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[1] Zhang Huasheng, , Zhang Kanjian, et al. Stability analysis of time-varying systemsvia parameter-dependent homogeneous Lyapunov functions [J]. Journal of Southeast University (English Edition), 2014, 30 (3): 302-305. [doi:10.3969/j.issn.1003-7985.2014.03.008]
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Stability analysis of time-varying systemsvia parameter-dependent homogeneous Lyapunov functions()
基于参数依赖齐次多项式的时变系统稳定性分析
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Journal of Southeast University (English Edition)[ISSN:1003-7985/CN:32-1325/N]

Volumn:
30
Issue:
2014 3
Page:
302-305
Research Field:
Automation
Publishing date:
2014-09-30

Info

Title:
Stability analysis of time-varying systemsvia parameter-dependent homogeneous Lyapunov functions
基于参数依赖齐次多项式的时变系统稳定性分析
Author(s):
Zhang Huasheng1;  2;  3;  Zhang Kanjian1;  2
1Key Laboratory of Measurement and Control of Complex Systems of Engineering of Ministry of Education, Southeast University, Nanjing 210096, China
2School of Automation, Southeast University, Nanjing 210096, China
3School of Mathematical Sciences, Liaocheng University, Liaocheng 252000, China
张化生1;  2;  3;  张侃健1;  2
1东南大学复杂工程系统测量与控制教育部重点实验室, 南京210096; 2东南大学自动化学院, 南京210096; 3聊城大学数学科学学院, 聊城252000
Keywords:
linear time-varying systems;  polytopic uncertainty;  robust stability;  linear matrix inequality
线性时变系统;  多面体不确定性;  鲁棒稳定性;  线性不等式
PACS:
TP202.1;TP271.7
DOI:
10.3969/j.issn.1003-7985.2014.03.008
Abstract:
This paper considers the stability analysis of linear continuous-time systems, and that the dynamic matrices are affected by uncertain time-varying parameters, which are assumed to be bounded, continuously differentiable, with bounded rates of variation. First, sufficient conditions of stability for time-varying systems are given by the commonly used parameter-dependent quadratic Lyapunov function. Moreover, the use of homogeneous polynomial Lyapunov functions for the stability analysis of the linear system subject to the time-varying parametric uncertainty is introduced. Sufficient conditions to determine the sought after Lyapunov function is derived via a suitable paramenterization of polynomial homogeneous forms. A numerical example is given to illustrate that the stability conditions are less conservative than similar tests in the literature.
基于齐次多项式Lyapunov函数这一新工具研究了时变不确定系统鲁棒稳定性问题.针对常见的含参数时变且有界连续可微线性系统的最大稳定区域问题, 首先构造常用的参数依赖二次Lyapunov函数, 进而给出一个时变系统稳定的充分条件.然后, 通过构造适合的参数依赖齐次Lyapunov函数, 并利用齐次多项式矩阵表示方法, 最终以线性不等式的形式给出系统全局渐近稳定的一个充分条件.数值仿真结果表明齐次Lyapunov函数方法得到的结论对于某些系统比之前类似文献具有更小的保守性.

References:

[1] El Ghaoui L, Niculescu S I. Advances in linear matrix inequality methods in control [M]. Philadelphia: Advances in Design and Control, 2000.
[2] Lacerda M J, Oliveira R C L F, Peres P L D. Robust H2 and H∞ filter design for uncertain linear systems via LMIs and polynomial matrices [J]. Signal Processing, 2011, 91(5): 1115-1122.
[3] Montagner V F, Oliveira R C L F, Peres P L D, et al. Stability analysis and gain scheduled state feedback control for continuous time systems with bounded parameter variations[J]. International Journal of Control, 2009, 82(6):1045-1059.
[4] Ambrosino R, Ariola M, Amato F. A convex condition for robust stability analysis via polyhedral Lyapunov functions [J]. SIAM J Control Optim, 2012, 50(1): 490-506.
[5] BenAbdallah A, Hammami M A, Kallel J. Robust stability of uncertain piecewise-linear systems: LMI approach [J]. Nonlinear Dyn, 2011, 63(1/2): 183-192.
[6] Chesi G, Garulli A, Tesi A, et al. Homogeneous Lyapunov functions for systems with structured uncertainties [J]. Automatic, 2003, 39(6): 1027-1035.
[7] Chesi G. Robust stability of time-varying uncertain systems with rational dependence on the uncertainty [J]. IEEE Transactions on Automatic Control, 2010, 55(10): 2353-2357.
[8] Brockett R W. Lie algebra and Lie groups in control theory [M]. Netherlands: Springer, 1973.
[9] Prajna S, Papachristodoulou A, Seiler P, et. al. New developments in sum of squares optimization and SOSTOOLS [C]//Proceedings of the 2004 American Control Conference. Boston, USA, 2004:5606-5611.
[10] Ahmadi A A, Parrilo P A. Converse results on existence of sum of squares Lyapunov functions [C]//IEEE Conference on Decision and Control and European Control Conference(CDC-ECC). Orlando, FL, USA, 2011: 6516-6521.
[11] Chesi G, Garulli A, Tesi A, et al. Homogeneous polynomial forms for robustness analysis of uncertain systems [M]. Heidelberg: Springer, 2009.
[12] Montagner V E, Peres P L D. A new LMI condition for the robust stability of linear time-varying systems [C]//Proceedings of the 42nd IEEE Conference on Decision and Control. Maui, Hawaii, USA, 2003:6133-6138.
[13] Montagner V E, Peres P L D. Robust stability and H1 performance of linear time-varying systems in polytopic domains [J]. Int J Control, 2004, 77(15): 1343-1352.
[14] Geromel J C, Colaneri P. Robust stability of time varying polytopic systems [J]. Systems and Control Letters, 2006, 55(1): 81-85.
[15] Aouani N, Salhi S, Garcia G, et al. New robust stability and stabilizability conditions for linear parameter time varying polytopic systems [C]//The 3rd International Conference on Signals, Circuits and Systems. Medenine, Tunisia, 2009: 1-6.

Memo

Memo:
Biographies: Zhang Huasheng(1978—), male, doctor, zhsh0510@163.com; Zhang Kanjian(1971—), male, doctor, professor, kjzhang@seu.edu.cn.
Foundation items: The Major Program of National Natural Science Foundation of China(No.11190015), the National Natural Science Foundation of China(No.61374006).
Citation: Zhang Huasheng, Zhang Kanjian. Stability analysis of time-varying systems via parameter-dependent homogeneous Lyapunov functions [J].Journal of Southeast University(English Edition), 2014, 30(3):302-305.[doi:10.3969/j.issn.1003-7985.2014.03.008]
Last Update: 2014-09-20