Evaluating channels for control: capacity reconsidered

We show that Shannon's classical notion of capacity is not enough to characterize a communication channel if we intend to use that channel as a part of a feedback loop. This is done by considering the stabilization problem for a simple discrete time linear system. While classical capacity is not enough, we show by example that the stricter notion of Shannon's zero-error capacity is conservative. We finish by introducing a new parametric notion of capacity that we call "any-time capacity" and evaluate it for two channels with feedback: the continuous valued additive white Gaussian noise channel and the binary erasure channel.