# 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  
In this research, the subject was a dual-inertia servo system, where the parameters of the friction model are either difficult to obtain or subject to variability. Considering friction as a disturbance, it was estimated and compensated through the use of a state observer. Figure 4: Feedforward compensation based on friction modeling. Figure 3: Curve fitting and iterative optimization of free deceleration. (**a**) Experiment 1. (**b**) Experiment 2. In recent years, active disturbance rejection control (ADRC) has been extensively studied and applied as an algorithm that does not depend on precise models and can effectively address system disturbances [18]. The core idea of ADRC is to conceptualize all uncertainties affecting the controlled system as the \"total disturbance\". This disturbance is then expanded into the system states and estimated in real-time through an extended state observer (ESO), the typical structure of which is depicted in Figure 5. The tracking differentiator (TD) plays a crucial role by enabling rapid and overshoot-free tracking of input signals, providing differential signals, and managing transition processes to prevent control saturation due to excessively large reference signals. The extended state observer is a significant component within ADRC, tasked with observing the output and its derivatives. The nonlinear state error feedback control law (NLESF) combines the errors of the extended states nonlinearly to derive the control values. Traditional ADRC is characterized by its minimal disturbance estimation error and rapid dynamic response. However, it encounters difficulties in parameter tuning and features complex nonlinear functions. For instance, ADRC utilizes the fast synthetic control function, _fhan_, for discretization to achieve swift tracking and differentiation while avoiding high-frequency oscillations; it also employs the _fal_ power function to establish a nonlinear feedback control law. The expressions for both functions, as shown in (7), are relatively complex. Therefore, this project adopted the linear ADRC (LADRC) approach proposed by the team led by [PERSON] [19]. \[\left\{ \begin{array}{l}{\it fhan}=-\left\{\begin{array}{l}{\it rsign}(a), \left|a\right|>d\\ {\it r}\frac{a}{d},\left|a\right|\leq d\\ \end{array}\right.\\ {\it fal}=\left\{\begin{array}{l}\frac{\pi}{\eta^{1-\alpha}},\left|x\right| <\eta\\ \left|x\right|_{\alpha}{\it sign}(x),\left|x\right|\geq\eta\end{array}\right.\\ \end{array} \right. \tag{7}\]  
In Equation (7), \(r\) represents the rate factor, which determines the response speed of the differentiator; \(a\) and \(d\) are mathematical expressions for control compensation and the rate factor and do not possess practical significance; \(x\) denotes the error or the error differential signal; \(\alpha\) and \(\eta\) are adjustable parameters of the controller, and they also do not have practical significance. To simplify the analysis, the initial consideration focused on the no-load steady-state phase where the steady-state torque was zero. The expression for the controlled object is as follows:  
\[ \begin{array}{l}\dot{\omega}=\frac{1}{{\it f}_{m}}[K_{T}({\it i}_{q}+{\it i }_{comp})-T_{f}]\\ =\frac{1}{{\it f}_{m}}{\it i}_{q}+\frac{1}{{\it f}_{m}}[K_{T}{\it i}_{comp}-T_{ f}]={\it bi}_{q}+f\end{array}  \tag{8}\]  
Figure 5: Block diagram of the classical structure of active disturbance rejection control. The selection of the system state variables is as follows:  
\[\dot{x}= \begin{bmatrix}\dot{\omega_{m}}\\ \dot{f}\end{bmatrix} =\underbrace{ \begin{bmatrix}0&1\\ 0&0\end{bmatrix} }_{A} \begin{bmatrix}\omega_{m}\\ f\end{bmatrix} + \begin{bmatrix}b\\ 0\end{bmatrix} \dot{i}_{q}+ \begin{bmatrix}0\\ 1\end{bmatrix} \dot{f} \tag{9}\]  
The selection of the output variables is as follows:  
\[y=\omega_{m}= \begin{bmatrix}1&0\\ &c\end{bmatrix}\begin{bmatrix}\omega_{m}\\ f\\ x\end{bmatrix}  \tag{10}\]  
The design of the second-order ESO observer is as follows. \[\left\{