Decomposition of Third-Order Linear Time-Varying Systems into Its Second- and First-Order Commutative Pairs

Decomposition is a common tool for the synthesis of many physical systems. It is also used for analyzing large-scale systems which are then known as tearing and reconstruction. On the other hand, commutativity of cascade-connected systems has gained a great deal of interest, and its possible benefits have been pointed out on the literature. In this paper, the necessary and sufficient conditions for decomposition of any third-order linear time-varying system as a commutative pair of first- and second-order systems of which parameters are also explicitly expressed, are investigated. Further, additional requirements in case of nonzero initial conditions are derived. This paper highlights the direct formulas for realization of any third-order linear time-varying systems as a series (cascade) connection of first- and second-order subsystems. This series connection is commutative so that it is independent from the sequence of subsystems in the connection. Hence, the convenient sequence can be decided by considering the overall performance of the system when the sensitivity, disturbance, and robustness effects are considered. Realization covers transient responses as well as steady-state responses.