On Enlarging the Domain of Attraction

Abstract

In this chapter, we will present a method for designing feedback gains that result in large domains of attraction. Our approach is to formulate the problem into a constrained optimization problem. Since the precise domain of attraction under a feedback law is hard to identify, we will first obtain invariant ellipsoids as estimates of the domain of attraction and then maximize the estimate over stabilizing feedback laws

In solving the optimization problem, we will also reveal a surprising aspect of the design for large domain of attraction. If our purpose is solely to enlarge the domain of attraction, we might as well restrict the invariant ellipsoid (an estimate of the domain of attraction) in the linear region of the saturation function, although allowing saturation will provide us more freedom in choosing controllers. Another interesting aspect is that, for a discrete-time system, the domain of attraction can be further enlarged if the design is performed on its lifted system