*3.1. Frequency-Domain (FD) Methods*

An FD representation of a signal is extracted from a TD response signal using a Fourier transform. A Fourier transform describes a harmonic function by a linear combination of complex exponentials. A number of enhanced algorithms have been developed in order to improve the computational efficiency of Fourier transform, which are collectively termed as a fast Fourier transforms (FFT). FFT is generally used for deriving frequency response functions (FRF), and it plays the key role for many FD damage detection approaches [44]. FD methods in VDD can be classified into three main categories, which include [26]:


PP methods, also referred to as basic frequency-domain (BFD), are the most typical approaches in modal testing and they initially rely on power spectral analysis and Fourier transforms. PP methods are qualitative in nature and founded upon the fact that FRF reaches an extreme approximately around the natural frequency. Furthermore, these methods could be accompanied with a half power bandwidth approach to extract damping ratio [45]. Frequency domain decomposition (FDD) is an accurate and simple technique for system identification that is widely used in modal analysis. FDD has been developed based on spectral density decomposition. The obtained spectra are a reduced form of a dynamic response for individual modes [46]. The enhanced frequency domain decomposition (EFDD) method is an extension of FDD for estimating reliable modal parameters [47].

CMIF methods can also be considered to be an extension of PP techniques. They have been widely used for the output-only identification of system parameters. CMIF is developed by performing SVD on a normal FRF matrix at each spectral line [48]. CMIF is combined with other algorithms to be used as a standalone model, such as the enhanced frequency response function (eFRF) and enhanced mode indicator function (EMIF). eFRF is the subsequent development of the CMIF method and it is used to estimate the frequencies that are associated with a particular peak in the CMIF [49]. The eFRF is rooted in the concept of physical coordinate transformation to enhance the estimation of modal parameters. EMIF could be considered as an extension of the CMIF/eFRF, which estimates modal parameters in several modes at one time. The distinctive property of this method is due to the fixity in the number of natural frequencies based on the peaks of CMIF plots.

LSCF is a fast and accurate method for estimating modal parameters. Originally, LSCF was applied to extract initial values in the maximum likelihood method. LSCF performs reliably due to its clear stabilization diagram [50]. The polyreference least-squares complex frequency-domain method (PolyMAX) is the polyreference version of the LSCF that takes advantage of the right matrix-fraction model. The main benefit of this method is that the closely spaced modal frequencies can be separated from each other [51]. El-Kafafy and Peeters [52] introduced the poly-reference least squares complex frequency-domain (pLSCF) for modal analysis. A two-step scheme is proposed to enhance the damping estimates. The proposed method can improve the processing time and accuracy of the modal identification, particularly for damping estimates.

FD methods are fast and accurate, but they suffer from some limitations in the frequency resolution of the estimated spectral data [53]. Conventional FD methods are not accurate and reliable for the analysis of non-linear and non-stationary signals. The resolution of the identified system parameters in low-frequency ranges or fewer numbers of incorporated modes is poor in these methods [54]. Moreover, the estimated damping coefficients are not accurate in the non-parametric FD methods [55]. The strong demand in the field of system identification to achieve higher accuracy and extract more information from the vibration responses led to the development of TD methods.

## *3.2. Time-Domain (TD) Methods*

TD techniques rely on the fact that the vibrational properties of structures can be captured through the time-history response of a dynamic system. Hence, the extracted response in a healthy state is different from that of one that is in a damaged state. Figure 2 shows the schematic architecture of TD methods that were used for the identification of dynamic systems. Different numerical techniques, for example, FFT, SVD, least squares (LS), QR decomposition, Eigen-vector decomposition (EVD) [56], and statistical methods, were used to develop these algorithms. Observer/Kalman filter identification (OKID), NExT, and random decrement (RD) are the most common TD methods for extracting the FRF when there is no access to the input data. The input signal can be estimated using an auto-correlation or cross-correlation function [57].

**Figure 2.** Schematic model for some of output-only time-domain modal identification methods.

The TD techniques in the health monitoring of civil structures can be classified into three widely known categories, which include [26]:


These methods are adopted for the identification of system parameters in civil engineering structures by introducing output-only extensions. NExT and SSI-COV methods are generally indirect where the SSI-DATA and ARMA methods are direct in nature, as can be seen in Figure 2. The TD methods are more appropriate for continuous monitoring compared to modal analysis methods due to direct use of response signals. Furthermore, the information extracted using TD methods is more complete than that of FD methods, particularly when a large number of modes or a large frequency range exist. A further explanation of the TD class of the identification methods are provided in the following subsections.

#### 3.2.1. Natural Excitation Technique (NExT) Methods

James et al. [58] first proposed the NExT for the modal analysis of systems exposed to noise uncertainty. The key idea behind the NExT scheme is that the cross-correlation of the response signal from random excitation can be considered as a summation of decaying sinusoids. The complex exponential (CE) model is based on Prony's method and it was the first modal estimation method used as single-input single-output (SISO). The CE algorithm fits the curves of analytical impulse

response into the experimental impulse response data [9]. The CE algorithm has been extended to the single-input multiple-output (SIMO) version by applying the LS technique. The algorithm was named the least-squares complex exponential (LSCE) [59]. The polyreference complex exponential (PRCE) is an extension of LSCE in the form of the multiple-input multiple-output (MIMO) identification of modal parameters [60].

Juang et al. [61] first proposed the Eigensystem realization algorithm (ERA) method for modal identification in the field of aerospace engineering. This method uses state-space forms together with Markov parameters in order to extract the coe fficients of a dynamic system. Juang et al. [62] introduced an extension of ERA while using data correlations (ERA/DC). In the proposed method, the ERA realization-based approach is combined with a state-space correlation fit. The extracted modal parameters in this method are less sensitive to noise corruption and less prone to bias error. The improved polyreference complex exponential (IPCE) [63] is another extension to ERA and it is specifically designed for enhancing the reliability of PRCE and reducing the influence of random noise in modal identification. The IPCE technique uses correlation filtering as a pre-processing step to reduce the noise e ffects on measured data and minimize system order.

The Ibrahim time-domain (ITD) was of the first SIMO algorithms for estimating eigenvalues and eigenvectors in one-step. The ITD technique is reported not to be appropriate for heavy damped systems or systems with low natural frequencies [64]. The multiple references Ibrahim time-domain (MRITD) is an enhanced extension of ITD for MIMO modal analysis. The method is a high-resolution modal decomposition approach that is based on eigenanalysis [65]. OKID was originally a companion of the ERA, being denoted as OKID-ERA. The method suggested establishing a non-recursive LS observer to relate input and output data [66]. In recent years, the OKID has been introduced as a separate class of algorithms that could be combined with other models in a pairwise basis. Output-only ERA-OKID [67] and output-only observer/Kalman filter identification (O3KID) [68] methods are two recent versions of OKID for experimental modal analysis of civil engineering structures. The RD method is a TD approach for modal analysis through transforming system responses into random decrement functions [69]. RD functions are considered to be the free-vibration responses of a system. With the assumption of a zero-mean stationary Gaussian stochastic process, RD functions are proportional to correlation functions. The proposed method uses the concept of averaging to extract the random decrement signatures (RDS) of structures. Brincker [70] presented a general overview of the random decrement application in the modal analysis of structures. NExT methods provide a reliable tool for the modal identification of civil engineering structures. These methods have been implemented on several real world structures. However, the obtained results for damping ratios were less accurate that those that were obtained with other counterparts, such as the SSI and FDD algorithms [55].

#### 3.2.2. Auto-Regressive Moving Average (ARMA) Methods

Auto-regressive moving average (ARMA)-based methods outperformed the purely statistical methods. The auto regressive (AR) part aims to model linear function time-history and the moving average (MA) part aims to determine the moving average of the time-series [71]. The general structure of the AR-based models depends on which of the <sup>A</sup>(q), <sup>B</sup>(q), <sup>C</sup>(q), <sup>D</sup>(q), and <sup>F</sup>(q) polynomials are used in the model. Equation (1) shows the general structure of AR-based models.

$$A(q)y(t) = \frac{B(q)}{F(q)}u(t) + \frac{C(q)}{D(q)}c(t) \tag{1}$$

Table 1 shows the most common ARMA models that were used in the system identification of civil engineering structures.


**Table 1.** Common auto-regressive moving average (ARMA) models shown with their associated polynomials.

Several studies have been conducted while using autoregressive-based methods for the identification of damage in structures. Table 2 presents a review of applied ARMA methods in SHM and modal analysis with a focus on the test methods, damage features, and the incorporated pattern recognition techniques.

**Table 2.** Autoregressive-based methods applied in the structural health monitoring (SHM) of civil engineering structures.


The structures range from a very simple model of a mass spring system to more elaborate real world structures, as shown in Table 2. In many cases, the coefficients of the incorporated AR-based algorithm are directly used as damage features. For modal analysis, the extracted system parameters are utilized for the evaluation of dynamic behavior in a structure. Using modal parameters as damage features, the variation of the extracted parameters for the reference and actual state could be considered for the condition assessment of a structure. The obtained results for modal parameters are more robust when compared to other TD methods. However, higher scatterings in damping ratios are observed and the reported error is much more than other TD methods, such as SSI.

3.2.3. Subspace System Identification (SSI) Methods

Parametric TD methods provide a powerful and versatile mathematical framework for analyzing dynamic systems. Among all system identification methods, subspace-based techniques are the most remarkable achievement in the field of control and system identification. Meanwhile, many research studies on system identification have been concentrated on subspace methods in recent years. The subspace approach is a suitable technique for estimating the state-space model of a dynamic system. The SSI is a numerically reliable algorithm and it estimates models with good quality, particularly for multivariable systems [85,86]. The state-space form of the equation of motion can be written as Equation (2).

$$\begin{cases} \mathbf{x}(t+1) = A\mathbf{x}(t) + Bu(t) + w(t) \\ y(t) = \mathbf{C}\mathbf{x}(t) + Du(t) + v(t) \end{cases} \tag{2}$$

where *x* (*t* + <sup>1</sup>), *u* (*t* + <sup>1</sup>), *<sup>x</sup>*(*t*), and *u*(*t*) are state vectors and scalars at time instant of *t* + 1 and *t*, respectively. yk is output vector, *A*, *B*, *C*, and *D* are system, control, output, and feedback matrices, respectively. *w*(*t*) and *v*(*t*) are measurement and process noise, respectively. Most subspace algorithms reported in the literature are closely related to the LS-based methods [87]. In the first step, an oblique projection is calculated and it is pre- and post-multiplied by appropriate weight matrices to infer the system order and state sequence. In the second step, a geometrical projection is adapted in order to eliminate the dependence of the SSI algorithm on future output. In the third step, LS is deployed to drive the A and C matrices. Finally, the Kalman predictor is used to estimate the system model by inferring the Kalman gain K of the state-space model. In a general sense, the most researched subspace methods in the field of system identification can be classified within the following three main categories [87]:


Larimore and Wallace [88] proposed CVA methods that are based on Markov parameters for TD system identification. The study continues with the same principles as the pioneering activities of Akaike [89] in a statistical setting. SVD is used as a tool to extract the incorporated canonical variates. Verhaegen [90] proposed the MOESP method for the identification of the multivariable state-space model from noisy input-output data. The MOESP subspace algorithm is known for two characteristics, those of the reduced-size Hankel matrix and the extended observability matrix. The method was not applicable for stochastic systems. Van Overschee and De Moor [91] unified proposed subspace schemes into a pragmatic approach, referred to as N4SID. The algorithm was analytically robust and reliable due to the use of SVD and QR decomposition. Based on the way, the subspace algorithm deals with the measurement time history data; they can be divided into the two categories of SSI-DATA and SSI-COV. In the next two subsections, a review on application of the SSI-DATA and SSI-COV algorithms in the SHM of civil engineering structures is outlined.

#### Data-Driven Stochastic Subspace Identification Method (SSI-DATA)

SSI-DATA is a method for identifying modal parameters by the direct use of measured response time-history [92]. Overschee et al. [93] introduced a subspace algorithm using power spectrum data. The state-space coe fficients were derived using inverse discrete Fourier transform. The computational complexity in this method is higher when compared to that of direct subspace methods or FDD.

Table 3 shows some examples of the methods that use SSI-DATA for damage detection and modal analysis.

**Table 3.** Some examples of schemes that use data-driven subspace system identification (SSI-DATA) in their damage identification process.


Covariance-Driven Stochastic Subspace Identification (SSI-COV) Method

SSI-COV is a parametric output-only method that is used for modal analysis. The method use the vibration response time-history to extract the state-space model of a dynamic system. SSI-COV is a two-step procedure that utilizes the correlation function of vibration time-history. The system order is the only user defined parameter in SSI and it must be carefully chosen to obtain meaningful results. In these methods, the applied excitation is considered as white noise and it is equal to the covariance of the measured response time-history.

Table 4 shows some SSI-COV algorithms that are applied for the damage detection of civil engineering structures. The damage indices in these methods are either modal parameters or a novel indicator for detecting changes that are caused by damage. In the previous two subsections, the application of SSI-DATA and SSI-COV algorithms for the damage detection of civil engineering structures was presented. In the following section, a comparison between the SSI algorithm and other key system identification methods in TD and FD is presented.



#### **4. Comparison between SSI and Other System Identification Algorithms**

The SSI-DATA is a direct method and it does not need any data pre-processing to calculate correlation functions or spectrum analysis. On the other hand, the subspace identification algorithm uses QR factorization, SVD decomposition, and LS robust numerical techniques in the analysis process [119]. Several comparative studies are presented below in order to evaluate the advantages and drawbacks of the subspace identification algorithm when compared to other time and frequency domain damage identification methods in SHM. Table 5 presents a comparison between SSI method and other system identification algorithms.




Table 5 shows the results of a comparison between several TD, FD, and time-frequency domain methods for damage detection and modal analysis of civil engineering structures. The obtained results confirm the reliable performance of the TD methods. It was shown that SSI is a powerful tool for modal identification, in which closely spaced frequencies are not of the same type (e.g., bending or torsion) [55]. It was reported that the identified mode shapes using SSI-DATA were the most reliable when compared to MNExT-ERA or EFDD [122]. Moreover, fewer errors were obtained in estimating the damping ratio using the SSI algorithm as compared to ARMA [124]. However, SSI has some disadvantages, such as requiring human judgment of system order for implementation [55]. On the other hand, the work burden of the SSI algorithm is large and it usually needs to perform multiple analyses [54].

#### **5. Challenges of SSI in Practical Application**

Research on the application of subspace methods for the damage detection of civil structures emerged in the mid-1990s. Most of the methods used in SHM presume a parametric model of a dynamic system in order to characterize structural behavior under an applied excitation load. However, civil engineering structures still have many challenges to achieve a robust SHM model. The size of civil structures does not permit a large number of sensors to be mounted on a structure. Moreover, forced-vibration is not considered to be practical due to interruption in the serviceability of structures. On the other hand, civil structures are complex in terms of geometry and their material properties involve a large range of uncertainty due to operational and environmental factors [125]. In Table 6, the problems that are faced with practical implementation of SHM systems using the SSI algorithm and the researches to resolve the associated problems are presented.

Table 6 reviews the challenges of implementing the subspace algorithm in real-world applications. In practical applications of SHM, the response signal of structures is generally in the form of a non-Gaussian random signal. In such conditions, deterministic techniques result in unreliable system models. On the other hand, data loss or corruption caused by failure or loss of sensing, transmission, or storage devices during their normal use is a concern for a reliable damage detection scheme. Consequently, appropriate procedures must be considered in order to deal with the uncertainty that is caused by such instrumental failures. The accuracy of an identification algorithm is due not only to its insensitivity to environmental variation and instrumental failure, but also to the inherent performance of the estimation scheme. Studies have been conducted to improve the performance of the SSI method and enhance the modal identification process itself. The inherent performance of an estimation scheme to deal with problems, such as the short-length of a signal, non-stationarity measurement data, system non-linearity, leakage error, or di fferent measurement setups remains a challenge for a reliable SHM. In some cases, resolving the problem and increasing the accuracy demands exhaustive expert assistance and time-consuming computation burdens. Gluing the non-simultaneously measured set-ups of sensor data is another controversial issue that needs to be considered before applying any identification platform. The aforementioned drawbacks are the topic of ongoing research in the field of SHM.


#### **Table 6.** The challenges of SSI in practical applications.

#### **6. Application of Subspace Identification in Civil Engineering Structures**

Currently, stochastic subspace methods are widely accepted tools in civil engineering communities. Large number of SHM methods is operating in structures that are subjected to dynamic vehicular, seismic, wind, or impact loading. In this study, the application of the SSI algorithm in civil engineering structures is investigated in the following three categories:


Beams and two-dimensional (2D) models attempt to explain the performance of the SHM algorithm while using numerical FE or laboratory-scale models [153]. Many of the studies on these structures are generally academic in nature. Three-dimensional (3D) frames and building structures are generally used to investigate the practical aspects of SHM. Bridges are spectacular due to their specific loading conditions when compared to building-type structures [154]. Vehicular and pedestrian loading, together with exposure to environmental variations due to temperature and wind precipitation, have been the focus of the analysis of bridge structures in many pioneering studies of SHM [155]. Tables 7–9 show some application of SSI algorithms for damage detection of 2D frames, 3D frames, and buildings and bridges, and other structures, respectively.


**Table 7.** Application of SSI algorithms for damage detection of beams and two-dimensional (2D) frames. **Table 8.** Application of SSI algorithms for damage detection of three-dimensional (3D) frames and buildings.




In most structures, damage occurs in the form of a reduction in the cross-section of structural members. Partial reductions in 2D structures are usually detectable with a high level of precision. The robust applicability of the proposed methods to solve the problem of detecting local damage in real and complicated civil engineering structures has not been validated, even though these methods work relatively well in simple structures. Three-dimensional frames and building structures are generally complex and they pose challenges for both practitioners and researchers. Damage in building structures is usually in the form of a partial reduction in the cross-sectional area in column elements. In some cases, damage is made by opening bolts in a beam-column connection. However, beam damages are less important and they require higher detectability resolution. Bridge structures are a very important element of transportation. An in-service bridge is subject to loads, such as traffic, temperature variation, wind loading, and deterioration, under aggressive environments. Applying SHM to bridge structures poses significant challenges due to the specific types of loading and complexity of the structure. As a general conclusion, it could be derived that damage detection strategies that use modal frequencies, mode shapes or mode shape curvatures as their damage sensitive features [180] are only efficient for the detection of global damage and are not generally sensitive enough to detect changes in the local elements of structures [181].

#### **7. Application of Subspace Identification in Civil Engineering Structures**

In general, an individual program or a combination of software packages implement the process of damage detection [182,183]. A structural monitoring program is considered an algorithm for analyzing response signals, extracting damage features and deploying pattern recognition paradigms, ultimately leading to damage identification [184]. Subspace methods have been used as the central part of many of the structural monitoring programs used in industry [185,186]. In this subsection, the industrial software packages used in modal identification and SHM, which have adapted SSI as their core identification process, are further investigated.

Table 10 provides some of the commercially available software that use subspace identification algorithms as their standard technique for SHM and modal analysis. Most of the available algorithms are generally used for modal analysis. The SHM algorithms are composed of (i) identification, (ii) feature extraction, and (iii) pattern recognition steps and the implementation of a unified algorithm for huge diversity of each category is quite challenging and rewarding.


**Table 10.** SSI-based commercially available software for SHM and modal analysis.

## **8. Future Research Directions**

In the future, researchers should focus on the identification of local damages by improving the accuracy and noise-robustness of damage identification algorithms [187,188]. Furthermore, they must think about introducing a novel platform for the implementation of commercialized SHM software that is versatile enough to deal with the diversity of techniques in a damage detection system. Providing a user-friendly platform for the implementation of the SHM algorithm will improve the general usage of SHM software in solving real-life engineering problems. Furthermore, the extensible architecture would enhance the applicability of the software by enabling users to modify the existing base code and add their own extensions. A modular and flexible architecture enables a wide variety of reported methods to deal with within an integrated framework. Enhancing the SSI properties to deal with real-world applications, such as noise inclusion, short length data, and gluing sensor data, will enhance accuracy to provide more reliable damage detection results.
