The Average Dwell Time Condition Is Necessary & Sufficient for Arbitrary Switching Stability of Switched Nonlinear Systems
Applied and Computational Mathematics
Volume 5, Issue 6, December 2016, Pages: 230-233
Received: Oct. 12, 2016; Accepted: Nov. 14, 2016; Published: Dec. 8, 2016
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Jiqiang Wang, Jiangsu Province Key Laboratory of Aerospace Power Systems, College of Energy & Power Engineering, Nanjing University of Aeronautics & Astronautics, Nanjing, P. R. China
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Constrained switching of switched nonlinear systems consists of many classes of switching signals with markedly different features. One of the most important ones is the average dwell time (ADT) switching. For switched systems, it is a well-known result that a switched nonlinear system is globally uniformly asymptotically stable under arbitrary switching sequence if the ADT satisfies the lower bound defined by a real constant value (). In this note, it will be shown that this ADT condition is also necessary.
Switched Nonlinear Systems, Multiple Lyapunov Functions, Average Dwell Time, Arbitrary Switching Stability
To cite this article
Jiqiang Wang, The Average Dwell Time Condition Is Necessary & Sufficient for Arbitrary Switching Stability of Switched Nonlinear Systems, Applied and Computational Mathematics. Vol. 5, No. 6, 2016, pp. 230-233. doi: 10.11648/j.acm.20160506.11
Copyright © 2016 Authors retain the copyright of this article.
This article is an open access article distributed under the Creative Commons Attribution License ( which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
J. C. Law, J. D. Mattingly, Aircraft Engine Controls: Design, System Analysis, and Health Monitoring, AIAA, Inc., 2009.
M. Lichtsinder and Y. Levy, Jet engine model for control and real-time simulations, Journal of Engineering for Gas Turbines and Power, 128: 745-753, 2006.
A. S. Morse, supervisory control of families of linear set-point controllers: part 1: exact matching, IEEE Transactions on Automatic Control, 41 (10): 1413-1431, 1996.
M. Johansson and A. Rantzer, computation of piecewise quadratic Lyapunov functions for hybrid systems, IEEE Transactions on Automatic Control, 43 (4): 555-559, 1998.
D. Liberzon, Switching in Systems and Control, Boston: Birkhäuser, 2003.
J. P. Hespanha, Uniform stability of switched linear systems extensions of Lassalle’s invariance principle, IEEE Transactions on Automatic Control, 49 (4): 470-482, 2004.
Z. Sun and S. S. Ge, Analysis and synthesis of switched linear control systems, Automatica, 41 (2): 181-195, 2005.
A. P. Molchanov and Y. S. Pyatnitskiy, Criteria of asymptotic stability of differential and difference inclusions encountered in control theory, Systems and Control Letters, 13: 59-64, 1989.
J. P. Hespanha, A. S. Morse, Stability of switched systems with average dwell time, Proceedings of 38th Conf. Decision & Control, AZ, 1999, 2655-2660.
H. Ye, A. N. Michel, L. Hou, Stability theory for hybrid dynamical systems, IEEE Transactions on Automatic Control, 43 (4): 461-474, 1998.
L. Zhang, H. Gao. Asynchronously switched control of switched linear systems with average dwell time, Automatica, 46: 953-958, 2010.
E. Virnik, Stability analysis of positive descriptor systems, Linear Algebra and its Applications, 429 (10): 2640–2659, 2008.
G. Zhai, X. Xu, A commutation condition for stability analysis of switched linear descriptor systems, Nonlinear Analysis: Hybrid Systems, 5 (3): 383–393, 2011.
J. Lian, J. Liu, New results on stability of switched positive systems: an average dwell-time approach, IET Control Theory & Applications, 7 (12): 1651–1658, 2013.
B. Xia, J. Lian, X. Yuan, Stability of switched positive descriptor systems with average dwell time switching, Journal of Shanghai Jiaotong University, 20 (2): 177-184, 2015.
X. Zhao, Y. Yin, X. Zheng, State-dependent switching control of switched positive fractional-order systems, ISA Transactions, 62: 103-108, 2016.
J. Zhang, X. Zhao, Y. Chen, Finite-time stability and stabilization of fractional order positive switched systems, Circuits, Systems, & Signal Processing, 35 (7): 2450-2470, 2016.
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