Many practical systems have inherent time delays that cannot be ignored; thus, their
dynamics are described using delay differential equations (DDEs). The Galerkin approximation
method is one strategy for studying the stability of time-delay systems (TDS). In
this work, we consider delays that are time-varying and, specifically, time-periodic.
The Galerkin method can be used to obtain a system of ordinary differential equations
(ODEs) from a second-order time-periodic DDE in two ways: either by converting the
DDE into a second-order time-periodic partial differential equation (PDE) and then
into a system of second-order ODEs, or by first expressing the original DDE as two
first-order time-periodic DDEs, then converting into a system of first-order time-periodic
PDEs, and finally converting into a first-order time-periodic ODE system. The difference
between these two formulations in the context of control is presented in this paper.
Specifically, we show that the former produces spurious Floquet multipliers at a spectral
radius of 1. We also propose an optimization-based framework to obtain feedback gains
that stabilize closed-loop control systems with time-periodic delays. The proposed
optimization-based framework employs the Galerkin method and Floquet theory and is
shown to be capable of stabilizing systems considered in the literature. Finally,
we present experimental validation of our theoretical results using a rotary inverted
pendulum apparatus with inherent sensing delays as well as additional time-periodic
state-feedback delays that are introduced deliberately.