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 timedelay 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 nonadaptive systems and generalize this phenomenon to a different class of feedback controllers.