Estimation of Multi‐Order Spectra for Nonlinear Closed‐Loop Systems

Abstract

In this paper, a multi-order spectra estimation method is proposed for a nonlinear closed-loop system based on the Volterra series. Owing to the correlation between the noise and the input, the estimation accuracy is poor when the nonlinear spectra of the plant is obtained using the traditional estimation method. In order to overcome this problem, a two-step scheme is used to estimate the multi-order spectrum of a nonlinear system operating in closed-loop. Firstly, the generalized frequency response functions (GFRFs) from the reference signal to the input of the plant are estimated, and they are used to simulate the noise-free input spectra of the plant. Secondly, the GFRFs of the controlled plant are estimated using the noise-free input spectra and the output spectra. Because the GFRFs are multidimensional functions, the required amount of calculation for the estimation is very large. To reduce computational complexity, a simplified GFRF model is adopted to estimate the multi-order nonlinear spectrum of the plant. In this model, the GFRF is transformed to a one-dimensional function. Two simulation experiments are provided to illustrate the proposed approach.

Numerical solution of optimal control problems governed by integro‐differential equations

“…We next approximate the performance index of the optimal control problem under study. Substituting Equations (17) and (18) into Equation (13) and using the properties of the Kronecker product gives that…”

“…Various types of optimal control problems have been investigated by these methods. Up to now, much effort has been committed to the development of various approaches for solving optimal control problems governed by ordinary differential equations [5][6][7][8][9][10][11][12][13][14]. However, a few research works have been devoted to the theoretical aspects and numerical investigation of optimal control problems described by integro-differential equations such as dynamic programming [15], direct methods based on Legendre's polynomials [16], Chebyshev's polynomials [17,18], Legendre pseudospectral approach [19], and Chebyshev pseudospectral method [20].…”

Fault mechanism analysis and diagnosis for closed-loop drive system of industrial robot based on nonlinear spectrum

“…The analysis method by nonlinear spectrum is to transform the Volterra kernel in time domain into frequency domain by multi-dimensional Fourier transformation, which can obtain the nonlinear spectrum transfer characteristics of the system. Theoretical and experimental results indicate that when the system state changes, the nonlinear spectrum characteristics will also change 15 , 16 .Therefore, the nonlinear spectrum characteristics can be adopted to analyze the fault mechanism of complex system. Up to now there are two models of nonlinear spectrum: generalized frequency response function (GFRF) and nonlinear output frequency response function (NOFRF).…”

Fault Diagnosis of Nonlinear Analog Circuit Based on Generalized Frequency Response Function and LSSVM Classifier Fusion

“…For the prediction of nonlinear aeroelastic vibration response and flutter, a frequency component identification method based on correlation analysis is used to estimate the GFRFs [19]. In [20], a two-step strategy is used to estimate the generalized frequency response function of the nonlinear controlled plant in the closed-loop state. For the above estimation methods, the input and output data need to be Fourier-transformed in the process of solving GFRFs.…”