基于Koopman算子的离散时间非线性系统计算与平衡Gramians方法
A Koopman Operator Approach for Computing and Balancing Gramians for Discrete Time Nonlinear Systems
AI总结：
本文提出利用Koopman算子计算和平衡非线性离散时间系统的Gramians，通过提升技术在全状态可观测空间中计算Gramians，并展示其与局部可控性和可观测性结果的关系，最后通过示例验证方法有效性。
摘要：
本文考虑量化非线性离散时间动力系统可控性和可观测性的任务。我们引入Koopman算子作为系统的标准表示，并应用提升技术在全状态可观测空间中计算Gramians。我们阐述了这些Gramians的性质，并识别了其与局部可控性和可观测性经典结果的关系。一旦定义，我们展示这些Gramians可通过可观测空间上的坐标变换进行平衡，从而允许直接应用平衡截断。本文通过一个非线性系统示例突显了我们的方法。
英文摘要：
In this paper, we consider the problem of quantifying controllability and observability of a nonlinear discrete time dynamical system. We introduce the Koopman operator as a canonical representation of the system and apply a lifting technique to compute gramians in the space of full-state observables. We illustrate the properties of these gramians and identify several relationships with canonical results on local controllability and observability. Once defined, we show that these gramians can be balanced through a change of coordinates on the observables space, which in turn allows for direct application of balanced truncation. Throughout the paper, we highlight the aspects of our approach with an example nonlinear system.
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