Dynamic mode decomposition with control (DMDc) is a powerful data-driven method for modelling dynamical systems with inputs and outputs. However, the inability to identify oscillations and the high sensitivity to noise limit the application of standard DMDc in structural dynamics. To address this challenge, this paper proposes a novel method called scaled derivative-based DMDc (sd-DMDc) to construct a discrete state-space model for mechanical systems using noisy data. This method requires only the displacement and excitation data, computes the scaled derivatives of displacement to extend the system states, and integrates the extended forward-backward strategy. Two crucial parameters are defined and optimized by minimizing the response error. The sd-DMDc algorithm is first verified through numerical simulations of a multiple-input multiple-output (MIMO) cantilever plate. The results indicate that sd-DMDc exhibits superior modelling accuracy across various noise levels and measurement points compared to the delay-DMDc method. It has also been proved that sd-DMDc can accurately identify dynamical systems from partial observations. Subsequently, vibration experiments are conducted on a MIMO cantilever plate. The sd-DMDc method is further verified using experimental data, and the conclusions align with those drawn from simulation studies. The sd-DMDc algorithm demonstrates great modelling accuracy, surpassing that of delay-DMDc, especially for partial observations. In addition, the model given by sd-DMDc can extract the modal parameters of the plate, which are consistent with the modal test results.

中文翻译：

---

一种基于比例导数的 DMDc 方法，用于对多输入多输出机械系统进行建模


带控制的动态模态分解 （DMDc） 是一种强大的数据驱动方法，用于对具有输入和输出的动态系统进行建模。然而，无法识别振荡和对噪声的高度敏感性限制了标准 DMDc 在结构动力学中的应用。为了应对这一挑战，本文提出了一种称为基于缩放导数的 DMDc （sd-DMDc） 的新方法，以使用噪声数据为机械系统构建离散状态空间模型。该方法只需要位移和激励数据，计算位移的缩放导数以扩展系统状态，并集成扩展的正向-后向策略。通过最小化响应误差来定义和优化两个关键参数。首先通过对多输入多输出 （MIMO） 悬臂板进行数值仿真来验证 sd-DMDc 算法。结果表明，与延迟 DMDc 方法相比，sd-DMDc 在各种噪声水平和测量点上表现出卓越的建模精度。研究还证明，sd-DMDc 可以从部分观测中准确识别动力学系统。随后，在 MIMO 悬臂板上进行振动实验。使用实验数据进一步验证了 sd-DMDc 方法，结论与仿真研究中得出的结论一致。sd-DMDc 算法表现出极高的建模精度，超过了 delay-DMDc，尤其是在部分观测方面。此外，sd-DMDc 给出的模型可以提取板的模态参数，这些参数与模态测试结果一致。