Fundamental limitations and differences of robust and adaptive control

Capability of robust and adaptive control in dealing with uncertainty is investigated in an information framework. For clarity and technical tractability, a simple first-order linear time-varying system is employed as a vehicle to obtain explicit performance lower bounds of robust and adaptive control. The results indicate that although robust control does not require information on the dynamics of system parameters, it can provide only very limited robustness. On the other hand, if additional information on parameter dynamics is available, such as systems whose parameters vary with time relatively slowly, adaptive control methodologies can substantially improve the capability of a feedback system in dealing with uncertainty beyond robust control. On the other hand, due to intimate interaction and inherent conflict between identification and control, a certain performance lower bound is shown to exist which is at least twice as big as the nominal performance even when the system varies very slowly.

1 Introduction

Information about a plant can be specified by an uncertainty set that contains the plant. Intuitively, it seems quite clear that to achieve a control objective for a plant that belongs to an uncertainty set, one can either utilize the capability of information acquisition in reducing uncertainty (identification), or that of control in providing robustness against uncertainty (robust control), or a combination of both (adaptation). However, precise characterization of this understanding turns out to be extremely challenging and has eluded rigorous analysis over a long period of time. Some of the challenges are inherent and stem from a lack of rigorous characterization of optimal robustness in terms of both stability and performance, and technical difficulties in clarifying interactions between identification and control.