## 2 Modeling and Analysis of Electric Actuator Control System
### Model Simplification and Active Disturbance Rejection Control (ADRC) Design  
In an actual system, the width of the dead-zone is unknown, so it is difficult to compensate. Therefore, an ESO is considered to treat the dynamic response caused by dead-zone non-linearity  
Figure 2: Block diagram of the non-linear actuator. Where \(U_{a}\) is the input voltage, and inside the dashed frame is the non-linear actuator model with deadzone. as an internal disturbance. In order to construct a second-order ADRC model, the model shown in Figure 2 needs to be further simplified. If the aerodynamic load of the actuator regarded as the external disturbance of the system, the current passing through the servo motor remains unchanged, when there is no disturbance in the system, that is, the rate of current is 0. From Equations (1)-(5), it can be simplified to:  
\[\ddot{\delta}+\frac{K_{T}K_{E}}{JR_{a}}\dot{\delta}=\frac{K_{T}}{i_{1}i_{2}J_{ R}}U_{a}-\frac{T_{d}}{i_{1}i_{2}j} \tag{7}\]  
Considering load disturbance and non-linear factors, Equation (7) can be rewritten as:  
\[\ddot{y}+a_{1}\dot{y}+a_{2}y=bu+w \tag{8}\]  
where \(w\) is an external disturbance containing the load disturbance term \(-\frac{T_{d}}{i_{1}i_{2}j}\), \(u\) means the input voltage \(U_{a}\), and \(y\) is the output deflection angle \(\delta\), \(a_{1}\) and \(a_{2}\) are unknown parameters considering the uncertainties of the model. Take the variables \(f\) and \(b_{0}\), make \(b_{0}\approx b\), and  
\[f=-a_{1}\dot{y}-a_{2}y+w+(b-b_{0})u \tag{9}\]  
where \(f\) is total disturbance including internal disturbance (such as un-modeled dynamics and dead-zone non-linearities) and external disturbance (including aerodynamic load), then (8) can be re-written as  
\[\ddot{y}=f+b_{0}u \tag{10}\]  
Take state vector \(\mathbf{x}=\left[x_{1}x_{2}x_{3}\right]^{T}\), then (10) can be expressed in state-space form as  
\[\left\{ \begin{array}{c}\dot{x}_{1}=x_{2}\\ \dot{x}_{2}=x_{3}=f+b_{0}u\\ \dot{x}_{3}=\dot{f}=k\\ y=x_{1}\end{array} \right. \tag{11}\]  
That is  
\[\left\{ \begin{array}{c}\dot{\mathbf{x}}=\mathbf{A}\mathbf{x}+\mathbf{B}u+ \mathbf{E}\\ \mathbf{y}\mathbf{=}\mathbf{C}\mathbf{x}\end{array} \right. \tag{12}\]  
where  
\[\mathbf{A}=\left[ \begin{array}{ccc}0&1&0\\ 0&0&1\\ 0&0&0\end{array} \right];\mathbf{B}=\left[ \begin{array}{ccc}0&b_{0}&0\end{array}  \right]^{T};\mathbf{C}=\left[ \begin{array}{ccc}1&0&0\end{array} \right];\]  
\[\mathbf{E}=\left[ \begin{array}{ccc}0&0&1\end{array} \right]^{T}\]  
According to the rank criterion of system observability, it is obvious that the system is observable. Designing ESO for (12), this can be expressed as  
\[\left\{ \begin{array}{c}\dot{\mathbf{z}}=\mathbf{A}\mathbf{z}+\mathbf{B} \mathbf{u}+\mathbf{L}(\mathbf{y}\mathbf{\hat{y}})\\ \mathbf{y}\mathbf{=}\mathbf{C}\mathbf{z}\end{array} \right. \tag{13}\]  
where, \(\mathbf{L}=\left[ \begin{array}{ccc}\beta_{1}&\beta_{2}&\beta_{3}\end{array}  \right]^{T}\) is the gain of ESO, the state vector \(\mathbf{z}=\left[ \begin{array}{ccc}z_{1}&z_{2}&z_{3}\end{array} \right]^{T}\) is the observed value of \(\mathbf{x}\). Then the extended state \(z_{3}\) is the estimated value \(f\) of the total disturbance \(f\). If the control quantity \(u\) is  
\[u=\frac{u_{0}-z_{3}}{b_{0}} \tag{14}\]and substitute (14) into (10), then (10) can be written as  
\[\ddot{y}=(f-\hat{f})+u_{0} \tag{15}\]  
Ideally, if the extended state \(z_{3}\) estimate value is accurate enough, then \(f-\hat{f}\), and the output value of (15) is directly dependent on \(u_{0}\). Let the state error feedback control law be  
\[u_{0}=\left[ \begin{array}{cc}k_{p}&k_{d}\end{array} \right[ \begin{array}{c}v -z_{1}\\ -z_{2}\end{array} \right] \tag{16}\]  
where \(k_{p},k_{d}\) are the control parameters. Equations (13), (14), and (16) are composed into ADRC, as shown in Figure 3, where \(\delta_{r}\) is the command value.