New commutative formulas for second-order linear time-varying systems

Abstract

This paper studies the commutativity of second-order linear time-varying systems. We present new and simple explicit commutative theories and conditions for second-order linear time varying systems, the commutative theories and conditions are derived and solved to simplify the use of commutativity for practical and industrial uses. Some second-order systems are commutative, some are not commutative, while some are commutative under certain conditions. Moreover, some second-order systems are sensitive toward change in initial conditions or parameters while orders are not, the effect due to disturbance also varies within systems depending on the systems. Furthermore, the stability and robustness of second-order systems has so many issues. But the proposed method provides simple explicit commutative theories and conditions for second-order systems that tackle these issues, simple in computations and accuracy. This finding will help to simplify and investigate the commutativity, sensitivity, and effects due to disturbance on the second-order linear-time varying systems, which has great important in science and engineering. An example is given to support our results.