# Structural Modal Time Domain Identification Method Based on the Bayesian Uncertain Quantification
## 1 Introduction
[16; 17] obtained the optimal estimates of the modal parameters based on the maximum likelihood estimation algorithm minimizing the error between the measurements and the estimated modal model and quantified the uncertainty through the covariance matrix of the modal parameters. [PERSON] et al. [18] gave the covariance matrices of the state space matrices \(\mathbf{(A,C)}\) and the modal parameters for the input-output subspace approach based on the perturbation method and further derived the covariance of the input matrices \(\mathbf{(B,D)}\) and the transfer function [19]. In Bayesian system identification methods, probability is used as a measure of the relative plausibility of the identification result given the system model and measurement data, and the goal of modal identification is to determine the posterior probability density function (PDF) of the modal parameters. [PERSON] et al. [20] proposed a fast Bayesian identification method for forced vibration tests with a single shaker input and multiple outputs. [PERSON] and [PERSON] [21; 22] further considered the effect of environmental factors on forced vibration modal tests. [PERSON] and [PERSON] [23; 24] also utilized the response under seismic excitation to identify the modal parameters of the structure and quantified the identification uncertainty through a posterior coefficient of variation. Recently, [PERSON] [25] developed an uncertainty law for modal parameters with well-separated modes with known single broadband inputs. All aforementioned Bayesian modal identification methods are performed in the frequency domain and necessitate the application of a Fast Fourier Transform (FFT) to the measured response data. This process introduces additional errors, particularly when the sample size of the measured data is limited. In contrast to the frequency domain approach, the time domain method facilitates the direct utilization of measured response data for modal parameter identification, which minimizes the uncertainty caused by data processing. According to the literature already mastered by the authors, there is currently no relevant work on Bayesian time domain methods for the uncertainty quantification of the structural modal parameter identification based on EMA under deterministic excitation. Assuming that the system input is band-limited Gaussian white noise, [PERSON] and [PERSON] [26] constructed a multivariate Gaussian probability density function through the autocorrelation function of structural response and established a Bayesian time domain approach for modal updating based on the approximation of a conditional probability expansion of the response. Numerical experiments were conducted on a single degree of freedom system and a four-bay truss to verify the proposed method. Inspired by the Bayesian FFT method, this paper proposes a Bayesian time domain modal identification uncertainty quantization method based on the modal state space model. This paper is organized as follows. In Section 2, based on the state space model, the functional expressions of modal displacement and acceleration with respect to dynamic spectral parameters are derived in modal space, and a predictive model of the theoretical response in physical space is given through the modal basis. In Section 3, firstly, the negative log-likelihood function (NLLF) is derived according to the Bayesian formula. Secondly, by vectorizing the spatial parameter matrix, a quadratic function involving the spatial parameter vector is derived. Furthermore, an optimization objective function solely focused on spectral parameters is established by applying the function extremum theory, and the spatial parameter can be derived analytically from the spectral parameter. Finally, the posterior covariance matrix of the modal parameters is solved by using the finite difference method. The implementation process of the proposed method is summarized at the end of the subsection.