论文标题
统一定理用于子空间识别和动态模式分解
Unifying Theorems for Subspace Identification and Dynamic Mode Decomposition
论文作者
论文摘要
本文为自主动力系统提供了子空间识别（SID）和动态模式分解（DMD）的统一结果。我们观察到，SID试图解决一个优化问题，以估计扩展的可观察性矩阵和状态序列，该序列可最大程度地减少状态空间模型的预测误差。此外，我们观察到DMD试图解决一个受限制的矩阵回归问题，从而最大程度地减少了扩展自回归模型的预测误差。我们证明，完美（无错误的）状态空间和低排名扩展自回归模型的存在条件是等效的，并且SID和DMD优化问题是等效的。我们利用这些结果提出了一种SID-DMD算法，该算法可提供可证明的最佳模型，并且易于实现。我们使用案例研究旨在直接从视频数据中构建动态模型来证明我们的发展。
This paper presents unifying results for subspace identification (SID) and dynamic mode decomposition (DMD) for autonomous dynamical systems. We observe that SID seeks to solve an optimization problem to estimate an extended observability matrix and a state sequence that minimizes the prediction error for the state-space model. Moreover, we observe that DMD seeks to solve a rank-constrained matrix regression problem that minimizes the prediction error of an extended autoregressive model. We prove that existence conditions for perfect (error-free) state-space and low-rank extended autoregressive models are equivalent and that the SID and DMD optimization problems are equivalent. We exploit these results to propose a SID-DMD algorithm that delivers a provably optimal model and that is easy to implement. We demonstrate our developments using a case study that aims to build dynamical models directly from video data.
