Stabilization and Disturbance Attenuation Over a Gaussian Communication Channel

We consider the problem of stabilizing an unstable system driven by a Gaussian disturbance using a feedback signal transmitted over a memoryless Gaussian communication channel. By applying the concept of entropy power, we show that the mean square norm of the state vector must satisfy a lower bound that holds for any causal, measurable communication and control strategies that result in signals having well defined differential entropy. In addition, we show that use of nonlinear, time varying strategies does not allow stabilization over a channel with a lower signal-to-noise ratio than that achievable with linear time invariant state feedback. Finally, we show that for scalar systems the lower bound on the mean square norm of the state is tight, and achievable using linear time invariant communication and control.