Reachability analysis of complex planar hybrid systems

We present a general framework for the formal speci cation and algorithmic analysis of hybrid systems. A hybrid system consists of a discrete program with an analog environment. We model hybrid systems as nite automata equipped with variables that evolve continuously with time according to dynamical laws. For veri cation purposes, we restrict ourselves to linear hybrid systems, where all variables follow piecewise-linear trajectories. We provide decidability and undecidability results for classes of linear hybrid systems, and we show that standard programanalysis techniques can be adapted to linear hybrid systems. In particular, we consider symbolic model-checking and minimization procedures that are based on the reachability analysis of an in nite state space. The procedures iteratively compute state sets that are de nable as unions of convex polyhedra in multidimensional real space. We also present approximation techniques for dealing with systems for which the iterative procedures do not converge.

Annual Reviews in Control, 2009

Safety verification and reachability analysis for hybrid systems is a very active research domain. Many approaches that seem quite different, have been proposed to solve this complex problem. This paper presents an overview of various approaches for autonomous, continuous-time hybrid systems and presents them with respect to basic problems related to verification.

In this paper we deal with a subclass of planar hybrid automata, denoted by SPDI, which can be represented by piecewise constant differential inclusions. The computation of certain objects of the phase portrait of an SPDI, namely the viability, controllability, invariance kernels and semi-separatrix curves have been shown to be efficiently decidable. On the other hand, although the reachability problem for SPDIs is known to be decidable, its complexity makes it unfeasible on large systems. We present and combine recent results on the use of the SPDI phase portraits for improving reachability analysis by (i) state-space reduction and (ii) decomposition techniques of the state space, enabling compositional parallelization of the analysis. Both techniques contribute to increase the feasibility of reachability analysis on large SPDI systems.

Lecture Notes in Computer Science, 2006

We describe two new ways of efficiently deriving continuous reach set information for hybrid systems. In both cases, we overapproximate the differential equations to constraints that are then solved using a solver for first-order predicate logical formulae over the reals. We prove some properties about the amount of introduced over-approximations. Moreover, we embed the results in our earlier method for verification of hybrid systems using abstraction refinement and provide some timings that document the resulting improvement.

International Journal of Modeling and Optimization, 2012

Hybrid systems are mathematical models of control systems whose safety verification is critical for many applications. In practice, a rigorous tool is still not available for verifying every class of hybrid systems. HyTech was the first attempt in this direction followed by PHaver, both restricted to Linear Hybrid Automata (LHA). HSolver is another successful contribution for verification of nonlinear systems. PHaver can efficiently verify safety properties with the help of piecewise constant bounds on derivatives. Its use is greatly motivated by on-the-fly over approximations of piecewise affine dynamics with various user-specified parameters. HSolver verifies safety of nonlinear systems using constraint propagation based abstraction refinement. We have evaluated a few examples and shown that both tools have their strengths and weaknesses. In all the examples, the approximation of nonlinear systems by linear systems is performed by the rate translation.

2015

With the ubiquitous use of computers in controlling physical systems, it requires to have a new formalism that could model both continuous flows and discrete jumps. Hybrid systems are introduced to this purpose. A hybrid system, which is modeled by a hybrid automaton in the thesis, is equipped with finitely many discrete modes and continuous real-valued variables. A state of it is then represented by a mode along with a valuation of the variables. Given that the system is in a mode , the variable values are changed continuously according to the Ordinary Differential Equation (ODE) associated to , or discretely by a jump starting from . The thesis focuses on the techniques to compute all reachable states over a bounded time horizon and finitely many jumps for a hybrid system with non-linear dynamics. The results of that can then be used in safety verification of the system. Flow* and SpaceEx. Legends: Var: number of variables, T : time horizon, δ: time step-size, k: TM order, P: precision, box: box over-approximation, octagon: octagon over-approximation, T.O.:

1997

Hybrid systems possess continuous dynamics defined within regions of state spaces and discrete transitions among the regions. Many practical control verification and synthesis tasks can be reduced to reach ability problems for these systems that decide if a particular state-space region is reachable from an initial operating region. In this paper, we present a computational analysis of the face reachability problem for a class of three-dimensional dynamical systems whose state spaces are defined by piecewise constant vector fields and whose trajectories never return to a state-space region once they exit the region. These systems represent a restricted class of control systems whose dynamics results from a juxtaposition of piecewise parameterized vector fields. We had previously developed a computational algorithm for synthesizing the desired dynamics of a system in phase space by piecing together vector fields geometrically. We demonstrate in this paper that the reachability problem for this class of systems is decidable while the computation is provably intractable (i.e., PSPACE-hard). We prove the intractability via a reduction of satisfiability of quantified boolean formulas to this reachability problem. This result sheds light on the computational complexity of phase-space based control synthesis methods and extends the work of Asarin, Maler, and Pnueli [2] that proves computational undecidability for three-dimensional constant-derivative systems.

Lecture Notes in Computer Science, 2010

We present a method to enhance the power of a given reachability analysis engine for hybrid systems. The method works by a new form of composition of reachability analyses, each on a different relaxation of the input hybrid system. We present preliminary experiments that indicate its practical potential for checking safety and stability.

1996

Hybrid systems possess continuous dynamics de ned within regions of state spaces and discrete transitions among the regions. Many practical control veri cation and synthesis tasks can be reduced to reachability problems for these systems that decide if a particular state-space region is reachable from an initial operating region. In this paper, we present a computational analysis of the face reachability problem for a class of three-dimensional dynamical systems whose state spaces are de ned by piecewise constant vector elds and whose trajectories never return to a state-space region once they exit the region. These systems represent a restricted class of control systems whose dynamics results from a juxtaposition of piecewise parameterized vector elds. We had previously developed a computational algorithm for synthesizing the desired dynamics of a system in phase space by piecing together vector elds geometrically. We demonstrate in this paper that the reachability problem for this class of systems is decidable while the computation is provably intractable (i.e., PSPACE-hard). We prove the intractability via a reduction of satis ability of quanti ed boolean formulas to this reachability problem. This result sheds light on the computational complexity of phase-space based control synthesis methods and extends the work of Asarin, Maler, and Pnueli 2] that proves computational undecidability for three-dimensional constant-derivative systems.

Springer eBooks, 1997

This paper addresses the exact computation of the set of reachable states of a strongly linear hybrid system. It proposes an approach that is an extension of classical state-space exploration. This approach uses a new operation, based on a cycle analysis in the control graph of the system, for generating sets of reachable states, as well as a powerful representation system for sets of values. The method broadens the range of hybrid systems for which a finite and exact representation of the set of reachable states can be computed. In particular, the state-space exploration may be performed even if the set of variable values reachable at a given control location cannot be expressed as a finite union of convex regions. The technique is illustrated on a very simple example.