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14
Approximate Reachability Analysis of PiecewiseLinear Dynamical Systems
, 2000
"... . 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 small ..."
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Cited by 139 (33 self)
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. 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 previouslyused methods. The d=dt system can, so far, treat nontrivial 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 piecewisetrivial dynamics. For computer scientists it poses new challenges in treating continuous functions and their approximations and in applying computational geometry...
OMinimal Hybrid Systems
, 2000
"... An important approach to decidability questions for verification algorithms of hybrid systems has been the construction of a bisimulation. Bisimulations are finite state quotients whose reachability properties are equivalent to those of the original infinite state hybrid system. In this paper we ..."
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Cited by 117 (10 self)
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An important approach to decidability questions for verification algorithms of hybrid systems has been the construction of a bisimulation. Bisimulations are finite state quotients whose reachability properties are equivalent to those of the original infinite state hybrid system. In this paper we introduce the notion of ominimal hybrid systems, which are initialized hybrid systems whose relevant sets and flows are definable in an ominimal theory. We prove that ominimal hybrid systems always admit finite bisimulations. We then present specific examples of hybrid systems with complex continuous dynamics for which finite bisimulations exist.
On the decidability of the reachability problem for planar differential inclusions
 In HSCC’2001, number 2034 in LNCS
, 2001
"... Abstract. In this paper we develop an algorithm for solving the reachability problem of twodimensional piecewise rectangular differential inclusions. Our procedure is not based on the computation of the reachset but rather on the computation of the limit of individual trajectories. A key idea is ..."
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Cited by 39 (15 self)
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Abstract. In this paper we develop an algorithm for solving the reachability problem of twodimensional piecewise rectangular differential inclusions. Our procedure is not based on the computation of the reachset but rather on the computation of the limit of individual trajectories. A key idea is the use of onedimensional affine Poincaré maps for which we can easily compute the fixpoints. As a first step, we show that between any two points linked by an arbitrary trajectory there always exists a trajectory without selfcrossings. Thus, solving the reachability problem requires considering only those. We prove that, indeed, there are only finitely many “qualitative types ” of those trajectories. The last step consists in giving a decision procedure for each of them. These procedures are essentially based on the analysis of the limits of extreme trajectories. We illustrate our algorithm on a simple model of a swimmer spinning around a whirlpool. 1
Widening the boundary between decidable and undecidable hybrid systems
, 2002
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Algorithmic Analysis of Polygonal Hybrid Systems, Part I: Reachability
, 2007
"... In this work we are concerned with the formal verification of twodimensional nondeterministic hybrid systems, namely polygonal differential inclusion systems (SPDIs). SPDIs are a class of nondeterministic systems that correspond to piecewise constant differential inclusions on the plane, for which ..."
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Cited by 14 (6 self)
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In this work we are concerned with the formal verification of twodimensional nondeterministic hybrid systems, namely polygonal differential inclusion systems (SPDIs). SPDIs are a class of nondeterministic systems that correspond to piecewise constant differential inclusions on the plane, for which we study the reachability problem. Our contribution is the development of an algorithm for solving exactly the reachability problem of SPDIs. We extend the geometric approach due to Maler and Pnueli [MP93] to nondeterministic systems, based on the combination of three techniques: the representation of the twodimensional continuoustime dynamics as a onedimensional discretetime system (using Poincaré maps), the characterization of the set of qualitative behaviors of the latter as a finite set of types of signatures, and acceleration used to explore reachability according to each of these types.
Model Checking Polygonal Differential Inclusions Using Invariance Kernels
 In VMCAI’04, number 2937 in LNCS
, 2004
"... Polygonal hybrid systems are a subclass of planar hybrid automata which can be represented by piecewise constant differential inclusions. Here, we identify and compute an important object of such systems' phase portrait, namely invariance kernels. An invariant set is a set of initial points of ..."
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Cited by 9 (9 self)
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Polygonal hybrid systems are a subclass of planar hybrid automata which can be represented by piecewise constant differential inclusions. Here, we identify and compute an important object of such systems' phase portrait, namely invariance kernels. An invariant set is a set of initial points of trajectories which keep rotating in a cycle forever and the invariance kernel is the largest of such sets. We show that this kernel is a nonconvex polygon and we give a noniterative algorithm for computing the coordinates of its vertices and edges. Moreover, we present a breadthfirst search algorithm for solving the reachability problem for such systems. Invariance kernels play an important role in the algorithm.
Computing Invariance Kernels of Polygonal Hybrid Systems
 NORDIC JOURNAL OF COMPUTING
, 2004
"... Polygonal hybrid systems are a subclass of planar hybrid automata which can be represented by piecewise constant differential inclusions. One way of analysing such systems (and hybrid systems in general) is through the study of their phase portrait, which characterise the systems’ qualitative behav ..."
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Cited by 7 (5 self)
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Polygonal hybrid systems are a subclass of planar hybrid automata which can be represented by piecewise constant differential inclusions. One way of analysing such systems (and hybrid systems in general) is through the study of their phase portrait, which characterise the systems’ qualitative behaviour. In this paper we identify and compute an important object of polygonal hybrid systems’ phase portrait, namely invariance kernels. An invariant set is a set of points such that any trajectory starting in such point keep necessarily rotating in the set forever and the invariance kernel is the largest of such sets. We show that this kernel is a nonconvex polygon and we give a noniterative algorithm for computing the coordinates of its vertexes and edges. Moreover, we show some properties of such systems’ simple cycles.
Reachability Analysis of Hybrid Systems Using Bisimulations
, 1998
"... A unified approach to decidability questions for the verification of hybrid systems is obtained by the construction of a bisimulation. These are finite state quotients whose reachability properties are equivalent to those of the original infinite state system. This approach has had some success in t ..."
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Cited by 4 (0 self)
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A unified approach to decidability questions for the verification of hybrid systems is obtained by the construction of a bisimulation. These are finite state quotients whose reachability properties are equivalent to those of the original infinite state system. This approach has had some success in the reachability analysis of timed automata and linear hybrid automata. In this paper, we use results from stratification theory, subanalytic sets and model theory of fields in order to extend earlier results on the existence of bismimulations for certain classes of analytic vector fields. 1 Introduction Hybrid systems consist of finite state machines interacting with differential equations. Various modeling formalisms, analysis, design and control methodologies, as well as applications, can be found in [2, 3, 4, 8, 13]. Formal verification is one of the main approaches for analyzing properties of hybrid systems. The system is first modeled as a hybrid automaton, and the property to be anal...
Invariance Kernels of Polygonal Differential Inclusions
, 2003
"... Polygonal hybrid systems are a subclass of planar hybrid automata which can be represented by piecewise constant differential inclusions. Here, we identify and compute an important object of such systems ’ phase portrait, namely invariance kernels. An invariant set is a set of initial points of traj ..."
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Cited by 3 (2 self)
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Polygonal hybrid systems are a subclass of planar hybrid automata which can be represented by piecewise constant differential inclusions. Here, we identify and compute an important object of such systems ’ phase portrait, namely invariance kernels. An invariant set is a set of initial points of trajectories which keep rotating in a cycle forever and the invariance kernel is the largest of such sets. We show that this kernel is a nonconvex polygon and we give a noniterative algorithm for computing the coordinates of its vertexes and edges. 1