Formal analysis of piecewise affine systems through formula-guided refinement

We present a computational framework for identifying a set of initial states from which all trajectories of a piecewise affine (PWA) system satisfy a Linear Temporal Logic (LTL) formula over a set of linear predicates in its state variables. Our approach is based on the construction and refinement of finite abstractions of infinite systems. We derive conditions guaranteeing the equivalence of an infinite system and its finite abstraction with respect to a specific temporal logic formula and propose methods aimed at the construction of such formula-equivalent abstractions. We show that the proposed procedure can be implemented using polyhedral operations and analysis of finite graphs. While provably correct, the overall method is conservative and expensive. The proposed algorithms have been implemented as a software tool that is available for download. An illustrative example for a PWA gene network model is included.