On the Absence of Limit Cycles in State-Space Digital Filters With Minimum $L_{2}$-Sensitivity

Abstract

This brief proposes a systematic approach to synthesis of limit cycle free state-space digital filters with minimum L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> -sensitivity. We synthesize the minimum L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> -sensitivity realization adopting the balanced realization as an initial realization. The coordinate transformation matrix which transforms the balanced realization into the minimum L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> -sensitivity realization is expressed as the product of a positive definite symmetric matrix and arbitrary orthogonal matrix. We show that the controllability and observability Gramians of the minimum L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> -sensitivity realization satisfy a sufficient condition for the absence of limit cycles when we select an appropriate orthogonal matrix. As a result, the minimum L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> -sensitivity realization without limit cycles can be synthesized by selecting an appropriate orthogonal matrix.