# Wavelet-Based Estimation of Damping from Multi-Sensor, Multi-Impact Data
## 3 Damping Extraction Methods
### Frequency-Domain Damping Extraction Methods
#### 3.1.3 Poly-Reference Least Squares Complex Frequency  
Similarly to LSRF and PP, the poly-reference least squares complex frequency (pLSCF) method uses measured frequency responses as primary data. This method, combined with maximum likelihood, is used to identify system poles with higher certainty [39; 40]. It should be noted that, while LSRF applies least squares to the real-valued model, pLSCF applies least squares to the complex-valued model. This method, also commercially known as PolyMAX, is commonly applied to operational data due to its fast computation. In this method, the following right matrix-fraction model is used to represent the measured data:  
\[\hat{H}(\Omega_{f},\theta)=B(\Omega_{f},\theta)A(\Omega_{f},\theta)^{-1}, \tag{10}\]  
where \(\hat{H}(\Omega_{f},\theta)\in\mathbb{C}^{N_{b}\times N_{i}}\) is the frequency response function (FRF) matrix, \(B(\Omega_{f},\theta)\in\mathbb{C}^{N_{b}\times N_{i}}\) is the numerator matrix polynomial, and \(A(\Omega_{f},\theta)\in\mathbb{C}^{N_{i}\times N_{i}}\) is the denominator matrix polynomial. The polynomial basis function \(\Omega_{f}=e^{-i\omega_{f}T_{s}}\) accounts for the discrete-time nature of the estimation process.  
The stabilization chart is used to distinguish between physical modes and spurious numerical modes. It is constructed using the pLSCF estimator, which employs measured FRFs as primary data and can also be applied to operational data. By examining the stability of identified poles across various model orders, the physical modes are selected. After determining the poles and the participation factors from the stabilization diagram, the mode shapes, denoted as \(\psi_{r}\), along with the upper and lower residual terms, can be estimated using the least squares frequency domain (LSFD) method [41]. The modal model in the frequency domain is then expressed as follows:  
\[H(s_{f})=\sum_{r=1}^{N_{m}}\!\left(\frac{\psi_{r}L_{r}}{s_{f}-\lambda_{r}}+ \frac{\psi_{r}^{*}L_{r}^{*}}{s_{f}-\lambda_{r}^{*}}\right)+\frac{[LR]}{s_{f}^{ 2}}+[UR], \tag{11}\]  
where \(H(s_{f})\in\mathbb{C}^{N_{b}\times N_{i}}\) is the FRF matrix, \(N_{m}\) is the number of identified modes, \(\psi_{r}\in\mathbb{C}^{N_{b}\times 1}\) is the \(r\)-th mode shape, \(\lambda_{r}\) is the pole, and \(L_{r}\in\mathbb{C}^{1\times N_{i}}\) is the participation factor. The terms \([LR]\) and \([UR]\) represent the lower and upper residual terms, modeling the influence of out-of-band modes in the considered frequency band. The interpretation of the stabilization diagram provides a set of physical poles and their corresponding participation factors. Since the mode shapes and residual terms are the only unknowns, they are estimated by solving Equation (11) in a linear least-squares sense.  
The next step is to formulate the modal model equation and fit this model to the measured data using maximum likelihood estimation. The maximum likelihood method allows us to account for the uncertainty in the measured frequency responses, leading to more accurate parameter estimates. In this stage, we will minimize the cost function of the frequency-domain maximum likelihood estimator, which employs the modal model instead of the rational fraction polynomial model to represent the measured FRFs. This minimization will be performed using a Gauss-Newton algorithm, starting from the initial values of the modal parameters provided by the pLSCF and LSFD estimators.