Hoang-Dung Tran

Output Reachable Set Estimation for Switched Linear Systems and Its Application in Safety Verification

This paper addresses the output reachable set estimation problem for continuous-time switched linear systems consisting of Hurwtiz stable subsystems. Based on a common Lyapunov function approach, the output reachable set is estimated by a union of bounding ellipsoids. Then, multiple Lyapunov functions with time-scheduled structure are employed to estimate the output reachable set for switched systems under dwell-time constraint. Furthermore, the safety verification problem of uncertain switched systems is investigated based on the result of output reachable set estimation. First, a sufficient condition ensuring the existence of an approximate bisimulation relation between two switched linear systems with a prescribed precision is proposed. Then, the safety verification for an uncertain switched system can be performed through an alternative safety verification for a switched system with exact parameters. Numerical examples are provided to illustrate our results.

On reachable set estimation for discrete-time switched linear systems under arbitrary switching

This paper addresses the problem of reachable set estimation for discrete-time switched systems under arbitrary switching. By introducing a novel conception called M-step sequence which is capable of characterizing all possible subsystem activation orders during M discrete-time steps, a Lyapunov function based approach is proposed to derive a set of bounding ellipsoids to estimate the reachable set. The proposed approach can cover the previous switched Lyapunov function approach and yields less conservativeness. Moreover, it can be shown that the M-step sequence method can also reduce the conservativeness in stability analysis for discrete-time switched systems under arbitrary switching in contrast to switched Lyapunov function method. Several numerical examples are provided to illustrate our approach.

Reachability Analysis for One Dimensional Linear Parabolic Equations

Abstract Partial differential equations (PDEs) mathematically describe a wide range of phenomena such as fluid dynamics, or quantum mechanics. Although great achievements have been accomplished in the field of numerical methods for solving PDEs, from a safety verification (or falsification) perspective, methods are still needed to verify (or falsify) a system whose dynamics is specified as a PDE that may depend not only on space, but also on time. As many cyber-physical systems (CPS) involve sensing and control of physical phenomena modeled as PDEs, reachability analysis of PDEs provides novel methods for safety verification and falsification. As a first step to address this challenging problem, we propose a reachability analysis approach leveraging the well-known Galerkin Finite Element Method (FEM) for a class of one-dimensional linear parabolic PDEs with fixed but uncertain inputs and initial conditions, which is a subclass of PDEs that is useful for modeling, for instance, heat flows. In particular, a continuous approximate reachable set of the parabolic PDE is computed using linear interpolation. Since a complete conservativeness is hardly achieved by using the approximate reachable set, to enhance the conservativeness, we investigate the error bound between the numerical solution and the exact analytically unsolvable solution to bloat the continuous approximate reachable set. This bloated reachable set is then used for safety verification and falsification. In the case that the safety specification is violated, our approach produces a numerical trace to prove that there exists an initial condition and input that lead the system to an unsafe state.