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The Complexity of Minimal Positive Realization

  • Zhaobo Liu
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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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Cite

BibTeXRIS
BibTeX
@misc{liu2026complexity,
  title = {{The Complexity of Minimal Positive Realization}},
  author = {Zhaobo Liu},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.27962},
  eprint = {2610.27962},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.27962},
}
RIS
TY  - UNPB
TI  - The Complexity of Minimal Positive Realization
AU  - Zhaobo Liu
PY  - 2026
DA  - 2026/10/20
PB  - arXiv
DO  - 10.48550/arxiv.2610.27962
UR  - https://arxiv.org/abs/2610.27962
AB  - 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.
ER  -