Necessary and sufficient conditions for the complete controllability and observability of systems in series using the coprime factorization of a rational matrix

The series connectionΣ₁₂linear time-invariant systemsΣ_1andΣ_2that have minimal state space system descriptions is considered. From these descriptions strict system equivalent polynomial matrix system descriptions in the manner of Rosenbrock are derived. They are based on the factorization of the transfer matrix of the subsystems as a "ratio" of two right or left coprime polynomial matrices. They give rise to a simple polynomial matrix system description of the tandem connection₁₂. Theorem 1 states that for the complete controllability and observability of the state space system description ofΣ_12it is necessary and sufficient that certain "denominator" and "numerator" groups are coprime. Consequences for feedback systems are drawn in Corollary 1. The role of pole-zero cancellations is explained by Lemma 3 and Corollaries 2 and 3.