399
1 INTRODUCTION
Dynamicpositioning(DP)systemformarinevehicles
isachallengingpracticalproblem.Itincludesstation
keeping,positionmooringandslowspeedreferences
tracking.Ofthatthree,themainpurposeofDPisto
maintain acertain accurate position and course,
regardlessoftheinterferencesuchaswaveandwind.
This ta
sk should only be achieved under its own
propulsion and using navigation systems. An
applicationoftheappropriatecontrolmethodforDP
isdirectlyrelatedtotheadoptedmodel,itspurpose,
structureandnumberoftheinstalledactuators.
The first DP systems were designed using
conventional PID controllers in cascade with low‐
pa
ss and notch filters. Here, the wave disturbances
werefilteredbeforefeedbackwasappliedinorderto
avoid unnecessary control action. Model‐based
controlsfordynamicpositioningincludesalsoLQG,
sliding mode control (Tomera 2010), robust H∞
control(Grimbleetal.1993,Messeretal.1993),non‐
linear ba
ckstepping method (Krstic et al. 1995) and
anotherstate‐spacetechniques(Fossenetal.2002).
Theartificialintelligence(Xuetal.2011),fuzzylogic
(Cao et al. 2001) and neural nets (Cao et al. 2000)
werealsousedforDP.Anumberofresearcheswere
carriedoutwithinthescopeofapplicat
ion.Itisvery
difficult to derive a relative simple controller based
on traditional methods for the vehicle modelwhich
represent such a complex system. Nowadays, PID
controller is one of the most popular industrial
controllers. It is because of its simple control
structure,easinessofdesign,andinexpensivecost.In
the last twenty years, fract
ional calculus have been
developedbyscientistsandengineersandappliedin
the area of control theory. In the literature a
generalization of integer‐order controllers by
corresponding fractional‐order controllers was
proposed.Themostpopularincludes(Efe2011),FO
PID (Podlubny 1999), FO MPC (Domek 2013), FO
ba
ckstepping method, FO sliding mode control
(Vinagreetal.2006).Fractional‐orderPI
λ
D
controller
was the first proposed by Podlubny in 1997
Fractional Order Dynamic Positioning Controller
A.Witkowska
GdańskUniversityofTechnology,Poland
ABSTRACT: Improving the per
f
ormance of Dynamic Positioning System in such applications as station
keeping, position mooring andslowspeed references trackingrequires improving the position and heading
controlprecision.Thesegoalscanbeachievedthroughtheimprovementoftheshipcontrolsystem.Fractional‐
ordercalculusisaveryusefultoolwhichextendsclassical,integer‐ordercalculusandisusedincontemporary
modeling and control applicat
ions. Fractional‐order PIλD controller, based on the added flexibility of
fractional‐orderoperators,arecapableofsuperiorperformancecomparedtotheirinteger‐ordercounterparts.
ThispaperpresentsthefractionalorderPIλDcontrollerdesignedtomaintaintheshippositionandheading
andtheresultswerecomparedwithclassicalint
egerorderPIDcontroller.
http://www.transnav.eu
the International Journal
on Marine Navigation
and Safety of Sea Transportation
Volume 9
Number 3
September 2015
DOI:10.12716/1001.09.03.13
400
(Podlubny et al. 1997). Fractional‐order PI
λ
D
controllers, based on the added flexibility of
fractional‐orderoperatorsinvolving an integratorof
orderλand a differentiator of order . They are
capable of superior performance compared to their
integer‐order counterparts. Fractional‐order calculus
isaveryusefultoolinsomefieldsofresearch.Buton
the study
of vehicle control system, only several
literatureswerefound(Nouillantet.al.2002,Zhang
et.al.2006).Therefore,thepurposeofthispaperisto
implement a fractional‐orderPID control scheme as
DPcontroller.Theproposedsolutionshavenotbeen
applied for controlling DP units to increase the
accuracy
ofkeepingitspositionanddirection.
Presently there exist several tools for working
withfractionalmodelsandcontrollers.Theyinclude
CRONE (Oustaloup et. al. 2000), Ninteger (Valerio
2005) and FOMCON (Tepljakov et. al. 2011)
MATLABtoolboxes.
A larger number of control parameters FOPID
(fiveparameterstuned,insteadofthree),anda
larger
space values for these parameters (set of real
numbers),providesgreaterflexibilityinthedesignof
the controller with respect to the standard PID
controller.However,thisalsoimpliesthatthetuning
ofthecontrollercanbemuchmorecomplex.
Different methods for the design of a FOPID
controller
have been proposed in the literature
(Monjeet.al.2008).Theyincludeafrequencydomain
approach (Vinagre et. al. 2000), Ziegler‐Nichols
tuningrulesand alsooptimizationmethods suchas
Particle Swarm Optimization (PSO) (Karimi et. al.
2009). In this paper the Genetic Algorithm method
wasusedtotunetheparameters
ofthecontroller.
2 DYNAMICPOSITIONINGSYSTEM
Themaincontrolloopofdynamicpositioningsystem
consists of the following modules: position and
heading controller, control allocation system,
dynamicsofpropellers,dynamicsoftheDPshipas
the object of control, and the observer of current
estimatedoutputquantitiesforposition,
course,and
longitudinal, lateral and angular speed components
asshowninFigure1.Basedonthecomparisonofthe
vectorofsetpositionanddirectionvalues
dwiththe
vectorofcurrentestimatedvalues
obtainedfrom
the observer. DP controller calculates the vector of
therequiredsurge,swayforcesandyawmoment
to
compensate deflections from the given values,
according to the assumed error of control. In this
system the ship position and direction controller
controls the motion of this object independently in
threedegreesoffreedom,calculatingthesetvaluesof
forthedriveallocationcontrolsystem.
Windmodel
State
Estimator
u
d
DPController
(position,heading)
Control
Allocation
Object
Figure1.Dynamicshippositioningsystem.
Basic problems to be solved when designing DP
systemsinclude:
filtration of signals and estimation of measured
andnon‐measuredquantities,
selection of the control method and the DP
controller,
allocationofdrivecontrols(Witkowska2014).
In the paper the second task will be the aim of
analysis.
During the study of controller properties it was
assumed that the control allocation system and
actuators were neglected, ie. produced by thrusters
resultant forces and moment are equal to the
calculatedbytheDPcontroller.
2.1 MathematicalmodelofDPvessel
Kinematic and dynamic properties of DP vessel on
the
water, are described using a non‐linear
differential equations in three degrees of freedom
takingintoaccountthesurge,swayandyawmotion.
Inthiscaseitispossibletocontrolindependentlythe
position components x, y and the rotation r of the
shipbychangingtheangleofship,
bowpositionwith
respect to North. Other movements of the ship:
rollingpitchingandheavingcanbeomittedprovided
that the ship is stable laterally and longitudinally
movesacrossthesurfaceofthewaters.Inaddition,at
low speed the Coriolis force and centripetal also
nonlinear hydrodynamic damping force can be
neglected. Given the above assumptions, the
mathematicalmodelofshipmotioninthehorizontal
plane is described by the following system of
differentialequations(Fossen2002):
d
Rv
dt
, (1)
dv
MDv
dt
(2)
where:
= [
x,
y,
z]
T
‐generalizedvectorofforces
andmoment,providedbyDPcontroller,
=[x,y,
]
T
‐
shippositionandheading0<
<2oftheshipinthe
earth‐fixed frame,
=[u,v,r]
T
‐ linear velocities in
surge,swayandangularvelocitycoordinatedinthe
body fixed frame, MɌ
3x3
, DɌ
3x3
and R(
)Ɍ
3x3
denotesrespectivelymatrix of inertia,dampingand
the transformation matrix of the coordinate system
associatedwiththecenterofgravityoftheshiptothe
coordinate system with thefixedpoint of the earth.
TherotationmatrixR(
)withthepropertyR
T
=R
‐1
is
givenby:
0
0
001
cos sin
Rsincos
(3)
The first‐order wave frequency (WF) part of
motion is modeled as a second order linear system
for each of the 3 DOF (i=1,2,3), producing signals
addedtothepositionandheadingmeasurements.
401
0
22
00
2
2
iii
i
ii i
hs
ss
(4)
where:
i=0.1‐relativedampingratio,0i=0.65rad/s
‐ dominating wave frequency,
i=0.5 m.‐wave
intensityparameter.
Transmittance input signal is a white noise with
zeromean.Asimplifiedwavedisturbancesmodelis
a linear approximation of the wave spectrum. The
outputofthetransmittanceisaddedtothemeasured
signals‐positionandheadingofthevesselinorder
tomodellinganimpact
ofhighfrequencycomponent
of the ship movement. Transmittance parameters
define the significant wave height about Hs 3m.,
which means sea state 5 degrees in the scale of
Douglas.Theabovewavefrequencymodeliswidely
usedintheliteratureforthesimulationtestsandto
study the
observer properties of filtration and
estimation.
2.2 Stateobserver
The DP control system mostly assumes that only
position and heading signals are available through
the navigation measurement systems such as GPS,
DGPS and gyro. In contrast, immeasurable ship
speed which is necessary to derive control laws are
estimatedfromthestate
observer.In theDPsystem
(Fig. 2) the nonlinear passive observer was
considered (Fossen et. al. 1999), which also makes
filteringofhighfrequencywavedisturbances.
2.3 DPcontroller
Nowadays,mostof DP systems which areapartof
the vessels equipment, use the classical PID
algorithmstocontrolthe
positionandheadingangle.
2.3.1 NonlinearPID
After 1995, nonlinear PID control design have
beenapplied(Fossen2002)toDPsystemswithgood
results.ThePIDcontrolconceptcanbegeneralizedto
nonlinear mechanical system by exploiting the
kinematicequationsofmotioninthedesign(5):
0
t
TTT
Idpdd
d
KR d KR KR
dt
(5)
In order to take advantages of the observer, the
control law was implemented using the estimated
statesinsteadofthetrueone, due to theabsenceof
required measurements, and because of the wave
filtering. Thus, the last component of (5) was
substitutedby(1)
0
t
TT
I
dp dd
KR d KR Kv
(6)
where
I, p, Kd Ɍ
3x3
‐matrixes of integral,
proportional and derivative gains, illustrating the
influenceofindividualcomponentsinthreedegrees
offreedom;
‐vectorofestimatedshippositionsand
heading,
d‐vectorofdesiredvariable,‐vectorof
estimated ship velocities. The Figure 2 presents a
block‐diagramconfigurationofnonlinearPID(6).
d
R
T
(
)
+
‐
+
1
K
i
R
T
(
)
+
‐
‐
K
p
K
d
v
+
+
‐
IntegralAction
DerivativeAction
ProportionalAction
Figure2.Block‐diagramofPID(6).
In this figure the symbol 1/s denotes the
Laplaceaʹsformofintegraloperator.
2.3.2 FractionalorderPID
All the classical types of PID controllers are the
special cases of the fractional PI
λ
D
controller
involvinganintegratoroforderanddifferentiator
oforder.
According to nonlinear PID controller (5) and
fractionalcalculusthevectorofcontrolsignals
can
thenbeexpressedinthetimedomainas:
TTT
Itdp dd t
K
RD KR KR D
(7)
where=[
x,y,
]>0isthevectorofintegralorders,
=[
x, y,
]>0 is the vector of derivative orders.
Here D
q
(q=‐, q=) is the differintegral operator
withthefractionalorderq,combineddifferentiation‐
integration operator commonly used in fractional
calculus. This operator isa notation for taking both
thefractionalderivativeandthefractionalintegralin
asingleexpressionandisdefinedby(8)
0
11
0
0
q
d
q
q
dt
q
t
q
dq
q
D
t
(8)
Clearly, selecting = [1,1,1] and =[1,1,1], a
nonlinearPIDcontroller(5)canberecovered.
According to nonlinear PID controller (6) and
fractionalcalculusthevectorofcontrolsignals
can
thenbeexpressedinthetimedomainas:
TT
I
tdp dd
KR D KR Kv
(9)
The Figure 3 presents a block‐diagram
configurationofFOPID(9).
402
d
R
T
(
)
+
‐
+
1
K
i
R
T
(
)
+
‐
‐
K
p
K
d
v
+
+
‐
IntegralAction
DerivativeAction
ProportionalAction
Figure3.Block‐diagramofFOPID(9).
The symbol s
‐
denotes the Laplaceaʹs form of
fractionalintegraloperator.
There are some definitions for fractional
derivatives.ThecommonlyusedincludeGrunwald–
Letnikov,Riemann–Liouville,andCaputodefinitions
(Podlubny 1999). Since most of the fractional‐order
differential equations do not have exact analytic
solutions, so approximation and numerical
techniques must be used. Several
analytical and
numericalmethodshavebeenproposedtosolvethe
fractional‐order differential equations. One of the
best‐known approximations is due to Oustaloup
(Oustaloup 2000).The method based on
approximatingafractional‐orderoperators
q
,where0
< q < 1, in a specified frequencyrangeω= (ω
b,ωh)
andoforderN.
Figure4presentsanexampleofapproximationof
integrals of constant function in time. There was
consideredintegralorders( =0.1,0.2,0.3,0.4,0.6,
0.8,1), zero initial conditions, frequency rangeω=
(0.01,1000)rad/sandorder5.
Figure4. The approximation of integrals of constant
functionfordifferentorders(=0.1,0.2,0.3,0.4,0.6,0.8,1)
infrequencyrangeω=(0.001,1000)rad/sandoforder5.
3 SIMULATIONTESTRESULTS
The mathematicalmodel of supply vesselwas used
asacasestudy(Godhavnet.al.1998)andnonlinear
passive observer (Witkowska 2013). The vessel
systemmatricesweregivenbelow.
1.1274 0 0
0 1.8902 0.0744
0 0.0744 0.1278
M
(9)
0.0358 0 0
0 0.1183 0.000124
0 0.000041 0.0308
D
(10)
A model of supply vessel was used to illustrate
theperformanceofDPsystem(Fig.1.)withfractional
order PID controllers (7), (9). For this purpose the
FOMCON toolbox for Matlab was used for
simulations. The FOMCON toolbox allow us to
implement, simulate and analyze FOPID controllers
easilyvia
itsfunctions.Alsointhislibrary, one can
find the Fractional PID block which implements
FOPID controllers in Simulink. In fractional order
PID controller there the only aproximation of
differintegral operator s
q
was needed with specified
frequencyrangeandorder.Inthesimulationstudies
the Oustaloup method was assumed within the
frequencyrange(0.01,1000)rad/sandthenumberof
zerosandpoles sitedto5.Simulationswerecarried
outintimedomain.
Thesimulationtestsaimatcheckingtheoperation
correctness of
DP system with fractional PID
controllers (7) and (9) in comparison with classical
PIDcontroller(5)tunedbyGA.Duringasimulation
tests the initial conditions were chosen as:
(t
0)=(0,0,0), (t0)=(0,0,0) and the initial values of all
estimates were set as zero. Desired position and
orientationweresetas
d(t)=(5,5,0).Thesimulation
studies were carried out in the presence of wave
disturbances (4). The amplitudes of the wave were
setas2m.,2m.,3
0
respectivelyforsurge,swayand
yawdirection.
Figure 5 presents ship trajectory in DP system
with PID (6) and FOPID (9) controllers and for
differentvectors.Ascanseethechangesofintegral
ordershaveansignificantinfluenceonshipposition
andheadingchanges.
Figure5.ShiptrajectoryinDPsystemfordifferentintegral
orders(
1=[0.1,0.1,0.1],2=[1,1,1],3=[0.5,0.3,0.1]).
The parameters of classical PID controller were
tunedbyGAmethodandreceivedparameterswere
nextsitedforfractionalcontrollers.Thusonlyvector
ofintegratorordersanddifferentiatororderswere
finallyselected.Suchanapproachiscommonlyused
in the literature. The controller parameters were
collected in Table1. It
can be noted that FOPID
controllerdescribedbyequation(9)hasbeenreduced
toPDcontroller.
403
Table1.Setparameters
_______________________________________________
Parameter Value[‐]Controllers
_______________________________________________
Kp(10)^4*[2*2.021300;02*PID,
1.7009900;002000.49]FOPID(7),
K
d(10)^8*[0.020700;00.01550.0439; FOPID(9)
00.04394.05]
K
I(10)^2*[1.0127400;00.89020;
000.1278]
[1,1,0]FOPID(9)
[0.93333,0.7333,0.86667]FOPID(7)
[110]
_______________________________________________
Figures 6‐9 present time‐series of ship position
and heading, estimated and measured velocities,
forces and moments acting on a hull and ship
trajectoryinDPsystemwithcontrollerswith(6),(7),
(9)aftertuning(PID(6)‐blacksolidline,FOPID(7)‐
greyline,FOPID(9)‐blackdottedline).
0 50 100 150 200 250 300
-5
0
5
10
x-position , m.
0 50 100 150 200 250 300
-5
0
5
10
y-position , m.
0 50 100 150 200 250 300
-1
0
1
heading , deg.
Time, sec
Figure6.MeasuredandfilteredpositionandheadinginDP
system(PID(6)‐blacksolid line, FOPID (7)‐grey line‐
onlymeasured,FOPID(9)‐blackdottedline).
0 50 100 150 200 250 300
-0.1
0
0.1
surge velocity u, m/s.
0 50 100 150 200 250 300
-0.1
0
0.1
sway velocity v , m/s.
0 50 100 150 200 250 300
-1
0
1
x 10
-3
yaw velocity r, rad/s.
Time, sec
Figure7. Estimated and measured surge, sway and yaw
velocitiesin DP system(PID (6)‐blacksolid line,FOPID
(7)‐greyline,FOPID(9)‐blackdottedline).
0 50 100 150 200 250 300
-500
0
500
surge forces, KN.
0 50 100 150 200 250 300
-500
0
500
sway forces, KN.
0 50 100 150 200 250 300
-4000
-2000
0
2000
yaw forces, KNm.
Time, sec
Figure8. Forces and moments in surge, sway and yaw
directioninDPsystem(PID(6)‐blacksolidline,FOPID
(7)‐greyline,FOPID(9)‐blackdottedline)
0 1 2 3 4 5 6
0
1
2
3
4
5
6
Y-position, m.
X-position, m.
Figure9.ShiptrajectoryinDPsystem(PID(6)‐blacksolid
line,FOPID(7)‐greyline,FOPID(9)‐blackdottedline).
The computer simulations shown in Figure 6
present the convergence of position and heading to
their desired values. The time‐histories confirm a
good ability of all controllersto keep fixedposition
andheading.IncomparisonwithPID(6),theFOPID
(7),(9)methods gives close time quality coefficients
ofposition
andheadingsuch as rise time, and time
control.
Thesurge,swayvelocitiesandyawangle(Fig.7)
wereestimatedfromobserverconsidering(6)and(9)
control laws other than (7) where the information
about velocities was calculated from position and
heading introducing undesirable oscillations. Not
includingtheobserverin
theequation(7)isalsothe
reason of disturbed forces and moment signals
(Fig.8)and shiptrajectory(Fig.9) incontrastto the
smoothforcesandmomentsignalsreceivedfrom(6)
and(9)controllaws.
4 CONCLUSIONS
PIDcontrolmethodhasbeenawidelyusedcontrol
techniqueduetoits
practicalityandsuitabilityfora
largeclassofsystemswhicharelinearornonlinear.
Thepaperpresentcoverstheintegerordercaseand
thepreliminarystudiesextendsfortheuseofFOPID
techniqueinclassical(nofractionalorder)DPsystem.
Two cases were considered. The first case FOPID
includeonly
integratorbecauseoftheneedtousethe
404
information about estimated velocities from the
observer.Forthiscasewhenweusednonefractional
DP system with observer, the results have not
substantiallyimproved.Itcanbenotedthatfractional
controllerhasbeenreducedtoclassicalPDcontroller.
The second case assumed the DP system without
observer. For this
case two integrator and
differentiator were tuned, the results have been
worseincomparisonwithclassicalPIDcontroller.
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