  In his paper, we present order reduction techniques for nonlinear
quasi-periodic systems subjected to external excitations. The order reduction
techniques presented here are based on the Lyapunov-Perrone (L-P)
Transformation. For a class of non-resonant quasi-periodic systems, the L-P
transformation can convert a linear quasi-periodic system into a linear
time-invariant one. This Linear Time-Invariant (LTI) system retains the
dynamics of the original quasi-periodic system. Once this LTI system is
obtained, the tools and techniques available for analysis of LTI systems can be
used, and the results could be obtained for the original quasi-periodic system
via the L-P transformation. This approach is similar to using the
Lyapunov-Floquet (L-F) transformation to convert a linear time-periodic system
into an LTI system and perform analysis and control.
  Order reduction is a systematic way of constructing dynamical system models
with relatively smaller states that accurately retain large-scale systems'
essential dynamics. In this work, reduced-order modeling techniques for
nonlinear quasi-periodic systems subjected to external excitations are
presented. The methods proposed here use the L-P transformation that makes the
linear part of transformed equations time-invariant. In this work, two order
reduction techniques are suggested. The first method is simply an application
of the well-known Guyan like reduction method to nonlinear systems. The second
technique is based on the concept of an invariant manifold for quasi-periodic
systems.
  The 'quasi-periodic invariant manifold' based technique yields' reducibility
conditions.' These conditions help us to understand the various types of
resonant interactions in the system. These resonances indicate energy
interactions between the system states, nonlinearity, and external excitation.
