Özet:

The realization of many engineering systems consists of cascade connection of systems of simple orders, which is very important in design of electrical and electronic systems. Although the order of connection of the systems mainly depends on the special design approach, engineering ingenuity, traditional synthetic methods, when the sensitivity, stability, linearity, noise disturbance, robustness effects are considered the change of the order of connection without changing the main function of the total systems (commutativity) may lead positive results. Therefore, the commutativity is very important from the practical point of view. In this study, commutativity conditions of one type of Lame’s differential equations are considered. In the sense of theoretical results for the commutativity of second-order continuous-time linear time-varying systems, it is proved that the system modeled by a Lame’s differential equation has commutative pairs depending on the parameters of the equation. Commutative conjugate of the system modeled by a Lame’s differential equation is constructed. To support the theoretical results, an illustrative example is considered for application. For the illustration, Simulink toolbox of MATLAB 2019b is used. Ode5 (Dormant-Prince) is used as the solver with a fixed step-length. Numerical results are presented. It is observed that the responses computed for x∈[0,120] are identical, which proves the validity of the commutativity results under zero initial conditions. The validity of commutativity with arbitrary initial conditions is also tested. It is observed that the commutativity is spoiled for arbitrarily chosen initial conditions which are not chosen appropriately. Theory of commutativity of the system modeled by Lame’s differential equation with non-zero initial conditions can be conducted in future work using the general formulas in (Koksal, 2019b).

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