Passive Linear Discrete-Time Systems: Characterization through Structure

We here show that the family of finite-dimensional, discrete-time, passive, linear time-invariant systems can be characterized through the structure of maximal, matrix-convex set, closed under multiplication among its elements. Moreover, this observation unifies three setups: (i) difference inclusions, (ii) matrix-valued rational functions, (iii) realization arrays associated with rational functions. It turns out that in the continuous-time case, the corresponding structure is of a maximal matrix-convex, cone, closed under inversion.