Exponential stability of nonlinear impulsive switched systems with stable and unstable subsystems

Xiao-li ZHANG; An-hui LIN; Jian-ping ZENG

doi:10.1631/jzus.C1300123

Your Location：

Home >

Browse articles >

Exponential stability of nonlinear impulsive switched systems with stable and unstable subsystems

Computer & Automation | Updated：2022-05-19

- Exponential stability of nonlinear impulsive switched systems with stable and unstable subsystems
- 有稳定和不稳定子系统的非线性脉冲切换系统的指数稳定性
- Frontiers of Information Technology & Electronic Engineering Vol. 15, Issue 1, Pages: 31-42(2014)
- Affiliations：
  
  School of Information Science and Technology, Xiamen University, Xiamen 361005, China
- Author bio：
  
  Xiao-li ZHANG, E-mail: zhxl@xmu.edu.cn
- Funds：
- DOI：10.1631/jzus.C1300123
  CLC： TP13
- Received：09 May 2013，
  
  Revised：2013-12-16，
  
  Published：2014-01
- Accepted：
Scan QR Code
Xiao-li ZHANG, An-hui LIN, Jian-ping ZENG. Exponential stability of nonlinear impulsive switched systems with stable and unstable subsystems[J]. Frontiers of Information Technology & Electronic Engineering, 2014, 15(1): 31-42.
DOI：

Xiao-li ZHANG, An-hui LIN, Jian-ping ZENG. Exponential stability of nonlinear impulsive switched systems with stable and unstable subsystems[J]. Frontiers of Information Technology & Electronic Engineering, 2014, 15(1): 31-42. DOI： 10.1631/jzus.C1300123.

摘要

研究目的

对一类同时包含稳定和不稳定子系统的切换非线性系统的稳定性及鲁棒稳定性进行研究。旨在针对在切换时刻包含非线性脉冲

且每个子系统都具有非线性级联形式的切换非线性系统

给出其稳定的充分条件

为此类系统的稳定性问题研究提供理论依据。

重要结论

将在切换线性系统切换时刻的非线性脉冲处理方法和手段推广至切换非线性系统中。在应用非线性系统的Lyapunov函数处理非线性脉冲时

适当地利用了矩阵不等式的相关方法。当不稳定子系统和稳定子系统的活跃时间小于一定比例

并且在切换时刻存在满足相应界的非线性脉冲时

切换非线性系统仍能保持其指数稳定性。在不确定性满足相应界的条件下

切换非线性系统也能保持其鲁棒指数稳定性。

Abstract

Exponential stability and robust exponential stability relating to switched systems consisting of stable and unstable nonlinear subsystems are considered in this study. At each switching time instant

the impulsive increments which are nonlinear functions of the states are extended from switched linear systems to switched nonlinear systems. Using the average dwell time method and piecewise Lyapunov function approach

when the total active time of unstable subsystems compared to the total active time of stable subsystems is less than a certain proportion

the exponential stability of the switched system is guaranteed. The switching law is designed which includes the average dwell time of the switched system. Switched systems with uncertainties are also studied. Sufficient conditions of the exponential stability and robust exponential stability are provided for switched nonlinear systems. Finally

simulations show the effectiveness of the result.

关键词

Keywords

references

M. de la Sen . . Stability of impulsive time-varying systems and compactness of the operators mapping the input space into the state and output spaces . . J. Math. Anal. Appl. , , 2006 . . 321 ( ( 2 ): ): 621 - - 650 . . DOI: 10.1016/j.jmaa.2005.08.038 http://doi.org/10.1016/j.jmaa.2005.08.038 . .

M. de la Sen , , , N.S. Luo . . A note on the stability of linear time-delay systems with impulsive inputs . . IEEE Trans. Circ. Syst. I , , 2003 . . 50 ( ( 1 ): ): 149 - - 152 . . DOI: 10.1109/TCSI.2002.807514 http://doi.org/10.1109/TCSI.2002.807514 . .

Z.H. Guan , , , D.J. Hill , , , X.M. Shen . . On hybrid impulsive and switching systems and application to nonlinear control . . IEEE Trans. Automat. Control , , 2005 . . 50 ( ( 7 ): ): 1058 - - 1062 . . DOI: 10.1109/TAC.2005.851462 http://doi.org/10.1109/TAC.2005.851462 . .

J.P. Hespanha , , , A.S. Morse . . Stability of switched systems with average dwell-time . . Proc. 38th IEEE Conf. on Decision and Control , , 1999 . . p.2655 - - 2660 . . DOI: 10.1109/CDC.1999.831330 http://doi.org/10.1109/CDC.1999.831330 . .

J.P. Hespanha , , , D. Liberzon , , , A.R. Teel . . Lyapunov conditions for input-to-state stability of impulsive systems . . Automatica , , 2008 . . 44 ( ( 11 ): ): 2735 - - 2744 . . DOI: 10.1016/j.automatica.2008.03.021 http://doi.org/10.1016/j.automatica.2008.03.021 . .

I.A. Hiskens . . Stability of hybrid system limit cycles: application to the compass gait biped robot . . Proc. 40th IEEE Conf. on Decision and Control , , 2001 . . p.774 - - 779 . . DOI: 10.1109/.2001.980200 http://doi.org/10.1109/.2001.980200 . .

S. Kim , , , S.A. Campbell , , , X.Z. Liu . . Stability of a class of linear switching systems with time delay . . IEEE Trans. Circ. Syst. I , , 2006 . . 53 ( ( 2 ): ): 384 - - 393 . . DOI: 10.1109/TCSI.2005.856666 http://doi.org/10.1109/TCSI.2005.856666 . .

B. Lennartson , , , M. Tittus , , , B. Egardt , , , 等 . . Hybrid systems in process control . . IEEE Control Syst. Mag. , , 1996 . . 16 ( ( 5 ): ): 45 - - 56 . . DOI: 10.1109/37.537208 http://doi.org/10.1109/37.537208 . .

D. Liberzon . . Switching in Systems and Control . . Birkh user, Boston , , 2003 . . DOI: 10.1007/978-1-4612-0017-8 http://doi.org/10.1007/978-1-4612-0017-8 . .

B. Liu , , , H.J. Marquez . . Controllability and observability for a class of controlled switching impulsive systems . . IEEE Trans. Automat. Control , , 2008 . . 53 ( ( 10 ): ): 2360 - - 2366 . . DOI: 10.1109/TAC.2008.2007476 http://doi.org/10.1109/TAC.2008.2007476 . .

J. Liu , , , X.Z. Liu , , , W.C. Xie . . Input-to-state stability of impulsive and switching hybrid systems with time-delay . . Automatica , , 2011 . . 47 ( ( 5 ): ): 899 - - 908 . . DOI: 10.1016/j.automatica.2011.01.061 http://doi.org/10.1016/j.automatica.2011.01.061 . .

V.M. Marchenko , , , Z. Zaczkiewicz . . Representation of solutions of control hybrid differential-difference impulse systems . . Differ. Equat. , , 2009 . . 45 ( ( 12 ): ): 1811 - - 1822 . . DOI: 10.1134/S0012266109120118 http://doi.org/10.1134/S0012266109120118 . .

I.R. Petersen , , , C.V. Hollot . . A Riccati equation approach to the stabilization of uncertain linear systems . . Automatica , , 1986 . . 22 ( ( 4 ): ): 397 - - 411 . . DOI: 10.1016/0005-1098(86)90045-2 http://doi.org/10.1016/0005-1098(86)90045-2 . .

S.Y. Qin , , , Y.H. Song . . The theory of hybrid control systems and its application perspective in electric power systems . . Proc. Int. Conf. on Info-Tech and Info-Net , , 2001 . . p.85 - - 94 . . DOI: 10.1109/ICII.2001.983729 http://doi.org/10.1109/ICII.2001.983729 . .

X.M. Sun , , , D. Wang , , , W. Wang , , , 等 . . Stability analysis and L2-gain of switched delay systems with stable and unstable subsystems . . IEEE 22nd Int. Symp. on Intelligent Control , , 2007 . . p.208 - - 213 . . DOI: 10.1109/ISIC.2007.4450886 http://doi.org/10.1109/ISIC.2007.4450886 . .

X.M. Sun , , , W. Wang , , , G.P. Liu , , , 等 . . Stability analysis for linear switched systems with time-varying delay . . IEEE Trans. Syst. Man Cybern. B , , 2008 . . 38 ( ( 2 ): ): 528 - - 533 . . DOI: 10.1109/TSMCB.2007.912078 http://doi.org/10.1109/TSMCB.2007.912078 . .

P. Varaiya . . Smart cars on smart roads: problems of control . . IEEE Trans. Automat. Control , , 1993 . . 38 ( ( 2 ): ): 195 - - 207 . . DOI: 10.1109/9.250509 http://doi.org/10.1109/9.250509 . .

M. Wang , , , G.M. Dimirovski , , , J. Zhao . . Average dwelltime method to stabilization and L2-gain analysis for uncertain switched nonlinear systems . . Proc. 17th IFAC World Congress , , 2008 . . p.7642 - - 7647 . . DOI: 10.3182/20080706-5-KR-1001.01292 http://doi.org/10.3182/20080706-5-KR-1001.01292 . .

M. Wicks , , , P. Peleties , , , R. DeCarlo . . Switched controller synthesis for the quadratic stabilization of a pair of unstable linear systems . . Eur. J. Control , , 1998 . . 4 ( ( 2 ): ): 140 - - 147 . . DOI: 10.1016/S0947-3580(98)70108-6 http://doi.org/10.1016/S0947-3580(98)70108-6 . .

H.L. Xu , , , K.L. Teo . . Exponential stability with L2-gain condition of nonlinear impulsive switched systems . . IEEE Trans. Automat. Control , , 2010 . . 55 ( ( 10 ): ): 2429 - - 2433 . . DOI: 10.1109/TAC.2010.2060173 http://doi.org/10.1109/TAC.2010.2060173 . .

H.L. Xu , , , X.Z. Liu , , , K.L. Teo . . Robust H ∞ stabilization with definite attendance of uncertain impulsive switched systems . . ANZIAM J. , , 2005 . . 46 ( ( 4 ): ): 471 - - 484 . . . .

H.L. Xu , , , K.L. Teo , , , X.Z. Liu . . Robust stability analysis of guaranteed cost control for impulsive switched systems . . IEEE Trans. Syst. Man Cybern. B , , 2008 . . 38 ( ( 5 ): ): 1419 - - 1422 . . DOI: 10.1109/TSMCB.2008.925747 http://doi.org/10.1109/TSMCB.2008.925747 . .

J. Yao , , , Z.H. Guan , , , G.R. Chen , , , 等 . . Stability, robust stabilization and H ∞ control of singular-impulsive systems via switching control . . Syst. Control Lett. , , 2006 . . 55 ( ( 11 ): ): 879 - - 886 . . DOI: 10.1016/j.sysconle.2006.05.002 http://doi.org/10.1016/j.sysconle.2006.05.002 . .

G.S. Zhai , , , B. Hu , , , K. Yasuda , , , 等 . . Disturbance attenuation properties of time-controlled switched systems . . J. Franklin Inst. , , 2001a . . 338 ( ( 7 ): ): 765 - - 779 . . DOI: 10.1016/S0016-0032(01)00030-8 http://doi.org/10.1016/S0016-0032(01)00030-8 . .

G.S. Zhai , , , B. Hu , , , K. Yasuda , , , 等 . . Stability analysis of switched systems with stable and unstable subsystems: an average dwell time approach . . Int. J. Syst. Sci. , , 2001b . . 32 ( ( 8 ): ): 1055 - - 1061 . . DOI: 10.1080/00207720116692 http://doi.org/10.1080/00207720116692 . .

G.S. Zhai , , , H. Lin , , , Y. Kim , , , 等 . . L2-gain analysis for switched systems with continuous-time and discrete-time subsystems . . Int. J. Control , , 2005 . . 78 ( ( 15 ): ): 1198 - - 1205 . . DOI: 10.1080/00207170500274966 http://doi.org/10.1080/00207170500274966 . .

W. Zhu . . Stability analysis of switched impulsive systems with time delays . . Nonl. Anal. Hybr. Syst. , , 2010 . . 4 ( ( 3 ): ): 608 - - 617 . . DOI: 10.1016/j.nahs.2010.03.009 http://doi.org/10.1016/j.nahs.2010.03.009 . .

G.D. Zong , , , S.Y. Xu , , , Y.Q. Wu . . Robust H ∞ stabilization for uncertain switched impulsive control systems with state delay: an LMI approach . . Nonl. Anal. Hybr. Syst. , , 2008 . . 2 ( ( 4 ): ): 1287 - - 1300 . . DOI: 10.1016/j.nahs.2008.09.018 http://doi.org/10.1016/j.nahs.2008.09.018 . .

Views

Downloads

CSCD

Alert me when the article has been cited

Submit

Tools

Publicity Resources

SRIS-Net: a robust image steganography algorithm based on feature score maps

Identity-based threshold proxy re-encryption scheme from lattices and its applications

Novel robust simultaneous localization and mapping for long-term autonomous robots

Mittag-Leffler stability analysis of multiple equilibrium points in impulsive fractional-order quaternion-valued neural networks

On robustness of an AMB suspended energy storage flywheel platform under characteristic model based all-coefficient adaptive control laws

Related Author

Haoyuan SUN

Long ZHENG

Guomei WANG

Zhi LI

Ai XIAO

Liqiang WU

Yiliang HAN

Xiaoyuan YANG

Related Institution

School of Computer Science and Technology, Guizhou University

Key Laboratory of Network and Information Security, Engineering University of People's Armed Police

MOE Key Laboratory of Computer Network and Information Security, Xidian University

School of Mechanical Engineering and Automation, Harbin Institute of Technology

Zhuhai Big Data Research Institute

Chat

Address：Zhejiang University Press, 148 Tianmushan Road, Hangzhou, China Postal code：310028
Tel：+86-571-88273162 Email：fitee@zju.edu.cn
It is recommended to read the content of this site in Chrome&IE9+. Please switch to extreme mode in browser 360.
Cookies We use cookies to help provide and enhance our service and tailor content. By continuing, you agree to the use of cookies.

⁰