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Abstract

Unmanned surface vehicle has the properties such as complexity, nonlinearity, time variability, and uncertainty, which lead to the difficulty of obtaining a precise kinematics model. A neural adaptive sliding mode controller for the unmanned surface vehicle steering system is developed based on the sliding mode control technique and the radial basis function neural network. In the new approach, two parallel radial basis function neural networks are used to reduce the influence of the system uncertainties and eliminate the dependency of the controller on the precise kinematics model of the system. Among these two radial basis function neural networks, one is used to approximate the unknown nonlinear yaw dynamics and the other is used to adjust the control gain as well as realize the variable gain sliding mode control. The weights of the two neural networks are trained online using the sliding surface variable and the control, where the Lyapunov method is used to derive the adaptive laws to ensure the stability of the whole closed-loop system. The proposed adaptive controller is suitable for the steering control at different cruising speeds with bounded external disturbances. The simulation results show that the proposed controller has a good control performance regarding the smooth control, fast response, and high accuracy.

Keywords

Steering system unmanned surface vehicle sliding mode control radial basis function neural network adaptive controller

Introduction

Unmanned surface vehicles (USVs) attract the attention of many countries and become a hot research spot in the field of marine equipment because of its low labor cost and strong maneuverability. In the future, USVs will play a vital role in anti-submarine warfare, mine countermeasures, environmental detection, water sampling, personnel search and rescue in the ocean, and so on.¹ As an intelligent surface body of autonomous navigation, an efficient steering system is an important issue. With the change of the parameters such as cruising speed, water depth, and ship loading, the hull of the wet area and drag characteristics will change significantly. This will lead to a change in various hydrodynamic coefficients of the USV model. As a result, the model of the USV is a time-varying system with parameter uncertainty. In addition, the disturbance of wind, wave, and current leads to a more complex control problem.² It is a challenge to design an effective adaptive controller with a strong robustness and anti-interference capability which is able to avoid excessive rudder motion for the nonlinear, time-varying, and uncertain USV.³

A lot of research has been done on the steering system of the USV. Several control methods have been developed for USV steering problem such as classic proportional–integral–derivative (PID) controllers, adaptive controllers, robust controllers, and intelligent controllers.⁴ PID controller is a conventional and common controller for feedback control systems, and it greatly improves the performance of the steering system. However, the performance of the control system is affected by the change in the parameters of the system.⁵ When the parameters of the system are variable or uncertain, the mathematical model of the plant should be identified to achieve better performance. The identification will increase the complexity and reduce the reliability of the control system.

By taking the uncertainty into account for robust ship control performance, adaptive and robust controllers have been proposed for the USV steering system.⁵ System identification technology and extended Kalman filter were integrated to design a steering controller of the vessel associated with unstructured uncertainties by Perera et al.⁶ The reduced-order model of the vessel steering can be successfully identified with the required states and parameters. Based on the theory of model reference adaptive systems (MRAS), an adaptive steering controller with input constraints was proposed by Kahveci and Ioannou.⁷ The adaptive law was combined with a control design including a linear quadratic (LQ) controller and a Riccati-based anti-windup compensator. In order to reduce the roll motion of surface ships, a robust fin controller based on L2 gain design was proposed by Hinostroza et al.⁸ Considering the state constraints in the presence of environmental disturbances, a disturbance compensating model predictive control (MPC) algorithm was applied to solve the ship heading control problem by Li and Sun.⁹ The simulation results show the good performance of the proposed controller in terms of reducing heading error and satisfying yaw velocity and actuator saturation constraints. A multivariable controller for the USV with two fixed thrusters was proposed by Park et al.¹⁰ It used system identification technique to estimate the dynamic model of the USV. A controller for a ship steering system with uncertain time delays was designed by Lei and Guo.¹¹ An extended state observer with the aid of a modified Smith predictor was constructed to estimate the total disturbances including the internal and the external disturbances.

The aforementioned adaptive controllers improved the robustness of the system, where a linear or partially linear model was adopted to describe the ship steering characteristics. However, in fact, the ship steering system is nonlinear and time-varying.¹² As the ship steering has nonlinear manipulation characteristics and uncertainties, intelligent control algorithms combined with other advanced control methods are also used in steering system.¹² A novel adaptive robust online constructive fuzzy control scheme for tracking surface vehicles was proposed by Wang et al.¹³ It employed an online constructive fuzzy approximator to deal with uncertainties and unknown disturbances. A model reference adaptive fuzzy control algorithm was applied in the steering control system of the ship by Yang et al.¹⁴ The external disturbances are supposed to be bounded. A performance constrained fuzzy controller design was investigated for the nonlinear uncertain discrete-time ship steering system with complex noises by Chang et al.¹⁵ The nonlinear discrete-time ship steering system was represented by a discrete-time fuzzy model with perturbations, additional noise, and multiplicative noise. Fuzzy autopilot significantly enhances the performance in a noisy environment with fewer rudder motions. However, the tuning of the fuzzy controller parameters is based on an extended trial and error procedure.¹⁶ The neural networks (NNs) are also used to approximate the nonlinearities of the ship steering dynamics and combined with other control technology to produce adaptive controllers.^17–21 A NN combined with a barrier Lyapunov function was used to approximate the system uncertainties and prevent states from violating the constraints.²² NN combined with the L2 gain theory was applied to the autopilot design of ship course-keeping with uncertain modeling errors and disturbances by Luo et al.²³ The NN is used to compensate the modeling errors, and the L2 gain is designed to suppress the uncertain external disturbances. The designed controller has strong robustness. A NN PID controller was designed to bring a better performance on the ship steering hydraulic system,²⁴ where the gradient descent algorithm is used to find the local optimal solution. Backstepping is a recursive design methodology in a systematic manner which is widely applied in the autopilot of the marine vehicles.^25,26 Adaptive robust controllers designed on the basis of backstepping method have the good adaptive ability and tracking performance.^27–29 However, backstepping algorithm suffers from the problem of “explosion of terms” because it requires repeated differentiation of the virtual control function, which results in the computation complexity of the actual control law.

Sliding mode controller (SMC) has the outstanding advantages in robustness, anti-disturbance, and response speed.³⁰ It is also used in the autopilot design of USV.^31–33 An adaptive sliding mode control algorithm and nonlinear disturbance observer method were developed for course-keeping maneuvers in vessel steering by Liu.³¹ It provides a robust performance against the environmental disturbance and the uncertainty in the rudder dynamics. To select a proper course controller, the proportional–derivative (PD), sliding mode, and fuzzy controllers were tested for three different changes in the course ranging from $0^{°}$ to $30^{°}$ , $90^{°}$ , and $180^{°}$ by Szymak.³⁴ The comparison results show that the SMC has the best control quality indices. However, the SMC suffers two main disadvantages. The first one is that the complete knowledge of the system dynamics is needed in the calculation of the equivalent control. The second is that it will cause system chattering because of the discontinuous switching characteristics in the nature of the sliding mode control.

In order to reduce the chattering of the SMC and increase the robustness of the system, second-order SMC has been studied in recent years.^35–38 More flexible sliding surface variables can be selected, and the chattering is reduced effectively in these methods. However, when the mathematical model of the plant is uncertain, the control performance is affected. The NN which can effectively solve the problems with regard to unknown nonlinear systems attracts many concerns from many researchers.^{17–24,39–42} Using the universal approximation property of the NNs, the influence of the external disturbance and uncertainty can be reduced, and the controller becomes independent of the precise system model information.³⁹ The neural adaptive SMCs are designed to improve the robustness of the system and reduce the system chattering.^40–42

In order to develop an efficient steering system for the nonlinear course control problem of USV with parameter uncertainties and external disturbances, a neural adaptive SMC based on radial basis function (RBF) NN is designed in this article. RBF NN has good generalization ability. It has the advantages of simple structure, avoiding unnecessary and redundant computation.^43,44 The research on RBF NN shows that it can approximate any nonlinear function in a compact set with arbitrary precision.^45,46 Therefore, RBF NN is chosen to design the adaptive controller. The weights of the NNs are derived using Lyapunov method to guarantee the stability of the whole closed-loop system. Using the saturation function instead of the sign function in the corrective control, the system chattering is eliminated.

The nonlinear and uncertain term of the USV’s kinematic model is approximated by the RBF NN to reduce the reliance of equivalent control on the accurate kinematics model of the system. The estimated value is used to calculate the equivalent control instead of using the actual value. In this way, the equivalent control can be calculated without the knowledge of the complete dynamics of the USV. The reliance of the equivalent control on the complete kinematic model is reduced, and the influence of the system uncertainties is reduced as well. In addition, the designed controller adopts a variable gain achieved through the RBF NN. It is adjusted dynamically using the system error and the control. The variable gain has two contributions. (1) The control gain is related to the system parameters which are difficult to be acquired. They change with the ship’s navigational states and external environment. The variable gain reduces the reliance on the system parameters and the influence of the system uncertainties is reduced as well. (2) To eliminate the system chattering, the saturation function is used in the corrective control instead of the sign function. It comes at a cost of the system’s robustness. The variable gain brings the variable boundary which shrinks with the convergence of the system states. It enhances the robustness of the system and accelerates the response speed of the system.

The remainder of this article is arranged as follows. In section “Problem formulation,” a brief description of the kinematics model of USV is proposed. In section “Controller design and stability analysis,” the design of the controller and the stability analysis is discussed. Simulations and analyses of the proposed control system and the conventional SMC are shown in section “Simulation studies.” In the last section, the article is concluded.

Problem formulation

In essence, the dynamic characteristics of USVs are nonlinear and uncertain. The nonlinearity of the steering system comes from the nonlinear characteristics of the fluid force while the uncertainties come from the changes in the cruising speed and the external environment. On the basis of the linear second-order K-T equation,⁴⁷ considering the external disturbances of wind, wave, and current, the nonlinear response equation of the USV can be described as

{\begin{matrix} \overset{\cdot}{ψ} (t) = r (t) \\ \overset{\cdot}{r} (t) = - \frac{K (t)}{T (t)} H (r) + \frac{K (t)}{T (t)} δ (t) + d (t) \end{matrix}

(1)

where $ψ (t)$ is the yaw angle; $r (t)$ is the yaw rate; $δ (t)$ is the rudder angle; $d (t)$ is the disturbance which stands for the unknown external disturbances; $K (t)$ and $T (t)$ are coefficients changing with the ships quality, shape, cruise speed, and external environment. $H (r)$ is an uncertain nonlinear function of $r (t)$ . According to the study by Peng et al.,⁴⁸ $H (r)$ can be described as

H (r) = α (t) r (t) + β (t) r^{3} (t)

(2)

where $α (t)$ and $β (t)$ are coefficients changing with the ships quality, shape, cruise speed, and external environment. These coefficients $K (t)$ , $T (t)$ , $α (t)$ , and $β (t)$ are positive for a stable linear movement of the ship.¹ Although they can be partially obtained by spiral and inverse spiral experiments in calm water, however, when the cruise speed and external environment change, they change. As the cruise speed increases, the inertia of the ship increases, $K (t)$ increases correspondingly, and $T (t)$ decreases correspondingly. On the contrary, as the cruise speed decreases, the inertia of the ship decreases, $K (t)$ decreases correspondingly, and $T (t)$ increases correspondingly. It is complicated to find the mathematical relationships between these coefficients and the cruise speed together with the environment. Thus, a controller which is independent of these parameters is preferable.

For the sake of writing conveniently, equation (1) can be rewritten as

{\begin{matrix} \overset{\cdot}{ψ} (t) = r (t) \\ \overset{\cdot}{r} (t) = f (t) + g (t) δ (t) + d (t) \end{matrix}

(3)

where $f (t) = - (K (t) / T (t)) H (r)$ is nonlinear and uncertain, and $g (t) = (K (t) / T (t))$ changes with the cruise speed and environment.

Controller design and stability analysis

SMC design

The task is to design a feedback control $δ (t)$ for the plant (equation (3)) to make the output error converge to zero. Assuming that the desired yaw angle is $ψ_{d} (t)$ , the heading tracking error is

e (t) = ψ_{d} (t) - ψ (t)

(4)

Define the output error as the heading tracking error $e (t)$ . Its time derivative can be expressed as

\overset{\cdot}{e} (t) = {\overset{\cdot}{ψ}}_{d} (t) - r (t)

(5)

According to conventional sliding mode control theory,^39,49 the sliding surface variable can be defined as

s (t) = \overset{\cdot}{e} (t) + γ e (t), γ > 0

(6)

where $γ$ is a design parameter to be determined. It can be seen from equation (6) that $e (t)$ and $\overset{\cdot}{e} (t)$ converge to zero when $s (t)$ equals zero. In order to achieve convergence of the state variables $e (t)$ and $\overset{\cdot}{e} (t)$ to zero, $s (t)$ must converge to zero in a finite time by means of the control $δ (t)$ . The aim of SMC is to force the system states to the sliding surface $s (t) = 0$ . The control $δ (t)$ should be chosen to make the candidate Lyapunov function satisfy the Lyapunov stability criteria. Consider the following Lyapunov function

V = \frac{1}{2} s^{2} (t)

(7)

The time derivative of $V$ is

\overset{\cdot}{V} = s (t) \overset{\cdot}{s} (t)

(8)

The derivative of the sliding surface variable is derived using equations (3)–(6)

\begin{matrix} \overset{\cdot}{s} (t) = \overset{\cdot\cdot}{e} (t) + γ \overset{\cdot}{e} (t) \\ = {\overset{\cdot\cdot}{ψ}}_{d} (t) - f (t) - g (t) δ (t) - d (t) + γ \overset{\cdot}{e} (t) \end{matrix}

(9)

The control $δ (t)$ is given by

δ (t) = δ_{eq} (t) + δ_{n} (t)

(10)

where $δ_{eq} (t)$ is the equivalent control which determines the behavior of the system constrained by the switching surface, $δ_{n} (t)$ is the corrective control which represents the nonlinear feedback term for suppression of the effect of uncertainty and drives the system trajectory toward the switching surface until it is intersected. Assuming that the disturbance is zero, and let $\overset{\cdot}{s} (t) = 0$ , the equivalent control can be obtained

δ_{eq} (t) = \frac{1}{g (t)} [{\overset{\cdot\cdot}{ψ}}_{d} (t) - f (t) + γ \overset{\cdot}{e} (t)]

(11)

Selecting the corrective control

δ_{n} (t) = \frac{λ}{g (t)} sgn (s (t))

(12)

where $λ$ is a design parameter to be determined, it should be large enough to ensure the stability of the system, $sgn (s (t))$ is a sign function, that is

sgn (s (t)) = {\begin{matrix} 1 if s (t) > 0 \\ 0 if s (t) = 0 \\ - 1 if s (t) < 0 \end{matrix}

(13)

Hence, the control $δ (t)$ can be designed as

δ (t) = \frac{1}{g (t)} [{\overset{\cdot\cdot}{ψ}}_{d} (t) - f (t) + γ \overset{\cdot}{e} (t) + λ sgn (s (t))]

(14)

Suppose that the disturbance is bounded, with $D$ is the upper bound of the disturbance, that is, $d (t) \leq D$ . The time derivative of $V$ can be derived from equations (5)–(8) and (14)

\begin{matrix} \overset{\cdot}{V} = s (t) \overset{\cdot}{s} (t) = s (t) [- d (t) - λ sgn (s (t))] \\ \leq D | s (t) | - λ | s (t) | = - | s (t) | (λ - D) \end{matrix}

(15)

Let the parameter $λ = D + (μ / \sqrt{2})$ , then

\overset{\cdot}{V} \leq - \frac{μ}{\sqrt{2}} | s (t) | = - μ V^{1 / 2}

(16)

Therefore, the sliding surface variable $s (t)$ is finite time stable and the control $δ (t)$ will drive $s (t)$ to the origin in a finite time.³⁵ $e (t)$ and $\overset{\cdot}{e} (t)$ will converge too. However, because of the discontinuity of the corrective control, the control is chattering. In order to eliminate the system chattering, the sign function can be replaced by the saturation function. The control is redesigned as

\bar{δ} (t) = \frac{1}{g (t)} [{\overset{\cdot\cdot}{ψ}}_{d} (t) - f (t) + γ \overset{\cdot}{e} (t) + λ sat (s (t))]

(17)

where $sat (s (t))$ stands for saturation function as follows

sat (s (t)) = {\begin{matrix} 1 & if s (t) > b \\ \frac{1}{b} s (t) & if | s (t) | \leq b, (0 < b < 1) \\ - 1 & if s (t) < - b \end{matrix}

(18)

where $b$ is the boundary of the saturation function. Unlike the sign function’s switch characteristics, the saturation function has a boundary layer. It is directly designed on the sliding surface and forms the sliding area. Outside the boundary layer, the switching control is adopted to make the system states approach the sliding mode fast. In the boundary layer, the linear feedback control is adopted to reduce the chattering caused by the fast switching of the SMC. The system states can reach the sliding area accurately, and the system chattering can be reduced effectively. Using saturation function instead of the sign function in the corrective control is an effective approach to solving the system chattering, while the fixed boundary layer will not lead the system states to converge to the sliding surface. Thus, the robustness of the system is reduced. In order to increase the system’s robustness, a neural adaptive SMC is designed.

Neural adaptive SMC design

The conventional SMC for the steering system of USV is designed in the previous section. It can be seen from equation (14) that $f (t)$ and $g (t)$ should be known to calculate the control law in the sliding mode control. However, $f (t)$ is nonlinear and uncertain, and $g (t)$ is changing with the cruising speed and external environment, both of $f (t)$ and $g (t)$ are not constant. Moreover, the saturation function is used instead of the sign function in the corrective control to eliminate the system chattering, which comes at a cost of system robustness. For the characteristics of nonlinearity, uncertainty, and complexity of USV, a variable gain adaptive SMC based on RBF NN is designed in this article. The block diagram of the control system is shown in Figure 1. The RBF NN1 is used to approximate $f (t)$ . The input of RBF NN1 is $x = [ψ (t), \overset{\cdot}{ψ} (t)]^{T}$ , and the output $\hat{f} (t)$ is used to design the control instead of actual value $f (t)$ . The weight vector $w_{1}$ of RBF NN1 is trained by the sliding surface variable $s (t)$ . The NN1 reduces the dependence of the controller on the kinematics model of the system, but comes at a cost of the response speed of the system. RBF NN2 is used to achieve variable gain and variable boundary of the saturation function to accelerate the response speed and enhance the robustness of the system. The input of RBF NN2 is $x = [ψ (t), \overset{\cdot}{ψ} (t)]^{T}$ too, and the output $Γ (t)$ is used to design the control instead of the uncertain parameter $g (t)$ . The weight vector $w_{2}$ of RBF NN2 is trained by the sliding surface variable and the control. Due to the saturation characteristic of the rudder during the rotation, the saturation unit is added to the output of the control.

Figure 1.

The block diagram of the control system.

Suppose that RBF NN1 has $n$ hidden layer nodes and RBF NN2 has $m$ hidden layer nodes, the outputs of the hidden layer neurons are

h_{1 i} = \exp (- \frac{∥ x - c_{1 i} ∥^{2}}{σ_{1}^{2}}), i = 1, 2, . . ., n

(19)

h_{2 j} = \exp (- \frac{∥ x - c_{2 j} ∥^{2}}{σ_{2}^{2}}), j = 1, 2, . . ., m

(20)

where $h_{1 i} (i = 1, 2, . . ., n)$ is the output of hidden layer neuron $i$ in RBF NN1, $c_{1 i} (i = 1, 2, \dots, n)$ is the center value vector of neuron $i$ in RBF NN1, $σ_{1}$ is the width of the hidden layer neurons in RBF NN1, $h_{1 j} (j = 1, 2, \dots, m)$ is the output of hidden layer neuron $j$ in RBF NN2, $c_{2 j} (j = 1, 2, \dots, m)$ is the center value vector of neuron $j$ in RBF NN2, and $σ_{2}$ is the width of the hidden layer neurons in RBF NN2. Then, the outputs of the RBF NN1 and RBF NN2 are

\hat{f} (t) = w_{1}^{T} h_{1}

(21)

Γ (t) = w_{2}^{T} h_{2}

(22)

where $\hat{f} (t)$ is the estimated value of $f (t)$ , $Γ (t)$ is the variable gain, $h_{1} = [h_{11} h_{12} \dots h_{1 n}]^{T}$ is the output vector of the hidden layer neurons in RBF NN1, and $h_{2} = [h_{21} h_{22} \dots h_{2 m}]^{T}$ is the output vector of the hidden layer neurons in RBF NN2. There is an ideal NN weight vector $w_{1}^{*}$ so that the RBF NN1 can approximate the given function with the minimum approximation error $ϵ_{1}$ , that is

f (t) = w_{1}^{* T} h_{1} + ϵ_{1}

(23)

The approximation error of $f (t)$ can be described as

\tilde{f} (t) = f (t) - \hat{f} (t) = w_{1}^{* T} h_{1} + ϵ_{1} - w_{1}^{T} h_{1} = {\tilde{w}}_{1}^{T} h_{1} + ϵ_{1}

(24)

where $\tilde{f} (t)$ is the approximation error between $f (t)$ and $\hat{f} (t)$ , and ${\tilde{w}}_{1} = w_{1}^{*} - w_{1}$ is the error between ideal weight vector and weight vector of RBF NN1. There is an ideal NN weight vector $w_{2}^{*}$ , such that the output of RBF NN2 and $g (t)$ with a minimum error $ϵ_{2}$ , that is

g (t) = w_{2}^{* T} h_{2} + ϵ_{2}

(25)

The error between $g (t)$ and $Γ (t)$ can be described as

χ (t) = g (t) - Γ (t) = w_{2}^{* T} h_{2} + ϵ_{2} - w_{2}^{T} h_{2} = {\tilde{w}}_{2}^{T} h_{2} + ϵ_{2}

(26)

where ${\tilde{w}}_{2} = w_{2}^{*} - w_{2}$ is the error between ideal weight vector and weight vector of RBF NN2. Due to the uncertainty of $f (t)$ and $g (t)$ in the system, the estimated value $\hat{f} (t)$ is used to design the control input instead of $f (t)$ and $Γ (t)$ is used to realize variable gain control. According to equation (14), the control can be designed as

\hat{δ} (t) = \frac{1}{Γ (t)} [{\overset{\cdot\cdot}{ψ}}_{d} (t) - \hat{f} (t) + γ \overset{\cdot}{e} (t) + λ sgn (s (t))]

(27)

Equation (27) shows that the control $\hat{δ} (t)$ is independent of the parameters of the system and the control gain is varying. The adaptive law is derived from the Lyapunov method. Consider the following Lyapunov function candidate

J = \frac{1}{2} s^{2} (t) + \frac{1}{2 τ_{1}} {\tilde{w}}_{1}^{T} {\tilde{w}}_{1} + \frac{1}{2 τ_{2}} {\tilde{w}}_{2}^{T} {\tilde{w}}_{2}

(28)

where $τ_{1}$ and $τ_{2}$ are positive parameters to be designed. The time derivative of $J$ is

\overset{\cdot}{J} = s (t) \overset{\cdot}{s} (t) - \frac{1}{τ_{1}} {\tilde{w}}_{1}^{T} {\overset{\cdot}{w}}_{1} - \frac{1}{τ_{2}} {\tilde{w}}_{2}^{T} {\overset{\cdot}{w}}_{2}

(29)

where $\overset{\cdot}{s} (t)$ can be derived from equations (9) and (21)–(27)

\begin{matrix} \overset{\cdot}{s} (t) = {\overset{\cdot\cdot}{ψ}}_{d} (t) - f (t) - (Γ (t) + χ (t)) \hat{δ} (t) - d (t) + γ \overset{\cdot}{e} (t) \\ = [\hat{f} (t) - f (t)] - λ sgn (s (t)) - χ (t) \hat{δ} (t) - d (t) \\ = - {\tilde{w}}_{1}^{T} h_{1} - ϵ_{1} - ({\tilde{w}}_{2}^{T} h_{2} + ϵ_{2}) \hat{δ} (t) \\ - λ sgn (s (t)) - d (t) \end{matrix}

(30)

The time derivative of $J$ can be derived from equations (29)–(30)

\begin{matrix} \overset{\cdot}{J} = s (t) [- {\tilde{w}}_{1}^{T} h_{1} - ϵ_{1} - ({\tilde{w}}_{2}^{T} h_{2} + ϵ_{2}) \hat{δ} (t) - λ sgn (s (t)) - d (t)] \\ - \frac{1}{τ_{1}} {\tilde{w}}_{1}^{T} {\overset{\cdot}{w}}_{1} - \frac{1}{τ_{2}} {\tilde{w}}_{2}^{T} {\overset{\cdot}{w}}_{2} \\ = - {\tilde{w}}_{1}^{T} (s (t) h_{1} + \frac{1}{τ_{1}} {\overset{\cdot}{w}}_{1}) - {\tilde{w}}_{2} [s (t) h_{2} \hat{δ} (t) + \frac{1}{τ_{2}} {\overset{\cdot}{w}}_{2}] \\ + s (t) [- ϵ_{1} - ϵ_{2} \hat{δ} (t) - d (t)] - λ | s (t) | \end{matrix}

(31)

To ensure the stability of the system, the appropriate control law and parameters are selected to make $\overset{\cdot}{J}$ negative definite. $ϵ_{1}$ and $ϵ_{2}$ are small real numbers. If we choose

{\begin{matrix} {\overset{\cdot}{w}}_{1} = - τ_{1} s (t) h_{1} \\ {\overset{\cdot}{w}}_{2} = - τ_{2} s (t) h_{2} \hat{δ} (t) \end{matrix}

(32)

The parameter $λ$ is designed as $λ \geq | ϵ_{1} | + | ϵ_{2} \hat{δ} (t) | + D$ . Substituting equation (32) into equation (31), the equation can be expressed as

\overset{\cdot}{J} = s (t) [- ϵ_{1} - ϵ_{2} \hat{δ} (t) - d (t)] - λ | s (t) | \leq 0

(33)

From equation (33), we can see, the closed-loop system is asymptotically stable, if $t \to \infty$ , $s (t) \to 0$ . Although $s (t) = 0$ , $\overset{\cdot}{J} = 0$ , but $J$ may not be 0, so it cannot guarantee that ${\tilde{w}}_{1}$ and ${\tilde{w}}_{2}$ converge to zero. The estimated value $\hat{f} (t)$ and dynamic gain $Γ (t)$ are not equal to $f (t)$ and $g (t)$ , respectively. But because $J \geq 0$ and $\overset{\cdot}{J} \leq 0$ , $J$ is bounded, ${\tilde{w}}_{1}$ and ${\tilde{w}}_{2}$ are bounded. The NNs are trained online according to equation (32), and the controller has online learning ability to deal with the system time-varying and nonlinear uncertainties.

The same as the traditional SMC, the control $\hat{δ} (t)$ will cause the high frequency of control action because of its discontinuous characters. Although the NNs are used to approximate the uncertain parts and the frequency of control action is reduced, the system chattering cannot be completely eliminated. Thus, the saturation function is used in the control law instead of sign function to eliminate the system chattering as well, the desired control is designed as

δ^{*} (t) = \frac{1}{Γ (t)} [{\overset{\cdot\cdot}{ψ}}_{d} (t) - \hat{f} (t) + γ \overset{\cdot}{e} (t) + λ sat (s (t))]

(34)

Comparing the control $δ^{*} (t)$ with $\bar{δ} (t)$ , $δ^{*} (t)$ is independent of the model parameters of the system and the control gain is variable. The boundary of the saturation function is variable too, and it is adjusted with the system states. As the states of the system converge, the boundary is gradually contracted. The systems robustness is improved.

Simulation studies

In this section, the proposed adaptive neural SMC is applied to the autopilot of USV first. A comparison study of the conventional SMC and the proposed neural sliding mode controller (NSMC) is presented to show the efficiency of the proposed method. The USV is given as DH-01, which is 110 cm long, 36 cm wide, and 5.4 kg weight. Its maximum power is 36 W. Assuming that the cruise speed is 7 knots, the model parameters study by Peng et al.⁴⁸ are as follows: $K (t) = 0.366$ , $T (t) = 0.24$ , $α (t) = 0.35$ , and $β (t) = 0.3$ . Suppose the input is a sine function, that is, $ψ_{d} (t) = 0.5 \sin 0.2 t$ . The parameters of the RBF NN are designed as $n = m = 5$ , $σ_{1} = σ_{2} = 5$ , $c_{1} = c_{2} = [- 1 - 0.5 0 0.5 1; - 1 - 0.5 0 0.5 1]$ , the initial weights of the NN1 are 0.12, and the initial weights of the NN2 are 0.1. The other parameters of the controller are designed as $γ = 5$ , $λ = 0.5$ , $τ_{1} = 10$ , $τ_{2} = 10$ , $b = 0.2$ . Owing to the saturation of the actuator, the control is limited to 35°, which is equivalent to 0.61 rad. To compare the performance of the proposed NSMC with the traditional SMC. The parameters of SMC are chosen as NSMC. Both of them adopt saturation function in the corrective control to avoid system chattering. The parameters of the two controllers are listed in Table 1.

Table 1.

Parameters of the two controllers.

Controller	b	$γ$	$λ$	$τ_{1}$	$τ_{2}$	n	m	$σ_{1}$	$σ_{2}$	Initialweightsof NN1	Initialweightsof NN2	$c_{1}$	$c_{2}$
SMC	0.2	5	0.5	–	–	–	–	–	–	–	–	–	–
NSMC	0.2	5	0.5	10	10	5	5	5	5	0.12	0.1	[−1 −0.5 0 0.5 1;−1 −0.5 0 0.5 1]	[−1 −0.5 0 0.5 1;−1 −0.5 0 0.5 1]

NN: neural network; SMC: sliding mode controller; NSMC: neural sliding mode controller.

Assuming that the parameters of the system are constant and there is no disturbance. The simulation results are provided in Figures 2 –9. Figures 2 and 3 depict the output responses of the steering system with NSMC and SMC. It can be observed from Figure 2 that both of the two controllers have good following characteristics. The adjustment time of each system is less than 1 s. The proposed NSMC can achieve a faster response speed than SMC without knowing the precise kinematic model of the system. Figure 3 depicts the speed tracking of the two controllers. Figures 4 and 5 depict the tracking errors of the two controllers, from which we can see that if the system model and parameters are certain and there is no external interference, both controllers can achieve the tracking error of the system converge quickly. Figure 6 depicts the controls of the two controllers, respectively. They are smooth in the extreme. Because the sign function in the corrective control is replaced by saturation function. Figure 7 depicts the sliding surface variables of the two controllers. They converge quickly. Figure 8 depicts the nonlinear $f (t)$ and its estimated value. It can be observed that the estimated values of $f (t)$ are not equal to its actual value, because the RBF NNs are trained by their system output error, respectively, rather than the approximation error. It is analyzed in section “Neural adaptive SMC design” of this article. Figure 9 depicts the variable gain which is adjusted online by the RBF NN2.

Figure 2.

The curves of the output $ψ (t)$ without disturbance.

Figure 3.

The curves of the output $r (t)$ without disturbance.

Figure 4.

The tracking error of $ψ (t)$ without disturbance.

Figure 5.

The tracking error of $r (t)$ without disturbance.

Figure 6.

The control without disturbance.

Figure 7.

The sliding surface variable without disturbance.

Figure 8.

$f (t)$ and its estimated value without disturbance.

Figure 9.

The variable gain without disturbance.

During the actual navigation, the cruise speed constantly changes and the disturbance of the external environment is inevitable. Suppose the cruise speed changes from 7 to 5 knots at 10 s. The hydrodynamic coefficients of the ship change with the cruise speed. When $t < 10 s$ , the cruise speed is 7 knots, and the system parameters are set to $K (t) = 0.366$ , $T (t) = 0.24$ , $α (t) = 0.35$ , and $β (t) = 0.3$ . When $t \geq 10 s$ , the system parameters are set to $K (t) = 0.266$ , $T (t) = 0.29$ , $α (t) = 0.4$ , and $β (t) = 0.35$ . Suppose the external disturbance $d (t) = 0.2 \sin t$ . The upper bound of the disturbance is 0.2, which is equivalent to about 7° rudder angle disturbance. The simulation results are provided in Figures 10 –17.

Figure 10.

The curves of the output $ψ (t)$ with disturbance.

Figure 11.

The curves of the output $r (t)$ with disturbance.

Figure 12.

The tracking error of $ψ (t)$ with disturbance.

Figure 13.

The tracking error of $r (t)$ with disturbance.

Figure 14.

The control with disturbance.

Figure 15.

The sliding surface variable with disturbance.

Figure 16.

$f (t)$ and its estimated value with disturbance.

Figure 17.

The variable gain with disturbance.

Figures 10 and 11 depict the output responses of the steering system. It can be observed that the SMC cannot track the input very well in presence of external disturbance, while the system with NSMC is almost unaffected. It has good tracking characteristics even in presence of external disturbance and parameters change. The adjustment time of the system with NSMC is less than 1 s. Figure 12 depicts the tracking errors of the two controllers. It can be seen that, if there is an external disturbance, the tracking error of SMC oscillates around zero. It is larger than the tracking error of NSMC, and it increases at 10 s as the USV speed reduces. It is easily affected by the external disturbance and parameters change because the sign function in the corrective control is replaced by saturation function to eliminate system chattering. The simulation results are consistent with the theoretical analysis in the previous section. However, the tracking error of NSMC is almost unaffected by the external disturbance and parameters change. Figure 13 depicts the tracking errors of $r (t)$ . It is similar to the tracking error of $ψ (t)$ . From the output characteristics of the system, it can be seen that the SMC cannot work well with the external disturbance and sudden changes of the system parameters, while NSMC is slightly affected. The NSMC has better robustness due to the online learning ability of the NN. Figure 14 depicts the controls of the two controllers, respectively. They are smooth in the extreme. Figure 15 depicts the sliding surface variables of the three controllers. It can be observed that the sliding surface variable of SMC oscillates around zero, and it converges to a very small range around zero in NSMC. Figure 16 depicts the nonlinear uncertainties $f (t)$ and its estimated value. Figure 17 depicts the variable gain which is adjusted online using the RBF NN2. It can be concluded from these simulations that the NSMC has very good control performance. Even if the system parameters and the external environment change, it is slightly affected because it has online learning ability to deal with the change of the system parameters and external disturbance.

Conclusion

This article presents an autopilot design method for USV steering system with parameter uncertainty and bounded disturbance. The nonlinear and uncertain part of the system is approximated using RBF NNs while a variable gain sliding mode control strategy is realized to reduce the influence of the system uncertainties and accelerate the system response. The saturation function is applied instead of a sign function to eliminate the system chattering. Simulation results show that the designed controller has an efficient performance in adaptability and course tracking. Compared to traditional SMC, the proposed controller is designed without knowing the precise kinematic model of the system. In addition, the robustness and tracking accuracy of the system is improved under the premise of eliminating the system chattering. The ongoing work is extending the proposed method to other ocean delivery systems.

Footnotes

Handling Editor: Guang Li

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 iDs

Lili Wan

Yongchuan Tang

References

Sun

Chen

Global generalised exponential/finite-time control for course-keeping of ships. Int J Control 2016; 89: 1169–1179.

Caccia

Bibuli

Bono

et al . Basic navigation, guidance and control of an unmanned surface vehicle. Auton Robot 2008; 25: 349–365.

Liu

Zhang

et al . Unmanned surface vehicles: an overview of developments and challenges. Annu Rev Control 2016; 41: 71–93.

Roberts

Sutton

Zirilli

et al . Intelligent ship autopilots: a historical perspective. Mechatronics 2003; 13: 1091–1103.

Das

Talole

SE.

Robust steering autopilot design for marine surface vessels. IEEE J Oceanic Eng 2016; 41: 913–922.

Perera

Oliveira

Soares

CG.

System identification of vessel steering with unstructured uncertainties by persistent excitation maneuvers. IEEE J Oceanic Eng 2015; 41: 515–528.

Kahveci

Ioannou

PA.

Adaptive steering control for uncertain ship dynamics and stability analysis. Automatica 2013; 49: 685–697.

Hinostroza

Luo

Soares

CG.

Robust fin control for ship roll stabilization based on L2-gain design. Ocean Eng 2015; 94: 126–131.

Sun

Disturbance compensating model predictive control with application to ship heading control. IEEE T Contr Syst T 2011; 20: 257–265.

10.

Park

Shim

Jun

et al . A model estimation and multi-variable control of an unmanned surface vehicle with two fixed thrusters. In: Oceans, Sydney, NSW, Australia, 24–27 May 2010, pp.1–5. New York: IEEE.

11.

Lei

Guo

Disturbance rejection control solution for ship steering system with uncertain time delay. Ocean Eng 2015; 95: 78–83.

12.

Sharma

Sutton

Motwani

et al . Non-linear control algorithms for an unmanned surface vehicle. Proc IMechE, Part M: J Engineering for the Maritime Environment 2014; 228: 146–155.

13.

Wang

Sun

et al . Adaptive robust online constructive fuzzy control of a complex surface vehicle system. IEEE T Cybernetics 2015; 46: 1511–1523.

14.

Yang

Zhou

Ren

Model reference adaptive robust fuzzy control for ship steering autopilot with uncertain nonlinear systems. Appl Soft Comp J 2003; 3: 305–316.

15.

Chang

Huang

BJ.

Multi-constrained fuzzy intelligent control for uncertain discrete systems with complex noises: an application to ship steering systems. J Mar Eng Technol 2016; 16: 11–21.

16.

Tang

Zhou

Wen

A new fuzzy-evidential controller for stabilization of the planar inverted pendulum system. PLoS ONE 2016; 11: e0160416.

17.

Cui

Yang

et al . Adaptive neural network control of AUVs with control input nonlinearities using reinforcement learning. IEEE T Syst Man Cyb 2017; 47: 1019–1029.

18.

Zhao

SS.

Adaptive neural network control of a fully actuated marine surface vessel with multiple output constraints. IEEE T Contr Syst T 2014; 22: 1536–1543.

19.

Wang

Zhang

Trajectory tracking control of an underactuated unmanned underwater vehicle synchronously following mother submarine without velocity measurement. Adv Mech Eng 2015; 7: 1687814015595340.

20.

Stoten

et al . Adaptive neural network feedforward control for dynamically substructured systems. IEEE T Contr Sys T 2014; 22: 944–954.

21.

Stoten

et al . Adaptive feedforward control for dynamically substructured systems based on neural network compensation. IEEE T Contr Sys T 2014; 22: 944–954.

22.

Yin

Yang

Tracking control of a marine surface vessel with full-state constraints. Int J Syst Sci 2017; 48: 535–546.

23.

Luo

Zou

Tei Shan

LI.

Neural network and L2 gain based autopilot design of ship course keeping. Shipbuild China 2009; 50: 40–45.

24.

Lin

Zeng

. Co-simulation of neural networks PID control for ship steering hydraulic system. In: The 2010 IEEE international conference on information and automation, Harbin, China, 20–23 June 2010, pp.2097–2100. New York: IEEE.

25.

Wei

Adaptive neural network control of a vessel with output constraints using the asymmetric barrier Lyapunov function. IFAC Proc Vol 2013; 46: 246–251.

26.

Fossen

Strand

JP.

Tutorial on nonlinear backstepping: applications to ship control. Model Ident Control 1999; 20: 83–135.

27.

Xia

Shao

Adaptive filtering backstepping for ships steering control without velocity measurements and with input constraints. Math Probl Eng 2014; 2014: 1–9.

28.

Witkowska

Śmierzchalski

Designing a ship course controller by applying the adaptive backstepping method. Int J Appl Math Comp 2012; 22: 985–997.

29.

Liao

Pang

Zhuang

JY.

Backstepping adaptive sliding mode control for unmanned surface vessel course tracking with water-jet-propelled. Appl Res Comp 2012; 29: 82–84.

30.

Cui

Chen

Yang

et al . Extended state observer-based integral sliding mode control for an underwater robot with unknown disturbances and uncertain nonlinearities. IEEE T Ind Electron 2017; 64: 6785–6795.

31.

Liu

Ship adaptive course keeping control with nonlinear disturbance observer. IEEE Access 2017; 5: 17567–17575.

32.

Wei

Chen

Yang

. Nonlinear sliding mode formation control for underactuated surface vessels. In: Proceedings of the 10th world congress on intelligent control and automation, Beijing, China, 6–8 July 2012, pp.1655–1660. New York: IEEE.

33.

Ashrafiuon

Muske

Mcninch

et al . Sliding-mode tracking control of surface vessels. IEEE T Ind Electron 2008; 55: 4004–4012.

34.

Szymak

Course control of unmanned surface vehicle. Sol St Phen 2013; 196: 117–123.

35.

Ding

Liu

Zheng

WX.

Sliding mode direct yaw-moment control design for in-wheel electric vehicles. IEEE T Ind Electron 2017; 64: 6752–6762.

36.

Ding

Second-order sliding mode controller design subject to mismatched term. Automatica 2017; 77: 388–392.

37.

Ding

Zheng

WX.

Robust control of multiple integrators subject to input saturation and disturbance. Int J Control 2015; 88: 844–856.

38.

Ding

Zheng

Sun

et al . Second-order sliding-mode controller design and its implementation for buck converters. IEEE T Ind Inform 2018; 14: 1990–2000.

39.

Fahimi

Sliding-mode formation control for underactuated surface vessels. IEEE T Robot 2007; 23: 617–622.

40.

Huang

Chiou

KC.

Development and application of a novel radial basis function sliding mode controller. Mechatronics 2003; 13: 313–329.

41.

Cai

Cui

et al . A neural network-based sliding mode controller of folding-boom aerial work platform. Adv Mech Eng 2017; 9: 1687814017720876.

42.

Fei

Adaptive RBF neural vibration control of flexible structure. Adv Mech Eng 2015; 7: 956026.

43.

Liu

Yang

Chen

et al . Neuro-adaptive observer based control of flexible joint robot. Neurocomputing 2018; 275: 73–82.

44.

Yang

Chen

et al . Discrete-time optimal adaptive RBFNN control for robot manipulators with uncertain dynamics. Neurocomputing 2017; 234: 107–115.

45.

Wang

Liu

et al . Neural-based adaptive output-feedback control for a class of nonstrict-feedback stochastic nonlinear systems. IEEE T Cybernetics 2015; 45: 1977–1987.

46.

Hajiketabi

Abbasbandy

Casas

The lie-group method based on radial basis functions for solving nonlinear high dimensional generalized Benjamin–Bona–Mahony–Burgers equation in arbitrary domains. Appl Math Comput 2018; 321: 223–243.

47.

Fossen

TI.

Handbook of marine craft hydrodynamics and motion control. Chichester: John Wiley and Sons Ltd, 2011.

48.

Peng

Wang

et al . Neural adaptive steering of an unmanned surface vehicle with measurement noises. Neurocomputing 2016; 186: 228–234.

49.

Shtessel

Edwards

Fridman

et al . Sliding mode control and observation. New York: Springer, 2014.

Neural adaptive sliding mode controller for unmanned surface vehicle steering system

Abstract

Keywords

Introduction

Problem formulation

Controller design and stability analysis

SMC design

Neural adaptive SMC design

Simulation studies

Conclusion

Footnotes

Declaration of conflicting interests

Funding

ORCID iDs

References