Fixed-Time Trajectory Tracking Control of Unmanned Surface Vessels with Prescribed Performance Constraints

Abstract

This study proposes a fixed-time precision tracking control scheme for unmanned surface vessels with complex disturbances and unknown system dynamics. The control scheme is based on a nonsingular fast terminal sliding mode and includes an adaptive fixed-time integral sliding mode lumped disturbance observer to precisely observe and compensate for lumped unknowns. To ensure system performance under both transient and steady-state conditions, a prescribed performance function was utilized to enable the trajectory tracking error to converge to a predetermined range. The proposed control scheme, called prescribed performance fixed-time precision tracking (PPFTPT), was designed to achieve precision tracking of a predetermined trajectory within a fixed time. The stability of the closed-loop system was analyzed using the Lyapunov stability theory. Simulation results using the Cybership II model confirmed the feasibility and superiority of the proposed PPFTPT scheme.

1. Introduction

Unmanned surface vessels (USVs) are an emerging technology equipment used for marine resource surveys, maritime defense, and other surface operation tasks. They play a vital role in various fields, including the military, people’s livelihoods, and scientific research [1,2]. Accurate trajectory tracking is crucial for reliable and autonomous USV operation in different scenarios which therefore has significant research value and practical significance. However, designing highly precise tracking control systems for USVs is challenging due to the complexity and variability of the marine environments in which they operate. Common issues include environmental disturbances and unidentified dynamics, which can negatively impact the performance of USV tracking control systems [3,4].

Unmanned systems include traditional trajectory tracking control methods such as proportional–integral–derivative (PID) control [5,6], backstepping control [7], sliding-mode control [8,9,10], model predictive control [11], and reinforcement learning [12]. Compared to traditional control methods, finite-time control [13,14] offers several advantages, such as quick convergence, high control accuracy, and robustness. It enables the system’s state to converge in a limited time and is widely used in automated unmanned system trajectory tracking control. For example, Wang et al. [15] proposed a finite-time strategy to accurately track the trajectory of USVs disturbed by complex marine environments by combining nonsingular terminal sliding mode and finite-time perturbation observer techniques. This scheme ensures that both perturbation observation and trajectory tracking errors converge to zero in a finite time. Similarly, Zhang et al. [16] recommended a finite-time trajectory tracking control method based on backstepping technique and finite-time control. To estimate unknown external disturbances and the unreconstructed portion of the neural network, they employed a multivariate sliding mode finite-time perturbation observer that incorporated a neural network for reconstructing dynamic uncertainty. An adaptive finite-time trajectory tracking control rule was then created, and the rigorous theory demonstrated that the states of a closed-loop trajectory tracking system may be constrained. Wang et al. [17] proposed a reinforcement learning-based finite-time trajectory tracking control scheme for USVs by combining a reinforcement learning mechanism with finite-time control theory. The proposed scheme was theoretically proven to ensure the semiglobal real finite-time stability of a closed-loop USV system, and the tracking error converges in a finite time to an arbitrary small neighborhood of the origin. Nevertheless, the highest limit of the state-convergence time in finite-time control depends on the initial state of the system. When the initial state is far from the equilibrium point, the convergence time may tend toward infinity, limiting its practical use.

To address the shortcomings of finite-time control schemes, Polyakov [18] proposed a fixed-time convergence approach. The fixed-time method has the benefit of an independent upper bound that is not influenced by the system’s initial state and a lower bound that is constrained by the time required for the system to converge to the equilibrium point from any initial state. Yao et al. [19] proposed a fixed-time sliding mode strategy based on the double-limit chi-square theory to achieve USV trajectory tracking within a fixed time, although it requires a known upper bound on the perturbation. Zhang et al. [20] combined a fixed-time observer to achieve fixed-time tracking control of USVs under unknown velocity and complex disturbances. Zhang et al. [21] proposed a nonsingular fixed-time terminal sliding mode control scheme using a fixed-time disturbance observer to enable USVs to track all globally stable reference trajectories in a fixed time.

To ensure system safety, USVs should not only achieve high-accuracy steady-state performance when managing challenging and complicated tasks, but also ensure that their performance is constrained during transient events. One approach to addressing this problem is predefined performance control, as suggested by Bechlioulis and Rovithakis [22]. This technique transforms errors to ensure system performance under both transient and steady-state conditions. An adaptive predefined performance dynamic surface control technique was proposed by Shen et al. [23] that uses a non-logarithmic function to ensure that the tracking error converges to a region close to the origin. A dynamic surface approach is then applied to prevent control input saturation. Ma et al. [24] proposed a USV finite-time tracking model with an asymmetric tracking error constraint by combining prescribed performance control with finite-time control. However, the convergence performance of this scheme was dependent on the initial state.

This study presents a prescribed performance fixed-time trajectory tracking approach to achieve precise USV tracking control with lumped uncertainty, constrained transient and steady-state responses, and fixed-time convergence. Compared to conventional USV tracking control approaches, this study introduces the following advancements:

(1) An adaptive fixed-time integral sliding mode lumped disturbance observer (AFISMLDO) was developed to achieve fast and accurate reconstruction and compensation of lumped nonlinearities containing external perturbations and unknown system dynamics. The proposed observer converges faster than that reported by Wang et al. [15], and the convergence does not depend on the initial state. Moreover, in contrast to the results of Yao et al. [19], the adaptive strategy proposed in this study accounts for external disturbances and ensures the boundedness of online estimated parameters without requiring a known upper limit, achieving quick convergence in a fixed time without being affected by the system’s initial state.

(2) A prescribed performance fixed-time precision tracking (PPFTPT) control scheme was developed by integrating the prescribed performance control, fixed-time method, and nonsingular terminal sliding mode. In contrast to the results of Zhang et al. [21], this controller ensures that the transient and steady-state responses of the closed-loop system always remain within the predefined region. Additionally, the system achieves fixed-time convergence independent of the initial condition.

(3) A nonsingular sliding mode fixed-time convergence law was developed to enhance the convergence performance of the sliding mode variables and prevent singularities. Finally, the stability of the closed-loop system was analyzed using the Lyapunov stability theory.

The remainder of this paper is organized as follows. Section 2 presents the problem formulation and preliminary findings. Section 3 presents the key findings of the study. Section 4 details the proposed control scheme and its verification through a simulation experiment. Finally, Section 5 presents the conclusions.

2. Problem Formulation

2.1. Lemmas and Assumptions

Lemma 1 

([25]). If ${\omega }_1,{\omega }_2,\cdots ,{\omega }_n\ge 0$, then

$$\left\{\begin{array}{l}{\displaystyle \sum _{i=1}^n{\omega }_i^k}\ge {\left({\displaystyle \sum _{i=1}^n{\omega }_i}\right)}^k,0<k<1\\ {\displaystyle \sum _{i=1}^n{\omega }_i^k}\ge n^{1-k}{\left({\displaystyle \sum _{i=1}^n{\omega }_i}\right)}^k,0<k<\infty \end{array}\right..$$

(1)

Lemma 2 

([26]). For any $c>0,a\ge 0,b>0$, the following inequality holds:

$$a^c(b-a)\le \frac{1}{1+c}(b^{1+c}-a^{1+c}).$$

(2)

Lemma 3 

([27]). For any $c>1,a>0,b\le a$, the following inequality holds:

$${(a-b)}^c\ge b^c-a^c.$$

(3)

Lemma 4 

([28]). Consider the following system:

$$\dot{x}=-l_{1}si{g}^{k_1}x-l_{2}si{g}^{k_2}x, x(0)=x_0,$$

(4)

where $si{g}^k(x)={\left|x\right|}^k\cdot sign(x)$,  $l_1>0,l_2>0$,  $k_1=(m_1+1)/2+sign(x-1)(m_1-1)/2$,  $k_2=(m_2+1)/2-sign(x-1)(m_2-1)/2$, and  $m_1>0,1/2<m_2<1$. This system is a fixed-time stable system. The convergence time is independent of the initial state of the system and satisfies the following:

$$T\le \frac{1}{l_1(m_1-1)}\ln (\frac{l_1+l_2}{l_1})+\frac{1}{l_2(1-m_2)}\ln (\frac{l_1+l_2}{l_2}).$$

(5)

Lemma 5 

([27]). Consider the following system:

$$\left\{\begin{array}{l}\dot{x}(t)=f(x(t),t)\\ x(0)=x_0\end{array}\right.,$$

(6)

where $x={[x_1,x_2,\cdots ,x_m]}^T\in R^m$ is a state vector. $f(x(t),t):R^m\times R^{+}\to R^m$ is a nonlinear function, and $f(0,t)=0$. If there exists a function such that $V(x):R^m\to R ,V(x)\ge 0$, $\forall x\in R^m$ or $V(x)=0\iff x=0$.

(1). For all $x(t)$, the system is globally fixed-time stable if $\dot{V}(x)\le -a{V}^p(x)-b{V}^q(x)$. In the case where $a$ and $b$ are positive constants, $0<p<1$ and $q>1$. Convergence time $T$ satisfies $T\le \frac{1}{a(1-p)}+\frac{1}{b(q-1)}$.

(2). For all  $x(t)$  , if  $\dot{V}(x)\le -a{V}^p(x)-b{V}^q(x)+\vartheta$  , where  $a$  and  $b$  denote positive constants,  $0<p<1$  and  $q>1$  , the system is fixed-time stable, and the system state can converge to a set of residuals at a fixed time.

$\left\{\underset{x\to T}{\lim }\left.x\right|V(x)\le \min \left\{a^{-\frac{1}{p}}{(\frac{\vartheta }{1-\theta })}^{\frac{1}{p}},b^{-\frac{1}{P}}{(\frac{\vartheta }{1-\theta })}^{\frac{1}{q}}\right\}\right\}$, where $0<\theta <1$, the convergence time  $T$ satisfies  $T\le \frac{1}{a\theta (1-p)}+\frac{1}{b\theta (q-1)}$.

Lemma 6 

([29]). Consider the following system:

$$\dot{V}=-k_1{V}^{2-a_1/a_2}-k_2{V}^{a_1/a_2},V(0)=V_0,$$

where $k_1>0$ and $k_2>0$. $a_1$ and $a_2$ are positive odd number and satisfy $a_1/a_2<1$, the system is fixed-time stable. Convergence time $T$ satisfies $T<T_{\max }=\frac{a_2\pi }{2\sqrt{k_1{k}_2}(a_2-a_1)}$.

2.2. USV Dynamic and Kinematic Model

In the horizontal plane, USV motion has three degrees of freedom. Figure 1 depicts two reference frames that are used to characterize the motion of USVs. Together with the earth-fixed coordinate $X_{0}O{Y}_0$ and the body-fixed coordinate $XOY$, the kinematic and dynamic model of USV with three degrees of freedom (3-DOF), i.e., surge, sway, and yaw, can be described as

$$\left\{\begin{array}{l}\dot{\mathit{\eta }}=\mathit{J}(\psi )\mathit{\nu }\\ \dot{\mathit{\nu }}={\mathit{M}}^{-1}(\mathit{\Xi }(\mathit{\eta },\mathit{\nu })+\mathit{\tau }+\mathit{d})\end{array}\right.,$$

(7)

where $\mathit{\Xi }(\mathit{\eta },\mathit{\nu })=-\mathit{C}(\mathit{\nu })\mathit{\nu }-\mathit{D}(\mathit{\nu })\mathit{\nu }$, $\mathit{\eta }={[x,y,\psi ]}^T$ denotes the position and heading angle of the USV in the inertial coordinate system. $\mathit{\nu }={[u,v,r]}^T$ demonstrates the surge, sway, and yaw velocities in the USV body coordinate system. $\mathit{d}=\mathit{M}{\mathit{J}}^T\mathit{\delta }$ and $\mathit{\delta }={[{\delta }_u,{\delta }_v,{\delta }_r]}^T$ indicate the external disturbances generated in the marine environment. The transformation matrix for the USV body coordinate system to the inertial coordinate system is denoted as $J(\psi )$, which can be expressed as $\mathit{J}(\psi )=\left[\begin{array}{ccc}\cos \psi & -\sin \psi & 0\\ \sin \psi & \cos \psi & 0\\ 0& 0& 1\end{array}\right]$, where ${\mathit{J}}^T(\psi )\mathit{J}(\psi )=\mathit{I}$, $‖\mathit{J}(\psi )‖=1$, $\forall \psi \in [0,2\pi ]$, and ${\mathit{J}}^T(\psi )\mathit{S}(r)\mathit{J}(\psi )=\mathit{J}(\psi )\mathit{S}(r){\mathit{J}}^T(\psi )=\mathit{S}(r)$, where $\mathit{S}(r)=\left[\begin{array}{ccc}0& -r& 0\\ r& 0& 0\\ 0& 0& 0\end{array}\right]$ and $M$ are the inertial matrices that satisfy $\mathit{M}={\mathit{M}}^T>\mathbf{0}$. $M$ can be expressed as $\mathit{M}=\left[\begin{array}{ccc}{m}_{11}& 0& 0\\ 0& m_{22}& m_{23}\\ 0& m_{32}& m_{33}\end{array}\right]$. The Coriolis and centripetal matrices are denoted by $\mathit{C}(\mathit{\nu })$ and satisfy $\mathit{C}(\mathit{\nu })=-\mathit{C}{(\mathit{\nu })}^T$. $\mathit{D}(\nu )$ represents the hydrodynamic damping matrix, which can be expressed as $\mathit{C}(\mathit{\nu })=\left[\begin{array}{ccc}0& 0& c_{13}(\nu )\\ 0& 0& c_{23}(\nu )\\ -c_{13}(\nu )& -c_{23}(\nu )& 0\end{array}\right]$ and $\mathit{D}(\nu )=\left[\begin{array}{ccc}{d}_{11}(\nu )& 0& 0\\ 0& d_{22}(\nu )& d_{23}(\nu )\\ 0& d_{32}(\nu )& d_{33}(\nu )\end{array}\right]$. The parameters are $m_{11}=m-X_{\dot{u}}$, $m_{22}=m-Y_{\dot{v}}$, $m_{23}=m{x}_g-Y_{\dot{r}}$, $m_{32}=m{x}_g-N_{\dot{v}}$, $m_{33}=I_z-N_{\dot{r}}$, $c_{13}(\nu )=-m_{11}v-m_{23}r$, $c_{23}(\nu )=-m_{11}u$, $d_{11}(\nu )=-X_u-X_{\left|u\right|u}\left|u\right|-X_{uuu}{\left|u\right|}^2$, $d_{22}(\nu )=-Y_v-Y_{\left|v\right|v}\left|v\right|$, $d_{23}(\nu )=-Y_r-Y_{\left|v\right|r}\left|v\right|-Y_{\left|r\right|r}\left|r\right|$, $d_{32}(\nu )=-N_v-N_{\left|v\right|v}\left|v\right|-N_{\left|r\right|v\left|r\right|}$, $d_{33}(\nu )=-N_r-N_{\left|v\right|r}\left|v\right|-N_{\left|r\right|r}\left|r\right|$, and $Y_{\dot{r}}=N_{\dot{v}}$. $m$ denotes the USV mass, $I_z$ denotes the moment of inertia, and $X_{*},Y_{*},N_{*}$ are the hydrodynamic parameters.

Furthermore, $\tau ={[{\tau }_u,{\tau }_v,{\tau }_r]}^T$ denotes the control forces and moments. From a realistic point of view, the propulsion system is physically constrained to provide a limited control force or moment. $\tau =sat({\tau }_c)$ and $sat({\tau }_c)$ can be described as

$$sat({\tau }_{c,i})=\left\{\begin{array}{l}{\tau }_{i,\max }sign({\tau }_{c,i}),\left|{\tau }_{c,i}\right|>{\tau }_{i,\max }\\ {\tau }_{c,i},\left|{\tau }_{c,i}\right|\le {\tau }_{i,\max }\end{array}\right.,$$

(8)

where ${\tau }_{i,\max }(i=u,v,r)$ indicates the maximum control force or moment provided by vessel propulsion system. ${\tau }_{c,i}$ is the command control vector calculated by the controller.

Consider the following desired trajectory,

$$\left\{\begin{array}{l}{\dot{\mathit{\eta }}}_d=\mathit{J}({\psi }_d){\mathit{\nu }}_d\\ {\dot{\mathit{\nu }}}_d={\mathit{M}}^{-1}({\Xi }_d({\mathit{\eta }}_d,{\mathit{\nu }}_d)+{\mathit{\tau }}_d)\end{array}\right.,$$

(9)

where ${\mathit{\Xi }}_d({\mathit{\eta }}_d,{\mathit{\nu }}_d)=-\mathit{C}({\mathit{\nu }}_d){\mathit{\nu }}_d-\mathit{D}({\mathit{\nu }}_d){\mathit{\nu }}_d$ and ${\mathit{\eta }}_d={[x_d,y_d,{\psi }_d]}^T$ denote the desired position and heading angle of the USV, respectively. ${\mathit{\nu }}_d={[u_d,v_d,r_d]}^T$ reflects the desired surge, sway, and yaw angular velocities in the USV body coordinate system. ${\tau }_d={[{\tau }_{ud},{\tau }_{vd},{\tau }_{rd}]}^T$ denotes the desired control force and torque.

Assumption 1 

([23]). The desired trajectory ${\mathit{\eta }}_d$ is smooth, differentiable, and bounded. Both  ${\dot{\eta }}_d$ and ${\ddot{\eta }}_d$ are assumed to exist and be bounded.

3. Fixed-Time Precision Tracking with Prescribed Performance Control Scheme

A fixed-time convergence dynamic trajectory tracking control law was developed by integrating the nonsingular terminal sliding mode control, prescribed performance control, and AFISMLDO, which was first proposed to estimate the overall disturbance.

To simplify the controller design, coordinates $\mathit{\nu },{\mathit{\nu }}_d$ were transformed as follows:

$$\mathit{\zeta }=\mathit{J}\mathit{\nu }.$$

(10)

$${\mathit{\zeta }}_d={\mathit{J}}_d{\mathit{\nu }}_d.$$

(11)

Integrating Equations (7) and (10), we obtain

$$\left\{\begin{array}{l}\dot{\mathit{\eta }}=\zeta \\ \dot{\zeta }=\mathit{\Gamma }(\mathit{\eta },\mathit{\nu },\mathit{\delta })+\mathit{J}{M}^{-1}\mathit{\tau }\end{array}\right.,$$

(12)

where $\mathit{\Gamma }(\mathit{\eta },\mathit{\nu },\mathit{\delta })=\mathit{S}\zeta +\mathit{J}{\mathit{M}}^{-1}\mathit{\Xi }(\mathit{\eta },\mathit{\nu })+\mathit{\delta }$ and $\mathit{S}=\mathit{S}({\zeta }_r)$.

Similarly, according to Equations (9) and (11), we obtain

$$\left\{\begin{array}{l}{\dot{\mathit{\eta }}}_d={\mathit{\zeta }}_d\\ {\dot{\mathit{\zeta }}}_d={\mathit{\Gamma }}_d({\mathit{\eta }}_d,{\mathit{\nu }}_d)+{\mathit{J}}_d{\mathit{M}}^{-1}{\mathit{\tau }}_d\end{array}\right.,$$

(13)

where ${\mathit{\Gamma }}_d({\mathit{\eta }}_d,{\mathit{\nu }}_d)={\mathit{S}}_d{\zeta }_d+{\mathit{J}}_d{\mathit{M}}^{-1}{\mathit{\Xi }}_d({\eta }_d,{\mathit{\nu }}_d)$ and ${\mathit{S}}_d=\mathit{S}({\zeta }_{d,r})$.

The tracking error system can be obtained by subtracting Equation (12) from Equation (13):

$$\left\{\begin{array}{l}{\dot{\mathit{\eta }}}_e={\zeta }_e\\ {\dot{\zeta }}_e=\mathit{J}{\mathit{M}}^{-1}{\mathit{\tau }}_c+\mathit{S}\zeta -{\mathit{S}}_d{\zeta }_d+\mathit{\Omega }(\cdot )\end{array}\right.,$$

(14)

where ${\dot{\mathit{\eta }}}_e=\dot{\mathit{\eta }}-{\dot{\mathit{\eta }}}_d={[{\dot{\eta }}_{eu},{\dot{\eta }}_{ev},{\dot{\eta }}_{ev}]}^T$ and ${\dot{\zeta }}_e=\dot{\zeta }-{\dot{\zeta }}_d={[{\dot{\zeta }}_{eu},{\dot{\zeta }}_{ev},{\dot{\zeta }}_{er}]}^T$.$\mathit{\Omega }(\cdot )$ denotes the lumped unknown terms containing the unknown system dynamics and external disturbances, which is described as follows:

$$\mathit{\Omega }(\cdot )=J{\mathit{M}}^{-1}\mathit{\Xi }(\mathit{\eta },\mathit{v})+\mathit{\delta }-{\mathit{J}}_d{\mathit{M}}^{-1}{\mathit{\tau }}_d-{\mathit{J}}_d{\mathit{M}}^{-1}{\mathit{\Xi }}_d({\mathit{\eta }}_d,{\mathit{v}}_d).$$

(15)

Assumption 2 

([23]). The lumped disturbance is a continuously differentiable function, and its derivative satisfies  $‖\dot{\mathit{\Omega }}(\cdot )‖\le {\mathit{\Omega }}_m$ and  $0<{\mathit{\Omega }}_m\le D$.  ${\mathit{\Omega }}_m$ is unknown, bounded, and its maximum upper bound is  $D$.

3.1. Nonlinear Model Transformation

Consider a tracking error system with state variables ${\eta }_e={[{\eta }_{eu},{\eta }_{ev},{\eta }_{er}]}^T$ and ${\zeta }_e={[{\zeta }_{eu},{\zeta }_{ev},{\zeta }_{er}]}^T$ subject to the following performance constraints:

$$\left\{\begin{array}{l}-{\overline{\eta }}_{ek}(t)<{\eta }_{ek}(t)<{\overline{\eta }}_{ek}(t)\\ -{\overline{\zeta }}_{ek}(t)<{\zeta }_{ek}(t)<{\overline{\zeta }}_{ek}(t)\end{array}\right.,k=u,v,r,$$

(16)

where ${\overline{\eta }}_{ek}(t)=({\overline{\eta }}_{ek,0}-{\overline{\eta }}_{ek,\infty })\exp (-\mu t)+{\overline{\eta }}_{ek,\infty }$,${\overline{\zeta }}_{ek}(t)=({\overline{\zeta }}_{ek,0}-{\overline{\zeta }}_{ek,\infty })\exp (-\mathcal{l}t)+{\overline{\zeta }}_{ek,\infty }$, ${\overline{\eta }}_{ek,0}>0$, ${\overline{\eta }}_{ek,\infty }>0$, ${\overline{\zeta }}_{ek,0}>0$, ${\overline{\zeta }}_{ek,\infty }>0$, $\mu >0$, and $\mathcal{l}>0$ are constants that determine the transient and steady-state responses of the controlled system. The designer can specify the desired response performance according to the task requirements, and prescribed performance control ensures that the system states stay within the preset boundaries. In order to prevent the specified limitations from being broken at the start of control, the following assumption is made before moving forward.

Assumption 3 

([30]). The initial conditions of USV satisfy $-{\overline{\eta }}_{ek}(0)<{\eta }_{ek}(0)<{\overline{\eta }}_{ek}(0)$

and  $-{\overline{\zeta }}_{ek}(0)<{\zeta }_{ek}(0)<{\overline{\zeta }}_{ek}(0)$.

The state transformation is defined as

$$\left\{\begin{array}{l}{\sigma }_{ek}(t)=\frac{1}{2}\ln (\frac{\frac{{\eta }_{ek}}{{\overline{\eta }}_{ek}}+1}{1-\frac{{\eta }_{ek}}{{\overline{\eta }}_{ek}}})\\ {\varpi }_{ek}(t)=\frac{1}{2}\ln (\frac{\frac{{\zeta }_{ek}}{{\overline{\zeta }}_{ek}}+1}{1-\frac{{\zeta }_{ek}}{{\overline{\zeta }}_{ek}}})\end{array}\right.,$$

(17)

where ${\sigma }_{ek}={[{\sigma }_{eu},{\sigma }_{ev},{\sigma }_{er}]}^T$ and ${\varpi }_{ek}={[{\omega }_{eu},{\omega }_{ev},{\omega }_{er}]}^T$ are the transformations of variables ${\eta }_{ek}$ and ${\zeta }_{ek}$, respectively. This transformation guarantees the same monotonicity of the system state before and after the transformation. Therefore, based on this transformation, the required performance of the original system is ensured as long as the converted system is stable.

The derivative of Equation (17) gives

$$\left\{\begin{array}{ll}{\dot{\sigma }}_{ek}(t)& =\frac{1}{2{\overline{\eta }}_{ek}}\left(\frac{1}{{\eta }_{ek}/{\overline{\eta }}_{ek}+1}-\frac{1}{{\eta }_{ek}/{\overline{\eta }}_{ek}-1}\right)\left({\dot{\eta }}_{ek}-\frac{{\eta }_{ek}{\dot{\overline{\eta }}}_{ek}}{{\overline{\eta }}_{ek}}\right)\\ & \triangleq {\alpha }_k{\dot{\eta }}_{ek}-{\beta }_k{\eta }_{ek}\\ {\dot{\varpi }}_{ek}(t)& =\frac{1}{2{\overline{\zeta }}_{ek}}\left(\frac{1}{{\zeta }_{ek}/{\overline{\zeta }}_{ek}+1}-\frac{1}{{\zeta }_{ek}/{\overline{\zeta }}_{ek}-1}\right)\left({\dot{\zeta }}_{ek}-\frac{{\zeta }_{ek}{\dot{\overline{\zeta }}}_{ek}}{{\overline{\zeta }}_{ek}}\right)\\ & \triangleq {\chi }_k{\dot{\zeta }}_{ek}-{\delta }_k{\zeta }_{ek}\end{array}\right..$$

(18)

where ${\alpha }_k=\frac{1}{2{\overline{\eta }}_{ek}}\left(\frac{1}{{\eta }_{ek}/{\overline{\eta }}_{ek}+1}-\frac{1}{{\eta }_{ek}/{\overline{\eta }}_{ek}-1}\right)$, ${\beta }_k=\frac{{\dot{\overline{\eta }}}_{ek}}{2{\overline{\eta }}^2{}_{ek}}\left(\frac{1}{{\eta }_{ek}/{\overline{\eta }}_{ek}+1}-\frac{1}{{\eta }_{ek}/{\overline{\eta }}_{ek}-1}\right)$,${\chi }_k=\frac{1}{2{\overline{\zeta }}_{ek}}\left(\frac{1}{{\zeta }_{ek}/{\overline{\zeta }}_{ek}+1}-\frac{1}{{\zeta }_{ek}/{\overline{\zeta }}_{ek}-1}\right)$, and ${\delta }_k=\frac{{\dot{\overline{\varpi }}}_{ek}}{2{\overline{\varpi }}^2{}_{ek}}\left(\frac{1}{{\varpi }_{ek}/{\overline{\varpi }}_{ek}+1}-\frac{1}{{\varpi }_{ek}/{\overline{\varpi }}_{ek}-1}\right)$.

Using Equations (14) and (18), we can obtain the following transformed system model:

$$\left\{\begin{array}{ll}{\dot{\sigma }}_e& ={\mathrm{\Pi }}_1\mathrm{\Lambda }\varphi +{\mathrm{\Pi }}_2\mathrm{\Psi }\phi \\ {\dot{\varpi }}_e& ={\mathrm{\Gamma }}_1(\mathit{J}{\mathit{M}}^{-1}{\mathit{\tau }}_c+\mathit{S}\zeta -{\mathit{S}}_d{\zeta }_d+\mathit{\Omega }(\cdot ))\\ & +{\mathrm{\Gamma }}_2\mathrm{\Psi }\phi \end{array}\right.,$$

(19)

where ${\mathrm{\Pi }}_1=diag\left\{{\alpha }_1,{\alpha }_2,{\alpha }_3\right\}$, ${\mathrm{\Pi }}_2=diag\left\{{\beta }_1,{\beta }_2,{\beta }_3\right\}$, ${\mathrm{\Gamma }}_1=diag\left\{{\chi }_1,{\chi }_2,{\chi }_3\right\}$, ${\mathrm{\Gamma }}_2=diag\left\{{\delta }_1,{\delta }_2,{\delta }_3\right\}$, $\mathrm{\Lambda }=diag\left\{{\overline{\eta }}_{eu},{\overline{\eta }}_{ev},{\overline{\eta }}_{er}\right\}$, $\mathrm{\Psi }=diag\left\{{\overline{\zeta }}_{eu},{\overline{\zeta }}_{ev},{\overline{\zeta }}_{er}\right\}$, $\varphi =\left[\frac{e^{2{\sigma }_{e1}}-1}{e^{2{\sigma }_{e1}}+1},\frac{e^{2{\sigma }_{e2}}-1}{e^{2{\sigma }_{e2}}+1},\frac{e^{2{\sigma }_{e3}}-1}{e^{2{\sigma }_{e3}}+1}\right]$, and $\phi =\left[\frac{e^{2{\varpi }_{e1}}-1}{e^{2{\varpi }_{e1}}+1},\frac{e^{2{\varpi }_{e2}}-1}{e^{2{\varpi }_{e2}}+1},\frac{e^{2{\varpi }_{e3}}-1}{e^{2{\varpi }_{e3}}+1}\right]$.

Based on definition $\phi =f({\varpi }_e)$, in Equation (17), we can rewrite Equation (19) as

$$\left\{\begin{array}{ll}{\dot{\sigma }}_e& ={\mathrm{\Pi }}_1\mathrm{\Lambda }\varphi +{\mathrm{\Pi }}_2\mathrm{\Psi }\phi \\ \dot{\phi }& =Y{\mathrm{\Gamma }}_1(\mathit{J}{\mathit{M}}^{-1}{\mathit{\tau }}_c+\mathit{S}\zeta -{\mathit{S}}_d{\zeta }_d+\mathit{\Omega }(\cdot ))\\ & +Y{\mathrm{\Gamma }}_2\mathrm{\Psi }\phi \end{array}\right.,$$

(20)

where $Y=diag\left\{\frac{4e^{2{\varpi }_{e_u}}}{{(1+e^{2{\varpi }_{eu}})}^2},\frac{4e^{2{\varpi }_{e_v}}}{{(1+e^{2{\varpi }_{ev}})}^2},\frac{4e^{2{\varpi }_{e_r}}}{{(1+e^{2{\varpi }_{er}})}^2}\right\}$.

Let $z_1={\sigma }_e$ and $z_2={\dot{\sigma }}_e$. Then, from Equation (20), we have

$$\left\{\begin{array}{l}{\dot{\mathit{z}}}_1={\mathit{z}}_2\\ {\dot{\mathit{z}}}_2=\hslash +{\mathrm{\Pi }}_2\mathrm{\Psi }Y{\mathrm{\Gamma }}_{1}J{M}^{-1}{\tau }_c+{\mathrm{\Pi }}_2\mathrm{\Psi }Y{\mathrm{\Gamma }}_1\mathit{\Omega }(\cdot )\end{array}\right.,$$

(21)

where $\hslash ={\dot{\mathrm{\Pi }}}_1\mathrm{\Lambda }\varphi +{\mathrm{\Pi }}_1\dot{\mathrm{\Lambda }}\varphi +{\mathrm{\Pi }}_1\mathrm{\Lambda }\dot{\varphi }+{\dot{\mathrm{\Pi }}}_2\mathrm{\Psi }\phi +{\mathrm{\Pi }}_2\dot{\mathrm{\Psi }}\phi +{\mathrm{\Pi }}_2\mathrm{\Psi }[Y{\mathrm{\Gamma }}_{1}S\zeta -Y{\mathrm{\Gamma }}_1{S}_d{\zeta }_d+Y{\mathrm{\Gamma }}_2\mathrm{\Psi }\phi ]$, ${\dot{\mathrm{\Pi }}}_1=diag\left\{{\dot{\alpha }}_1,{\dot{\alpha }}_2,{\dot{\alpha }}_3\right\}$, ${\dot{\mathrm{\Pi }}}_2=diag\left\{{\dot{\beta }}_1,{\dot{\beta }}_2,{\dot{\beta }}_3\right\}$, $\dot{\mathrm{\Lambda }}=\left\{{\dot{\overline{\eta }}}_{eu},{\dot{\overline{\eta }}}_{ev},{\dot{\overline{\eta }}}_{er}\right\}$, $\dot{\mathrm{\Psi }}=diag\left\{{\dot{\overline{\zeta }}}_{eu},{\dot{\overline{\zeta }}}_{ev},{\dot{\overline{\zeta }}}_{er}\right\}$, ${\dot{\alpha }}_k=\frac{-{\dot{\overline{\eta }}}_{ek}({\eta }_{ek}^2+{\overline{\eta }}_{ek}^2)+2{\overline{\eta }}_{ek}{\eta }_{ek}{\dot{\eta }}_{ek}}{{({\eta }_{ek}^2-{\overline{\eta }}_{ek}^2)}^2}$, and ${\dot{\beta }}_k=\frac{-{\ddot{\overline{\eta }}}_{ek}({\eta }_{ek}^2-{\overline{\eta }}_{ek}^2)+2{\dot{\overline{\eta }}}_{ek}({\eta }_{ek}{\dot{\eta }}_{ek}-{\overline{\eta }}_{ek}{\dot{\overline{\eta }}}_{ek})}{{({\eta }_{ek}^2-{\overline{\eta }}_{ek}^2)}^2}$.

3.2. Adaptive Fixed-Time Lumped Disturbance Observer

To efficiently estimate and correct the lumped disturbances in the system described by Equation (14), we developed a fixed-time lumped disturbance observer that mitigates their impact. By incorporating auxiliary variables into the design of the sliding mode surface, we integrated the fixed-time control theory with the disturbance observer technique to create a lumped disturbance observer with fixed-time convergence performance. This resulted in an accurate estimation of the set total disturbance of the system.

The sliding surface is defined as $s_0={[s_{0u},s_{0v},s_{0r}]}^T$,

$$s_0={\zeta }_e-Z,$$

(22)

where $Z$ is an auxiliary variable and can be expressed as

$$\dot{Z}=J{M}^{-1}{\tau }_c+S\zeta -S_d{\zeta }_d+\widehat{\mathit{\Omega }},$$

(23)

where $\widehat{\mathit{\Omega }}$ is the estimation of $\Omega$.

We then selected the second sliding surface $s_1={[s_{1u,}{s}_{1v},s_{1r}]}^T$:

$$s_1={\dot{s}}_0+{\displaystyle \int [l_{1}si{g}^{k_1}(s_0)+l_{2}si{g}^{k_2}(s_0)}]ds,$$

(24)

where $k_1=\frac{m_1+1}{2}+\frac{m_1-1}{2}sign(\left|s_0\right|-1),m_1>0$, $k_2=\frac{m_2+1}{2}+\frac{1-m_2}{2}sign(\left|s_0\right|-1),\frac{1}{2}<m_2<1$.

The observer’s derivative is obtained as follows:

$$\begin{array}{ll}\dot{\widehat{\mathit{\Omega }}}& =l_{1}si{g}^{k_1}(s_0)+l_{2}si{g}^{k_2}(s_0)\\ & +{\Phi }_{1}si{g}^{a_1}(s_1)+{\Phi }_{2}si{g}^{b_1}(s_1),\\ & +{\Phi }_{0}sign(s_1)+\widehat{D}sign(s_1)\end{array}$$

(25)

where ${\Phi }_i(i=0,1,2)$ are positive design parameters and $0<a_1<1$, $b_1>1$.

Considering Assumption 2, we know that $‖\dot{\mathit{\Omega }}‖\le {\mathit{\Omega }}_m\le D$. $D$ is subject to adaptive estimation, and the estimated error is defined as follows:

$$\tilde{D}=D-\widehat{D},$$

(26)

where $\widehat{D}$ is the estimation error of $D$ and $\tilde{D}$.

Deriving Equations (22) and (24) yields the following expression:

$$\left\{\begin{array}{ll}{\dot{s}}_0& =\mathit{\Omega }-\widehat{\mathit{\Omega }}\\ {\ddot{s}}_0& =\dot{\mathit{\Omega }}-\dot{\widehat{\mathit{\Omega }}}\\ {\dot{s}}_1& =\dot{\mathit{\Omega }}-{\Phi }_{1}si{g}^{a_1}(s_1)-{\Phi }_{2}si{g}^{b_1}(s_1)\\ & -{\Phi }_{0}sign(s_1)-\widehat{D}sign(s_1)\end{array}\right..$$

(27)

The following inequalities can be deduced from Lemmas 2 and 3:

$$\begin{array}{l}\tilde{D}{\widehat{D}}^p={\widehat{D}}^p(D-\widehat{D})\\ \le \frac{1}{1+p}(D^{p+1}-{\widehat{D}}^{p+1})\\ =\frac{1}{1+p}(D^{p+1}-{(D-\tilde{D})}^{p+1}).\\ \le \frac{1}{1+p}(D^{p+1}-({\tilde{D}}^{p+1}-D^{p+1}))\\ =\frac{1}{1+p}(2D^{p+1}-{\tilde{D}}^{p+1})\end{array}$$

(28)

Similarly, we can obtain

$$\begin{array}{l}\tilde{D}{\widehat{D}}^q={\widehat{D}}^q(D-\widehat{D})\\ \le \frac{1}{1+q}(D^{q+1}-{\widehat{D}}^{q+1})\\ =\frac{1}{1+q}(D^{q+1}-{(D-\tilde{D})}^{q+1}),\\ \le \frac{1}{1+q}(D^{q+1}-({\tilde{D}}^{q+1}-D^{q+1}))\\ =\frac{1}{1+q}(2D^{q+1}-{\tilde{D}}^{q+1})\end{array}$$

(29)

where $0<p<1$ and $q>1$ are the design parameters.

The sliding mode disturbance observer adaptive law is selected as follows:

$$\dot{\widehat{D}}=\gamma (-{\widehat{D}}^p-{\widehat{D}}^q+‖s_1‖).$$

(30)

Theorem 1.

For the USV trajectory tracking error control system in Equation (14) in the presence of lumped disturbances, if the adaptive law defined in Equation (30) and the adaptive fixed-time lumped disturbance observer defined in Equations (22)–(25) are selected, it is ensured that the overall disturbances are approximated within a fixed time, that is, the estimation error  $\tilde{\mathit{\Omega }}=\mathit{\Omega }-\widehat{\mathit{\Omega }}$ is confined to a bounded region within a fixed time.

Proof. 

To verify Theorem 1, the following candidate Lyapunov function is chosen:

$$V_d=\frac{1}{2}{s}_1^T{s}_1+\frac{1}{2\gamma }{\tilde{\mathit{\Omega }}}^2.$$

(31)

The derivative of Equation (31) is as follows:

$$\begin{array}{l}{\dot{V}}_d=s_1^T{\dot{s}}_1+\frac{1}{\gamma }\tilde{\mathit{\Omega }}\dot{\tilde{\mathit{\Omega }}}=\\ -{\Phi }_1{‖s_1‖}^{a_1+1}-{\Phi }_2{‖s_1‖}^{b_1+1}+\\ s_1^T(-{\Phi }_{0}sign(s_1)-\widehat{D}sign(s_1)+\dot{\mathit{\Omega }})-\frac{1}{\gamma }\tilde{D}\dot{\widehat{D}}\le .\\ -{\Phi }_1{‖s_1‖}^{a_1+1}-{\Phi }_2{‖s_1‖}^{b_1+1}-\\ ‖s_1‖({\Phi }_0+\widehat{D}-D)-\frac{1}{\gamma }\tilde{D}\dot{\widehat{D}}\end{array}$$

(32)

By substituting the adaptive law into Equation (30), the inequalities in Equations (28) and (29) are combined into Equation (32). According to Lemma 1, the following inequalities are obtained:

$$\begin{array}{l}{\dot{V}}_d\le -{\Phi }_1{‖s_1‖}^{a_1+1}-{\Phi }_2{‖s_1‖}^{b_1+1}+\\ \frac{1}{1+p}(2D^{p+1}-{\tilde{D}}^{p+1})+\frac{1}{1+q}(2D^{q+1}-{\tilde{D}}^{q+1})\le \\ -{\Phi }_1{‖s_1‖}^{2\frac{a_1+1}{2}}-{\Phi }_2{‖s_1‖}^{2\frac{b_1+1}{2}}-\frac{1}{1+p}{\tilde{D}}^{2\frac{p+1}{2}}-\\ \frac{1}{1+q}{\tilde{D}}^{2\frac{q+1}{2}}+\frac{1}{1+p}2D^{p+1}+\frac{1}{1+q}2D^{q+1}\le \\ -\min \left\{{\Phi }_1{2}^{\frac{p+1}{2}},\frac{1}{1+p}{(2\gamma )}^{\frac{p+1}{2}}\right\}\times \\ {(\frac{1}{2}{‖s_1‖}^2+\frac{1}{2\gamma }{\tilde{D}}^2)}^{\frac{p+1}{2}}+\vartheta -\\ \min \left\{{\Phi }_2{2}^{\frac{q+1}{2}},\frac{1}{1+q}{(2\gamma )}^{\frac{q+1}{2}}\right\}{2}^{\frac{1-q}{2}}\times \\ {(\frac{1}{2}{‖s_1‖}^2+\frac{1}{2\gamma }{\tilde{D}}^2)}^{\frac{q+1}{2}}\le \\ -{\gamma }_1{V}_d{}^{\frac{p+1}{2}}-2^{\frac{1-q}{2}}{\gamma }_2{V}_d{}^{\frac{q+1}{2}}+\vartheta \end{array},$$

(33)

where $\vartheta =\frac{1}{1+p}2D^{p+1}+\frac{1}{1+q}2D^{q+1}$.

Applying Lemma 5, the parameters are chosen as

$$\begin{array}{ccc}0<p=a_1<1, & q=b_1>1,& {\gamma }_1=\min ({\Phi }_1{2}^{\frac{p+1}{2}},\frac{1}{1+p}{(2\gamma )}^{\frac{p+1}{2}}), \\ & & {\gamma }_2=\min ({\Phi }_2{2}^{\frac{q+1}{2}},\frac{1}{1+q}{(2\gamma )}^{\frac{q+1}{2}}).\end{array}$$

Then, Equation (33) is a fixed-time convergence. By selecting the following parameters

$A_0={\gamma }_1$, $B_0=2^{\frac{1-q}{2}}{\gamma }_2$, $k_{01}=(p+1)/2$, and $k_{02}=(q+1)/2$, the convergence time of the fixed-time observer is satisfied as

$$T_1\le \frac{1}{A_0\theta (1-k_{01})}+\frac{1}{B_0\theta (k_{02}-1)}.$$

(34)

where $0<\theta <1$. Therefore, the system states converge within a bounded region at a fixed time $T_1$.

Subsequently, we ascertain the convergence of $s_1$. Choosing a Lyapunov function $V_d=\frac{1}{2}{s}_1^T{s}_1$ and determining its derivative, we have

$$\begin{array}{l}{\dot{V}}_d=s_1^T(\dot{\Omega }-{\Phi }_{1}si{g}^{a_1}(s_1)-{\Phi }_{2}si{g}^{b_1}(s_1)-\\ {\Phi }_{0}sign(s_1)-\widehat{D}sign(s_1))\le \\ -{\Phi }_1{‖s_1‖}^{a_1+1}-{\Phi }_2{‖s_1‖}^{b_1+1}-{\Phi }_0‖s_1‖+\\ ‖s_1‖D-\widehat{D}‖s_1‖\le \\ -{\Phi }_1{‖s_1‖}^{a_1+1}-{\Phi }_2{‖s_1‖}^{b_1+1}-\\ ({\Phi }_0-\tilde{D})‖s_1‖\le \\ -{\Phi }_1{‖s_1‖}^{a_1+1}-{\Phi }_2{‖s_1‖}^{b_1+1}\end{array}.$$

(35)

When parameter ${\Phi }_0\ge \tilde{D}$ is chosen, it is ensured that Equation (35) converges at a fixed time.

Consequently, the state of the system approaches sliding mode $s_1$ at a set time $s_1=0$ in Equation (24), which can be written as

$${\ddot{s}}_{0i}=-l_{1}si{g}^{k_1}(s_{0i})-l_{2}si{g}^{k_2}(s_{0i}),i=u,v,r.$$

(36)

From Lemma 4, it can be inferred that $s_0$ converges to zero at a fixed time. Consequently, when ${\dot{s}}_0=\mathit{\Omega }-\widehat{\mathit{\Omega }}$, the observation error also converges to zero at a fixed time $t\ge T_1$, $\tilde{\mathit{\Omega }}=0$. Therefore, the overall convergence time $T_d$ of the designed AFISMLDO includes both the sliding mode arrival time $T_1$ and the sliding mode moving time $T_2$ and can be expressed as $T_d\le T_1+T_2$. Using Equation (36) and Lemma 4, the mathematical expression for $T_2$ can be obtained as

$$T_2\le \frac{1}{l_1(m_1-1)}\ln (\frac{l_1+l_2}{l_1})+\frac{1}{l_2(1-m_2)}\ln (\frac{l_1+l_2}{l_2}).$$

(37)

☐

Remark 1.

The fixed-time convergence of the lumped disturbance observer is ensured by the integration term in Equation (24). Additionally, including the sign function inside the integral term of the disturbance observer can significantly reduce the jitter phenomena observed in the sliding mode method.

Figure 2 shows a block diagram of the proposed control scheme.

3.3. Controller Design and Stability Analysis

For fixed-time control technique, faster transient response results in greater control force or moment to drive USV to track the preset trajectory. Therefore, thruster saturation is more likely to occur. To alleviate the input saturation problem, the following adaptive auxiliary system is designed [23]:

$$\begin{array}{l}{\dot{\mathit{\chi }}}_1=-c_1{\mathit{\chi }}_1+{\mathit{\chi }}_2,\\ {\dot{\mathit{\chi }}}_2=-c_2{\mathit{\chi }}_2-J{\mathit{M}}^{-1}\Delta \mathit{\tau },\end{array}$$

(38)

where ${\mathit{\chi }}_1={[{\chi }_{1,u},{\chi }_{1,v},{\chi }_{1,r}]}^T$ and ${\mathit{\chi }}_2={[{\chi }_{2,u},{\chi }_{2,v},{\chi }_{2,r}]}^T$ are auxiliary variables. $c_1$ and $c_2$ are appropriate parameter matrices. $\Delta \mathit{\tau }={\mathit{\tau }}_c-\mathit{\tau }$ is the difference between the command control input and the actual control input.

A fixed-time nonsingular terminal sliding mode is constructed for variables ${\mathit{e}}_1$ and ${\mathit{e}}_2$ as follows:

$$\varsigma ={\mathit{e}}_2+{\lambda }_{1}si{g}^{a_1}({\mathit{e}}_1)+{\lambda }_2\mathit{f}({\mathit{e}}_1),$$

(39)

where $\varsigma ={[{\varsigma }_u,{\varsigma }_v,{\varsigma }_r]}^T$, ${\mathit{e}}_1={\mathit{z}}_1-{\mathit{\chi }}_1={[e_{1,u},e_{1,v},e_{1,r}]}^T$, ${\mathit{e}}_2={\mathit{z}}_2-{\dot{\mathit{\chi }}}_1={[e_{2,u},e_{2,v},e_{2,r}]}^T$, ${\lambda }_1,{\lambda }_2>0$,$a_1>1$. $\mathit{f}({\mathit{e}}_1)$ can be given by the following equation:

$$\mathit{f}({\mathit{e}}_1)=\left\{\begin{array}{l}si{g}^{a_2}({\mathit{e}}_1),‖{\mathit{e}}_1‖>\epsilon \\ {\lambda }_{3}si{g}^{a_3}({\mathit{e}}_1)+{\lambda }_{4}si{g}^{a_4}({\mathit{e}}_1),‖{\mathit{e}}_1‖\le \epsilon \end{array}\right.,$$

(40)

where $\epsilon$ is a minimal positive number, $0<a_2<1$, $1<a_3<a_4$, and $a_1,a_2,a_3,a_4,{\lambda }_1,{\lambda }_2$ satisfy the following equations:

$${\lambda }_3=\frac{1-a_3}{a_4-a_3}{\epsilon }^{a_2-a_3},{\lambda }_4=\frac{a_4-1}{a_4-a_3}{\epsilon }^{a_2-a_4}.$$

(41)

The time derivative of (39) is given as

$$\dot{\varsigma }={\mathit{e}}_2+{\lambda }_1{a}_{1}diag({‖{\mathit{e}}_1‖}^{a_1-1}){\mathit{e}}_2+{\lambda }_2\dot{\mathit{f}}({\mathit{e}}_1),$$

(42)

where $diag({‖{\mathit{e}}_1‖}^{a_1-1})=diag({\left|{\mathit{e}}_{1,u}\right|}^{a_1-1},{\left|{\mathit{e}}_{1,v}\right|}^{a_1-1},{\left|{\mathit{e}}_{1,r}\right|}^{a_1-1})$.

In order to improve the convergence speed of sliding mode variables and suppress chattering, a fixed-time reaching law based on arctangent function is designed as follows:

$$\dot{\varsigma }=-\frac{1}{g(\varsigma )}[{\alpha }_{1}si{g}^{\frac{2b_1-b_2}{b_1}}(\varsigma )+{\alpha }_{2}si{g}^{\frac{b_2}{b_1}}(\varsigma )]-{\alpha }_3\varsigma ,$$

(43)

where $g(\varsigma )=(\omega -\arctan \rho {‖\varsigma ‖}^{\iota })\gamma$, $\omega =\frac{\pi }{2}$, $\gamma >\frac{2}{\pi }$, $0<\iota <1$. $b_1$ and $b_2$ are positive odd number and $b_1>b_2$. ${\alpha }_1,{\alpha }_2,{\alpha }_3>0$.

Combining Equations (21)~(43), the PPFTPT control strategy is designed as follows:

$$\begin{array}{l}{\mathit{\tau }}_c=-\mathit{M}{\mathit{J}}^{-1}{\left(\mathit{Y}{\mathit{\Gamma }}_1\right)}^{-1}{({\mathit{\Pi }}_2\mathit{\Psi })}^{-1}[c_1{\dot{\mathit{\chi }}}_1+c_2{\mathit{\chi }}_2+{\mathit{\Pi }}_2\mathit{\Psi }\mathit{Y}{\mathit{\Gamma }}_1\mathit{\Omega }(\cdot )\\ +\hslash +{\lambda }_1{a}_{1}diag({‖{\mathit{e}}_1‖}^{a_1-1}){\mathit{e}}_2+{\lambda }_2\dot{f}({\mathit{e}}_1)+\frac{{\alpha }_1}{g(\varsigma )}si{g}^{\frac{2b_1-b_2}{b_1}}(\varsigma )\\ +\frac{{\alpha }_2}{g(\varsigma )}si{g}^{\frac{b_2}{b_1}}(\varsigma )+{\alpha }_3\varsigma ]\end{array}.$$

(44)

Theorem 2.

For the system described in Equation (21), under Assumptions 1–3, the PPFTPT control scheme in Equation (44) ensures that the USV follows the desired trajectory rapidly and precisely while meeting the transient and steady-state responses. It ensures that the actual position and velocity errors of the USV are stabilized within a small neighborhood of the equilibrium point in a fixed time, and the convergence time is independent of the system’s initial state. The convergence time  $T\le T_d+T_c$, and  $T_c=T_3+T_4$. where $T_3$ and $T_4$ are positive constants,  $T_3\le \frac{{ƛ}_2\pi }{2\sqrt{{\beta }_1{\beta }_2}({ƛ}_2-{ƛ}_1)},T_4\le \frac{1}{{\lambda }_1(a_1-1)}+\frac{1}{{\lambda }_2(1-a_2)}$, 

${\beta }_1=\frac{{\alpha }_1}{\omega \phi }{2}^{\frac{3-b_2/b_1}{2}},{\beta }_2=\frac{{\alpha }_2}{\omega \phi }{2}^{\frac{1+b_2/b_1}{2}},{ƛ}_1=b_1+b_2,{ƛ}_2=2b_1$.

Proof. 

The proof of Theorem 2 consists of two phases: the sliding mode arrival phase and the sliding phase.

Step 1. We select the following Lyapunov function:

$$V_2=\frac{1}{2}{\varsigma }^T\varsigma .$$

(45)

Differentiating Equation (45) yields

$$\dot{V}={\varsigma }^T({\dot{\mathit{z}}}_2-{\ddot{\mathit{\chi }}}_1+{\lambda }_1{a}_{1}diag({‖{\mathit{e}}_1‖}^{a_1-1}){\mathit{e}}_2+{\lambda }_2\dot{\mathit{f}}({\mathit{e}}_1)).$$

(46)

Substituting Equation (21) into Equation (46),

$$\begin{array}{l}\dot{V}={\varsigma }^T(\hslash +{\mathrm{\Pi }}_2\mathrm{\Psi }Y{\mathrm{\Gamma }}_{1}J{M}^{-1}{\tau }_c+{\mathrm{\Pi }}_2\mathrm{\Psi }Y{\mathrm{\Gamma }}_1\mathit{\Omega }(\cdot )+c_1{\dot{\mathit{\chi }}}_1\\ +c_2{\mathit{\chi }}_2+J{M}^{-1}\Delta \tau +{\lambda }_1{a}_{1}diag({‖e_1‖}^{a_1-1})z_2+{\lambda }_2\dot{f}(e_1))\\ ={\varsigma }^T(\hslash +{\mathrm{\Pi }}_2\mathrm{\Psi }Y{\mathrm{\Gamma }}_{1}J{M}^{-1}{\tau }_c+{\mathrm{\Pi }}_2\mathrm{\Psi }Y{\mathrm{\Gamma }}_1\mathit{\Omega }(\cdot )+c_1{\dot{\mathit{\chi }}}_1\\ +c_2{\mathit{\chi }}_2+{\lambda }_1{a}_{1}diag({‖e_1‖}^{a_1-1})e_2+{\lambda }_2\dot{f}(e_1))\end{array}.$$

(47)

Substituting the control strategy in Equation (44) into Equation (47),

$$\begin{array}{l}\dot{V}=-{\varsigma }^T(\frac{{\alpha }_1}{g(\varsigma )}si{g}^{\frac{2b_1-b_2}{b_1}}\varsigma )+\frac{{\alpha }_2}{g(\varsigma )}si{g}^{\frac{b_2}{b_1}}(\varsigma )+{\alpha }_3\varsigma )\\ =-\frac{{\alpha }_1}{g(\varsigma )}{\left|\varsigma \right|}^{\frac{3b_1-b_2}{b_1}}-\frac{{\alpha }_2}{g(\varsigma )}{\left|\varsigma \right|}^{\frac{b_2+b_1}{b_1}}\le -\frac{{\alpha }_1}{\omega \beta }{2}^{\frac{3-b_2/b_1}{2}}{(\frac{1}{2}{\varsigma }^2)}^{\frac{3-b_2/b_1}{2}}\\ -\frac{{\alpha }_2}{\omega \beta }{2}^{\frac{1+b_2/b_1}{2}}{(\frac{1}{2}{\varsigma }^2)}^{\frac{1+b_2/b_1}{2}}\\ \le -k_1{V}^{2-\frac{{ƛ}_1}{{ƛ}_2}}-k_2{V}^{\frac{{ƛ}_1}{{ƛ}_2}}\end{array}.$$

(48)

In the light of Lemma 6, $e_1$ and $e_2$ can reach the sliding surface in a fixed time, and the convergence time satisfies

$$T_3\le \frac{{ƛ}_2\pi }{2\sqrt{{\beta }_1{\beta }_2}({ƛ}_2-{ƛ}_1)}.$$

(49)

Step 2. When the sliding mode attains the sliding surface, we have

$$\varsigma \equiv 0,\dot{\varsigma }\equiv 0.$$

(50)

From Equations (38) and (50), we can obtain

$${\mathit{e}}_2+{\lambda }_{1}si{g}^{a_1}({\mathit{e}}_1)+{\lambda }_{2}f({\mathit{e}}_1)=0.$$

(51)

When $\varsigma \equiv 0,\dot{\varsigma }\equiv 0$, the analysis is divided into two cases, namely $‖{\mathit{e}}_1‖\le \epsilon ,‖{\mathit{e}}_1‖>\epsilon$.

(1)

When $‖{\mathit{e}}_1‖\le \epsilon$ substituting Equation (39) into Equation (51), the following can be obtained:

$$\begin{array}{l}{\mathit{e}}_2=-{\lambda }_{1}si{g}^{a_1}({\mathit{e}}_1)-{\lambda }_2\frac{1-a_3}{a_4-a_3}{\epsilon }^{a_2-a_3}si{g}^{a_3}({\mathit{e}}_1)\\ -{\lambda }_2\frac{a_4-1}{a_4-a_3}{\epsilon }^{a_2-a_4}si{g}^{a_3}({\mathit{e}}_1)\\ \le -{\lambda }_{1}si{g}^{a_1}({\mathit{e}}_1)-{\lambda }_{2}si{g}^{a_2}({\mathit{e}}_1)\end{array}.$$

(52)

That is, Equation (52) satisfies

$${\dot{\mathit{e}}}_1\le -{\lambda }_{1}si{g}^{a_1}({\mathit{e}}_1)-{\lambda }_{2}si{g}^{a_2}({\mathit{e}}_1).$$

(53)

According to Lemma 4, it can be seen that System (53) is stable in a fixed time, and the convergence time of the system satisfies

$$T_4\le \frac{1}{{\lambda }_1(a_1-1)}+\frac{1}{{\lambda }_2(1-a_2)}.$$

(54)

(2)

When $\epsilon <‖{\mathit{e}}_1‖$, the following can also be obtained:

$${\dot{\mathit{e}}}_1=-{\lambda }_{1}si{g}^{a_1}({\mathit{e}}_1)-{\lambda }_{2}si{g}^{a_2}({\mathit{e}}_1).$$

(55)

Similarly, System (55) is stable in $T_4$.

In summary, the proposed PPFTPT control scheme based on an adaptive integral sliding mode lumped observer, prescribed performance control, and fixed-time control can enable dynamic error system Equation (21) to be fixed-time stable with a maximum upper bound on the convergence time $T\le T_d+T_c=T_1+T_2+T_3+T_4$.

☐

Remark 2.

From Lemma 5, it is clear that the system in Equation (21) is fixed-time stable because the transformation used to obtain this system ensures the same monotonicity of the system state before and after transformation. Therefore, based on this transformation, the predetermined performance of the original system is ensured as long as the transformed system is stable.

4. Simulation Results

Table 1. Hydrodynamic parameters of the Cybership Ⅱ.

Symbol	Values	Symbol	Values	Symbol	Values
$m$	23.8	$Y_{\dot{r}}$	0.0	$Y_r$	0.1079
$I_z$	1.76	$N_{\dot{v}}$	0.0	$Y_{\dot{v}}$	−10.0
$x_g$	0.046	$N_{\dot{r}}$	−1.0	$N_v$	0.1052
$X_u$	−0.7225	$Y_v$	−0.8612	$X_{\dot{u}}$	−2.0
$X_{\left|u\right|u}$	−1.3274	$Y_{\left|v\right|v}$	−36.2823	$N_{\left|v\right|v}$	5.0437
$X_{uuu}$	−5.8664	-	-	-	-

presents the model parameters for the Cybership II USV created by the Norwegian University of Science and Technology, which was utilized for simulation verification to assess the effectiveness of the proposed PPFTPT control system in terms of accuracy and speed.

The wind, wave, and current disturbances acting on the USV are assumed to be as follows:

$$\delta =\left[\begin{array}{l}4\cos (0.8\pi t)+0.1\sin (0.5\pi t)\\ \cos (0.2\pi t)+4\sin (\pi t)\\ 2\cos (2\pi t)+2\sin (0.1\pi t)+0.5\end{array}\right].$$

(56)

The control input for the desired trajectory is set to ${\tau }_d={[4,2{\cos }^2(0.01t),2{\sin }^2(0.01t)]}^T$.

The initial position and velocity of the desired trajectory are set to ${\eta }_d(0)={[15,16,3\pi /4]}^T$ and ${\nu }_d={[1,0,0]}^T$, respectively. The initial state of the USV is set to $\eta (0)={[15,14,2\pi /3]}^T$ and $\nu (0)={[0,0,0]}^T$.

The observer parameters are $l_1=l_2=2$, ${\Phi }_1={\Phi }_2=0.5$, ${\Phi }_0=0.2$, $p=0.8$ and $q=1.1$.

The prescribed performance control parameters are ${\overline{\eta }}_{ek,0}={[2,2,2]}^T$, ${\overline{\eta }}_{ek,\infty }={[0.1,0.1,0.1]}^T$, ${\overline{\zeta }}_{ek,0}={[1,1,1]}^T$, ${\overline{\zeta }}_{ek,\infty }={[0.1,0.1,0.1]}^T$, $\mu =0.8$, $\mathcal{l}=0.3$. The controller parameters are ${\kappa }_1={\kappa }_2=1$, ${\kappa }_3={\kappa }_4=1$, $m_1=9/5$, $m_2=5/9$, $n_1=9/5$, and $n_2=5/9$. The maximum values of force and moment of the propulsion system are set to ${\tau }_{u,\max }={\tau }_{v,\max }={\tau }_{r,\max }=50N$. Auxiliary system parameter matrix $c_1=diag(10,10,10),c_2=diag(5,5,5)$.

Additionally, the nonsingular fixed-time sliding mode (NFTSM) control scheme developed by Zhang et al. [21] and the finite-time control (FTC) scheme presented by Wang et al. [15] were selected for comparison to evaluate the performance of the proposed scheme. The relevant parameters were selected to be consistent with the original literature.

Figure 3 depicts the trajectory diagram of USV tracking, a predefined trajectory driven by three different control strategies. The PPFTPT approach developed in this study achieves quicker and more precise tracking while ensuring that the initial steady-state and transient responses of the system satisfy the preset requirements. Figure 4 and Figure 5 show the tracking error curves of the USV for the desired positions and velocities, respectively, of the three control schemes. The curve of rose red color represent the prescribed performance constraint. The findings demonstrate that the proposed PPFTPT scheme ensures the rapid convergence of the tracking error to the equilibrium point and obtains an instantaneous response with a smaller error overshoot, even under the dual influence of complex ocean disturbances and unknown system dynamics. By contrast, the convergence performance of the FTC scheme is affected by the initial state of the system. When the states are away from equilibrium and in a state of iterative correction, the tracking error is significant. The NFTSM control scheme can converge in a fixed time; however, it cannot guarantee the transient and steady-state responses of the system and has a large error overshoot.

Figure 6 displays the observation curves of the designed adaptive fixed-time lumped disturbance observer. As demonstrated, compared to the finite-time disturbance observer (FDO) proposed by Wang et al. [15], the AFISMLDO designed in this study accurately reconstructs and compensates for the lumped nonlinearity of the system and converges in a fixed time. Additionally, the presence of a symbolic function within the integral term of the interference observer effectively attenuates the jitter phenomenon in the sliding mode algorithm. Figure 7 shows the control input curve for a suitable control gain. It can be seen from the figure that control input in the initial stage of tracking does not exceed the maximum limit value under the action of the auxiliary system.

In addition, the control performance of the three control schemes was evaluated using two performance indicators: the absolute error integration criterion, $IA{E}_i$, and the time-multiplied absolute error integration criterion, $ITA{E}_i$. These are mathematically expressed as follows:

$$\begin{array}{l}IA{E}_i={\displaystyle \underset{0}{\overset{t}{\int }}\left|e_i(t)\right|}dt\\ ITA{E}_i={\displaystyle \underset{0}{\overset{t}{\int }}t\left|e_i(t)\right|}dt\end{array}.$$

(57)

As shown in

Table 2. Integrated absolute errors (IAEs) of the three control schemes.

	$\mathit{I}\mathit{A}{\mathit{E}}_{\mathit{x}}$	$\mathit{I}\mathit{A}{\mathit{E}}_{\mathit{y}}$	$\mathit{I}\mathit{A}{\mathit{E}}_{\mathit{\psi }}$	$\mathit{I}\mathit{A}{\mathit{E}}_{\mathit{u}}$	$\mathit{I}\mathit{A}{\mathit{E}}_{\mathit{v}}$	$\mathit{I}\mathit{A}{\mathit{E}}_{\mathit{r}}$
FTC	10.6960	9.4076	4.9003	5.8903	4.8753	4.6619
NFTSM	9.4515	8.0515	4.9516	3.6386	4.5563	4.2502
PPFTPT	6.2814	6.8879	4.0054	2.3791	1.5636	3.5810

and

Table 3. Integrated time absolute errors (ITAEs) of the three control schemes.

	$\mathit{I}\mathit{T}\mathit{A}{\mathit{E}}_{\mathit{x}}$	$\mathit{I}\mathit{T}\mathit{A}{\mathit{E}}_{\mathit{y}}$	$\mathit{I}\mathit{T}\mathit{A}{\mathit{E}}_{\mathit{\psi }}$	$\mathit{I}\mathit{T}\mathit{A}{\mathit{E}}_{\mathit{u}}$	$\mathit{I}\mathit{T}\mathit{A}{\mathit{E}}_{\mathit{v}}$	$\mathit{I}\mathit{T}\mathit{A}{\mathit{E}}_{\mathit{r}}$
FTC	31.1470	29.6483	23.8410	24.5132	23.5192	25.3359
NFTSM	29.5980	28.1374	24.8864	23.2581	23.3564	25.3644
PPFTPT	7.5815	8.4894	5.8946	3.8440	2.8446	3.1908

, the PPFTPT control strategy presented in this paper achieves the best index values and obtains the highest transient and steady-state accuracy in the presence of unknown dynamic characteristics of the system and complex external disturbances. In summary, the PPFTPT control strategy suggested in this paper can ensure that the USV completes accurate tracking of the desired trajectory and significantly improves the convergence speed and steady-state accuracy of the USV tracking control system.

To further verify the fixed-time preset performance convergence characteristics of the established PPFTPT scheme, three different initial states were set for the simulation study, and the initial state configurations are listed in

Table 4. Initial position and velocity.

	Initial Position	Initial Velocity
Case 1	${\eta }_0={[14,15,\pi /3]}^T$	${\nu }_0={[0.2,0.5,0.1]}^T$
Case 2	${\eta }_0={[15.5,15.5,\pi /2]}^T$	${\nu }_0={[0.6,0.4,0.3]}^T$
Case 3	${\eta }_0={[16,17,\pi ]}^T$	${\nu }_0={[0.5,0.3,0.2]}^T$

. The relevant parameters were selected as in previous experiments. As shown in Figure 8, for the three different initial states, the PPFTPT control strategy successfully drove the USV to track the predetermined trajectory almost simultaneously and accurately, and the relative motion states were maintained within the range set by the preset performance function during the entire tracking process. Figure 9 and Figure 10 show the variation curves of the USV position- and velocity-tracking errors, respectively, using the PPFTPT strategy. The results show that the position- and velocity-tracking errors always converged to the equilibrium point quickly within a fixed upper time bound T. Therefore, the influence of the initial state on the tracking performance of the controller was significantly reduced.

5. Conclusions

In this study, a novel prescribed performance fixed-time sliding mode control method was proposed to improve trajectory tracking of USVs within an expected time. An adaptive fixed-time lumped disturbance observer was developed to approximate the unknown environmental disturbances and system uncertainties, enhance the robustness and immunity of the control system, and achieve fixed-time convergence of the estimation errors. Subsequently, a PPFTPT control scheme was introduced with a prescribed performance function to enforce a full-state constraint on the system. The control scheme ensures the transient and steady-state responses of the system, eliminates the singularity potential, and achieves the fixed-time convergence of the system, independent of the initial state. Finally, simulation experiments were conducted using the Cybership II ship model, and a comparative analysis of the experimental results verified that the proposed PPFTPT scheme can accurately and quickly track the intended trajectory with a convergence time independent of the initial state of the system. Compared to traditional trajectory tracking methods, the proposed PPFTPT scheme in this study can achieve higher tracking accuracy and stronger robustness and can be stable in fixed time, which is more advantageous in practical applications. However, this study only investigates the trajectory tracking of a single USV, and in the future, we aim to study the cooperative formation tracking control of multiple USVs.