On extended (J, J′)-lossless factorization

The extended (J, J′)-lossless factorization for discrete-time rational matrix functions with zeros on the unit circle is considered. A necessary and sufficient condition for the existence of the extended (J, J′)-lossless factorization is obtained. The condition is given in terms of the original system parameters in the form of a generalized eigenvalue problem and a discrete-time algebraic Riccati equation. State-space representations for the extended (J, J′)-lossless factorization are provided.