Statistical System Identification Based On The Analytical Probabilistic Model Of Transmissibility Function

System identification is of primary importance in the studies of catastrophism mechanism and structural health monitoring for engineering structures.Unfortunately,most of the system identification approaches available require measuring the system input precisely.As a mathematical representation of the output-to-output relationship,transmissibility functions have proven to be an effective and promising tool in system identification.To avoid measuring system input,this thesis aims to utilize the unique property of dynamic transmissibility which is independent of the input as well as the advantage of Bayes’ theorem in quantifying the uncertainties so that multiple uncertainties such as measurement noise and modelling error existing in system identification can be well accommodated.Issues of theoretical,computational and practical realization natures with computer are investigated deeply,drawing experiences from theoretical,numerical and experimental studies.Under the support of the National Science Foundation of China entitled “Application of Statistical Properties of Transmissibility function to Modal Identification of Bridge Engineering”,the following work has been done: 1.Before applying the transmissibility function in statistical system identification,one primary concern is to propose a new probabilistic model for it.Defined as the ratio of Fourier transform(FFT)coefficients of two measurements,transmissibility function can be modeled as a ratio random variable.When multiple response measurements are obtained simultaneously,one can formulate ratio random vector.On the basis of statistics of raw FFT coefficients and circularly-symmetric complex normal ratio distribution,explicit closed-form probabilistic model is established for scalar transmissibility functions.The statistical structure of the probabilistic model is concise,compact and easy-implemented.2.The Bayesian system identification approach is proposed based on the probabilistic model of transmissibility.In the Bayesian framework,the likelihood function can be derived,while the statistical system identification is converted to an optimization problem by updating the posterior probability distribution of the parameters.Also,two kinds of algorithms including Laplace approximation approach and stochastic sampling approach are introduced to identify the optimal values of the physical parameters as well as their uncertainties.3.The objective function of the Bayesian system identification involves calculating thedeterminant and the inverse of a covariance matrix which is used to characterize the variance between the measured response and the theoretical response.When approaching the system poles,the covariance matrix is ill-posed.Therefore,small perturbation will cause significant computation error,leading to diverge of the problem concerned.To address the critical issue,the determinant and inverse of the covariance matrix are derived analytically through employing advanced linear algebra techniques,thus one can avoid calculating these values using direct numerical algorithm.4.In the optimization process using either Laplace approximation approach or stochastic sampling approach,the identification results of the parameters are affected by the initial values preset.Given that the initial values deviate the true values too much,the computation efficiency can be affected significantly.Sometimes,it may cause divergence of the problem concerned.To address this issue,the asymptotic expression of the initial value of prediction error is estimated by deriving the first derivative of the objective function with respect to the parameters to be identified using approximate analysis theory.5.Based on the responses of a numerical example and an experimental study,the accuracy and robustness of the proposed model updating methodology are validated.Results show that,by comparing with the method based on frequency response function which requires measuring the input,the method based on transmissibility function can obtain satisfactory results with similar accuracy but avoid measuring the input information.Furthermore,the effects of sampling time length,selection of reference point,noise level and frequency band selection on the identification results are discussed,thereby shedding some insights into the uncertainty propagation law of the proposed method.