Observer-Based Adaptive Control for Trajectory Tracking of AUVs with Input Saturation

Abstract

In this paper, an observer-based adaptive control method is investigated for the horizontal trajectory tracking of autonomous underwater vehicles (AUV) with input saturation and system disturbances. Firstly, the desired surge speed and trajectory angle are established, which could decouple the tracking error subsystem and avoid the complex form. Secondly, the input saturation is approximated by a smooth function, and a nonlinear extended states observer (NESO) is designed for estimating system disturbances. Based on the command filtered backstepping technique, which can avoid the explosion caused by the derivative of the virtual control, an observer-based adaptive output feedback control method is developed, and an auxiliary system is applied to compensate for filtered tracking errors, input saturation bias, and observer errors. Finally, simulation results show the proposed method has good robustness in the face of system uncertainties, and the error is nearly 33.3% smaller than that of other control methods when meeting sudden trajectory changes. A good control performance is guaranteed.

1. Introduction

With the ongoing exploration and exploitation of oceans, autonomous underwater vehicles (AUV) are widely applied in underwater searching, ocean sampling, seafloor mapping, resource exploration, and other fields, and significant progress has been made with these applications [1,2]. However, the strong-coupled and highly nonlinear system, together with complex, unpredictable disturbances, creates some challenges in the trajectory of tracking controller design [3,4].

During the trajectory tracking stage, the main challenge is designing a control framework to drive the AUV following a given path. Therefore, various advanced control theories and techniques have been studied in designing higher performance controllers, e.g., adaptive control, sliding mode control, robust control, prescribe performance, and model predictive control, to name a few. Among these control schemes, sliding mode control provides an effective method with significant advantages of invariance to parameter uptake and external environmental disturbances [5,6]. However, high-frequency dithering is an unavoidable problem that may cause rapid actuator wear. Robust control can effectively suppress disturbance, but it has strong conservative properties [7,8]. Prescribed performance control sets the dynamic process and steady-state process, but it may cause control singularity problems when the system suffers from actuator saturation, disturbance, and other problems [9,10]. Model predictive control is capable of handling input and state constraints and is suitable for dealing with complex underwater missions. Nevertheless, the control signal of MPC is an implicit function, presenting a challenge to the closed-loop stability of the system [11,12]. Compared with these control methods, adaptive control has been a meaningful subject for nonlinear systems which is suitable for the trajectory tracking control of underactuated AUVs [13,14]. In particular, backstepping-based adaptive control has become one of the common methods to solve problems of nonlinear systems. However, many adaptive control schemes cannot achieve satisfactory results due to two reasons: one is complex uncertainties, such as system parameter uncertainties and external disturbances. The other is the complexity explosion caused by the derivation of the virtual controller. In short, adaptive control is more suitable for trajectory tracking control of the nonlinear AUV system, with two issues being handled.

A key technique of trajectory tracking control is to adopt a suitable method to resolve the effects of uncertainties. In practical engineering applications, uncertainties and disturbances are inevitable in AUV systems due to unmodeled dynamics, model uncertainties, and environmental impacts, such as ocean currents and waves [15,16]. Most of these uncertainties and disturbances are hard to measure and adversely affect tracking performance, rendering great challenges in controller design [17]. Estimating these uncertainties and compensating for them in controller design has drawn great attention. A high-gain observer is employed for unknown internal and external disturbances, which is compensated for by output feedback control law [18]. An extended states observer is developed for system uncertainties in path-following control of under-actuated AUVs [19,20]. However, it may cause differential peak problems due to high gain, which affects control performance. Neural network approximation capabilities are utilized to estimate uncertainties and environmental disturbances, which are compensated for by adaptive control methods [21,22]. However, the performance of the neural network is heavily dependent on the number of neural network nodes, resulting in a computationally intensive system that is not conducive to practical engineering applications. The super twisting algorithm is adopted to solve the influence of nonlinear environmental disturbances in controller design [23]. However, controller parameter adjustment is complicated.

Another key technique of trajectory tracking control is to design a controller to handle the input constraints, i.e., input saturation, which can occur due to the limitation of physical conditions in practical applications. The input constraints may make the controller unable to achieve the corresponding output, which may result in system instability and even control failure. Usually, two common approaches are used to tackle the input constraints: compensation with an auxiliary term [24] and approximate with a smooth function [25]. As to the input saturation caused by differentiating a composite reference trajectory (i.e., virtual input) during the application of the backstepping technique, using command/integral filters is an effective way to deal with input saturation and guarantee good performance [26,27,28].

In addition to the adaptive control method used for under-actuated AUV, controller design and system stability analysis may suffer from computational difficulty as the increasing of system order. The development of the backstepping algorithm makes the feedback adaptive control systematic and simplifies the complex controller design [29,30,31]. However, the required form of strict feedback is difficult for the under-actuated AUV system. Regarding this problem, a composite system, such as a singularly perturbed system and cascade system, in the form of lower order subsystems and their interconnecting structures, is adopted to simplify the control system [32,33].

Thus, based on the above discussion, a backstepping-based adaptive control strategy employing disturbance observer and command filtered technique is proposed for trajectory tracking of AUVs with system uncertainties and environmental disturbances. They are summarized below.

(1) A composite system, which consists of two subsystems with an interconnection function, is established instead of a kinematic tracking error dynamic. Then, it decouples the tracking error subsystem from the surge force and the yaw torque to the traditional solution.

(2) A novel control strategy is proposed based on the disturbance observer and auxiliary compensator. A nonlinear extended states observer (NESO) is designed to estimate the unknown disturbances, and command-filtered backstepping is designed to stabilize the surge speed and trajectory angle independently. During this process, the command-filtered backstepping technique can avoid the explosion caused by the derivative of the virtual control, and an auxiliary system is applied to compensate for the filter tracking errors, input saturation bias, and observer errors.

The rest of this paper is arranged as follows: Section 2 presents the mathematical model and horizontal trajectory tracking problems. Section 3 introduces the nonlinear observer-based adaptive control method and stability analysis. Simulation results and conclusion are provided in Section 4 and Section 5, respectively.

2. System Modeling

2.1. AUV Modeling in Horizontal Plane

A mathematical model for the horizontal motion of underactuated AUVs can be examined in two separate reference frames: the inertial frame and the body-fixed frame, as shown in Figure 1. The inertial frame is defined relative to the Earth’s reference ellipsoid, and the body-fixed frame is a moving coordinate frame that is fixed to the craft.

In this paper, we consider the horizontal motion of under-actuated AUVs, which could neglect the dynamics of heave, roll, and pitch. Additionally, the kinematic equation could be expressed as follows:

$$\left\{\begin{array}{l}\dot{x}=u\cos (\psi )-v\sin (\psi )\\ \dot{y}=u\sin (\psi )+v\cos (\psi )\\ \dot{\psi }=r\end{array}\right..$$

(1)

Additionally, the dynamic equation is provided as follows:

$$\left\{\begin{array}{l}\dot{u}=M_1(X_{u}u+a_{23}vr+T+d_1)\\ \dot{v}=M_2(Y_{v}v+a_{13}ur)\\ \dot{r}=M_3(N_{r}r+a_{12}uv+{\tau }_r+d_2)& .\end{array}\right.$$

(2)

The main model parameters are shown in

Table 1. The model parameters.

Terms	Description
$x$, $y$	position of the AUV in the earth-fixed frame
$\psi$	yaw angle
$u$	surge velocity
$v$	sway velocity
$r$	yaw angular velocity
$T$	surge force
${\tau }_r$	yaw torque
$m$	mass terms
$d_1$$,d_2$	unknown disturbances
$X_u$$, Y_v$$, N_r$	hydrodynamic coefficients of the AUV model

, and some parameters are calculated as $M_1=1/(m-X_{\dot{u}})$, $M_2=1/(m-Y_{\dot{v}})$, $M_3=1/(I_{zz}-N_{\dot{r}})$, $a_{12}=Y_{\dot{v}}-X_{\dot{u}}$, and $a_{13}=X_{\dot{u}}-m$, $a_{23}=m-Y_{\dot{v}}$.

2.2. Problem Formulation and Control Objective

First, the position tracking errors $x_e$ and $y_e$ are defined as follows:

$$\left\{\begin{array}{l}{x}_e=x-x_t\\ y_e=y-y_t& ,\end{array}\right.$$

(3)

where $x_t$ and $y_t$ are the time-varying positions with respect to the earth-fixed frame. The derivative of the position tracking error can be expressed by

$$\left\{\begin{array}{l}{\dot{x}}_e=\dot{x}-{\dot{x}}_t={\dot{x}}_d-{\dot{x}}_t+(\dot{x}-{\dot{x}}_d)\\ {\dot{y}}_e=\dot{y}-{\dot{y}}_t={\dot{y}}_d-{\dot{y}}_t+(\dot{y}-{\dot{y}}_d)& ,\end{array}\right.$$

(4)

where ${\dot{x}}_d$ and ${\dot{y}}_d$ are designed desired velocities in the earth-fixed frame.

Proposition 1 [34].

System (4) will be input-to-state stable if the desired velocities ${\dot{x}}_d$ and ${\dot{y}}_d$ are chosen as:

$$\left\{\begin{array}{l}{\dot{x}}_d=-k_x\tanh (x_e)+{\dot{x}}_t\\ {\dot{y}}_d=-k_y\tanh (y_e)+{\dot{y}}_t& ,\end{array}\right.$$

(5)

where $k_x>0$ and $k_y>0$. Note that the choice of controller gains $k_x$ and $k_y$ will determine how fast the trajectories converge to zero and can be tuned on the basis of the system’s desired output behavior.

Based on Proposition 1, $x_e$ and $y_e$ will be ultimately bounded while $\dot{x}-{\dot{x}}_d$ and $\dot{y}-{\dot{y}}_d$ converge to zero. Then, the system control object is converted to design $T$ and ${\tau }_r$ in system (2) to guarantee $\dot{x}-{\dot{x}}_d$ and $\dot{y}-{\dot{y}}_d$ converge to zero. That means that the vehicle needs to stabilize to the desired velocities $u_d$ and $v_d$ in the body-fixed frame.

$$\left[\begin{array}{l}{u}_d\\ v_d\end{array}\right]=\left[\begin{array}{cc}\cos (\psi )& \sin (\psi )\\ -\sin (\psi )& \cos (\psi )\end{array}\right]\left[\begin{array}{c}{\dot{x}}_d\\ {\dot{y}}_d\end{array}\right].$$

(6)

The sway speed $v_d$ cannot be directly controlled due to the under-actuated characteristic of the system. Define $u_d=\sqrt{{\dot{x}}_d^2+{\dot{y}}_d^2-v^2}$ and ${\gamma }_d=\arctan ({\dot{y}}_d/{\dot{x}}_d)$. Then, the control requirement is transformed to make surge speed $u$ and the trajectory angle $\gamma =\psi +\beta$ to the desired ones, and $\beta =\arctan (v/u)$, ${\dot{x}}_d\ne 0$. The tracking errors $u_e$ and ${\gamma }_e$ are obtained as follows:

$$\left\{\begin{array}{l}{\gamma }_e=\psi +\beta -{\gamma }_d\\ u_e=u-u_d& .\end{array}\right.$$

(7)

The derivatives of ${\gamma }_e$ and $u_e$ are

$$\left\{\begin{array}{l}{\dot{\gamma }}_e=r+\dot{\beta }-{\dot{\gamma }}_d\\ {\dot{u}}_e=M_1(X_{u}u+a_{23}vr+T+d_1)-{\dot{u}}_d& .\end{array}\right.$$

(8)

The trajectory tracking control problem can be reformulated in the new mathematical model as follows:

$$\left\{\begin{array}{l}{\dot{\gamma }}_e=r+\dot{\beta }-{\dot{\gamma }}_d\\ {\dot{u}}_e=M_1(X_{u}u+a_{23}vr+T+d_1)-{\dot{u}}_d\\ \dot{v}=M_2(Y_{v}v+a_{13}ur)\\ \dot{r}=M_3(N_{r}r+a_{12}uv+{\tau }_r+d_2)& .\end{array}\right.$$

(9)

Remark 1.

As an illustration, the position trajectory tracking control for $x_t$ and $y_t$ is transformed into the stabilization problem of $u_e$ and ${\gamma }_e$. It could decouple the tracking error subsystem and avoid the complex form caused by the iterative solution controller of the Lyapunov function for $x_e$ and $y_e$. The control objective is to design the control law of $T$ and ${\tau }_r$.

The following assumption holds, considering the AUV’s physical limitation during the trajectory tracking control.

Assumption 1.

The under-actuated AUV’s position, yaw angle, and velocities are measurable.

Assumption 2.

Assume that both the unknown disturbances and their derivatives are bound. $\left|d_i\right|\le {\overline{d}}_i$, ${\dot{d}}_i\le {\overline{\omega }}_i$, where ${\overline{d}}_i>0$ and ${\overline{\omega }}_i>0$ are unknown constants, $i=1,2$.

Remark 2.

In practice, the inertial navigation system can measure the acceleration of the carrier under the inertial reference frame. Then, it could obtain the required position, yaw angle, and velocity information by integrating the acceleration under the navigation coordinate system. Therefore, Assumption 1 is reasonable. In the problem formulation of most practical systems, environmental disturbances, i.e., ocean currents and entanglements in the water, are assumed bounded with an upper bound. The internal uncertainties are also specified. Thus, Assumption 2 could be depicted according to this mentioned tip.

Based on Assumptions 1 and 2 and considering a horizontal curvilinear path to be tracked, the control objective is to design a control law for $T$ and ${\tau }_r$ to stabilize the surge speed tracking error $u_e$ and angle tracking error ${\gamma }_e$ to zeros.

3. Methodology

In this section, an observer-based adaptive control scheme is proposed to solve the trajectory tracking control problem of an underactuated AUV and the block diagram of the proposed controller is shown in Figure 2.

3.1. NESO Design

As is well known, ESO can estimate lumped disturbances effectively. However, ESO can achieve fast convergence by selecting high gain, which may also cause differential peak problems. To overcome the mentioned restriction, a NESO combines the advantages of the hyperbolic tangent function and high-gain structure to estimate the unknown disturbances $d_1$ and $d_2$ in the system (9). The upper bound of the estimation error depends on the high gain value, and the hyperbolic tangent function is employed to reduce the effect of the differential peak problem.

Let $x_{11}=u_e$, $x_{12}=d_1$, $x_{21}=r$, $x_{22}=d_2$, ${\omega }_1={\dot{d}}_1$, and ${\omega }_2={\dot{d}}_2$, then there are two subsystems such that:

Subsys1:

$$\left\{\begin{array}{l}{\dot{x}}_{11}=M_1{x}_{12}+M_1(X_{u}u+a_{23}vr+T)-{\dot{u}}_d\\ {\dot{x}}_{12}={\omega }_1& .\end{array}\right.$$

(10)

Subsys2:

$$\left\{\begin{array}{l}{\dot{x}}_{21}=M_3{N}_r{x}_{21}+M_3{x}_{22}+M_3(a_{12}uv+{\tau }_r)\\ {\dot{x}}_{22}={\omega }_2& .\end{array}\right.$$

(11)

Additionally, the NESOs are designed as follows:

NESO1:

$$\left\{\begin{array}{l}{\dot{\widehat{x}}}_{11}={\widehat{x}}_{12}-g{k}_{11}^{eso}{\tilde{x}}_{11}+M_1(X_{u}u+a_{23}vr+T)-{\dot{u}}_d\\ {\dot{\widehat{x}}}_{12}=-g{k}_{12}^{eso}\tanh \left(g{k}_{th}{\tilde{x}}_{11}\right)& .\end{array}\right.$$

(12)

NESO2:

$$\left\{\begin{array}{l}{\dot{\widehat{x}}}_{21}=M_3{N}_r{\widehat{x}}_{21}+M_3{\widehat{x}}_{22}+M_3(a_{12}uv+{\tau }_r)-g{k}_{21}^{eso}{\tilde{x}}_{21}\\ {\dot{\widehat{x}}}_{22}=-g{k}_{22}^{eso}\tanh \left(g{k}_{th}{\tilde{x}}_{21}\right)& ,\end{array}\right.$$

(13)

where ${\widehat{x}}_{ij}$ is the estimated value of $x_{ij}$, ${\tilde{x}}_{ij}={\widehat{x}}_{ij}-x_{ij}$ is the estimation error, $k_{ij}^{eso}>0$ is the observer gain coefficient, $i=1,2$, $j=1,2$, $g>1$ is the high gain coefficient, and $k_{th}>0$ is the hyperbolic tangent function $\tanh (x)=(e^x-e^{-x})/(e^x+e^{-x})$.

Then, the estimated value of the disturbance could be obtained, ${\widehat{d}}_i={\widehat{x}}_{i2}$, $i=1,2$.

Theorem 1.

Under the condition of satisfying Assumption 1, the NESOs (12) and (13) designed for subsystems (10) and (11) can ensure the consistent final bound convergence of the estimation errors, and the upper bound of the errors depends on the high gain coefficient $g$.

Proof. 

Since the two NESOs have the same structure, only the NESO1 convergence analysis is provided below.

Let $z_{1j}(t)=g^{2-j}{\tilde{x}}_{1j}(\epsilon t)$, $j=1,2$, and ${\xi }_1=z_{11}$, ${\xi }_2=z_{12}-k_{11}^{eso}{z}_{11}$. Then, we can obtain the following from (10) and (12):

$$\left\{\begin{array}{l}{\dot{\xi }}_1={\xi }_2\\ {\dot{\xi }}_2=-k_{12}^{eso}\tanh \left(k_{th}{\xi }_1\right)-k_{11}^{eso}{\xi }_2-g^{-1}{\omega }_1& .\end{array}\right.$$

(14)

On the basis of this observation, the following Lyapunov function is considered.

$$V_{\xi 1}({\xi }_1,{\xi }_2)={\displaystyle {\int }_0^{{\xi }_1(t)}\tanh \left(k_{th}{\xi }_1(\tau )\right)d{\xi }_1(\tau )}+\frac{1}{2k_{12}^{eso}}{\xi }_2^2.$$

(15)

The derivative of $V_{\xi 1}({\xi }_1,{\xi }_2)$ along the system (15) is:

$${\dot{V}}_{\xi 1}\le -\frac{k_{11}^{eso}}{k_{12}^{eso}}{\left[{\xi }_2+\frac{{\omega }_1}{2g{k}_{11}^{eso}}\right]}^2+\frac{{\overline{\omega }}_1^2}{4g{k}_{11}^{eso}{k}_{12}^{eso}}.$$

(16)

From (16), it follows that ${\xi }_1$ and ${\xi }_2$ are bound. Additionally, according to (14), when the system reaches a steady state, it can be obtained that

$$\left\{\begin{array}{l}\left|{\xi }_1\right|\le \frac{1}{k_{th}}{\tanh }^{-1}\left(\frac{2{\overline{\omega }}_1}{g{k}_{12}^{eso}}\right)\\ \left|{\xi }_2\right|\le \frac{{\overline{\omega }}_1}{g{k}_{11}^{eso}}& .\end{array}\right.$$

(17)

Additionally,

$$\left\{\begin{array}{l}\left|{\tilde{x}}_{11}\right|=\frac{\left|{\xi }_1\right|}{g}\le \frac{1}{g{k}_{th}}{\tanh }^{-1}\left(\frac{2{\overline{\omega }}_1}{k_{12}^{eso}}\right)\\ \left|{\tilde{x}}_{12}\right|=\left|z_{12}\right|\le \left|{\xi }_2\right|+k_{11}^{eso}\left|{\xi }_1\right|\le \frac{{\overline{\omega }}_1}{g{k}_{11}^{eso}}+\frac{k_{11}^{eso}}{g{k}_{th}}{\tanh }^{-1}\left(\frac{2{\overline{\omega }}_1}{k_{12}^{eso}}\right)& .\end{array}\right.$$

(18)

Thus, the estimated errors ${\tilde{x}}_{11}$ and ${\tilde{x}}_{12}$ are bound, and the upper bounds depend on the high gain coefficient $g$.

Similarly, it can be concluded that the estimation errors ${\tilde{x}}_{21}$ and ${\tilde{x}}_{22}$ are bound, and the upper bound depends on the high gain coefficient $g$. □

Remark 3.

The designed NESOs (12) and (13) are based on the proposed hyperbolic tangent function and the high-gain ESO. High gain can improve the estimation accuracy of ESO, and the hyperbolic tangent function can reduce the differential peak using its saturation property.

3.2. Input Saturation Processing

In practice, input saturation is one of the constraints which may degrade system performance or even affect system stability. Input saturation is described by

$$u(\zeta (t))=\mathrm{sat}(\zeta (t))=\left\{\begin{array}{l}\mathrm{sgn}(\zeta (t))M,\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\left|\zeta (t)\right|\ge M\\ \zeta (t),\hspace{1em}\hspace{0.33em}\hspace{0.33em}\left|\zeta (t)\right|\le M\hspace{0.33em}& ,\end{array}\right.$$

(19)

where $M$ is the saturation amplitude.

In this paper, the hyperbolic tangent function is used to approximate the saturation function:

$$H_M(\zeta )=M*\tanh (\zeta /M)=M*\frac{e^{\zeta /M}-e^{-\zeta /M}}{e^{\zeta /M}+e^{-\zeta /M}}.$$

(20)

Then, $\mathrm{sat}(\zeta (t))$ can be expressed as:

$$\mathrm{sat}(\zeta )=H_M(\zeta )+\rho (\zeta )=M*\tanh (\zeta /M)+\rho (\zeta ),$$

(21)

where $\rho (\zeta )=\mathrm{sat}(\zeta )-H_M(\zeta )$ is a bound function in time, and its bound can be obtained:

$$\left|\rho (\zeta )\right|=\left|\mathrm{sat}(\zeta )-H_M(\zeta )\right|\le M*(1-\tanh (1))=D.$$

(22)

Note that in the section $0\le \left|\zeta \right|\le M$, the bound of $\rho (\zeta )$ is between 0 and $D$.

Considering the input saturation and state limitation of the system, the command integral filter is designed as follows:

$$\left[\begin{array}{c}{\dot{\zeta }}_1\\ {\dot{\zeta }}_2\end{array}\right]=\left[\begin{array}{c}{\zeta }_2\\ 2\xi {\omega }_n[H_R(\frac{{\omega }_n^2}{2\xi {\omega }_n^{}}[H_M({\alpha }_i)-{\zeta }_1])-{\zeta }_2]\end{array}\right],$$

(23)

where the input is the virtual control quantity ${\alpha }_i$, and the output is ${\overline{x}}_i$ and its first derivative. ${\tau }_1$ and ${\tau }_2$ are the filter states, $\xi$ is the damping ratio, ${\omega }_n$ is the natural frequency, and $H_M$ and $H_R$ are the amplitude and rate saturation function, respectively. Using (23) with proper designed parameters, the following inequalities hold:

$$\left|H_M({\alpha }_i)-{\zeta }_1\right|\le l_1\vartheta ,$$

(24)

where $l_1>0$, $\vartheta =1/{\omega }_n>0$.

3.3. Adaptive Controller Design with Input Saturation

An adaptive controller using backstepping technology is designed for under-actuated AUVs in this section.

For simplicity, a dynamic system for (9) is represented by:

$$\left\{\begin{array}{l}\dot{y}=f(y,z)\\ \dot{z}=g(y,z)& ,\end{array}\right.$$

(25)

where $y={[{\gamma }_e;u_e]}^{\mathrm{T}}$ and $z={[v;r]}^{\mathrm{T}}$. $f(\cdot )$ and $g(\cdot )$ are corresponding differentiable functions.

Let $z_s(y)=(v_s;r_s)$ represent the designed space configuration. $z_c(y)=(v_c;r_c)$ represents the shift from $z_s$ to $z$, and $z_c=z-z_s$. Following that, system (9) is shown as follows:

$$\left\{\begin{array}{l}\dot{y}=f(y,z_s)+[f(y,z)-f(y,z_s)]\\ {\dot{z}}_c=g(y,z_s+z_c)-{\dot{z}}_s=g(y,z_s+z_c)-\frac{\mathsf{\partial }{z}_s(y)}{\mathsf{\partial }y}\dot{y}-\frac{\mathsf{\partial }{z}_s(y)}{\mathsf{\partial }t}& .\end{array}\right.$$

(26)

In (26), there are two lower-order subsystems:

$$\dot{y}=f(y,z_s),$$

(27)

$${\dot{z}}_c=g(y,z_s+z_c).$$

(28)

Additionally, $f(y,z)-f(y,z_s)$ and ${\dot{z}}_s$ are regarded as the interconnections of two subsystems.

Part 1: Controller design for the subsystem (27)

In practice, considering the input saturation, the system could be described as follows:

$$\left\{\begin{array}{l}{\dot{\gamma }}_e=r+\dot{\beta }-{\dot{\gamma }}_d\\ {\dot{u}}_e=f_u+M_1{u}_1+M_1{d}_1-{\dot{u}}_d\\ {\dot{u}}_1=-c_1{u}_1+\mathrm{sat}({\tau }_1)& ,\end{array}\right.$$

(29)

where $f_u=M_1(X_{u}u+a_{23}vr)$, $u_1=T$. To determine $u_1$, it is designed as follows:

$$r_s=-k_r{\gamma }_e-\dot{\beta }+{\dot{\gamma }}_d,$$

(30)

where $k_r>0$. Then, the tracking error of the control system is introduced as follows:

$$u_e=u-u_d,$$

(31)

$$z_1=u_1-{\alpha }_1^c,$$

(32)

where $u_d$ is the desired surge speed, and ${\alpha }_1^c$ is filtered signal of the virtual control quantity ${\alpha }_1^{co}$ for $u_1$.

The compensation signal of the filter error is defined as:

$$v_1=u_e-{\xi }_1,$$

(33)

$$v_2=z_1-{\xi }_2.$$

(34)

The residual compensations of the filtering error and the saturation control quantity are constructed as:

$${\dot{\xi }}_1=-k_1{\xi }_1+M_1({\alpha }_1^c-{\alpha }_1^{co})+M_1{\xi }_2,$$

(35)

$${\dot{\xi }}_2=-k_2{\xi }_2+D_1,$$

(36)

where $k_1>0$, $k_2>0$;$D_1=\mathrm{sat}({\tau }_1)-{\tau }_1+M_1{\tilde{d}}_1$, ${\xi }_1(0)=0$, ${\xi }_2(0)=0$, and ${\tilde{d}}_1=d_1-{\widehat{d}}_1$ represent observation error.

The control signals could be designed as follows:

$${\alpha }_1^{co}=(-k_1{u}_e-f_u-M_1{\widehat{d}}_1+{\dot{u}}_d)/M_1,$$

(37)

$${\tau }_1=c_1{u}_1-k_2{z}_1+{\dot{\alpha }}_1^c-M_1{v}_1+(M_1-1)D_1.$$

(38)

Theorem 2.

Considering the subsystem (27), select the filter as shown in (23), construct a compensation system as shown in (35) and (36), and the control signals are designed as (37) and (38), thus the filter compensation errors (33) and (34) are asymptotically stable, and the tracking errors (31) and (32) are bound and convergent.

Proof. 

Using (30), (31), and (32), system (29) is put into

$$\left\{\begin{array}{l}{\dot{\gamma }}_e=-k_r{\gamma }_e\\ {\dot{u}}_e=-k_1{u}_e+M_1[({\alpha }_1^c-{\alpha }_1^{co})+z_1]\\ {\dot{z}}_1=-k_2{z}_1+M_1{D}_1-M_1{v}_1& .\end{array}\right.$$

(39)

Step 1: According to (33) and (34), a Lyapunov function is designed:

$$V_1=\frac{1}{2}(v_1^2+v_2^2).$$

(40)

The derivative of $V_1$ is

$$\begin{array}{l}{\dot{V}}_1=v_1^{}(-k_1{v}_1+M_1{v}_2)+v_2^{}(-k_2{v}_2-M_1{v}_1)\\ \le -k_1{v}_1^2-k_2{v}_2^2\le -n_1{V}_1& ,\end{array}$$

(41)

where $n_1=2\min \{k_1,k_2\}$,$k_1>0$,$k_2>0$.

Multiply both sides of (41) by $e^{n_{1}t}$, and then integrate both sides to obtain the following:

$$V_1(t)\le V_1(0)e^{-n_{1}t}\le V_1(0).$$

(42)

Additionally, we can determine that $\left|v_i\right|\le B_1$, $i=1,2$, $B_1=\sqrt{2V_1(0)}$. This indicates that $v_1$ and $v_2$ converge to ${\Omega }_1:=\{v_i\in R,i=1,2\left|\left|v_i\right|\le B_1\right.\}$.

Step 2: Then, consider a Lyapunov function as

$$V_2=\frac{1}{2}({\gamma }_e^2+u_e^2+z_1^2).$$

(43)

The derivative of $V_2$ is

$$\begin{array}{l}{\dot{V}}_2={\gamma }_e{\dot{\gamma }}_e+u_e^{}{\dot{u}}_e^{}+z_1^{}{\dot{z}}_1^{}\\ \le -k_r{\gamma }_e^2-\left(k_1-M_1\right)u_e^2-\left(k_2-1.5M_1\right)z_1^2+0.5M_1({\left|l_1\vartheta \right|}^2+{\overline{d}}_1^2+B_1^2)\\ \le -n_2{V}_2+{\delta }_2& ,\end{array}$$

(44)

where $n_2=2\min \left\{k_r,k_1-M_1,k_2-1.5M_1\right\}$, $k_r>0$, $k_1>M_1$, $k_2>1.5M_1$, and ${\delta }_2=0.5M_1({\left|l_1\vartheta \right|}^2+{\overline{d}}_1^2+B_1^2)$.

Multiply both sides of (44) by $e^{n_{2}t}$, and then integrate both sides to obtain the following:

$$V_2(t)\le V_2(0)e^{-n_{2}t}+{\delta }_2(1-e^{-n_{2}t})/n_2\le V_2(0)+{\delta }_2/n_2.$$

(45)

Additionally, we can determine that $\left|{\gamma }_e\right|\le B_2$, $\left|u_e\right|\le B_2$, $\left|z_1\right|\le B_2$, and $B_2=\sqrt{2V_2(0)+2{\delta }_2/n_2}$.

This indicates that ${\gamma }_e$, $u_e$, and $z_1$ converge to a compact set: ${\Omega }_2:=\{u_e,z_1\in R,\left|\left|{\gamma }_e\right|\le B_2,\left|u_e\right|\le B_2,\left|z_1\right|\le B_2\right.\}$. □

Part 2: Controller design for the subsystem (28)

To determine ${\tau }_r$, according to the relationship $v=v_c+v_s$, $r=r_c+r_s$, we further design

$$v_s=-\frac{a_{13}u{r}_s}{Y_v}.$$

(46)

This accounts for the fact that the vehicle model herein is under-actuated. Accordingly, subsystem (28) is taken as:

$$\left\{\begin{array}{l}{\dot{v}}_c=g_v{v}_c+g_r{r}_c\\ {\dot{r}}_c=f_r+M_3{u}_2+M_3{d}_2\\ {\dot{u}}_2=-c_2{u}_2+\mathrm{sat}({\tau }_2)& ,\end{array}\right.$$

(47)

where $f_r=M_3(N_r(r_c+r_s)+a_{12}uv)$, $g_v=M_2{Y}_v$, and $g_r=M_2{a}_{13}u$.

The error variables of the system are introduced as follows:

$$\left\{\begin{array}{l}{v}_c=v-v_s\\ z_2=r_c-{\alpha }_2^c\\ z_3=u_2-{\alpha }_3^c& ,\end{array}\right.$$

(48)

where ${\alpha }_2^c$ is the filtered signal of the virtual control quantity ${\alpha }_2^{co}$ for $r_c$. ${\alpha }_3^c$ is the filtered signal of the virtual control quantity ${\alpha }_3^{co}$ for $u_2$.

Define the compensation signal of the filter error:

$$v_3=v_c-{\xi }_3,$$

(49)

$$v_4=z_2-{\xi }_4,$$

(50)

$$v_5=z_3-{\xi }_5.$$

(51)

Construct the residual compensations of the filtering error and the saturation control quantity as follows:

$${\dot{\xi }}_3=-k_3{\xi }_3+[({\alpha }_2^c-{\alpha }_2^{co})+{\xi }_4],$$

(52)

$${\dot{\xi }}_4=-k_4{\xi }_4+M_3({\alpha }_3^c-{\alpha }_3^{co})+M_3{\xi }_5,$$

(53)

$${\dot{\xi }}_5=-k_5{\xi }_5+D_2,$$

(54)

where, $k_3>0$, $k_4>0$, $k_5>0$;${\xi }_3(0)=0$, ${\xi }_4(0)=0$, ${\xi }_5(0)=0$;$D_2=\mathrm{sat}({\tau }_2)-{\tau }_2+M_3{\tilde{d}}_2$, and ${\tilde{d}}_2=d_2-{\widehat{d}}_2$.

Additionally, the control signals are designed as follows:

$${\alpha }_2^{co}=-k_3{v}_c/g_v+g_r{r}_c,$$

(55)

$${\alpha }_3^{co}=(-k_4{z}_2-f_r-M_3{\widehat{d}}_2+{\dot{\alpha }}_2^c-v_3)/M_3,$$

(56)

$${\tau }_2=c_2{u}_2-k_5{z}_3+{\dot{\alpha }}_3^c-M_3{v}_4+(M_3-1)D_2.$$

(57)

Theorem 3.

Considering the subsystem (28), select the filter as shown in (23), construct a compensation system as shown in (52)~(54), and the control signals are designed as (55)~(57); the filter compensation errors (49)~(51) are asymptotically stable, and the tracking errors (48) are bound and convergent.

Proof. 

Taking the derivative of (48), we could obtain the following:

$$\left\{\begin{array}{l}{\dot{v}}_c=[{\alpha }_2^{co}+({\alpha }_2^c-{\alpha }_2^{co})+z_2]-g_r{r}_c\\ {\dot{z}}_2=f_r+M_3[{\alpha }_3^{co}+({\alpha }_3^c-{\alpha }_3^{co})+z_3]+M_3{d}_2-{\dot{\alpha }}_2^c\\ {\dot{z}}_3=-c_2{u}_2+{\tau }_2+D_2-{\dot{\alpha }}_3^c& .\end{array}\right.$$

(58)

Then, from (55)~(57), we obtain

$$\left\{\begin{array}{l}{\dot{v}}_c=-k_3{v}_c+[({\alpha }_2^c-{\alpha }_2^{co})+z_2]\\ {\dot{z}}_2=-k_4{z}_2+M_3[({\alpha }_3^c-{\alpha }_3^{co})+z_3]-v_3\\ {\dot{z}}_3=-k_5{z}_3+M_3{D}_2-M_3{v}_4& .\end{array}\right.$$

(59)

Step 1: According to (49)–(51), a Lyapunov function is designed:

$$V_3=\frac{1}{2}(v_3^2+v_4^2+v_5^2).$$

(60)

The derivative of $V_3$ is

$${\dot{V}}_3=v_3{\dot{v}}_3+v_4{\dot{v}}_4+v_5{\dot{v}}_5\le -k_3{v}_3^2-k_4{v}_4^2-k_5{v}_5^2\le -n_3{V}_3,$$

(61)

where $n_3=2\min \{k_3,k_4,k_5\}$,$k_3>0$,$k_4>0$,$k_5>0$.

Multiply both sides of (61) by $e^{n_{3}t}$, and then integrate both sides to obtain the following:

$$V_3(t)\le V_3(0)e^{-n_{3}t}\le V_3(0).$$

(62)

Additionally, then we can determine that $\left|v_i\right|\le B_3$, $i=3,4,5$, $B_3=\sqrt{2V_3(0)}$. This indicates that $v_3$, $v_4$, and $v_5$ converge to ${\Omega }_3:=\{v_i\in R,i=3,4,5\left|\left|v_i\right|\le B_3\right.\}$.

Step 2: Consider a Lyapunov function candidate:

$$V_4=\frac{1}{2}({\dot{v}}_c^2+{\dot{z}}_2^2+{\dot{z}}_3^2).$$

(63)

The derivation of equation (63) is provided by

$$\begin{array}{l}{\dot{V}}_4=(v_c^{}{\dot{v}}_c^{}+z_2^{}{\dot{z}}_2^{}+z_3^{}{\dot{z}}_3^{})\\ \le -(k_3-1)v_c^2-\left(k_4-M_3-1\right)z_2^2-\left(k_5-1.5M_3\right)z_3^2\\ +\frac{1}{2}{\left|l_1\vartheta \right|}^2+0.5M_3{\left|l_1\vartheta \right|}^2+\frac{1}{2}{M}_3{\overline{D}}_2^2+\left(\frac{1}{2}+\frac{1}{2}{M}_3\right)B_3^2\\ \le -n_4{V}_4+{\delta }_4& ,\end{array}$$

(64)

where $n_4=2\min \left\{k_3-1,k_4-M_3-1,k_5-1.5M_3\right\}$, $k_3-1>0$, $k_4-M_3-1>0$, and $k_5-1.5M_3>0$.

$${\delta }_4=\frac{1}{2}{\left|l_1\vartheta \right|}^2+0.5M_3{\left|l_1\vartheta \right|}^2+\frac{1}{2}{M}_3{\overline{D}}_2^2+\left(\frac{1}{2}+\frac{1}{2}{M}_3\right)B_3^2.$$

Multiply both sides of (64) by $e^{n_{4}t}$, and then integrate both sides to obtain the following:

$$V_4(t)\le V_4(0)e^{-n_{4}t}+{\delta }_4(1-e^{-n_{4}t})/n_4\le V_2(0)+{\delta }_4/n_4.$$

(65)

Additionally, then we can determine that $\left|v_c\right|\le B_4$, $\left|z_2\right|\le B_4$, $\left|z_3\right|\le B_4$, and $B_4=\sqrt{2V_4(0)+2{\delta }_4/n_4}$.

This indicates that $v_c$, $z_2$, and $z_3$ converge to a compact set: ${\Omega }_4:=\{v_c,z_2,z_3\in R,\left|\left|v_c\right|\le B_4,\left|z_2\right|\le B_4,\left|z_3\right|\le B_4\right.\}$. □

3.4. Asymptotic Stability Analysis

Theorem 4.

Assume that the trajectory tracking of the underactuated AUV is described by the differential model Equation (9) and Assumptions 1–2 are fulfilled. If the disturbances are estimated by Equations (12) and (13), the command integral filter is designed as Equation (23), and the control strategy is obtained from Equations (37) and (38) and Equations (55)–(57). The system (9) is asymptotically stable.

Proof. 

In order to analyze the asymptotic stability analysis of the system (9), a Lyapunov function is chosen:

$$U=V+W=\frac{1}{2}y{y}^{\mathrm{T}}+\frac{1}{2}{z}_c{z}_c^{\mathrm{T}}.$$

(66)

Additionally, the derivative of (66) is

$$\dot{U}=\frac{\mathsf{\partial }V}{\mathsf{\partial }y}f(y,z_s)+\frac{\mathsf{\partial }W}{\mathsf{\partial }{z}_c}g(y,z)+\frac{\mathsf{\partial }V}{\mathsf{\partial }y}[f(y,z)-f(y,z_s)]+\left[\frac{\mathsf{\partial }W}{\mathsf{\partial }y}-\frac{\mathsf{\partial }W}{\mathsf{\partial }{z}_c}\frac{\mathsf{\partial }{z}_c}{\mathsf{\partial }y}\right]f(y,z)-\frac{\mathsf{\partial }W}{\mathsf{\partial }{z}_s}\frac{\mathsf{\partial }{z}_s}{\mathsf{\partial }t}.$$

(67)

In (67), because $-\ddot{\beta }+{\ddot{\gamma }}_d$ is usually very small in practice, $-\dot{\beta }+{\dot{\gamma }}_d$ can be treated as a constant and $\frac{\mathsf{\partial }W}{\mathsf{\partial }{z}_s}\frac{\mathsf{\partial }{z}_s}{\mathsf{\partial }t}$ could be negligible.

$$\frac{\mathsf{\partial }V}{\mathsf{\partial }y}f(y,z_s)={\dot{V}}_2\le -n_2{V}_2+{\delta }_2=-n_2{‖y‖}^2+{\delta }_2,$$

(68)

$$\frac{\mathsf{\partial }W}{\mathsf{\partial }{z}_c}g(y,z)={\dot{V}}_4\le -n_4{V}_4+{\delta }_4=-n_4{‖z_c‖}^2+{\delta }_4,$$

(69)

$$\frac{\mathsf{\partial }V}{\mathsf{\partial }y}[f(y,z)-f(y,z_s)]\le \left|r_c\right|\left|{\gamma }_e\right|+M_1(-X_u+k_2)\left|v_c\right|\left|u_e\right|\le {\lambda }_1‖y‖‖z_c‖,$$

(70)

where ${\lambda }_1=\left[1+M_1(k_2-X_u)\right]/B_2{B}_4$.

$$\left[\frac{\mathsf{\partial }W}{\mathsf{\partial }y}-\frac{\mathsf{\partial }W}{\mathsf{\partial }{z}_c}\frac{\mathsf{\partial }{z}_s}{\mathsf{\partial }y}\right]f(y,z)=-k_1\left[\frac{a_{13}}{Y_v}u{v}_c-v_c\right](r_c-k_1{\gamma }_e)\le ({\lambda }_2‖y‖+{\lambda }_3‖z_c‖)‖z_c‖,$$

(71)

where ${\lambda }_2=k_1^2(1+\left|a_{13}/Y_v\right|{\left|u\right|}_{\max })/B_2{B}_4$,${\lambda }_3=(k_1+\left|a_{13}/Y_v\right|{\left|u\right|}_{\max })/B_4^2$.

Substituting conditions (68)~(71) into (67) can yield the inequality.

$$\begin{array}{ll}\dot{U}& =\frac{\mathsf{\partial }V}{\mathsf{\partial }y}f(y,z_s)+\frac{\mathsf{\partial }W}{\mathsf{\partial }{z}_c}g(y,z)+\frac{\mathsf{\partial }V}{\mathsf{\partial }y}[f(y,z)-f(y,z_s)]+\left[\frac{\mathsf{\partial }W}{\mathsf{\partial }y}-\frac{\mathsf{\partial }W}{\mathsf{\partial }{z}_c}\frac{\mathsf{\partial }{z}_c}{\mathsf{\partial }y}\right]f(y,z)\\ & \le -n_2{‖y‖}^2+{\delta }_2-n_4{‖z_c‖}^2+{\delta }_4+{\lambda }_1‖y‖‖z_c‖+({\lambda }_2‖y‖+{\lambda }_3‖z_c‖)‖z_c‖\\ & \le -\overline{n}U& .\end{array}$$

(72)

Inequality (72) is satisfied with $\overline{n}=\min \left\{n_2(1-{\theta }_1),n_4(1-{\theta }_2)\right\}$ for $\forall ‖y‖\ge {\mu }_1=\left(\frac{{\lambda }_1{B}_4+{\delta }_2/B_2}{n_2{\theta }_1}\right)>0$, $\forall ‖z_c‖\ge {\mu }_2=\left(\frac{{\lambda }_2{B}_2+{\lambda }_3{B}_4+{\delta }_4/B_4}{n_4{\theta }_2}\right)>0$, and $0<{\theta }_1<1$, $0<{\theta }_2<1$.

According to the converse Lyapunov theorem, the origin of the full system is asymptotically stable if inequality (72) holds. Namely, the compact sets ${\Omega }_2$ and ${\Omega }_4$ provide a mathematical bound on the design parameters. □

4. Simulation

In this section, some simulations are carried out using the proposed control method in two different cases to verify the effectiveness.

The AUV mathematical model parameters are set as shown in

Table 2. AUV mathematical model parameters.

Parameters	Values	Units
M1	0.25	${\mathrm{kg}}^{-1}$
M2	0.2	${\mathrm{kg}}^{-1}$
M3	0.2	${\mathrm{kg}}^{-1}$
Xu	20	$\mathrm{kg}/\mathrm{s}$
Yv	45	$\mathrm{kg}/\mathrm{s}$
Nr	30	$\mathrm{kg}\cdot {\mathrm{m}}^2/\mathrm{s}$
a12	20	$\mathrm{kg}\cdot {\mathrm{m}}^2$
a13	40	$\mathrm{kg}\cdot {\mathrm{m}}^2$
a23	60	$\mathrm{kg}\cdot {\mathrm{m}}^2$

.

The control method design parameters are selected as shown in

Table 3. Parameters in NESO and command integral filter.

Description	Symbol
Parameters of the NESO	$g=40$, $k_{th}=100$, $k_{11}^{eso}=100$, $k_{12}^{eso}=200$, $k_{21}^{eso}=100$, $k_{22}^{eso}=400$
Parameters of the command integral filter	$\xi =0.9$, ${\omega }_n=20$, $H_M=1000$, $H_R=500$
Parameters of the control law	$c_1=c_2=0.2$, $k_x=0.1$, $k_y=0.1$, $k_r=0.7$, $k_1=10$, $k_2=0.1$, $k_3=10$, $k_4=20$, $k_5=0.1$

, including parameters for NESO, the command integral filter, and control law.

Case 1: Smooth path tracking and the reference trajectory is chosen as $x_t=-100\sin (0.01t)$ and $y_t=-100\cos (0.01t)$. The system disturbances are chosen as follows:

$$d_1=0.2+0.3\sin (0.05t)+0.2M_1(X_{u}u+a_{23}vr)$$

$$d_2=0.2+0.1\sin (0.05t)+0.2M_3(N_{r}r+a_{23}uv),$$

which denote the external disturbances and system uncertainties.

In this part, the simulation mainly demonstrates the effectiveness of the designed controller concerning system disturbance, comparing the designed controller to the one without NESO.

Simulation results are presented in Figure 3, Figure 4, Figure 5 and Figure 6. Figure 3 plots the trajectory tracking results. Figure 4 plots trajectory tracking errors referring to the trajectory angle, surge speed, sway speed, and pitch rate. Figure 5 plots the results of disturbance observation, and Figure 6 plots the corresponding control input in the surge direction and in the yaw direction.

From the simulation results provided in Figure 3, Figure 4, Figure 5 and Figure 6, it can be seen that the NESO could estimate the system uncertainties effectively, and the AUV could track the desired trajectory in the presence of system uncertainties. Moreover, was found that the NESO-based adaptive controller performed better than the one without NESO in trajectory tracking. Therefore, the proposed method has good robustness in the face of system uncertainties.

Case 2: Piece-wise continuous path tracking and the reference trajectory are chosen as follows:

$$\left\{\begin{array}{l}{x}_t(t)=t,y_t(t)=t+50,(0\le t<50)\\ x_t(t)=t,y_t(t)=100,(50\le t<100)\\ x_t(t)=100\sin ((t-100)\pi /200)+100y_t(t)=100\cos ((t-100)\pi /200)+100,(100\le t<200)\\ x_t(t)=50\sin ((t-250)\pi /100)+250y_t(t)=-50\cos ((t-250)\pi /100)+250,(200\le t<250)\\ x_t(t)=250+1.5(t-250)y_t(t)=-50,(250\le t<300)\\ x_t(t)=25+t{y}_t(t)=t-350,(300\le t<400)\end{array}\right..$$

In this part, the simulation mainly demonstrates the effectiveness of the designed controller for nonlinear trajectory tracking with sudden changes, which compared the designed controller and the sliding mode control used in the literature [35].

The simulation results are presented in Figure 7 and Figure 8. Figure 7 shows that the actual trajectory could follow the reference trajectory with the position error in a small range. Figure 8 plots trajectory tracking errors referring to the trajectory angle, surge speed, sway speed, and pitch rate.

From the comparison results in Figure 7, it can be seen that the tracking ability of the designed controller is better than the sliding mode control method. Owing to the sudden trajectory change, the largest error of the designed controller is nearly 33.3% smaller than the sliding mode control. From Figure 8, it could be concluded that a slight tremble is unavoidable due to the sudden change in the speed at the join point of the piece-wise trajectory. Referring to Figure 7 and Figure 8, it is obvious that the designed controller is fast enough to make tracking errors small.

In short, the backstepping-based adaptive controller with NESO has superior performance compared to the other controllers, which verifies the validity of the proposed control strategy. The advantages of tracking accuracy and speed, as mentioned in the contributions of the paper, are confirmed.

5. Conclusions

This paper considered the horizontal trajectory tracking control problem of under-actuated AUVs subject to input saturation and system disturbances. In the guidance loop, the kinematic tracking error system was converted to the tracking error subsystem of the surge force and the yaw torque, which realizes system decoupling and simplifies the controller design. Based on the backstepping recursive design technique, an observe-based adaptive control approach was developed. The main advantages of the proposed control method are that it cannot only solve the input saturation and system disturbances but it also simplifies the control design process with two decoupled subsystems. The simulation results show that the proposed method has good robustness in the face of system uncertainties, and the error of the designed controller is nearly 33.3% smaller than the sliding mode control when meeting sudden trajectory changes. This demonstrates that the under-actuated AUV was capable of trajectory tracking even under unknown disturbances.

In future research, the proposed control method would be tested in actual simulation, which would provide far more convincing proof of the practical impact. Furthermore, optimization algorithms will be studied to select reasonable parameters that could promote controller performance.