## 3 LADRC Controller Design
### Linear Extended State Observer  
LESO improves the performance of control systems by estimating and compensating for disturbances and unknown state variables. This section outlines the design process of the LESO scheme based on the mathematical model of the ROV system. The kinematics expression of the ROV is \(\dot{\eta}=J(\eta)\
u\)[25], from which we can derive Formula (9):  
\[ \begin{split}\ddot{\eta}&=J(\eta)\dot{\
u}+\dot{J}( \eta)\
u\\ &=J(\eta)M_{0}^{-1}(\Delta f+\tau_{\Delta}-C_{0}(\
u)\
u-D_{0}( \
u)\
u+g(\eta))+\dot{J}(\eta)\
u+J(\eta)M_{0}^{-1}\tau\\ &\hskip 142.26378 pt=f+X+J(\eta)M_{0}^{-1}\tau\end{split}  \tag{9}\]  
In Formula (9), \(X=j(\eta)\
u-J(\eta)M_{0}^{-1}(C_{0}(\
u)\
u-D_{0}(\
u)\
u+g(\eta)),f=J(\eta)M_{ 0}^{-1}(\Delta f+\tau_{\Delta})\) represents the total uncertainty of the unmodeled dynamics and external disturbances. For the convenience of designing the LESO, let \(f=\left[f_{x},f_{y},f_{z},f_{\phi}\right]\), \(\tau=\left[\tau_{x},\tau_{y},\tau_{z},\tau_{\phi}\right]\), then Formula (9) can be written as [26]:  
\[ \begin{split}\ddot{x}&=a_{1}\tau_{x}+X_{1}+f_{x}\\ \ddot{y}&=a_{2}\tau_{y}+X_{2}+f_{y}\\ \ddot{z}&=a_{3}\tau_{z}+X_{3}+f_{z}\\ \ddot{\psi}&=a_{6}\tau_{\psi}+X_{6}+f_{\psi}\end{split}  \tag{10}\]  
\[ \begin{split}\left\{\begin{array}{c}X_{1}=\frac{cos\psi}{m-X_{ \dot{x}}}\Big{(}X_{u}+X_{u|u|}\mid u\mid-(m-Y_{\dot{v}})ur\Big{)}\\ -\frac{sin\psi}{m-Y_{\dot{v}}}\Big{(}Y_{v}+Y_{\psi|\mid}\mid v\mid-(m-X_{\dot{u }})ur\Big{)}\\ -usint\psi-cos\psi\\ A_{1}=\Big{(}\frac{cos\psi}{m-X_{\dot{x}}}-\frac{sin\psi}{m-Y_{\dot{v}}} \Big{)}\\ X_{2}=\frac{sin\psi}{m-X_{\dot{u}}}\Big{(}X_{u}+X_{u|u|}\mid u\mid-(m-Y_{ \dot{v}})ur\Big{)}+\frac{cos\psi}{m-Y_{\dot{v}}}\\ \Big{(}Y_{v}+Y_{\psi|\mid}\mid v\mid\mid-(m-X_{\dot{u}})ur\Big{)}+ucos\psi-psin \psi\\ a_{2}=\Big{(}\frac{sin\psi}{m-X_{\dot{u}}}+\frac{cos\psi}{m-Y_{\dot{v}}} \Big{)}\\ X_{3}=\frac{\big{(}Z_{u}+Z_{u|u|}|w\big{)}w}{m-Z_{\dot{w}}}\\ a_{3}=\frac{1}{m-Z_{\dot{w}}}\\ X_{4}=\frac{\big{(}N_{r}+N_{r|r|}|r\big{)}r\big{)}}{I_{z}-N_{\dot{r}}},a_{ 4}=\frac{1}{I_{z}-N_{\dot{r}}}\end{split} \right.\end{split} \tag{11}\]  
The total uncertainty \(f\), which represents the unmodeled dynamics and external disturbances, is defined as the total disturbance. To achieve an accurate estimation of the total disturbance \(f\) experienced in the control underwater robots [19], the dynamics model of the ROV is rewritten as follows. \[\ddot{\psi}=\frac{1}{I_{z}-N_{r}}\tau_{\psi}+\frac{\Big{(}N_{r}+N_{r|r|}|r| \Big{)}r}{I_{z}-N_{r}}+f_{\psi} \tag{12}\]  
Using the dynamic expression of heading angle \(\psi\) from Equations (10) and (11) as an example for controller design, we provide a detailed design explanation for LESO and LSEFC. In Formula (12), \(\frac{1}{I_{z}-N_{r}}\) is a constant term whose value is determined by the system's inertia parameter \(I_{z}\) and damping coefficient \(N_{r}\). To simplify the formula, \(b=\frac{1}{I_{z}-N_{r}}\) is used to replace this complex coefficient. The second term \(\frac{\big{(}N_{r}+N_{r|r|}|r\big{)}r}{I_{z}-N_{r}}\) describes the nonlinear disturbances caused by hydrodynamics, which are typically regarded as a type ofinterference or unmodeled dynamics. In the design of LESO, this part is incorporated into the total uncertainty \(f_{\psi}\) for unified treatment. Therefore, Formula (12) can be simplified to:  
\[\ddot{\psi}=b\tau_{\psi}+f_{\psi} \tag{13}\]  
Taking the state variables \(\psi_{1},\psi_{2},\psi_{3}\), among them \(\psi_{3}=f_{\psi}\) as the extended state, the ROV heading angle \(\psi\) control model can be expressed as:  
\[\left\{ \begin{array}{c}\psi=\psi_{1}\\ \dot{\psi}_{1}=\psi_{2}\\ \dot{\psi}_{2}=b\tau_{\psi}+\psi_{3}\\ \dot{\psi}_{3}=D\end{array} \right. \tag{14}\]  
Let \(D=\dot{f}_{\psi^{\prime}}\), \(\psi_{2}=\dot{\psi}\). A linear expansion state observer can be established for the system (13) [19]:  
\[\left\{ \begin{array}{c}e_{1}=\psi-z_{1}\\ \dot{z}_{1}=z_{2}+\beta_{1}e_{1}\\ \dot{z}_{2}=z_{3}+\beta_{2}e_{1}+b\tau_{\psi}\\ \dot{z}_{3}=\beta_{3}e_{1}\end{array} \right.