## 1 Introduction
There exists a one-to-one relationship between the modal shapes of dynamic mechanical systems and the linear compound matrix, and a one-to-one relationship between the modal coordinate responses and principal components (PCs) [25]. This hypothesis has been confirmed also in [26; 27]. However, classical PCA or POD models are batch-wise multivariate statistical methods that are only suitable for stationary signals. In other words, they are time-invariant and cannot identify the operational modal parameters of LTV structures. Recently, some adaptive PCA methods have been developed for nonstationary signal processing. In [28], [PERSON] combined an exponentially weighted moving average with PCA. The moving window PCA method combats the time-varying nature of the structural parameters using a fixed-length window. A PCA model is established in the old moving window and reconstructed in the new window each time a new data sample is received [29]. Therefore, this method is computationally expensive and memory-intensive. Recursive PCA (RPCA) employs a recursive technique to update the reference model for every new sample [30]. Depending on the required parameters (autocorrelation matrix and eigenvector), multiple RPCA algorithms may be used, such as autocorrelation matrix RPCA (ARPCA) [31] and eigenvector RPCA (ERPCA) [32]. Most RPCA techniques are based on updating a rank-one matrix. As a result, RPCA increasingly includes old data and cannot represent the current features. This problem can be solved by introducing a forgetting factor to reduce the impact of old data. [PERSON] et al. [33] developed a limited memory PCA (LMPCA)-based OMA method to identify both the modal shapes and frequencies of SLTV structures. However, this method has a long runtime,high memory requirements, pathological matrix problems, and suffers from modal exchange. To reduce the runtime and memory requirements of LMPCA-based OMA, [PERSON] et al. [34] proposed an OMA method based on limited memory autocorrelation matrix RPCA. This method updates the autocorrelation matrix in every moving window, rather than rebuilding the PCA model. Therefore, the computation time and memory requirements of LMRPCA-based OMA are much less than moving window PCA-based OMA. To avoid pathological matrix problems, [PERSON] et al. [35] described a new OMA method using moving window self-iteration principal component extraction (MWSIPCE). However, these moving window-based methods have an inherent delay and should be confined to SLTV structures. [PERSON] et al. [36] developed an ARPCA with a forgetting factor (ARPCAWF) for OMA, and identified transient modal parameters for a beam with time-varying density by updating the autocorrelation matrix online. This method is suitable for both SLTV and fast linear time-varying (FLTV) structures, but suffers from pathological matrix problems and modal exchange, and has high computation times and memory requirements. In this paper, an ERPCA with a forgetting factor (ERPCAWF) based OMA method for the online identification of transient modal parameters in weakly damped LTV structures is considered. ERPCA uses all of the old data and a rank-one matrix to update the eigenvalue and eigenvector. This eigenvector corresponds to the modal shape vector. However, the presence of old data makes it difficult to extract the current features. Thus, a forgetting factor is introduced to reduce the weight of the old data. In our method, the forgetting factor has a fixed value; determining this value according to the time-varying speed of the structures remains an open problem. In addition, the proposed method is currently confined to LTV structures. In summary, our contributions are as follows:  
1. We simplify the model and coefficients, and describe a transient representation of LTV structures. Furthermore, we design the steps of the PCA-based OMA vibration control method to reduce amplitude of the vibration response. 2.