Optimal signal to noise ratio in feedback over communication channels with memory

Communication channels impose a number of obstacles to feedback control, such as delay, noise, and constraints in communication data-rate. One alternate line of recent work considers the problem of feedback stabilization subject to a constraint in the signal-to-noise ratio (SNR). It has been shown for continuous-time systems that the optimal control problem arising in achieving minimal SNR can be formulated as a linear quadratic Gaussian (LQG) control problem with weights chosen as in the loop transfer recovery (LTR) technique. The present paper extends such LQG/LTR formulation to discrete-time systems with feedback over channels with memory. By using such formulation, we derive exact expressions for the LTI controller and loop sensitivity functions that achieve minimal SNR under the effect of time-delay, non minimum phase zeros and colored additive noise. For the minimum-phase case with white noise and no time delay, we show that the optimal feedback loop obtained after applying LTR has a structure equivalent to that of a communication channel with feedback from the output to the input.