Feedback stabilization over signal-to-noise ratio constrained channels

There has recently been significant interest in feedback stabilization problems over communication channels, including several with bit rate limited feedback. Motivated by considering one source of such bit rate limits, we study the problem of stabilization over a signal-to-noise ratio (SNR) constrained channel. We discuss both continuous and discrete time cases, and show that for either state feedback, or for output feedback delay-free, minimum phase plants, there are limitations on the ability to stabilize an unstable plant over an SNR constrained channel. These limitations in fact match precisely those that might have been inferred by considering the associated ideal Shannon capacity bit rate over the same channel.