## 1 Introduction  
Structural health monitoring (SHM) is continuously emerging in the field of civil and structural engineering as it is an efficient way to monitor the current state of infrastructure. A well-developed SHM technique can provide useful information about the existence, location, and severity of damage [1], especially during and after a destructive input. For example, as one of the most catastrophic natural disasters worldwide, earthquakes often caused serious damage to structures in a moment. Therefore, the development of real-time or online processing is essential for SHM systems under seismic excitations. Vibration-based SHM methods are the most common SHM techniques; they are based on the conceptual idea of a modal domain that changes in modal parameters, such as modal frequencies, damping ratios, and mode shapes, are a result of changes in the physical mass, damping, and stiffness properties of the structure [2]. To track the variation and detect the damage, vibration-based SHM methods need to be continuous and autonomous during seismic events for providing timely information and warnings. In the 1990s, [PERSON] et al. introduced least-square estimation (LSE) to detect the changes in structural parameters under seismic excitations [3, 4]. For nonlinear hysteretic structural systems, LSE approaches have been employed to identify cumulative damage [5, 6]. Later, sequential non-linear LSE (SNLSE) has been developed based on acceleration responses, which is more practical in the field application [7]. Apart from LSE approaches, [PERSON] and [PERSON] first adopted a filtering approach to iteratively identify structural systems [8]. In 2004, [PERSON] et al. extended and used Kalman filter to adaptively track parameter changes [9, 10]. Then, Kalman filter approach was applied to nonlinear systems [11, 12] and verified via experiments [7, 8]. Moreover, other filtering approaches, such as H\({}_{\infty}\) filter [13] and Monte Carlo filter [14], have been proposed to track parameters over time. During the past few decades, subspace identification (SI) and its relative approaches have attracted substantial interest in system identification, damage detection, and structural control since they can directly identify the system matrices from input and output data. Several well-known SI algorithms, including CVA (canonical variate analysis) [15], N4 SID (numerical algorithms for subspace state space system identification) [16], PAST (projection approximation subspace tracking) [17], and so on, have been developed to identify structural systems and detect structural damage. To accommodate for real-time or online processing, various recursive forms have been proposed using either the matrix inversion lemma [18; 19; 20; 21] or [PERSON]'s criterion [22; 23; 24; 25]. One of the major applications in the field of civil and structural engineering is to extract time-varying dynamic characteristics during seismic excitations. Compared to the LSE and filtering approaches, SI and its relative approaches have stable performance owing to the usage of geometry projection and less limitation on measurement because of the explicit input-output relationship. Consequently, SI based on the recursive formulation has been demonstrated to identify states of structures, track modal parameters, provide a warning message about damage, and help post-earthquake reconnaissance and rehabilitation [26; 27; 28; 29; 30]. In this study, a recursive subspace identification (RSI) algorithm originally developed in the field of control and automation is introduced to investigate the time-varying dynamic characteristics under seismic excitations. This algorithm is derived with a fixed window and, compared to the above-mentioned algorithms, the window length can be used to indicate extra time information. In Section 2, SI is first introduced to extract the system matrices and the modal parameters from input and output data. Furthermore, the proposed algorithm is carefully derived as a recursive form of SI to facilitate real-time or online processing.