Integrated symbolic control design for nonlinear systems with infinite states specifications

Discrete abstractions of continuous and hybrid systems have recently been the topic of great interest from both the control systems and the computer science communities, because they provide a sound mathematical framework for analysing and controlling embedded systems. In this paper we give a further contribution to this research line, by addressing the problem of symbolic control design of nonlinear systems with infinite states specifications, modelled by differential equations. We first derive the symbolic controller solving the control design problem, given in terms of discrete abstractions of the plant and the specification systems. We then present an algorithm which integrates the construction of the discrete abstractions with the design of the symbolic controller. Space and time complexity analysis of the proposed algorithm is performed and a comparison with traditional approaches currently available in the literature for symbolic control design, is discussed. Some examples are included, which show the interest and applicability of our results.

[1]  Paulo Tabuada,et al.  Approximately bisimilar symbolic models for nonlinear control systems , 2007, Autom..

[2]  Calin Belta,et al.  Temporal Logic Control of Discrete-Time Piecewise Affine Systems , 2012, IEEE Transactions on Automatic Control.

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

[4]  Jörg Raisch,et al.  Discrete Supervisory Control of Hybrid Systems Based on l-Complete Approximations , 2002, Discret. Event Dyn. Syst..

[5]  Christos G. Cassandras,et al.  Introduction to Discrete Event Systems , 1999, The Kluwer International Series on Discrete Event Dynamic Systems.

[6]  Paulo Tabuada,et al.  Verification and Control of Hybrid Systems - A Symbolic Approach , 2009 .

[7]  Paulo Tabuada,et al.  Symbolic Models for Nonlinear Control Systems: Alternating Approximate Bisimulations , 2007, SIAM J. Control. Optim..

[8]  David Angeli,et al.  A Lyapunov approach to incremental stability properties , 2002, IEEE Trans. Autom. Control..

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

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

[11]  Paulo Tabuada,et al.  Approximately Bisimilar Symbolic Models for Incrementally Stable Switched Systems , 2008, IEEE Transactions on Automatic Control.

[12]  M Zamani,et al.  Symbolic models for unstable nonlinear control systems , 2010, Proceedings of the 2010 American Control Conference.

[13]  Stavros Tripakis,et al.  On-the-Fly Controller Synthesis for Discrete and Dense-Time Systems , 1999, World Congress on Formal Methods.

[14]  Paulo Tabuada,et al.  Symbolic models for nonlinear time-delay systems using approximate bisimulations , 2010, Syst. Control. Lett..

[15]  Pierre Wolper,et al.  Memory-efficient algorithms for the verification of temporal properties , 1990, Formal Methods Syst. Des..

[16]  Antonio Bicchi,et al.  On the reachability of quantized control systems , 2002, IEEE Trans. Autom. Control..

[17]  Jan H. van Schuppen,et al.  Reachability and control synthesis for piecewise-affine hybrid systems on simplices , 2006, IEEE Transactions on Automatic Control.

[18]  B. Anderson,et al.  Digital control of dynamic systems , 1981, IEEE Transactions on Acoustics, Speech, and Signal Processing.

[19]  Eduardo Sontag,et al.  Forward Completeness, Unboundedness Observability, and their Lyapunov Characterizations , 1999 .

[20]  Calin Belta,et al.  Controlling a Class of Nonlinear Systems on Rectangles , 2006, IEEE Transactions on Automatic Control.

[21]  Moshe Y. Vardi,et al.  Memory-Ecient Algorithms for the Verication of Temporal Properties , 1992 .

[22]  Stephan Merz,et al.  Model Checking , 2000 .

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

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

[25]  Antoine Girard.  Approximately Bisimilar Finite Abstractions of Stable Linear Systems , 2007, HSCC.

[26]  Jun-ichi Imura,et al.  Discrete-State Abstractions of Nonlinear Systems Using Multi-resolution Quantizer , 2009, HSCC.

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

[28]  Jan Lunze,et al.  A discrete-event model of asynchronous quantised systems , 2002, Autom..

[29]  Paulo Tabuada.  An Approximate Simulation Approach to Symbolic Control , 2008, IEEE Transactions on Automatic Control.

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

[31]  Paulo Tabuada,et al.  Linear Time Logic Control of Discrete-Time Linear Systems , 2006, IEEE Transactions on Automatic Control.