仿射互质因子段上的精确判定与满射稳定化图册
Exact Decision and a Surjective Stabilizer Atlas for Affine Coprime-Factor Segments
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中文总结 AI 辅助
针对仿射互质因子段，提出基于双Cayley分解和量词消去的精确存在性判定，并构造所有公共镇定器的满射图册。
中文摘要 AI 辅助
我们考虑对任意但固定的有限输入输出维数$n$的方阵真有理实有理被控对象，在显式非退化假设下，将其排列为仿射右互质因子段，并研究其同时内部镇定问题。经过经典归一化后，剩余的全局谱切割约束以及有限极点和无穷远处的重构约束，通过函数级双Cayley分解，被无损地编码为有限维半代数种子。对于有效呈现的实闭域中的输入系数，该种子产生一个精确的存在性判定，该判定不枚举控制器的McMillan度；判定引擎是经典的量词消去。同一编码，现在遍历有理半径、容许种子以及剩余幺模和有理Schur参数，产生该段的所有真有理公共镇定器的满射图册。这些结果涉及这一结构化类，并与三个一般被控对象的同时镇定的有理不可判定性相容。
英文摘要
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.
发表机构
- School of Mathematical Sciences（数学科学学院）
- Dalian University of Technology（大连理工大学）
机构由 AI 辅助整理，请以论文原文为准。
