Decomposition is a common tool for synthesis of many physical systems. It isalso used for analyzing large scale systems which then known as tearing andreconstruction. On the other hand, commutativity of cascade connected systemshave gained a grade deal of interest and its possible benefits have beenpointed out on the literature. In this paper, the necessary and sufficientconditions for decomposition of any third order linear time-varying system as acommutative pair of first and second order systems of which parameters are alsoexplicitly expressed, are investigated. Further, additional requirements incase of non-zero initial conditions are derived. This paper highlights thedirect formulas for realization of any third order linear time-varying systemas a series (cascade) connection of first and second order subsystems. Thisseries connection is commutative so that it is independent from the sequence ofsubsystems in the connection. Hence, the convenient sequence can be decided byconsidering the overall performance of the system when the sensitivity,disturbance and robustness effects are considered. Realization covers transientresponses as well as steady state responses.
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