A note on quadratic stability of uncertain linear discrete-time systems

In this paper we consider a linear discrete-time system depending on a vector of uncertain parameters. Assuming that the system matrix depends on parameters as the ratio of a multiaffine matrix-valued function and a multiaffine polynomial and that the parameters range in a hyper-rectangle, we show that the uncertain system is quadratically stable if and and only if the set of vertex systems is quadratically stable. This allows us to state a necessary and sufficient condition for quadratic stability in terms of the solvability of a feasibility problem involving linear matrix inequalities.