Finite-time tracking control for autonomous underwater vehicles: An impulsive communication scheme

Abstract

This article investigates the finite-time dynamical tracking for autonomous underwater vehicles by introducing an impulsive communication scheme. More precisely, the communication of the autonomous underwater vehicles is conducted by the prescribed impulsive instants, which is more applicable for the underwater communication. Based on finite-time control theory and impulsive control theory, the tracking controller for each autonomous underwater vehicle is designed and sufficient conditions are developed for achieving the dynamical tracking cooperation. In the end, a simulation example is given to demonstrate the effectiveness and advantage of the proposed approach.

Introduction

With the fast development of computer and intelligence science, the autonomous underwater vehicles have received a lot of attention due to their wide engineering applications, such as sea investigation, environmental monitoring, and localization.^1–5 Recently, cooperation control for autonomous underwater vehicles is designed to achieve complex underwater missions. Compared with a single autonomous underwater vehicle, cooperation control can obtain advantages in increasing the mission robustness and decreasing the cost. As a result, there have been many research works in tracking problems of autonomous underwater vehicles,^6–8 which means that all the autonomous underwater vehicles can finally converge for tracking.

Note that for the control issues of multiple autonomous underwater vehicles, most existing literatures focus on the control design, which are mainly based on continuous-time assumptions. However, it should be pointed out that the continuous-time underwater communication model is not practical for the multiple autonomous underwater vehicles due to underwater communication limitations and would lead to design conservatism in the real-world applications.^9–13 It is well known that an effective underwater communication scheme is the impulsive communication by underwater acoustic devices.^14,15 Unfortunately, the impulsive communication method would increase the difficulty of the analysis and synthesis of control designs.

However, finite-time stability is becoming a hot research area due to its practical background. Different from traditional asymptotical stability with infinite time, finite-time stability can achieve the stability within a given time interval such that it is more applicable for real-world control problems.^16–19 As a result, many remarkable results addressed the finite-time control problems can be found in the literature and references therein.^20–23 Although there are some encouraging results for finite-time converging problems, the finite-time tracking cooperation problem of autonomous underwater vehicles still remains open and challenging.

Till now, there is little concern on the finite-time dynamical tracking cooperation of autonomous underwater vehicles with impulsive communication scheme, which motivates us for this article. In this study, we address the dynamical tracking cooperation for a group of leader and follower autonomous underwater vehicles under directed information exchange topology, which means that all the followers can track the leader in a prescribed finite time. Compared with the existing literature, the main contributions of our article can be summarized as twofold: (1) a novel finite-time dynamical tracking cooperation strategy for autonomous underwater vehicles with impulsive communication is introduced. (2) By developing an impulsive information exchange framework, tracking controllers are associated with impulse time sequences. Based on finite-time stability theory and impulsive system theory, tracking criteria are derived for ensuring that dynamical tracking cooperation can be accomplished over the finite-time interval. With the designed controllers, not only the communication scheme can be more practical but also the communication resources can be saved in comparison with most continuous-time communication approaches in the open papers.

The rest of our article is organized as follows. We first introduce the model of multiple autonomous underwater vehicles and formulate the tracking problem. Then, the main theoretical analysis is presented. Next, we provide numerical simulation results to show the effectiveness and the benefit of our design scheme. Finally, concluding remarks are drawn and future prospects of research are given.

Notation: ${\mathbb{R}}^n$ represents n-dimensional Euclidean space. ${\mathbb{R}}^{m\times n}$ stands for the set of $m\times n$ real matrices. $\otimes \mathrm{denotes}$ the Kronecker product. $P>0$ means P is real symmetric and positive definite. For simplicity, denote $s_{\alpha }=\sin \alpha$ and $c_{\alpha }=\cos \alpha$ .

Preliminaries and problem formulation

Model of multiple autonomous underwater vehicles

Consider a group of $N+1$ autonomous underwater vehicles labeled by N followers and $1$ leader, which are described by the following dynamics and depicted in Figure 1 ²⁴

$$\{\begin{array}{c}{\stackrel{·}{p}}_i=J_i\left({\mathrm{\Theta }}_i\right)v_i\hfill \\ \multicolumn{1}{c}{M_i{\stackrel{·}{v}}_i=-D_i\left(v_i\right)v_i-g_i\left({\mathrm{\Theta }}_i\right)+{\tau }_i}\end{array}$$

(1)

where $p_i=[x_i,y_i,z_i{]}^T$ denotes the generalized position, $v=[u_i,{\upsilon }_i,{\omega }_i{]}^T$ denotes the generalized linear velocities, ${\mathrm{\Theta }}_i=[{\varphi }_i,{\theta }_i,{\psi }_i{]}^T$ denotes the generalized attitude, $\{B\}$ is body-fixed reference frame, $\{E\}$ is global coordinate frame, $J_i({\mathrm{\Theta }}_i)$ is the kinematic transformation matrix from $\{B\}$ to $\{E\}$ , ${\tau }_i=[{\tau }_{i1},{\tau }_{i2},{\tau }_{i3}{]}^T$ denotes the control forces, $M_i$ is inertia matrix, $D_i(v_i)$ is damping matrix, and $g_i({\mathrm{\Theta }}_i)$ is restoring force, which are given as

$$M_i=\mathrm{diag}\{m_{i1},m_{i2},m_{i3}\}$$

(2)

$$D_i\left(v_i\right)=\mathrm{diag}\{d_{L_{i1}}+d_{Q_{i1}}\left|u_i\right|,d_{L_{i2}}+d_{Q_{i2}}\left|{\upsilon }_i\right|,d_{L_{i3}}+d_{Q_{i3}}\left|{\omega }_i\right|\}$$

(3)

$$g_i\left({\mathrm{\Theta }}_i\right)={\left[\left(W_i-B_i\right)s_{{\theta }_i},-\left(W_i-B_i\right)c_{{\theta }_i}{s}_{{\varphi }_i},-\left(W_i-B_i\right)c_{{\theta }_i}{c}_{{\varphi }_i}\right]}^T$$

(4)

where $m_{\mathrm{ij}}$ denotes the mass inertia, $d_{L_{\mathrm{ij}}}$ and $d_{Q_{\mathrm{ij}}}$ denote the classical hydrodynamic derivatives, $W_i$ denotes the submerged weight, $B_i$ denotes the buoyancy force, $m_{\mathrm{ij}},$ $d_{L_{\mathrm{ij}}},$ $d_{Q_{\mathrm{ij}}}>0,$ $j=1,2,3$ . $J_i({\mathrm{\Theta }}_i)$ is given as

$$J_i\left({\mathrm{\Theta }}_i\right)=\left[\begin{array}{cc}{J}_i{\left({\mathrm{\Theta }}_i\right)}_1\hfill & J_i{\left({\mathrm{\Theta }}_i\right)}_2\hfill \\ \multicolumn{1}{c}{-s_{{\theta }_i}}& J_i{\left({\mathrm{\Theta }}_i\right)}_3\hfill \end{array}\right]$$

(5)

$$J_i{\left({\mathrm{\Theta }}_i\right)}_1=\left[\begin{array}{cc}{c}_{{\phi }_i}{c}_{{\theta }_i}\hfill & -s_{{\phi }_i}{c}_{{\varphi }_i}+c_{{\phi }_i}{s}_{{\theta }_i}{s}_{{\varphi }_i}\hfill \\ \multicolumn{1}{c}{s_{{\phi }_i}{c}_{{\theta }_i}}& c_{{\phi }_i}{c}_{{\varphi }_i}+s_{{\varphi }_i}{s}_{{\theta }_i}{s}_{{\phi }_i}\hfill \end{array}\right]$$

(6)

$$J_i{\left({\mathrm{\Theta }}_i\right)}_2=\left[\begin{array}{c}{s}_{{\phi }_i}{s}_{{\varphi }_i}+c_{{\phi }_i}{c}_{{\varphi }_i}{s}_{{\theta }_i}\hfill \\ \multicolumn{1}{c}{-c_{{\phi }_i}{s}_{{\varphi }_i}+s_{{\theta }_i}{s}_{{\phi }_i}{c}_{{\varphi }_i}}\end{array}\right]$$

(7)

$$J_i{\left({\mathrm{\Theta }}_i\right)}_3=\left[\begin{array}{cc}{c}_{{\theta }_i}{s}_{{\varphi }_i}\hfill & c_{{\theta }_i}{c}_{{\varphi }_i}\hfill \end{array}\right]$$

(8)

Figure 1.

The reference frames of autonomous underwater vehicle.

In order to achieve dynamical tracking, a directed graph $\mathcal{G}=\{\mathcal{V},\mathcal{E},\mathcal{A}\}$ is used for denoting underwater information exchanges among the followers, which implies that the information exchanges have directions. More precisely, $\mathcal{V}(\mathcal{G})=\{v_1,\dots ,v_N\}$ is the set of nodes, $\mathcal{E}$ is the set of edges, and $\mathcal{A}=[a_{\mathrm{ij}}]\in {\mathbb{R}}^{N\times N}$ is the weighted adjacency matrix. $a_{\mathrm{ij}}$ is defined by the rule that $a_{\mathrm{ij}}>0$ when $(v_i,v_j)\in \mathcal{E}$ , which means that there are information exchanges between the followers i and j. Otherwise, $a_{\mathrm{ij}}=0$ . The Laplacian matrix $L=(l_{\mathrm{ij}}{)}_{N\times N}$ of $\mathcal{G}$ is defined as $l_{\mathrm{ij}}=-a_{\mathrm{ij}}$ , $i\ne j$ and $l_{\mathrm{ii}}=-\sum _{j=1,i\ne j}^N{a}_{\mathrm{ij}}$ . For a directed graph, a directed path from node i to node j is a sequence of edges of the form $(v_{i1},v_{i2}),(v_{i2},v_{i3}),\dots$ . In addition, $B=\mathrm{diag}\{b_1,b_2,\dots ,b_N\}$ denotes the leader adjacency matrix with $b_i>0(b_i=0)$ , which means that the leader has a directed path to the follower i (otherwise).

Before proceeding, the following lemmas are introduced.

Lemma 1

If $\mathcal{G}$ has a directed spanning tree, $L+B$ has eigenvalues with positive real parts. ²⁵

Lemma 2

Consider the following system ²⁶

$$\stackrel{·}{x}(t)=\mathrm{Ax}(t)$$

(9)

$$x\left(t_0\right)=x_0$$

(10)

Given two positive real numbers $c1$ , $c2$ and a non-negative definite matrix Q, system (9) is said to be finite-time stable with respect to $(c1,c2,T,Q)$ if and only if

$$\Vert x\left(t_0\right){\Vert }_Q^2<c1\Rightarrow \Vert x(t){\Vert }_Q^2<c2,\forall t\in [0,T]$$

(11)

Control objective

The control aim is to develop the tracking controllers for multiple autonomous underwater vehicles (equation (1)) so that in a finite-time interval $[0,T]$ , that is

$$\underset{t\to T}{lim}\Vert p_i-p_0\Vert =0$$

(12)

$$\underset{t\to T}{lim}\Vert v_i-v_0\Vert =0,i=1,2,\dots ,N$$

(13)

Main results

In this section, the impulsive communication scheme is first established. Then, the tracking controllers are designed based on the obtained results.

Define an impulsive sequence in $[0,\infty )$ , that is, $\{t_k\}$ with $0=t_0<t_1<\cdots <t_k<\cdots ,\underset{k\to \infty }{lim}{t}_k=\infty$ . Then, for $t=t_k$ , let Dirac function $\delta (t)$ denote the impulsive effects, such that $\Delta t_k=t_k-t_{k-1},k=1,2,\dots$ . It is assumed that the information exchanges among the autonomous underwater vehicles are conducted with the impulsive sequence.

By defining ${\sigma }_i:=J_i({\mathrm{\Theta }}_i)v_i$ , one has

$$\{\begin{array}{c}{\stackrel{·}{p}}_i={\sigma }_i\hfill \\ \multicolumn{1}{c}{{\stackrel{·}{\sigma }}_i=-J_i^{-1}\left({\mathrm{\Theta }}_i\right)M_i^{-1}{D}_i\left(J_i^{-1}\left({\mathrm{\Theta }}_i\right){\upsilon }_i\right)J_i^{-1}\left({\mathrm{\Theta }}_i\right){\upsilon }_i}\\ \multicolumn{1}{c}{-J_i^{-1}\left({\mathrm{\Theta }}_i\right)M_i^{-1}{g}_i\left({\mathrm{\Theta }}_i\right)+J_i^{-1}\left({\mathrm{\Theta }}_i\right)M_i^{-1}{\tau }_i}\end{array}$$

(14)

Consequently, the tracking controller for each autonomous underwater vehicle can be designed as

$${\tau }_0=D_i\left(J_i^{-1}\left({\mathrm{\Theta }}_i\right){\upsilon }_i\right)J_i^{-1}\left({\mathrm{\Theta }}_i\right){\upsilon }_i+g_i\left({\mathrm{\Theta }}_i\right)$$

(15)

$$\begin{array}{c}{\tau }_i=D_i\left(J_i^{-1}\left({\mathrm{\Theta }}_i\right){\upsilon }_i\right)J_i^{-1}\left({\mathrm{\Theta }}_i\right){\upsilon }_i+g_i\left({\mathrm{\Theta }}_i\right)\hfill \\ -\sum _{k=1}^{\infty }\left\{\rho \sum _{j=1,j\ne i}^N{a}_{\mathrm{ij}}[(p_i-p_j)+({\upsilon }_i-{\upsilon }_j)]\right\}\delta (t-t_k)\hfill \\ -\sum _{k=1}^{\infty }\left\{\rho b_i[(p_i-p_0)+({\upsilon }_i-{\upsilon }_0)]\right\}\delta (t-t_k),\hfill \\ i=1,2,\dots ,N\hfill \end{array}$$

(16)

where $\rho >0$ is a constant.

Define $e_{\mathrm{pi}}:=p_i-p_0$ and $e_{\mathrm{vi}}:={\sigma }_i-{\sigma }_0$ , $i=1,2,\dots ,N$ . Then, the closed-loop dynamics of the multiple autonomous underwater vehicles can be derived as follows

$$\{\begin{array}{c}{\stackrel{·}{e}}_p=e_v\hfill \\ \multicolumn{1}{c}{{\stackrel{·}{e}}_v=0,t\ne t_k}\\ \multicolumn{1}{c}{\Delta e_v=-[\rho (L+B)\otimes I_3]e_p-[\rho (L+B)\otimes I_3]e_v,t=t_k}\end{array}$$

(17)

where

$$e_p:={[e_{p1}^T,e_{p2}^T,\dots ,e_{\mathrm{pN}}^T]}^T$$

(18)

$$e_v:={[e_{v1}^T,e_{v2}^T,\dots ,e_{\mathrm{vN}}^T]}^T$$

(19)

Without loss of generality, assume that $e_p(t_0^{+})=e_p(t_0)$ and $e_v(t_0^{+})=e_v(t_0)$ .

Based on the designed control input, the following theorem is established.

Theorem 1

The dynamical tracking of the multiple autonomous underwater vehicles with $1$ leader and N followers can be achieved in finite-time T, $0=t_0<t_1<\cdots <t_l=T<t_{l+1}<\cdots$ if $\mathcal{G}$ has a directed spanning tree and there exist matrices $P>0$ , $Q>0$ , and a constant $\theta >0$ , such that the following inequalities hold

$$\mathrm{\Pi }:=\left[\begin{array}{cc}-\theta Q^{1/2}P{Q}^{1/2}\hfill & Q^{1/2}P{Q}^{1/2}/2\hfill \\ \multicolumn{1}{c}{*}& -\theta Q^{1/2}P{Q}^{1/2}\hfill \end{array}\right]<0$$

(20)

$$c_1<\frac{{\lambda }_{min}(P)}{{\lambda }_{max}(P)\exp \left(\theta t_1\right)}{c}_2$$

(21)

$$\mu c_1<c_2$$

(22)

where

$$\begin{array}{l}\mu :=\frac{{\lambda }_{\max }(P)exp\left(\theta t_1\right)}{{\lambda }_{\min }(P)}\\ \times \max {\left\{\frac{\sigma {\lambda }_{\max }\left(P_1\right)exp\left(\theta \left(t_{k+1}-t_k\right)\right)}{{\lambda }_{\min }(P)}\right\}}^{k-1}\end{array}$$

(23)

$$\begin{array}{c}\sigma :={\lambda }_{max}\left({(I_2\otimes Q\otimes I_3)}^{-1}\right)\hfill \\ {(I_{6N}-\rho \left[\begin{array}{cc}0\hfill & 0\hfill \\ \multicolumn{1}{c}{(L+B)\otimes I_3}& (L+B)\otimes I_3\hfill \end{array}\right])}^T\hfill \\ \times (I_2\otimes Q\otimes I_3)\hfill \\ (I_{6N}-\rho \left[\begin{array}{cc}0\hfill & 0\hfill \\ \multicolumn{1}{c}{(L+B)\otimes I_3}& (L+B)\otimes I_3\hfill \end{array}\right]),\hfill \\ k=2,3,\dots ,l\hfill \end{array}$$

(24)

Proof

By Lemma 1, since $\mathcal{G}$ has a directed spanning tree, the eigenvalues of $(L+B)$ are positive. Then, it can be verified that the dynamical tracking control can be achieved in finite time if system (17) is the finite-time stable.

Construct the following Lyapunov function

$$V(t)=\frac{1}{2}\left[\begin{array}{cc}{e}_p^T\hfill & e_v^T\hfill \end{array}\right](\mathcal{P}\otimes I_3)\left[\begin{array}{c}{e}_p\hfill \\ \multicolumn{1}{c}{e_c}\end{array}\right]$$

(25)

where

$$\mathcal{P}=\mathrm{diag}\{Q^{1/2}P{Q}^{1/2},Q^{1/2}P{Q}^{1/2}\}\in {\mathbb{R}}^{2M\times 2M}$$

(26)

$$P>0,Q>0$$

(27)

The Dini derivative of $V(t)$ is defined as $D^{+}V(t):=\underset{h\to 0^{+}}{lim}sup[V(t+h)-V(t)]/h$ .

For $t\in [t_0,t_1)$ , one obtains

$$\begin{array}{c}\multicolumn{1}{c}{D^{+}V(t)=\frac{1}{2}\left[\begin{array}{cc}{\stackrel{·}{e}}_p^T\hfill & {\stackrel{·}{e}}_v^T\hfill \end{array}\right](\mathcal{P}\otimes I_3)\left[\begin{array}{c}{e}_p\hfill \\ \multicolumn{1}{c}{e_v}\end{array}\right]}\\ +\frac{1}{2}\left[\begin{array}{cc}{e}_p^T\hfill & e_v^T\hfill \end{array}\right](\mathcal{P}\otimes I_3)\left[\begin{array}{c}{\stackrel{·}{e}}_p\hfill \\ \multicolumn{1}{c}{{\stackrel{·}{e}}_v}\end{array}\right]\hfill \\ \multicolumn{1}{c}{=\left[\begin{array}{cc}{e}_v^T\hfill & 0\hfill \end{array}\right](\mathcal{P}\otimes I_3)\left[\begin{array}{c}{e}_p\hfill \\ \multicolumn{1}{c}{e_v}\end{array}\right]}\end{array}$$

(28)

Then, one obtains $D^{+}V(t)<\theta V(t)$ if equation (20) holds, which implies that

$$V(t)<\exp (\theta t)V\left(t_0\right),[t_0,t_1)$$

(29)

Consequently, we have

$$\begin{array}{c}{\lambda }_{min}(P)\left[\begin{array}{cc}{e}_p^T\hfill & e_v^T\hfill \end{array}\right](I_2\otimes Q\otimes I_3)\left[\begin{array}{c}{e}_p\hfill \\ \multicolumn{1}{c}{e_v}\end{array}\right]\le V(t)<\exp (\theta t)V\left(t_0\right)\hfill \\ \le {\lambda }_{max}\left(P_1\right)\exp (\theta t)\left[\begin{array}{cc}{e}_p^T\left(t_0\right)\hfill & e_v^T\left(t_0\right)\hfill \end{array}\right]\hfill \\ \times (I_2\otimes Q\otimes I_3)\left[\begin{array}{c}{e}_p\left(t_0\right)\hfill \\ \multicolumn{1}{c}{e_v\left(t_0\right)}\end{array}\right]\hfill \end{array}$$

(30)

Then, there exists a constant $c_1>0$ , such that

$$\begin{array}{l}{\lambda }_{\max }(P)exp(\theta t)\left[\begin{array}{cc}{e}_p^T\left(t_0\right)& e_v^T\left(t_0\right)\end{array}\right]\left(I_2\otimes Q\otimes I_3\right)\\ \times \left[\begin{array}{c}{e}_p\left(t_0\right)\\ e_v\left(t_0\right)\end{array}\right]<c_1{\lambda }_{\max }(P)exp(\theta t)\end{array}$$

(31)

which yields

$$\left[\begin{array}{cc}{e}_p^T& e_v^T\end{array}\right]\left(I_2\otimes Q\otimes I_3\right)\left[\begin{array}{c}{e}_p\\ e_v\end{array}\right]<\frac{c_1{\lambda }_{\max }(P)exp(\theta t)}{{\lambda }_{\min }(P)}\mathrm{,}\forall t\in \left[t_0\mathrm{,}{t}_1\right)$$

(32)

Next, for $t=t_1^{-}$ , since $t_1=t_1^{-}$ , we have

$$\left[\begin{array}{cc}{e}_p^T\left(t_1^{-}\right)& e_v^T\left(t_1^{-}\right)\end{array}\right]\left(I_2\otimes Q\otimes I_3\right)\left[\begin{array}{c}{e}_p\left(t_1^{-}\right)\\ e_v\left(t_1^{-}\right)\end{array}\right]\le \frac{c_1{\lambda }_{\max }(P)exp\left(\theta t_1\right)}{{\lambda }_{\min }(P)}$$

(33)

Then, there exists a constant $c_2>0$ , such that

$$\left[\begin{array}{cc}{e}_p^T\hfill & e_v^T\hfill \end{array}\right](I_2\otimes Q\otimes I_3)\left[\begin{array}{c}{e}_p\hfill \\ \multicolumn{1}{c}{e_v}\end{array}\right]<c_2,\forall t\in [t_0,t_1]$$

(34)

if equation (21) holds.

For $t\in (t_1,t_2)$ , one has

$$V(t)<\exp \left(\theta (t-t_1)\right)V\left(t_1^{+}\right)$$

(35)

which implies that

$${\lambda }_{\min }(P)\left[\begin{array}{cc}{e}_p^T& e_v^T\end{array}\right]\left(I_2\otimes Q\otimes I_3\right)\left[\begin{array}{c}{e}_p\\ e_v\end{array}\right]\le V(t)<exp\left(\theta \left(t-t_1\right)\right)V\left(t_1^{+}\right)$$

(36)

For $t=t_1^{+}$ , we have

$$\begin{array}{c}V\left(t_1^{+}\right)=\frac{1}{2}\left[\begin{array}{cc}{e}_p^T\left(t_1^{+}\right)\hfill & e_v^T\left(t_1^{+}\right)\hfill \end{array}\right](\mathcal{P}\otimes I_3)\left[\begin{array}{c}{e}_p\left(t_1^{+}\right)\hfill \\ \multicolumn{1}{c}{e_v\left(t_1^{+}\right)}\end{array}\right]\hfill \\ =\frac{1}{2}\left[\begin{array}{cc}{e}_p^T\left(t_1\right)\hfill & e_v^T\left(t_1\right)\hfill \end{array}\right][I_{6N}-\rho (L+B)\hfill \\ {\otimes I_3\left[\begin{array}{cc}{I}_N\hfill & I_N\hfill \end{array}\right]]}^T\hfill \\ (\mathcal{P}\otimes I_3)[I_{6N}-\rho (L+B)\otimes I_3\left[\begin{array}{cc}{I}_N\hfill & I_N\hfill \end{array}\right]]\left[\begin{array}{c}{e}_p\left(t_1\right)\hfill \\ \multicolumn{1}{c}{e_v\left(t_1\right)}\end{array}\right]\hfill \\ \le \sigma {\lambda }_{max}(P)\left[\begin{array}{cc}{e}_p^T\left(t_1\right)\hfill & e_v^T\left(t_1\right)\hfill \end{array}\right](I_2\otimes Q\otimes I_3)\left[\begin{array}{c}{e}_p\left(t_1\right)\hfill \\ \multicolumn{1}{c}{e_v\left(t_1\right)}\end{array}\right]\hfill \end{array}$$

(37)

where $\sigma$ is defined in equation (24).

For $t\in (t_1,t_2)$ , it follows that

$$\begin{array}{c}\left[\begin{array}{cc}{e}_p^T\hfill & e_v^T\hfill \end{array}\right](I_2\otimes Q\otimes I_3)\left[\begin{array}{c}{e}_p\hfill \\ \multicolumn{1}{c}{e_v}\end{array}\right]\hfill \\ <\frac{\sigma {\lambda }_{max}(P)\exp \left(\theta (t-t_1)\right)}{{\lambda }_{min}(P)}\left[\begin{array}{cc}{e}_p^T\left(t_1\right)\hfill & e_v^T\left(t_1\right)\hfill \end{array}\right]\hfill \\ \times (I_2\otimes Q\otimes I_3)\left[\begin{array}{c}{e}_p\left(t_1\right)\hfill \\ \multicolumn{1}{c}{e_v\left(t_1\right)}\end{array}\right]\hfill \end{array}$$

(38)

Repeating the same process, we have the following inequality

$$\begin{array}{c}\left[\begin{array}{cc}{e}_p^T\hfill & e_v^T\hfill \end{array}\right](I_2\otimes Q\otimes I_3)\left[\begin{array}{c}{e}_p\hfill \\ \multicolumn{1}{c}{e_v}\end{array}\right]\hfill \\ <\frac{\sigma {\lambda }_{max}(P)\exp \left(\theta (t-t_k)\right)}{{\lambda }_{min}(P)}\left[\begin{array}{cc}{e}_p^T\left(t_k\right)\hfill & e_v^T\left(t_k\right)\hfill \end{array}\right]\hfill \\ \times (I_2\otimes Q\otimes I_3)\left[\begin{array}{c}{e}_p\left(t_k\right)\hfill \\ \multicolumn{1}{c}{e_v\left(t_k\right)}\end{array}\right],\forall t\in (t_k,t_{k+1}],k<l\hfill \end{array}$$

(39)

Furthermore, one has

$$\begin{array}{c}\left[\begin{array}{cc}{e}_p^T\hfill & e_v^T\hfill \end{array}\right](I_2\otimes Q\otimes I_3)\left[\begin{array}{c}{e}_p\hfill \\ \multicolumn{1}{c}{e_v}\end{array}\right]\hfill \\ \le \frac{c_1{\lambda }_{max}(P)\exp \left(\theta t_1\right)}{{\lambda }_{min}(P)}\hfill \\ \times max{\left\{\frac{\sigma {\lambda }_{max}(P)\exp \left(\theta (t_{k+1}-t_k)\right)}{{\lambda }_{min}(P)}\right\}}^k,\forall t\in (t_k,t_{k+1}]\hfill \end{array}$$

(40)

such that if equations (21) and (22) hold, it can be obtained that

$$\left[\begin{array}{cc}{e}_p^T\hfill & e_v^T\hfill \end{array}\right](I_2\otimes Q\otimes I_3)\left[\begin{array}{c}{e}_p\hfill \\ \multicolumn{1}{c}{e_v}\end{array}\right]<c_2,\forall t\in (0,T]$$

(41)

which means that system (17) is finite-time stable with respect to $(c_1,c_2,T,(I_2\otimes Q\otimes I_3))$ and completes the proof.

Remark 1

The proposed finite-time impulsive tracking control method is more practical than the continuous-time cases since the underwater communication constrains are considered.

Remark 2

The developed control method can also be applied to switching communication topologies. Define a finite set of graphs $\{{\mathcal{G}}_{\omega }\}$ with $\omega \in \Omega :=\{1,\dots ,\omega \}$ , where $\{L_{\omega }\}$ denotes the corresponding Laplacian matrices. For $t^{+}$ , the switching is piecewise constant from the right and non-chattering. With infinite switching times, the proposed method can also be applied.^27,28

Numerical example

In the following section, we provide the simulation results to demonstrate the above results.

Example 1

Consider a group of five autonomous underwater vehicles with the same structure and parameters as $M_i=\mathrm{diag}\{120,100,100\},$ $D_i(v_i)=\mathrm{diag}\{100+80|u_i|,+60|{\upsilon }_i|,80+60|{\omega }_i|\}$ , and ${\varphi }_i=-\pi /8,{\theta }_i=\pi /6,{\psi }_i=\pi /4$ . The communication topology is shown in Figure 2 with

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

Figure 2.

The communication topology.

The initial conditions are chosen as follows

$$\begin{array}{c}{p}_0(0)=[0,0,0{]}^T,v_0(0)=[1,1,1{]}^T\hfill \\ p_1(0)=[-20,-20,20{]}^T,p_2(0)=[-20,30,25{]}^T\hfill \\ p_3(0)=[20,-15,30{]}^T,p_4(0)=[15,10,-15{]}^T\hfill \\ v_1(0)=v_2(0)=v_3(0)=v_4(0)=[0,0,0{]}^T\hfill \end{array}$$

The parameters are set as $P=Q=I_5$ and $\Delta t_k=0.5\mathrm{s}$ with $T=10\mathrm{s}$ , which implies that the tracking can be achieved at $t_{20}$ with $k=20$ .

It can be obtained by Theorem 1 that

$$[e_p^T\left(t_0\right),e_v^T\left(t_0\right)](I_2\otimes Q\otimes I_3)\left[\begin{array}{c}{e}_p\left(t_0\right)\hfill \\ \multicolumn{1}{c}{e_v\left(t_0\right)}\end{array}\right]=5.212\mathrm{e}3$$

Then, let $\rho =5$ , $\theta =0.02$ , and $c_2=3\mathrm{e}50$ , such that $\mu =5.8530\mathrm{e}44$ and $\sigma =224.7016$ , which means that the dynamical tracking can be achieved in finite-time by $(c_1=5.212\mathrm{e}3,c_2=3\mathrm{e}50,T=10,Q=I_5)$ . As a result, the positions and the velocities of the tracking procedure can be seen in Figures 3 and 4, respectively.

Figure 3.

The generalized positions.

Figure 4.

The generalized linear velocities.

Example 2

Another example is given with different structure and parameters as $M_i=\mathrm{diag}\{140,120,120\},$ $D_i(v_i)=\mathrm{diag}\{120+100|u_i|,+80|{\upsilon }_i|,100+80|{\omega }_i|\}$ , and ${\varphi }_i=-\pi /6,{\theta }_i=\pi /6,{\psi }_i=\pi /6$ . The initial conditions are chosen as follows

$$\begin{array}{c}{p}_0(0)=[5,5,5{]}^T,v_0(0)=[0,0,0{]}^T\hfill \\ p_1(0)=[-20,-20,20{]}^T,p_2(0)=[-20,30,25{]}^T\hfill \\ p_3(0)=[20,-15,30{]}^T,p_4(0)=[15,10,-15{]}^T\hfill \\ v_1(0)=v_2(0)=v_3(0)=v_4(0)=[0,0,0{]}^T\hfill \end{array}$$

The parameters are set as $P=Q=I_5$ and $\Delta t_k=0.5\mathrm{s}$ with $T=10\mathrm{s}$ . Following similar design procedure, it can be obtained by Theorem 1 that the dynamical tracking can be achieved in finite time by $(c_1=4900,c_2=2.8\mathrm{e}50,T=10,Q=I_5)$ . The positions and the velocities of the tracking procedure can be seen in Figures 5 and 6, respectively.

Figure 5.

The generalized positions.

Figure 6.

The generalized linear velocities.

Example 3

Moreover, in order to show the advantages of our proposed impulsive communication scheme, the comparison between impulsive communication and continuous-time communication strategies is given. By choosing the same parameters in Example 1, the positions and the velocities of the tracking procedure with continuous-time communication strategy can be seen from Figures 7 and 8, respectively. It can be found that the proposed impulsive communication does not affect the tracking trajectory much compared with the continuous-time communication but is more practical and applicable for the underwater communication applications.

Figure 7.

The generalized positions.

Figure 8.

The generalized linear velocities.

Conclusion and discussions

We study the finite-time dynamical tracking of multiple autonomous underwater vehicles by developing an impulsive information exchange strategy. Based on the results of model conversion, the finite-time tracking conditions are derived. Furthermore, by constructing the Lyapunov function, we establish the sufficient conditions and design the distributed controllers for the leader and follower autonomous underwater vehicles. We present an illustrative example to show the efficiency of our findings. The developed results can be extended to the case with transmission delays in the information exchanges in the future.