Volterra Series Based Nonlinear System Identification And Its Application

As we all know, the actual system always contain a variety of nonlinear factors, such as the gap, dry friction, oil film in mechanical system; large deformation in structural system, the nonlinear constitutive relationship related with stresses and strains; nonlinear control strategy in control system, and so on. Generally speaking, linear analysis can be adopted in such a case where precision requirement of system is low or the nonlinearity has little effect on the performance of system. Essentially, linear analysis is simplified from the actual system. In general, the linear model can almost capture the actual behavior of a system, however, in recent years, with the development and progress of science and technology, the linear approximation is not always reliable for the increasing requirements. If the nonlinear factors are overlooked in the analysis and calculation, it can result in unacceptable error. Moreover, more and more nonlinear phenomena have attracted people’s attention in the practical engineering, nonlinear problems have become a hot research topic. Therefore, we need to study nonlinear systems, reveal the nature of nonlinear systems, which have a great significance for analysis and design of nonlinear system. Therefore, people developed a variety of mathematical theories and methods for the modeling, solution and analysis of nonlinear system, among which the Volterra series is one of the most widely used methods.Volterra series is a description of the nonlinear relationship between system input and output, and it is a powerful mathematical tool for the analysis of nonlinear system, which is essentially an extension of the standard convolution description of linear system to nonlinear system. The applications of Volterra series widely range from aero-elastic system, biomedical engineering, fluid dynamics, control engineering, electrical engineering to mechanical engineering, etc.The contents of this thesis overall involve two aspects: the identification and application of nonlinear system based on Volterra series, which also can be divided into the following five parts in detail.Firstly, a wavelet balance method based approach is proposed to identify the Volterra kernel functions from observations of the in- and outgoing signals. The basic routine of the approach is that, from the system outputs under multilevel excitations, the Volterra series outputs of different order are first estimated with the wavelet balance method, and then the Volterra kernel functions of different order are separately estimated through their corresponding Volterra series outputs by expanding them with four order B-spline wavelet on the interval(BSWI).The identification method proposed in this paper convert the Volterra kernel function identification problem into an estimation problem of a few expansion coefficient, which can effectively reduce the number of unknown parameters in Volterra kernel function identification, and can improve the identification efficiency.Secondly, the thesis used the spatio-temporal Volterra series to model the nonlinear distributed parameter system(DPS). The key issue involved in modeling nonlinear DPS using spatio-temporal Volterra series is the identification of its kernel functions. In order to reduce the difficulty in modeling the nonlinear DPS, from the system outputs, the Karhunen-Loève(KL)decomposition is first used for the time/space separation, a low-order model can be derived; second, according to the wavelet basis expansion-based Volterra kernel function identification method through multilevel excitations proposed in this thesis, model the low-order nonlinear systems, respectively; Finally, using the time/space synthesis, the spatio-temporal Volterra model can be reconstructed.Thirdly, based on Volterra series analytical method, the block-oriented nonlinear systems are identified in this thesis. The relationships between Hammerstein, Wiener models and cascade of Hammerstein model and their associated Volterra models are firstly presented in this thesis. We prove that all these three block-oriented nonlinear systems can be represented by Volterra series. The basic routine of the identification approach for Hammerstein and Wiener model is that, from the system outputs under multilevel excitations, the Volterra series outputs of different order are estimated with the wavelet balance method. According to the first-order Volterra output and the input data, the impulse response function of the linear subsystem of Hammerstein and Wiener models can be obtained. Then, based on each higher order Volterra outputs and the impulse response function of the linear subsystem, the coefficients of the polynomial nonlinear subsystem of Hammerstein and Wiener models can be estimated, respectively. The basic routine of the identification approach for the cascade of Hammerstein model is that, from the system outputs under multilevel excitations, the Volterra series outputs of different order are first estimated with the wavelet balance method. Then, through each order Volterra outputs and input, the impulse response functions of each order linear subsystems can be estimated, respectively.Fourthly, based on nonlinear output frequency response function developed from Volterra series, a novel approach is developed to detect the position of nonlinear components in two dimensional structures. On account of important properties about the NOFRF of 2D structures with local nonlinearity in row and column directions, the position of nonlinear components in 2D structures are determined in row and column directions, respectively; then combining with the location information in row and column directions, the position of nonlinear components in 2D structures can be determined.Finally, in this thesis, a novel approach is proposed to detect damage in structures based on the Volterra kernel functions identification. The damage detection method is extremely sensitive to the appearance of crack in structures, and can therefore be used as crack detection indicator to indicate the existence and the size of crack. The new damage detection approach mainly includes three steps. First, the Volterra kernel functions are identified from the input-output data. Then, the Volterra kernel functions-based indexes are calculated. Finally, damage detection is conducted by comparing the values of the Volterra kernel functions-based indexes of the inspected structure with the values of the indexes for a uncrack structure. In addition, based on the important properties about the NOFRF of 2D structures with local nonlinearity in row and column directions, a novel approach for structural damage localization is proposed.