The Information cost of sensitivity shaping over Gaussian channels

In this paper, we extend the results of our previous work on feedback stabilization and feedback communication to include the presence of an external input acting as an output disturbance on the feedback system. Rather than on the communication system, we focus on the control system. We show that the Directed Information still characterizes the information rate flowing through the channel, and that it is constant for a given open loop Degree of Instability and channel model. This indicates that, in our case if the degree of instability is kept constant, loop shaping does not affect the rate through the channel but only the transmission power. For the memoryless AWGN channel, we characterize the required transmission power in a closed form which relates a function of the required transmission power to the sum two rates: one given by the Directed Information, the other due to the non flat frequency shaping of the Sensitivity and expressed in term of a relative entropy. This result provides a new interpretation of the H∞ Sensitivity minimization problem based on Information theory concepts.