The problem of course control for underactuated surface ship is addressed in this paper. Firstly, neural networks are adopted to determine the parameters of the unknown part of ideal virtual backstepping control, even the weight values of neural network are updated by adaptive technique. Then uniform stability for the convergence of course tracking errors has been proven through Lyapunov stability theory. Finally, simulation experiments are carried out to illustrate the effectiveness of proposed control method.

Tracking control performance for surface vessel along the predefined route has been an essential control problem for marine autopilot system design, and it has received considerable attractions from control community. In 1922, proportional-integral-derivative (PID) autopilot for ship steering was presented by Nicholas Minosky [

For the ship with nonlinear maneuvering characteristics and without uncertainties, a state feedback linearization control law was designed [

Therefore, a solution to the course control of underactuated surface vessel is addressed in this paper. In view of the characteristics of the underactuated performance, the backstepping control method is used to deal with above problem. The direct adaptive neural network is adopted to design control law by using the RBF neural network to overcome the problem that the ideal virtual control cannot be used directly in practice. The weights of the neural network are updated by adaptive technique to guarantee the stability of the closed-loop system through Lyapunov stability theory. Simulation results are illustrated to verify the performance of the proposed adaptive neural network controller with good precision.

Consider the following nonlinear systems:

There is an unknown constant

In view of the problems and solutions described in the last section, the direct adaptive neural network controller for nonlinear systems with RBF neural network is chosen. Detailed design steps will be described in the following.

Let

Consider the following Lyapunov function:

According to Assumption

There is an ideal virtual feedback control law:

Because of the unknown smooth functions

RBF neural network

Because

Adaptive law can be chosen as follows:

Let

According to the complete square formula,

Because

The cross coupling

Let

From (

Consider the following Lyapunov function:

Then the derivation of

According to Assumption

There is an ideal feedback control law:

Because of the unknown smooth functions

Because

Adaptive law can be chosen as

Let

According to the complete square formula,

Because

The cross coupling

Consider the following Lyapunov function:

Then the derivation of

According to Assumption

There is an ideal feedback control law as

Because of the unknown smooth functions

By introducing the direct variable

Because

The following adaptive law can be selected as

Let

According to the complete square formula,

Because

The cross coupling

Consider the following Lyapunov function:

According to Assumption

There is an ideal feedback control law as

Because of the unknown smooth functions

RBF neural network

Because

The following adaptive law can be selected as

Let

According to the complete square formula,

Because

Let

The stability and control performance of the closed-loop adaptive system are demonstrated by the following theorem.

In the initial conditions, by formula (

The signal of the whole closed-loop system is bounded, and the state variable

By choosing the proper control parameters, the output tracking error

This section introduces a simplified dynamic model of an underactuated surface vehicle with only one control input

The control objective is to design the controller

Selection of coordinate transformation is as follows:

The original system can be transformed into a single input single output system:

For system model (

Let

For the subsystem

Let

Select the following virtual control law:

Select Lyapunov function as

The adaptive law of neural network can be designed as

Furthermore,

Finally we can get

Let

Because

Let

Equation (

In the same way we use RBF NN estimate

The actual use of the NN for the system and controller can be expressed as

Select Lyapunov function as

The derivation of

Therefore, all signals in the close loop of course tracking system are stable, and the tracking errors can be made arbitrarily small by selecting appropriate controller parameters. So the final control law can be designed as

The simulation experiment can be operated based on an experimental ship. The nonlinear mathematical model for the ship has been presented in [

The characteristic parameters of the ship used in the simulation are given as

In order to further verify the validity of the proposed control method, the algorithm of this paper is compared with the simulation results in [

Ship tracking performance of proposed control method.

Tracking errors of surge and sway.

Control force and torque of ship.

State changing curves of ship.

In this paper, we proposed a solution to the course control of underactuated surface vessel. Firstly, the direct adaptive neural network control and its application are introduced. Then the backstepping controller with robust neural network is designed to deal with the uncertain and underactuated characteristics for the ship. Neural networks are adopted to determine the parameters of the unknown part of the ideal virtual control and the ideal control; even the weights of neural network are updated by using adaptive technique. Finally uniform stability for the convergence of tracking errors has been proven through Lyapunov stability theory. The simulation results illustrate the performance of the proposed course tracking controller with good precision.

The authors declare that there is no conflict of interests regarding the publication of this paper.

This work was supported by the National Natural Science Foundation of China, under Grant 51309067/E091002.