On nonnegative realizations of rational matrix functions and nonnegative input-output systems

We consider realizations (A, B, C) of rational l × κ matrix functions W, i.e. W(λ) = C(λ — A)-1 B if λ ∉ σ(A). Generalizing the scalar case we prove that W has a realization with nonnegative matrices iff for each of its minimal realizations (A 0, B 0, C 0) there exists an A 0-invariant polyhedral proper cone P 0 with B 0ℝκ + ⊂ P 0 and C 0 P 0 ⊂ ℝl +. For such functions W the McMillan degree is less than or equal to the size of nonnegative-minimal realizations, and the latter is less than or equal to the minimum of the number of extreme rays of polyhedral cones P 0 with the properties mentioned above. We give an example for which both these inequalities are strict.

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