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In order to track the desired path under unknown parameters and environmental disturbances, an adaptive backstepping sliding mode control algorithm with a neural estimator is proposed for underactuated ships considering both ship-bank interaction effect and shift angle. Using the features of radial basis function neural network, which can approximate arbitrary function, the unknown parameters of the ship model and environmental disturbances are estimated. The trajectory tracking errors include stabilizing sway and surge velocities errors. Based on the Lyapunov stability theory, the tracking error will converge to zero and the system is asymptotically stable. The controlled trajectory is contractive and asymptotically tends to the desired position and attitude. The results show that compared with the basic sliding mode control algorithm, the overshoot of the adaptive backstepping sliding mode control with neural estimator is smaller and the regulation time of the system is shorter. The ship can adjust itself and quickly reach its desired position under disturbances. This shows that the designed RBF neural network observer can track both the mild level 3 sea state and the bad level 5 sea state, although the wave disturbance has relatively fast time-varying disturbance. The algorithm has good tracking performance and can realize the accurate estimation of wave disturbance, especially in bad sea conditions.

In ship heading control, the steering motion makes the bow produce a small lateral drift angle and changes the fluid distribution on both sides of the hull. This will result in the difference between the actual motion direction and the expected ship heading. The current steering control algorithm ignores the effect of the drift angle, but the existence of the drift angle will increase overshoot and reduce the steering control accuracy and performance. In bad sea conditions, the model dynamic uncertainty, time-varying parameters, and drift angle correction need to be considered. The shift angle was proposed by Yu in 2008 [

For large ships, the channel width is narrow and the water depth is shallow. The maneuverability of ships navigating in restricted waters is quite different from that in open waters, mainly due to the influence of the bottom and bank wall, and the hydrodynamic force is usually greater. For the ships running near the shore, the effect of the quay wall is a potentially unsafe factor, which often leads to collision accidents caused by ships too close to the shore wall, resulting in casualties and a large number of property losses. Therefore, it is necessary to study the ship-bank interaction effect [

Recently, backstepping control is used for uncertain nonlinear systems to improve the global ultimate asymptotic stability. Chu proposed an adaptive global sliding mode fuzzy control using a radial basis function neural network based on the backstepping technique [

For dealing with systems with uncertainty, several methods have been used, such as fuzzy logic, neural network, and so on. Cheng presented a novel indirect neural observer with the ADALINE network incorporated into the conventional sliding mode term and utilized a radial basis function neural network approximation utilized to handle the system uncertainties [

Sliding mode control is a branch of variable structure control proposed by Soviet scholars in the 1950s. Guo proposed a hybrid control algorithm based on backstepping control and nonsingular fast terminal sliding mode control to solve the tracking problem of the mobile robot [

However, how to design a robust controller for ships is still a challenging problem because of model errors, parameter changes, and external disturbances. First of all, the existence of a drift angle is usually ignored in course control, but the actual drift angle is not zero, which will cause a drift angle difference between the actual motion direction of the ship and the expected course. If not corrected, the performance of course control will be reduced. Secondly, the effect of ship-bank interaction is a common phenomenon in berthing, crossing bridge, and near-shore navigation. The large ship-bank interaction effect can even lead to ship capsizing. However, in ship motion control, the existence of the ship-bank interaction effect is usually ignored. Thirdly, the sudden disturbance will cause the control force to be too large, which will lead to the excessive driving power of the actuator. Any actuator has a certain range of execution; once the input exceeds the limit value, it will affect the operation of the actuator, resulting in system performance degradation, affecting the control effect, and maintaining too large control input for a long time will also increase the loss of rudder.

In this paper, the small angle approximation and slow time-varying assumption are avoided. The unknown disturbances such as drift angle and ship-bank interaction effect are dynamically estimated by the neural network. The time-varying large drift angle is accurately compensated by using the weight parameter adaptive law of the RBF neural network. Based on backstepping control and adaptive sliding mode technology, the surge speed and heading tracking controllers are designed, respectively, to realize the precise path tracking control with unknown time-varying large drift angles and the influence of the bank wall effect. The stability of the closed-loop system is proved by the Lyapunov theory. The experimental results show that the algorithm can reduce the heading error, the overshoot, adjustment time, and the control torque of the actuator. It can improve the robustness and economy of path tracking control for underactuated ships under unknown sea conditions.

The coordinate system is established as follows: The stationary observer on the shore is defined as the origin _{0}. The positive east direction is defined as the _{0} axis and the positive north direction is defined as the _{0} axis. The three-degree-of-freedom motion of surge, sway, and yaw are considered. The motion model of the dynamic positioning system is shown in Figure

Ship motion.

The mathematical model of the dynamic positioning ship is as follows:

The dynamic models of underactuated ships can be calculated as follows:

The velocity equation is as follows:

Denote the following:

Then (

In the system model, the number of system state variables is 3 and the number of input control variables is 2, so the system is underactuated.

The disturbance of external environment includes the disturbance of wind, sea wave, and current.

According to Isherwoodʼs research, the force and torque of wind disturbance can be calculated as follows:

The forces and moments acting on ships by ocean currents can be calculated as follows:

The forces and moments produced by wave disturbance can be calculated as follows:

When the ship is sailing near the bank of a channel or the pier of a bridge, the water flow near the bank accelerates and the pressure decreases. This results in the additional force which makes the ship close to the bank. This force is called the bank suction. Shore suction may cause the ship to touch the shore. At the same time, there is a moment which makes the bow deviate from the shore; that is, the shore thrust moment. Bank suction and bank thrust moment are generally called ship-bank interaction effects. The ship-bank interaction effect of a vertical wall is shown in Figure

Ship-bank interaction effect.

The force and moment of a vertical bank are calculated by Norrbin’s formula [

In 1985, Powell proposed a radial basis function (RBF) method for multivariate interpolation. The most frequently used Radial Basis Function is the Gauss function:

Figure

RBF structure.

In this system, the input of the RBF neural network is system state

The shift angle is computed as follows:

The heading error is computed as follows:

Taking the derivative of (

Construct new variables:

The following sliding surface functions are constructed:

The control rate is designed as follows:

Denote

Subtracting (

Denote the following:

Equation (

An adaptive control law is constructed as follows:

Based on the Lyapunov stability theory, for the ship dynamic model (

The Lyapunov function is constructed as follows:

Taking the derivative of (

Substituting (

The derivative of (

Substituting (

Taking the derivative of (

Substituting

Taking the derivative of (

Substituting

Substituting

Substituting

Substituting (

The derivative of (

Substituting (

Substituting (

When the conditions are established:

We have the following:

Based on the Lyapunov stability theory, the tracking error converges to zero and the system is asymptotically stable.

In order to verify the control effect of the adaptive backstepping sliding mode control algorithm on the ship, a supply ship [_{11} = 1.1274, _{22} = 1.8902, _{23} = −0.0744, _{32} = −0.0744, _{33} = 0.1278, _{11} = 0.0358, _{22} = 0.1183, _{23} = −0.0124, _{32} = −0.0041, _{33} = 0.0308. The initial position is [−2 m, 2 m,

The experiment was conducted in Intel(R) Core(TM) i7-3612QE CPU @ 2.10 GHz 2.10 GHz, 16.0 GB memory, and 64 bit OS.

The RBF neural network is composed of three layers, that is the input layer, the hidden layer, and the output layer. There are six nodes in the input layer, that is

The initial forward position is −2 m, the initial sway position is 2 m, the initial yaw angle is −1.5 rad, the initial forward velocity is 0 m/s, the initial yaw velocity is 0 m/s, and the initial yaw rate is −1.5 rad/s. The desired forward position is set to 0 M. The desired sway position is 0 M. The expected yaw angle is 0 rad. The expected forward speed is 0 m/s. The desired yaw velocity is set to 0 m/s. The expected yaw rate is 0 rad/s.

Figure

Ship position response curve.

Figure

Speed velocity curve.

Figure

Shift angle and heading angle curve.

Figure

Control input curve.

Figures

The control algorithm is performed at the third level of sea conditions, and other conditions are the same as those in

Position response at the third level of sea conditions.

Figure

Speed response at the third level of sea conditions.

Figure

Shift angle and heading angle curve at the third level of sea conditions.

Figure

Control input at the third level of sea conditions.

The control algorithm is performed at the fifth level of sea conditions, and other conditions are the same as those in

Position response at the fifth level of sea conditions.

Figure

Speed response at the fifth level of sea condition.

Figure

Shift angle and heading angle curve at the fifth level of sea conditions.

Figure

Control input at the fifth level of sea conditions.

Figures

The Ship-bank interaction effect is considered, and other conditions are the same as those in

Position response with ship-bank interaction.

Figure

Speed response with ship-bank interaction.

Figure

Shift angle and heading angle with ship-bank interaction.

Figure

Control input with ship-bank interaction.

Figures

Different parameters of a ship are also tested, and other conditions are the same as those in _{11} = 0.0084, _{22} = 0.0155, _{23} = −0.00009, _{32} = −0.00023, _{33} = 0.00083, _{11} = 0.0018, _{22} = 0.0116, _{23} = 0.0050, _{32} = 0.0026, _{33} = 0.0017.

Figure

Position response of a mariner class vessel.

Figure

Speed response of a mariner class vessel.

Figure

Shift angle and heading angle of a mariner class vessel.

Figure

Control input of a mariner class vessel.

Figures

Different initial conditions are also tested, and other conditions are the same as those in

Position response with different initial conditions.

Figure

Speed response with different initial conditions.

Figure

Shift angle and heading angle with different initial conditions.

Figure

Control input with different initial conditions.

Figures

In order to verify the effectiveness of the algorithm, the control effects of different algorithms are compared, including basic sliding mode control (SMC), backstepping sliding mode control (BPSMC), and adaptive neural backstepping sliding mode control (ANBPSMC). The remaining parameters remain unchanged as in

Surge response of different algorithms.

Figure

Sway response of different algorithms.

Figure

Yaw response of different algorithms.

Figure

Surge speed response of different algorithms.

Figure

Sway speed response of different algorithms.

Figure

Yaw angle speed response of different algorithms.

Figure

Input

Figure

Input

Figures

This paper presents an adaptive backstepping sliding mode control algorithm with a neural estimator for underactuated ships considering both ship-bank interaction effect and shift angle. Based on the feature that the RBF neural network can approach any function, the unknown parameters of the ship model and environmental disturbance are estimated. Based on the Lyapunov stability theory, the tracking error will converge to zero and the system is asymptotically stable. The design of the switching function makes the system robust to uncertainties and external disturbances and avoids chattering. The results show that compared with the basic sliding mode control algorithm and backstepping sliding mode control, the neural estimator adaptive backstepping sliding mode control has less overshoot, shorter system regulation time, less heading error, and less control torque of the actuator. The technical contributions of the control method are summarized as follows:

The Ship-bank interaction effect is considered in underactuated ship motion control, especially in berthing, crossing bridge, and near-shore navigation.

The time-varying shift angle is estimated by a radial basis function (RBF) neural network with weight parameter adaptive law in an underactuated ship heading control.

An adaptive backstepping sliding mode control with a neural estimator is designed. Based on the Lyapunov stability theory, the tracking error converges to zero and the system is asymptotically stable.

The next step is to improve the control algorithm to improve the control accuracy and robustness of the underactuated ship motion control.

The data used to support the findings of this study are included within the article.

The author declares no financial and personal relationships with other people or organizations that can inappropriately influence this work. There is no professional or other personal interest of any nature or kind in any product, service, and company that could be construed as influencing the position presented in the manuscript.

The work was supported by the National Natural Science Foundation of China (51879119), Natural Science Foundation of Fujian Province (2018J05085), and the high level research and cultivation fund of transportation engineering discipline in Jimei University (202003).