LMI-BASED H 2 AND H STATE-FEEDBACK CONTROLLER DESIGN FOR FIN STABILIZER OF NONLINEAR ROLL MOTION OF A FISHING BOAT

This paper presents the analyses of nonlinear roll responses of a fishing boat in waves. In addition to roll damping nonlinearity, the nonlinear roll restoring which has seventh order equation has been taken into consideration to accurate control application. To overcome nonlinearity and the effects of uncertainties, LMI (Linear Matrix Inequality) based H2 and H State Feedback Control are applied for the fin roll stabilizer of a fishing boat. The fin characteristics are calculated by Star CCM+ package software. Finally, utilising the studies presented to illustrate the feasibility and efficiency of the H2 and H control methods, the results of the simulations are demonstrated the performance of fin roll stabilizer.


Introduction
Although significant advances have been made in fishing boats, fishing still remains a highly dangerous operation.While a fishing boat sailing through waves, a large rolling motion and capsizing may occur due to both resonance and inadequate potential energy.The operation of fishing boats can be improved by reducing the roll motion in roll stabilizer.The benefits of a stabilization device are clear: an increase in personnel safety, higher individual productivity and extended operating for fishing boats.In the literature, there are a number of roll stabilizer system applications such as rudder roll stabilizer [1]; u-tube [2]; fin roll stabilizers [3]; [4] and an integrated fin and rudder roll stabilization [5], [6] etc.
A rudder stabilizer achieves roll reduction through rudder deflection.Hence, the performance of a rudder roll stabilizer decreases at low speeds.Lauvdal and Fossen [7] used to design a stable controller for roll damping and course keeping using sliding mode control.angles.A nonlinear backstepping controller is developed to provide global exponential stability of roll motion.
Hakan Demirel, Fuat Alarçin LMI -based H2 and H state feedback controller design for fin stabilizer of nonlinear roll motion of a fishing boat 92 Alarcin and Gulez [8] described the use of a neural network controller for roll stabilization of a fishing boat using the rudder to control roll and yaw motions.
Roll reduction can also be achieved by using u-tanks in the range of 20-70% (RMS).Holden et al. [9] carried out active u-tank stabilizer using Lyapunov theory for large roll If possible, the paper should be written on an even number of pages.Extra space between paragraphs, figures and tables should not be gained by inserting blank paragraphs.If in doubt, place and number the figures at the end of the paper and put captions at desired positions in the text.
Alternatively, the fin can be used to provide an active dynamic stabilization to roll motions.Galeazzi et al. [10] analyzed the stability of the nonlinear system by using Lyapunov methods and studied fin stabilizers to stabilize the roll resonance condition.
Guan and Zhang [11], who proposed backstepping-based controller for ship roll motion, controlled the nonlinear fin roll stabilizer by ignoring the high order nonlinear terms.Ghassemi et al. [12] expressed active fin stabilizers are effective devices to reduce the roll motion of a ship againist wave movements.Therefore, he used PID and Neural Network combination to control attack angle.Karakaş et al. [13] designed a controller based upon Lyapunov's Direct Method to decrease roll motion.Kose [14] modelled with mechanical and electrical dynamics using a general back stepping method.Su [15] used genetic algorithm to fin controller for adjust the angle of attack.
A number of problems related to ship roll motion are investigated by many researchers.In these applications, ( [16], [17], [18], [19]) different nonlinear models of roll motion have been used.Taylan [20] studied a nonlinear roll motion model and solution of this model through Krylov-Bogoliubov asymptotic method on a time line.To overcome undesirable problems, it was presented the model predictive control by Liu et al. [21] In this paper, the problem of a nonlinear roll motion of fishing boat is considered.Section 2 deals with the fin roll stabilizer system based on nonlinear restoring moment and damping effect.Section 3 deals with the fin lifting force caused by the rotation and angle of attack.Section 4 describes designing LMI -based H2 and H State -Feedback controller.Section 5 and 6 discusses results and conclusion

The Nonlinear Model for Roll Motion
In the cause of simplify to using mathematical model of ship motions, some significant suppositions are made in modelling a fishing boat rolling motion, ignoring all other degrees of freedom of ships, xz-plane symmetry, rigid body and homogenous mass distribution.Considering these simplifications, ship rolling behavior was represented by Taylan [22] the following nonlinear expression, Where are angle, angular velocity and angular acceleration of roll motion respectively, the mass moment of inertia is shown I Where, Inertia moment originates owing to the ship's reaction to a motion and is proportional to the acceleration of motion.Added inertia moment means a reaction of sea water to the ship motion.
These Inertia values can be expressed depending on weight displacement of ship (  ), breadth (B) and the vertical distance of the center of gravity (KG), [23] ) 4 ( g 12 The nonlinear roll damping coefficients for ship types are calculated by theoretical and semi-empirical methods considering lift effect, bilge keel, skin friction and free surface effect, [11].A non-dimensional damping coefficient is shown as follows.
These coefficients, given a and d in Table 1, are directly related to a linear damping coefficients L B and a non-linear damping coefficient N B .
Table 1.The non-dimensional damping coefficients for three different types of the ship, [11].When a ship travels through waves, the hydrostatic properties and the restoring moment vary due to the varying surface elevations.For rolling motion, the restoring moment can be approximated by the polynomial.The roll restoring moment coefficients are defined as follows: Angle of vanishing stability v  and area under the GZ curve v A  , dynamic characteristics of the GZ curve such as metacentric height GM.Based on the above-mentioned coefficients, numerical calculations were carried out for a fishing boat, whose body plan is given Fig. 1.  is the wave encounter angle of the ship.It can be envisaged that the wave excitation will depend not only on amplitude and frequency of the waves but also on encountered angle and speed.

Roll Fin Actuator
The motion of a ship can be affected by fin actuators that impart forces and moments.Actuators play a very important role in the control system structure.When the roll fin stabilizers attack to the fluid, it can be seen that the surface of fins lifting force caused by the rotation and angle of attack.The lift force and the lift in non-dimensional form are as in the following form, [3] Where L is the lifting force (N),  is the density of fluid (t/m 3 ), F A is the fins area (m 2 ), L C is the lift coefficient of the fin, V is the ship speed (m/s).
It is required to select the fin stabilizer having larger roll reduction performance with the smallest area.The area can be calculated by the following equation, [ Where wsc  is the wave slope capacity.General formula of fin roll moment is expressed as the following equation: [11]   Where F M , fin roll stabilizer moment; F l , distance from the center of the fin to the mass center of the ship;  is the total fin angle of attack, and it includes the virtual fin angle f  and the oblique flow fin angle '  a ; f  express actuator dynamics.
A control system changes the mechanical angle of the fins, f  , according to the ship roll angle  and roll rate  .A positive angle of attack is produce a positive roll moment produced by the fin.Lift of the fin is dependent on its geometric parameters.The fin stabilizer geometry is chosen as NACA 0015 foil section for calculations.The flow analyses are performed by using a commercial computational fluid dynamics (CFD) software based on finite volume method .
A result for the lift coefficient is displayed in the figure 3. Figure 3 shows the correlation between the lift coefficient and the angle of attack.The flow is considered as steady, incompressible and fully turbulent.The numerical flow analyses are performed at different angles of attack.The fin stabilizer reaches the maximum lift coefficient at the angle of 17 o .H2 and H State Feedback Controllers for the roll motion are modeled for the case which the fin creates the maximum lift.

LMI Based State -Feedback H2 and H Controller Design
H control depends upon minimizing the infinitive norm of transfer function matrix which is written from controlled output to disturbance input in order to avoiding the disturbance input to affect the system.Thus, H control is very convenient control algorithm for the structural systems which are under effect of disturbance inputs.In other words, H∞ control design is mostly related to frequency domain performance and does not assure good transient behaviors for the closed-loop system.H2 control provides more practical performance on system transient behaviors.In this study to obtain desired frequency and transient response performance, H and H2 controller are applied for control problem by the use of LMIs.Doyle et al. derived simple state-space formulas for all controllers solving the following standard H∞ problem and Francis examined mathematics of H∞ control problem.[24,25].Yazıcı and Güçlü designed LMI based state -feedback mixed H2 / H∞ controller for seismic excited structural system.[26].
Let us consider a nominal linear time invariant system revealed by; is the state vector.The state-space matrices and vectors of the closed-loop system are expressed as below.P >0 ,  > 0, such that for all t .If we assume the initial conditions are zero, the Hamiltonian of the system (28) becomes.[26] and this Hamiltonian must be negative definite for all x and w.The following equation can be gained by subrogating the Equation ( 28) into Equation (29).
Arranging the inequality (30) the following matrix inequality can be revealed as LMI -based H2 and H state feedback controller design for fin stabilizer of nonlinear roll motion of a fishing boat Hakan Demirel, Fuat Alarçin 99 The inequality (31) takes the latest form using the Schur complement.Suppose P and R symmetric matrices.The condition, Pre-and post multiplying (31) by 1  P and using the Schur complement, inequality (34) can be expressed, Using the variable change X∞= 1  P the following LMI's can be obtained.
Using the Schur complement over again, the following inequality can be revealed as H∞ constraint of the closed-loop system (3) for X∞ > 0, Hakan Demirel, Fuat Alarçin LMI -based H2 and H state feedback controller design for fin stabilizer of nonlinear roll motion of a fishing boat 100 The optimal H2 controller can be expressed by researching the minimum η which satisfies the above mentioned LMIs for 2 X = T 2 X and Q = T 2 Q Note that the H2 norm of After finding a solution of this control problem, the state-feedback control matrixes for the closed-loop system is obtained as All the simulations and computations are done using Matlab with Simulink.For the solution of the resulting LMIs, Yalmip parser and Sedumi solver are used [28].
LMI -based H2 and H state feedback controller design for fin stabilizer of nonlinear roll motion of a fishing boat Hakan Demirel, Fuat Alarçin 101

Simulation
The nonlinear roll model of the dynamic characteristics of the fishing boat and the model of the fin stabilizer particulars are shown in Table 2.The simulations were performed using the fishing boat model given in ref. [29].First, the nonlinear roll motion of the fishing boat was considered.Then, by adding fin roll stabilizer, active system was formed.After designing H and H2 controller, the roll angle and the roll rate were calculated.The simulated time solutions of nonlinear roll motion controlled with LMI -based H2 and H State Feedback control methods and of the uncontrolled nonlinear roll motion were illustrated in Figure 4 -9, respectively.The disturbance was accepted as a sinusoidal wave form.To show effectiveness of the H2 and H control methods, the nonlinear simulation results were compared with uncontrolled system.As can be seen from the figures, H State Feedback controller results achieved better performance.It is clear from the above that the effectiveness of the fin controller will depend on the H and H2 controller performance.It is also seen that in general a greater reduction in roll is always associated with larger fin actions.Figure 10 shows the frequency responses of the roll motion for both controlled and uncontrolled cases.Since the system has one degree of freedom, there is a resonance frequency points at 1.31 Hz.When the response plots of the roll motion with uncontrolled and controlled cases are compared, good improvement of the resonance values is observed with the proposed controllers.

Conclusions
In this study, nonlinear roll motion of a fishing boat, which will be capable of operating in the Black Sea and the Mediterranean Sea, was controlled by fin stabilizer system.The Computational Fluid Dynamics (CFD) calculations for the lift are carried out using Star CCM + software.A random sea is assumed for the input to cause roll.The H and H2 controller algorithm for a fishing boat have been designed and simulation results have been presented.The control problem was formulated that considered the angles of both fins as control commands.Regarding the rolling motion as the H and H2 control variable, it is shown that the fin roll stabilizer performance is the effective roll stabilized by H and H2 control methods.As can be seen in the simulation results, H controller is achieved better performance by comparison with H2 controller concerning roll reduction.

7 
for an intact vessel.

Figure 1 .Figure 2 .M
Figure 1.Body plan of the fishing boatThe right arm curve is a graphical representation of the fishing boat's stability on Fig.2.This figure shows that the linear manner depends on the nature of restoring moment and angle of attack.The area under the curve represents the fishing boat's ability to counter the capsizing moments acting on the boat.

Figure 3 .
Figure 3. Lift coefficient -Angle of attack (AoA) graph output vector.Suppose that the control input is linear function of the state, i.e.B K)x B w z (C D K)x D w z (C D K)x D w H2 and H state feedback controller design for fin stabilizer of nonlinear roll motion of a fishing boat 98 closed-loop transfer functions, respectively from disturbance (w) to measurements (z1 and z2).Find a state-feedback control law u = Kx such that minimizes .In order to determine a state-feedback control law, H2 and H∞ designed objectives have been severally obtained by LMIs.The Bounded Real Lemma plays a central role to obtain the H∞ constraint.There exists a quadratic Lypunov function

Figure 10 .
Figure 10.Frequency responses of the roll motion

Table 2 .
The fishing boat and the fin particulars