Abstract. The paper presents the new optimal real-time control algorithm of the power source. The minimum of the square-instantaneous current was assumed as an optimal criterion, with an additional constraint on source instantaneous power. The mathematical model of a multiphase source was applied as a voltage-current convolution in the discrete time domain. The resulting control algorithm was the recursive digital filter with infinite recursion.

The paper presents a modification of the time-invatiant linear operators to adapt them to the almost-periodically modulated signal processing. In particular, the distribution of the operator to a sum of operators: even and odd. These operators were realized using cyclic convolution type digital filters with addition of modulation disorder. (The G, B distribution of modulation disordered operators-digital filters realization.

The article presents the elementary theory of differential and integral operators of the fractional order in a discrete time approach. A notion of a simple proper fraction operator has been introduced. It has been done for the time equivalent by applying the Taylor series. On this basis a new theory of a certain complexity operators has been formed which includes differential operators of the fractional order. Somewhat more general approach has been presented in the later part of the article by introducing a rational power of the convolution operator. Both approaches to the fractional operators are realized by non–recursive digital filters of infinite impulse responses. The stability of such filters is also being considered. The article also contains the application to the distributed parameters electrical circuits theorem.

This paper is a study on the synthesis of digital filters used to control active or self-excited systems. The active two-terminal branch implements a nonconvolution-type immittance operator, which generates a current waveform depending on the given impedance or admittance operator. In this article, for the first time, the method of how to construct immittance operators for linear time-variant (nonconvolution-type) two-terminal circuits, over discrete time, is presented. These operators are useful when calculating periodic steady-state signals of a parametric circuit. The formula for the duty cycle is derived based on the current generated by this branch, assuming a known branch voltage or vice versa. This formula allows us to make a direct calculation of the duty-cycle in an analytical manner and does not refer to any auxiliary signals, e.g., sawtooth signals, or to any control systems, e.g., PI controller. The determined duty-cycles allow us to select the appropriate switching frequency and voltage value for the switched voltage source. With this method, it is also possible to assess the parameters of the current signal that would be generated in the actual active filter branch due to the calculated PWM voltage. The presented method can be an alternative to commonly used PI controllers in feedback for controlling active power filters/inverters.