How much uncertainty can be dealt with by feedback?

Feedback is used primarily for reducing the effects of the plant uncertaint y on the performance of control systems, and as such understanding the foll owing questions is of fundamental importance: How much uncertainty can be d ealt with by feedback? What are the limitations of feedback? How does the f eedback performance depend quantitatively on the system uncertainty? How ca n the capability of feedback be enhanced if a priori information about the system structure is available? As a starting point toward answering these q uestions, a typical class of first-order discrete-time dynamical control sy stems with both unknown nonlinear structure and unknown disturbances is sel ected for our investigation, and some concrete answers are obtained in this paper. In particular, we find that in the space of unknown nonlinear funct ions, the generalized Lipschitz norm is a suitable measure for characterizi ng 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 wit h radius 3/2 + root2 in this normed function space.