Preservation of quadratic stability under various common approximate discretization methods

In this paper we prove the following result. If AA is a Hurwitz matrix and ff is a rational function that maps the open left half of the complex plane into the open unit disc, then any Hermitian matrix P>0P>0 which is a Lyapunov matrix for AA (that is, PA+A∗P<0PA+A∗P<0) is also a Stein matrix for f(A)f(A) (that is, f(A)∗Pf(A)−P<0f(A)∗Pf(A)−P<0).We use this result to prove that all A-stable approximations for the matrix exponential preserve quadratic Lyapunov functions for any stable linear system. The importance of this result is that it implies that common quadratic Lyapunov functions for switched linear systems are preserved for all step sizes when discretising quadratically stable switched systems using A-stable approximations.Examples are given to illustrate our results.