# Health Monitoring of Civil Infrastructures by Subspace System Identification Method:
## Vibration-Based Damage Detection (VDD)
### Time-Domain (TD) Methods
#### 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).  
\[\left\{\begin{array}{l}x(t+1)=Ax(t)+Bu(t)+w(t)\\ y(t)=Cx(t)+Du(t)+v(t)\end{array}\right. \tag{2}\]  
where \(x\ (t+1),\ u\ (t+1)\), \(x(t)\), and \(u(t)\) are state vectors and scalars at time instant of \(t+1\) and \(t\), respectively. \(y_{k}\) 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]:  
* CVA methods;
* MOESP methods; and,
* N4 SID methods.  
[PERSON] and [PERSON] [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 [PERSON] [89] in a statistical setting. SVD is used as a tool to extract the incorporated canonical variates. [PERSON] [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. [PERSON] and [PERSON] [91] unified proposed subspace schemes into a pragmatic approach, referred to as N4 SID. 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]. [PERSON] et al. [93] introduced a subspace algorithm using power spectrum data. The state-space coefficients 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.