Motion Control of Autonomous Underwater Vehicle Based on Fractional Calculus Active Disturbance Rejection

Abstract

An active disturbance rejection control based on fractional calculus is proposed to improve the motion performance and robustness of autonomous underwater vehicle (AUV). The active disturbance rejection control (ADRC) method can estimate and compensate the total disturbance of AUV automatically. The fractional-order PID (proportional integral derivative) has fast dynamic response, which can eliminate the estimation error of extended state observer. The fractional calculus active disturbance rejection strategy combines the advantages of the above two algorithms, and it is designed for AUV heading and pitch subsystems. In addition, the stability of fractional calculus ADRC heading subsystem is proven by Lyapunov stability theorem. The numerical simulations and experimental results document that the superior performance has been achieved. The fractional calculus ADRC strategy has more excellent abilities for disturbance rejection, performs better than ADRC and PID, and has important theoretical and practical value.

1. Introduction

Autonomous underwater vehicles (AUV) are playing an increasingly important role in developing unknown ocean and accomplishing different military missions in recent years. AUV motion control has attracted many researchers’ interest. Nevertheless, the characteristics of strong coupling and high nonlinearity make it more challenging. Different approaches have been implemented to solve this problem, such as adaptive controller [1,2], sliding mode controller [3,4,5], H-infinity controller [6], and PID controller [7,8].

Prof. Han put forward the active disturbance rejection control (ADRC) method in the 1990s [9]. It was independent of model and it could actively reject disturbances. In recent decades, ADRC method has been widely used in motion control of AUV. An ADRC method based on fuzzy adaptive ESO (extended state observer) was proposed for trajectory tracking of AUV. The simulations indicated that the method outperformed others in terms of tracking accuracy and robustness [10]. A depth ADRC method for AUV was feasible and effective [11]. An ADRC method was used to control the speed and heading of AUV. The simulation results showed that the nonlinear extended state observer could effectively observe the position, speed and disturbance [12]. A diving ADRC method based on improved tracking differentiator for AUV was proposed, and the simulations verified the effectiveness [13].

In 1695, Leibniz proposed fractional calculus in a letter to L’Hôpital [14]. The research on fractional calculus and its application in automatic control are increasing. In the 1990s, Podlubny conducted essential research on fractional-order proportional integral derivative (FOPID) strategy and established the beginning of fractional-order control [15]. Subsequently, FOPID controller has been widely applied in robotic manipulator, automatic voltage regulator and hydraulic turbine regulating system, hydraulic-loading system and many other fields [16,17,18].

In the last few years, some achievements have been reported in the theory and application of active disturbance rejection control based on fractional-order PID in engineering, for example the position loop control of servo system [19], the hypersonic vehicle [20,21], the speed servo system of permanent magnet synchronous motor [22], the ship servo system [23], the hydroturbine speed governor system [24], and the linear motor [25]. The mentioned literature inspires fractional calculus ADRC scheme for AUV motion control. However, this strategy has not previously been reported about motion control of AUV in the reviewed literature. The major contributions of this paper are the following:

• The fractional calculus active disturbance rejection strategy is proposed to improve the performance of AUV motion. It combines the advantages of ADRC and fractional calculus;
• The stability of fractional calculus active disturbance rejection system is proven by Lyapunov stability theorem;
• The fractional calculus active disturbance rejection is sufficiently simulated and successfully employed in heading and pitch subsystems. Experimental results and analysis demonstrate the advantages of the proposed strategy.

The rest of this paper is structured as follows. In Section 2, the motion equations of AUV in six degrees of freedom are discussed, and the decoupled equations of horizontal and vertical subsystems are derived. Section 3 is dedicated to the controller design and the stability analysis of fractional calculus ADRC heading and pitch subsystems. In Section 4, a series of numerical simulations and experiments are presented, and the comparison is analyzed and discussed in detail. Finally, Section 5 draws the key conclusions.

2. Materials and Methods

2.1. Reference Frames

The reference frames and notations are sketched in Figure 1 and

Table 1. The notations for AUV.

Vectors	x-Axis	y-Axis	z-Axis
Linear velocities	u	$\upsilon$	$\omega$
Angular velocities	p	q	r
Euler angles	$\phi$	$\theta$	$\psi$
Forces	X	Y	Z
Moments	K	M	N
Positions	$\xi$	$\eta$	$\zeta$

. A more detailed explanation of notations and terms is available in the literature [26,27].

2.2. Transformations between Body-Fixed and World-Fixed

The linear velocity transformation and the Euler angle transformation between body-fixed and world-fixed reference frames are defined as Equation (1) [28].

$$\begin{array}{c}\hfill \left[\begin{array}{c}\dot{\xi }\\ \dot{\eta }\\ \dot{\zeta }\\ \dot{\phi }\\ \dot{\theta }\\ \dot{\psi }\end{array}\right]=\left[\begin{array}{cccccc}c\psi c\theta & c\psi s\theta s\phi -s\psi c\phi & s\psi s\theta s\phi +c\psi c\phi & 0& 0& 0\\ s\psi c\theta & s\psi s\theta s\phi +c\psi c\phi & s\psi s\theta c\phi -c\psi s\phi & 0& 0& 0\\ -s\theta & c\theta s\phi & c\theta c\phi & 0& 0& 0\\ 0& 0& 0& 1& s\phi t\theta & c\phi t\theta \\ 0& 0& 0& 0& c\phi & -s\phi \\ 0& 0& 0& 0& s\phi /c\theta & c\phi /c\theta \end{array}\right]\left[\begin{array}{c}u\\ v\\ w\\ p\\ q\\ r\end{array}\right]\end{array}$$

(1)

where t(·) represents tan(·), s(·) represents sin(·), c(·) represents cos(·), $\psi$ represents the heading angle, $\theta$ represents the pitch angle, and $\phi$ represents the roll angle. Note that the pitch angle is not defined as $\theta =\pm {90}^{°}$.

2.3. Motion Equations in Six Degrees of Freedom

The motion equations have been obtained by combining the rigid-body dynamics and kinematics equations. When deriving the motion equation it will be assumed:

• AUV is rigid;
• The Earth-fixed frame is inertial;
• AUV has homogeneous mass distribution and xz-plane symmetry;
• The origin of the body-fixed coordinate system coincides with the center of gravity.

The AUV equations of motion can be formulated in Equation (2).

$$\left\{\begin{array}{c}X=m(\dot{u}-vr+wq)\hfill \\ Y=m(\dot{v}-wp+ur)\hfill \\ Z=m(\dot{w}-uq+vp)\hfill \\ K=I_x\dot{q}+(I_x-I_y)qr\hfill \\ M=I_y\dot{q}+(I_x-I_z)rp\hfill \\ N=I_z\dot{r}+(I_y-I_x)pq\hfill \end{array}\right.$$

(2)

The right-hand side of these formulas stands for the external forces and moments acting on the AUV. By contrast, the left-hand side stands for the AUV motion [29].

Surge equation along the x-axis is described as Equation (3).

$$\begin{array}{cc}\hfill m(\dot{u}-vr+wq)=& \frac{1}{2}\rho L^4[X_{qq}{q}^2+X_{rr}{r}^2+X_{rp}rp]+\frac{1}{2}\rho L^3[X_{\dot{u}}\dot{u}+X_{vr}vr+X_{wq}wq]\hfill \\ \hfill & +\frac{1}{2}\rho L^2[X_{uu}{u}^2+X_{vv}{v}^2+X_{ww}{w}^2]+X_{prop}\hfill \end{array}$$

(3)

The sway equation along the y-axis is described in Equation (4).

$$\begin{array}{cc}\hfill m(\dot{v}-wp+vr)=& \frac{1}{2}\rho L^4[Y_{\dot{r}}\dot{r}+Y_{\dot{p}}\dot{p}+Y_{p|p|}p|p|+Y_{pq}pq]\hfill \\ \hfill & +\frac{1}{2}\rho L^3[Y_{\dot{v}}\dot{v}+Y_{vq}vq+Y_{wp}wp+Y_{wr}wr]\hfill \\ \hfill & +\frac{1}{2}\rho L^3[Y_{r}ur+Y_{p}up+Y_{v|r|}\frac{v}{|v|}|{(v^2+w^2)}^{\frac{1}{2}}||r|]\hfill \\ \hfill & +\frac{1}{2}\rho L^2[Y_0{u}^2+Y_{v}uv+Y_{v|v|}|{(v^2+w^2)}^{\frac{1}{2}}|]+\frac{1}{2}\rho L^2{Y}_{vw}vw+Y_{\delta }\hfill \end{array}$$

(4)

The heave equation along the z-axis is described in Equation (5).

$$\begin{array}{cc}\hfill m(\dot{w}-uq+vp)=& \frac{1}{2}\rho L^4[Z_{\dot{q}}\dot{q}+Z_{pp}{p}^2+Z_{rr}{r}^2+Z_{rp}rp]\hfill \\ \hfill & +\frac{1}{2}\rho L^3[Z_{\dot{w}}\dot{w}+Z_{vr}vr+Z_{vp}vp]\hfill \\ \hfill & +\frac{1}{2}\rho L^3[Z_{q}uq+Z_{w|q|}\frac{w}{|w|}|{(v^2+w^2)}^{\frac{1}{2}}||q|]\hfill \\ \hfill & +\frac{1}{2}\rho L^2[Z_0{u}^2+Z_{w}uw+Z_{w|w|}w|{(v^2+w^2)}^{\frac{1}{2}}|]\hfill \\ \hfill & +\frac{1}{2}\rho L^2[Z_{|w|}u|w|+Z_{ww}|w{(v^2+w^2)}^{\frac{1}{2}}|]+\frac{1}{2}\rho L^2{Z}_{vv}{v}^2+Z_{\delta }\hfill \end{array}$$

(5)

Te roll equation about the x-axis is described in Equation (6).

$$\begin{array}{cc}\hfill I_x\dot{p}+(I_z-I_y)qr=& \frac{1}{2}\rho L^5[K_{\dot{p}}\dot{p}+K_{\dot{r}}\dot{r}+K_{qr}qr+K_{pq}pq+K_{p|p|}p|p|]\hfill \\ \hfill & +\frac{1}{2}\rho L^4[K_{p}up+K_{r}ur+K_{\dot{v}}\dot{v}]\hfill \\ \hfill & +\frac{1}{2}\rho L^4[K_{vq}vq+K_{wp}wp+K_{wr}^{{}^{\prime }}wr]\hfill \\ \hfill & +\frac{1}{2}\rho L^3[K_0{u}^2+K_{v}uv+K_{v|v|}v|{(v^2+w^2)}^{\frac{1}{2}}|]\hfill \\ \hfill & +\frac{1}{2}\rho L^3{K}_{vw}vw-hWcos\phi sin\phi \hfill \end{array}$$

(6)

The pitch equation about the y-axis is described in Equation (7).

$$\begin{array}{cc}\hfill I_y\dot{q}+(I_x-I_z)rp=& \frac{1}{2}\rho L^5[M_{\dot{q}}\dot{q}+M_{pp}{p}^2+M_{rr}{r}^2+M_{rp}rp+M_{q|q|}q|q|]\hfill \\ \hfill & +\frac{1}{2}\rho L^4[M_{\dot{w}}\dot{w}+M_{vr}vr+M_{vp}vp]\hfill \\ \hfill & +\frac{1}{2}\rho L^4[M_{q}uq+M_{|w|q}^{{}^{\prime }}|{(v^2+w^2)}^{\frac{1}{2}}q]\hfill \\ \hfill & +\frac{1}{2}\rho L^3[M_0{u}^2+M_{w}uw+M_{w|w|}w|{(v^2+w^2)}^{\frac{1}{2}}|]\hfill \\ \hfill & +\frac{1}{2}\rho L^3[M_{|w|}u|w|+M_{ww}|w{(v^2+w^2)}^{\frac{1}{2}}|]\hfill \\ \hfill & +\frac{1}{2}\rho L^3{M}_{vv}{v}^2-hWsin\theta +M_{\delta }\hfill \end{array}$$

(7)

The yaw equation about the z-axis is described in Equation (8).

$$\begin{array}{cc}\hfill I_z\dot{r}+(I_y-I_x)pq=& \frac{1}{2}\rho L^5[N_{\dot{r}}\dot{r}+N_{\dot{p}}\dot{p}+N_{pq}pq+N_{qr}qr+N_{r|r|}r|r|]\hfill \\ \hfill & +\frac{1}{2}\rho L^4[N_{wr}wr+N_{wp}wp+N_{vq}vq]\hfill \\ \hfill & +\frac{1}{2}\rho L^4[N_{p}up+N_{r}ur+N_{|v|r}|{(v^2+w^2)}^{\frac{1}{2}}r]\hfill \\ \hfill & +\frac{1}{2}\rho L^3[N_0{u}^2+N_{v}uv+N_{v|v|}v|{(v^2+w^2)}^{\frac{1}{2}}|]\hfill \\ \hfill & +\frac{1}{2}\rho L^3{N}_{vw}vw+N_{\delta }\hfill \end{array}$$

(8)

where m is AUV mass, L is the length of AUV, W is the gravity of AUV, h is the distance between the center of gravity and the center of buoyancy in z axis, $X_{prop}$ is the propulsion force, $Y_{\delta },Z_{\delta },M_{\delta },N_{\delta }$ are the rudder forces and moments, $I_x$, $I_y$, $I_z$ are the moments of inertia about x, y, and z axes, and the hydrodynamic coefficients ($X_{\dot{u}}$, $Y_{\dot{v}}$, $\cdots \cdots$) are listed in Appendix A.

The acceleration terms are separated from the motion equations of AUV, as shown in Equation (9) [30].

$$\begin{array}{c}\hfill \left[\begin{array}{c}\dot{u}\\ \dot{v}\\ \dot{w}\\ \dot{p}\\ \dot{q}\\ \dot{r}\end{array}\right]={\left[\begin{array}{cccccc}m-X_{\dot{u}}& 0& 0& 0& 0& 0\\ 0& m-Y_{\dot{v}}& 0& -Y_{\dot{p}}& 0& -Y_{\dot{r}}\\ 0& 0& m-Z_{\dot{w}}& 0& -Z_{\dot{q}}& 0\\ 0& -Y_{\dot{p}}& 0& I_x-K_{\dot{p}}& 0& -K_{\dot{r}}\\ 0& 0& -Z_{\dot{q}}& 0& I_y-M_{\dot{q}}& 0\\ 0& -Y_{\dot{r}}& 0& -K_{\dot{r}}& 0& I_z-N_{\dot{r}}\end{array}\right]}^{-1}\left[\begin{array}{c}X\\ Y\\ Z\\ K\\ M\\ N\end{array}\right]\end{array}$$

(9)

2.4. Motion Equations in the Horizontal Plane

The motion equations are usually divided into two independent subsystems, the longitudinal one and the lateral one [31]. Omitting the heave, roll, and pitch modes, the motion equations in the horizontal plane are sketched in Equation (10).

$$\left\{\begin{array}{c}m(\dot{u}-vr)=X\hfill \\ m(\dot{v}+ur)=Y\hfill \\ I_z\dot{r}=N\hfill \end{array}\right.$$

(10)

Under the assumption that AUV moves at a fixed velocity, the first equation will be eliminated. For the reason that starboard-port and fore-aft of AUV are symmetric, $Y_{\dot{r}}=0$ and $N_{\dot{v}}=0$. The linearized maneuvering equations are generalized in Equation (11).

$$\begin{array}{c}\hfill \left[\begin{array}{c}\dot{v}\\ \dot{r}\\ \dot{\psi }\end{array}\right]=\left[\begin{array}{ccc}\frac{Y_v}{m-Y_{\dot{v}}}& \frac{Y_r-mu}{m-Y_{\dot{v}}}& 0\\ \frac{N_v}{I_z-N_{\dot{r}}}& \frac{N_r}{I_z-N_{\dot{r}}}& 0\\ 0& 1& 0\end{array}\right]\left[\begin{array}{c}v\\ r\\ \psi \end{array}\right]+\left[\begin{array}{c}\frac{Y_{{\delta }_r}}{m-Y_{\dot{v}}}\\ \frac{N_{{\delta }_r}}{I_z-N_{\dot{r}}}\\ 0\end{array}\right]{\delta }_r\end{array}$$

(11)

where ${\delta }_r$ is the rudder angle, $Y_v,Y_{\dot{v}},Y_r,N_v,N_{\dot{r}},N_r,Y_{{\delta }_r},N_{{\delta }_r}$ are the hydrodynamic coefficients [32].

2.5. Motion Equations in the Vertical Plane

It is assumed that u is a constant and the sway, roll, and yaw modes are neglected. Hence, the motion equations in the vertical plane are sketched in Equation (12).

$$\left\{\begin{array}{c}m(\dot{u}-vr)=X\hfill \\ m(\dot{v}+ur)=Z\hfill \\ I_y\dot{q}=M\hfill \end{array}\right.$$

(12)

The X equation can be removed since $\dot{u}=0$. Accordingly, the linearized maneuvering equation can be generalized in Equation (13) [33].

$$\begin{array}{c}\hfill \left[\begin{array}{cccc}m-Z_{\dot{w}}& -Z_{\dot{q}}& 0& 0\\ -M_{\dot{w}}& I_y-M_{\dot{q}}& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]\left[\begin{array}{c}\dot{w}\\ \dot{q}\\ \dot{\theta }\\ \dot{\zeta }\end{array}\right]=\left[\begin{array}{cccc}{Z}_w& -mu+Z_q& 0& 0\\ M_w& M_q& -M_{\theta }& 0\\ 0& 1& 0& 0\\ 1& 0& -u& 0\end{array}\right]\left[\begin{array}{c}w\\ q\\ \theta \\ \zeta \end{array}\right]+\left[\begin{array}{c}{Z}_{{\delta }_s}\\ M_{{\delta }_s}\\ 0\\ 0\end{array}\right]{\delta }_s\end{array}$$

(13)

In view of symmetry of starboard-port, $Z_{\dot{q}}$ and $M_{\dot{v}}$ are zero. Assume that the heave speed w is small. Therefore, a simplified form of linear model is depicted as Equation (14).

$$\begin{array}{c}\hfill \left[\begin{array}{c}\dot{q}\\ \dot{\theta }\\ \dot{\zeta }\end{array}\right]=\left[\begin{array}{ccc}\frac{M_q}{I_y-M_{\dot{q}}}& \frac{-M_{\theta }}{I_y-M_{\dot{q}}}& 0\\ 1& 0& 0\\ 0& -u& 0\end{array}\right]\left[\begin{array}{c}q\\ \theta \\ \zeta \end{array}\right]+\left[\begin{array}{c}\frac{M_{{\delta }_s}}{I_y-M_{\dot{q}}}\\ 0\\ 0\end{array}\right]{\delta }_s\end{array}$$

(14)

where ${\delta }_s$ is the stern plane angle, $Z_{\dot{w}},M_{{\delta }_s},Z_{{\delta }_s},M_w,Z_w,M_{\dot{q}},M_q,Z_q$ are the hydrodynamic coefficients, $M_{\theta }=-mgh$ is righting moment, h presents the distance from the barycenter to the buoyant center in the z axis.

3. Controller Design

3.1. Fractional Calculus and FOPID

Fractional calculus is an extension of traditional differential and integral to arbitrary (non-integer) order. There are several definitions of fractional calculus such as Riemann–Liouville, Grünwald–Letnikov and Cauchy. Detailed explanation of fractional calculus can be learned from the literature [14]. Owing to the importance of applications, the Grünwald–Letnikov’s definition is taken into consideration, as illustrated in Equation (15). Equation (16) is an approximation of Equation (15). The coefficient $\alpha$ is calculated by Equations (17) and (18) [26].

$${}_a^{GL}{\mathcal{D}}_t^{\alpha }f\left(t\right)=\underset{\widehat{h}\to 0}{lim}{\widehat{h}}^{-\alpha }\sum _{i=0}^{[\frac{t-a}{\widehat{h}}]}{(-1)}^i\left(\begin{array}{c}a\\ i\end{array}\right)f(t-i\widehat{h})(i=1,2,\cdots )$$

(15)

$${\mathcal{D}}_t^{\alpha }f\left(t\right)\approx {\mathcal{D}}_t^{\alpha }f\left(t\right)\approx {\widehat{h}}^{-\alpha }\sum _{i=0}^{N\left(t\right)}{\omega }_i^{\alpha }f(t-i\widehat{h})$$

(16)

$${\omega }_i^{\alpha }={(-1)}^i\left(\begin{array}{c}a\\ i\end{array}\right)$$

(17)

$${\omega }_0^{\alpha }=1;{\omega }_i^{\alpha }=(1-\frac{\alpha +1}{i}){\omega }_{i-1}^{\alpha }(i=1,2,\cdots .)$$

(18)

where $\widehat{h}$ is the step size, $[\frac{t-a}{\widehat{h}}]$ denotes the integer part of $[\frac{t-a}{\widehat{h}}]$, $\alpha$ is the integral and differential orders ($\alpha \in (0,1)$).

The PID strategy is used widely in engineering, however, it is only a particular control point of the quarter-plane as shown in Figure 2. Clearly, $\lambda$ = 1 and $\mu$ = 1 give a traditional PID controller. The description of fractional order PID controller in time domain is given by Equation (19).

$$u\left(t\right)=K_{fp}e\left(t\right)+K_{fi}{\mathcal{D}}^{-\lambda }e\left(t\right)+K_{fd}{\mathcal{D}}^{\mu }e\left(t\right)$$

(19)

where $K_{fp},K_{fi},K_{fd}$ are the proportional, integral, and differential coefficients of fractional order PID controller, respectively. $\lambda$ and $\mu$ are the integral and differential order, respectively.

Substituting Equation (15) into Equation (19), the numerical form of fractional order PID controller is given by Equation (20).

$$u\left(m\right)=K_{fp}e\left(m\right)+K_{fi}{\widehat{h}}^{\lambda }\sum _{i=0}^{N\left(t\right)}{\omega }_i^{\alpha }e(m-i)+K_{fd}{\widehat{h}}^{-\mu }\sum _{i=0}^{N\left(t\right)}{\omega }_i^{\alpha }e(m-i)$$

(20)

3.2. ADRC Method

ADR controller consists of three parts: tracking differentiator (TD), extended state observer (ESO), and nonlinear state error feedback (NLSEF) [34], as illustrated in Figure 3.

TD deals with the transition process and attains an approximation of differentiation by solving differential equations (see Equation (21)).

$$\left\{\begin{array}{c}e=x_1-x\hfill \\ x_1=x_1+\tau x_2\hfill \\ x_2=x_2+\tau fhan(e,x_2,r,h)\hfill \end{array}\right.$$

(21)

The function of $fhan(·)$ is explained in detail in Equation (22).

$$\left\{\begin{array}{c}d=r\tau \hfill \\ d_0=\tau d\hfill \\ y=e+\tau x_2\hfill \\ a_0=\sqrt{d^2+8r|y|}\hfill \\ a=\left\{\begin{array}{c}{x}_2+\frac{a_0-d}{2}sgn\left(y\right),|y|>d_0\hfill \\ x_2+\frac{y}{\tau },|y|\le d_0\hfill \end{array}\right.\hfill \\ fhan(e,x_2,r,\tau )=\left\{\begin{array}{c}-rsgn\left(a\right),|a|>d\hfill \\ -r\frac{a}{d},|a|\le d\hfill \end{array}\right.\hfill \end{array}\right.$$

(22)

where $sng(·)$ is the sign function, r is the parameter of controller, and $\tau$ is the sampling period. r and $\tau$ can be adjusted separately in terms of the rise time and smoothness.

The system state and total disturbance are estimated according to the input and output information of plant from inside or outside in real time. The description of ESO is given by Equation (23).

$$\begin{array}{c}\hfill \left\{\begin{array}{c}e=z_1-y\hfill \\ {\dot{z}}_1=z_2-{\beta }_{1}e\hfill \\ {\dot{z}}_2=z_3-{\beta }_{2}fal(e,0.5,\sigma )+b_0{u}_0\hfill \\ {\dot{z}}_3=-{\beta }_{3}fal(e,0.25,\sigma )\hfill \end{array}\right.\end{array}$$

(23)

where $z_1\left(t\right)\to x_1\left(t\right),z_2\left(t\right)\to x_2\left(t\right)$, $z_3\left(t\right)$ is employed to evaluate the total disturbance, ${\beta }_1,{\beta }_2,{\beta }_3$ are the observer coefficients. (${\beta }_1=100$, ${\beta }_2=300$, ${\beta }_3=1000$, $\sigma =0.0025$).

The nonlinear function $fal(·)$ is explained in Equation (24).

$$fal(\epsilon ,\gamma ,\sigma )=\left\{\begin{array}{cc}\frac{ϵ}{{\sigma }^{1-\gamma }}\hfill & |ϵ|\le \sigma \\ {|ϵ|}^{\gamma }sign\left(ϵ\right)\hfill & |ϵ|>\sigma \end{array}\right.$$

(24)

where $\gamma ,\sigma$ are variable parameters.

Equation (25) presents the nonlinear state error feedback law.

$$\left\{\begin{array}{c}{e}_1=x_1-z_1,e_2=x_2-z_2\hfill \\ u_0=K_{p}fal(e_1,{\gamma }_1,\delta )+K_{d}fal(e_2,{\gamma }_2,\delta )\hfill \\ u=u_0-\frac{z_3}{b_0}\hfill \end{array}\right.$$

(25)

where $b_0$ is the compensation factor; $K_p$ and $K_d$ are the coefficients (${\gamma }_1=0.75$, ${\gamma }_2=1.5$).

3.3. Fractional Calculus ADRC Strategy

In order to overcome the shortcomings of slow closed-loop response and long adjustment time, the fractional calculus is introduced to the nonlinear state error feedback. The heading and pitch subsystems of fractional calculus ADRC are designed in the following sections.

3.3.1. Heading Subsystem of Fractional Calculus ADRC

The structure of heading system of fractional calculus ADRC is described in Figure 4.

The transient process is arranged with respect to desired heading ${\psi }_d$. ${\psi }_1$ follows the desired heading ${\psi }_d$ and ${\psi }_2$ is the differentiation of ${\psi }_1$ [12].

$$\left\{\begin{array}{c}{\psi }_1={\psi }_1+\tau {\psi }_2\hfill \\ {\psi }_2={\psi }_2+\tau fhan({\psi }_1-{\psi }_d,{\psi }_2,r,\tau )\hfill \end{array}\right.$$

(26)

According to the output signal $\psi$ and the input signal ${\delta }_r$ (rudder angle) of AUV, state ${\psi }_1$ is estimated by $z_{h1}$, state ${\psi }_2$ is estimated by $z_{h2}$, and $z_{h3}$ estimates the total disturbance acting on AUV.

$$\left\{\begin{array}{c}{e}_h=z_1-\psi \hfill \\ {\dot{z}}_{h1}=z_{h2}-{\beta }_{h1}{e}_h\hfill \\ {\dot{z}}_{h2}=z_{h3}-{\beta }_{h2}fal(e_h,0.5,\sigma )+b_{h0}{\delta }_{r0}\hfill \\ {\dot{z}}_{h3}=-{\beta }_{h3}fal(e_h,0.25,\sigma )\hfill \end{array}\right.$$

(27)

The fractional calculus is introduced into nonlinear state error feedback law due to $e_{h1}$. ${\delta }_{r0}$ is compensated by $z_{h2}$.

$$\left\{\begin{array}{c}{e}_{h1}=z_{h1}-{\psi }_1,e_{h2}=z_{h2}-{\psi }_2\hfill \\ {\delta }_{r0}=K_{fp}^h{e}_{h1}+K_{fi}^h{\mathcal{D}}^{-{\lambda }_h}{e}_{h1}+K_{fd}^h{\mathcal{D}}^{{\mu }_h}{e}_{h2}\hfill \\ {\delta }_r={\delta }_{r0}-\frac{z_3}{b_{h0}}\hfill \end{array}\right.$$

(28)

where ${\psi }_d$ is the desired heading angle (input), ${\delta }_{r0}$ is the nonlinear states error feedback, $K_{fp}^h$ is the proportional coefficient, $K_{fi}^h$ is the integral coefficient, $K_{fd}^h$ is the differential coefficient, ${\lambda }_h$ and ${\mu }_h$ are the integral and differential orders, respectively.

3.3.2. Pitch Subsystem of Fractional Calculus ADRC

It is similar to the heading subsystem. Based on the diving model Equation (14), the pitch subsystem is designed in Figure 5.

$$\left\{\begin{array}{c}{\theta }_1={\theta }_1+\tau {\theta }_2\hfill \\ {\theta }_2={\theta }_2+\tau fhan({\theta }_1-{\theta }_d,{\theta }_2,r,\tau )\hfill & \hfill TD\\ e_p=z_{p1}-\theta \hfill \\ {\dot{z}}_{p1}=z_{p2}-{\beta }_{p1}{e}_p\hfill \\ {\dot{z}}_{p2}=z_{p3}-{\beta }_{p2}fal(e_p,0.5,\sigma )+b_{p0}{\delta }_{s0}\hfill \\ {\dot{z}}_{p3}=-{\beta }_{p3}fal(e_p,0.25,\sigma )\hfill & \hfill ESO\\ e_{p1}={\theta }_1-z_{p1},e_{p2}={\theta }_2-z_{p2},\hfill \\ {\delta }_{s0}=K_{fp}^p{e}_{p1}+K_{fi}^p{\mathcal{D}}^{-{\lambda }_p}{e}_{p1}+K_{fd}^p{\mathcal{D}}^{{\mu }_p}{e}_{p2}\hfill \\ {\delta }_s={\delta }_{s0}-\frac{z_{p3}}{b_{p0}}\hfill & \hfill NLSEF\end{array}\right.$$

(29)

where ${\theta }_d$ is the desired pitch angle (input), ${\delta }_{s0}$ is the nonlinear states error feedback, $K_{fp}^p$ is the proportional coefficient, $K_{fi}^p$ is the integral coefficient, $K_{fd}^p$ is the differential coefficient, ${\lambda }_p$ and ${\mu }_p$ are the integral and differential orders, respectively.

3.4. Stability of Fractional Calculus ADRC Subsystem

For the heading subsystem of fractional calculus ADRC (see Equation (11)), $e_{h1}=z_{h1}-{\psi }_1,e_{h2}=z_{h2}-{\psi }_2$, the equations of nonlinear state error feedback can be illustrated in Equation (30).

$$\left\{\begin{array}{c}{\dot{e}}_{h1}=e_{h2}-{\beta }_{h1}{e}_{h1}\hfill \\ {\dot{e}}_{h2}=w\left(t\right)-{\beta }_{h2}fal(e_{h1},0.5,\sigma )\hfill \end{array}\right.$$

(30)

Assuming that $b_{h0}=1$, $\dot{\psi }=f_w\left(t\right)-z_{h2}+{\delta }_{r0}$. $f_w\left(t\right)$ represents the unknown disturbances ($f_w\left(t\right)={\psi }_2$).

Lyapunov function of heading system of fractional calculus ADRC is defined in Equation (31).

$$\begin{array}{c}{V}_1=\frac{1}{2}{g}_2^2(e_{h1},e_{h2})\hfill \end{array}$$

(31)

$$\begin{array}{cc}\hfill & g_2(e_{h1},e_{h2})=\left\{\begin{array}{cc}|h_2(e_{h1},e_{h2})|,\hfill & \hfill |h_2(e_{h1},e_{h2})|>g_1\left(e_{h1}\right)\\ g_1\left(e_{h1}\right),\hfill & \hfill |h_2(e_{h1},e_{h2})|\le g_1\left(e_{h1}\right)\end{array}\right.\hfill \end{array}$$

(32)

$$\begin{array}{cc}\hfill & h_2(e_{h1},e_{h2})=e_{h2}-{\beta }_{h1}{e}_{h1}+k{g}_1\left(e_{h1}\right)sign\left(e_{h1}\right)\hfill \end{array}$$

(33)

$$\begin{array}{cc}\hfill & g_1\left(e_{h1}\right)=\frac{{\beta }_{h1}}{k{\beta }_{h2}}|fal(e_{h1},0.5,\sigma ))|,k>1\hfill \end{array}$$

(34)

Lemma 1.

$G_2=\{(e_{h1},e_{h2}):|h_2(e_{h1},e_{h2})|\le g_1\left(e_{h1}\right)\}$, $G_2$ is the self-stable region [35].

Proof. 

If $|h_2(e_{h1},e_{h2})|\le g_1\left(e_{h1}\right)$, such that $-k{g}_{1}sign\left(e_{h1}\right)-g_1\le e_{h2}-{\beta }_{h1}{e}_{h1}\le$

$-k{g}_{1}sign\left(e_{h1}\right)+g_1$. Time differentiation of $V_1$ yields:

$$\begin{array}{cc}\hfill {\dot{V}}_1& =g_1\left(e_{h1}\right){\dot{g}}_1\left(e_{h1}\right)\hfill \\ \hfill & ={\left(\frac{{\beta }_{h1}}{k{\beta }_{h2}}\right)}^2|fal\left(e_{h1}\right)||\dot{fal}\left(e_{h1}\right)|{\dot{e}}_{h1}sign\left(e_{h1}\right)\hfill \\ \hfill & ={\left(\frac{{\beta }_{h1}}{k{\beta }_{h2}}\right)}^2|fal\left(e_{h1}\right)||\dot{fal}\left(e_{h1}\right)|\frac{{\dot{e}}_{h1}{e}_{h1}}{|e_{h1}|}\hfill \\ \hfill & ={\left(\frac{{\beta }_{h1}}{k{\beta }_{h2}}\right)}^2\frac{|fal(e_{h1})|}{|e_{h1}|}|\dot{fal}\left(e_{h1}\right)|e_{h1}(e_{h2}-{\beta }_{h1}{e}_{h1})\hfill \end{array}$$

$$\begin{array}{cc}\hfill {\dot{V}}_1& \le {\left(\frac{{\beta }_{h1}}{k{\beta }_{h2}}\right)}^2\frac{|fal(e_{h1})|}{|e_{h1}|}|\dot{fal}\left(e_{h1}\right)|e_{h1}(-k{g}_{1}sign\left(e_{h1}\right)+g_1)\hfill \\ \hfill & \le -{\left(\frac{{\beta }_{h1}}{k{\beta }_{h2}}\right)}^2\frac{|fal(e_{h1})|}{|e_{h1}|}|\dot{fal}\left(e_{h1}\right)|(k-1)g_1|e_{h1}|<0\hfill \end{array}$$

□

Hence, Lemma 1 is complete.

Theorem 1.

For the heading subsystem of fractional calculus ADRC, if the system satisfies

$$\begin{array}{c}\hfill {\beta }_{h1}>(k+1)c_2{\beta }_{h2}|\frac{dfal}{d{e}_{h1}}|\end{array}$$

(35)

$$\begin{array}{c}\hfill {\beta }_{h1}{g}_2\le \frac{c_2}{c_2-1}W\end{array}$$

(36)

then the system is stable.

Proof. 

If $|h_2(e_{h1},e_{h2})|>g_1\left(e_{h1}\right)$, time differentiation of $V_1$ yields

$$\begin{array}{cc}\hfill {\dot{V}}_1& =g_2(e_{h1},e_{h2}){\dot{g}}_2(e_{h1},e_{h2})=h_2(e_{h1},e_{h2}){\dot{h}}_2(e_{h1},e_{h2})\hfill \\ \hfill & =h_2[\frac{\partial h_2}{\partial e_{h2}}{\dot{e}}_{h2}+\frac{\partial h_2}{\partial e_{h1}}{\dot{e}}_{h1}]\hfill \\ \hfill & =h_2[W-{\beta }_{h2}fal\left(e_{h1}\right)+\frac{\partial h_2}{\partial e_{h1}}(h_2-k{g}_1\left(e_{h1}\right)sign\left(e_{h1}\right))]\hfill \\ \hfill & =h_{2}W+\frac{\partial h_2}{\partial e_{h1}}{h}_2^2-h_2({\beta }_{h2}\frac{|fal|}{g_1}+k\frac{\partial h_2}{\partial e_{h1}})g_{1}sign\left(e_{h1}\right)\hfill \\ \hfill & =h_{2}W+\frac{\partial h_2}{\partial e_{h1}}{h}_2^2+h_2[k{\beta }_{h}1e_{h1}+k(-{\beta }_{h1}{e}_{h1}+k\frac{d\left(g_{1}sign\left(e_{h1}\right)\right)}{d{e}_{h1}})]g_{1}sign\left(e_{h1}\right)\hfill \\ \hfill & =h_{2}W+\frac{\partial h_2}{\partial e_{h1}}{h}_2^2+k^2{h}_2{g}_1|\frac{d{g}_1}{d{e}_{h1}}|\hfill \\ \hfill & \le h_{2}W-{\beta }_{h1}{h}_2^2+k{h}_2^2|\frac{d{g}_1}{d{e}_{h1}}|+k^2{h}_2{g}_1|\frac{d{g}_1}{d{e}_{h1}}|\hfill \end{array}$$

Substituting Equation (36) into the above expression yields:

$$\begin{array}{cc}\hfill {\dot{V}}_1& \le \frac{c_2-1}{c_2}{\beta }_{h1}{g}_2{h}_2-{\beta }_{h1}{h}_2^2+(k^2+k)h_2^2|\frac{d{g}_1}{d{e}_{h1}}|\hfill \\ \hfill & =-\frac{1}{c_2}{\beta }_{h1}{h}_2^2+(k^2+k)h_2^2\frac{{\beta }_{h2}}{k{\beta }_{h1}}|\frac{dfal}{d{e}_{h1}}|\hfill \\ \hfill & =[-\frac{{\beta }_{h1}}{c_2}+(1+k)\frac{{\beta }_{h2}}{k{\beta }_{h1}}|\frac{dfal}{d{e}_{h1}}|]h_2^2\hfill \end{array}$$

Substituting Equation (35) into the above expression yields ${\dot{V}}_1<0$. □

Consider the Lyapunove function candidate:

$$V_{fh}=\frac{1}{2}{({\psi }_d-\psi )}^T({\psi }_d-\psi )$$

(37)

Differentiating $V_{fh}$ with respect to time yields:

$$\begin{array}{cc}\hfill {\dot{V}}_{fh}& ={({\dot{\psi }}_d-\dot{\psi })}^T({\psi }_d-\psi )\hfill \\ \hfill & ={({\dot{\psi }}_d-f_w\left(t\right)-{\delta }_{r0}+z_{h2})}^T({\psi }_d-\psi )\hfill \\ \hfill & =-{{\delta }_{r0}}^T({\psi }_d-\psi )+{({\dot{\psi }}_d-w\left(t\right)+z_{h2})}^T({\psi }_d-\psi )\hfill \\ \hfill & =-{{\delta }_{r0}}^T({\psi }_d-\psi )+{({\dot{\psi }}_d+e_{h2})}^T({\psi }_d-\psi )\hfill \end{array}$$

If $|{\delta }_{r0}|>|{\dot{\psi }}_d+e_{h2}|$, such that ${\dot{V}}_{fh}<0$, the heading subsystem of fractional calculus ADRC is stable.

The stability condition of the heading system of fractional calculus ADRC is explained in Equation (38).

$$|{\delta }_{r0}|=|K_{fp}^h{e}_{h1}+K_{fi}^h{\mathcal{D}}^{-{\lambda }_h}{e}_{h1}+K_{fd}^h{\mathcal{D}}^{{\mu }_h}{e}_{h1}|>|{\dot{\psi }}_d+e_{h1}|$$

(38)

The proofs of the pitch subsystem of fractional calculus ADRC are notably similar to that of the heading subsystem, and they are omitted.

4. Experiment Result and Analysis

The computer simulations and experiments have been used to evaluate the performance and robustness of PID, ADRC, and fractional calculus ADRC. A more detailed discussion on these three methods is presented in the following sections.

4.1. Simulations of Heading Subsystem

The comparisons of step response for heading subsystem using PID, ADRC, and fractional calculus ADRC are illustrated in Figure 6. The results are summarized in

Table 2. Parameter index of step and pulse for heading subsystem.

Method	Overshoot (%)	Rise Time (s)	Settling Time (s)	SSE (${}^{°}$)
PID	1.9	1.696	4.056	0
ADRC	1.4	0.864	1.176	0
FCADRC	1.4	0.215	0.511	0

. Note that the resulting notions of root mean square and steady-state error will be abbreviated as RMSE and SSE, respectively. The fractional calculus ADRC strategy has shorter settling time and rise time than ADRC and PID. From this, we can see that fractional calculus ADRC can achieve fast response speed under the same overshoot and SSE.

The white noise response of heading subsystem is illustrated in Figure 7. When the input signal changes continuously, ADRC and fractional calculus ADRC strategies can rapidly track the desired heading. From the plot of partial enlargement, the results clearly demonstrate that PID cannot satisfy the tracking objective. The performance of ADRC is worse than fractional calculus ADRC.

The disturbances should be considered when designing motion controllers for AUV, which is particularly important to evaluate performance and robustness. Therefore, the wave forces are introduced into the heading subsystem [27]. Figure 8 shows the result of computer simulation of the wave forces.

Table 3. Parameter index of controllers in presence of disturbance for heading subsystem (Unit: ${}^{°}$).

Method	Maximum	Minimum	RMSE
PID	8.7441	7.2291	0.4121
ADRC	8.3081	7.6562	0.1699
FCADRC	8.1169	7.8919	0.0585

reports the maximum, minimum, and RMSE of the heading subsystem with these three methods. As shown in Figure 9, the heading angle of PID varies from 7.2291${}^{°}$ to 8.7441${}^{°}$, which fluctuates by as much as 1.515${}^{°}$. PID is more sensitive to the external disturbance. The heading angle of ADRC ranges from 7.6562${}^{°}$ to 8.3081${}^{°}$, and the heading angle of fractional calculus ADRC ranges from 7.8919${}^{°}$ to 8.1169${}^{°}$. They fluctuate within a narrow range. The RMSE decreases by approximately $58\%$ and $85\%$, respectively. ADRC is less sensitive to the external disturbance than PID, whereas fractional calculus ADRC is less sensitive than ADRC. The simulations indicate that both ADRC and fractional calculus ADRC can effectively reject the disturbance.

4.2. Simulations of Pitch Subsystem

The step response of pitch subsystem using PID, ADRC, and fractional calculus ADRC is sketched in Figure 10.

Table 4. Parameter index of step response for pitch subsystem.

Method	Overshoot (%)	Rise Time (s)	Settling Time (s)	SSE (m)
PID	2.4	1.65	4	0
ADRC	2.4	1.65	3	0
FCADRC	2.2	1.21	3	0

lists the overshoot, rise time, settling time, and SSE. To compare with the performance of controllers, the overshoot and SSE should be maintained at the same value. It is easily obtained that the fractional calculus ADRC strategy has the shortest rise time and settling time.

The white noise response of pitch subsystem using PID, ADRC, and fractional calculus ADRC is illustrated in Figure 11. When the input signal changes continuously, the FOADRC strategy can rapidly track the desired pitch angle. From the plot of partial enlargement, the fractional calculus clearly improves the response speed.

The same disturbance is introduced into pitch subsystem (see Figure 8). The maximum, minimum and RMSE of PID, ADRC, and fractional calculus ADRC are reported in

Table 5. Performance index of controllers in the presence of disturbance for pitch subsystem (Unit: ${}^{°}$).

Method	Maximum	Minimum	RMSE
PID	5.3304	4.5560	0.1872
ADRC	5.1641	4.7102	0.1076
FCADRC	5.1371	4.8422	0.0709

. The desired pitch angle is 5${}^{°}$. Figure 12 provides a whole picture of comparison between controllers. The pitch angle of PID varies from 4.5560${}^{°}$ to 5.3304${}^{°}$, the pitch angle of ADRC ranges from 4.7102${}^{°}$ to 5.1641${}^{°}$, and the pitch angle of FOADRC ranges from 4.8422${}^{°}$ to 5.1371${}^{°}$. The fluctuation of pitch angle decreases by less than $42\%$ and $62\%$, respectively. The results of pitch and heading subsystems are similar. The fractional calculus ADRC is the least sensitive to the external disturbance. It is further confirmed that fractional calculus ADRC has excellent performance and robustness for pitch subsystems.

4.3. Experiments

According to the computer simulations, the disturbance rejection capability of PID is inferior to ADRC and fractional calculus ADRC. Therefore, the ADRC and fractional calculus ADRC strategies were tested on the $Sailfish$ AUV in sea state 4. The weight of Sailfish is 260 kg. The length and diameter are 3.8 m and 32.4 cm, respectively. It is equipped with AHRS (Attitude and Heading Reference System), DVL (Doppler Velocity Log), GPS (Global Positioning System), SONAR (Sound Navigation and Ranging), and so on, as shown in Figure 13. The hardware of control system is an Intel(R) core(TM) i7 Industrial Personal Computer CM920 that has 4 GB of ram, Ubuntu 16.04 operating system, and MOOS-IvP (Mission Oriented Operating Suite-Interval Programming) software system. The sea trial was conducted on 29 October 2019, in Qingdao. It was sunny, the velocity of north wind was 10–12 km, and the ocean waves were 1.25–2.5 m high.

4.3.1. Heading Subsystem

The fractional calculus ADRC strategy was validated experimentally in the heading subsystem of AUV. The desired heading angle was 120${}^{°}$. The cruise speed of AUV was 1.5 m/s, and the propeller speed was about 3300 r, as shown in Figure 14. The maximum of rudder angles was 115${}^{°}$, and the minimum is 35${}^{°}$. At the beginning, the rudder reached the maximum and maintained it for a few seconds. The rudder angle decreased to approximately 75${}^{°}$ gradually when the current heading closed to the desired heading (see Figure 15). The control parameters of ADRC and fractional calculus ADRC are listed in

Table 6. Control parameters of heading angle via ADRC.

${\mathit{\beta }}_{\mathit{h}1}$	${\mathit{\beta }}_{\mathit{h}2}$	${\mathit{\beta }}_{\mathit{h}3}$	$\mathit{\sigma }$	${\mathit{b}}_{\mathit{h}0}$	${\mathit{K}}_{\mathit{p}}^{\mathit{h}}$	${\mathit{K}}_{\mathit{d}}^{\mathit{h}}$
100	300	1000	0.0025	100	8	1.5

and

Table 7. Control parameters of heading angle via fractional calculus ADRC.

${\mathit{\beta }}_{\mathit{h}1}$	${\mathit{\beta }}_{\mathit{h}2}$	${\mathit{\beta }}_{\mathit{h}3}$	$\mathit{\sigma }$	${\mathit{b}}_{\mathit{h}0}$	${\mathit{K}}_{\mathit{f}\mathit{p}}^{\mathit{h}}$	${\mathit{K}}_{\mathit{f}\mathit{i}}^{\mathit{h}}$	${\mathit{K}}_{\mathit{f}\mathit{d}}^{\mathit{h}}$	${\mathit{\lambda }}_{\mathit{h}}$	${\mathit{\mu }}_{\mathit{h}}$
100	300	1000	0.0025	100	0.6	0.01	0.01	0.85	0.9

.

The results of ADRC and fractional calculus ADRC on sea trial are depicted in

Table 8. Parameter list of heading angle via ADRC and fractional calculus ADRC.

Method	Settling Time (s)	Overshoot (%)	RMS (${}^{°}$)	SSE (${}^{°}$)
ADRC	11	4.5	116.77	3.595
FCADRC	11	3.1	117.11	2.291

and Figure 16. There was a decrease in steady-state error of $36.3\%$ and overshoot of $31.1\%$. The results indicate the better performance and robustness of fractional calculus ADRC.

Table 9. Steady-state error of heading angle in simulation and sea trial (Unit: ${}^{°}$).

Method	Simulation	Sea Trial
ADRC	0.1699	3.595
FCADRC	0.0585	2.291

reports the results of steady-state errors in simulation and sea trial. It is the ideal condition in simulation, but not on sea trial. The assumptions in Section 2.3 cannot be satisfied absolutely. It is similar to depth subsystem.

4.3.2. Depth Subsystem

The fractional calculus ADRC strategy was validated experimentally in the depth subsystem of AUV. The desired depth is 3 m. The control parameters of ADRC and fractional calculus ADRC are reported in

Table 10. Control parameters of depth via ADRC.

${\mathit{\beta }}_{\mathit{p}1}$	${\mathit{\beta }}_{\mathit{p}2}$	${\mathit{\beta }}_{\mathit{p}3}$	$\mathit{\sigma }$	${\mathit{b}}_{\mathit{p}0}$	${\mathit{K}}_{\mathit{f}\mathit{p}}^{\mathit{p}}$	${\mathit{K}}_{\mathit{f}\mathit{d}}^{\mathit{p}}$
100	300	1000	0.0025	100	20.5	1.4

and

Table 11. Control parameters of depth via fractional calculus ADRC.

${\mathit{\beta }}_{\mathit{p}1}$	${\mathit{\beta }}_{\mathit{p}2}$	${\mathit{\beta }}_{\mathit{p}3}$	$\mathit{\sigma }$	${\mathit{b}}_{\mathit{p}0}$	${\mathit{K}}_{\mathit{f}\mathit{p}}^{\mathit{p}}$	${\mathit{K}}_{\mathit{f}\mathit{i}}^{\mathit{p}}$	${\mathit{K}}_{\mathit{f}\mathit{d}}^{\mathit{p}}$	${\mathit{\lambda }}_{\mathit{p}}$	${\mathit{\mu }}_{\mathit{p}}$
100	300	1000	0.0025	100	3.3	0.042	0.01	0.8	0.8

. The results of fractional calculus ADRC and ADRC on sea trial were sketched in Figure 17. The overshoot, settling time, RMS, and SSE were tabulated in

Table 12. Parameter list of depth via ADRC and fractional calculus ADRC.

Method	Settling Time (s)	Overshoot (%)	RMS (m)	SSE (m)
ADRC	35	0	2.5187	0.5070
FCADRC	35	0	2.8799	0.1807

. The steady-state error of ADRC was more than twice that of fractional calculus ADRC, which decreased by 64%. The sea trial substantiated that fractional calculus ADRC is superior to ADRC.

Figure 18 and Figure 19 show the comparisons of pitch angle and stern rudder using ADRC and FOADRC. For AUV sailing at constant depth, the pitch angle and stern rudder should be kept constant or changed in a small range. In the beginning, the pitch angle should be large enough to dive. However, the pitch angle of ADRC was too small to dive. It cannot reach the desired depth. It is clear that fractional calculus ADRC works better than ADRC.

5. Conclusions

This paper proposed a fractional calculus ADRC strategy for AUV heading and pitch subsystems, and the stability of heading system of fractional calculus ADRC is proven by Lyapunov stability theorem. This strategy is sufficiently simulated and successfully employed in heading and pitch subsystems on sea trial. This method has smaller steady-state error, settling time, and overshoot than ADRC and PID. The numerical simulations and experimental result indicated the better performance and robustness of fractional calculus ADRC. It can restrain the disturbance effectively and improve the response speed.

In the future, the fractional calculus ADRC statergy will be designed for path following, and the controller parameters can be adjusted automatically.