O-Minimal Hybrid Systems

Abstract. 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 o-minimal hybrid systems, which are initialized hybrid systems whose relevant sets and flows are definable in an o-minimal theory. We prove that o-minimal hybrid systems always admit finite bisimulations. We then present specific examples of hybrid systems with complex continuous dynamics for which finite bisimulations exist.

[1]  S. Shankar Sastry,et al.  Subanalytic Stratifications and Bisimulations , 1998, HSCC.

[2]  E. Bierstone,et al.  Semianalytic and subanalytic sets , 1988 .

[3]  Panos J. Antsaklis,et al.  Hybrid System Modeling and Autonomous Control Systems , 1992, Hybrid Systems.

[4]  Patrick Speissegger,et al.  The Pfaffian closure of an o-minimal structure , 1997, math/9710220.

[5]  S. Shankar Sastry,et al.  Hybrid Systems with Finite Bisimulations , 1997, Hybrid Systems.

[6]  P. Caines,et al.  Hierarchical hybrid control systems: a lattice theoretic formulation , 1998, IEEE Trans. Autom. Control..

[7]  Robert L. Grossman,et al.  Timed Automata , 1999, CAV.

[8]  L. Dries,et al.  Geometric categories and o-minimal structures , 1996 .

[9]  A. Pillay,et al.  Definable sets in ordered structures , 1984 .

[10]  Lou van den Dries,et al.  T-Convexity and Tame Extensions , 1995, J. Symb. Log..

[11]  Thomas A. Henzinger,et al.  The Algorithmic Analysis of Hybrid Systems , 1995, Theor. Comput. Sci..

[12]  Rajeev Alur,et al.  A Theory of Timed Automata , 1994, Theor. Comput. Sci..

[13]  Angus Macintyre,et al.  On the decidability of the real exponential field , 1996 .

[14]  George J. Pappas,et al.  Subanalytic Stratiications and Bisimulations ? , 1998 .

[15]  Gerardo Lafferriere,et al.  A New Class of Decidable Hybrid Systems , 1999, HSCC.

[16]  Thomas A. Henzinger,et al.  Hybrid Systems: Computation and Control , 1998, Lecture Notes in Computer Science.

[17]  Pravin Varaiya,et al.  What's decidable about hybrid automata? , 1995, STOC '95.

[18]  B. Krogh,et al.  Synthesis of supervisory controllers for hybrid systems based on approximating automata , 1995, Proceedings of 1995 34th IEEE Conference on Decision and Control.

[19]  K. A. Cer.  Deciding Reachability for Planar Multi-polynomial Systems ? , 1996 .

[20]  Peter E. Caines,et al.  Under Revision for System and the Hierarchical Lattices of a Finite Machine , 1995 .

[21]  George J. Pappas,et al.  Hierarchically consistent control systems , 1998, Proceedings of the 37th IEEE Conference on Decision and Control (Cat. No.98CH36171).

[22]  A. Tarski.  A Decision Method for Elementary Algebra and Geometry , 2023 .

[23]  Eduardo D. Stonag.  Critical points for least-squares problems involving certain analytic functions, with applications to sigmoidal nets , 1996, Adv. Comput. Math..

[24]  Pravin Varaiya,et al.  Decidability of Hybrid Systems with Rectangular Differential Inclusion , 1994, CAV.

[25]  Oded Maler,et al.  Hybrid and Real-Time Systems , 1997 .

[26]  A. Wilkie.  Model completeness results for expansions of the ordered field of real numbers by restricted Pfaffian functions and the exponential function , 1996 .

[27]  Jörg Raisch,et al.  Discrete approximation and supervisory control of continuous systems , 1998, IEEE Trans. Autom. Control..

[28]  Thomas A. Henzinger,et al.  Hybrid Automata with Finite Bisimulatioins , 1995, ICALP.

[29]  Wilfrid Hodges,et al.  A Shorter Model Theory , 1997 .

[30]  Dirk van Dalen,et al.  Logic and structure , 1980 .

[31]  L. Dries,et al.  On the real exponential field with restricted analytic functions , 1994 .

[32]  Thomas A. Henzinger,et al.  Hybrid Systems III , 1995, Lecture Notes in Computer Science.