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: jiabzh@gmail.com
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
C
opyright © 2012 SciRes. JTTs
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 ABCD, where B and C are course-
changing points. The setting of controller is constant
during the straight AB, BC, CD, 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,
 
1,
,
NE PE

 




(1)

,
1,
ZE






(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.
Copyright © 2012 SciRes. JTTs
B. Z. JIA ET AL.
Copyright © 2012 SciRes. JTTs
336
Figure 3. Structure of control system.
uzzy inference, and antecedent variables employ output
3.2. Fuzzy Logic Inference Rules
rating regime em-
f
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.
d
e
 (3)
where,
is the actual course angle, d
of
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
ij
p
d
K
uKee
wu and Triang member-
troller in steady operating regime
is
t
(4)
The Fuzzy inference rules of global controller as fol-
lo
3.2.1. Transi ent Operating Regime
i je
3.2.2. Steady Operating Regime
here, A, Bj is fzzy sets of ele
i
ship function is used to describe NE and PE operating
regime as in Figure 2.
Conventional PI con
e
.
described as,
d
sp i
uKeKe
ij ijij
Tp d
uKeK
If
is ZE th n s
uKee
te the fuzzy sets of e
ly. The output of steady local control-
(5)
where,
d
s
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
pd
ij ij
TT
pd
wK ewK e
Ke Ke
 
 






 
T
p
uK
, T
d
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
11
T
p
P
ij
K
wK ,
33
Tijij
dd
 11
ij
K
wK (6)

wij is ijth fuzzy inferencrule starting stren
ing “max-min” de-fuzzed mthod,
e gth, by us-
e
 

123 123
ij eee eee
 


w
(7)
Obviously, TS-PD controller has variable gain charac-
teristics [6].
(8)
where, is output of steady operating regime,
Global controller output is,
3

k
S
u
1
T
k
uwu
S
u
B. Z. JIA ET AL. 337
d
SS
Sp i
uKeKe
t
(9)
By using “max-min” de-fuzzed method, there is,


k
wNE NE PE

 (10)
By combining (4) and (5),
(11)
where,


1
11
1
33
11
d
d
kT TS S
pd pi
k
S
pp
kij
ij ijS
di
ij
uwKe KeKe Ket
Ke
wKe Ket


 



3
333
ki
jij
ww
K



333
11 1
kij Sij
P
pp
ki j
K
ww KK
 

 (12)
3
1
kS
I
i
k
K
wK
(13)
333
11 1
kij ij
D
d
ki j
K
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]
 
1k
ttt
 

  (16)
and τ
are model parameters. Assuming zero initial conditions,
(16) can be written as
t
where φ is ship heading of ship, δ is rudder angle, k
 

21Gsss k s

  (17)
d/dt
1/s
Ship
e
d
Steering
gear
FS-PID
Controller
δ
-1
u
Figure 4. Closed loop system.
The parameter K, τ1, τ2, τ3 is the function of ship speed
u and length l

0
0
123
,1,2,3
ii
KKul
lu
i

 

(18)
The paramehip’s forward
ve
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-
cl
r con-
uding steering gear can be written as

 

21
E
K
s
(20)
where,
1
Gs Us s


s
E
K
kk
, 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
1
2deg7 deg
3
s
s
 (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
 
PI
(22)
According Routh stability criterio
conditions can be written as f
n, system stability
ollows,

0
0
E
0
1
DE
DE
P
DEPIP
KK
KK KK
KKKKKK KK
10
0
P
I
KK
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)
Copyright © 2012 SciRes. JTTs
B. Z. JIA ET AL.
Copyright © 2012 SciRes. JTTs
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,

1
1
p
DE
IEp I
DEp
KK
KK
KKK KKp
KK



 
(24)
response,
sl
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