Koopman Operators for Estimation and Control of Dynamical Systems
Koopman Operators for Estimation and Control of Dynamical Systems
复制标题
用于动力系统估计和控制的库普曼算子
DOI：
10.1146/annurev-control-071020-010108
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发表时间：
2021
期刊：
影响因子：
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通讯作者：
C. Rowley
中科院分区：
文献类型：
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作者：
Samuel E. Otto;C. Rowley
表示系统动态的一种常见方法是指定状态如何随时间演变。另一种观点是指定国家的职能是如何随时间演变的。函数的这种演化是由一个称为库普曼算子的线性算子控制的，它的光谱特性揭示了系统的内在特征。例如，它的本征函数决定了动力学线性演化的坐标。本文讨论了库普曼算符方法的理论基础，以及过去二十年发展起来的从数据中近似库普曼算符的数值方法，对于有和没有驱动的系统都是如此。我们特别关注遍历系统，对它有特别有效的数值方法可用。对于具有仿射控制输入的非线性系统，Koopman形式自然而然地导致了状态和输入双线性的系统，这种结构可以用于控制器和估值器的设计。
A common way to represent a system's dynamics is to specify how the state evolves in time. An alternative viewpoint is to specify how functions of the state evolve in time. This evolution of functions is governed by a linear operator called the Koopman operator, whose spectral properties reveal intrinsic features of a system. For instance, its eigenfunctions determine coordinates in which the dynamics evolve linearly. This review discusses the theoretical foundations of Koopman operator methods, as well as numerical methods developed over the past two decades to approximate the Koopman operator from data, for systems both with and without actuation. We pay special attention to ergodic systems, for which especially effective numerical methods are available. For nonlinear systems with an affine control input, the Koopman formalism leads naturally to systems that are bilinear in the state and the input, and this structure can be leveraged for the design of controllers and estimators.