Digital Filter Realizations with Minimum L2-Sensitivity and No Limit Cycles

This paper proposes a systematic approach to synthesis of limit cycle free state-space digital filters with minimum L2-sensitivity. We synthesize the minimum L2-sensitivity real-ization adopting the balanced realization as an initial realiza-tion. The coordinate transformation matrix which transforms the balanced realization into the minimum L2-sensitivity real-ization is expressed as the product of a positive definite sym-metric matrix and an arbitrary orthogonal matrix. We show that the controllability and observability Gramians of the min-imum L2-sensitivity realization satisfy a sufficient condition for the absence of limit cycles when we select an appropriate orthogonal matrix. As a result, the minimum L2-sensitivity realization without limit cycles can be synthesized. 1