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54
Dynamical systems revisited: Hybrid systems with zeno executions
 Proc. Third International Workshop on Hybrid Systems: Computation and Control, volume 1790 of Lecture Notes in Computer Science
, 2000
"... Abstract. Results from classical dynamical systems are generalized to hybrid dynamical systems. The concept of ω limit set is introduced for hybrid systems and is used to prove new results on invariant sets and stability, where Zeno and nonZeno hybrid systems can be treated within the same framewor ..."
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Cited by 27 (3 self)
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Abstract. Results from classical dynamical systems are generalized to hybrid dynamical systems. The concept of ω limit set is introduced for hybrid systems and is used to prove new results on invariant sets and stability, where Zeno and nonZeno hybrid systems can be treated within the same framework. As an example, LaSalle’s Invariance Principle is extended to hybrid systems. Zeno hybrid systems are discussed in detail. The ω limit set of a Zeno execution is characterized for classes of hybrid systems. 1
Widening the boundary between decidable and undecidable hybrid systems
, 2002
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Lecture notes on hybrid systems
, 2004
"... The aim of this course is to introduce some fundamental concepts from the area of hybrid systems, that is dynamical systems that involve the interaction of continuous (real valued) states and discrete (finite valued) states. Applications where these types of dynamics play a prominent role will be hi ..."
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Cited by 22 (0 self)
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The aim of this course is to introduce some fundamental concepts from the area of hybrid systems, that is dynamical systems that involve the interaction of continuous (real valued) states and discrete (finite valued) states. Applications where these types of dynamics play a prominent role will be highlighted. We will introduce general methods for investigating properties such as existence of solutions, reachability and decidability of hybrid systems. The methods will be demonstrated on the motivating applications. Students who successfully complete the course should be able to appreciate the diversity of phenomena that arise in hybrid systems and how discrete “discrete ” entities and concepts such as automata, decidability and bisimulation can coexist with continuous entities and
Towards Computing Phase Portraits of Polygonal Differential Inclusions
, 2002
"... Polygonal hybrid systems are a subclass of planar hybrid automata which can be represented by piecewise constant differential inclusions. Here, we study the problem of defining and constructing the phase portrait of such systems. We identify various important elements of it, such as viability an ..."
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Cited by 19 (13 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 study the problem of defining and constructing the phase portrait of such systems. We identify various important elements of it, such as viability and controllability kernels, and propose an algorithm for computing them all. The algorithm is based on a geometric analysis of trajectories.
A homology theory for hybrid systems: Hybrid homology
 Lect. Notes in Computer Science 3414
, 2005
"... Abstract. By transferring the theory of hybrid systems to a categorical framework, it is possible to develop a homology theory for hybrid systems: hybrid homology. This is achieved by considering the underlying “space ” of a hybrid system—its hybrid space or Hspace. The homotopy colimit can be appl ..."
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Cited by 18 (9 self)
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Abstract. By transferring the theory of hybrid systems to a categorical framework, it is possible to develop a homology theory for hybrid systems: hybrid homology. This is achieved by considering the underlying “space ” of a hybrid system—its hybrid space or Hspace. The homotopy colimit can be applied to this Hspace to obtain a single topological space; the hybrid homology of an Hspace is the homology of this space. The result is a spectral sequence converging to the hybrid homology of an Hspace, providing a concrete way to compute this homology. Moreover, the hybrid homology of the Hspace underlying a hybrid system gives useful information about the behavior of this system: the vanishing of the first hybrid homology of this Hspace—when it is contractible and finite—implies that this hybrid system is not Zeno. 1
Linear complementarity systems: Zeno states
 SIAM J. Control Optim
, 2005
"... Abstract. A linear complementarity system (LCS) is a hybrid dynamical system defined by a linear timeinvariant ordinary differential equation coupled with a finitedimensional linear complementarity problem (LCP). The present paper is the first of several papers whose goal is to study some fundame ..."
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Cited by 16 (2 self)
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Abstract. A linear complementarity system (LCS) is a hybrid dynamical system defined by a linear timeinvariant ordinary differential equation coupled with a finitedimensional linear complementarity problem (LCP). The present paper is the first of several papers whose goal is to study some fundamental issues associated with an LCS. Specifically, this paper addresses the issue of Zeno states and the related issue of finite number of mode switches in such a system. The cornerstone of our study is an expansion of a solution trajectory to the LCS near a given state in terms of an observability degree of the state. On the basis of this expansion and an inductive argument, we establish that an LCS satisfying the Pproperty has no strongly Zeno states. We next extend the analysis for such an LCS to a broader class of problems and provide sufficient conditions for a given state to be weakly nonZeno. While related modeswitch results have been proved by Brunovsky and Sussmann for more general hybrid systems, our analysis exploits the special structure of the LCS and yields new results for the latter that are of independent interest and complement those by these two and other authors.
Conewise linear systems: nonZenoness and observability
 SIAM J. Control Optim
"... Abstract. Conewise linear systems are dynamical systems in which the state space is partitioned into a finite number of nonoverlapping polyhedral cones on each of which the dynamics of the system is described by a linear differential equation. This class of dynamical systems represents a large numbe ..."
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Cited by 15 (3 self)
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Abstract. Conewise linear systems are dynamical systems in which the state space is partitioned into a finite number of nonoverlapping polyhedral cones on each of which the dynamics of the system is described by a linear differential equation. This class of dynamical systems represents a large number of piecewise linear systems, most notably, linear complementarity systems with the Pproperty and their generalizations to affine variational systems, which have many applications in engineering systems and dynamic optimization. The challenges of dealing with this type of hybrid system are due to two major characteristics: mode switchings are triggered by state evolution, and states are constrained in each mode. In this paper, we first establish the absence of Zeno states in such a system. Based on this fundamental result, we then investigate and relate several state observability notions: shorttime and Ttime (or finitetime) local/global observability. For the shorttime observability notions, constructive, finitely verifiable algebraic (both sufficient and necessary) conditions are derived. Due to their longtime modetransitional behavior, which is very difficult to predict, only partial results are obtained for the Ttime observable states. Nevertheless, we completely resolve the Ttime local observability for the bimodal conewise linear system, for finite T, and provide numerical examples to illustrate the difficulty associated with the longtime observability.
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.
Characterization of zeno behavior in hybrid systems using homological methods
 In 24th American Control Conference
, 2005
"... Abstract — It is possible to associate to a hybrid system a single topological space–its underlying topological space. Simultaneously, every hybrid system has a graph as its indexing object–its underlying graph. Here we discuss the relationship between the underlying topological space of a hybrid sy ..."
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Cited by 12 (3 self)
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Abstract — It is possible to associate to a hybrid system a single topological space–its underlying topological space. Simultaneously, every hybrid system has a graph as its indexing object–its underlying graph. Here we discuss the relationship between the underlying topological space of a hybrid system, its underlying graph and Zeno behavior. When each domain is contractible and the reset maps are homotopic to the identity map, the homology of the underlying topological space is isomorphic to the homology of the underlying graph; the nonexistence of Zeno is implied when the first homology is trivial. Moreover, the first homology is trivial when the null space of the incidence matrix is trivial. The result is an easy way to verify the nonexistence of Zeno behavior. I.
Analysis of Zeno behaviors in hybrid systems
 In: Proceedings of the 41st IEEE Conference on Decision and Control, Las Vagas, NV (2002
, 2002
"... In this paper, we investigate conditions for existence of Zeno behaviors in hybrid systems. These are behaviors that occur in a hybrid system when the system undergoes an unbounded number of discrete transitions in a finite and bounded length of time. Zeno behavior occurs, for example, when a contro ..."
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Cited by 11 (0 self)
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In this paper, we investigate conditions for existence of Zeno behaviors in hybrid systems. These are behaviors that occur in a hybrid system when the system undergoes an unbounded number of discrete transitions in a finite and bounded length of time. Zeno behavior occurs, for example, when a controller unsuccessfully attempts to satisfy an invariance specification by switching the system among different configurations faster and faster. Two types of Zeno systems will be investigated: (1) strongly Zeno systems where all runs of the system are Zeno; and (2) (weakly) Zeno systems where only some runs of the system are Zeno. We derive necessary and sufficient conditions for both strong Zenoness and Zenoness, under certain assumptions. Our analysis is based on studying the trajectory set of a certain “equivalent ” continuoustime system that is associated with the dynamic equations of the hybrid system. We also study the relation between the possibility of existence of Zeno behaviors in a system and the problem of existence of nonZeno safety controllers that prevent the system from entering a suitably defines illegal region of its operating space. In particular, we show