Contractibility of dynamic LTI controllers using complementary matrices

A generalized structure of complementary matrices involved in the input-state-output inclusion principle for linear time-invariant systems including contractibility conditions for static state feedback controllers is well known. In this note, it is shown how to further extend this structure when considering contractibility of dynamic controllers. Necessary and sufficient conditions for contractibility are proved in terms of both unstructured and block structured complementary matrices for general expansion/contraction transformation matrices. Explicit sufficient conditions for blocks of complementary matrices ensuring contractibility are proved for general expansion/contraction transformation matrices. Moreover, these conditions are further specialized for a particular class of transformation matrices.