Despite the simplicity of the Duffing oscillator, its dynamical behaviour is extremely rich. Hence, the Duffing equations are used to describe the dynamic behaviour of many real-world nonlinear systems for a wide range of frequency bands and amplitude of the excitation signal in basic sciences and engineering. For example, the Duffing oscillator has been successfully used to model a variety of physical processes such as stiffening springs, beam buckling, nonlinear electronic circuits, superconducting Josephson parametric amplifiers, and ionisation waves in plasmas etc. Therefore, the identification of the Duffing oscillator model/parameters directly from the measured input–output data is a topic of active research in many scientific fields In this paper, we use the concept of stochastic differential equations (SDEs) to identify a model of the Duffing oscillator. SDE-based grey-box models allow us to capture the underlying mathematical structure describing the physics of the system (e.g. the original Duffing equations) using the drift term and explicitly handling of model uncertainty (or the process noise) using the diffusion term whereas the measurement uncertainty is modelled using the measurement noise term respectively. In this paper, we propose a slight variation of the maximum likelihood estimation framework used for the identification of SDEs based grey-box models yielding improved performance for long-term predictions. The proposed framework is combined with an iterative residual analysis to develop a grey-box model of the forced Duffing oscillator. The benchmark data from the so-called Brussels “Silverbox system”, which is an electrical circuit mimicking the forced Duffing oscillator dynamics is used for the identification purpose. Finally, the identified model performance (the simulation errors) is compared with the existing results available in the literature.

尽管 Duffing 振荡器很简单，但它的动态行为却非常丰富。因此，Duffing 方程用于描述许多现实世界非线性系统在基础科学和工程中的广泛频带和激励信号幅度的动态行为。例如，Duffing 振荡器已成功用于模拟各种物理过程，如加劲弹簧、梁屈曲、非线性电子电路、超导 Josephson 参量放大器和等离子体中的电离波等。因此，Duffing 振荡器模型的识别/parameters 直接来自测量的输入-输出数据是许多科学领域积极研究的主题在本文中，我们使用随机微分方程 (SDE) 的概念来识别 Duffing 振荡器的模型。基于 SDE 的灰盒模型允许我们使用漂移项捕获描述系统物理特性的底层数学结构（例如原始 Duffing 方程），并使用扩散项明确处理模型不确定性（或过程噪声），而测量不确定性分别使用测量噪声项建模。在本文中，我们提出了用于识别基于 SDE 的灰盒模型的最大似然估计框架的轻微变化，从而提高了长期预测的性能。所提出的框架与迭代残差分析相结合，以开发强制 Duffing 振荡器的灰盒模型。来自所谓的布鲁塞尔“Silverbox 系统”的基准数据，这是一个模拟强制 Duffing 振荡器动力学的电路，用于识别目的。最后，将确定的模型性能（模拟误差）与文献中现有的结果进行比较。