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.
@article{bwmeta1.element.bwnjournal-article-amcv24i2p341bwm,
author = {John Leth and Rafael Wisniewski},
title = {Local analysis of hybrid systems on polyhedral sets with state-dependent switching},
journal = {International Journal of Applied Mathematics and Computer Science},
volume = {24},
year = {2014},
pages = {341-355},
zbl = {1293.93681},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-amcv24i2p341bwm}
}
John Leth; Rafael Wisniewski. Local analysis of hybrid systems on polyhedral sets with state-dependent switching. International Journal of Applied Mathematics and Computer Science, Tome 24 (2014) pp. 341-355. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-amcv24i2p341bwm/
[000] 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. | Zbl 1078.93014
[001] 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. | Zbl 1113.93080
[002] Blanchini, F. and Miani, S. (2008). Set-theoretic Methods in Control, Systems & Control: Foundations & Applications, Birkhäuser, Boston, MA. | Zbl 1140.93001
[003] Bredon, G.E. (1993). Topology and Geometry, Graduate Texts in Mathematics, Vol. 139, Springer-Verlag, New York, NY. | Zbl 0791.55001
[004] 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. | Zbl 1130.93048
[005] 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.
[006] 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.
[007] 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. | Zbl 1106.93042
[008] Grünbaum, B. (2003). Convex Polytopes, 2nd Edn., Graduate Texts in Mathematics, Vol. 221, Springer-Verlag, New York, NY. | Zbl 1033.52001
[009] 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.
[010] 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. | Zbl 1114.34001
[011] Heemels, W.P.M.H., De Schutter, B. and Bemporad, A. (2001). Equivalence of hybrid dynamical models, Automatica 37(7): 1085-1091. | Zbl 0990.93056
[012] Hudson, J.F.P. (1969). Piecewise Linear Topology, University of Chicago Lecture Notes, W.A. Benjamin, Inc., New York, NY/Amsterdam. | Zbl 0189.54507
[013] Johansson, M. and Rantzer, A. (1998). Computation of piecewise quadratic Lyapunov functions for hybrid systems, IEEE Transactions on Automatic Control 43(4): 555-559. | Zbl 0905.93039
[014] Kunze, M. (2000). Non-smooth Dynamical Systems, Lecture Notes in Mathematics, Vol. 1744, Springer-Verlag, Berlin. | Zbl 0965.34026
[015] Lang, S. (1999). Fundamentals of Differential Geometry, Graduate Texts in Mathematics, Vol. 191, Springer-Verlag, New York, NY. | Zbl 0932.53001
[016] 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. | Zbl 1068.70003
[017] Leth, J. and Wisniewski, R. (2012). On formalism and stability of switched systems, Journal of Control Theory and Applications 10(2): 176-183.
[018] Liberzon, D. (2003). Switching in Systems and Control, Systems & Control: Foundations & Applications, Birkhäuser, Boston, MA. | Zbl 1036.93001
[019] 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.
[020] Munkres, J.R. (1975). Topology: A First Course, Prentice-Hall, Englewood Cliffs, NJ. | Zbl 0306.54001
[021] Pettersson, S. and Lennartson, B. (2002). Hybrid system stability and robustness verification using linear matrix inequalities, International Journal of Control 75(16): 1335-1355. | Zbl 1017.93027
[022] Rantzer, A. and Johansson, M. (2000). Piecewise linear quadratic optimal control, IEEE Transactions on Automatic Control 45(4): 629-637. | Zbl 0969.49016
[023] 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. | Zbl 1293.93723
[024] 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. | Zbl 1090.37013
[025] Sontag, E.D. (1981). Nonlinear regulation: The piecewise linear approach, IEEE Transactions on Automatic Control 26(2): 346-358. | Zbl 0474.93039
[026] Tabuada, P. (2009). Verification and Control of Hybrid Systems, Springer, New York, NY. | Zbl 1195.93001
[027] 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. | Zbl 0904.90113
[028] 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. | Zbl 0940.93004
[029] Wisniewski, R. (2006). Towards modelling of hybrid systems, 45th IEEE Conference on Decision and Control, San Diego, CA, USA, pp. 911-916.
[030] 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.
[031] 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. | Zbl 1280.93007
[032] 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.
[033] 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. | Zbl 1257.93030