## 2 Principle
### Linear Situation with Coefficient  
With the change of experimental condition, the TF deviation ratio may not satisfy the above convergence condition \(|I-P|<I\). If so, the iteration diverges, which should be solved.  
For a linear system, this is mainly caused by the estimation error between the linear TF \(H_{L}\) and the actual TF \(H\), especially when the noise of data acquisition is great. For most inertial systems, the abnormal high frequency response caused by noise may override the relatively small response characteristics of the system itself in the high frequency part.  
Normally. the drive correction is scaled by a factor called iteration coefficient which is represented by \(\mu\) in this paper. The \(\mu_{k}\) represents a series of constants used in each iteration step.  
Thus, the initial conditions are changed. The initial drive, the first response error and first drive correction are:  
\[D_{0}=\mu_{0}H_{L}^{-1}R_{0} \tag{15}\]  
\[\Delta_{1}=R_{0}-HD_{0}=(I-\mu_{0}P)R_{0} \tag{16}\]  
\[\Delta_{D1}=H_{L}^{-1}\Delta_{1}=H_{L}^{-1}(I-\mu_{0}P)R_{0} \tag{17}\]  
The response error after \(k\) times of iteration is:  
\[\Delta_{k}=R_{0}-HD_{k-1}=(I-\mu_{k-1}P)\Delta_{k-1} \tag{18}\]  
The drive after \(k\) times correction is:  
\[D_{k}=D_{k-1}+\mu_{k}\Delta_{Dk} \tag{19}\]  
The \(i\)th drive correction and the drive after \(k\) times correction are:  
\[\Delta_{Di}=H_{L}^{-1}\left[\prod_{m=0}^{i-1}(I-\mu_{m}P)\right]R_{0},\ i\geq 1 \tag{20}\]  
\[D_{k}=D_{0}+\sum_{i=1}^{k}\mu_{i}\Delta_{Di} \tag{21}\]  
In the case of a linear system, P stays the same. But in practice, the system is always non-linear, and the difference is the degree of non-linear. When the system is less nonlinear, P is close to I, so the convergence condition \(|I-\mu_{i}P|<I\) can be satisfied when \(\mu_{i}\) is close to 1.  
According to engineering experience, the iteration coefficient is chosen to be \(0\leq\mu_{i}\leq 1\). At the beginning of the iteration, because the drive and response of the iteration differ greatly from the desired value, so the drive correction will be large. Therefore, the selection of the iteration coefficient is small to avoid driving overshoot.  
As a controlling method, if the TF deviation ratio without iteration coefficient is out of the convergence condition (\(|I-P|\geq I\)), then iteration coefficient is used to make the convergence condition satisfy (\(|I-\mu_{i}P|<I\)). If the convergence condition is achieved, the drive correction will trend to zero based on the expression of drive correction in equation (20).  
\[\lim_{i\rightarrow\infty}\Delta_{Di}=0 \tag{22}\]Based on the expression of drive and drive correction in equations (20) and (21), the last drive after infinite steps is  
\[D_{\infty}\triangleq H_{L}^{-1}S_{\infty}R_{0} \tag{23}\]  
where \(S_{\infty}\) stands for composite gain and is expressed as:  
\[S_{\infty}=\mu_{0}+\sum_{i=1}^{\infty}\mu_{i}\left[\prod_{m=0}^{i-1}\left(I- \mu_{m}P\right)\right] \tag{24}\]  
It is easy to approve that  
\[1-PS_{\infty}=\prod_{i=0}^{\infty}(I-\mu_{i}P) \tag{25}\]  
If the convergence condition satisfies, then  
\[1-PS_{\infty}=0 \tag{26}\]  
The composite gain is the inverse of the TF deviation ratio \(P\):  
\[S_{\infty}=P^{-1} \tag{27}\]  
Finally, it can be proved that the last drive after infinite steps will be equal to the road drive:  
\[D_{\infty}=H_{L}^{-1}S_{\infty}R_{0}=H_{L}^{-1}P^{-1}R_{0}=H^{-1}R_{0} \tag{28}\]