Towards Automatic Detection of Implicit Equality Constraints in Stability Verification of Hybrid Systems

We present a powerful heuristic that detects implicit equality constraints that may occur in the specification of systems of constraints in order to find Lyapunov-based certificates of stability for hybrid systems. A hybrid system is a fusion of systems exhibiting discrete-time as well as continuous-time behavior, e.g. embedded systems within a physical environment. Stability is a property which ensures that a system starting in any possible state will reach a desired state and remain there. Such systems are particularly useful where a certain autonomous operation is required, e.g. keeping a certain temperature or speed of a chemical reaction or steering a vehicle over a predefined track. Stable hybrid systems are extremely valuable because after an error has disturbed their normal operation, they automatically “steer back” to normal operation. Stability can be certified by finding a so-called Lyapunov function. The search for this kind of function usually involves solving systems of inequality constraints. We have identified and implemented a heuristic that detects implicitly specified equality constraints and tries to resolve them by substitution. Keywords—Hybrid Systems; Automatic Verification; Stability; Lyapunov Theory; Optimization; Sums-of-Squares