Nonlinear extension of Bode's integral based on an information-theoretic interpretation

For linear systems, we show that the constraint known as Bode's sensitivity integral has an information-theoretic interpretation in terms of the difference in the entropy rates between the input and output of the systems. For nonlinear system, we use this interpretation to show that, if the open loop system is globally exponentially stable and has fading memory, this difference is zero. For nonlinear systems that are not stable, we investigate the method to calculate this difference using new results on the inner/outer factorization of nonlinear systems.