The behavior of the rightmost characteristic roots of linear time-delay systems as a function of the delay parameter is studied for small values of a gain parameter. The results explain the qualitative behavior of the stability regions in the delay parameter space of oscillatory systems subjected to large delays, and shed new light on the corresponding stabilization problem. In particular, for a chain of multiple oscillators the determination of a stabilizing value of the delay parameter is interpreted as a phase synchronization problem. The results are illustrated with the output feedback stabilization of an oscillator and the stability analysis of a gyroscopic system.
Abdallah, C., Dorato, P., Benitez-Read, J., and Byrne, R., 1993, "Delayed positive feedback can stabilize oscillatory systems ," in Proceedings of the 1993 American Control Conference, IEEE, San Francisco, CA, pp. 3106- 3310.
2.
Atay, F., 2003, "Distributed delays facilitate amplitude death of coupled oscillators," Physical Review Letters91, 094101.
3.
Atay, F., 2006, "Oscillator death in coupled functional differential equations near hopf bifurcation," Journal of Differential Equations221, 190-209.
4.
Bellman, R. and Cooke, K., 1963, Differential-Difference Equations, Academic Press, New York.
5.
Cooke, K. and van den Driessche, P. , 1986, "On zeroes of some transcendental equations," Funkcialaj Ekvacioj29, 77-90.
6.
Das, S. and Chatterjee, A., 2005, "Second order multiple scales for oscillators with large delays," Nonlinear Dynamics39, 375-394.
7.
Engelborghs, K., Luzyanina, T., and Samaey, G., 2001, "DDE-BIFTOOL v. 2.00: a Matlab package for bifurcation analysis of delay differential equations," TW Report 330, Department of Computer Science, Katholieke Universiteit Leuven, Belgium.
8.
Freitas, P., 2000, "Delay-induced instabilities in gyroscopic systems ," SIAM Journal on Control and Optimization39(1), 196-207.
9.
Gu, K., Niculescu, S.-I., and Chen, J., 2005, "On stability of crossing curves for general systems with two delays," Journal of Mathematical Analysis and Applications311(1), 231-253.
10.
Haegeman, B., Engelborghs, K., Roose, D., Pieroux, D., and Erneux, T., 2002, "Stability and rupture of bifurcation bridges in semiconductor lasers," Physical Review E66, 046216.
11.
Hardy, G. and Wright, E., 1968, An Introduction to the Theory of Numbers, Oxford University Press.
12.
Heil, T., Fischer, I., Elsäßer, W., Krauskopf, B., Green, K., and Gavrielides, A., 2003, "Delay dyanamics of semiconductor lasers with short external cavities: bifurcation scenarios and mechanisms," Physical Review E67, 066214.
13.
Kharitonov, V., Niculescu, S.-I., Moreno, J., and Michiels, W., 2005, "Static output feedback stabilization: necessary conditions for multiple delay controllers," IEEE Transactions on Automatic Control50(1), 82-86.
14.
Kokame, H., Hirata, K., Konishi, K., and Mori, T., 2001, "Difference feedback can stabilize unstable steady states ," IEEE Transactions on Automatic Control46(12), 1908-1913.
15.
Kolmanovskii, V. and Myshkis, A., 1999, Introduction to the Theory and Applications of Functional Differential Equations (Mathematics and its Applications, volume 463), Kluwer Academic Publishers, Dortrecht.
16.
Krantz, S., 1999, Handbook of Complex Variables, Birkhäuser , Boston.
17.
Manitius, A. and Olbrot, A., 1979, "Finite spectrum assignment problem for systems with delays ," IEEE Transactions on Automatic Control24(4), 541-553.
18.
Mazenc, F., Mondié, S., and Niculescu, S.-I. , 2004, "Global stabilization of oscillators with bounded delayed input," Systems & Control Letters53(5), 415-422.
19.
Michiels, W., Niculescu, S.-I., and Moreau, L., 2004, "Using delays and time-varying gains to improve the output feedback stabilizability of linear systems: a comparison," IMA Journal of Mathematical Control and Information21(4), 393-418.
20.
Niculescu, S.-I., 2001, Delay Effects on Stability. A Robust Control Approach (Lecture Notes in Control and Information Sciences, volume 269), Springer, Berlin.
21.
Niculescu, S.-I. and Michiels, W., 2004, "Stabilizing a chain of integrators using multiple delays ," IEEE Transactions on Automatic Control49(5), 802-807.
22.
Niculescu, S.-I., Michiels, W., Gu, K., and Abdallah, C.T., 2009, "Delay effects on output feedback control of dynamical systems," to appear in Complex Time-Delay Systems (Lecture Notes on Control and Information Sciences), Atay, F. M. (ed), Springer, Berlin.
23.
Olgac, N., Ergenc, A., and Sipahi, R., 2005, "‘Delay Scheduling’, stabilization in multiple-delay systems," Journal of Vibration and Control11, 1159-1172.
24.
Pyragas, K., 1992, "Continuous control of chaos by self-controlling feedback ," Physics Letters A170, 421-428.
25.
Seydel, R., 1994, Practical Bifurcation and Stability Analysis: From Equilibrium to Chaos (Interdisciplinary Applied Mathematics, volume 5) , Springer, Berlin.
26.
Sipahi, R. and Olgac, N., 2005, "Complete stability robustness of third-order LTI multiple time-delay systems," Automatica41(8), 1413-1422.
27.
Smith, O.J., 1957, "Closer control of loops with dead time," Chemical Engineering Progress53, 217-219.
28.
Stépán, G., 1989, Retarded Dynamical Systems: Stability and Characteristic Function (Research Notes in Mathematics, volume 210), Longman Scientific, London.
29.
Wahi, P. and Chatterjee, A., 2005, "Asymptotics for the characteristic roots of delayed dynamic systems," Journal of Applied Mechanics72, 475-483.
30.
Wolfrum, M. and Yanchuk, S., 2006, "Eckhaus instability in systems with large delays ," Physical Review Letters96, 220201.
31.
Yanchuk, S., Wolfrum, M., Hövel, P., and Schöll, E., 2006, "Control of unstable steady states by long delay feedback ," Physical Review E74, 026201.
