Some performance measures useful in the design of controllers for two dimensional dynamical systems subject to uncertain inputs

Abstract

Classical control theory has always been concerned with uncertain inputs. However, this concern has generally been implicit rather than explicit. Modern control methods tend to be more explicit. In particular, methods based on game theory and methods based on a Lyapunov type of analysis have been proposed. One question which seems to be largely ignored is: how do these various methods compare when applied to a typical control problem? Indeed what performance measures can we use to compare various control designs dealing with uncertain inputs? This paper addresses both of these questions. We define performance measures called the v-reachable set and the v-attractive set. The v-reachable set measures the extent that the uncertain input can drive the system away from the target set and the v-attractive set measures the extent that the uncertainty diminishes the size of the controllable set to the target. An ideal control law would be one in which the v-reachable set is the target set and the v-attractive set is the controllable set. Five different controllers based on five different design philosophies are applied to two different control systems. We find that there can be marked differences in the above performance measures. However using these performance measures as guide to choosing feedback parameters, the differences between the various design methods can be minimized. It is found that for these problems a modified game theoretic approach and the Lyapunov type of approach tend to provide the best designs.