Invariant zeros and input-output structure of linear, time-invariant systems

We define the concepts of conditional reachability and observability. Necessary and sufficient conditions are obtained for a linear, time-invariant system to be conditionally reachable and observable. Some new results on the perfect reachability and observability are also presented. It is shown that the rank of the transfer function matrix and the geometric multiplicity of the invariant zeros play a key role in these conditions.