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.