Abstraction of Continuous-Time Systems Based on Feedback Controllers and Mixed Monotonicity

In this article, we consider the problem of the computation of efficient symbolic abstractions for continuous-time control systems. The new abstraction algorithm builds 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 righthand 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. For a certain class of control systems, with a suitable set of discretization parameters, the new algorithm may even produce deterministic abstract systems or systems with a singleton input alphabet. The approach is illustrated by examples that compare the two abstraction algorithms