# Robust Static Structural System Identification Using Rotations
## 2 Methodology
### Structural System Identification Using the Constrained Observability Method  
In a previous study [20], the observability of structural parameters was determined symbolically by checking the null space of the coefficient matrix of a system of equations adapted from equilibrium equations. The null space for a \(m\times n\) matrix \([A]\) is the vector space whose vectors satisfy the homogeneous equation as shown in Equation (1). \[[A]\{x\}=\{0\} \tag{1}\]  
The null space for matrix \([A]\) is defined in Equation (2). \[Null([A])=\{\{x\}\in R^{n}|[A]\{x\}=0\} \tag{2}\]  
The null space matrix \([N]\) for the matrix \([A]\) is a matrix whose columns form a basis of the null space \(Null([A])\), which can be computed by Gaussian elimination. In fact,any vector in the null space is a linear combination of the basis vectors for \(Null([A])\), i.e., columns of the matrix \([N]\). In structural system identification using both OM and NOM, the identification of structural parameters is a recursive process in which the parameters that are identified in the last recursive step are incorporated into the input of the current recursive step to enable the identification of other parameters. In each recursive step, the nonlinear variables appearing in the system of equations are treated as linearized ones. However, the assumption of linearity might reduce the number of observable parameters with given measurement sets [39]. Then, two-stage structural system identification using the constrained OM (COM) was proposed. In Stage 1, the original structural system identification using NOM is carried out until no more parameters are observable. In Stage 2, the system of equations from the last recursive step in Stage 1 is reformulated as a constrained optimization problem, minimizing the square sum of the residuals of this system. The procedure to carry out structural system identification using COM is described in the seven steps below, and all the seven steps are the novelty of this method. Stage 1 is related to Steps 2-5, while Stage 2 is related to Step 7. Step 1: Define an initial FEM using 2D beam elements for the structure to be analyzed. Generate the system of equilibrium equations for this FEM, as shown by Equation (3). \[[K]\{\delta\}=\{f\} \tag{3}\]  
The matrix \([K]\) is the global stiffness matrix that includes the information of length \(L_{j}\), elastic moduli \(E_{j}\), area \(A_{j}\) and inertia \(I_{j}\) of element \(j\) (\(j\) = 1, 2... \(N_{e}\)). The displacement vector \(\{\delta\}\) comprises horizontal deflection \(u_{i}\), vertical deflection \(v_{i}\) and rotation \(w_{i}\) of node \(i\) (\(i\) = 1, 2... \(N_{n}\)). The force vector \(\{f\}\) comprises horizontal force \(H_{i}\), vertical force \(V_{i}\) and moment \(M_{i}\) applied on node \(i\) (\(i\) = 1, 2... \(N_{n}\)). The numbers of elements and nodes in the FEM are denoted by \(N_{e}\) and \(N_{n}\). Step 2: Introduce the boundary conditions and values of the increments in displacements and forces during the static test to obtain the observability equations. The entries in the matrix \([K]\) are sums of monomial ratios, which are the ratios between axial or bending stiffnesses and the square (or cubic) of the length, i.e., \(\frac{E_{j}A_{j}}{L_{j}}\), \(\frac{E_{j}J_{j}}{L_{j}^{2}}\) or \(\frac{E_{j}J_{j}}{L_{j}^{3}}\). In static structural system identification, it is assumed that \(L_{j}\) is known and \(E_{j}\), \(A_{j}\) and \(I_{j}\) are unknown parameters to be estimated. Note that \(E_{j}\), \(A_{j}\), \(I_{j}\) and \(L_{j}\) from different elements might appear in the same entry due to element connectivity. To separate these parameters, each column of matrix \([K]\) is divided into multiple columns such that any resulting column is uniquely related to one monomial (stiffness), \(E_{j}A_{j}\) or \(E_{j}I_{j}\). Meanwhile, the displacement vector \(\{\delta\}\) is expanded correspondingly. Then, these stiffnesses are extracted from the matrix \([K]\) and the expanded displacement vector is multiplied by them.