Feedback Stabilization Over a First Order Moving Average Gaussian Noise Channel

Abstract

Recent developments in information theory by Y.-H. Kim have established the feedback capacity of a first order moving average additive Gaussian noise channel. Separate developments in control theory have examined linear time invariant feedback control stabilization under signal to noise ratio (SNR) constraints, including colored noise channels. This note considers the particular case of a minimum phase plant with relative degree one and a single unstable pole at <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">z</i> =phi (with |phi| > 1) over a first order moving average Gaussian channel. SNR constrained stabilization in this case is possible precisely when the feedback capacity of the channel satisfies <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">CFB</i> ges log <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> |phi|. Furthermore, using the results of Kim we show that there exist linear encoding and decoding schemes that achieve stabilization within the SNR constraint precisely when <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">CFB</i> ges log <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> |phi|.