Abstract
In this article we study ways of constructing meaningful operational models of piecewise-smooth systems (PWS). The systems we consider are described by polynomial vector fields defined on non-overlapping semi-algebraic sets, which form a partition of the state space. Our approach is to give meaning to motion in systems of this type by automatically synthesizing operational models in the form of hybrid automata (HA). Despite appearances, it is in practice often difficult to arrive at satisfactory HA models of PWS. The different ways of building operational models that we explore in our approach can be thought of as defining different semantics for the underlying PWS. These differences have a number of interesting nuances related to phenomena such as chattering, non-determinism, so-called mythical modes and sliding behaviour.
- Rajeev Alur, Costas Courcoubetis, Thomas A. Henzinger, and Pei-Hsin Ho. 1992. Hybrid Automata: An Algorithmic Approach to the Specification and Verification of Hybrid Systems. In Hybrid Systems (Lecture Notes in Computer Science), Robert L. Grossman, Anil Nerode, Anders P. Ravn, and Hans Rischel (Eds.), Vol. 736. Springer, 209--229. Google Scholar
Digital Library
- Aaron D. Ames, Alessandro Abate, and Shankar Sastry. 2005. Sufficient Conditions for the Existence of Zeno Behavior. In Proceedings of the 44th IEEE Conference on Decision and Control. 696--701.Google Scholar
Cross Ref
- Saugata Basu, Richard Pollack, and Marie-Françoise Roy. 2006. Algorithms in Real Algebraic Geometry (Algorithms and Computation in Mathematics) (2 ed.). Springer. Google Scholar
Digital Library
- Bob F. Caviness and Jeremy R. Johnson. 1998. Quantifier Elimination and Cylindrical Algebraic Decomposition. Springer.Google Scholar
- Xin Chen, Erika Ábrahám, and Sriram Sankaranarayanan. 2013. Flow*: An Analyzer for Non-linear Hybrid Systems. In Proceedings of the 25th International Conference on Computer Aided Verification (CAV’13). Springer, 258--263.Google Scholar
Cross Ref
- George E. Collins. 1975. Quantifier elimination for real closed fields by cylindrical algebraic decompostion. Lecture Notes in Computer Science, Vol. 33. Springer, 134--183.Google Scholar
- Charles C. Conley. 1978. Isolated invariant sets and the Morse index. Conference Board of the Mathematical Sciences.Google Scholar
Cross Ref
- Jorge Cortés. 2008. Discontinuous dynamical systems: A tutorial on solutions, non-smooth analysis and stability. IEEE Control Systems 28, 3 (2008), 36--73.Google Scholar
Cross Ref
- David Cox, John Little, and Donal O’Shea. 2010. Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra. Springer. Google Scholar
Digital Library
- James H. Davenport and Joos Heintz. 1988. Real Quantifier Elimination is Doubly Exponential. J. Symb. Comput. 5, 1--2 (Feb. 1988), 29--35. Google Scholar
Digital Library
- Jennifer M. Davoren and Anil Nerode. 2000. Logics for hybrid systems. Proc. IEEE 88, 7 (2000), 985--1010.Google Scholar
Cross Ref
- Magnus Egerstedt. 2000. Behavior Based Robotics Using Hybrid Automata. In Proceedings of the Third International Workshop on Hybrid Systems: Computation and Control (HSCC’00). Springer, 103--116. Google Scholar
Digital Library
- Goran Frehse. 2015. An Introduction to Hybrid Automata, Numerical Simulation and Reachability Analysis. Springer, 50--81.Google Scholar
- Goran Frehse, Colas Le Guernic, Alexandre Donzé, Scott Cotton, Rajarshi Ray, Olivier Lebeltel, Rodolfo Ripado, Antoine Girard, Thao Dang, and Oded Maler. 2011. SpaceEx: Scalable Verification of Hybrid Systems. In Proceedings of the 23rd International Conference on Computer Aided Verification (CAV’11). Springer, 379--395. Google Scholar
Digital Library
- Nathan Fulton, Stefan Mitsch, Jan-David Quesel, Marcus Völp, and André Platzer. 2015. KeYmaera X: An Axiomatic Tactical Theorem Prover for Hybrid Systems. In Automated Deduction - CADE-25: 25th International Conference on Automated Deduction, Berlin, Germany, August 1-7, 2015, Proceedings (Lecutre Notes in Computer Science), Amy P. Felty and Aart Middeldorp (Eds.), Vol. 9195. Springer.Google Scholar
Cross Ref
- Khalil Ghorbal and André Platzer. 2014. Characterizing Algebraic Invariants by Differential Radical Invariants. In Tools and Algorithms for the Construction and Analysis of Systems - 20th International Conference, TACAS 2014, Held as Part of the European Joint Conferences on Theory and Practice of Software, ETAPS 2014, Grenoble, France, April 5-13, 2014. Proceedings. 279--294.Google Scholar
- Khalil Ghorbal, Andrew Sogokon, and André Platzer. 2017. A hierarchy of proof rules for checking positive invariance of algebraic and semi-algebraic sets. Computer Languages, Systems 8 Structures 47 (2017), 19--43.Google Scholar
- Rafal Goebel, Ricardo G. Sanfelice, and Andrew R. Teel. 2009. Hybrid dynamical systems. IEEE Control Systems 29, 2 (2009), 28--93.Google Scholar
Cross Ref
- Otomar Hájek. 1979. Discontinuous differential equations I. Journal of Differential Equations 32, 2 (1979), 149--170.Google Scholar
Cross Ref
- Jack K. Hale and Joseph P. LaSalle. 1963. Differential Equations: Linearity vs. Nonlinearity. SIAM Rev. 5, 3 (July 1963), 249--272.Google Scholar
Digital Library
- Philip Hartman. 1964. Ordinary Differential Equations. John Wiley 8 Sons, Inc., New York.Google Scholar
- Thomas A. Henzinger. 1996. The Theory of Hybrid Automata. In Proceedings of the 11th Annual IEEE Symposium on Logic in Computer Science (LICS’96). IEEE Computer Society, 278--292. Google Scholar
Digital Library
- Karl H. Johansson, Magnus Egerstedt, John Lygeros, and Shankar Sastry. 1999. On the regularization of Zeno hybrid automata. Systems 8 Control Letters 38, 3 (1999), 141--150.Google Scholar
- Soonho Kong, Sicun Gao, Wei Chen, and Edmund M. Clarke. 2015. dReach: -Reachability Analysis for Hybrid Systems. In Tools and Algorithms for the Construction and Analysis of Systems - 21st International Conference, TACAS 2015, Held as Part of the European Joint Conferences on Theory and Practice of Software, ETAPS 2015, London, UK, April 11-18, 2015. Proceedings. Springer, 200--205. Google Scholar
Digital Library
- Jiang Liu, Naijun Zhan, and Hengjun Zhao. 2011. Computing Semi-algebraic Invariants for Polynomial Dynamical Systems. In Proceedings of the Ninth ACM International Conference on Embedded Software (EMSOFT’11). ACM, 97--106. Google Scholar
Digital Library
- John Lygeros, Karl H. Johansson, Shankar Sastry, and Magnus Egerstedt. 1999. On the existence of executions of hybrid automata. In the 38th IEEE Conference on Decision and Control, Phoenix, AZ. IEEE, 2249--2254.Google Scholar
Cross Ref
- John Lygeros, Karl H. Johansson, Slobodan N. Simić, Jun Zhang, and S. Shankar Sastry. 2003. Dynamical properties of hybrid automata. IEEE Trans. Automat. Control 48, 1 (Jan 2003), 2--17.Google Scholar
Cross Ref
- Bhubaneswar Mishra. 1993. Algorithmic Algebra. Springer. Google Scholar
Digital Library
- Pieter J. Mosterman. 2003. Mode Transition Behavior in Hybrid Dynamic Systems. In Proc. of the 2003 Winter Simulation Conference, S. Chick, P. J. Sánchez, D. Ferrin, and D. J. Morrice (Eds.). 623--631. Google Scholar
Digital Library
- Pieter J. Mosterman and Gautam Biswas. 1998. A theory of discontinuities in physical system models. Journal of the Franklin Institute 335, 3 (1998), 401--439.Google Scholar
Cross Ref
- Pieter J. Mosterman, Feng Zhao, and Gautam Biswas. 1998. An Ontology for Transitions in Physical Dynamic Systems. In Proceedings of the Fifteenth National Conference on Artificial Intelligence and Tenth Innovative Applications of Artificial Intelligence Conference, AAAI 98, IAAI 98, July 26-30, 1998, Madison, Wisconsin, USA. 219--224. http://www.aaai.org/Library/AAAI/1998/aaai98-030.php Google Scholar
Digital Library
- Eva M. Navarro-López and Rebekah Carter. 2011. Hybrid automata: an insight into the discrete abstraction of discontinuous systems. International Journal of Systems Science 42, 11 (2011), 1883--1898. Google Scholar
Digital Library
- Dmitri Novikov and Sergei Yakovenko. 1999. Trajectories of polynomial vector fields and ascending chains of polynomial ideals. In Annales de l’institut Fourier, Vol. 49. 563--609.Google Scholar
- André Platzer. 2008. Differential Dynamic Logic for Hybrid Systems.J. Autom. Reas. 41, 2 (2008), 143--189. Google Scholar
Digital Library
- André Platzer. 2010. Logical Analysis of Hybrid Systems: Proving Theorems for Complex Dynamics. Springer. Google Scholar
Digital Library
- Ricardo G. Sanfelice, Rafal Goebel, and Andrew R. Teel. 2008. Generalized solutions to hybrid dynamical systems. ESAIM: Control, Optimisation and Calculus of Variations 14 (10 2008), 699--724. Issue 4.Google Scholar
- Abraham Seidenberg. 1954. A new decision method for elementary algebra. Annals of Mathematics (1954), 365--374.Google Scholar
- Alfred Tarski. 1951. A decision method for elementary algebra and geometry. Bull. Amer. Math. Soc. 59 (1951).Google Scholar
- Gerald Teschl. 2012. Ordinary Differential Equations and Dynamical Systems. Graduate Studies in Mathematics, Vol. 140. American Mathematical Society.Google Scholar
- Ashish Tiwari. 2008. Abstractions for hybrid systems. Formal Methods in System Design 32, 1 (2008), 57--83. Google Scholar
Digital Library
- Vadim I. Utkin. 1992. Sliding Modes in Control and Optimization. Springer.Google Scholar
- Arjan J. Van Der Schaft and Hans Schumacher. 2000. An introduction to hybrid dynamical systems. Lecture Notes in Control and Information Sciences, Vol. 251. Springer.Google Scholar
- Shuling Wang, Naijun Zhan, and Liang Zou. 2015. An Improved HHL Prover: An Interactive Theorem Prover for Hybrid Systems. Springer, 382--399.Google Scholar
- Hans S. Witsenhausen. 1966. A class of hybrid-state continuous-time dynamic systems. IEEE Trans. Automat. Control 11, 2 (Apr 1966), 161--167.Google Scholar
Cross Ref
- Feng Zhao and Vadim I. Utkin. 1996. Adaptive simulation and control of variable-structure control systems in sliding regimes. Automatica 32, 7 (1996), 1037--1042. Google Scholar
Digital Library
Index Terms
Operational Models for Piecewise-Smooth Systems
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