Optimal and robust control for linear state-delay systems. (English) Zbl 1119.49021

Summary: This paper presents the optimal regulator for a linear system with state delay and a quadratic criterion. The optimal regulator equations are obtained using the maximum principle. Performance of the obtained optimal regulator is verified in the illustrative example against the best linear regulator available for linear systems without delays. Simulation graphs demonstrating better performance of the obtained optimal regulator are included. The paper then presents a robustification algorithm for the obtained optimal regulator based on integral sliding mode compensation of disturbances. The general principles of the integral sliding mode compensator design are modified to yield the basic control algorithm oriented to time-delay systems, which is then applied to robustify the optimal regulator. As a result, the sliding mode compensating control leading to suppression of the disturbances from the initial time moment is designed. The obtained robust control algorithm is verified by simulations in the illustrative example.


49K15 Optimality conditions for problems involving ordinary differential equations
49K40 Sensitivity, stability, well-posedness
93C05 Linear systems in control theory
Full Text: DOI


[1] Kwakernaak, H.; Sivan, R., Linear optimal control systems, (1972), Wiley-Interscience New York · Zbl 0276.93001
[2] Fleming, W.H.; Rishel, R.W., Deterministic and stochastic optimal control, (1975), Springer New York · Zbl 0323.49001
[3] Basin, M.V.; Rodriguez-Gonzalez, J.; Martinez-Zuniga, R., Optimal control for linear systems with time delay in control input, J. franklin inst., 341, 267-278, (2004) · Zbl 1073.93055
[4] Basin, M.V.; Rodriguez-Gonzalez, J., A closed-form optimal control for linear systems with equal state and input delays, Automatica, 41, 915-921, (2005) · Zbl 1087.49028
[5] Basin, M.V.; Fridman, L.M.; Acosta, P.; Rodriguez-Gonzalez, J., Robust integral sliding mode regulator for linear stochastic time-delay systems, Int. J. robust nonlinear control, 15, 407-421, (2005) · Zbl 1100.93012
[6] Malek-Zavarei, M.; Jamshidi, M., Time-delay systems: analysis, optimization and applications, (1987), North-Holland Amsterdam · Zbl 0658.93001
[7] Kolmanovskii, V.B.; Shaikhet, L.E., Control of systems with aftereffect, (1996), American Mathematical Society Providence
[8] J.L. Dugard, E.I. Verriest (Eds.), Stability and Control of Time-Delay Systems, Springer, Berlin, 1998.
[9] Kolmanovskii, V.B.; Myshkis, A.D., Introduction to the theory and applications of functional differential equations, (1999), Kluwer New York · Zbl 0907.39012
[10] J.M. Dion, J.L. Dugard, M. Fliess (Eds.), Linear Time Delay Systems, Pergamon, London, 1999.
[11] Mahmoud, M.S., Robust control and filtering for time-delay systems, (2000), Marcel Dekker New York · Zbl 0969.93002
[12] Niculescu, S., Delay effects on stability: A robust control approach, (2001), Springer Heidelberg · Zbl 0997.93001
[13] Boukas, E.-K.; Liu, Z.-K., Deterministic and stochastic time-delay systems, (2002), Birkhauser Boston · Zbl 1052.93059
[14] Gu, K.; Niculescu, S.-I., Survey on recent results in the stability and control of time-delay systems, ASME trans. J. dyn. syst. meas. control, 125, 158-165, (2003)
[15] Richard, J.-P., Time-delay systems: an overview of some recent advances and open problems, Automatica, 39, 1667-1694, (2003) · Zbl 1145.93302
[16] Oguztoreli, M.N., Time-lag control systems, (1966), Academic Press New York · Zbl 0143.12101
[17] Eller, D.H.; Aggarwal, J.K.; Banks, H.T., Optimal control of linear time-delay systems, IEEE trans. autom. control, AC-14, 678-687, (1969)
[18] Alekal, Y.; Brunovsky, P.; Chyung, D.H.; Lee, E.B., The quadratic problem for systems with time delays, IEEE trans. autom. control, AC-16, 673-687, (1971)
[19] Delfour, M.C., The linear quadratic control problem with delays in space and control variables: a state space approach, SIAM. J. control optim., 24, 835-883, (1986) · Zbl 0606.93037
[20] Uchida, K.; Shimemura, E.; Kubo, T.; Abe, N., The linear-quadratic optimal control approach to feedback control design for systems with delay, Automatica, 24, 773-780, (1988) · Zbl 0659.93028
[21] Pontryagin, L.S.; Boltyanskii, V.G.; Gamkrelidze, R.V.; Mishchenko, E.F., The mathematical theory of optimal processes, (1962), Interscience New York · Zbl 0102.32001
[22] Kharatashvili, G.L., A maximum principle in external problems with delays, ()
[23] Utkin, V.I.; Guldner, J.; Shi, J., Sliding mode control in electromechanical systems, (1999), Taylor and Francis London
[24] W.-J. Cao, J.-X. Xu, Nonlinear integral type sliding surfaces for both matched and unmatched uncertain systems, in: Proceedings of the American Control Conference 2001, Arlington, VA, June 2001, pp. 4369-4374.
[25] Shtessel, Y.B.; Zinober, A.S.I.; Shkolnikov, I., Sliding mode control for nonlinear systems with output delay via method of stable system center, ASME trans. J. dyn. syst. meas. control, 125, 253-257, (2003)
[26] Poznyak, A.S.; Shtessel, Y.B.; Gallegos, C.J., Min – max sliding-mode control for multimodel linear time varying systems, IEEE trans. autom. control, AC-48, 2141-2150, (2003) · Zbl 1364.93128
[27] Basin, M.V.; Rodriguez-Gonzalez, J.G.; Martinez-Zuniga, R., Optimal filtering for linear state delay systems, IEEE trans. autom. control, AC-50, 684-690, (2005) · Zbl 1365.93496
[28] Basin, M.V.; Rodriguez-Gonzalez, J.G.; Martinez-Zuniga, R., Optimal controller for linear systems with time delays in input and observations, Dyn. continuous discrete impulsive syst., 12B, 1-11, (2005) · Zbl 1106.93055
[29] Filippov, A.F., Differential equations with discontinuous right-hand sides, (1989), Kluwer New York · Zbl 1098.34006
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