Advantages of Bilinear Koopman Realizations for the Modeling and Control of Systems With Unknown Dynamics
Advantages of Bilinear Koopman Realizations for the Modeling and Control of Systems With Unknown Dynamics
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DOI：
10.1109/lra.2021.3068117
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发表时间：
2020-10
影响因子：
5.2
通讯作者：
Daniel Bruder;Xun Fu;Ram Vasudevan
中科院分区：
文献类型：
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作者：
Daniel Bruder;Xun Fu;Ram Vasudevan
通过将非线性动力系统提升到可观测函数的空间中，可以使非线性动力系统更容易控制，其中它们的演化由线性Koopman算子描述。这封信描述了如何使用Koopman算子从数据中生成近似的线性、双线性和非线性模型实现，并主张用双线性实现来表征具有未知动态的系统。给出了一个动力系统在给定的可观测函数集上有一个有效的线性或双线性实现的充要条件，并用来证明每个控制仿射系统都有一个无限维的双线性实现，但不一定有一个线性实现。因此，从通用基函数集合构造的近似双线性实现倾向于随着基函数数量的增加而改进，而近似线性实现可能不会。为了证明双线性Koopman实现控制的优点，从数据中构建了模拟机器人手臂的线性，双线性和非线性Koopman模型实现。在轨迹跟踪任务中，当结合到模型预测控制框架中时，双线性实现超过线性实现的预测精度和非线性实现的计算效率。
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 letter 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 is 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.