Adaptive-Fuzzy Control Compensation Design for Direct Adaptive Fuzzy Control

For direct adaptive fuzzy control of perturbed uncertain nonlinear systems, proportional-integral-derivative control compensation usually has to be used to guarantee <inline-formula><tex-math notation="LaTeX">$H_{{{\infty }}}$ </tex-math></inline-formula> tracking performance or <inline-formula><tex-math notation="LaTeX">$L_{2}$</tex-math> </inline-formula>-gain property of the closed-loop system. One common problem is that the control compensation often introduces unwanted high gain at the control input. This paper proves that classical direct adaptive fuzzy control with adaptive-fuzzy control compensation not only is capable of solving the problem of high gain, but also can guarantee <inline-formula><tex-math notation="LaTeX">$L_{2}$</tex-math></inline-formula>-gain property of the closed-loop system. A sliding-surface-based adaptive fuzzy control is designed for control compensation. By using a class of triangular membership functions nearest to the origin and a simple adaptive law, the adaptive fuzzy control term is converted to an equivalence control term, which is used to facilitate stability analysis. Moreover, the system output can track not only the constant-period periodic signals but also the constant signal. Compared with previous direct adaptive fuzzy control approaches that can guarantee <inline-formula><tex-math notation="LaTeX">$H_{{{\infty }}}$</tex-math></inline-formula> tracking performance or <inline-formula><tex-math notation="LaTeX">$L_{2}$</tex-math> </inline-formula>-gain property, in addition to the <inline-formula><tex-math notation="LaTeX">$L_{2}$</tex-math> </inline-formula>-gain property of common tracking error rather than modified tracking error, the major advantages of our approach are that the assumptions on control gain function are relaxed and a less conservative control design on robustness is achieved. Simulation results are given to demonstrate the effectiveness of the proposed approach.