## 3 Method
### Control Strategy  
The purpose of control is to make the series of drive converge with good stability and fast convergence rate.  
The convergence condition is mentioned before, which makes the drive correction tend to zero:  
\[\Delta_{Dk}\to 0 \tag{37}\]  
For the nonlinear system in Equation (31), if the error ratio \(\Psi_{k}\) is nearly zero:  
\[\Psi_{k}=H_{k}H_{k-1}^{-1}-\mu_{k}\bigg{(}P_{k}+\frac{\partial H_{k}}{ \partial D_{k}}H_{L}^{-1}H_{k-1}^{-1}R_{0}\bigg{)}\to 0 \tag{38}\]  
Then the convergence will be fast and robust. The \(\Psi_{k}\) could be divided into 2 parts, the TF ratio \(\mathcal{L}_{k}\), and effective ratio \(\mathcal{M}_{k}\):  
\[\mathcal{L}_{k}=H_{k}H_{k-1}^{-1} \tag{39}\]\[\mathcal{M}_{k}=P_{k}+\frac{\partial H_{k}}{\partial D_{k}}H_{L}^{-1}H_{k-1}^{-1}R _{0} \tag{40}\]  
Then set \(\Psi_{k}\) as 0, the ideal coefficient will be:  
\[\mu_{k}=\frac{\mathcal{L}_{k}}{\mathcal{M}_{k}} \tag{41}\]  
While \(\mu_{k}\) is numerical coefficient, and \(H_{k}\) are complex sequence with frequency \(f\) as independent variable, so \(\mu_{k}\) is calculated as:  
\[\frac{1}{\mu_{k}}=\|\frac{\mathcal{M}_{k}}{\mathcal{L}_{k}}\|=\|\frac{P_{k}+ \frac{\partial H_{k}}{\partial D_{k}}H_{L}^{-1}H_{k-1}^{-1}R_{0}}{H_{k}H_{k-1} ^{-1}}\|=\|P_{k-1}+\frac{\partial H_{k}}{\partial D_{k}}H_{L}^{-1}H_{k}^{-1}R _{0}\| \tag{42}\]  
This iteration coefficient is calculated with vector norm to ensure stability, which makes the \(\mu_{k}\) as small as possible. The vector norm \(\|\quad\|\) is vector norm of complex sequence. The vector norm used here is the Maximum norm of the complex vector. For complex vector, first it takes the absolute value of complex numbers to get the real vector, and then takes the maximum norm of the real vector to get the maximum. The expression of the norm is shown as:  
\[\|\Phi_{k}\|=max\{|\Phi_{k}(f)|\}=max\{|\Phi_{k}(f_{1})|,|\Phi_{k}(f_{2})|, \ldots,|\Phi_{k}(f_{n})|\} \tag{43}\]  
The vector norm is also shown in Figure 3:  
As shown in Figure, the \(\Phi_{k}\) is complex sequence with the frequency as the independent variable, and the complex point \(\Phi_{k}(f_{m})\) has the maximum magnitude, which is the value of the vector norm \(\|\Phi_{k}\|\).  
In short, the purpose of control strategy is to make the iterative process meet the convergence condition, as shown in Figure 4 below:  
As shown in Figure 4, the error ratio \(\Psi_{k}^{s}\) without iteration coefficient \(\mu_{k}\) may be out of the Unit Circle, and the iteration coefficient \(\mu_{k}\) can make the error ratio \(\Psi_{k}\) be in the Unit Circle to meet the convergence condition and be as small as possible to make convergence fast. The Unit Circle is the complex circle with radius 1.  
If the system is close to linear system, then the TF will basically remain unchanged, the TF ratio \(\mathcal{L}_{k}\) will be close to 1 at the real axis. While the effective ratio \(\mathcal{M}_{k}\) contains 2 parts, TF deviation ratio \(P_{k}\) and the relative change amount of the TF. In addition, if the system is close to linear system, then the TF deviation ratio \(P_{k}\) is close to 1, and the relative change amount of the TF is a small area around the original point.  
Figure 3: Vector norm sketch.