Symbolic Models for a Class of Impulsive Systems

Symbolic models have been used as the basis of a systematic framework to address control design of several classes of hybrid systems with sophisticated control objectives. However, results available in the literature are not concerned with impulsive systems which are an important modeling framework of many applications. In this letter, we provide an approach for constructing symbolic models for a class of impulsive systems possessing some stability properties. We formally relate impulsive systems and their symbolic models using a notion of so-called alternating simulation function. We show that behaviors of the constructed symbolic models are approximately equivalent to those of the impulsive systems. Finally, we illustrate the effectiveness of our results through a case study.

[1]  George J. Pappas,et al.  Hierarchical control system design using approximate simulation , 2001 .

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

[3]  Sôichi Kakeya,et al.  On Differential Inequalities , 1918 .

[4]  Majid Zamani,et al.  Compositional Synthesis of Symbolic Models for Networks of Switched Systems , 2019, IEEE Control Systems Letters.

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

[6]  Majid Zamani,et al.  SCOTS: A Tool for the Synthesis of Symbolic Controllers , 2016, HSCC.

[7]  David Angeli,et al.  Further Results on Incremental Input-to-State Stability , 2009, IEEE Transactions on Automatic Control.

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

[9]  Christel Baier,et al.  Principles of Model Checking (Representation and Mind Series) , 2008 .

[10]  Manuel Mazo,et al.  Symbolic Abstractions of Networked Control Systems , 2018, IEEE Transactions on Control of Network Systems.

[11]  Majid Zamani,et al.  Compositional Abstractions of Interconnected Discrete-Time Switched Systems , 2019, 2019 18th European Control Conference (ECC).

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

[13]  Calin Belta,et al.  Formal Methods for Discrete-Time Dynamical Systems , 2017 .

[14]  D. Bainov,et al.  Strong stability of impulsive systems , 1988 .

[15]  Kexue Zhang,et al.  Impulsive Systems on Hybrid Time Domains , 2019 .

[16]  Laurentiu Hetel,et al.  Nonlinear impulsive systems: 2D stability analysis approach , 2017, Autom..

[17]  Antoine Girard,et al.  Symbolic models for stochastic switched systems: A discretization and a discretization-free approach , 2014, Autom..

[18]  Antoine Girard.  Approximately bisimilar abstractions of incrementally stable finite or infinite dimensional systems , 2014, 53rd IEEE Conference on Decision and Control.

[19]  John Lygeros,et al.  Symbolic Control of Stochastic Systems via Approximately Bisimilar Finite Abstractions , 2013, IEEE Transactions on Automatic Control.

[20]  H. Fédérer.  Geometric Measure Theory , 1969 .

[21]  Maria Domenica Di Benedetto,et al.  Symbolic Models for Networks of Control Systems , 2016, IEEE Transactions on Automatic Control.

[22]  Antoine Girard,et al.  Optimal multirate sampling in symbolic models for incrementally stable switched systems , 2018, Autom..

[23]  Joseph Sifakis,et al.  On the Synthesis of Discrete Controllers for Timed Systems (An Extended Abstract) , 1995, STACS.

[24]  Pushpak Jagtap,et al.  Symbolic models for retarded jump-diffusion systems , 2020, Autom..

[25]  Antoine Girard,et al.  Compositional Abstraction and Safety Synthesis Using Overlapping Symbolic Models , 2017, IEEE Transactions on Automatic Control.

[26]  Compositional Synthesis of Symbolic Models for Infinite Networks , 2020 .

[27]  Andrew R. Teel,et al.  On assigning the derivative of a disturbance attenuation CLF , 1998, Proceedings of the 37th IEEE Conference on Decision and Control (Cat. No.98CH36171).

[28]  Gunther Reissig,et al.  Feedback Refinement Relations for the Synthesis of Symbolic Controllers , 2015, IEEE Transactions on Automatic Control.

[29]  Antoine Girard,et al.  From dissipativity theory to compositional synthesis of symbolic models , 2017, 2018 Indian Control Conference (ICC).

[30]  R. Sanfelice,et al.  Hybrid dynamical systems , 2009, IEEE Control Systems.

[31]  Sergey Dashkovskiy,et al.  Input-to-state Stability of Impulsive Systems with Different Jump Maps , 2016, ArXiv.

[32]  Boris M. Miller,et al.  Impulsive Control in Continuous and Discrete-Continuous Systems , 2003 .

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

[34]  Manuel Mazo,et al.  Symbolic Models for Nonlinear Control Systems Without Stability Assumptions , 2010, IEEE Transactions on Automatic Control.

[35]  Majid Zamani,et al.  Compositional Synthesis of Finite Abstractions for Networks of Systems: A Small-Gain Approach , 2018, Autom..

[36]  Björn Rüffer,et al.  Incremental stability properties for discrete-time systems , 2016, 2016 IEEE 55th Conference on Decision and Control (CDC).

[37]  Maria Domenica Di Benedetto,et al.  Symbolic Models for Nonlinear Time-Varying Time-Delay Systems via Alternating Approximate Bisimulation , 2010 .

[38]  Maria Domenica Di Benedetto,et al.  Design of Symbolic Controllers for Networked Control Systems , 2014, IEEE Transactions on Automatic Control.

[39]  João Pedro Hespanha,et al.  Lyapunov conditions for input-to-state stability of impulsive systems , 2008, Autom..