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
For control theorists and engineers, this work provides fundamental theoretical limits on feedback system performance, though it is largely theoretical and incremental in nature.
This paper derives three universal laws for feedback systems—entropy rate never decreases, generalized Bode integral, and absolute lower bound in variance minimization—using information theory. These laws characterize fundamental limitations imposed by feedback, with the entropy rate law showing that feedback cannot reduce the entropy rate of the output.
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.