Adaptive finite-time sliding mode control of coupled motion in ships with liquid tank dynamics using neural networks

Abstract

This paper proposes an adaptive finite-time sliding-mode control approach based on neural networks to address the coupled motion of ships with liquid tank dynamics. Firstly, a brief analysis of ship sloshing dynamics is conducted to derive the impact of sloshing forces on the governing equations of ship dynamics. Secondly, to achieve effective tracking control, a dynamic positioning model of the ship is established using the Froude-Kryloff assumption, simplifying the handling of complex terms. Next, based on analysis of ship mechanics, the control equations for liquid-carrying ships are derived. Then, a finite-time hierarchical sliding-mode controller is designed to stabilize the system within a global finite time using nonlinear sliding surfaces. Subsequently, to address the unknown upper bounds of the nonlinear sloshing forces and second-order wave forces, an RBF neural network along with an adaptive control method is employed. In addition, the global finite-time stability of the system is proven using Lyapunov stability theory and homogeneity. The significance of these findings lies in their potential to enhance operational stability and safety in maritime applications, providing valuable insights for both research and practical implementation. Finally, numerical simulations are conducted to confirm the effectiveness of the proposed control approach, demonstrating its potential to significantly improve operational stability in various maritime operations.

Introduction

With the rapid development of artificial intelligence and unmanned technologies, the application of unmanned ships is becoming increasingly widespread. There is a growing body of research on ships, for example, Long et al. conducted a series of control studies on surface boats.^1,2 The maritime conveyance of substantial volumes of fluids, including petroleum, liquefied natural gas, and liquid hydrogen, is gaining increased attention.

When large unmanned liquid carriers transport liquids in their own tanks, due to the significant volume of liquids, the ship will swing violently when it encounters adverse sea conditions, and the wide range of swinging of the ship will cause the liquid tank to swing strongly, thus generating a strong and repeated swinging force acting on the bulkhead, which seriously affects the ship’s motion. When the frequency of the ship’s hull movement approaches the natural frequency of the liquid cargo hold, it may induce resonance between them, causing the vessel to sway further and potentially leading to capsizing, which could result in serious safety hazards.^3,4 Therefore, research on the control of liquid-carrying ships is of great significance to the marine transport industry. In order to efficiently achieve liquid transport as well as the stability of unmanned liquid-carrying ship navigation, it needs to rely on efficient and accurate ship six-degree-of-freedom trajectory tracking control. The impact of liquid tanks on liquid-carrying ships can be summarized as follows: liquids will sway back and forth in the tanks when the ship is in motion, thus changing the position of the ship’s center of gravity all the time; the liquids are powerful to the walls of the tanks and the transported liquids may be very viscous, such as petroleum. ⁵ Thus, achieving six-degree-of-freedom tracking control for large unmanned liquid-carrying vessels presents significant challenges.

Based on the existing literature, it can be roughly classified that the research on the coupled motion of ships and liquid tanks falls into two categories: numerical analysis of the coupled motion of ships and the design of motion controllers for liquid-carrying ships or liquid tanks. Currently, there is limited research in the control of liquid-carrying ships, leaving substantial room for further progress in this area. Using the inverse model of a tracked object, Noda et al. ⁶ proposed a 2-DOF system with a feed-forward controller, in which the sloshing control of the tracked object is conducted by a feedback controller designed by the hybrid shape method. Hu et al. ⁷ proposed a rolling motion control algorithm combining MPC and a gyro stabilizer and designed a combined controller. Qi et al. ⁸ proposed a new constrained attitude tracking control and active wobble suppression scheme that uses the forces and torques generated by liquid wobble as nonlinear functions of the spacecraft state, and suppresses liquid wobble by limiting the angular velocity and controlling the torque and its rate of change. Kurode et al. ⁹ designed a nonlinear sliding mode observer to estimate the tilt angle and loss rate of liquid sloshing in a flowing vessel for a high gain approach to a simplified subsystem. Song et al. ¹⁰ investigated liquid-filled spacecraft with significant liquid sloshing by designing a predefined-time attitude controller with moment constraints, followed by the development of a new control algorithm that combines sliding mode control with predefined-time stabilization. Deng and Yue ¹¹ designed a hybrid controller that integrates sliding mode control with an adaptive algorithm for spacecraft with partially filled liquid propellant tanks to achieve attitude tracking. Most of the previous work on liquid-carrying control has focused on applying controllers to liquid tanks or on attitude trajectory tracking done by spacecraft or robots, highlighting the considerable innovative interest in studying trajectory tracking control for automated liquid-carrying ships.

In this paper, an innovative set of six-degree-of-freedom power-positioned motion control equations for liquid-carrying ships is proposed, based on an in-depth mechanical analysis of ship-liquid coupled motion. In the numerical analysis of the coupled motion of the ship and liquid tank sloshing, the radial force in the first-order wave force, as a function of the wave frequency, when converted from the frequency domain to the time domain by Fourier transform, produces a convolutional integral term that is computationally complex and requires high accuracy, which makes the numerical computation face great challenges. Additionally, computing the forces generated by the liquid chamber wakes is also highly complex, further increasing the analytical challenges. ¹² In order to solve these computational challenges, this study adopts the Froude-Kryloff hypothesis, which successfully circumvents the complex convolutional integral treatment and directly derives the analytical expression for the first-order radiative force under regular waves. ¹³ As noted in Fossen TI and Fjellstad ¹⁴ , the nonlinear positioning power ship model without liquid tanks is obtained, and the swaying force is reasonably added to the right end of the ship’s equation of motion according to the analysis of the coupled equation mechanics thus obtaining the final motion control equation. Secondly, the forces generated by liquid sloshing in the equations are approximated using a neural network, ¹⁵ and an adaptive law is designed for the network weights with the second-order wave forces with an unknown upper bound. ¹⁶ Subsequently, a finite-time hierarchical sliding mode controller is designed using a nonlinear sliding mode surface, which not only ensures that the state of the system can converge to 0 in finite time with systematic error and avoids the singularity phenomenon by adding an integral term in the sliding mode surface. The introduction of a convergence law containing a nonlinear term allows the system to reach the sliding mode surface in finite time, and the global finite time convergence of the system is proved by the homogeneity theory ¹⁷ and the finite-time Lyapunov stability theory, which enables the system to achieve global finite time stability.

In this paper, an adaptive finite-time sliding-mode control method is proposed based on the derivation of the coupled motion control equations for liquid-carrying ships, enabling efficient control of these vessels. The main contributions of this study can be summarized as follows:

• (1) The research involves an in-depth mechanical analysis of liquid-carrying ships, leading to the development of control equations that facilitate the precise positioning of these vessels.

• (2) To address the unknown elements within the control equations, this study employs adaptive techniques and utilizes Radial Basis Function (RBF) neural networks to approximate the nonlinear swaying forces associated with liquid motion.

• (3) A cutting-edge nonlinear sliding mode control law has been crafted for liquid-carrying ships to regulate their six-degree-of-freedom motion, ensuring they can swiftly and reliably achieve the intended attitude and trajectory within a finite time frame.

• (4) Last but not least, simulations validate the control equations, refines neural networks, and ensures the robustness of ship control strategies.

This paper is organized as follows: The problem statement section presents the mechanical analysis of the ship and derives the control equations for the coupled motion of a liquid-carrying vessel. The controller design section details the main results, including the establishment of a single-degree-of-freedom tracking objective, the design of a term-specific adaptive law, and the development of an adaptive nonlinear sliding mode controller for trajectory tracking. The stability analysis section performs stability analysis using Lyapunov stability theory and homogeneity theory. The simulation section presents the simulation results and conducts correlation analyses. The conclusion section summarizes the conclusions drawn from this study.

Problem statement

To validate the plausibility of the subsequent mechanistic interpretations and the resulting equations, two fundamental assumptions are posited as follows.

• Assumption 1. The ship is a rigid body.

• Assumption 2. Simplify the hull to a box ship.

To begin with, envision a ship that lacks liquid tanks. From Newton’s Second Law of Motion, the ship’s six-degree-of-freedom equation of motion can be expressed as

$$\mathbf{M}\stackrel{··}{\mathit{\zeta }}={\mathit{F}}_S(t)+{\mathit{F}}_H(t)+{\mathit{F}}_L(t),$$

(1)

where $\mathbf{M}$ denotes the mass matrix of the ship. As shown in Figure 1, $\mathit{\zeta }$ represents six-degree-of-freedom motion of the ship, which includes roll, surge, pitch, sway, yaw and heave. ${\mathit{F}}_S\left(t\right)=-\mathbf{K}\mathit{\zeta }$ denotes ship resilience, which $\mathbf{K}$ stands for stiffness matrix. ${\mathit{F}}_H\left(t\right)={\mathit{F}}_1\left(t\right)+{\mathit{F}}_2\left(t\right)$ denotes the wave forces acting on the ship, which is divided into first-order wave force and second-order wave force. ${\mathit{F}}_1\left(t\right)$ can be divided into the sum of the first-order radiative force ${\mathit{F}}_R\left(t\right)$ , the first-order incident force ${\mathit{F}}_I\left(t\right)$ and the first-order wrap-around force ${\mathit{F}}_D\left(t\right)$ . ${\mathit{F}}_L\left(t\right)$ denotes other external forces on the ship. Using the IRF method, ¹⁸ the first-order radiative force ${\mathit{F}}_R\left(t\right)$ can be expressed as

$${\mathit{F}}_R\left(t\right)=-{\mathbf{M}}^a\left(\infty \right)\stackrel{··}{\mathit{\zeta }}-\underset{-\infty }{\overset{t}{\int }}\mathit{R}(t-\rho )\stackrel{·}{\mathit{\zeta }}\left(\rho \right)d\rho ,$$

(2)

where ${\mathbf{M}}^a\left(\infty \right)$ stands for additional mass of the ship at infinite frequency of encounters, $\mathit{R}\left(t\right)$ is the hysteresis equation, and $\rho$ is the time-integrated variable.

Figure 1.

6DOF motion for liquid-carrying ship.

According to (1), (2) and the above analysis, the time-domain equations of motion for the ship without liquid tanks can be obtained as follows:

$$\begin{array}{c}[\mathbf{M}+{\mathbf{M}}^a\left(\infty \right)]\stackrel{··}{\mathit{\zeta }}+\underset{-\infty }{\overset{t}{\int }}\mathit{R}(t-\rho )\stackrel{·}{\mathit{\zeta }}\left(\rho \right)d\rho +\mathbf{K}\mathit{\zeta }\hfill \\ ={\mathit{F}}_I\left(t\right)+{\mathit{F}}_D\left(t\right)+{\mathit{F}}_2\left(t\right)+{\mathit{F}}_L\left(t\right).\hfill \end{array}$$

(3)

Considering the effects of liquid tanks on ship movements, liquid chamber actions may exhibit some of the following phenomena:

• The liquid in the compartment possesses its own mass and, being a fluid, necessitates the introduction of both a mass matrix for the liquid and an additional mass matrix;

• Liquid-carrying ships in motion within tanks will experience shaking due to the presence of a free liquid surface. Consequently, the position of the ship’s center of gravity will shift back and forth, leading to changes in the ship’s restoring force. However, when the liquid tanks sway violently, it becomes challenging to determine a correction for the ship’s restoring force coefficient;

• Liquid exerts hydrostatic, hydrodynamic, and viscous forces on the bulkheads of the liquid tank.

Incorporating the effects of liquid tank sloshing into the ship’s equation of motion involves adding the force of the liquid tank, denoted as ${\mathit{F}}_{\mathrm{sa}}\left(t\right)$ , to the right-hand side of equation (3):

$${\mathit{F}}_{\mathrm{sa}}\left(t\right)={\mathbf{M}}^{\mathrm{sa}}\stackrel{··}{\mathit{\zeta }}+{\mathit{F}}_b\left(t\right),$$

(4)

where ${\mathit{F}}_{\mathrm{sa}}\left(t\right)$ denotes the force in the liquid compartment, ${\mathbf{M}}^{\mathrm{sa}}$ stands for the sum of the liquid mass matrix and its added mass, and ${\mathit{F}}_b\left(t\right)$ is the sum of hydrostatic, hydrodynamic, viscous, and restoring forces on the bulkheads.

The equation of motion for the ship coupled to the liquid-tank sloshing in the time domain can be obtained by simply adding the liquid-tank force to the right end of equation (3). ⁵ It can be written as

$$\begin{array}{c}[\mathbf{M}+{\mathbf{M}}^a\left(\infty \right)-{\mathbf{M}}^{\mathrm{sa}}]\stackrel{··}{\mathit{\zeta }}+\underset{-\infty }{\overset{t}{\int }}\mathit{R}(t-\rho )\stackrel{·}{\mathit{\zeta }}\left(\rho \right)d\rho +\mathbf{K}\mathit{\zeta }\hfill \\ ={\mathit{F}}_I\left(t\right)+{\mathit{F}}_D\left(t\right)+{\mathit{F}}_2\left(t\right)+{\mathit{F}}_L\left(t\right)+{\mathit{F}}_b\left(t\right).\hfill \end{array}$$

(5)

To further explain the plausibility of the control equation model derived below, the Assumption 3 can be given.

• Assumption 3. The control equations (6) and (7) given later do not take into account other secondary effects such as wind, friction, etc.

The convolution term in equation (5) poses a challenge in determining whether it is bounded or not. Several external forces also complicate the determination process. Equation (5) represents the motion of the ship with the ship itself as the center, which does not allow for the determination of the ship’s position. To realize control over the ship’s motion, a six-degree-of-freedom power-positioned ship motion model must be introduced as follows:

$$\{\begin{array}{c}\stackrel{·}{\mathit{\eta }}=\mathit{J}\left(\mathit{\eta }\right)\mathit{\sigma }\hfill \\ \mathbf{M}\stackrel{·}{\sigma }+\mathit{G}\left(\mathit{\sigma }\right)\mathit{\sigma }+\mathit{I}\left(\mathit{\sigma }\right)\mathit{\sigma }+\mathit{g}\left(\mathit{\eta }\right)=\mathit{\tau }+\mathit{\beta }.\hfill \end{array}$$

(6)

$\mathit{\eta }={(x,y,z,\phi ,\theta ,\psi )}^T$ stands for the ship’s moving position and angle of rotation in a fixed coordinate system in space, which includes longitudinal position $x$ , transverse position $y$ , vertical position $z$ , transverse angle $\phi$ , longitudinal angle $\theta$ , bow angle $\psi$ . The vector $\mathit{\sigma }={(u,v,w,p,q,r)}^T$ represents the translational and rotational velocities of the ship in the ship-following coordinate system. Specifically, $u$ is the longitudinal oscillation velocity, $v$ is the transverse oscillation velocity, $w$ is the vertical oscillation velocity, $p$ is the transverse oscillation angular velocity, $q$ is the longitudinal oscillation angular velocity, and $r$ is the bow-rock angular velocity. $\tau ={(X,Y,Z,K,M,N)}^T$ is control input, which indicates the forces and moments on the ship in the coordinate system of ship motion, that is, longitudinal force X, transverse force Y, vertical force Z, transverse rocking moment K, longitudinal rocking moment M and bow rocking moment N. $\mathit{\beta }={\mathit{\beta }}_1+{\mathit{\beta }}_2$ denotes wave force, which can be categorized into first order wave forces ${\mathit{\beta }}_1$ and second order wave forces ${\mathit{\beta }}_2$ . $\mathbf{M}$ , $\mathit{J}\left(\mathit{\eta }\right)$ , $\mathit{G}\left(\mathit{\sigma }\right)$ , $\mathbf{I}\left(\mathit{\sigma }\right)$ , and $\mathit{g}\left(\mathit{\eta }\right)$ denote ship inertia matrix with additional mass, coordinate rotation matrix, Coriolis centripetal matrix, damping matrix and hull restoring force matrix, respectively.

Using the same method as used to obtain the coupled ship-tank motion in equation (5), and adding the tank forces to the dynamics of equation (6), the final six-degree-of-freedom model of a positioned ship in a wave with control inputs can be obtained as

$$\{\begin{array}{c}\stackrel{·}{\eta }=\mathbf{J}\left(\mathit{\eta }\right)\mathit{\sigma }\hfill \\ [\mathbf{M}-{\mathbf{M}}^s]\stackrel{·}{\sigma }+\mathit{G}\left(\mathit{\sigma }\right)\mathit{\sigma }+\mathit{I}\left(\mathit{\sigma }\right)\mathit{\sigma }+\mathit{g}\left(\mathit{\eta }\right)\hfill \\ =\mathit{\tau }+{\mathit{\beta }}_1+{\mathit{\beta }}_2+\mathit{F},\hfill \end{array}$$

(7)

where ${\mathbf{M}}^s$ denotes liquid tanks and their added mass’s matrix, $\mathit{F}$ stands for the sum of the hydrostatic force of the liquid on the bulkhead, the hydrodynamic viscous force and the change in the hull restoring force.

Obviously, the combined force $\mathit{F}$ of these four forces is nonlinear. By applying the Froude-Kryloff hypothesis, an expression for the first-order wave force on a ship in regular waves can be obtained, while the first-order wave force under irregularity can be regarded as the sum of the first-order forces on a ship in regular waves of different frequencies and amplitudes. Therefore, ${\mathit{\beta }}_1$ is calculable as a known term.

Based on the discussion above, the following two remarks are presented:

• Remark 1. Since the hull is assumed to be rigid in Assumption 1, it is only under this assumption that the forces acting on the ship can be analyzed using Newton’s second law.

• Remark 2. The purpose of simplifying the hull to a box ship is to use Froude-Kryloff hypothesis to obtain the first order radiation force.

Controller design

Figure 2 presents a simple structural block diagram of the control system discussed in this paper. Here, ${\mathit{\eta }}_d$ and ${\stackrel{·}{\eta }}_d$ represent the desired trajectories of the system, while $\mathit{\eta }$ and $\stackrel{·}{\eta }$ denote the actual trajectories. The difference between the desired and actual trajectories is calculated to obtain the tracking errors $E_1$ and $E_2$ , which are then used to derive the sliding surface $\mathit{S}$ . Subsequently, the control input $\mathit{\tau }$ is designed using the sliding surface $\mathit{S}$ along with adaptive neural networks, which then act on the system.

Figure 2.

Schematic diagram of control system.

Processing of control equations

In order to achieve efficient control, it is wanted that $\mathit{\eta }\to {\mathit{\eta }}_d={({\eta }_{1d},{\eta }_{2d},{\eta }_{3d},{\eta }_{4d},{\eta }_{5d},{\eta }_{6d})}^T$ , $\stackrel{·}{\eta }\to {\stackrel{·}{\eta }}_d$ in finite time, where ${\eta }_d$ presents the expected value. $\stackrel{··}{\mathit{\eta }}=\stackrel{.}{\mathbf{J}}\mathit{\sigma }+\mathbf{J}\stackrel{.}{\sigma }$ can be derived by (7):

$$\begin{array}{c}\stackrel{··}{\mathit{\eta }}=\stackrel{·}{\mathit{J}}\mathit{\sigma }+\mathit{J}{(\mathbf{M}-{\mathbf{M}}^s)}^{-1}[\mathit{\tau }+{\mathit{\beta }}_1+{\mathit{\beta }}_2+\mathit{F}]\hfill \\ \mathit{G}\left(\mathit{\sigma }\right)\mathit{\sigma }-\mathit{I}\left(\mathit{\sigma }\right)\mathit{\sigma }-\mathit{g}\left(\mathit{\eta }\right)].\hfill \end{array}$$

(8)

For the known terms in it, an equivalent controller is designed to carry out the processing. Command. $\mathit{\tau }={\mathit{\tau }}_a+{\mathit{\tau }}_b$ The expression for the equivalent controller is as follows:

$$\begin{array}{c}{\mathit{\tau }}_b=-{\mathit{\beta }}_1+\mathit{G}\left(\mathit{\sigma }\right)\mathit{\sigma }+\mathit{I}\left(\mathit{\sigma }\right)\mathit{\sigma }+\mathit{g}\left(\mathit{\eta }\right)\hfill \\ -{(\mathit{M}-{\mathit{M}}^s)}^{-1}{\mathit{J}}^{-1}\stackrel{·}{\mathit{J}}\mathit{\sigma }.\hfill \end{array}$$

(9)

From (7), (8) and (9), it follows

$$\{\begin{array}{c}\stackrel{·}{\mathit{\eta }}=\mathbf{J}\left(\mathit{\eta }\right)\mathit{\sigma }\hfill \\ \multicolumn{1}{c}{\stackrel{··}{\mathit{\eta }}=\mathit{J}{(\mathit{M}-{\mathit{M}}^s)}^{-1}[{\mathit{\tau }}_a+{\mathit{\beta }}_2+\mathit{F}].}\end{array}$$

(10)

Define ${\mathit{E}}_1=\mathit{\eta }-{\mathit{\eta }}_d={(e_{11},e_{21},e_{31},e_{41},e_{51},e_{61})}^T$ , ${\mathit{E}}_2=\stackrel{·}{\eta }-{\stackrel{·}{\eta }}_d={(e_{12},e_{22},e_{32},e_{42},e_{52},e_{62})}^T$ . Then ${\mathit{E}}_1$ and ${\mathit{E}}_2$ are the systematic errors, so the control objectives can be transformed as ${\mathit{E}}_1\to 0$ , ${\mathit{E}}_2\to 0$ .

Command

$$\{\begin{array}{c}\mathit{J}{(\mathit{M}-{\mathit{M}}^s)}^{-1}{\mathit{\tau }}_a={({\tau }_1,{\tau }_2,{\tau }_3,{\tau }_4,{\tau }_5,{\tau }_6)}^T\hfill \\ \mathit{J}{(\mathit{M}-{\mathit{M}}^s)}^{-1}{\mathit{\beta }}_2={({\beta }_{21},{\beta }_{22},{\beta }_{23},{\beta }_{24},{\beta }_{25},{\beta }_{26})}^T\hfill \\ \mathit{J}{(\mathit{M}-{\mathit{M}}^s)}^{-1}\mathit{F}={(f_1,f_2,f_3,f_4,f_5,f_6)}^T\hfill \\ \mathit{J}\left(\mathit{\eta }\right)\mathit{\sigma }={(u_a,v_a,w_a,p_a,q_a,r_a)}^T.\hfill \end{array}$$

It is clear that equation (10) can be divided into six independent equations, and these independent equations are of the same form. The first one of these can be written as

$$\{\begin{array}{c}\stackrel{·}{\mathrm{x}}=u_a\hfill \\ \multicolumn{1}{c}{{\stackrel{·}{u}}_a={\tau }_1+{\beta }_{21}+f_1.}\end{array}$$

(11)

Formulation of control laws

For the design of the following adaptive law, the sliding surface $S_i$ is chosen to be

$$\begin{array}{c}\multicolumn{1}{c}{S_i=e_{i2}+\underset{0}{\overset{t}{\int }}{m}_{i1}{\left|e_{i1}\left(p\right)\right|}^{{\alpha }_{i1}}\mathrm{sgn}\left(e_{i1}\left(p\right)\right)}\\ +m_{i2}{\left|e_{i2}\left(p\right)\right|}^{{\alpha }_{i2}}\mathrm{sgn}\left(e_{i2}\left(p\right)\right)\mathrm{dp},\hfill \end{array}$$

which $0<{\alpha }_{i1}<1$ , ${\alpha }_{i2}=\frac{2{\alpha }_{i1}}{1+{\alpha }_{i1}}$ , $m_{i1}>0$ , $m_{i2}>0$ , $i=1,2,\dots ,6$ . $p$ is a time-integrated variable, $t$ is the time in the current state.

Second-order wave force ${\mathit{\beta }}_2$ is bounded, therefore, there exists a unknown positive number ${\beta }_{m}s.t$ .

$$\parallel \mathit{J}{(\mathit{M}-{\mathit{M}}^s)}^{-1}{\mathit{\beta }}_2{\parallel }_{\infty }\le {\beta }_m.$$

${\hat{\beta }}_{\mathrm{im}}$ is an estimate of ${\beta }_m$ , estimation error ${\tilde{\beta }}_{\mathrm{im}}={\beta }_m-{\hat{\beta }}_{\mathrm{im}}$ , then choose adaptive law ${\stackrel{·}{\hat{\beta }}}_{\mathrm{im}}={\gamma }_{i\beta }{S}_i$ , $i=1,2,\dots ,6$ , ${\gamma }_{i\beta }>0$ .

Radial Basis Function Neural Networks (RBFNNs) are a class of artificial neural networks that use radial basis functions as their activation functions. They are particularly effective in function approximation and pattern recognition tasks due to their capability to model complex, nonlinear relationships. The architecture of an RBFNN typically consists of an input layer, a hidden layer where radial basis functions are employed, and an output layer that aggregates contributions from the hidden neurons.

Using RBFNNs to approximate the nonlinear term $\mathit{J}{(\mathit{M}-{\mathit{M}}^s)}^{-1}\mathit{F}$ , the network input-output algorithm is given as follows:

$$\{\begin{array}{c}{h}_{\mathrm{ij}}=\exp (-\frac{\parallel {\mathit{x}}_i-{\mathit{c}}_{\mathrm{ij}}{\parallel }^2}{2b_{\mathrm{ij}}^2}),i=1,2...6\hfill \\ f_i={{\mathit{W}}_i}^{*T}{\mathit{h}}_i\left({\mathit{x}}_i\right)+{ϵ}_i,i=1,2...6,\hfill \end{array}$$

where ${\mathit{x}}_i={(e_{i1},e_{i2})}^T$ denotes network input, $j$ is the network hidden layer node, $h_{\mathrm{ij}}$ and ${\mathit{c}}_{\mathrm{ij}}$ are the $j$ th neuron of the $f_i$ hidden layer’s output and Gaussian function extended width constant. The ideal weight of the network is ${{\mathit{W}}_i}^{*}$ , which its estimate is ${\hat{\mathit{W}}}_i$ , ${\tilde{\mathit{W}}}_i={\mathit{W}}_i-{\hat{\mathit{W}}}_i$ , then choose network updating law of adaptation ${\stackrel{·}{\hat{\mathit{W}}}}_i={\gamma }_i{S}_i{\mathit{h}}_i$ , ${\gamma }_i>0$ . ${ϵ}_i$ is Neural network approximation error. Moreover, $\exists {ϵ}_k>0$ , $\forall i$ , $\left|{ϵ}_i\right|\le {ϵ}_k$ . Network output ${\hat{f}}_i={\hat{\mathit{W}}}_i^T{\mathit{h}}_i$ , and the estimation error ${\tilde{f}}_i=f_i-{\hat{f}}_i$ .

The following designed control law carries the second order derivative term ${\stackrel{··}{\eta }}_d$ of the desired function, and the control law cannot be infinite in real engineering, so the following assumption can be given.

• Assumption 4. ${\stackrel{··}{\eta }}_d$ is bounded.

To facilitate the design of the total controller, we first design the control rate for system (11). Based on the above treatment of ${\mathit{\beta }}_2$ and $f_i$ and the design of the S-surface, we can give the following equivalent control law ${\tau }_{11}$ :

$$\begin{array}{c}\multicolumn{1}{c}{{\tau }_{11}=-m_{11}{\left|e_{11}\right|}^{{\alpha }_{11}}\mathrm{sgn}\left(e_{11}\right)-m_{12}{\left|e_{12}\right|}^{{\alpha }_{12}}\mathrm{sgn}\left(e_{12}\right)}\\ +{\stackrel{··}{\eta }}_{1d}-{\hat{\mathit{W}}}_1^T{\mathit{h}}_1-{\hat{\beta }}_{1m}.\hfill \end{array}$$

Design $S_i$ ’s approach law:

$$\begin{array}{c}\multicolumn{1}{c}{{\stackrel{·}{\mathrm{S}}}_i=-k_{i1}{\left|S_i\right|}^{{\mu }_{i1}}\mathrm{sgn}\left(S_i\right)-k_{i2}{\left|S_i\right|}^{{\mu }_{i2}}\mathrm{sgn}\left(S_i\right)}\\ -k_{i3}{S}_i-k_4\mathrm{sgn}\left(S_i\right),\hfill \end{array}$$

(12)

where $k_4\ge {ϵ}_k$ , $k_{i1}$ , $k_{i2}$ , $k_{i3}>0$ , $0<{\mu }_{i1}<1$ , ${\mu }_{i2}>1$ .

In order to enable the system to maintain asymptotic stability on the sliding surface in a dynamic situation, it is necessary to introduce a switching control component, and from the reaching law of above given $S_i$ , the switching control law ${\tau }_{12}$ can be designed as

$$\begin{array}{c}\multicolumn{1}{c}{{\tau }_{12}=-k_{11}{\left|S_1\right|}^{{\mu }_{11}}\mathrm{sgn}\left(S_1\right)-k_{12}{\left|S_1\right|}^{{\mu }_{12}}\mathrm{sgn}\left(S_1\right)}\\ -k_{13}{S}_1-k_4\mathrm{sgn}\left(S_1\right).\hfill \end{array}$$

Then the control input ${\tau }_1$ in system (11) is

$$\begin{array}{c}\multicolumn{1}{c}{{\tau }_1={\tau }_{11}+{\tau }_{12}}\\ =-m_{11}{\left|e_{11}\right|}^{{\alpha }_{11}}\mathrm{sgn}\left(e_{11}\right)-m_{12}{\left|e_{12}\right|}^{{\alpha }_{12}}\mathrm{sgn}\left(e_{12}\right)\hfill \\ +{\stackrel{··}{\eta }}_{1d}-k_{11}{\left|S_1\right|}^{{\mu }_{11}}\mathrm{sgn}\left(S_1\right)-k_{12}{\left|S_1\right|}^{{\mu }_{12}}\mathrm{sgn}\left(S_1\right)\hfill \\ -k_{13}{S}_1-k_4\mathrm{sgn}\left(S_1\right)-{\hat{\mathit{W}}}_1^T{\mathit{h}}_1-{\hat{\beta }}_{1m}.\hfill \end{array}$$

(13)

Define $g\left(i\right)$ as follows

$$\begin{array}{c}\multicolumn{1}{c}{g\left(i\right)=-m_{i1}{\left|e_{i1}\right|}^{{\alpha }_{i1}}\mathrm{sgn}\left(e_{i1}\right)-m_{i2}{\left|e_{i2}\right|}^{{\alpha }_{i2}}\mathrm{sgn}\left(e_{i2}\right)}\\ +{\stackrel{··}{\eta }}_{\mathrm{id}}-k_{i1}{\left|S_i\right|}^{{\mu }_{i1}}\mathrm{sgn}\left(S_i\right)-k_{i2}{\left|S_i\right|}^{{\mu }_{i2}}\mathrm{sgn}\left(S_i\right)\hfill \\ -k_{i3}{S}_i-k_4\mathrm{sgn}\left(S_i\right)-{\hat{\mathit{W}}}_i^T{\mathit{h}}_i-{\hat{\beta }}_{\mathrm{im}}.\hfill \end{array}$$

Then ${\tau }_i$ can be described as ${\tau }_i=g\left(i\right)$ , define $G$ as follows

$$\mathit{G}={(g\left(1\right),g\left(2\right),g\left(3\right),g\left(4\right),g\left(5\right),g\left(6\right))}^T.$$

So $\mathit{J}{(\mathit{M}-{\mathit{M}}_s)}^{-1}{\mathit{\tau }}_a=\mathit{G}$ . The ultimate control law $\mathit{\tau }$ :

$$\mathit{\tau }=(\mathit{M}-{\mathit{M}}^S){\mathit{J}}^{-1}\mathit{G}+{\mathit{\tau }}_b.$$

(14)

Stability analysis

To ensure brevity in the proof, a stability analysis of system (11) is conducted.

• Lemma 1. $\mathrm{Barbalat}$ ¹⁹ : If $X:[0,\infty ]\to R$ is uniformly continuous,

$$\underset{t\to \infty }{\lim }\underset{0}{\overset{t}{\int }}X\left(\tau \right)d\tau$$

is existing and bounded, then $\underset{t\to \infty }{\lim }X\left(t\right)=0$ .

• Lemma 2. If Lyapunov function $V\left(t\right)\ge 0$ , $\stackrel{·}{V}\le {\alpha }_1{V}^{{\mu }_1}+{\alpha }_2{V}^{{\mu }_2}$ , then $V\to 0$ in finite time ( ${\alpha }_1<0$ , ${\alpha }_2<0$ , $0<{\mu }_1<1$ , ${\mu }_2>1$ ).

Choose Lyapunov function

$$V_1=\frac{1}{2}{S}_1^2+\frac{1}{2{\gamma }_1}{\tilde{\mathit{W}}}_1^T{\mathit{W}}_1+\frac{1}{2{\gamma }_{1\beta }}{\tilde{\beta }}_{1m}^2.$$

Taking the derivative of $V_1$

$$\begin{array}{c}\multicolumn{1}{c}{{\stackrel{·}{V}}_1=S_1{\stackrel{·}{\mathrm{S}}}_1-\frac{1}{{\gamma }_1}{\tilde{\mathit{W}}}_1^T{\stackrel{·}{\hat{\mathit{W}}}}_1-\frac{1}{{\gamma }_{1\beta }}{\tilde{\beta }}_{1m}{\stackrel{·}{\hat{\beta }}}_{1m}}\\ =S_1[-k_{11}{\left|S_1\right|}^{{\mu }_{11}}\mathrm{sgn}\left(S_1\right)-k_{12}{\left|S_1\right|}^{{\mu }_{12}}\mathrm{sgn}\left(S_1\right)-\hfill \\ -k_{13}{S}_1-{\hat{\mathit{W}}}_1^T{\mathit{h}}_1-{\hat{\beta }}_{1m}]-({\beta }_{21}+f_1)S_1\hfill \\ -\frac{1}{{\gamma }_1}{\gamma }_1{\tilde{\mathit{W}}}_1^T{\mathit{h}}_1{S}_1-\frac{1}{{\gamma }_{1\beta }}{\gamma }_{1\beta }{\tilde{\beta }}_{1m}{S}_1\hfill \\ \le -k_{11}{\left|S_1\right|}^{{\mu }_{11}+1}-k_{12}{\left|S_1\right|}^{{\mu }_{12}+1}-k_{13}{S}_1^2\hfill \\ -k_4\left|S_1\right|+{ϵ}_k\left|S_1\right|\hfill \\ \le -k_{11}{\left|S_1\right|}^{{\mu }_{11}+1}-k_{12}{\left|S_1\right|}^{{\mu }_{12}+1}-k_{13}{S}_1^2.\hfill \end{array}$$

(15)

According to Lyapunov stability theory, the system is determined to be stable.

Integrating both sides of inequality (15) yields:

$$\underset{0}{\overset{t}{\int }}{\stackrel{·}{V}}_1\mathrm{dp}\le -\underset{0}{\overset{t}{\int }}{k}_{11}{\left|S_1\right|}^{{\mu }_{11}+1}+k_{12}{\left|S_1\right|}^{{\mu }_{12}+1}+k_{13}{S}_1^2\mathrm{dp},$$

then

$$\begin{array}{c}\multicolumn{1}{c}{V_1\left(t\right)-V_1\left(0\right)\le -\underset{0}{\overset{t}{\int }}{k}_{11}{\left|S_1\right|}^{{\mu }_{11}+1}+k_{12}{\left|S_1\right|}^{{\mu }_{12}+1}}\\ +k_{13}{S}_1^2\mathrm{dp}\le 0.\hfill \end{array}$$

The initial state $V_1\left(0\right)$ is bounded, and since $V_1\ge 0$ , it follows that $V_1\left(t\right)$ is also bounded. From this, if $t\to \infty$ , $\underset{0}{\overset{t}{\int }}{k}_{11}{\left|S_1\right|}^{{\mu }_{11}+1}+k_{12}{\left|S_1\right|}^{{\mu }_{12}+1}+k_{13}{S}_1^2\mathrm{dp}$ is uniformly continuous and bounded.

According to Lemma 1, as $t\to \infty$ , $S_1\to 0$ . $S_1$ is asymptotic stable. Moreover, the estimation errors of the neural network and the second-order wave force on the last session converge to $0$ as they converge to the sliding surface.

Also choose Lyapunov function $V_2$ :

$$V_2=S_1^2.$$

Taking the derivative of $V_2$ :

$$\begin{array}{c}\multicolumn{1}{c}{{\stackrel{·}{V}}_2=2S_1{\stackrel{·}{S}}_1}\\ =-2k_{11}{\left|S_1\right|}^{{\mu }_{11}+1}-2k_{12}{\left|S_1\right|}^{{\mu }_{12}+1}-2k_{13}{S}_1^2\hfill \\ -2k_4\left|S_1\right|\hfill \\ \le -2k_{11}{\left|S_1\right|}^{{\mu }_{11}+1}-2k_{12}{\left|S_1\right|}^{{\mu }_{12}+1}\hfill \\ \le -2k_{11}{\left|V_2\right|}^{\frac{{\mu }_{11}+1}{2}}-2k_{12}{\left|V_2\right|}^{\frac{{\mu }_{12}+1}{2}}.\hfill \end{array}$$

Since $0<{\mu }_{11}<1$ and ${\mu }_{12}>1$ , it follows that $0<\frac{{\mu }_{11}+1}{2}<1$ and $\frac{{\mu }_{12}+1}{2}>1$ . From Lemma 2, $S_1\to 0$ in finite time.

The above proves that the system represented by (11) arrives at the $S_1$ surface in finite time. After the system reaches the $S_1$ surface, $S_1=0$ .

• Lemma 3. If time-varying homogeneous vector function $\mathit{F}={(f_1,f_2,\dots ,f_n)}^T$ s.t.

• (i)  $\forall \lambda >0,f_i({\lambda }^{r_1}{x}_1,{\lambda }^{r_2}{x}_2,\dots ,{\lambda }^{r_n}{x}_n)={\lambda }^{r_i+m}{f}_i(x_1,x_2,\dots ,x_n)$ , $m<0,i=1,2,\dots ,n$ ,

• (ii) The origin $0$ is the asymptotic stability point of $\mathit{F}$ , then $\mathit{F}\to 0$ in finite times.

When $S_1=0$ ,

$${\stackrel{·}{\mathrm{e}}}_{12}=-m_{11}{\left|e_{11}\right|}^{{\alpha }_{11}}\mathrm{sgn}\left(e_{11}\right)-m_{12}{\left|e_{12}\right|}^{{\alpha }_{12}}\mathrm{sgn}\left(e_{12}\right).$$

(16)

Choose Lyapunov function $V_3$ :

$$V_3=m_{11}\underset{0}{\overset{e_{11}}{\int }}{\left|p\right|}^{{\alpha }_{11}}\mathrm{sgn}\left(p\right)\mathrm{dp}+\frac{1}{2}{e}_{12}^2.$$

Take the derivative of ${\stackrel{·}{V}}_3$ :

$$\begin{array}{c}\multicolumn{1}{c}{{\stackrel{·}{V}}_3=m_{11}{\left|e_{11}\right|}^{{\alpha }_{11}}\mathrm{sgn}\left(e_{11}\right){\stackrel{·}{e}}_{11}+e_{12}{\stackrel{·}{e}}_{12}}\\ =-m_{12}{\left|e_{12}\right|}^{{\alpha }_{12}+1}\hfill \\ \le 0.\hfill \end{array}$$

When ${\stackrel{·}{V}}_3=0$ , $e_{12}=0,{\stackrel{·}{e}}_{12}=0$ . Then from (16), $e_{11}$ =0. According to $\mathrm{Laselle}$ , ²⁰ when $t\to \infty ,(e_{11},e_{12})\to (0,0)$ . It indicates that the system is asymptotic stable.

Define ${\mathit{N}}_1={(n_{11},n_{12})}^T$ , where

$$\{\begin{array}{c}{n}_{11}(e_{11},e_{12})={\stackrel{·}{e}}_{11}\hfill \\ n_{12}(e_{11},e_{12})={\stackrel{·}{e}}_{12}.\hfill \end{array}$$

The above system of equations can be written as

$$\{\begin{array}{c}{n}_{11}(e_{11},e_{12})=\stackrel{·}{x}-{\stackrel{·}{\eta }}_{1d}\hfill \\ n_{12}(e_{11},e_{12})=-m_{11}{\left|e_{11}\right|}^{{\alpha }_{11}}\mathrm{sgn}\left(e_{11}\right)-\hfill \\ m_{21}{\left|e_{12}\right|}^{{\alpha }_{12}}\mathrm{sgn}\left(e_{12}\right),\hfill \end{array}$$

by changing the independent variable to $({\lambda }^{r_1}{e}_{11},{\lambda }^{r_2}{e}_{12})$ , $\lambda >0$ , then the following system of equations can be obtained:

$$\{\begin{array}{c}{n}_{11}({\lambda }^{r_1}{e}_{11},{\lambda }^{r_2}{e}_{12})={\lambda }^{r_2}(\stackrel{·}{x}-{\stackrel{·}{\eta }}_{1d})\hfill \\ n_{12}({\lambda }^{r_1}{e}_{11},{\lambda }^{r_1}{e}_{12})=-m_{11}{\left|{\lambda }^{r_1}{e}_{11}\right|}^{{\alpha }_1}\mathrm{sgn}\left(e_{11}\right)-\hfill \\ m_{12}{\left|{\lambda }^{r_2}{e}_{12}\right|}^{{\alpha }_2}\mathrm{sgn}\left(e_{12}\right).\hfill \end{array}$$

(17)

Command $m=\frac{{\alpha }_{11}-1}{{\alpha }_{11}+1}<0$ , $r_1=\frac{2}{1+{\alpha }_{11}}$ , $r_2=1$ . Then the system of equation (17) can be transformed into

$$\{\begin{array}{c}{n}_{11}({\lambda }^{r_1}{e}_{11},{\lambda }^{r_2}{e}_{12})={\lambda }^{r_1+m}{n}_{11}(e_{11},e_{12})\hfill \\ n_{12}({\lambda }^{r_1}{e}_{11},{\lambda }^{r_2}{e}_{12})={\lambda }^{r_2+m}{n}_{11}(e_{11},e_{12}).\hfill \end{array}$$

(18)

From the above proof, Lemma 3 and (18), it can be shown that the tracking error of system (11) can converge to $0$ in finite time $t_1$ under the defined control law (13).

• Remark 3. $t_1$ denotes the upper bound on convergence time of system (11).

Here, $t_1$ satisfies an upper time limit for:

$$\begin{array}{c}\multicolumn{1}{c}{t_1\le \frac{2V_2{\left(0\right)}^{\frac{1-{\mu }_{11}}{2}}}{k_{11}(1-{\mu }_{11})}+\frac{1}{k_{11}}\underset{0}{\overset{V_2\left(0\right)}{\int }}\frac{1}{-k_{11}-k_{12}{V}_2^{\frac{{\mu }_{12}-{\mu }_{11}}{2}}}d{V}_2}\\ +\frac{{\left|e_{11}\right|}_{\max }^{1-{\alpha }_{11}}}{m_{11}(1-{\alpha }_{11})}\hfill \end{array}$$

In the following section, a brief analysis of global finite-time stability will be conducted, defining $\mathit{S}$ as:

$$\mathit{S}={(S_1,S_2,S_3,S_4,S_5,S_6)}^T.$$

Choose Lyapunov function $V_4$ :

$$V_4=\frac{1}{2}{\mathit{S}}^T\mathit{S}+\sum _{i=1}^6\frac{1}{2{\gamma }_i}{\widetilde{{\mathit{W}}_{\mathit{i}}}}^T{\mathit{W}}_{\mathit{i}}+\sum _{i=1}^6\frac{1}{2{\gamma }_{i\beta }}{\tilde{\beta }}_{\mathrm{im}}^2.$$

Modeled on the (15), take the derivative of $V_4$ :

$$\begin{array}{c}\multicolumn{1}{c}{{\stackrel{·}{V}}_4=\sum _{i=1}^6{S}_i{\stackrel{·}{\mathrm{S}}}_i-\frac{1}{{\gamma }_i}{\tilde{\mathit{W}}}_i^T\widehat{{\mathit{W}}_i}-\frac{1}{{\gamma }_{i\beta }}{\tilde{\beta }}_{\mathrm{im}}{\stackrel{·}{\hat{\beta }}}_{\mathrm{im}}}\\ \le \sum _{i=1}^6(-k_{i1}{\left|S_i\right|}^{{\mu }_{i1}+1}-k_{i2}{\left|S_i\right|}^{{\mu }_{i2}+1}-k_{i3}{S}_i^2).\hfill \end{array}$$

Similarly, according to Lemma 2 and (15), it can be demonstrated that both the neural network estimation error ${\tilde{\mathit{W}}}_i$ and the estimation error of ${\beta }_m$ converge to $0$ . Then by Lemma 2, it is easy to show that all defined sliding surfaces $S_i$ converge in finite time to 0 under its approach law (12). Thus, $\mathit{S}$ converges to $0$ in finite time.

Next, define Lyapunov function $V_{5i}$ as follows:

$$V_{5i}=m_{i1}\underset{0}{\overset{e_{i1}}{\int }}{\left|p\right|}^{{\alpha }_{i1}}\mathrm{sgn}\left(p\right)\mathrm{dp}+\frac{1}{2}{e}_{i2}^2,i=1,2,\dots ,6.$$

Similarly, the homogeneous vector functions ${\mathit{N}}_2$ , ${\mathit{N}}_3$ , ${\mathit{N}}_4$ , ${\mathit{N}}_5$ , and ${\mathit{N}}_6$ can be defined for the remaining five degrees of freedom equations in system (10).

It is subsequently established that system (11) reaches the stability point (0,0) within a finite time of systematic errors after reaching the sliding surface. Based on this proof, it can be demonstrated that the tracking errors of the other five subsystems, as delineated by system (10), converge to $0$ in finite time $t_i$ for $i=2,3,\dots ,6$ under the influence of the respective control laws ${\tau }_i$ for $i=2,3,\dots ,6$ .

In conclusion, under the influence of the designed control law (14), the tracking error $({\mathit{E}}_1,{\mathit{E}}_2)$ of system (10) converges to 0. Upon conclusion of the proof, an upper bound on the time $T$ can be established, within which the tracking error of the entire system converges to 0 in finite time, satisfying the following inequality:

$$\begin{array}{c}\multicolumn{1}{c}{T\le \underset{i}{\max }[\frac{2{\left|S_i\left(0\right)\right|}^{1-{\mu }_{i1}}}{k_{i1}(1-{\mu }_{i1})}+\frac{{\left|e_{i1}\right|}_{\max }^{1-{\alpha }_{i1}}}{m_{i1}(1-{\alpha }_{i1})}}\\ +\frac{1}{k_{i1}}\underset{0}{\overset{S_i^2\left(0\right)}{\int }}\frac{2S_i}{-k_{i1}-k_{i2}{\left|S_i\right|}^{{\mu }_{i2}-{\mu }_{i1}}}d{S}_i].\hfill \end{array}$$

In practical engineering, in order to avoid highly oscillating situations, the $\mathrm{sat}$ function instead of the symbolic function $\mathrm{sgn}$ is introduced, which is defined as follows:

$$\begin{array}{c}\multicolumn{1}{c}{\mathrm{sat}\left(\frac{S_i}{{\mathrm{\Delta }}_i}\right)=\{\begin{array}{c}1,S_i>{\mathrm{\Delta }}_i\hfill \\ \multicolumn{1}{c}{S_i/{\mathrm{\Delta }}_i,|S_i/{\mathrm{\Delta }}_i|\le 1}\\ \multicolumn{1}{c}{-1,S_i<{\mathrm{\Delta }}_i.}\end{array}}\end{array}$$

Simulations

Simulation and results

In order to verify the effectiveness of the proposed control rate for liquid-laden vessels, the following simulations are conducted in MATLAB. This paper simulates system (11) with the ultimate tracking objectives being the actual position and velocity (the derivative of position). In this paper, the system (11) is simulated. Given that the ultimate tracking objectives are the actual position and velocity (the derivative of position), a time-varying function with non-zero derivatives is selected for tracking purposes. To validate the performance of the control law, a time-varying trigonometric function is specifically chosen as the desired function. The pair $({\eta }_{1d},{\stackrel{·}{\eta }}_{1d})$ is selected to be $(A\sin \left(\omega t\right),A\omega \cos \left(\omega t\right))$ .

In the expectation function of the trace, we choose $A=1,\omega =1$ . The initial error is chosen to be (2,-1).

Figure 3 illustrates the systematic error when the control rate is absent, demonstrating that the ship’s trajectory does not converge to the desired path, while the velocity error shows significant oscillations.

Figure 3.

Systematic errors without control laws.

Firstly, an initial selection of the controller parameters was made as follows, $m_{11}=1$ , $m_{12}=1$ , ${\alpha }_{11}=0.5$ , ${\alpha }_{12}=\frac{2}{3}$ , $k_{11}=2$ , $k_{12}=2$ , $k_{13}=1$ , $k_4=2$ , ${\gamma }_{1\beta }=2$ , ${\gamma }_1=1$ , ${ϵ}_k=0.05,{\mu }_{11}=0.5$ , ${\mu }_{12}=1.5$ . Figure 4 displays the simulation of control inputs and control laws over time, based on the initial conditions and expected tracking targets. Using this control input, the position and velocity errors are shown in Figure 5. As seen in Figure 5, the systematic error approaches nearly 0. Figure 6 provides a close-up of the systematic error from Figure 5. The numerical results indicate that the position error satisfies ( $|x-{\eta }_{1d}|\le 0.01$ ) after 7.08 s, and the velocity error satisfies ( $|\stackrel{·}{x}-{\stackrel{·}{\eta }}_{1d}|\le 0.02$ ) after 8.65 s.

Figure 4.

Control law.

Figure 5.

Position error-1 and velocity error-1.

Figure 6.

Image enlargement of error-1.

In order to test the performance of the given control rate, based on the previously given parameters of the controller, several parameters are modified. The results shown in Figures 7 and 8 are obtained by increasing both $m_1$ and $m_2$ to $2$ on the basis of the initial control parameters, and it can indicate that $|x-{\eta }_{1d}|\le 0.01$ after $5.2s$ and $|\stackrel{·}{x}-{\stackrel{·}{\eta }}_{1d}|\le 0.02$ after $5.7s$ . This suggests that with appropriate increases in $m_1$ , $m_2$ , the controller will cause the system to converge more quickly to values approximating 0. In fact, the appropriate increase of parameters $k_{11}$ , $k_{12}$ , $k_{13}$ , ${\alpha }_{12}$ , ${\mu }_{12}$ , the closer the value of $k_4$ is to the magnitude of the neural network’s weights error, and the closer the value of ${\alpha }_{11}$ , ${\mu }_{11}$ is to 1, the better the final control is. However, it should be noted that excessively large parameters may cause the systematic error not to converge to 0 in the end or produce a high degree of oscillations.

Figure 7.

Position error-2 and velocity error-2.

Figure 8.

Image enlargement of error-2.

In the Stability Analysis section, it is demonstrated that, under the designed control law, the system error can converge to 0 in finite time. In the numerical results, the system error is generally reduced to less than 2% of its original value relative to the initial value. However, in the actual numerical simulations, the use of the sign function ( $\mathrm{sgn}$ ) in sliding mode control generates discontinuous signals during control switching. When the system state is near the sliding surface, frequent switching of control inputs between positive and negative directions can trigger system oscillations. As a result, the final system error may vary continuously in a small neighbourhoodneighborhood of 0.

This research employs a control law based on the signum function, which is set-valued at zero ( $\mathrm{sgn}\left(0\right)=[-1,1]$ ). This feature is vital for the stability and robustness of the closed-loop system. Although this paper does not specifically address robustness against external disturbances and parameter uncertainties, the set-valued control input enhances adaptability in practical scenarios. Additionally, its set-valued nature aids in time-discretization, ensuring the effectiveness of the discretized control law. ²¹ The well-posedness of the closed-loop system’s solutions is analyzed using Filippov’s differential inclusions, providing a rigorous mathematical framework to handle discontinuities in the control law. This framework ensures finite-time convergence to the desired trajectory, as shown by numerical simulations.

The errors exhibited in Figures 6 and 8 converge to 0 in finite time; however, both the position and velocity errors oscillate during this convergence, with the velocity error displaying more pronounced oscillations. Consequently, a preliminary attempt is made to apply the control law involving a saturation function ( $\mathrm{sat}$ ). ²² Different values of $\mathrm{\Delta }$ are simulated while maintaining the controller parameters at their initial values outlined in the simulation setup. The selection of optimal parameters for $\mathrm{\Delta }$ is challenging; therefore, an analysis will be conducted numerically within the interval [0, 2]. The analysis commences with $\mathrm{\Delta }=0.1$ and increment by 0.1 to simulate the amplitudes of the position and velocity errors after a sufficiently lengthy period. As illustrated in Figure 9, the simulation results indicate that $\mathrm{\Delta }$ is relatively small across three ranges: 0.1 to 0.2, 0.4 to 1.0, and 1.5 to 2.0. After more careful selection of $\mathrm{\Delta }$ , we further investigate $\mathrm{\Delta }=0.11$ , obtaining the simulation results shown in Figure 10. A comparison of Figure 10 with Figure 6 demonstrates that the oscillations in both the position and velocity errors are significantly reduced, thereby illustrating the effectiveness of the saturation function introduced herein. It is important to note that the numerical simulation of the closed-loop system described here does not accurately represent real marine conditions. In actual six-degree-of-freedom ship control systems, the accuracy of position error is often prioritized over that of velocity error (to mitigate the ship’s own oscillations), making it necessary to further refine and adjust more detailed parameters in future studies.

Figure 9.

Max system error versus $\mathrm{\Delta }$ .

Figure 10.

Position error-3 and velocity error-3.

To further validate the effectiveness of the proposed control method, a comparison is conducted with the traditional PID controller. Figures 11 and 12 illustrate the changes in position error and velocity error over time for both control methods under the same initial conditions. As can be seen from the figures, the control method proposed in this paper outperforms the traditional PID controller in terms of convergence speed and error accuracy. Specifically, the proposed method can reduce the position error and velocity error to a lower level in a shorter time, with smaller error fluctuations. Figure 13 further compares the time required for both control methods to achieve the same level of precision (position error $\le 0.01$ ). The results indicate that the method proposed in this paper can reach this precision in 1.5 s, while the PID controller requires 3.5 s. This comparison fully demonstrates the advantages of the control method proposed in this paper in terms of speed and accuracy.

Figure 11.

Position error over time.

Figure 12.

Velocity error over time.

Figure 13.

Comparison of time required to achieve the same accuracy.

Impact of RBFNN accuracy on closed-loop performance

The influence of Radial Basis Function Neural Network (RBFNN) accuracy on closed-loop performance was investigated through a numerical study. The study adjusted the RBFNN accuracy by varying its parameters, specifically the number of hidden neurons and the spread of the radial basis functions. The system configuration and control parameters described in Section were utilized, with a desired trajectory of ${\eta }_d\left(t\right)=\sin \left(t\right)$ and a desired velocity of ${\stackrel{·}{\eta }}_d\left(t\right)=\cos \left(t\right)$ . RBFNNs with 5, 10, and 20 hidden neurons were set up, and the spread of the radial basis functions was adjusted accordingly. Under each configuration, the tracking errors (both position and velocity errors) and convergence time of the closed-loop system were recorded.

The results as shown in Figure 14 indicate that higher RBFNN accuracy leads to smaller tracking errors and shorter convergence times. Specifically, increasing the number of hidden neurons from 5 to 20 reduced the position error by 80%, the velocity error by 60%, and the convergence time by 33%. However, further increases in the number of hidden neurons yield diminishing returns in performance improvement while significantly increasing computational complexity. Therefore, in practical applications, it is essential to select RBFNN parameters based on the specific requirements and computational resources of the system to achieve a balance between accuracy and computational efficiency.

Figure 14.

Comparison of neural networks with different accuracies.

Conclusion

In this paper, a six-degree-of-freedom finite-time trajectory tracking sliding-mode control method for an unmanned powered ship with liquid-tank sloshing is proposed. A method of avoiding complex terms is carried out for the coupled equations of the ship with liquid tank sway, and the motion control equations of the liquid-carrying ship are derived. An adaptive control law is designed by constructing a nonlinear sliding mode surface and adopting an adaptive control method, which ensures that the ship attitude as well as the trajectory can be adjusted to the desired value in a finite time. The design of the controller is proved to be globally stable using Lyapunov stability theory and homogeneity theory. Simulation results show that the liquid-laden ship achieves the finite-time trajectory tracking control objective within the constraints of the controller. Meanwhile, in this study, the designed adaptive neural network nonlinear sliding mode controller requires a very high approximation accuracy of the nonlinear wobble force, as well as the difficulty in parameter design of this sliding mode controller at the time of use, which are the problems to be solved in the future.