Delay-Independent Closed-Loop Stabilization of Neutral System with Infinite Delays

Authors: I. Davies, O. L. C. Haas

Abstract:

In this paper, the problem of stability and stabilization for neutral delay-differential systems with infinite delay is investigated. Using Lyapunov method, new delay-independent sufficient condition for the stability of neutral systems with infinite delay is obtained in terms of linear matrix inequality (LMI). Memory-less state feedback controllers are then designed for the stabilization of the system using the feasible solution of the resulting LMI, which are easily solved using any optimization algorithms. Numerical examples are given to illustrate the results of the proposed methods.

Keywords: Infinite delays, Lyapunov method, linear matrix inequality, neutral systems, stability.

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1108426

References:

[1] I. Davies, “Euclidean Null Controllability of Infinite Neutral Differential Systems,” Anziam Journal. vol. 48, pp. 285-294, 2006.
[2] V. B. Kolmanovskii, and A. Myshkis, Applied Theory of Functional Differential Equations. Kluwer Academic Publishers, Boston, 1992.
[3] A. Ardjouni, and A. Djoudi, “Fixed Points and Stability in Linear Neutral Differential Equations with Variable Delays,” Nonlinear Analysis: Theory, Methods & Applications. Vol.74, pp. 2062-2070, 2011.
[4] M. A. Cruz, and J. K. Hale, “Stability of Functional Differential Equations of Neutral Type,” Journal of Differential Equations. Vol. 7, pp. 334-355, 1970.
[5] Y. He, M. Wu, and J. She, “Delay Dependent Robust Stability and Stabilization of Uncertain Neutral Systems,” Asian Journal of Control. vol. 10, pp. 376-383, 2008.
[6] P. Liu, “On the Stability of Neutral-Type Uncertain Systems with Multiple Time Delays,” International Journal of Applied Mathematics and Computer Science. Vol. 15, pp. 221-229, 2005.
[7] W. K. Liu, and W. Jiang, “Stability of Fractional Neutral Systems,” Advances in Differential Equations. Vol. 2014: 78, pp. 1-9, 2014.
[8] J. H. Park, and S. Won, “Stability Analysis for Neutral Delay- Differential Systems”, Journal of the Franklin Institute. Vol. 337, 1-9, 2000.
[9] F. Qiu and Q. Zhang, “Robust Stability of Neutral System with Mixed Time-Varying Delays and Nonlinear Perturbations Using Delay Decomposition Approach,” Abstract and Applied Analysis. Vol. 2014, pp. 1-11, 2014.
[10] E. I. Verriest, and S. I. Niculescu, Delay-Independent Stability of Linear Neuytral Systems: A Riccati Equation Approach, European Control Conference, Brussels, Belgium. pp. 3632-3636, July, 1997.
[11] T. Vyhlídal, N. Olgac, and V. Kučera, “Delayed Resonator with Acceleration Feedback–Complete Stability Analysis by Spectral Methods and Vibration Absorber Design,” Journal of Sound and Vibration. Vol. 333, pp. 6781-6795, 2014.
[12] H. Xia, P. Zhao, L. Li, Y, Wang, and A. Wu, “Improved Stability Criteria for Linear Neutral Time-Delay Systems,” Asian Journal of Control. Vol. 17, pp. 343-351, 2015.
[13] B. Yang, J. Wang and C. Zhong, “A New Delay Dependent Stability Criterion for Neutral Delay Differential Systems,” Asian Journal of Control. Vol. 5, pp. 433-436, 2003.
[14] C. Briat, (Ed.), “Robust Stability Analysis in the *-Norm and Lyapunov- Razumikhin Functions for the Stability Analysis of Time-Delay Systems,” 50th IEEE Conference on Decision and Control and European Control Conference (CDC-ECC). Orlando, FL, USA. pp. 12-15, 2011.
[15] F. El Haoussi, E. Tissir, A. Hmamed, and F. Tadeo, “Stabilization of Neutral Systems with Saturating Actuators,” Journal of Control Science and Engineering. Vol. 2012, pp. 1-8, 2012.
[16] K. Balachandran, and J. Dauer, “Null Controllability of Nonlinear Infinite Delay Systems with Time Varying Multiple Delays in Control,” Applied Mathematics Letters. Vol. 9, pp. 115-121, 1996.
[17] P. P. Khargonekar, I. R. Petersen, and K. Zhou, “Robust Stabilization of Uncertain Linear Systems: Quadratic Stabilizability and h∞ Control Theory,” IEEE Transactions on Automatic Control. Vol. 35, pp. 356- 361, 1990.
[18] S. P. Boyd, L. El Ghaoui, E. Feron, and V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory, SIAM, Philadelphia, 1994.
[19] K. Gu, J. Chen, and V. L. Kharitonov, Stability of Time-Delay Systems. Springer, Birkhäuser, 2003.
[20] J. K Hale, and S. M. Verduyn Lunel, Introduction to Functional Differential Equations. Springer, 1993.
[21] P. Gahinet, A. Nemirovskii, A. J. Laub, and M. Chilali (eds.), The LMI Control Toolbox for User’s Guide, Mathworks, Natick, MA, 1995.
[22] L. M. Li, “Stability of Linear Neutral Delay-Differential Systems,” Bull. Austral. Math. Soc. Vol. 38, pp. 339-344, 1988.
[23] G. Di Hu, G. Da Hu, “Some Simple Stability Criteria of Neutral Delay- Differential Systems,” Appl. Math. Comput. Vol. 80, pp. 257-271, 1996.