Are phase and time-delay margins always adversely affected by high-gain?

This paper addresses the important issue of robustness for design of feedback control systems. A well-known fact in robust control is that a high gain in the feedback loop leads to increased control effort and reduced phase margin. In an adaptive control framework, a high adaptive gain, implying fast adaptation, often leads to undesirable high-frequency oscillations in the control signal and sensitivity to time-delays. Since adaptive controllers are nonlinear, the notion of the phase margin cannot be defined for these architectures. A more generalized notion is the time-delay margin that can serve as a measure for the system robustness. In this paper, we consider a linear system under constant disturbance in the presence of some type of a high-gain in the feedback loop. We explore two different adaptive control architectures, conventional model reference adaptive control (MRAC) and an ℒ₁ adaptive controller. Since the closed-loop retains the linear structure, one can explicitly compute the corresponding gain and phase margins. We further consider the time-delay margin of these feedback structures and reveal that the classical definition of the time-delay margin does not hold for the ℒ₁ adaptive controller. Moreover, while the phase margin of the ℒ₁ adaptive controller is independent of the adaptation rate, its time-delay margin is guaranteed to be bounded away from zero as one increases the speed of adaptation. The message is twofold: first, the time-delay margin cannot be related to the phase margin straightforwardly and therefore constitutes a broader concept for measuring system robustness; secondly, high gain can improve robustness if it is internal to the controller computation block. For the sake of completeness, we present also two non-adaptive systems and generalize this phenomenon to a different class of feedback controllers.