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

• Autorzy: John Leth , Rafael Wisniewski
• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

stability; switched systems; hybrid systems; differential inclusions

• Wydawca: University of Zielona Gora Press
• Czasopismo: International Journal of Applied Mathematics and Computer Science
• Rocznik: 2014
• Tom: 24
• Numer: 2
• Strony: 341-355

Daty

wydano

2014

otrzymano

2013-04-10

poprawiono

2013-11-11

Twórcy

autor

John Leth

• Department of Electronic Systems, Automation and Control, Aalborg University, Fredrik Bajers Vej 7 C, 9220 Aalborg East, Denmark

autor

Rafael Wisniewski

• Department of Electronic Systems, Automation and Control, Aalborg University, Fredrik Bajers Vej 7 C, 9220 Aalborg East, Denmark

Bibliografia

• 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.
• 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.
• Blanchini, F. and Miani, S. (2008). Set-theoretic Methods in Control, Systems & Control: Foundations & Applications, Birkhäuser, Boston, MA.
• Bredon, G.E. (1993). Topology and Geometry, Graduate Texts in Mathematics, Vol. 139, Springer-Verlag, New York, NY.
• 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.
• 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.
• 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.
• 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.
• Grünbaum, B. (2003). Convex Polytopes, 2nd Edn., Graduate Texts in Mathematics, Vol. 221, Springer-Verlag, New York, NY.
• 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.
• 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.
• Heemels, W.P.M.H., De Schutter, B. and Bemporad, A. (2001). Equivalence of hybrid dynamical models, Automatica 37(7): 1085-1091.
• Hudson, J.F.P. (1969). Piecewise Linear Topology, University of Chicago Lecture Notes, W.A. Benjamin, Inc., New York, NY/Amsterdam.
• Johansson, M. and Rantzer, A. (1998). Computation of piecewise quadratic Lyapunov functions for hybrid systems, IEEE Transactions on Automatic Control 43(4): 555-559.
• Kunze, M. (2000). Non-smooth Dynamical Systems, Lecture Notes in Mathematics, Vol. 1744, Springer-Verlag, Berlin.
• Lang, S. (1999). Fundamentals of Differential Geometry, Graduate Texts in Mathematics, Vol. 191, Springer-Verlag, New York, NY.
• 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.
• Leth, J. and Wisniewski, R. (2012). On formalism and stability of switched systems, Journal of Control Theory and Applications 10(2): 176-183.
• Liberzon, D. (2003). Switching in Systems and Control, Systems & Control: Foundations & Applications, Birkhäuser, Boston, MA.
• 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.
• Munkres, J.R. (1975). Topology: A First Course, Prentice-Hall, Englewood Cliffs, NJ.
• Pettersson, S. and Lennartson, B. (2002). Hybrid system stability and robustness verification using linear matrix inequalities, International Journal of Control 75(16): 1335-1355.
• Rantzer, A. and Johansson, M. (2000). Piecewise linear quadratic optimal control, IEEE Transactions on Automatic Control 45(4): 629-637.
• 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.
• 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.
• Sontag, E.D. (1981). Nonlinear regulation: The piecewise linear approach, IEEE Transactions on Automatic Control 26(2): 346-358.
• Tabuada, P. (2009). Verification and Control of Hybrid Systems, Springer, New York, NY.
• 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.
• 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.
• Wisniewski, R. (2006). Towards modelling of hybrid systems, 45th IEEE Conference on Decision and Control, San Diego, CA, USA, pp. 911-916.
• 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.
• 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.
• 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.
• 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.

Identyfikatory

DOI

10.2478/amcs-2014-0026