On admissibility of pseudoobservability and pseudocontrollability indexes

It has been shown recently that a given MIMO system may be represented by several equivalent state-space representations, referred to as overlapping or pseudocanonical forms, instead of the unique observability (controllability) canonical form. The structure of a pseudocanonical form is determined by a set of pseudoobservability (-controllability) indexes, analogous to the observability (controllability) indexes determining the structure of the canonical form. A MIMO system can be represented by several different pseudocanonical forms, specified by their structure indexes. In this note, we give an upper bound on the admissible structure indexes of a given system in terms of a new set of invariants of this system. An illustrative example is included.