Design and Stability Analysis of Fuzzy Switched PID Controller for Ship Track-Keeping

doi:10.4236/jtts.2012.24036

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Journal of Transportation Technologies, 2012, 2, 334-338

http://dx.doi.org/10.4236/jtts.2012.24036 Published Online October 2012 (http://www.SciRP.org/journal/jtts)

Design and Stability Analysis of Fuzzy Switched PID

Controller for Ship Track-Keeping*

Baozhu Jia1,2, Hui Cao2, Jie Ma1

1State Key Laboratory of Ocean Engineering, Shanghai Jiao Tong University, Shanghai, China

2Marine Engineering College, Dalian Maritime University, Dalian, China

Email: [email protected]

Received July 25, 2012; revised August 23, 2012; accepted September 10, 2012

ABSTRACT

The fuzzy switched PID controller which combines fuzzy PD and conventional PI controller is proposed for ship

track-keeping autopilot In this paper. By using rudder angle, the whole voyage is divided into two operating regimes

which named transient operating regime and steady operating regime respectively. The fuzzy PD controller is employed

in transient operating regime for increasing response, reducing overshoot and shorting transition time. And conventional

PI controller is used to improve the stable accuracy in steady operating regime. The global controller is achieved by

fuzzy blending of all local controllers. Routh stability criterion is utilized to obtain the stability condition of closed-loop

system. The simulation results show the effectiveness of proposed method.

Keywords: Ship Control; Fuzzy Switched PID; Track-Keeping; Fuzzy Operating Regime; Stability Condition

1. Introduction

Ship autopilot mainly used to keep the course at designed

trajectory within minimum deviation. In terms of charac-

teristics of nonlinear and underactuated of cargo ship,

finding an effective autopilot control algorithm has been

the significant topic in area of ship steering control. This

has evoked interest into the field of autopilot in marine

cargo ship controlling community recently. Fossen, et al.

proposed adaptive controller to improve performance and

reduce fuel consumption on both of course-keeping and

course-changing maneuvers [1]. In [2], the problem is

solved by the adaptive robust fuzzy method, which called

state input-output theory is adopted. The work in [3] has

proposed adjust ship course by using track error-driven

control algorithm.

Ship voyage involves several straight-line segmenta-

tions. During straight-line voyage, the control require-

ment mainly focuses on stable accuracy, which are called

tack-keeping. Rudder angle changes frequently and the

controller output is small. Most of time, autopilot oper-

ating in the mode of track-keeping process, so we called

is as steady operating regime in this paper. Whereas,

when the ship changing into a new direction from origi-

nal course, the control requirements mainly depends on

response, overshoot and transition time, what are named

course-changing process, the controller provides a larger

control signal to achieve faster response. There are do-

zens of course-changing points in whole voyage, what

depends on the ocean environment and port of destina-

tion as well. When course error decreasing, the controller

outputs would gradually reduce according to control laws.

The course-changing process is generally short-term and

countable, so it can be called transient operating regime.

The good control strategy could adaptively switch to

the most matching control law based on operating regime

according to switching rules. The fuzzy switching PID

control algorithm combines Takagi-Sugeno PD controller

with general PI controller to improve the control per-

formance in terms of fast response and high stable accu-

racy throughout the voyage. During track-keeping proc-

ess, the so called TS-PD controller is employed to keep

the stability accuracy, and when course-changing process,

the general PD controller is used to improve the response

speed. The TS-PD controller and general PI controller

switched according to the operating regime switching

rules.

Fuzzy logic has been proved to be an universal ap-

proximator for any real continuous function. It can be

constructed in many different configurations. However,

since Takagi-Sugeno fuzzy inference is nonlinear in na-

ture, we utilize it to construct the supervisory rules. The

global controller output is used to be antecedent variable

of supervisory rules. The stability of fuzzy control sys-

tems may not be easily analyzed. Lyapunov approach

and variable structure or the phase-plane approach nor-

mally be employed. Besides, it is difficult to find a

common Lyapunov’s function for a fuzzy controller. In

B. Z. JIA ET AL. 335

this paper, the Routh stability criterion is used to achieve

stability condition for closed loop system. The fuzzy swit-

ched inference guarantees the smooth characteristic be-

tween adjacent controllers.

2. Basic Problem

2.1. Problem Describe

Between departure port and destination port, the sailing

route is planned considering the factors of ocean envi-

ronment, navigational safety and economy. It usually com-

bined with several straight lines as shown in Figure 1.

From departure port A to destination port D, the de-

signed route is A→B→C→D, where B and C are course-

changing points. The setting of controller is constant

during the straight A→B, B→C, C→D, the function is

used to keep trajectory at predetermined routs under ex-

ternal disturbance. When sailing to B and C point, the

course would changes to a new course from initial, con-

troller used to drive the ship into new course within

smaller deviation and shorter response time.

2.2. Fuzzy Operating Regime

The concept of operating regime is firstly proposed by

Johansen [4]. Any models and controllers have its spe-

cific period and space range. Only within this ranges, the

model and controller have sufficient accuracy and effi-

ciency to accomplish designed purpose. The operating

regime can be defined by multiple factors, such as model

hypothesis, linearization validity, stability condition and

experiment restriction, etc.

Local controller is determined by both the steady op-

erating regime and transient operating regime. How to

find a variable can be used to divide the discourse do-

main into several local operating regimes is the key issue.

The experience shows that the rudder angle is usually

small during straight voyage. On the contrary, when the

course changed, rudder angle would change by a large

scale. So in this paper, the output of controller, i.e. the

Figure 1. Route planning diagram.

rudder angle order δ, is employed to divide the operating

regime.

Fuzzy operating regime would guarantee the smooth

switching of multiple controllers [5]. Figure 2 shows the

fuzzy partition of operating regime in this paper, the

membership functions are assumed to be trapezoid.

Where fuzzy set ZE is fuzzy set used to describe transient

operating regime, and NE and PE are fuzzy set used to

describe steady operating regime related to positive and

negative course error, respectively. Generally, starboard

tuning of ship is defined as positive direction. The OM is

the axis of symmetry of fuzzy set ZE, NE and PE. The

cross-shadow area is overlap region of adjacent fuzzy

operating regime. λ and μ are shape parameter of fuzzy

sets. Assuming = 2









, λ and μ can be calculated

indirectly according to given ε.

For any rudder angle



, the degree of membership

may be represented as,

 

NE PE



 























(1)

























(2)

3. Fuzzy Switched System

3.1. System Structure

The structure of closed-loop control system with rudder

angle feedback and vessel position feedback is shown as

Figure 3. PID control methods are widely used in ship

autopilot. There are three performance indexes for clos-

ed-loop system, overshoot, transient time, and steady acc-

uracy, these indexes are mutual constrained.

The controller output is related to the error changing

because of fuzzy PD algorithm during running in tran-

sient operating regime. Otherwise, when running under

steady operating regime, the controller would switch to

PI controller to reduce the static deviation of track-

keeping. The supervisory rules employ Takagi-Sugeno





Figure 2. Fuzzy partition of operating regime.

B. Z. JIA ET AL.

336

Figure 3. Structure of control system.

uzzy inference, and antecedent variables employ output

3.2. Fuzzy Logic Inference Rules

rating regime em-

of global controller. The global controller can be achi-

eved by using the so called “product-sum” inference.

Fuzzy PD controller in transient ope

ployed Takagi-Sugeno fuzzy inference rule, abbreviated

as TS-PD controller in this paper. Course angle error e

and its differential e

 are used as antecedent input vari-

able.



 (3)

where,



is the actual course angle, d



is the de-

signed course angle, The counterclockwise north head-

ing direction is defined as positive. The course angles are

expressed with radian. Fuzzy inference rules of TS-PD

controller can be described as follows,

If e is A and e

 is B then ij ij

i j

K

 uKee

wu and Triang member-

troller in steady operating regime

(4)

The Fuzzy inference rules of global controller as fol-

3.2.1. Transi ent Operating Regime

i je

3.2.2. Steady Operating Regime

here, A, Bj is fzzy sets of ele

ship function is used to describe NE and PE operating

regime as in Figure 2.

Conventional PI con

.

described as,

sp i

uKeKe



ij ijij

Tp d

uKeK





is ZE th n s

uKee

te the fuzzy sets of e

ly. The output of steady local control-

(5)

where,

Sp i

Ket



where, I = 1, 2, 3, j = 1, 2, 3, deno

and e

 respective

ler T

u is,

33 33

()

ij ijijijij

Tpd

uwuw KeKe 

 

11 11

33 33

11 11

ij ij

ij ijij ij

ij ij

wK ewK e

Ke Ke

 











  



, T

is proportional gain of transient operating

regime

is integral gain of transient operating re-

gime,

33 ijij

ws.

If δ is NE or PE then

If e is A and e

 is B then

wK ,

Tijij

 11

wK (6)



wij is ijth fuzzy inferencrule starting stren

ing “max-min” de-fuzzed mthod,

e gth, by us-

















 







123 123

ij eee eee

 





(7)

Obviously, TS-PD controller has variable gain charac-

teristics [6].

(8)

where, is output of steady operating regime,

Global controller output is,



u



uwu



B. Z. JIA ET AL. 337

Sp i

uKeKe

t

(9)

By using “max-min” de-fuzzed method, there is,







wNE NE PE





 (10)

By combining (4) and (5),

(11)

where,





kT TS S

pd pi

kij

ij ijS

uwKe KeKe Ket

wKe Ket







 



















 





333

jij













333

11 1

kij Sij

ki j

ww KK

 



 (12)



 (13)

333

11 1

kij ij

ki j

wwK

 

 (14)

are nonlinear proportional gain, integral gain and differ-

ential gain of global controller’s output respectively.

The transfer function of equation (11

as,

) can be rewritten



cPID

Gs KKsKs (15)

3.3. Plant Models

By considering only the closed loop sys

error to ship heading, the Figure 3 can be simplified as

model relating the rudder angle of

tem from course

Figure 4.

The mathematical

the ship to the heading was proposed by Norbin [7]

 

ttt

 



  (16)

and τ

are model parameters. Assuming zero initial conditions,

(16) can be written as



where φ is ship heading of ship, δ is rudder angle, k

 



21Gsss k s



  (17)

d/dt

1/s

Ship



Steering

gear

FS-PID

Controller

-1

Figure 4. Closed loop system.

The parameter K, τ1, τ2, τ3 is the function of ship speed

u and length l



123

,1,2,3

KKul



 





(18)

The paramehip’s forward

odel can be written as

ters u and l are function of s

locity and its length.

Steer gear m

















 





EdEE E

tktttkutt

 





(19)

kE is steering gear gain coefficient, τE is steer gea

stant time coefficient. The transfer function of plant in-

r con-

uding steering gear can be written as







 



s

(20)

where,

Gs Us s







s



, is generalized controller amplifica-

tion coefficient. For majority of cargo ships the rudder

angle and the rudder rate are confined to be in the ranges



35deg





max

 

max

2deg7 deg





 (21)

It is uuired that rudder should move from 35˚

port to 35˚ starboard within 30 s.

sually req

3.4. Stability Conditions

By considering the closed-loop control system constr-

ucted by controller (15) and generalized

characteristic equation can be written as

plant (20), its



 

432 10

EDE

ssKKsKKs KK

 



(22)

According Routh stability criterio

conditions can be written as f

n, system stability

ollows,



DEPIP

KK KK

KKKKKK KK





















where we assume that for a cargo ship, K0 = −3.86, τ10 =

5.66, τ20 = 0.38, τ30 = 0.89, l = 161 m. Also, we assume

that the ship is traveling in the x direction at a velocity of

10 knots (about 7.8 m/s). In Equation (23), parameter k =

−0.187, τ = 106.2. For pump-controlled hydraulic steer-

(23)

B. Z. JIA ET AL.

338

ing gear, kE = 1.4, τE = 0.04 s, K = −0.26 < 0.

To solve Equation (23) obtained the stability condi-

tions for closed-loop control system,



IEp I

DEp

KKK KKp









 



(24)

response,

4. Simulation Results

In Figure 2, λ = 3, μ = 5. Assuming heading be changed

by 30˚ to right direction, the curve of track-keeping, cou-

rse error and rudder angle changing have been shown in

Figures 5 and 6 respectively, they illuminate that the pro-

posed fuzzy switched PID controller has fast

ight overshoot and high accuracy in whole voyage.

Figure 5. Simulation result of 30˚ course changing.

Figure 6. Rudder angle curve.

5. Conclusion

This paper proposes a performance-oriented control al-

gorithm that will improve the efficiency and accuracy of

whole voyage track-keeping. There are hardly any exist-

ing control laws that could meet the above needs. The ad-

vantages of fuzzy PD controller and conventional PI

controller are combined to overcome the shortages of tra-

ditional PID controller. The proposed control strategy is

easy to be performed. The simulation results on cargo

ship prove this control strategy can achieve the desired

consequent.

6. Acknowledgements

incial Natural

n of China (No. 201202017) and the

ce on Control Applications, Dayton, 13-16 Sep-

, pp. 1076-1081

Q. Wenming, “ISS-Based Ro-

thm for Maintaining a Ship’s

This work supported b

Science Foundatio

y the Liaoning Prov

Fundamental Research Funds for the Central Universities

(No. 2011QN111), the authors should like to express our

deepest gratitude to all those whose kindness and advice

have made this work possible.

REFERENCES

[1] T. I. Fossen and M. J. Paulsen. “Adaptive Feedback Lin-

earization Applied to Steering for Ships”, The 1st IEEE

Conferen

tember 1992

[2] L. Tieshan, Y. Shujia and

bust Adaptive Fuzzy Algori

Track,” Journal of Marine Science and Application, Vol.

6, No. 4, 2007, pp. 1-7. doi:10.1007/s11804-007-7027-z

[3] Q. M. Chen, “The Study on Indirect Multi-Mode Control

Method for Ship Track-Keeping Autopilot,” Journal of

Applied Sciences, Vol. 19, No. 2, 2001, pp. 153-156.

[4] T. A. Johansen, “Operating Regime Based Process Mod-

eling and Identification,” Ph.D. Thesis, University of

Trondheim, Norway

[5] B. Z. Jia, G. Ren and G. Long, “Design and Stability

S. Shao. “Typical Takagi-

, 1994.

Analysis of Fuzzy Switching PID Controller,” Proceed-

ings of the 6th World Congress on Intelligent Control and

Automation, Dalian, 21-23 June, 2006, pp. 3934-3938.

[6] Y. S. Ding, H. Ying and

Sugeno PI and PD Fuzzy Controller, Analytical Struc-

tures and Stability Analysis,” Information Sciences, Vol.

151, 2003, pp. 245-262.

doi:10.1016/S0020-0255(02)00302-X

[7] J. R. Layne and K. M. Passino, “Fuzzy Model Reference

Learning Control for Cargo Ship Steering,” IEEE Control

Systems Magazine, Vol. 13, No. 6, 1993, pp. 23-33.

doi:10.1109/37.248001