Abstraction of Monotone Systems Based on Feedback Controllers

Abstract

Abstract In this paper, we consider the problem of computation of efficient symbolic abstractions for a certain subclass of continuous-time monotone control systems. The new abstraction algorithm utilizes the properties of such systems to build symbolic models with the same number of states but fewer transitions in comparison to the one produced by the standard algorithm. At the same time, the new abstract system is at least as controllable as the standard one. The proposed algorithm is based on the solution of a region-to-region control synthesis problem. This solution is formally obtained using the theory of viscosity solutions of the dynamic programming equation and the theory of differential equations with discontinuous right-hand side. In the new abstraction algorithm, the symbolic controls are essentially the feedback controllers that solve this control synthesis problem. The improvement in the number of transitions is achieved by reducing the number of successors for each symbolic control. The approach is illustrated by an example that compares the two abstraction algorithms.