# Robust Static Structural System Identification Using Rotations
## 1 Introduction  
### Existing Structural System Identification Method  
Potential catastrophic events due to the failure or malfunction of civil infrastructures (e.g., bridges, high-rise buildings, dams) might claim people's lives and cause substantial economic losses. To avoid this undesirable consequence, it is vital to know the current condition of structures. For this reason, structural health monitoring (SHM) and structural system identification have attracted much attention in recent decades [1, 2, 3]. A basic assumption in structural system identification is that the deterioration or the damage of structures is reflected in the change in structural parameters (such as bending and axial stiffness). These parameters can be estimated by various structural system identification methods using a measured response from structures under external excitations. Dynamic-based methods account for the majority of structural system identification methods. Depending on the source of excitation, dynamic structural system identification methods are categorized as input-output methods and output-only methods. In the input-output methods, the time history of the excitation and the response are measured to compute frequency response functions. These functions are used to determine the natural frequencies, mode shapes, and damping ratios of the structure using well-established methods [4, 5, 6]. In the output-only methods [7, 8], instead of using heavy, cumbersome and expensive devices to excite large-scale structures in a controlled way, the ambient (wind-induced, traffic-induced and subtle seismic) vibrations are measured convenientlyand economically. The modal information of structures can be obtained by analyzing the correlation functions or the spectral density matrices computed from the operating response data. Among these methods, the Bayesian methods [8] can obtain the posterior distributions of the target parameters by combining the assumed distributions of the parameters in the priors and the evidence from the measurements. The error-domain model [9; 10; 11] can identify parameter values in the existence of systematic errors and detect the presence of unrecognized systematic errors. The residual minimization method [12] can localize model errors and estimate physical parameters by minimizing the force unbalance in the equations of motion. Compared with dynamic methods, static structural system identification methods are much less developed. [PERSON] et al. [13] identified the plate-bending stiffness of a reinforced concrete pier deck with incomplete static test data using accurate forces and displacements. Later, [PERSON] et al. [14] formulated structural system identification as a constrained nonlinear least squares problem by minimizing the discrepancies between the measured and predicted displacements or forces. [PERSON] et al. [15] incorporated an adaptive parameter grouping strategy with the work from [PERSON]. The proposed method can locate and assess damage even under sparse and noisy measurements. [PERSON] et al. [16] were able to locate and quantify damages in a cantilever beam and two trusses using a flexibility disassembly technique with a baseline model and all DOFs measured. [PERSON] et al. [17] approximated the beam curvatures by the second-order difference of deflections. This work was able to locate damage from the abnormality in the curvature curves while the damage extent could not be quantified. Recently, observability methods [18; 19] were applied in the field of state estimation. This method was able to identify whether a set of available measurements was sufficient to estimate the state of systems. [PERSON] et al. [20] applied the observability method (OM) to determine the observability of structural parameters in 2D beam models symbolically with static measurements. Later, structural system identification by numerical OM (NOM) was proposed by [PERSON] [21] to carry out numerical estimation of target parameters. Recently, [PERSON] et al. [22] and [PERSON] et al.