Optimal LQG coding and control over communication channels: An existence and an infeasibility result

This paper studies the problem of joint coding for and control of a plant over a discrete or continuous noisy memoryless channel. One main result the paper presents is that under certain assumptions, optimal coding and control policies exist and are stationary. It is also shown that for the remote control of an open-loop unstable system over a discrete noisy communication channel, if the system is driven by a Brownian disturbance, stationary memoryless coding schemes or innovation coding schemes lead to an unstable system. This result is true even when the capacity of the discrete channel is arbitrarily high, but finite. In particular, sensor and controller designs for control of systems driven by noise over communication networks when modeled as continuous alphabet erasure channels, lead to instability when are applied to realistic channels. Conditions ensuring stability leading to a recurrent Markov chain are presented. To achieve this, one needs to use coding policies with non-trivial memory constructions.