Control-Theoretic Approach to Communication With Feedback

Feedback communication is studied from a control-theoretic perspective, mapping the communication problem to a control problem in which the control signal is received through the same noisy channel as in the communication problem, and the (nonlinear and time-varying) dynamics of the system determine a subclass of encoders available at the transmitter. The MSE exponent is defined to be the exponential decay rate of the mean square decoding error and is used for analysis of the reliable rate of communication. A sufficient condition is provided under which the MMSE capacity, the supremum achievable MSE exponent, is equal to the information-theoretic capacity, the supremum achievable rate. For the special class of stationary Gaussian channels and linear time-invariant systems, a simple application of Bode´s integral formula shows that the feedback capacity, recently characterized by Kim, is equal to the maximum instability that can be tolerated by any linear controller under a given power constraint. Finally, the control mapping is generalized to the N-sender AWGN multiple access channel. It is shown that Kramer´s code for this channel, which is known to be sum rate optimal in the class of generalized linear feedback codes, can be obtained by solving a linear quadratic Gaussian control problem.