Robust stability of fuzzy logic control systems

Abstract

An approximate method is formulated for analyzing the performance of a broad class of linear and nonlinear systems controlled using fuzzy logic. It is assumed that an approximate model of the nominal plant is available. Stability in the presence of small, bounded parametric uncertainty and external disturbance of a known origin is considered. The method is based on an appropriate formulation of a performance measure as a Lyapunov-like function of the nominal plant. A heuristic measure of the system's error sensitivity with respect to the parameters of interest is incorporated into this performance expression which is to be minimized. By applying a Lyapunov-like stability condition, the robustness of the system is analyzed by observing a definiteness condition of a simple matrix. The theory is developed in two parts. First, it is assumed that the states are decoupled in the sense that a parameter perturbation in a particular state has no effect on other states. The main robust stability result is developed based on this assumption. In the second part, some interaction is allowed to exist between states as a result of parameter variation. Estimates of the measure of the interactions are derived on stability grounds. A more general result for robust stability is then given. The point of stability convergence is the system's specified target point. Thus, stability convergence to this point, in the presence of parameter perturbations and external load inherently addresses the robust stability of the closed-loop system. Finally, inequality bounds are derived for the sensitivity, error deviation and parameter deviations in terms of fuzzy quantities. A measure for robustness is then formulated using singular values. A fuzzy automotive engine idle speed controller is used to illustrate the methodology.