APPLICATION OF MODEL PREDICTIVE CONTROL

APPLICATION OF MODEL PREDICTIVE CONTROL (MPC) FOR SHIP HEADING CONTROL Alwi Husein Mulachela, Wawan Hafid Syaifudin, Prof. Dr. Basuki Widodo, M.Sc. Mathematics, Faculty of Mathematics and Natural Sciences, Sepuluh Nopember Institute of Technology Jl. Arief Rahman Hakim, Surabaya 60111 E-mail: [email protected]

Abstract Ship heading control has been a representative control problem for marine application and has attracted considerable attention from the control community. More recently, given that the large yaw velocity can produce other motions (such as sway and roll) that can cause sea sickness and cargo damage, enforcing yaw velocity constraints while maneuvering in seaways becomes an important design consideration in surface vessel . In this final project, the sigma class corvet ship is adopted as an example. To address the constraint violation for ship heading control in wave fields, the Model Predictive Control (MPC) controller has been proposed to satisfy the state constraints in the presence of environmental disturbances. The simulation results show good performance of the proposed controller in terms of satisfying yaw velocity and actuator saturation constraint. Keywords: Model Predictive Control (MPC), Ship Heading Control

1. Introduction 1.1

Background Indonesia is a maritime country and a the largest archipelago in the world, two-thirds of its territory is the ocean. In an effort to maintain the integrity of the waters we need a strong security defense system. False an effort that has been done is to increase patrols in the waters of Indonesia. To support the efforts of the patrol Indonesian waters course required platforms (primary equipment defense system) is adequate, for example, is a ship. False a security system for vessels on waterways can performed by applying technology to overcome the factors environmental disturbances include waves, winds and currents sea in control of the ship's bow. Marine vessel is a vehicle which has 6 degrees freedom (DOF) in a move that surge, sway, heave, roll, pitch, and yaw. The sixth movement experienced by a ship caused by interference from the outside, such as the current sea, waves and wind [1]. In this final project variables controlled in only one degree of freedom are yaw assuming motion sway, surge, heave, roll, pitch not effect on maneuvering the ship. Ship heading control, or commonly referred to as course keeping, the main system under autopilot control. Ship heading control is one of the problem in control problems ship in applications in the field of marine [1]. Nomoto model that used in ship heading control only consider one degree of freedom (DOF) ships dynamic system, ie r (yaw rate), and an input control [2]. In many cases, the speed yaw that is too big can produce other movements (such as sway and roll) that can cause damage to cargo ship. Therefore, when the vessel yaw speed constraints maneuver in the sea lanes into something quite important needs to be considered in the design of the control of the ship.

Several methods have been used in the system Automatic control on the bow of the ship is to use fuzzy logic [3]. Another method that can be used in the system ship heading control is Model Predictive Control (MPC). The controller types are categorized based controllers process models, namely process model is used explicitly for designing controllers, by minimizing a the criterion function [4]. In addition, the MPC can incorporate all the goal of becoming a single objective function and optimization provided very effective for dealing with systems that have constraints on input and state space [2]. MPC is widely used in industry. One of The main reason for the success of MPC in industrial applications is the ability to apply various types of constraints on process. However, in some cases the optimization constraints on MPC can not be resolved, this is the case for their mismatch model or interference from outside. This matter causing no solution could be found to fulfill all the constraints given [2]. With the interference of environmental factors on ship, it is essential to this factor taken into account in the design control of the ship, in this case is a ship heading control. However in this final project, the environmental nuisance factor is limited to interference coming from waves alone. With assumption that the interference factor derived from the wind and the wind currents which caused a wave is assumed to be interference of waves. In this final project will be used algorithm models predictive control (MPC) for control on a ship heading control.

1.2

Purpose The goal in the final project is : 1. Applying the model predictive control (MPC) in control bow of the ship (ship heading control).

2. Problem Statement The problem discussed in the final project is : 1. How the application of the model predictive control (MPC) on control of the bow of the ship (ship heading control).

3. Mathematical Modelling 3.1

Model Predicting Control (MPC) Model Predicting Control is a method advanced control process which is widely used in process industry. Of the many multivariable control algorithms, MPC is one of them [5]. Five concepts of the MPC are : 1. Modeling process and disturbance 2. Performance index 3. Constraint Control 4. Optimization 5. Receding horizon principle MPC methodology described below

Figure 3.1 MPC Data Structure [6] The work of the MPC step as shown in figure is Initially, the system has been has a model of the plant. Data input and output previous MPC entered through the input port. Input and output of this estimate is based on a model plant that has defined previously. They derive the output later called the predicted output value than with reference trajectory. From the results of this comparison, will generating an error called futures error. Future error This then goes into a block optimizer. Here, the optimizer function to work within the range constraint that has given with the aim to minimize the cost function, which an established criterion function of a quadratic function the error between the predicted output signal with a reference trajectory. MPC then took the decision to The error minimizing future with decision still in the constraint set. Result of This block is called futures input returned together with input and output data prior to the estimated back. This calculation takes place on and repeatedly. Because of their input correction based on output Predicted that makes MPC can work generate response is increasingly approaching the reference trajectory [7]. And controlling output processes predictable on MPC uses the concept of prediction horizon is how far predictions of the future are expected. This can be illustrated in figure below.

Figure 3.2 Process Output Calculation and Control Predicted [7]

This figure shows the system response when given MPC. It can be seen that the response capable generate value the better, this is because the signal control change at any time based errors that occur [7].

3.2

Linier Model Predicting Control (MPC) This modeling used in the state space is discrete and linear. Discrete state space equation used is as follows [8]:

̃(k 1 | k ) A𝒙 ̃ (k | k ) B𝒖 ̃ (k | k ) 𝒙 ̃ (k | k ) 𝒙 ̃ (k | k ) 𝒚

(2.1) (2.2)

With

̃(k|k) 𝒙 ̃(k|k) 𝒚 ̃ (k|k) 𝒖 𝐀 𝐁

: n-state space dimention vector : n-output scaleable vector : m-input dimention vector : nxn dimensional matrix state : dimensional input matrix nxm

State space equation above is an ideal condition, where there is no interference (disturbance). In this thesis, the system used a disturbance (disturbance) of the surrounding environment, in the form of ocean waves. so later there is a disturbance in the matrix state space equation. To further facilitate the ̃(k|k) can written in the form 𝒙 ̃(k). In calculating the predicted output writing, 𝒙 ̃ (k). By because the state space equation (2.1) with MPC, which used input signal is 𝒖 ̃ (k) in it. The first thing do is look for should be transformed so there is an element 𝒖 the prediction of the state space equation (2.1) by iterating the equation as follows:

̃ (k 1 | k ) A𝒙 ̃ (k ) B𝒖 ̃ (k ) 𝒙 ̃ (k 2 | k ) A𝒙 ̃ (k 1| k ) B𝒖 ̃ (k 1| k ) 𝒙 𝟐 ̃ ̃ 𝑨 𝒙̃ (k) AB𝒖 (k | k) B𝒖 (k 1| k)

(2.3) (2.4)

⁞

̃ (k N | k ) A𝒙 ̃ (k N 1| k ) B𝒖 ̃ (k N 1| k ) 𝒙 ̃ (k | k) 𝑨𝑵−𝟐 B𝒖 ̃ (k 1| k) 𝑨𝑵 𝒙̃ (k) 𝑨𝑵−𝟏 B𝒖 ̃ (k N 1 | k ) ... B𝒖

3.3

(2.5)

Optimization on Linier Model Predicting Control (MPC) with Disturbance Optimal control method used in MPC is a linear quadratic programming (QP). QP used because the MPC is one of the closed-loop control. On linear MPC case taking into account the interference, then form of equation (2.1) becomes [2]:

̃ (k 1 | k ) A𝒙 ̃ (k ) B𝒖 ̃ (k ) 𝒘 ̃ (k ) 𝒙 ̃ are n-dimentional disturbance vector With 𝒘 3.3.1

Stage Cost and Constraint Each k Time At each k time, stage cost defined [9] :

(2.6)

̃(𝑘), 𝒖 ̃ (𝑘)) = [ 𝒔(𝒙

̃ 𝒙(𝑘) 𝑇 𝑸 𝟎 ̃ 𝒙(𝑘) ] [ ][ ] ̃ (𝑘) 𝟎 𝑩 𝒖 ̃ (𝑘) 𝒖

(2.7)

Based on the equation (2.7), Q is a definite matrix which is a positive weighting matrix errors on state-space dimension nxn, whereas R is a positive definite matrix is a matrix of dimension mxm weight in control. In each k time there are obstacles, which are defined ̃(𝑘) ≤ 𝐟𝟏 𝐅𝟏 𝒙 ̃ (𝑘) ≤ 𝐟𝟐 𝐅𝟐 𝒖

(2.8) (2.9)

With 𝐅𝟏 lxn dimentional matrix, 𝐅𝟐 oxm dimentional matrix, and 𝐟𝟏 lxl dimentional vector with 𝐟𝟐 oxl dimentional vector. Equation (2.8) is a constraint on the state, while equation (2.9) is a constraint on the control system. 3.3.2

Optimization on Linier MPC By using equation (2.7), then the control value at each k step time can be generalized and searchable by completed the optimization of the following [9]: 𝑘+𝑁−1 Minimize 𝐽 = ∑𝑗=𝑘 𝒔(𝒙 ̃(𝑗 + 1), 𝒖̃ (𝑗))

(2.10)

With obstacle ̃(𝑘) ≤ 𝐟𝟏 𝑗 = 𝑘, 𝑘 + 1, … , 𝑘 + 𝑁 − 1 𝐅𝟏 𝒙 (2.11) ̃ (𝑘) ≤ 𝐟𝟐 𝑗 = 𝑘, 𝑘 + 1, … , 𝑘 + 𝑁 − 1 𝐅𝟐 𝒖 (2.12) ̃ (k) + 𝐰 ̃(k) 𝐱̃(k + 1|k) = 𝐀𝐱̃(𝑘) + 𝐁𝐮 ̃ (j + 1) + 𝐰 ̃(j + 1) 𝑗 = 𝑘, 𝑘 + 1, … , 𝑘 + 𝑁 − 1 𝐱̃(j + 2|j) = 𝐀𝐱̃(𝑗 + 1) + 𝐁𝐮 (2.13) The result from the optimization above is : ̃(𝑘 + 1|𝑘), 𝒙 ̃(𝑘 + 2|𝑘), … , 𝒙 ̃(𝑘 + 𝑁|𝑘) 𝒙 ̃ (𝑘|𝑘), 𝒖 ̃ (𝑘 + 1|𝑘), … , 𝒖 ̃ (𝑘 + 𝑁 − 1|𝑘) 𝒖

and

If the above optimization J is feasible, then the problem minimizing the equation (2.10) which was completed in each step k produces optimal solution: ̃ ∗ (𝑘|𝑘), 𝒖 ̃ ∗ (𝑘 + 1|𝑘), … , 𝒖 ̃ ∗ (𝑘 + 𝑁 − 1|𝑘), 𝒙 ̃∗ (𝑘 + 1|𝑘), 𝒙 ̃∗ (𝑘 + 2|𝑘), … , 𝒙 ̃∗ (𝑘 + 𝑁|𝑘) 𝒖 (2.14) By using the principle of receding horizon on the MPC, the optimal control value given to the system is optimal initial vector of the settlement, then the value of control given the system (2.6) are:

̃ (𝑘|𝑘) = 𝒖 ̃ ∗ (𝑘|𝑘) 𝒖

(2.15)

̃ (𝑘|𝑘) is the value of vector control at the time to k, while 𝒖 ̃ ∗ (𝑘|𝑘) is the with 𝒖 optimal control value at k time. To complete the optimization problem in equation (2.10), overall optimization variables are defined as [9]: Z =( ~ u (k | k), ~x (k 1| k), ~u( k 1| k)... ,~u (k  N 1| k), ~x ( k  N | k) ) (2.16) By using equation (2.16), then we can explain again the objective function in equation (2.10), as well as constraints in equation (2.11), (2.12) and (2.13) in the form: Minimize 𝑇

𝒖 ̃(𝑘|𝑘) 𝑅 ̃( 𝒙 𝑘 + 1|𝑘) 0 0 ( | ) 𝒖 ̃ 𝑘+1 𝑘 𝑗= ⋮ ⋮ 0 𝒖 ̃(𝑘 + 𝑁 − 1|𝑘) [ 𝒙̃ (𝑘 + 𝑁|𝑘) ] [ 0

0 𝑄 0 ⋮ 0 0

0 0 𝑅 ⋮ 0 0

… … … ⋱ … …

0 0 0 ⋮ 𝑅 0

𝒖 ̃(𝑘|𝑘) 0 ̃( 𝒙 𝑘 + 1|𝑘) 0 0 𝒖 ̃(𝑘 + 1|𝑘) ⋮ ⋮ 0 𝒖 ̃(𝑘 + 𝑁 − 1|𝑘) 𝑄] [ 𝒙̃ (𝑘 + 𝑁|𝑘) ] (2.17)

With obstacle

𝐅𝟏 0 0 ⋮ 0 [0

0 𝐅𝟐 0 ⋮ 0 0

0 … 0 … 𝐅𝟏 … ⋱ ⋮ … 0 0 …

−𝐁 𝐈 0 0 −𝐀 −𝐁 0 0 0 ⋮ ⋮ ⋮ 0 0 0 [ 0 0 0

0 0 0 ⋮ 𝐅𝟏 0

𝒖 ̃(𝑘|𝑘) 0 𝐅𝟐 ̃( 𝒙 𝑘 + 1|𝑘) 𝐅𝟏 0 ( | ) 𝐅 0 𝒖 ̃ 𝑘+1 𝑘 ≤ 𝟐 ⋮ ⋮ ⋮ 𝐅𝟐 0 𝒖 ̃(𝑘 + 𝑁 − 1|𝑘) [𝐅𝟏 ] 𝐅𝟐 ] [ 𝒙̃ (𝑘 + 𝑁|𝑘) ]

(2.18)

𝒖 ̃(𝑘|𝑘) 0 ⋯ 0 0 𝐀𝐱̃(k) + 𝐰 ̃(k) ̃( 𝒙 𝑘 + 1|𝑘) 𝐈 ⋯ 0 0 0 −𝐀 ⋯ 0 0 𝒖 ̃(𝑘 + 1|𝑘) 0 ≤ ⋮ ⋱ ⋮ ⋮ ⋮ ⋮ 0 ⋯ −𝐁 𝐈 𝒖 0 ̃(𝑘 + 𝑁 − 1|𝑘) [ ] 0 ⋯ 0 −𝐀] [ 𝒙̃ (𝑘 + 𝑁|𝑘) ] 0 (2.19)

The objective function in equation (2.17) as well as constraints on Equation (2.18) and (2.19) can be written in the form quadratic programming [9]: Minimize

J = zTHz + gTz

(2.20)

With Obstacle Pz ≤ h Yz = b Where

(2.21) (2.22)

−𝐁 0 0 ⋮ 0 [ 0 3.4

𝑅 0 0 … 0 0 0 0 𝑄 0 … 0 0 0 0 0 0 𝑅 … 0 0 ,𝑔 = ⋱ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ 0 … 𝑅 0 0 0 0 [0]2Nx1 [ 0 0 0 … 0 𝑄]2Nx2N 𝐅𝟏 0 0 … 0 0 𝐅𝟐 𝐅 0 𝐅𝟏 0 … 0 0 𝟐 𝐅 0 0 𝐅𝟏 … 0 0 ,ℎ = 𝟐 ⋱ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ 𝐅 0 0 0 … 𝐅𝟏 0 𝟐 [𝐅𝟏 ]2Nx1 [ 0 0 0 … 0 𝐅𝟐 ]2Nx2N 0 ⋯ 0 0 𝐈 0 𝐀𝐱̃(k) + 𝐰 ̃(k) 𝐈 ⋯ −𝐀 −𝐁 0 0 0 −𝐀 ⋯ 0 0 0 0 0 ,𝑏 = ⋮ ⋱ ⋮ ⋮ ⋮ ⋮ ⋮ 0 ⋯ −𝐁 𝐈 0 0 0 [ ]2Nx1 0 ⋯ 0 −𝐀]2Nx2N 0 0 0

Mathematical Model From the Ship Heading Control When the ship cruised in the seas, in general, there are six kind of movement is experienced, ie yaw, heave, surge, sway, roll and pitch. These movements also called 6 Degree of Freedom. The general form equation expressed in the ship maneuvering form [1]: 𝑀𝑣̇ + 𝐷𝑣 = 𝜏𝐿

(2.23)

With ν = [u, v, r]T a velocity vector M and D as the inertia and damping matrices obtained from the linearized equation of forces and moments on the direction of surge, sway and yaw. Full bow of the ship (ship heading control) is the main components of the autopilot system of maneuvers ship. The main task of this system is to maintain the ship's bow to match the desired conditions. Control objective of control of the bow of the ship (ship heading control) is [2]: 𝜓 → 𝜓𝑑

(2.24)

Where 𝜓 is the angle of the ship's bow that is true, and 𝜓𝑑 is the desired angle bow of the ship. With notes that 𝜓 = r, where r is the yaw speed. In ship heading control, Nomoto models is the most common model used. This model only considers one degree of freedom (DOF) ships dynamic system, ie r (yaw speed), and one control inputs, namely δ as steering angle. Large corner bow of the ship (𝜓) calculated the x-axis Earth (Xe), while the ship steering angle (δ) amount is calculated against the hull. Dynamic model of the ship's bow was obtained from approach to the transfer function of a second order Nomoto model of [1]:

𝑟(𝑠) δ(s)

𝐾 (1+𝑇3 𝑠)

= (1+𝑇𝑅

(2.25)

1 𝑠)(1+𝑇2 𝑠)

Where r is the yaw rate and steering angle δ is a ship. The values of the parameters of the equation (2.24) meet [1] :

𝑇1 𝑇2 =

det(𝑀) det(𝑁) 𝑛11 𝑚22 + 𝑛22 𝑚11 − 𝑛12 𝑚21 − 𝑛21 𝑛12

𝑇1 + 𝑇2 = 𝐾𝑅 =

det(𝑁)

𝑛21 𝑏1 − 𝑛11 𝑏2

𝐾𝑅 𝑇3 =

det(𝑁) 𝑚21 𝑏1 − 𝑚11 𝑏2 det(𝑁)

(2.26)

(2.27) (2.28) (2.29)

For element 𝑚𝑖𝑗 , 𝑛𝑖𝑗 and b (i = 1,2 and j = 1,2) we get the matrix above :

𝑚 − 𝑌𝑣̇ 𝑚𝑥𝐺 − 𝑌𝑣̇ ] 𝑚𝑥𝐺 − 𝑁𝑣̇ 𝐼𝑍 − 𝑁𝑣̇ −𝑌 𝑚𝑢0 − 𝑌𝑟 𝑁(𝑢0 ) = [ 𝑣 ] 𝑁𝑣 𝑚𝑥𝐺 𝑢0 − 𝑁𝑟 𝑌 𝑏 = [ δ] 𝑁δ 𝑀= [

(2.30) (2.31) (2.32)

With 𝑌v̇ = derivative style directions sway against 𝑣̇ , 𝑌𝑣̇ = derivative yaw force against 𝑟̇ , 𝑁𝑟̇ = derivative of the yaw moment 𝑟̇ , 𝑌v = derivative of sway direction against 𝑣, 𝑌𝑟 = derivative yaw direction against 𝑟, 𝑁𝑣 = derivative sway moment to 𝑣, 𝑁𝑣̇ = derivative sway moment to 𝑣̇ , 𝑁𝑟 = derivative yaw moment against 𝑟. The parameters in determining the gain control is derived Nomoto based linearization of the model Davidson and Schiff (1946) as follows [1]:

𝐾=

𝑛21 𝑏2 − 𝑛11 𝑏1

(2.33)

det(𝑁)

With det(𝑁) = 𝑌𝑣 (𝑁𝑟 − 𝑚𝑥𝐺 𝑢) − 𝑁𝑣 (𝑌𝑟 − 𝑚𝑢) (2.34) det(𝑀) = (𝑚 − 𝑌𝑣̇ )(𝐼𝑧 − 𝑁𝑟̇ ) − (𝑚𝑥𝐺 − 𝑁𝑣̇ )(𝑚𝑥𝐺 − 𝑌𝑟̇ ) (2.35) And 𝑛11 = −𝑌𝑣 , 𝑛21 = −𝑁𝑣

𝑏1 = 𝑏2 =

(𝐼𝑧 − 𝑁𝑟̇ )𝑌𝛿 − (𝑚𝑥𝐺 − 𝑌𝑟̇ )𝑁𝛿 det 𝑀 (𝑚− 𝑌𝑣̇ )𝑁𝛿 − (𝑚𝑥𝐺 − 𝑁𝑣̇ )𝑌𝛿 det 𝑀

(2.36)

(2.37)

Slender body approaches derived coefficients strip hydrodynamics can be expressed as a function of the ratio of length the width of the vessel multiplied by a certain constant. Hydrodynamic coefficients in this equation is non-dimensional form which lowered the prime systems I. To obtain the forces multiplied by 1⁄2 𝜌𝑈 2 𝐿2 and moment multiplied by 1⁄2 𝜌𝑈 2 𝐿3. Where ρ = sea water mass density (1024 kg/𝑚3 ), L = vessel lenght, U = vessel service velocity, B = vessel width, T = vessel depth, 𝐶𝐵 = block coefficient. Thus obtained : −𝑌′𝑣̇̇ 𝜋(𝑇/𝐿)2

−𝑌′𝑟̇ 𝜋(𝑇/𝐿)2

= 1 + 0,16

𝐶𝐵 𝐵 𝑇

𝐵 2

− 5,1 ( 𝐿 )

𝐵

𝐵 2

𝐿

𝑇

= 0,67 ( ) − 0,0033 ( )

−𝑁′𝑣̇

𝐵

−𝑁′𝑟̇

−𝑌′𝑣 𝜋(𝑇/𝐿)2 −𝑌′𝑟

=

1 12

= 1 + 0,4

𝜋(𝑇/𝐿)2 −𝑁′𝑟

𝐶𝐵 𝐵 𝑇

(2.40) 𝐵

− 0,33 ( 𝐿 )

𝐶𝐵 𝐵

1

=

=

𝐵

𝐵

𝑇

+ 2,4 (𝐿 ) 2 1 4

(2.41)

(2.42)

𝑇

= − 2 + 2,2 ( 𝐿 ) − 0,08 (𝑇 )

−𝑁′𝑣

𝜋(𝑇/𝐿)2

𝐵

+ 0,017

1

𝜋(𝑇/𝐿)2

(2.39)

= 1,1 ( 𝐿 ) − 0,0033 (𝑇 )

𝜋(𝑇/𝐿)2

𝜋(𝑇/𝐿)2

(2.38)

𝐵

(2.43)

(2.44) 𝐵

+ 0,039 (𝑇 ) − 0,56 ( 𝐿 )

(2.45)

In control of the bow of the ship, there are constraints on the limit The maximum speed limit on the yaw and steering angle ship. To constraints on the yaw speed, advance calculated yaw moment (τ) [1]:

τ=

𝐿

(2.46)

0,514𝑈0

with : τ L 𝑈0

= Yaw moment = Vessel length in meter = Vessel velocity maximum (knot) 2100

With the yaw rate maximum value = τ ( /̊ minutes )

3.5

Disturbance On the Ship

(2.47)

On the ship, a disorder that is given comes from sea waves (waves). This disturbance in the form of a wave sinusoidal on the yaw movement of the vessels which meet equation [1]: 𝜓 = 𝜓𝑎 sin(𝜔𝑡) 𝜓̇ = 𝜔 𝜓𝑎 sin(𝜔𝑡)

(2.48) (2.49)

From both equation above (2.48) and (2.49), we can get the disturbance vector of the dynamical system ship. ̃ (𝑘) = [𝜓̇ ; 𝜓] = [𝜔 𝜓𝑎 sin(𝜔𝑡) ; 𝜓𝑎 sin(𝜔𝑡)] 𝒘

(2.50)

Where 𝜔 sea wave frequency against ship dynamic system, 𝜓𝑎 is amplitude value from the wave after being multiplied by RAO factor.

4. Numerical / Analogical Solution 4.1

Modelling In Ship Control Policy

The ship we used as a model is Warship Class Corvet Sigma. These ships are working on sea state 5, which corresponds to marine conditions in the territory of Indonesia. Table below shows the data parameters. Parameter Ρ(Kg/m3 ) L(m) U(m/s) B(m)

Value 1024 101,07 15,4 14

Parameter T(m) 𝐶𝐵 𝑋𝐺 (m) M(Ton)

Value 3,7 0,65 5,25 2423

By performing calculations using the equations (2.38) to the equation (2.45), is obtained : 𝑌′𝑣̇ = −0,005452 ; 𝑌′𝑟̇ = −0,000192 ; 𝑁′𝑣̇ = 1,2(10−5 ) 𝑁′𝑟̇ = −0,000334 ; 𝑌′𝑣 = −0,008348 ; 𝑌′𝑟 = 0,0021 𝑁′𝑣 = −0,002474 ; 𝑁′𝑟 = −0,001347 Furthermore, based on the parameters that have been obtained at the top and do the calculation, then based on the equation (2.25) obtained the transfer function of order 2 Model Nomoto as follows : 𝑟(𝑠) 2035,906768 ∙ 𝑠 + 758,9931779 = 𝛿𝑅 (𝑠) 19,87637646𝑠 2 + 7,370705305𝑠 + 1 From the transfer function, we change it to the state space function : 1 ̃̇ = [−0,3708 −0,0503] 𝒙 ̃ + [ ]𝒖 ̃ 𝒙 0 1 0

̃̇ (𝑘) ̃(𝑘) 𝒙 1 −0,3708 −0,0503 𝒙 ̃ (𝑘) [ 𝟏 ]= [ ][ 1 ] + [ ]𝒖 ̃(𝑘) 𝒙 0 1 0 ̃𝟐̇ (𝑘) 𝒙 2 ̃̇ using advanced finite difference schemes, we obtained: by approaching 𝒙 ̃𝟏̇ (𝑘 + 1) − 𝒙 ̃𝟏̇ (𝑘) 𝒙 ̃̇ (𝑘) 1 −0,3708 −0,0503 𝒙 ∆𝑡 ̃ (𝑘) = [ ][ 𝟏 ] + [ ]𝒖 0 1 0 ̃𝟐̇ (𝑘 + 1) − 𝒙 ̃𝟐̇ (𝑘) ̃𝟐̇ (𝑘) 𝒙 𝒙 [ ] ∆𝑡 ̃̇ (𝑘 + 1) − 𝒙 ̃𝟏̇ (𝑘) 𝒙 −0,3708∆𝑡 [ 𝟏 ]= [ ∆𝑡 ̃𝟐̇ (𝑘 + 1) − 𝒙 ̃𝟐̇ (𝑘) 𝒙 ̃̇ (𝑘 + 1) 𝒙 −0,3708∆𝑡 [ 𝟏 ]= [ ∆𝑡 ̃𝟐̇ (𝑘 + 1) 𝒙

̃̇ (𝑘) ∆𝑡 −0,0503∆𝑡 𝒙 ̃ (𝑘) ][ 𝟏 ] + [ ]𝒖 0 0 ̃𝟐̇ (𝑘) 𝒙

̃̇ (𝑘) ̃̇ (𝑘) 𝒙 ∆𝑡 −0,0503∆𝑡 𝒙 ̃ (𝑘) ][ 𝟏 ] + [ 𝟏 ] + [ ]𝒖 0 0 ̃𝟐̇ (𝑘) ̃𝟐̇ (𝑘) 𝒙 𝒙

̃̇ (𝑘 + 1) 𝒙 1 − 0,3708∆𝑡 [ 𝟏 ]= [ ̇ ∆𝑡 ̃𝟐 (𝑘 + 1) 𝒙

̃̇ (𝑘) ∆𝑡 −0,0503∆𝑡 𝒙 ̃ (𝑘) ][ 𝟏 ] + [ ]𝒖 ̇ 0 0 ̃𝟐 (𝑘) 𝒙

by taking the sampling time Δt = 0.1, and rounding up to 5 digits behind the comma, we get: ̃̇ (𝑘) ̃̇ (𝑘 + 1) 𝒙 0,9629 −0,00503 𝒙 0,1 ̃ (𝑘) [ 𝟏 ]= [ ][ 𝟏 ] + [ ]𝒖 ̇ ̇ 0,1 0 0 ̃𝟐 (𝑘 + 1) ̃𝟐 (𝑘) 𝒙 𝒙 We can write as : ̃(𝑘 + 1) = 𝑨𝑥̃(𝑘) + 𝑩𝒖 ̃ (𝑘) 𝒙

(4.1)

With 𝑟 0,9629 −0,00503 0,1 ̃ = [𝜓] 𝑨= [ ] , 𝑩 = [ ] , and 𝒙 0,1 1 0 On a given vessel disorder that comes from sea wave (waves). This disturbance in the form of a wave sinusoidal on the yaw movement of the vessels which refers to Equation (2.50). From equation (4.1) can be written back to : ̃(𝑘 + 1|𝑘) = 𝑨𝑥̃(𝑘) + 𝑩𝒖 ̃ (𝑘) + 𝒘 ̃ (𝑘) 𝒙

(4.4)

In the study of the control system, a system is said controlled if for any X0 any circumstances, no input u(t) which is not limited to any transfer state X0 the final state Xk with k finite end time. On equation (4.1) with indigo A and B are known, can shown if rank [B | AB] = 2, which means that the system in controlled circumstances. This is what allows application of linear MPC in control of the ship's bow.

4.2

Application of Model Predictive Control In Control Policy Ship At MPC, predictive control is carried out. As described in the previous chapter 2, the MPC will be estimated a value of a when m is in n stage. The following will be given some steps to solve the problem of control on the prow of a ship with MPC uses linear optimization.

4.2.1

Early Initialization Bow Ships At the initialization stage of first values, input the first ship control values u(0) (the steering angle (δ)) and the initial position of the ship bow x(0) (yaw speed boat (r) and the angle of the bow of the ship (ψ)). On This initialization we use prediction horizon N, ie, optimal control when the predicted value k, k + 1, ..., k + N at the time step k.

4.2.2

Calculation of Constraints On Policy Control Ship Constraints on the control of the bow of the ship, as it has been described in sub-section 2.4 is a limit on the maximum value of the steering angle (δ) and the maximum value of yaw velocity. The maximum limit for the steering angle (δ) amounted 35 °. While the maximum limit of the speed yaw based calculation of the equation (2:46) and (2:47) where the value U0 = 30 knots, maximum values obtained yaw rate is 0.0932 rad / sec. By using equation (2:34) and (2:35), then the constraints on the bow of the ship's control system can written back into ̃(𝑘) ≤ 𝐟𝟏 𝐅𝟏 𝒙 ̃ (𝑘) ≤ 𝐟𝟐 𝐅𝟐 𝒖 With

35𝜋/180 0,0932 1 0 1 𝑭𝟏 = [ ] , 𝒇𝟏 = [ ] ; 𝑭𝟐 = [ ] , 𝒇𝟐 = [ ] 0,0932 35𝜋/180 −1 0 −1 4.2.3

Optimal Control Prediction With MPC At MPC predictive control values can be searched by completed the optimization in the form of quadratic programming :

J = zTHz + gTz Minimize with constraint Pz ≤ h Yz = b Where ̃ (𝑘|𝑘), 𝒙 ̃(𝑘 + 1|𝑘), 𝒖 ̃ (𝑘 + 1|𝑘) … 𝒖 ̃ (𝑘 + 𝑁 − 1|𝑘), 𝒙 ̃(𝑘 + 𝑁|𝑘)) 𝒛 = (𝒖

By using the principle of receding horizon on the MPC, the optimal control value given to the system is optimal initial vector of the settlement, then the value of control given the system (4.4) are: ̃ (𝑘|𝑘) = 𝒖 ̃ ∗ (𝑘|𝑘) 𝒖 ̃ (𝑘|𝑘) is control vektor value at k, whereas 𝒖 ̃ ∗ (𝑘|𝑘) is optimal With 𝒖 control value at k. 4.2.4

Prediction Position Of Ship Bow Having obtained the optimal value of the control that comes from the calculation using quadratic programming, the next step to enter into the control values equation (4.4) : ̃(𝑘 + 1|𝑘) = 𝑨𝑥̃(𝑘) + 𝑩𝒖 ̃ (𝑘) + 𝒘 ̃ (𝑘) 𝒙 ̃ (𝑘) = 0. Based on MPC If the system subject are not interference, then 𝒘 algorithm, with the inclusion of control values into the state, then we obtaine the ̃(𝑘 + 1|𝑘) is inserted into prediction values in the next step. After that the value of 𝒙 in the subsequent optimization calculations to get the value the next optimal control. And so on until the output of the system has followed the trajectory of reference desired.

4.3

Simulation Analysis Report On Policy Control Ship

From the analysis of MPC in sub-section 4.2 above, will be simulated in this ̃(0) section use the software MATLAB. In this simulation given initial initialization) 𝒙 ̃ (0) = 0. The purpose of controlling the ship's bow the MPC method is = [0.30 °], and 𝒖 to make the state and the yaw speed ship steering angle control is within the constraint have been determined as well as the angle of the ship toward 0 °. ships assumed moving at a speed of 30 knots constant surge. By using a large angle of reference that the ship's bow (ψ) calculated the x-axis of the earth (Xe), meaning in this case the ship will be controlled to move in the right direction to position the vessel parallel to the x-axis of the earth (Xe), or in other words the angle bow of the ship had reached 0 °. When the angle of the ship's bow was is 0 °, then ship only perform straight movement constant surge forward with a speed of 30 knots. 4.3.1

Effect of Value Prediction Horizon Variable To Conditions Without Disruption Variations prediction horizon used in the simulation test is 20. Simulation time used is 20 seconds, with a sampling time Δt = 0.1 sec. The given condition is the ship is at sea state 0, where there are no waves in the ocean (wave height 0 meters), so it could be said that there is no disruption to work on the ship.

Yaw Velocity (rad/sec)

Time (second)

Yaw Velocity (rad/sec)

Figure 4.1 Yaw Velocity On MPC Without Disturbance on N = 20

Time (second)

Yaw Velocity (rad/sec)

Figure 4.2 The Bow Angle Ships On MPC Without Disruption For N = 20

Time (second) Figure 4.3 Steering angle Ships On MPC Without Disruption For N = 20

The simulation results for the prediction horizon value (N) = 20 indicates that the yaw speed of the ship has been exceeded constraint limits that have been granted by 0.0932 rad / sec. This seen from the simulation results in figure 4.1, where the value yaw rate continues to move down until it reaches a value his maximum in the second to 0.5 by 0.2809 rad / sec, then gradually follow the boundary constraint granted, until finally the whole value of the yaw rate meet the constraint began seconds to 2.3. Negative yaw rate value indicates that the ship moving in the right direction. Yaw velocity value reaches 0 rad / sec the second to 7.3, which means that the ship did not make a move yaw again or in other words boats just do straight motion towards the front (to do surge movements only) with speed of 30 knots. Based on Figure 4.2, the angle of the bow ship reaches a value of 0 ° in the second to 7.3. Value corner bow ships continue to fall until it reaches 0 ° indicates that bow of the ship is moving from the initial position of 30 ° toward 0 °, which means that the ship has reached a reference trajectory that has set. For a ship steering angle, based on drawings 4.3, looks that large steering angle has exceeded the constraint of 35 °. Using the MPC controller, motion steering the boat pressed to the maximum possible so that the corner of the ship's bow as soon as possible to the position of 0 °. It can be seen that the angle steering reached 49.58 °, then gradually follow boundary constraint, until finally meet the constraint entirely from this point to 0.5. Large yaw rate and the steering angle which is outside constraint limits can cause the movement of ships be stable and the resulting wear on steering the boat. With using value prediction horizon of 20 controllers MPC without interruption can not operate optimally because the system is still outside the boundary constraint set. 4.3.2

Effect of Value Prediction Horizon Variable To Conditions With Disruption Variation prediction horizon used in the simulation test is by 20 with time simulation used was 20 seconds, with a time of sampling Δt = 0.1 sec. The given condition is the ship is at sea state 5 with wave height is 3.25 meters and the direction coming wave of 90 °. After multiplied by a factor RAO multiplier into a dynamic ̃ (𝑘) = system vessel values obtained Ψa = 0,001, and the value obtained ω = 0.1 𝒘 [0.0001 0.1 cos (t) ; 0.001 0.1 sin (t)]. This disturbance will given the dynamic system of the ship. These are following test results of the simulation :

Yaw Velocity (rad/sec)

Time (second)

Yaw Velocity (rad/sec)

Figure 4.4 Yaw Velocity On MPC With Disturbance on N = 20

Time (second)

Yaw Velocity (rad/sec)

Figure 4.5 The Bow Angle Ships On MPC With Disruption For N = 20

Time (second) Figure 4.6 Steering angle Ships On MPC With Disruption For N = 20

Based on a simulation for prediction horizon value (N) = 20 with the condition of the vessel given image noise 4,5, seen that the yaw speed of the ship has been exceeded constraint that has been given by 0.0932 rad / sec. Value yaw rate continues to move down until it reaches a value his maximum in the second to 0.5 by 0.2818 rad / sec, then gradually follow the boundary constraint granted, until finally the whole value of the yaw rate meet the constraint began seconds to 2.4. Yaw velocity reaches a value of 0 rad / sec at the second to 5.5, after which the oscillating yaw speed value to the value The reach 0.0076 rad / sec. Yaw motion of the ship follow the movement of interference given in the form sinusoidal wave. Based on the pictures 4.5, corner bow ship reaches a value of 0 ° in seconds to reach a value 5,5.Setelah 0 °, the angular position prow of a ship also oscillates with the greatest value reached 0.44 °. From the results of this simulation show that the movement of the ship can not converge to 0 °, bow angle values ship oscillating also follow the movement of the waves the sea in the form of a sinusoidal wave. For a ship steering angle, based on pictures 4,6, seen that large steering angle has exceeded the constraint at 35 °. Using the MPC controller, motion steering the ship is pressed to the maximum possible in order to angle the bow ship as soon as possible to the position of 0 °. It can be seen that the value 49.59 ° steering angle is reached, then gradually follow the boundary constraint, until finally meet the constraint entirely from this point to 0.5. Large yaw rate and the steering angle which is in outside the boundary constraint can cause movement of ships into unstable and result in premature wear on steering the boat. With using value prediction horizon of 20 controllers MPC with the disorder can not work maximum because the system is still outside the boundary constraint set.

5. Conclusion it can be seen that by providing different disorders, MPC can still work well. It is seen from the simulation results that indicate that the system is still being within the limits of a given constraint.

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