  In this paper, we revisit the concepts and tools of harmonic analysis and
control and provide a rigorous mathematical answer to the following question:
when does an harmonic control has a representative in the time domain ? By
representative we mean a control in the time domain that leads by sliding
Fourier decomposition to exactly the same harmonic control. Harmonic controls
that do not have such representatives lead to erroneous results in practice.
The main results of this paper are: a one-to-one correspondence between ad hoc
functional spaces guaranteeing the existence of a representative, a strict
equivalence between the Carath{\'e}orody solutions of a differential system and
the solutions of the associated harmonic differential model, and as a
consequence, a general harmonic framework for Linear Time Periodic (LTP)
systems and bilinear affine systems. The proposed framework allows to design
globally stabilizing harmonic control laws. We illustrate the proposed approach
on a single-phase rectifier bridge. Through this example, we show how one can
design stabilizing control laws that guarantee periodic disturbance rejection
and low harmonic content.
