**1. Introduction**

In the recent years, there has been a considerable interest in "Smart City" concept and the monitoring, controlling, and preservation of the health state of critical infrastructures, like roads, buildings, bridges, and tunnels. Structural health is required to be diagnosed at every moment during a structure life cycle in order to provide high quality services. Civil engineering structures are designed for the lifetime of the occupants or facilities and their failure might lead to catastrophic consequences in terms of human life and economic assets [1]. Structural health monitoring (SHM) is one of the main stakeholders of the said Smart City concept [2]. SHM is an effective solution contributing to the need for safety, lowering the maintenance costs and reliable condition assessment of structures [3,4].

SHM is an interdisciplinary subject that incorporates knowledge and experiences from synergetic technologies in civil, mechanical, control, and computer engineering to deal with the health assessment of structures. The health monitoring of structures has been the subject of many studies for the past three decades [5,6]. The development of a reliable SHM method for civil structures is a challenging task due to ambient-induced uncertainty and the associated complexity measures. Four analysis levels of damage detection are applied in the context of SHM that include: (1) detection, (2) localization, (3) quantification, and (4) prediction of the remaining life, whereas the first three levels are more explicitly reported in literature [7,8].

Based on the acting load, SHM methods can be divided into two classes of static and dynamic-based methods [9]. The methods that use vibration characteristics of structures to assess the health state of structures, so-called vibration-based damage detection (VDD) [10,11]. The key premise of VDD is to estimate the modal parameters of a structure while using the analytical model that was constructed by system identification methods [12,13]. Static-based damage detection (SDD) methods rely on measuring the change in static response of structure, such as load bearing capacity, strain, deflection, and stiffness. Posenato et al. and Wu et al. [14] used strain data for damage detection of structure. Chen et al. [15] proposed a method to take advantages of stay cable force measurements and structural temperatures for damage detection. Yu et al. [16] used deflection data for damage identification in structures. Zhu et al. [17] introduced a temperature-driven method while using strain information of structures for anomaly detection.

Weigh in motion (WIM) is a widely used vehicle classification method for the health monitoring of structures. Bridge weigh in motion (BWIM) is a type of WIM technology that can identify traffic data, including speed, number of axles, axles spacing, and gross and axle weight of the passing vehicles using a series of conventional strain gauges. BWIM is particularly suitable for short-term measurements of traffic data, as it can be easily installed and detached from the bridge. Cardini and Dewolf [18] applied BWIM through using strain gauges to gain information on the quantity and weights of the trucks crossing highway bridges. Cantero et al. [19] proposed a BWIM-based damage identification method through introducing the concept of 'Virtual Axle' for deriving a damage indicator. Gonzalez and Karoumi [20] proposed a model-free damage detection method using deck accelerations response and BWIM. Kalyankar and Uddin [21] developed a three dimensional finite element model to estimate multi-vehicles–bridge interaction in a BWIM. The environmental factors are a weak point of vibrationand static-based damage detection [22]. However, in some cases, the temperature based methods shown higher sensitivity when compared to vibration based methods.

Estimations of the modal parameters in SHM are generally performed by system identification methods [23]. System identification is a mathematical procedure for establishing an analytical model based on experimental data. System identification is a mature field in SHM to extract modal parameters in VDD methods [24]. System identification methods in SHM can be classified into three categories based on their domains, including: time-domain (TD), frequency-domain (FD), and time/frequency domain (TFD) [25]. TD methods are more attractive for monitoring of civil structures, owing to the direct use of vibration signals. TD methods are generally classified within three groups of subspace system identification (SSI), natural excitation technique (NExT), and auto-regressive moving average (ARMA) [26]. The premise of the NExT method is that the response signals of a structure for ambient excitation and free-vibration have the same analytical form [27]. ARMA-based methods are popular statistical strategies for VDD of civil engineering structures. The auto regressive (AR) part ARMA models a linear function for the response time-history and the moving average (MA) section determines the moving average of the measurement response. The SSI algorithm presents a harmonious combination of algebraic, mathematical, statistical, and geometrical tools to identify the system parameters. SSI takes advantages of LS, angles between subspaces, QR decomposition, singular value decomposition (SVD), Kalman filter, and stochastic realization theory to deal with the problem of modal parameter identification. Subspace-based methods that are used for the parameter identification of civil structures are mainly one of the following two methods of data-driven subspace

system identification (SSI-DATA) and covariance-driven subspace system identification (SSI-COV) approaches [28]. The ARMA model has the most similarity to the SSI-COV model, as both methods use the correlation function of the vibration measurement in their preprocessing stage.

Recently, a large number of subspace-based methods have been applied in VDD. However, the previously conducted surveys have not kept pace with the changing environment and diversity in this field. Therefore, there is a need for a review focusing on the most important recent studies conducted in the considered area. The presented review attempts to address the available studies that employed subspace-based techniques in VDD of civil structures. This paper provides an overview of the background and new findings in SSI with a focus on both theory and practice. In addition, it describes some contributions toward the development and application of the SSI algorithm in recent years.

This review study is organized, as follows. In Section 2, SHM methods are outlined and the strength and drawbacks of each class are highlighted. Section 3 evaluated vibration-based damage detection (VDD) methods. The focus is on subspace system identification in the three later sections (Sections 4–6). Section 7 introduces some commercially available software that use subspace as their main constituent. Future research directions and conclusions are provided in Sections 8 and 9, respectively.

#### **2. Structural Health Monitoring (SHM)**

An SHM system implements strategies for the damage detection of structures [29–31]. Currently, SHM is known as a well-established tool for the diagnosis of damages in civil engineering communities and it is employed in a number of di fferent structures, such as buildings, bridges, and dams. Structural data are collected from several points through installed sensors and they are analyzed to evaluate the health of a structure. Figure 1 shows the categorization of SHM methods with a focus on addressing key subspace-based algorithms. SHM methods are divided into local methods, which mainly rely on non-destructive evaluation (NDE) strategies and global methods. Based on the incorporated domain, the VDD strategies could be categorized as TD, FD and TFD methods. The TD methods are generally from one of the auto-regressive moving average (ARMA), natural excitation technique (NExT), or subspace system identification (SSI) families. The SSI methods are an important class of algorithm and they can be divided into three categories of canonical variate analysis (CVA), numerical algorithms for state-space subspace system identification (N4SID), and multivariable output error state-space (MOESP). N4SID is of the most favored SSI algorithm in SHM due to its capability to cope with output-only data (stochastic, unknown input). Two classes of N4SID algorithms are commonly practiced in SHM of civil engineering structures, namely the SSI-DATA method and the SSI-COV method.

NDE is a local SHM method used to perform constrained random tests to diagnose the state and severity of the possible defects. Some NDE methods are concerned with measuring defects in steel components, whereas others are designed for concrete substructures. Several NDE methods are available to identify defects in steel structures such as the ultrasonic test (UT), radiographic test (RT) [32], and eddy current test (ET) [33]. UT is an acoustic NDE method that uses ultrasonic waves passing through a structure for detecting defects. The phased array ultrasonic test (PAUT) method is a more reliable type of UT that uses a greater number of arrays in order to reliably simulate a specimen's profile [33]. A variety of methods is applied for NDE of concrete components. These methods range from the very simple strength evaluation methods, such as using rebound hammer [34], to more complex methods, such as impact-echo [35] and radiography testing [36]. Even though local SHM methods yield excellent performance for detection and localization of damages, they have some limitations and drawbacks. The main disadvantage of these techniques is that the evaluation process cannot be implemented without any prior knowledge of the approximate damage location. Moreover, in many cases, access to below of the test area is an essential requirement that is not always a ffordable or practical [37]. Further detailed information on local SHM methods in civil engineering structures are provided in [38].

**Figure 1.** Classification of the structural health monitoring (SHM) methods with focus on subspace system identification (SSI) algorithm.

Researchers have proposed using global damage detection methods to identify damages in structures in order to overcome the previously mentioned limitations. Global methods are very effective choices to overcome the limitation of local SHM methods in civil engineering. In global methods, there is no limitation regarding the location of damage or even access to and preparation of the damaged area. These methods can localize and estimate the extent of damages while using the global characteristics of structures. The SDD methods require large measurement datasets for a reliable damage detection process. The targeted structure must generally be removed from its normal service in order to implement loading tests in SDD methods. Therefore, the SDD methods are not appropriate for continuous monitoring applications. When compared to VDD, the number of studies that employed SDD techniques are limited [39,40]. SDD methods are beyond the scope of this review paper and, for the sake of brevity, will not be covered in this study. A full review of the SDD has been well-documented elsewhere (for example [41]). VDD methods will be studied in the proceeding subsection.

## **3. Vibration-Based Damage Detection (VDD)**

VDD is considered to be the most popular methodology in global SHM. VDD methods rely on changes in dynamic properties as an indicator of damage existence. These methods exploit observable variations in modal parameters, such as resonant frequency, damping, and mode shape or their derivative as indicators of change in physical properties of a structure. A thorough review of the system identification methods in VDD and modal analysis is provided in Song et al. [42] and Reynders [43].
