Gerardo Lafferriere, George J. Pappas, and Shankar Sastry. Hybrid systems with finite bisimulations. In P. Antsaklis, W. Kohn, M. Lemmon, A. Nerode, and S. Sastry, editors, Hybrid Systems V, Lecture Notes in Computer Science. Springer Verlag, New York, 1998. To appear.  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 ....

G.J. Pappas, G. Lafferriere, and S. Sastry. Hybrid Systems with Finite Bisimulations. In Hybrid Systems V, Lecture Notes in Computer Science, Springer Verlag, 1999.

.... of hybrid systems [1] Formalisms for input output hybrid automata have been also proposed in [26, 41, 23] A related approach to the work presented in this paper is based on the modeling formalism of hybrid automata and uses bisimulations to study the decidability of verification algorithms [14, 21, 2]. Bisimulations are quotient systems that preserve the reachability properties of the original hybrid system and therefore, problems related to the reachability of the original system can be solved by studying the quotient system. The idea of using finite bisimulations for the analysis and ....

....it is desirable to induce dynamical systems in finite quotient spaces that preserve the properties of interest and then study the simplified models. In general, piecewise linear hybrid dynamical systems cannot be induced in finite quotient spaces by preserving the reachability properties [21]. The solution we propose is to take advantage of the available control inputs in order to simplify the system. More specifically, we want to formulate conditions on the available control inputs in order to construct meaningful discrete abstractions of the hybrid system. The main mathematical tool ....

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G. Lafferriere, G. Pappas, and S. Sastry. Hybrid systems with finite bisimulations. In P. Antsaklis, W. Kohn, M. Lemmon, A. Nerode, and S. Sastry, editors, Hybrid Systems V, volume 1567 of Lecture Notes in Computer Science, pages 186--203. Springer, 1999. 37

....to the open loop hybrid system with discrete input signal u and the binary vector output signal v in Fig. 2 as HO in the sequel. 4 Transition System Semantics for Hybrid Systems Transition systems have been an effective formalism to define semantics for hybrid systems modeled by hybrid automata [14, 15, 12]. In this section we introduce transition systems with inputs and outputs to define the semantics for the open loop hybrid system HO in Fig. 2. We are interested in the sequential input output behavior of HO , rather than the details of the continuous trajectories between threshold events, since ....

....property. Definition 13. Output consistent partition OCP) Given a transition system T = Q; U; V; Q 0 ) a partition P of Q [ Q 0 is an output consistent partition if for any P 2 P either P Q or P Q 0 and is constant on any P ae Q. Definition 14. Quotient Transition System) [14, 12, 13] Given a transition system T = Q; U; V; Q 0 ) and an OCP P of Q, the quotient transition system of T is defined as T=P = P ; U; V; P ; Q 0 =P) where for all P; P 0 2 P , and u 2 U , P u P P 0 iff there exist q 2 P and q 0 2 P 0 such that q u q 0 , or equivalently P ost ....

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S. Sastry G. Lafferriere, G. J. Pappas. Hybrid systems with finite bisimulations. Technical Report UCB/ERL M98/15, University of California at Berkeley, April 1998.

....shown that if the refinement procedure reaches a fixed point, then the language of the automaton is exactly the language of the hybrid system. Such a fixed point may not exist, however, since an infinite state discrete model may be needed to realize the event language of the hybrid system (see [8] and the references therein for recent results regarding finite state representations of hybrid systems) The major obstacle towards applying this automata construction and refinement methodology to any real system is the lack of efficient methods for computing the mappings of families of ....

G. Lafferriere, G.J. Papps, and S. Sastry. Hybrid systems with finite bisimulations. Technical Report UCB/ERL M98/15, University of California at Berkeley, April 1998.

....individual continuous component can be represented by a discrete abstraction preserving the behavior of interest. Several computational approaches have been proposed to determine finite bisimulations that are essentially equivalence relations on the state space and define continuous partitions [11, 16]. Methods for computing continuous partitions based on approximations of the flow have also been considered [9] Computational algorithms for the use of the phasespace geometric description of dynamics have been developed in [29] Related approaches using a feedback architecture of a continuous ....

G. La#erriere, G. Pappas, and S. Sastry. Hybrid systems with finite bisimulations. In P. Antsaklis, W. Kohn, M. Lemmon, A. Nerode, and S. Sastry, editors, Hybrid Systems V, volume 1567 of Lecture Notes in Computer Science, pages 186--203. Springer, 1999.

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Gerardo Lafferriere, George J. Pappas, and Shankar Sastry. Hybrid systems with finite bisimulations. In P. Antsaklis, W. Kohn, M. Lemmon, A. Nerode, and S. Sastry, editors, Hybrid Systems V, Lecture Notes in Computer Science. Springer Verlag, New York, 1998. To appear.

No context found.

G. Lafferriere, G. J. Pappas, and S. Sastry. Hybrid systems with finite bisimulations. In P. Antsaklis, W. Kohn, M. Lemmon, A. Nerode, and S. Sastry, editors, Hybrid Systems V, Lecture Notes in Computer Science. Springer Verlag, New York, 1998.

....[9] 13] 21] 30] Depending on the property, special graph quotients which preserve the property of interest are constructed. More recently, a methodology for constructing finite graph quotients which have equivalent reachability properties with analytic vector fields is presented in [19], 20] A similar construction which characterizes reachability of a continuous system in terms of an associated discrete system may be found in [8] In this spirit, and after having characterized consistent linear abstractions, we obtain a hierarchical controllability criterion which has ....

....and hybrid abstractions of continuous systems. A very interesting problem, however, remains the construction of finite and consistent state space partitions, given a continuous control system. An algorithm for constructing finite reachability preserving quotients of vector fields is proposed in [19], 20] and [39] APPENDIX MATLAB IMPLEMETATION OF ALGORITHMS 6.1 AND 6.4 function [controllable] HCA(A,B,k,tol) Controllability Algorithms 6.1 and 6.4 Required Inputs: System Matrices A, B, Integer ( is Algorithm 6.4) Optional Inputs: Tolerance ....

G. Lafferriere, G. J. Pappas, and S. Sastry, "Hybrid systems with finite bisimulations," in Hybrid Systems V. ser. Lecture Notes in Computer Science, P. Antsaklis, W. Kohn, M. Lemmon, A. Nerode, and S. Sastry, Eds. New York: Springer-Verlag, 1998, vol. 1567, pp. 186--203.

....linear vector fields with purely imaginary eigenvalues and all relevant sets are definable in this structure. Since the restriction of on is definable, the operator corresponding to is definable. This leads to the following corollary of Theorem 5. 4, which generalizes to the planar results in [17] [43], and [47] Corollary 5.5: Let be a hybrid system for which all relevant sets (guards, invariants, initial conditions) are finitely subanalytic and all vector fields are diagonalizable linear vector fields with purely imaginary eigenvalues. Let be a finite collection of finitely subanalytic sets. ....

....forms: linear vector fields with real eigenvalues; diagonalizable linear vector fields with purely imaginary eigenvalues. Let be a finite collection of finitely subanalytic sets. Then the transition system has a finite bisimulation quotient. The above theorem extends the planar results in [43] to . Note that relaxations of Corollary 5.6 would allow spiraling, linear vector fields, which are not definable in this structure. As was shown by Example 5.1, such systems, in general, do not admit finite bisimulations. This shows that even though the conditions of Theorem 5.4 are sufficient, ....

G. Lafferriere, G. J. Pappas, and S. Sastry, "Hybrid systems with finite bisimulations," in Hybrid Systems V, P. Antsaklis, W. Kohn, M. Lemmon, A. Nerode, and S. Sastry, Eds. New York: SpringerVerlag, 1998, Lecture Notes in Computer Science.

....claim: Claim: At each step of the bisimilarity algorithm, S is compatible with M= The claim shows that S is finer than all partitions obtained at each step. Since S is finite this clearly shows that the algorithm terminates. The details of the proof of the claim will appear in [11]. The proof for the case of imaginary eigenvalues is, a special case of Theorem 6.2 below. Theorem 6.2 If X is an analytic vector field in R 2 which admits an analytic family of first integrals, then the bisimilarity algorithm terminates. Here, by an analytic family of first integrals we mean ....

Gerardo Lafferriere, George J. Pappas, and Shankar Sastry. Hybrid systems with finite bisimulations. In P. Antsaklis, W. Kohn, M. Lemmon, A. Nerode, and S. Sastry, editors, Hybrid Systems V, Lecture Notes in Computer Science. Springer Verlag, New York, 1998. To appear.

....purely imaginary eigenvalues and all relevant sets are definable in this structure. Since the restriction of sin on [ Gamma; is definable, the P re operator corresponding to F is definable. This leads to the following corollary of Theorem 5. 4, which generalizes to R n the planar results in [17, 43, 47]. Corollary 5.5. Let H be a hybrid system for which all relevant sets (guards, invariants, initial conditions) are finitely subanalytic and all vector fields are diagonalizable linear vector fields with purely imaginary eigenvalues. Let Sigma be a finite collection of finitely subanalytic sets. ....

....linear vector fields with purely imaginary eigenvalues DISCRETE ABSTRACTIONS OF HYBRID SYSTEMS 23 Let Sigma be a finite collection of finitely subanalytic sets. Then the transition system TH; Sigma has a finite bisimulation quotient. The above theorem extends the planar results in [43] to R n . Note that relaxations of Corollary 5.6 would allow spiraling, linear vector fields which are not definable in this structure. As was shown by Example 5.1, such systems, in general, do not admit finite bisimulations. This shows that even though the conditions of Theorem 5.4 are ....

G. Lafferriere, G. J. Pappas, and S. Sastry. Hybrid systems with finite bisimulations. In P. Antsaklis, W. Kohn, M. Lemmon, A. Nerode, and S. Sastry, editors, Hybrid Systems V, Lecture Notes in Computer Science. Springer Verlag, New York, 1998.

....purely imaginary eigenvalues and all relevant sets are definable in this theory. Since the restriction of sin on [ Gamma ; is definable, the P re operator corresponding to F is definable. This leads to the following corollary of Theorem 5. 4, which generalizes to R n the planar results in [20, 23]. Corollary 5.5. Let H be a hybrid system for which all relevant sets (guards, invariants, initial conditions) are bounded subanalytic and all vector fields are diagonalizable linear vector fields with purely imaginary eigenvalues. Then H admits a finite bisimulation. R; Gamma; Delta; e ....

....sets are bounded subanalytic and all vector fields are of one of the following two forms: ffl linear vector fields with real eigenvalues ffl diagonalizable linear vector fields with purely imaginary eigenvalues Then H admits a finite bisimulation. The above theorem extends the planar results in [20] to R n . Note that relaxations of Theorem 5.6 would allow spiraling, linear vector fields which are not definable in this theory. As was shown by Example 5.1, such systems, in general, do not admit finite bisimulations. This shows that even though the conditions of Theorem 5.4 are sufficient, ....

G. Lafferriere, G. J. Pappas, and S. Sastry. Hybrid systems with finite bisimulations. In P. Antsaklis, W. Kohn, M. Lemmon, A. Nerode, and S. Sastry, editors, Hybrid Systems V, Lecture Notes in Computer Science. Springer Verlag, New York, 1998.

....of hybrid systems 2. Generation of more o minimal theories immediately leads to new classes of o minimal hybrid systems 3. Constructive results within o minimal theories immediately lead to decidability results By providing a purely model theoretic framework, we also extend the planar results of [19] and [20] The outline of the paper is as follows: In Section 2 we review the notion of bisimulations of transitions systems. In Section 3 we define a general class of hybrid systems and describe the bisimulation algorithm as it applies to hybrid systems. Section 4 presents the notion of ....

....involved in the description of the hybrid system have rational coefficients, we obtain a new class of decidable hybrid systems. The o minimality of this structure can also be used to show the existence of finite bisimulations in special cases when the flow is not definable. This was illustrated in [19] for the case of planar hybrid systems whose vector fields admit definable Hamiltonians. This captures the decidability result of [10] 6.3. R an = R; Gamma; Theta; 0; 1; f fg) In order to describe the definable sets in this theory, we need the notions of semianalytic and subanalytic ....

[Article contains additional citation context not shown here]

G. Lafferriere, G.J. Pappas, and S. Sastry. Hybrid systems with finite bisimulations. Technical Report UCB/ERL M98/15, University of California at Berkeley, Berkeley, CA, April 1998.

No context found.

G. La#erriere, G. Pappas, and S. Sastry. Hybrid systems with finite bisimulations. In P. Antsaklis, W. Kohn, M. Lemmon, A. Nerode, and S. Sastry, editors, Hybrid Systems V, volume 1567 of Lecture Notes in Computer Science, pages 186--203. Springer, 1999.

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