On the analysis of direct adaptive fuzzy controllers

The control problem of using adaptive fuzzy controllers for nonlinear unstable systems is reported. In this study, how the parameter matrix P required in the Lyapunov equation for guaranteeing the system stability can affect the system performance is discussed. It can be found that even though the Lyapunov synthesis approach has already proven that the system is asymptotically stable, the parameter matrix P still needs to be selected appropriately besides the symmetric positive definite property. In our simulations, it can also be found that this kind of controller does not converge to a fixed controller, but adaptively adjusts its parameters according to the errors. As a consequence, when the considered system has a sensory noise, the system may gradually become unstable. Ways of restraining such an unbounded phenomenon are proposed in the paper.