Disturbance Attenuation Trajectory Tracking Control of Unmanned Surface Vessel Subject to Measurement Biases

Abstract

This article addresses trajectory tracking control of unmanned surface vessels (USVs) subject to position and velocity measurement biases. Unlike model uncertainties and external disturbances, measurement biases can lead to mismatched disturbances in system kinematics, rendering great difficulty to the USV control system design. To overcome this problem, a disturbance attenuation controller was recursively synthesized by incorporating two disturbance observers into the backstepping control design. The stability argument shows that all error signals in the closed-loop system can regulate to the small neighborhoods about the origin. The proposed controller has two remarkable features: (1) By adopting two disturbance observers to estimate the mismatched and matched lumped disturbances, the proposed controller is robust against model uncertainties and external disturbances and insensitive to measurement biases. (2) Meanwhile, the proposed controller is structurally simple and user friendly. Lastly, comparative simulations were conducted to validate the obtained results.

1. Introduction

Nowadays, unmanned surface vessels (USVs) play an important role in offshore transportation, oceanic monitoring, maritime rescue, resource exploration, and other missions [1,2,3]. Due to the complex operating environment, USVs are affected by external disturbances involving wind, waves, and currents. Additionally, USVs also suffer from model uncertainties, since the hydrodynamic coefficients cannot be accessed exactly. Given this situation, a highly reliable and strongly robust control system is urgently required for the trajectory tracking of USVs. Many advanced control methods, such as proportional-differential (PD) control [4], backstepping control [5,6,7,8,9,10], sliding mode control [11,12,13,14,15,16,17,18,19,20], and intelligent control [21,22,23,24,25,26,27,28], have been utilized for USV control system designs.

Measurement biases exist extensively because measurement devices on USVs are far from perfect. However, measurement biases have rarely been considered in previous relevant studies. Unlike model uncertainties and external disturbances, measurement biases can not only bring matched disturbances into the system dynamics, but also create mismatched disturbances in system kinematics. The mismatched disturbance cannot be directly compensated for, since it appears in a different channel from that of the control inputs, which renders great difficulty to the USV control system design. Many output feedback control approaches have been applied for trajectory tracking of USVs by integrating them with velocity filters or observers [29,30,31,32,33,34,35,36]. Since only position information is required for feedback, these controllers are still applicable when the USV suffers from velocity measurement biases. However, when the USV is affected by position and velocity measurement biases simultaneously, the tracking performance of these controllers cannot be guaranteed. In fact, the trajectory tracking control of USVs subject to position and velocity measurement biases remains an open problem.

It is noteworthy that disturbance observer-based control is an efficient and commonly used method to tackle mismatched disturbances [37,38,39,40,41,42]. The basic idea is adopting the disturbance observer to estimate the mismatched disturbance and then compensating the estimation in the feedforward loop. Benefiting from this design, disturbance observer-based control can acquire strong robustness against mismatched disturbances. Such an approach can also be applied to the trajectory tracking control of USVs subject to measurement biases. In [43], a robust finite-time controller was developed for trajectory tracking of USVs with measurement biases by introducing a dual disturbance observer-based structure.

Inspired by the aforementioned discussions, this article proposes a disturbance attenuation controller for the trajectory tracking control of USVs subject to position and velocity measurement biases. The main contributions are threefold:

• The proposed controller was recursively synthesized under the backstepping control framework. Particularly, two disturbance observers were incorporated to estimate the mismatched and matched lumped disturbances. In this way, the proposed controller is robust against model uncertainties and external disturbances and insensitive to measurement biases.
• Meanwhile, the proposed controller is structurally simple and user friendly. Only four parameters need to be adjusted to achieve satisfactory tracking performance.
• The uniform ultimate boundedness of the closed-loop system is strictly proven. The stability argument shows that all error signals under the proposed controller can regulate to the small neighborhoods about the origin.

The remainder of this article is organized as follows. Section 2 describes the problem. Section 3 presents the control design and stability argument. Section 4 provides the simulation results. Finally, Section 5 concludes this research. Throughout this article, ${ℝ}^n$ and ${ℝ}^{m\times n}$ stand for the sets of $n\times 1$ vectors and $m\times n$ matrices, respectively.

2. Problem Description

Consider the planar motion of a fully actuated USV, as depicted in Figure 1. Referring to [44], the kinematics and dynamics of the USV can be formulated as:

$$\dot{\eta }=R\left(\eta \right)\upsilon ,$$

(1)

$$M\dot{\upsilon }+C\left(\upsilon \right)\upsilon +D\left(\upsilon \right)\upsilon =\mathrm{sat}\left(\tau \right)+d,$$

(2)

where $\eta ={\left[x,y,\psi \right]}^{\mathrm{T}}\in {ℝ}^3$ and $\upsilon ={\left[u,v,r\right]}^{\mathrm{T}}\in {ℝ}^3$ are the generalized position and velocity of the USV. $R\left(\eta \right)\in {ℝ}^{3\times 3}$ is the transformation matrix, denoted as:

$$R\left(\eta \right)=\left[\begin{array}{ccc}\cos \psi & -\sin \psi & 0\\ \sin \psi & \cos \psi & 0\\ 0& 0& 1\end{array}\right].$$

(3)

$M\in {ℝ}^{3\times 3}$ is the inertia matrix, $C\left(\upsilon \right)\in {ℝ}^{3\times 3}$ is the Coriolis and centripetal matrix, $D\left(\upsilon \right)\in {ℝ}^{3\times 3}$ is the damping matrix, $\tau \in {ℝ}^3$ is the control inputs, and $d\in {ℝ}^3$ is the external disturbances. Due to model uncertainties, the matrices $M$, $C\left(\upsilon \right)$, and $D\left(\upsilon \right)$ can be rewritten as:

$$M=M_0+M_{\Delta },\hspace{0.17em}\hspace{0.17em}C\left(\upsilon \right)=C_0\left(\upsilon \right)+C_{\Delta }\left(\upsilon \right),\hspace{0.17em}\hspace{0.17em}D\left(\upsilon \right)=D_0\left(\upsilon \right)+D_{\Delta }\left(\upsilon \right),$$

(4)

where $M_0$, $C_0\left(\upsilon \right)$, and $D_0\left(\upsilon \right)$ are the nominal parts and $M_{\Delta }$, $C_{\Delta }\left(\upsilon \right)$, and $D_{\Delta }\left(\upsilon \right)$ are the unknown parts. The input saturation is taken into account as $\mathrm{sat}\left(\tau \right)={\left[\mathrm{sat}\left({\tau }_1\right),\mathrm{sat}\left({\tau }_2\right),\mathrm{sat}\left({\tau }_3\right)\right]}^{\mathrm{T}}$, where $\mathrm{sat}\left({\tau }_i\right)$ is defined as:

$$\mathrm{sat}\left({\tau }_i\right)=\{\begin{array}{l}{\tau }_{\mathrm{m}},\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\tau }_i\ge {\tau }_{\mathrm{m}},\hfill \\ {\tau }_i,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}-{\tau }_{\mathrm{m}}\le {\tau }_i<{\tau }_{\mathrm{m}},\hfill \\ -{\tau }_{\mathrm{m}},\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\tau }_i<-{\tau }_{\mathrm{m}},\hfill \end{array}$$

(5)

where ${\tau }_{\mathrm{m}}$ is the maximum acceptable control value. Then, the saturated control inputs can be rewritten as:

$$\mathrm{sat}\left(\tau \right)=\tau +{\tau }_{\Delta },$$

(6)

where ${\tau }_{\Delta }$ is the control deviation. Moreover, the position and velocity measurement biases are also involved. The measured position and velocity can be expressed as:

$${\eta }_c=\eta +{\zeta }_1,$$

(7)

$${\upsilon }_c=\upsilon +{\zeta }_2,$$

(8)

where ${\zeta }_1\in {ℝ}^3$ and ${\zeta }_2\in {ℝ}^3$ are the position and velocity measurement biases, respectively. Combining (1) and (7), we have:

$$\begin{array}{ll}{\dot{\eta }}_c& =R\left(\eta \right)\upsilon +{\dot{\zeta }}_1\\ & =R\left({\eta }_c\right){\upsilon }_c+{\dot{\zeta }}_1-R\left({\eta }_c\right){\upsilon }_c+R\left({\eta }_c-{\zeta }_1\right)\left({\upsilon }_c-{\zeta }_2\right)\\ & =R\left({\eta }_c\right){\upsilon }_c+{\delta }_1,\end{array}$$

(9)

where ${\delta }_1$ is the mismatched lumped disturbance, denoted as:

$${\delta }_1={\dot{\zeta }}_1-R\left({\eta }_c\right){\upsilon }_c+R\left({\eta }_c-{\zeta }_1\right)\left({\upsilon }_c-{\zeta }_2\right).$$

(10)

Combining (2) and (8), we have:

$$\begin{array}{cc}\hfill M_0{\dot{\upsilon }}_c+C_0\left({\upsilon }_c\right){\upsilon }_c+D_0\left({\upsilon }_c\right){\upsilon }_c=& M_0{\dot{\upsilon }}_c+C_0\left({\upsilon }_c\right){\upsilon }_c+D_0\left({\upsilon }_c\right){\upsilon }_c-M\left({\dot{\upsilon }}_c-{\dot{\zeta }}_2\right)\hfill \\ \hfill & -C\left({\upsilon }_c-{\zeta }_2\right)\left({\upsilon }_c-{\zeta }_2\right)-D\left({\upsilon }_c-{\zeta }_2\right)\left({\upsilon }_c-{\zeta }_2\right)\hfill \\ \hfill & +\tau +{\tau }_{\Delta }+d\hfill \\ \hfill =& \tau +{\delta }_2,\hfill \end{array}$$

(11)

where ${\delta }_2$ is the matched lumped disturbance, denoted as:

$$\begin{array}{cc}{\delta }_2=\hfill & M_0{\dot{\upsilon }}_c+C_0\left({\upsilon }_c\right){\upsilon }_c+D_0\left({\upsilon }_c\right){\upsilon }_c-M\left({\dot{\upsilon }}_c-{\dot{\zeta }}_2\right)\hfill \\ \hfill & -C\left({\upsilon }_c-{\zeta }_2\right)\left({\upsilon }_c-{\zeta }_2\right)-D\left({\upsilon }_c-{\zeta }_2\right)\left({\upsilon }_c-{\zeta }_2\right)+{\tau }_{\Delta }+d.\hfill \end{array}$$

(12)

Thus, the kinematics and dynamics of the USV based on the measured states ${\eta }_c$ and ${\upsilon }_c$ can be expressed as:

$${\dot{\eta }}_c=R\left({\eta }_c\right){\upsilon }_c+{\delta }_1,$$

(13)

$$M_0{\dot{\upsilon }}_c+C_0\left({\upsilon }_c\right){\upsilon }_c+D_0\left({\upsilon }_c\right){\upsilon }_c=\tau +{\delta }_2.$$

(14)

Remark 1.

It should be noticed that the lumped disturbance ${\delta }_1$ induced by measurement biases belongs to mismatched disturbances. It appears in a different channel from that of the control inputs $\tau$ and thus cannot be directly compensated through feedback. Unlike model uncertainties and external disturbances, the measurement biases ${\zeta }_1$ and ${\zeta }_2$ can not only bring the matched disturbance into the system dynamics, but also offer the mismatched disturbance in the system kinematics. This renders great difficulty to the USV control system design.

Let ${\eta }_{\mathrm{d}}$ and ${\upsilon }_{\mathrm{d}}$ be the desired position and velocity of the USV. Define the measured position and velocity tracking errors ${\eta }_e={\eta }_c-{\eta }_{\mathrm{d}}$ and ${\upsilon }_e={\upsilon }_c-{\upsilon }_{\mathrm{d}}$ and the actual position and velocity tracking errors ${{\eta }^{\prime }}_e=\eta -{\eta }_{\mathrm{d}}$ and ${{\upsilon }^{\prime }}_e=\upsilon -{\upsilon }_{\mathrm{d}}$, respectively. The goal is to design a controller $\tau$ to track the desired position and velocity of the USV even in the presence of model uncertainties, external disturbances, and measurement biases. The following assumptions are made:

Assumption 1.

The model uncertainties $\Delta M$, $\Delta C\left(\upsilon \right)$, and $\Delta D\left(\upsilon \right)$ are bounded.

Assumption 2.

The external disturbance $d$ and its first-order time derivative are bounded.

Assumption 3.

The measurement biases ${\zeta }_1$ and ${\zeta }_2$ and their first two-order time derivatives are bounded.

3. Control Design and Stability Argument

3.1. Control Design

Under the backstepping control framework, introduce the following error signals:

$$x_1={\eta }_c-{\eta }_{\mathrm{d}},\hspace{0.17em}\hspace{0.17em}{x}_2={\upsilon }_c-\alpha ,$$

(15)

where $\alpha$ is the virtual control signal designed later. Then, the control design procedure contains the following two steps.

Step 1: Virtual control signal design. Evaluating the time derivative of $x_1$ leads to:

$${\dot{x}}_1=R\left({\eta }_c\right)\left(x_2+\alpha \right)-{\dot{\eta }}_{\mathrm{d}}+{\delta }_1,$$

(16)

Then, the virtual control signal is designed as:

$$\alpha =-R^{-1}\left({\eta }_c\right)\left(k_1{x}_1-{\dot{\eta }}_{\mathrm{d}}+{\widehat{\delta }}_1\right),$$

(17)

where $k_1>0$ and ${\widehat{\delta }}_1$ is the estimation of the mismatched lumped disturbance ${\delta }_1$. ${\widehat{\delta }}_1$ can be generated through the following disturbance observer:

$$\{\begin{array}{l}{\widehat{\delta }}_1={\sigma }_1+h_1{x}_1,\\ {\dot{\sigma }}_1=-h_1{\sigma }_1-h_1\left(R\left({\eta }_c\right)\left(x_2+\alpha \right)-{\dot{\eta }}_{\mathrm{d}}+h_1{x}_1\right),\end{array}$$

(18)

where $h_1>0$. Substituting (17) into (16) gives:

$${\dot{x}}_1=R\left({\eta }_c\right)x_2-k_1{x}_1+{\tilde{\delta }}_1,$$

(19)

where ${\tilde{\delta }}_1={\delta }_1-{\widehat{\delta }}_1$ is the estimation error of the mismatched lumped disturbance ${\delta }_1$. Evaluating the time derivative of ${\tilde{\delta }}_1$ and substituting (16) and (18) into it, we have:

$$\begin{array}{ll}{\dot{\tilde{\delta }}}_1& ={\dot{\delta }}_1-{\dot{\sigma }}_1-h_1{\dot{x}}_1\\ & ={\dot{\delta }}_1+h_1\left({\widehat{\delta }}_1-h_1{x}_1\right)+h_1\left(R\left({\eta }_c\right)\left(x_2+\alpha \right)+h_1{x}_1-{\dot{\eta }}_{\mathrm{d}}\right)-h_1\left(R\left({\eta }_c\right)\left(x_2+\alpha \right)+{\delta }_1-{\dot{\eta }}_{\mathrm{d}}\right)\\ & ={\dot{\delta }}_1-h_1{\tilde{\delta }}_1.\end{array}$$

(20)

Choose the following Lyapunov function:

$$V_1=\frac{1}{2}{x}_1^{\mathrm{T}}{x}_1+\frac{1}{2}{\tilde{\delta }}_1^{\mathrm{T}}{\tilde{\delta }}_1.$$

(21)

Evaluating the time derivative of $V_1$ and substituting (19) and (20) into it yields:

$$\begin{array}{ll}{\dot{V}}_1& =x_1^{\mathrm{T}}{\dot{x}}_1+{\tilde{\delta }}_1^{\mathrm{T}}{\dot{\tilde{\delta }}}_1\\ & =-k_1{x}_1^{\mathrm{T}}{x}_1+x_1^{\mathrm{T}}R\left({\eta }_c\right)x_2+x_1^{\mathrm{T}}{\tilde{\delta }}_1+{\tilde{\delta }}_1^{\mathrm{T}}{\dot{\delta }}_1-h_1{\tilde{\delta }}_1^{\mathrm{T}}{\tilde{\delta }}_1.\end{array}$$

(22)

Step 2: Real control signal design. Evaluating the time derivative of $x_2$ leads to:

$$\begin{array}{ll}{\dot{x}}_2& =M_0^{-1}\left(-C_0\left({\upsilon }_c\right){\upsilon }_c-D_0\left({\upsilon }_c\right){\upsilon }_c+\tau +{\delta }_2\right)-\dot{\alpha }\\ & =M_0^{-1}\tau +N+{\delta }_c,\end{array}$$

(23)

where $N$ is the model-based nonlinear term and ${\delta }_c$ is the matched lumped disturbance, denoted as:

$$\begin{array}{ll}N=& M_0^{-1}\left(-C_0\left({\upsilon }_c\right){\upsilon }_c-D_0\left({\upsilon }_c\right){\upsilon }_c\right)+{\dot{R}}^{-1}\left({\eta }_c\right)\left(k_1{x}_1+{\widehat{\delta }}_1-{\dot{\eta }}_{\mathrm{d}}\right)\\ & +\left(k_1+h_1\right)\left(x_2+\alpha -R^{-1}\left({\eta }_c\right){\dot{\eta }}_{\mathrm{d}}\right)+R^{-1}\left({\eta }_c\right){\dot{\sigma }}_1-R^{-1}\left({\eta }_c\right){\ddot{\eta }}_{\mathrm{d}},\end{array}$$

(24)

$${\delta }_c=M_0^{-1}{\delta }_2+\left(k_1+h_1\right)R^{-1}\left({\eta }_c\right){\delta }_1.$$

(25)

Then, the real control signal is designed as:

$$\tau =M_0\left(-k_2{x}_2-R^{\mathrm{T}}\left({\eta }_c\right)x_1-N-{\widehat{\delta }}_c\right),$$

(26)

where $k_2>0$ and ${\widehat{\delta }}_c$ is the estimation of the mismatched lumped disturbance ${\delta }_c$. ${\widehat{\delta }}_c$ can be generated through the following disturbance observer:

$$\{\begin{array}{l}{\widehat{\delta }}_c={\sigma }_2+h_2{x}_2,\\ {\dot{\sigma }}_2=-h_2{\sigma }_2-h_2\left(M_0^{-1}\tau +N+h_2{x}_2\right),\end{array}$$

(27)

where $h_2$ is a positive design parameter. Substituting (26) into (23) gives:

$${\dot{x}}_2=-k_2{x}_2-R^{\mathrm{T}}\left({\eta }_c\right)x_1+{\tilde{\delta }}_c,$$

(28)

where ${\tilde{\delta }}_c={\delta }_c-{\widehat{\delta }}_c$ is the estimation error of the matched lumped disturbance ${\delta }_c$. Evaluating the time derivative of ${\tilde{\delta }}_c$ and substituting (23) and (27) into it, we have:

$$\begin{array}{ll}{\dot{\tilde{\delta }}}_c& ={\dot{\delta }}_c-{\dot{\sigma }}_2-h_2{\dot{x}}_2\\ & ={\dot{\delta }}_c+h_2\left({\widehat{\delta }}_c-h_2{x}_2\right)+h_2\left(M_0^{-1}\tau +N+h_2{x}_2\right)-h_2\left(M_0^{-1}\tau +N+{\delta }_c\right)\\ & ={\dot{\delta }}_c-h_2{\tilde{\delta }}_c.\end{array}$$

(29)

Choose the following Lyapunov function:

$$V_2=V_1+\frac{1}{2}{x}_2^{\mathrm{T}}{x}_2+\frac{1}{2}{\tilde{\delta }}_c^{\mathrm{T}}{\tilde{\delta }}_c.$$

(30)

Evaluating the time derivative of $V_2$ and substituting (22), (28), and (29) into it yields:

$$\begin{array}{ll}{\dot{V}}_2& ={\dot{V}}_1+x_2^{\mathrm{T}}{\dot{x}}_2+{\tilde{\delta }}_c^{\mathrm{T}}{\dot{\tilde{\delta }}}_c\\ & ={\dot{V}}_1-k_2{x}_2^{\mathrm{T}}{x}_2-x_2^{\mathrm{T}}{R}^{\mathrm{T}}\left({\eta }_c\right)x_1+x_2^{\mathrm{T}}{\tilde{\delta }}_c+{\tilde{\delta }}_c^{\mathrm{T}}{\dot{\delta }}_c-h_2{\tilde{\delta }}_c^{\mathrm{T}}{\tilde{\delta }}_c\\ & =-k_1{x}_1^{\mathrm{T}}{x}_1-k_2{x}_2^{\mathrm{T}}{x}_2+x_1^{\mathrm{T}}{\tilde{\delta }}_1+{\tilde{\delta }}_1^{\mathrm{T}}{\dot{\delta }}_1-h_1{\tilde{\delta }}_1^{\mathrm{T}}{\tilde{\delta }}_1+x_2^{\mathrm{T}}{\tilde{\delta }}_c+{\tilde{\delta }}_c^{\mathrm{T}}{\dot{\delta }}_c-h_2{\tilde{\delta }}_c^{\mathrm{T}}{\tilde{\delta }}_c.\end{array}$$

(31)

3.2. Stability Argument

After the above preparations, we can attain the main theorem as follows:

Theorem 1.

For the USV system (1) and (2), when the virtual control signal (17), the real control signal (26), and the two disturbance observers (18) and (27) are employed, the closed-loop system is uniformly ultimately bounded, and all error signals can regulate to the small neighborhoods about the origin.

Proof. 

From Assumptions 1–3, it follows that the time derivatives of the mismatched and matched lumped disturbances ${\dot{\delta }}_1$ and ${\dot{\delta }}_c$ are bounded with $\Vert {\dot{\delta }}_1\Vert \le {\overline{b}}_1$ and $\Vert {\dot{\delta }}_c\Vert \le {\overline{b}}_c$, where ${\overline{b}}_1>0$ and ${\overline{b}}_c>0$ are unknown constants. Construct the following inequalities:

$$x_1^{\mathrm{T}}{\tilde{\delta }}_1\le \frac{k_1}{2}{\Vert x_1\Vert }^2+\frac{1}{2k_1}{\Vert {\tilde{\delta }}_1\Vert }^2,$$

(32)

$${\tilde{\delta }}_1^{\mathrm{T}}{\dot{\delta }}_1\le \frac{h_1}{2}{\Vert {\tilde{\delta }}_1\Vert }^2+\frac{1}{2h_1}{\overline{b}}_1^2,$$

(33)

$$x_2^{\mathrm{T}}{\tilde{\delta }}_c\le \frac{k_2}{2}{\Vert x_2\Vert }^2+\frac{1}{2k_2}{\Vert {\tilde{\delta }}_c\Vert }^2,$$

(34)

$${\tilde{\delta }}_c^{\mathrm{T}}{\dot{\delta }}_c\le \frac{h_2}{2}{\Vert {\tilde{\delta }}_c\Vert }^2+\frac{1}{2h_2}{\overline{b}}_c^2.$$

(35)

Substituting (32)–(36) into (31), we have:

$$\begin{array}{ll}{\dot{V}}_2& \le -\frac{k_1}{2}{\Vert x_1\Vert }^2-\frac{k_2}{2}{\Vert x_2\Vert }^2-\left(\frac{h_1}{2}-\frac{1}{2k_1}\right){\Vert {\tilde{\delta }}_1\Vert }^2-\left(\frac{h_2}{2}-\frac{1}{2k_2}\right){\Vert {\tilde{\delta }}_c\Vert }^2+\frac{1}{2h_1}{\overline{b}}_1^2+\frac{1}{2h_2}{\overline{b}}_c^2\\ & \le -\rho V_2+\vartheta ,\end{array}$$

(36)

where $\rho$ and $\vartheta$ are defined as:

$$\rho =\min \left\{\frac{k_1}{2},\frac{k_2}{2},\frac{h_1}{2}-\frac{1}{2k_1},\frac{h_2}{2}-\frac{1}{2k_2}\right\},\hspace{0.17em}\hspace{0.17em}\vartheta =\frac{1}{2h_1}{\overline{b}}_1^2+\frac{1}{2h_2}{\overline{b}}_c^2.$$

(37)

To ensure $\rho >0$, the parameters should be selected satisfying $h_1{k}_1>1$ and $h_2{k}_2>1$. Solving (36), we further have:

$$V_2\le \frac{\vartheta }{\rho }+\left(V_2\left(0\right)-\frac{\vartheta }{\rho }\right){\mathrm{e}}^{-\rho t}.$$

(38)

Thus, $V_2$ is uniformly ultimately bounded. Combining with the definition of $V_2$, the error signals $x_1$, $x_2$, ${\tilde{\delta }}_1$, and ${\tilde{\delta }}_c$ can regulate to the small neighborhoods about the origin. Subsequently, it follows that ${\eta }_e$, ${\upsilon }_e$, ${{\eta }^{\prime }}_e$, and ${{\upsilon }^{\prime }}_e$ can also regulate to the small neighborhoods about the origin. The proof is finished. □

Remark 2.

The proposed disturbance attenuation controller was exploited by integrating two disturbance observers with the backstepping control design. To facilitate the readers, the structure of the proposed controller is presented in Figure 2.

Remark 3.

The proposed controller is structurally simple and user friendly. Only four parameters need to be adjusted by the users. They are $k_1$ for the virtual control signal (17), $k_2$ for the real control signal (26), $h_1$ for the disturbance observer (18), and $h_2$ for the disturbance observer (27). Specifically, large $k_1$, $k_2$, $h_1$, and $h_2$ can lead to a relatively fast convergence rate and high tracking accuracy. However, large $k_1$ and $k_2$ can also result in relatively large control inputs, whereas large $h_1$ and $h_2$ can in turn cause a relatively poor transient response. Altogether, these parameters should be properly adjusted to achieve satisfactory tracking performance.

Remark 4.

When we do not utilize the two disturbance observers (18) and (27), the proposed controller reduces to the traditional backstepping controller in [9], in which the virtual and real control signals are designed as (39) and (40), respectively.

$$\alpha =-R^{-1}\left({\eta }_c\right)\left(k_1{x}_1-{\dot{\eta }}_{\mathrm{d}}\right),$$

(39)

$$\tau =M_0\left(-k_2{x}_2-R^{\mathrm{T}}\left({\eta }_c\right)x_1-N\right).$$

(40)

Note that the traditional backstepping controller (40) is unable to guarantee the tracking performance of the USV in the presence of model uncertainties, external disturbances, and measurement biases. Alternatively, the proposed controller (26) can solve this problem well by adopting two disturbance observers to estimate the mismatched and matched lumped disturbances.

4. Simulation Results

In this section, comparative simulations are conducted in the MATLAB/Simulink environment to validate the obtained results. A benchmark USV model named CyberShip II [7] was used for the simulations. The nominal matrices of the USV are given as:

$$M_0=\left[\begin{array}{ccc}25.8000& 0& 0\\ 0& 24.6612& 1.0948\\ 0& 1.0948& 2.7600\end{array}\right],$$

(41)

$$C_0\left(\upsilon \right)=\left[\begin{array}{ccc}0& 0& -24.6612v-1.0948r\\ 0& 0& 25.8u\\ 24.6612v+1.0948r& -25.8u& 0\end{array}\right],$$

(42)

$$D_0\left(\upsilon \right)=\left[\begin{array}{ccc}0.7225+1.3274\left|u\right|+5.8664u^2& 0& 0\\ 0& 0.8612+36.2823\left|v\right|+8.05\left|r\right|& 0.845\left|v\right|+3.45\left|r\right|-0.1079\\ 0& -0.1052-5.0437\left|v\right|-0.13\left|r\right|& 1.9-0.08\left|v\right|+0.75\left|r\right|\end{array}\right].$$

(43)

The model uncertainties are chosen as $\Delta M=0.1M_0$, $\Delta C\left(\upsilon \right)=0.1C_0\left(\upsilon \right)$, and $\Delta D\left(\upsilon \right)=0.1D_0\left(\upsilon \right)$. The external disturbances are chosen as $d={\left[5.4\sin \left(0.3t\right),3.2\cos \left(0.6t\right),0.8\sin \left(0.2t\right)\right]}^{\mathrm{T}}$. The position and velocity measurement biases are chosen as ${\zeta }_1={\zeta }_2={\left[0.001\sin \left(0.3t\right),0.001\sin \left(0.2t\right),0.001\sin \left(0.1t\right)\right]}^{\mathrm{T}}$. The maximum acceptable control value is set as ${\tau }_{\mathrm{m}}=50$.

In simulations, two typical cases are considered. In Case 1, the USV is expected to track a circular trajectory, whereas in Case 2, the USV is expected to track a lemniscate trajectory.

4.1. Circular Trajectory Tracking

In Case 1, the USV is expected to track a circular trajectory. The desired trajectory was set as ${\eta }_{\mathrm{d}}={\left[2\sin \left(0.1t\right),2\cos \left(0.1t\right),0.1t\right]}^{\mathrm{T}}$. The initial position and velocity of the USV were chosen as ${\eta }_0={\left[1,1,\pi /4\right]}^{\mathrm{T}}$ and ${\upsilon }_0={\left[0,0,0\right]}^{\mathrm{T}}$.

Besides the proposed disturbance attenuation controller (26), the traditional backstepping controller (39), which was originally designed in [9], was also used for comparisons. The parameters of the proposed controller were selected as $k_1=1$, $k_2=6$, $h_1=4$, and $h_2=4$. For fair comparisons, the parameters that both controllers share were selected with the same values. Keeping this in mind, the parameters of the traditional backstepping controller were selected as $k_1=1$ and $k_2=6$.

The simulation results of Case 1 are provided in Figure 3, Figure 4, Figure 5, Figure 6, Figure 7, Figure 8, Figure 9 and Figure 10. Figure 3 shows the planar trajectory tracking of the USV. The time profiles of the position and velocity tracking are given in Figure 4 and Figure 5, respectively. It is desirable that both controllers can accomplish the trajectory tracking task successfully. In particular, Figure 6 and Figure 7 present the time profiles of the position and velocity tracking errors, respectively. Obviously, the position and velocity steady-state errors under the traditional backstepping controller were much larger than those under the proposed controller. Therefore, the proposed controller can acquire superior tracking performance compared to the traditional backstepping controller. Owing to the utilization of two disturbance observers, the proposed controller is robust against model uncertainties and external disturbances and insensitive to measurement biases. The time profile of the control inputs is depicted in Figure 8. The control inputs under both controllers can meet the input saturation constraints. Figure 9 shows the mismatched lumped disturbance and its estimation, and Figure 10 shows the matched lumped disturbance and its estimation. It is clear that the two disturbance observers are efficient in estimating the mismatched and matched lumped disturbances.

4.2. Lemniscate Trajectory Tracking

In Case 2, the USV was expected to track a lemniscate trajectory. The desired trajectory was set as ${\eta }_{\mathrm{d}}={\left[1.5\sin \left(0.2t\right),2\cos \left(0.1t\right),0.1t\right]}^{\mathrm{T}}$. The initial position and velocity of the USV were chosen the same as those in Case 1. Moreover, the parameters of the proposed controller (26) and the traditional backstepping controller (39) were also selected the same as those in Case 1.

The simulation results of Case 2 are presented in Figure 11, Figure 12, Figure 13, Figure 14, Figure 15, Figure 16, Figure 17 and Figure 18. Likewise, the proposed controller can still fulfil the high-performance trajectory tracking in Case 2. Note that circle and lemniscate are two representative trajectories for the USV to be tracked in practice. The simulation results of Cases 1 and 2 indicate that the proposed controller is universal for tracking different types of reference trajectories and extremely suitable for practical engineering.

5. Conclusions

This article addresses the trajectory tracking control of USVs subject to position and velocity measurement biases by proposing a disturbance attenuation controller. The proposed controller was designed by integrating two disturbance observers under a backstepping control framework. The proposed controller can guarantee all error signals in the closed-loop system and regulate to the small neighborhoods about the origin. Lastly, the obtained results were validated through comparative simulations. In fact, relatively few existing studies have focused on the trajectory tracking control of USVs with the consideration of model uncertainties, external disturbances, and measurement biases simultaneously. The current work extends the relevant observer and controller designs in this field. Future research will focus on extending the proposed controller to the finite-time or fixed-time convergence scenarios.