Global fast terminal sliding mode control based on radial basis function neural network for course keeping of unmanned surface vehicle

Abstract

A scheme to solve the course keeping problem of the unmanned surface vehicle with nonlinear and uncertain characteristics and unknown external disturbances is investigated in this article. The chattering existing in global fast terminal sliding mode controller in solving the course keeping problem of the unmanned surface vehicle with external disturbance is analyzed. To reduce the chattering and eliminate the influence of the unknown disturbance, an adaptive global fast terminal sliding mode controller based on radial basis function neural network is developed. The equivalent control that usually requires a precise model information of the system is computed using the radial basis function neural network. The weights of the neural network are online adjusted according to the adaptive law that is derived using Lyapunov method to ensure the stability of the closed-loop system. Using the online learning of the neural network, the nonlinear uncertainty of the system and the unknown disturbance of external environment are compensated, and the system chattering is reduced effectively as well. The simulation results demonstrate that the proposed controller can achieve a good performance regarding the fast response and smooth control.

Keywords

Unmanned surface vehicle course keeping RBF global fast terminal sliding mode control finite-time control

Introduction

Unmanned surface vehicle (USV) plays a vital role in anti-submarine warfare, mine countermeasures, environmental detection, water sampling, personnel search and rescue in the ocean, and so on.^1
–3 An effective course keeping controller is essential for the highly autonomous USV in navigation.⁴ Because of the complex, nonlinear, and uncertain characteristics of the ship,⁵ coupled with the complex and changeable external navigation environment such as wind, wave, and currents, the course keeping problem becomes more complicated. It is a challenge to design a high-performance course keeping controller that has strong robustness and fast response speed.

Many studies of the USV’s course keeping have been done by scholars. The conventional proportional–integral–derivative (PID) controller is a simple and easily constructed controller. However, when the external environment changes, the performance of the PID controller is affected.⁶ Considering the uncertainty and external disturbance of the system, some new control strategies have been proposed by scholars to obtain better control performance in recent years. The linear quadratic regulator theory,⁷ model predictive control (MPC) technology,⁸ backstepping method,^9
–11 dynamic surface control technology,¹² fuzzy control,^13

–16 neural network,^17
–19 sliding mode control,^20

–23 and so on, are all used to design the controller for course keeping of the ship. A linear quadratic adaptive controller combined with a Riccati-based anti-windup compensator has been designed for course keeping of the ship with uncertain dynamics using the Certainty Equivalence Principle.⁷ The stability of the system under arbitrary disturbance is guaranteed with consideration of the input saturation of the rudder angle. A disturbance compensating MPC algorithm has been applied to solve the course keeping problem of ship as well.⁸ The state constraints in the presence of external environmental disturbances are satisfied. The aforementioned two controllers have good performance in terms of reducing heading error and satisfying yaw velocity and actuator saturation constraints. They adopt linear models to describe the steering characteristics of the ship in the control law designs, but in fact, the steering system of the ship has the characteristics of nonlinear uncertainties. There are many control strategies developed to deal with the uncertain nonlinear systems in recent years.²⁴ For example, an innovative adaptive approximation-based regulation control scheme is developed for a general class of uncertain nonlinear systems, including unknown dynamics, which are not feedback linearizable.²⁵ A backpropagating constraints-based trajectory tracking controller is developed for trajectory tracking of a quadrotor with complex unknowns and cascade constraints arising from constrained actuator dynamics, including saturations and dead zones.²⁶ For the ship with nonlinear manipulation characteristics and uncertainties, intelligent control algorithms combined with some advanced control methods are also used in the controller design for course keeping. A backstepping controller combined with radial basis function (RBF) neural network has been proposed for course keeping of the ship with uncertainties and unknown external disturbances.⁹ An adaptive nonlinear control strategy combined dynamic surface control and Nussbaum gain function with backstepping algorithm has been proposed for the course keeping of ships with parameter uncertainties and completely unknown control coefficient.¹² Fuzzy control is also used to deal with the system with unknown dynamics.²⁷ A model reference adaptive robust fuzzy control algorithm has been presented for course keeping of the ship.¹³ Considering the unknown yaw dynamics and measurement noises, a robust adaptive course keeping controller of USV has been developed with the aid of a predictor, neural networks, and a modified dynamic surface control technique.¹⁷ An integrated nonlinear feedback course keeping controller is developed to improve the robustness of motion control in heavy sea states.²⁸ These aforementioned control strategies effectively improve the control performance of the nonlinear system and reduce the impact of system uncertainties and external disturbance on the system. However, these algorithms are computationally complex and difficult to implement. Although the stability of the systems is discussed, they cannot guarantee the infinite time convergence of the system states and the convergence speed cannot be guaranteed.

As a real-time control system, the response speed is an important control quality index to evaluate its performance . The finite-time control can solve the problem of finite-time convergence of the system states.^29
–31 Homogeneous dynamic system theory,³² finite-time Lyapunov theory,³³ terminal sliding mode (TSM) control,^34
–36 and so on, are the main methods of the finite-time control. The TSM control introduces a nonlinear switching surface, and the system state can slide to the equilibrium point in finite time after reaching the sliding surface. However, when the system state is close to the equilibrium point, the convergence speed is slower than that of the linear sliding surface. To improve the convergence speed of the system states, the global fast terminal sliding mode (GFTSM) controller is chosen for the course keeping.³⁷ GFTSM control technology can ensure that the tracking error converges to the equilibrium point at a faster speed in finite time.³⁸ Some scholars have applied the GFTSM control technology to solve the problem of ship’s course keeping.^39,40 But GFTSM control has some disadvantages. Firstly, the system dynamics is needed in the calculation of the equivalent control that limits its application. Especially, some parameters are difficult to obtain because the navigation environment is complex and the cruising state is changeable. Secondly, the GFTSM control requires the external disturbance to be bounded. Actually, it is hard to estimate the upper bound of its disturbance in real applications. Thirdly, the system chattering is inevitable with the consideration of the external disturbance and modeling error because of the discontinuous item in the switch control. It is aggravated as the disturbance increases. The chattering will accelerate the wear of the mechanical system and it is not suitable for application in the actual mechanical systems.

The neural network that can effectively solve the problems with regard to unknown nonlinear systems attracts many concerns from researchers.^41,42 The combination of the neural network and GFTSM control technology not only compensates the approximation error of the neural network but also reduces the dependence of the GFTSM controller on the system parameters.^43,44 The system chattering is greatly reduced as well. A novel adaptive GFTSM controller for course keeping of the USV based on RBF neural network is proposed in this article. The RBF neural network is used to compute the equivalent control, and the GFTSM control technology is combined to achieve heading angle tracking in a fast speed without knowing the upper bound of the disturbance and the precise model information of the system.

The remainder of this article is arranged as follows. In the section “problem formulation,” a brief description of the dynamics model of USV is proposed. The section “GFTSM controller design for course keeping” presents the GFTSM controller design for course keeping of the USV. The novel adaptive GFTSM controller combined with RBF neural network is proposed in the section “GFTSM-NN controller design for course keeping.” Simulation results and analyses of the proposed control system are shown in section “Simulation results.” In the last section, the whole article is concluded.

Problem formulation

The motion of the USV in six degrees of freedom (DOF) is shown in Figure 1. The body-fixed reference frame oxyz is fixed to the USV. The USV can move and rotate along the tree coordinate axes. The earth-fixed $o_{0} x_{0} y_{0} z_{0}$ is an inertial reference frame fixed to the earth’s surface. Assuming that the USV is homogeneous mass distributed and xz-plane symmetrical, when the origin of the body-fixed reference frame is chosen to be coincident with the center of the gravity and the dynamics associated with the motion in heave, roll, and pitch are neglected, the MMG model of the USV can be written as⁴⁵

{\begin{array}{l} Surge : m (\dot{u} - v r - x_{C} r^{2}) = X \\ Sway : m (\dot{v} + u r + x_{C} \dot{r}) = Y \\ Yaw : I_{Z} \dot{r} + m x_{C} (\dot{v} + u r) = N \end{array}

where u, v, and r stand for the surge velocity, sway velocity, and yaw rate, respectively; X, Y, and N are external forces with respect to surge, sway, and yaw; m is the mass of the USV; I_Z is the rotational inertias; and x_C is the coordinate of the ship center in the body-fixed reference frame. Assuming that the mean forward speed is constant U for constant thrust, and the sway velocity v, the yaw rate r, the rudder angle δ, and the perturbations are small, the steering equations of motion are completely decoupled from the speed equation as

{\begin{array}{l} m (\dot{v} + U r + x_{C} \dot{r}) = Y \\ I_{Z} \dot{r} + m x_{C} (\dot{v} + U r) = N \\ \dot{ψ} = r (t) \end{array}

where ψ is the heading angle. The hydrodynamic force and moment can be modeled as

{\begin{matrix} Y = Y_{\dot{v}} \dot{v} + Y_{\dot{r}} \dot{r} + Y_{v} v + Y_{r} r + Y_{δ} δ \\ N = N_{\dot{v}} \dot{v} + N_{\dot{r}} \dot{r} + N_{v} v + N_{r} r + N_{δ} δ \end{matrix}

where $Y_{\dot{v}}$ , $Y_{\dot{r}}$ , Y_v, Y_r, $Y_{δ}$ , $N_{\dot{v}}$ , $N_{\dot{r}}$ , N_v, N_r, and $N_{δ}$ are non-dimensional coefficients of hydrodynamics. The heading angle ψ of the USV is controlled by the rudder angle δ when it sails. On the basis of the ship’s Norrbin nonlinear mathematical model¹⁰ and taking the disturbance of the external environment into account, the nonlinear kinematic equation of the ship is described as

{\begin{array}{l} \dot{ψ} = r \\ \dot{r} ​ = - \frac{K}{T} H (r) + \frac{K}{T} δ + d \end{array}

where d is the disturbance term that includes the unknown external disturbances and the modeling errors, and K and T are hydrodynamic coefficients related to the parameters such as quality and shape of the ship, cruise speed, and the depth of the water. They are positive when the ship is line movement stable and difficult to obtain.²⁹ $H (r)$ is a nonlinear function of yaw rate which can be written as

H (r) = a_{1} r + a_{2} r^{3} + ...

where a₁ and a₂ are Norrbin coefficients related to the parameters such as quality and shape of the ship, cruise speed, and the depth of the water. For the sake of writing conveniently, equation (4) can be rewritten as

{\begin{array}{l} \dot{ψ} = r \\ \dot{r} = f + G δ + d \end{array}

where $f = - \frac{K}{T} H (r)$ is nonlinear and parameters uncertainty, and $G = \frac{K}{T}$ is positive when the ship is line movement stable. During navigation, the course keeping controller of the ship is designed based on equation (6). The task is to seek an adaptive feedback control law such that the desired heading angle of the USV can be tracked quickly and accurately.

Figure 1.

Six DOF motion of the USV. DOF: degree of freedom; USV: unmanned surface vehicle.

GFTSM controller design for course keeping

A course keeping control algorithm based on the GFTSM control theory is proposed to accomplish the heading angle tracking of the USV in this section. Assume that the desired heading angle is $ψ_{d}$ , which is sufficiently smooth and has the second-order derivative almost everywhere. The heading tracking error is defined as

e = ψ - ψ_{d}

The task of the controller design is to make the tracking error e converge to zero in finite time. According to GFTSM control theory, design the global fast sliding surface variable as

s = \dot{e} + α e + β e^{q_{0} / p_{0}}

where $α > 0$ , $β > 0$ , p₀, and q₀ are odd integers that satisfy $p_{0} > q_{0}$ , which are parameters to be determined. The tracking error e is the state variable. Select the sliding surface variable $s = 0$ , then $\dot{e} = - α e - β e^{q_{0} / p_{0}}$ . Define the Lyapunov function $V_{1} = \frac{1}{2} e^{2}$ , and take the time derivative of it, that is, ${\dot{V}}_{1} = e \dot{e} = - α e^{2} - β e^{\frac{p_{0} + q_{0}}{p_{0}}} \leq 0$ . It can be seen that e converges to zero when s equals zero. To achieve convergence of the tracking error e to zero, s must converge to zero by means of the control.

When the state of the system is far away from the equilibrium, the convergence rate is mainly determined by the fast terminal attractor $\dot{e} = - β e^{q_{0} / p_{0}}$ . On the contrary, when the state of the system is close to the equilibrium, the convergence rate is mainly determined by the item $\dot{e} = - α e$ , and the state variable decays exponentially. By solving the differential equation $s = \dot{e} + α e + β e^{q_{0} / p_{0}} = 0$ , it can be obtained that the time of the system state converges from any state that $e (0) \neq 0$ to $e (t_{s e}) = 0$ on the selected sliding surface is

t_{s e} = \frac{p_{0}}{α (p_{0} - q_{0})} ln \frac{α e {(0)}^{(p_{0} - q_{0}) / p_{0}} + β}{β}

Take the time derivative of equation (8) along with equations (6) and (7), the following equation is obtained

\begin{matrix} \dot{s} = \ddot{e} + α \dot{e} + \frac{β q_{0}}{p_{0}} e^{(q_{0} - p_{0}) / p_{0}} \dot{e} \\ = f + G δ + d - {\ddot{ψ}}_{d} + α \dot{e} + \frac{β q_{0}}{p_{0}} e^{(q_{0} - p_{0}) / p_{0}} \dot{e} \end{matrix}

Based on sliding mode control theory, the control δ consists of the equivalent control $δ_{eq}$ and the switch control $δ_{n}$ two parts, that is, $δ = δ_{eq} + δ_{n}$ . Assume that the disturbance is zero and $\dot{s} = 0$ , the equivalent control is obtained

δ_{eq} = \frac{1}{G} ({\ddot{ψ}}_{d} - f - α \dot{e} - \frac{β q_{0}}{p_{0}} e^{(q_{0} - p_{0}) / p_{0}} \dot{e})

Design the switch control $δ_{n}$ as

δ_{n} = - φ s - γ s^{q / p}

where φ and γ are positive parameters to be determined, and p and q are odd integers which satisfy $p > q$ . Then, the control δ is written as

δ = \frac{1}{G} ({\ddot{ψ}}_{d} - f - α \dot{e} - \frac{β q_{0}}{p_{0}} e^{(q_{0} - p_{0}) / p_{0}} \dot{e}) - φ s - γ s^{q / p}

Take equation (13) into equation (10), the derivative of s is rewritten as

\dot{s} = - G φ s - s^{q / p} (G γ - \frac{d}{s^{q / p}})

Define the Lyapunov function $V_{2} = \frac{1}{2} s^{2}$ and take the time derivative of it, that is

\begin{matrix} {\dot{V}}_{2} = s \dot{s} = - G φ s^{2} - s^{(p + q) / p} (G γ - \frac{d}{s^{q / p}}) \\ = - φ' s^{2} - γ' s^{(p + q) / p} \end{matrix}

where $φ' = G φ > 0$ , $γ' = G γ - \frac{d}{s^{q / p}}$ . If d has an upper bound D and satisfies the following equation

γ > \frac{D}{G | s^{q / p} |}

then $γ' > 0$ and ${\dot{V}}_{2} = - φ' s^{2} - γ' s^{(p + q) / p} \leq 0$ . According to Lyapunov stability theory, the system is stable. The equation (16) can be rewritten as $| s | > {(\frac{D}{G γ})}^{p / q}$ , from which we can see that the state of the system can reach the neighborhood of the sliding mode surface small enough as long as the values of γ and $p / q$ are large enough. The differential equation (14) can be rewritten as

\dot{s} = - φ' s - γ' s^{q / p}

By solving the differential equation (17), the time of the sliding surface variable converges from any state that $s (0) \neq 0$ to $s (t_{s}) = 0$ is obtained

t_{s} = \frac{p}{φ' (p - q)} ln \frac{φ' s {(0)}^{(p - q) / p} + γ'}{γ'}

The equation (16) can be rewritten as

γ = \frac{D}{G | s^{q / p} |} + \frac{σ}{G}

where σ is a positive real number. Because $L, γ'$ , then the arriving time in equation (18) can be rewritten as

t_{s} \leq \frac{p}{φ' (p - q)} ln \frac{φ' s {(0)}^{(p - q) / p} + σ}{σ}

It shows that the sliding surface variable can converge to zero in finite time as long as the parameters are properly selected. It also can be seen from equation (19) that because $D > 0$ , the parameter γ is discontinuous, the switch control and the control of the system are discontinuous as well. It will lead to the system chattering. The bigger the upper bound of the disturbance, the bigger the amplitude of γ and greater the amplitude of the chattering.

GFTSM-NN controller design for course keeping

In the GFTSM controller design, the precise dynamics model of the system and the upper bound of the disturbance should be known. While the dynamics model of the USV contains nonlinear uncertainty and the parameters are difficult to acquire. They change with the cruise speed and external environment during the actual navigation. The upper bound of the disturbance is hard to estimate as well because of the complex and changeable navigation environment. In addition, the system chattering is inevitable as analyzed in the last section. To solve these problems, an adaptive GFTSM controller based on RBF neural network (GFTSM-NN) is proposed for course keeping control in this section. It integrates GFTSM control technology and RBF neural network to achieve an improved performance.

The controller design and stability analysis

The block diagram of the GFTSM-NN control system for course keeping is shown in Figure 2. From which we can see that the control $\hat{δ}$ is equal to the sum of equivalent control ${\hat{δ}}_{eq}$ and switch control $δ_{n}$ . The equivalent control is computed by the RBF neural network to compensate the external disturbances and system uncertainties. The input of the RBF neural network is the sliding surface variable s which is designed as the GFTSM controller, and the output of the RBF neural network is the equivalent control ${\hat{δ}}_{eq}$ . The weights of the RBF neural network are adjusted online, and the adaptive law is derived using Lyapunov method. The switch control $δ_{n}$ is designed as the GFTSM controller which is shown in equation (12). To ensure that the system state converges to zero in finite time, define the sliding surface variable as equation (8). Considering that the rudder angle of the ship has saturation characteristics during its rotation, a saturation function is added to the control to limit the amplitude of the motion.

Figure 2.

Block diagram of the GFTSM-NN control system. GFTSM-NN: global fast terminal sliding mode controller based on RBF neural network; RBF: radial basis function.

To compensate the external disturbance, the control is designed as

δ^{*} = δ_{eq}^{*} + δ_{n}

where

δ_{e q}^{*} = \frac{1}{G} ({\ddot{ψ}}_{d} - f - α \dot{e} - \frac{β q_{0}}{p_{0}} e^{(q_{0} - p_{0}) / p_{0}} \dot{e} - d)

Compared the equation (22) with equation (11), the added term “ $- d$ ” is used to compensate the unknown external disturbance. Take equations (21) and (22) into equation (10), the derivative of the sliding surface variable can be written as

\dot{s} = - G φ s - G γ s^{q / p}

Since $s \dot{s} = - G φ s^{2} - G γ s^{(p + q) / p} \leq 0$ , the reaching condition is satisfied. The sliding surface variable and the system state converge to zero. However, the control δ* is impossible to obtain because $δ_{eq}^{*}$ depends on the dynamics model of the system that are difficult to acquire and the disturbance is unknown. To reduce the dependence of control on the mathematical model of the system and the external disturbance, the RBF neural network is used to approximate $δ_{eq}^{*}$ and the equivalent control is computed. By this way, there is no need to know the complete dynamics of the system, and the external disturbance and the system uncertainties are compensated. Design the actual control as

\hat{δ} = {\hat{δ}}_{eq} + δ_{n}

and submit it to equation (10), the derivative of the sliding surface variable can be written as

\dot{s} = G (- φ s - γ s^{q / p} + \hat{δ} - δ^{*})

Assume that the RBF neural network has n hidden layer nods and the input of the neural network is s. Then the output of the neural network ${\hat{δ}}_{eq}$ can be written as

{\hat{δ}}_{eq} = w^{T} h

where $w = [w_{1} w_{2} ... w_{n}]^{T}$ is the weight vector of the neural network and $h = [h_{1} h_{2} ... h_{n}]^{T}$ is the output vector of the hidden layer neurons in the RBF neural network. The output of the ith hidden layer neuron $h_{i}$ is

h_{i} = e x p (- \frac{{‖ s - c_{i} ‖}^{2}}{b_{i}^{2}}), (i = 1, 2, ..., n)

where c_i is the center value of the i th hidden layer neuron and b_i is the width of the i th hidden layer neuron. There is an ideal neural network weight vector $w^{*}$ so that the RBF neural network can approximate $δ_{eq}^{*}$ with a minimum error ε, that is

δ_{eq}^{*} = w^{* T} h + ε

where ε is a very small positive number. The approximate error can be described as

{\tilde{δ}}_{eq} = {\hat{δ}}_{eq} - δ_{eq}^{*} = w^{T} h - w^{*T} h - ε = {\tilde{w}}^{T} h - ε

where ${\tilde{w}}^{T} = w^{T} - w^{*T}$ . Then $\dot{s}$ can be derived from equations (25) and (29)

\dot{s} = G (- φ s - γ s^{q / p} + {\tilde{w}}^{T} h - ε)

The adaptive law can be derived using the Lyapunov method to guarantee the system stability. The sliding surface variable is designed as equation (8). It has been proved in the previous section that the tracking error e will converge to zero when the sliding surface variable s is zero. Then, it should be proved that s and $\tilde{w}$ are convergent to zero. Consider the following Lyapunov function candidate

V_{3} = \frac{1}{2} s^{2} + \frac{1}{2 τ} {\tilde{w}}^{T} \tilde{w}

The time derivative of $V_{3}$ can be derived from equations (30) and (31)

\begin{array}{l} {\dot{V}}_{3} = s \dot{s} + \frac{1}{τ} {\tilde{w}}^{T} \dot{w} \\ = - G φ s^{2} - G γ s^{(p + q) / p} + G s ({\tilde{w}}^{T} h - ε) + \frac{1}{τ} {\tilde{w}}^{T} \dot{w} \\ = - G φ s^{2} - G s^{(p + q) / p} (γ + \frac{ε}{s^{q / p}}) + {\tilde{w}}^{T} (G s h + \frac{1}{τ} w) \end{array}

To ensure the system stability, the appropriate adaptive law and parameters are selected to make the ${\dot{V}}_{3}$ negative definite. Design the adaptive law

\dot{w} = - τ G s h

The equation (32) can be written as

{\dot{V}}_{3} = - G φ s^{2} - G s^{(p + q) / p} (γ + \frac{ε}{s^{q / p}})

If the parameter γ satisfies $γ > \frac{ε}{| s^{q / p} |}$ , then ${\dot{V}}_{3} \leq 0$ . The sliding surface variable s and the $w$ converge to zero. The tracking error converges to zero too. The closed-loop system is stable. The RBF neural network is learning online according to equation (33). As long as the parameters γ and $p / q$ are large enough, the state of the system can reach the neighborhood of $s = 0$ small enough. The parameter γ can be rewritten as $γ = \frac{ε}{| s^{q / p} |} + \hat{σ}$ , where $\hat{σ}$ is a positive real number. Compared with equation (19), the amplitude of discontinuous item in γ is decreased greatly because ε is a very small positive number. Thus, the system chattering is reduced greatly. The analysis of the arriving time is similar to the GFTSM controller.

The steps of the GFTSM-NN control algorithm

There are six steps to compute the control as shown in Figure 3.

Step 1: Parameter initialization. Initialize the parameters of the controller, such as p₀, q₀, p, q, α, β, φ, $\hat{σ}$ , τ as well as the number of the hidden layer nodes, the center values, width, and initial weights of the neural network. This is a very important step, because controller parameters have great impact on system performance.

Step 2: Detect the actual heading angle and its yaw rate of the USV in real-time and subtract the desired values from them to obtain the position error e and velocity error $\dot{e}$ . The control objective is to make the tracking error converge to zero.

Step 3: Compute the sliding surface variable s according to equation (8). Compared with the conventional sliding mode control method, the added exponential term will speed up the system convergence.

Step 4: Update the weights of the RBF neural network according to equation (33) which is derived using Lyapunov method.

Step 5: Compute the output of the RBF neural network ${\hat{δ}}_{eq}$ according to equation (26). This step aims at computing the equivalent control of the control and compensate the external disturbance.

Step 6: Compute the control according to equation (24). It consists of two parts: the output of the RBF neural network and the switch control.

Figure 3.

The steps of the GFTSM-NN control algorithm. GFTSM-NN: global fast terminal sliding mode controller based on RBF neural network; RBF: radial basis function.

Simulation results

In this section, a comparison study of the GFTSM controller and the proposed GFTSM-NN controller is presented to show the feasibility and effectiveness of the proposed method. Given the USV is DH-01, which is 110-cm long, 36-cm wide, 5.4-kg weight. Its max power is 36 W. The coefficients of the USVs kinematic model are listed in Table 1.¹⁷

Table 1.

Coefficients of the USV’s kinematic model

Parameters	Value
K	0.366
T	0.24
a ₁	0.35
a ₂	0.3

USV: unmanned surface vehicle.

The parameters of the RBF neural network are designed as $n = 5$ , $b = 2$ , and c = [c ₁ c ₂ c ₃ c ₄ c ₅] = [0.2 0.4 0.6 0.8 1]. The initial weights are 0.01, and the learning rate $τ = 10$ . The other parameters of the controller are designed as $p_{0} = 7$ , $q_{0} = 5$ , $p = 7$ , $q = 3$ , $α = 4$ , $β = 1$ , $φ = 300$ , and $\hat{σ} = 2$ . Owing to the saturation of the actuator, the control is no more than 35°, which is equivalent to 0.61 rad. Assume that the external disturbance $d = 0.3 sin t$ . To compare the performance of the designed GFTSM-NN controller with the GFTSM controller, the parameters of the GFTSM controller are chosen the same as the GFTSM-NN controller. The following figures show the comparison of the two controllers for the tracking of different desired courses. All simulations are run in Matlab 2014b.

The straight line tracking

The simulation results of the two control systems for straight line tracking are provided in Figures 4 to 10. The desired input is a constant. Figures 4 and 5 depict the heading angle tracking characteristics and tracking errors of the two controllers, respectively. It can be observed that both outputs of the two control systems can track the linear input quickly and accurately, and the tracking errors converge quickly. The GFTSM-NN controller can follow the desired heading angle in 3 s while the GFTSM controller needs nearly 3.5 s. The former one has a faster tracking speed. Figure 6 depicts the tracking characteristics of the yaw rate, from which we can see that the speed signal can be tracked quickly by both controllers as well, and the GFTSM-NN controller has a faster tracking speed. From Figures 4 to 6, it can be seen that both systems have good performance of straight line tracking, and the GFTSM-NN control system designed in this article has a faster response speed. Figure 7 depicts the curves of the sliding surface variable s, they converge in both control systems. The sliding surface variable s of GFTSM-NN converge faster than GFTSM.

Figure 4.

The heading angle tracking of the straight line. GFTSM-NN: global fast terminal sliding mode controller based on RBF neural network; RBF: radial basis function.

Figure 5.

The tracking error of the heading angle (straight line). GFTSM-NN: global fast terminal sliding mode controller based on RBF neural network; RBF: radial basis function.

Figure 6.

The yaw rate tracking of the straight line. GFTSM-NN: global fast terminal sliding mode controller based on RBF neural network; RBF: radial basis function.

Figure 7.

The sliding surface variable s of the straight line tracking. GFTSM-NN: global fast terminal sliding mode controller based on RBF neural network; RBF: radial basis function.

Figure 8.

The rudder angle of the straight line tracking. GFTSM-NN: global fast terminal sliding mode controller based on RBF neural network; RBF: radial basis function.

Figure 9.

The weights of the neural network (straight line).

Figure 10.

The switch control of the straight line tracking. GFTSM-NN: global fast terminal sliding mode controller based on RBF neural network; RBF: radial basis function.

Figure 8 depicts the rudder angle of the USV. It can be observed that the rudder angle of the GFTSM controller is chattering. The high-frequency action accelerates the wear of the mechanical system, and it is not suitable for application in the actual mechanical system. Meanwhile, the rudder angle of the GFTSM-NN controller is smooth in the extreme. Because there is a disturbance in the system, the switching control of the GFTSM controller must continuously switch to eliminate the influence of the disturbance, and the greater the disturbance, the greater the system chattering. The related theory has been explained in “GFTSM controller design for course keeping” section. When the neural network whose weights are drawn in Figure 9 is employed to compute the equivalent control in the GFTSM-NN controller design, not only the controller is independent of the systems precise model information but also the system disturbance is compensated, and the system chattering is greatly reduced as well. Simulation results are consistent with the theoretical analysis. The curves of the switch control $δ_{n}$ and the parameter γ are drawn in Figures 10 and 11, respectively, from which we can see that the vibration amplitude of the parameter γ in the GFTSM-NN controller is much smaller than the GFTSM controller, and the chattering of the switch control is reduced greatly correspondingly.

Figure 11.

The parameter γ of the straight line tracking. GFTSM-NN: global fast terminal sliding mode controller based on RBF neural network; RBF: radial basis function.

The curve tracking

The simulation results of the two control systems for curve tracking are provided in Figures 12 to 20. The desired input is a sine function, that is, $ψ_{d} = 0.5 sin 0.2 t$ . Figures 12 and 13 depict the heading angle tracking characteristics and tracking errors of the two controllers, respectively. It can be observed that both outputs of the two control systems can track the sine signal quickly and accurately, and the tracking errors converge quickly. The GFTSM-NN control system has a faster response speed. Figure 14 depicts the tracking characteristic of the yaw rate, from which we can see that the speed signal can be tracked quickly as well. It is similar to the heading angle tracking curves. The GFTSM-NN controller has a faster tracking speed and higher tracking accuracy. From Figures 12 to 14, it can be seen that both systems have good output characteristics, and the tracking error of the GFTSM-NN control system designed in this article converges faster than GFTSM control system. Figure 15 depicts the curves of the sliding surface variable s, they converge in both control systems.

Figure 12.

The heading angle tracking of the curve. GFTSM-NN: global fast terminal sliding mode controller based on RBF neural network; RBF: radial basis function.

Figure 13.

The tracking error of the heading angle (curve). GFTSM-NN: global fast terminal sliding mode controller based on RBF neural network; RBF: radial basis function.

Figure 14.

The yaw rate tracking of the curve. GFTSM-NN: global fast terminal sliding mode controller based on RBF neural network; RBF: radial basis function.

Figure 15.

The sliding surface variable s of the curve tracking. GFTSM-NN: global fast terminal sliding mode controller based on RBF neural network; RBF: radial basis function.

Figure 16.

The rudder angle of the curve tracking. GFTSM-NN: global fast terminal sliding mode controller based on RBF neural network; RBF: radial basis function.

Figure 17.

The equivalent control for the curve tracking. GFTSM-NN: global fast terminal sliding mode controller based on RBF neural network; RBF: radial basis function.

Figure 18.

The weights of the neural network (straight line). GFTSM-NN: global fast terminal sliding mode controller based on RBF neural network; RBF: radial basis function.

Figure 19.

The switch control of the curve tracking. GFTSM-NN: global fast terminal sliding mode controller based on RBF neural network; RBF: radial basis function.

Figure 20.

The parameter γ of the curve tracking. GFTSM-NN: global fast terminal sliding mode controller based on RBF neural network; RBF: radial basis function.

Figure 16 depicts the rudder angles of the two control systems. It is chattering in the GFTSM controller and smooth in the GFTSM-NN controller, which is similar to the straight line tracking. Figure 17 depicts $δ_{eq}^{*}$ and the equivalent control of the GFTSM-NN controller and GFTSM controller, respectively. The equivalent control of the GFTSM-NN controller is the output of the neural network whose weights are drawn in Figure 18. It has online learning ability and it is consistent with the trend of $δ_{e q}^{*}$ . The equivalent control of the GFTSM controller does not have online learning ability. The external disturbance is overcome and the system state is pulled to the equilibrium by switch control. The curves of the switch control $δ_{n}$ and the parameter γ are drawn in Figures 19 and 20, respectively, which are similar to the straight line tracking. In the GFTSM controller, the values of γ are between 2.4184 to 919.9749. The vibration amplitude is very large. The maximum value 919.9749 appears at around 4 s, because the $| s |$ has the minimum value $2.8563 \times 10^{- 9}$ at around 4 s. The jumping of the γ leads to the chattering of the switch control. It is a fault of the GFTSM controller. The values of γ in the GFTSM-NN controller are between 2.0019 to 3.1095. The vibration amplitude is reduced greatly, and the chattering of the switch control is reduced effectively correspondingly. The simulation results are consistent with the previous analysis.

Conclusions

An adaptive GFTSM-NN is proposed to realize the course keeping of USV with unknown disturbance and nonlinear uncertainty in this article. The equivalent control of the GFTSM controller is computed by the RBF neural network. It is adjusted online according to the adaptive law which is derived using the Lyapunov method to ensure the closed-loop stability of the system. A contrastive study of the proposed GFTSM-NN controller and the GFTSM controller is conducted. Compared with traditional GFTSM controller, the proposed GFTSM-NN controller is independent of the nonlinear dynamics of the system and free of upper bound of the external disturbance. Simulation results show that the proposed control system has a faster response speed than GFTSM control system and the system chattering is eliminated successfully.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Lili Wan

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