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January  2016, 12(1): 303-315. doi: 10.3934/jimo.2016.12.303

## Finite-time stabilization and $H_\infty$ control of nonlinear delay systems via output feedback

 1 Institute of Mathematics, VAST, 18 Hoang Quoc Viet Road, Hanoi 10307, Vietnam, Vietnam 2 Université de Limoges, Laboratoire XLIM, 123, avenue Albert Thomas, 87060 Limoges CEDEX, France

Received  November 2014 Revised  January 2015 Published  April 2015

This paper studies the robust finite-time $H_\infty$ control for a class of nonlinear systems with time-varying delay and disturbances via output feedback. Based on the Lyapunov functional method and a generalized Jensen integral inequality, novel delay-dependent conditions for the existence of output feedback controllers are established in terms of linear matrix inequalities (LMIs). The proposed conditions allow us to design the output feedback controllers which robustly stabilize the closed-loop system in the finite-time sense. An application to $H_\infty$ control of uncertain linear systems with interval time-varying delay is also given. A numerical example is given to illustrate the efficiency of the proposed method.
Citation: Ta T.H. Trang, Vu N. Phat, Adly Samir. Finite-time stabilization and $H_\infty$ control of nonlinear delay systems via output feedback. Journal of Industrial and Management Optimization, 2016, 12 (1) : 303-315. doi: 10.3934/jimo.2016.12.303
##### References:
 [1] F. Amato, M. Ariola and C. Cosentino, Finite-time stabilization via dynamic output feedback, Automatica, 42 (2006), 337-342. doi: 10.1016/j.automatica.2005.09.007. [2] F. Amato, G. De Tommasi and A. Pironti, Necessary and sufficient conditions for finite-time stability of impulsive dynamical linear systems, Automatica, 49 (2013), 2546-2550. doi: 10.1016/j.automatica.2013.04.004. [3] E. K. Boukas, Static output feedback control for stochastic hybrid systems: LMI approach, Automatica, 42 (2006), 183-188. doi: 10.1016/j.automatica.2005.08.012. [4] S. Boyd, L. El. Ghaoui and V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory, SIAM, Philadelphia, PA, 1994. doi: 10.1137/1.9781611970777. [5] P. Dorato, Short time stability in linear time-varying systems, In Proc IRE Int Convention Record, 4 (1961), 83-87. [6] E. Fridman and U. Shaked, Delay-dependent stability and $H_{\infty}$control: constant and time-varying delays, International Journal of Control, 76 (2003), 48-60. doi: 10.1080/0020717021000049151. [7] P. Gahinet, A. Nemirovskii, A. J. Laub and M. Chilali, LMI Control Toolbox For use with MATLAB, The MathWorks, Inc, 1995. [8] G. Garcia, S. Tarbouriech and J. Bernussou, Finite-time stabilization of linear time-varying continuous systems, IEEE Transactions on Automatic Control, 54 (2009), 364-369. doi: 10.1109/TAC.2008.2008325. [9] L. Gollmann and H. Maurer, Theory and applications of optimal control problems with multiple time-delays, Journal of Industrial and Management Optimization, 10 (2014), 413-441. [10] V. Kharitonov, Time-Delay Systems: Lyapunov Functionals and Matrices, Control Engineering. Birkhäuser/Springer, New York, 2013. [11] O. M. Kwon, J. H. Park and S. M. Lee, Exponential stability for uncertain dynamic systems with time-varying delays: LMI optimization approach, Journal of Optimization Theory and Applications, 137 (2008), 521-532. doi: 10.1007/s10957-008-9357-7. [12] H. Liu, Y. Shen and X. Zhao, Delay-dependent observer-based $H_\infty$ finite-time control for switched systems with time-varying delay, Nonlinear Analysis: Hybrid Systems, 6 (2012), 885-898. doi: 10.1016/j.nahs.2012.03.001. [13] Q. Y. Meng and Y. J Shen, Finite-time $H_\infty$ control for linear continuous system with norm-bounded disturbance, Communications in Nonlinear Science and Numerical Simulation, 14 (2009), 1043-1049. doi: 10.1016/j.cnsns.2008.03.010. [14] E. Moulay, M. Dambrine, N. Yeganefar and W. Perruquetti, Finite-time stability and stabilization of time-delay systems, Systems and Control Letters, 57 (2008), 561-566. doi: 10.1016/j.sysconle.2007.12.002. [15] T. Senthilkumar and P. Balasubramaniam, Delay-dependent robust stabilization and $H_\infty$ control for nonlinear stochastic systems with Markovian jump parameters and interval time-varying delays, Journal of Optimization Theory and Applications,151 (2011), 100-120. doi: 10.1007/s10957-011-9858-7. [16] T. Senthilkumar and P. Balasubramaniam, Delay-dependent robust stabilization and $H_\infty$ control for nonlinear stochastic systems with Markovian jump parameters and interval time-varying delays, Journal of Optimization Theory and Applications, 151 (2011), 100-120. doi: 10.1007/s10957-011-9858-7. [17] A. Seuret and F. Gouaisbaut, Wirtinger-based integral inequality: Application to time-delay systems, Automatica, 49 (2013), 2860-2866. doi: 10.1016/j.automatica.2013.05.030. [18] L. Wu, J. Lam and C. Wang, Robust $H_{\infty}$ dynamic output feedback control for 2D linear parameter-varying systems, IMA journal of mathematical control and information, 26 (2009), 23-44. doi: 10.1093/imamci/dnm028. [19] Z. Xiang, Y. N. Sun and M. S. Mahmoud, Robust finite-time $H_\infty$ control for a class of uncertain switched neutral systems, Communications in Nonlinear Science Numerical Simulations, 17 (2012), 1766-1778. doi: 10.1016/j.cnsns.2011.09.022. [20] W. Xiang and J. Xiao, $H_{\infty}$ finite-time control for nonlinear switched discrete-time systems with norm-bounded disturbance, Journal of the Franklin Institute, 348 (2011), 331-352. doi: 10.1016/j.jfranklin.2010.12.001. [21] H. Xu and K. L. Teo, $H_\infty$ optimal stabilization of a class of uncertain impulsive systems: An LMI approach, Journal of Industrial and Management Optimization, 5 (2009), 153-159. doi: 10.3934/jimo.2009.5.153. [22] Y. Zhang, C. Liu and X. Mu, Robust finite-time $H_\infty$ control of singular stochastic systems via static output feedback, Applied Mathematics and Computation, 218 (2012), 5629-5640. doi: 10.1016/j.amc.2011.11.057.

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##### References:
 [1] F. Amato, M. Ariola and C. Cosentino, Finite-time stabilization via dynamic output feedback, Automatica, 42 (2006), 337-342. doi: 10.1016/j.automatica.2005.09.007. [2] F. Amato, G. De Tommasi and A. Pironti, Necessary and sufficient conditions for finite-time stability of impulsive dynamical linear systems, Automatica, 49 (2013), 2546-2550. doi: 10.1016/j.automatica.2013.04.004. [3] E. K. Boukas, Static output feedback control for stochastic hybrid systems: LMI approach, Automatica, 42 (2006), 183-188. doi: 10.1016/j.automatica.2005.08.012. [4] S. Boyd, L. El. Ghaoui and V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory, SIAM, Philadelphia, PA, 1994. doi: 10.1137/1.9781611970777. [5] P. Dorato, Short time stability in linear time-varying systems, In Proc IRE Int Convention Record, 4 (1961), 83-87. [6] E. Fridman and U. Shaked, Delay-dependent stability and $H_{\infty}$control: constant and time-varying delays, International Journal of Control, 76 (2003), 48-60. doi: 10.1080/0020717021000049151. [7] P. Gahinet, A. Nemirovskii, A. J. Laub and M. Chilali, LMI Control Toolbox For use with MATLAB, The MathWorks, Inc, 1995. [8] G. Garcia, S. Tarbouriech and J. Bernussou, Finite-time stabilization of linear time-varying continuous systems, IEEE Transactions on Automatic Control, 54 (2009), 364-369. doi: 10.1109/TAC.2008.2008325. [9] L. Gollmann and H. Maurer, Theory and applications of optimal control problems with multiple time-delays, Journal of Industrial and Management Optimization, 10 (2014), 413-441. [10] V. Kharitonov, Time-Delay Systems: Lyapunov Functionals and Matrices, Control Engineering. Birkhäuser/Springer, New York, 2013. [11] O. M. Kwon, J. H. Park and S. M. Lee, Exponential stability for uncertain dynamic systems with time-varying delays: LMI optimization approach, Journal of Optimization Theory and Applications, 137 (2008), 521-532. doi: 10.1007/s10957-008-9357-7. [12] H. Liu, Y. Shen and X. Zhao, Delay-dependent observer-based $H_\infty$ finite-time control for switched systems with time-varying delay, Nonlinear Analysis: Hybrid Systems, 6 (2012), 885-898. doi: 10.1016/j.nahs.2012.03.001. [13] Q. Y. Meng and Y. J Shen, Finite-time $H_\infty$ control for linear continuous system with norm-bounded disturbance, Communications in Nonlinear Science and Numerical Simulation, 14 (2009), 1043-1049. doi: 10.1016/j.cnsns.2008.03.010. [14] E. Moulay, M. Dambrine, N. Yeganefar and W. Perruquetti, Finite-time stability and stabilization of time-delay systems, Systems and Control Letters, 57 (2008), 561-566. doi: 10.1016/j.sysconle.2007.12.002. [15] T. Senthilkumar and P. Balasubramaniam, Delay-dependent robust stabilization and $H_\infty$ control for nonlinear stochastic systems with Markovian jump parameters and interval time-varying delays, Journal of Optimization Theory and Applications,151 (2011), 100-120. doi: 10.1007/s10957-011-9858-7. [16] T. Senthilkumar and P. Balasubramaniam, Delay-dependent robust stabilization and $H_\infty$ control for nonlinear stochastic systems with Markovian jump parameters and interval time-varying delays, Journal of Optimization Theory and Applications, 151 (2011), 100-120. doi: 10.1007/s10957-011-9858-7. [17] A. Seuret and F. Gouaisbaut, Wirtinger-based integral inequality: Application to time-delay systems, Automatica, 49 (2013), 2860-2866. doi: 10.1016/j.automatica.2013.05.030. [18] L. Wu, J. Lam and C. Wang, Robust $H_{\infty}$ dynamic output feedback control for 2D linear parameter-varying systems, IMA journal of mathematical control and information, 26 (2009), 23-44. doi: 10.1093/imamci/dnm028. [19] Z. Xiang, Y. N. Sun and M. S. Mahmoud, Robust finite-time $H_\infty$ control for a class of uncertain switched neutral systems, Communications in Nonlinear Science Numerical Simulations, 17 (2012), 1766-1778. doi: 10.1016/j.cnsns.2011.09.022. [20] W. Xiang and J. Xiao, $H_{\infty}$ finite-time control for nonlinear switched discrete-time systems with norm-bounded disturbance, Journal of the Franklin Institute, 348 (2011), 331-352. doi: 10.1016/j.jfranklin.2010.12.001. [21] H. Xu and K. L. Teo, $H_\infty$ optimal stabilization of a class of uncertain impulsive systems: An LMI approach, Journal of Industrial and Management Optimization, 5 (2009), 153-159. doi: 10.3934/jimo.2009.5.153. [22] Y. Zhang, C. Liu and X. Mu, Robust finite-time $H_\infty$ control of singular stochastic systems via static output feedback, Applied Mathematics and Computation, 218 (2012), 5629-5640. doi: 10.1016/j.amc.2011.11.057.
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