## 3 The Gain Parameter Adaption and Scheduling Control Method
### The Gain Scheduling Control Principle
\begin{array}{l}K_{1}(\omega)=\frac{\omega_{j}-\omega}{\omega_{j}- \omega_{j-1}}K_{1}(\omega_{j-1})+\frac{\omega-\omega_{j-1}}{\omega_{j}-\omega _{j-1}}K_{1}(\omega_{j})\\ K_{2}(\omega)=\frac{\omega_{j}-\omega_{j}}{\omega_{j}-\omega_{j-1}}K_{2}( \omega_{j-1})+\frac{\omega-\omega_{j-1}}{\omega_{j}-\omega_{j-1}}K_{2}( \omega_{j})\\ \end{array} \right. \tag{15}\]  
Equation (14) indicates that the accuracy of the influence coefficient has a significant influence on the performance and stability of the dynamic balance control system. However, the influence coefficient error between the estimation value and the real value is inevitable on account of the balance system errors and working condition transformation. Here we discuss the effect of the influence coefficient error on the stability of the control system. Assuming that the acceleration of the spindle is very small, the damping factor is very large and the discrete speed point is very dense in a certain range, we can take the dynamic characteristics of the system as close to steady state at various discrete speeds. If the system is stable at all discrete speeds, it can be assumed that the system is stable during the entire transmission period. When the speed is constant, Equation (10) is substituted to Equation (14) to obtain the following formula:  
\[P_{k+1}=K_{2}P_{k}-K_{1}RP_{k}-K_{1}V_{k}=\left[K_{2}-K_{1}R\right]P_{k}-K_{1} V_{k} \tag{16}\]  
\[P_{k+1}=\left[\overline{R}Q\hat{R}+H\right]^{-1}\overline{R}Q[\hat{R}-R]P_{k}- \left[\overline{R}Q\hat{R}+H\right]^{-1}\overline{R}QV_{k} \tag{17}\]  
In Equation (17), \(\hat{R}\) is the estimated value of \(R\), and \(\hat{R}-R\) is the error matrix of the influence coefficient. For stability control, the input of \(P_{k+1}\) must converge to a constant vector. At the balance speed, the original unbalance vibration \(V_{k}\) of the rotor system is a constant vector. Because the estimated influence coefficient matrix is a constant matrix, the second item on the right of Equation (17) is the constant vector. Therefore the first item on the right of Equation (17) is related to stability. That is to say the condition for system stability is:  
\[\gamma=\delta_{\max}[\overline{R}Q\hat{R}+H]^{-1}\overline{R}Q[\hat{R}-R]<1 \tag{18}\]  
In Equation (18), \(\gamma\) represents the stability of the control system, \(\delta_{\max}\) represents the maximum singular value of the matrix. Equation (18) can be translated as follows. \[ \begin{array}{l}\gamma=\delta_{\max}[\overline{R}Q\hat{R}+H]^{-1}\overline{R }Q[\hat{R}-R]\leq\frac{\delta_{\max}\overline{R}Q\delta_{\max}[\hat{R}-R]}{ \delta_{\min}[\overline{R}Q\hat{R}]}\\ \leq\frac{\delta_{\max}\overline{R}Q\delta_{\max}[\hat{R}-R]}{\delta_{\min}[ \overline{R}Q\hat{R}]}=\xi\\ \end{array}  \tag{19}\]  
In Equation (19), \(\delta_{\min}\) represents the minimum singular value of the matrix. \(\zeta\) represents the stability of the traditional weighted least square method (WLSM) when \(H\) = 0, where the robustness of WLSM had been proved in paper reference [16]. It is obvious that the stability of the system can be increased by the unbalance correction penalty factor H.