## 3 The Gain Parameter Adaption and Scheduling Control Method
### The Adaptive Principle of the Influence Coefficient and Gain
#### 3.3.1 The Self-Adaption Methods of the Influence Coefficient  
Based on the preceding text analysis, the \(k\)th time testing and calculating value (\(R_{k}\)) of the influence coefficient at the same balance speed can be expressed as.  
\[R_{k}=[V_{k}-V_{k-1}][P_{k}-P_{k-1}]^{-1} \tag{20}\]  
The \(k\)th time estimated value (\(\hat{R}_{k}\)) of influence coefficient can be obtained by the weighting average between the \(k\)th time testing and calculating value and the (\(k-1\))th time estimated value (\(\hat{R}_{k-1}\)).  
\[\hat{R}_{k}=AR_{k}B+C\hat{R}_{k-1}D \tag{21}\]  
In Equation (21), \(A\), \(B\), \(C\), and \(D\) are a 2 \(\times\) 2 diagonal matrix and named as forgetting factors. The diagonal values of \(A\), \(B\), \(C\), and \(D\) belong to (0,1). The parameters in Equation (21) are assumed as follows.  
\[A=\left|\begin{array}{cc}a_{11}&0\\ 0&a_{22}\end{array}\right|_{,}B=\left|\begin{array}{cc}b_{11}&0\\ 0&b_{22}\end{array}\right|_{,}C=\left|\begin{array}{cc}c_{11}&0\\ 0&c_{22}\end{array}\right|_{,}D=\left|\begin{array}{cc}d_{11}&0\\ 0&d_{22}\end{array}\right|_{,}R_{k}=\left|\begin{array}{cc}R_{k11}&R_{k12} \\ R_{k21}&R_{k22}\end{array}\right|\]  
Based on the above hypotheses, the influence coefficient of correcting face \(i\) relative to testing point \(j\) after \(k\)th time estimated can be expressed as follows.  
\[\hat{R}_{kij}=a_{ii}b_{jj}R_{kij}+c_{ii}d_{jj}\hat{R}_{(k-1)ij} \tag{22}\]  
The real influence coefficients at the same balance speeds are assumed as constant, the \(k\)th time testing and calculating value of influence coefficient can be expressed as Equation (23) in consideration of the random error.  
\[R_{kij}=R_{ij}+e \tag{23}\]  
In Equation (23), \(R_{ij}\) is the real influence coefficient, \(e\) is the uniform white noise signal. The transfer function of Equation (22) can be expressed as Equation (24) if \(R_{kij}\) is taken as the input signal and \(\hat{R}_{kij}\) is taken as the output signal.  
\[G(Z)=\frac{a_{ii}b_{jj}}{1-c_{ii}d_{jj}z^{-1}} \tag{24}\]  
To ensure the signal is undistorted, \(G\)(1) = 1 when \(\omega\) = 0. So, the relation among the forgetting factors can be obtained from Equation (24).  
\[\begin{array}{c}a_{ii}b_{jj}=1-c_{ii}d_{jj}\\ \mu_{ij}=a_{ii}b_{jj}\end{array} \tag{25}\]The variance ratio before and after the signal filtered can be obtained from Equations (24) and (25).  
\[\frac{\delta_{1}^{2}}{\delta_{2}^{2}}=\frac{2-\mu_{ij}}{\mu_{ij}} \tag{26}\]  
Equation (26) indicates that the filtering effect will be better when \(\mu_{ij}\) tends to zero. A small value is suitable for \(\mu_{ij}\) if the kinetic parameters of the spindle system change slowly, and a large value is suitable for \(\mu_{ij}\) on the contrary. Based on the expected variance ratio, the value of \(\mu_{ij}\) can be calculated, and then the coefficient of \(A\), \(B\), \(C\), and \(D\) can be calculated. The adaptive relation equation of the influence coefficient as Equation (22) can be set up between the current testing and caculating value and the original estimated value.