Course Control of Underactuated Ship Based on Nonlinear Robust Neural Network Backstepping Method

Abstract

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.

Academic Editor:Chaomin Luo

College of Automation, Harbin Engineering University, Harbin 150001, China

Received 25 August 2015; Revised 25 November 2015; Accepted 31 January 2016

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

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 [1]. PID controller greatly improved the performance of autopilots. Until the 1980s almost all makes of autopilots were based on these controllers. One challenge for tracking control of surface vessel based on above method is that the systems are often underactuated by the sway motion due to weight, complexity, and efficiency considerations and exhibit nonholonomic constraints, which meets Brocket's theorem that there is no continuous or even smooth time-invariant state feedback law that can stabilize the system to the origin [2]. Another challenge is that the vessel model itself exhibits severe nonlinear characteristic and model uncertainties induced by the ocean environment [3, 4].

For the ship with nonlinear maneuvering characteristics and without uncertainties, a state feedback linearization control law was designed [5], while feedback linearization with saturation and slew rate limiting actuators was discussed [6]. Later, combined with a genetic algorithm, the backstepping method was employed to develop a nonlinear ship course controller by Witkowska and Smierzchalski [7], where the ship course parameters were automatically tuned to the optimal values with the aid of a genetic algorithm. Even considering the ship steering model with both constant parametric uncertainties and input disturbance with unknown bound, a robust adaptive nonlinear control law was presented based on projection approach and Lyapunov stability theory [8]. Recently many papers have tackled these problems based on Lyapunov theory [9-12]. In [13-15] a global tracking controller for underactuated ship is addressed with nonzero off-diagonal terms, the reference trajectory is generated by using a virtual target guidance algorithm, and the controller designed is facilitated by an introduction of changing the ship outputs, several coordinate transformations, and backstepping method. And the controller design is heavily depending on accurate dynamic model; the robustness against disturbance has not been addressed. A method using backstepping adaptive dynamical sliding mode control is presented for path following control of USV in [16], the control system takes account of the modeling errors and disturbances, and simplified tracking error dynamics are obtained by assuming that the sway velocity is small which can be neglected in the controller design and only for straight line path tracking can be achieved. The LOS based guidance law is also used in the controller design which causes the complexity of computing high-order derivative of virtual control. In [17], a transformation of vessel kinematics to the Serret-Frenet frame is introduced by exploring an extra degree of freedom by controlling explicitly the progression rate of the virtual target along the path and overcomes the major singular problem; approach angle is introduced for controller design via backstepping method. Neural networks are introduced to enhance system stability and transient performance, which can handle the known dynamics and uncertainties of systems well [18-20]. Particularly in [12] a single hidden layer neural network (SHLNN) is adopted to obtain the adaptive signal online, but the choice of the single hidden layer neural network is limited by the number of hidden layer node selections that will affect the online learning speed and accuracy and cannot produce a better estimation effect on the fast changing disturbances.

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.

2. Adaptive Robust Neural Network Controller Design

2.1. Problem Description

Consider the following nonlinear systems: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is system state, [figure omitted; refer to PDF] is control input, and [figure omitted; refer to PDF] is system output. The control objective is to design an adaptive neural network controller and make [figure omitted; refer to PDF] track [figure omitted; refer to PDF] . [figure omitted; refer to PDF] meets the smooth bounded reference model as follows: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is state constant, [figure omitted; refer to PDF] represents system output, and [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , denote nonlinear function, assuming that the reference model for each state is bounded as [figure omitted; refer to PDF] , [figure omitted; refer to PDF] .

Assumption 1.

There is an unknown constant [figure omitted; refer to PDF] to meet, [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] is a known positive smooth function.

2.2. Direct Adaptive Neural Network Controller Design

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.

Step 1.

Let [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and then [figure omitted; refer to PDF]

Consider the following Lyapunov function: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] , [figure omitted; refer to PDF] represents the ideal weight vector of neural network, [figure omitted; refer to PDF] represents the estimated value of the neural network weight vector, [figure omitted; refer to PDF] represents the estimation error of weight vector, [figure omitted; refer to PDF] is the adaptive gain matrix, and the derivation of [figure omitted; refer to PDF] can be computed as [figure omitted; refer to PDF]

According to Assumption 1, we can get [figure omitted; refer to PDF]

There is an ideal virtual feedback control law: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is designed controller parameter.

Because of the unknown smooth functions [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , we cannot actually get the ideal feedback control law [figure omitted; refer to PDF] ; from (7) we can see that the unknown part [figure omitted; refer to PDF] is smooth function of [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , so that [figure omitted; refer to PDF]

RBF neural network [figure omitted; refer to PDF] is used to approximate the unknown function [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] can be expressed as [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is estimated error and meets [figure omitted; refer to PDF] .

Because [figure omitted; refer to PDF] is unknown, the virtual control law is selected as follows: [figure omitted; refer to PDF] and then [figure omitted; refer to PDF]

Adaptive law can be chosen as follows: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] and then [figure omitted; refer to PDF]

Let [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , and then the upper equation becomes [figure omitted; refer to PDF]

According to the complete square formula, [figure omitted; refer to PDF]

Because [figure omitted; refer to PDF] , we can make [figure omitted; refer to PDF] by choosing the appropriate [figure omitted; refer to PDF] and obtain the following inequality: [figure omitted; refer to PDF]

The cross coupling [figure omitted; refer to PDF] in (16) will be eliminated in the next step.

Step 2.

Let [figure omitted; refer to PDF] ; then [figure omitted; refer to PDF]

From (10) we can see that [figure omitted; refer to PDF] is a function of [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] can be written as [figure omitted; refer to PDF] where [figure omitted; refer to PDF] can be calculated.

Consider the following Lyapunov function: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is an adaptive gain matrix.

Then the derivation of [figure omitted; refer to PDF] can be calculated as [figure omitted; refer to PDF]

According to Assumption 1 we can get [figure omitted; refer to PDF]

There is an ideal feedback control law: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is a designed controller parameter.

Because of the unknown smooth functions [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , we cannot actually get the ideal feedback control law [figure omitted; refer to PDF] ; from (22) we can see that the unknown part is a smooth function of [figure omitted; refer to PDF] and [figure omitted; refer to PDF] ; let [figure omitted; refer to PDF] where [figure omitted; refer to PDF] . RBF neural network [figure omitted; refer to PDF] is used to approximate the unknown function [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] can be expressed as [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is expressed as the ideal constant weight vector and [figure omitted; refer to PDF] is the estimated error and meets [figure omitted; refer to PDF] .

Because [figure omitted; refer to PDF] is unknown, select the following virtual control law: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is the estimated value of [figure omitted; refer to PDF] ; then [figure omitted; refer to PDF] where [figure omitted; refer to PDF] .

Adaptive law can be chosen as [figure omitted; refer to PDF] where [figure omitted; refer to PDF] ; then [figure omitted; refer to PDF]

Let [figure omitted; refer to PDF] , [figure omitted; refer to PDF] ; then the upper equation becomes [figure omitted; refer to PDF]

According to the complete square formula, [figure omitted; refer to PDF]

Because [figure omitted; refer to PDF] , then we can make [figure omitted; refer to PDF] by selecting the proper [figure omitted; refer to PDF] ; then [figure omitted; refer to PDF]

The cross coupling [figure omitted; refer to PDF] in (31) will be eliminated in the next step.

Step [figure omitted; refer to PDF] . The derivative of [figure omitted; refer to PDF] can be calculated as [figure omitted; refer to PDF] where [figure omitted; refer to PDF]

Consider the following Lyapunov function: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is an adaptive gain matrix.

Then the derivation of [figure omitted; refer to PDF] can be calculated as [figure omitted; refer to PDF]

According to Assumption 1 we can get [figure omitted; refer to PDF]

There is an ideal feedback control law as [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is designed controller parameter.

Because of the unknown smooth functions [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , we cannot actually get the ideal feedback control law [figure omitted; refer to PDF] ; from (37) we can see that the unknown part is a smooth function of [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , and let [figure omitted; refer to PDF] where [figure omitted; refer to PDF]

By introducing the direct variable [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , we can make the number of neural networks minimized. RBF neural network [figure omitted; refer to PDF] is used to approximate the unknown function [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] can be expressed as [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is estimated error and meets [figure omitted; refer to PDF] .

Because [figure omitted; refer to PDF] is unknown, select the following virtual control law: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is the estimated value of [figure omitted; refer to PDF] ; then [figure omitted; refer to PDF] where [figure omitted; refer to PDF] .

The following adaptive law can be selected as [figure omitted; refer to PDF] where [figure omitted; refer to PDF] ; then [figure omitted; refer to PDF]

Let [figure omitted; refer to PDF] , [figure omitted; refer to PDF] ; then (44) can be rewritten as [figure omitted; refer to PDF]

According to the complete square formula, [figure omitted; refer to PDF]

Because [figure omitted; refer to PDF] , then we can make [figure omitted; refer to PDF] by selecting the proper [figure omitted; refer to PDF] ; then [figure omitted; refer to PDF]

The cross coupling [figure omitted; refer to PDF] in (47) will be eliminated in the next step.

Step [figure omitted; refer to PDF] . The derivative of [figure omitted; refer to PDF] can be calculated as [figure omitted; refer to PDF] where [figure omitted; refer to PDF] where [figure omitted; refer to PDF]

Consider the following Lyapunov function: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is an adaptive gain matrix. Then the derivation of [figure omitted; refer to PDF] can be calculated as [figure omitted; refer to PDF]

According to Assumption 1 we can get [figure omitted; refer to PDF]

There is an ideal feedback control law as [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is designed controller parameter.

Because of the unknown smooth functions [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , we cannot actually get the ideal feedback control law [figure omitted; refer to PDF] ; from (54) we can see the unknown part is a smooth function of [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , and let [figure omitted; refer to PDF] where [figure omitted; refer to PDF] .

RBF neural network [figure omitted; refer to PDF] is used to approximate the unknown function [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] can be expressed as [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is estimated error and meets [figure omitted; refer to PDF] .

Because [figure omitted; refer to PDF] is unknown, select the following virtual control law: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is the estimated value of [figure omitted; refer to PDF] ; then [figure omitted; refer to PDF] where [figure omitted; refer to PDF] .

The following adaptive law can be selected as [figure omitted; refer to PDF] where [figure omitted; refer to PDF] ; then [figure omitted; refer to PDF]

Let [figure omitted; refer to PDF] , [figure omitted; refer to PDF] ; (60) can be rewritten as [figure omitted; refer to PDF]

According to the complete square formula, [figure omitted; refer to PDF]

Because [figure omitted; refer to PDF] , then we can make [figure omitted; refer to PDF] by selecting the proper [figure omitted; refer to PDF] ; then [figure omitted; refer to PDF]

Let [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] ; then [figure omitted; refer to PDF]

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

Theorem 2.

In the initial conditions, by formula (1), reference model (2), control law (57), and neural network weight update rate in (12), (27), (43), and (59), supposing that there is a large enough set of closed sets [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , for any given moment [figure omitted; refer to PDF] , making [figure omitted; refer to PDF] , the following conclusions can be obtained as follows:

(1) The signal of the whole closed-loop system is bounded, and the state variable [figure omitted; refer to PDF] and the neural network estimation errors [figure omitted; refer to PDF] will eventually converge to the closed set as follows: [figure omitted; refer to PDF]

(2) By choosing the proper control parameters, the output tracking error [figure omitted; refer to PDF] is close to a small neighborhood of zero [21].

3. Adaptive Robust Neural Network Control for Ship Course

3.1. Problem Formulation

This section introduces a simplified dynamic model of an underactuated surface vehicle with only one control input [figure omitted; refer to PDF] for heading control. A surface ship usually has three degrees of freedom for path following control in horizontal plane. Assuming that the vessel has three planes of symmetry, for most underactuated vessels have port/starboard symmetry, it can be neglected to simplify the vessel model for controller design. The detailed model which considers the environment disturbances can be set as follows: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] denotes transverse displacement in the earth inertial coordinates; [figure omitted; refer to PDF] is resultant velocity of ship; [figure omitted; refer to PDF] is course angle; [figure omitted; refer to PDF] is yawing angular velocity; [figure omitted; refer to PDF] represent performance index for ship steering; [figure omitted; refer to PDF] is coefficient of nonlinear term; [figure omitted; refer to PDF] is control rudder angle; [figure omitted; refer to PDF] represent system output.

The control objective is to design the controller [figure omitted; refer to PDF] to make the control output [figure omitted; refer to PDF] , [figure omitted; refer to PDF] achieve the setting value [figure omitted; refer to PDF] . Because the dimension of the system control input is less than the degree of freedom of the system, it is an underactuated system.

3.2. Dynamic Controller Design

Selection of coordinate transformation is as follows: [figure omitted; refer to PDF]

The original system can be transformed into a single input single output system: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and the output of whole system is [figure omitted; refer to PDF] .

For system model (67) and (68), the controller design is carried out by using backstepping method.

Step 1.

Let [figure omitted; refer to PDF] , [figure omitted; refer to PDF] ; then [figure omitted; refer to PDF]

For the subsystem [figure omitted; refer to PDF] , [figure omitted; refer to PDF] is chosen as virtual control input. Select the Lyapunov function [figure omitted; refer to PDF] , and there is [figure omitted; refer to PDF]

Let [figure omitted; refer to PDF] ; then [figure omitted; refer to PDF] , [figure omitted; refer to PDF]

Select the following virtual control law: [figure omitted; refer to PDF]

[figure omitted; refer to PDF] , because [figure omitted; refer to PDF] is unknown function, [figure omitted; refer to PDF] , and we will adopt RBF NN to estimate [figure omitted; refer to PDF] and get [figure omitted; refer to PDF] . But the actual use of the NN for the system is [figure omitted; refer to PDF] . Actual virtual control input is [figure omitted; refer to PDF] ; then [figure omitted; refer to PDF] where [figure omitted; refer to PDF] .

Select Lyapunov function as [figure omitted; refer to PDF] then [figure omitted; refer to PDF]

The adaptive law of neural network can be designed as [figure omitted; refer to PDF] where [figure omitted; refer to PDF] . Let [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] .

Furthermore, [figure omitted; refer to PDF] then [figure omitted; refer to PDF] because [figure omitted; refer to PDF]

Finally we can get [figure omitted; refer to PDF]

Step 2.

Let [figure omitted; refer to PDF] ; derivation of [figure omitted; refer to PDF] can be calculated as [figure omitted; refer to PDF]

Because [figure omitted; refer to PDF] , then [figure omitted; refer to PDF] where [figure omitted; refer to PDF] , [figure omitted; refer to PDF] is unknown parameter, [figure omitted; refer to PDF] is known nonlinear function, and then [figure omitted; refer to PDF]

Let [figure omitted; refer to PDF]

Equation (83) can be rewritten as [figure omitted; refer to PDF]

In the same way we use RBF NN estimate [figure omitted; refer to PDF] : [figure omitted; refer to PDF]

The actual use of the NN for the system and controller can be expressed as [figure omitted; refer to PDF]

Select Lyapunov function as [figure omitted; refer to PDF]

The derivation of [figure omitted; refer to PDF] can be calculated as [figure omitted; refer to PDF]

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 [figure omitted; refer to PDF]

4. Numerical Simulations and Analysis

The simulation experiment can be operated based on an experimental ship. The nonlinear mathematical model for the ship has been presented in [22], which captures the fundamental characteristics of dynamics and offers good maneuverability in the open-loop test. To illustrate the effectiveness of the theoretical results, the proposed control scheme is implemented and simulated with the above nonlinear model with tracking task.

The characteristic parameters of the ship used in the simulation are given as [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] . Neural network contains 25 neurons; that is, [figure omitted; refer to PDF] ; the center vector [figure omitted; refer to PDF] is uniformly distributed in the width [figure omitted; refer to PDF] . Neural network [figure omitted; refer to PDF] contains 135 neurons; that is, [figure omitted; refer to PDF] ; the center vector [figure omitted; refer to PDF] is uniformly distributed in the width [figure omitted; refer to PDF] . The controller design parameters are given as follows which satisfy the condition mentioned in design procedure: [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] , [figure omitted; refer to PDF] . The initial linear and angular velocity of ship used in the simulation are given as [figure omitted; refer to PDF] , [figure omitted; refer to PDF] is the initial position and orientation vector of ship, and the desired velocity of ship is given as [figure omitted; refer to PDF] (m/s). We choose the reference trajectory as [figure omitted; refer to PDF] .

In order to further verify the validity of the proposed control method, the algorithm of this paper is compared with the simulation results in [12]. So the robustness of trajectory tracking controller against the disturbance and model uncertainties can be evaluated. All the simulation results are depicted in Figures 1-4. Figure 1 shows the trajectory tracking of ship with the given path, and the ship can track and converge to the reference path with more accuracy in [12]. Figure 2 plots the position tracking errors; the along-track and cross-track errors asymptotically converge to zero faster. Figure 3 gives the control inputs response. Surge, sway, yaw velocities, and orientation of ship during the trajectory tracking control process are plotted in Figure 4, which gives a clear insight into the model response involved in nonlinear dynamics.

Figure 1: Ship tracking performance of proposed control method.

[figure omitted; refer to PDF]

Figure 2: Tracking errors of surge and sway.

[figure omitted; refer to PDF]

Figure 3: Control force and torque of ship.

[figure omitted; refer to PDF]

Figure 4: State changing curves of ship.

[figure omitted; refer to PDF]

5. Conclusions

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.

Acknowledgment

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

Conflict of Interests

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

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