Revisiting the Mit Rule for Adaptive Control

Abstract

The MIT rule is a scalar parameter adjustment law which was proposed in 1961 for the model reference adaptive control of linear systems modeled as the cascade of a known stable plant and a single unknown gain. This adjustment law was derived by approximating a gradient descent procedure for an integral error squared performance criterion. For the early part of the 1960s this rule was the basis of many adaptive control schemes and a considerable wealth of practical experience and engineering folklore was amassed.The MIT rule is in general not globally convergent nor stable but has a performance determined by several factors such as algorithm gain, reference input magnitude and frequency, and the particular transfer function appearing in the cascade. These restrictions on the MIT rule slowly came to be discerned through experimentation and simulation but effectively were without theoretical support until some novel algorithm modifications and stability analysis, so-called Lyapunov redesign, due to Parks. Our aim in this paper is to pursue a theoretical analysis of the original MIT rule to support the existing simulation evidence and to indicate mechanisms for treating questions of robustness of MIT-rule-based adaptive controllers with undermodelling effects.The techniques that we apply to this problem centre on root locus methods, Nyquist methods and the application of the theory of averaging. Stability and instability results are presented and, using pertinent theories for different regimes of the gain-frequency plane, we approximate the experimentally derived stability margins, but for a broader signal class than simply periodic inputs. The mechanisms of instability and stability for these adaptive systems are highlighted and allow us to enunciate guidelines for the MIT rule to work. It is a pleasing by-product of this theoretical analysis that these guidelines coincide to a large degree with those advanced in earlier times on experimental and heuristic grounds.