Weiming Xiang

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.

Order-reduction abstractions for safety verification of high-dimensional linear systems

Order-reduction is a standard automated approximation technique for computer-aided design, analysis, and simulation of many classes of systems, from circuits to buildings. To be used as a sound abstraction for formal verification, a measure of the similarity of behavior must be formalized and computed, which we develop in a computational way for a class of asymptotic stable linear systems as the main contributions of this paper. We have implemented the order-reduction as a sound abstraction process through a source-to-source model transformation in the HyST tool and use SpaceEx to compute sets of reachable states to verify properties of the full-order system through analysis of the reduced-order system. Our experimental results suggest systems with thousand of state variables can be reduced to systems with tens of state variables such that the order-reduction overapproximation error is small enough to prove or disprove safety properties of interest using current reachability analysis tools. Our results illustrate this approach is effective in tackling the state-space explosion problem for verification of high-dimensional linear systems.

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.

Reachable set estimation and control for switched linear systems with dwell-time restriction

The reachable set estimation and control problems for continuous-time switched linear systems are addressed in this paper. First, a general result on reachable set estimation for switched system is proposed based on a Lyapunov function approach. Then, with the help of a class of time-scheduled Lyapunov functions, a numerically tractable sufficient condition ensuring the system state bounded in a prescribed set is derived for switched systems under dwell time constraint. Moreover, a time-scheduled state feedback controller is designed to ensure the state trajectories of the closed-loop system are confined in a prescribed set. Finally, a networked control system subject to packet dropouts is modeled as a switched system with dwell time constraints, and the controller design problem is studied as an application of our results.

Tutorial: Software tools for hybrid systems verification, transformation, and synthesis: C2E2, HyST, and TuLiP

Hybrid systems have both continuous and discrete dynamics and are useful for modeling a variety of control systems, from air traffic control protocols to robotic maneuvers and beyond. Recently, numerous powerful and scalable tools for analyzing hybrid systems have emerged. Several of these tools implement automated formal methods for mathematically proving a system meets a specification. This tutorial session will present three recent hybrid systems tools: C2E2, HyST, and TuLiP. C2E2 is a simulated-based verification tool for hybrid systems, and uses validated numerical solvers and bloating of simulation traces to verify systems meet specifications. HyST is a hybrid systems model transformation and translation tool, and uses a canonical intermediate representation to support most of the recent verification tools, as well as automated sound abstractions that simplify verification of a given hybrid system. TuLiP is a controller synthesis tool for hybrid systems, where given a temporal logic specification to be satisfied for a system (plant) model, TuLiP will find a controller that meets a given specification.

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.