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Bundle Reduction and the Alignment Distance on Spaces of State-Space LTI Systems

机译：束减少和状态空间LTI系统在空间上的对准距离

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This paper introduces a large class of differential-geometric distances between finite-dimensional linear dynamical systems, collectively called the alignment distance. Contrary to the existing distances, the alignment distance is based on the state-space description of dynamical systems, and is defined on the manifolds of systems of fixed order and fixed input–output dimension under a matrix rank constraint (e.g., minimality, controllability, or observability). While the quotient topology and principal fiber bundle structure associated with such manifolds have been known since the early days of modern control theory, distances natural to this structure have not been studied. The starting point for defining such a distance is to identify a linear system of order with its equivalence class of state-space realizations, all related by the so-called similarity action, i.e., state-space change of basis under , the Lie group of nonsingular matrices. The main idea of the alignment distance is to first find the best state-space change of basis that brings a realization of a system “as close as possible” to a realization of another system (the alignment step), and then compare the aligned realizations. A direct implementation of this idea, due to noncompactness of , is complicated. However, using the notion of “reduction of the structure group” of a principal bundle, we show that the change of basis can be restricted to an orthogonal change of basis, provided one uses realizations in a reduced subbundle. This key observation brings about significant computational benefits. As a technical contribution (possibly of independent interest), we show that several forms of realization balancing available in the control literature have differential-geometric significance, and are, indeed, examples of reducing the structure group from to its subgroup of orthogonal matrices . The alignment distance can be defined for stable and unstable systems, discrete or continuous-time, and stochastic systems.
机译：本文介绍了一类有限维线性动力学系统之间的一类微分几何距离，统称为对准距离。与现有距离相反，对准距离基于动力学系统的状态空间描述，并且在矩阵秩约束（例如，最小性，可控制性，或可观察性）。尽管自现代控制理论的早期以来就已经知道与这种歧管相关的商拓扑和主要纤维束结构，但尚未研究该结构的自然距离。定义这种距离的出发点是，用状态空间实现的等价类来识别一个线性的阶次系统，所有这些都与所谓的相似性动作有关，即相似性作用，即非奇异矩阵。对齐距离的主要思想是首先找到基础的最佳状态空间变化，以使一个系统的实现“尽可能接近”另一个系统的实现（对齐步骤），然后比较对齐的实现。由于的不紧凑性，直接实现此想法很复杂。但是，使用主体束的“结构团的还原”的概念，我们表明，只要在缩减的子束中使用实现，则基础的变化可以限制为基础的正交变化。此关键观察结果带来了显着的计算优势。作为一项技术贡献（可能具有独立利益），我们证明了控制文献中可用的几种实现平衡形式具有微分几何意义，并且确实是将结构组从其简化为正交矩阵的子组的示例。可以为稳定和不稳定系统，离散或连续时间以及随机系统定义对齐距离。

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