Applications of stability criteria to time delay systems

Stability and stabilization of time delay systems (even of the linear ones) is again in the mainstream of the research. A most recent example is the stability analysis of feedback control loops containing a rst order controlled object with pure delay and a standard PID controller, thus generating a system with a second degree quasi-polynomial as characteristic equation. Since the classical memoir of Cebotarev and Meiman (1949) up to the more recent monographs by Stepan (1989) and G orecki et al (1989) several approaches to this problem have been given, aiming to nd the most complete Routh{Hurwitz type conditions for this case. In fact the main problem is here a missing case in the original memoir of Cebotarev and Meiman and its signicance within the framework of the most recent analysis of G orecki et al. The present paper aims to a fairly complete analysis of the problem combined with some hints for the nonlinear case (Aizerman problem). State feedback stabilization based on Artstein reduction of a system with input delay to a system without delay is also considered.

[1]  Shankar P. Bhattacharyya,et al.  New results on the synthesis of PID controllers , 2002, IEEE Trans. Autom. Control..

[2]  Henryk Górecki,et al.  Analysis and Synthesis of Time Delay Systems , 1989 .

[3]  Shankar P. Bhattacharyya,et al.  Stabilization of first-order systems with time delay using the PID controller , 2001, Proceedings of the 2001 American Control Conference. (Cat. No.01CH37148).

[4]  Shankar P. Bhattacharyya,et al.  On the stability and controller robustness of some popular PID tuning rules , 2003, IEEE Trans. Autom. Control..

[5]  Jack K. Hale,et al.  Introduction to Functional Differential Equations , 1993, Applied Mathematical Sciences.

[6]  Anand Pillay,et al.  Introduction to stability theory , 1983, Oxford logic guides.

[7]  Dan Popescu,et al.  Improved dynamic properties by feedback for systems with delay in control , 2002, Analysis and Optimization of Differential Systems.

[8]  G. Stépán.  Retarded dynamical systems : stability and characteristic functions , 1989 .

[9]  Dan Popescu,et al.  Control of Systems with Input Delay—An Elementary Approach , 2004 .

[10]  A. Halanay,et al.  General theory of linear hybrid control , 1977 .

[11]  F. Zheng,et al.  Feedback stabilization of linear systems with distributed delays in state and control variables , 1994, IEEE Trans. Autom. Control..

[12]  Z. Artstein.  Linear systems with delayed controls: A reduction , 1982 .

[13]  Alexander P. Seyranian,et al.  Introduction to Stability Theory , 2003 .

[14]  Shankar P. Bhattacharyya,et al.  PI stabilization of first-order systems with time delay , 2001, Autom..