# Genetic Algorithm-Based Model Updating in a Real-Time Digital Twin for Steel Bridge Monitoring
## 3 Real Bridge: Dynamic Identification
### Operational Modal Analysis Using SSI  
The Stochastic Subspace Identification (SSI) procedure is a widely used operational modal analysis (OMA) technique for extracting the modal characteristics (natural frequencies, damping ratios, and mode shapes) of structures, such as bridges, using output-only data from accelerometers. Unlike traditional methods that require input excitation measurements, SSI relies solely on the response data (e.g., accelerations) recorded by sensors installed on the structure.  
To obtain the modal parameters, a discrete-time state-space representation is used:  
\[\hat{x}_{k+1}=[A]\cdot\hat{x}_{k}+[B]\cdot u_{k} \tag{2}\]  
\[y_{k}=[C]\cdot\hat{x}_{k}+[D]\cdot u_{k} \tag{3}\]  
where \(x_{k}=x_{k}(\Delta_{t})\) is the discrete-time state vector containing displacement and velocities of the system, \(y_{k}\) is the vector of system response, \(u_{k}\) is the vector of input, \([A]\) and \([B]\) are the matrices of input that contain, respectively, the physical information and the statistical parameters, \([C]\) is the output matrix, and \([D]\) is the direct transmission matrix in discrete time.  
The primary challenge in estimating this parametric model lies in the uncertainty of the true model order, which is influenced by the presence of noise in the recorded signals or the limited availability of recordings. To address this, a strategy is employed where multiple models of increasing order are evaluated to capture all physical modes within the frequency range of interest. However, this approach also introduces numerous numerical modes (spurious modes) that lack physical significance and arise solely from the noise in the recordings.  
The SSI method is applied to the free vibration segments extracted from six accelerometers strategically positioned on each span of the bridge. This approach utilizes the free vibrations of the structure to accurately estimate its dynamic properties. To differentiate between physical modes (true structural dynamics) and spurious modes (numerical artifacts or noise), a stabilization diagram (Figure 4) is employed. This diagram serves as a powerful visualization tool, plotting the system's natural frequencies across multiple model orders. Stable modes, representing consistent physical behavior, appear as vertically aligned points, while spurious modes scatter randomly. Through this analysis, two dominant frequencies associated with the vertical modes of the bridge were identified. These frequencies correspond to the fundamental dynamic characteristics of the structure, providing critical insights into its vibrational behavior.  
Figure 4: Stabilization diagram for frequency identification. **(a)** Span 1; **(b)** span 2.  
The identified frequencies are further validated by examining their corresponding modal shapes, as illustrated in Figure 5. These shapes depict the spatial deformation patterns of the bridge at each natural frequency, offering a deeper understanding of the structure's dynamic response. For instance, the first mode typically represents the global bending or sway of the bridge, and the second mode often captures higher order bending or torsional behavior. By correlating the frequencies from the stabilization diagram with their respective modal shapes, the analysis confirms the physical significance of the identified modes. This step is crucial for ensuring that the results align with the expected dynamic behavior of the bridge. The identified frequencies and modal shapes serve as baseline data for structural health monitoring, enabling the detection of changes in the bridge's dynamic properties that may indicate damage or degradation.