An impossibility theorem on feedback based on stochastic embedding

Feedback is a key concept in control systems. A fundamental theoretical problem has been to understand the maximum capability of feedback in controlling uncertain dynamical systems. This needs not only to answer what the feedback can do, but also to answer, the more difficult and basic question, what feedback cannot do. An impossibility theorem on feedback will describe the fundamental limitations of the feedback principle in dealing with uncertainties by all possible feedbacks laws. In this paper, we will use a stochastic embedding approach to address this problem. This approach is based on extensions of the Cramer-Rao inequality to uncertain nonlinearly parameterized dynamical systems, which will then lead to a new impossibility theorem on the feedback capability for the control of a basic class of discrete-time nonlinearly parameterized uncertain dynamical systems.