Control over Gaussian channels with and without source–channel separation

We consider the problem of controlling an unstable linear plant with Gaussian disturbances over an additive white Gaussian noise channel with an average transmit power constraint, where the signaling rate of communication may be different from the sampling rate of the underlying plant. Such a situation is quite common since sampling is done at a rate that captures the dynamics of the plant and that is often lower than the signaling rate of the communication channel. This rate mismatch offers the opportunity of improving the system performance by using coding over multiple channel uses to convey a single control action. In a traditional, separation-based approach to source and channel coding, the analog message is first quantized down to a few bits and then mapped to a channel codeword whose length is commensurate with the number of channel uses per sampled message. Applying the separation-based approach to control meets its challenges: first, the quantizer needs to be capable of zooming in and out to be able to track unbounded system disturbances, and second, the channel code must be capable of improving its estimates of the past transmissions exponentially with time, a characteristic known as anytime reliability. We implement a separated scheme by leveraging recently developed techniques for control over quantized-feedback channels and for efficient decoding of anytime-reliable codes. We further propose an alternative, namely, to perform analog joint source–channel coding, by this avoiding the digital domain altogether. For the case where the communication signaling rate is twice the sampling rate, we employ analog linear repetition as well as Shannon–Kotel’nikov maps to show a significant improvement in stability margins and linear-quadratic costs over separation-based schemes. We conclude that such analog coding performs better than separation, and can stabilize all moments as well as guarantee almost-sure stability.