How much uncertainty can be dealt with by feedback?

Feedback is used primarily for reducing the effects of the plant uncertainty on the performance of control systems, and as such understanding the following questions is of fundamental importance: How much uncertainty can be dealt with by feedback? What are the limitations of feedback? How does the feedback performance depend quantitatively on the system uncertainty? How can the capability of feedback be enhanced if a priori information about the system structure is available? As a starting point toward answering these questions, a typical class of first-order discrete-time dynamical control systems with both unknown nonlinear structure and unknown disturbances is selected for our investigation, and some concrete answers are obtained in this paper. In particular, we find that in the space of unknown nonlinear functions, the generalized Lipschitz norm is a suitable measure for characterizing the size of the structure uncertainty, and that the maximum uncertainty that can be dealt with by the feedback mechanism is described by a ball with radius 3/2 + root2 in this normed function space.