Graphical Stabilization Approach for Unstable First Order Linear Systems with Time Delay

Abstract

In this paper, a graphical stabilization approach is proposed and analyzed for a class of unstable first order linear systems with time delay. We first show that the control designs based on time invariant models are unable to guarantee stability and asymptotic tracking for unstable first order linear systems in general case. So, the condition stability is analysed graphically by computing the first derivative and plotting the graph of a function with precision; the first derivative allows us to determine the critical points and several conditions of stability. Therefore, it’s important to note that the method can guarantee the existence of a proportional gain to ensure the stability of the closed-loop system such that the time delay is small relatively to the time constant. Finally, a numerical example illustrates the efficiency and performances of the proposed approach.

Keywords

• asymptotic stability,linear system,output feedback,proportional gain,time delay

References

1. S. Xu and T. Chen, “Robust H∞ control for uncertain stochastic systems with state delay”, IEEE Transactions on Automatic Control, 47(12): pp. 2089–2094, 2002.
2. V. Kharitonov and A. Zhabko “Lyapunov-Krasovskii approach to the robust stability analysis of time delay systems”, Automatica, 39: pp. 15–20, 2003.
3. C. Hwang and JH. Hwang, “Stabilization of first order plus dead time unstable processes using PID controllers”, IEEE Proc. Control Theory and Applications, 2004, 151(1): pp. 89–94.
4. T. Liu, W. Zhang and D. Gu, “Analytical design of two degree of freedom control scheme for open loop unstable processes with time delay”, Journal of Process Control, 2005, 5(15): pp. 559–572.
5. J. E. Marshall, H. Gorecki, K. Walton and A. Korytowski, “Time-delay Systems, stability and performance criteria with applications”, Ellis Horwood, 1992.
6. G. Abdallah, P. Dorato, J. Benitez-Read, and R. Byrne, “Delayed positive feedback can stabilize oscillatory systems”, ACC’93, American Control Conference, pp. 3106–3107, 1993.
7. B. Del-Muro-Cuellar, JF. Marquez-Rubio, M. Velasco-Villa and J. Alvarez-Ramirez, “On the Control of Unstable First Order Linear Systems with Large Time Lag: Observer Based Approach”, European Journal of Control, 2012; 18(5): pp. 439–451.
8. D. Peaucelle, D. Henrion and D. Arzelier, “Quadratic separation for feedback connection of an uncertain matrix and an implicit linear transformation”, 16th IFAC World Congress, Prague, Czech Republic, July 2005.

9. L. Orihuela, P. Millan, C. Vivas and F. Rubio, “Delay-dependent robust stability analysis for systems with interval delays”, 2010 American Control Conference, Marriott Waterfront, Baltimore, MD, USA, June 30-July 02, 2010, pp. 4993–4998.