# Robust Static Structural System Identification Using Rotations
## 2 Methodology
### Structural System Identification Using the Constrained Observability Method
Hence, a (constant) modified matrix \([K^{*}]\) and a modified vector \(\{\delta^{*}\}\) composed of nonlinear products, e.g., \(E_{j}A_{j}u_{i}\), \(E_{j}I_{j}w_{i}\), \(E_{j}I_{j}v_{i}\), are obtained, as shown in Equation (4). \[[K^{*}]\cdot\{\delta^{*}\}=\{f\} \tag{4}\]  
Once the boundary conditions and the forces applied in the non-destructive static test have been defined, it is assumed that a subset of increments in deflections \(\{\delta_{1}^{*}\}\) of \(\{\delta^{*}\}\) are known and a subset of forces \(\{f_{1}\}\) are known, while the remaining parts \(\{\delta_{0}^{*}\}\) and \(\{f_{0}\}\) are unknown. By static condensation, Equation (4) can be partitioned as Equation (5). \[[K^{*}]\{\delta^{*}\}=\left[ \begin{array}{cc}K_{00}^{*}&K_{01}^{*}\\ K_{10}^{*}&K_{11}^{*}\end{array} \right]\left\{ \begin{array}{c}\delta_{0}^{* }\\ \delta_{1}^{*}\end{array} \right\}=\left\{ \begin{array}{c}f_{0}\\ f_{1}\end{array} \right\}=\{f\} \tag{5}\]To cluster the unknowns \(\{\delta_{0}^{*}\}\) and \(\{f_{0}\}\), Equation (5) can be rewritten equivalently as the observability equation, Equation (6). \(I\) and \(0\) are the null and the identity matrices. \[[B]\{z\}=\left[ \begin{array}{cc}K_{10}^{*}&0\\ K_{00}^{*}&-I\end{array} \right]\left\{ \begin{array}{c}\delta_{0}^{*}\\ f_{0}\end{array} \right\}=\left\{ \begin{array}{c}f_{1}-K_{11}^{*}\delta_{1}^{* }\\ -K_{01}^{*}\delta_{1}^{*}\end{array} \right\}=\{D\} \tag{6}\]  
Step 3: Check the null space matrix \([N]\) of the coefficient matrix \([B]\) to determine the observability of unknowns in \(\{z\}\) and obtain the numerical estimations for observable variables. The structure of the general solution \(\{z_{g}\}\) to Equation (6) is given by Equation (7). \[\{z_{g}\}=\{z_{p}\}+\{z_{h}\}=\left\{ \begin{array}{c}\delta_{0}^{*}\\ f_{0}\end{array} \right\}_{p}+[N]\cdot\{\rho\} \tag{7}\]  
\(\{z_{p}\}\) is a particular solution to Equation (6) and \(\{z_{h}\}\) is a solution to the homogeneous version of Equation (6) (i.e., the vector \([D]\) is replaced by zeros). \(\{z_{h}\}\) is a linear combination of the columns of the matrix \([N]\). The coefficients of the linear combination are denoted by the vector \(\{\rho\}\). The observability of unknowns in the vector \(\{z\}\) is determined by checking null rows in the matrix \([N]\). If any row in the matrix \([N]\) is null, then the same row of \(\{z_{h}\}\) is also null. Then, the associated variable in \(\{z_{g}\}\) is determined by the particular solution \(\{z_{p}\}\). Hence, the variable is uniquely determined and observable. The calculation of the null space matrix \([N]\) should be carried out symbolically to avoid omitting observable variables, because close-to-zero values from numerical calculations might appear in those null rows. The numerical values of the observable variables are obtained from the particular solution of Equation (6) using least squares methods. Step 4: Check whether new variables in the unknown vector \(\{z\}\) were identified or not. If identified, go to Step 5. Otherwise, go to Step 6. Step 5: Any identified variables will be incorporated into the preceding input to form the subsequent input and initiate a new recursive step. This may enable the identification of other unknowns in \(\{z\}\). Steps 2-4 are repeated until no more variables can be identified. Step 6: Check whether all variables are identified or not. If all the parameters are identified, then end the procedure. Otherwise, go to Step 7. Step 7: Obtain the estimations of parameters by solving the constrained optimization problem that is adapted from the observability equations in the last recursive step. Since the linearization of the unknowns leads to the reduction in observable unknowns, the nonlinearity in structural system identification is regained by means of solving an optimization problem with the constraints that the linearized products are equal to the product of associated single variables, e.g., \(EI_{2}v_{2}=EI_{2}\cdot v_{2}\).