Robust Stability of Mode-to-Mode Fuzzy Controllers

Abstract

An approximate method is formulated for analyzing the performance of nonlinear systems controlled by modeto-mode fuzzy controllers. It is assumed that an approximate model of the nominal plant is available and the nominal mode-to-mode trajectory converges asymptotically from the start mode to the target mode of operation. Stability of the nominal mode-to-mode trajectory in the presence of small, bounded variations of the system’ s parametersortheinitialconditions isconsidered. Themethod is based on formulating a performance measureas a Lyapunov function of the error between the nominal and perturbed mode-to-mode trajectories. The sensitivity of thedeviationsfrom the nominalmode-to-mode trajectory with respect to parametricor initial conditionvariations is incorporated into the total differential of this performance measure. Using a Lyapunov stability condition, the robustness of the closed-loop system is analyzed by observing a dee niteness condition of a time-varying matrix. A measure of robustness is then formulated using the largest singular value of a time-varying matrix. A hover mode to forward e ight mode fuzzy controller is used to illustrate the methodology. I. Introduction A LTHOUGH an often-mentioned reason for using fuzzy logic controllers is the lack of an accurate system model, the availability ofan approximate dynamicalmodel ofthe plant has not been fully explained when addressing issues ofrobust performance in the design of such control strategies. In Refs. 1 and 2, an attempt was made to bridge the gap between precise performance specie cations andthe use of fuzzy logiccontrollersalong with an approximate dynamical model of the plant to be controlled. This attempt involved formulating a performance measure as a Lyapunov-like function of the error dynamics of the nominal plant. This performance measure alsoincorporatedaheuristicmeasureofthesystem’ serrorsensitivity with respect to parametric variations. The robust stability addressed in this context was stability convergence to an equilibrium point in the presence of small parametric perturbations. Inequality bounds were derived for the sensitivity,errordeviation, and parameter deviations in terms of fuzzy quantities. Finally, a measure of robustness was formulated using singular values. This methodology was used to analyze the robustness of an automotive engine idle speed fuzzy controller. 1 In this paper,the formulation in Refs. 1 and 2 is extended tostudy the robust stability of the nominal trajectory of a mode-to-mode fuzzy controller where the trajectory is considered to be feasible (or realistic). Given a nominal mode-to-mode trajectory that converges asymptotically from the equilibrium point of the start mode to the equilibrium point of the target mode of operation, the stability of this nominal trajectory, in the presence of variations of the system’ s parameters or initial conditions, inherently addresses the robust stability of the closed-loop system. The robust stability of the nominal mode-to-mode trajectory is considered because perturbed trajectories that remain close to the nominal trajectory will also transition from the start mode to a small neighborhood around the equilibrium point of the target mode. Because the robust stability of the nominal trajectory is considered, the performance measure is formulated as a Lyapunov function of the error between the nominal and perturbed trajectories. The total differential of this performance measure also incorporates the sensitivity of the trajectory error with respect to parametric or initial condition variations. The robustness of the closed-loop system is analyzed by observing a dee niteness condition of a time-varying matrix. A measure of robustness is then formulated using the ine nity norm of a time-varying matrix. Fi