Three laws of feedback systems: entropy rate never decreases, generalized Bode integral, absolute lower bound in variance minimization, Gaussianity-whiteness measure (joint Shannon-Wiener entropy), Gaussianing-whitening control, and beyond

TL;DR

This paper derives universal laws and fundamental lower bounds for feedback systems using information theory, introducing new concepts like negentropy rate and Gaussianity-whiteness measure to characterize system limitations.

Contribution

It presents three fundamental laws of feedback systems in different domains and introduces novel measures for analyzing Gaussianity and whiteness, extending understanding of feedback system limitations.

Findings

Entropy rate of feedback systems never decreases.

Generalized Bode integral imposes fundamental frequency domain constraints.

Absolute lower bounds in variance minimization are established.

Abstract

This paper aims at obtaining universal laws and absolute lower bounds of feedback systems using information theory. The feedback system setup is that with causal plants and causal controllers. Three laws (entropy rate never decreases, generalized Bode integral, and absolute lower bound in variance minimization) are obtained, which are in entropy domain, frequency domain, and time domain, respectively. Those laws characterize the fundamental limitations of such systems imposed by the feedback mechanism. Two new notions, negentropy rate and Gaussianity-whiteness measure (joint Shannon-Wiener entropy), are proposed to facilitate the analysis. Topics such as whiteness-Gaussianity-variance decomposition, Gaussianing-whitening control (the maximum Gaussianity-whiteness measure principle), whitening control (spectrum/spectral flattening control), generalized Bode plot, and so on are also discussed. The special case of linear time-invariant feedback systems is considered in the end.