离散时间多项式系统及其输入输出：结构、可达性、可观测性与最小实现
Discrete-time polynomial systems with inputs and outputs: structure, reachability, observability, and minimal realizations
- Northeastern University（东北大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文研究离散时间多项式系统的实现问题，提出规范实现并证明其唯一性，通过线性化与代数准则刻画有限维性，并探讨有界映射与状态仿射系统的等价性。
AI中文摘要：
我们研究了离散时间输入/输出多项式系统的实现问题。这些系统利用交换代数和代数几何的工具进行形式化，其状态空间是代数簇，或者更抽象地，是仿射k-概形的k-点集，其中k是任意无限域。此类系统的输入/输出行为由“多项式响应映射”描述，其中输出是过去输入的多项式函数。主要结果表明，每个多项式响应映射都承认一个规范（准可达且代数可观测的）实现，该实现在同构意义下是唯一的。该方法的关键在于通过考虑一个“对偶”系统来线性化动力学，其中状态是定义在状态上的函数。规范实现的有限维性及其多项式性，通过映射的可观测空间、代数和域，以及代数输入/输出差分方程和雅可比秩准则来刻画。一个特殊的子类由我们称为“有界”的映射组成，其定义性质是它们在过去输入中的次数一致有界。有界映射被证明是有限可实现的，当且仅当它们可由有限维状态仿射系统实现，而后者的理论又归结为有理形式幂级数的理论。我们还研究了给定映射的准可达实现的格，包括正规实现。这项工作是作者于1976年撰写的博士论文的更新；与近期工作的联系在最后一节简要讨论。
英文摘要：
We study the realization problem for discrete-time input/output polynomial systems. These are formalized using tools from commutative algebra and algebraic geometry as systems whose state spaces are algebraic varieties, or more abstractly the set of k-points of an affine k-scheme, where k is an arbitrary infinite field. The input/output behaviors of such systems are described by "polynomial response maps" in which outputs are polynomial functions of past inputs. The main results show that every polynomial response map admits a canonical (quasi-reachable and algebraically observable) realization, which is unique up to isomorphism. The key to the approach is to linearize dynamics by considering a "dual" system in which states are functions defined on states. Finite dimensionality of the canonical realization, and its polynomiality, are characterized in terms of the space, algebra, and field of observables of the map, as well as in terms of algebraic input/output difference equations, and by a Jacobian rank criterion. A particular subclass consists of the maps that we call "bounded," defined by the property that their degree in the past inputs is uniformly bounded. Bounded maps are shown to be finitely realizable if and only if they are realizable by finite-dimensional state-affine systems, whose theory in turn reduces to that of rational formal power series. We also study the lattice of quasi-reachable realizations of a given map, including normal realizations. This work is an update of the PhD thesis written by the author in 1976; connections to recent work are briefly discussed in the last section.
