论文标题
双线性Koopman实现的优势用于建模和控制具有未知动态的系统
Advantages of Bilinear Koopman Realizations for the Modeling and Control of Systems with Unknown Dynamics
论文作者
论文摘要
通过将它们提升到可观察到的功能的空间，可以使非线性动力系统更易于控制，在该函数中，线性Koopman操作员描述了它们的演变。本文介绍了如何使用Koopman运算符来从数据中生成近似线性，双线性和非线性模型实现，并提出支持双线性实现的，以表征具有未知动力学的系统。提出了在给定的一组可观察功能上具有有效的线性或双线性实现的必要条件，并显示出每个控制范围系统的无限二维双线性实现，但不一定要接收线性。因此，随着基本函数的数量增加，基于基础函数的通用集构建的近似双线性实现往往会改善，而近似线性实现可能不会。为了证明双线性Koopman实现对控制的优势，由数据构建了模拟机器人组的线性，双线性和非线性Koopman模型实现。在以下任务的轨迹中，双线性实现超过了线性实现的预测准确性以及将非线性实现的计算效率纳入模型预测性控制框架时。
Nonlinear dynamical systems can be made easier to control by lifting them into the space of observable functions, where their evolution is described by the linear Koopman operator. This paper describes how the Koopman operator can be used to generate approximate linear, bilinear, and nonlinear model realizations from data, and argues in favor of bilinear realizations for characterizing systems with unknown dynamics. Necessary and sufficient conditions for a dynamical system to have a valid linear or bilinear realization over a given set of observable functions are presented and used to show that every control-affine system admits an infinite-dimensional bilinear realization, but does not necessarily admit a linear one. Therefore, approximate bilinear realizations constructed from generic sets of basis functions tend to improve as the number of basis functions increases, whereas approximate linear realizations may not. To demonstrate the advantages of bilinear Koopman realizations for control, a linear, bilinear, and nonlinear Koopman model realization of a simulated robot arm are constructed from data. In a trajectory following task, the bilinear realization exceeds the prediction accuracy of the linear realization and the computational efficiency of the nonlinear realization when incorporated into a model predictive control framework.
