This paper proposes a novel nonlinear dynamical system identification method based on the sparse regression algorithm and the separable least squares method. To effectively avoid solving the second derivative of the displacement signal and reduce the effect of noise, the Duhamel's integral is adopted to represent the dynamic relationship between the system input and output. In the expression form of Duhamel's integral, nonlinear dynamical system identification can be cast as a separable least squares problem. Thus, the separable least squares method is leveraged to separately identify the parameters of the linear subsystem and the coefficients corresponding to nonlinearities among the nonlinear dynamical system. During the identification process of nonlinear restoring forces, one complete set of nonlinear basis functions are used to represent the nonlinear restoring forces. Not all the candidate nonlinear terms are contributing, however, thus the sparse regression algorithm is adopted to select the actual contributing nonlinear components in the candidate nonlinear terms and eliminate the non-contributing nonlinear components, and then the corresponding parameters of contributing nonlinear components are estimated by the unbiased least squares method. Finally, one RKHS (Reproducing Kernel Hilbert Space)-based non-parametric de-noise method is further proposed to reduce the noise in the vibration displacement and obtain the noise-reduced velocity from the displacement signal. The numerical simulation about the identification of the rotating blade-casing system and the dynamic experiment of the HSLDS (high-static-low-dynamic stiffness) isolator system verify the effectiveness of the new identification method for nonlinear dynamical systems proposed in this paper.

提出了一种基于稀疏回归算法和可分离最小二乘法的非线性动力学系统辨识方法。为了有效避免求解位移信号的二阶导数并减少噪声的影响，采用杜哈默尔积分来表示系统输入与输出之间的动态关系。在Duhamel积分的表达形式中，非线性动力学系统的辨识可以看作是一个可分离的最小二乘问题。因此，利用可分离的最小二乘法来分别识别线性子系统的参数和非线性动力学系统中与非线性相对应的系数。在非线性恢复力的识别过程中，一组完整的非线性基函数用于表示非线性恢复力。并不是所有的候选非线性项都起作用，因此，采用稀疏回归算法在候选非线性项中选择实际的贡献非线性分量，并消除非贡献的非线性分量，然后估算出贡献的非线性分量的对应参数。通过无偏最小二乘法。最后，进一步提出了一种基于RKHS（再现核希尔伯特空间）的非参数降噪方法，以减少振动位移中的噪声并从位移信号中获得降噪的速度。