An abstraction and refinement computational approach to safety verification of discrete time nonlinear systems

This paper addresses safety verification of nonlinear systems through invariant set computation. More precisely, our goal is verifying if the state of a given discrete time nonlinear system will keep evolving within a safe region, starting from a given set of initial conditions. To this purpose, we introduce a conformant PieceWise Affine (PWA) abstraction of the nonlinear system, which is instrumental to computing a conservative approximation of its maximal invariant set within the safe region. If the obtained set covers the set of initial conditions, safety is proven. Otherwise, subsequent refinements of the PWA abstraction are performed, either on the whole safe region or on some appropriate subset identified through a guided refinement approach and containing the set of initial conditions. Some numerical examples demonstrate the effectiveness of the approach

Robust constrained control of piecewise affine systems through set-based reachability computations

This article addresses finite-horizon robust control of a piecewise affine system affected by uncertainty and characterized by different affine dynamics (modes) associated with a polyhedral partition of the state space. The goal is to design a static state-feedback control law that maintains the state of the system within given—possibly time-varying—sets, subject to actuation constraints. The proposed approach rests on two phases: a reference mode sequence with a sufficiently large robustness level is determined first, and then a tracking state-feedback control law defined on the reach sets of the controlled system is designed to counteract uncertainty and maintain the reach sets within the reference sequence. If this is not possible and the reach sets split over different modes, then, further reference mode sequences and tracking controllers are computed. The designed state-feedback control law is represented through a collection of controllers defined on precomputed reach sets of the closed-loop control system. Performance of the approach is shown on some numerical examples