  Decomposition is a common tool for synthesis of many physical systems. It is
also used for analyzing large scale systems which then known as tearing and
reconstruction. On the other hand, commutativity of cascade connected systems
have gained a grade 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 non-zero initial conditions are derived. This paper highlights the
direct formulas for realization of any third order linear time-varying system
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.
