Realization of a Fourth-Order Linear Time-Varying Differential System with Nonzero Initial Conditions by Cascaded two Second-Order Commutative Pairs

Decomposition is an important tool that is used in many differential systems for solving real engineering problems and improving the stability of a system. It involves breaking down of high-order linear systems into lower-order commutative pairs. Commutativity plays an essential role in mathematics, and its applications are extended in physical science and engineering. This paper explicitly expresses all form of necessary and sufficient conditions for decomposition of any kind of fourth-order linear time-varying system as commutative pairs of two second-order systems. Regarding the nonzero initial conditions, additional requirements are derived in order to satisfy the decomposition process. In this paper, explicit method for reducing fourth-order linear time-varying systems (LTVS) into two second-order commutative pairs is derived and solved. The method points out the effect of disturbance and sensitivity on the systems and also highlights the necessary and sufficient conditions for commutativity of the decomposed systems. The results are illustrated by solving some examples.