This paper studies commutativity of sixth-order differential systems with zero and nonzero initial conditions (ICs). Given a sixth-order differential system \(A\), we find its commutative pair, that is a new differential system \(B\) of order \(m \le 6\); the obtained result will be used to show the input–output equivalency between the cascaded-connected systems \(AB\) and \(BA\). Necessary and sufficient conditions for commutativity of sixth-order continuous-time linear time-varying systems (CTLTVSs) are derived. Based on our findings, we discovered that only a small group of sixth-order CTLTVSs has commutative pairs other than their feedback conjugates obtained by constant feedback and forward path gains. The explicit results obtained in this paper will help to fill the gap on commutativity theory, system behaviors which has significant application to science and play an important role in engineering. This research will also address the problems of sensitivity due to changes in ICs or parameters, and also disturbance as a result of external noise. The results are well verified by examples as well as validated by Matlab Simulink tool and Wolfram Mathematica 11.

本文研究了具有零和非零初始条件 (IC) 的六阶微分系统的交换性。给定一个六阶微分系统\(A\)，我们找到它的交换对，即一个新的\(m \le 6\)阶微分系统\(B \)；所得结果将用于显示级联系统\(AB\)和\(BA\)之间的输入-输出等效性. 推导了六阶连续时间线性时变系统（CTLTVS）交换性的充要条件。根据我们的发现，我们发现只有一小群六阶 CTLTVS 具有交换对，而不是通过恒定反馈和前向路径增益获得的反馈共轭。本文所获得的明确结果将有助于填补交换性理论、系统行为的空白，这些理论对科学具有重要的应用意义，在工程中具有重要的作用。这项研究还将解决由于 IC 或参数变化引起的灵敏度问题，以及由于外部噪声引起的干扰问题。结果通过示例得到了很好的验证，并得到了 Matlab Simulink 工具和 Wolfram Mathematica 11 的验证。