## 3 The Gain Parameter Adaption and Scheduling Control Method
### The Gain Scheduling Control Principle  
When the influence coefficients of the balance speed have been estimated, a series of iteration controls are used to minimize the unbalance vibration amplitude, and the \(k\)th time iteration equation can be expressed as  
\[V_{k}=RP_{k}+V_{0} \tag{8}\]  
In Equation (8), \(V_{k}\) is the residual unbalance vibration matrix after the \(k\)th time iteration control. \(R\) is the influence coefficient matrix at the balance speed. \(P_{k}\) is the correcting matrix of the \(k\)th time correcting operation. \(V_{0}\) is the initial unbalance vibration matrix. The vector matrixes of \(V_{k}\), \(R\), \(P_{k}\) and \(V_{0}\) are expressed as follows  
\[V_{k}=\left[ \begin{array}{cc}\overset{\rightarrow}{V_{k1}}&\overset{ \rightarrow}{V_{k2}}\\ \overset{\rightarrow}{R_{12}}&\overset{\rightarrow}{R_{22}}\\ \end{array} \right]^{T},\,R=\left| \begin{array}{cc}\overset{\rightarrow}{R _{11}}&\overset{\rightarrow}{R_{21}}\\ \overset{\rightarrow}{R_{12}}&\overset{\rightarrow}{R_{22}}\\ \end{array} \right|,\,P_{k}=\left[ \begin{array}{cc}\overset{\rightarrow}{ P_{k1}}&\overset{\rightarrow}{P_{k2}}\\ \end{array} \right]^{T},\,V_{0}=\left[ \begin{array}{cc}\overset{\rightarrow}{ V_{01}}&\overset{\rightarrow}{V_{02}}\\ \end{array} \right]^{T}\]  
Based on Equation (8), the (\(k+1\))th time iteration equation can be expressed as:  
\[V_{k+1}=RP_{k+1}+V_{0} \tag{9}\]  
\(V_{k+1}\) can be expressed by subtracting Equation (8) from Equation (9) as the following  
\[V_{k+1}=R[P_{k+1}-P_{k}]+V_{k} \tag{10}\]  
According to the least square method, the generalized linear quadratic objective function can be built and shown as Equation (11). \[J_{k+1}=\frac{1}{2}\overline{V_{k+1}^{T}}QV_{k+1}+\frac{1}{2}\overline{P_{k+1 }^{T}}HP_{k+1} \tag{11}\]  
In Equation (11), \(H\) is a \(2\times 2\) positive definite real diagonal weighted matrix and is applied to improve the robustness and stability of the control system. \(Q\) is a \(2\times 2\) positive definite real diagonal weighted matrix and is applied to make the residual unbalance vibration of the testing points tend to equilibrium, so the diagonal values of \(Q\) should be set properly corresponding to the measuring points. Equation (12) can be derived by substituting Equation (10) into Equation (11) and then simplifying the equation. \[J_{k+1}=\frac{1}{2}\overline{\{R[P_{k+1}-P_{k}]\}^{T}}Q\{R[P_{k+1}-P_{k}]+V_{ k}]\}+\frac{1}{2}\overline{P_{k+1}^{T}}HP_{k+1} \tag{12}\]  
The function (13) can be derived by minimizing the objective function (12). \[\frac{\partial J_{k+1}}{\partial P_{k+1}}=\overline{R^{T}}Q\{R[P_{k+1}-P_{k}]+ V_{k}]\}+HP_{k+1}=0 \tag{13}\]Based on Equation (13), the following control law can be extracted. \[\left\{ \begin{array}{l}P_{k+1}=\left[\overline{R^{T}}QR+H\right]^{-1}\overline{ R^{T}}QRP_{k}-\left[\overline{R^{T}}QR+H\right]^{-1}\overline{R^{T}}QV_{k}=K_{2}P_{k}-K _{1}V_{k}\\ K_{1}=\left[\overline{R^{T}}QR+H\right]^{-1}\overline{R^{T}}Q,K_{2}=K_{1}R\\ \end{array} \right. \tag{14}\]  
In Equation (14), \(K_{1}\) and \(K_{2}\) are named as gain matrixes, which can be calculated when the influence coefficient matrix has been obtained. The gain scheduling table can be built and applied in the balance control when all the gain values have been calculated corresponding to each balance speed. In practical application, the work speed of the spindle may be not be equal to any balance speed with gain prameters, but the corresponding gain matrixes can be calculated by the linear interpolation method. If the work speed is assumed as \(\omega\), the two nearist balance speeds are assumed as \(\omega_{j}\) and \(\omega_{j-1}\). The gain matrixes corresponding to \(\omega\) can be calculated as follows. \[\left\{