A method to identify the dynamic parameters of ships

In this paper, a method to identify the dynamic parameters of ships is proposed. Accordingly, firstly, the parallel model and the Lyapunov method to identify the parameters of the model are used. Then, the proposed method to adjust the dynamic parameters of the model that ensures convergence of parameters of model on the dynamic parameters of the ship is used.

A METHOD TO IDENTIFY THE DYNAMIC PARAMETERS OF SHIPS

Do Cong Thang*

Abstract: In this paper, a method to identify the dynamic parameters of ships is

proposed Accordingly, firstly, the parallel model and the Lyapunov method to identify the parameters of the model are used Then, the proposed method to adjust the dynamic parameters of the model that ensures convergence of parameters of model on the dynamic parameters of the ship is used Experimental results on our datasets show that the proposed approach can speed up identifying time compared

to previous works The above proposed algorithms can be applied to construct adaptive control systems for steerring the heading angle of marine vessels

Keywords: Ships, Dynamic parameters, Identity, Parallel model, Lyapunov

1 INTRODUCTION

Marine ships are important means of transportation by sea To operate and exploit ships in high efficiency, it requires the use of equipment and automation systems, in which the automatic control system of the ship is essential To design the controller, it is necessary to define the parameter of the controlled object

Nowaday, modern ships are equipped with automatic navigation systems (these are referred to as automatic steering systems) The most common control law in these systems is the PID and its variants In general, they work well under ideal conditions: quiet ocean (external noise is not significant), unchanging load, constant speed, and so on When the effect of waves, winds up or when the load changes, the operator must adjust the parameters for the controller Although there

is human participation, driver quality is reduced, increasing fuel consumption, increasing journey time, reducing vessel efficiency

There have been a lot of research with efforts to improve the quality of ship direction control systems [1, 2, 3, 4, 5] Accordingly, the main ways are: pay attention to the nonlinearity in ship dynamics [1, 6], using modern control rules such as fuzzy control [3, 4], adaptive control [ 2, 3, 5, 6], model reference adaptive control [3, 5] Accordingly, the system is able to self-adjust the parameters of the system on the basis of comparison with the parameters of the standard model (reference model) Then, while the state of the system converges to the state of the standard model, it is not optimally correlated with the actual values of the dynamic parameters of the ship and to the actual effect of the action outside

Therefore, this article will propose a method for dynamically identifying vessel dynamics using a parallel model and is an essential part of the ship adaptive control problem

2 PROBLEM IDENTIFICATION DYNAMIC PARAMETERS

OF MARINE SHIP 2.1 Set the problem

Assume that dynamic of the ship are described by the equation [1], and by the expression (5.1) of [2]

TT  T T    K T (1) Where:  is the direction of movement of the ship (heading angle);

1, 2, 3,

For all types of ships, T T T and K are variable parameters depend on the 1, 2, 3 speed of movement, on the load that the ship is carrying However, when going out

to the ocean, the ships are operating at maximum speed in order to quickly reach the destination, shorten the journey time, ensure economic efficiency The change

in dynamic parameters depends on the load being an uncertainty factor, and hence, the uncertainty of these parameters From that it is necessary to recognize the parameters of (1), although T T T1, 2, 3, K the dynamic parameters are unknown

and/or vary depending on the load and in turn the load is also uncertain

2.2 Establishment of ship dynamic parameter identification law

For convenience of presentation, equation (1) is written as:

2 (3) (i)

i=1

ψ +a ψ =K δ+K δ (2)

KT

T +T

a = ; a = ; K = ; K =

Parameters a , a , K , K vary depending on the load of the ship It can be 1 2 1 2 said that these parameters change in wide band, in dependence on the load is the uncertainty factor For stable ships, although ai changes with load, the dynamics of (2) remains stable

To solve the problem, we need to identify the dynamic parameters of the ship according to equation (2), which means that we must obtain evaluations for the parameters To identify the parameters of the object (2) we use a parallel model with dynamics of the form:

2

i=1

ψ +a ψ =K δ+K δ (3)

It is necessary to determine the correcting rules for the dynamic parameters of the model (3), namely, the correction law of

m m m1 m2

a , a , K , K so that

1

a a

;

2

a a ; Km1K , K1 m2K2and ensure that the motion of the model (3) converges on the motion of the object (2), which means that ψmψ, ψ(i)m ψ(i) it

is mean that errors: ε = ψ-ψm 0, ε=ψ-ψ  m 0, m 0

From (2) and (3) we have:

ε +a ε +α ψ =γ δ + γ δ (4)

With

ε=ψ-ψ ; α =a -a , i=1,2; γ =K -K ; γ =K -K

1 2 3

x =ε, x =ε, x =ε , Equation (4) can be rewritten as:

1 2

2 3

3 32 2 33 3 1 m 2 m 1 2

x =x

x =x

x =a x +a x -α ψ -α ψ +γ δ+γ δ

















 



(5)

Inside, a =-a , a =- a 32 1 33 2

In the form of a vector - matrix, Equation (5) will be:

= + ψ +1 2ψ + δ+1 2δ

With

= 0 ; = 0 ; = 0 ; = 0

32 33

0 1 0

= x x x ; = 0 0 1

0 a a

The next issue is to determine the parameter correction law

a , K , i=1,2

such that the vector deviates X between the state vector of the object and the state

vector of the model convergence to the origin X0, that is

ε0, ε 0, ε0, which is equivalent to the adequate condition specified for system (6) stable

The following theorem establishes adequate condition for dynamics system (6)

is stable, It is implied that condition for the motion of the model (3) converging on the motion of the object (2)

Theorem: Suppose A is the Hurwitz matrix In order for the system (6) to

stabilize and accordingly, the motion of the model (3) converges on the motion of the object (2), the model parameter modification law must satisfy the following condition:

3

1 m 3i i

i=1 1 3

i=1 2 3

i=1 3 3

i=1 4

1

α = ψ p x ; l

1

α = ψ p x ; l

1

-γ = δ p x ;

l 1

-γ = δ p x ;

l









 

 







In which l l l l are positive coefficients, 1, , ,2 3 4 p , i=1, 2, 3 which are the third 3i

line components of the positive definite symmetric matrix P

Prove:

To prove the theorem, we use the Lyapunov method For system (6) we choose Lyapunov function of the form:

1 1 2 2 3 1 4 2

= + [l α +l α +l γ +l γ ]

V X PX (8) With l , l , l , l are positive coefficients; P is positive definite symmetric matrix 1 2 3 4

The derivative V of the Lyapunov function (8) along the orbit of (6) would be:

1

2

T T

m i i m i i i i

3

2 3 1 1 1 2 2 2 3 1 1 4 2 2 1

i i i



          (9)

Note that the matrix A is derived from equation (2) in which the ship object is assumed to be stable So matrix A is a durable matrix, also known as the Hurwitz matrix, where P is the positive definite symmetric matrix, so we have [7]:

T

In that matrix Q is positive definite symmetry matrix

From (9) and (10) we have:

T

1 m 3i i 1 1 2 m 3i i 2 2

1

= +α (-ψ p x +l α )+α (-ψ p x +l α )

2

1 3i i 3 1 2 3i i 4 2

+γ (δp x +l γ )+γ (δ p x +l γ ) (11) From (11) we have the condition that the derivative V is always negative which means that the condition is sufficient for the system (6) to be stable:

3

m 3i i 1 1 i=1

3

m 3i i 2 2 i=1

3

3i i 3 1 i=1

3

3i i 4 2 i=1

ψ p x +l α =0

ψ p x +l α =0

δ p x +l γ =0

δ p x +l γ =0















From here we obtain the laws that adjust the parameters of (7) of the model, which is also sufficient condition for the motion of the model (3) to converge on the motion of the object (2) That is something to prove

Note that

α =a -a ,i=1, 2; γ =K -K ;γ =K -K from (4), and furthermore, the parameters ai, K1, K2 are unknown predetermined parameters, from (7) we obtain the law modifying the parameters of the model:

1

2

1

2

3

i=1 1 3

i=1 2 3

i=1 3 3

i=1 4

1

a = α = ψ p x

l 1

a = α = ψ p x

l 1

K = γ = δ p x

l 1

K = γ = δ p x

l

 

 















Thus, the evaluation of the dynamic parameters of the object will take the form:

1

0

2

0

1

o

2

o

i=1

1 t

i=1

2 t

i=1

3 t

i=1

4 t

1

a (t)=a (t)= ψ (σ) p x (σ)dσ+a (t );

l 1

a (t)=a (t)= ψ (σ) p x (σ)dσ+a (t );

l 1

K (t)=K (t)= δ(σ) p x (σ)dσ+K (t )

l 1

K (t)=K (t)= δ(σ) p x (σ)dσ+K (t

l



























o) (14)

Where a (t ), a (t ), K (t ), K (t ) ˆ1 0 ˆ2 0 ˆ1 0 ˆ2 0 are the initial values of the process of

identification

Assessment rules (14) are technically feasible and easily feasible The results of the evaluation (14) of the dynamic parameters of the object used to synthetize the control rule for the object (2)

3 SIMULATION AND DISCUSSION 3.1 Simulation data

The following table shows the parameters used to simulate

Table 1 Parameters used for simulation

T1(s) T2(s) T3(s) K(1/s) l1 l2 l3 l4 p31 p32 p33

118 7,8 18,5 0,185 8.104 20,5 4 0,05 10-5 10-3 10-2

3.2 Method, simulation tool

From above, the structure of the ship's dynamic parameter identification system (2) consists of two main blocks, the parallel model block (3) and the identity block

(14) The following is a simulation program written on the Matlab&Simulink

software, which consists of three basic blocks: SHIP dynamic (which need to be parameterized) describing equation (2), parallel model MODEL present (3) and parameter dentify block IDENTIFIER (14)

Figure 1 Matlab & Simulink simulation program

for vessel parameter identification system

Where the symbols and variables are as follows: Delta = δ; PSI=ψ; PSI1.= ψ ; PSI2.= ψ ; PSI3.= ψ; PSIm=ψ ; PSIm.=im ψm; PSIm =ψm; Alpha1= a1-a1m=α1; Alpha2

2

=α =a2-a2m=α2; Gama1=γ1=K1-K1m, Gama2=γ2=K2-K2m

3.3 Simulation results and comments

0 200 400 600 800 1000 1200 1400 1600 1800 2000

-2

0

2

4

6x 10

-3



th i gian (giây)

2 )

a) Time(second)

0 200 400 600 800 1000 1200 1400 1600 1800 2000 -0.05

0 0.05 0.1 0.15 0.2

b) Time(second)

0 200 400 600 800 1000 1200 1400 1600 1800 2000

-0.3

-0.2

-0.1

0

0.1

0.2

t(s)

-3 )

c) Time(second)

0 200 400 600 800 1000 1200 1400 1600 1800 2000 -2

-1.5 -1 -0.5 0 0.5

2 )

d) Time(second)

Figure 2 Parameter errors between object and identification model: α =a -a1 1 m1 (a),

2 2 m2

α =a -a (b), γ =K -K1 1 m1 (c), γ =K -K2 2 m2 (d)

Based on the simulation results shown in Figure 2, we find that the difference between the parameters of the object model (SHIP) and the parallel model is large

at the beginning time, then the errors are very small, almost zero That is, the parameter of the parallel model already holds the parameter of the object model Furthermore, to ensure that the parameters of the object and the model are

identical, we check the difference of the experimental error X

0 200 400 600 800 1000 1200 1400 1600 1800 2000

0

200

400

600

800

a) Time(second)

0 200 400 600 800 1000 1200 1400 1600 1800 2000 -30

-20 -10 0 10 20 30

t(s)

b) Time(second)

0 200 400 600 800 1000 1200 1400 1600 1800 2000 -4

-2 0 2 4

t(s)

2 )

c) Time(second)

Figure 3 Changing of variables in the identification process: variable x1 (a),

variable x2 (b), variable x3 (c)

Figure 3 shows that the solution X converges to zero That means the errors

m

ε = ψψ 0, ε=ψ  ψm 0, m 0

Thus, we affirm with certainty that the motion of the model (3) converges on the motion of the object (2), the dynamic parameters of the ship has been identified through the parallel model and parameter identification law

4 CONCLUSIONS

The above method has developed for identifying dynamic parameters of ships based on the parallel model and Lyapunov method Proven theorem on sufficient conditions for the parameters of convergent model of parameters of ships

Normally, the ship's load is changed during port time (at the port and at the port departure) That will result in a change in the dynamic parameters of the ship At the departure point, the system performs the identification, and hence, adjusts the control parameters to ensure adaptation to the new dynamic parameters of the vessel, and also means adapting to the new load of the ship Then, throughout the journey to the new port next loading of the ship is almost unchanged The proposed identification laws are easy to implement technically Simulation results have demonstrated the convergence and effectiveness of the proposed algorithm

REFERENCES

[1] Thor I Feossen "Guidance and Control of Ocean Vehicles" John Wiley &

Sons, Chichester, New York, Brisbane, Toronto, Singapore (1994)

[2] Åström K.J "Why use adaptive techniques for steering large tankers?" Int J

control, (1980), vol 32, no 4, 689-708

[3] Jeffery Layne and Kevin M Passino "Fuzzy Model Reference Learning

Control for Cargo Ship steering" 0272-1708, (1993)IEEE

[4] Junsheng Ren, PSIank zhang "Fuzzy-Approps-Based Adaptive Controller

Design for Ship Course-Keeping Steering in Strict-Feedback Forms," ISSN:

2040-7459; e-ISSN: 2040-7467, Maxwell Scientific Organization

[5] J Van Amerongen, A J Udink, "Model Reference Adaptive Autopilots for

Ships", Automatica, Vol 11, pp 441-449

[6] Jialu DU, Chen Guo, Yongsheng Zhao, Yingjun Bi, "Adaptive Robust

Nonlinear Design of Course Keeping Ship Steering Autopilot" 2004 8th

International Conference on Control, Automation, Robotics and Vision Kunming, China, 6-9th December 2004

[7] Granthmakher, "Matrix theory", Moscow 1982

[8] Andrew P Sage, Chelsea C White, "Optimal Control System", prentice-Hall

Inc, 1982

TÓM TẮT

MỘT PHƯƠNG PHÁP NHẬN DẠNG THAM SỐ ĐỘNG HỌC CỦA TÀU THUỶ

Bài báo đề xuất phương pháp nhận dạng tham số động học của tàu thuỷ

Sử dụng mô hình song song và phương pháp Lyapunov, đã xây dựng được các thuật toán tự hiệu chỉnh các tham số của mô hình, đảm bảo cho các tham số động học của mô hình hội tụ về các tham số động học của tàu Các thuật toán đề xuất trên đây có thể áp dụng để xây dựng các hệ thống điều khiển thích nghi để điều khiển tàu theo hướng

Từ khóa: Tàu thuỷ, Tham số động học, Nhận dạng, Mô hình song song, Lyapunov

Author affiliations:

Hung Yen University of Technology and Education;

*

Corresponding author: docongthang77@gmail.com.