Preservation of piecewise-linear Lyapunov function under Padé discretization

In this paper we show that certain piecewise-linear Lyapunov functions are preserved for LTI systems under Padé approximations. In particular, we present a simple method to find a piecewise-linear Lyapunov function that is so preserved under the Padé discretization of any order and sampling time. This result may be of interest in the discretisation of switched linear systems for both simulation and control design.