In recent years, formation tracking control of underactuated surface vessels (USVs) has gained compelling interests, as they can be widely used in ocean engineering fields [1]. Compared with a single vessel, formation has many advantages. In the course of ocean exploration, the efficiency of a single vessel is low. Beyond that, formation can perform the complex tasks such as pursuing tasks that usually cannot accomplished by a single vessel. There are various formation control schemes include leader–follower approach [2][3], virtual structure [4], graph-based mechanism [5]. In particular, the leader-follower strategy is preferred in ocean engineering applications. The follower maintains a desired relative distance and angle with the leader, a reference trajectory can be defined by the leader. Therefore, the group behavior is directed by the leader, and the formation stability can be induced by the control law of single vessel.

There are two main challenges which are still worth mentioning. The first challenge is actuator saturation, it can severely degrade the closed-loop system performance [6]. In practice, each actuator of vessel can only provide a limited force or torque. To reduce the risk of actuator saturation and enhance system performance, some methods are proposed in recent years. To avoid a poor tracking performance in formation control, generalized saturation functions combined with formation tracking errors are employed to handle the input saturation problem [7]. To solve the input saturation and underactuated problems simultaneously, additional variables are induced to velocity errors in the body-fixed frame [8] [9]. Nonlinear saturation of actuator is approximated by a Gauss error function [10], such that anti-saturation controller can be designed. An auxiliary dynamic system is constructed to solve the input saturation problem [11]. The auxiliary dynamic system is designed as a second-order system, and it provides a conservative input signal. A first-order auxiliary system is designed [12], it is employed to deal with input saturation of rudder. However, the limited acting frequency of actuator is not considered in the above articles.

High-frequency action may lead not only to mechanical abrasion but also to a short service life [13]. Input saturation and actuator rate limits are considered in [14], and a nonlinear controller is designed based on dynamic surface control and a backstepping technology. [15] consider actuator saturation containing limited magnitude and rate, and design a sliding model controller. In [14] and [15], Zeno behavior is avoided because the acting frequency of actuator is bounded. A controller combined with an event-triggered condition is proposed in [16], and the triggered condition is designed based on the flow time and error function. It still causes a large action times of actuator. [17] propose an event-triggered control law, and an event-triggered mechanism is used to reduce the execution frequency of actuators. The event-triggered mechanism is designed in the controller-to-actuator channel, the execution rate of the actuator are reduced.

The second significant challenge is the uncertainties. Because of the working condition of surface vessel, the uncertainties contain model uncertainties and environmental disturbances. In [18], the proposed controller is robust to the parameter uncertainties of the nonlinear terms and exogenous disturbances. Unknown plant parameters and environment disturbances are solved by using a parameter estimation and upper bound estimation in [19]. In [20], a robust adaptive control algorithm is proposed to solve uncertain parameters. Adaptive estimators are designed to estimate unknown time-varying functions of vessel model in [21]. Considering the leader-follower formation tracking problem, a neural network is used to approximate the uncertainties, and the uncertainties are compensated by on line learning [22]. A cooperative controller is proposed by using position–heading measurements only in [23], where an extended state observer (ESO) is constructed to provide the estimations of velocity, yaw rate and the uncertainties. However, the existing observers presented in [9,11,13,17,23,24] rely on continuous sampling. In each execution cycle signals are transmitted, which means that many network resources and calculation resources are consumed. To remove unnecessary calculation and communication, an event-triggered extended state observer (ETESO) is proposed in [25]. In [26], an ETESO is proposed for a class of nonlinear networked control systems, and the state estimation problem is solved. As for USVs, when their working mode is tracking control, in most cases, yaw angle is unchanging. In the common continuous observer, the signals have to be transmitted during each execution cycle, which will consume many network resources and calculation resources. Therefore, it is rewarding to design a leader-follower tracking controller with light transmission load and less action times of actuator.

Based on the above research background, this paper investigates leader–follower formation tracking control of USVs subject to input saturation, model uncertainties and environmental disturbances. Inspired by the works in [23] and [16], an improved output feedback controller is proposed. Firstly, an ETESO is used to recover the velocity data, model uncertainties and disturbances. In each execution cycle, an event-triggered strategy is monitored to determine whether or not to transmit the position and head informations to the observer. Sensor-to-observer communication costs are drastically reduced. Secondly, an estimator is employed to estimate velocity of the leader and reduce the amount of data exchanged in formation. Thirdly, auxiliary variables provide a solution of input saturation and underactuated problem. Then, an event-triggered controller (ETC) is designed based on the ETESO, the estimator and auxiliary variables. Finally, the stability of closed-loop system is proved. Simulations demonstrate the proposed control strategy.

Two salient features of the proposed control are summarized as follows. First, an estimator is employed to estimate the velocity of the leader, the embedded processor of each follower does reading computations only use the position and yaw of the leader, the transmission loads between formation are saved. Second, an event-triggered mechanism is designed in the controller-to-actuator channel, action times of actuator are reduced.

Compared with the existing related results, the novelty of this paper is summarized as follows. First, different from the existing formation control approaches [23] and [17], velocity of the leader is estimated by an estimator, which reduce the amount of data exchanged between USVs. Second, compared with the observer proposed in [23], the ETESO can reduce sensor-to-observer communication costs. Third, compared with the strategy proposed in [16], the proposed control law can largely reduce the action times of actuator.

The rest sections are organized as follows. A table of notations containing some used variables in the paper is presented. Section II describes the preliminaries and problem formulation. Section III describes observer design, Section IV gives controller design and stability analysis. Simulation results are presented in Section V. Section VI concludes this paper.

A group of USVs consists N USVs, and the model of the ith USV is given as [27] and where ${(x_i,y_i)}^T\in {\mathbb{R}}^2$ is the position, and ${\psi }_i$ is the yaw angle. $u_i$ is the surge velocity, $v_i$ is the sway velocity, $r_i$ is the yaw rate. $m_{i,11}$ and $m_{i,33}$ denote the inertia mass. The uncertainties ${\sigma }_{i,1}$, ${\sigma }_{i,2}$ and ${\sigma }_{i,3}$ contain Coriolis parameters, hydrodynamic damping parameters and time-varying disturbances. ${\tau }_{i,u}$ and ${\tau }_{i,r}$ are input signals.

To facilitate observer design and analysis, the ith USV model is written as where ${\eta }_i={[x_i,y_i,{\psi }_i]}^T$, ${\nu }_i={[u_i,v_i,r_i]}^T$, ${\tau }_i={[{\tau }_{i,u},0,{\tau }_{i,r}]}^T$, $M_i=diag\{m_{i,11},m_{i,22},m_{i,33}\}$, $m_{i,22}$ is the inertia mass. ${\sigma }_i={[{\sigma }_{i,1},{\sigma }_{i,2},{\sigma }_{i,3}]}^T$,

The mismatch function between input without saturation and with saturation is described as ${\varpi }_i={[{\varpi }_{i,u},0,{\varpi }_{i,r}]}^T$$={\tau }_{ic}-{\tau }_i$, where ${\tau }_{ic}={[{\tau }_{i,uc},0,{\tau }_{i,rc}]}^T$ with ${\tau }_{i,uc}$ and ${\tau }_{i,rc}$ are surge force and yaw moment calculated by the proposed controller, respectively. ${\tau }_i={[{\tau }_{i,u},0,{\tau }_{i,r}]}^T$. Then, the saturated control is given by ${\tau }_i={\tau }_{ic}-{\varpi }_i$.

Assumption 1: The unknown function ${\sigma }_i$ satisfies ${\dot{\sigma }}_i\le {\sigma }_i^{*}$ with ${\sigma }_i^{*}$ is a positive constant.

${\sigma }_i$ a vector containing velocity, Coriolis parameters, hydrodynamic damping parameters and time-varying disturbances. It is natural to assume that their derivatives are bounded. Control inputs to drive USVs are bounded, Assumption 1 is reasonable.

Assumption 2: Velocity of the ith USV is bounded, such that $\parallel {\nu }_i\parallel \le {\nu }_i^{*}$, ${\nu }_i^{*}$ is a positive constant.

The control inputs to drive USVs are bounded, the energy of external disturbances is limited. Therefore, Assumption 2 is reasonable.

Assumption 3: The desired distance ${\rho }_{ij,d}$ and angle ${\lambda }_{ij,d}$ are bounded.

Assumption 3 means that the desired value is reasonable.

The following definitions are made.

The subscript j is the index of the leader j. ${\eta }_j={[x_j,y_j,{\psi }_j]}^T$ is used to denote the position and yaw of the leader j. ${\nu }_j={[u_j,v_j,r_j]}^T$ is the velocity and yaw rate of the leader j.

The relative distance ${\rho }_{ij}$ and angle ${\lambda }_{ij}$ between the ith USV and the leader j are expressed as

The differential equations of ${\rho }_{ij}$ and ${\lambda }_{ij}$ are given by:

From $x_j-x_i={\rho }_{ij}cos\left({\lambda }_{ij}\right)$ and $y_j-y_i={\rho }_{ij}sin\left({\lambda }_{ij}\right)$, then

The control objective is to design an event-trigged control law for USVs to track the leader with desired distance ${\rho }_{ij,d}$ and angle ${\lambda }_{ij,d}$.

Remark 1. In the controller design process, the ith USV track the the jth USV by using the informations $x_j$, $y_j$ and ${\psi }_j$. It means that the communication costs between the multiple USVs are not saved.

An ETESO is developed for estimating the uncertainties containing model uncertainties and time-varying disturbances, and an event-triggered condition is employed to avoid unnecessary communications. Inspired by [25], an ETESO is designed as follow: where ${\widehat{\eta }}_i={[{\widehat{x}}_i,{\widehat{y}}_i,{\widehat{\psi }}_i]}^T$, ${\widehat{\nu }}_i={[{\widehat{u}}_i,{\widehat{v}}_i,{\widehat{r}}_i]}^T$ and ${\widehat{\sigma }}_i={[{\widehat{\sigma }}_{i,1},{\widehat{\sigma }}_{i,2},{\widehat{\sigma }}_{i,3}]}^T$. ${\widehat{x}}_i,{\widehat{y}}_i,{\widehat{\psi }}_i,{\widehat{u}}_i,{\widehat{v}}_i,{\widehat{r}}_i,{\widehat{\sigma }}_{i,1},{\widehat{\sigma }}_{i,2}$ and ${\widehat{\sigma }}_{i,3}$ are the estimates of $x_i,y_i,{\psi }_i,u_i,v_i,r_i,{\sigma }_{i,1},{\sigma }_{i,2}$ and ${\sigma }_{i,3}$, respectively. ${ϵ}_i$ is a positive constant and ${\xi }_i$ is the transmitted signal given by

$t_k$ is the previous event-triggering time instant and $d\left(t\right)$ is an event-triggered mechanism designed as

$c_q>0$ is design threshold and

Remark 2. The continuous-time control scheme is emulated by computer with sampler and zero-order holder, and the control command is updated periodically. At each sampling instant, $q\left(t\right)$ is monitored to determine whether or not to transmit the information ${\eta }_i\left(t\right)$ to the observer. If $\parallel q\left(t\right)\parallel \ge c_q$, the event-triggering time instant $t_k$ is updated to the current time t, and ${\eta }_i\left(t\right)$ is transmitted to the observer. Otherwise, only ${\eta }_i\left(t_k\right)$ is available to the observer. The first event-triggering time is $t_1=0$.

To facilitate controller design and stability analysis, define ${\xi }_i={[{\xi }_{i,1},{\xi }_{i,2},{\xi }_{i,3}]}^T$, () can be written as with

Define ${\tilde{x}}_i={\widehat{x}}_i-x_i$, ${\tilde{y}}_i={\widehat{y}}_i-y_i$, ${\tilde{\psi }}_i={\widehat{\psi }}_i-{\psi }_i$, ${\tilde{u}}_i={\widehat{u}}_i-u_i$, ${\tilde{v}}_i={\widehat{v}}_i-v_i$, ${\tilde{r}}_i={\widehat{r}}_i-r_i$,

It follows that

Theorem 1. Consider the USVs model (7) and (8), the ETESO (15), (16) and (17), and the event-triggering condition (19). If Assumptions 1 and 2 are satisfied, there exist ${ϵ}_i>0$ and $\iota >0$ such that for any $k>0$,

Proof. Along with (18) and (19), the event-triggering mechanism can be described as

When $t\in [t_k,t_{k+1})$, the sample error ${\eta }_i\left(t_k\right)-{\eta }_i\left(t\right)$ becomes

Using Assumption 2, it obtains that

There exists $\iota ={ϵ}_i^2{c}_q/{\nu }^{*}$ and $\iota =t-t_k$, such that $\parallel {\eta }_i\left(t_k\right)-{\eta }_i\left(t\right)\parallel \le c_q{ϵ}_i^2$. Notes that $t\in [t_k,t_{k+1})$, so it can draw a conclusion that $min\{t_{k+1}-t_k\}\ge \iota$.

Theorem 1 shows that Zeno behavior can be avoided by using the proposed observer. Next, the error dynamics of the proposed ETESO are investigated.

To facilitate the stability analysis, define $E_i\left(t\right)={[e_{i,1}{\left(t\right)}^T,e_{i,2}{\left(t\right)}^T,e_{i,3}{\left(t\right)}^T]}^T\in {\mathbb{R}}^9$, and the error dynamics (30)-(32) can be expressed as and $H_i=diag\{I_3,R_i^T\left({\psi }_i\left(t\right)\right),R_i^T\left({\psi }_i\left(t\right)\right)\}$.

Using a transformation $E_t\left(t\right)=T{E}_i\left(t\right)$ with $T=diag\{R_i^T\left({\psi }_i\left(t\right)\right),I_3,I_3\}$, it follows that where ${\overline{H}}_i=diag\{R^T\left({\psi }_i\left(t\right)\right),R^T\left({\psi }_i\left(t\right)\right),R^T\left({\psi }_i\left(t\right)\right)\}$, $S_T=diag\{S^T,0_3,0_3\}$, and

Theorem 3. The observer error dynamics are bounded, if there exist a positive definite matrix P satisfying the inequalities used in stability analysis, the inequalities are defined as where $\varrho$ is a positive constant, ${\overline{r}}_i$ is the upper bound of $r_i$.

Proof. Consider the Lyapunov function

Differentiating $V_o$ with respect to time

Notes that $2{ϵ}_i\parallel P{B}_i\parallel \parallel {\dot{\sigma }}_i\left(t\right)\parallel +2\parallel P{D}_i\parallel \parallel q\left(t\right)\parallel \le \varrho c_{o_1}\parallel E_t\left(t\right)\parallel$, $0<c_{o_1}<1$. It leads to where $c_{o_2}=\varrho (1-c_{o_1})/{\lambda }_{max}\left(P\right)$.

Then the state $E_t\left(t\right)$ is bounded. Noticing $\parallel T^{-1}\parallel =1$ and $E_i\left(t\right)=T^{-1}{E}_t\left(t\right)$, the estimation error $E_i\left(t\right)$ is bounded.

An output feedback controller combined with the ETESO is developed based on an event-triggered strategy. As shown in Figure 1, the characteristic of the proposed controller is as follows. The estimator is used to provide the estimation of velocity of the leader; the ETESO is designed to provide the velocity, yaw rate and the uncertainties; the transmitted signal contains an event-trigged strategy which reduce sensor-to-observer communication costs; the triggering condition of controller can largely reduce the action times of actuator. The control law design process is divided into the following steps.

Step 1: Define the formation tracking error vector where ${\rho }_{ij,e}={\rho }_{ij}-{\rho }_{ij,d}$, ${\lambda }_{ij,e}={\lambda }_{ij}-{\lambda }_{ij,d}$.

Using (13) and (14), the time derivative of $P_{ij}$ is given as where $B_{ij}=diag\{1,\frac{1}{{\rho }_{ij}}\}$ ,

$z_i={[u_i,v_i]}^T$, $z_j={[u_j,v_j]}^T$ and $D_{ij,d}={[{\dot{\rho }}_{ij,d},{\dot{\lambda }}_{ij,d}]}^T$.

Define $f_i=\left[f_{i,1}{f}_{i,2}\right]=P_{ij}^T{B}_{ij}{\Xi }_{ij,1}$, $\parallel f_i\parallel \le f_i^{*}$, $f_i^{*}$ is an unknown positive constant.

Remark 3: From the definition of ${\Xi }_{ij,1}$, $P_{ij}$ and $B_{ij}$, one notes that $\parallel {\Xi }_{ij,1}\parallel =1$, $P_{ij}$ and $B_{ij}$ are bounded, so $\parallel f_i\parallel \le f_i^{*}$ is reasonable.

Choosing ${\beta }_i$ as a virtual controller, ${\beta }_i$ is designed as: where ${\overline{\Xi }}_{ij,1}^{-1}={\Xi }_{ij,1}^T{\left({\Xi }_{ij,1}{\Xi }_{ij,1}^T\right)}^{-1}$, $K_{i,1}\in {\mathbb{R}}^{2\times 2}$ is a positive definite matrix, ${\overline{f}}_{i,2}={[0,tanh(f_{i,2}/{ϵ}_f)]}^T$, ${ϵ}_f>0$ is a constant. ${\widehat{\Theta }}_i$ is the estimator of ${\Theta }_i$, ${\Theta }_i={\Xi }_{ij,2}{z}_j$, and it updates as

${\overline{P}}_{ij}=B_{ij}{[{\rho }_{ij,e}tanh({\rho }_{ij,e}/{ϵ}_{\rho }),{\lambda }_{ij,e}tanh({\lambda }_{ij,e}/{ϵ}_{\lambda })]}^T$, $K_{\Theta }$ is a positive definite matrix, ${ϵ}_{\rho }$ and ${ϵ}_{\lambda }$ are positive constants.

Remark 4: The adaptive term ${\widehat{\Theta }}_i$ is employed to reduce the volume of data on the formation communication [24], and estimate the information ${\Theta }_i$. ${\Theta }_i$ can be estimated because ${\Xi }_{ij,2}{z}_j\le \left|z_j\right|$ and $z_j$ is bounded. It makes sure that formation tracking control is achieved without velocity of the leader, the amount of data exchange between USVs can be reduced. Unlike the controller presented in [24], the proposed controller is designed without velocities of follower; the ETESO can reduce sensor-to-observer communication costs; the ETC can largely reduce the action times of actuator.

Step 2: Define a error as with where

${\gamma }_{ij}$ is a positive constant.

Using the the USV’s dynamic, the time derivative of ${\psi }_{i,e}$ is

The virtual control ${\beta }_{i,3}$ is chosen as: where $k_{i,2}$ is a positive constant.

Step 3: Define error functions as follow: where ${\widehat{z}}_i={[{\widehat{u}}_i,{\widehat{v}}_i]}^T$, ${\overline{\beta }}_i={[{\overline{\beta }}_{i,1},{\overline{\beta }}_{i,2}]}^T$, ${\gamma }_1$ is a positive constant. ${\overline{\beta }}_{i,h}$, $h=1,2,3,$ is derived from the first order filter $l_{i,h}{\dot{\overline{\beta }}}_{i,h}+{\overline{\beta }}_{i,h}={\beta }_{i,h}$, $l_{i,h}>0$ is a constant. ${\overline{\alpha }}_i={[{\alpha }_{i,1},tanh\left({\alpha }_{i,2}\right)/{\gamma }_1]}^T$, ${\alpha }_{i,h}$, $h=1,2,3,$ is the auxiliary variable derived to deal with the underactuated problem and input saturation. ${\omega }_i={[{\omega }_{i,1},{\omega }_{i,2}]}^T$, ${\omega }_{i,h}$, $h=1,2,3,$ is the error caused by the first order filter.

The update laws of extra variables are given by where $T_u$, $k_{i,4}$ and $T_r$ are positive constants.

Step 4: A control law at the kinetic level is designed as where $k_{i,3}$ and $k_{i,5}$ are positive constants.