Although the explicit commutativitiy conditions for second-order linear time-varying systems have been appeared in some literature, these are all for initially relaxed systems. This paper presents explicit necessary and sufficient commutativity conditions for commutativity of second-order linear time-varying systems with non-zero initial conditions. It has appeared interesting that the second requirement for the commutativity of non-relaxed systems plays an important role on the commutativity conditions when non-zero initial conditions exist. Another highlight is that the commutativity of switched systems is considered and spoiling of commutativity at the switching instants is illustrated for the first time. The simulation results support the theory developed in the paper.

In this paper, a DC power supply is presented that, in addition to a power factor correction, is equipped with an active power filter function. This function enables compensation of both the reactive and the distortion power, generated by an external load, which is connected to the same power line node as the presented device. A tuneable inductive filter is added at the input of the power electronics controlled current source, which constitutes the main block of the power supply. This filter allows for a visible improvement in the quality of the current source control, compared to a similar device with a fixed inductive filter. This improvement depends on extending the “frequency response” of the current source, which facilitates an increase in the dynamics of changes in the input current of the power supply. The actual modification to the presented device is related to its control section, which is equipped with analogue regulators. The main purpose of this work is to present the results from a simulation model of an electrical system with a power supply, especially compared to those from a similar device but with a discrete control. The work represents a continuation of a research cycle on DC power supplies that are equipped with a power compensation function and are based on tuneable magnetic devices.