Problem 8.3

Determining the least upper bound on

the achievable delay margin

Daniel E. Davison and Daniel E. Miller

Department of Electrical and Computer Engineering

University of Waterloo

Waterloo, Ontario N2L 3G1

Canada

[email protected] and [email protected]

1 MOTIVATION AND PROBLEM STATEMENT

Control engineers have had to deal with time delays in control processes for decades and, consequently, there is a huge literature on the topic, e.g., see [1] or [2] for collections of recent results. Delays arise from a variety of sources, including physical transport delay (e.g., in a rolling mill or in a chemical plant), signal transmission delay (e.g., in an earth-based satellite control system or in a system controlled over a network), and computational delay (e.g., in a system which uses image processing). The problems posed here are concerned in particular with systems where the time delay is not known exactly: such uncertainty exists, for example, in a rolling mill system where the physical speed of the process may change day-to-day, or in a satellite control system where the signal transmission time between earth and the satellite changes as the satellite moves, or in a control system implemented on the internet where the time delay is uncertain because of unknown traffic load on the network.

Motivated by the above examples, we focus here on the simplest problem that captures the difficulty of control in the face of uncertain delay. Specifically, consider the classical linear time-invariant (LTI) unity-feedback control system with a known controller and with a plant that is known except for an uncertain output delay. Denote the plant delay by , the plant transfer function by , and the controller by C(s). Assume the feedback system is internally stable when = 0. Let us define the delay margin (DM) to be the largest time delay such that, for any delay less than

Computation of DM(P₀, C) is straightforward. Indeed, the Nyquist stability criterion can be used to conclude that the delay margin is simply the phase margin of the undelayed system divided by the gain crossover frequency of the undelayed system. Other techniques for computing the delay margin for LTI systems have also been developed, e.g., see [3], [4], [5], and [6], just to name a few.

In contrast to the problem of computing the delay margin when the controller is known, the design of a controller to achieve a prespecified delay margin is not straightforward, except in the trivial case where the plant is open-loop stable, in which case the zero controller achieves DM(P₀, C) = To the best of the authors’ knowledge, there is no known technique for designing a controller to achieve a prespecified delay margin. Moreover, the fundamental question of whether or not there exists a finite upper bound on the delay margin that is achievable by a LTI controller has not even been addressed.

Hence, there are three unsolved problems:

Problem 1: Does there exist an (unstable) LTI plant, P0, for which there is a finite upper bound on the delay margin that is achievable by a LTI controller? In other words, does there exist a for which

Problem 2: If the answer to Problem 1 is affirmative, devise a computationally feasible algorithm that, given (s), computes to a given prescribed degree of accuracy.

Problem 3: If the answer to Problem 1 is affirmative, devise a computationally feasible algorithm that, given (s) and a value T in the range constructs a C(s) that satisfies

2 RELATED RESULTS

It is natural to attempt to use robust control methods to solve these problems (e.g., see [7] or [8]). That is, construct a plant uncertainty “ball” that includes all possible delayed plants, then design a controller to stabilize every plant within that ball. To the best of the authors’ knowledge, such techniques always introduce conservativeness, and therefore cannot be used to solve the problems stated above.

Alternatively, it has been established in the literature that there are upper bounds on the gain margin and phase margin if the plant has poles and zeros in the open right-half plane [9], [7]. These bounds are not conservative, but it is not obvious how to apply the same techniques to the delay margin problem.

As a final possibility, performance limitation integrals, such as those described in [10], may be useful, especially for solving Problem 1.

BIBLIOGRAPHY

[1] M. S. Mahmoud, Robust Control and Filtering for Time-Delay Systems, New York: Marcel Dekker, 2000.

[2] L. Dugard and E. Verriest, eds., Stability and Control of Time-Delay Systems, Springer-Verlag, 1997.

[3] J. Zhang, C. R. Knospe, and P. Tsiotras, “Stability of linear time-delay systems: A delay-dependent criterion with a tight conservatism bound, ” In: Proceedings of the American Control Conference, Chicago, Illinois, pp. 1458–1462, June 2000.

[4] J. Chen, G. Gu, and C. N. Nett, “A new method for computing delay margins for stability of linear delay systems, ” In: Proceedings of the 33rd Conference on Decision and Control, Lake Buena Vista, Florida, pp. 433–437, Dec. 1994.

[5] J. Chiasson, “A method for computing the interval of delay values for which a differential-delay system is stable, ” IEEE Transactions on Automatic Control, vol. 33, pp. 1176–1178, Dec. 1988.

[6] K. Walton and J. Marshall, “Direct method for TDS stability analysis, ” IEE Proceedings, Part D, vol. 134, pp. 101–107, Mar. 1987.

[7] J. C. Doyle, B. A. Francis, and A. R. Tannenbaum, Feedback Control Theory, New York, NY: Macmillan, 1992.

[8] K. Zhou, J. C. Doyle, and K. Glover, Robust and Optimal Control, Upper Saddle River, NJ: Prentice Hall, 1996.

[9] P. P. Khargonekar and A. Tannenbaum, “Non-euclidean metrics and the robust stabilization of systems with parameter uncertainty, ” IEEE Transactions on Automatic Control, vol. AC-30, pp. 1005–1013, Oct. 1985.

[10] J. Freudenberg, R. Middleton, and A. Stefanopoulou, “A survey of inherent design limitations, ” In: Proceedings of the American Control Conference, Chicago, Illinois, pp. 2987–3001, June 2000.