DESIGN CONTROL OF SURFACE MARINE VEHICLE USING DISTURBANCE COMPENSATING MODEL PREDICTIVE CONTROL (DC-MPC)

This research studied ship motion control by considering four degrees of freedom (DoF): yaw, roll, sway, and surge in which comprehensive mathematical modeling forming a nonlinear differential equation. Furthermore, this research also investigated solutions for fundamental yet challenging steering problems of ship maneuvering using advanced control method: Disturbance Compensating Model Predictive Control (DC-MPC) method, which based on Model Predictive Control (MPC). The DC-MPC allows optimizing a compensated control then consider sea waves as the environmental disturbances. Those sea waves influence the control and also becomes one of the constraints for the system. The simulation compared the varying condition of Horizon Prediction (Np) and another method showing that the DC-MPC can manage well the given disturbances while maneuvering in certain Horizon Prediction. The results revealed that the ship is stable and follows the desired trajectory.


INTRODUCTION
The surface marine vehicle is one of the best choices for either industrial transportation or military purposes since it is designed to operate with adequate reliability and economy [1]. Indonesia is an archipelago country, which is two of third of its territory consists of water. Nowadays, Indonesian government concerns in Maritime's development, such as surface and underwater vessels. Those vehicles, generally, are used to protect illegal exploiting of its natural capital [2]. The moving object, particularly the surface marine vehicle, has six degrees of freedom: surging, swaying, yawing, rolling, pitching, and heaving [3]. Those movements are centered on three main axes. Some studies have presented mathematical modeling of ship maneuvering, especially in controlling the flow [4,5]. Ship control has been a popular theme in current research, mostly in control steering to get optimal performance [1,6] purposing to design a trajectory tracking of the ship while maneuvering on the sea. Further ship problems were investigated by an adaptive control such as adaptive control design with extending Z-filter using error estimator for ship's path [7] and adaptive fuzzy robust control for ship steering autopilot [8]. Simultaneously, Nomoto's Model was also controlled using control steering in Nomoto's Model to get the autopilot design of the time-varying system [9].
Commonly mathematical modeling of ship maneuvering is very complicated. It has significant inertia, non-linearity, parameter perturbations, and random external disturbances such as wave, wind, and ocean current, uncertainty in course control. In detail, the mathematical modeling in this research forms nonlinear differential equations by considering four degrees of freedom: surging, swaying, yawing, and rolling with hydrodynamics derivatives factors [3] [10]. It is assumed that pitching and heaving do not influence ship maneuvers. Interestingly, it is the most comprehend and covers the fundamental characteristic of a dynamic system. The problem of ship maneuvering is mostly about controlling the system with a nonlinear equation, especially in the auto-pilot system. It should be controlled by an appropriate method that can suit the unstable condition. Various advanced control methods are applied to solve the problems: Model Predictive Control (MPC), Disturbance Compensating Model Predictive Control (DC-MPC), Adaptive robust control, backstepping approach, and genetic algorithm [7,9,11,12,13,14]. However, some factors influencing the maneuvering should be calculated selected controller, degree of freedom, and environmental disturbances.
Control theory, interestingly, is improving along the time and comes with advanced methods followed by its application in diverse fields. One popular method, Pontryagin's Minimum Principle, is used to minimize nonlinear dynamical systems [15,16] and advancement with stochastic MPC to control energy consumption [17]. In 2019, research about sliding mode control can provide a stability control system and fast dynamic response [18] and combine stochastic predictive control and max-plus algebra [19]. Simultaneously, a practical MPC enhances the performance of induction motor drive [20], the framework for modeling and representation of hybrid Model predictive control [21]. Those various predictive control applications allow this method to enforce in such ship heading control with its complexity. Furthermore, in 2019 Ueno et.al studied model ship control and estimation through the propeller and Haseltalab &Negenborn determined control maneuvring and energy using MPC [22,23].
On the other hand, the environmental disturbance is one of the factors that could influence ship motion control. Waves, winds, and currents are the main types of natural disorders. Commonly, many kinds of research are not considering natural disturbances that will form a more complex structure of ship maneuvering. However, this research found ocean waves created by the wind on which has significant interaction while shipping maneuvers. Many control methods are used to control ship performance, especially against disturbance such as Model predictive control for a ship heading control [24,25,26,27]. This paper considered ocean waves formed by the wind on which has significant interaction while maneuvering. The wave itself can be approached by sinusoidal [28].
Motivated by these issues, this paper proposes the Disturbance Compensating Model Predictive Control (DC-MPC) method, the development of Model Predictive Control (MPC), in the crewless vehicle's application. The MPC itself is a strategy on designing control to gain a signal input by minimizing an objective function [20], while DC-MPC is advanced design system controls for handling disturbance directly based on feedback control. There are several main steps to control using this method. Firstly, the disturbance, in this case, is the wave, should be defined. Secondly, the system can be optimized and form disturbance compensating, distinguished between ordinary Model Predictive Control (MPC). Finally, it is used as an input in MPC's algorithm. DC-MPC purposed to fix disturbance that should be considered in a system properly [29]. This research develops ship motion control, which moved on the sea surface with high-speed maneuvering. This problem is solved using Disturbance Compensating Model Predictive Control. This research aims to give an alternative method that may be better than previous ones, particularly in controlling marine surface vehicles by calculating natural disturbance. It also can be used as fundamental for conducting advanced research in ship control design.

RESEARCH METHOD
This research aims a sophisticated control in surface marine vehicle by formulating a dynamical modeling of the maneuvering considering surge, sway, yaw, and roll. In this part, we present the algorithm of DC-MPC and apply into derived model.

Mathematical Modelling
The Surface marine vehicle is a moving object in which the performance can be represented by using Newton's law. Several papers have been proposed the mathematics model of ship maneuvering [2], in which ship maneuvering was generally represented by six DOF rigid-body equation of motion: While this research is used Ship container as a model in this research to get the exact parameter and coefficients, Another approach to formulate modeling by using system identifying [30]. The formulation of dynamical modeling considers four degrees of freedom: surge, sway, yaw, and roll and forms nonlinear differential equation that should be linearized for getting a new system which is simpler and more natural to be analyzed and simulated.
The angle definition of ship maneuvering in this condition can be illustrated as in Figure 1.
where , , , denote surge, sway, yaw and roll velocity respectively. Variable , , , are the surge displacement, sway displacement, yaw angle, and roll angle in the earth fixed frame. Design Control of Surface Marine Vehicle Using…….
Equation (3) can be rewritten into a differential equation describing the rate of the surge, sway, yaw, and roll respect to time (s). In mathematics the formulation follows Where ̇,̇ are surge and sway velocities respectively, while ,̇ are yaw and roll angular velocities with = ′ + ′ , = ′ + ′ , = ′ ′ , = ′ ′ , = ′ + ′ , = ′ ′ , = ′ + ′ meanwhile, ′ , ′ , ′ , ′ denotes hydrodynamic force and moment of ship: the normal rudder force, , can be defined by: where is a rudder area, Δ is a ratio of the rudder, is rudder longitudinal, and is rudder lateral. The Equation (3) and (4) are mathematics models of ship motion system that should be controled and should be linearized before applying DC-MPC using the expansion of Jacobian matrix around stabilization points( 0 , 0 , 0 , 0 , 0 , 0 , 0 ). The initial velocity of the surge influences resultant ship manuvering:  Added inertia moment in the y-direction Coefficient of Hydrodynamic differential moment along z axes respect to Coefficient of Hydrodynamic differential moment along z axes respect to Coefficient of Hydrodynamic differential moment along z axes respect to Coefficient of Hydrodynamic differential moment along z axes respect to Coefficient of Hydrodynamic differential force along y-axes respect to Coefficient of Hydrodynamic differential force along y-axes respect to Coefficient of Hydrodynamic differential force along y-axes respect to Coefficient of Hydrodynamic differential force along y-axes respect to Coefficient of Hydrodynamic differential moment along with y-axes respect to Coefficient of Hydrodynamic differential moment along with y-axes respect to Coefficient of Hydrodynamic differential moment along with y-axes respect to Coefficient of Hydrodynamic differential moment along with y-axes respect to gravity without being influenced by other velocities such as yaw and sways while stable = √ 2 + 2 = √( 0 + Δ ) 2 + Δ 2 Ship maneuvering is stable when it constantly moves toward reference that have defined. There is no change both for surge velocity and sway velocity, so the stability point for surge velocity can be chosen, 0 = 15 knot, and sway velocity = 0. Consequently, other stable points are defined by 0 = 0, 0 = 0, 0 = 0, 0 = 0, and the rudder angle is 0 = 0.

Method
The difference between MPC and DC-MPC is in their ability to process disturbance. While DC-MPC allows to minimize disturbance and create compensating to the plant, the MPC added the disturbance directly in state space. This paper focuses on the linear DC-MPC so that the very first step is linearizing Equation (2) and (4) using the Taylor series at initial values that have been defined.
The DC-MPC proposes a computationally, efficient two-step algorithm to handle disturbance by exploring the disturbance information [17]. The algorithm of DC-MPC follows several steps: Step 1 Estimate the disturbance ̂( − 1) of the previous time step − 1 The disturbance at time step( − 1), ̂( − 1) can be estimated follow: with the assumption, the disturbance at time step , ( ) can be estimated by Step 2 The influence of disturbance is modeled as a first-order sea wave. Calculate the disturbance compensating control by solving the optimization problem Where = max( ), which ∈ is the difference between wave and estimation wave − 1. The optimization output is the optimal disturbance compensating ∆ * and it will be used as a factor that influences the boundary constraint of control input in MPC.
Step 3 Solve the optimization problem ( ( ), det * ) as follows number of state ( × ). Then = dimension of matrix R depends of the number of control input ( × ). Q and R are parameters under assume [16] [ 0 0 ] ≤ 0 Step 4 Implementation of the following control system: It means that ( ) = * ( | ) + ∆ * is control optimal in this process.
The further process completes the model's mathematical analysis into computational using some coefficients and parameters of container ships shown in [3]. The simulation aims to get the best performance of maneuver in various Horizon Prediction.

RESULT AND DISCUSSION
Several steps of the analyzed method should be completed by the simulation to prove the hypotheses regarding DC-MPC. Simulation of ship maneuvering has been investigated using the nonlinear model and prediction of the empiric method [31,32]. The given initial values in simulation are ̅ (0) = [ , , , , , ] = [5,15 ; 0; 0,0001 ; 0; 0] and (0) = 0, while horizon value is randomly chosen in 25 times simulation. The purpose of the DC-MPC method controls the stability of ship motion. Therefore the angular velocities of yaw, roll, and rudder angle should be limited into boundary constraints. The heading control angle velocity is close to 0 unless there is disturbance such as sea wave.
The case is assumed that ship maneuvering has 10 knots of surge velocity. Meanwhile, reference of heading angle is calculated toward earth fixed axis( ), ship motion is controlled for maneuvering parallel to axis-, or on the other words heading angle reach0 . It is continued by controlling ship motion to move forward, which has 10 knots for surge velocity.
It is interesting to note that variation of prediction horizon values can influence the heading ship's position. Consequently, the simulation involves many kinds of prediction horizon to know how big it will affect the ship maneuvering when other parameters are fixed. The validation of the mathematical analysis in design control of the system applies in different Horizon prediction ( ) using a random integer from 1 until 100. However, this paper shows some comparisons of those result = 40, 60, and 80. Those three conditions performed differently showed by Fig. 2, 3, and 4.
To further validate its performance, the DC-MPC scheme is evaluated and compared with M-MPC, which the disturbances do not before estimate.   Figure 3 shows the behavior of roll velocity for difference prediction horizon in which = 60 reaches the reference at 1.6 ℎ time. Meanwhile, the = 40 needs the time to be stable after handling the disturbance wisely in 1.2 ℎ . Besides, roll velocity can be well-controlled within the already given constraint, which isp≤0.0106 rad/s.  Figure 4 compares the rudder angle with the variation of prediction horizon values: 40, 60, and 80. It can be seen that those three prediction horizons move within the constraint (0.175rad/s), whereas prediction horizons below 25 perform passing the constraint. Conversely, the heading angle behavior for prediction horizon, 40 and 60, can reach the reference (nearly zero) before 5 units of time, which were at 4.8 and 3.8, respectively.
It is noticeable that the behavior of the rudder angle, yaw, and roll velocity remains stable.  Figures 5-7 show that the states and control input has closely behaved and moved around the reference for both methods. Besides, both methods are working very well under the given constraints. However, as shown in Figures 6 and 7, the performance of roll velocity and rudder angle using DC-MPC reaches the references relative faster than MPC, which are in 1.8 ℎ and 2 times.

CONCLUSION
In this paper, the mathematical modeling of sea surface vehicle considering four degrees of freedoms, namely surging, swaying, yawing, and rolling forming nonlinear differential equations is proposed. The formulation consists of ten state variables and one control input. Based on the discussion mentioned above, we conclude some interesting points: a. The first order waves that influence each axis ordinate of the moving object (ship) will be used as environmental disturbances. b. The DC-MPC controller system analysis showed that it works well for the ship heading control with disturbance. c. The DC-MPC method is used since it can control the ship's maneuvering while considering the ocean waves. Design Control of Surface Marine Vehicle Using…….
d. The simulation result of container ship shows that ship maneuvering's stability depends on the number of prediction horizons in which closed to Np=60, so the optimum stabilization was gained. It means that this method can minimize error, and the disturbance compensating could control the environment disturbance that had given. e. Comparing the controlling of ship maneuvring using DC-MPC and MPC with disturbance (MPC-D) showed that DC-MPC is better than MPC.