非线性系统的带控制的Deep Koopman算子
Deep Koopman Operator with Control for Nonlinear Systems
摘要：
近年来，Koopman算子已成为一个很有前途的数据驱动工具，以促进实时控制的未知非线性系统。它将非线性系统映射到嵌入空间中的等价线性系统，为实时线性控制方法做好了准备。然而，设计合适的Koopman嵌入函数仍然是一项具有挑战性的任务。此外，大多数基于Koopman的算法只考虑具有线性控制输入的非线性系统，当系统与控制输入完全非线性时，导致预测和控制性能较差。在这项工作中，我们提出了一个端到端的深度学习框架来一起学习Koopman嵌入函数和Koopman算子，以减轻这些困难。首先用神经网络对嵌入函数和Koopman算子进行参数化，然后用K步损失函数对它们进行端到端的训练。然后，一个辅助控制网络被增加到编码的非线性状态相关的控制项，以模拟控制输入中的非线性。然后在线性嵌入空间中引入线性二次型调节器（LQR），推导出最优控制策略，并从控制网络中解码出实际的控制输入。实验结果表明，我们的方法优于其他现有的方法，减少了数量级的预测误差，并在几个非线性动力学系统，如阻尼摆，CartPole，和七自由度机器人机械手实现优越的控制性能。
英文摘要：
Recently Koopman operator has become a promising data-driven tool to facilitate real-time control for unknown nonlinear systems. It maps nonlinear systems into equivalent linear systems in embedding space, ready for real-time linear control methods. However, designing an appropriate Koopman embedding function remains a challenging task. Furthermore, most Koopman-based algorithms only consider nonlinear systems with linear control input, resulting in lousy prediction and control performance when the system is fully nonlinear with the control input. In this work, we propose an end-to-end deep learning framework to learn the Koopman embedding function and Koopman Operator together to alleviate such difficulties. We first parameterize the embedding function and Koopman Operator with the neural network and train them end-to-end with the K-steps loss function. Then, an auxiliary control network is augmented to encode the nonlinear state-dependent control term to model the nonlinearity in the control input. This encoded term is considered the new control variable instead to ensure linearity of the modeled system in the embedding system.We next deploy Linear Quadratic Regulator (LQR) on the linear embedding space to derive the optimal control policy and decode the actual control input from the control net. Experimental results demonstrate that our approach outperforms other existing methods, reducing the prediction error by order of magnitude and achieving superior control performance in several nonlinear dynamic systems like damping pendulum, CartPole, and the seven DOF robotic manipulator.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
