Robust composite neural dynamic surface control for the path following of unmanned marine surface vessels with unknown disturbances

Abstract

This article presents a robust composite neural-based dynamic surface control design for the path following of unmanned marine surface vessels in the presence of nonlinearly parameterized uncertainties and unknown time-varying disturbances. Compared with the existing neural network-based dynamic surface control methods where only the tracking errors are commonly used for the neural network weight updating, the proposed scheme employs both the tracking errors and the prediction errors to construct the adaption law. Therefore, faster identification of the system dynamics and improved tracking accuracy are achieved. In particular, an outstanding advantage of the proposed neural network structure is simplicity. No matter how many neural network nodes are utilized, only one adaptive parameter that needs to be tuned online, which effectively reduces the computational burden and facilitates to implement the proposed controller in practice. The uniformly ultimate boundedness stability of the closed-loop system is established via Lyapunov analysis. Comparison studies are presented to demonstrate the effectiveness of the proposed composite neural-based dynamic surface control architecture.

Introduction

In recent years, the development of unmanned marine surface vessels (MSVs) has received increased attention in many marine activities, including military reconnaissance, environmental monitoring, seabed mapping, and submarine pipelines inspection. ^{1 –3} To accomplish these tasks, path-following operation is usually needed for MSVs. However, the critical working environment where wind, waves, and ocean currents are existing can significantly affect the speed and maneuverability of the vessels. ⁴ This makes precise path following challenging for the control system design. On the other hand, as one type of marine robots, unmanned MSV is required to have automatic control ability in complex marine environment. Therefore, advanced control strategy should be further investigated for the path following of MSVs.

The main purpose of path following is to make the controlled plant reach and follow a predefined path without temporal constraint. ⁵ So far numerous research results have been presented focus on path following of wheeled mobile robots, ^{6 –9} aerial vehicles, ^10,11 underwater snake robots, ¹² autonomous underwater vehicles (AUVs), ^13,14 and MSVs. ^{15 –20} The literature on path following control of land and aerial vehicles is much richer; however, the obtained results cannot directly work for the marine robots. The main difficulty is due to the fact that most marine vehicles are underactuated with nonintegrable dynamics which is not transformable into a driftless system. For the marine robots, the main disturbance for underwater vehicles is ocean current while MSVs can suffer from all the wind, waves, and ocean currents that are more fascinating for researchers. In the presented path-following control results for MSVs, ^{15 –20} backstepping technique has been widely addressed because it provides a systematic framework for the design of tracking and regulation strategies. It should be noted that these controllers were under the assumption that the hydrodynamic parameters were precisely known, ^{16 –20} or the parameters are unknown but linear with respect to some known nonlinear functions. ^15,17 However, it is difficult to obtain the mathematical model of the vessel accurately. Meanwhile, marine control applications are characterized by time-varying external disturbances and complex sea conditions. Therefore, considering that the vessel model is precisely known or the system uncertainties are in linearly parameterized forms is restrictive in practice. For system with high uncertainties, fuzzy logic system (FLS) and neural network (NN) techniques with the universal approximation ability have been used for backstepping-based tracking control of mobile robots, ^21,22 AUVs, ^23,24 and MSVs. ^25,26 By employing FLS, backstepping and sliding mode control techniques, an adaptive robust path-following control scheme was proposed for an underactuated AUVs with parameter uncertainties and current disturbances. ²³ FLS control in combination with a nonlinear guidance law was proposed by Yu et al. ²⁴ for three-dimensional (3-D) path following of AUVs subject to uncertainties. A robust adaptive radial basis function NN (RBF NN) control approach was presented by Zheng and Sun ²⁵ for the path following of an unmanned MSV with system uncertainties and input saturation. In the work of Zhang et al., ²⁶ the robust NN control method was extend to the waypoints-based path-following control of underactuated surface ships with obstacles avoidance.

With conventional backstepping design, there exists the problem of explosion of complexity. It is caused by the repeated differentiations of the virtual controls. Furthermore, when the RBF NN technique is introduced, one needs to take derivatives of these RBFs, which will further increase the computational burden of the control system. This drawback can be overcome using the dynamic surface control (DSC) method, in which the derivative of the virtual control is obtained through a first-order filter. ²⁷ By employing the NN-based DSC (NDSC) method, a class of pure-feedback nonlinear system with unknown dead zone and perturbed uncertainties was studied. ²⁸ A fuzzy DSC (FDSC) controller was proposed by Chen et al. ²⁹ for nonlinearly parameterized systems using a Fourier series expansion to model unknown periodic disturbances. The control problem of maintaining prescribed performance for multiple input and multiple output systems was addressed by Sun et al. ³⁰ using a modified NDSC approach. For surface vessels and other vehicles, DSC technique has been applied to several control scenarios. ^{31 –34} By introducing the DSC method in each of the kinematic and dynamic linearization law design, a global tracking control structure for underactuated ships with input and velocity constraints was presented by Chwa. ³¹ An adaptive leader–follower formation controller using NDSC technique was presented by Peng et al. ³² for autonomous surface vehicles with uncertain dynamics. Based on a robust adaptive NDSC, the path-following control problem of underactuated ships was addressed by Zhang and Zhang. ³³ Later, the DSC technique was extend to the containment maneuvering of marine surface vehicles moving along multiple parameterized paths. ³⁴

It is worth mentioning that, although significant development has been made in the FDSC/NDSC designs, ^{28 –34} the original intention using FLS/NN to approximate the system uncertainty is missing. In order to obtain a better control performance, the nonlinear functions should be approximated by the FLS/NN more precisely. However, the abovementioned literatures mainly focus on the asymptotic stability and tracking, and little attention has been paid to the precision improvement of identifying models. On the other hand, a common weakness of these approximation-based control approaches ^{23 –34} is that the scale of the adaptation laws depends on the number of the FLS rule bases or the NN nodes. To guarantee a better approximation result, the number of the parameters to be tuned online can be very large, which will make the learning time become unacceptably long for the implement in practice. Therefore, a more concise approximation-based control structure should be developed to effectively reduce the number of the learning parameters.

Motivated by the abovementioned observations, this article addresses both the performance improvement and the realizability for path-following control of unmanned MSVs in the presence of system uncertainties and unknown time-varying disturbances. To simultaneously solve the two problems, a new kind of composite NDSC (CNDSC) path-following control scheme is proposed. The main contributions of this article are summarized as follows: (1) By employing the DSC technique, the problem of explosion of complexity is eliminated effectively. (2) Different from the existing NDSC methods in which only the tracking errors defined between the system states and their desired values are commonly used for NN weight updating, the proposed CNDSC uses both the tracking errors and the prediction errors to update the NN parameters. The prediction errors are derived from the differences between the states and the state estimation models, which can provide extra information for NN parameter learning. Thus, better tracking performance is achieved. (3) Although the RBFs are introduced in the NN control structure, only one adaptive parameter is needed to be updated online. This makes the path-following controller easier to be implemented from the practical application viewpoint. (4) Closed-loop system stability is established using the Lyapunov method, which shows that all the signals are uniformly ultimate boundedness (UUB).

The rest of this article is organized as follows: “Problem formulation” section presents the model of the unmanned MSV, briefly introduces the RBF NNs, and states the control objective. In the “The CNDSC scheme” section, a robust CNDSC path-following scheme is proposed, and the stability analysis is given. “Simulation results and discussion” section provides the comparative simulations to verify the proposed control approach. “Conclusion” section concludes this article.

Problem formulation

The unmanned MSV model

To address the motions of the MSV on the horizontal plane, the states of the vessel are given by the vector ${\left[{\mathit{\eta }}^T,{\mathit{v}}^T\right]}^T$ , where $\mathit{\eta }\triangleq {\left[x,y,\psi \right]}^T$ represents the position and yaw angle of the vessel in the earth-fixed inertial frame {n}, and $\mathit{v}\triangleq {\left[u,v,r\right]}^T$ describes the surge, sway, and yaw rate velocities in the body-fixed frame {b}. Then, the three degree-of-freedom maneuvering model of the vessel can be described as ^4,35

$\dot{\mathit{\eta }}=\mathit{R}(\psi )\mathit{v}$

1

$\mathit{M}\dot{\mathit{v}}+\mathit{C}(\mathit{v})\mathit{v}+\mathit{D}\mathit{v}=\mathit{B}\mathit{f}+\mathit{d}$

2

where R (ψ) is the rotation matrix from {b} to {n} and is defined as

$\mathit{R}(\psi )\triangleq \left[\begin{array}{ccc}\text{cos}(\psi )& -\text{sin}(\psi )& 0\\ \text{sin}(\psi )& \text{cos}(\psi )& 0\\ 0& 0& 1\end{array}\right]$

3

The matrix $\mathit{M}\in {\Re }^{3\times 3}$ denotes the rigid-body mass and inertia matrix, $\mathit{C}(\mathit{v})\in {\Re }^{3\times 3}$ represents the Coriolis and centripetal matrix, $\mathit{D}\in {\Re }^{3\times 3}$ is the hydrodynamic damping matrix, and $\mathit{B}\in {\Re }^{3\times 2}$ is the actuator configuration matrix. The vector $\mathit{f}\triangleq {\left[f_u,f_r\right]}^T$ is the control input vector with the thruster force f_u and the rudder angle f_r. The vector $\mathit{d}\triangleq {\left[d_u,d_{\mathit{v}},d_r\right]}^T$ describes the environmental disturbances induced by wind, waves, and ocean currents. For maneuvering control purposes, the matrices M , C ( v ), D , and B are considered to be the following structures

$\mathit{M} \triangleq \left[\begin{array}{ccc}{m}_{11}& 0& 0\\ 0& m_{22}& m_{23}\\ 0& m_{32}& m_{33}\end{array}\right]$

4

$\mathit{C}(\mathit{v}) \triangleq \left[\begin{array}{ccc}0& 0& -m_{22}v-m_{23}r\\ 0& 0& m_{11}u\\ m_{22}v+m_{23}r& -m_{11}u& 0\end{array}\right]$

5

$\mathit{D} \triangleq \left[\begin{array}{ccc}{d}_{11}& 0& 0\\ 0& d_{22}& d_{23}\\ 0& d_{32}& d_{33}\end{array}\right]$

6

$\mathit{B} \triangleq \left[\begin{array}{cc}{b}_{11}& 0\\ 0& b_{22}\\ 0& b_{32}\end{array}\right]$

7

Notice that the unmanned MSV studied in this article is equipped with one thruster and one rudder; therefore, the vessel is underactuated because it lacks a direct lateral control input. Furthermore, for all port-starboard symmetric vessels, the location of the body-fixed frame {b} can be chosen such that ${\mathit{M}}^{-1}\mathit{B}\mathit{f}={\left[{\tau }_u\mathrm{,0},{\tau }_r\right]}^T$ , where τ_u and τ_r are the control force and moment in the surge and yaw directions, respectively. Then, we define the transformed external disturbance vector as ${\mathit{M}}^{-1}\mathit{d}={\left[d_{wu},d_{w\mathit{v}},d_{wr}\right]}^T$ , thus the component form of the vessel model can be written as

$\{\begin{array}{l}\dot{x}=u\text{cos}\left(\psi \right)-\mathit{v}\text{sin}\left(\psi \right)\hfill \\ \dot{y}=u\text{sin}\left(\psi \right)+\mathit{v}\text{cos}\left(\psi \right)\hfill \\ \dot{\psi }=r\hfill \\ \dot{u}=F_u\left(\mathit{v},r\right)-\frac{d_{11}}{m_{11}}u+{\tau }_u+d_{wu}\hfill \\ \dot{\mathit{v}}=X\left(u\right)r+Y\left(u\right)\mathit{v}+d_{w\mathit{v}}\hfill \\ \dot{r}=F_r\left(u,\mathit{v},r\right)+{\tau }_r+d_{wr}\hfill \end{array}$

8

with

$F_u\left(v,r\right)\triangleq \frac{1}{m_{11}}\left(m_{22}v+m_{23}r\right)r$

9

$X(u)\triangleq \frac{m_{23}^2-m_{11}{m}_{33}}{m_{22}{m}_{33}-m_{23}^2}u+\frac{d_{33}{m}_{23}-d_{23}{m}_{33}}{m_{22}{m}_{33}-m_{23}^2}$

10

$Y(u)\triangleq \frac{(m_{22}-m_{11})m_{23}}{m_{22}{m}_{33}-m_{23}^2}u-\frac{d_{22}{m}_{33}-d_{32}{m}_{23}}{m_{22}{m}_{33}-m_{23}^2}$

11

$\begin{array}{c}{F}_r\left(u,v,r\right)\triangleq \frac{m_{23}{d}_{22}-m_{22}\left(d_{32}+\left(m_{22}-m_{11}\right)u\right)}{m_{22}{m}_{33}-m_{23}^2}v\\ +\frac{m_{23}\left(d_{23}+m_{11}u\right)-m_{22}\left(d_{33}+m_{23}u\right)}{m_{22}{m}_{33}-m_{23}^2}r\end{array}$

12

where m₁₁, m₂₂, m₂₃, m₃₂, m₃₃, d₁₁, d₂₂, d₂₃, d₃₂, and d₃₃ are the vessel inertia and hydrodynamic damping parameters, respectively, which are all assumed to be completely unknown; $F_u\left(v,r\right),X\left(u\right),Y\left(u\right)$ , and $F_r\left(u,v,r\right)$ are considered to be not linearly parameterized nonlinear functions; d_wu, d_wv, and d_wr are the unknown and time-varying external disturbance forces and moment that acting on the surge, sway, and yaw directions.

It is worth noting that the expression of the vessel model (8)–(12) is different from that in the study by Zeng et al. ¹³ and Moe et al. ³⁵ In the previous works, ^35,13 only the ocean current disturbance is considered and assumed to be constant and irrotational, thus the kinematic and dynamic equations can be represented by the relative velocities. However, this assumption is too idealized for the real marine environment to satisfy because the sea condition is complex and changeable. In this article, we consider a more strict and practical environmental condition that containing wind, waves, and ocean currents; therefore, absolute velocities are used for vessel modeling.

To facilitate the control system design, the following assumptions are introduced and will be used in the subsequent developments.

Assumption 1

The external disturbances are bounded by unknown positive constants $d_{wu}^{*},d_{w\mathit{v}}^{*}$ , and $d_{wr}^{*}$ , that is, $|d_{wu}|\le d_{wu}^{*},|d_{w\mathit{v}}|\le d_{w\mathit{v}}^{*}$ , and $|d_{wr}|\le d_{wr}^{*}$ .

Remark 1

This article mainly considers the natural environment disturbances induced by wind, waves, and ocean currents, while the disturbance caused by other objects such as a sudden passing by unknown objects ahead of the vessel is not considered. In practical marine environment, the external disturbances $d_{wu},d_{w\mathit{v}}$ , and d_wr can be largely attributed to the exogenous effects and have finite energy. Therefore, they cannot be infinitely large and are bounded.

Assumption 2

The function Y(u) is negative definite, thus there exists positive constant Y_min such that $Y(u)\le -Y_{\text{min}}<0,\forall u\in \left[0,+\infty \right]$ .

Remark 2

Assumption 2 is certain for commercial vessels by design, since Y(u) ≥ 0 would imply that the sway direction of the vessel is undamped or nominally unstable.

Assumption 3

The sway velocity v of the vessel is passive-bounded. ¹⁷

Remark 3

This assumption is always satisfied for common surface vessels, and the passive-boundedness of the sway motion has been proofed and discussed systematically according to different types of vessels’ sway dynamics in previous literature. ¹⁷

The NNs

In this research, since the nonlinear functions in equation (8) are considered to be completely unknown and cannot be linearly parameterized, RBF NNs will be employed to deal with the system uncertainties.

As a powerful tool for function approximation, RBF NN can approximate any unknown and continuous function f( Z ) over a compact set ${\mathit{\Omega }}_{\mathit{Z}}\in {\Re }^n$ as follows ³⁶

$f(\mathit{Z})={\mathit{W}}^{*T}\mathit{H}(\mathit{Z})+\epsilon ,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\forall \mathit{Z}\in {\mathit{\Omega }}_{\mathit{Z}}$

13

where $\mathit{Z}={\left[z_1,\hspace{0.17em}{z}_2,\hspace{0.17em}\hspace{0.17em}\dots \hspace{0.17em},\hspace{0.17em}{z}_n\right]}^T\in {\Re }^n$ is the input vector, ε is the approximation error that satisfies $|\epsilon |\le \overline{\epsilon }$ with $\overline{\epsilon }$ being a bounded unknown parameter, and W * is the ideal weight vector and is defined as

${\mathit{W}}^{*}:=\text{arg}\hspace{0.17em}\underset{\mathit{W}\in {\Re }^l}{\text{min}}\left\{\underset{\mathit{Z}\in {\mathit{\Omega }}_{\mathit{Z}}}{\text{sup}}\hspace{0.17em}|f(\mathit{Z})-{\mathit{W}}^T\mathit{H}(\mathit{Z})|\right\}$

14

with the weight vector $\mathit{W}={\left[w_1,\hspace{0.17em}{w}_2,\hspace{0.17em}\dots ,\hspace{0.17em}{w}_l\right]}^T\in {\Re }^l$ and the basis function vector $\mathit{H}(\mathit{Z})=[h_1(\mathit{Z}),h_2(\mathit{Z}),$ ${\dots ,\hspace{0.17em}{h}_l(\mathit{Z})]}^T\in {\Re }^l$ . In this article, the radial basis function is chosen to be the following Gaussian function form

$\begin{array}{c}{h}_j(\mathit{Z})=\frac{1}{\sqrt{2\pi }{b}_j}\text{exp}\left(-\frac{{\left(\mathit{Z}-c_j\right)}^T\left(\mathit{Z}-c_j\right)}{2b_j^2}\right)\\ j=1,\hspace{0.17em}2,\hspace{0.17em}\dots \hspace{0.17em}l\end{array}$

15

where $c_j=[c_{j1},\hspace{0.17em}{c}_{j2},\hspace{0.17em}\dots ,\hspace{0.17em}{c}_{jn}{]}^T\in {\Re }^n$ is the center of the receptive field, and b_j is the width of the Gaussian function.

The control objective

In this article, the desired path is assumed to be generated by a virtual vessel without considering the obstacle avoidance problem, which can be formulated as

$\{\begin{array}{l}{\dot{x}}_d=u_d\text{cos}\left({\psi }_d\right)-v_d\text{sin}\left({\psi }_d\right)\hfill \\ {\dot{y}}_d=u_d\text{sin}\left({\psi }_d\right)\hspace{0.17em}+\hspace{0.17em}{v}_d\text{cos}\left({\psi }_d\right)\hfill \\ {\dot{\psi }}_d=r_d\hfill \\ {\dot{v}}_d=X\left(u_d\right)r_d\hspace{0.17em}+\hspace{0.17em}Y\left(u_d\right)v_d\hfill \end{array}$

16

where all the states have the same meanings as system (8)–(12), and the variables $x_d,\hspace{0.17em}{\dot{x}}_d,\hspace{0.17em}{\ddot{x}}_d,\hspace{0.17em}{y}_d,\hspace{0.17em}{\dot{y}}_d,\hspace{0.17em}{\ddot{y}}_d,\hspace{0.17em}{\dot{\psi }}_d$ , and ${\ddot{\psi }}_d$ exist and are bounded.

Considering that an underactuated MSV in the presence of system uncertainties and unknown external disturbances is represented by equations (8) –(12), and the reference path is generated by equation (16), the path-following control objective is to develop a robust CNDSC scheme for the surge force τ_u and yaw torque τ_r such that the MSV can follow closely a virtual vessel that moves on the predefined path, and all the signals of the closed-loop system are guaranteed to be UUB.

The CNDSC scheme

In this section, a robust CNDSC design for the path following of unmanned MSVs is proposed. For the control system, virtual controls for u and r are firstly designed on the kinematic level and are treated as the reference signals to the dynamic subsystem. Then, dynamic controllers for τ_u and τ_r are developed using DSC and CNN techniques. In the end, the stability of the closed-loop system is analyzed.

Control system design

Step 1. Define the position and heading track error variables between the virtual vessel and the real vessel as follows

$\begin{array}{c}{x}_e=x_d-x,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{y}_e=y_d-y\\ z_e=\sqrt{x_e^2+y_e^2},\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\psi }_e={\psi }_r-\psi \end{array}$

17

where x_e and y_e represent the components of the position track error along O_n − x_n and O_n − y_n axes in the inertial frame {n}, z_e denotes the resultant position track error, that is, the distance from the target point to the real vessel, and ψ_e is the heading track error with ${\psi }_r\in \left(-\pi ,\hspace{0.17em}\pi \right]$ being the vessel’s azimuth angle relative to the virtual vessel. The graphical description for the path following of MSVs is depicted in Figure 1. From it, one can easily get the calculation of ψ_r as

${\psi }_r=\frac{1}{2}(1-\text{sgn}\left(x_e\right))\text{sgn}\left(y_e\right)\pi +\text{arctan}\left(\frac{y_e}{x_e}\right)$

18

and it follows that

$x_e=z_e\text{cos}\left({\psi }_r\right),\hspace{0.17em}\hspace{0.17em}{y}_e=z_e\text{sin}\left({\psi }_r\right)$

19

Figure 1.

Principle of the path following for MSVs. MSV: marine surface vessel.

Considering equations (8), (17), and (19), and differentiating z_e and ψ_e with respect to time, it gives

$\begin{array}{c}{\dot{z}}_e={\dot{x}}_d\text{cos}\left({\psi }_r\right)\hspace{0.17em}+\hspace{0.17em}{\dot{y}}_d\text{sin}\left({\psi }_r\right)-u\text{cos}\left({\psi }_e\right)\hspace{0.17em}-\hspace{0.17em}\mathit{v}\text{sin}\left({\psi }_e\right)\\ {\dot{\psi }}_e={\dot{\psi }}_r-r\end{array}$

20

In order to eliminate the position and heading track errors, u and r are considered as immediate control signals, and the corresponding virtual controls for u and r are designed as

$\begin{array}{c}{\alpha }_u=(k_1\left(z_e-{\delta }_e\right)+{\dot{x}}_d\text{cos}\left({\psi }_r\right)+{\dot{y}}_d\text{sin}\left({\psi }_r\right)\hspace{0.17em}-\hspace{0.17em}\mathit{v}\text{sin}\left({\psi }_e\right))\\ \times \text{sec}\left({\psi }_e\right)\\ {\alpha }_r=k_2{\psi }_e+{\dot{\psi }}_r\end{array}$

21

where k₁, k₂, and δ_e are positive design parameters.

Remark 4

It can be seen that the virtual control α_u in equation (21) is not defined at ${\psi }_e=\pm \frac{1}{2}\pi$ . However, by introducing the positive constant δ_e, the convergence of $z_e-{\delta }_e$ will make the real vessel always follow behind the virtual vessel, and the condition $|{\psi }_e|<\frac{1}{2}\pi ,\hspace{0.17em}\forall 0\le t\le \infty$ is guaranteed to be satisfied. Therefore, the singularity will not occur.

To avoid repeatedly differentiating α_u and α_r, which leads to the so-called explosion of complexity in the following step, the DSC technique is employed here. Let α_u and α_r pass through two first-order filters β_u and β_r, respectively, it follows that

$T_i{\dot{\beta }}_i+{\beta }_i={\alpha }_i,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\beta }_i\left(0\right)={\alpha }_i\left(0\right),\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}i=u,\hspace{0.17em}r$

22

where T_i is the time constant. Define $y_i={\beta }_i-{\alpha }_i$ and $i_e=i-{\beta }_i$ , then we have ${\dot{\beta }}_i=-y_i/T_i$ and

$\begin{array}{c}{\dot{y}}_u={\dot{\beta }}_u-{\dot{\alpha }}_u=-\frac{y_u}{T_u}+(-\frac{\partial {\alpha }_u}{\partial {\dot{x}}_d}{\ddot{x}}_d-\frac{\partial {\alpha }_u}{\partial {\dot{y}}_d}{\ddot{y}}_d-\frac{\partial {\alpha }_u}{\partial z_e}{\dot{z}}_e\\ -\frac{\partial {\alpha }_u}{\partial {\psi }_r}{\dot{\psi }}_r-\frac{\partial {\alpha }_u}{\partial {\psi }_e}{\ddot{\psi }}_e-\frac{\partial {\alpha }_u}{\partial v}\dot{v})\\ =-\frac{y_u}{T_u}+B_u({\dot{x}}_d,\hspace{0.17em}{\ddot{x}}_d,\hspace{0.17em}{\dot{y}}_d,\hspace{0.17em}{\ddot{y}}_d,\hspace{0.17em}{z}_e,\hspace{0.17em}{\dot{z}}_e,\hspace{0.17em}{\psi }_r,{\dot{\psi }}_r,\hspace{0.17em}{\dot{\psi }}_e,\hspace{0.17em}{\ddot{\psi }}_e,\hspace{0.17em}v,\hspace{0.17em}\dot{v})\end{array}$

23

$\begin{array}{c}{\dot{y}}_r={\dot{\beta }}_r-{\dot{\alpha }}_r=-\frac{y_r}{T_r}+\left(-\frac{\partial {\alpha }_r}{\partial {\psi }_e}{\dot{\psi }}_e-\frac{\partial {\alpha }_r}{\partial {\dot{\psi }}_r}{\ddot{\psi }}_r\right)=-\frac{y_r}{T_r}+B_r({\psi }_e,\hspace{0.17em}{\dot{\psi }}_e,\hspace{0.17em}{\dot{\psi }}_r,\hspace{0.17em}{\ddot{\psi }}_r)\end{array}$

24

with B_i being the continuous function and has a maximum value M_i. ³⁷ Then, substituting equation (21) into equation (20), it gives

$\begin{array}{c}{\dot{z}}_e={\dot{x}}_d\text{cos}\left({\psi }_r\right)+{\dot{y}}_d\text{sin}\left({\psi }_r\right)-\left(u_e+{\alpha }_u+y_u\right)\text{cos}\left({\psi }_e\right)-\mathit{v}\text{sin}\left({\psi }_e\right)\\ =-\left(u_e+y_u\right)\text{cos}\left({\psi }_e\right)-k_1\left(z_e-{\delta }_e\right)\\ {\dot{\psi }}_e={\dot{\psi }}_r-\left(r_e+{\alpha }_r+y_r\right)\\ =-r_e-y_r-k_2{\psi }_e\end{array}$

25

Considering the position track error, the heading track error, and the filtering errors, choose the Lyapunov function candidate as

$V_1=\frac{1}{2}\left({\left(z_e-{\delta }_e\right)}^2+{\psi }_e^2+y_u^2+y_r^2\right)$

26

Then, considering equations (23) to (25), the time derivative of V₁ is obtained as

$\begin{array}{c}{\dot{V}}_1\hspace{0.17em}=-k_1{\left(z_e-{\delta }_e\right)}^2-\left(z_e-{\delta }_e\right)\left(u_e+y_u\right)\text{cos}\left({\psi }_e\right)\\ -k_2{\psi }_e^2-\left(r_e+y_r\right){\psi }_e+B_u{y}_u-\frac{y_u^2}{T_u}+B_r{y}_r-\frac{y_r^2}{T_r}\\ \le -k_1{\left(z_e-{\delta }_e\right)}^2+\frac{1}{2}\left({\left(z_e-{\delta }_e\right)}^2+u_e^2\right)\\ +\frac{1}{2}\left({\left(z_e-{\delta }_e\right)}^2+y_u^2\right)-k_2{\psi }_e^2+{\psi }_e^2\hspace{0.17em}+\hspace{0.17em}\frac{1}{2}\left(r_e^2+y_r^2\right)\\ +\hspace{0.17em}\left(\frac{1}{2}{M}_u^2{y}_u^2\hspace{0.17em}+\hspace{0.17em}\frac{1}{2}\right)-\frac{y_u^2}{T_u}+\left(\frac{1}{2}{M}_r^2{y}_r^2\hspace{0.17em}+\hspace{0.17em}\frac{1}{2}\right)-\frac{y_r^2}{T_r}\\ =-\left(k_1-1\right){\left(z_e-{\delta }_e\right)}^2-\left(k_2-1\right){\psi }_e^2\\ +\sum _{i\hspace{0.17em}=\hspace{0.17em}u,\hspace{0.17em}r}\left(-{\rho }_i{y}_i^2+\frac{1}{2}{i}_e^2\right)\end{array}$

27

where ${\rho }_i=\frac{1}{T_i}-\frac{1}{2}\left(M_i^2+1\right)$ .

Step 2. Differentiating u_e and r_e with respect to time, it follows that

$\begin{array}{c}{\dot{u}}_e=\dot{u}-{\dot{\beta }}_u=F_u\left(v,\hspace{0.17em}r\right)-\frac{d_{11}}{m_{11}}u+{\tau }_u+d_{wu}-{\dot{\beta }}_u\\ {\dot{r}}_e=\dot{r}-{\dot{\beta }}_r=F_r\left(u,\hspace{0.17em}v,\hspace{0.17em}r\right)+{\tau }_r+d_{wr}-{\dot{\beta }}_r\end{array}$

28

Defining two functions ${\varphi }_u\left({\mathit{Z}}_u\right)\hspace{0.17em}=\hspace{0.17em}{F}_u\left(v,\hspace{0.17em}r\right)\hspace{0.17em}-\hspace{0.17em}\frac{d_{11}}{m_{11}}u$ and ${\varphi }_r\left({\mathit{Z}}_r\right)\hspace{0.17em}=\hspace{0.17em}{F}_r\left(u,\hspace{0.17em}v,\hspace{0.17em}r\right)$ , with ${\mathit{Z}}_u={\mathit{Z}}_r={\left[u,\hspace{0.17em}v,\hspace{0.17em}r\right]}^T$ . Since the parameters of the MSV are considered to be unknown, ${\varphi }_u\left({\mathit{Z}}_u\right)$ and ${\varphi }_r\left({\mathit{Z}}_r\right)$ are unknown smooth functions. Using RBF NN to approximate ${\varphi }_i\left({\mathit{Z}}_i\right),\hspace{0.17em}i=u,\hspace{0.17em}r$ , the error dynamical system (28) can be rewritten in the following regular form

$\begin{array}{c}{\dot{u}}_e={\mathit{W}}_u^{*T}{\mathit{H}}_u\left({\mathit{Z}}_u\right)+{\epsilon }_u+{\tau }_u+d_{wu}-{\dot{\beta }}_u\\ {\dot{r}}_e={\mathit{W}}_r^{*T}{\mathit{H}}_r\left({\mathit{Z}}_r\right)+{\epsilon }_r+{\tau }_r+d_{wr}-{\dot{\beta }}_r\end{array}$

29

where ${\mathit{W}}_i^{*}$ is the ideal weight vector and ε_i is the NN approximation error. Then, the desired feedback control law ${\tau }_i^{*}$ can be designed as

$\begin{array}{c}{\tau }_u^{*}=-\left(k_u+\frac{1}{2}\right)u_e-{\mathit{W}}_u^{*T}{\mathit{H}}_u\left({\mathit{Z}}_u\right)-{\epsilon }_u+{\dot{\beta }}_u\\ {\tau }_r^{*}=-\left(k_r+\frac{1}{2}\right)r_e-{\mathit{W}}_r^{*T}{\mathit{H}}_r\left({\mathit{Z}}_r\right)-{\epsilon }_r+{\dot{\beta }}_r\end{array}$

30

Since ${\mathit{W}}_i^{*}$ and ε_i are unknown, the dynamic controllers for surge force τ_u and yaw moment τ_r are proposed as

$\begin{array}{l}{\tau }_u=-\left(k_u+\frac{1}{2}\right)u_e-\frac{1}{2}{u}_e{\widehat{\vartheta }}_u{\mathit{H}}_u^T\left({\mathit{Z}}_u\right){\mathit{H}}_u\left({\mathit{Z}}_u\right)+{\dot{\beta }}_u,\\ {\tau }_r=-\left(k_r+\frac{1}{2}\right)r_e-\frac{1}{2}{r}_e{\widehat{\vartheta }}_r{\mathit{H}}_r^T\left({\mathit{Z}}_r\right){\mathit{H}}_r\left({\mathit{Z}}_r\right)+{\dot{\beta }}_r\end{array}$

31

where k_i > 0 is a design parameter. ${\widehat{\vartheta }}_i$ is the estimation of the unknown positive constant ${\vartheta }_i^{*}={\Vert {\mathit{W}}_i^{*}\Vert }^2$ with $\Vert {\mathit{W}}_i^{*}\Vert$ being the norm of the ideal weight vector of the ith NN. Then, substituting the dynamic control law (equation (31)) into equation (29), it gives

$\begin{array}{c}{\dot{u}}_e=-\left(k_u+\frac{1}{2}\right)u_e-\frac{1}{2}{u}_e{\widehat{\vartheta }}_u{\mathit{H}}_u^T\left({\mathit{Z}}_u\right){\mathit{H}}_u\left({\mathit{Z}}_u\right)\\ +{\mathit{W}}_u^{*T}{\mathit{H}}_u\left({\mathit{Z}}_u\right)+{\epsilon }_u+d_{wu}\\ {\dot{r}}_e=-\left(k_r+\frac{1}{2}\right)r_e-\frac{1}{2}{r}_e{\widehat{\vartheta }}_r{\mathit{H}}_r^T\left({\mathit{Z}}_r\right){\mathit{H}}_r\left({\mathit{Z}}_r\right)\\ +{\mathit{W}}_r^{*T}{\mathit{H}}_r\left({\mathit{Z}}_r\right)+{\epsilon }_r+d_{wr}\end{array}$

32

Define $z_{inn}=i-\widehat{i}$ as the prediction error of the serial–parallel estimation model, and design the update law of the NN modeling information as follows

$\begin{array}{c}\dot{\widehat{u}}=\frac{1}{2}{z}_{unn}{\widehat{\vartheta }}_u{\mathit{H}}_u^T\left({\mathit{Z}}_u\right){\mathit{H}}_u\left({\mathit{Z}}_u\right)+{\tau }_u+{\lambda }_u{z}_{unn}\\ \dot{\widehat{r}}=\frac{1}{2}{z}_{rnn}{\widehat{\vartheta }}_r{\mathit{H}}_r^T\left({\mathit{Z}}_r\right){\mathit{H}}_r\left({\mathit{Z}}_r\right)+{\tau }_r+{\lambda }_r{z}_{rnn}\end{array}$

33

where λ_i > 0 is a design parameter. The prediction error signal z_inn employed here is aimed to construct the update law design of the CNN parameter, and the learning algorithm for ${\widehat{\vartheta }}_i$ is proposed as

$\begin{array}{c}{\dot{\widehat{\vartheta }}}_u={\eta }_u\left(\frac{1}{2}\left(u_e^2+{\mu }_u{z}_{unn}^2\right){\mathit{H}}_u^T\left({\mathit{Z}}_u\right){\mathit{H}}_u\left({\mathit{Z}}_u\right)-{\kappa }_u{\widehat{\vartheta }}_u\right)\\ {\dot{\widehat{\vartheta }}}_r={\eta }_r\left(\frac{1}{2}\left(r_e^2+{\mu }_r{z}_{rnn}^2\right){\mathit{H}}_r^T\left({\mathit{Z}}_r\right){\mathit{H}}_r\left({\mathit{Z}}_r\right)-{\kappa }_r{\widehat{\vartheta }}_r\right)\end{array}$

34

where ${\eta }_i>0,\hspace{0.17em}{\mu }_i>0$ , and κ_i > 0 are design parameters. The visualization of the proposed CNDSC path-following control system is given in Figure 2.

Figure 2.

Visualization of the path-following control system.

Define the estimation error of the NN parameter to be ${\tilde{\vartheta }}_i={\widehat{\vartheta }}_i-{\vartheta }_i^{*}$ . Then, consider the Lyapunov function candidate as follows

$V_2=\frac{1}{2}\sum _{i=u,\hspace{0.17em}r}\left(i_e^2+\frac{1}{{\eta }_i}{\tilde{\vartheta }}_i^2+{\mu }_i{z}_{inn}^2\right)$

35

Invoking equations (32) to (34), it follows that

$\begin{array}{l}{i}_e{\dot{i}}_e=-\left(k_i+\frac{1}{2}\right)i_e^2-\frac{1}{2}{i}_e^2{\widehat{\vartheta }}_i{\mathit{H}}_i^T\left({\mathit{Z}}_i\right){\mathit{H}}_i\left({\mathit{Z}}_i\right)+\hspace{0.17em}{i}_e{\mathit{W}}_i^{*T}{\mathit{H}}_i\left({\mathit{Z}}_i\right)\hspace{0.17em}+\hspace{0.17em}{i}_e{\epsilon }_i+\hspace{0.17em}{i}_e{d}_{wi}\end{array}$

36

$\begin{array}{l}{\tilde{\vartheta }}_i{\dot{\tilde{\vartheta }}}_i={\tilde{\vartheta }}_i{\dot{\widehat{\vartheta }}}_i=\frac{1}{2}{\eta }_i{i}_e^2{\tilde{\vartheta }}_i{\mathit{H}}_i^T\left({\mathit{Z}}_i\right){\mathit{H}}_i\left({\mathit{Z}}_i\right)+\frac{1}{2}{\eta }_i{\mu }_i{z}_{inn}^2{\tilde{\vartheta }}_i{\mathit{H}}_i^T\left({\mathit{Z}}_i\right){\mathit{H}}_i\left({\mathit{Z}}_i\right)-{\eta }_i{\kappa }_i{\tilde{\vartheta }}_i{\widehat{\vartheta }}_i\end{array}$

37

$\begin{array}{l}{z}_{inn}{\dot{z}}_{inn}=z_{inn}\left(\dot{i}-\dot{\widehat{i}}\right)=-\frac{1}{2}{z}_{inn}^2{\widehat{\vartheta }}_i{\mathit{H}}_i^T\left({\mathit{Z}}_i\right){\mathit{H}}_i\left({\mathit{Z}}_i\right)+z_{inn}{\mathit{W}}_i^{*T}{\mathit{H}}_i\left({\mathit{Z}}_i\right)-{\lambda }_i{z}_{inn}^2+z_{inn}{\epsilon }_i+z_{inn}{d}_{wi}\end{array}$

38

Then, considering equations (36) to (38), the derivative of V₂ is obtained as

${\dot{V}}_2=\sum _{i=u,\hspace{0.17em}r}\left(i_e{\dot{i}}_e+\frac{1}{{\eta }_i}{\tilde{\vartheta }}_i{\dot{\tilde{\vartheta }}}_i+{\mu }_i{z}_{inn}{\dot{z}}_{inn}\right)=\sum _{i=u,\hspace{0.17em}r}(-\left(k_i+\frac{1}{2}\right)i_e^2-\frac{1}{2}{i}_e^2{\widehat{\vartheta }}_i{\mathit{H}}_i^T\left({\mathit{Z}}_i\right){\mathit{H}}_i\left({\mathit{Z}}_i\right)+i_e{\mathit{W}}_i^{*T}{\mathit{H}}_i\left({\mathit{Z}}_i\right)+\frac{1}{2}{i}_e^2{\tilde{\vartheta }}_i{\mathit{H}}_i^T\left({\mathit{Z}}_i\right){\mathit{H}}_i\left({\mathit{Z}}_i\right)+\frac{1}{2}{\mu }_i{z}_{inn}^2{\tilde{\vartheta }}_i{\mathit{H}}_i^T\left({\mathit{Z}}_i\right){\mathit{H}}_i\left({\mathit{Z}}_i\right)+{\mu }_i{z}_{inn}{\mathit{W}}_i^{*T}{\mathit{H}}_i\left({\mathit{Z}}_i\right)-\frac{1}{2}{\mu }_i{z}_{inn}^2{\widehat{\vartheta }}_i{\mathit{H}}_i^T\left({\mathit{Z}}_i\right){\mathit{H}}_i\left({\mathit{Z}}_i\right)-{\mu }_i{\lambda }_i{z}_{inn}^2-{\kappa }_i{\tilde{\vartheta }}_i{\widehat{\vartheta }}_i+i_e{\epsilon }_i+i_e{d}_{wi}+{\mu }_i{z}_{inn}{\epsilon }_i+{\mu }_i{z}_{inn}{d}_{wi})$

39

According to the following facts

$\begin{array}{c}{i}_e{\mathit{W}}_i^{*T}{\mathit{H}}_i\left({\mathit{Z}}_i\right)\le \frac{1}{2}\left(i_e^2{\Vert {\mathit{W}}_i^{*T}{\mathit{H}}_i\left({\mathit{Z}}_i\right)\Vert }^2+1\right)=\frac{1}{2}\left(i_e^2{\vartheta }_i^{*}{\mathit{H}}_i^T\left({\mathit{Z}}_i\right){\mathit{H}}_i\left({\mathit{Z}}_i\right)+1\right)\\ z_{inn}{\mathit{W}}_i^{*T}{\mathit{H}}_i\left({\mathit{Z}}_i\right)\le \frac{1}{2}\left(z_{inn}^2{\Vert {\mathit{W}}_i^{*T}{\mathit{H}}_i\left({\mathit{Z}}_i\right)\Vert }^2+1\right)=\frac{1}{2}\left(z_{inn}^2{\vartheta }_i^{*}{\mathit{H}}_i^T\left({\mathit{Z}}_i\right){\mathit{H}}_i\left({\mathit{Z}}_i\right)+1\right)\end{array}$

40

one has

${\dot{V}}_2\le {\displaystyle \sum _{i=u\mathrm{,}\hspace{0.17em}r}}(-\left(k_i+\frac{1}{2}\right)i_e^2-\frac{1}{2}{i}_e^2{\widehat{\vartheta }}_i{\mathit{H}}_i^T\left({\mathit{Z}}_i\right){\mathit{H}}_i\left({\mathit{Z}}_i\right)+\frac{1}{2}{i}_e^2{\vartheta }_i^{*}{\mathit{H}}_i^T\left({\mathit{Z}}_i\right){\mathit{H}}_i\left({\mathit{Z}}_i\right)+\frac{1}{2}{i}_e^2{\tilde{\vartheta }}_i{\mathit{H}}_i^T\left({\mathit{Z}}_i\right){\mathit{H}}_i\left({\mathit{Z}}_i\right)+\frac{1}{2}{\mu }_i{z}_{inn}^2{\tilde{\vartheta }}_i{\mathit{H}}_i^T\left({\mathit{Z}}_i\right){\mathit{H}}_i\left({\mathit{Z}}_i\right)-{\kappa }_i{\tilde{\vartheta }}_i{\widehat{\vartheta }}_i+\frac{1}{2}{\mu }_i{z}_{inn}^2{\vartheta }_i^{*}{\mathit{H}}_i^T\left({\mathit{Z}}_i\right){\mathit{H}}_i\left({\mathit{Z}}_i\right)-{\mu }_i{\lambda }_i{z}_{inn}^2-\frac{1}{2}{\mu }_i{z}_{inn}^2{\widehat{\vartheta }}_i{\mathit{H}}_i^T\left({\mathit{Z}}_i\right){\mathit{H}}_i\left({\mathit{Z}}_i\right)+i_e{\epsilon }_i+i_e{d}_{wi}+{\mu }_i{z}_{inn}{\epsilon }_i+{\mu }_i{z}_{inn}{d}_{wi}+\frac{1}{2}{\mu }_i+\frac{1}{2})={\displaystyle \sum _{i=u\mathrm{,}\hspace{0.17em}r}}(-\left(k_i+\frac{1}{2}\right)i_e^2-\frac{1}{2}{i}_e^2{\tilde{\vartheta }}_i{\mathit{H}}_i^T\left({\mathit{Z}}_i\right){\mathit{H}}_i\left({\mathit{Z}}_i\right)+\frac{1}{2}{i}_e^2{\tilde{\vartheta }}_i{\mathit{H}}_i^T\left({\mathit{Z}}_i\right){\mathit{H}}_i\left({\mathit{Z}}_i\right)+\frac{1}{2}{\mu }_i{z}_{inn}^2{\tilde{\vartheta }}_i{\mathit{H}}_i^T\left({\mathit{Z}}_i\right){\mathit{H}}_i\left({\mathit{Z}}_i\right)-{\kappa }_i{\tilde{\vartheta }}_i{\widehat{\vartheta }}_i-\frac{1}{2}{\mu }_i{z}_{inn}^2{\tilde{\vartheta }}_i{\mathit{H}}_i^T\left({\mathit{Z}}_i\right){\mathit{H}}_i\left({\mathit{Z}}_i\right)-{\mu }_i{\lambda }_i{z}_{inn}^2+i_e{\epsilon }_i+i_e{d}_{wi}+{\mu }_i{z}_{inn}{\epsilon }_i+{\mu }_i{z}_{inn}{d}_{wi}+\frac{1}{2}{\mu }_i+\frac{1}{2})={\displaystyle \sum _{i=u\mathrm{,}\hspace{0.17em}r}}(-\left(k_i+\frac{1}{2}\right)i_e^2-{\kappa }_i{\tilde{\vartheta }}_i{\widehat{\vartheta }}_i-{\mu }_i{\lambda }_i{z}_{inn}^2+i_e{\epsilon }_i+i_e{d}_{wi}+{\mu }_i{z}_{inn}{\epsilon }_i+{\mu }_i{z}_{inn}{d}_{wi}+\frac{1}{2}{\mu }_i+\frac{1}{2})$

41

Remark 5

The adaptive law of the composite NN is constructed by the tracking error i_e and the prediction error z_inn. Since an extra information provided by the prediction error is used to identify the system uncertainties, the improved function approximation and fast adaption ability can be obtained.

Remark 6

In the proposed NN structure, the norm of the weight vector is chosen as the adaptive parameter instead of the elements of the weight vector, thus the number of the parameters need to learn online for each NN is reduced to one. Therefore, the calculation burden is lightened considerably.

Main result

So far, we have finished the CNDSC path-following controller design. The main result of the closed-loop system performance can be summarized in the following theorem.

Theorem 1

For an underactuated MSV given by equations (8) to (12) with system uncertainties and external disturbances satisfying Assumptions 1 to 3, the predefined path is generated by equation (16), the kinematic controller is developed as equation (21), the dynamic control law is proposed as equation (31), and the composite NN adaptive laws are designed as equations (33) and (34), then all the closed-loop system signals $z_e-{\delta }_e,\hspace{0.17em}{\psi }_e,\hspace{0.17em}{y}_i,\hspace{0.17em}{i}_e,\hspace{0.17em}{\tilde{\vartheta }}_i$ , and z_inn are UUB.

Proof

Consider the Lyapunov function candidate of the whole control system as

$V=V_1+V_2$

42

Differentiating V with respect to time and considering equations (27) and (41), it follows that

$\begin{array}{c}\dot{V}={\dot{V}}_1+{\dot{V}}_2\\ \le -\left(k_1-1\right){\left(z_e-{\delta }_e\right)}^2-\left(k_2-1\right){\psi }_e^2\\ +\sum _{i=u,\hspace{0.17em}r}(-{\rho }_i{y}_i^2-k_i{i}_e^2-{\kappa }_i{\tilde{\vartheta }}_i{\widehat{\vartheta }}_i-{\mu }_i{\lambda }_i{z}_{inn}^2\\ +i_e{\epsilon }_i+i_e{d}_{wi}+{\mu }_i{z}_{inn}{\epsilon }_i+{\mu }_i{z}_{inn}{d}_{wi}+\frac{1}{2}{\mu }_i+\frac{1}{2})\end{array}$

43

It is true that

$\begin{array}{c}-k_i{i}_e^2+i_e{\epsilon }_i=-k_i{\left(i_e-\frac{{\epsilon }_i}{2k_i}\right)}^2+\frac{1}{4k_i}{\epsilon }_i^2\\ -{\lambda }_i{z}_{inn}^2+z_{inn}{\epsilon }_i=-{\lambda }_i{\left(z_{inn}-\frac{{\epsilon }_i}{2{\lambda }_i}\right)}^2+\frac{1}{4{\lambda }_i}{\epsilon }_i^2\\ -{\tilde{\vartheta }}_i{\widehat{\vartheta }}_i=-{\tilde{\vartheta }}_i^2-{\tilde{\vartheta }}_i{\vartheta }_i^{*}=-{\left({\tilde{\vartheta }}_i+\frac{{\vartheta }_i^{*}}{2}\right)}^2+\frac{1}{4}{\vartheta }_i^{*2}\\ i_e{d}_{wi}\le \frac{{\xi }_i}{4}{i}_e^2+\frac{1}{{\xi }_i}{d}_{wi}^2\\ {\mu }_i{z}_{inn}{d}_{wi}\le \frac{{\varpi }_i}{4}{z}_{inn}^2+\frac{1}{{\varpi }_i}{\mu }_i^2{d}_{wi}^2\end{array}$

44

where the quantities ξ_i and ϖ_i are small positive constants. Then, we have

$\dot{V}\le -\left(k_1-1\right){\left(z_e-{\delta }_e\right)}^2-\left(k_2-1\right){\psi }_e^2+{\displaystyle \sum _{i=u\mathrm{,}\hspace{0.17em}r}}(-{\rho }_i{y}_i^2-k_i{\left(i_e-\frac{{\epsilon }_i}{2k_i}\right)}^2+\frac{1}{4k_i}{\epsilon }_i^2-{\mu }_i{\lambda }_i{\left(z_{inn}-\frac{{\epsilon }_i}{2{\lambda }_i}\right)}^2+\frac{{\mu }_i}{4{\lambda }_i}{\epsilon }_i^2-{\kappa }_i{\left({\tilde{\vartheta }}_i+\frac{{\vartheta }_i^{*}}{2}\right)}^2+\frac{{\kappa }_i}{4}{\vartheta }_i^{*2}+\frac{{\xi }_i}{4}{i}_e^2+\frac{1}{{\xi }_i}{d}_{wi}^2+\frac{{\varpi }_i}{4}{z}_{inn}^2+\frac{1}{{\varpi }_i}{\mu }_i^2{d}_{wi}^2+\frac{1}{2}{\mu }_i+\frac{1}{2})=-\left(k_1-1\right){\left(z_e-{\delta }_e\right)}^2-\left(k_2-1\right){\psi }_e^2-{\displaystyle \sum _{i=u\mathrm{,}\hspace{0.17em}r}}({\rho }_i{y}_i^2+k_i{\left(i_e-\frac{{\epsilon }_i}{2k_i}\right)}^2+{\mu }_i{\lambda }_i{\left(z_{inn}-\frac{{\epsilon }_i}{2{\lambda }_i}\right)}^2+{\kappa }_i{\left({\tilde{\vartheta }}_i+\frac{{\vartheta }_i^{*}}{2}\right)}^2)+{\displaystyle \sum _{i=u\mathrm{,}\hspace{0.17em}r}}\left(\frac{{\xi }_i}{4}{i}_e^2+\frac{{\varpi }_i}{4}{z}_{inn}^2\right)+{\displaystyle \sum _{i=u\mathrm{,}\hspace{0.17em}r}}(\frac{1}{4k_i}{\epsilon }_i^2+\frac{{\mu }_i}{4{\lambda }_i}{\epsilon }_i^2+\frac{{\kappa }_i}{4}{\vartheta }_i^{*2}+\frac{1}{{\xi }_i}{d}_{wi}^2+\frac{1}{{\varpi }_i}{\mu }_i^2{d}_{wi}^2+\frac{1}{2}{\mu }_i+\frac{1}{2})\le -\left(k_1-1\right){\left(z_e-{\delta }_e\right)}^2-\left(k_2-1\right){\psi }_e^2-{\displaystyle \sum _{i=u\mathrm{,}\hspace{0.17em}r}}\left({\rho }_{\text{min}}{y}_i^2+{\kappa }_{\text{min}}{\left({\tilde{\vartheta }}_i+\frac{{\vartheta }_i^{*}}{2}\right)}^2\right)-{\displaystyle \sum _{i=u\mathrm{,}\hspace{0.17em}r}}(\left(k_{\text{min}}-\frac{{\xi }_{\text{max}}}{4}\right){\left(i_e-\frac{2}{4k_i-{\xi }_i}{\epsilon }_i\right)}^2+\left({\mu }_{\text{min}}{\lambda }_{\text{min}}-\frac{{\varpi }_{\text{max}}}{4}\right){\left(z_{inn}-\frac{2}{4{\mu }_i{\lambda }_i-{\varpi }_i}{\epsilon }_i\right)}^2)+P$

45

with

$\begin{array}{c}P=(\frac{1}{2k_{\text{min}}}+\frac{{\mu }_{\text{max}}}{2{\lambda }_{\text{min}}}+\frac{2}{4k_{\text{min}}-{\xi }_{\text{max}}}\\ +\frac{2}{4{\mu }_{\text{min}}{\lambda }_{\text{min}}-{\varpi }_{\text{max}}}){\epsilon }_{\text{max}}^2+\frac{{\kappa }_{\text{max}}}{2}{\vartheta }_{\text{max}}^2+\left(\frac{2}{{\xi }_{\text{min}}}+\frac{2}{{\varpi }_{\text{min}}}{\mu }_{\text{max}}^2\right)d_{\text{max}}^{*2}+\frac{1}{2}{\mu }_{\text{max}}+\frac{1}{2}\end{array}$

46

where the positive constants ${\rho }_{\text{min}}=\text{min}\left[{\rho }_i\right]$ , ${\kappa }_{\text{min}}=\text{min}\left[{\kappa }_i\right]$ , $k_{\text{min}}=\text{min}\left[k_i\right]$ , ${\mu }_{\text{min}}=\text{min}\left[{\mu }_i\right]$ , ${\lambda }_{\text{min}}=\text{min}\left[{\lambda }_i\right]$ , ${\xi }_{\text{min}}=\text{min}\left[{\xi }_i\right]$ , ${\varpi }_{\text{min}}=\text{min}\left[{\varpi }_i\right]$ , $T_{\text{max}}=\text{max}\left[T_i\right]$ , $M_{\text{max}}=\text{max}\left[M_i\right]$ , ${\kappa }_{\text{max}}=\text{max}\left[{\kappa }_i\right]$ , ${\mu }_{\text{max}}=\text{max}\left[{\mu }_i\right]$ , ${\xi }_{\text{max}}=\text{max}\left[{\xi }_i\right]$ , ${\varpi }_{\text{max}}=\text{max}\left[{\varpi }_i\right]$ , ${\vartheta }_{\text{max}}=\text{max}\left[{\vartheta }_i^{*}\right]$ , $d_{\text{max}}^{*}=\text{max}\left[d_i^{*}\right]$ , and ${\epsilon }_{\text{max}}=\text{max}\left[{\epsilon }_i\right]$ .

Therefore, if the control parameters are appropriately chosen such that $k_1-1>0,\hspace{0.17em}{k}_2-1>0,\hspace{0.17em}{\rho }_{\text{min}}>0,\hspace{0.17em}{\kappa }_{\text{min}}>0,\hspace{0.17em}{k}_{\text{min}}-\frac{{\xi }_{\text{max}}}{4}>0$ , and ${\mu }_{\text{min}}{\lambda }_{\text{min}}-\frac{{\varpi }_{\text{max}}}{4}>0$ , and if $|z_e-{\delta }_e|\ge \sqrt{\frac{P}{k_1-1}}$ or $|{\psi }_e|\ge \sqrt{\frac{P}{k_2-1}}$ or $|y_i|\ge \sqrt{\frac{P}{{\rho }_{\text{min}}}}$ or $|{\tilde{\vartheta }}_i+\frac{{\vartheta }_i^{*}}{2}|\ge \sqrt{\frac{P}{{\kappa }_{\text{min}}}}$ or $|i_e-\frac{2}{4k_i-{\xi }_i}{\epsilon }_i|\ge \sqrt{\frac{P}{k_{\text{min}}-\frac{{\xi }_{\text{max}}}{4}}}$ or $|z_{inn}-\frac{2}{4{\mu }_i{\lambda }_i-{\varpi }_i}{\epsilon }_i|\ge \sqrt{\frac{P}{{\mu }_{\text{min}}{\lambda }_{\text{min}}-\frac{{\varpi }_{\text{max}}}{4}}}$ , thus $\dot{V}\le 0$ . Then, it can be concluded that the error states $z_e,\hspace{0.17em}{\psi }_e,\hspace{0.17em}{y}_i,\hspace{0.17em}{\tilde{\vartheta }}_i,\hspace{0.17em}{i}_e$ , and z_inn are invariant to the following defined sets

${\mathit{\Omega }}_{z_e}=\left(z_e||z_e|\le \sqrt{\frac{P}{k_1-1}}+{\delta }_e\right)$

47

${\mathit{\Omega }}_{{\psi }_e}=\left({\psi }_e||{\psi }_e|\le \sqrt{\frac{P}{k_2-1}}\right)$

48

${\mathit{\Omega }}_{y_i}=\left(y_i||y_i|\le \sqrt{\frac{P}{{\rho }_{\text{min}}}}\right)$

49

${\mathit{\Omega }}_{{\tilde{\vartheta }}_i}=\left({\tilde{\vartheta }}_i||{\tilde{\vartheta }}_i|\le \sqrt{\frac{P}{{\kappa }_{\text{min}}}}+\frac{{\vartheta }_{\text{max}}}{2}\right)$

50

$\begin{array}{l}{\Omega }_{i_e}=(i_e|\left|i_e\right|\le \sqrt{\frac{P}{k_{\text{min}}-\frac{{\xi }_{\text{max}}}{4}}}\hfill \\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hfill \end{array}+\frac{2}{4k_{\text{min}}-{\xi }_{\text{max}}}{\epsilon }_{\text{max}})$

51

${\Omega }_{z_{inn}}=(z_{inn}|\left|z_{inn}\right|\le \sqrt{\frac{P}{{\mu }_{\text{min}}{\lambda }_{\text{min}}-\frac{{\varpi }_{\text{max}}}{4}}}+\frac{2}{4{\mu }_{\text{min}}{\lambda }_{\text{min}}-{\varpi }_{\text{max}}}{\epsilon }_{\text{max}})$

52

Remark 7

In the developed CNDSC scheme, the design parameters k₁, k₂, ρ_i, k_i, λ_i, κ_i, and η_i should be tuned to obtain good transient and closed-loop stable performances. By increasing k₁, k₂, ρ_i, k_i, and λ_i, the tracking errors and prediction errors can be made arbitrarily small. As for the control gains of the NN adaptation law, decreases in κ_i or increases in η_i will achieve a better tracking performance.

Simulation results and discussion

In this section, comparison numerical simulation results are presented to illustrate the performance of the proposed path-following control strategy. The model of the vessel is given by Fredriksen and Pettersen, ³⁸ and the control objective is to make the MSV follow closely a virtual vessel along the desired path. The virtual vessel is characterized by the predefined surge speed u_d and yaw rate speed r_d, where u_d is the desired cruise speed of the vessel, and r_d is mainly used for shaping the path. For the computer simulation purpose, the unknown time-varying disturbances induced by wind, waves, and ocean currents are produced by means of the following first-order Gauss–Markov processes ³⁹

$\{\begin{array}{c}{\dot{d}}_{wu}+c_u{d}_{wu}=w_u\\ {\dot{d}}_{wv}+c_v{d}_{wv}=w_v\\ {\dot{d}}_{wr}+c_r{d}_{wr}=w_r\end{array}$

53

where w_u, w_v, and w_r are zero-mean Gaussian white noise processes, c_u, c_v, and c_r are positive constant parameters. For the generated disturbances d_wu, d_wv, and d_wr, it can be validated that Assumption 1 is satisfied.

To better show the effectiveness of the controller proposed in this article, the comparison studies are performed with the basic DSC (BDSC) and the NDSC methods. The BDSC is a model-based approach, which depends on the accurate mathematical model of the vessel, and the control system are designed as

$\begin{array}{l}{\alpha }_u=(k_1\left(z_e-{\delta }_e\right)+{\dot{x}}_d\text{cos}\left({\psi }_r\right)\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}+{\dot{y}}_d\text{sin}\left({\psi }_r\right)-v\text{sin}\left({\psi }_e\right))\times \text{sec}\left({\psi }_e\right)\\ {\alpha }_r=k_2{\psi }_e+{\dot{\psi }}_r\\ T_u{\dot{\beta }}_u+{\beta }_u={\alpha }_u\hspace{0.17em}\\ T_r{\dot{\beta }}_r+{\beta }_r={\alpha }_r\\ {\tau }_u=-k_u{u}_e-F_u\left(v,\hspace{0.17em}r\right)+\frac{d_{11}}{m_{11}}u+{\dot{\beta }}_u\\ {\tau }_r=-k_r{r}_e-F_r\left(u,v,\hspace{0.17em}r\right)+{\dot{\beta }}_r\end{array}$

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The NDSC method employs DSC and RBF NN techniques, where the NN adaption law is built based on the tracking error and using the weight vector. The controller using the NDSC scheme is formulated as follows

$\begin{array}{l}{\alpha }_u=(k_1\left(z_e-{\delta }_e\right)+{\dot{x}}_d\text{cos}\left({\psi }_r\right)\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}+{\dot{y}}_d\text{sin}\left({\psi }_r\right)-v\text{sin}\left({\psi }_e\right))\times \text{sec}\left({\psi }_e\right)\\ {\alpha }_r=k_2{\psi }_e+{\dot{\psi }}_r\\ T_u{\dot{\beta }}_u+{\beta }_u={\alpha }_u\\ T_r{\dot{\beta }}_r+{\beta }_r={\alpha }_r\\ {\dot{\widehat{\mathit{W}}}}_u={\eta }_u\left({\mathit{H}}_u\left({\mathit{Z}}_u\right)u_e-{\kappa }_u{\widehat{\mathit{W}}}_u\right)\\ {\dot{\widehat{\mathit{W}}}}_r={\eta }_r\left({\mathit{H}}_r\left({\mathit{Z}}_r\right)r_e-{\kappa }_r{\widehat{\mathit{W}}}_r\right)\\ {\tau }_u=-\left(k_u+\frac{1}{2}\right)u_e-{\widehat{\mathit{W}}}_u^T{\mathit{H}}_u\left({\mathit{Z}}_u\right)+{\dot{\beta }}_u\\ {\tau }_r=-\left(k_r+\frac{1}{2}\right)r_e-{\widehat{\mathit{W}}}_r^T{\mathit{H}}_r\left({\mathit{Z}}_r\right)+{\dot{\beta }}_r\end{array}$

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For the proposed CNDSC scheme, the kinematic controllers are designed as equation (21), the first-order filters are chosen as equation (22), the dynamic controllers are proposed as equation (31), the state predictors are developed as equation (33), and the NN updating laws are designed as equation (34).

Comparative analysis

In the following simulations, the initial states of the virtual vessel are set as ${\left[x_d(0),\hspace{0.17em}{y}_d(0),\hspace{0.17em}{\psi }_d(0),\hspace{0.17em}{v}_d(0)\right]}^T={\left[0,\hspace{0.17em}0,\hspace{0.17em}0,\hspace{0.17em}0\right]}^T$ . The initial position and heading angle of the real vessel are selected as ${\left[x\left(0\right),\hspace{0.17em}y\left(0\right),\hspace{0.17em}\psi \left(0\right)\right]}^T={\left[-15,\hspace{0.17em}10,\hspace{0.17em}0\right]}^T$ , and the speeds are chosen as ${\left[u\left(0\right),v\left(0\right),\hspace{0.17em}r\left(0\right)\right]}^T={\left[0,\hspace{0.17em}0,\hspace{0.17em}0\right]}^T$ . According to the abovementioned control system design and stability analysis, the positive constant parameters of the CNDSC scheme are chosen as $k_1=1.2,\hspace{0.17em}{k}_2=2,\hspace{0.17em}{T}_u=T_r=0.05,\hspace{0.17em}{k}_u=4,\hspace{0.17em}{k}_r=10,\hspace{0.17em}{\lambda }_u=10,\hspace{0.17em}{\lambda }_r=50,\hspace{0.17em}{\eta }_u={\eta }_r=1,\hspace{0.17em}{\mu }_u={\mu }_r=3$ , and ${\kappa }_u={\kappa }_r=1$ . It is noted that these positive parameters should be tuned according to the stability analysis result, such as k₁ − 1 > 0 and ${\kappa }_{\text{min}}=\text{min}[{\kappa }_i]>0$ . The above selected values are a compromise between the error convergence rates and the control efforts. If the tracking errors are desired to be lower, $k_1,\hspace{0.17em}{k}_2,\hspace{0.17em}{k}_i,\hspace{0.17em}{\kappa }_i,\hspace{0.17em}{\lambda }_i,\hspace{0.17em}{\eta }_i$ , and 1/T_i should be chosen larger. Increases in μ_i can provide more information of the prediction error for the CNDSC, which will improve the tracking accuracy. The small positive value δ_e is selected as ${\delta }_e=\text{exp}(-5z_e)$ . The Gaussian function is chosen as the form of equation (15), and the RBF NNs for approximating ${\varphi }_u\left({\mathit{Z}}_u\right)$ and ${\varphi }_r\left({\mathit{Z}}_r\right)$ all contain 21 nodes (i.e. l = 21), with centers spaced evenly in the interval $\left[-2,\hspace{0.17em}2\right]\hspace{0.17em}\times \hspace{0.17em}\left[-1,\hspace{0.17em}1\right]\hspace{0.17em}\times \hspace{0.17em}\left[-1,\hspace{0.17em}1\right]$ and widths $b_j=2\hspace{0.17em}\left(j=1,\hspace{0.17em}2,\hspace{0.17em}\hspace{0.17em}\dots \hspace{0.17em},\hspace{0.17em}l\right)$ . To ensure fairness, the main corresponding control parameters of the BDSC and NDSC approaches are chosen as the same values of the CNDSC method.

In order to sufficiently verify the performance of the proposed path-following control scheme, two types of paths are considered. The first one is the path with straight line and circle, which has been commonly used to test the path-following controllers. ^17,25,26 The second one is an S-curve path proposed in this article, which has a time-varying curvature. This is a stricter condition than the straight line or circle path since both the magnitude and direction of the path curvature are changeable.

Case 1. In this case, the desired surge and yaw rate speeds of the virtual vessel that produce the reference path are chosen as follows

$\begin{array}{l}{u}_d=1.5\hspace{0.17em}\text{m/s},\\ r_d=\{\begin{array}{l}0,\hspace{0.17em}0\le t<100\hfill \\ 0.03,\hspace{0.17em}100\le t\le 300\hfill \end{array}\end{array}$

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The path-following control results of the BDSC, NDSC, and CNDSC approaches are presented in Figures 3 and 4, showing the tracking performance. Among them, Figure 4 also shows the abilities of the three controllers to deal with the parameter uncertainties that is the MSV has 30% parameters perturbation after t = 100 s. From Figures 3 and 4 with local zooms, it is clearly demonstrated that the proposed CNDSC scheme achieves faster convergence and improved tracking accuracy. Especially in Figure 4, one can see that the NDSC and CNDSC methods are less sensitive to parameters than the BDSC method, where the CNDSC still obtains a better tracking accuracy. The “blue vessel” intuitively describes the real-time position and heading angle of the MSV with the CNDSC scheme. It can be seen that the MSV can produce an adaptive sideslip angle to compensate for the drift force induced by the time-varying environmental disturbance, and its position is always kept on the reference path. This validates that the proposed control scheme is capable of enhancing the robustness of the closed-loop system. The similar conclusions can be verified from the simulation results in case 2.

Figure 3.

Path-following control results comparison: path with straight line and circle.

Figure 4.

Path-following control results comparison: MSV with parameter perturbation. MSV: marine surface vessel.

Notice that the traditional NDSC method uses the weight vector ${\widehat{\mathit{W}}}_i$ as the learning parameter, then, the number of adaptive parameters will be 42. However, since the proposed CNDSC updating law (34) uses ${\widehat{\vartheta }}_i$ as the learning parameter, the number of online adaptive parameters can be compressed to 2. The comparison of the computer burdens for NDSC and CNDSC in case 1 is given in Table 1. Where the RAM of the computer for simulation is 8 GB, and the platform is MATLAB. It can be seen from Table 1 that the proposed CNDSC can effectively reduce the computational burden of the path-following controller.

Table 1.

The computer burdens of the NDSC and CNDSC.

	NDSC with ${\widehat{\mathit{W}}}_i$	CNDSC with ${\widehat{\vartheta }}_i$
Average memory (MB)	158.6	74.1
Simulation time (s)	139.45	11.32

CNDSC: composite neural-based dynamic surface control; NDSC: neural network-based dynamic surface control.

Case 2. This case is to confirm the efficiency of the proposed control scheme for arbitrary curved path. To generate a reference path with a changeable curvature, the desired surge and yaw rate speeds are carefully selected as

$\begin{array}{l}{u}_d=1.5\hspace{0.17em}\text{m/s},\\ r_d=\{\begin{array}{l}0.1(\text{exp}\left(0.004t\right)-1),\hspace{0.17em}0\le t<100\hfill \\ 0.03(\text{tanh}(0.01\left(t-100\right))-1),\hspace{0.17em}100\le t\le 300\hfill \end{array}\end{array}$

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The comparative simulation results and the closed-loop system signals are shown in Figures 5 to 10. Figure 5 shows the S-curve path path-following control results of the three controllers. It reveals that the proposed CNDSC scheme achieves the fastest response and least tracking errors. This conclusion is further confirmed from the position variations in Figure 6 and the position and heading track errors in Figure 7. Since the prediction errors in the CNDSC can provide an extra information for identifying the system uncertainties, the improved transient performance and better tracking accuracy are obtained in contrast to the NDSC method. It is noticed that the model-based BDSC approach also can follow the reference path. However, due to the fact that the unknown time-varying environmental disturbances cannot be efficiently compensated, it yields a larger position track error and a higher oscillation for the heading track error, as shown in Figure 7. In Figure 8, the state prediction values of the CNDSC approach are plotted, which can track the real states accurately. Figure 9 shows the CNDSC adaptation parameters which are convergent. In the end, Figure 10 gives the control inputs which are in reasonable ranges.

Figure 5.

Path-following control results comparison: S-curve path.

Figure 6.

Positions comparison of the S-curve path path following.

Figure 7.

Position and heading track errors comparison of the S-curve path path following.

Figure 8.

State predictions of the proposed controller for the S-curve path path following.

Figure 9.

NN adaptive parameters of the proposed controller for the S-curve path path following. NN: neural network.

Figure 10.

Control inputs of the proposed controller for the S-curve path path following.

Conclusions

In this article, the robust CNDSC scheme has been developed for the path following of MSVs with model uncertainties and time-varying external disturbances. The proposed new control method has the ability to eliminate the problem of explosion of complexity in the backstepping design and deal with the system uncertainties and unknown environmental disturbances. Moreover, by employing a predictor for each dynamical subsystem and incorporating the prediction error into the NN adaption law, better tracking accuracy and improved control performance for the path following have been achieved. In particular, the number of the NN adaptive adjusting parameter has been reduced to one, which makes the NN controller concise and easier to apply in practice. Stability analysis shows that all the closed-loop signals are guaranteed to be UUB. Finally, comparative simulation results have been given to illustrate the effectiveness of the proposed control approach. Future works include extending the capability of the CNDSC to handle the problems of input time delays and input saturations, and further performing verifications in practical applications.