Robustness Issues with Adaptive Control

Abstract

This chapter discusses limitations and weaknesses of model-reference adaptive control. Parameter drift is the result of the lack of a mathematical guarantee of boundedness of adaptive parameters. Systems with bounded external disturbances under feedback control actions using model-reference adaptive control can experience a signal growth of a control gain or an adaptive parameter even though both the state and control signals remain bounded. This signal growth associated with the parameter drift can cause instability of adaptive systems. Model-reference adaptive control for non-minimum phase systems presents a major challenge. Non-minimum phase systems have unstable zeros in the right half plane. Such systems cannot tolerate large control gain signals. Model-reference adaptive control attempts to seek the ideal property of asymptotic tracking. In so doing, an unstable pole-zero cancelation occurs that leads to instability. For non-minimum phase systems, adaptive control designers generally have to be aware of the limiting values of adaptive parameters in order to prevent instability. Time-delay systems are another source of challenge for model-reference adaptive control. Many real systems have latency which results in a time delay at the control input. Time delay is caused by a variety of sources such as communication bus latency, computational latency, transport delay, etc. Time-delay systems are a special class of non-minimum phase systems. Model-reference adaptive control of time-delay systems is sensitive to the amplitude of the time delay. As the time delay increases, robustness of model-reference adaptive control decreases. As a consequence, instability can occur. Model-reference adaptive control is generally sensitive to unmodeled dynamics. In a control system design, high-order dynamics of internal states of the system sometimes are neglected in the control design. The neglected internal dynamics, or unmodeled dynamics, can result in loss of robustness of adaptive control systems. The mechanism of instability for a first-order SISO system with a second-order unmodeled actuator dynamics is presented. The instability mechanism can be due to the frequency of a reference command signal or an initial condition of an adaptive parameter that coincides with the zero phase margin condition. Fast adaptation is referred to the use of a large adaptation rate to achieve the improved tracking performance. An analogy of an integral control action of a linear time-invariant system is presented. As the integral control gain increases, the cross-over frequency of the closed-loop system increases. As a consequence, the phase margin or time-delay margin of the system decreases. Fast adaptation of model-reference adaptive control is analogous to the integral control of a linear control system whereby the adaptation rate plays the equivalent role as the integral control gain. As the adaptation rate increases, the time-delay margin of an adaptive control system decreases. In the limit, the time-delay margin tends to zero as the adaptation rate tends to infinity. Thus, the adaptation rate has a strong influence on the closed-loop stability of an adaptive control system.