This paper focuses on the study and the characterization of stability regions of discrete time systems with a time varying state delay subjected to actuator saturation through anti-windup strategies. Delay-dependent stability conditions are stated in the local as well as global context. An optimization procedure to maximize the estimate of domain of attraction is given. The proposed technique is illustrated by means of numerical examples.
J. Richard, “Time-Delay System: An Overview of Some Recent Advances and Open Problems,” Automatica, Vol. 39, No. 10, 2003, pp. 1667-1697. doi:10.1016/S0005-1098(03)00167-5
U. Shaked, I. Yaesh and C. E. DeSouza, “Bounded Real Criteria for Linear Time Delay Systems,” IEEE Transactions on Automatic Control, Vol. 43, No. 7, 1998, pp. 1016-1022. doi:10.1109/9.701117
H. H. Choi and M. J. Chung, “An LMI Approach to H∞ Controller Design for Linear Time Delay Systems,” Automatica, Vol. 33, No. 4, 1997, pp. 737-739. doi:10.1016/S0005-1098(96)00242-7
E. Fridman, A. Pila and U. Shaked, “Regional Stabilization and H∞ Control of Time-Delay Systems with Saturating Actuators,” International Journal of Robust and & Nonlinear Control, Vol. 13, No. 29, 2003, pp. 885-907. doi:10.1002/rnc.852
B. Lehman and K. Shujaee, “Delay Independent Stability Conditions and Decay Estimates for Time Varying Functional Differential Equations,” IEEE Transactions on Automatic Control, Vol. 39, No. 8, 1994, pp. 1673-1676. doi:10.1109/9.310048
M. S. Mahmooud, “Robust Control and Filtering for Timedelay System,” Maecel-Dekker, New York, 2000.
S. F. Chen, “Asymptotic Stability of Discrete-Time System with Time-Varying Delay Subject to Saturation Nonlinearities,” Chaos, Solitons and Fractals, Vol. 42, No. 2, 2009, pp. 1251-1257. doi:10.1016/j.chaos.2009.03.026
S.-I. Niculescu, “Delay Effects on Stability. A Robust Control Approach,” Springer-Verlag, Berlin, 2000.
V. L. Kharitonov and S.-I. Niculescu, “On the Stability of Linear Systems with Uncertain Delay,” IEEE Transactions on Automatic Control, Vol. 48, No. 1, 2003, pp. 127132. doi:10.1109/TAC.2002.806665
K. Gu and S.-I. Niculescu, “Survey on Recent Results in the Stability and Control of Time Delay Systems,” Journal of Dynamic Systems, Measurement and Control, Vol. 125, No. 2, 2003, p. 158.
V. K. R. Kandanvli and H. Kar, “An LMI Condition for Robust Stability of Discrete-Time State-Delayed System Using Quantization/Overflow Nonlinearities,” Signal Processing, Vol. 89, No. 11, 2009, pp. 2092-2102. doi:10.1016/j.sigpro.2009.04.024
V. K. R. Kandanvli and H. Kar, “Robust Stability of Discrete-Time State-Delayed Systems with Saturation Nonlinearities; Linear Matrix Inequality Approach,” Signal Processing, Vol. 89, No. 2, 2009, pp. 161-173. doi:10.1016/j.sigpro.2008.07.020
V. K. R. Kandanvli and H. Kar, “Delay-Dependent LMI Condition for Global Asymptotic Stability of Discrete Time Uncertain State Delayed Systems Using Quantization/Overflow Nonlinearities,” International Journal of Robust Nonlinear Control, Vol. 21, No. 14, 2011, pp. 1611-1622. doi:10.1002/rnc.1654
S. Oucherih, “Global Stabilization of a Class of Linear Continuous Time Delay System with Saturating Controls,” IEEE Transactions on Circuit and System I: Fundamental Theory and Applications, Vol. 43, No. 12, 1996, pp. 1012-1015.
S. Tarbourieh and J. M. Gomes da Silva Jr., “Synthesis of Controllers for Continuous-Time Delay Systems with Saturating Control via LMI,” IEEE Transactions on Automatic Control, Vol. 45, No. 1, 2000, pp. 105-111. doi:10.1109/9.827364
Y. Y. Cao, Z. L. Lin and T. Hu, “Stability Analysis of Linear Time-Delay Systems Subject to Input Saturation,” IEEE Transactions on Circuits and System (II), Vol. 49, No. 2, 2002, pp. 233-240. doi:10.1109/81.983870
O. Said, “Synthesis of Controllers for Time-Delay Systems Subject to Actuator Saturation and Disturbance,” Journal of Dynamic Systems Measurement, and Control, Vol. 125, No. 2, 2003, pp. 244-249. doi:10.1115/1.1570450
E. Tissir and A. Hmamed, “Further Results on The stabilization of Time Delay Systems Containing Saturating Actuators,” International Journal of Systems Science, Vol. 23, No. 4, 1992, pp. 615-622.
J. M. Gomes da Silva Jr. and S. Tarbouriech, “Using Anti-Windup Loops for Enlarging the Stability Regions of Time-Delay Systems Subject to Input Saturation,” Proceedings of the 2004 American Control Conference, Boston, 30 June-2 July 2004, pp. 4819-4824.
W. Y. Qiang, Y. Y. Cao and Y. X. Sun, “Anti-Windup Compensator Gain Design for Time Delay Systems with Constraints,” Acta Automatica Sinica, Vol. 32, No. 1, 2006, pp. 1-8.
S. Tarbouriech, J. M. Gomes Da Silva Jr. and G. Garcia, “Delay-Dependent Anti-Windup Loops for Enlarging the Stability Region of Time Delay Systems with Saturating Inputs,” ASME Journal of Dynamic Systems, Measurment, and Control, Vol. 125, No. 2, 2003, pp. 265-267. doi:10.1115/1.1569953
J. M. Gomes Da Silva Jr. and S. Tarbouriech, “AntiWindup Design with Guaranteed Regions of Stability: An LMI-Based Approach,” IEEE Transactions on Automatic Control, Vol. 50, No. 1, 2005, pp. 106-111. doi:10.1109/TAC.2004.841128
L. Zhang, E. K. Boukas and A. Haidar, “Delay Range Dependent Control Synthesis for Time Delay Systems with Actuator Saturation,” Automatica, Vol. 44, No. 10, 2008, pp. 2691-2695. doi:10.1016/j.automatica.2008.03.009
A. Ahmed, M. Rehan and N. Iqbal, “Delay-Dependent Anti-Windup Synthesis for Stability of Constrained State Delay Systems Using Pole-Constraints,” ISA Transactions, Vol. 50, No. 2, 2011, pp. 249-255. doi:10.1016/j.isatra.2010.11.003
J. M. Gomes da Silva Jr. and S. Tarbouriech, “AntiWindup Design with Guaranteed Regions of Stability for Discrete Time Linear Systems,” System and Control Letters, Vol. 55, No. 3, 2006, pp. 184-192. doi:10.1016/j.sysconle.2005.07.001
Y. He, M. Wu, G. P. Liu and J. H. She, “Output Feedback Stabilization for a Discrete-Time System with a TimeVarying Delay,” IEEE Transactions on Automatic Control, Vol. 53, No. 10, 2008, pp. 2372-2377. doi:10.1109/TAC.2008.2007522
K. F. Chen and I. K. Fong, “Stability Analysis and Output-Feedback Stabilization of Discrete Time Systems with an Interval Time-Varying State Delay,” IET Control Theory and Application, Vol. 4, No. 4, 2010, pp. 563-572. doi:10.1049/iet-cta.2009.0100
P. Gahinet, A. Nemirovski, A. J. Laub and M. Chilali, “LMI Control Toolbox—For Use with MATLAB,” MATH Works Inc., Natic, 1995.
S. Boyd, L. EI-Ghaousi, E. Feron and V. Balakrishnan, “Linear Matrix Inequalities in Systems and Control Theory,” SIAM, Philadelphia, 1994. doi:10.1137/1.9781611970777
X. Meng, J. Lam, B. Du and H. Gao, “A Delay Partitioning Approach to the Stability Analysis of Discrete Time Systems,” Automatica, Vol. 46, No. 3, 2010, pp. 610-614. doi:10.1016/j.automatica.2009.12.004