Stability and Robust Stability of Switched Positive Linear Systems With All Modes Unstable

Shaoxia Feng; Juan Wang; Jun Zhao

doi:10.1109/JAS.2017.7510718

Volume 6 Issue 1

Jan. 2019

IEEE/CAA Journal of Automatica Sinica

JCR Impact Factor: 15.3, Top 1 (SCI Q1)

CiteScore: 23.5, Top 2% (Q1)
Google Scholar h5-index: 77， TOP 5

Turn off MathJax

Article Contents

Article Navigation > IEEE/CAA Journal of Automatica Sinica > 2019 > 6(1): 167-176

Shaoxia Feng, Juan Wang and Jun Zhao, "Stability and Robust Stability of Switched Positive Linear Systems With All Modes Unstable," IEEE/CAA J. Autom. Sinica, vol. 6, no. 1, pp. 167-176, Jan. 2019. doi: 10.1109/JAS.2017.7510718

Citation:

Shaoxia Feng, Juan Wang and Jun Zhao, "Stability and Robust Stability of Switched Positive Linear Systems With All Modes Unstable," IEEE/CAA J. Autom. Sinica, vol. 6, no. 1, pp. 167-176, Jan. 2019. doi: 10.1109/JAS.2017.7510718

Citation:

PDF( 978 KB)

Stability and Robust Stability of Switched Positive Linear Systems With All Modes Unstable

doi: 10.1109/JAS.2017.7510718

Shaoxia Feng^{1, 2
,},
Juan Wang^{3
,
,},
Jun Zhao^4
,

1.
Shanxi Institute of Mechanical and Electrical Engineering, Changzhi 046011
2.
College of Information Science and Engineering, Northeastern University, Shenyang 110819, China
3.
Faculty of Information and Control Engineering, Shenyang Jianzhu University, Shenyang 110168, China
4.
State Key Laboratory of Synthetical Automation for Process Industries (Northeastern University), and also with the College of Information Science and Engineering, Northeastern University, Shenyang 110819, China

Funds:

National Natural Science Foundation of China 61703288

National Natural Science Foundation of China 61603079

National Natural Science Foundation of China 61873174

More Information

Author Bio:
Shaoxia Feng received the B.S. degree in mathematics and applied mathematics from China University of Mining and Technology in 2014, and M.S. degree in control theory and control engineering from Northeastern University in 2017. Her research interests include switched systems, switched positive system, and robust control. (e-mail: fengshaoxianeu@163.com)

Juan Wang is a Lecturer for the Faculty of Information and Control Engineering of Shenyang Jianzhu University, China. She received the B.S. degree in mathematics and applied mathematics from Bohai University, China, in 2007, and the M.S. degree in operational research and cybernetics from Northeastern University, China, in 2009. She received the Ph.D. degree in control theory and applications in 2016 at Northeastern University, China. Her research interests include switched systems, positive system, and actuator saturation

Jun Zhao received the B.S. and M.S. degrees in mathematics in 1982 and 1984, respectively, both from Liaoning University, China. He received the Ph.D. degree in control theory and applications in 1991 at Northeastern University, China. From 1992 to 1993 he was a Postdoctoral Fellow at the same university. Since 1994, as a Professor, he has been with the College of Information Science and Engineering, Northeastern University, China. From 1998 to 1999, he was a Senior Visiting Scholar at the Coordinated Science Laboratory, University of Illinois at Urbana-Champaign, USA. From November 2003 to May 2005, he was a Research Fellow in the Department of Electronic Engineering, City University of Hong Kong. During 2007-2010, he was a Fellow at the School of Engineering, Australian National University. His research interests include switched systems, nonlinear systems and network synchronization. (e-mail: zhaojun@mail.neu.edu.cn)
Corresponding author: Juan Wang, e-mail: wangjuanneu@163.com
Received Date: 2016-09-23
Revised Date: 2017-05-27
Accepted Date: 2017-07-06

Abstract

Abstract

This paper is concerned with the stability and robust stability of switched positive linear systems (SPLSs) whose subsystems are all unstable. By means of the mode-dependent dwell time approach and a class of discretized co-positive Lyapunov functions, some stability conditions of switched positive linear systems with all modes unstable are derived in both the continuous-time and the discrete-time cases, respectively. The copositive Lyapunov functions constructed in this paper are timevarying during the dwell time and time-invariant afterwards. In addition, the above approach is extended to the switched interval positive systems. A numerical example is proposed to illustrate our approach.
- Discretized co-positive Lyapunov functions,
- mode-dependent dwell time,
- robust stability,
- switched positive linear systems (SPLSs),
- unstable subsystem

FullText(HTML)

References(30)

References

[1]	R. Shorten, F. Wirth, and D. Leith, "A positive systems model of TCP-like congestion control: asymptotic results, " IEEE/ACM Trans. Netw., vol. 14, no. 3, pp. 616-629, Jun. 2006. http://www.researchgate.net/publication/2938477_A_positive_systems_model_of_TCP-like_congestion
[2]	A. Jadbabaie, J. Lin, and A. S. Morse, "Coordination of groups of mobile autonomous agents using nearest neighbor rules, " IEEE Trans. Autom. Control, vol. 48, no. 6, pp. 988-1001, Jun. 2003. https://ieeexplore.ieee.org/document/1205192
[3]	M. Bolajraf, F. Tadeo, T. Alvarez, and M. A. Rami, "State-feedback with memory for controlled positivity with application to congestion control, " IET Control Theory Appl., vol. 4, no. 10, pp. 2041-2048, Oct. 2010. http://www.researchgate.net/publication/224185829_State-feedback_with_memory_for_controlled_positivity_with_application_to_congestion_control
[4]	J. Zhao and D. J. Hill, "On stability, L$_2$-gain and H$_{\infty}$ control for switched systems, " Automatica, vol. 44, no. 5, pp. 1220-1232, May 2008.
[5]	J. Wang and J. Zhao, "Stability analysis and control synthesis for a class of cascade switched nonlinear systems with actuator saturation, " Circ. Syst. Signal Process., vol. 33, no. 9, pp. 2961-2970, Sep. 2014. doi: 10.1007/s00034-014-9776-7
[6]	X. M. Sun, J. Zhao, and D. J. Hill, "Stability and L$_2$-gain analysis for switched delay systems: A delay-dependent method, " Automatica, vol. 42, no. 10, pp. 1769-1774, Oct. 2006.
[7]	E. Fornasini and M. E. Valcher, "Asymptotic stability and stabilizability of special classes of discrete-time positive switched systems, " Linear Algebra Appl., vol. 438, no. 4, pp. 1814-1831, Feb. 2013. https://www.sciencedirect.com/science/article/pii/S0024379511006112
[8]	F. Blanchini, P. Colaneri, and M. E. Valcher, "Co-positive Lyapunov functions for the stabilization of positive switched systems, " IEEE Trans. Automat. Control, vol. 57, no. 12, pp. 3038-3050, Dec. 2012. http://www.researchgate.net/publication/260661819_Co-Positive_Lyapunov_Functions_for_the_Stabilization_of_Positive_Switched_Systems
[9]	E. Fornasini and M. E. Valcher, "On the stability of continuous-time positive switched systems, " in Proc. American Control Conference (ACC), Baltimore, MD, USA, 2010, pp. 6225-6230. http://www.researchgate.net/publication/224162584_On_the_stability_of_continuous-time_positive_switched_systems
[10]	L. Gurvits, R. Shorten, and O. Mason, "On the stability of switched positive linear systems, " IEEE Trans. Automat. Control, vol. 52, no. 6, pp. 1099-1103, Jun. 2007. https://www.researchgate.net/publication/3032757_On_the_Stability_of_Switched_Positive_Linear_Systems
[11]	Z. Liu, Y. Z. Wang, and G. D. Zhao, "Exponential stability for positive switched linear systems with delays and impulses, " in Proc. 33rd Chinese Control Conference (CCC), Nanjing, China, 2014, pp. 2469-2474. https://www.researchgate.net/publication/286595317_Exponential_stability_for_positive_switched_linear_systems_with_delays_and_impulses
[12]	X. Y. Ding, L. Shu, and Z. H. Wang, "On stability for switched linear positive systems, " Math. Comput. Modell., vol. 53, no. 5-6, pp. 1044- 1055, Mar. 2011. https://www.sciencedirect.com/science/article/pii/S0895717710005534
[13]	X. D. Zhao, L. X. Zhang, and P. Shi, "Stability of a class of switched positive linear time-delay systems, " Int. J. Robust Nonlinear Control, vol. 23, no. 5, pp. 578-589, Mar. 2013. https://www.researchgate.net/publication/259717122_Stability_of_a_Class_of_Switched_Positive_Linear_Time-Delay_Systems
[14]	X. D. Zhao, L. X. Zhang, P. Shi, and H. R. Karimi, "Stabilization of a class of slowly switched positive linear systems: state-feedback control, " in Proc. American Control Conference (ACC), Montreal, QC, Canada, 2012, pp. 5282-5286. Stabilization of a class of slowly switched positive linear systems: state-feedback control
[15]	O. Mason and R. Shorten, "On linear copositive Lyapunov functions and the stability of switched positive linear systems, " IEEE Trans. Automat. Control, vol. 52, no. 7, pp. 1346-1349, Jul. 2007. http://www.researchgate.net/publication/3032814_On_Linear_Copositive_Lyapunov_Functions_and_the_Stability_of_Switched_Positive_Linear_Systems?ev=sim_pub
[16]	X. W. Liu, "Stability analysis of switched positive systems: a switched linear copositive Lyapunov function method, " IEEE Trans. Circ. Syst. Ⅱ: Express Briefs, vol. 56, no. 5, pp. 414-418, May 2009. http://www.researchgate.net/publication/224441998_Stability_Analysis_of_Switched_Positive_Systems_A_Switched_Linear_Copositive_Lyapunov_Function_Method
[17]	X. D. Zhao, L. X. Zhang, P. Shi, and M. Liu, "Stability of switched positive linear systems with average dwell time switching, " Automatica, vol. 48, no. 6, pp. 1132-1137, Jun. 2012. https://www.sciencedirect.com/science/article/pii/S0005109812001045
[18]	H. Yang, V. Cocquempot, and B. Jiang, "On stabilization of switched nonlinear systems with unstable modes, " Syst. Control Lett., vol. 58, no. 10-11, pp. 703-708, Oct.-Nov. 2009. doi: 10.1007%2F978-3-319-07884-7
[19]	J. Lian and J. Liu, "New results on stability of switched positive systems: an average dwell-time approach, " IET Control Theory Appl., vol. 7, no. 12, pp. 1651-1658, Aug. 2013. http://www.researchgate.net/publication/260586544_New_results_on_stability_of_switched_positive_systems_an_average_dwell-time_approach
[20]	R. Q. Shi, X. M. Tan, X. D. Zhao, and X. L. Zheng, "Stability and l$_1$-gain analysis for switched delay positive systems with stable and unstable subsystems, " Circ. Syst. Signal Process., vol. 34, no. 5, pp. 1683-1696, May 2015.
[21]	W. M. Xiang and J. Xiao, "Stabilization of switched continuous-time systems with all modes unstable via dwell time switching, " Automatica, vol. 50, no. 3, pp. 940-945, Mar. 2014. https://www.sciencedirect.com/science/article/pii/S0005109813005876
[22]	K. Q. Gu, V. L. Kharitonov, and J. Chen, Stability of Time-Delay Systems. Boston, USA: Birkhäuser, 2003.
[23]	L. Allerhand and U. Shaked, "Robust stability and stabilization of linear switched systems with dwell time, " IEEE Trans. Automat. Control, vol. 56, no. 2, pp. 381-386, Feb. 2011. http://www.researchgate.net/publication/224208117_Robust_stability_and_stabilization_of_linear_switched_systems_with_dwell_time
[24]	L. G. Wu, X. J. Su, and P. Shi, "Sliding mode control with bounded $\ell_2$ gain performance of Markovian jump singular time-delay systems, " Automatica, vol. 48, no. 8, pp. 1929-1933, Aug. 2012.
[25]	L. G. Wu, X. J. Su, and P. Shi, "Output feedback control of Markovian jump repeated scalar nonlinear systems, " IEEE Trans. Automat. Control, vol. 59, no. 1, pp. 199-204, Jan. 2014. doi: 10.1007%2F978-3-319-31915-5_12
[26]	L. X. Zhang and B. Jiang, "Stability of a class of switched linear systems with uncertainties and average dwell time switching, " Int. J. Innov. Comput. Inform. Control, vol. 6, no. 2, pp. 667-676, Feb. 2010. https://www.researchgate.net/publication/279546251_Stability_of_a_class_of_switched_linear_systems_with_uncertainties_and_average_dwell_time_switching
[27]	J. J. Liu, K. J. Zhang, and H. K. Wei, "Robust stability of positive switched systems with dwell time, " Int. J. Syst. Sci., vol. 47, no. 11, pp. 2553-2562, Aug. 2016. http://www.researchgate.net/publication/273168026_robust_stability_of_positive_switched_systems_with_dwell_time
[28]	J. F. Zhang, Z. Z. Han, H. Wu, and J. Huang, "Robust stabilization of discrete-time positive switched systems with uncertainties and average dwell time switching, " Circ. Syst. Signal Process., vol. 33, no. 1, pp. 71 -95, Jan. 2014. doi: 10.1007/s00034-013-9632-1
[29]	J. F. Zhang, W. Zhang, X. S. Cai, and Z. Z. Han, "Stability and stabilization of positive switched systems under asynchronous switching, " in Proc. International Conference on Mechatronics and Control (ICMC), Jinzhou, China, 2014, pp. 347-352. https://www.researchgate.net/publication/283302183_Stability_and_stabilization_of_positive_switched_systems_under_asynchronous_switching
[30]	Y. Li, H. B. Zhang, and Q. X. Zheng, "Robust stability and $L_1$-gain analysis of interval positive switched T-S fuzzy systems with mode-dependent dwell time, " Neurocomputing, vol. 235, pp. 90-97, Apr. 2017.

Supplements(0)

Cited By

Proportional views

Proportional views

通讯作者: 陈斌, bchen63@163.com

1.
沈阳化工大学材料科学与工程学院沈阳 110142

Figures(10)

Get Citation

PDF

XML

Article Metrics

Article views (1633) PDF downloads(87)

Stability and Robust Stability of Switched Positive Linear Systems With All Modes Unstable

doi: 10.1109/JAS.2017.7510718

Abstract

References

Proportional views

Catalog

通讯作者: 陈斌, bchen63@163.com

Article Metrics

Export File

Citation

Format

Content