Stabilization of linear systems with distributed input delay and input saturation. (English) Zbl 1348.93224

Summary: This paper is concerned with stabilization of a linear system with distributed input delay and input saturation. Both constant and time-varying delays are considered. In the case that the input delay is constant, under the stabilizability assumption on an auxiliary system, it is shown that the system can be stabilized by state feedback for an arbitrarily large delay as long as the open-loop system is not exponentially unstable. In the case that the input delay is time-varying, but bounded, it is shown that the system can be stabilized by state feedback if the non-asymptotically stable poles of the open-loop system are all located at the origin. In both cases, stabilizing controllers are explicitly constructed by utilizing the parametric Lyapunov equation based low gain design approach we recently developed. It is also shown that in the presence of actuator saturation and under the same assumptions on the system, these controllers achieve semi-global stabilization. Some discussions on the assumptions we impose on the system are given. A numerical example illustrates the effectiveness of the proposed stabilization approach.


93D15 Stabilization of systems by feedback
93C23 Control/observation systems governed by functional-differential equations


Full Text: DOI


[1] Artstein, Z., Linear systems with delayed controls: a reduction, IEEE transactions on automatic control, 27, 4, 869-879, (1982) · Zbl 0486.93011
[2] Barbalat, I., Système déquations différentielle d’oscillations non linéaires, Revue roumaine de mathématiques pures et appliquées, 4, 2, 267-270, (1959) · Zbl 0090.06601
[3] Engelborghs, K.; Luzyanina, T.; Samaey, G., DDE-BIFTOOL v.2.00: A Matlab package for bifurcation analysis of delay differential equation, Department of computer science, K.U. Leuven, T.W. reports, 330, (2001)
[4] Fiagbedzi, Y.A.; Pearson, A.E., A multistage reduction technique for feedback stabilizing distributed time-lag systems, Automatica, 23, 311-326, (1987) · Zbl 0629.93046
[5] Gao, H.; Chen, T.; Lam, J., A new delay system approach to network-based control, Automatica, 44, 1, 39-52, (2008) · Zbl 1138.93375
[6] Gu, K., An improved stability criterion for systems with distributed delays, International journal of robust nonlinear control, 13, 819-831, (2003) · Zbl 1039.93031
[7] Gu, K. (2000). An integral inequality in the stability problem of time-delay systems. In Proceedings of the 39th IEEE conference on decision and control (pp. 2805-2810). Sydney, Australia December.
[8] Gu, K.; Han, Q.L.; Luo, A.C.J.; Niculescu, S.I., Discretized Lyapunov functional for systems with distributed delay and piecewise constant coefficients, International journal of control, 74, 7, 737-744, (2001) · Zbl 1015.34061
[9] Hale, J.; Verduyn, L.S.M., Introduction to functional differential equations, (1993), Springer New York
[10] Hu, T.; Lin, Z., Control systems with actuator saturation: analysis and design, (2001), Birkhäuser Boston · Zbl 1061.93003
[11] Ichikawa, A., Null controllability with vanishing energy for discrete-time systems, Systems & control letters, 57, 34-38, (2008) · Zbl 1129.93322
[12] Kolmanovskii, V.B., & Richard, J.P. (1997). Stability of some systems with distributed delays. In JESA, special issue on ‘analysis and control of time-delay systems’ (pp. 971-982). Vol. 31.
[13] Lam, J.; Gao, H.; Wang, C., \(H_\infty\) model reduction of linear systems with distributed delay, IEE Proceedings-control theory and applications, 152, 6, 662-674, (2005)
[14] Lin, Z.; Fang, H., On asymptotic stabilizability of linear systems with delayed input, IEEE transactions on automatic control, 52, 6, 998-1013, (2007) · Zbl 1366.93581
[15] Lin, Z.; Saberi, A., Semiglobal exponential stabilization of linear systems subject to input saturation via linear feedbacks, Systems & control letters, 21, 3, 225-239, (1993) · Zbl 0788.93075
[16] MacDonald, N., ()
[17] Mazenc, F.; Mondié, S.; Niculescu, S.-I., Global stabilization of oscillators with bounded delayed input, Systems & control letters, 53, 5, 415-422, (2004) · Zbl 1157.93490
[18] Olbrot, A.W., Stabilizability, detectability, and spectrum assignment for linear autonomous systems with general time delays, IEEE transactions on automatic control, 23, 5, 887-890, (1978) · Zbl 0399.93008
[19] Ozbay, H., Bonnet, C., & Clairambault, J. (2008). Stability analysis of systems with distributed delays and applications to hematopoietic cell maturation dynamics. In Proceedings of the 47th IEEE conference on decision and control (pp. 2050-2055).
[20] Richard, J.P., Time-delay systems:an overview of some recent advances and open problems, Automatica, 39, 1667-1694, (2003) · Zbl 1145.93302
[21] Sussmann, H.J.; Sontag, E.D.; Yang, Y., A general result on the stabilization of linear systems using bounded controls, IEEE transactions on automatic control, 39, 12, 2411-2425, (1994) · Zbl 0811.93046
[22] Tarbouriech, S.; da Silva, J.M.G., Synthesis of controllers for continuous-time delay systems with saturating controls via LMI’s, IEEE transactions on automatic control, 45, 1, 105-111, (2000) · Zbl 0978.93062
[23] VanAssche, V., Dambrine, M., Lafay, J.F., Richard, J.P., &, J.P (1999). Some problems arising in the implementation of distributed-delay control laws. In Proceedings of the 38th IEEE conference on decision and control (pp. 4668-4672).
[24] Wu, M.; He, Y.; She, J.-H., New delay-dependent stability criteria and stabilizing method for neutral systems, IEEE transactions on automatic control, 49, 12, 2266-2271, (2004) · Zbl 1365.93358
[25] Xu, S.; Chen, T., An LMI approach to the \(H_\infty\) filter design for uncertain systems with distributed delays, IEEE transactions on circuits and systems — II: express briefs, 51, 4, 195-201, (2004)
[26] Ye, H.; Wang, H.; Wang, H., Stabilization of a PVTOL aircraft and an inertia wheel pendulum using saturation technique, IEEE transactions on control systems technology, 15, 6, 1143-1150, (2007)
[27] Zhou, B.; Duan, G.; Lin, Z., A parametric Lyapunov equation approach to the design of low gain feedback, IEEE transactions on automatic control, 53, 6, 1548-1554, (2008) · Zbl 1367.93553
[28] Zhou, B.; Li, Z.; Duan, G., On improving transient performance in global control of multiple integrators system by bounded feedback, Systems & control letters, 57, 20, 867-875, (2008) · Zbl 1162.93032
[29] Zhou, B.; Lin, Z.; Duan, G., Properties of the parametric Lyapunov equation based low gain design with applications in stabilization of time-delay systems, IEEE transactions on automatic control, 54, 7, 1698-1704, (2009) · Zbl 1367.93554
[30] Zhou, B.; Lin, Z.; Duan, G., Stabilization of linear systems with input delay and saturation — a parametric Lyapunov equation approach, International journal of robust and nonlinear control, 20, 1502-1519, (2010) · Zbl 1204.93098
[31] Zhou, B.; Lin, Z.; Duan, G., Global and semi-global stabilization of linear systems with multiple delays and saturations in the input, SIAM journal on control and optimization, 48, 8, 5294-5332, (2010) · Zbl 1218.34093
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