Purpose of Model Updating

There are two main purposes of model updating or system identification. One common goal is to identify physical parameters, e.g., stiffness of a structural element or the diffusion rate of an air pollutant. These identified parameters can be further used as indicator for the status of the system or phenomenon. For example, the stiffness parameter of a structural member can be monitored from time to time and an abnormal reduction indicates possible damage of the member. However, reduction may be simply due to statistical uncertainty. Therefore, it is necessary to quantify the uncertainty of the estimation so that one can distinguish whether the parameter change is due to deterioration of the structural member. In this case, it is desirable to obtain a narrow distribution of the parameter so that small changes can be detected with a high level of confidence. 

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. 

Using the quantified uncertainty obtained from Bayesian methods, there are two important types of applications. The first category is robust reliability analysis. Under severe earthquake excitations, buildings and bridges may exhibit significant nonlinear behavior. With a stochastic representation of the anticipated ground motions [96,130,237], one important reliability problem is to determine the first passage probability of some response quantities of interest in a 

The aforementioned methods can be applied to evaluate the reliability of engineering systems subjected to stochastic input with a given mathematical model. On the other hand, if a parametric model of the underlying system is available and the probability density function of these parameters is obtained by Bayesian methods, the uncertain parameter vector can be augmented to include the model parameters and the uncertain input components. Then, robust reliability analysis can proceed for stochastic excitation with an uncertain mathematical model. This allows for more realistic reliability evaluation in practice so that the modeling error and other types of uncertainty of the mathematical model can be taken into account. 

Over the last two decades, there has been increasing interest in probabilistic, or stochastic, robust control theory. Monte Carlo simulation methods have been used to synthesize and analyze controllers for uncertain systems [170,255], First- and second-order reliability methods were incorporated to compute the probable performance of linear-quadratic-regulator 

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