## 3 Optimization Methods for Feedback Controllers
\tag{11}\]  
In the first term on the right-hand side of (10), \(\mathrm{[f}_{\mathrm{start}},\,\mathrm{f}_{\mathrm{stop}}]\) is the target frequency band where the primary noise should be attenuated with the designed controller, and \(\mathrm{G}(\mathrm{f}_{\mathrm{k}})\) is the value of the sensitivity function at \(\mathrm{f_{k}}\)  
\[\mathrm{G}(\mathrm{f}_{\mathrm{k}})=\left.\mathrm{G}(\mathrm{z})\right|_{ \mathrm{z}=\mathrm{e}\mathrm{i}^{2\mathrm{z}\mathrm{f}_{\mathrm{k}}/\mathrm{ f}_{\mathrm{k}}}} \tag{12}\]  
As the magnitude of sensitivity function represents the NR performance, the first term on the right-hand side of (10) gives a weighted total NR performance with \(\mathrm{w_{k}}\) as the weight coefficients for frequency \(\mathrm{f}_{\mathrm{k}}\). In order for a better NR performance, this term should be minimized. The secondary term on the right-hand side of (10) is a punishment term corresponding to the stability margins, where \(\mathrm{L}(\mathbf{X})\), \(\mathrm{\gamma}(\mathbf{X})\) are the magnitude and phase margins with the designed controller, which is determined by \(\mathbf{X}\), and \(\mathrm{L}_{\mathrm{min}}\), \(\mathrm{\gamma}_{\mathrm{min}}\) are their corresponding lower bounds. If the designed controller does not have enough stability margins, this term would address a positive punishment on the objective function and the punishment intensity is \(\mathrm{\xi}_{1}\). The third term on the right-hand side of (10) represents the constraint for noise enhancement at frequencies higher than \(\mathrm{f}_{\mathrm{stop}}\). If the noise enhancement at \(\mathrm{f}_{\mathrm{k}}\) (>\(\mathrm{f}_{\mathrm{stop}}\)) is larger than a predefined upper bound \(\mathrm{\delta}_{\mathrm{k}}\), this term would address a positive punishment value \(\mathrm{\xi}_{2}\) on the objective function. Similar to the case for feedforward controller design [20], an upper bound \(\mathbf{ub}\) as well as a lower bound \(\mathbf{lb}\) are imposed on the parametric vector \(\mathbf{X}\), so that the parametric space could be reduced and the optimization process could be eased. The fourth term on the right-hand side of (10) gives the punishment if **X** exceeds its predefined bounds, where sum(\(\cdot\)) denotes for the summation over a vector and \(\xi_{3}\) is the punishment intensity. It can be found that in order to minimize the objective function defined by (10), all the punishment terms should be forced to 0 if proper intensity values are chosen. This indicates that the feedback control system has enough stability margin and the noise enhancement is also limited for all frequencies. Thus, two optimization methods are proposed, as follows:  
\[\text{Method1}:\ \ \ \textbf{X}_{\text{o}}=\ \min\{\text{Obj}|\textbf{X}\}_{| \text{G}(\text{z})=\text{G}_{\text{mic}}(\text{z})} \tag{13}\]  
\[\text{Method2}:\ \ \ \textbf{X}_{\text{o}}=\ \min\{\text{Obj}|\textbf{X}\}_{| \text{G}(\text{z})=\text{G}_{\text{cur}}(\text{z})} \tag{14}\]  
For Method 1, the NR performance and noise enhancement are evaluated at the feedback microphone, which is in accordance with previous works. However, if the headphone presents large FR mismatches, the actual performance perceived by the human ear would deteriorate, as analyzed in Section 2. Instead, the evaluation point is set to the human ear in Method 2 and a better actual performance of the feedback ANC headphone could be expected. It should be noted that in Method 2 the FR corresponding to \(\Delta\)(z) should be tested and known as a priori. In this work, DE algorithm is used to solve the optimization problems shown by (13) and (14). DE algorithm is considered to be powerful and efficient to find the global minimum with respect to optimization problems with large scale parametric spaces [19]. Unlike the genetic algorithm, which needs to discretize the space of variables, the DE algorithm operates directly over the continuous space, which is more straightforward to find the global minimum and the combination with other optimizers [14] could be avoided.