On Improved Delay-dependent Robust Stability Criteria for Uncertain Systems with Interval Time-varying Delay

Jun-Jun Hui; He-Xin Zhang; Xiang-Yu Kong; Xin Zhou

doi:10.1007/s11633-014-0822-5

February 2015

Article Contents

Jun-Jun Hui, He-Xin Zhang, Xiang-Yu Kong and Xin Zhou. On Improved Delay-dependent Robust Stability Criteria for Uncertain Systems with Interval Time-varying Delay. International Journal of Automation and Computing, vol. 12, no. 1, pp. 102-108, 2015. doi: 10.1007/s11633-014-0822-5

Cite as: Jun-Jun Hui, He-Xin Zhang, Xiang-Yu Kong and Xin Zhou. On Improved Delay-dependent Robust Stability Criteria for Uncertain Systems with Interval Time-varying Delay. International Journal of Automation and Computing, vol. 12, no. 1, pp. 102-108, 2015. doi: 10.1007/s11633-014-0822-5

On Improved Delay-dependent Robust Stability Criteria for Uncertain Systems with Interval Time-varying Delay

The Xi an Research Institute of High Technology, Xi an 710025, China;
The Baoji Research Institute of High Technology, Baoji 721013, China

Received: 2013-08-10

Fund Project:

This work was supported by National Natural Science Foundation of China (No. 61074072).

Abstract

This paper considers the problem of delay-dependent robust stability for uncertain systems with interval time-varying delays. By using the direct Lyapunov method, a new Lyapunov-Krasovskii (L-K) functional is introduced based on decomposition approach, when dealing with the time derivative of L-K functional, a new tight integral inequality is adopted for bounding the cross terms. Then, a new less conservative delay-dependent stability criterion is formulated in terms of linear matrix inequalities (LMIs), which can be easily solved by optimization algorithms. Numerical examples are given to show the effectiveness and the benefits of the proposed method.
- Lyapunov-Krasovskii (L-K) functional,
- delay decomposition approach,
- linear matrix inequality (LMI),
- interval timevarying delay,
- robust stability

References

[1]	K. Q. Gu, V. L. Kharitonov, J. Chen. Stability of Timedelay Systems, Basel: Birkhauser, pp. 1-17, 2003.
[2]	M. Wu, Y. He, J. H. She, G. P. Liu. Delay-dependent criteria for robust stability of time-varying delay systems. Automatica, vol. 40, no. 8, pp. 1435-1439, 2004.
[3]	Y. He, Q. G. Wang, C. Lin, M. Wu. Delay-range-dependent stability for systems with time-varying delay. Automatica, vol. 43, no. 2, pp. 371-376, 2007.
[4]	P. Park, J. W. Ko. Stability and robust stability for systems with a time-varying delay. Automatica, vol. 43, no. 10, pp. 1855-1858, 2007.
[5]	C. Jeong, P. Park, S. H. Kim. Improved approach to robust stability and H_∞ performance analysis for systems with an interval time-varying delay. Applied Mathematics and Computation, vol. 218, no. 21, pp. 10533-10541, 2012.
[6]	K. Q. Gu. An integral inequality in the stability problem of time-delay systems. In Proceedings of the 39th IEEE Conference on Decision and Control, IEEE, Sydney, Australia, pp. 2805-2810, 2000.
[7]	Q. L. Han. Absolute stability of time-delay systems with sector-bounded nonlinearity. Automatica, vol. 41, no. 12, pp. 2171-2176, 2005.
[8]	X. M. Zhang, Q. L. Han. New Lyapunov-krasovskii functionals for global asymptotic stability of delayed neural networks. IEEE Transactions on Neural Networks, vol. 20, no. 3, pp. 533-539, 2009.
[9]	J. Sun, G. P. Liu, J. Chen, D. Rees. Improved delayrange-dependent stability criteria for linear systems with time-varying delays. Automatica, vol. 46, no. 2, pp. 466-470, 2010.
[10]	O. M. Kwon, J. H. Park, S. M. Lee. An improved delaydependent criterion for asymptotic stability of uncertain dynamic systems with time-varying delays. Journal of Optimization Theory and Applications, vol. 145, no. 2, pp. 343-353, 2010.
[11]	K. Ramakrishnan, G. Ray. Delay-range-dependent stability criterion for interval time-delay systems with nonlinear perturbations. International Journal of Automation and Computing, vol. 8, no. 1, pp. 141-146, 2011.
[12]	W. Zhang, X. S. Cai, Z. Z. Han. Robust stability criteria for systems with interval time-varying delay and nonlinear perturbations. Journal of Computational and Applied Mathematics, vol. 234, no. 1, pp. 174-180, 2010.
[13]	X. F. Jiang, Q. L. Han. New stability criteria for linear systems with interval time varying delay. Automatica, vol. 44, no. 10, pp. 2680-2685, 2008.
[14]	H. Y. Shao. New delay-dependent stability criteria for systems with interval delay. Automatica, vol. 45, no. 3, pp. 744-749, 2009.
[15]	X. L. Zhu, Y. Wang, G. H. Yang. New stability criteria for continuous-time systems with interval time-varying delay. IET Control Theory and Applications, vol. 4, no. 6, pp. 1101-1107, 2010.
[16]	C. Peng, Y. C. Tian. Improved delay-dependent robust stability criteria for uncertain systems with interval timevarying delay. IET Control Theory and Application, vol.2, no. 9, pp. 752-761, 2008.
[17]	Q. L. Han. A delay decomposition approach to stability of linear neutral systems. In Proceedings of the 17th World Congress, The International Federation of Automatic Control, IEEE, Seoul, Korea, pp. 2607-2612, 2008.
[18]	Q. L. Han. A discrete delay decomposition approach to stability of linear retarded and neutral systems. Automatica, vol. 45, no. 2, pp. 517-524, 2009.
[19]	E. Fridman, U. Shaked, K. Liu. New conditions for delayderivative-dependent stability. Automatica, vol. 45, no. 11, pp. 2723-2727, 2009.
[20]	C. Peng, M. R. Fei. A refined delay-partitioning approach to the stability of linear systems with interval time-varying delays. In Proceedings of the 2013 IEEE International Symposium on Industrial Electronics, IEEE, Taipei, China, pp. 1-5, 2013.
[21]	K. Ramakrishnan, G. Ray. Delay-dependent robust stability criteria for linear uncertain systems with interval time varying delay. In Proceedings of IEEE Region 10 Conference, IEEE, Singapore, pp. 1-6, 2009.
[22]	K. Ramakrishnan, G. Ray. Robust stability criteria for uncertain linear systems with interval time-varying delay. Journal of Control Theory and Applications, vol. 9, no. 4, pp. 559-566, 2011.
[23]	J. M. Jiao. Robust stability and stabilization of discrete singular systems with interval time-varying delay and linear fractional uncertainty. International Journal of Automation and Computing, vol. 9, no. 1, pp. 8-15, 2012.
[24]	P. Balasubramaniam, T. Senthilkumar. Delay-dependent robust stabilization and H_∞ control for uncertain stochastic T-S fuzzy systems with discrete interval and distributed time-varying delays. International Journal of Automation and Computing, vol. 10, no. 1, pp. 18-31, 2013.
[25]	D. Yue, E. G. Tian, Y. J. Zhang. A piecewise analysis method to stability analysis of linear continuous/discrete systems with time-varying delay. International Journal Robust Nonlinear Control, vol. 19, no. 13, pp. 1493-1518, 2009.
[26]	L. H. Xie. Output feedback control of systems with parameter uncertainty. International Journal of Control, vol. 63, no. 4, pp. 741-750, 1996.

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On Improved Delay-dependent Robust Stability Criteria for Uncertain Systems with Interval Time-varying Delay

The Xi an Research Institute of High Technology, Xi an 710025, China;

The Baoji Research Institute of High Technology, Baoji 721013, China

Received: 2013-08-10

Fund Project:

This work was supported by National Natural Science Foundation of China (No. 61074072).

Key words:

Abstract: This paper considers the problem of delay-dependent robust stability for uncertain systems with interval time-varying delays. By using the direct Lyapunov method, a new Lyapunov-Krasovskii (L-K) functional is introduced based on decomposition approach, when dealing with the time derivative of L-K functional, a new tight integral inequality is adopted for bounding the cross terms. Then, a new less conservative delay-dependent stability criterion is formulated in terms of linear matrix inequalities (LMIs), which can be easily solved by optimization algorithms. Numerical examples are given to show the effectiveness and the benefits of the proposed method.

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Reference (26)

[1]	K. Q. Gu, V. L. Kharitonov, J. Chen. Stability of Timedelay Systems, Basel: Birkhauser, pp. 1-17, 2003.
[2]	M. Wu, Y. He, J. H. She, G. P. Liu. Delay-dependent criteria for robust stability of time-varying delay systems. Automatica, vol. 40, no. 8, pp. 1435-1439, 2004.
[3]	Y. He, Q. G. Wang, C. Lin, M. Wu. Delay-range-dependent stability for systems with time-varying delay. Automatica, vol. 43, no. 2, pp. 371-376, 2007.
[4]	P. Park, J. W. Ko. Stability and robust stability for systems with a time-varying delay. Automatica, vol. 43, no. 10, pp. 1855-1858, 2007.
[5]	C. Jeong, P. Park, S. H. Kim. Improved approach to robust stability and H_∞ performance analysis for systems with an interval time-varying delay. Applied Mathematics and Computation, vol. 218, no. 21, pp. 10533-10541, 2012.
[6]	K. Q. Gu. An integral inequality in the stability problem of time-delay systems. In Proceedings of the 39th IEEE Conference on Decision and Control, IEEE, Sydney, Australia, pp. 2805-2810, 2000.
[7]	Q. L. Han. Absolute stability of time-delay systems with sector-bounded nonlinearity. Automatica, vol. 41, no. 12, pp. 2171-2176, 2005.
[8]	X. M. Zhang, Q. L. Han. New Lyapunov-krasovskii functionals for global asymptotic stability of delayed neural networks. IEEE Transactions on Neural Networks, vol. 20, no. 3, pp. 533-539, 2009.
[9]	J. Sun, G. P. Liu, J. Chen, D. Rees. Improved delayrange-dependent stability criteria for linear systems with time-varying delays. Automatica, vol. 46, no. 2, pp. 466-470, 2010.
[10]	O. M. Kwon, J. H. Park, S. M. Lee. An improved delaydependent criterion for asymptotic stability of uncertain dynamic systems with time-varying delays. Journal of Optimization Theory and Applications, vol. 145, no. 2, pp. 343-353, 2010.
[11]	K. Ramakrishnan, G. Ray. Delay-range-dependent stability criterion for interval time-delay systems with nonlinear perturbations. International Journal of Automation and Computing, vol. 8, no. 1, pp. 141-146, 2011.
[12]	W. Zhang, X. S. Cai, Z. Z. Han. Robust stability criteria for systems with interval time-varying delay and nonlinear perturbations. Journal of Computational and Applied Mathematics, vol. 234, no. 1, pp. 174-180, 2010.
[13]	X. F. Jiang, Q. L. Han. New stability criteria for linear systems with interval time varying delay. Automatica, vol. 44, no. 10, pp. 2680-2685, 2008.
[14]	H. Y. Shao. New delay-dependent stability criteria for systems with interval delay. Automatica, vol. 45, no. 3, pp. 744-749, 2009.
[15]	X. L. Zhu, Y. Wang, G. H. Yang. New stability criteria for continuous-time systems with interval time-varying delay. IET Control Theory and Applications, vol. 4, no. 6, pp. 1101-1107, 2010.
[16]	C. Peng, Y. C. Tian. Improved delay-dependent robust stability criteria for uncertain systems with interval timevarying delay. IET Control Theory and Application, vol.2, no. 9, pp. 752-761, 2008.
[17]	Q. L. Han. A delay decomposition approach to stability of linear neutral systems. In Proceedings of the 17th World Congress, The International Federation of Automatic Control, IEEE, Seoul, Korea, pp. 2607-2612, 2008.
[18]	Q. L. Han. A discrete delay decomposition approach to stability of linear retarded and neutral systems. Automatica, vol. 45, no. 2, pp. 517-524, 2009.
[19]	E. Fridman, U. Shaked, K. Liu. New conditions for delayderivative-dependent stability. Automatica, vol. 45, no. 11, pp. 2723-2727, 2009.
[20]	C. Peng, M. R. Fei. A refined delay-partitioning approach to the stability of linear systems with interval time-varying delays. In Proceedings of the 2013 IEEE International Symposium on Industrial Electronics, IEEE, Taipei, China, pp. 1-5, 2013.
[21]	K. Ramakrishnan, G. Ray. Delay-dependent robust stability criteria for linear uncertain systems with interval time varying delay. In Proceedings of IEEE Region 10 Conference, IEEE, Singapore, pp. 1-6, 2009.
[22]	K. Ramakrishnan, G. Ray. Robust stability criteria for uncertain linear systems with interval time-varying delay. Journal of Control Theory and Applications, vol. 9, no. 4, pp. 559-566, 2011.
[23]	J. M. Jiao. Robust stability and stabilization of discrete singular systems with interval time-varying delay and linear fractional uncertainty. International Journal of Automation and Computing, vol. 9, no. 1, pp. 8-15, 2012.
[24]	P. Balasubramaniam, T. Senthilkumar. Delay-dependent robust stabilization and H_∞ control for uncertain stochastic T-S fuzzy systems with discrete interval and distributed time-varying delays. International Journal of Automation and Computing, vol. 10, no. 1, pp. 18-31, 2013.
[25]	D. Yue, E. G. Tian, Y. J. Zhang. A piecewise analysis method to stability analysis of linear continuous/discrete systems with time-varying delay. International Journal Robust Nonlinear Control, vol. 19, no. 13, pp. 1493-1518, 2009.
[26]	L. H. Xie. Output feedback control of systems with parameter uncertainty. International Journal of Control, vol. 63, no. 4, pp. 741-750, 1996.

On Improved Delay-dependent Robust Stability Criteria for Uncertain Systems with Interval Time-varying Delay

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