Representations of linear time-invariant composite systems

This paper studies representations in the frequency domain as well as in the time domain of linear time-invariant composite systems. Given two systems which are completely characterized by their transfer-function matrices, or equivalently, whose state equation representations are controllable and observable, the question is: Under what conditions will the transfer-function matrix of a composite connection of these two systems characterize the composite system completely, or equivalently, when will the state equation of the composite system be controllable and observable? The following results are presented. 1) The characteristic polynomial of a rational function matrix is introduced. If a state equation representation of a system is controllable and observable, then the characteristic polynomial of the A matrix is equal to the characteristic polynomial of the transfer-function matrix of the system. 2) The controllability of any composite system is a property of the transfer-function matrices of its subsystems. 3) The necessary and sufficient conditions are stated explicitly in terms of transfer-function matrices and controllability λ subspaces which are uniquely determined by transfer-function matrices. These conditions are obtained based on the results of Chen and Desoer.