  Stability margins for linear time-varying (LTV) and switched-linear systems
are traditionally computed via quadratic Lyapunov functions, and these
functions certify the stability of the system under study. In this work, we
show how the more general class of homogeneous polynomial Lyapunov functions is
used to compute stability margins with reduced conservatism, and we show how
these Lyapunov functions aid in the search for periodic trajectories for
marginally stable LTV systems. Our work is premised on the recent observation
that the search for a homogeneous polynomial Lyapunov function for some LTV
systems is easily encoded as the search for a quadratic Lyapunov function for a
related LTV system, and our main contribution is an intuitive algorithm for
generating upper and lower bounds on the system's stability margin. We show
also how the worst-case switching scheme - which draws an LTV system closest to
a periodic orbit - is generated. Three numerical examples are provided to aid
the reader and demonstrate the theoretical contributions of the work.
