Empirical Differential Gramians for Nonlinear Model Reduction
Empirical Differential Gramians for Nonlinear Model Reduction
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DOI：
10.1016/j.automatica.2021.109534
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发表时间：
2019-02
期刊：
影响因子：
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通讯作者：
Y. Kawano;J. Scherpen
中科院分区：
文献类型：
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作者：
Y. Kawano;J. Scherpen
在本文中，我们提出了一种用于输入向量场为常数的非线性系统的经验平衡截断方法。首先，我们定义差分可达性和可观测性格拉米亚。它们是状态轨迹（即初始状态和输入轨迹）的矩阵值函数，很难找到它们作为初始状态和输入的函数。本文的主要结果是表明，对于固定状态轨迹，可以通过使用变分系统的脉冲和初始状态响应来计算这些格拉米亚式的值。因此，与传统的非线性平衡方法不同，平衡截断可以沿着固定状态轨迹进行，而无需求解非线性偏微分方程。我们进一步开发了一种近似方法，它只需要原始非线性系统的轨迹。
In this paper, we present an empirical balanced truncation method for nonlinear systems whose input vector fields are constants. First, we define differential reachability and observability Gramians. They are matrix valued functions of the state trajectory (i.e. the initial state and input trajectory), and it is difficult to find them as functions of the initial state and input. The main result of this paper is to show that for a fixed state trajectory, it is possible to compute the values of these Gramians by using impulse and initial state responses of the variational system. Therefore, balanced truncation is doable along the fixed state trajectory without solving nonlinear partial differential equations, differently from conventional nonlinear balancing methods. We further develop an approximation method, which only requires trajectories of the original nonlinear systems.