Equivalence of dynamical systems by bisimulation

A general notion of bisimulation is defined for linear input-state-output systems, using analogies with the theory of concurrent processes. A characterization of bisimulation and an algorithm for computing the maximal bisimulation relation is derived using geometric control theory. Bisimulation is shown to be a notion which unifies the concepts of state-space equivalence and state-space reduction, and which allows to study equivalence of systems with nonminimal state-space dimension. The notion of bisimulation is especially powerful for "nondeterministic" dynamical systems, and leads in this case to a notion of equivalence which is finer than equality of external behavior. For abstractions of systems it is shown how the results specialize to previously obtained results by other authors. Extensions of the main results to the nonlinear case are provided.

[1]  Anthony M. Bloch,et al.  Nonlinear Dynamical Control Systems (H. Nijmeijer and A. J. van der Schaft) , 1991, SIAM Review.

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

[3]  Jan C. Willems,et al.  Introduction to Mathematical Systems Theory. A Behavioral , 2002 .

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

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

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

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

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

[9]  Arjan van der Schaft,et al.  Equivalence of hybrid dynamical systems , 2004 .

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

[11]  Jean-Pierre Aubin,et al.  Impulse differential inclusions: a viability approach to hybrid systems , 2002, IEEE Trans. Autom. Control..

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

[13]  A.J. van der Schaft,et al.  Achievable behavior by composition , 2002, Proceedings of the 41st IEEE Conference on Decision and Control, 2002..

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

[15]  C. Desoer,et al.  Linear System Theory , 1963 .

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

[17]  M. Sain.  Finite dimensional linear systems , 1972 .

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

[19]  Arjan van der Schaft,et al.  Bisimulation of Dynamical Systems , 2004, HSCC.

[20]  George J. Pappas,et al.  Bisimilar control affine systems , 2002, Proceedings of the 41st IEEE Conference on Decision and Control, 2002..

[21]  Robin Milner,et al.  Communicating and mobile systems - the Pi-calculus , 1999 .

[22]  G. Basile,et al.  Controlled and conditioned invariants in linear system theory , 1992 .

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

[24]  A. Isidori.  Nonlinear Control Systems , 1985 .

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

[26]  J. M. Schumacher,et al.  Compositionality issues in discrete, continuous, and hybrid systems , 2001 .

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

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

[29]  S. Shankar Sastry,et al.  O-Minimal Hybrid Systems , 2000, Math. Control. Signals Syst..

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