Control and Communication with Signal-to-Noise Ratio Constraints

This thesis is about two problems in the intersection of communication and control theory. Their common feature is that they involve communication over an additive white noise channel with a signal-to-noise ratio (SNR) constraint. The first problem concerns the transmission of a real-valued signal from a partially observed Markov source. The distortion criterion is the mean squared error and the transmission is subject to a delay constraint, which introduces the need for real-time coding. The problem is first considered for scalar-valued signals when the channel has no feedback and then, in turn, generalized to each of the cases with non-white channel noise, vector-valued signals or channel feedback. It is shown that jointly optimal encoders and decoders within the linear time-invariant (LTI) class can be obtained by solving a convex optimization problem and performing a spectral factorization. The functional to minimize is the sum of the well-known cost in a corresponding Wiener filtering problem and a new term that is induced by the channel noise. The second problem, which can be viewed as a generalization of the first problem, concerns a networked control system where an LTI plant, subject to a stochastic disturbance, is to be controlled over the channel. The controller is based on output feedback and consists of an encoder/observer that measures the plant output and transmits over the channel, and a decoder/controller that receives the channel output and issues the control signal. The objective is to stabilize the plant, satisfy the SNR constraint and minimize the variance of the disturbance response. The problem is studied for channels without and with feedback. In both cases, it is shown that optimal controllers within the LTI class can be obtained by solving a convex optimization problem and performing a spectral factorization. Previously known conditions on the SNR for stabilizability follow directly from the constraints of these optimization problems.