Abstract
In this paper, the problem of stability for linear systems with interval time-varying delays is investigated. By constructing a suitable augmented Lyapunov-Krasovskii functional and utilizing Wirtinger-based integral inequality, two sufficient conditions for guaranteeing the asymptotic stability of the concerned systems are derived within the framework of linear matrix inequalities (LMIs). The superiority and validity of the proposed criteria are verified by comparing maximum delay bounds under various conditions via two numerical examples.
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References
S. I. Niculescu, Delay Effects on Stability: A Robust Control Approach, Springer, Berlin, 2001. [click]
J. P. Richard, “Time-delay systems: an overview of some recent advances and open problems,” Automatica, vol. 39, pp. 1667–1694, 2003. [click]
K. Gu, “A further refinement of discretized Lyapunov functional method for the stability of time-delay systems,” International Journal of Control, vol. 74, pp. 967–976, 2001. [click]
Y. S. Moon, P. G. Park,W. H. Kwon, and Y. S. Lee, “Delaydependent robust stabilization of uncertain state delayed system,” International Journal of Control, vol. 74, pp. 1447–1455, 2001. [click]
E. Fridman and U. Shaked, “An improved stabilization method for linear time delay system,” IEEE Transactions on Automatic Control, vol. 47, pp. 1931–1937, 2002. [click]
M. Wu, Y. He, and J.H. She, “New delay-dependent stability criteria and stabilizing method for neutral systems,” IEEE Transactions on Automatic Control, vol. 49, pp. 2266–2270, 2004. [click]
Y.S. Lee, Y.S. Moon, and W.H. Kwon “Delay-dependent Guaranteed Cost Control for Uncertain State-delayed Systems,” Proceedings of American Control Conference, Arlington, USA, pp. 3376–3381, 2001. [click]
M. N. Alpaslan Parlakçı, “Improved robust stability criteria and design of robust stabilizing controller for uncertain linear time-delay systems,” International Journal of Robust and Nonlinear Control, vol. 16, pp. 599–636, 2006. [click]
C. Briat, “Convergence and Equivalence Results for the Jensen’s Inequality-Application to Time-Delay and Sampled-Data Systems,” IEEE Transactions on Automatic Control, vol. 56, pp. 1660–1665, 2011. [click]
S. H. Kim, P. Park, and C. K. Jeong, “Robust H ∞ stabilisation of networks control systems with packet analyser,” IET Control Theory & Applications, vol. 4, pp. 1828–1837, 2010. [click]
O. M. Kwon, J. H. Park, S. M. Lee, and E. J. Cha, “New augmented Lyapunov-Krasovskii functional approach to stability analysis of neural networks with time-varying delays,” Nonlinear Dynamics, vol. 76, pp. 221–236, 2014. [click]
O. M. Kwon, M. J. Park, J. H. Park, S. M. Lee, and E. J. Cha, “Stability and H ∞ performance analysis for Markovian jump systems with time-varying delays,” Journal of the Franklin Institute, vol. 351, pp. 4724–4748, 2014. [click]
O. M. Kwon, M. J. Park, J. H. Park, S. M. Lee, and E. J. Cha, “Improved results on stability of linear systems with time-varying delays via Wirtinger-based integral inequality,” Journal of the Franklin Institute, vol. 351, pp. 5386, 2014. [click]
Y. He, Q. G. Wang, C. Lin, and M. Wu, “Delay-rangedependent stability for systems with time-varying delay,” Automatica, vol. 43, pp. 371–376, 2007. [click]
P. Park, J. W. Ko, and C. K. Jeong, “Reciprocally convex approach to stability of systems with time-varying delays,” Automatica, vol. 47, pp. 235–238, 2011. [click]
W. I. Lee and P. Park, “Second-order reciprocally convex approach to stability of systems with interval time-varying delays,” Applied Mathematics and Computation, vol. 229, pp. 245–253, 2014. [click]
W. Qian, T. Li, S. Cong, and S. Fei, “Stability analysis for interval time-varying delay systems based on time-varying bound integral method,” Journal of the Franklin Institute, vol. 351, pp. 4892–4903, 2014. [click]
T. Li, L. Guo, and Y. Zhang, “Delay-range-dependent robust stability and stabilization for uncertain systems with time-varying delay,” International Journal of Robust and Nonlinear Control, vol. 18, pp. 1372–1387, 2008. [click]
J. Sun, G. P. Liu, and J. Chen, “Delay-dependent stability and stabilization of neutral time-delay systems,” International Journal of Robust and Nonlinear Control, vol.15, pp. 1364–1375, 2009. [click]
P. G. Park and J. W. Ko, “Stability and robust stability for systems with a time-varying delay,” Automatica, vol. 43, pp. 1855–1858, 2007. [click]
R. Dey, S. Ghosh, G. Ray, and A. Rakshit, “State feedback stabilization of uncertain linear time-delay systems: A nonlinear matrix inequality approach,” Numerical Linear Algebra with Applications, vol. 18, pp. 351–361, 2011. [click]
J. H. Kim, “Note on stability of linear systems with timevarying delay,” Automatica, vol. 47, pp. 2118–2121, 2011. [click]
J.W. Ko and P. G. Park, “Delay-dependent stability criteria for systems with asymmetric bounds on delay derivative,” Journal of the Franklin Institute, vol. 348, pp. 2674–2688, 2011. [click]
C.-K. Zhang, Y. He, L. Jiang, and M. Wu, “Stability analysis for delayed neural networks considering both conservativeness and complexity,” IEEE Transactions on Neural Networks and Learning Systems, 2015. [click]
W. Qian and Juan Liu, “New stability for systems with interval time-varying delay,” Journal of the Franklin Institute, vol. 350, pp. 890–897, 2013. [click]
H.-B. Zeng, Y. He, M. Wu, and S.-P. Xiao, “Less conservative results on stability for linear systems with a timevarying delay,” Optimal Control Applications and Methods, vol. 34, pp. 670–679, 2013. [click]
A. Seuret and F. Gouaisbaut, “Wirtinger-based integral inequality: application to time-delay systems,” Automatica, vol. 49, pp. 2860–2866, 2013. [click]
É. Gyurkovics, “A note on Wirtinger-type integral inequalities for time-delay systems,” Automatica, vol. 61, pp. 44–46, 2015. [click]
M. J. Park, O. M. Kwon, J. H. Park, S. M. Lee, and E. J. Cha, “Stability of time-delay systems via Wirtinger-based double integral inequality,” Automatica, vol. 55, pp. 204–208, 2015. [click]
P. G. Park, W. I. Lee, and S. Y. Lee, “Auxiliary functionbased integral inequalities for quadratic functions and their applications to time-delay systems,” Journal of the Franklin Institute, vol. 352, pp. 1378–1396, 2015. [click]
L. V. Hien and H. Trinh, “Refined Jensen-based inequality approach to stability analysis of time-delay systems,” IET Control Theory & Applications, vol. 9, no. 14, pp. 2188–2194, 2015. [click]
J. Sun, G. P. Liu, J. Chen, and D. Rees, “Improved delayrange- dependent stability criteria for linear systems with time-varying delays,” Automatica, vol. 46, pp. 466–470, 2010. [click]
Y. Liu, L. S. Hu, and P. Shi, “A novel approach on stabilization for linear systems with time-varying input delay,” Applied Mathematics and Computations, vol. 218, pp. 5937–5947, 2012. [click]
W. I. Lee, S. Y. Lee, and P. G. Park, “Improved criteria on robust stability and H ∞ performance for linear systems with interval time-varying delays via new triple integral functionals,” Applied Mathematics and Computation, vol. 243, pp. 570–577, 2014. [click]
C. Jeong, P. Park, and 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, pp. 10533–10541, 2012. [click]
M.C. de Oliveira and R.E. Skelton, Stability Tests for Constrained Linear Systems, Springer-Verlag, Berlin, 2001. [click]
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Recommended by Guest Editor PooGyeon Park. This work was supported by the intramural research grant of Chungbuk National University in 2015. This research was also supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2011-0009273) and the Human Resources Development of the Korea Institute of Energy Technology Evaluation and Planning (KETEP) grant funded by the Korea government Ministry of Trade, Industry & Energy (no. 20144030200450).
Oh Min Kwon received the B.S. degree in Electronic Engineering from Kyungbuk National University, Daegu, Korea, in 1997, and Ph.D. degree in Electrical and Electronic Engineering from POSTECH, Pohang, Korea, in 2004. From February 2004 to January 2006, he was a senior researcher in Mechatronics Center of Samsung Heavy Industries. He is currently working as an associate professor in School of Electrical Engineering, Chungbuk National University. His research interests include timedelay systems, cellular neural networks, robust control and filtering, large-scale systems, secure communication through synchronization between two chaotic systems, complex dynamical networks, multi-agent systems, and so on. He has presented more than 130 international papers in these areas. He is a member of KIEE, ICROS, and IEEK. Currently, he serves as an editorial member of ICROS, Nonlinear Analysis: Hybrid Systems, and The Scientific World Journal.
Myeong Jin Park received the B.S. and Ph.D. degrees both in Electrical Engineering from Chungbuk National University, Cheongju, Korea, in 2009 and 2015, respectively. His current research interests include consensus problem in multi-agent systems and stability analysis for systems with time-delay.
Ju H. Park received the Ph.D. degree in Electronics and Electrical Engineering from POSTECH, Pohang, Republic of Korea, in 1997. From May 1997 to February 2000, he was a Research Associate in ERC-ARC, POSTECH. In March 2000, he joined Yeungnam University, Kyongsan, Republic of Korea, where he is currently a Full Professor. From December 2006 to December 2007, he was a Visiting Professor in the Department of Mechanical Engineering, Georgia Institute of Technology. His research interests include robust control and filtering, neural networks, complex networks, and chaotic systems. He has published a number of papers in these areas. He serves as an Editor of International Journal of Control, Automation and Systems. He is also an Associate Editor/Editorial Board member for several international journals, including IET Control Theory and Applications, Applied Mathematics and Computation, Journal of The Franklin Institute, Journal of Applied Mathematics and Computing, etc.
Sang Moon Lee received the B.S. degree in Electronic Engineering from Kyungpook National University, and M.S. and Ph.D. degrees at Department of Electronic Engineering from POSTECH, Korea. Currently, he is an assistant professor at Division of Electronic Engineering in Daegu University. His main research interests include robust control theory, nonlinear systems, model predictive control and its industrial applications.
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Kwon, O.M., Park, M.J., Park, J.H. et al. Enhancement on stability criteria for linear systems with interval time-varying delays. Int. J. Control Autom. Syst. 14, 12–20 (2016). https://doi.org/10.1007/s12555-015-2003-x
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DOI: https://doi.org/10.1007/s12555-015-2003-x