The minimal signal-to-noise ratio required to stabilize over a noisy channel

We consider the problem of stabilizing a discrete-time plant over a noisy channel that is subject to a signal-to-noise ratio (SNR) constraint. It has previously been shown that a necessary condition for stabilization with linear time-invariant state feedback is that the SNR satisfy a lower bound whose value depends solely upon the unstable plant poles. The same lower bound is also necessary for stabilization using linear time invariant output feedback for a minimum phase plant with relative degree one. Following Nair and Evans, we use the concept of entropy power to show that the same bound is also necessary when nonlinear time-varying control laws are applied. Another previous result is that the aforementioned bound on the SNR is also sufficient for stabilization. We apply the concept of entropy rate to provide an alternate proof of this result that also demonstrates an intimate connection between the discrete Bode sensitivity integral and the problem of disturbance-free estimation