# A Study of Friction Nonlinearity and Compensation for Turntable Servo Systems
## 3 Compensation Based on the Friction Model
### Friction Compensation Based on Extended State Observer
\begin{array}{l}\dot{\tilde{x}}=A\hat{x}+Bu+L(y-\hat{y})\Rightarrow \left[\begin{array}{c}\dot{\omega}_{m}\\ \hat{f}\end{array}\right]=\underbrace{\begin{bmatrix}0&1\\ 0&0\end{bmatrix}}_{A}\begin{bmatrix}\omega_{m}\\ f\end{bmatrix}+\begin{bmatrix}b\\ 0\end{bmatrix}\dot{i}_{q}+\begin{bmatrix}l_{1}\\ l_{2}\end{bmatrix}\begin{bmatrix}\omega_{m}-\hat{\omega}_{m}\\ f-\hat{f}\end{array} \right]\\ \\ \dot{y}=C\hat{x}\Rightarrow\hat{\omega}_{m}= \begin{bmatrix}1&0\\ &c\end{bmatrix}\begin{bmatrix}\hat{\omega}_{m}\\ \hat{f}\end{bmatrix} \end{array}\right. \tag{11}\]  
In Equation (11), \(L(y-\hat{y})\) represents the correction error term. Let the state estimation error be \(\widetilde{x}=x-\hat{x}\). By combining (9), (10), and (11), the state error equation can be derived. \[\frac{d\widetilde{x}}{dt}=\dot{x}-\dot{\hat{x}}=(A-LC)\widetilde{x}+\begin{bmatrix }0\\ 1\end{bmatrix}\dot{f} \tag{12}\]  
In the equation above, if \(f\) is bounded, then when the matrix \((A-LC)\) is a Hurwitz matrix (where all the roots of the polynomial have negative real parts), the observation error remains bounded [20]. By setting the characteristic polynomial in the form of (13) to ensure that all characteristic roots are in the left half-plane, the parameters are given as \(l_{1}=2\omega_{0}\) and \(l_{2}={\omega_{0}}^{2}\), where \(\omega_{0}\) represents the bandwidth of the observer. A higher \(\omega_{0}\) reduces the observation error but amplifies the impact of noise; conversely, a lower \(\omega_{0}\) results in a delay in the overall disturbance observation. In this study, \(\omega_{0}\) = 500 rad/s. \[|sI-(A-LC)|= \begin{vmatrix}s+l_{1}&-1\\ l_{2}&s\end{vmatrix} =s^{2}+l_{1}s+l_{2}=(s+\omega_{o})^{2} \tag{13}\]  
In the control law designed (14), \(\omega_{v}\) represents the desired bandwidth of the speed loop. For this study, \(\omega_{v}\) is set at 100 rad/s. \[i_{q0}=\omega_{v}({\omega_{r}}^{*}-\hat{\omega}_{m}) \tag{14}\]  
When the input is controlled in the form of (15), the output satisfies (16). \[i_{q}=\frac{i_{q0}-\hat{f}}{b} \tag{15}\]  
\[\dot{\omega}_{m}=bi_{q}+f=i_{q0}-\hat{f}+f\cong i_{q0} \tag{16}\]  
A simulation was conducted on the speed loop control based on the second-order extended state observer (ESO), aiming to assess the disturbance observation effects. The control diagram is depicted in Figure 6. Under no-load conditions, a step speed command of \(\omega_{r}{}^{*}\) = 30 rad/s was applied. It was observed that the motor speed was able to track the actual speed, stabilizing around 0.2 s after an overshoot. The friction torque, as an expanded state variable, eventually converged to the actual value. However, during the speed ramp-up phase, the observed values failed to track the real-time changes in friction torque. For applications requiring high dynamic response, it might be beneficial to also treat the dynamic response as an expanded state in the system, thus increasing the order of the ESO to reduce the observation error [21]. Replacing the speed loop PI controller with the aforementioned LADRC speed loop controller yielded discernible changes in the system dynamics. Upon imposing a predefined ramp position input, the system response, depicted in Figure 7, exhibited oscillations in the load position. The subsequent elimination of oscillatory behavior through clearance adjustment to zero underscores its pivotal role in system stabilization. These simulation findings highlight the diminished efficacy of the state feedback compensation algorithm for clearance after the adoption of the LADRC speed loop controller instead of the PI speed regulator. According to (8), the state feedback compensation current, _i_comp, is categorized as part of the total disturbance and is eliminated via feedforward compensation by the LADRC controller. An improved LADRC controller configuration is depicted in Figure 8a, where its output includes the compensation current. In Figure 8b, both the load position and speed are stable, with no oscillations observed.