Simulation Studies Relating to Rudder Roll Stabilization of a Container Ship Using Neural Networks

RRS (Rudder Roll Stabilization) of Ships is a difficult problem because of its associated non-linear dynamics, coupling effects and complex control requirements. This paper proposes a solution of this stabilization problem that is based on an ANN (Artificial Neural Network) controller. The controller has been trained using supervised learning. The simulation studies have been carried out using MATLAB and a non-linear model of a container ship. It has been demonstrated that the proposed controller regulates heading and also controls roll angle very successfully.


INTRODUCTION
network was first proposed by Zhang et. al. [7]. An alternative approach involving the use of RBF (Radial Basis Function) Networks was first reported by Unar and Murray-Smith [8][9] and the use of recurrent NNs has also been investigated [10][11]. Recent work has been based either on MLP networks [12][13], RBF networks [14][15][16] or on a combination of neural networks and fuzzy logic [17][18][19]. It has been found that feedforward neural networks (MLP as well as RBF) are more suitable than recurrent networks for control applications. In all of the above mentioned work, ANNs have been used as coursechanging or course-keeping/track-keeping controllers. This paper presents a neural network controller which controls the course as well as the roll response of a ship simultaneously. In other words, it behaves as a RRS.
Roll is one of the most critical ship motions because of the possibility of extremely large amplitudes and even situations leading to a capsize. Roll dynamics is also important for offshore barges in determining the loads on deck cargo. Large roll motions not only affect the comfort and efficiency of crew members but also the accuracy of electrical mechanisms and the accuracy of control for the ship course [20].
Some devices such as bilge keels, anti-roll tanks, gyroscopic stabilizers, moving weights and fins [21][22][23] can be used to reduce the roll motion but they increase the cost, occupy useful space and increase the overall weight of the ship. Also the speed and overall performance of the vessel may be affected because of these additional appendages [24]. Since every ship has a steering system, the employment of an RRS to simultaneously control the roll and heading motions is a cost effective approach and improves the ship's capabilities to perform assigned tasks in rough weather conditions. Moreover, the RRS has approximately the same effectiveness as can be achieved with the fin stabilizers [22].

PREVIOUS WORK ON RUDDER ROLL STABILISATION
The non-linear behavior of ship roll motion has been the subject of extensive research since the mid of 19 th century, pioneered by the work of Froude [25]. However, Chadwick [26] was among the first to investigate the roll stabilization problem from control point of view. He discussed in detail the possible inputs for roll stabilization systems and proposed a fin-roll stabilization system which was essentially a proportional controller. The idea of using the rudder as a roll stabilizer was first suggested by and Lauvdal and Fossen [40][41]. A detailed review of their work can be found in Perez [42].  The BP algorithm is based on supervised learning of ANNs. In such an approach an ANN is trained to behave like a specific form of a conventional controller. During the training phase, the same input is applied to the supervisor and to the ANN which is to be trained. The error between the output of the supervisor and the ANN is calculated, as shown in Fig. 1.
A cost function based on this error signal is minimized by using the error BP algorithm. Training is said to be successful when the error reaches to a pre-defined minimum value. More than one supervisor may be used to train a controller to perform multiple tasks. In this way, a single ANN controller will perform like multiple conventional controllers. In this paper, a single MLP controller is proposed which mimics the dynamics of two different supervisors. Details are presented in Section 5.

RUDDER ROLL STABILIZATION SYSTEM
The block diagram of a RRS system, as suggested by Fossen [22], is shown in Fig. 3, where Y

Reference Model
The primary purpose of a reference model is to generate desired behavior which is to be compared with the actual behavior of the system. In Fig. 2 where  is the damping ratio and  n is the undamped natural frequency in radian per seconds. In this study,  = 0.8 and w n = 0.15 radians per second has been chosen for the heading subsystem. The desired roll angle has been assumed to be zero.

Steering Machine
The function of the steering machine is to move the rudder to a desired angular position when demanded by the control system or by the helmsmen. In this paper, a simplified model proposed by Van Amerongen [54] has been used and is shown in Fig. 3

Heading and Roll Controllers
The heading controller (also known as the ship autopilot) is an

Ship Model
The model used in this paper represents a high speed container ship [22,56]. The main data of the ship are given in where x is the state vector and u represents the input to the ship. The state variables include the surge velocity u, sway velocity v, yaw velocity r, roll velocity p, heading angle  and roll angle f. The input is the rudder angle.
The actual non-linear equations of motion and the values of hydrodynamic coefficients can be found in Fossen [22].

Wave Model
Sea waves have a profound impact on the motion of a ship. One major cause of ship capsizing is the abnormal roll motion due to waves [57]. It is therefore very important to ensure robustness of a controller during seakeeping.
In this study, the Pierson-Markowitz [22,58] wave model has been used. This model describes a spectrum for fully developed wind generated seas and has the following form: where  = 8.1x10 -3 g 2 and  = 0.74(g/v 19.4 ) 4 . v 19.4 is the wind speed at a height of 19.4 m over the sea surface and g is the acceleration due to gravity (i.e. 9.8 m/s 2 ).

DEVELOPMENT OF ANN CONTROLLER
Conventionally, two separate controllers are designed, one as an autopilot (i.e. heading controller) and the other as a RRS [22,24,41,59]. The outputs of both the controllers are added to produce net control signal (i.e. rudder angle), as shown in Fig. 2. To minimize the coupling effects between roll and yaw dynamics, the well-known frequency separation method is used which ensures that the bandwidth of the heading subsystem is less than that of the roll subsystem.

Simulation Studies Relating to Rudder Roll Stabilization of a Container Ship Using Neural Networks
In this paper a single feedforward NN has been developed which behaves both as an autopilot as well as a roll stabilizer. For training of the controller two well optimized decoupled SM (Sliding Mode) controllers have been used as supervisors. One SM controller is a heading controller and the other one is a roll stabilizer. It has been ensured that the roll sub-system does not degrade the optimal performance of the heading sub-system. In these figures, the solid line and the dashed line represents the ANN controller and SM controller respectively. Fig. 4 clearly indicates that both the ANN and SM controller turns the ship successfully without any overshoot. The corresponding heading rate is also satisfactory. Fig. 5 shows that the rudder response of both controllers during the turn is almost the same. Fig. 6 depicts the roll angle and roll rate in presence of wind generated waves. It can be seen that the performance of the proposed ANN controller provides improved roll stabilization when compared to the SM controller.

CONCLUSION
This paper has presented a NN based controller for RRS for a simulated container ship. A single ANN controller has been developed to behave both as a heading and a roll control system. Simulation results show that the controller has successfully captured the dynamics of the separate heading and roll controllers. The results are satisfactory and meet the performance requirements suggested for the combined system. In addition, the ANN has been shown to provide improved roll stabilization.
Overall this study has shown that an ANN can be trained successfully to replace conventionally designed control systems.