The Average Dwell Time Condition <img width="63" height="41" src="http://article.sciencepublishinggroup.com/journal/289/2890005/image001.png" /> Is Necessary & Sufficient for Arbitrary Switching Stability of Switched Nonlinear Systems

Jiqiang Wang

doi:doi:10.11648/j.acm.20160506.11

| Peer-Reviewed

The Average Dwell Time Condition Is Necessary & Sufficient for Arbitrary Switching Stability of Switched Nonlinear Systems

Jiqiang Wang

Published in Applied and Computational Mathematics (Volume 5, Issue 6)

Received: 12 October 2016 Accepted: 14 November 2016 Published: 8 December 2016

Views: Downloads:

Download PDF

Share This Article

Twitter
Linked In
Facebook

Abstract

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.

Published in	Applied and Computational Mathematics (Volume 5, Issue 6)
DOI	10.11648/j.acm.20160506.11
Page(s)	230-233
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2016. Published by Science Publishing Group

Keywords

Switched Nonlinear Systems, Multiple Lyapunov Functions, Average Dwell Time, Arbitrary Switching Stability

References

[1]	J. C. Law, J. D. Mattingly, Aircraft Engine Controls: Design, System Analysis, and Health Monitoring, AIAA, Inc., 2009.
[2]	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.
[3]	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.
[4]	M. Johansson and A. Rantzer, computation of piecewise quadratic Lyapunov functions for hybrid systems, IEEE Transactions on Automatic Control, 43 (4): 555-559, 1998.
[5]	D. Liberzon, Switching in Systems and Control, Boston: Birkhäuser, 2003.
[6]	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.
[7]	Z. Sun and S. S. Ge, Analysis and synthesis of switched linear control systems, Automatica, 41 (2): 181-195, 2005.
[8]	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.
[9]	J. P. Hespanha, A. S. Morse, Stability of switched systems with average dwell time, Proceedings of 38th Conf. Decision & Control, AZ, 1999, 2655-2660.
[10]	H. Ye, A. N. Michel, L. Hou, Stability theory for hybrid dynamical systems, IEEE Transactions on Automatic Control, 43 (4): 461-474, 1998.
[11]	L. Zhang, H. Gao. Asynchronously switched control of switched linear systems with average dwell time, Automatica, 46: 953-958, 2010.
[12]	E. Virnik, Stability analysis of positive descriptor systems, Linear Algebra and its Applications, 429 (10): 2640–2659, 2008.
[13]	G. Zhai, X. Xu, A commutation condition for stability analysis of switched linear descriptor systems, Nonlinear Analysis: Hybrid Systems, 5 (3): 383–393, 2011.
[14]	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.
[15]	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.
[16]	X. Zhao, Y. Yin, X. Zheng, State-dependent switching control of switched positive fractional-order systems, ISA Transactions, 62: 103-108, 2016.
[17]	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.

Cite This Article

Plain Text BibTeX RIS

APA Style

Jiqiang Wang. (2016). The Average Dwell Time Condition Is Necessary & Sufficient for Arbitrary Switching Stability of Switched Nonlinear Systems. Applied and Computational Mathematics, 5(6), 230-233. https://doi.org/10.11648/j.acm.20160506.11

Copy | Download

ACS Style

Jiqiang Wang. The Average Dwell Time Condition Is Necessary & Sufficient for Arbitrary Switching Stability of Switched Nonlinear Systems. Appl. Comput. Math. 2016, 5(6), 230-233. doi: 10.11648/j.acm.20160506.11

Copy | Download

AMA Style

Jiqiang Wang. The Average Dwell Time Condition Is Necessary & Sufficient for Arbitrary Switching Stability of Switched Nonlinear Systems. Appl Comput Math. 2016;5(6):230-233. doi: 10.11648/j.acm.20160506.11

Copy | Download

@article{10.11648/j.acm.20160506.11,
  author = {Jiqiang Wang},
  title = {The Average Dwell Time Condition  Is Necessary & Sufficient for Arbitrary Switching Stability of Switched Nonlinear Systems},
  journal = {Applied and Computational Mathematics},
  volume = {5},
  number = {6},
  pages = {230-233},
  doi = {10.11648/j.acm.20160506.11},
  url = {https://doi.org/10.11648/j.acm.20160506.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20160506.11},
  abstract = {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.},
 year = {2016}
}

Copy | Download

TY  - JOUR
T1  - The Average Dwell Time Condition  Is Necessary & Sufficient for Arbitrary Switching Stability of Switched Nonlinear Systems
AU  - Jiqiang Wang
Y1  - 2016/12/08
PY  - 2016
N1  - https://doi.org/10.11648/j.acm.20160506.11
DO  - 10.11648/j.acm.20160506.11
T2  - Applied and Computational Mathematics
JF  - Applied and Computational Mathematics
JO  - Applied and Computational Mathematics
SP  - 230
EP  - 233
PB  - Science Publishing Group
SN  - 2328-5613
UR  - https://doi.org/10.11648/j.acm.20160506.11
AB  - 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.
VL  - 5
IS  - 6
ER  -

Copy | Download

Author Information

Jiqiang Wang

Jiangsu Province Key Laboratory of Aerospace Power Systems, College of Energy & Power Engineering, Nanjing University of Aeronautics & Astronautics, Nanjing, P. R. China

Download PDF

Sections

Plain Text BibTeX RIS

APA Style

Jiqiang Wang. (2016). The Average Dwell Time Condition Is Necessary & Sufficient for Arbitrary Switching Stability of Switched Nonlinear Systems. Applied and Computational Mathematics, 5(6), 230-233. https://doi.org/10.11648/j.acm.20160506.11

Copy | Download

ACS Style

Jiqiang Wang. The Average Dwell Time Condition Is Necessary & Sufficient for Arbitrary Switching Stability of Switched Nonlinear Systems. Appl. Comput. Math. 2016, 5(6), 230-233. doi: 10.11648/j.acm.20160506.11

Copy | Download

AMA Style

Jiqiang Wang. The Average Dwell Time Condition Is Necessary & Sufficient for Arbitrary Switching Stability of Switched Nonlinear Systems. Appl Comput Math. 2016;5(6):230-233. doi: 10.11648/j.acm.20160506.11

Copy | Download

@article{10.11648/j.acm.20160506.11,
  author = {Jiqiang Wang},
  title = {The Average Dwell Time Condition  Is Necessary & Sufficient for Arbitrary Switching Stability of Switched Nonlinear Systems},
  journal = {Applied and Computational Mathematics},
  volume = {5},
  number = {6},
  pages = {230-233},
  doi = {10.11648/j.acm.20160506.11},
  url = {https://doi.org/10.11648/j.acm.20160506.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20160506.11},
  abstract = {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.},
 year = {2016}
}

Copy | Download

TY  - JOUR
T1  - The Average Dwell Time Condition  Is Necessary & Sufficient for Arbitrary Switching Stability of Switched Nonlinear Systems
AU  - Jiqiang Wang
Y1  - 2016/12/08
PY  - 2016
N1  - https://doi.org/10.11648/j.acm.20160506.11
DO  - 10.11648/j.acm.20160506.11
T2  - Applied and Computational Mathematics
JF  - Applied and Computational Mathematics
JO  - Applied and Computational Mathematics
SP  - 230
EP  - 233
PB  - Science Publishing Group
SN  - 2328-5613
UR  - https://doi.org/10.11648/j.acm.20160506.11
AB  - 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.
VL  - 5
IS  - 6
ER  -

Copy | Download