A discrete-time strong implication-form Lyapunov function for ISS systems with respect to positive semidefinite measurement functions

In this paper, we study the notion of ISS with respect to two measurement functions for discrete-time systems. An example in previous work shows that, in the two measurement function case, the existence of an implication-form ISS-Lyapunov function does not imply ISS. Here, we define a "strong" ISS-Lyapunov function and demonstrate that its existence is equivalent to ISS with respect to two measurement functions.