## 1 Introduction  
Structural health monitoring (SHM) processes are essential for maintaining the safety and performance of significant structures such as bridges, dams, and buildings. By continuously monitoring these structures, potential damage can be detected early, allowing for preventive measures to be taken to avoid catastrophic failures. Detecting damage in structures is a critical task in engineering that involves analyzing data from various sensors and using algorithms to detect changes in dynamic characteristics relative to the health state of the structure. The recorded responses from a multi-channel network of accelerometers are processed using various stochastic or deterministic techniques to detect the eigenfrequencies, mode shapes, and modal damping ratios of existing structures. The \"frequency domain decomposition\" (FDD) technique has been employed in \"operational modal analysis\" (OMA) [1, 2, 3, 4, 5]. Similarly, the \"stochastic subspace identification\" process [6] is utilized in OMA-based techniques [7, 8, 9], integrating three distinct analysis algorithms: principal component, canonical variate, and unweighted principal component. Based on these techniques, the \"modal time-histories method\" [10] was developed for structures subjected to seismic ground excitation or strong wind loads [11]. Regarding damage identification, the \"minimum rank perturbation theory\" (MRPT) [12, 13] interprets non-zero terms in a structure's damage matrix as indicators of damage location. Other techniques rely on the concept of \"damage stiffness matrix\" for detecting structural damage [3; 14; 15]. Additionally, some techniques [16; 17] use discontinuities in structural mode shapes as indicators of damage. Several techniques apply the \"modal strain energy-based damage index\" [18; 19; 20; 21] or \"modal flexibility\" [21; 22; 23; 24] to identify structural damage. A comprehensive review of damage indices for rapid seismic structural health monitoring is provided in [25]. Furthermore, various approaches incorporate artificial neural networks and machine learning tools for damage detection [21; 26; 27]. The integration of monitoring processes with pushover analysis has been explored in several damage-detection techniques [28; 29]. Acoustic emission (AE) measurements are also employed in damage-detection methodologies [30]. It is worth noting that many of the aforementioned techniques combine multiple damage indices and machine learning tools for the identification of damage in reinforced concrete (RC), steel, or composite steel-concrete elements, as well as in entire structures. Recently, [PERSON] [31] proposed an alternative technique for damage identification in existing, ductile, planar RC frames. This is the hybrid \"\(M\) and \(P\)\" technique, where \"\(M\)\" is for \"monitoring\" and \"\(P\)\" for \"pushover\" analysis. Subsequently, the \"\(M\) and \(P\)\" technique was validated by examining a five-story RC frame [32] in a seismic case. The same technique was suitably adapted by [PERSON] et al. [33] for identifying seismic damage along the longitudinal direction of RC bridges. An example is provided to better illustrate the application of the \"\(M\) and \(P\)\" technique on existing damaged structures. Although this example involves a seismic case, the technique is also applicable to other dynamic excitations, such as wind loads. Consider a scenario where a strong main earthquake occurs, causing initial damage to an existing structure. After a short interval, a strong aftershock strikes, further damaging the already-weakened structure. This pattern repeats a few more times, with each aftershock causing progressively more and larger damage. Following these events, the goal is to identify the damage that has accumulated. However, it is evident that nonlinear response history analysis is not feasible in this case, as we lack both information about the seismic ground excitations and the current condition of the structure at that moment.