Approximate Bisimulations for Nonlinear Dynamical Systems

The notion of exact bisimulation equivalence for nondeterministic discrete systems has recently resulted in notions of exact bisimulation equivalence for continuous and hybrid systems. In this paper, we establish the more robust notion of approximate bisimulation equivalence for nondeterministic nonlinear systems. This is achieved by requiring that a distance between system observations starts and remains, close, in the presence of nondeterministic system evolution. We show that approximate bisimulation relations can be characterized using a class of functions called bisimulation functions. For nondeterministic nonlinear systems, we show that conditions for the existence of bisimulation functions can be expressed in terms of Lyapunov-like inequalities, which for deterministic systems can be computed using recent sum-of-squares techniques. Our framework is illustrated on a safety verification example.

[1]  P. Kokotovic,et al.  Inverse Optimality in Robust Stabilization , 1996 .

[2]  Jos F. Sturm,et al.  A Matlab toolbox for optimization over symmetric cones , 1999 .

[3]  Ian M. Mitchell,et al.  Level Set Methods for Computation in Hybrid Systems , 2000, HSCC.

[4]  P. Parrilo Structured semidefinite programs and semialgebraic geometry methods in robustness and optimization , 2000 .

[5]  Edmund M. Clarke,et al.  Model Checking , 1999, Handbook of Automated Reasoning.

[6]  Kim-Chuan Toh,et al.  SDPT3 — a Matlab software package for semidefinite-quadratic-linear programming, version 3.0 , 2001 .

[7]  Daniel Liberzon,et al.  Universal construction of feedback laws achieving ISS and integral-ISS disturbance attenuation , 2002, Syst. Control. Lett..

[8]  Peter J Seiler,et al.  SOSTOOLS: Sum of squares optimization toolbox for MATLAB , 2002 .

[9]  A. Papachristodoulou,et al.  On the construction of Lyapunov functions using the sum of squares decomposition , 2002, Proceedings of the 41st IEEE Conference on Decision and Control, 2002..

[10]  J. Marsden,et al.  A subspace approach to balanced truncation for model reduction of nonlinear control systems , 2002 .

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

[12]  Luca de Alfaro,et al.  Linear and Branching Metrics for Quantitative Transition Systems , 2004, ICALP.

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

[14]  B. Krogh,et al.  Reachability analysis of hybrid control systems using reduced-order models , 2004, Proceedings of the 2004 American Control Conference.

[15]  George J. Pappas,et al.  Geometric programming relaxations for linear system reachability , 2004, Proceedings of the 2004 American Control Conference.

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

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

[18]  Radha Jagadeesan,et al.  Metrics for labelled Markov processes , 2004, Theor. Comput. Sci..

[19]  Peter J Seiler,et al.  SOSTOOLS and its control applications , 2005 .

[20]  Paulo Tabuada,et al.  Bisimulation relations for dynamical, control, and hybrid systems , 2005, Theor. Comput. Sci..

[21]  George J. Pappas,et al.  Approximate Bisimulations for Constrained Linear Systems , 2005, Proceedings of the 44th IEEE Conference on Decision and Control.

[22]  Antoine Girard,et al.  Reachability of Uncertain Linear Systems Using Zonotopes , 2005, HSCC.

[23]  Antoine Girard,et al.  Approximation Metrics for Discrete and Continuous Systems , 2006, IEEE Transactions on Automatic Control.