Multilevel Selective Harmonic Modulation via Optimal Control

TL;DR

This paper introduces an optimal control framework for designing staircase control signals with specific harmonic components, enabling automatic determination of switching angles and waveform shape.

Contribution

It formulates the SHM problem as an optimal control problem with a novel penalization approach, ensuring control uniqueness, continuity, and eliminating the need for predefining switch points.

Findings

The method successfully designs control signals with prescribed harmonic content.

It guarantees unique and continuous solutions for the control problem.

Numerical examples demonstrate the effectiveness of the proposed approach.

Abstract

We consider the Selective Harmonic Modulation (SHM) problem, consisting in the design of a staircase control signal with some prescribed frequency components. In this work, we propose a novel methodology to address SHM as an optimal control problem in which the admissible controls are piecewise constant functions, taking values only in a given finite set. In order to fulfill this constraint, we introduce a cost functional with piecewise affine penalization for the control, which, by means of Pontryagin's maximum principle, makes the optimal control have the desired staircase form. Moreover, the addition of the penalization term for the control provides uniqueness and continuity of the solution with respect to the target frequencies. Another advantage of our approach is that the number of switching angles and the waveform need not be determined a priori. Indeed, the solution to the optimal control problem is the entire control signal, and therefore, it determines the waveform and the location of the switches. We also provide several numerical examples in which the SHM problem is solved by means of our approach.