## 3 Proposed Identification Methods
### Quasi-Newton Method  
The quasi-Newton (QN) method is a powerful gradient-based optimization algorithm that leverages information about the curvature of a cost function to iteratively find its minimum value. It approximates the Hessian matrix at each iteration and performs a cubic line search procedure to enhance the solution obtained. The QN method is particularly useful when the direct calculation of the Hessian matrix is difficult or impractical [54; 55; 56; 57]. By incorporating the QN method into the optimization process, the algorithm benefits from improved local optimization capabilities, increasing the likelihood of finding an optimal solution. At each iteration, the QN method constructs a quadratic model using curvature information. The optimal solution of this quadratic model denoted as \(x^{*}\) can be represented as follows:  
\[x^{*}=-H^{-1}\cdot\
abla f_{k} \tag{14}\]  
where \(H\) is a positive definite symmetric matrix (the Hessian), and \(\
abla f_{k}\) is the gradient of \(f\) at the point \(x_{k}\)[55; 56]. To update the Hessian matrix \(H\), the QN method uses the Broyden-Fletcher-Goldfarb-Shanno (BFGS) method, which gives the following formula [55]:  
\[H_{k+1}=H_{k}+\frac{y_{k}y_{k}^{T}}{y_{k}^{T}s_{k}}-\frac{H_{k}s_{k}s_{k}^{T} H_{k}^{T}}{s_{k}^{T}H_{k}s_{k}} \tag{15}\]  
where \(s_{k}\) and \(y_{k}\) are calculated as follows:\[s_{k} =x_{k+1}-x_{k} \tag{16}\] \[y_{k} =\
abla f_{k+1}-\
abla f_{k}\]  
Here, \(s_{k}\) represents the change in the independent variable \(x\), and \(y_{k}\) represents the change in the gradient. The Hessian matrix \(H\) can be initialized as any symmetric positive definite matrix, such as the identity matrix \(I\). To avoid inverting \(H\), the BFGS method and the Davidon-Fletcher-Powell (DFP) method use formulas that approximate the inverse Hessian at each update rather than inverting \(H\) directly. The BFGS method uses the updating formula shown above to approximate the Hessian matrix \(H\), while the DFP method uses the same updating formula but substitutes \(y_{k}\) for \(s_{k}\) to calculate the approximate inverse Hessian \(H^{-1}\)[55]. The inverse Hessian matrix \(H^{-1}_{k+1}\) can be calculated as follows:  
\[H^{-1}_{k+1}=H^{-1}_{k}+\frac{s_{k}s_{k}^{T}}{s_{k}^{T}y_{k}}-\frac{H^{-1}_{k }y_{k}y_{k}^{T}H^{-1}_{k}}{y_{k}^{T}H^{-1}_{k}y_{k}} \tag{17}\]  
Equation (17) approximates the inverse Hessian matrix in the DFP method. The BFGS method uses a similar updating formula but substitutes \(y_{k}\) for \(s_{k}\) in the equation. Both the BFGS and DFP methods avoid directly inverting the Hessian matrix \(H\) by employing these formulas to approximate the inverse Hessian at each iteration [55; 56]. At each \(k\)-th iteration, a line search is performed in the direction \(d_{k}\), which is calculated as follows [55; 56]:  
\[d_{k}=-H^{-1}_{k}\cdot\
abla f(x_{k}) \tag{18}\]  
The next solution \(x_{k+1}\) is calculated as follows:  
\[x_{k+1}=x_{k}+\alpha_{k}d_{k} \tag{19}\]  
where \(x_{k}\) represents the current iterate, \(d_{k}\) is the search direction, and \(\alpha_{k}\) is a scalar step length parameter chosen to satisfy the Wolfe conditions [55; 56]. The pseudo-code of the QN (BFGS) method is presented in the method in Algorithm 2. It utilizes the BFGS updating formula to approximate the Hessian matrix \(H\), and the convergence tolerance \(\epsilon\) is used to check the stopping criteria. The line search procedure determines the step size \(\alpha_{k}\) at each iteration. It is worth noting that QN methods exhibit a fast convergence rate and are relatively easy to implement. However, they require a good initial guess and are limited to finding the local minima in multimodal problems [55; 56].