Delay Margin of Low-Order Systems Achievable by PID Controllers

• Published in: IEEE transactions on automatic control Vol. 64; no. 5; pp. 1958 - 1973
• Main Authors: Ma, Dan, Chen, Jie
• Format: Journal Article
• Language: English
• Published: New York IEEE 01.05.2019; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Controllers; Delay effects; Delay margin; Delays; Linear systems; PD control; PID controller; Poles and zeros; Process control; Proportional integral derivative; Robust stabilization; Robustness; Time lag; time-varying delay; uncertain time delay; Upper bounds
• ISSN: 0018-9286; 1558-2523
• DOI: 10.1109/TAC.2018.2853567

This paper concerns the delay margin achievable using proportional-integral-derivative (PID) controllers for linear time-invariant (LTI) systems subject to variable, unknown time delays. The basic issue under investigation addresses the question: What is the largest range of time delay so that there exists a single PID controller to stabilize the delay plants within the entire range? Delay margin is a fundamental measure of robust stabilization against uncertain time delays and poses a fundamental, longstanding problem that remains open except in simple, isolated cases. In this paper, we develop explicit expressions of the exact delay margin and its upper bounds achievable by a PID controller for low-order delay systems, notably the first- and second-order unstable systems with unknown constant and possibly time-varying delays. The effect of nonminimum phase zeros is also examined. PID controllers have been extensively used to control and regulate industrial processes that are typically modeled by first- and second-order dynamics. While furnishing the fundamental limits of delay within which a PID controller may robustly stabilize a delay process, our results should also provide useful guidelines in tuning PID parameters and in the analytical design of PID controllers.