# Robust Static Structural System Identification Using Rotations
## 2 Methodology
### Procedure for the Statistical Analysis of the Distribution of Estimations  
In Step 3 (Figure 1) of the structural system identification using COM described in Section 2.1, instead of obtaining numerical estimations of the variables, the analytical expressions can be obtained symbolically using Gaussian elimination. The obtained analytical expression of the structural parameters is in the form of a fraction with the measurements in the denominator. In this paper, it is assumed that measurement errors follow a normal distribution. Random errors are added to theoretical displacements (usually generated by the FEM) in a proportional manner, as indicated by Equation (10). \[\widetilde{\delta}=\delta_{r}\cdot(1+E_{level}\cdot\xi) \tag{10}\]  
where \(\delta_{r}\) is the displacement obtained from FEM and \(E_{level}\) is the error level; \(\xi\) is a random number that follows a normal distribution with zero mean and standard deviation 0.5. The setting of this value is to be consistent with previous research. The effect of choosing such a value is the decrease in the dispersion of the distribution of errors. Hence, it reduces the chance of obtaining more extreme errors. As the random variables are in the denominator, the definition of inverse distribution is introduced here. Let \(X\) be a random variable and the random variable \(Y\) be the inverse of \(X\), i.e., \(Y=1/X\). Then, the distribution of \(Y\) is the inverse distribution of \(X\). A closed-form solution for the probability density function (PDF) of the distribution of \(Y\) is available when the random variable \(X\) follows a normal distribution [45]. If the normally distributed  
Figure 1: Flowchart of the algorithm for structural system identification by COM. variable \(X\) has a mean of \(\mu\) and standard deviation of \(\sigma\), i.e., \(X\)\(\sim\)\(N(\mu,\sigma^{2})\), the PDF of the random variable \(Y=1/X\) (or the inverse distribution of \(X\)) is given by Equation (11). \[p_{Y}(y|\mu,\sigma)=\frac{1}{\sqrt{2\pi}\sigma y^{2}}e^{-\frac{\left|\frac{ \left|\frac{\left|\theta\right|}{2}\right|-\mu^{2}}{2\sigma^{2}}}} \tag{11}\]  
The distribution of \(Y\) is bimodal, and it has a negative mode at \(y_{1}\) and a positive mode at \(y_{2}\). \[y_{1}=-\frac{\mu+\sqrt{\mu^{2}+8\sigma^{2}}}{4\sigma^{2}}\,\ y_{2}=\frac{-\mu+ \sqrt{\mu^{2}+8\sigma^{2}}}{4\sigma^{2}} \tag{12}\]  
The mean and standard deviation for the distribution of \(Y\) at a specified interval can be calculated by associated integrations of the PDF \(p_{Y}\). These formulas can be found in classic statistic books [45]. With Equations (11) and (12) and the analytical expression of the parameters, the procedure to carry out statistical analysis of the distribution of estimations can be summarized below, and is depicted in Figure 2. Illustrative examples are provided and discussed in detail in Section 3. Step 1: Define a FEM for the structure to be analyzed according to the targeted accuracy of estimations. Step 2: Choose a measurement set to obtain the analytical expression for the target parameter \(\theta\), employing structural system identification using COM (see Section 2.1). Rewrite this expression for \(\theta\) as the reciprocal of an expression denoted by \(D_{demom}\), i.e., \(\theta=1/D_{demom}\). Step 3: Calculate the theoretical displacements of the structure using the finite element method. Step 4: Analyze the distribution of \(D_{demom}\) using Equation (10) and the theoretical values obtained in Step 3. Step 5: Analyze the distribution of \(\theta=1/D_{demom}\) using Equations (11) and (12). In the parameter estimation problem in power systems, if a measurement set of \(n\) measurements is able to identify all the \(n\) parameters and the drop in any measurement fails to do so, then this set is defined as an essential set [47]. The availability of analytical expressions for static structural system identification using COM depends on the number and the type of measurements in Step 2.