  Polyhedral Lyapunov functions can approximate any norm arbitrarily well.
Because of this, they are used to study the stability of linear time varying
and linear parameter varying systems without being conservative. However, the
computational cost associated with using them grows unbounded as the size of
their representation increases. Finding them is also a hard computational
problem.
  Here we present an algorithm that attempts to find polyhedral functions while
keeping the size of the representation fixed, to limit computational costs. We
do this by measuring the gap from contraction for a given polyhedral set. The
solution is then used to find perturbations on the polyhedral set that reduce
the contraction gap. The process is repeated until a valid polyhedral Lyapunov
function is obtained.
  The approach is rooted in linear programming. This leads to a flexible method
capable of handling additional linear constraints and objectives, and enables
the use of the algorithm for control synthesis.
