Local analysis of hybrid systems on polyhedral sets with state-dependent switching

John Leth; Rafael Wisniewski

International Journal of Applied Mathematics and Computer Science (2014)

  • Volume: 24, Issue: 2, page 341-355
  • ISSN: 1641-876X

Abstract

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This paper deals with stability analysis of hybrid systems. Various stability concepts related to hybrid systems are introduced. The paper advocates a local analysis. It involves the equivalence relation generated by reset maps of a hybrid system. To establish a tangible method for stability analysis, we introduce the notion of a chart, which locally reduces the complexity of the hybrid system. In a chart, a hybrid system is particularly simple and can be analyzed with the use of methods borrowed from the theory of differential inclusions. Thus, the main contribution of this paper is to show how stability of a hybrid system can be reduced to a specialization of the well established stability theory of differential inclusions. A number of examples illustrate the concepts introduced in the paper.

How to cite

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John Leth, and Rafael Wisniewski. "Local analysis of hybrid systems on polyhedral sets with state-dependent switching." International Journal of Applied Mathematics and Computer Science 24.2 (2014): 341-355. <http://eudml.org/doc/271865>.

@article{JohnLeth2014,
abstract = {This paper deals with stability analysis of hybrid systems. Various stability concepts related to hybrid systems are introduced. The paper advocates a local analysis. It involves the equivalence relation generated by reset maps of a hybrid system. To establish a tangible method for stability analysis, we introduce the notion of a chart, which locally reduces the complexity of the hybrid system. In a chart, a hybrid system is particularly simple and can be analyzed with the use of methods borrowed from the theory of differential inclusions. Thus, the main contribution of this paper is to show how stability of a hybrid system can be reduced to a specialization of the well established stability theory of differential inclusions. A number of examples illustrate the concepts introduced in the paper.},
author = {John Leth, Rafael Wisniewski},
journal = {International Journal of Applied Mathematics and Computer Science},
keywords = {stability; switched systems; hybrid systems; differential inclusions},
language = {eng},
number = {2},
pages = {341-355},
title = {Local analysis of hybrid systems on polyhedral sets with state-dependent switching},
url = {http://eudml.org/doc/271865},
volume = {24},
year = {2014},
}

TY - JOUR
AU - John Leth
AU - Rafael Wisniewski
TI - Local analysis of hybrid systems on polyhedral sets with state-dependent switching
JO - International Journal of Applied Mathematics and Computer Science
PY - 2014
VL - 24
IS - 2
SP - 341
EP - 355
AB - This paper deals with stability analysis of hybrid systems. Various stability concepts related to hybrid systems are introduced. The paper advocates a local analysis. It involves the equivalence relation generated by reset maps of a hybrid system. To establish a tangible method for stability analysis, we introduce the notion of a chart, which locally reduces the complexity of the hybrid system. In a chart, a hybrid system is particularly simple and can be analyzed with the use of methods borrowed from the theory of differential inclusions. Thus, the main contribution of this paper is to show how stability of a hybrid system can be reduced to a specialization of the well established stability theory of differential inclusions. A number of examples illustrate the concepts introduced in the paper.
LA - eng
KW - stability; switched systems; hybrid systems; differential inclusions
UR - http://eudml.org/doc/271865
ER -

References

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  1. Ames, A. and Sastry, S. (2005). A homology theory for hybrid systems: Hybrid homology, in M. Morari and L. Thiele (Eds.), Hybrid Systems: Computation and Control, Lecture Notes in Computer Science, Vol. 3414, Springer-Verlag, Berlin/Heidelberg, pp. 86-102. Zbl1078.93014
  2. Balluchi, A., Benvenuti, L. and Sangiovanni-Vincentelli, A. (2005). Hybrid systems in automotive electronics design, 44th IEEE Conference on Decision and Control & 2005/2005 European Control Conference, CDC-ECC '05, Seville, Spain, pp. 5618-5623. Zbl1113.93080
  3. Blanchini, F. and Miani, S. (2008). Set-theoretic Methods in Control, Systems & Control: Foundations & Applications, Birkhäuser, Boston, MA. Zbl1140.93001
  4. Bredon, G.E. (1993). Topology and Geometry, Graduate Texts in Mathematics, Vol. 139, Springer-Verlag, New York, NY. Zbl0791.55001
  5. Bujorianu, M.L. and Lygeros, J. (2006). Toward a general theory of stochastic hybrid systems, in H. Bloom and J. Lygeros (Eds.), Stochastic Hybrid Systems, Lecture Notes in Control and Information Sciences, Vol. 337, Springer, Berlin, pp. 3-30. Zbl1130.93048
  6. Ding, J., Gillulay, J.H., Huang, H., Vitus, M.P., Zhang, W. and Tomlin, C. (2011). Hybrid systems in robotics, IEEE Robotics & Automation Magazine 18(3): 33-43. 
  7. Goebel, R., Sanfelice, R.G. and Teel, A.R. (2009). Hybrid dynamical systems: Robust stability and control for systems that combine continuous-time and discrete-time dynamics, IEEE Control Systems Magazine 29(2): 28-93. 
  8. Goebel, R. and Teel, A. R. (2006). Solutions to hybrid inclusions via set and graphical convergence with stability theory applications, Automatica 42(4): 573-587. Zbl1106.93042
  9. Grünbaum, B. (2003). Convex Polytopes, 2nd Edn., Graduate Texts in Mathematics, Vol. 221, Springer-Verlag, New York, NY. Zbl1033.52001
  10. Habets, L.C.G.J.M. and van Schuppen J.H. (2005). Control to facet problems for affine systems on simplices and polytopes-with applications to control of hybrid systems, Proceedings of the 44th IEEE Conference on Decision and Control, Seville, Spain, pp. 4175-4180. 
  11. Haddad, W.M., Chellaboina, V. and Nersesov, S.G. (2006). Impulsive and Hybrid Dynamical Systems, Princeton Series in Applied Mathematics, Princeton University Press, Princeton, NJ. Zbl1114.34001
  12. Heemels, W.P.M.H., De Schutter, B. and Bemporad, A. (2001). Equivalence of hybrid dynamical models, Automatica 37(7): 1085-1091. Zbl0990.93056
  13. Hudson, J.F.P. (1969). Piecewise Linear Topology, University of Chicago Lecture Notes, W.A. Benjamin, Inc., New York, NY/Amsterdam. Zbl0189.54507
  14. Johansson, M. and Rantzer, A. (1998). Computation of piecewise quadratic Lyapunov functions for hybrid systems, IEEE Transactions on Automatic Control 43(4): 555-559. Zbl0905.93039
  15. Kunze, M. (2000). Non-smooth Dynamical Systems, Lecture Notes in Mathematics, Vol. 1744, Springer-Verlag, Berlin. Zbl0965.34026
  16. Lang, S. (1999). Fundamentals of Differential Geometry, Graduate Texts in Mathematics, Vol. 191, Springer-Verlag, New York, NY. Zbl0932.53001
  17. Leine, R.I. and Nijmeijer, H. (2004). Dynamics and Bifurcations of Non-smooth Mechanical Systems, Lecture Notes in Applied and Computational Mechanics, Vol. 18, Springer-Verlag, Berlin. Zbl1068.70003
  18. Leth, J. and Wisniewski, R. (2012). On formalism and stability of switched systems, Journal of Control Theory and Applications 10(2): 176-183. 
  19. Liberzon, D. (2003). Switching in Systems and Control, Systems & Control: Foundations & Applications, Birkhäuser, Boston, MA. Zbl1036.93001
  20. Lygeros, J., Johansson, K.H., Simić, S.N., Zhang, J. and Sastry, S.S. (2003). Dynamical properties of hybrid automata, IEEE Transactions on Automatic Control 48(1): 2-17. 
  21. Munkres, J.R. (1975). Topology: A First Course, Prentice-Hall, Englewood Cliffs, NJ. Zbl0306.54001
  22. Pettersson, S. and Lennartson, B. (2002). Hybrid system stability and robustness verification using linear matrix inequalities, International Journal of Control 75(16): 1335-1355. Zbl1017.93027
  23. Rantzer, A. and Johansson, M. (2000). Piecewise linear quadratic optimal control, IEEE Transactions on Automatic Control 45(4): 629-637. Zbl0969.49016
  24. Rienmüller, T., Hofbaur, M., Travé-Massuyès, L. and Bayoudh, M. (2013). Mode set focused hybrid estimation, International Journal of Applied Mathematics and Computer Science 23(1): 131-144, DOI: 10.2478/amcs-2013-0011. Zbl1293.93723
  25. Simić, S.N., Johansson, K.H., Lygeros, J. and Sastry, S. (2005). Towards a geometric theory of hybrid systems, Dynamics of Continuous, Discrete & Impulsive Systems B: Applications & Algorithms 12(5-6): 649-687. Zbl1090.37013
  26. Sontag, E.D. (1981). Nonlinear regulation: The piecewise linear approach, IEEE Transactions on Automatic Control 26(2): 346-358. Zbl0474.93039
  27. Tabuada, P. (2009). Verification and Control of Hybrid Systems, Springer, New York, NY. Zbl1195.93001
  28. Tomlin, C., Pappas, G.J. and Sastry, S. (1998). Conflict resolution for air traffic management: A study in multiagent hybrid systems, IEEE Transactions on Automatic Control 43(4): 509-521. Zbl0904.90113
  29. van der Schaft, A. and Schumacher, H. (2000). An Introduction to Hybrid Dynamical Systems, Lecture Notes in Control and Information Sciences, Vol. 251, Springer-Verlag, London. Zbl0940.93004
  30. Wisniewski, R. (2006). Towards modelling of hybrid systems, 45th IEEE Conference on Decision and Control, San Diego, CA, USA, pp. 911-916. 
  31. Wisniewski, R. and Leth, J. (2011). Convenient model for systems with hystereses-control, Proceedings of the 50th IEEE Conference on Decision and Control, Orlando, FL, USA. 
  32. Yang, H., Jiang, B., Cocquempot, V. and Chen, M. (2013). Spacecraft formation stabilization and fault tolerance: A state-varying switched system approach, System & Control Letters 62(9): 715-722. Zbl1280.93007
  33. Yang, H., Jiang, B., Cocquempot, V. and Zang, H. (2011). Stabilization of switched nonlinear systems with all unstable modes: Application to multi-agent systems, IEEE Transactions on Automatic Control 56(9): 2230-2235. 
  34. Yordanov, B., Tůmová, J., Černá I., Barnat, J. and Belta, C. (2012). Temporal logic control of discrete-time piecewise affine systems, IEEE Transactions on Automatic Control 57(6): 1491-1504. Zbl1257.93030

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