## 2 Overview and Modeling of the AUH
### Leso  
The LESO is a linear observer that extends the state-space of the system to include the disturbance variables. It estimates the state and disturbance variables simultaneously, which improves the accuracy of the control system. LADRC has been applied in various fields, such as motor control, robotics, and power electronics. It is a promising control method for systems with uncertainties and disturbances.  
In this paper, the controllers designed an all-adopt second-order LESO, and the same is true for the heading controller and depth controller. Hence, the design of the depth controller is taken as an example here.  
First, \(\zeta_{1}\), \(u_{1}\), \(b_{1}\) are designed as below.  
\[\left\{\begin{array}{l}\zeta_{1}=\frac{Z_{w}w-Z_{[w]w}|w|w+\tau_{\text{env}} +W-B}{m+Z_{w}}\\ u_{1}=\tau_{Z}\\ b_{1}=\frac{1}{m+Z_{w}}\end{array}\right. \tag{3}\]  
Afterwards, Equation (2) can be written as follows.  
\[\ddot{w}=b_{1}u_{1}+\zeta_{1} \tag{4}\]  
where, \(\zeta_{1}\) refers to the generalized total disturbance of the system, including internal uncertainty and unknown external disturbance. Later, a second-order state-space expression can be obtained by extending \(\zeta_{1}\) to a state variable \(x_{3}\). In stable systems, it is generally assumed that the differential of \(x_{3}\) is bounded and defined that \(\dot{x}_{3}=k\).  
\[\left\{\begin{array}{l}y_{1}=x_{1}\\ \dot{x}_{1}=x_{2}\\ \dot{x}_{2}=x_{3}+b_{1}u\\ \dot{x}_{3}=k\end{array}\right. \tag{5}\]  
In Equation (5), \(y_{1}\) represents the AUH's depth in the current state, which can be detected by a depth altimeter. The observer [25] is established based on the above state-space equation.  
\[\left\{\begin{array}{l}e=\hat{x}_{1}-y_{1}\\ \hat{x}_{1}=\hat{x}_{2}-l_{1}e\\ \hat{x}_{2}=\hat{x}_{3}-l_{2}e+b_{1}u\\ \hat{x}_{3}=-l_{3}e\end{array}\right. \tag{6}\]  
In Equation (6), \(\hat{x}_{1}\), \(\hat{x}_{2}\), and \(\hat{x}_{3}\) refer to the estimations of \(x_{1}\), \(x_{2}\), and \(x_{3}\), respectively. \(l_{1}\), \(l_{2}\), and \(l_{3}\) represent the gains of the observer. To sum up, Equation (6) is the second-order extended state observer used by the depth controller. Moreover, to simplify the tuning process, the observer gains are parameterized as [24]  
\[L=[l_{1},l_{2},l_{3}]^{T}=[3w_{0},3w_{0}^{2},w_{0}^{3}]^{T} \tag{7}\]  
where \(w_{0}\) refers to the bandwidth of the system.  
The linear state error feedback (LSEF) control law is mainly used to generate actual control signals and improve control accuracy by combining with the compensation quantity of the disturbance estimation above. In this paper, the PD control law is adopted, and the output of each propeller is calculated [24].  
\[\left\{\begin{array}{l}e_{1}=\overline{v}-\hat{x}_{1}\\ e_{2}=\hat{x}_{2}\\ u_{0}=K_{p}e_{1}-K_{d}e_{2}\\ u=\frac{u_{0}-\hat{x}_{3}}{b_{1}}\end{array}\right. \tag{8}\]  
where \(\overline{v}\) refers to the desired signal and \(u\) denotes the output of the controller.  
Generally, LADRC is composed of LESO and LSEF. However, classical LADRC methods often exhibit overshoots when working with large spans, which can be addressed by incorporating a tracking differentiator (TD). In order to mitigate this issue, this paper applies signal processing techniques and a TD in the control system's input signal to enhance the transition process and improve control performance.