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

This brief proposes a systematic approach to synthesis of limit cycle free state-space digital filters with minimum L
₂
-sensitivity. We synthesize the minimum L
₂
-sensitivity realization adopting the balanced realization as an initial realization. The coordinate transformation matrix which transforms the balanced realization into the minimum L
₂
-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
₂
-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
₂
-sensitivity realization without limit cycles can be synthesized by selecting an appropriate orthogonal matrix.