partial-derivative term is contained in controller and virtual controllers, only one adaptive law needs to be used regardless of the increase of fuzzy rules, and the time derivative of Lyapunov function can be ensured negative semidefinite also. It is proved that all closed-loop signals are bounded, and the tracking error converges to a prescribed interval asymptotically with time converging to infinity. The transient performance and robustness of the proposed scheme are also discussed. Two practical control systems are used to illustrate the effectiveness of the obtained results.