A set of discrete-time linear systems which have a common Lyapunov function and its extension

The common Lyapunov function problem studied is to find a set of systems which has a common quadratic Lyapunov function guaranteeing asymptotic stability of every member system. For both continuous-time and discrete-time systems, some sets having this property have been known. These results run parallel with each other. The aim of this paper is to complete this parallelism by providing the discrete-time counterpart for a recently obtained continuous-time result. We show that a set of discrete-time systems has a common quadratic Lyapunov function if every system matrix is transformed into complex triangular matrices by a common complex nonsingular matrix. The obtained class includes the known results. We also show an attempt to enlarge the obtained class