Comparing Explicit and Implicit Methods for Estimating the Region of Attraction

Abstract

The region of attraction (ROA) is an important tool in investigating control and dynamic systems. Numerous computational methods have been proposed for estimating the ROA within the Mathematics, Power systems, and Control community. These methods employ different methods such as invariant manifolds, linear differential inclusion, level set methods, the sum of squares decomposition, and many others. This paper provides new insights into these methods and divides them into two classes: explicit and implicit. The explicit methods discretize the ROA (or the boundary of the ROA) into a finite union of geometric elements and store them directly. The essential idea of the implicit methods is to present the ROA implicitly, meaning that it is feasible to find the ROA as the level set of some function of the state space variables. The key point is the iteration and optimization of this implicit function. The explicit and implicit methods are compared and discussed by investigating some numerical examples. In particular, we consider the determination of the boundary of the ROA using the stable manifolds, and the estimation of the ROA as backward reachable sets using a Hamilton-Jacobi-Isaacs partial differential equation.