APPLICATION OF MPC ALGORITHM FOR TRACKING CONTROL

APPLICATION OF MPC ALGORITHM FOR TRACKING CONTROL ỨNG DỤNG ĐIỀU KHIỂN DỰ BÁO TRONG ĐIỀU KHIỂN BÁM QUỸ ĐẠO Lam Chuong Vo1a, Luan Vu Truong Nguyen1b, Hieu Giang Le 1c 1Ho Chi Minh City U

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APPLICATION OF MPC ALGORITHM FOR TRACKING CONTROL

ỨNG DỤNG ĐIỀU KHIỂN DỰ BÁO TRONG ĐIỀU KHIỂN BÁM QUỸ ĐẠO

Lam Chuong Vo1a, Luan Vu Truong Nguyen1b, Hieu Giang Le 1c

1Ho Chi Minh City University of Technology and Education, Ho Chi Minh City, Vietnam

a chuongvl@hcmute.edu.vn; b vuluantn@hcmute.edu.vn; c gianglh@hcmute.edu.vn

ABSTRACT

In this work, the model predictive control (MPC) is utilized for the design of vessel models (IFAC Benchmark models) in terms of the tracking control problem Besides, the linearized and non-linear models are also used as the vessel models Accordingly, MPC algorithm is first developed to get an optimal control sequence for the linearized models In case of the non-linear models, the linearization technique must be first applied at the current states and the control output given by MPC algorithm with linearized model is then calculated

in the systematic way Furthermore, the wave disturbances and some constraints on variables are also introduced to guarantee a realistic evaluation The extended Kalman Filter (EKF) is applied to estimate the state variables in the presence of the disturbances The results are demonstrated that it still can guarantee the recursive feasibility and convergence due to computationally cheap

Keywords: Model predictive control (MPC), Non-linear model predictive control,

Wave spectra formulation, Extended Kalman Filter (EKF), IFAC Benchmark model

TÓM TẮT

Bài báo này khảo sát đầy đủ ứng dụng của điều khiển dự báo (MPC) trong điều khiển bám quỹ đạo của các mô hình tàu (mô hình chuẩn IFAC) Trong bài báo, hai mô hình toán phi tuyến và tuyến tính hóa của tàu được sử dụng Đầu tiên, giải thuật MPC được sử dụng để đạt được tín hiệu điều khiển tối ưu cho mô hình tuyến tính hóa Kế tiếp, trong trường hợp mô hình phi tuyến, chúng tôi thực hiện việc tuyến tính hóa tại các trạng thái hiện tại, và sau đó tính toán tín hiệu điều khiển sử dụng giải thuật MPC cho hệ tuyến tính ở bước trước Nhiễu

do sóng biển và giới hạn của một số biến trong hệ cũng được khảo sát để đảm bảo quá trình đánh giá phù hợp thực tế Bộ lọc Kalman mở rộng (EKF) được dùng để ước lượng các biến trạng thái trong trường hợp có nhiễu tác động

Từ khóa: Điều khiển dự báo, Điều khiển dự báo hệ phi tuyến, Mô hình phổ sóng biển,

Bộ lọc Kalman mở rộng, Mô hình chuẩn IFAC

1 INTRODUCTION

Recently, the model predictive control (MPC) is widely used in the industry, which is so-called the advanced control techniques that is tremendously successful in practical applications [1] The MPC is established by considering the current control action, as well as solving an optimization problem on-line at each sampling instantly This method uses an explicit model and feedback states to predict the future behavior of a plant on a finite horizon The MPC algorithm yields an optimal control sequence and only the first control in this sequence is applied to the plant [1], [2], [3] The MPC algorithm is also an efficient algorithm that deals with multivariable system and constraints on system variables [1] The MPC is first introduced for the linear system and then extended to the non-linear system

The IFAC benchmark models, which consist of linearized and non-linear mathematical models of a ship, have been used to evaluate the performance of MPC controller [4], [5] The

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wave disturbances (Fossen, 1994) are implemented within the system simulation to investigate the disturbance rejection capabilities of this control theory

This paper is organized as follows Section 2 is briefly introduced the simulation models for both the linearized and non-linear problems The dynamic model of wave disturbances is also discussed in this section The MPC algorithm is presented in section 3 In next section, all the results of simulation without the presence of wave disturbances are shown for several case studies Finally, conclusions are given in section 5

2 SIMULATION MODEL

2.1 Vessel models

2.1.1 Linearized vessel models

Two linearized models representing the heading dynamics of cargo vessel and an oil tanker are used as the control objects for this research These are defined as IFAC benchmark

models (Källström et al.,) [6], [7]

x

y v

Direction Fixed

: turning yaw rate (rad/s)

ψ

: sway velocity of ship (m/s)

v

: heading angle of ship (rad) ψ

: rudder angle of ship (rad)

δ

Figure 1: Notation is used to describe ship’s motion

The linearized model of a ship moving under constant velocity (in a straight line motion) is described as follows:

Cx

u y



 =





(1) where x∈R3, x=[v ψ ψ , is the system states vector;  ]T 1

u∈R , u=δ , is the control input A B C, , are system, input and output matrix respectively The structure of A B C, , is given by:

0

(2)

where a11, , , , , a12 a21 a22 b b are constant parameters 1 2

2.1.2 Non-linear vessel models

The non-linear model of ships is described as follows:

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1 11 1 12 2 1 1 1

2 21 1 22 2 2

3 2



 =





3

where c is also a constant parameter

All constant parameters of linearized and non-linear model are shown in the IFAC Benchmark problems [7], [8]

Both linearized and non-linear systems have to meet some constraints on input, rate of change of input and output (Källström et al., 1979) [7]

(a) The rudder motion is constrained: u ≤400 (5) (b) The rate of change of rudder motion is constrained: u ≤10 / sec0 (6) (c) No overshoot occurs in the output response (7)

To satisfy these design specifications, the desired response of the heading angle, ψ, is designed to be a zig-zag critically damped second order response (Fossen, 1994) [9]

2.2 Wave disturbances

In order to simulate the motion of ocean vehicles and the robustness of controller in the presence of irregular waves, we will consider the effect of wave disturbances The earliest spectral formulation is due to Neumann (1952) who proposed the one-parameter spectrum Pierson and Moskowitz (1963) developed a wave spectra formulation for fully developed wind-generated seas from analyses of wave spectra in the North Atlantic Ocean The Pierson-Moskowitz (PM) spectrum, [9], is written as follows:

S ω =Aω− −Bω− (8)

2 0.0081

A= g , B=3.11 /H s2 (9) where H is the significant wave height (m) s

For simulating the time domain of wave disturbances, we use the linear wave spectrum approximation (Balchen 1976):

o o

( )

2

w

K s

h s

=

Rewriting (10) in state-space form:

c x

w y







(11) where

2

w

K

[0 1]

cw = K w =2λω σo w (gain constant) (12) Here σ is the constant describing the wave intensity, λ is the damping coefficient w while ω is the dominating wave frequency o

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3 MODEL PREDICTIVE CONTROLLER

Given a specific sampling time T , the plant is easily transformed into its discrete-time s

version:

C x

d



Then, the MPC online optimization problem can be formulated as follows: at each time

instant k, we will find the optimal control sequence {u k( ), (u k+1), (u k+N u−1)} to

minimize the following quadratic cost function:

1

( ) ( )

sp

p p

−

Subject to:

( | ) 0,1, , 1 ( | ) 1, 2, ,

u u p





(15)

where equation (15) stands for constraints of the IFAC benchmark problem (5), (6), (7); y is sp

the reference trajectory; Ψ Λ, are the corresponding weighting matrices N p, N are the u

prediction and control horizon respectively Only the first value of the control sequence,

( )

u k , is applied to the system

Let us define vectors:

From the state space equations of the system (13), we calculate the state vector

predicted at sampling instant k for the future instant k+ p in the prediction horizon:

( ) ( ) ( ) ( 1)

X k =Ax k +Mx∆U k +Vu k− (16) where

2

0

u

d

i

A A I B A I B A

A

A B A B A B







The cost function can also be written as follows:

( ) Ysp( ) Ypred( ) U( )

Ψ ∆

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Replace (16) into (18), we have:

Where

( ) ( ) ( 1)

o

( ) ( )

and

C C C

C

d

=







Since the cost function is a quadratic form and all constraints are linear, we can use

quadratic programming (QP) to solve the optimization problem The problem (14), (15) can

be easily written in an equivalent form which is standard for quadratic programming:

1 min ( )

2

J

subject to: xmin x xmax

Ax b

≤ ≤



 where x= ∆U( ),k xmin = −∆Umax, xmax = ∆Umax, H=2(MTΨM+Λ)

Ψ

min

max

min

max

( 1) ( 1) ( ) ( )

o

J

M

k k k k

where Y k is according to the equation (19), o( ) M= CMx

∆

,

and

( 1)

u k k

u k

−





   

4 SIMULATION RESULTS

A zig-zag maneuver (±45o) is chosen as the desired heading or yaw angle trajectory

The zig-zag test is a standard maneuver used to compare the maneuvering properties and

control characteristics of a ship with those of other ships In all simulation results, the

sampling time T is chosen as 1 (second) s

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4.1 Linearized system with wave disturbances

0 100 200 300 400 500 600 700 800 900 1000

-50

0

50

Response of Ship Heading(deg)

reference output

actual output

0 100 200 300 400 500 600 700 800 900 1000

-0.1

-0.05

0

0.05

0.1

Error of Trajectory Tracking(deg)

0 100 200 300 400 500 600 700 800 900 1000

-10

-5

0

5

10

The rudder motion(deg)

Time(s)

0 100 200 300 400 500 600 700 800 900 1000 -50

0

50

reference output actual output

0 100 200 300 400 500 600 700 800 900 1000 -0.1

-0.05 0 0.05 0.1

0 100 200 300 400 500 600 700 800 900 1000 -40

-20 0 20 40

Time(s)

Cargo #2: l=161( ), 15(m v= knot) Tanker #2: l=322( ), 16(m v= knot)

Figure 4: The responses of cargo vessel and oil tanker with wave disturbances

For the linearized model, it can be seen that the controller provides an excellent tracking ability, both heading errors of cargo vessel and oil tanker are in the range of o

0.1

± In the case

of the cargo vessel, we choose N p =35, 4N u = For the oil tanker, we need to adapt these parameters because the oil tanker is bigger and faster than the cargo vessel, andN p =40, 5N u = are chosen In the presence of wave disturbances, we use Kalman filter to

estimate the states of the system and filter this disturbances out of the output of system [10] 4.2 Nonlinear system with wave disturbances

4.2.1 MPC for nonlinear system with successive linearization (MPC-NSL)

In order to use the MPC algorithm above for the nonlinear system, we need to do some modifications for better performances At each sampling instant, we perform a linearization of the nonlinear model at current process states, and then calculate the control input using the linear MPC algorithm (section 3) with linearized model It is called MPC Nonlinear with Successive Linearization (MPC-NSL) [1] The responses of nonlinear models are as follows

0 100 200 300 400 500 600 700 800 900 1000

-50

0

50

reference output

actual output

0 100 200 300 400 500 600 700 800 900 1000

-0.2

-0.1

0

0.1

0.2

0 100 200 300 400 500 600 700 800 900 1000

-40

-20

0

20

40

Time(s)

0 100 200 300 400 500 600 700 800 900 1000 -50

0

50

0 100 200 300 400 500 600 700 800 900 1000 -0.1

-0.05 0 0.05 0.1

0 100 200 300 400 500 600 700 800 900 1000 -10

0 10

Time(s)

Figure 5: The responses of nonlinear models of the vessels using MPC-NSL

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In this case, N p =35, 7N u = and N p =42, 8N u = are chosen for cargo and tanker respectively That means we need a longer prediction and control horizon to keep an excellent performance of the tracking control The heading errors of this case are even better than those

of linearized model

4.2.2 Nonlinear system with wave disturbances

In the presence of wave disturbances in the nonlinear model, the extended Kalman filter

[3] is adopted to estimate the states of the system and reject those disturbances

Similar to the case of no disturbances, the errors of tracking are still in the range of o

0.05

± for cargo vessel and ±0.1o for oil tanker However, the rudder, in this case, has to fluctuate constantly to keep the ship for tracking the reference

0 100 200 300 400 500 600 700 800 900 1000

-50

0

50

reference output

actual output

0 100 200 300 400 500 600 700 800 900 1000

-0.05

0

0.05

0 100 200 300 400 500 600 700 800 900 1000

-20

-10

0

10

20

Time(s)

0 100 200 300 400 500 600 700 800 900 1000 -50

0

50

0 100 200 300 400 500 600 700 800 900 1000 -0.1

-0.05 0 0.05 0.1

0 100 200 300 400 500 600 700 800 900 1000 -40

-20 0 20 40

Time(s)

Figure 6: The responses of nonlinear model with wave disturbances using MPC-NSL

CONCLUSION

In this paper, the MPC algorithm is applied in the ship’s heading control for the course keeping It is investigated both case of the linear and non-linear models, and the wave disturbances as well The output responses of the plants are excellent for all cases The errors

of tracking are in the range of ±0.1o for both the cargo vessel and oil tanker in terms of the linearized and non-linear models, in which the errors are only in the range of ±0.05o for the cargo vessel and ±0.1o for the oil tanker In accordance with all case studies, the control parameters of the MPC algorithm, such as N p, N u is adjusted to keep the excellent performance of the control system Moreover, the rudder is also operated constantly for tracking exactly in the presence of wave disturbances

REFERENCES

[1] Piotr Tatjewski, Advanced Control of Industrial Processes: Structures and Algorithms,

Springer-Verlag London Limited, 2007

[2] Maciejowski J M Predictive Control with Constraints, Prentice Hall, 2000

[3] Bemporad A., Borrelli F., Morari M., Model Predictive Control Based on Linear

Programming - The Explicit Solution, IEEE Transactions on Automatic Control, Dec

2002, Vol 47, No 12

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[4] Zhen Li, Jing Sun, Soryeok Oh, Path Following for Marine Surface Vessels with Rudder

and Roll Constraints: an MPC Approach, American Control Conference, June 2009

[5] Zhen Li, Jing Sun, Disturbance Compensating Model Predictive Control with Application

to Ship Heading Control, IEEE Transactions on Control Systems Technology

[6] Åström K J., Källström C G., Identification of Ship Steering Dynamics, Automatica,

1976, Vol 12, p 9-22

[7] Källström C.G., Åström K.J., Thorell N.E., Eriksson J and Sten L., Adaptive Autopilots

for Tankers, Automatica, 1979, Vol 15, No 3, p 241-254

[8] Goclowski J., Gelb A., Dynamics of an Automatic Ship Steering System, IEEE Transactions on Automatic Control, July 1966, Vol 11, No 3

[9] Fossen T I Guidance and Control of Ocean Vehicles, John Wiley & Sons, 1994

[10] Robert G B., Patrick Y C H., Introduction to Random Signals and Applied Kalman Filtering, John Wiley & Sons, 1997

AUTHOR’S INFORMATION

1 Võ Lâm Chương, Ms.E., Mechatronics Department, Faculty of Mechanical Engineering,

Ho Chi Minh City University of Technology and Education

Email: chuongvl@hcmute.edu.vn Phone: 0909110407

2 Trương Nguyễn Luân Vũ, Assoc Prof., Mechanical Technology Department, Faculty of Mechanical Engineering, Ho Chi Minh City University of Technology and Education

Email: vuluantn@hcmute.edu.vn Phone: 0909011136

3 Lê Hiếu Giang, Assoc Prof., Mechanical Technology Department, Faculty of Mechanical Engineering, Ho Chi Minh City University of Technology and Education

Email: gianglh@hcmute.edu.vn Phone: 0938308141

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