Delay Margin Achievable by PID Control

PID 控制可達致的時滯裕度

Student thesis: Doctoral Thesis

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Award date5 Nov 2019

Abstract

The thesis contributes to delay robustness of PID controllers in stabilizing systems containing uncertain and varying delays.

We first consider second-order systems and seek analytical characterization and exact computation of the PID delay margin. The PID delay margin refers to the maximal range of delay values within which the system can be robustly stabilized by a PID controller. Our primary contribution is threefold. First, we show that the delay margin achieved by PID control coincides with that by PD controllers. Second, we show that the proportional control contributes no action to increase the delay margin. Finally, we show that the PID delay margin can be computed efficiently by solving a unimodal problem, that is, a univariate convex optimization problem that admits a unique maximum. As such, from a computational standpoint, the PID delay margin problem can be considered resolved. Additionally, we provide the a priori upper and lower bounds of the corresponding PID delay margins, which provide an intrinsic bound-independent PID controller design, and in turn an estimate that can be used to guide numerical optimization.

Next, we demonstrate analytically the tradeoff between achieving delay margin and asymptotic tracking, showing that integral control actually reduces the delay mar- gin. Except for the tradeoff between achieving delay margin and asymptotic tracking, the tradeoff between achieving delay margin and the consideration of disturbance and noise rejection has also been investigated and the PID delay margin can be actually enlarged with no performance implication.

In the sequel, the thesis studies the robust consensus problems for continuous-time first-order multi-agent systems (MAS) with delays. We consider dynamic output feed- back control protocols in the form of PID control and seek to determine the delay consensus margin achievable by PID feedback protocols; the delay consensus margin is a robustness measure that defines the maximal range of delay within which robust consensus can be achieved despite the variation and uncertainty in the delay. Both undirected and directed network communication graphs are considered. The delay consensus margin in general constitutes a non-smooth max-min problem. In an appealing discovery, we find that the delay consensus margin achieved by PID protocols coincides with that by PD protocols, both for undirected and directed graphs. With an undirected graph, we show that this delay consensus margin can be found by solving a unimodal concave optimization problem, and with a directed graph, it can be computed approximately via an iterative algorithm each step of which amounts to solving a unimodal quasi-concave problem. Both cases can then be solved based on convex optimization or gradient-based numerical methods. The results show how the agent dynamics and graph connectivity may fundamentally limit the range of delay tolerance, so that consensus can or cannot be maintained with the variation of the delay.

All results in this thesis not only ensure that the PID delay margin problem be readily solvable, but also provide fundamental conceptual insights into the PID control of delay systems, and analytical justifications to long-held engineering intuitions and heuristics, thus lending useful guidelines in the tuning and analytical design of PID controllers.