Three Laws of Multivariable Feedback Systems, Extended Spectral Flatness (Extended Wiener Entropy), 'Uncertainty Principles' in Variance Minimization, and Performance Limitations in Minimum Variance Estimation/Filtering

TL;DR

This paper establishes fundamental performance limits for multivariable feedback systems across entropy, frequency, and time domains, introducing new measures and principles that deepen understanding of system constraints.

Contribution

It introduces new notions like negentropy rate and extended spectral flatness, and derives performance limitations and uncertainty principles for causal MIMO feedback systems.

Findings

Performance limits characterized in entropy, frequency, and time domains.

Uncertainty principles in minimum variance control.

Performance bounds in estimation and filtering.

Abstract

In this paper, three laws are obtained for multiple-input multiple-output feedback systems, which are in entropy domain, frequency domain, and time domain, respectively. The system setup is that with causal plants and causal controllers. Those laws characterize the performance limitations of such systems imposed by the feedback mechanism. Some new notions are proposed to facilitate the analysis: negentropy rate, extended spectral flatness (extended Wiener entropy), Gaussianity-whiteness measure (joint Shannon-Wiener entropy), etc. Two approaches are adopted: the integrated approach and the divided approach. And 'uncertainty principles' are found in minimum variance control. Besides, performance limitations in minimum variance estimation and filtering are obtained. In the end, the special case of linear time-invariant feedback systems is discussed.