Distributed adaptive sliding mode control for attitudes synchronization of multiple autonomous underwater vehicles

Abstract

This article is concerned with attitudes synchronization of networked autonomous underwater vehicles, which aims to enable all the autonomous underwater vehicles tracking the desired attitude coordinately, when the desired attitude information can only be achieved by one or one subset autonomous underwater vehicles. A Lyapunov technique is presented for designing an adaptive synchronization control protocol using sliding mode and consensus theory. The uncertainty of the autonomous underwater vehicle dynamics, and the unknown external disturbance and input saturation were both considered in our controller proposed; moreover, only the neighbors’ information were used by the controller, so it is totally distributed. Simulations were performed to validate the theoretical results.

Introduction

As an important member of the robot family, autonomous underwater vehicle (AUV) has been employed widely in sub-sea exploration. Due to the complexity of the ocean environment and difficulty of the underwater tasks, we usually need multiple AUVs work cooperatively as a group. Coordinated detection, coordinated guidance, and coordinated control are the key cooperative behaviors for the multiple AUVs. To achieve these cooperative behaviors, we usually need AUVs move in a formation, which is called formation control problem. The following control strategies are the typical existing works for formation: the leader–follower, the virtual structure method, artificial potential field, the method based on behavior, and the method based on consensus. In 1989, leader–follower method is used for the formation control of mobile robot for the first time in the study of Wang. ¹ In the study of Fahimi, ² a leader–follower strategy was designed and applied to keep multiple underactuated surface ships moving in formation using sliding mode method. In the study of Kowdiki et al., ³ the leader–follower method was combined with artificial potential field method; this compound formation strategy was used to control mobile robots in their work. Nair et al. ⁴ proposed a fuzzy control algorithm by combining the artificial potential field method and synovial control to cope with the uncertain model parameters. The behavior method was proposed in the study of Mataric ⁵ in the earliest.

Attitude is the direction of a robot in each dimension of its moving space; so in the formation control problems, it is important to make the attitudes of each robot track the desired attitude coordinately, which is called synchronized attitudes problem. Synchronized attitudes control has wide implications in the field of multi-robots and especially in the field of teleoperation for robots when they are executing tasks. For attitudes synchronization control, Zhang et al. ⁶ considered model parameter uncertainty and external disturbance, and designed attitudes coordinate tracking controller for spacecrafts. Cong et al. ⁷ also considered spacecraft attitudes synchronized control problem based on time-varying sliding mode control methods. Zou and Kumar ⁸ proposed an attitude synchronization control based on terminal sliding mode control and adaptive control method to ensure that the coordinated error convergence to zero in finite time. A distributed adaptive synchronization controller is proposed using the state information of adjacent agents in the study of Peng et al. ⁹ The design does not require an accurate model of each agent, and developed controller ensures that the state of each agent is synchronized with the state of the leader.

In the practical coordination control of multi-robots, we also have to consider the input saturation problems. Li et al. ¹⁰ studied the consensus of first-order model in the case of input saturation and Ren ¹¹ studied the second-order dynamic model under condition that the limited input saturation, respectively. In practical applications, the actuator of the system exists as saturated nonlinear problem; for this problem, there are two common methods normally, one is the direct method ¹² and the other is anti-windup method. ¹³ Li et al. ¹⁴ uses the direct method combined with the fuzzy control algorithm to solve the saturation limited problem in robot control. As for a single attitude tracking problem of AUV, Cui et al. ¹⁵ proposed an anti-saturation compensator to solve the problem of saturated non-linearity; the stability of the closed-loop system is proved by the Lyapunov equation. In the control design of Cui et al., ¹⁶ the external interference, the control input non-linearity, and the model uncertainty are taken into account. The two neural networks (NN), including the commenter and the action NN, are integrated into the adaptive control design. In order to eliminate the non-linearity of the control input of the AUV, a compensation item is also designed in the adaptive control. In the study of Cui et al., ¹⁷ a general underwater robot integrated sliding mode controller (ISMC) based on multi-input and multi-output–extended state observer (MIMO-ESO) has been developed. In the control design, it has been successfully solved with the unmeasured velocity, unknown disturbance. And it is shown that the proposed method is better than the conventional potential difference (PD) control method.

In this article, we considered the attitudes synchronization of multiple AUVs with input saturation and model uncertainty. The remainder of this article is organized as follows: in section “Introduction,” we introduce the problem of AUV attitude synchronization control and formulate the control objective of this work. In section “Problem formulation,” we establish a dynamic model of the AUV, and the communication extension graph is introduced. In section “Control design without input nonlinearities,” an adaptive sliding mode control is proposed for the AUV heading synchronized control. In section “Control design with input saturation,” we introduce the input saturation cases with multiple AUVs attitude synchronization control. The simulation experiment results are presented and discussed in section “Simulation results.” Conclusion and future research works are given in section “Conclusion.”

Problem formulation

For a group of $n$ robots in one group, it is a conventional way to model the interaction between them by a directed graph adjacency matrix and topology graph. ¹⁸ First of all, we introduce a virtual agent, denoted as $v_0$ , which denotes the desired state strictly. For the $n$ agents system, the graph $G=(V,A^{*})$ contains a node set $V=\{v_0,v_1,\dots ,v_n\}$ , and a weighted adjacent matrix $A^{*}=[a_{\mathrm{ij}}^{*}]$ , where $a_{\mathrm{ij}}^{*}>0$ represents that agent i can obtain the status information from agent j, otherwise $a_{\mathrm{ij}}^{*}=0$ . Then, define a diagonal matrix $\mathrm{\Delta }$ whose elements are ${\delta }_{\mathrm{jj}}={\Sigma }_k{a}_{\mathrm{jk}}^{*}$ and the normalized Laplacian of $G$ is $L=I-A$ , where A is the normalized adjacent matrix which defined as $a_{\mathrm{ij}}=a_{\mathrm{ij}}^{*}/{\delta }_{\mathrm{jj}}$ when ${\delta }_{\mathrm{jj}}\ne 0$ , otherwise $a_{\mathrm{ij}}=a_{\mathrm{ij}}^{*}$ . With this added virtual agent in the system, we call the graph $G$ the extended communication graph. Figure 1 is a case extended communication graph and the according Laplace matrix L under this definition. We can see from this figure that only agent 1 can get information from the virtual leader, agent 2 can get information from both agents 1 and 3, and agent 3 can get information from agent 2.

Figure 1.

Example graph and its Laplacian.

Because the virtual agent cannot get any information from others, the row sum of the matrix A is equal to 1, except for the first row. It is obvious that all the diagonal elements of L are 1, we can know that L is a diagonal dominant matrix, positive define, and reversible; its inverse matrix is $L^{-1}=\sum _{l=0}^{\infty }{A}^l$ .

Normally, the Euler angles $\theta ,\psi ,\phi$ are used to describe the spatial posture of AUV, which are called pitch, yaw, and roll angles, respectively. Due to the same form of the dynamics of yaw, pitch, and roll in the attitude control of AUV, we study the heading control of AUV as an example. If the interactions between pitch and roll motion are neglected, as well as the external disturbance and non-linearity are taken into account, the kinematic model of the yaw motion of multi-AUVs is written in the following form

$$\{\begin{array}{c}\stackrel{·}{\psi }={\omega }_y\hfill \\ {\stackrel{·}{\omega }}_y=C_1{\omega }_y+C_2\beta +C_3{\delta }_r+D\hfill \end{array}$$

(1)

where $\psi =[{\psi }_0,{\psi }_1,\dots ,{\psi }_n{]}^T$ is the yaw angle collection of the $n$ AUVs, ${\omega }_y=[{\omega }_{y0},{\omega }_{y1},\dots ,{\omega }_{\mathrm{yn}}{]}^T$ is the angle velocity collection in the body-referenced frame, $\beta =[{\beta }_0,{\beta }_1,\dots ,{\beta }_n{]}^T$ is sideslip angle, ${\delta }_r=[{\delta }_{r0},{\delta }_{r1},\dots ,{\delta }_{\mathrm{rn}}{]}^T$ is the control input, and $D=[d_0,d_1,\dots ,d_n{]}^T$ denotes the external disturbance and non-linearity. Specially, $C_1=\mathrm{diag}(c_{10},c_{11},\dots ,c_{1n})$ and $C_2=\mathrm{diag}(c_{20},c_{21},\dots ,c_{2n})$ are parameters depending on the hydrodynamics coefficients, and their diagonal elements are unknown, $C_3=\mathrm{diag}(c_{30},c_{31},\dots ,c_{3n})$ denote open-loop control gains, respectively.

To make the attitude of each AUV track the desired attitude ${\psi }_0$ synchronously, define the ith tracking error as

$$e_i=\sum _{j=0}^n{a}_{\mathrm{ij}}({\psi }_i-{\psi }_j)$$

(2)

where $a_{\mathrm{ij}}$ is the ijth element of the normalized adjacent matrix A. According the definition of the normalized adjacent matrix, this tracking error can also be described as

$$e_i={\psi }_i-{\psi }_{\mathrm{di}}$$

(3)

where ${\psi }_{\mathrm{di}}=\sum _{j=0}^n{a}_{\mathrm{ij}}{\psi }_j$ . For example, the synchronized attitude error for Figure 1 is

$$\{\begin{array}{c}{e}_0=0\hfill \\ \multicolumn{1}{c}{e_1={\psi }_1-{\psi }_0}\\ \multicolumn{1}{c}{e_2={\psi }_2-\frac{1}{2}{\psi }_1-\frac{1}{2}{\psi }_3}\\ \multicolumn{1}{c}{e_3={\psi }_3-{\psi }_2}\end{array}$$

(4)

As so far, the control objective of this article can be described as follows: by designing the control inputs ${\delta }_{\mathrm{ri}}$ for each AUV to make yaw angle ${\psi }_i$ track the desired attitude ${\psi }_{\mathrm{di}}$ as time goes on, that is to say, to enable the tracking error $e=\psi -{\psi }_d$ tends to zero asymptotically, where $e=[e_0,e_1,\dots ,e_n{]}^T$ , ${\psi }_d=[{\psi }_{d0},{\psi }_{d1},\dots ,{\psi }_{\mathrm{dn}}{]}^T$ .

Before going on, we introduce the following assumptions.

Assumption 1

The yaw angle $\psi$ and its time derivative $\stackrel{·}{\psi }$ are measurable outputs.

Assumption 2

The external disturbance $D_i$ is bounded and satisfies $\parallel D_i{\parallel }_{\infty }\le {\rho }_i$ , where $P=[{\rho }_0,{\rho }_1,\dots ,{\rho }_n{]}^T$ is the unknown upper bound of disturbance and ${\rho }_i>0$ .

Assumption 3

The control gain $C_3$ is unknown, but its sign and upper bound are known and satisfy $0<c_{3i}<{\overline{c}}_{3i}$ , where ${\overline{C}}_3=[{\overline{c}}_{30},{\overline{c}}_{31},\dots ,{\overline{c}}_{3n}{]}^T$ is a known constant, and its diagonal elements are positive.

Control design without input nonlinearities

The time derivative of the tracking error is

$$\stackrel{·}{e}=\stackrel{·}{\psi }-{\stackrel{·}{\psi }}_d={\omega }_y-{\stackrel{·}{\psi }}_d$$

(5)

According to the definition of Laplace matrix L and equation (5), we have

$$\stackrel{·}{e}=L{\omega }_y$$

(6)

In this section, we consider the unknown disturbance and propose an adaptive controller based on sliding mode control for the AUV heading synchronization control.

Define the sliding mode surface as follows

$$S=\stackrel{·}{e}+\mathrm{\Lambda }e$$

(7)

where $S=[s_0,s_1,\dots ,s_n{]}^T$ , $\mathrm{\Lambda }=\mathrm{diag}({\lambda }_0,{\lambda }_1,\dots ,{\lambda }_n)$ is a design parameter and ${\lambda }_i>0$ .

Substituting equation (5) into equation (7), we have

$$S={\omega }_y-{\stackrel{·}{\psi }}_d+\mathrm{\Lambda }(\psi -{\psi }_d)$$

(8)

Its time derivative is

$$\stackrel{·}{S}={\stackrel{·}{\omega }}_y-{\stackrel{··}{\psi }}_d+\mathrm{\Lambda }L{\omega }_y$$

(9)

Then, substituting equation (1) into equation (9), we have

$$\stackrel{·}{S}=C_1{\omega }_y+C_2\beta +C_3{\delta }_r+D-{\stackrel{··}{\psi }}_d+\mathrm{\Lambda }L{\omega }_y$$

(10)

Choose the exponential reaching law ¹⁵ as

$$\stackrel{·}{S}=-\mathrm{TS}-\mathrm{Ksgn}(S)$$

(11)

where $T=\mathrm{diag}({\tau }_0,{\tau }_1,\dots ,{\tau }_n)$ , $K=\mathrm{diag}(k_0,k_1,\dots ,k_n)$ , and ${\tau }_i>0,k_i>0$ . In the exponential approximation, the approach speed is gradually reduced from a larger value to zero, not only the approaching time but also the speed at which the moving point reaches the sliding mode surface is reduced. In the exponential reaching law, in order to ensure rapid approaching at the same time to reduce chattering, we would better increase ${\tau }_i$ while reducing $k_i$ . That means a large ${\tau }_i$ and a small $k_i$ would be a better choice.

The control law for AUVs is proposed as follows

$$\begin{array}{c}{\delta }_r=B[-\mathrm{TS}-\mathrm{Ksgn}(S)-C_1{\omega }_y-C_2\beta \hfill \\ \multicolumn{1}{c}{-\mathrm{Psgn}(S)+{\stackrel{··}{\psi }}_d-\mathrm{\Lambda }L{\omega }_y]}\end{array}$$

(12)

where $B=\mathrm{diag}(b_0,b_1,\dots ,b_n)=C_3^{-1}$ and $b_i>0$ . Because $C_1,C_2,B,P$ are all unknown, we use their estimations ${\hat{C}}_1,{\hat{C}}_2,\hat{B},\hat{P}$ instead. Then, the control law becomes

$$\begin{array}{c}{\delta }_r=\hat{B}[-\mathrm{TS}-\mathrm{Ksgn}(S)-{\hat{C}}_1{\omega }_y-{\hat{C}}_2\beta \hfill \\ -\hat{P}\mathrm{sgn}(S)+{\stackrel{··}{\psi }}_d-\mathrm{\Lambda }L{\omega }_y]=\hat{B}Y\hfill \end{array}$$

(13)

where $Y=-\mathrm{TS}-\mathrm{Ksgn}(S)-{\hat{C}}_1{\omega }_y-{\hat{C}}_2\beta -\hat{P}\mathrm{sgn}(s)+{\stackrel{··}{\psi }}_d-\mathrm{\Lambda }L{\omega }_y$ .

The according adaptation law is chosen as

$$\{\begin{array}{c}{\stackrel{·}{\hat{c}}}_{1i}={\kappa }_{1i}{S}^T{L}^{-1}{\omega }_y\hfill \\ \multicolumn{1}{c}{{\stackrel{·}{\hat{c}}}_{2i}={\kappa }_{2i}{S}^T{L}^{-1}\beta }\\ \multicolumn{1}{c}{{\stackrel{·}{\hat{b}}}_i=-{\kappa }_{3i}{S}^T{L}^{-1}Y}\\ \multicolumn{1}{c}{{\stackrel{·}{\hat{\rho }}}_i={\kappa }_{4i}{S}^T{L}^{-1}\mathrm{sgn}(S)}\end{array}$$

(14)

where $K_1=\mathrm{diag}(k_{10},k_{11},\dots ,k_{1n})$ , $K_2=\mathrm{diag}(k_{20},k_{21},\dots ,k_{2n})$ , $K_3=\mathrm{diag}(k_{30},k_{31},\dots ,k_{3n})$ , and $K_4=\mathrm{diag}(k_{40},k_{41},\dots ,k_{4n})$ are design parameters, and their diagonal elements are all positive.

Theorem 1

For the heading control system of an AUV in equation (1), the sliding mode is chosen as equation (7), control law is equation (13), and adaptation laws is equation (14). The closed-loop system converges to the sliding mode surface $S(t)=0$ in finite time, and when $t\to \infty$ , all errors will eventually converge to zero.

Proof

Select the following Lyapunov candidate function

$$\begin{array}{cc}\multicolumn{1}{c}{V=}& \frac{1}{2}{S}^T{L}^{-1}S+\sum _{i=0}^n\frac{1}{2{\kappa }_{1i}}{\tilde{c}}_{1i}^2+\sum _{i=0}^n\frac{1}{2{\kappa }_{2i}}{\tilde{c}}_{2i}^2+\sum _{i=0}^n\frac{1}{2{\kappa }_{3i}{b}_i}{{\tilde{b}}_i}^2\hfill \\ \multicolumn{1}{c}{}& +\sum _{i=0}^n\frac{1}{2{\kappa }_{4i}}{{\tilde{\rho }}_i}^2\hfill \end{array}$$

(15)

where ${\tilde{c}}_{1i}={\hat{c}}_{1i}-c_{1i},{\tilde{c}}_{2i}={\hat{c}}_{2i}-c_{2i},{\tilde{b}}_i={\hat{b}}_i-b_i$ , and ${\tilde{\rho }}_i={\hat{\rho }}_i-{\rho }_i$ are estimation errors of $c_{1i}$ , $c_{2i}$ , $b_i$ , and ${\rho }_i$ .

Its time derivative is written as follows

$$\begin{array}{cc}\multicolumn{1}{c}{\stackrel{·}{V}=}& S^T{L}^{-1}\stackrel{·}{S}+\sum _{i=0}^n\frac{1}{k_{1i}}{\tilde{c}}_{1i}{\stackrel{·}{\hat{c}}}_{1i}+\sum _{i=0}^n\frac{1}{k_{2i}}{\tilde{c}}_{2i}{\stackrel{·}{\hat{c}}}_{2i}+\sum _{i=0}^n\frac{1}{k_{3i}{b}_i}{\tilde{b}}_i{\stackrel{·}{\hat{b}}}_i\hfill \\ \multicolumn{1}{c}{}& +\sum _{i=0}^n\frac{1}{k_{4i}}{\tilde{\rho }}_i{\stackrel{·}{\hat{\rho }}}_i\hfill \end{array}$$

(16)

Substituting equations (10) and (14) into equation (16), we have

$$\begin{array}{cc}\multicolumn{1}{c}{\stackrel{·}{V}=}& S^T{L}^{-1}[C_1{\omega }_y+C_2\beta +C_3{\delta }_r+D-{\stackrel{··}{\psi }}_d+\mathrm{\Lambda }L{\omega }_y]\hfill \\ \multicolumn{1}{c}{}& +\sum _{i=0}^n{\tilde{c}}_{1i}{S}^T{L}^{-1}{\omega }_y+\sum _{i=0}^n{\tilde{c}}_{2i}{S}^T{L}^{-1}\beta \hfill \\ \multicolumn{1}{c}{}& +\sum _{i=0}^n\frac{{\tilde{b}}_i}{b_i}(-S^T{L}^{-1}Y)+\sum _{i=0}^n{\tilde{\rho }}_i{S}^T{L}^{-1}\mathrm{sgn}(S)\hfill \end{array}$$

(17)

Then, substituting equation (13) into equation (17), we have

$$\begin{array}{cc}\multicolumn{1}{c}{\stackrel{·}{V}=}& S^T{L}^{-1}[C_1{\omega }_y+C_2\beta -\mathrm{TS}-\mathrm{Ksgn}(s)-{\hat{C}}_1{\omega }_y\hfill \\ \multicolumn{1}{c}{}& -{\hat{C}}_2\beta -\hat{\rho }\mathrm{sgn}(S)+{\stackrel{··}{\psi }}_d-\mathrm{\Lambda }L{\omega }_y+D-{\stackrel{··}{\psi }}_d+\mathrm{\Lambda }L{\omega }_y]\hfill \\ \multicolumn{1}{c}{}& +\sum _{i=0}^n{\tilde{c}}_{1i}{S}^T{L}^{-1}{\omega }_y+\sum _{i=0}^n{\tilde{c}}_{2i}{S}^T{L}^{-1}\beta \hfill \\ \multicolumn{1}{c}{}& +\sum _{i=0}^n{\tilde{\rho }}_i{S}^T{L}^{-1}\mathrm{sgn}(S)\hfill \end{array}$$

(18)

Then, by simplifying calculations in equation (18) we can get

$$\begin{array}{cc}\multicolumn{1}{c}{\stackrel{·}{V}=}& -S^T{L}^{-1}\mathrm{TS}-S^T{L}^{-1}\mathrm{Ksgn}(S)-S^T{L}^{-1}\tilde{\rho }\mathrm{sgn}(S)\hfill \\ \multicolumn{1}{c}{}& +S^T{L}^{-1}D\hfill \end{array}$$

(19)

According to Assumption 2, we can get

$$\stackrel{·}{V}\le -S^T{L}^{-1}\mathrm{TS}-S^T{L}^{-1}\mathrm{Ksgn}(S)$$

(20)

We can find that $\stackrel{·}{V}<0$ for all $s_i\ne 0$ . So the system state would converge to the sliding mode surface $S(t)=0$ in finite time; all errors converge to zero and the closed-loop system is stable, and this completes the proof.

It is worth noting that since the control law contains the sign function $\mathrm{sgn}(s)$ , the control law is discontinuous on both sides of the sliding surface $s(0)$ , so there is a chattering phenomenon. In practical applications, the boundary layer method is usually used to reduce the chattering. Define the boundary layer as

$$B_d(t)=\{x|s(x,t)|\le \varphi \},\varphi >0$$

(21)

where $\varphi$ is the thickness of the set boundary layer. In the boundary layer, replace the sign function $\mathrm{sgn}(s)$ with the saturation function $\mathrm{sat}(s)$ ¹⁵

$$\mathrm{sat}(s)=\{\begin{array}{cc}1,\hfill & s>\varphi \hfill \\ \multicolumn{1}{c}{1/\varphi ,}& |s|\le \varphi \hfill \\ \multicolumn{1}{c}{-1,}& s<-\varphi \hfill \end{array}$$

(22)

Inside the $B_d(t)$ , to make smooth process to control the discontinuity, the essence is to add a similar structure of the filter, to eliminate the chattering.

Next, we will prove that in the case of that only part of the AUVs can get the desired yaw angle; all the yaw angle of AUVs can track this desired attitude.

Define the error between the AUV heading angle and the desired attitude as

$$y=y-y_{d}1=y-y_{0}1$$

(23)

where $y$ is equal to $\psi$ and $1=[1,1,\dots ,1{]}^T$ . From the previously defined sliding mode surface, we can get

$$S=\stackrel{·}{e}+\mathrm{\Lambda }e=\mathrm{Ly}+\mathrm{\Lambda }\mathrm{Ly}$$

(24)

introducing auxiliary variables

$$\xi =y+\mathrm{\Lambda }y$$

(25)

where $\xi =[{\xi }_0,{\xi }_1,\dots ,{\xi }_n{]}^T$ and according tracking error is

$$\tilde{\xi }=\xi -{\xi }_{d}1=\xi -{\xi }_{0}1$$

(26)

then

$$\tilde{\xi }=A\xi +S-{\xi }_{0}1$$

(27)

where A is the normalized weighted adjacency matrix which has been introduced before. Due to the first row elements of matrix A are zero, and the rest of the rows elements’ summations are 1, we can get

$$\tilde{\xi }=A\tilde{\xi }+S$$

(28)

then we have

$$\tilde{\xi }=(I-A{)}^{-1}S=L^{-1}S$$

(29)

the time derivative of equation (23) is

$$y=y-y_{0}1=\tilde{\xi }-\mathrm{\Lambda }y$$

(30)

define ${\overline{a}}_y=-\mathrm{\Lambda }$ , we have

$$y={\overline{a}}_{y}y+\tilde{\xi }$$

(31)

because of equation (29), we can get

$$y={\overline{a}}_{y}y+L^{-1}S$$

(32)

Lemma 1

For the time constant $T_1$ , define $s_{i,\max }=\mathrm{su}{p}_{0\le \tau \le t}|s_i|$ , ${\beta }_{\mathrm{si}}=\mathrm{su}{p}_{t>T_1}|s_i|$ , $s_{\max ,i}(t)=\mathrm{ma}{x}_i\mathrm{su}{p}_{0\le \tau \le t}|s_i|$ , and the following inequalities are established ¹⁹

$$\parallel y(t)\parallel \le k_0{e}^{-{\lambda }_{0}t}\parallel y(0)\parallel +\frac{k_0}{{\lambda }_0}[N{\lambda }_{\max }]s_{\max ,i(t)}$$

(33)

$$\parallel y(t)\parallel \le k_0{e}^{-{\lambda }_{0}t}(\parallel y(0)\parallel +\frac{e^{{\lambda }_0{T}_1}}{{\lambda }_0}{\beta }_s(T_1))+\frac{k_0}{{\lambda }_0}{\beta }_{\mathrm{sT}}$$

(34)

where ${\lambda }_{\max }$ is the maximum eigenvalue of $L^{-1}$ , ${\beta }_s(t)=N{\lambda }_{\max }{s}_{\max ,i}(t)$ , ${\beta }_{\mathrm{sT}}=N{\lambda }_{\max }\mathrm{su}{p}_{T_1<t}{s}_{\max ,i}(t)$ , the constant ${\lambda }_0>0$ , $k_0>0$ . The tracking error $y$ converge to the neighborhood of the origin, the system to achieve consistent attitude information.

Proof

The solution of equation (32) is

$$y(t)=y(0)e^{{\overline{a}}_{y}t}+\underset{0}{\overset{t}{\int }}{e}^{{\overline{a}}_y(t-\tau )}{L}^{-1}\mathrm{Sd}\tau$$

(35)

and $\parallel e^{{\overline{a}}_{y}t}\parallel \le k_0{e}^{-{\lambda }_{0}t}$ .

Because $L$ is positive definite, $L^{-1}$ is positive definite too, then

$$\begin{array}{c}\parallel y(t)\parallel \hfill \\ \le k_0{e}^{-{\lambda }_{0}t}\parallel y(0)\parallel +\underset{0}{\overset{t}{\int }}{e}^{{\overline{a}}_y(t-\tau )}{L}^{-1}\mathrm{Sd}\tau \hfill \\ \le k_0{e}^{-{\lambda }_{0}t}\parallel y(0)\parallel +k_0{e}^{-{\lambda }_{0}t}[N{\lambda }_{\max }]s_{\max ,i}(t)\frac{e^{{\lambda }_{0}t}-1}{{\lambda }_0}\hfill \\ \le k_0{e}^{-{\lambda }_{0}t}\parallel y(0)\parallel +\frac{k_0}{{\lambda }_0}[N{\lambda }_{\max }]s_{\max ,i}(t)\hfill \end{array}$$

(36)

In addition, due to the following equation

$$\begin{array}{c}\underset{0}{\overset{t}{\int }}{e}^{{\overline{a}}_y(t-\tau )}{L}^{-1}\mathrm{Sd}\tau \hfill \\ =\underset{0}{\overset{T_1}{\int }}{e}^{{\overline{a}}_y(t-\tau )}{L}^{-1}\mathrm{Sd}\tau +\underset{T_1}{\overset{t}{\int }}{e}^{{\overline{a}}_y(t-\tau )}{L}^{-1}\mathrm{Sd}\tau \hfill \end{array}$$

(37)

we rewrite equation (36) as follows

$$\begin{array}{c}y(t)\le k_0{e}^{-{\lambda }_{0}t}\parallel y(0)\parallel +k_0{e}^{-{\lambda }_{0}t}\frac{e^{{\lambda }_0{T}_1}-1}{{\lambda }_0}{\beta }_s(T_1)\hfill \\ +k_0{e}^{-{\lambda }_{0}t}\frac{e^{{\lambda }_{0}t}-e^{{\lambda }_0{T}_1}}{{\lambda }_0}{\beta }_{\mathrm{sT}}\hfill \\ \le k_0{e}^{-{\lambda }_{0}t}(\parallel y(0)\parallel +\frac{e^{{\lambda }_0{T}_1}}{{\lambda }_0}{\beta }_s(T_1))+\frac{k_0}{{\lambda }_0}{\beta }_{\mathrm{sT}}\hfill \end{array}$$

(38)

This completes the proof.

Control design with input saturation

In the practical applications of AUV, the control system of the actuator exists as saturated nonlinear problems usually. If these problems were ignored, the control performance and stability will be affected. Then attitude control of AUV is achieved by manipulating the rudder and the rudder also exists as saturation, which is shown in Figure 2.

Figure 2.

Saturation properties of the rudder.

The saturation characteristic of the input rudder angle can be expressed as

$${\delta }_r=\mathrm{sat}(u)=\{\begin{array}{cc}{\delta }_m,\hfill & u>{\delta }_m\hfill \\ \multicolumn{1}{c}{u,}& -{\delta }_m\le u\le {\delta }_m\hfill \\ \multicolumn{1}{c}{-{\delta }_m,}& u<{\delta }_m\hfill \end{array}$$

(39)

where $u=[u_0,u_1,\dots ,u_n{]}^T$ is the designed control input, ${\delta }_r$ is the actual control input, that is, the output of the actuator, and ${\delta }_m$ is the upper bound of the actuator.

In this article, we use the anti-windup method to cope with the saturation problem. This method is divided into three steps: first of all, ignore the input saturation limitation and design the control law which satisfies the system performance. Then, the output of the actuator is compared with the value of the control law, and input the obtained deviation signal to the auxiliary anti-windup compensator. Finally, the output of the compensator is fed back to the controller, and further compensates for input restrictions. It is worth noting that this compensator only works when the rudder angle reaches saturation. At this time, AUV closed-loop system block diagram is shown in Figure 3.

Figure 3.

AUV anti-windup compensation control system.

The anti-windup compensator is designed as

$$\stackrel{·}{w}=-K^{*}w-\frac{|S^T{\overline{C}}_3\mathrm{\Delta }u|+\frac{1}{2}(\mathrm{\Delta }{u}^T\mathrm{\Delta }u)}{w^{T}w}w+\mathrm{\Delta }u$$

(40)

when $|w_i|\ge l$ , where $l$ is a small positive constant, otherwise, $\stackrel{·}{w}=0$ , in addition, $\mathrm{\Delta }u={\delta }_r-u$ , $w=[w_0,w_1,\dots ,w_n{]}^T$ is the state of the compensator, $K^{*}=\mathrm{diag}(k_0^{*},k_1^{*},\dots ,k_n^{*})$ is a design parameter and $k_i^{*}>0$ .

Considering the actuator saturation, the control law is designed as

$$u=\hat{B}{Y}_1$$

(41)

where $B=C_3^{-1}$ , $Y_1=-T(S-w)-\mathrm{Ksgn}(S)-\hat{P}\mathrm{sgn}(S)-{\hat{C}}_1{\omega }_y-{\hat{C}}_2\beta -\hat{D}+{\stackrel{··}{\psi }}_d-\mathrm{\Lambda }L{\omega }_y$ .

We also consider the unknown disturbance to the AUV in this article. Here, we design a nonlinear disturbance observer to estimate the unknown disturbance to compensate for the impact on the AUV system. The following nonlinear disturbance observer was proposed

$$\{\begin{array}{c}\stackrel{·}{\hat{D}}=-K_5{\tilde{\omega }}_y\hfill \\ {\stackrel{·}{\hat{\omega }}}_y=\hat{D}+{\hat{C}}_1{\omega }_y+{\hat{C}}_2\beta +{\hat{C}}_3{\delta }_r-K_6{\tilde{\omega }}_y\hfill \end{array}$$

(42)

where ${\tilde{\omega }}_y=\hat{\omega }-{\omega }_y$ , $\tilde{D}=\hat{D}-D$ , $K_5=\mathrm{diag}(k_{50},k_{51},\dots ,k_{5n})$ , and $K_6=\mathrm{diag}(k_{60},k_{61},\dots ,k_{6n})$ are designed parameters whose diagonal elements are positive.

The adaptation law is designed as follows

$$\{\begin{array}{c}{\stackrel{·}{\hat{c}}}_{1i}=k_{1i}(S^T-{\tilde{\omega }}_y^T)L^{-1}{\omega }_y\hfill \\ {\stackrel{·}{\hat{c}}}_{2i}=k_{2i}(S^T-{\tilde{\omega }}_y^T)L^{-1}\beta \hfill \\ {\stackrel{·}{\hat{b}}}_i=-k_{3i}{S}^T{L}^{-1}{Y}_1\hfill \\ {\stackrel{·}{\hat{\rho }}}_i=k_{4i}{S}^T{L}^{-1}\mathrm{sgn}(S)\hfill \\ {\stackrel{·}{\hat{c}}}_{3i}=-{\tilde{\omega }}_y^T{L}^{-1}{\delta }_r\hfill \end{array}$$

(43)

where $K_1=\mathrm{diag}(k_{10},k_{11},\dots ,k_{1n})$ , $K_2=\mathrm{diag}(k_{20},k_{21},\dots ,k_{2n})$ , $K_3=\mathrm{diag}(k_{30},k_{31},\dots ,k_{3n})$ , and $K_4=\mathrm{diag}(k_{40},k_{41},\dots ,k_{4n})$ are designed parameters, and their diagonal elements are all positive.

Theorem 2

For the heading control system of an AUV in equation (1), the sliding mode is chosen as equation (7), control law is equation (41), adaptation laws is equation (43), the anti-windup compensator is equation (40), and disturbance observer is equation (42). The closed-loop system converges to the sliding mode surface $S(t)=0$ in finite time, and the synchronized attitude tracking error converges to the neighborhood of the origin.

Proof

Consider the following Lyapunov candidate function

$$\begin{array}{cc}\multicolumn{1}{c}{V=}& \frac{1}{2}{S}^T{L}^{-1}S+\sum _{i=0}^n\frac{1}{2k_{1i}}{\tilde{c}}_{1i}^2+\sum _{i=0}^n\frac{1}{2k_{2i}}{\tilde{c}}_{2i}^2+\sum _{i=0}^n\frac{1}{2k_{3i}{b}_i}{\tilde{b}}_i^2\hfill \\ \multicolumn{1}{c}{}& +\sum _{i=0}^n\frac{1}{2k_{4i}}{\tilde{\rho }}_i^2+\sum _{i=0}^n\frac{1}{2}{\tilde{c}}_{3i}^2+\frac{1}{2}{w}^T{L}^{-1}w+\frac{1}{2K_5}{\tilde{D}}^T{L}^{-1}\tilde{D}\hfill \\ \multicolumn{1}{c}{}& +\frac{1}{2}{\tilde{\omega }}_y^T{L}^{-1}{\tilde{\omega }}_y\hfill \end{array}$$

(44)

The time derivative of equation (44) is

$$\begin{array}{cc}\multicolumn{1}{c}{\stackrel{·}{V}=}& S^T{L}^{-1}\stackrel{·}{S}+\sum _{i=0}^n\frac{1}{k_{1i}}{\tilde{c}}_{1i}{\stackrel{·}{\hat{c}}}_{1i}+\sum _{i=0}^n\frac{1}{k_{2i}}{\tilde{c}}_{2i}{\stackrel{·}{\hat{c}}}_{2i}+\sum _{i=0}^n\frac{1}{k_{3i}{b}_i}{\tilde{b}}_i{\stackrel{·}{\hat{b}}}_i\hfill \\ \multicolumn{1}{c}{}& +\sum _{i=0}^n\frac{1}{k_{4i}}{\tilde{\rho }}_i{\stackrel{·}{\hat{\rho }}}_i+\sum _{i=0}^n{\tilde{c}}_{3i}{\stackrel{·}{\hat{c}}}_{3i}+w^T{L}^{-1}\stackrel{·}{w}+\frac{1}{K_5}{\tilde{D}}^T{L}^{-1}\stackrel{·}{\hat{D}}\hfill \\ \multicolumn{1}{c}{}& +{\tilde{\omega }}_y^T{L}^{-1}({\stackrel{·}{\hat{\omega }}}_y-{\stackrel{·}{\omega }}_y)\hfill \end{array}$$

(45)

Substituting equations (10) and (43) into equation (45), we can get

$$\begin{array}{c}\stackrel{·}{V}=S^T{L}^{-1}[C_1{\omega }_y+C_2\beta +C_3{\delta }_r+D-{\stackrel{··}{\psi }}_d+\mathrm{\Lambda }L{\omega }_y]\hfill \\ +\sum _{i=0}^n{\tilde{c}}_{1i}(S^T-{\tilde{\omega }}_y^T)L^{-1}{\omega }_y+\sum _{i=0}^n{\tilde{c}}_{2i}(S^T-{\tilde{\omega }}_y^T)L^{-1}\beta \hfill \\ +\sum _{i=0}^n\frac{{\tilde{b}}_i}{b_i}(-S^T{L}^{-1}Y)+\sum _{i=0}^n{\tilde{\rho }}_i{S}^T{L}^{-1}\mathrm{sgn}(S)\hfill \\ +\sum _{i=0}^n{\tilde{c}}_{3i}(-{\tilde{\omega }}_y^T{L}^{-1}{\delta }_r)-w^T{L}^{-1}{K}^{*}w-|S^T{L}^{-1}{\overline{C}}_3\mathrm{\Delta }u|\hfill \\ -\frac{1}{2}(\mathrm{\Delta }{u}^T{L}^{-1}\mathrm{\Delta }u)+w^T{L}^{-1}\mathrm{\Delta }u-{\tilde{D}}^T{L}^{-1}{\tilde{\omega }}_y\hfill \\ +{\tilde{\omega }}_y^T{L}^{-1}[\hat{D}+{\hat{C}}_1{\omega }_y+{\hat{C}}_2\beta +{\hat{C}}_3{\delta }_r-K_6{\tilde{\omega }}_y-C_1{\omega }_y\hfill \\ -C_2\beta -C_3{\delta }_r-D]\hfill \end{array}$$

(46)

Simplifying calculations from equation (46), we can get

$$\begin{array}{c}\stackrel{·}{V}=S^T{L}^{-1}[C_1{\omega }_y+C_2\beta +C_3{\delta }_r+D-{\stackrel{··}{\psi }}_d+\mathrm{\Lambda }L{\omega }_y]\hfill \\ +\sum _{i=0}^n{\tilde{c}}_{1i}{S}^T{L}^{-1}{\omega }_y+\sum _{i=0}^n{\tilde{c}}_{2i}{S}^T{L}^{-1}\beta \hfill \\ -\sum _{i=0}^n\frac{{\tilde{b}}_i}{b_i}{S}^T{L}^{-1}Y+\sum _{i=0}^n{\tilde{\rho }}_i{S}^T{L}^{-1}\mathrm{sgn}(S)\hfill \\ -w^T{L}^{-1}{K}^{*}w-|S^T{L}^{-1}{\overline{C}}_3\mathrm{\Delta }u|-\frac{1}{2}(\mathrm{\Delta }{u}^T{L}^{-1}\mathrm{\Delta }u)\hfill \\ +w^T{L}^{-1}\mathrm{\Delta }u-{\tilde{\omega }}_y^T{L}^{-1}{K}_6{\tilde{\omega }}_y\hfill \end{array}$$

(47)

and according to Assumption 3, we have

$$\begin{array}{c}\stackrel{·}{V}\le S^T{L}^{-1}[C_1{\omega }_y+C_2\beta -T(S-w)-\mathrm{Ksgn}(S)\hfill \\ -\hat{P}\mathrm{sgn}(S)-{\hat{C}}_1{\omega }_y-{\hat{C}}_2\beta -\hat{D}+{\stackrel{··}{\psi }}_d\hfill \\ -\mathrm{\Lambda }L{\omega }_y+D-{\stackrel{··}{\psi }}_d+\mathrm{\Lambda }L{\omega }_y]+\sum _{i=0}^n{\tilde{c}}_{1i}{S}^T{L}^{-1}{\omega }_y\hfill \\ +\sum _{i=0}^n{\tilde{c}}_{2i}{S}^T{L}^{-1}\beta +\sum _{i=0}^n{\tilde{\rho }}_i{S}^T{L}^{-1}\mathrm{sgn}(S)\hfill \\ -w^T{L}^{-1}{K}^{*}w-\frac{1}{2}(\mathrm{\Delta }{u}^T{L}^{-1}\mathrm{\Delta }u)+w^T{L}^{-1}\mathrm{\Delta }u\hfill \\ -{\tilde{\omega }}_y^T{L}^{-1}{K}_6{\tilde{\omega }}_y\hfill \\ =-S^T{L}^{-1}\mathrm{TS}+S^T{L}^{-1}\mathrm{Tw}-S^T{L}^{-1}\mathrm{Ksgn}(S)\hfill \\ -S^T{L}^{-1}\mathrm{Psgn}(S)-S^T{L}^{-1}\tilde{D}-w^T{L}^{-1}{K}^{*}w\hfill \\ -\frac{1}{2}(\mathrm{\Delta }{u}^T{L}^{-1}\mathrm{\Delta }u)+w^T{L}^{-1}\mathrm{\Delta }u-{\tilde{\omega }}_y^T{L}^{-1}{K}_6{\tilde{\omega }}_y\hfill \end{array}$$

(48)

According to Assumption 2, we can get

$$\begin{array}{c}\multicolumn{1}{c}{\stackrel{·}{V}\le -S^T{L}^{-1}\mathrm{TS}+S^T{L}^{-1}\mathrm{Tw}-S^T{L}^{-1}\mathrm{Ksgn}(S)}\\ \multicolumn{1}{c}{-w^T{L}^{-1}{K}^{*}w-\frac{1}{2}(\mathrm{\Delta }{u}^T{L}^{-1}\mathrm{\Delta }u)+w^T{L}^{-1}\mathrm{\Delta }u}\end{array}$$

(49)

Due to the following inequalities

$$S^T{L}^{-1}\mathrm{Tw}\le \frac{1}{2}(S^T{L}^{-1}\mathrm{TS}+w^T{L}^{-1}\mathrm{Tw})$$

(50)

$$w^T{L}^{-1}\mathrm{\Delta }u\le \frac{1}{2}(w^T{L}^{-1}w+\mathrm{\Delta }{u}^T{L}^{-1}\mathrm{\Delta }u)$$

(51)

Substituting equations (50) and (51) into equation (49), we can get

$$\begin{array}{cc}\multicolumn{1}{c}{\stackrel{·}{V}\le }& -S^T{L}^{-1}\mathrm{TS}+\frac{1}{2}(S^T{L}^{-1}\mathrm{TS}+w^T{L}^{-1}\mathrm{Tw})\hfill \\ \multicolumn{1}{c}{}& -S^T{L}^{-1}\mathrm{Ksgn}(S)-w^T{L}^{-1}{K}^{*}w-\frac{1}{2}(\mathrm{\Delta }{u}^T{L}^{-1}\mathrm{\Delta }u)\hfill \\ \multicolumn{1}{c}{}& +\frac{1}{2}(w^T{L}^{-1}w+\mathrm{\Delta }{u}^T{L}^{-1}\mathrm{\Delta }u)\hfill \\ \multicolumn{1}{c}{}& =-\frac{1}{2}{S}^T{L}^{-1}\mathrm{TS}-S^T{L}^{-1}\mathrm{Ksgn}(S)-w^T{L}^{-1}(K^{*}\hfill \\ \multicolumn{1}{c}{}& -\frac{1}{2}T-\frac{1}{2}I)w\hfill \\ \multicolumn{1}{c}{}& \le -\frac{1}{2}{S}^T{L}^{-1}\mathrm{TS}-S^T{L}^{-1}\mathrm{Ksgn}(S)\hfill \end{array}$$

(52)

The design of the parameter needs to satisfy the condition that each element of the diagonal matrix $(K^{*}-(T/2)-(I/2))$ is positive.

So the system state would converge to the sliding mode surface $S(t)=0$ in finite time, and all errors will converge to zero, and the closed-loop system is stable; this completes the proof.

In addition, according to Lemma 1, there exists a time constant $T_2$ that makes the following inequality holds

$$\parallel y(t)\parallel \le k_0{e}^{-{\lambda }_{0}t}(\parallel y(0)\parallel +\frac{e^{{\lambda }_0{T}_2}}{{\lambda }_0}{\beta }_s(T_2))+\frac{k_0}{{\lambda }_0}{\beta }_{\mathrm{sT}}$$

(53)

Simulation results

To illustrate the dynamic performance of the proposed distributed attitudes synchronization control scheme, consider a group of three AUVs with a communication network that induces a graph shown in Figure 4, where the 0-node denotes the virtual AUV, and its attitude is the desired head for each AUV. To illustrate the results proposed simply, we only consider the case without input nonlinearities.

Figure 4.

The extended communication graph.

The corresponding Laplacian matrix L is

$$L=\left[\begin{array}{cccc}1\hfill & 0\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{-1}& 1\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& -1\hfill & 1\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & -1\hfill & 1\hfill \end{array}\right]$$

(54)

We assume that three AUVs are identical, and their model parameters are the same. We also assume that they are subject to the same external disturbance. Select the model parameter values $c_1=-4.5$ , $c_2=10.7$ , $c_3=2.6$ , and $b=0.385$ , and the initial states of the AUV are set as $\psi (0)=[\begin{array}{ccc}1.5\hfill & 0.7\hfill & 0.3\hfill \end{array}]$ . Assume that the desired yaw angle is ${\psi }_d=2\sin 0.5t$ , the external disturbance is chosen as $D=0.55\sin (0.01t)+0.33\sin (0.1t)+0.25\cos (0.02t)+0.3\sin (0.1t)\cos (0.5t)$ . The parameters in the sliding mode controller are selected as $\tau =50,k=1,\rho =1,\lambda =1$ .

The simulation results were shown in Figures 5–10. From Figures 5 and 6 we can find that all the yaw angles of AUVs can track for the desired attitudes coordinately, and the according synchronization error goes to zero; moreover, the sliding mode of each one converges to zero asymptotically from Figure 8. We can also find that the yaw angular velocity of each AUV can reach consensus from Figure 7. Figure 9 depicts the rudder inputs of each AUV; we can find they also realize synchronization, moreover, without saturation. Figure 10 is the external disturbance and its estimation.

Figure 5.

Synchronized attitudes of yaw angles of each AUV.

Figure 6.

The synchronized errors of yaw tracking.

Figure 7.

Synchronized angular velocity of yaw angles of each AUV.

Figure 8.

The sliding mode of each controller.

Figure 9.

The rudder input of each AUV.

Figure 10.

The disturbance.

Conclusion

In this article, we studied the synchronized attitudes control problem of multiple AUVs. We proposed a distributed adaptive control law using the attitudes tracking error based on sliding method. Our strategy is fully distributed because each controller needs only its neighbors’ information; moreover, they can track the desired attitude when this attitude can be reached only by one or one subset AUVs. The uncertainty of the models, and the input saturation and unknown external disturbance are both considered by adaptive laws and commentators. The stability of the control law proposed was analyzed and proved theoretically. The simulation results validate the effectiveness of algorithm proposed. Attitudes synchronization of AUVs in three dimensions will pose great challenges and will warrant our further research in the future.