Parametric identification

Abstract. The thin-film protective coat of titanium nitride (TiN) plotted to stainless steel (brand 12X18H10T) is explored. The mathematical model and methods of parametric identification are described. Kinetic parameters of hydrogen permeability through stainless steel membrane with TiN protective coat are determined. 

However kind of solutions depends on parameters of model and initial conditions. That is why while studying the processes proceeding in real systems by methods of mathematical modelling the problems of parametrical identification, i.e. problems of model s parameters determination by observations data are appeared. 

Structure of the differential operator B 0 is determined (Block 6) on the basis of the physical laws, describing the thermo-mass-transfer processes (analytical method), or, alternatively, as a result of the structure-parametric identification tasks (experimental-cj ulative method). 

Another purpose of model updating is to obtain a mathematical model to represent the underlying system for future prediction. Even though there are also parameters to be identified as in the previous case, these parameters may not necessarily be physical, e.g., coefficients of auto-regressive models. In this situation, the identified parameters are not necessarily as important as the previous case provided that the identified model provides an accurate prediction for the system output. It will be shown in the following chapters that there is no direct relationship between satisfactory model predictions and small posterior uncertainty of the parameters. This point will be further elaborated in Chapter 6. Nevertheless, no matter for which purpose, quantification of the parametric uncertainty is useful for further processing. For example, it can be utilized for comparison of the identified parameter values at different stages or for uncertainty analysis of the output of the identified model. Furthermore, it will be demonstrated in Chapter 6 that quantification of the posterior uncertainty allows for the selection of a suitable class of models for parametric identification. 

What are the properties of a suitable model class for parametric identification ... 

In science and engineering problems, there are various uncertain parameters necessary to be determined for modeling and other purposes. The Bayes theorem offers the possibility for inferencing uncertain models/systems from their measurements. There are two levels of system identification. The first is parametric identiflcation, in which a class of mathematical models for a particular physical phenomenon or system is given with unknown parameters to be identified. The second level deals with the selection of a suitable class of mathematical models for parametric identification. This is significantly more difficult but more important than the first level since parametric identification results will be by no means meaningful if one fails to obtain a suitable class of models. However, due to the difficulty of this problem, it is usually determined by user s judgement. Chapters 2-5 focus on parametric identification and Chapter 6 addresses the problem of model class selection. 

Once the conditional optimal values for b and are obtained, the value of the goodness-of-flt function can be computed by Equation (2.121) whereas the normalizing constant in the posterior PDF does not affect the parametric identification results. By maximizing the goodness-of-fit function with respect to n, the updated model parameters can be obtained. Therefore, the closed-form solution of the conditional optimal parameters reduces the dimension of the original optimization problem from - - - - 1 to N only. 

The Bayesian spectral density approach for parametric identification and model updating regression analysis are applied. During the monitoring period, four typhoons flitted over Macao. The structural behavior under such violent wind excitation is treated as discordance and the measurements obtained under these events are not taken into account for the analysis. By excluding these fifteen days of measurements, there are 168 pairs of identified squared fundamental frequency and measured temperature in the data set. Figure 2.28(a) shows the variation of the identified squared fundamental frequencies with their associated uncertainties represented by a confidence interval that is bounded by the plus or minus three standard derivations from the estimated values. It is noticed that this confidence interval contains 99.7% of the probability. Since the confidence intervals are narrow compared with the variation... 

The problem of parametric identification for mathematical models using input-output or output-only dynamic measurements has received much attention over the years. One important special case is modal identification, in which the parameters for identification are the small-amplitude modal frequencies, damping ratios, mode shapes and modal participation factors of the lower modes of the dynamical system. In other words, the model class in modal identification is the class of linear modal models. Many time-domain and frequency-domain methodologies have been formulated for input excitation and output response measurements [24,48,75,81,187],... 

Chapter 3 presented the Bayesian spectral density approach for the parametric identification of the multi-degree-of-freedom dynamical model using the measured response time history. The methodology is applicable for linear models and can also be utilized for weakly nonlinear models by obtaining the mean spectrum with equivalent linearization or strongly nonlinear models by obtaining the mean spectrum with simulations. The stationarity assumption in modal/model identification for an ambient vibration survey is common but there are many cases where the response measurements are better modeled as nonstationary, e.g., the structural response due to a series of wind gusts or seismic responses. In the literature, there are very few approaches which consider explicitly nonstationary response data, for example, [226,229]. Meanwhile, extension of the Bayesian spectral density approach for nonstationary response measurement is difficult since construction of the likelihood function is nontrivial in the frequency domain. Estimation of the time-dependent spectrum requires a number of data sets, which are associated with the same statistical time-frequency properties but this is impossible to achieve in practice. 

The Bayesian time-domain approach presented in this chapter addresses this problem of parametric identification of linear dynamical models using a measured nonstationary response time history. This method has an explicit treatment on the nonstationarity of the response measurements and is based on an approximated probability density function (PDF) expansion of the response measurements. It allows for the direct calculation of the updated PDF of the model parameters. Therefore, the method provides not only the most probable values of the model parameters but also their associated uncertainty using one set of response data only. It is found that the updated PDF can be well approximated by an appropriately selected multi-variate Gaussian distribution centered at the most probable values of the parameters if the problem is... 

In Chapter 2, Section 2.4, parametric identification was introduced for linear and nonlinear regression problems. In this section, the Bayesian model class selection is applied to these problems. In order to smooth the presentation, some of the equations from Section 2.4 are repeated in this section. 

The Bayesian spectral density approach in Chapter 3 is used for parametric identification. The spectral density estimator is utilized up to 8 Hz to include all the peaks so No, = 480. Table 6.2 shows the optimal modal frequencies for model classes with different number of modes. There is in general no difficulty in identifying the first five modes but it is not the... 

Prior distribution does not significantly affect the parametric identification (both identified values and associated uncertainty) results if it is sufficiently fiat in the range with significant likelihood values. Therefore, it is common to absorb the prior distribution into the normalizing constant and the results are equivalent to the maximum likelihood solution. However, it is not appropriate to absorb the prior distribution into the normalizing constant for model class... 

Ashrafi, S.A., Smyth, A.W. Betti, R. 2005. A parametric identification scheme for nondeteriorating and deteriorating non-linear hysteretic behavior. Structural Control and Health Monitoring, 13,108-131. 

Smyth, A.W., Masri, S.R, Chassiakos, A.C. Caughey, T.K. 1999. On-line parametric identification of MDOF non-linear hysteretic systems, Journal of Engineering Mechanics, 125, 133-142. 

The third issue to be introduced is on the online model class selection. The usual approach in system identificatimi is to find the best/optimal model in a prescribed class of models, e.g., class of shear building models with uncertain interstory stiffnesses. This problem is commonly referred to as parametric identification. The more general problem of model class selection has not been explored as intensively as parametric identification. It is well known that a more complicated model class often fits the data better than one which has fewer adjustable parameters. 

In section Structural Parametric Identification by Extended Kalman Filter, online structural parametric identification using the EKF will be briefly reviewed. In section Online Identification of Noise Parameters, an online identification algorithm for the noise parameters in the EKF is introduced. Then, in section Outlier-Resistant Extended Kalman Filter, an online outlier detection algorithm is presented, and it is embedded into the EKF. This algorithm allows for robust structural identification in the presence of possible outliers. In section Online Bayesian Model Class Selection, a recursive Bayesian model class section method is presented for non-parametric identification problems. 

Structural Parametric Identification by Extended Kalman Filter... 

In this section, the outlier-resistant extended Kalman filter (OR-EKF) is introduced for online outlier detection and robust structural parametric identification using dynamic response data. In this algorithm, an online outlier detection algorithm is embedded into the EKF. Section Online Outlier Detection by Outlier Probability introduces the concept of outlier probability, and it will be utilized for online outlier detection. This outlier detection algorithm is embedded in the EKF for online structural identification. Section Procedure of the Outlier-Resistant Extended Kalman Filter summarizes the procedure of the OR-EKF algorithm. 

First, a distinction can be made between non-parametric and parametric identification. Non-parametric system identification involves the estimation of an impulse response function, frequency response function (FRF), correlation function, or power spectral density (PSD), not as a mathematical function depending on a few parameters, but as a set of tabulated values for each considered time lag or frequency. Although nonparametric models are sometimes directly used for modal analysis, they are most often used as preprocessed data for parametric identification since the estimation accuracy of parametric approaches is much higher than that of nonparametric approaches (Peeters and De Roeck 2001 Reynders 2012). 

The long-term activity at the University of L Aquila aims to develop innovative automated techniques, which permit to automatically update the appropriately selected models, which in turn may address questions regarding the seismic safety of the monitored structures. In particular, following a parametric identification approach, using the distributed available measurements from sensors embedded in the... 

Early studies of parametric identification of hysteresis typically employed the non-degrading classical differential model containing only five parameters (Kyprianou et al. 2001 Ni... 

The model is obtained through the approach of a set of linear transfer functions that includes the nonlinearities of the whole system. The parametric identification process is based on black box models (Ljung, 1987 Norton, 1986). The nonholonomic system dealt with in this work is considered initially to be a MIMO (Multiple-Input Multiple-Output) system, as shown in Figure 4, due to the d5mamic influence between two DC motors. This MIMO system is composed of a set of SISO (single input single output) subsystems with coupled connections. 

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