Hostname: page-component-cd9895bd7-q99xh Total loading time: 0 Render date: 2024-12-28T04:10:50.927Z Has data issue: false hasContentIssue false
  • English
  • Français

DELAY-DEPENDENT ROBUST H∞ CONTROL FOR SINGULAR SYSTEMS WITH MULTIPLE DELAYS

Part of: Control systems Functional-differential and differential-difference equations

Published online by Cambridge University Press:  01 October 2008

SHUQIAN ZHU ,
CHENGHUI ZHANG ,
XINZHI LIU  and
ZHENBO LI
SHUQIAN ZHU
Affiliation:
School of Mathematics, Shandong University, Jinan 250100, P. R. China (email: sqzhu@sdu.edu.cn)
CHENGHUI ZHANG
Affiliation:
School of Control Science and Engineering, Shandong University, Jinan 250061, P. R. China (email: zchui@sdu.edu.cn)
XINZHI LIU*
Affiliation:
Department of Applied Mathematics, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada (email: xzliu@uwaterloo.ca)
ZHENBO LI
Affiliation:
School of Statistics and Mathematics, Shandong Economic University, Jinan 250014, P. R. China (email: lizhenbo@126.com)
*
✉For correspondence; e-mail: xzliu@uwaterloo.ca
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

This paper studies the problem of delay-dependent robust H∞ control for singular systems with multiple delays. Based on a Lyapunov–Krasovskii functional approach, an improved delay-dependent bounded real lemma (BRL) for singular time-delay systems is established without using any of the model transformations and bounding techniques on the cross product terms. Then, by applying the obtained BRL, a delay-dependent condition for the existence of a robust state feedback controller, which guarantees that the closed-loop system is regular, impulse free, robustly stable and satisfies a prescribed H∞ performance index, is proposed in terms of a nonlinear matrix inequality. The explicit expression for the H∞ controller is designed by using linear matrix inequalities and the cone complementarity iterative linearization algorithm. Numerical examples are also given to illustrate the effectiveness of the proposed method.

Keywords

singular time-delay systemmultiple delaysdelay-dependent robust H∞ controlbounded real lemmalinear matrix inequality

MSC classification

Secondary: 93C15: Systems governed by ordinary differential equations 34K06: Linear functional-differential equations 34K20: Stability theory
Type
Research Article
Information
The ANZIAM Journal , Volume 50 , Issue 2 , October 2008 , pp. 209 - 230
Copyright
Copyright © Australian Mathematical Society 2009

References

[1]Basin, M. V. and Rodkina, A. E., “On delay-dependent stability for a class of nonlinear stochastic delay-difference equations”, Dyn. Contin. Discrete Impuls. Syst. 12A (2005) 663–675.Google Scholar
[2]El Ghaoui, L., Oustry, F. and Ait Rami, M., “A cone complementarity linearization algorithm for static output-feedback and related problems”, IEEE Trans. Automat. Control 42 (1997) 1171–1176.CrossRefGoogle Scholar
[3]Feng, J., Zhu, S. and Cheng, Z., “Guaranteed cost control of linear uncertain singular time-delay systems”, in Proceedings of 41st IEEE Conference on Decision and Control, 2002, 1802–1807.Google Scholar
[4]Fridman, E., “Stability of linear descriptor systems with delay: a Lyapunov-based approach”, J. Math. Anal. Appl. 273 (2002) 24–44.Google Scholar
[5]Fridman, E. and Shaked, U., “A descriptor system approach to H ∞ control of linear time-delay systems”, IEEE Trans. Automat. Control 47 (2002) 253–270.CrossRefGoogle Scholar
[6]Fridman, E. and Shaked, U., “H ∞-control of linear state-delay descriptor systems: an LMI approach”, Linear Algebra Appl. 351–352 (2002) 271–302.Google Scholar
[7]Hale, J. K. and Verduyn Lunel, S. M., Introduction to functional differential equations (Springer, New York, 1993).CrossRefGoogle Scholar
[8]Kim, J. K., “New design method on memoryless H ∞ control for singular systems with delayed state and control using LMI”, J. Franklin Inst. 342 (2005) 321–327.CrossRefGoogle Scholar
[9]Kim, J. K., Lee, J. H. and Park, H. B., Robust H ∞ control of singular systems with time delays and uncertainties via LMI approach, in Proceedings of American Control Conference, 2002, 620–621.Google Scholar
[10]Kolmanovskii, V. B. and Nosov, V. R., Stability of functional differential equations (Academic Press, New York, 1986).Google Scholar
[11]Li, X. and de Souza, C. E., “Criteria for robust stability and stabilization of uncertain linear systems with state-delay”, Automatica 33 (1997) 1657–1662.CrossRefGoogle Scholar
[12]Logemann, H., “Destabilizing effects of small time delays on feedback-controlled descriptor systems”, Linear Algebra Appl. 272 (1998) 131–153.CrossRefGoogle Scholar
[13]Moon, Y. S., Park, P., Kwon, W. H. and Lee, Y. S., “Delay-dependent robust stabilization of uncertain state-delayed systems”, Internat. J. Control 74 (2001) 1447–1455.Google Scholar
[14]Park, P., “A delay-dependent stability criterion for systems with uncertain time-invariant delays”, IEEE Trans. Automat. Control 44 (1999) 876–877.CrossRefGoogle Scholar
[15]Petersen, I. R., “A stabilization algorithm for a class of uncertain linear systems”, Syst. Control Lett. 8 (1987) 351–357.CrossRefGoogle Scholar
[16]Rodkina, A. E. and Basin, M. V., “On delay-dependent stability for a class of nonlinear stochastic delay-differential equations”, Math. Control Signals Systems 18 (2006) 187–197.Google Scholar
[17]Rodkina, A. E. and Basin, M. V., “On delay-dependent stability for vector nonlinear stochastic delay-difference equations with Volterra diffusion term”, Syst. Control Lett. 56 (2006) 423–430.CrossRefGoogle Scholar
[18]Xu, S., Dooren, P. V., Stefan, R. and Lam, J., “Robust stability and stabilization for singular systems with state delay and parameter uncertainty”, IEEE Trans. Automat. Control 47 (2002) 1122–1128.Google Scholar
[19]Xu, S., Lam, J. and Yang, C., “H ∞ control for uncertain singular systems with state delay”, Internat. J. Robust Nonlinear Control 13 (2003) 1213–1223.CrossRefGoogle Scholar
[20]Xu, S., Lam, J. and Zou, Y., “Simplified descriptor system approach to delay-dependent stability and performance analyses for time-delay systems”, IEE Proc. Control Theory Appl. 152 (2005) 147–151.CrossRefGoogle Scholar
[21]Xu, S., Lam, J. and Zou, Y., “New results on delay-dependent robust H ∞ control for systems with time-varying delays”, Automatica 42 (2006) 343–348.CrossRefGoogle Scholar
[22]Xu, S., Lam, J. and Zou, Y., “An improved characterization of bounded realness for singular delay systems and its applications”, Internat. J. Robust Nonlinear Control 18 (2008) 263–277.Google Scholar
[23]Yue, D. and Han, Q., “H ∞ filter design of uncertain descriptor systems with discrete and distributed delays”, IEEE Trans. Signal Process. 52 (2003) 3200–3212.CrossRefGoogle Scholar
[24]Yue, D. and Han, Q., “A delay-dependent stability criterion of neutral systems and its application to a partial element equivalent circuit model”, IEEE Trans. Circuits Systems II 51 (2004) 685–689.Google Scholar
[25]Yue, D. and Han, Q., “Delay-dependent robust H ∞ controller design for uncertain descriptor systems with time-varying discrete and distributed delays”, IEE Proc. Control Theory Appl. 152 (2005) 628–638.Google Scholar
[26]Zhou, S. and Lam, J., “Robust stabilization of delayed singular systems with linear fractional parametric uncertainties”, Circuits Systems Signal Process 22 (2003) 579–588.CrossRefGoogle Scholar
[27]Zhu, S., Cheng, Z. and Feng, J., “Delay-dependent robust stability criterion and robust stabilization for uncertain singular time-delay systems”, in Proceedings of American Control Conference, 2005, 2839–2844.Google Scholar
[28]Zhu, S., Li, Z. and Cheng, Z., “Delay-dependent robust resilient H ∞ control for uncertain singular time-delay systems”, in Proceedings of IEEE Conference on Decision and Control and European Control Conference, 2005, 1373–1378.Google Scholar
[29]Zhu, S., Zhang, C., Cheng, Z. and Feng, J., “Delay-dependent robust stability criteria for two classes of uncertain singular time-delay systems”, IEEE Trans. Automat. Control 52 (2007) 880–885.Google Scholar