Synthesizing Switching Logic Using Constraint Solving

Abstract

For a system that can operate in multiple different modes, we define the switching logic synthesis problem as follows: given a description of the dynamics in each mode of the system, find the conditions for switching between the modes so that the resulting system satisfies some desired properties. In this paper, we present an approach for solving the switching logic synthesis problem in the case when (1) the dynamics in each mode of the system are given using differential equations and, hence, the synthesized system is a hybrid system, and (2) the desired property is a safety property. Our approach for solving the switching logic synthesis problem, called the constraint-based approach, consists of two steps. In the first constraint generation step, the synthesis problem is reduced to satisfiability of a quantified formula over the theory of reals. In the second constraint solving step, the quantified formula is solved. This paper focuses on constraint generation. The constraint generation step is based on the concept of a controlled inductive invariant. The search for controlled inductive invariant is cast as a constraint solving problem. The controlled inductive invariant is then used to arrive at the maximally liberal switching logic. We prove that the synthesized switching logic always gives us a well-formed and safe hybrid system. When the system, the safety property, and the controlled inductive invariant are all expressed only using polynomials, the generated constraint is an ∃∀ formula in the theory of reals, whose satisfiability is decidable.

Keywords: Formal methods, Controller synthesis, Hybrid systems.