## 2 Mathematical Model
### Broyden-Fletcher-Goldfarb-Shanno (BFGS) Algorithm  
The BFGS method [20] is an unrestricted optimization method that has been recently acknowledged as the most effective algorithm for solving unrestricted nonlinear optimization problems by the iterative method. The BFGS method was used to adjust the \(\textbf{P}_{(\mathrm{n})}\) value in the following manner:  
\[\textbf{P}_{(\mathrm{n})}=-\Big{(}\
abla J_{(\mathrm{n})}\textbf{A}_{(\mathrm{ n})}\Big{)}=-\Bigg{(}\Bigg{(}\frac{\partial J}{\partial\textbf{w}}\Bigg{)}_{( \mathrm{n})}\textbf{A}_{(\mathrm{n})}\Bigg{)} \tag{10}\]  
\[\textbf{A}_{(\mathrm{n})}=\textbf{A}_{(\mathrm{n-1})}+\textbf{E}_{(\mathrm{n- 1})} \tag{11}\]  
\[\textbf{E}_{(\mathrm{n})}= \frac{1}{(\Delta\textbf{w}_{(\mathrm{n})})^{T}\Delta\textbf{g}_{( \mathrm{n})}}\times \tag{12}\] \[\Bigg{[}\big{(}\Delta\textbf{w}_{(\mathrm{n})}\big{)}^{T}\Delta \textbf{w}_{(\mathrm{n})}\big{)}\bigg{(}1+\frac{(\Delta\textbf{g}_{(\mathrm{n })})^{T}\textbf{A}_{(\mathrm{n})}\Delta\textbf{g}_{(\mathrm{n})}}{\Delta \textbf{w}\Delta\textbf{g}_{(\mathrm{n})}}\bigg{)}-\textbf{A}_{(\mathrm{n})} \Delta\textbf{g}_{(\mathrm{n})}(\Delta\textbf{x}_{(\mathrm{n})})^{T}-\Delta \textbf{x}_{(\mathrm{n})}(\Delta\textbf{g}_{(\mathrm{n})})^{T}\textbf{A}_{( \mathrm{n})}\Bigg{]}\]  
where \(\textbf{A}_{(0)}=\textbf{I}\) is the identity matrix. \(\Delta\textbf{w}_{(\mathrm{n})}=\textbf{w}_{(\mathrm{n})}-\textbf{w}_{( \mathrm{n-1})}\) and \(\Delta\textbf{g}_{(\mathrm{n})}=\textbf{g}_{(\mathrm{n})}-\textbf{g}_{( \mathrm{n-1})}\) are the differences between iteration values and \(\textbf{g}_{(\mathrm{n})}=\
abla J(\textbf{w})_{(\mathrm{n})}\) is the gradient function, which can be calculated, as follows:  
\[\
abla J(\textbf{w})_{(\mathrm{n})}=\Big{(}\frac{\partial J}{\partial\textbf{ w}}\Big{)}_{(\mathrm{n})}=\Bigg{(}\frac{\partial J}{\partial\textbf{M}_{m} },\Big{[}\frac{\partial J}{\partial\textbf{A}_{N}^{M}}\Big{]}_{N_{\mathrm{H} (\mathrm{A})}},\Big{[}\frac{\partial J}{\partial\textbf{A}_{N}^{M}}\Big{]}_{ N_{K_{\mathrm{H}(\mathrm{A})}}},\Big{[}\frac{\partial J}{\partial\textbf{A}_{N}^{M}} \Big{]}_{N_{K_{\mathrm{H}(\mathrm{A})}}},\Big{[}\frac{\partial J}{\partial \textbf{A}_{N}^{M}}\Big{]}_{N_{L_{\mathrm{e}(\mathrm{A})}}}\Bigg{)}_{(\mathrm{ n})} \tag{13}\]  
where \(\partial J/\partial\textbf{w}\) is the Jacobian matrix of the target function \(J(\textbf{w})\), it can be obtained by using the forward difference method, as follows:  
\[\Bigg{(}\frac{\partial J}{\partial\textbf{w}}\Bigg{)}_{(\mathrm{n})}=\Bigg{(} \frac{\textbf{x}(t;\textbf{w}+\Delta h)-\textbf{x}(t;\textbf{w})}{\Delta h}+ \frac{\textbf{i}(t;\textbf{w}+\Delta h)-\textbf{i}(t;\textbf{w})}{\Delta h}\Bigg{)} \tag{14}\]  
where \(\Delta h\) is set to a small number as \(1\times 10^{-8}\) to get the approximation of \(\textbf{g}_{(n)}\). After determination of the search direction \(\textbf{P}_{(\mathrm{n})}\) and \(\textbf{w}\) minor deviation, a corresponding minor deviation \(\Delta\textbf{x}(t;\textbf{w})\) is produced for displacement \(\textbf{x}(t;\textbf{w})\). Thus, substituting \(\Delta\textbf{x}(t;\textbf{w})\) and \(\Delta\textbf{i}(t;\textbf{w})\) into Equation (8) leads:  
\[ \begin{array}{l}J(\textbf{w})_{(\mathrm{n+1})}=\int_{0}^{t_{f}}\Big{[} \textbf{x}(t;\textbf{w}_{(\mathrm{n})}-\beta_{1(\mathrm{n})}\textbf{P}_{( \mathrm{n})})-\textbf{x}_{nea}(t)\Big{]}^{2}dt+\mu\int_{0}^{t_{f}}\Big{[} \textbf{i}(t;\textbf{w}_{(\mathrm{n})}-\beta_{2(\mathrm{n})}\textbf{P}_{( \mathrm{n})})-\textbf{i}_{nea}(t)\Big{]}^{2}dt\\ =\int_{0}^{t_{f}}\Big{[}\textbf{x}(t;\textbf{w}_{(\mathrm{n})})-\beta_{1( \mathrm{n})}\Delta\textbf{x}(t;\textbf{P}_{(\mathrm{n})})-\textbf{x}_{nea}(t) \Big{]}^{2}dt+\mu\int_{0}^{t_{f}}\Big{[}\textbf{i}(t;\textbf{w}_{(\mathrm{n}) })-\beta_{2(\mathrm{n})}\Delta\textbf{i}(t;\textbf{P}_{(\mathrm{n})})-\textbf{ i}_{nea}(t)\Big{]}^{2}dt\end{array}