Dynamic Mode Decomposition with Control Liouville Operators
Dynamic Mode Decomposition with Control Liouville Operators
复制标题
DOI：
10.1016/j.ifacol.2021.06.133
复制
发表时间：
2021-01
影响因子：
--
通讯作者：
Joel A. Rosenfeld;R. Kamalapurkar
中科院分区：
文献类型：
--
作者：
Joel A. Rosenfeld;R. Kamalapurkar
利用向量值再生核希尔伯特空间（RKHS）理论，建立了控制仿射动力系统的动力学模式分解（DMD）的理论基础.具体而言，控制Liouville算子和控制占用核的引入，从输入动态分离的漂移动态。给定的反馈控制器通过一个乘法算子来表示，控制Liouville算子和乘法算子的组合用于将非线性闭环系统表示为RKHS上的线性全导数算子.全导数算子的有限秩表示的谱分解产生闭环系统的DMD。DMD生成可用于预测闭环系统的轨迹的模型。对于一类大的系统，全导算子被证明是紧凑的，提供的域和范围RKHS被适当地选择。序列的模型，从增加秩有限秩表示的紧凑的总导数算子，收敛到真正的系统动态，提供足够丰富的数据。数值实验，以证明所开发的技术的有效性。
This article builds the theoretical foundations for dynamic mode decomposition (DMD) of control-affine dynamical systems by leveraging the theory of vector-valued reproducing kernel Hilbert spaces (RKHSs). Specifically, control Liouville operators and control occupation kernels are introduced to separate the drift dynamics from the input dynamics. A given feedback controller is represented through a multiplication operator, and a composition of the control Liouville operator and the multiplication operator is used to express the nonlinear closed-loop system as a linear total derivative operator on RKHSs. A spectral decomposition of a finite-rank representation of the total derivative operator yields a DMD of the closed-loop system. The DMD generates a model that can be used to predict the trajectories of the closed-loop system. For a large class of systems, the total derivative operator is shown to be compact provided that the domain and the range RKHSs are selected appropriately. The sequence of models, resulting from increasing-rank finite-rank representations of the compact total derivative operator, is shown to converge to the true system dynamics, provided that sufficiently rich data are available. Numerical experiments are included to demonstrate the efficacy of the developed technique.