Exact Decision and a Surjective Stabilizer Atlas for Affine Coprime-Factor Segments
Abstract
We consider simultaneous internal stabilization of square proper real-rational plants of arbitrary but fixed finite input-output dimension n, arranged as an affine right-coprime factor segment, under explicit nondegeneracy hypotheses. After a classical normalization, the remaining global spectral-cut constraint and the reconstruction constraints at finite poles and at infinity are encoded, without loss, as a finite-dimensional semialgebraic seed, using a function-level double-Cayley factorization. For input coefficients in an effectively presented real closed field, that seed yields an exact existence decision that does not enumerate controller McMillan degree; the decision engine is classical quantifier elimination. The same encoding, now ranging over rational radii, admissible seeds, and residual unimodular and rational Schur parameters, yields a surjective atlas of all proper real-rational common stabilizers of the segment. The results concern this structured class and are compatible with the rational undecidability of simultaneous stabilization of three general plants.
Related
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BibTeX
@misc{qiu2026exact,
title = {{Exact Decision and a Surjective Stabilizer Atlas for Affine Coprime-Factor Segments}},
author = {Junkai Qiu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.07733},
eprint = {2609.07733},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.07733},
}
RIS
TY - UNPB
TI - Exact Decision and a Surjective Stabilizer Atlas for Affine Coprime-Factor Segments
AU - Junkai Qiu
PY - 2026
DA - 2026/09/07
PB - arXiv
DO - 10.48550/arxiv.2609.07733
UR - https://arxiv.org/abs/2609.07733
AB - We consider simultaneous internal stabilization of square proper real-rational plants of arbitrary but fixed finite input-output dimension n, arranged as an affine right-coprime factor segment, under explicit nondegeneracy hypotheses. After a classical normalization, the remaining global spectral-cut constraint and the reconstruction constraints at finite poles and at infinity are encoded, without loss, as a finite-dimensional semialgebraic seed, using a function-level double-Cayley factorization. For input coefficients in an effectively presented real closed field, that seed yields an exact existence decision that does not enumerate controller McMillan degree; the decision engine is classical quantifier elimination. The same encoding, now ranging over rational radii, admissible seeds, and residual unimodular and rational Schur parameters, yields a surjective atlas of all proper real-rational common stabilizers of the segment. The results concern this structured class and are compatible with the rational undecidability of simultaneous stabilization of three general plants.
ER -