FINITE-TIME STABILITY ANALYSIS OF DISCRETE TIME-DELAY SYSTEMS USING DISCRETE CONVOLUTION OF DELAYED STATES

Sreten B. Stojanović, Dragutin Lj. Debeljković, Dragan S. Antić

DOI Number
-
First page
147
Last page
158

Abstract


Finite-time stability for the linear discrete-time system with state delay was investigated in this article. Stability of the system was analyzed using both the Lyapunov-like approach and the discrete Jensen’s inequality. A novel Lyapunov-like functional with a discrete convolution of delayed states was proposed and used for the derivation of the sufficient stability conditions of the investigated system. As a result, the novel stability conditions guarantee that the states of the systems do not exceed the predefined boundaries on a finite time interval. The proposed methodology was illustrated with a numerical example. A computer simulation was performed for the analysis of the dynamical behavior of this system.

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References


G. Kamenkov, "On stability of motion over a finite interval of time", Journal of Applied Mathematics and Mechanics, vol. 17, no. 2, pp. 529–540, 1953.

P. Dorato, "Short time stability in linear time-varying system", in Proceedings book of IRE International Convention Record, Part IV, (New York, USA), pp. 83–87, 1961.

L. Weiss, F. Infante, “Finite-time stability under perturbing forces and on product spaces”, IEEE Transaction on Automatic Control, vol. 12, no. 1, 1967, pp. 54-59. [Online]. Available: http://dx.doi.org/10.1109/TAC.1967.1098483

F. Amato, M. Ariola, P. Dorato, "Finite-time control of linear systems subject to parametric uncertainties and disturbances", Automatica, vol. 37, no. 9, pp. 1459-1463, 2001. [Online]. Available: http://dx.doi.org/10.1016/S0005-1098(01)00087-5

F. Amato, M. Ariola, P. Dorato, "Finite-time stabilization via dynamic output feedback", Automatica, vol. 42, no. 2, pp. 337-342, 2006. [Online]. Available: http://dx.doi.org/10.1016/j.automatica.2005.09.007

E. Moulay, W. Perruquetti, "Finite-time stability and stabilization of a class of continuous systems", Journal of Mathematical Analysis and Applications, vol. 323, no. 2, pp. 1430–1443, 2006. [Online]. Available: http://dx.doi.org/10.1016/j.jmaa.2005.11.046

Q. Ming, Y. Shen, "Finite-time H∞ control for linear continuous system with norm-bounded disturbance", Communi¬cations in Nonlinear Science and Numerical Simulation, vol. 14, no. 4, pp. 1043-1049, 2009. [Online]. Available: http://dx.doi.org/10.1016/j.cnsns.2008.03.010

G. Garcia, S. Tarbouriech, J. Bernussou, "Finite-time stabilization of linear time-varying continuous systems", IEEE Transaction on Automatic Control, vol. 54, no. 2, pp. 364–369, 2009. [Online]. Available: http://dx.doi.org/10.1109/TAC.2008.2008325

F. Amato, M. Ariola, "Finite-Time Control of Discrete-Time Linear Systems", IEEE transactions on automatic control, vol. 50, no. 5, pp. 724-729, 2005. [Online]. Available: http://dx.doi.org/10.1109/TAC.2005.847042

F. Amato, M. Carbone, M. Ariola, C. Cosentino, "Finite-time stability of discrete-time systems", in Proceedings book of the American Control Conference 2004, (Boston, USA), pp. 1440-1444, June 2004.

L. Zhu, Y. Shen, C. Li, "Finite-time control of discrete-time systems with time-varying exogenous disturbance", Communic¬ations in Nonlinear Science and Numerical Simulation, vol. 14, no. 2, pp. 361–370, 2009. [Online]. Available: http://dx.doi.org/10.1016/j.cnsns.2007.09.013

I. Hiroyuki, H. Katayama, "Necessary and sufficient conditions for finite-time boundedness of linear discrete-time systems", in Proceedings book of the Joint 48th IEEE Conference on Decision and Control and 28th Chinese Control Conference, (Shanghai, China), pp. 3226- 3231, December 2009.

F. Amato, R. Ambrosino, M. Ariola, F. Calabrese, "Finite-Time Stability Analysis of Linear Discrete-Time Systems via Polyhedral Lyapunov Functions", in Proceedings book of the 2008 American Control Conference, (Seattle, USA), pp. 1656-1660, June 2008.

I. Hiroyuki, H. Katayama, Finite-time control for linear discrete-time systems with input constraints, in Proceedings book of the 2009 American Control Conference, (St. Louis, USA), pp. 1171-1176, June 2009.

F. Amato, R. Ambrosino, M. Ariola, G. De Tommasi, "Input to output finite-time stabilization of discrete-time linear systems", in Proceedings book of the 18th IFAC World Congress (Milano, Italy), pp. 156-161, August 2011.

F. Amato, M. Ariola, C. Cosentino, "Finite-time control of discrete-time linear systems: Analysis and design conditions", Automatica, vol. 46, no. 5, pp. 919-924, 2010. [Online]. Available: http://dx.doi.org/10.1016/j.automatica.2010.02.008

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. [Online]. Available: http://dx.doi.org/10.1016/j.automatica.2007.02.022

E. Shustin, E. Fridman, "On delay-derivative-dependent stability of systems with fast-varying delays", Automatica, vol. 43, no. 9, pp. 1649–1655, 2007. [Online]. Available: http://dx.doi.org/10.1016/j.automatica.2007.02.009

S. Xu, J. Lam, "Improved delay-dependent stability criteria for time-delay systems", IEEE Transactions on Automatic Control, vol. 50, no. 3, pp. 384-387, 2005. [Online]. Available: http://dx.doi.org/10.1109/TAC.2005.843873

O.M. Kwon, J.H. Park, S.M. Lee, "On robust stability criterion for dynamic systems with time-varying delays and nonlinear perturbations", Applied Mathematics and Computation, vol. 203, no. 2, pp. 937-942, 2008. [Online]. Available: http://dx.doi.org/10.1016/j.amc.2008.05.097

O.M. Kwon, J.H. Park, "Exponential stability for time delay systems with interval time-varying delays and nonlinear perturbations", Journal of Optimization Theory and Applications, vol. 139, no. 2, pp. 277-293, 2008. [Online]. Available: http://dx.doi.org/10.1007/s10957-008-9417-z

X. Sun, Q.L. Zhang, C.-Y. Yang, Y.-Y. Shao, Z. Su, "Delay-dependent stability analysis and stabilization of discrete-time singular delay systems", Acta Automatica Sinica, vol. 36, no. 10, pp. 1477–1483, 2010. [Online]. Available: http://dx.doi.org/10.1016/S1874-1029(09)60061-6

M.P. Lazarevic, D.Lj. Debeljkovic, Z.Lj. Nenadic, S.A. Milinkovic, "Finite-time stability of delayed systems", IMA Journal of Mathematical Control and Information, vol. 17, no. 2, pp. 101–109, 2000. [Online]. Available: http://dx.doi.org/10.1093/imamci/17.2.101

D.Lj. Debeljkovic, M.P. Lazarevic, Dj. Koruga, S.A. Milinkovic, M.B. Jovanovic, "Further results on the stability of linear nonautonomous systems with delayed state defined over finite time interval", in Proceedings book of the 2000 American Control Conference, (Chicago, USA), pp. 1450–1451, June 2000.

D.Lj. Debeljkovic, I.M. Buzurovic, T. Nestorovic, D. Popov, On finite and practical stability of time delayed systems: Lyapunov-Krassovski ap¬proach: delay dependent criteria, in Proceedings book of the 23rd IEEE Chinese Control and Decision Conference, (Mianyang, China), pp. 331–337, 2011.

S.B. Stojanovic, D.Lj. Debeljkovic, D.S. Antic, "Finite time stability and stabilization of linear time delay systems", Facta Universitatis, Series Automatic Control and Robotics, vol.11, no.1, pp. 25–36, 2012. [Online]. Available: http://facta.junis.ni.ac.rs/acar/acar201201/acar20120103.pdf

S.B. Stojanovic, D.Lj. Debeljkovic, D.S. Antic, "Robust finite-time stability and stabilization of linear uncertain time-delay systems", Asian Journal of Control, vol. 15, no. 5, pp. 1548–1554, 2013. [Online]. Available: http://dx.doi.org/10.1002/asjc.689

S.B. Stojanovic, D.Lj. Debeljkovic, N. Dimitrijevic, "Finite-time stability of discrete–time systems with time-varying delay", Chemical Industry and Chemical Engineering Quarterly, vol. 18, no 4/I, pp. 525-533, 2012. [Online]. Available: http://dx.doi.org/10.2298/CICEQ120126026S

L.L. Hou, G.D. Zong, Y.Q. Wu, "Finite-time control for discrete-time switched systems with time delay, International Journal of Control, Automation and Systems, vol. no. 4, pp. 855-860, 2012. [Online]. Available: http://dx.doi.org/10.1007/s12555-012-0424-3

D.Lj. Debeljkovic, S.B. Stojanovic, A.M. Jovanovic, "Finite-time stability of continuous time delay systems: Lyapunov-like approach with Jensen’s and Coppel’s inequality", Acta Polytechnica Hungarica, vol. 10, No. 7, pp. 135-150, 2013. [Online]. Available: http://dx.doi.org/10.12700/APH.10.07.2013.7.10

S.B. Stojanovic, D.Lj. Debeljkovic, “Delay-dependent stability of linear time delay systems: necessary and sufficient conditions", Dynamics of Continuous, Discrete and Impulsive Systems, Series B: Applications & Algorithms, vol. 16, no. 6, pp. 887-900, 2009. [Online]. Available: http://monotone.uwaterloo.ca/


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