M. Greenstreet and I. Mitchell, "Integrating Projections," in Hybrid Systems: Computation and Control (S. Sastry and T. Henzinger, eds.), no. 1386 in LNCS, pp. 159--174, Springer Verlag, 1998. 4  Home/Search   Document Details and Download   Summary   Related Articles   Check

....based control system, but webelievethatthehybrid automata formalism is well suited for this. It allows us to incorporate continuous and discrete phenomena into the model in a systematic way.At the same it gives us access to classes of decidability and reachability results for hybrid automata [10,23, 40, 53, 54, 61, 65]. These can be used as theoretical or numerical tools for formal verification of the safety or other performance aspects of a given hybrid system. However, before weintroduce these hybrid automata, we start this section with an introductory discussion about the behavior based control architecture ....

....the different behaviors in a behavior based architecture as distinct nodes in ahybrid automaton is that features about the system could be proved. There are today theories dealing with safety or performance verification of hybrid automata, even though muchwork still remains to be done in this area [10, 23,40,53, 54, 61, 65]. Another benefit from modeling the system as a hybrid automaton is that it provides us with a framework for dealing with the interaction between discrete and continuous dynamics in a systematic way. In the following subsection we will thus sketch some of the most basic ideas behind these hybrid ....

M.R. Greenstreet and I. Mitchell. Integrating Projections. In Hybrid Systems: Computation and Control, S. Sastry and T.A. Henzinger, Eds., number 1386 in LCNS, pp. 159-174. Springer Verlag, 1998.

....the loss of precision caused by the accumulation of round off errors in the numerical integration process may lead the analyzer, whether human or computer, to overlook potentially hazardous events. In contrast to hybrid systems simulators (see [21] for a survey) and reachset computation tools [3, 4, 6, 8, 26] which use traditional (not interval based) numerical methods, HyperTech is guaranteed not to miss any events. Thus, if an unsafe (target) state of a hybrid automaton is reachable, HyperTech is guaranteed to note its reachability. 3 The rest of the paper is organized as follows. In Section 2, we ....

M.R. Greenstreet and I. Mitchell. Integrating projections. In HSCC 98: Hybrid Systems: Computation and Control, LNCS 1386, pages 159--174. Springer-Verlag, 1998.

....so far, treat non trivial continuous systems, hybrid systems, convex di erential 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 veri cation methodology outside discrete systems or hybrid systems with piecewise trivial dynamics. For computer scientists it poses new challenges in treating continuous functions and their ....

M.R. Greenstreet and I. Mitchell, Integrating Projections, in T.A. Henzinger and S. Sastry (Eds), Hybrid Systems: Computation and Control, LNCS 1386, 159-174, Springer, 1998.

....computations may be infeasible in practice, and approximation methods must be used. Some approximation methods bound the dynamics by piecewise constant inclusions, e.g. rate translation (Henzinger, Ho, Wong Toi 1998) others use numerical integration (Chutinan Krogh 1998; Dang Maler 1998; Greenstreet Mitchell 1998; Tomlin 1998) Unfortunately, traditional numerical integration methods suffer from roundoff errors, and do not necessarily yield validated results. On the other hand, accurate rate translation suffers from state ex plosion. In contrast to traditional numerical methods, interval numerical ....

Greenstreet, M. R., and Mitchell, I. 1998. Integrating projections. In HSCC 98: Hybrid Systems--- Computation and Control, LNCS 1386. SpringerVerlag. 159--174.

....in their own right, this class of hybrid systems can be used to approximate hybrid systems with differential equations. Indeed, most of the techniques that have been proposed for reachability computations for general continuous dynamics involve some form of discretization of the continuous space [10, 13, 28], followed by a reachability computation on the resulting discrete time system. In this paper we study the problem of controller synthesis for hybrid systems whose continuous dynamics are given by difference equations under reachability specifications. In Section 2, we formulate the problem, ....

M.R. Greenstreet and I. Mitchell. Integrating projections. In S. Sastry and T.A. Henzinger, editors, Hybrid Systems: Computation and Control, number 1386 in LNCS, pages 159--174. Springer Verlag, 1998.

....which reachability analysis of continuous systems is carried out by shifting outwards the boundaries of rectangular parts of the state space depending on the continuous flow on these boundaries. This procedure is called face lifting . A similar concept is the so called bloating by Greenstreet [4, 5]. Here, the shifting of the boundaries is determined by integrating the original di#erential equations. Both approaches and the one presented here have in common, that the reachable regions in each iteration step are computed only from the bounds of the predecessor regions and the continuous flow. ....

....presented here have in common, that the reachable regions in each iteration step are computed only from the bounds of the predecessor regions and the continuous flow. A major di#erence is that in our procedure the step size is determined by the partitioning of the state space, whereas in [2] and [4, 5] time is discretized. Similarities also exist with the work of Krogh and Chutinan [1] who compute polyhedral approximations of the continuous system s reachable state space (called the flow pipe ) 8 Conclusions We have presented and discussed a reachability analysis for switched continuous ....

M.R. Greenstreet and I. Mitchell. Integrating projections. In T.A. Henzinger and S. Sastry, editors, HSCC 98: Hybrid Systems---Computation and Control, Lecture Notes in Computer Science 1386, pages 159--174. Springer-Verlag, 1998.

....by DARPA under contract F33615 98 C 3614, and by ARO under grant MURI DAAH04 96 1 0341. differential equations. Indeed, most of the techniques that have been proposed for reachability computations for general continuous dynamics involve some form of discretization of the continuous space [8, 12, 26], followed by a reachability computation on the resulting discrete time system. In Sect. 2, we formulate the problem of controller synthesis for discrete time systems under reachability specifications, introduce the concepts of maximal controlled invariant set and least restrictive controller, ....

M.R. Greenstreet and I. Mitchell. Integrating projections. In Hybrid Systems: Computation and Control, vol. 1386 of LNCS, pp. 159--174. Springer Verlag, 1998.

....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 ....

M.R. Greenstreet and I. Mitchell, Integrating Projections, in T.A. Henzinger and S. Sastry (Eds), Hybrid Systems: Computation and Control, LNCS 1386, 159-174, Springer, 1998.

....Though the approximations are accurate in two dimensions, the approximation technique seems to be of limited use in higher dimensions. The idea of computing reachable regions only from the bounds of the predecessor regions, instead of looking at the whole regions, is not new. It also appears in [4, 5], where the bounding edges of a region are bloated outward with an integration routine on the original differential equations of a system to compute the edges of the successor region. Reachable sets may be represented by a set of non convex polyhedral 2 dimensional projections. The idea of ....

M.R. Greenstreet and I. Mitchell. Integrating projections. In T.A. Henzinger and S. Sastry, editors, HSCC 98: Hybrid Systems---Computation and Control, Lecture Notes in Computer Science 1386, pages 159--174. Springer-Verlag, 1998.

....inclusions are non deterministic , in the sense that many executions may exist for a single initial condition. ffl Differential inclusions are commonly used as: 1. Abstractions of differential equations, such as x(t) f(x) 2 F (x) for example for reachability or viability computations [1, 2]. In fact one can show that the reachable set of a Lipschitz differential equation over a finite time horizon can be approximated arbitrarily closely by the reach set of a rectangular automaton [3] 2. Abstractions of control systems [1] x(t) f(x(t) v(t) with v(t) 2 V (x(t) R m is ....

M.R. Greenstreet and I. Mitchell, "Integrating projections", in Hybrid Systems: Computation and Control, S. Sastry and T.A. Henzinger, Eds., number 1386 in LNCS, pp. 159--174. Springer Verlag, 1998.

....the ideas underlying face lifting appear already in [KM91] where the authors try to prove a homomorphism from a transistor level differential model into an automaton. While doing so they also cut the space into a grid and try to calculate the reachability relation among cubes. Finally, in [G96] GM98] the authors try to extend face lifting to higher dimensions using another strategy. They restrict themselves to polyhedra which can be written as intersections of cylindrifications of two dimensional (arbitrary) polygons. This way all the operations are performed on the two dimensional ....

M.R. Greenstreet and I. Mitchell, Integrating Projections, these proceedings.

....the approximate system can be computed in a finite number of steps. However, the amount of preprocessing required to initially approximate the dynamics may be formidable, especially if the time horizon over which the reach set needs to be calculated is large. A related approach is that of [18] and [19]. Here the approximation of the dynamics is not carried out a priori over the entire space, but only locally around the boundary of the reachable set as the latter propagates in time. This could potentially lead to substantial computational savings. The class of sets generated by these techniques ....

M.R. Greenstreet and I. Mitchell, "Integrating projections", in Hybrid Systems: Computation and Control, S. Sastry and T.A. Henzinger, Eds., number 1386 in LNCS, pp. 159--174. Springer Verlag, 1998.

....which asks whether some 2 G. LAFFERRIERE, G. PAPPAS, AND S. YOVINE unsafe region is reachable from an initial region. For purely continuous systems, the reachability problem is known to be a very difficult problem with a few exceptions. Optimal control [41] game theoretic [34] and other [13, 33] approaches have been used to calculate reachable sets for some systems. For purely discrete systems, one of the most successful methods is model checking which essentially explores all possible system trajectories. The lack of structure of purely discrete problems makes computational approaches ....

M.R. Greenstreet and I. Mitchell. Integrating projections. In T. Henzinger and S. Sastry, editors, Hybrid Systems : Computation and Control, volume 1386 of Lecture Notes in Computer Science, pages 159--174. Springer Verlag, Berlin, 1998.

....means. A number of approaches to verification exist for simple linear systems. Recent work has focused on developing methodology for more complex systems such as hybrid systems (see e.g. Alur et al. 1996; Bouajjani et al. 1993) The approaches in (Henzinger et al. 1995; Puri et al. 1996; Greenstreet and Mitchell, 1998; Dang and Maler, 1998) use a polygonal or grid based representation for phase space regions of a hybrid system during verification, and share several similar concepts with the algorithm presented here. Our algorithm evolves from earlier work in phase space control synthesis (Bradley and Zhao, ....

....a more straightforward representation of real world uncertainties. Furthermore, our algorithm is guaranteed to terminate, and allows for a systematic refinement of approximation of dynamics. Recently, several researchers suggested verification techniques based on projecting phase space regions. In (Greenstreet and Mitchell, 1998), the authors described a method for computing projection polyhedra obtained from integrating initial regions. While an efficient and exact algorithm exists for two dimensional linear systems with convex spaces, a general higher dimensional polyhedra have to be reconstructed from a series of ....

Greenstreet, M.R. and I. Mitchell (1998). Integrating projections. Hybrid Systems: computation and control, Springer-Verlag Lecture Notes in Computer Science.

No context found.

M. Greenstreet and I. Mitchell, "Integrating Projections," in Hybrid Systems: Computation and Control (S. Sastry and T. Henzinger, eds.), no. 1386 in LNCS, pp. 159--174, Springer Verlag, 1998. 4

....preparing a publication describing the novel localized level set al..gorithm [9] which reduces the computation and storage requirements of my reachability algorithm. I am also investigating whether the use of projections in my tool might allow it to tackle systems of up to perhaps a dozen dimensions [1, 2, 8]. As a faculty member, I propose to expand my research in several directions. Verification: My next step in verification is the development of a toolkit for hybrid systems analysis, aimed at industrial and academic systems engineers. Such a tool would help designers verify, and possibly ....

M. Greenstreet and I. Mitchell, "Integrating Projections," in Hybrid Systems: Computation and Control (S. Sastry and T. Henzinger, eds.), no. 1386 in LNCS, pp. 159--174, Springer Verlag, 1998.

....expense of overapproximation. However, because we are interested in verifying system safety, computing an overapproximation of the set of states which evolve into an unsafe set and then proving that the system never enters that overapproximation is su#cient. Our work is inspired by the ideas of [27, 28] for continuous systems and those of [29, 30, 31] for discrete systems, as well as research which uses intersections and projections to treat curves [32] and geometric optics [33, 34] The outline of this paper is as follows. In section 2, we define the reachability problem and its solution, and ....

M. Greenstreet and I. Mitchell, "Integrating projections," in Hybrid Systems: Computation and Control (S. Sastry and T. Henzinger, eds.), no. 1386 in Lecture Notes in Computer Science, pp. 159--174, Springer Verlag, 1998.

.... are used in [35] VeriSHIFT [36] uses ellipsoidal overapproximations of reach sets for linear systems with linear input; it implements techniques developed in [37] Polygonal overapproximations of reachable sets for some classes of nonlinear systems is treated in [38] The tool Coho, developed in [39, 40] uses as set representation two dimensional projections of higher dimensional non convex polyhedra, and evolves these projectagons under affine over approximations of nonlinear dynamics using linear programming. In [41] the authors present a solution using sets specified by linear inequalities, ....

M. Greenstreet and I. Mitchell, "Integrating projections," in Hybrid Systems: Computation and Control (S. Sastry and T. Henzinger, eds.), no. 1386 in LNCS, pp. 159-174, Springer Verlag, 1998.

....and time dynamical systems theory to incorporate discrete switches [7 11] However, the efficient computation of reachable sets for hybrid systems with nonlinear dynamics remains a difficult problem to solve. Numerical techniques which over approximate the nonlinear dynamics with linear dynamics [12], or which over approximate the reachable sets [13 16] have recently been developed. Research supported by DARPA under the Software Enabled Control Program (AFRL contract F33615 99 C 3014) and by a Frederick E. Terman Faculty Award. In this paper, we present an implementation of an exact ....

M. Greenstreet and I. Mitchell, "Integrating projections," in Hybrid Systems: Computation and Control (S. Sastry and T. Henzinger, eds.), no. 1386 in LNCS, pp. 159-- 174, Springer Verlag, 1998.

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