  Quadratic Lyapunov function has been widely used in the analysis of linear
time invariant (LTI) systems ever since it has shown that the existence of such
quadratic Lyapunov function certifies the stability of the LTI system. In this
work, the problem of finding upper and lower bounds for the
$L_{\infty}$-induced norm of the LTI system is considered. Quadratic Lyapunov
functions are used to find the star norm, the best upper on the
$L_{\infty}$-induced norm, by bounding the unit peak input reachable sets by
inescapable ellipsoids. Instead, a more general class of homogeneous Lyapunov
functions is used to get less conservative upper bounds on the
$L_{\infty}$-induced norm and better conservative approximations for the
reachable sets than those obtained using standard quadratic Lyapunov functions.
The homogeneous Lyapunov function for the LTI system is considered to be a
quadratic Lyapunov function for a higher-order system obtained by Lifting the
LTI system via Kronecker product. Different examples are provided to show the
significant improvements on the bounds obtained by using Homogeneous Lyapunov
functions.
