Reachability of Uncertain Linear Systems Using Zonotopes

Abstract. In this paper we propose a trajectory based reachability analysis by using local finite-time invariance property. Trajectory based analysis are based on the execution traces of the system or the simulation thereof. This family of methods is very appealing because of the simplicity of its execution, the possibility of having a partial verification, and its highly parallel structure.The key idea in this paper is the construction of local barrier functions with growth bound in local domains of validity. By using this idea, we can generalize our previous method that is based on the availability of global bisimulation functions. We also propose a computational scheme for constructing the local barrier functions and their domains of validity, which is based on the S-procedure. We demonstrate that our method subsumes some other existing methods as special cases, and that for polynomial systems the computation can be implemented using sum-ofsquares programming.

Section: Introductionmentioning

confidence: 99%

Trajectory Based Verification Using Local Finite-Time Invariance

Julius

Pappas

2009

“…This has resulted in a variety of computationally intensive approaches for hybrid system verification using predicate abstraction [4,5], barrier certificates [6], level sets [7], and exact arithmetic [8]. Even though these approaches can handle low-dimensional hybrid systems, for the class of uncertain linear systems, promising scalable results have been obtained using zonotope computations [9].…”

Section: Introductionmentioning

confidence: 99%

Verification Using Simulation

Girard

Pappas

2006

Self Cite

Abstract. Verification and simulation have always been complementary, if not competing, approaches to system design. In this paper, we present a novel method for so-called metric transition systems that bridges the gap between verification and simulation, enabling system verification using a finite number of simulations. The existence of metrics on the system state and observation spaces, which is natural for continuous systems, allows us to capitalize on the recently developed framework of approximate bisimulations, and infer the behavior of neighborhood of system trajectories around a simulated trajectory. For nondeterministic linear systems that are robustly safe or robustly unsafe, we provide not only a completeness result but also an upper bound on the number of simulations required as a function of the distance between the reachable set and the unsafe set. Our framework is the first simulation-based verification method that enjoys completeness for infinite-state systems. The complexity is low for robustly safe or robustly unsafe systems, and increases for nonrobust problems. This provides strong evidence that robustness dramatically impacts the complexity of system verification and design.

“…The reachability problem for continuous systems described by differential equations has motivated much research both for theoretical problems, such as computability (see for example [2]), and for the development of computation methods and tools. If the goal is to compute exactly the reachable set or approximate it as accurately as possible, one can use a variety of methods for tracking the evolution of the reachable set under the continuous flows using some set represention (such as polyhedra, ellipsoids, level sets) [20,10,8,21,3,30,23,19]. Since high quality approximations are hard to compute, other methods seek approximations that are sufficiently good to prove the property of interest 1 (such as barrier certificates [24], polynomial invariants [29]).…”

Section: Introductionmentioning

confidence: 99%

Approximate Reachability Computation for Polynomial Systems

Dang

2006

Abstract. In this paper we propose an algorithm for approximating the reachable sets of systems defined by polynomial differential equations. Such systems can be used to model a variety of physical phenomena. We first derive an integration scheme that approximates the state reachable in one time step by applying some polynomial map to the current state. In order to use this scheme to compute all the states reachable by the system starting from some initial set, we then consider the problem of computing the image of a set by a multivariate polynomial. We propose a method to do so using the Bézier control net of the polynomial map and the blossoming technique to compute this control net. We also prove that our overall method is of order 2. In addition, we have successfully applied our reachability algorithm to two models of a biological system.