Adaptive Event-Triggered Control Strategy for Ensuring Predefined Three-Dimensional Tracking Performance of Uncertain Nonlinear Underactuated Underwater Vehicles

Abstract

This paper presents an adaptive event-triggered control strategy for guaranteeing predefined tracking performance of uncertain nonlinear underactuated underwater vehicles (UUVs) in the three-dimensional space. Compared with the related results in the literature, the main contribution of this paper is to develop a nonlinear error transformation approach for ensuring predefined three-dimensional tracking performance under the underactuated property of 6-DOF UUVs and limited network resources. A nonlinear tracking error function is designed using a linear velocity rotation matrix and a time-varying performance function. An adaptive event-triggered control scheme using the nonlinear tracking error function and neural networks is constructed to ensure the practical stability of the closed-loop system with predefined three-dimensional tracking performance. In the proposed control scheme, auxiliary stabilizing signals are designed to resolve the underactuated problem of UUVs. Simulation results are presented to illustrate the effectiveness of the theoretical methodology.

1. Introduction

Control of nonlinear underwater vehicles is attracting much research attention recently, due to its practical application in submarine survey, exploration, oceanographic mapping, and region search for deep-sea wrecks [1,2,3,4,5,6]. The initial studies have focused on the planar or depth control design of underwater vehicles in the two-dimensional space [7,8,9,10,11]. However, these studies in the two-dimensional space provide limited solutions to various tracking problems in the practical three-dimensional underwater environment. For more practical application, three-dimensional control approaches have been studied for nonlinear underwater vehicles described by 5-degrees-of-freedom (5-DOF) or 6-DOF kinematics and dynamics. In [12], a path following controller was designed for 5-DOF underwater vehicles with ocean current disturbances. A ocean current observer to detect an external current was designed for trajectory tracking of 5-DOF underactuated underwater vehicles (UUVs) [13]. In [14,15], fuzzy-based or neural-network control techniques were developed for uncertain 5-DOF UUVs with external disturbances. To deal with uncertain 6-DOF models with the roll motion, three-dimensional trajectory tracking control designs were developed using several control techniques such as backstepping control [16,17] and sliding mode control [18]. However, in the aforementioned results, the transient and steady-state performance metrics of tracking errors cannot be designed a priori. To preselect the tracking performance metrics of underwater vehicles, the prescribed performance design technique [19] has been combined with the control methodologies of underwater vehicles [20,21,22,23]. In [21], a region tracking controller with predefined transient performance was designed for fully actuated underwater vehicles in presence of an ocean current and a thruster fault. Neural-network-based prescribed performance control designs were investigated for uncertain 5-DOF UUVs [22] and 6-DOF fully actuated underwater vehicles [20]. In [23], a prescribed performance control design was developed for 6-DOF UUVs. Despite these efforts, the controllers designed in [20,21,22,23] should be continuously updated and thus cannot be used in the network-based control environment with limited communication bandwidth. Since underwater acoustic communication has limited bandwidth, low propagation speed, and high energy consumption, the data transmission for the network-based control should be kept to a minimum amount [24]. Thus, it is significant to investigate an event-triggered control design issue for ensuring predefined three-dimensional tracking performance under the underactuated property of 6-DOF UUVs and limited network resources.

Event-triggered control strategies have been proposed to address control problems of linear and nonlinear systems under capacity-limited networks [25,26,27,28]. In the event-triggered control, the signal transmission burden can be reduced in the communication network because control inputs are executed only when certain triggering conditions are satisfied. Owing to this advantage, adaptive event-triggered control approaches have been actively developed for uncertain nonlinear systems [29,30,31,32]. However, the event-triggered control of underwater vehicles was only studied to design a depth controller in the two-dimensional space [33]. To the best of our knowledge, no studies have been reported thus far on the event-triggered control problem for ensuring predefined three-dimensional tracking performance of uncertain 6-DOF UUVs.

On the basis of the above discussion, the purpose of this paper is to present an adaptive event-triggered control strategy with predefined three-dimensional tracking performance for uncertain nonlinear 6-DOF UUVs. It is assumed that all nonlinearities in the dynamics of the UUV are unknown. Compared with the related results [20,21,22,23,33] in the literature, the main contribution of this study is to develop an error-transformation-based adaptive event-triggered tracking law for achieving predefined three-dimensional tracking performance while overcoming the underactuated problem of the nonlinear 6-DOF dynamics. To this end, a nonlinearly transformed tracking error function using a linear velocity rotation matrix and a time-varying performance function is presented. A neural-network-based adaptive event-triggered control scheme is recursively designed to ensure predefined three-dimensional tracking performance of the uncertain UUV where neural networks are employed to approximate unknown nonlinearities. In the proposed control scheme, auxiliary stabilizing signals using neural networks are derived to deal with the underactuated control design problem. We rigorously prove that the resulting event-triggered tracker ensures the practical stability of the closed-loop system and the exclusion of Zeno behavior.

The rest of this paper is outlined as follows. In Section 2, we introduce the 6-DOF kinematics and dynamics of uncertain nonlinear UUVs and formulate the predefined three-dimensional tracking performance control problem. A neural-network-based adaptive event-triggered tracker is constructed using the nonlinearly transformed tracking error function and the auxiliary stabilizing signals, and the closed-loop stability is analyzed in Section 3. Simulation studies are given in Section 4. Finally, we conclude in Section 5.

2. Problem Formulation

The kinematics for the position and attitude of an UUV can be described by

$$\begin{array}{c}\hfill \begin{array}{c}\dot{\eta }=R_1\left(\zeta \right)\upsilon \hfill \\ \dot{\zeta }=R_2\left(\zeta \right)\omega \hfill \end{array}\end{array}$$

(1)

where $\eta ={[x,y,z]}^{\top }$; x, y, and z are the positions of the center of gravity in an inertial coordinate frame, $\zeta ={[\varphi ,\theta ,\psi ]}^{\top }$; $\varphi$, $\theta$, and $\psi$ denote roll, pitch, and yaw angles, respectively, $\upsilon ={[u,v,w]}^{\top }$; u, v, and w are surge, sway, and heave velocities, respectively, and $\omega ={[p,q,r]}^{\top }$; p, q, and r denote the roll, pitch, and yaw angular velocities in the body-fixed frame, respectively. Here, the linear velocity rotation matrix $R_1\left(\zeta \right)$ and and the angular velocity transformation matrix $R_2\left(\zeta \right)$ are given by

$$\begin{array}{cc}\hfill & R_1\left(\zeta \right)=\left[\begin{array}{ccc}{c}_{\theta }{c}_{\psi }& s_{\theta }{c}_{\psi }{s}_{\varphi }-s_{\psi }{c}_{\varphi }& s_{\theta }{c}_{\psi }{c}_{\varphi }+s_{\psi }{s}_{\varphi }\\ c_{\theta }{s}_{\psi }& s_{\theta }{s}_{\psi }{s}_{\varphi }+c_{\psi }{c}_{\varphi }& s_{\theta }{s}_{\psi }{c}_{\varphi }-c_{\psi }{s}_{\varphi }\\ -s_{\theta }& c_{\theta }{s}_{\varphi }& c_{\theta }{c}_{\varphi }\end{array}\right]\hfill \\ \hfill & R_2\left(\zeta \right)=\left[\begin{array}{ccc}1& t_{\theta }{s}_{\varphi }& t_{\theta }{c}_{\varphi }\\ 0& c_{\varphi }& -s_{\varphi }\\ 0& s_{\varphi }/c_{\theta }& c_{\varphi }/c_{\theta }\end{array}\right]\hfill \end{array}$$

with $s_{(·)}=sin(·)$, $c_{(·)}=cos(·)$, and $t_{(·)}=\tan (·)$. The structure of the neutrally buoyant UUV concerned in this paper is depicted in Figure 1.

The dynamics of the UUV is given by

$$\begin{array}{c}\hfill M\left[\begin{array}{c}\dot{\upsilon }\\ \dot{\omega }\end{array}\right]=C(\upsilon ,\omega )+G\left(\zeta \right)+\tau \end{array}$$

(2)

where $M=M_1+M_2$; $M_1\in {\mathbb{R}}^{6\times 6}$ and $M_2\in {\mathbb{R}}^{6\times 6}$ are the matrix of the rigid-body mass and the added mass, respectively, $C(\upsilon ,\omega )\in {\mathbb{R}}^6$ is a vector derived by the Coriolis and damping matrices, $G\left(\zeta \right)\in {\mathbb{R}}^6$ is a vector induced from the gravitation and the buoyancy of the UUV, and $\tau =[{\tau }_X,{\tau }_Y,{\tau }_Z,{\tau }_K,{\tau }_M,{\tau }_N]$ is a vector denoting the control forces and moments. In this paper, the torpedo-shaped UUV model is considered to deal with the tracking problem. The torpedo-shaped UUV cannot move directly to the y- or z-direction in the body reference frame and the roll movement is undesirable in the practical UUV [34]. Thus, the underactuated torque vector $\tau =[{\tau }_X,0,0,0,{\tau }_M,{\tau }_N]$ is considered in this paper. The detailed definitions of M, C, and G are presented in Appendix A. For more details for the model of UUVs, see [35,36].

Assumption 1.

The nonlinear function vectors$C(\upsilon ,\omega )$and$G\left(\zeta \right)$are unknown for the control design.

Assumption 2.

The desired three-dimensional trajectory${\eta }_d\in {\mathbb{R}}^3$and its derivatives${\dot{\eta }}_d\in {\mathbb{R}}^3$and${\ddot{\eta }}_d\in {\mathbb{R}}^3$are bounded.

Problem 1.

Our problem is to design an adaptive event-triggered control law τ for ensuring predefined three-dimensional tracking performance of the uncertain UUV described by (1) and (2) so that the position trajectory η of the UUV follows the desired trajectory ${\eta }_d$ in the three-dimensional space.

3. Adaptive Event-Triggered Control with Predefined Three-Dimensional Tracking Performance

In this section, an adaptive event-triggered control methodology using an error transformation function and stabilizing auxiliary signals is established to ensure predefined three-dimensional tracking performance of the UUV. The dynamic surface design procedure using the predefined performance concept is derived step by step.

Step 1: Let us consider the kinematics (1) and define the position errors $s={[s_1,s_2,s_3]}^{\top }=\eta -{\eta }_d$. Then, to ensure predefined three-dimensional tracking performance under the underactuated property, we define the nonlinearly transformed error surface ${\mathsf{\Gamma }}_1={[{\mathsf{\Gamma }}_{1,1},{\mathsf{\Gamma }}_{1,2},{\mathsf{\Gamma }}_{1,3}]}^{\top }$ as

$$\begin{array}{c}\hfill {\mathsf{\Gamma }}_1=R_1^{-1}\left(\zeta \right)\mathsf{\Phi }-\rho \end{array}$$

(3)

where $\mathsf{\Phi }={[{\mathsf{\Phi }}_1,{\mathsf{\Phi }}_2,{\mathsf{\Phi }}_3]}^{\top }$, and $\rho ={[\varrho ,0,0]}^{\top }$ denotes the radius of error surface with the design constant $\varrho$. Here, the design constant $\varrho$ is selected relatively small compared to the length of the UUV, and ${\mathsf{\Phi }}_i$, $i=1,2,3$, is defined as

$$\begin{array}{c}\hfill {\mathsf{\Phi }}_i\left(\frac{s_i}{{\mu }_i}\right)=ln\left(\frac{{\delta }_{1,i}{\delta }_{2,i}+{\delta }_{2,i}(s_i/{\mu }_i)}{{\delta }_{1,i}{\delta }_{2,i}-{\delta }_{1,i}(s_i/{\mu }_i)}\right)\end{array}$$

(4)

where $0<{\delta }_{1,i}\le 1$ and $0<{\delta }_{2,i}\le 1$ are design constants, and ${\mu }_i\left(t\right)=({\mu }_{i,0}-{\mu }_{i,\infty })e^{-g_{i}t}+{\mu }_{i,\infty }$ is the performance function with design parameters $g_i>0$, ${\mu }_{i,0}>0$, and ${\mu }_{i,\infty }>0$ satisfying ${\mu }_{i,0}>{\mu }_{i,\infty }$ and $-{\delta }_{1,i}{\mu }_i\left(0\right)<s_i\left(0\right)<{\delta }_{2,i}{\mu }_i\left(0\right)$.

Lemma 1.

If${\mathsf{\Gamma }}_{1,i}\in {\mathcal{L}}_{\infty }$, $-{\delta }_{1,i}{\mu }_i\left(t\right)<s_i\left(t\right)<{\delta }_{2,i}{\mu }_i\left(t\right)$is ensured for all$t\ge 0$where$i=1,2,3$.

Proof. 

Let us define $\mathsf{\Theta }=R_1\left(\zeta \right)({\mathsf{\Gamma }}_1+\rho )$ where $\mathsf{\Theta }={[{\mathsf{\Theta }}_1,{\mathsf{\Theta }}_2,{\mathsf{\Theta }}_3]}^{\top }$. From the definition of $R_1\left(\zeta \right)$, there exists a constant ${\overline{R}}_1$ such that $\parallel R_1\left(\zeta \right)\parallel \le {\overline{R}}_1$. Then, from ${\mathsf{\Gamma }}_{1,i}\in {\mathcal{L}}_{\infty }$, ${\mathsf{\Theta }}_i$ are bounded where $i=1,2,3$. Thus, there exist constants ${\underline{\mathsf{\Theta }}}_i$ and ${\overline{\mathsf{\Theta }}}_i$ such that ${\underline{\mathsf{\Theta }}}_i<{\mathsf{\Theta }}_i\left(t\right)<{\overline{\mathsf{\Theta }}}_i,\forall t\ge 0$. Using the bijective property ${\mathsf{\Phi }}_i:(-{\delta }_{1,i},{\delta }_{2,i})\mapsto (-\infty ,\infty )$ [37], it holds that ${\mathsf{\Phi }}_i^{-1}\left({\underline{\mathsf{\Theta }}}_i\right)<s_i/{\mu }_i<{\mathsf{\Phi }}_i^{-1}\left({\overline{\mathsf{\Theta }}}_i\right)$. Owing to $-{\delta }_{1,i}<{\mathsf{\Phi }}_i^{-1}<{\delta }_{2,i}$, we have $-{\delta }_{1,i}{\mu }_i\left(t\right)<s_i\left(t\right)<{\delta }_{2,i}{\mu }_i\left(t\right)$ for all $t\ge 0$. □

Remark 1.

In (3), $R_1^{-1}\left(\zeta \right)$ and ρ are combined with the nonlinear error function vector Φ in order to design the underactuated control scheme with the predefined three-dimensional tracking performance. From Lemma 1, the boundedness of the error surface vector leads to the satisfaction of the inequality $-{\delta }_{1,i}{\mu }_i\left(t\right)<s_i\left(t\right)<{\delta }_{2,i}{\mu }_i\left(t\right)$ for all $t\ge 0$ where $i=1,2,3$. That is, the bounds of the transient and steady-state performance of the position errors $s_i\left(t\right)$ can be predefined by selecting the design parameters ${\delta }_{1,i}$, ${\delta }_{2,i}$, and functions ${\mu }_i\left(t\right)$. Thus, the predefined three-dimensional tracking performance is ensured provided that ${\mathsf{\Gamma }}_{1,i}\in {\mathcal{L}}_{\infty }$. Accordingly, the primary focus of this study is to design an adaptive event-triggered control scheme for ensuring the boundedness of ${\mathsf{\Gamma }}_{1,i}$.

The time derivative of $\mathsf{\Gamma }$ is represented by

$$\begin{array}{cc}\hfill {\dot{\mathsf{\Gamma }}}_1=& {\dot{R}}_1^{-1}\left(\zeta \right)\mathsf{\Phi }+R_1^{-1}\left(\zeta \right)\dot{\mathsf{\Phi }}\hfill \\ \hfill =& -K({\mathsf{\Gamma }}_1+\rho )+R_1^{-1}A(\dot{\eta }-{\dot{\eta }}_d-\dot{\mu }{\mu }^{-1}{s}_1)\hfill \end{array}$$

(5)

where $K=-{\dot{R}}_1^{-1}{R}_1$, $A=\mathrm{diag}[A_1,A_2,A_3]$ with $A_i=(1/\left(s_i+{\delta }_{1,i}{\mu }_i\right))-(1/\left(s_i-{\delta }_{2,i}{\mu }_i\right))$, $i=1,2,3$, and $\mu =\mathrm{diag}[{\mu }_1,{\mu }_2,{\mu }_3]$. Here, $\mathrm{diag}[·]$ is the diagonal matrix.

Then, we have

$$\begin{array}{c}\hfill K=-{\dot{R}}_1^{-1}{R}_1=\left[\begin{array}{ccc}0& -r& q\\ r& 0& -p\\ -q& p& 0\end{array}\right].\end{array}$$

(6)

Using (6), we obtain that

$$\begin{array}{cc}\hfill {\dot{\mathsf{\Gamma }}}_1& =-K{\mathsf{\Gamma }}_1-K\rho +R_1^{-1}A{R}_1\upsilon -R_1^{-1}A({\dot{\eta }}_d+\dot{\mu }{\mu }^{-1}{s}_1)\hfill \\ \hfill & =-K{\mathsf{\Gamma }}_1-\left[\begin{array}{ccc}0& -r& q\\ r& 0& -p\\ -q& p& 0\end{array}\right]\left[\begin{array}{c}\varrho \\ 0\\ 0\end{array}\right]+H\left[\begin{array}{c}u\\ v\\ w\end{array}\right]-R_1^{-1}A({\dot{\eta }}_d+\dot{\mu }{\mu }^{-1}{s}_1)\hfill \\ \hfill & =-K{\mathsf{\Gamma }}_1+N\left[\begin{array}{c}u\\ q\\ r\end{array}\right]+H\left[\begin{array}{c}0\\ v\\ w\end{array}\right]-R_1^{-1}A({\dot{\eta }}_d+\dot{\mu }{\mu }^{-1}{s}_1)\hfill \end{array}$$

(7)

where $H=R_1^{-1}A{R}_1$ and

$$\begin{array}{c}\hfill N=\left[\begin{array}{ccc}{H}_{1,1}& 0& 0\\ H_{2,1}& 0& -\varrho \\ H_{3,1}& \varrho & 0\end{array}\right].\end{array}$$

Here, $H_{m,n}$ means the $(m,n)$ element of the matrix H.

Using the dynamic surface design concept [38], we define the error surface vector $e={[e_u,e_q,e_r]}^{\top }$ with $e_u=u-{\overline{\alpha }}_u$, $e_q=q-{\overline{\alpha }}_q$, and $e_r=r-{\overline{\alpha }}_r$, and the boundary layer error vector $c=[c_1,c_2,c_3]=\overline{\alpha }-\alpha$ where $\alpha ={[{\alpha }_u,{\alpha }_q,{\alpha }_r]}^{\top }$ is the virtual control vector and $\overline{\alpha }={[{\overline{\alpha }}_u,{\overline{\alpha }}_q,{\overline{\alpha }}_r]}^{\top }$ is the filtered signal vector of virtual control laws ${\alpha }_u$, ${\alpha }_q$, and ${\alpha }_r$ that is obtained by the first-order low-pass filter

$$\begin{array}{c}\xi \dot{\overline{\alpha }}+\overline{\alpha }=\alpha ,\overline{\alpha }\left(0\right)=\alpha \left(0\right)\hfill \end{array}$$

(8)

where $\xi >0$ is the small constant.

Using the error surface vector e and the boundary layer error c, (7) becomes

$$\begin{array}{cc}\hfill {\dot{\mathsf{\Gamma }}}_1& =-K{\mathsf{\Gamma }}_1+N(e+\alpha +c)+H\left[\begin{array}{c}0\\ v\\ w\end{array}\right]-R_1^{-1}A({\dot{\eta }}_d+\dot{\mu }{\mu }^{-1}{s}_1).\hfill \end{array}$$

(9)

The virtual control vector $\alpha ={[{\alpha }_u,{\alpha }_q,{\alpha }_r]}^{\top }$ is presented as

$$\begin{array}{c}\hfill \alpha =N^{-1}\left(-{\gamma }_1{\mathsf{\Gamma }}_1-H\left[\begin{array}{c}0\\ v\\ w\end{array}\right]+R_1^{-1}A({\dot{\eta }}_d+\dot{\mu }{\mu }^{-1}{s}_1)\right)\end{array}$$

(10)

where ${\gamma }_1=\mathrm{diag}\left[{\gamma }_{1,1}{\gamma }_{1,2}{\gamma }_{1,3}\right]$; ${\gamma }_{1,i}$, $i=1,\cdots ,3$, are positive constants.

Substituting (10) to (9) gives

$$\begin{array}{cc}\hfill {\dot{\mathsf{\Gamma }}}_1& =-K{\mathsf{\Gamma }}_1-{\gamma }_1{\mathsf{\Gamma }}_1+N(e+c).\hfill \end{array}$$

(11)

We choose a Lyapunov function $V_1={\mathsf{\Gamma }}_1^{\top }{\mathsf{\Gamma }}_1/2$. Then, the time derivative of $V_1$ is represented by

$$\begin{array}{cc}\hfill {\dot{V}}_1& =-{\mathsf{\Gamma }}_1^{\top }{\gamma }_1{\mathsf{\Gamma }}_1+{\mathsf{\Gamma }}_1^{\top }N(e+c).\hfill \end{array}$$

(12)

where ${\mathsf{\Gamma }}_1^{\top }K{\mathsf{\Gamma }}_1=0$ due to the skew symmetric matrix K

Step 2: To design the underactuated torque vector $\tau =[{\tau }_X,0,0,0,{\tau }_M,{\tau }_N]$, we define the error surface vector ${\mathsf{\Gamma }}_2={[{\mathsf{\Gamma }}_{2,1},\cdots ,{\mathsf{\Gamma }}_{2,6}]}^{\top }=\varpi -\chi$ where $\varpi ={[{\upsilon }^{\top },{\omega }^{\top }]}^{\top }$ and $\chi ={[{\overline{\alpha }}_u,{\beta }_1,{\beta }_2,{\beta }_3,{\overline{\alpha }}_q,{\overline{\alpha }}_r]}^{\top }$. Here, ${\beta }_1$, ${\beta }_2$, and ${\beta }_3$ in $\chi$ are the auxiliary stabilizing signals to be designed later.

Using (2), the time derivative of ${\mathsf{\Gamma }}_2$ is obtained as

$$\begin{array}{cc}\hfill {\dot{\mathsf{\Gamma }}}_2& =M^{-1}(F(\varpi ,\zeta )+\tau )-\dot{\chi }\hfill \end{array}$$

(13)

where $F(\varpi ,\zeta )=C\left(\varpi \right)+G\left(\zeta \right)$.

For the online approximation of unknown nonlinear function vector $F(\varpi ,\zeta )$, radial basis function neural networks [39] are employed. Then, F can be approximated over the compact set $\mathsf{{\rm Y}}$ as follows

$$F\left(\overline{x}\right)=W^{\ast \top }\mathsf{\Omega }\left(\overline{x}\right)+\epsilon$$

(14)

where $\overline{x}={[{\varpi }^{\top },{\zeta }^{\top }]}^{\top }\in \mathsf{{\rm Y}}\subset {\mathbb{R}}^6$ denotes the input vector of radial basis function neural networks, the optimal weighting matrix $W^{\ast }$ is defined as $W^{\ast }=\mathrm{diag}[W_1^{\ast },\cdots ,W_6^{\ast }]$ satisfying $\parallel W^{\ast }{\parallel }_F\le \overline{W}$ with an unknown constant $\overline{W}>0$, $W_f^{\ast }={[W_{f,1}^{\ast },\cdots ,W_{f,n}^{\ast }]}^{\top }$ with $f=1,\cdots ,6$, ${\parallel ·\parallel }_F$ is the Frobenius norm, $\mathsf{\Omega }$ denotes the Gaussian function vector $\mathsf{\Omega }={[{\mathsf{\Omega }}_1^{\top },\cdots ,{\mathsf{\Omega }}_6^{\top }]}^{\top }$; ${\mathsf{\Omega }}_f={[{\mathsf{\Omega }}_{f,1},\cdots ,{\mathsf{\Omega }}_{f,n}]}^{\top }$ with $f=1,\cdots ,6$, and $\epsilon \in {\mathbb{R}}^6$ is a reconstruction error vector such as $\parallel \epsilon \parallel \le \overline{\epsilon }$ with an unknown constant $\overline{\epsilon }>0$.

An adaptive event-triggered tracking law is presented as

$$\begin{array}{cc}\hfill \tau \left(t\right)=& \breve{\tau }\left(t_j\right),\forall t\in [t_j,t_{j+1})\hfill \end{array}$$

(15)

$$\begin{array}{cc}\hfill t_{j+1}=& \inf \{t>t_j|\parallel S_{\tau }\left(t\right)\parallel \ge {\kappa }_1\parallel e\left(t\right)\parallel +{\kappa }_2\}\hfill \end{array}$$

(16)

where $t_j$ denotes the update time of the control torque $\tau$ with $j\in {\mathbb{Z}}^{+}$, $S_{\tau }\left(t\right)=\breve{\tau }\left(t\right)-\tau \left(t\right)$, and ${\kappa }_1>0$ and ${\kappa }_2>0$ are design constants. When the event-triggering condition (16) is satisfied, the tracking law $\tau$ is updated at $t_{j+1}$ and is set to $\breve{\tau }={[{\breve{\tau }}_1,0,0,0,{\breve{\tau }}_5,{\breve{\tau }}_6]}^{\top }$ given by

$$\begin{array}{cc}\hfill \left[\begin{array}{c}{\breve{\tau }}_1\\ {\breve{\tau }}_5\\ {\breve{\tau }}_6\end{array}\right]=& -{\overline{\gamma }}_{2,1}e+\left[\begin{array}{c}{B}_1\\ B_5\\ B_6\end{array}\right]-\left[\begin{array}{c}{\widehat{W}}_1^{\top }{\mathsf{\Omega }}_1\\ {\widehat{W}}_5^{\top }{\mathsf{\Omega }}_5\\ {\widehat{W}}_6^{\top }{\mathsf{\Omega }}_6\end{array}\right]-(\widehat{\epsilon }+{\kappa }_2)tanh\left(\frac{e}{\vartheta }\right)-{\kappa }_{1}e-N^{\top }{\mathsf{\Gamma }}_1\hfill \end{array}$$

(17)

where ${\overline{\gamma }}_{2,1}=\mathrm{diag}[{\gamma }_{2,1},{\gamma }_{2,5},{\gamma }_{2,6}]$; ${\gamma }_{2,1}$, ${\gamma }_{2,5}$, and ${\gamma }_{2,6}$ denote positive design parameters, ${\widehat{W}}_1$, ${\widehat{W}}_5$, and ${\widehat{W}}_6$ are estimates of $W_1^{\ast }$, $W_5^{\ast }$, and $W_6^{\ast }$, respectively, $tanh(e/\vartheta )={[tanh({\mathsf{\Gamma }}_{2,1}/\vartheta ),tanh({\mathsf{\Gamma }}_{2,5}/\vartheta ),tanh({\mathsf{\Gamma }}_{2,6}/\vartheta )]}^{\top }$ with a constant $\vartheta >0$, and $\widehat{\epsilon }$ denotes an estimate of $\overline{\epsilon }$. Here, $B_1$, $B_5$, $B_6$ are elements of the vector $B={[B_1,\cdots ,B_6]}^{\top }=M\dot{\chi }$ with $\dot{\chi }={[({\alpha }_u-{\overline{\alpha }}_u)/\xi ,{\dot{\beta }}_1,{\dot{\beta }}_2,{\dot{\beta }}_3,({\alpha }_q-{\overline{\alpha }}_q)/\xi ,({\alpha }_r-{\overline{\alpha }}_r)/\xi ]}^{\top }$. The dynamics of the auxiliary stabilizing signals ${\beta }_1$, ${\beta }_2$, and ${\beta }_3$ are designed as

$$\begin{array}{cc}\hfill \left[\begin{array}{c}{\dot{\beta }}_1\\ {\dot{\beta }}_2\\ {\dot{\beta }}_3\end{array}\right]=& {\overline{M}}_2^{-1}\left({\overline{\gamma }}_{2,2}\left[\begin{array}{c}{\mathsf{\Gamma }}_{2,2}\\ {\mathsf{\Gamma }}_{2,3}\\ {\mathsf{\Gamma }}_{2,4}\end{array}\right]-{\overline{M}}_1\left[\begin{array}{c}({\alpha }_u-{\overline{\alpha }}_u)/\xi \\ ({\alpha }_q-{\overline{\alpha }}_q)/\xi \\ ({\alpha }_r-{\overline{\alpha }}_r)/\xi \end{array}\right]+\left[\begin{array}{c}{\widehat{W}}_2^{\top }{\mathsf{\Omega }}_2\\ {\widehat{W}}_3^{\top }{\mathsf{\Omega }}_3\\ {\widehat{W}}_4^{\top }{\mathsf{\Omega }}_4\end{array}\right]+\widehat{\epsilon }\left[\begin{array}{c}tanh\left(\frac{{\mathsf{\Gamma }}_{2,2}}{\vartheta }\right)\\ tanh\left(\frac{{\mathsf{\Gamma }}_{2,3}}{\vartheta }\right)\\ tanh\left(\frac{{\mathsf{\Gamma }}_{2,4}}{\vartheta }\right)\end{array}\right]\right)\hfill \end{array}$$

(18)

where ${\overline{\gamma }}_{2,2}=\mathrm{diag}[{\gamma }_{2,2},{\gamma }_{2,3},{\gamma }_{2,4}]$; ${\gamma }_{2,2}$, ${\gamma }_{2,3}$, and ${\gamma }_{2,4}$ are the positive design parameters, and

$$\begin{array}{cc}\hfill {\overline{M}}_1& =\left[\begin{array}{ccc}0& 0& m{x}_g-Y_{\dot{r}}\\ 0& -m{x}_g-z_{\dot{q}}& 0\\ 0& 0& 0\end{array}\right]\hfill \\ \hfill {\overline{M}}_2& =\left[\begin{array}{ccc}m-Y_{\dot{v}}& 0& -m{z}_g\\ 0& m-Z_{\dot{w}}& m{y}_g\\ -m{z}_g& m{y}_g& I_{xx}-K_{\dot{p}}\end{array}\right].\hfill \end{array}$$

The adaptive laws for ${\widehat{W}}_f$, $f=1,\cdots ,6$, and $\widehat{\epsilon }$ are designed as

$$\begin{array}{cc}\hfill {\dot{\widehat{W}}}_f=& {\varsigma }_f({\mathsf{\Gamma }}_{2,f}{\mathsf{\Omega }}_f-{\sigma }_1{\widehat{W}}_f)\hfill \end{array}$$

(19)

$$\begin{array}{cc}\hfill \dot{\widehat{\epsilon }}=& \iota \left({\mathsf{\Gamma }}_2^{\top }tanh\left(\frac{{\mathsf{\Gamma }}_2}{\vartheta }\right)-{\sigma }_2\widehat{\epsilon }\right)\hfill \end{array}$$

(20)

where $f=1,\cdots ,6$, $tanh({\mathsf{\Gamma }}_2/\vartheta )={[tanh({\mathsf{\Gamma }}_{2,1}/\vartheta ),\cdots ,tanh({\mathsf{\Gamma }}_{2,6}/\vartheta )]}^{\top }$, ${\varsigma }_f=\mathrm{diag}[{\varsigma }_{f,1},\cdots ,{\varsigma }_{f,n}]$; ${\varsigma }_{f,l}>0$, $l=1,\cdots ,n$ is a constant, $\iota$, ${\sigma }_1$ and ${\sigma }_2$ are positive design constants.

Let us consider a Lyapunov function $V_2$ as

$$\begin{array}{cc}\hfill V_2& =\frac{1}{2}{\mathsf{\Gamma }}_2^{\top }M{\mathsf{\Gamma }}_2+\frac{1}{2}tr\left({\tilde{W}}^{\top }{\varsigma }^{-1}\tilde{W}\right)+\frac{1}{2\iota }{\tilde{\epsilon }}^2\hfill \end{array}$$

(21)

where $\tilde{W}=W^{\ast }-\widehat{W}$ and $\tilde{\epsilon }=\overline{\epsilon }-\widehat{\epsilon }$ are estimation errors, $\varsigma =\mathrm{diag}[{\varsigma }_1,\cdots ,{\varsigma }_6]$, and $tr(·)$ means the trace of the matrix.

By substituting (13) and (14) into the time derivative of (21), ${\dot{V}}_2$ is obtained as

$$\begin{array}{cc}\hfill {\dot{V}}_2=& {\mathsf{\Gamma }}_2^{\top }(W^{\ast \top }\mathsf{\Omega }+\epsilon +\breve{\tau }-S_{\tau })-{\mathsf{\Gamma }}_2^{\top }B-tr\left({\tilde{W}}^{\top }{\varsigma }^{-1}\dot{\widehat{W}}\right)-\frac{1}{\iota }\tilde{\epsilon }\dot{\widehat{\epsilon }}.\hfill \end{array}$$

(22)

It holds that ${\mathsf{\Gamma }}_2^{\top }{S}_{\tau }=e^{\top }{\overline{S}}_{\tau }$ and $\parallel S_{\tau }\parallel =\parallel {\overline{S}}_{\tau }\parallel$ with $S_{\tau }={[{\breve{\tau }}_1-{\tau }_X,0,0,0,{\breve{\tau }}_5-{\tau }_M,{\breve{\tau }}_6-{\tau }_N]}^{\top }$ and ${\overline{S}}_{\tau }={[{\breve{\tau }}_1-{\tau }_X,{\breve{\tau }}_5-{\tau }_M,{\breve{\tau }}_6-{\tau }_N]}^{\top }$. Then, using the property $S_{\tau }\left(t_j\right)=0$ for $j\in {\mathbb{Z}}^{+}$ and the event-triggering condition (16), we have $-{\mathsf{\Gamma }}_2^{\top }{S}_{\tau }\le \parallel e\parallel ({\kappa }_1\parallel e\parallel +{\kappa }_2)$.

Then, using $B={[B_1,\cdots ,B_6]}^{\top }=M\dot{\chi }$ and the definition of the matrix M, we have

$$\begin{array}{c}\hfill {\mathsf{\Gamma }}_2^{\top }B={\mathsf{\Gamma }}_2^{\top }\left[\begin{array}{c}{B}_1\\ 0\\ 0\\ 0\\ B_5\\ B_6\end{array}\right]+{\mathsf{\Gamma }}_2^{\top }\left[\begin{array}{c}0\\ {\overline{M}}_1\left[\begin{array}{c}({\alpha }_u-{\overline{\alpha }}_u)/\xi \\ ({\alpha }_q-{\overline{\alpha }}_q)/\xi \\ ({\alpha }_r-{\overline{\alpha }}_r)/\xi \end{array}\right]+{\overline{M}}_2\left[\begin{array}{c}{\dot{\beta }}_1\\ {\dot{\beta }}_2\\ {\dot{\beta }}_3\end{array}\right]\\ \\ 0\\ 0\end{array}\right].\end{array}$$

(23)

Substituting $-{\mathsf{\Gamma }}_2^{\top }{S}_{\tau }\le \parallel e\parallel ({\kappa }_1\parallel e\parallel +{\kappa }_2)$ and (23) into (22) and using (17) and (18), we obtain

$$\begin{array}{cc}\hfill {\dot{V}}_2\le & -{\mathsf{\Gamma }}_2^{\top }{\gamma }_2{\mathsf{\Gamma }}_2+{\mathsf{\Gamma }}_2^{\top }{\tilde{W}}^{\top }\mathsf{\Omega }-\frac{1}{\iota }\tilde{\epsilon }\dot{\widehat{\epsilon }}-tr\left({\tilde{W}}^{\top }{\varsigma }^{-1}\dot{\widehat{W}}\right)+\parallel e\parallel {\kappa }_2-{\kappa }_2{e}^{\top }tanh\left(\frac{e}{\vartheta }\right)\hfill \\ \hfill & +\parallel {\mathsf{\Gamma }}_2\parallel \overline{\epsilon }-\overline{\epsilon }{\mathsf{\Gamma }}_2^{\top }tanh\left(\frac{{\mathsf{\Gamma }}_2}{\vartheta }\right)+\tilde{\epsilon }{\mathsf{\Gamma }}_2^{\top }tanh\left(\frac{{\mathsf{\Gamma }}_2}{\vartheta }\right)-e^{\top }{N}^{\top }{\mathsf{\Gamma }}_1\hfill \end{array}$$

(24)

where ${\gamma }_2=\mathrm{diag}[{\gamma }_{2,1},\cdots ,{\gamma }_{2,6}]$.

Lemma 2.

[39]$0\le \left|s\right|-stanh(s/\vartheta )\le 0.2785\vartheta$for$s\in \mathbb{R}$and any positive constant ϑ.

By substituting (19) and (20) into (24) and using Lemma 2, ${\dot{V}}_2$ becomes

$$\begin{array}{cc}\hfill {\dot{V}}_2\le & -{\mathsf{\Gamma }}_2^{\top }{\gamma }_2{\mathsf{\Gamma }}_2+{\sigma }_{1}tr\left({\tilde{W}}^{\top }\widehat{W}\right)+{\sigma }_2\tilde{\epsilon }\widehat{\epsilon }-e^{\top }{N}^{\top }{\mathsf{\Gamma }}_1+1.671\overline{\epsilon }\vartheta +0.8355{\kappa }_2\vartheta .\hfill \end{array}$$

(25)

Remark 2.

In the dynamics (2) of the torpedo-shaped UUV, the underactuated control torque vector $\tau =[{\tau }_X,0,0,0,{\tau }_M,{\tau }_N]$ should be designed. That is, the first, fifth, and sixth dynamic equations in (2) only have the control torques ${\tau }_X$, ${\tau }_M$, and ${\tau }_N$, respectively. Thus, the the auxiliary stabilizing signals are required for the state equations for v, ω, and p (i.e., the second, third, and fourth dynamic equations in (2)). In this study, the auxiliary stabilizing signals ${\beta }_1$, ${\beta }_2$, and ${\beta }_3$ in (18) are presented to design the underactuated control torque vector $\tau =[{\tau }_X,0,0,0,{\tau }_M,{\tau }_N]$ while ensuring the predefined three-dimensional tracking performance and the stability of the closed-loop system. Because of these auxiliary stabilizing signals, the UUV dynamics (2) is stably controlled by using the only three control inputs ${\tau }_X$, ${\tau }_M$, and ${\tau }_N$.

We analyze the predefined three-dimensional tracking performance and stability of the closed-loop system and the exclusion of Zeno behavior of the proposed event-triggered scheme in the following theorem.

Theorem 1.

Consider the kinematics and dynamics of the uncertain UUV (i.e., (1) and (2)). For initial conditions satisfying $V\left(0\right)\le \Delta$ with a constant $\Delta >0$, the adaptive event-triggered tracking law (17) guarantees that

(i) 

all the closed-loop signals are semi-globally uniformly ultimately bounded;

(ii) 

the predefined three-dimensional tracking performance is ensured (i.e.,$-{\delta }_{1,i}{\mu }_i\left(t\right)<s_i\left(t\right)<{\delta }_{2,i}{\mu }_i\left(t\right)$, $i=1,2,3$, $\forall t\ge 0$);

(iii) 

there exists an inter-event time$T_m>0$such that$|t_{j+1}-t_j|\ge T_m$.

Proof. 

The dynamics of the boundary layer error c is represented by

$$\begin{array}{c}\hfill \dot{c}=-\frac{1}{\xi }c+\mathsf{\Lambda }({\mathsf{\Gamma }}_1,{\mathsf{\Gamma }}_2,c,\widehat{W},\widehat{\epsilon },{\eta }_0)\end{array}$$

(26)

where ${\eta }_0={[{\eta }_d^{\top },{\dot{\eta }}_d^{\top },{\ddot{\eta }}_d^{\top }]}^{\top }$ and $\mathsf{\Lambda }={\dot{N}}^{-1}\{-{\gamma }_1{\mathsf{\Gamma }}_1-H{\left[0vw\right]}^{\top }+R_1^{-1}A({\dot{\eta }}_d+{\mu }^{-1}\dot{\mu }{s}_1)\}+N^{-1}\{-{\gamma }_1\dot{\mathsf{\Gamma }}-H{\left[0\dot{v}\dot{w}\right]}^{\top }-\dot{H}{\left[0vw\right]}^{\top }+{\dot{R}}_1^{-1}A({\dot{\eta }}_d+{\mu }^{-1}\dot{\mu }{s}_1)+R_1^{-1}\dot{A}({\dot{\eta }}_d+{\mu }^{-1}\dot{\mu }{s}_1)+R_1^{-1}A({\ddot{\eta }}_d+{\mu }^{-1}\dot{\mu }{\dot{s}}_1+{\dot{\mu }}^{-1}\dot{\mu }{s}_1+{\mu }^{-1}\ddot{\mu }{s}_1)\}$ is a continuous function.

We define the Lyapunov function $V=V_1+V_2+\left(c^{\top }c\right)/2$ to prove this theorem. Using (12), (25), and (26), $\dot{V}$ becomes

$$\begin{array}{cc}\hfill \dot{V}& \le -{\mathsf{\Gamma }}_1^{\top }{\gamma }_1{\mathsf{\Gamma }}_1-{\mathsf{\Gamma }}_2^{\top }{\gamma }_2{\mathsf{\Gamma }}_2-\frac{1}{\xi }{c}^{\top }c+c^{\top }\overline{\mathsf{\Lambda }}+{\sigma }_{1}tr\left({\tilde{W}}^{\top }\widehat{W}\right)\hfill \\ \hfill & -{\sigma }_2{\tilde{\epsilon }}^2+{\sigma }_2\tilde{\epsilon }\overline{\epsilon }+1.671\overline{\epsilon }\vartheta +0.8355{\kappa }_2\vartheta .\hfill \end{array}$$

(27)

where $\overline{\mathsf{\Lambda }}=\mathsf{\Lambda }+N^{\top }{\mathsf{\Gamma }}_1$ is a continuous function. Using the inequalities $tr\left({\tilde{W}}^{\top }\widehat{W}\right)\le -{\parallel \tilde{W}\parallel }_F^2/2+{\overline{W}}^2/2$, $\tilde{\epsilon }\overline{\epsilon }\le {\tilde{\epsilon }}^2/2+{\overline{\epsilon }}^2/2$, and $c^{\top }\overline{\mathsf{\Lambda }}\le {\parallel c\parallel }^2{\parallel \overline{\mathsf{\Lambda }}\parallel }^2$$/\left(2\phi \right)+\phi /2$ with constants $\phi >0$, (27) becomes

$$\begin{array}{cc}\hfill \dot{V}\le & -{\mathsf{\Gamma }}_1^{\top }{\gamma }_1{\mathsf{\Gamma }}_1-{\mathsf{\Gamma }}_2^{\top }{\gamma }_2{\mathsf{\Gamma }}_2-\frac{{\parallel c\parallel }^2}{\xi }+\frac{{\parallel c\parallel }^2{\parallel \overline{\mathsf{\Lambda }}\parallel }^2}{2\phi }-\frac{{\sigma }_1}{2}{\parallel \tilde{W}\parallel }_F^2-\frac{{\sigma }_2}{2}{\tilde{\epsilon }}^2+D\hfill \end{array}$$

(28)

where $D=({\sigma }_1/2){\overline{W}}^2+({\sigma }_2/2){\overline{\epsilon }}^2+\phi /2+1.671\overline{\epsilon }\vartheta +0.8355{\kappa }_2\vartheta$.

Let us define compact sets $\mathsf{\Pi }=\{{\mathsf{\Gamma }}_1^{\top }{\mathsf{\Gamma }}_1+{\mathsf{\Gamma }}_2^{\top }M{\mathsf{\Gamma }}_2+c^{\top }c+tr\left({\tilde{\mathsf{\Phi }}}^{\top }{\varsigma }^{-1}\tilde{\mathsf{\Phi }}\right)+(1/\iota ){\tilde{\epsilon }}^2\le 2\Delta \}$ and $\mathsf{\Xi }=\{{\eta }_d^{\top }{\eta }_d+{\dot{\eta }}_d^{\top }{\dot{\eta }}_d+{\ddot{\eta }}_d^{\top }{\ddot{\eta }}_d\le {\overline{\eta }}_0\}$ with a constant ${\overline{\eta }}_0>0$. Then, there exists a constant ${\mathsf{\Lambda }}^{\ast }$ such that $\parallel \overline{\mathsf{\Lambda }}\parallel \le {\mathsf{\Lambda }}^{\ast }$ on $\mathsf{\Pi }\times \mathsf{\Xi }$. By selecting $1/\xi ={\xi }^{\ast }+{\left({\mathsf{\Lambda }}^{\ast }\right)}^2/\left(2\phi \right)+1$ with a constant ${\xi }^{\ast }>0$, we have

$$\begin{array}{c}\hfill \dot{V}\le -\mathsf{\Psi }V-\left(1-\frac{\parallel \overline{\mathsf{\Lambda }}{\parallel }^2}{{\left({\mathsf{\Lambda }}^{\ast }\right)}^2}\right)\frac{{\parallel c\parallel }^2{\left({\mathsf{\Lambda }}^{\ast }\right)}^2}{2\phi }+D\end{array}$$

(29)

where $\mathsf{\Psi }=\min [2{\gamma }_1,2{\gamma }_2,2{\xi }^{\ast },{\sigma }_1{\varsigma }_{min},{\sigma }_2\iota ]$; ${\varsigma }_{min}$ is the minimum eigenvalue of $\varsigma$. Because of $\parallel \mathsf{\Lambda }\parallel \le {\mathsf{\Lambda }}^{\ast }$ on $V=\Delta$, $\dot{V}\le -\mathsf{\Psi }V+D$ is satisfied on $V=\Delta$. Then, it is ensured that $\dot{V}<0$ on $V=\Delta$ when $\mathsf{\Psi }>D/\Delta$ and thus $V\le \Delta$ denotes an invariant set. It is proved that all closed-loop signals are semi-global uniform ultimate bounded. This completes the proof of Theorem 1—(i).

By integrating $\dot{V}\le -\mathsf{\Psi }V+D$ with respect to time, we have $V\left(t\right)\le e^{-\mathsf{\Psi }t}V\left(0\right)+(D/\mathsf{\Psi })[1-e^{-\mathsf{\Psi }t}]$. Using $(1/2)\parallel {\mathsf{\Gamma }}_1{\parallel }^2\le V\left(t\right)$, it holds that ${lim}_{t\to \infty }\parallel {\mathsf{\Gamma }}_1\parallel \le \sqrt{2D/\mathsf{\Psi }}$. That is, ${\mathsf{\Gamma }}_1$ is bounded (i.e., ${\mathsf{\Gamma }}_{1,i}\in {\mathcal{L}}_{\infty }$, $i=1,2,3$). From Lemma 1, the predefined three-dimensional tracking performance is ensured (i.e., $-{\delta }_{1,i}{\mu }_i\left(t\right)<s_i\left(t\right)<{\delta }_{2,i}{\mu }_i\left(t\right)$, $i=1,2,3$, $\forall t\ge 0$). This completes the proof of Theorem 1—(ii).

To show the exclusion of Zeno behavior of the proposed event-triggering scheme, we prove that there exists a minimum value $T_m$ of inter-event times such that $|t_{j+1}-t_j|\ge T_m$ for $j\in {\mathbb{Z}}^{+}$.

For all $t\in [t_j,t_{j+1})$, we consider

$$\begin{array}{c}\hfill \frac{d}{dt}\parallel S_{\tau }\parallel =\frac{d}{dt}{\left(S_{\tau }^{\top }{S}_{\tau }\right)}^{\frac{1}{2}}=\frac{S_{\tau }^{\top }{\dot{S}}_{\tau }}{\parallel S_{\tau }\parallel }\le \parallel \frac{d}{dt}\left(\breve{\tau }\right)\parallel \end{array}$$

(30)

where $\frac{d}{dt}\left(\breve{\tau }\right)={[{\dot{\breve{\tau }}}_1,0,0,{\dot{\breve{\tau }}}_5,{\dot{\breve{\tau }}}_6]}^{\top }$; ${\dot{\breve{\tau }}}_1$, ${\dot{\breve{\tau }}}_5$, and ${\dot{\breve{\tau }}}_6$ are given by

$$\begin{array}{cc}\hfill \left[\begin{array}{c}{\dot{\breve{\tau }}}_1\\ {\dot{\breve{\tau }}}_5\\ {\dot{\breve{\tau }}}_6\end{array}\right]=& -{\overline{\gamma }}_{2,1}\dot{e}+\left[\begin{array}{c}{\dot{B}}_1\\ {\dot{B}}_5\\ {\dot{B}}_6\end{array}\right]-\left[\begin{array}{c}{\dot{\widehat{W}}}_1^{\top }{\mathsf{\Omega }}_1+{\widehat{W}}_1^{\top }{\dot{\mathsf{\Omega }}}_1\\ {\dot{\widehat{W}}}_5^{\top }{\mathsf{\Omega }}_5+{\widehat{W}}_5^{\top }{\dot{\mathsf{\Omega }}}_5\\ {\dot{\widehat{W}}}_6^{\top }{\mathsf{\Omega }}_6+{\widehat{W}}_6^{\top }{\dot{\mathsf{\Omega }}}_6\end{array}\right]-\dot{\widehat{\epsilon }}tanh\left(\frac{e}{\vartheta }\right)\hfill \\ \hfill & -(\widehat{\epsilon }+{\kappa }_2)\left(1-{tanh}^2\left(\frac{e}{\vartheta }\right)\right)\frac{\dot{e}}{\vartheta }-{\kappa }_1\dot{e}-{\dot{N}}^{\top }{\mathsf{\Gamma }}_1-N^{\top }{\dot{\mathsf{\Gamma }}}_1.\hfill \end{array}$$

(31)

From the boundedness of all the closed-loop signals, there exists a constant $d>0$ such that $\parallel \frac{d}{dt}\left(\breve{\tau }\right)\parallel \le d$. Integrating $\frac{d}{dt}\parallel S_{\tau }\parallel \le d$ during $t\in [t_j,t_{j+1})$ and using the event-triggering condition (16) yield $|t_{j+1}-t_j|\ge ({\kappa }_1\parallel e\left(t\right)\parallel +{\kappa }_2)/d\ge {\kappa }_2/d$. By defining $T_m={\kappa }_2/d$, it holds that $|t_{j+1}-t_j|\ge T_m$. This completes the proof of Theorem 1—(iii). □

4. Simulation Examples

The parameters of the UUV are borrowed from [35]. The reference signal is given by ${\eta }_d\left(t\right)={[30cos\left(0.15t\right),30sin\left(0.15t\right),2t]}^{\top }$ and the initial position of the UUV is set to $\eta \left(0\right)=[45,5,5]$. The control parameters of the UUV are chosen as ${\gamma }_1=\mathrm{diag}[1.0,1.0,1.0]$, ${\gamma }_2=\mathrm{diag}[50,30,30,30,50,50]$, ${\varsigma }_{f,l}=30$, $\iota =30$, ${\sigma }_1=0.05$, ${\sigma }_2=0.05$, $\xi =0.05$, $\vartheta =0.1$, and $\varrho =-0.1$ where $f=1,\cdots ,6$ and $l=1,\cdots ,n$. The parameters for performance functions are selected as ${\delta }_{1,1}=0.75$, ${\delta }_{2,1}=1.0$, ${\delta }_{1,2}=0.7$, ${\delta }_{2,2}=0.7$, ${\delta }_{1,3}=0.7$, ${\delta }_{2,3}=0.7$, $g_1=1.1$, $g_2=1.05$, $g_3=1.0$, ${\mu }_{1,0}=30$, ${\mu }_{1,\infty }=2$, ${\mu }_{2,0}=10$, ${\mu }_{2,\infty }=1$, ${\mu }_{3,0}=10$, and ${\mu }_{3,\infty }=1$. For the event-triggering condition, we set ${\kappa }_1=100$ and ${\kappa }_2=1$. The event-triggering condition is checked every the sampling time 0.005 s.

The three-dimensional tracking result is shown in Figure 2. In Figure 2, the UUV follows the desired trajectory with good performance. Figure 3 depicts the position tracking errors. Figure 3 reveals that the time responses of the position errors $s_i$ remain within the predefined time-varying performance bounds $-{\delta }_{1,i}{\mu }_i$ and ${\delta }_{2,i}{\mu }_i$ for all $t\ge 0$. The mean square errors of the position errors $s_i$ at the steady-state response are presented in

Table 1. Mean square errors of $s_1\left(t\right)$, $s_2\left(t\right)$, and $s_3\left(t\right)$ at the steady-state response.

${\mathit{s}}_1$	${\mathit{s}}_2$	${\mathit{s}}_3$
$0.0177$	$0.0052$	$4.0268\times {10}^{-4}$

where the steady-state response is set to the position errors $s_i\left(t\right)$ for $t\ge 10$ s. The outputs of the neural networks are displayed in Figure 4. The event-triggered underactuated control inputs for the UUV are shown in Figure 5. Figure 6 shows the triggered time intervals and the cumulative number of events of the proposed event-triggered control laws. The number of total events of the proposed event-triggered tracker is 2164. Thus, the data required for implementing the proposed tracker are only $18.03\%$ of the total sampled data 12,000 during 60 s. This implies that the proposed event-triggered control scheme can save the signal transmission burden. From these results, we can see that predefined three-dimensional tracking performance under the underactuated property of the uncertain nonlinear 6-DOF UUV can be achieved by the proposed adaptive event-triggered tracking methodology.

5. Conclusions

We presented an adaptive event-triggered control method for ensuring predefined three-dimensional tracking performance of uncertain nonlinear 6-DOF UUVs. The nonlinearly transformed error function and the auxiliary stabilizing signals were derived for achieving predefined three-dimensional tracking performance while overcoming the underactuated problem of the nonlinear 6-DOF dynamics. It was shown that the adaptive event-triggered tracker using neural networks achieves the practical stability and predefined tracking performance of the closed-loop system. The main contribution of this work to the event-triggered control of uncertain nonlinear 6-DOF UUVs is that the predefined three-dimensional tracking performance under the underactuated dynamics can be admitted than the conventional event-triggered techniques of underwater vehicles in the two-dimensional space.