The Complexity of Minimal Positive Realization

Zhaobo Liu · 2026 · arXiv

DOI: 10.48550/arXiv.2610.27962 · open-access PDF

Abstract

We study the complexity of minimal positive realization for discrete-time systems. We prove that, given a strictly proper transfer matrix with rational coefficients and an order bound encoded in unary, deciding whether a positive realization exists within that bound is complete for the existential theory of the reals. We further prove hardness for stable single-input single-output transfer functions that admit positive realizations, even when the bound equals the McMillan degree.

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