A nonlinear subspace-prediction error method for identification of nonlinear vibrating structures

Abstract

The industrial structural systems always contain various kinds of nonlinear factors. Recently, a number of new approaches have been proposed to identify those nonlinear structures. One of the promising methods is the nonlinear subspace identification method (NSIM). The NSIM is derived from the principals of the stochastic subspace identification method (SSIM) and the internal feedback formulation. First, the nonlinearities in the system are regarded as internal feedback forces to its underlying linear dynamic system. The linear and nonlinear components of the identified system can be decoupled. Second, the SSIM is employed to identify the nonlinear coefficients and the frequency response functions of the underlying linear system. A typical SSIM always consists of two steps. The first step makes a projection of certain subspaces generated from the data to identify the extended observability matrix. The second one is to estimate the system matrices from the identified observability matrix. Since the calculated process of the NSIM is non-iterative and this method poses no additional problems on the part of parameterization, the NSIM becomes a promising approach to identify nonlinear structural systems. However, the result generated by the NSIM has its deficiency. One of the drawbacks is that the identified results calculated by the NSIM are not the optimal solutions which reduce the identified accuracy. In this study, a new time-domain subspace method, namely the nonlinear subspace-prediction error method (NSPEM), is proposed to improve the identified accuracy of nonlinear systems. In the improved version of the NSIM, the prediction error method (PEM) is used to reestimate those estimated coefficient matrices of the state-space model after the application of NSIM. With the help of the PEM, the identified results obtained by the NSPEM can truly become the optimal solution in the least square sense. Two numerical examples with local nonlinearities are provided to illustrate the effectiveness and accuracy of the proposed algorithm, showing advantages with respect to the NSIM in a noise environment.