Biblio

In this paper, we present a fundamental limitation of disturbance attenuation in discrete-time single-input single-output (SISO) feedback systems when the controller has delayed side information about the external disturbance. Specifically, we assume that the delayed information about the disturbance is transmitted to the controller across a finite Shannon-capacity communication channel. Our main result is a lower bound on the log sensitivity integral in terms of open-loop unstable poles of the plant and the characteristics of the channel, similar to the classical Bode integral formula. A comparison with prior work that considers the effect of preview side information of the disturbance at the controller indicates that delayed side information and preview side information play different roles in disturbance attenuation. In particular, we show that for open-loop stable systems, delayed side information cannot reduce the log integral of the sensitivity function whereas it can for open-loop unstable systems, even when the disturbance is a white stochastic process.