Adaptive Sliding Mode Control for Unmanned Surface Vehicle Trajectory Tracking Based on Event-Driven and Control Input Quantization

Abstract

This primary study aims to optimize network resource utilization efficiency in marine control systems. A novel event-triggering condition is proposed to significantly reduce communication traffic, where the error norm is squared while the input norm remains linear. To simulate realistic environmental disturbances, bounded unknown parameters are incorporated. Within the networked transmission architecture, input quantization is introduced, enabling the design of a quantized feedback controller without prior knowledge of quantization parameters. By integrating the event-triggering mechanism with sliding mode control, a quantized feedback control system is developed. The closed-loop system’s stability is rigorously proven via Lyapunov theory, with guaranteed boundedness of trajectory tracking errors. Numerical simulations validate the effectiveness of the proposed method for marine vehicle trajectory control under environmental disturbances.

1. Introduction

In the field of marine engineering, unmanned surface vehicles (USVs) play an indispensable role. Owing to the high degree of autonomy of USVs in the marine underwater environment, they are widely utilized in many fields, such as sea surveys, emergency rescues, and military applications [1,2,3,4]. In terms of these application levels, trajectory tracking control is a key fundamental control challenge for USVs and plays an extremely important role. Trajectory tracking means that USVs start from any initial position, sail into the desired route under the driving effect of the control system, and track the desired trajectory with a small error. During actual navigation, ships are susceptible to multiple external interferences, including wind, waves, and currents. These disturbances pose significant challenges to trajectory tracking control [5,6,7].

In recent years, trajectory tracking of unmanned surface vehicles has attracted growing research attention. Recent advances in USV trajectory tracking encompass multiple control strategies, including sliding mode [8,9,10,11,12,13], fuzzy [14,15,16,17,18,19,20], neural networks [21,22,23,24,25,26], and robust [27,28,29,30] controls. The study in [31] presents a hybrid control architecture that merges discrete sliding mode control techniques with predictive control mechanisms. The dynamics of USVs are discretized using the forward Euler method to facilitate trajectory tracking control in the presence of time-varying communication delays. The study [32] proposes an end-sliding mode control framework for the trajectory tracking of underactuated autonomous underwater vehicles (AUVs). This framework is designed to explicitly address nonlinear coupling, model uncertainties, and bounded environmental disturbances, thereby achieving finite-time convergence under nominal conditions. Reference [33] addresses the path-tracking performance improvement of unmanned surface vessels through a novel adaptive immersion- and invariance-based robust adaptive control algorithm. Reference [34] introduces an actor–critic learning-enhanced adaptive model predictive control (MPC) framework that automates parameter tuning for USV trajectory tracking, enabling robust performance across diverse marine environments. However, most studies have focused on designing better control schemes to improve the accuracy and consistency of USV trajectory tracking. Unlike conventional surface vessels, the communication system of USVs relies entirely on wireless channels and therefore must consider the communication constraints and actuator execution frequency in nautical practice.

The limited bandwidth of marine communication channels necessitates the consideration of input quantization in USV control design to efficiently manage communication resources [35,36,37]. The design of the input requires the coordination of various types of sensors [38]. Reference [39] proposes a specified-performance trajectory tracking control scheme for a symmetric USV subject to both model uncertainties and input quantization constraints. Reference [40] presents a quantized-state prescribed-performance control methodology that ensures robust USV tracking without dependence on either a priori dynamic models or function approximation schemes. Reference [41] develops a distributed quantized attitude control framework using neural networks to solve the formation tracking problem for networked unmanned vessels under model uncertainties, employing only output feedback. Reference [42] proposes a dynamic uniform quantizer with an adjustable sensitivity range to mitigate quantization effects in unmanned marine vehicle (UMV) remote control communication, along with a quantization sliding mode controller and a dynamic parameter adjustment strategy. It is worth noting that most of the current studies consider the quantized variable as an ingestion of the unquantized variable and analyze the boundary of this perturbation for compensation when. Therefore, it is imperative to investigate controller designs that incorporate input quantization.

Event-driven mechanisms can effectively conserve network resources. Therefore, event-triggered mechanisms have received considerable attention in recent years [43,44,45,46,47]. The integrated consideration of event-triggered control and signal quantization has been established as a practicable methodology for modern control systems [48]. In [49], an attack-resistant event-triggered security control scheme is proposed, which introduced a relative threshold event-triggered control strategy to save communication resources, including network bandwidth and computational power. Reference [50] introduces an adaptive control strategy that incorporates three specialized quantizers and neural network compensators to address input quantization and system uncertainties. In [51], a distributed dynamic event-triggered control scheme is proposed to address the swarm control problem for a multi-unmanned surface vehicle system with a virtual pilot. Reference [52] proposed a periodic event-triggered adaptive neural tracking control scheme for unmanned surface vehicles (USVs) to address their control challenges under replay attacks. Reference [53] couples roll dynamics with network delays via event-triggered linear parameter-varying linear matrix inequality path tracking, guiding our delay-robust marine design. Reference [54] integrates motion planning and output-feedback intersection tracking, inspiring multi-USV formation under partial-state feedback. Reference [55] delivers topology-adaptive platoons resilient to link losses, a fault-tolerant paradigm we translate to open-water fleets. Building on these insights, we propose an event-triggered sliding-mode scheme that unites input quantization and unknown wave disturbances for USVs. It is worth noting that the above studies mainly consider the communication resource saving brought by the introduction of the event-triggered mechanism, so it seems more critical to consider the event-triggered control mechanism with input quantization.

Based on the preceding analysis, this study investigates a trajectory-tracking control strategy for the USV that incorporates both input quantization and an event-triggering mechanism. The proposed USV trajectory tracking controller, which integrates event-triggering and input quantization, achieves satisfactory control performance without requiring prior knowledge of quantization parameters. Compared to existing approaches, the main contributions of this work are summarized as follows:

(1)

Unlike conventional USV trajectory tracking methods relying on continuous control signals [56,57,58], this study proposes a novel event-triggering mechanism. By dynamically adjusting the triggering threshold, the communication network bandwidth usage is significantly reduced while maintaining system stability. This design is especially suited for maritime applications with limited bandwidth, thereby enhancing the practicality of the strategy.

(2)

Unlike existing controller designs [59,60], this work proposes a linear analytical model of the input quantization process. This model removes the need for prior quantization knowledge during controller design, thereby improving its real-world applicability.

Notation: Let $\eta ={[x,y,\phi ]}^{\mathrm{T}}\in {\mathbb{R}}^3$ denote the USV position and yaw vector; $v={[u,\upsilon ,r]}^{\mathrm{T}}\in {\mathbb{R}}^3$ denote the body-fixed velocity vector; $\tau$ denote the control force vector; ${\tau }_w$ denote the bounded environmental disturbance vector; ${\eta }_d$ denote the desired reference trajectory and its velocity; $Q(\tau )$ denote quantized control input after processing by the quantizer; $q_1$ denote scaling parameter in the linear quantization model; $q_2$ denote the offset parameter in the linear quantization model; $\mu$ denote the lower bound of $q_1$; $\widehat{\mu }$ denote an estimate of $\mu$ in the adaptive law; $\tilde{\mu }$ denote the adaptive estimation error of $\mu$; $w$ denote intermediate control input signal in the controller design; $e_{\eta }$ denote the trajectory tracking error vector; $s$ denote the sliding surface; $c$ denote the sliding-surface design parameter; $\gamma$ denote the adaptive gain; $\sigma$ denote the regularization parameter in the adaptive law; $\epsilon$ denote a small positive constant used in the smooth $\tanh (\cdot )$ function; $sign(\cdot )$ denote the sign function; $‖\cdot ‖$ denote the Euclidean norm; $\beta$ denote the event-triggered parameter; $g$ denote the event-triggered threshold parameter.

2. Problem Formulation

Although USVs exhibit six degrees of freedom in water, the vertical, longitudinal, and transverse motions typically demonstrate relatively small amplitudes in practical applications. Therefore, these three degrees of freedom are commonly neglected in kinematic and dynamic modeling; instead, the USV’s motion under environmental disturbances is described by the following model [61]. The motion model of the USV is shown in Figure 1.

$$\left\{\begin{array}{l}\dot{\eta }=R(\phi )v\\ M\dot{v}=-C(v)-D(v)v+Q(\tau )+{\tau }_w\end{array}\right.$$

(1)

with

$$\begin{array}{l}R\left(\phi \right)=\left[\begin{array}{ccc}\cos \left(\phi \right)& -\sin \left(\phi \right)& 0\\ \sin \left(\phi \right)& \cos \left(\phi \right)& 0\\ 0& 0& 1\end{array}\right], M=\left[\begin{array}{ccc}{m}_{11}& 0& 0\\ 0& m_{22}& m_{23}\\ 0& m_{23}& m_{33}\end{array}\right],\\ C(v)=\left[\begin{array}{ccc}0& 0& -c_1\\ 0& 0& c_2\\ c_1& -c_2& 0\end{array}\right], D(v)=-\left[\begin{array}{ccc}{d}_{11}& 0& 0\\ 0& d_{22}& d_{23}\\ 0& -d_{32}& d_{33}\end{array}\right],\end{array}$$

(2)

where $\eta ={[x,y,\phi ]}^{\mathrm{T}}\in {\mathbb{R}}^3$, $x$, $y$, and $\phi$ denote surge position, sway position, and yaw angle of the underwater vehicle in a fixed coordinate system; the velocity vector $v={[u,\upsilon ,r]}^{\mathrm{T}}\in {\mathbb{R}}^3$ consists of the surge $(u)$, sway $(\upsilon )$, and yaw $(r)$ velocities. $Q(\tau )={[Q({\tau }_u),Q({\tau }_{\upsilon }),Q({\tau }_r)]}^{\mathrm{T}}\in {\mathbb{R}}^3$ indicates the quantized value of the control input.

Matrices $R(\phi )$, $M$, $C(v)$ and $D(v)$ represent the rotation matrix, inertia matrix, Coriolis, centripetal term matrices, and damping matrix, respectively, where $m_{11}=m-X_{\dot{u}}$, $m_{22}=m-Y_{\dot{\upsilon }}$, $m_{23}=m{x}_g-Y_{\dot{r}}$, $m_{33}=I_Z-Z_{\dot{r}}$, $c_1=m_{22}\upsilon +m_{23}r$, $c_2=m_{11}u$, $d_{11}=X_u+X_{u\left|u\right|}\left|u\right|+X_{uuu}{u}^2$, $d_{22}=Y_{\upsilon }+Y_{\upsilon \left|\upsilon \right|}\left|v\right|+Y_{\upsilon \left|r\right|}\left|r\right|$, and $d_{33}=Z_r+Z_{r\left|\upsilon \right|}\left|\upsilon \right|+Z_{r\left|r\right|}\left|r\right|$. $m$ is the mass of USV. $X_{\dot{u}}$, $Y_{\dot{\upsilon }}$, and $Z_{\dot{r}}$ denote the added-mass coefficients that, respectively, quantify the inertial reaction forces generated by surge acceleration, sway acceleration, and yaw angular acceleration of the vehicle. $X_u$, $X_{u\left|u\right|}$, $X_{uuu}$, $Y_{\upsilon }$, $Y_{\upsilon \left|\upsilon \right|}$, $Y_{\upsilon \left|r\right|}$, $Y_r$, $Y_{r\left|\upsilon \right|}$, $Y_{r\left|r\right|}$, $Z_{\upsilon }$, $Z_{\upsilon \left|\upsilon \right|}$, $Z_r$, $Z_{r\left|\upsilon \right|}$, and $Z_{r\left|r\right|}$ represent the hydrodynamic coefficients. $x_g$ denotes the center of gravity in the fixed coordinate system of the hull, and $I_Z$ denotes the moment of inertia of the shaft. The physical meanings of the parameters of the USV model are presented in

Table 1. USV model parameters.

Symbol	Physical Meaning
$m$	Total mass
$x_g$	Center of gravity of the ship’s fixed coordinate system
$I_Z$	Yaw moment of inertia
$X_{\dot{u}},Y_{\dot{\upsilon }},Y_{\dot{r}},Z_{\dot{r}}$	Added-mass coefficients
$X_u,Y_{\upsilon },Y_r,Z_{\upsilon },Z_r$	Linear hydraulic damping Coefficient
$X_{u\left|u\right|},Y_{\upsilon \left|\upsilon \right|},Y_{\upsilon \left|r\right|},Y_{r\left|\upsilon \right|},Y_{r\left|r\right|},Z_{\upsilon \left|\upsilon \right|},Z_{r\left|r\right|}$	Second-order nonlinear damping coefficient
$X_{uuu}$	High-order nonlinear damping coefficient
$m_{11},m_{22},m_{23},m_{33}$	Elements of inertia matrix
$c_1,c_2$	Coriolis and centripetal terms
$d_{11},d_{22},d_{33}$	Linear damping coefficients

.

This paper addresses the quantization of control inputs using a uniform quantizer, defined as follows:

$$Q(\tau )=kround({\displaystyle \frac{\tau }{k}})$$

(3)

where $k$ is the quantization level, and the control signal $\tau$ passes through the quantizer and becomes $Q(\tau )$.

Let $Q(\tau )=q_1\tau +q_2$,

$$\left\{\begin{array}{l}{q}_1=\left\{\begin{array}{cc}{\displaystyle \frac{‖Q(\tau )‖}{‖\tau ‖}},& ‖\tau ‖\ge o\\ 1,& ‖\tau ‖<o\end{array}\right.\\ q_2=\left\{\begin{array}{cc}0,& ‖\tau ‖\ge o\\ Q(\tau )-\tau ,& ‖\tau ‖<o\end{array}\right.\end{array}\right.,$$

(4)

where $q_1$ is an unknown parameter. As the sign remains unaltered throughout the quantization process, we can conclude that $q_1>0$. Moreover, if $‖\tau ‖<o$, $q_2$ is bounded as well due to the boundedness of $Q(\tau )$. Let $‖q_2‖\le {\overline{q}}_2$, where ${\overline{q}}_2>0$ is the bound of $q_2$.

Assumption 1

([62]). The environmental disturbance ${\tau }_w={[{\tau }_{wu},{\tau }_{w\upsilon },{\tau }_{wr}]}^{\mathrm{T}}\in {\mathbb{R}}^3$ is bounded but unknown, and $\left|{\tau }_{wu}\right|<{\vartheta }_1$, $\left|{\tau }_{w\upsilon }\right|<{\vartheta }_2$, $\left|{\tau }_{wr}\right|<{\vartheta }_3$, ${\vartheta }_1>0$, ${\vartheta }_2>0$, ${\vartheta }_3>0$ are upper bounds on the unknown external time-varying disturbance variables.

Consider a reference trajectory:

$${\eta }_d={[x_d,y_d,{\phi }_d]}^{\mathrm{T}}\in {\mathbb{R}}^3$$

(5)

where $x_d$, $y_d$, and ${\phi }_d$ denote the reference surge position, sway position, and yaw angle of the USV in the body-fixed coordinate system, respectively.

Assumption 2. 

The vector  ${\eta }_d$ is continuously differentiable and its time derivatives are all bounded.

The control objective of this paper is to create a reference trajectory ${\eta }_d$, and design a quantized feedback adaptive trajectory tracking control law $\tau$ for the USV such that its actual trajectory $\eta$ tracks the reference trajectory ${\eta }_d$, namely

$$\underset{t\to \infty }{\lim }‖\eta -{\eta }_d‖\le {\epsilon }_1$$

(6)

where ${\epsilon }_1$ is an arbitrarily small positive constant.

3. Design of Quantitative Feedback Controller

The controller is designed under the full-state feedback assumption. Each component of the state vector-position, heading, linear velocity, and angular velocity is obtained by fusing measurements from a dedicated sensor suite. Position $(x,y)$ is measured by the position sensor; heading $\phi$ is measured by the heading sensor; linear velocity $(u,\upsilon )$ is measured by the velocity sensor, and the angular velocity $r$ is measured by the rate sensor. Future work will extend the results to output-feedback schemes [63,64,65] to relax the full state measurement requirement. In this section, an input-quantized adaptive sliding mode control (SMC) method is designed for USVs. A depiction of the control scheme is presented in Figure 2.

Define the unknown vector $F\in {\mathbb{R}}^3$ as

$$F(v)=M^{-1}(-C(v)v-D(v)v+{\tau }_w)$$

(7)

Substituting Equation (6) into Equation (1), the simplified kinematics and dynamics of the USV can be expressed as

$$\left\{\begin{array}{l}\dot{\eta }=R(\phi )v\\ \dot{v}=F+M^{-1}Q(\tau )\end{array}\right.$$

(8)

The trajectory tracking error is defined as

$$e_{\eta }=\eta -{\eta }_d$$

(9)

The trajectory tracking error is derived as follows:

$${\dot{e}}_{\eta }=\dot{\eta }-{\dot{\eta }}_d$$

(10)

Since $\dot{\eta }=R(\phi )v$ in Equation (7), substituting it into Equation (10) gives

$${\dot{e}}_{\eta }=\dot{\eta }-{\dot{\eta }}_d=R(\phi )v-{\dot{\eta }}_d$$

(11)

The derivative of the trajectory tracking error is again derived as

$${\ddot{e}}_{\eta }=\ddot{\eta }-{\ddot{\eta }}_d$$

(12)

The derivation of $\dot{\eta }=R(\phi )v$ gives the following equation:

$$\ddot{\eta }=\dot{R}(\phi )v+R(\phi )\dot{v}$$

(13)

Substituting Equation (13) into Equation (12), we obtain

$${\ddot{e}}_{\eta }=\dot{R}(\phi )v+R(\phi )\dot{v}-{\ddot{\eta }}_d$$

(14)

Since $\dot{v}=F+M^{-1}Q(\tau )$ in Equation (7), substituting it into Equation (14) gives

$${\ddot{e}}_{\eta }=\dot{R}(\phi )v+R(\phi )\dot{v}-{\ddot{\eta }}_d=\dot{R}(\phi )v+R(\phi )(F+M^{-1}Q(\tau ))-{\ddot{\eta }}_d$$

(15)

Lemma 1

(Application of Young’s Inequality). Let  $a,b>0$, $p,q>1$ and ${\displaystyle \frac{1}{p}}+{\displaystyle \frac{1}{q}}=1$, so that $ab\le {\displaystyle \frac{a^p}{p}}+{\displaystyle \frac{b^q}{q}}$.

Define the sliding mold surface as

$$s={\dot{e}}_{\eta }+c{e}_{\eta }$$

(16)

where c is a positive parameter that is crucial for the stability of the sliding surface. The time derivative of the sliding surface Equation (6) is given by

$$\dot{s}={\ddot{e}}_{\eta }+c{\dot{e}}_{\eta }$$

(17)

Substituting Equation (13) into Equation (17), we can obtain

$$\dot{s}={\ddot{e}}_{\eta }+c{\dot{e}}_{\eta }=\dot{R}(\phi )v+R(\phi )(F+M^{-1}Q(\tau ))-{\ddot{\eta }}_d+c{\dot{e}}_{\eta }$$

(18)

It follows that

$$\begin{array}{cc}\hfill s^{\mathrm{T}}\dot{s}& =s^{\mathrm{T}}[\dot{R}(\phi )v+R(\phi )\left[F+M^{-1}Q(\tau )\right]-{\ddot{\eta }}_d+c{\dot{e}}_{\eta }]\hfill \\ & =s^{\mathrm{T}}\dot{R}(\phi )v+s^{\mathrm{T}}R(\phi )F+s^{\mathrm{T}}R(\phi )M^{-1}Q(\tau )-s^{\mathrm{T}}{\ddot{\eta }}_d+s^{\mathrm{T}}c{\dot{e}}_{\eta }\hfill \end{array}$$

(19)

Substituting $Q(\tau )=q_1\tau +q_2$ into Equation (19) gives

$$\begin{array}{cc}\hfill s^{\mathrm{T}}\dot{s}& =s^{\mathrm{T}}\dot{R}(\phi )v+s^{\mathrm{T}}R(\phi )F+s^{\mathrm{T}}R(\phi )M^{-1}(q_1\tau +q_2)-s^{\mathrm{T}}{\ddot{\eta }}_d+s^{\mathrm{T}}c{\dot{e}}_{\eta }\hfill \\ & =s^{\mathrm{T}}\dot{R}(\phi )v+s^{\mathrm{T}}R(\phi )F+s^{\mathrm{T}}R(\phi )M^{-1}{q}_1\tau +s^{\mathrm{T}}R(\phi )M^{-1}{q}_2-s^{\mathrm{T}}{\ddot{\eta }}_d+s^{\mathrm{T}}c{\dot{e}}_{\eta }\hfill \end{array}$$

(20)

From Lemma 1 it follows that

$$s^{\mathrm{T}}R(\phi )M^{-1}{q}_2\le {\displaystyle \frac{1}{2}}{s}^{\mathrm{T}}R(\phi )M^{-1}s+{\displaystyle \frac{1}{2‖M‖}}{‖q_2‖}^2$$

(21)

where $a$ is defined as $s^{\mathrm{T}}R(\phi )M^{-1}s$ and $b$ as ${\displaystyle \frac{1}{‖M‖}}{‖q_2‖}^2$, rotation matrix $R(\phi )$ satisfying $R{(\phi )}^{\mathrm{T}}R(\phi )=I$ and $‖R(\phi )‖=1$, and a symmetric positive definite matrix $M$ with $M^T=M$.

Knowing that $‖q_2‖\le {\overline{q}}_2$, we obtain

$$s^{\mathrm{T}}R(\phi )M^{-1}{q}_2\le {\displaystyle \frac{1}{2}}{s}^{\mathrm{T}}R(\phi )M^{-1}s+{\displaystyle \frac{1}{2‖M‖}}{{\overline{q}}_2}^2$$

(22)

Substituting inequality Equation (22) into Equation (20) yields the inequality as follows:

$$\begin{array}{cc}\hfill s^{\mathrm{T}}\dot{s}& =s^{\mathrm{T}}\dot{R}(\phi )v+s^{\mathrm{T}}R(\phi )F+s^{\mathrm{T}}R(\phi )M^{-1}{q}_1\tau +s^{\mathrm{T}}R(\phi )M^{-1}{q}_2-s^{\mathrm{T}}{\ddot{\eta }}_d+s^{\mathrm{T}}c{\dot{e}}_{\eta }\hfill \\ & \le s^{\mathrm{T}}\dot{R}(\phi )v+s^{\mathrm{T}}R(\phi )F+s^{\mathrm{T}}R(\phi )M^{-1}{q}_1\tau +{\displaystyle \frac{1}{2}}{s}^{\mathrm{T}}R(\phi )M^{-1}s+{\displaystyle \frac{1}{2‖M‖}}{\overline{q}}_2^2-s^{\mathrm{T}}{\ddot{\eta }}_d+s^{\mathrm{T}}c{\dot{e}}_{\eta }\hfill \\ & =s^{\mathrm{T}}(-sl+sl+\dot{R}(\phi )v+R(\phi )F+{\displaystyle \frac{1}{2}}R(\phi )M^{-1}s+c{\dot{e}}_{\eta }-{\ddot{\eta }}_d)+s^{\mathrm{T}}R(\phi )M^{-1}{q}_1\tau +{\displaystyle \frac{1}{2‖M‖}}{\overline{q}}_2^2\hfill \end{array}$$

(23)

where $l$ is a positive constant, $\overline{u}=sl+\dot{R}(\phi )v+R(\phi )F+{\displaystyle \frac{1}{2}}R(\phi )M^{-1}s+c{\dot{e}}_{\eta }-{\ddot{\eta }}_d$, and we can obtain

$$\begin{array}{cc}{s}^{\mathrm{T}}\dot{s}\hfill & \le s^{\mathrm{T}}(-sl+sl+\dot{R}(\phi )v+R(\phi )F+{\displaystyle \frac{1}{2}}R(\phi )M^{-1}s+c{\dot{e}}_{\eta }-{\ddot{\eta }}_d)+s^{\mathrm{T}}R(\phi )M^{-1}{q}_1\tau +{\displaystyle \frac{1}{2‖M‖}}{\overline{q}}_2^2\hfill \\ & =-s^{\mathrm{T}}sl+s\overline{u}+s^{\mathrm{T}}R(\phi )M^{-1}{q}_1\tau +{\displaystyle \frac{1}{2‖M‖}}{\overline{q}}_2^2\hfill \end{array}$$

(24)

Remark 1. 

In the process of designing the control law of the system, $q_1$ must be known because $Q(\tau )=q_1\tau +q_2$, and $\tau$ must also be known. $q_1$ acts as an adaptive estimator, enabling the quantization feedback control law to be applied without prior knowledge of the quantization parameters. The estimation is carried out using the lower bound of $q_1$ to prevent singularities in the case of zero estimation. The time-varying parameter $\mu ={\displaystyle \frac{1}{q_{1\min }}}$ denotes the lower limit of $q_1$.

Here, the intermediate control input signal is

$$w=-\overline{m}\tanh ({\displaystyle \frac{s^{T}R(\phi )M^{-1}\overline{m}}{\epsilon }})-{\displaystyle \frac{M{R}^{\mathrm{T}}(\phi ){\widehat{\mu }}^2{\overline{u}}^{\mathrm{T}}\overline{u}s}{\left|\widehat{\mu }\right|‖s‖‖\overline{u}‖+\rho }}$$

(25)

The adaptive rate is

$$\dot{\widehat{\mu }}=s^{\mathrm{T}}\gamma \overline{u}-\gamma \sigma \widehat{\mu }$$

(26)

where $\epsilon$, $\rho$, $\gamma$, and $\sigma$ are positive parameters; $\tilde{\mu }=\widehat{\mu }-\mu$, and from $q_1>0$, it follows that $\mu >0$.

Remark 2. 

The adaptive law Equation (26) aims at estimating the bounds of the quantization coefficients. Therefore, controller design does not require prior knowledge of the quantization parameters. Of course, the quantization parameters can be adapted to the specific performance needs of the system.

Based on the quantification and also reducing the communication burden, we developed a time-varying threshold event triggering control strategy, which is as follows:

$$\tau =w(t_k),\forall t\in [t_k,t_{k+1})$$

(27)

$$t_{k+1}=\inf \{t\in \mathrm{R},{‖e(t)‖}^2\ge \beta ‖\tau ‖+g\}$$

(28)

where the event trigger threshold $g$ is a positive parameter, $e(t)=w(t)-\tau (t)$.

Remark 3. 

The following observations can be made about the time-varying threshold-based event-triggering strategy: during the time interval $t\in [t_k,t_{k+1})$, define the specific values of $\tau =w(t_k)$, $t=t_{k+1}$ to be determined by the condition ${‖e(t)‖}^2\ge \beta ‖\tau ‖+g$, whereas for ${‖e(t)‖}^2<\beta ‖\tau ‖+g$, the control input signal is not passed, and the value of the controller remains unchanged. In order to save communication, a larger event trigger threshold $g$ is used when the control input signal is large. Conversely, when the control input signal tends to zero, a smaller threshold $g$ is used to improve the control accuracy.

From Equation (28), the existence of a continuously varying parameter ${\lambda }_2(t)$, satisfies ${\lambda }_2(t_k)=0$, ${\lambda }_2(t_{k+1})=\pm 1$ and $\left|{\lambda }_2(t_k)\right|\le 1$. It follows that

$$\begin{array}{cc}\hfill w(t)& =\tau (t)+e(t)\hfill \\ & =\tau (t)+{\lambda }_2(t)\sqrt{\beta ‖\tau ‖+g}\hfill \\ & =\tau (t)+{\lambda }_2(t)\sqrt{\beta \tau sign(\tau (t))+g}\hfill \end{array}$$

(29)

Let ${\lambda }_2^2(t)={\lambda }_3(t)$ and ${\lambda }_3(t)sign(\tau (t))={\lambda }_1(t)$, we can obtain

$$w^2(t)-2w(t)\tau (t)+{\tau }^2(t)-{\lambda }_1(t)\beta \tau ={\lambda }_3(t)g$$

(30)

Let $-{\lambda }_1(t)\beta =\chi$ and ${\lambda }_3(t)g=d$, then we obtain

$$w^2(t)-2w(t)\tau (t)+{\tau }^2(t)+\chi \tau (t)=d$$

(31)

Since $2w(t)\tau (t)\le w^2(t)+{\tau }^2(t)$ holds, $2w^2(t)+2{\tau }^2(t)+\chi \tau (t)=\zeta$, $\zeta \ge d$ follows, we can obtain

$$2w^2(t)+2{(\tau (t)+{\displaystyle \frac{\chi }{4}})}^2-{\displaystyle \frac{{\chi }^2}{8}}=\zeta$$

(32)

$$w^2(t)+{(\tau (t)+{\displaystyle \frac{\chi }{4}})}^2={\displaystyle \frac{\zeta }{2}}+{\displaystyle \frac{{\chi }^2}{16}}$$

(33)

Let ${\displaystyle \frac{\zeta }{2}}+{\displaystyle \frac{{\chi }^2}{16}}={\theta }^2$, and we can obtain

$$\begin{array}{l}\sqrt{w^2(t)-{\theta }^2}=\tau (t)+{\displaystyle \frac{\chi }{4}}\\ \tau (t)\le w(t)-{\displaystyle \frac{\chi }{4}}\end{array}$$

(34)

Define $\tau (t)=w(t)-\overline{m}$, where $\overline{m}\ge {\displaystyle \frac{\chi }{4}}$.

4. Stability Analysis

This section presents the stability analysis of the proposed quantized feedback adaptive control method.

Theorem 1.

Considering the presence of environmental disturbances in the state and input quantization of the USV, the proposed quantized feedback adaptive trajectory tracking control strategy ensures that all signals in the closed-loop control system are ultimately bounded.

Proof. 

Define the Lyapunov function as

$$V={\displaystyle \frac{1}{2}}{s}^{\mathrm{T}}s+{\displaystyle \frac{1}{2\gamma \mu }}{\tilde{\mu }}^2$$

(35)

Taking the derivative of Equation (35),

$$\dot{V}=s^{\mathrm{T}}\dot{s}+{\displaystyle \frac{1}{\gamma \mu }}\tilde{\mu }\dot{\widehat{\mu }}$$

(36)

Substituting Equation (24) into Equation (36) yields

$$\begin{array}{cc}\dot{V}\hfill & \le -s^{\mathrm{T}}sl+s^{\mathrm{T}}\overline{u}+s^{\mathrm{T}}R(\phi )M^{-1}{q}_1\tau +{\displaystyle \frac{1}{2‖M‖}}{\overline{q}}_2^2+{\displaystyle \frac{1}{\gamma \mu }}\tilde{\mu }\dot{\widehat{\mu }}\hfill \\ & =-s^{\mathrm{T}}sl+s^{\mathrm{T}}\overline{u}+s^{\mathrm{T}}R(\phi )M^{-1}{q}_1(w(t)-\overline{m})+{\displaystyle \frac{1}{2‖M‖}}{\overline{q}}_2^2+{\displaystyle \frac{1}{\gamma \mu }}\tilde{\mu }\dot{\widehat{\mu }}\hfill \end{array}$$

(37)

Bring $w(t)$ into Equation (37) to obtain

$$\begin{array}{cc}\dot{V}\hfill & \le -s^{\mathrm{T}}sl+s^{\mathrm{T}}\overline{u}+s^{\mathrm{T}}R(\phi )M^{-1}{q}_1(-\overline{m}\tanh ({\displaystyle \frac{s^{T}R(\phi )M^{-1}\overline{m}}{\epsilon }})-{\displaystyle \frac{R^{\mathrm{T}}(\phi )Ms{\widehat{\mu }}^2{\overline{u}}^{\mathrm{T}}\overline{u}}{\left|\widehat{\mu }\right|‖s‖‖\overline{u}‖+\rho }})\hfill \\ & -s^{\mathrm{T}}R(\phi )M^{-1}{q}_1\overline{m}+{\displaystyle \frac{1}{2‖M‖}}{\overline{q}}_2^2+{\displaystyle \frac{1}{\gamma \mu }}\tilde{\mu }\dot{\widehat{\mu }}\hfill \\ & \le -s^{\mathrm{T}}sl+s^{\mathrm{T}}\overline{u}+s^{\mathrm{T}}R(\phi )M^{-1}{q}_1(-\overline{m}\tanh ({\displaystyle \frac{s^{\mathrm{T}}R(\phi )M^{-1}\overline{m}}{\epsilon }})-{\displaystyle \frac{R^{\mathrm{T}}(\phi )Ms{\widehat{\mu }}^2{\overline{u}}^{\mathrm{T}}\overline{u}}{\left|\widehat{\mu }\right|‖s‖‖\overline{u}‖+\rho }})\hfill \\ & +‖s^{\mathrm{T}}R(\phi )M^{-1}\overline{m}‖q_1+{\displaystyle \frac{1}{2‖M‖}}{\overline{q}}_2^2+{\displaystyle \frac{1}{\gamma \mu }}\tilde{\mu }\dot{\widehat{\mu }}\hfill \\ & =-s^{\mathrm{T}}sl+s^{\mathrm{T}}\overline{u}-s^{\mathrm{T}}R(\phi )M^{-1}{q}_1\overline{m}\tanh ({\displaystyle \frac{s^{\mathrm{T}}R(\phi )M^{-1}\overline{m}}{\epsilon }})-q_1{\displaystyle \frac{s^{\mathrm{T}}s{\widehat{\mu }}^2{\overline{u}}^{\mathrm{T}}\overline{u}}{\left|\widehat{\mu }\right|‖s‖‖\overline{u}‖+\rho }}\hfill \\ & +‖s^{\mathrm{T}}R(\phi )M^{-1}\overline{m}‖q_1+{\displaystyle \frac{1}{2‖M‖}}{\overline{q}}_2^2+{\displaystyle \frac{1}{\gamma \mu }}\tilde{\mu }\dot{\widehat{\mu }}\hfill \end{array}$$

(38)

□

Lemma 2.

Let $x\ge 0$ and $\epsilon >0$, and the following inequality holds [66]:

$$0\le x-x\tanh ({\displaystyle \frac{x}{\epsilon }})\le \kappa \epsilon$$

(39)

where $\kappa =1.2785\ln (1+e^{-1.2785})\approx 0.2785$ is a universal constant.

Since $\left|a\right|-{\displaystyle \frac{a^2}{\rho +\left|a\right|}}={\displaystyle \frac{\rho \left|a\right|}{\rho +\left|a\right|}}<\rho$, it follows that $-{\displaystyle \frac{a^2}{\rho +\left|a\right|}}\le \rho \pm a$. Consequently,

$$-{\displaystyle \frac{{\widehat{\mu }}^2{s}^{\mathrm{T}}s{\overline{u}}^{\mathrm{T}}\overline{u}}{\left|\widehat{\mu }\right|‖s‖‖\overline{u}‖+\rho }}\le \rho -\widehat{\mu }{s}^{\mathrm{T}}\overline{u}$$

(40)

Consider $q_1>q_{1\min }={\displaystyle \frac{1}{\mu }}>0$/. Consequently,

$$-q_1{\displaystyle \frac{{\widehat{\mu }}^2{s}^{\mathrm{T}}s{\overline{u}}^{\mathrm{T}}\overline{u}}{\left|\widehat{\mu }\right|‖s‖‖\overline{u}‖+\rho }}\le {\displaystyle \frac{1}{\mu }}(\rho -\widehat{\mu }{s}^{\mathrm{T}}\overline{u})$$

(41)

Substituting Equation (41) into Equation (38) yields

$$\begin{array}{cc}\dot{V}\hfill & \le -s^{\mathrm{T}}sl+s^{\mathrm{T}}\overline{u}-s^{\mathrm{T}}R(\phi )M^{-1}{q}_1\overline{m}\tanh ({\displaystyle \frac{s^{T}R(\phi )M^{-1}\overline{m}}{\epsilon }})+{\displaystyle \frac{1}{\mu }}(\rho -\widehat{\mu }{s}^{\mathrm{T}}\overline{u})\hfill \\ & +‖s^{\mathrm{T}}R(\phi )M^{-1}\overline{m}‖q_1+{\displaystyle \frac{1}{2‖M‖}}{\overline{q}}_2^2+{\displaystyle \frac{1}{\gamma \mu }}\tilde{\mu }\dot{\widehat{\mu }}\hfill \end{array}$$

(42)

Substituting the adaptive rate Equation (26) into Equation (42) yields

$$\begin{array}{cc}\dot{V}\hfill & \le -s^{\mathrm{T}}sl+s^{\mathrm{T}}\overline{u}-s^{\mathrm{T}}R(\phi )M^{-1}{q}_1\overline{m}\tanh ({\displaystyle \frac{s^{T}R(\phi )M^{-1}\overline{m}}{\epsilon }})+{\displaystyle \frac{1}{\mu }}(\rho -\widehat{\mu }{s}^{\mathrm{T}}\overline{u})\hfill \\ & +‖s^{\mathrm{T}}R(\phi )M^{-1}\overline{m}‖q_1+{\displaystyle \frac{1}{2‖M‖}}{\overline{q}}_2^2+{\displaystyle \frac{1}{\gamma \mu }}\tilde{\mu }(s^{\mathrm{T}}\gamma \overline{u}-\gamma \sigma \widehat{\mu })\hfill \\ & =-s^{\mathrm{T}}sl+s^{\mathrm{T}}\overline{u}-s^{\mathrm{T}}R(\phi )M^{-1}{q}_1\overline{m}\tanh ({\displaystyle \frac{s^{T}R(\phi )M^{-1}\overline{m}}{\epsilon }})+{\displaystyle \frac{1}{\mu }}(\rho -\widehat{\mu }{s}^{\mathrm{T}}\overline{u})\hfill \\ & +‖s^{\mathrm{T}}R(\phi )M^{-1}\overline{m}‖q_1+{\displaystyle \frac{1}{2‖M‖}}{\overline{q}}_2^2+{\displaystyle \frac{1}{\mu }}\tilde{\mu }(s^{\mathrm{T}}\overline{u}-\sigma \widehat{\mu })\hfill \end{array}$$

(43)

due to

$$-\tilde{\mu }\widehat{\mu }=-\tilde{\mu }(\tilde{\mu }+\mu )\le -{\tilde{\mu }}^2+{\displaystyle \frac{1}{2}}{\tilde{\mu }}^2+{\displaystyle \frac{1}{2}}{\mu }^2=-{\displaystyle \frac{1}{2}}{\tilde{\mu }}^2+{\displaystyle \frac{1}{2}}{\mu }^2$$

(44)

then $-{\displaystyle \frac{1}{\mu }}\tilde{\mu }\sigma \widehat{\mu }=-{\displaystyle \frac{\sigma }{2\mu }}{\tilde{\mu }}^2+{\displaystyle \frac{\sigma }{2}}\mu$ and thus

$$\begin{array}{cc}{\displaystyle \frac{1}{\mu }}(\rho -\widehat{\mu }{s}^{\mathrm{T}}\overline{u})+{\displaystyle \frac{1}{\mu }}\tilde{\mu }(s^{\mathrm{T}}\overline{u}-\sigma \widehat{\mu })\hfill & ={\displaystyle \frac{\rho }{\mu }}-{\displaystyle \frac{\widehat{\mu }{s}^{\mathrm{T}}\overline{u}}{\mu }}+{\displaystyle \frac{\tilde{\mu }{s}^{\mathrm{T}}\overline{u}}{\mu }}-{\displaystyle \frac{\tilde{\mu }\sigma \widehat{\mu }}{\mu }}\hfill \\ & ={\displaystyle \frac{\rho }{\mu }}-{\displaystyle \frac{\widehat{\mu }{s}^{\mathrm{T}}\overline{u}}{\mu }}+{\displaystyle \frac{(\widehat{\mu }-\mu )s^{\mathrm{T}}\overline{u}}{\mu }}-{\displaystyle \frac{\tilde{\mu }\sigma \widehat{\mu }}{\mu }}\hfill \\ & ={\displaystyle \frac{\rho }{\mu }}-s^{\mathrm{T}}\overline{u}-{\displaystyle \frac{\sigma }{2\mu }}{\tilde{\mu }}^2+{\displaystyle \frac{\sigma }{2}}\mu \hfill \end{array}$$

(45)

Substituting Equation (45) into Equation (42) yields

$$\begin{array}{cc}\dot{V}\hfill & \le -s^{\mathrm{T}}sl+s^{\mathrm{T}}\overline{u}-s^{\mathrm{T}}R(\phi )M^{-1}{q}_1\overline{m}\tanh ({\displaystyle \frac{s^{T}R(\phi )M^{-1}\overline{m}}{\epsilon }})\hfill \\ & +‖s^{\mathrm{T}}R(\phi )M^{-1}\overline{m}‖q_1+{\displaystyle \frac{1}{2‖M‖}}{\overline{q}}_2^2+{\displaystyle \frac{\rho }{\mu }}-s^{\mathrm{T}}\overline{u}-{\displaystyle \frac{\sigma }{2\mu }}{\tilde{\mu }}^2+{\displaystyle \frac{\sigma }{2}}\mu \hfill \\ & =-s^{\mathrm{T}}sl-s^{\mathrm{T}}R(\phi )M^{-1}{q}_1\overline{m}\tanh ({\displaystyle \frac{s^{T}R(\phi )M^{-1}\overline{m}}{\epsilon }})\hfill \\ & +‖s^{\mathrm{T}}R(\phi )M^{-1}\overline{m}‖q_1+{\displaystyle \frac{1}{2‖M‖}}{\overline{q}}_2^2+{\displaystyle \frac{\rho }{\mu }}-{\displaystyle \frac{\sigma }{2\mu }}{\tilde{\mu }}^2+{\displaystyle \frac{\sigma }{2}}\mu \hfill \end{array}$$

(46)

Considering $q_1>0$, the inequality is obtained from Lemma 2 as follows:

$$0\le ‖s^{\mathrm{T}}R(\phi )M^{-1}\overline{m}‖q_1-s^{\mathrm{T}}R(\phi )M^{-1}{q}_1\overline{m}\tanh ({\displaystyle \frac{s^{T}R(\phi )M^{-1}\overline{m}}{\epsilon }})\le 0.2785\epsilon q_1$$

(47)

We can obtain

$$\begin{array}{cc}\dot{V}\hfill & \le -s^{\mathrm{T}}sl-s^{\mathrm{T}}R(\phi )M^{-1}{q}_1\overline{m}\tanh ({\displaystyle \frac{s^{T}R(\phi )M^{-1}\overline{m}}{\epsilon }})\hfill \\ & +‖s^{\mathrm{T}}R(\phi )M^{-1}\overline{m}‖q_1+{\displaystyle \frac{1}{2‖M‖}}{\overline{q}}_2^2+{\displaystyle \frac{\rho }{\mu }}-{\displaystyle \frac{\sigma }{2\mu }}{\tilde{\mu }}^2+{\displaystyle \frac{\sigma }{2}}\mu \hfill \\ & =-s^{\mathrm{T}}sl+0.2785\epsilon q_1+{\displaystyle \frac{1}{2‖M‖}}{\overline{q}}_2^2+{\displaystyle \frac{\rho }{\mu }}-{\displaystyle \frac{\sigma }{2\mu }}{\tilde{\mu }}^2+{\displaystyle \frac{\sigma }{2}}\mu \hfill \end{array}$$

(48)

Subsequently, we can obtain

$$\begin{array}{cc}\dot{V}\hfill & \le -s^{\mathrm{T}}sl-0.2785\epsilon {\overline{q}}_1+{\displaystyle \frac{\rho }{\mu }}-{\displaystyle \frac{\sigma }{2\mu }}{\tilde{\mu }}^2+{\displaystyle \frac{\sigma }{2}}\mu +{\displaystyle \frac{1}{2‖M‖}}{\overline{q}}_2^2\hfill \\ & \le -s^{\mathrm{T}}sl-{\displaystyle \frac{\sigma }{2\mu }}{\tilde{\mu }}^2+l_1\hfill \\ & \le -{\rho }_{1}V+l_1\hfill \end{array}$$

(49)

where $l_1=-0.2785\epsilon {\overline{q}}_1+{\displaystyle \frac{\rho }{\mu }}+{\displaystyle \frac{\sigma }{2}}\mu +{\displaystyle \frac{1}{2‖M‖}}{\overline{q}}_2^2$, $q_1\le {\overline{q}}_1$, ${\rho }_1=\min \{2l,\gamma \sigma \}$.

Solving the inequality $\dot{V}\le -{\rho }_{1}V+l_1$ yields

$$0\le V(t)\le (V(0)-{\displaystyle \frac{l_1}{{\rho }_1}})e^{-{\rho }_{1}t}+{\displaystyle \frac{l_1}{{\rho }_1}}$$

(50)

It follows that

$$\underset{t\to +\infty }{\lim }V(t)\le {\displaystyle \frac{l_1}{{\rho }_1}}$$

(51)

If ${\rho }_1$ is taken sufficiently large so that ${\rho }_1\ge l_1$ holds, then it can be realized that when $t\to \infty$, $V(t)\to 0$, $s\to 0$, $e\to 0$, and $\dot{e}\to 0$, the error consistent is eventually bounded.

Theorem 2.

In the time-varying threshold event-triggered driving strategies, Equation (27) and Equation (28), a lower bound is observed between two consecutive trigger intervals. This feature ensures that there are no infinite triggers in a finite time.

Proof. 

Since $\forall t\in [t_k,t_{k+1})$, $\tau =w(t_k)$ and $e(t)=w(t)-\tau (t)=w(t)-w(t_k)$, we can obtain $\dot{e}(t)=\dot{w}(t)$ and

$${\displaystyle \frac{d}{dt}}‖e‖=sign(e)\dot{e}\le ‖\dot{\omega }‖,\forall t\in [t_k,t_{k+1})$$

(52)

It follows from Equation (25) that $w$ is continuous. Moreover, $w$ is a function of $s$ and $\mu$. Because $s$ and $\mu$ are bounded based on Theorem 1, $w$ is also bounded. There exists a positive constant ${\Omega }_w$ satisfying $\dot{w}\le {\Omega }_w$.

$$e(t_k)=0,\underset{t\to t_{k+1}}{\lim }‖e(t)‖\ge {(\beta ‖\tau ‖+g)}^{{\displaystyle \frac{1}{2}}}$$

(53)

From Equation (52), it follows that ${\displaystyle \frac{{\lim }_{t\to t_{k+1}}‖e(t)‖-e(t_k)}{t_{k+1}-t_k}}={\displaystyle \frac{{\lim }_{t\to t_{k+1}}‖e(t)‖}{t_{k+1}-t_k}}\le {\Omega }_{\omega }$. Thus,

$$\begin{array}{l}{\Omega }_{\omega }(t_{k+1}-t_k)\ge \underset{t\to t_{k+1}}{\lim }‖e(t)‖\ge (\beta ‖\tau ‖+g{)}^{{\displaystyle \frac{1}{2}}}\\ t_{k+1}-t_k\ge {\displaystyle \frac{{(\beta ‖\tau ‖+g)}^{\frac{1}{2}}}{{\Omega }_{\omega }}}\end{array}$$

(54)

There exists a positive constant $\overline{t}$ satisfying $\overline{t}\ge {\displaystyle \frac{{(\beta ‖\tau ‖+g)}^{\frac{1}{2}}}{{\Omega }_{\omega }}}$, so that $t_{k+1}-t_k\ge \overline{t}$. This mechanism avoids infinite triggering, thereby eliminating the possibility of Zeno behavior. This completes the proof of Theorem 2. □

5. Simulation Results

The purpose of this section is to verify the efficiency of the proposed USV trajectory tracking control scheme by performing two sets of simulations. The mass of the USV is $m=2500$ kg. The parameters of the USV are shown below:

$$\begin{array}{l}M=\left[\begin{array}{ccc}2081& 0& 0\\ 0& 2726& 88.3\\ 0& 88.3& 222.6\end{array}\right]C(v)=\left[\begin{array}{ccc}0& 0& -2276\upsilon -88.3r\\ 0& 0& 2887u\\ 2726\upsilon +88.3r& -2887u& 0\end{array}\right]\\ D(v)=-\left[\begin{array}{ccc}-107.3\left|u\right|-474.4u^2+58.1& 0& 0\\ 0& -2945\left|\upsilon \right|-65\left|r\right|+71.7& 0\\ 0& 0& 6.45\left|\upsilon \right|-60.5\left|r\right|+153.2\end{array}\right]\end{array}$$

Case 1.

The reference position of the USV is set to $x_d=sin(0.5t(k))-t(k)$, $y_d=t(k)+2sin(0.3t(k))$, and the reference yaw angle is set to ${\phi }_d=0.6\sin (t(k))$. The initial position $x$, $y$, and yaw angle $\phi$ of the actual USV are $(0.5,0.4)$ and $0.3$. The parameters of the proposed trajectory tracking control strategy are designed as $\epsilon =10$, $\rho =0.01$, and the quantization parameter is selected as $h=0.2$.

The simulation results are shown in Figure 3, Figure 4, Figure 5, Figure 6, Figure 7 and Figure 8. Figure 3 illustrates the trajectory tracking performance of the USV, where the proposed control law enables the USV to closely follow the desired trajectory from its initial position. Figure 4 shows the tracking effectiveness of the USV in terms of position and heading. Figure 5 shows the total speed, indicating that it is suitable for use at slightly higher speeds. Figure 6 shows that despite the presence of external disturbances and quantization errors, the proposed controller has an impressive tracking performance with errors converging quickly to a range that tends to zero. Figure 7 illustrates the comparison curves between the variation curves of the USV control law with and without the input quantization, with the control signal remaining within a reasonable controllable range. Figure 8 depicts the time interval between two consecutive event-triggered samples, confirming the absence of Zeno behavior. Simulation Case 1 indicates that a total of 291 events were triggered during the 20 s simulation period, demonstrating its effectiveness in reducing communication bandwidth.

Case 2.

Case 2 has more complex maneuvering conditions compared to Case 1. The reference surge position is set to $x_d=sin(0.3t(k))-1.2\cos (0.8t(k))-t(k)$, the reference sway position is set to, and the reference yaw angle of the USV is set to ${\phi }_d=-0.2\sin (t(k))+0.6t(k)$. The initial position, $x$, $y$, and yaw angle, $\phi$, of the actual USV are $(-1.0,1.2)$ and $0.2$. The remaining control design parameters remain consistent with those in Case 1.

The simulation outcomes are displayed in Figure 9, Figure 10, Figure 11, Figure 12, Figure 13 and Figure 14. Figure 9 demonstrates the USV’s trajectory tracking performance, highlighting that the proposed control strategy enables the USV to navigate along the desired route starting from its initial position. Figure 10 presents the time histories of the surge $x$, sway $y$, and yaw $\phi$ under more complex maneuvering conditions. Figure 11 illustrates the vessel’s total speed, indicating that the USV maintains a consistent medium-speed operation throughout the simulation period. It is noteworthy that, as shown in Figure 12, despite external disturbances and quantization errors, the proposed controller demonstrates excellent tracking performance, with the error rapidly diminishing to a range approaching zero. Figure 13 shows the original and quantized control inputs under event-triggered conditions. When the USV encounters more complex maneuvering conditions, the frequency of control signal changes increases but remains within manageable limits. Figure 14 illustrates the intervals between successive event-triggered instances, confirming the non-existence of Zeno behavior and demonstrating the adaptability of the event-triggered mechanism under dynamic conditions specific to Case 2. Simulations of Case 2 indicate that the total number of events triggered within the 20 s simulation period is 376.

6. Conclusions

This paper investigates trajectory tracking control for USVs incorporating event-triggered mechanisms and input quantization. A linear analytical model characterizing the input quantization process is developed, enabling the design of an event-triggered tracking control law without requiring prior knowledge of quantization parameters. The proposed control system demonstrates effective trajectory tracking performance. This method significantly reduces communication bandwidth requirements. The stability of the closed-loop system is rigorously verified through Lyapunov analysis, and the effectiveness of the method is confirmed by the simulation results. Overall, the method proposed in this paper demonstrates good applicability for unmanned surface vehicle trajectory tracking under strict bandwidth budget constraints. Future work will extend the framework to output-feedback scenarios and sea trial validation. However, the system also faces major technical limitations, including real-time algorithmic guarantees under dynamic working conditions, stability maintenance of the triggering mechanism under extreme conditions, and suppression of quantization noise interference with system performance.