On stability of linear and weakly nonlinear switched systems with time delay. (English) Zbl 1187.34067

Summary: This paper studies time-delayed switched systems that include both stable and unstable modes. By using multiple Lyapunov-functions technique and a dwell-time approach, several criteria on exponential stability for both linear and nonlinear systems are established. It is shown that by suitably controlling the switching between the stable and unstable modes, exponential stabilization of the switched system can be achieved. Some examples and numerical simulations are provided to illustrate our results.


34D20 Stability of solutions to ordinary differential equations
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[1] Bellen, A.; Zennaro, M., Numerical methods for delay differential equations, (2003), Oxford University Press New York · Zbl 0749.65042
[2] Bellman, R.; Cooke, K.L., Differential-difference equations, (1963), Academic Press New York · Zbl 0118.08201
[3] Dayawansa, W.P.; Martin, C.F., A converse Lyapunov theorem for a class of dynamical systems which undergo switching, IEEE trans. automat. control, 44, 4, 751-759, (1999) · Zbl 0960.93046
[4] Driver, R.D., Ordinary and delay differential equations, (1977), Springer-Verlag New York Inc. · Zbl 0374.34001
[5] El’sgol’ts, L.E.; Norkin, S.B., Introduction to the theory and application of differential equations with deviating arguments, (1973), Academic Press New York · Zbl 0287.34073
[6] Gopalsamy, K., Stability and oscillations in delay differential equations of population dynamics, (1992), Kluwer Academic Publishers Netherlands · Zbl 0752.34039
[7] Halanay, A., Differential equations: stability, oscillations, time lags, (1966), Academic Press, Inc. New York · Zbl 0144.08701
[8] Hale, J.K.; Lunel, S.M.V., Introduction to functional differential equations, (1993), Springer-Verlag New York
[9] J.P. Hespanha, A.S. Morse, Stability of switched systems with average dwell-time, in: Proc. of the 38th IEEE Conf. on Decision and Control, Phoenix, AR, 1999, pp. 2655-2660
[10] B. Hu, X. Xu, A.N. Michel, P.J. Antsaklis, Stability analysis for a class of nonlinear switched systems, in: Proc. of the 38th IEEE Conf. on Decision and Control, Phoenix, AR, 1999, pp. 4374-4379
[11] Kim, S.; Campbell, S.A.; Liu, X.Z., Stability of a class of linear switching systems with time delay, IEEE trans. circuits systems I, 53, 2, 384-393, (1999)
[12] Krasovskii, N.N.; Brenner, J.L., Stability of motion: applications of lyapunov’s second method to differential systems and equations with delay, (1963), Stanford University Press Stanford, CA · Zbl 0109.06001
[13] Kuang, Y., Delay differential equations with applications in population dynamics, (1993), Academic Press Inc. CA · Zbl 0777.34002
[14] MacDonald, N., Biological delay systems: linear stability theory, (1989), Cambridge University Press Cambridge · Zbl 0669.92001
[15] Minorsky, N., Self-excited oscillations in dynamical systems, J. appl. mech., 9, 65-71, (1942)
[16] Morse, A.S., Supervisory control of families of linear set-point controllers-part I: exact matching, IEEE trans. automat. control, 41, 10, 1413-1431, (1996) · Zbl 0872.93009
[17] ()
[18] X.-M. Sun, G.M. Mimirovski, J. Zhao, W. Wang, Exponential stability for switched delay systems based on average dwell time technique and Lyapunov function method. in: Proc. of the 2006 American Control Conference, Minneapolis, Minnesota, USA, June 14-16, 2006, pp. 1539-1543
[19] Wang, R.; Liu, X.; Guan, Z., Robustness and stability analysis for a class of nonlinear switched systems with impulse effects, Dynam. syst. appl., 14, 233-248, (2004) · Zbl 1093.34026
[20] S. Yang, X. Zhengrong, C. Qingwei, H. Weili, Dynamical output feedback control of discrete switched system with time delay in: 2004 IEEE, Proc. of the 5th World Congress on Intelligent Control and Automation, Hangzhou, China, 2004, pp. 1088-1091
[21] M. Žefran, J.W. Burdick, Design of switching controllers for systems with changing dynamics, in: Proc. of the 37th IEEE Conf. on Decision and Control, Tampa, FL, 1998, pp. 2113-2118
[22] Y. Zhang, Stability of hybrid systems with time delay, Ph.D. Thesis, University of Waterloo, Ontario, Canada, 2004
[23] Zhai, G.; Hu, B.; Yasuda, K.; Michel, A.N., Stability analysis of switched systems with stable and unstable subsystems: an average Dwell time approach, Internat. J. systems sci., 32, 8, 1055-1061, (2001) · Zbl 1022.93043
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