Feedback control over signal-to-noise ratio constrained communication channels with channel input quantisation

Communication channels impose a number of obstacles to feedback control, such as delay, noise, and constraints in the communication transmission data rate. One line of research considers the problem of feedback stabilisation of a linear time invariant (LTI) plant subject to a constraint on the channel signal-to-noise ratio (SNR) of a communication channel located in the same loop. The present paper extends such formulation to discrete-time systems with feedback over memoryless additive white Gaussian noise (AWGN) channels and additive coloured Gaussian noise (ACGN) channels with memory subject to uniform quantisation of the channel input. By using a uniform distribution formulation for the quantisation error, we derive closed-form expressions for the infimal LTI SNR for stabilisability subject to quantisation error and the channel capacity in the case of the memoryless AWGN channel and an exact expression for the optimal LTI controller that achieves such infimal LTI SNR for stabilisability subject to quantisation error in the case of the ACGN channel with memory.