Feedback Control in the Presence of Noisy Channels: “Bode-Like” Fundamental Limitations of Performance

This paper addresses fundamental limitations of feedback using information theoretic conservation laws and flux arguments. The paper has two parts. In the first part, we derive a conservation law dictating that causal feedback cannot reduce the differential entropy inserted in the loop by external sources. An interpretation of this result is that the total randomness induced by disturbances, as measured by differential entropy, cannot be reduced by causal feedback; it can only be re-allocated in time or in frequency (if well defined). Under asymptotic stationarity assumptions, this result has a spectral representation which constitutes an extension of Bode's inequality for arbitrary feedback. Our proofs make clear the role of causality, as well as how stability assumptions impact the final result. In the second part, we derive an inequality unveiling that the feedback loop must be able to convey information originating from two independent sources: 1) initial states of the physical plant; 2) exogenous disturbance signals. By using such principle, we construct a variety of information rate (information flux) inequalities. Furthermore, we derive a universal performance bound which is parameterized solely by the feedback capacity and the parameters of the plant. The latter is a new fundamental limitation, which is different from Bode's classical result, indicating that finite feedback capacity brings a new type of performance bound.