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.

分解是许多差分系统中使用的重要工具，可以解决实际的工程问题并提高系统的稳定性。它涉及将高阶线性系统分解为低阶可交换对。可交换性在数学中起着至关重要的作用，其应用扩展到物理科学和工程学中。本文明确地将任何形式的四阶线性时变系统分解的所有形式的必要条件和充分条件，表达为两个二阶系统的交换对。关于非零初始条件，为了满足分解过程，导出了其他要求。本文提出并求解了将四阶线性时变系统（LTVS）简化为两个二阶交换对的显式方法。该方法指出了扰动和灵敏度对系统的影响，并突出了分解系统可交换性的必要和充分条件。通过举例说明了结果。