“Delay Scheduling”: A New Concept for Stabilization in Multiple Delay Systems

Abstract

A trajectory-tracking problem is considered for a linear time invariant (LTI) dynamics with a fixed control law. However, the feedback line is affected by multiple time delays. The stability of the dynamics becomes a complex problem. It is well known that time-delayed LTI systems may exhibit multiple stable operating zones (which we call pockets) in the space of the delays. Our aim in this paper is to locate and experimentally validate these pockets. For the analytical determination of the pockets we utilize a new methodology, the cluster treatment of characteristic roots (CTCR). The study results in several interesting conclusions. (i) The systems may exhibit better control performance (for instance, faster disturbance rejection) for larger time delays. (ii) Consequently, we propose a unique and interesting utilization of the time delays as agents to enhance the control performance, the delay scheduling technique.

Keywords

Delay scheduling multiple time delay system pendulum control stability

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References

Anderson, R. J. and Spong, M. W. , 1989, “Bilateral control of teleoperators with time delay,” IEEE Transactions on Automatic Control 34(5), 494-501 .

Campbell, S. A. , 1999, “Stability and bifurcation of a simple neural network with multiple time delays,” Fields Institute Communications 21, 65-79 .

Chen, J. et al. , 1995, “A new method for computing delay margins for stability of linear delay systems,” Systems and Control Letters 26, 107-117 .

Cooke, K. L. and van den Driessche, P. , 1986, “On zeros of some transcendental equations,” Funkcialaj Ekvacioj 29, 77-90 .

Engelborghs, K. , 2000, “DDE-BIFTOOL: a MATLAB package for bifurcation analysis of delay differential equations,” TW305, Department of Computer Science, Katholieke Universiteit Leuven, Leuven, Belgium .

Filipovic, D. and Olgac, N. , 2002, “Delayed resonator with speed feedback-design and performance analysis,” Mechatronics 12(3), 393-413 .

Grimble, M. J. , 1998, “Polynomial systems approach to continuous-time weighted optimal linear filtering and prediction,” IEEE Transactions on Signal Processing 46(11), 3123-3127 .

Hale, J. K. and Huang, W. , 1993, “Global geometry of the stable regions for two delay differential equations,” Journal of Mathematical Analysis and Applications 178, 344-362 .

Kalman, R. E. , 1960, “A new approach to linear filtering and prediction problems,” ASME Transactions, Journal of Basic Engineering 82D.

10.

Kang, B. S. et al. , 1999, “Robust tracking control of a direct drive robot,” ASME Journal of Dynamics, System, Measurement and Control 121(2), 261-269 .

11.

Koivo, A. J. and Repperger, D. W. , 1973, “On a game problem involving systems with time delay,” IEEE Transactions on Automatic Control 18, 149-152 .

12.

Masoud, Z. N. and Nayfeh, A. H. , 2003, “Sway reduction on container cranes using delayed feedback controller,” Nonlinear Dynamics 34, 347-358 .

13.

Niculescu, S.-I. , 2001, Delay Effects on Stability, Springer-Verlag, Berlin .

14.

Olgac, N. et al. , 1985, “The impact of costly observations and observation delay in stochastic optimal control problems,” International Journal of Control 41(3), 769-785 .

15.

Olgac, N. and Sipahi, R. , 2002, “An exact method for the stability analysis of time delayed LTI systems,” IEEE Transactions on Automatic Control 47(5), 793-797 .

16.

Rekasius, Z. V. , 1980, “A stability test for systems with delays,” in Proceedings of the Joint Automatic Control Conference, San Francisco, CA, August, Paper No. TP9-A.

17.

Sipahi, R. and Olgac, N. , 2003a, “Degenerate cases in using direct method,” Transaction of ASME, Journal of Dynamic Systems, Measurement, and Control 125(2), 194-201 .

18.

Sipahi, R. and Olgac, N. , 2003b, “Stability analysis of multiple time delayed systems using the direct method,” ASME IMECE 41495.

19.

Sipahi, R. and Olgac, N. , 2004, “A novel stability study on multiple time-delay systems using the root clustering paradigm,” American Control Conference, Boston, MA.

20.

Stepan, G. , 1989, Retarded Dynamical Systems: Stability and Characteristic Function, Wiley, New York .

21.

Subramanian, A. and Sayed, A. H. , 2004, “Multi-objective filter design for uncertain stochastic time-delay systems,” IEEE Transactions on Automatic Control 49(1), 149-154 .

22.

Thomopoulos, S. C. A. , 1994, “Decentralized filtering and control in the presence of delays - discrete-time and continuous-time case,” Information Sciences 81, 133-153 .

23.

Yang, J. M. and Kim, J. H. , 1999, “Sliding mode control for trajectory tracking of non-holonomic wheeled mobile robots,” IEEE Transactions on Robotics and Automation 15(3), 578-587 .