Bisimulation of Dynamical Systems

A general notion of bisimulation is studied for dynamical systems. An algebraic characterization of bisimulation together, with an algorithm for computing the maximal bisimulation relation is derived using geometric control theory. Bisimulation of dynamical systems is shown to be a concept which unifies the system-theoretic concepts of state space equivalence and state space reduction, and which allows to study equivalence of systems with non-minimal state space dimension. The notion of bisimulation is especially powerful for 'non-deterministic' dynamical systems, and leads in this case to a notion of equivalence which is finer than equality of external behavior. Furthermore, by merging bisimulation of dynamical systems with bisimulation of concurrent processes a notion of structural bisimulation is developed for hybrid systems with continuous input and output variables.

[1]  S. Shankar Sastry,et al.  Hierarchically consistent control systems , 2000, IEEE Trans. Autom. Control..

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

[3]  Thomas Brihaye,et al.  On O-Minimal Hybrid Systems , 2004, HSCC.

[4]  Arjan van der Schaft,et al.  Equivalence of dynamical systems by bisimulation , 2004, IEEE Trans. Autom. Control..

[5]  A. Vanecek,et al.  Controlled and conditioned invariants in linear system theory: Guiseppe Basile and Giovanni Marro , 1994, at - Automatisierungstechnik.

[6]  M. Ben-Ari,et al.  Principles of Concurrent and Distributed Programming (2nd Edition) (Prentice-Hall International Series in Computer Science) , 2006 .

[7]  David Park,et al.  Concurrency and Automata on Infinite Sequences , 1981, Theoretical Computer Science.

[8]  George J. Pappas,et al.  Consistent abstractions of affine control systems , 2002, IEEE Trans. Autom. Control..

[9]  Robin Milner,et al.  Communication and concurrency , 1989, PHI Series in computer science.

[10]  Robin Milner,et al.  On Observing Nondeterminism and Concurrency , 1980, ICALP.

[11]  George J. Pappas,et al.  Discrete abstractions of hybrid systems , 2000, Proceedings of the IEEE.

[12]  Paulo Tabuada,et al.  Composing Abstractions of Hybrid Systems , 2002, HSCC.

[13]  Holger Hermanns,et al.  Interactive Markov Chains , 2002, Lecture Notes in Computer Science.

[14]  Maria Domenica Di Benedetto,et al.  Bisimulation theory for switching linear systems , 2004, 2004 43rd IEEE Conference on Decision and Control (CDC) (IEEE Cat. No.04CH37601).

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

[16]  W. Wonham.  Linear Multivariable Control: A Geometric Approach , 1974 .

[17]  Pedro R. D'Argenio,et al.  Algebras and Automata for Timed and Stochastic Systems , 1999 .

[18]  Paulo Tabuada,et al.  Bisimilar control affine systems , 2004, Syst. Control. Lett..

[19]  Johannes Schumacher,et al.  An Introduction to Hybrid Dynamical Systems, Springer Lecture Notes in Control and Information Sciences 251 , 1999 .

[20]  Panos J. Antsaklis,et al.  Hybrid Systems V , 1999, Lecture Notes in Computer Science.

[21]  George J. Pappas.  Bisimilar linear systems , 2003, Autom..

[22]  George J. Pappas,et al.  Unifying bisimulation relations for discrete and continuous systems , 2002 .

[23]  Christiaan Heij,et al.  Introduction to mathematical systems theory , 1997 .

[24]  Davide Sangiorgi,et al.  Communicating and Mobile Systems: the π-calculus, , 2000 .

[25]  A. J. van der Schaft.  Equivalence of dynamical systems by bisimulation , 2004, IEEE Transactions on Automatic Control.

[26]  Arjan van der Schaft,et al.  An Introduction to Hybrid Dynamical Systems, Springer Lecture Notes in Control and Information Sciences 251 , 1999 .

[27]  Arjan van der Schaft,et al.  Achievable behavior of general systems , 2003, Syst. Control. Lett..

[28]  J. Pearson.  Linear multivariable control, a geometric approach , 1977 .

[29]  Paulo Tabuada,et al.  Bisimulation Relations for Dynamical and Control Systems , 2003, CTCS.

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