Abstract not found

. In this paper we describe an experimental system called d=dt for approximating reachable states for hybrid systems whose continuous dynamics is defined by linear differential equations. We use an approximation algorithm whose accumulation of errors during the continuous evolution is much smaller than in previously-used methods. The d=dt system can, so far, treat non-trivial continuous systems, hybrid systems, convex differential inclusions and controller synthesis problems. 1 Introduction The problem of calculating reachable states for continuous and hybrid systems has emerged as one of the major problems in hybrid systems research [G96,GM98,DM98,KV97,V98,GM99,CK99,PSK99,HHMW99]. It constitutes a prerequisite for exporting algorithmic verification methodology outside discrete systems or hybrid systems with piecewise-trivial dynamics. For computer scientists it poses new challenges in treating continuous functions and their approximations and in applying computational geometry...

This paper offers a synthetic overview of, and original contributions to, the use of logics and formal methods in the analysis of hybrid systems

In this paper we describe the tool d=dt which provides automatic safety veri cation of hybrid systems with linear continuous dynamics with uncertain input. The veri cation procedure is based on a method for overapproximating reachable sets by orthogonal polyhedra.

The notion of bisimulation in theoretical computer science is one of the main complexity reduction methods for the analysis and synthesis of labeled transition systems. Bisimulations are special quotients of the state space that preserve many important properties expressible in temporal logics, and, in particular, reachability. In this paper, the framework of bisimilar transition systems is applied to various transition systems that are generated by linear control systems. Given a discrete-time or continuous-time linear system, and a finite observation map, we characterize linear quotient maps that result in quotient transition systems that are bisimilar to the original system. Interestingly, the characterizations for discrete-time systems are more restrictive than for continuous-time systems, due to the existence of an atomic time step. We show that computing the coarsest bisimulation, which results in maximum complexity reduction, corresponds to computing the maximal controlled or reachability invariant subspace inside the kernel of the observations map. These results establish strong connections between complexity reduction concepts in control theory and computer science.

In the literature, we nd several formulations of the control problem for timed and hybrid systems. We argue that formulations where a controller can cause an action at any point in dense (rational or real) time are problematic, by presenting an example where the controller must act faster and faster, yet causes no Zeno eects (say, the control actions are at times 0; 1 2 ; 1; 1 3 4 ; 2; 2 7 8 ; 3; 3 15 16 ; : : :). Such a controller is, of course, not implementable in software. Such controllers are avoided by formulations where the controller can cause actions only at discrete (integer) points in time. While the resulting control problem is well-understood if the time unit, or \sampling rate" of the controller, is xed a priori, we dene a novel, stronger formulation: the discrete-time control problem with unknown sampling rate asks if a sampling controller exists for some sampling rate. We prove that, surprisingly and unfortunately, this problem is undecidable even in the special case of timed automata. 1

In this paper, we introduce a parametric semantics for timed controllers called the Almost ASAP semantics. This semantics is a relaxation of the usual ASAP semantics (also called the maximal progress semantics) which is a mathematical idealization that can not be implemented by any physical device no matter how fast it is. On the contrary, any correct Almost ASAP controller can be implemented by a program on a hardware if this hardware is fast enough. We study the properties of this semantics, show how it can be analyzed using the tool HyTech, and illustrate its practical use on examples.

Abstract. Impulse differential inclusions are introduced as a framework for modelling hybrid phenomena. Connections to standard problems in area of hybrid systems are discussed. Conditions are derived that allow one to determine whether a set of states is viable or invariant under the action of an impulse differential inclusion. For sets that violate these conditions, methods are developed for approximating their viability and invariance kernels, that is the largest subset that is viable or invariant under the action of the impulse differential inclusion. The results are demonstrated on examples. 1.

Abstract. Computing reachable sets is an essential step in most analysis and synthesis techniques for hybrid systems. The representation of these sets has a deciding impact on the computational complexity and thus the applicability of these techniques. This paper presents a new approach for approximating reachable sets using oriented rectangular hulls (ORHs), the orientations of which are determined by singular value decompositions of sample covariance matrices for sets of reachable states. The orientations keep the over-approximation of the reachable sets small in most cases with a complexity of low polynomial order with respect to the dimension of the continuous state space. We show how the use of ORHs can improve the efficiency of reachable set computation significantly for hybrid systems with nonlinear continuous dynamics.

We develop a general framework for solving the hybrid system reachability problem, and indicate how several published techniques fit into this framework. The key unresolved need of any hybrid system reachability algorithm is the computation of continuous reachable sets; consequently, we present new results on techniques for calculating numerical approximations of such sets evolving under general nonlinear dynamics with inputs. Our tool is based on a local level set procedure for boundary propagation in continuous state space, and has been implemented using numerical schemes of varying orders of accuracy. We demonstrate the numerical convergence of these schemes to the viscosity solution of the Hamilton-Jacobi equation, which was shown in earlier work to be the exact representation of the boundary of the reachable set. We then describe and solve a new benchmark example in nonlinear hybrid systems: an auto-lander for a commercial aircraft in which the switching logic and continuous control laws are designed to maximize the safe operating region across the hybrid state space.