Disturbance-Observer-Based Adaptive Prescribed Performance Formation Tracking Control for Multiple Underactuated Surface Vehicles

Abstract

This study proposes a new disturbance-observer-based adaptive distributed formation control scheme for multiple underactuated surface vehicles (USVs) subject to unknown synthesized disturbances under prescribed performance constraints. A modified sliding mode differentiator (MSMD) is applied as a nonlinear disturbance observer to estimate unknown synthesized disturbances, which contain unknown environmental disturbances and system modelling uncertainties, thus enhancing the robustness of the system. Based on this, we impose the time-varying performance constraints on the position tracking error between the neighboring USVs. A novel differentiable error transformation equation is embedded in the prescribed performance control, and an adaptive prescribed performance controller is constructed by employing the backstepping method to ensure that the position tracking error remains within the prescribed transient and steady performance, and each USV realizes collision-free formation motion. Furthermore, a novel second-order nonlinear differentiator is introduced to extract the derivative information of the virtual control law. Finally, the numerical simulation results verify the effectiveness of the proposed control scheme.

1. Introduction

Underactuated surface vehicles (USVs) have recently shown considerable advantages in marine engineering applications [1,2,3,4], such as resource exploitation, local data acquisition, and marine environmental surveillance. Simultaneously, research on consensus issues in USVs formation control has become a prominent research area due to its potential benefits, including enhanced efficiency, improved fault tolerance and adaptability, and reduced operational costs [5,6,7,8]. Increasing efforts are focused on enhancing the accuracy and robustness of USV formation control.

As of now, numerous cooperative control methods for USVs have been extensively documented [9,10,11]. It is evident that, as a formation control framework, distributed control proves more effective in marine environments characterized by limited communication resources, avoiding the necessity for global communication and minimizing bandwidth usage. For instance, a distributed formation controller was developed for USVs under a directed interaction topology in [12], demonstrating its efficacy. In [13], a distributed control strategy for multiple unmanned surface vehicles was proposed, where only a subset of vehicles could access information from the leader vehicle amidst unknown external disturbances. Additionally, a new robust output feedback distributed controller was introduced in [14], leveraging backstepping and anti-saturation compensators to guide USVs towards the neighborhood of the convex hull formed by leaders.

It is inevitable to face system uncertainties and external environmental disturbances for the vessels; to further maintain stability and enhance the robustness of the closed-loop system in the presence of uncertain external disturbances, various methods have been proposed. The nonlinear extended state observer (ESO) based on fractional power function was established in [15,16], and the estimation of the extended state was used to compensate the uncertain dynamics in real time. A nonlinear sliding surface was designed to stabilize underactuated nonlinear systems in the presence of unknown bounded external disturbances [17,18], Additionally, an adaptive barrier function method was employed to estimate the unknown upper bounds of disturbances. A novel control strategy based on model-free control principles was proposed [19], combining intelligent PID and PD feedforward controllers to achieve effective disturbance rejection. This ensures stable and precise trajectory tracking performance for AUVs in complex underwater environments. On the other hand, a neural-network-based disturbance observer was created to estimation unidentified external disturbances in [20,21,22]. A fast fixed-time disturbance observer was designed to compensate for unknown synthesized disturbances in [23,24], which improved the robustness of the formation control system. Consequently, it is very important and practical to construct an efficient observer in order to increase perturbation estimation accuracy. Nevertheless, the majority of the aforementioned control strategies and techniques can only ensure the steady-state performance of USV formation control systems. They cannot guarantee that the transient performance complies with formation practices.

Incorporating prescribed performance control (PPC) constraints into controller design can significantly enhance the control performance and ensure the safe operation of USVs. Specifically, in [25,26,27], motion controllers were designed with PPC constraints, enforcing asymptotically stable boundaries on tracking errors. Additionally, Ref. [28] proposed a finite-time control algorithm based on prescribed performance to achieve rapid convergence and meet specified performance criteria. Moreover, in [29], a neural network controller for USV formation tracking employed error transformations using prescribed performance functions to address collision and connectivity constraints. However, the complexity of the error transformation equations in these studies may lead to tracking errors exceeding prescribed performance bounds under external disturbances, necessitating further advancements in controller design. Therefore, there remains a need for continued innovation in distributed formation control of USVs with transient performance constraints to effectively manage such complexities and enhance overall performance.

Based on the above observations, the objective of this study is to develop a distributed formation control scheme for the USVs in the presence of unknown synthesized disturbances, which satisfies prescribed performance requirements and has the following novelties and advantages over the existing solutions:

• A novel error transformation equation is proposed and integrated with a predefined performance function to ensure that the orientation of each USV meets the prescribed safe attitude requirement, thereby achieving collision-free formation motion for each USV. In contrast to [30], the proposed error transformation equation is differentiable. Additionally, compared to [25,26,27], the formulation of the error transformation equation in this study is simpler, thereby reducing the complexity of the formation controller.
• A modified sliding mode differentiator (MSMD) disturbance observer is proposed to estimate the unknown synthesized disturbances in a USV formation system due to its merits of no chattering in disturbance estimation, better robustness and simple. Additionally, the adaptive estimator provides a straightforward and efficient approach independent of prior knowledge to approximate the upper bound of disturbances, making it more universally applicable compared to the approach detailed in [31].
• Moreover, an adaptive prescribed performance controller is formulated using the backstepping control method. This approach incorporates a new second-order nonlinear differentiator (NLD) to enhance the precision of derivative extraction from the virtual control law, surpassing the capabilities of conventional first-order filters.

The remainder of the study is organized as follows: The preliminary and the problem formation are discussed in Section 2. Section 3 contains the MSMD disturbance-observer-based adaptive distributed formation control strategy for USV formation control. The stability analysis is displayed in Section 4. The simulation experiences are depicted in Section 5 to prove the availability of the raised control scheme. Finally, some conclusions are made in Section 6.

2. Preliminary and Problem Formation

2.1. Preliminary

This study considers the directed connected graph to describe the communication topology of formation members. The graph $G=G\left(\nu ,\epsilon ,A\right)$, in which the node set $\nu =\left\{{\nu }_1,{\nu }_2,\cdots {\nu }_n\right\}$, represents all USVs in the formation; the edge set $\epsilon \in n\times n$ represents the communication paths among USVs; $A=\left[a_{ij}\right]\in {ℝ}^{n\times n}$ is the weighted adjacency matrix. The element $a_{ij}$ satisfies $a_{ij}>0$ when $\left({\nu }_i,{\nu }_j\right)\in \epsilon$, and $a_{ij}=0$ otherwise. Generally, it is assumed that each node does not exchange information with itself, i.e., $a_{ij}=0, \left(i=j\right)$. The in-degree $d_i$ of node $i$ is defined as $d_{ij}={\displaystyle {\sum }_{i=1}^N{a}_{ij}}$, and the in-degree matrix $D=diag\left(d_{ij}\right)\in {ℝ}^{n\times n}$. Then, the Laplacian matrix is defined as $L=\left[l_{ij}\right]\in {ℝ}^{n\times n}$, where $L=D-A$.

Lemma 1 

[32]. For the system described by

$$\left\{\begin{array}{l}{\dot{y}}_1(t)=y_2(t)\\ {\dot{y}}_2(t)=f(y_1(t),y_2(t))\end{array}\right.$$

(1)

whenever the solutions of the system satisfy $y_1\to 0$ and $y_2\left(t\right)\to 0\left(t\to \infty \right)$, then for arbitrary bounded and integral function $\alpha \left(t\right)$ and a constant $t_1>0$, the solution of the following system

$$\left\{\begin{array}{l}{\dot{z}}_1\left(t\right)=z_2\left(t\right)\\ {\dot{z}}_2\left(t\right)=-{\mathcal{l}}^{2}f\left(z_1\left(t\right)-\alpha \left(t\right),z_2\left(t\right)/\mathcal{l}\right)\end{array}\right.$$

(2)

satisfies

$$\underset{\mathcal{l}\to \infty }{\lim }{\displaystyle {\int }_0^{t_1}\left|z_1\left(t\right)-\alpha \left(t\right)\right|}dt=0$$

(3)

that is understood as follows: $z_1\left(t\right)$ medially converges to the input signal $\alpha \left(t\right)$, $z_2\left(t\right)$ converges to the derivative of $\alpha \left(t\right)$ and $\mathcal{l}>0$ is known as the acceleration factor.

Lemma 2 

[33]. ${\left|a+b\right|}^l\le {\left|a\right|}^l+{\left|b\right|}^l, \forall 0<l<1, \mathrm{a}\in ℝ,\mathrm{b}\in ℝ$.

2.2. Mathematical Model of USVs

As Figure 1 shows us, $\left(O-XYZ\right)$ indicates the earth-fixed frame. $O$ always indicate the position of the gravity center for USVs. $OX$, $OY$ and $OZ$ point to the north, east and the center of the earth, respectively. $\left(o-x_b{y}_b{z}_b\right)$ indicates the body-fixed frame. $o{x}_b$ points to the bow and $o{y}_b$ points to the starboard side of USVs. $o{z}_b$ points toward bilge keel. Considering the system consisting of $M$ USVs in this work, the structural form of formation is shown in Figure 2. The kinematic equations for the three degrees of freedom (surge, sway and yaw) of the $ith$ USV are outlined below

$$\left\{\begin{array}{l}{\dot{\chi }}_i=\overline{R}\left({\psi }_i\right){\left[u_i,v_i\right]}^T\\ {\psi }_i=r_i\end{array}\right.$$

(4)

where ${\chi }_i={\left[x_i,y_i\right]}^T$ and ${\psi }_i$ represent the position and heading angle of the $ith$ USV in the earth-fixed reference frame, respectively; ${\left[u_i,v_i\right]}^T$ and $r_i$ successively contain the linear velocity and yaw velocity of the $ith$ USV in the body-fixed reference frame, respectively; $\overline{R}\left({\psi }_i\right)$ is the rotation matrix linked to the heading angle ${\psi }_i$, which is specifically expressed as

$$\overline{R}\left({\psi }_i\right)=\left[\begin{array}{ll}\cos {\psi }_i& -\sin {\psi }_i\\ \sin {\psi }_i& \cos {\psi }_i\end{array}\right]$$

(5)

To facilitate the design of the formation controller, the dynamic equations of the $ith$ USV can be expressed as follows:

$$\left\{\begin{array}{l}{\dot{u}}_i=\frac{1}{m_{i11}}{\tau }_{iu}+W_{iu}\\ {\dot{v}}_i=W_{iv}\\ {\dot{r}}_i=\frac{1}{m_{i33}}{\tau }_{ir}+W_{ir}\end{array}\right.$$

(6)

where $m_{i11}$, $m_{i22}$ and $m_{i33}$ are nominal mass and inertia; ${\tau }_{iu}$ and ${\tau }_{ir}$ are the control inputs; ${\tau }_{iw}={\left[{\tau }_{iwu},{\tau }_{iwv},{\tau }_{iwr}\right]}^T$ denotes the unknown environmental disturbances; $d_{i11}$, $d_{i22}$ and $d_{i33}$ are hydrodynamic damping factors; $W_{i\upsilon }, \left(\upsilon =u,v,r\right)$ is considered to be an unknown synthesized disturbances, which is represented by

$$\begin{array}{l}{W}_{iu}=\frac{m_{i22}}{m_{i11}}{v}_i{r}_i-\frac{d_{i11}}{m_{i11}}{u}_i+\frac{1}{m_{i11}}{\tau }_{iwu}\\ W_{iv}=-\frac{m_{i11}}{m_{i22}}{u}_i{r}_i-\frac{d_{i22}}{m_{i22}}{v}_i+\frac{1}{m_{i11}}{\tau }_{iwv}\\ W_{ir}=\frac{m_{i11}-m_{i22}}{m_{i33}}{v}_i{u}_i-\frac{d_{i33}}{m_{i33}}{r}_i+\frac{1}{m_{i33}}{\tau }_{iwr}\end{array}$$

(7)

2.3. Control Objective

Designing the distributed controller ${\tau }_i\left(i=1,\cdots ,M\right)$ for each USV with the unknown synthesized disturbances to achieve formation control for a set of a virtual leader and $M$ followers. Simultaneously, the steady-state performance and transient performance of the formation control system can be ensured, as outlined below:

$$\underset{t\to \infty }{\lim }‖{\overline{e}}_{i\chi }‖<F_{i\chi }$$

(8)

where ${\overline{e}}_{i,\chi }={\left[{\overline{e}}_{ix},{\overline{e}}_{iy}\right]}^T$ means the formation position tracking error; $F_{i,\chi }={\left[F_{ix},F_{iy}\right]}^T$ represents an indirectly adjustable constraint function.

Assumption 1. 

The directed graph  $G$  is connected, and the information of the virtual leader could be accessed by at least one USV at all times.

Assumption 2. 

The position, heading angle and velocity information of the virtual leader are available, derivable and bounded.

Assumption 3. 

The unknown synthesized disturbances  $W_{i\upsilon }$  is bounded, and the derivative of  $W_{i\upsilon }$  is also bounded.

3. Adaptive Formation Controller Design

First, in this section, we convert the formation control problem of systems (4) and (6) into an error stability problem with transient performance requirements in addition to stable performance constraints. Following that, the virtual control law is created by building the Lyapunov function and using a new second-order filter in the backstepping control method. Ultimately, an adaptive formation control law with disturbance compensation is produced by using the MSMD disturbance observer to estimate disturbances. The controller architecture is depicted in Figure 3.

3.1. MSMD Disturbance Observer Design

In this subsection, in order to realize the effective estimation of unknown synthesized disturbance, the MSMD disturbance observer is designed as follows

$$\left\{\begin{array}{l}{\dot{\widehat{V}}}_i={\overline{M}}_i^{-1}{\overline{\tau }}_i+{\zeta }_i\\ {\zeta }_i=-{\beta }_{i,1}si{g}^{\frac{1}{2}}\left({\widehat{V}}_i-V_i\right)+{\widehat{\overline{W}}}_i\\ {\dot{\widehat{\overline{W}}}}_i=-{\beta }_{i,2}si{g}^{\frac{p}{q}}\left({\widehat{\overline{W}}}_i-{\zeta }_i\right)\end{array}\right.$$

(9)

where $si{g}^a\left(x\right)={\left|x\right|}^a\mathrm{sgn}\left(x\right)$, where ${\beta }_{i,1}$ and ${\beta }_{i,2}$ are positive definite diagonal matrices, ${\overline{M}}_i=diag\left(m_{i11},m_{i33}\right)$, ${\overline{\tau }}_i={\left[{\tau }_{iu},{\tau }_{ir}\right]}^T$, ${\overline{W}}_i={\left[{\overline{W}}_{iu},{\overline{W}}_{ir}\right]}^T$, $V_i={\left[u_i,r_i\right]}^T$, ${\widehat{V}}_i$ and ${\widehat{\overline{W}}}_i$ are estimation values of $V_i$ and ${\overline{W}}_i$, respectively. $p>0$ and $q>0$ are terminal attractor design parameters and satisfy $1<p/q<2$.

Define observation errors $e_{i,1}={\widehat{V}}_i-V_i$, $e_{i,2}={\widehat{\overline{W}}}_i-{\overline{W}}_i$; referring to the system (9), the derivatives of $e_{i,1}$ and $e_{i,2}$ can be represented by

$$\left\{\begin{array}{l}{\dot{e}}_{i,1}=-{\beta }_{i,1}si{g}^{\frac{1}{2}}\left(e_{i,1}\right)+e_{i,2}\\ {\dot{e}}_{i,2}=-{\beta }_{i,2}si{g}^{\frac{p}{q}}\left(e_{i,2}-{\dot{e}}_{i,1}\right)-{\dot{\overline{W}}}_i\end{array}\right.$$

(10)

Lemma 3. 

Based on Assumption 3, we can make  ${\dot{\overline{W}}}_i$  satisfy  ${\displaystyle {\int }_{\delta }\left|{\dot{\overline{W}}}_i\right|}dt<K$  over a time interval  $\delta$  and $K>0$. Then, for any constants   $0<{\varsigma }_{i,k}<{\varsigma }_{i,k}^{\prime }, \left(k=1,2\right)$, each trajectory of  (10) starting from the region  $\left|e_{i,k}\right|\le {\varsigma }_{i,k}$  does not leave the region  $\left|e_{i,k}\right|\le {\varsigma }_{i,k}^{\prime }$  during this time interval if  $\delta$   is sufficiently small.

Proof of Lemma 3 

 

Let ${\varsigma }_{i,M1}$ satisfy ${\varsigma }_{i,1}<{\varsigma }_{i,1}^{\prime }<{\varsigma }_{i,M1}$, and for $\left|e_{i,1}\right|<{\varsigma }_{i,1}$, it can be obtained by Lemma 3

$$\left|{\dot{e}}_{i,1}\right|\le {\beta }_{i,1}{\left|e_{i,1}\right|}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}+\left|e_{i,2}\right|\le {\beta }_{i,1}{\varsigma }_{i,M1}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}+{\varsigma }_{i,M2}$$

(11)

then, by Lemma 2 and the Holder inequality, we obtain

$${\displaystyle {\int }_{\delta }\left|{\dot{e}}_{i,1}\right|}dt\le {\displaystyle {\int }_{\delta }\left({\beta }_{i,1}{\varsigma }_{i,M1}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}+{\varsigma }_{i,M2}\right)}dt\le \delta {\beta }_{i,1}{\varsigma }_{i,M1}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}+\delta {\varsigma }_{i,M2}\triangleq K_{i,1}$$

(12)

Hence, as long as the value of $\delta$ is sufficiently small, the rate change of $\left|e_{i,1}\right|$ is sufficiently small to ensure $\left|e_{i,1}\right|<{\varsigma }_{i,1}^{\prime }$.

For $\left|e_{i,2}\right|<{\varsigma }_{i,2}$, when ${\varsigma }_{i,M2}$ satisfies ${\varsigma }_{i,2}<{\varsigma }_{i,2}^{\prime }<{\varsigma }_{i,M2}$, combining with ${\displaystyle {\int }_{\delta }\left|{\dot{\overline{W}}}_i\right|}dt<K$, the following can be obtained:

$$\begin{array}{ll}\left|{\dot{e}}_{i,2}\right|& =\left|-{\beta }_{i,2}{\left|e_{i,2}-{\dot{e}}_{i,1}\right|}^{\raisebox{1ex}{$p$}\!\left/ \!\raisebox{-1ex}{$q$}\right.}\mathrm{sgn}\left(e_{i,2}-{\dot{e}}_{i,1}\right)-{\dot{\overline{W}}}_i\right|\\ & \le \left|{\beta }_{i,2}{\left|e_{i,2}-{\dot{e}}_{i,1}\right|}^{\raisebox{1ex}{$p$}\!\left/ \!\raisebox{-1ex}{$q$}\right.}+{\dot{\overline{W}}}_i\right|\le \left|{\beta }_{i,2}{\left|e_{i,2}-{\dot{e}}_{i,1}\right|}^{\raisebox{1ex}{$p$}\!\left/ \!\raisebox{-1ex}{$q$}\right.}\right|+\left|{\dot{\overline{W}}}_i\right|\\ & \le {\beta }_{i,2}{\left|e_{i,2}\right|}^{\raisebox{1ex}{$p$}\!\left/ \!\raisebox{-1ex}{$q$}\right.}+{\beta }_{i,2}{\left|{\dot{e}}_{i,1}\right|}^{\raisebox{1ex}{$p$}\!\left/ \!\raisebox{-1ex}{$q$}\right.}+\left|{\dot{\overline{W}}}_i\right|\le {\beta }_{i,2}{\varsigma }_{i,M2}^{\raisebox{1ex}{$p$}\!\left/ \!\raisebox{-1ex}{$q$}\right.}+{\beta }_{i,2}{\left|{\dot{e}}_{i,1}\right|}^{\raisebox{1ex}{$p$}\!\left/ \!\raisebox{-1ex}{$q$}\right.}+\left|{\dot{\overline{W}}}_i\right| \end{array}$$

(13)

Then, according to the Holder inequality, we can obtain

$$\begin{array}{ll}{\displaystyle {\int }_{\delta }\left|{\dot{e}}_{i,2}\right|}dt& \le {\displaystyle {\int }_{\delta }\left({\beta }_{i,2}{\varsigma }_{i,M2}^{\raisebox{1ex}{$p$}\!\left/ \!\raisebox{-1ex}{$q$}\right.}+{\beta }_{i,2}{\left|{\dot{e}}_{i,1}\right|}^{\raisebox{1ex}{$p$}\!\left/ \!\raisebox{-1ex}{$q$}\right.}+\left|{\dot{\overline{W}}}_i\right|\right)}dt\\ & \le {\displaystyle {\int }_{\delta }\left|{\dot{\overline{W}}}_i\right|}dt+{\displaystyle {\int }_{\delta }\left({\beta }_{i,2}{\varsigma }_{i,M2}^{\raisebox{1ex}{$p$}\!\left/ \!\raisebox{-1ex}{$q$}\right.}+{\beta }_{i,2}{\left|{\dot{e}}_{i,1}\right|}^{\raisebox{1ex}{$p$}\!\left/ \!\raisebox{-1ex}{$q$}\right.}\right)}dt\\ & \le K+\delta {\beta }_{i,2}{\varsigma }_{i,M2}^{\raisebox{1ex}{$p$}\!\left/ \!\raisebox{-1ex}{$q$}\right.}+\delta {\beta }_{i,2}{\varsigma }_{i,M1}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\triangleq K_{i,2}\end{array}$$

(14)

Hence, $\left|e_{i,2}\right|\le {{\varsigma }^{\prime }}_{i,M2}$ with small $\delta$. On the other hand, ${\dot{e}}_{i,1}$ serves as the input disturbance for the (10) and satisfies the conditions of Lemma 3, thus due to the induction assumption $\left|e_{i,k}\right|\le {\varsigma }_{i,k}^{\prime }, \left(k=1,2\right)$ with small $\delta$.

According to Lemma 3, the following theorem can be easily obtained. □

Theorem 1. 

By choosing appropriate positive constants  ${\beta }_{i,1}$  and  ${\beta }_{i,2}$, the MSMD disturbance observer  (9) is convergent in finite time.   $K_{i,1}$   and  $K_{i,2}$  are the positive constants and exist to satisfy the following formulas:

$$\left|{\widehat{V}}_i-V_i\right|\le K_{i,1}$$

(15)

$$\left|{\widehat{\overline{W}}}_i-{\overline{W}}_i\right|\le K_{i,2}$$

(16)

It is known from Assumption 3 and Theorem 1 that the observation error of (9) can be ensured to converge to the bounded region in finite time $T$, i.e., ${\overline{W}}_i$ can be precisely estimated by ${\widehat{\overline{W}}}_i$ in $T$ satisfies $0<T<\infty$.

Remark 1. 

In  (9), the terminal attractor function   $-{\beta }_{i,2}si{g}^{\raisebox{1ex}{$p$}\!\left/ \!\raisebox{-1ex}{$q$}\right.}\left({\widehat{\overline{W}}}_i-{\zeta }_i\right)$  can cause  ${\widehat{\overline{W}}}_i-{\zeta }_i$  to converge to zero in finite time,  and   $-{\beta }_{i,2}si{g}^{\raisebox{1ex}{$p$}\!\left/ \!\raisebox{-1ex}{$q$}\right.}\left({\widehat{\overline{W}}}_i-{\zeta }_i\right)$  converges to zero while   ${\widehat{\overline{W}}}_i-{\zeta }_i$   tends to zero. Thus,  ${\widehat{\overline{W}}}_i$  is smooth; consequently, all outputs of the MSMD are smooth with no chattering.

Remark 2. 

The observation error system (10) satisfies when the disturbance is fast-varying, the observation error converges to a closed sphere containing the origin after  $t>T$; when the disturbance is slow-varying, the observation error asymptotically converges to the zero point, and the convergence process oscillates and decays.

3.2. Adaptive Distributed Formation Controller Design

The unknown synthesized disturbances in (7) are estimated by using the MSMD disturbance observer (9). The design process of the adaptive distributed formation controller can be divided into the following two steps.

Step 1. In order to obtain the synchronization performance among USVs, the position tracking error $e_{i,\chi }={\left[e_{ix},e_{iy}\right]}^T$ for the $ith$ USV is formulated as follows:

$$e_{i,\chi }={\overline{R}}^T\left({\psi }_i\right)\left\{{\displaystyle \sum _{j=1}^M{a}_{ij}\left({\chi }_i-{\chi }_j-{\vartheta }_{ijd}\right)+a_{i0}\left({\chi }_i-{\chi }_d-{\vartheta }_{id}\right)}\right\}$$

(17)

where ${\chi }_d$ indicates the expected position of the virtual leader; ${\vartheta }_{id}$ indicates the expected relative position between the $ith$ USV and the virtual leader; ${\vartheta }_{ijd}$ is the expected relative position between the $ith$ USV and the $jth$ USV.

Taking the derivative of (17), we obtain:

$$\begin{array}{ll}{\dot{e}}_{i,\chi }=& -r_{i}H{e}_{i,\chi }+a_{id}{\left[u_i,v_i\right]}^T-{\displaystyle \sum _{j=1}^M{a}_{ij}}{\overline{R}}^T\left({\psi }_i\right)\overline{R}\left({\psi }_i\right){\left[u_j,v_j\right]}^T\\ & -a_{i0}{\overline{R}}^T\left({\psi }_i\right){\dot{\chi }}_d-{\displaystyle \sum _{j=0}^M{a}_{ij}\overline{R}\left({\psi }_i\right)}{\dot{\vartheta }}_{ijd}\end{array}$$

(18)

where $a_{id}={\displaystyle \sum _{j=1}^M{a}_{ij}+a_{i0}}$, and the matrix $H$ is defined as $H=\left[\begin{array}{ll}0& -1\\ 1& 0\end{array}\right]$.

In this subsection, the underactuated problem is resolved by using an auxiliary variable approach [34], introducing an auxiliary variable $\lambda ={\left[-{\lambda }_0,0\right]}^T, {\lambda }_0\in R^{+}$, and redefining the formation position tracking error as follows:

$${\overline{e}}_{i,\chi }=e_{i,\chi }-\lambda$$

(19)

Recalling the model (18), it follows that

$$\begin{array}{ll}{\dot{\overline{e}}}_{i,\chi }=& -r_{i}H{\overline{e}}_{i,\chi }+s_i{V}_i-{\displaystyle \sum _{j=1}^M{a}_{ij}}{\overline{R}}^T\left({\psi }_i\right)\overline{R}\left({\psi }_i\right){\left[u_j,v_j\right]}^T\\ & -a_{i0}{\overline{R}}^T\left({\psi }_i\right){\dot{\chi }}_d-{\displaystyle \sum _{j=0}^M{a}_{ij}\overline{R}\left({\psi }_i\right)}{\dot{\vartheta }}_{ijd}+{\left[0,a_{id}{v}_i\right]}^T\end{array}$$

(20)

where the matrix $s_i=diag\left(a_{id},{\lambda }_0\right)$.

To achieve the formation control objective, each component of the error ${\overline{e}}_{i,\chi }$ is imposed with the following constraints:

$$-{\underset{\_}{\kappa }}_{i,\chi }{F}_{i,\chi }\left(t\right)<{\overline{e}}_{i,\chi }<{\overline{\kappa }}_{i,\chi }{F}_{i,\chi }\left(t\right)$$

(21)

where ${\underset{\_}{\kappa }}_{i,\chi }$ and ${\overline{\kappa }}_{i,\chi }, \left(\chi =\mathrm{x},\mathrm{y}\right)$ are positive constants, and $F_{i,\chi }\left(t\right)$ is a smooth hyperbolic cosecant prescribed performance function, and the concrete form is as follows.

$$F_{i,\chi }\left(t\right)=\left(F_{i,\chi }^0-F_{i,\chi }^{\infty }\right)\exp \left(-{\gamma }_{i}t\right)+F_{i,\chi }^{\infty }$$

(22)

where ${\gamma }_i$ is the positive constant regulating the convergence time; $F_{i,\chi }^0$ is the initial value of $F_{i,\chi }\left(t\right)$; $F_{i,\chi }^{\infty }$ is the steady state value of $F_{i,\chi }\left(t\right)$, and $F_{i,\chi }^0>F_{i,\chi }^{\infty }>0$.

A novel error transformation equation is defined as follows

$${\xi }_{i,\chi }\left(t\right)=\frac{{\overline{e}}_{i,x}}{\sqrt{F_{i,\chi }^T{F}_{i,\chi }-{\overline{e}}_{i,\chi }^T{\overline{e}}_{i,\chi }}}$$

(23)

Remark 3. 

In comparison with the transformed error equation given as  ${\xi }_i=\left(e_i/\left[F_i-e_i\right]\right)$  in [30], the proposed transformed error equation in this paper is differentiable and meets the requirement of backstepping control. In addition, compared with [25,26,27], the expression of error transformation equation in this study is simpler, which reduces the complexity of the formation controller.

The derivative of (23) can be written as follows

$$\begin{array}{ll}{\dot{\xi }}_{i,\chi }& =\frac{F_{i,\chi }^T{F}_{i,\chi }}{\sqrt{{\left(F_{i,\chi }^T{F}_{i,\chi }-{\overline{e}}_{i,\chi }^T{e}_{i,\chi }\right)}^3}}\left({\dot{\overline{e}}}_{i,\chi }-\frac{{\overline{e}}_{i,\chi }{\dot{F}}_{i,\chi }}{{\dot{F}}_{i,\chi }}\right)\\ & ={\theta }_{i,\chi }\left\{-r_{i}H{\overline{e}}_{i,\chi }+s_i{V}_i-{\displaystyle \sum _{j=1}^M{a}_{ij}}{\overline{R}}^T\left({\psi }_i\right)\overline{R}\left({\psi }_i\right){\left[u_j,v_j\right]}^T\right.\\ & \hspace{1em}\hspace{1em}\left.-a_{i0}{\overline{R}}^T\left({\psi }_i\right){\dot{\chi }}_d-{\displaystyle \sum _{j=0}^M{a}_{ij}\overline{R}\left({\psi }_i\right)}{\dot{\vartheta }}_{ijd}+{\left[0,a_{id}{v}_i\right]}^T-\frac{{\dot{F}}_{i,\chi }}{F_{i,\chi }}{\overline{e}}_{i,\chi }\right\}\end{array}$$

(24)

where ${\theta }_{i,\chi }=diag\left(F_{i,x}^2/\sqrt{{\left(F_{i,x}^2-{\overline{e}}_{ix}^2\right)}^3},F_{i,y}^2/\sqrt{{\left(F_{i,y}^2-{\overline{e}}_{i,y}^2\right)}^3}\right)$.

In order to stabilize ${\xi }_i$, the virtual control law of the $ith$ USV is designed as follows

$$\begin{array}{ll}{\alpha }_i=& \frac{1}{s_i}\left\{-\frac{K_{i,\chi }}{{\theta }_i}{\xi }_{i,\chi }+{\displaystyle \sum _{j=1}^M{a}_{ij}}{\overline{R}}^T\left({\psi }_i\right)\overline{R}\left({\psi }_i\right){\left[u_j,v_j\right]}^T\right.\\ & \hspace{1em}\hspace{1em}\left.+a_{i0}{\overline{R}}^T\left({\psi }_i\right){\dot{\chi }}_d+{\displaystyle \sum _{j=0}^M{a}_{ij}\overline{R}\left({\psi }_i\right)}{\dot{\vartheta }}_{ijd}-{\left[0,a_{id}{v}_i\right]}^T+\frac{{\dot{F}}_{i,\chi }}{F_{i,\chi }}{\overline{e}}_{i,\chi }\right\}\end{array}$$

(25)

where ${\alpha }_i={\left[{\alpha }_{iu},{\alpha }_{ir}\right]}^T$, and $K_{i,\chi }$ is a positive definite diagonal matrix.

In order to avoid the repeated derivation of the above virtual control law, which causes the phenomenon of “differential explosion”, a novel second-order NLD is designed, which has higher precision compared to the first-order filter. The filtered virtual control law $z_{i,1}$ and its derivative $z_{i,2}$ are received by making the virtual control law ${\alpha }_i$ pass the novel second-order NLD.

$$\left\{\begin{array}{l}{\dot{z}}_{i,1}=z_{i,2}\\ {\dot{z}}_{i,2}=-{\mathcal{l}}_i^2\left\{\left(z_{i,1}-{\alpha }_i\right)+\left(\frac{z_{i,2}}{{\mathcal{l}}_i}\right)\right\}\end{array}\right.$$

(26)

Theorem 2. 

For the second-order NLD, if  ${\mathcal{l}}_i>0$, then for any bounded input,   ${\alpha }_i$  and constant  $t_1>0$, such that the output of   the second-order NLD satisfies the following condition:

$$\underset{\mathcal{l}\to \infty }{\lim }{\displaystyle {\int }_0^{t_1}\left|z_{i,1}-{\alpha }_i\right|}dt=0$$

(27)

Proof of Theorem 2 

 

In accordance with the equivalence principle and Lemma 1, distinctly, the system (26) is globally asymptotically stable. As time progresses to infinity, $z_{i,1}=0$, and $z_{i,2}=0$, and the solution of (26) satisfy $\underset{\mathcal{l}\to \infty }{\lim }{\displaystyle {\int }_0^{t_1}\left|z_{i,1}-{\alpha }_i\right|}dt=0$, which means that the estimation value $z_{i,1}$ can converge to the input signal ${\alpha }_i$ and the differential estimation value $z_{i,2}$ can converge to ${\dot{\alpha }}_i$. □

Remark 4. 

According to the above analysis, when  $\mathcal{l}\to \infty$, the error between the filter error value can be constrained to zero during the whole working period, which means that  $z_{i,1}$  can replace  ${\alpha }_i$  to be applied in the controller. Therefore, the whole closed-loop system with the second-order NLD can be obviously proven to be asymptotically stable.

Step 2. Next, the dynamic control law of the $ith$ USV will be designed based on the MSMD disturbance observer (9) and the second-order NLD (26). The velocity tracking error is defined as follows:

$$e_{i,V}=V_i-z_{i,1}$$

(28)

Taking the time derivative of (28) along (6), we obtain

$${\dot{e}}_{i,V}={\overline{M}}_i^{-1}{\overline{\tau }}_i+{\widehat{\overline{W}}}_i-z_{i,2}+\mathsf{\Delta }{\overline{W}}_i$$

(29)

where $\mathsf{\Delta }{\overline{W}}_i={\overline{W}}_i-{\widehat{\overline{W}}}_i$ indicates observation error; similarly, there is a design-positive parameter matrix $w_i\in {ℝ}^2$, satisfying the condition of $\left|\mathsf{\Delta }{\overline{W}}_i\right|\le w_i$.

The formation control law is related to the bound of $\mathsf{\Delta }{\overline{W}}_i$ and unknown synthesized disturbances ${\overline{W}}_i$. According to Assumption 3, ${\overline{W}}_i$ is unknown and bounded, but the related upper-bound value is difficult to be measured in advance, which makes it difficult to realize the formation control law. Hence, the adaptive technology is adopted to estimate the unknown disturbances is compensated by the proposed formation controller.

Based on (29), the adaptive formation controller is designed to function as

$${\overline{\tau }}_i=\overline{M}\left(-K_{i,V}{e}_{i,V}+z_{i,2}-{\widehat{\overline{W}}}_i-{\widehat{w}}_i\frac{e_{i,V}}{\left|e_{i,V}\right|}-{\theta }_i{s}_i{\xi }_{i,\chi }\right)$$

(30)

where $K_{i,V}$ is a positive definite diagonal matrix, and ${\widehat{w}}_i$ is the estimation value of $w_i$. To obtain ${\widehat{w}}_i$, specifically, the homologous adaptive law is designed as follows:

$${\dot{\widehat{w}}}_i=k_{i,1}\left(\left|e_{i,V}\right|-2k_{i,2}{\widehat{w}}_i\right)$$

(31)

where $k_{i,1}>0$ and $k_{i,2}>0$ are the control parameters to be designed.

Remark 5. 

The adaptive law deals with unknown synthesized disturbances by estimating an upper bound on the correlation term of the unknown synthesized disturbance  $\mathsf{\Delta }{\overline{W}}_i$. This approach does not require precise identification of the disturbance dynamics, but only needs to provide a threshold value that satisfies  $\left|\mathsf{\Delta }{\overline{W}}_i\right|\le w_i$  to ensure the stability of the control system and achieve the desired control objective. Thus, the provided formation controller is able to handle unknown disturbances effectively.

4. Stability Analysis

Theorem 3. Take into consideration the USV formation system (4) and (6) with Assumptions 1–3. The time-varying formation control is presented by the adaptive formation controller (30) and (31), which can be achieved with the prescribed performance. In that case, all closed-loop signals of the USVs formation control system are bounded.

Proof of Theorem 3 

 

The following Lyapunov function is selected:

$$V_i=\frac{1}{2}{\xi }_{i,\chi }^T{\xi }_{i,\chi }+\frac{1}{2}{e}_{i,V}^T{e}_{i,V}-\frac{1}{k_{i,1}}\mathsf{\Delta }{w}_i{\dot{\widehat{w}}}_i$$

(32)

where  $\mathsf{\Delta }{w}_i=w_i-{\widehat{w}}_i$, and the derivative of (32) is calculated as follows:

$$\begin{array}{ll}{\dot{V}}_i& ={\xi }_{i,\chi }^T{\dot{\xi }}_{i,\chi }+e_{i,V}^T{\dot{e}}_{i,V}-\frac{1}{k_{i,1}}\mathsf{\Delta }{w}_i{\dot{\widehat{w}}}_i\\ & ={\xi }_{i,\chi }^T{\theta }_{i,\chi }\left\{-r_{i}H{\overline{e}}_{i,\chi }+s_i{V}_i-{\displaystyle \sum _{j=1}^M{a}_{ij}}{\overline{R}}^T\left({\psi }_i\right)\overline{R}\left({\psi }_i\right){\left[u_j,v_j\right]}^T\right.\\ & \left.-a_{i0}{\overline{R}}^T\left({\psi }_i\right){\dot{\chi }}_d-{\displaystyle \sum _{j=0}^M{a}_{ij}\overline{R}\left({\psi }_i\right)}{\dot{\vartheta }}_{ijd}+{\left[0,a_{id}{v}_i\right]}^T-\frac{{\dot{F}}_{i,\chi }}{F_{i,\chi }}{\overline{e}}_{i,\chi }\right\}\\ & +e_{i,V}^T\left({\overline{M}}_i^{-1}{\overline{\tau }}_i+{\widehat{\overline{W}}}_i-z_{i,2}+\mathsf{\Delta }{\overline{W}}_i\right)-\frac{1}{k_{i,1}}\mathsf{\Delta }{w}_i{\dot{\widehat{w}}}_i\end{array}$$

(33)

Substituting (25), (30) and (31) into (33), we can obtain

$$\begin{array}{ll}{\dot{V}}_i& =-K_{i,\chi }{\xi }_{i,\chi }^T{\xi }_{i,\chi }-K_{i,V}{e}_{i,V}^T{e}_{i,V}+e_{i,V}^T\mathsf{\Delta }{\overline{W}}_i-w_i\left|e_{i,V}\right|+2k_{i,2}\mathsf{\Delta }{w}_i{\widehat{w}}_i\\ & \le -K_{i,\chi }{\xi }_{i,\chi }^T{\xi }_{i,\chi }-K_{i,V}{e}_{i,V}^T{e}_{i,V}+2k_{i,2}\mathsf{\Delta }{w}_i{\widehat{w}}_i\end{array}$$

(34)

Moreover, according to Young’s inequality

$$k_{i,2}\mathsf{\Delta }{w}_i{\widehat{w}}_i\le -k_{i,2}\left(1-\frac{1}{2{\varpi }_i}\right)\mathsf{\Delta }{w}_i^2+k_{i,2}{\varpi }_i{w}_i^2$$

(35)

where ${\varpi }_i>\frac{1}{2}$, substituting (35) into (34), and (34) is rearranged into the following expression:

$$\begin{array}{ll}{\dot{V}}_i& \le -K_{i,\chi }{\xi }_{i,\chi }^T{\xi }_{i,\chi }-K_{i,V}{e}_{i,V}^T{e}_{i,V}-k_{i,2}\left(1-\frac{1}{2{\varpi }_i}\right)\mathsf{\Delta }{w}_i^2+k_{i,2}{\varpi }_i{w}_i^2\\ & \le -{\mathsf{\Theta }}_{i,1}{V}_i+{\mathsf{\Theta }}_{i,2}\end{array}$$

(36)

where ${\mathsf{\Theta }}_{i,1}=2\min \left\{\lambda \left(K_{i,\chi }\right),\lambda \left(K_{i,V}\right),k_{i,2}\left(1-\frac{1}{2{\varpi }_i}\right)\right\}$, and ${\mathsf{\Theta }}_{i,2}\ge k_{i,2}{\varpi }_i{w}_i^2$. Solving inequality (36) can be obtained as follows:

$$V_i\le \left\{V_i\left(0\right)-\frac{{\mathsf{\Theta }}_{i,2}}{{\mathsf{\Theta }}_{i,1}}\right\}{e}^{-{\mathsf{\Theta }}_{i,1}t}+\frac{{\mathsf{\Theta }}_{i,2}}{{\mathsf{\Theta }}_{i,1}}$$

(37)

It can be seen from (37) that $V_i$ is uniformly ultimate bounded. Moreover, according to Theorems 1 and 2, the MSMD disturbance observer and the second-order NLD are both asymptotically stable and the correlated error values are bounded. Thus, all closed-loop signals of the USVs formation control system are bounded. Theorem 3 is illustrated to be complete. In conclusion, a new disturbance-observer-based adaptive formation controller in this study is designed to ensure that the formation system is asymptotically stable, and the formation system error is bounded. □

5. Simulation

In this section, in order to demonstrate the effectiveness and advantages of the proposed MSMD disturbance-observer-based adaptive-prescribed performance formation control strategy, the subsequent numerical simulation experiments are carried out. There are five USVs under the identical dynamics that are considered to track the virtual leader in this simulation.

A directed connectivity graph communication topology as shown in Figure 4 is employed between USVs, and only the $1st$ USV can access the desired trajectory information of the virtual leader. The Laplacian matrix of the graph is defined as follows:

$$L=\left[\begin{array}{lllll}0& 0& 0& 0& 0\\ -1& 1& 0& 0& 0\\ 0& -1& 1& 0& 0\\ 0& 0& -1& 1& 0\\ -1& 0& 0& -1& 1\end{array}\right]$$

(38)

The USV model in [35] is used as the research object in the simulation test, the initial positions of each USV are chosen as ${\eta }_1={\left[-22,-18,-5\right]}^T$, ${\eta }_2={\left[-15,-12,-10\right]}^T$, ${\eta }_3={\left[-2,-2,-20\right]}^T$, ${\eta }_4={\left[-18,7,15\right]}^T$, and ${\eta }_5={\left[-25,15,5\right]}^T$. The initial velocities of each USV are chosen as $V_i={\left[0,0\right]}^T$ and $r_i=0$. The expected relative position of each USV are chosen as follows: ${\vartheta }_1={\left[0,-20\right]}^T$, ${\vartheta }_2={\left[0,-10\right]}^T$, ${\vartheta }_3={\left[0,0\right]}^T$, ${\vartheta }_4={\left[0,10\right]}^T$ and ${\vartheta }_5={\left[0,20\right]}^T$. To validate the effectiveness of the proposed control algorithm and disturbance observer, the unknown and external disturbances are set as follows: ${\tau }_{iw}={\left[2\sin \left(t/20\right)+0.1\sin \left(t/30\right),1.2\sin \left(t/20\right),\sin \left(t/20\right)+1.2\sin \left(t/40\right)\right]}^T$. In this simulation, setting up $t^{\prime }=200s$ and the desired trajectory for the virtual leader ${\chi }_d$ is defined as follows:

(1)

When $t<t^{\prime }$, we choose

$$\left\{\begin{array}{l}x\left(t\right)=2t\\ y\left(t\right)=0\end{array}\right.$$

(39)

(2)

When $t^{\prime }\le t\le 800$, we choose

$$\left\{\begin{array}{l}x\left(t\right)=2t^{\prime }+150\sin \left(2\left(t-t^{\prime }\right)/150\right)\\ y\left(t\right)=150-150\cos \left(2\left(t-t^{\prime }\right)/150\right)\end{array}\right.$$

(40)

To implement the proposed formation control strategy, the parameters of disturbance observer (9) are selected as follows: $p=5$, $q=7$, ${\beta }_{i,1}=diag\left(1,2\right)$, and ${\beta }_{i,1}=diag\left(0.5,1.5\right)$. The predefined performance functions in (4) are specified as follows: $F_{i,\chi }=\left(10-0.1\right)\exp \left(-0.5t\right)+0.1$, ${\underset{\_}{\kappa }}_{i,x}=3$, ${\underset{\_}{\kappa }}_{i,y}=1$, and ${\overline{\kappa }}_{i,x}=1$, $\chi =x,y$. The pertinent parameters are listed below: ${\mathcal{l}}_i=3$, $K_{i,\chi }=diag\left(0.1,1\right)$, $K_{i,\chi }=diag\left(1,5\right)$, $k_{i,1}=k_{i,2}=0.01$.

The simulation results are presented in Figure 5, Figure 6, Figure 7, Figure 8, Figure 9, Figure 10, Figure 11, Figure 12 and Figure 13. In Figure 5, by using the proposed controller in this paper, five USVs are able to rapidly attain the expected formation configuration and sail along the desired trajectory in spite of unknown synthesized disturbances and performance constraints. The tracking errors are illustrated in Figure 6 and Figure 7, which demonstrate that their whole profile is limited to the range of the prescribed performance function and eventually converges to a predetermined location. Figure 8 shows the control inputs of the five following USVs. Figure 9 shows the velocity change curves of the five following USVs. Figure 10 shows the velocity error for five USVs, which eventually converges to zero, thus validating the effectiveness of the proposed controller.

It is fully shown in Figure 11 that the proposed MSMD disturbance observer in this study implements the smooth estimation of the unknown synthesized disturbances, thus obtaining better control effect. Additionally, Figure 12 shows the observation errors of the traditional disturbance observer; it can be seen that the steady-state accuracy in Figure 11 is higher than that in Figure 12. Additionally, the formation trajectory changes at 200s; compared with Figure 12, the response curves of the observation error in Figure 11 do not fluctuate significantly, indicating that the MSMD disturbance observer has better robustness. According to (30), the adaptive estimator is designed to approximate the upper bound of the unknown synthesized disturbances error $\mathsf{\Delta }{\overline{W}}_i$, and it should satisfy $w_i\ge \left|\mathsf{\Delta }{\overline{W}}_i\right|$ to make the system stable. Taking the $3th$ USV as an example, Figure 13 illustrates that the adaptive estimates quickly converge to a stable value, and it is larger than the maximum value of $\mathsf{\Delta }{\overline{W}}_i$. This validates the effectiveness of the adaptive formation control to mitigate the disturbance from unknown factors for the controller.

6. Conclusions

The study addresses the challenge of USV formation control under unknown synthesized disturbances and prescribed performance constraints. An adaptive prescribed performance formation control scheme is proposed, utilizing a novel error transformation equation designed to exhibit specific transient characteristics. This scheme effectively ensures that the formation error promptly enters and remains within the prescribed positional error bounds. For the control innovation, benefiting from the MSMD disturbance observer, the unknown synthesized disturbances within the formation system can be estimated accurately. Additionally, the suggested method eradicates the requirement for the knowledge of upper bounds of synthesized disturbances which are frequently required in the backstepping scheme, thereby reducing the burden of disturbance estimation and computational load; moreover, a new second-order nonlinear filter solves the “differential explosion” problem in the backstepping control process. Simulation results and comparative experiments validate the effectiveness and superiority of the proposed formation control method.

Although this work does not address every detail of the control task, such as the algorithm’s inability to specifically account for unnecessary communication burden in the sensor-to-controller channel, this remains a future problem to be addressed in subsequent research.