Research On A Nonlinear Auto-regressive Time Series Model And Its Application

As a branch of mathematical statistics,time series analysis is widely used in social sciences,natural sciences,management engineering,engineering technology,etc.In the last half century,its application in mechanical engineering has developed extremely rapidly,especially in prediction,spectral analysis,fault diagnosis,surface topography recognition,modal parameter estimation,structural damage detection,etc.Now,theoretical research of linear time series model has already been quite mature.Besides,nonlinear time series analysis develops well in the recent years but still has great development space and research value.On the basis of general expression for linear and nonlinear auto-regressive models(GNAR),a GNAR model with external inputs(GNARX)is proposed.Additionally,GNARX model's basic characteristics,parameter estimation and structure identification,applicability test,application to pattern recognition and fault diagnosis,and application to structural damage detection are systematically researched.The main research content of this paper can be summarized as follows:(1)For the point that parts of inputs of the system are known,compared with traditional system identification and general time series modeling strategy,the superiority of modeling strategy for time series with external inputs is researched.Then,GNARX model is proposed and its expression is deduced.With analysis of relationship between GNARX model and other models,i.e.,GNAR model,traditional input-output model,Volterra series model,and Sigma-Pi neural network,implication of GNARX model is elaborated.With modeling of AR,ARX,GNAR,and GNARX data,general applicability of GNARX model is verified.With modeling of data under different kinds of white noise and inputs,good robustness of GNARX model's linear and nonlinear parameters is proved.Compared with other models from two perspectives,theoretical analysis and data verification,good modeling and forecasting ability of GNARX model is expounded.(2)Deficiencies of least square(LS)and weighted least square(WLS)method are researched.The common problem of the two is that both of them only consider the mean property of data but do not consider the other statistical property of data.Thus,on the basis of consideration of total data's secondary moment property,modified Mahalanobis distances least square(MMDLS)method is proposed.It is used for GNARX model's parameter estimation,of which the effectiveness is better than LS and WLS.For the structure features of GNARX model,complexity of structure exhaustive search and instability of random pruning method are researched.And structure pruning algorithm based on parameters' rate of standard deviation is proposed.It is used for GNARX model's structure identification.With modeling of simulation and real data,compared with other algorithm,superiority of structure pruning algorithm based on parameters' rate of standard deviation is verified.(3)As correlation test,the problem is researched that autocorrelation coefficient criterion can be used for applicability test of linear time series model but turn out to be ineffective for nonlinear time series model.Thus,multi-cross-correlation coefficient criterion is proposed for applicability test of GNARX model.With simulation data comparison,superiority of the criterion is proved.As frequency domain test,the problem is researched that residual power spectrum test is suitable for linear model but not nonlinear model.The method of bispectrum test with standard residual is proposed for applicability test of GNARX model.This method calculates the integral bispectrum in triangular primary domain of standard residual and compares it with that of Gaussian white noise with the same data length.The data simulation validates this method.(4)With engineering examples,the general steps of time series method applied to pattern recognition and fault diagnosis are researched.Pattern vector selection method based on multi-state model structure identification is proposed.It effectively solves the problem that is difficult to use the same GNARX model structure to model data of one system but in different states.With simulation data,the feasibility of this method is verified.As GNARX model parameters of different within-class distances make different contribution to the classification accuracy,parameter importance is defined.Different importance coefficients are assigned to different parameters.Then,feature vector extraction method based on parameter importance is proposed,which further improves the fault diagnosis accuracy.With simulation data,the effectiveness of this method is verified.(5)With simulation of spring-mass-damper system,GNARX modeling mechanism for vibration signal is researched,which is also generalized for crane girder vibration signal.Further,structural damage detection method without baseline model is proposed.Then finite element model of bridge crane girder is developed.When heavy objects are lifted,the impact load on girder is simulated.Vibration signals at different positions are acquired to verify feasibility of the method.Bridge crane model is set up in the laboratory,girder models with different section dimensions and cracks with different positions and different depths are manufactured.When the crane lifts or lowers heavy objects,girder vibration signals at different positions are collected.From an experimental point of view,effectiveness of GNARX model applied to structural damage detection is further verified.