On the existence of a common quadratic Lyapunov function for two stable second order LTI discrete-time systems

In this paper, necessary and sufficient conditions for the existence of a common quadratic Lyapunov function for two stable second order discrete-time systems are derived. These are expressed in terms of the Schur stability of the matrix pencils. Methods for determining whether the conditions are satisfied can be carried out algebraically or by using root-loci. Work is currently in progress on the extension of the results to higher order systems.