Neural Koopman Lyapunov Control
Neural Koopman Lyapunov Control
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DOI：
10.1016/j.neucom.2023.01.029
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发表时间：
2022-01
期刊：
影响因子：
6
通讯作者：
Vrushabh Zinage;E. Bakolas
中科院分区：
文献类型：
--
作者：
Vrushabh Zinage;E. Bakolas
学习和合成未知非线性控制系统的稳定控制器是现实世界和工业应用中的一个具有挑战性的问题。库普曼算子理论允许人们通过线性系统的透镜来分析非线性系统，通过双线性控制系统的透镜来分析非线性控制系统。这些方法的关键思想在于将非线性系统的坐标转换为Koopman可观测值，这些坐标允许将原始系统（控制系统）表示为高维线性（双线性控制）系统。然而，对于非线性控制系统，采用基于Koopman算子的学习方法得到的双线性控制模型不一定是稳定的。可稳定升降双线性控制系统及其相关库普曼观测量的同时辨识仍然是一个有待解决的问题。在本文中，我们提出了一个框架，通过同时学习未知控制仿射非线性系统的双线性Koopman嵌入和基于Koopman的双线性模型的控制Lyapunov函数（CLF），来构建这种可稳定的双线性模型并从数据中识别其相关的可观测值。因此，我们提出的方法为未知控制仿射非线性控制系统作为双线性系统的基于Koopman表示的渐近稳定性提供了可证明的保证。数值仿真验证了所提出的一类稳定反馈控制器对未知控制仿射非线性系统的有效性。
Learning and synthesizing stabilizing controllers for unknown nonlinear control systems is a challenging problem for real-world and industrial applications. Koopman operator theory allows one to analyze nonlinear systems through the lens of linear systems and nonlinear control systems through the lens of bilinear control systems. The key idea of these methods lies in the transformation of the coordinates of the nonlinear system into the Koopman observables, which are coordinates that allow the representation of the original system (control system) as a higher dimensional linear (bilinear control) system. However, for nonlinear control systems, the bilinear control model obtained by applying Koopman operator based learning methods is not necessarily stabilizable. Simultaneous identification of stabilizable lifted bilinear control systems as well as the associated Koopman observables is still an open problem. In this paper, we propose a framework to construct such stabilizable bilinear models and identify their associated observables from data by simultaneously learning a bilinear Koopman embedding for the underlying unknown control affine nonlinear system as well as a Control Lyapunov Function (CLF) for the Koopman based bilinear model using a learner and falsifier. Our proposed approach thereby provides provable guarantees of asymptotic stability for the Koopman based representation of the unknown control affine nonlinear control system as a bilinear system. Numerical simulations are provided to validate the efficacy of our proposed class of stabilizing feedback controllers for unknown control-affine nonlinear systems.