41
1 INTRODUCTION
Nonlinearcontrolsystemsarecommonlyencountered
inmanydifferentareasofscienceandtechnology.In
particular,problemsdifficulttosolveariseinmotion
and/or position control of various vessels such as
drilling platforms and ships, sea ferries, container
ships, etc. Complex motions and/or complex‐shaped
bodiesmovinginthewater,andinthecaseofships
alsoatthebounda
rybetweenwaterandair,giverise
toresistanceforcesdependentinanonlinearwayon
velocities and positions, thus causing the floating
bodiestobecomestronglynonlineardynamicplants.
Ingeneral,therearetwobasicapproachestosolve
the control problems for nonlinear pla
nts. The first
one called “nonlinear” includes synthesizing a
nonlinear controller that would meet certain
requirementsovertheentirerangeofcontrolsignals
variability (Fabri & Kadrikamanathan 2001; Huba et
al.2011;Khalil2001;Tzirkel‐Hancock&Fallside1992;
Witkowska et al. 2007). The popular methods of
predictivecontrol(MPC)employnonlinearoron‐line
linearized models of the pla
nt (Maciejowski, 2002;
Rawlings & Mayne, 2009; Limon et al., 2005; Qin &
Badgwell, 2003). However, in the case of MIMO
nonlinearprocessessuchnonlinearcontrolalgorithms
aretoocomplexforcomputationstobeperformedon‐
line. Such ta
sks are particularly difficult when
additional constraints on the control signals are
considered, which demands using some numerical
procedures to solve optimization problems with
constraints.Whenanonlineardescriptionoftheplant
is not known accurately, predictive controllers
employing artificial intelligence, for example neural
Multivariable Adaptive Controller for the Nonlinear
MIMO Model of a Container Ship
M.Brasel&P.Dworak
FacultyofElectricalEngineering,WestPomeranianUniversityofTechnology,Szczecin,Poland
ABSTRACT: The paper presents an adaptive multivariable control system for a Multi‐Input, Multi‐Output
(MIMO)nonlineardynamicprocess.Theproblemsunderstudyareexemplifiedbyasynthesisofacourseangle
andforwardspeedcontrolsystemforthenonlinearfour‐Degrees‐of‐Freedom(4‐DoF)m
athematicalmodelofa
single‐screw, high‐speed container ship. The paper presents the complexity of the assumed model to be
analyzedandasynthesismethodforthemultivariableadaptivemodalcontroller.Duetoastronglynonlinear
natureoftheshipmovementsequationsamult
ivariableadaptivecontrolleristunedinrelationtochangeable
hydrodynamicoperating conditionsoftheship.Inaccordancewiththegivenoperatingconditionscontroller
parametersarechosenonthebasisoffourmeasuredauxiliarysignals.Thesystemsynthesisiscarriedoutby
linearizationofthenonlinearmodeloftheshipatitsnominaloperatingpointsinthesteady‐stateandbymea
ns
of a pole placement control method. The final part of the paper includes results of simulation tests of the
proposedcontrolsystemcarriedoutintheMATLAB/Simulinkenvironmentalongwithconclusionsandfinal
remarks.
http://www.transnav.eu
the International Journal
on Marine Navigation
and Safety of Sea Transportation
Volume 8
Number 1
March 2014
DOI:10.12716/1001.08.01.05
42
networks (Akesson & Tojvonen, 2006; Lawrynczuk,
2010;vanderBoometal.,2005)canbeused.
The second approach called “linear” consists in
designinganadaptivelinearcontroller with varying
parameters to be systematically tuned up
correspondingtochangingplantoperatingconditions
determined by system nominal operating points.
Here, linearization
of nonlinear MIMO plants is a
prerequisite for the methods to be employed. As a
results of the linearization local linear models are
obtainedandtheyarevalidforsmalldeviationsfrom
operatingpointsoftheplant.
Since properties exhibited by linear models at
different(distant)“operatingpoints”ofthe
plantmay
vary substantially the controllers used should be
eitherrobust(Ioannou&Sun1996)(usuallyofavery
highorderashasbeenobservedby(Gierusz2005))or
adaptivewithparametersbeingtunedintheprocess
ofoperation(Äström&Wittenmark1995).
If the mathematical description of the nonlinear
plant
is known, then it is possible to make use of
systems with linear controllers prepared earlier for
possibly all operating points of the plant. Such
controllers can create either a set of controllers with
switchableoutputsfromamongwhichonecontroller
designedforthegivensystemoperatingpoint(Bańka
et al. 2010a; Bańka et al. 2010b; Dworak &
Pietrusewicz 2010) is chosen, or multi‐controller
structuresfromwhichthecontrolsignalcomponents
are formed. One example is weighted means of
outputs of a selected controller group according to
Takagi‐Sugeno‐Kang (TSK) rules, i.e. with weights
beingproportionalto
thedegreeoftheirmembership
of appropriately fuzzyfied areas of plant outputs or
other auxiliary signals (Tanaka & Sugeno 1992;
Tatjewski 2007; Dworak et al. 2012a; Dworak et al.
2012b).
What all the above‐mentioned multi‐controller
structures, have in common is that all controllers
employed in these structures must
be stable by
themselves, in distinction to a single adaptive
controller with varying (tuned) parameters. This
meansthatsystemstrongstabilityconditionsshould
befulfilled(Vidyasagar1985).
InthepresentedpaperanadaptivemodalMIMO
controller with (stepwise) varying parameters in the
processofoperationisstudied.Thecontrollercanbe
physically realized as a multi‐controller structure of
modal controllers with switchable outputs. The
consideredadaptivecontrolsystemwill be designed
forallpossible“operatingpoints”oftheplant.Inthe
simulation studies a 4 ‐DoF nonlinear model of a
single‐screw high‐speed container vessel has been
usedas
anonlinearMIMOplant.Themaingoalofthe
paper is a synthesis of the course‐keeping adaptive
control system for a container vessel assuming two
controlledvariables:yawangleandforwardspeedof
theshiprelativetowater.
2 NONLINEARMODELOFACONTAINERSHIP
2.1 Shipdynamics
The considered
course‐keeping control system
structure has been studied by means of a 4‐DOF
nonlinear mathematical model of a container vessel
(Son&Nomoto1981,Fossen1994Thevesselis175m
long (L), 25.4m wide in beam (B) with an average
draughtof8.5m(H).Inorder
todescribemovements
of the ship two reference systems are defined. The
yaw angle and the ship position are defined in an
Earth‐basedfixedreferencesystem.Onthe contrary,
forceandspeedcomponentswithrespecttowaterare
determined in a moving system related with the
ship’sbodyandthe
axesdirectedtothefrontandthe
starboard of the ship with the origin placed in its
gravitycenter(G)(showninFig.1).
Designations for the linear and angular speed of
theship,intheconsidereddegreesofshipmotionare
as follows:
u
(surge velocity),
v
(sway velocity),
p
(roll rate) and r (yaw rate). Corresponding
designations of the position coordinates of the ship
are as follows:
o
x
(ship position in N‐S),
o
y
(ship
positioninW‐E),
(rollangle),
(yawangle).
Figure1.Ship’sco‐ordinatesystems.
General nonlinear equations of motion in surge,
sway,rollandyaw(Son&Nomoto1981,Fossen1994)
areasfollows:
.
xy
yxyyyy
xx yy xx
zz yy G
mmu mmvr X
mmv mmurm rmlpY
I
JpmlvmlurWGM K
IJrmvNYx
(1)
Here
m
denotes the ship mass. The
x
m ,
y
m ,
x
J ,
z
J denotetheadded massand addedmoment
ofinertiainthe
x
and
y
directionsandaboutthe
x
‐axes and
z
‐ axes, respectively.
x
I
and
z
I
denotemoment of inertia aboutthe
x
‐axes and z ‐
axes, respectively. Furthermore,
y
denotes the
x
‐
43
coordinatesofthecenterof
y
m
,whereas
x
l and
y
l
denote the
z ‐coordinates of the centers of
x
m and
y
m ,respectively.
G
x
isthelocationofthecenterof
gravity in the
x
‐axes, GM is the metacentric
heightand
W istheshipdisplacement.
The hydrodynamic forces
X
,
Y
and moments
K , N inaboveequationsaregivenas:
22
2
1
sin ,
uu vr vv rr
RX N
X
=X uu t T X vr X v X r
XcF
(2)
33
222 2
22
1cos,
vr p vvv rrr
vvr vrr vv v
rr r H N
Y=Yv Yr Y p Y Y v Y r
YvrYvr Yv Yv
Yr Yr a F
(3)
33
222 2
22
cos ,
vr p vvv rrr
vvr vrr vv v
rr r R H H N
N=N v N r N p N N v N r
NvrNvr Nv Nv
Nr Nr x ax F
(4)
33
222 2
22
1cos.
v r p vvv rrr
vvr vrr vv v
rr r H R N
K=K v K r K p K K v K r
KvrKvr Kv Kv
Kr Kr a zF
(5)
Here,therudderforce
N
F canberesolvedinto:
22
2
6.13
sin ,
2.25
R
NRRR
A
Fuv
L
(6)
where:
1
tan / ,
RRR
vu
(7)
2
18 / ,
RP T
uu kK J
(8)
32
,
R Rr Rrrr Rrrv
vvcrcrcrv
(9)
where:
/,
P
JuVnD (10)
0.527 0.455 ,
T
KJ (11)
2
cos 1 .
Ppppvpr
uvwvxrcvcr
(12)
Theremainingcoefficients and model parameters
usedintheequations(1)aregivenby(Fossen1994).
The actual speed of the vessel is designated as
22
Vuv
. The control signals of the nonlinear
MIMO model of the ship (1) are:
(rudder angle)
and
n (propellershaftspeed).
2.2 Actuatorsdynamics
Inordertosynthesizethecontrolsystem,thesteering
machinemodelbasedon(Fossen1994)isrepresented
bythefirst‐orderdynamicsystemwithtimeconstant
1.8
T sandgain 1K
,whereastheshaftmodel
isrepresentedbythelinearmodelwithaveragetime
constant
10.48
m
T s and gain 1
m
K . Therefore,
theactuatorsblockshowninFig.6canbedescribed
inthestate‐spaceform:
1111
11
() () ()
() (),
c
ttt
tt
xAxBu
yx
(13)
where:
11
0.556 0 0.556 0
,.
0 0.095 0 0.095
AB
Here
T
ccc
ttnt
u is a vector of
commanded control signals and
1
T
tt tnt
xu is a vector of control
signals.Inthesimulationsthefollowinglimitationsof
control signals are assumed: maximum speed of the
screw
max
160
n
rpm, maximum rudder angle
max
15
degandmaximumrudderangularvelocity
max
5
deg/s.
3 MULTIVARIABLEADAPTIVECONTROL
SYSTEM
The dynamic model of the container ship (1) can be
describedinthestate‐spacenonlinearform:
22
2
,
,,
t
t
x
f
xu
yg
xu
(14)
with the semi‐state vector
2
()tx defined as shown
inFig1.
2
T
tuvpr
x (15)
andoutputandcontrolsignalsdefinedas:
.
T
T
tut t
ttnt
y
u
(16)
In order to synthesize the control system the
result
ingmodelislinearizedinthenominaloperating
points of the ship, defined as
22
()
n
ttxx. The
nominalstatevectorofthemodel(1)inthenominal
operatingregimesisdefinedas:
44
2
0var.
T
nnnnn
tuv r
x (17)
Thevaluesofstatevariables(
n
u ,
n
v ,
n
r ,
n
)are
defined in the turning circle simulation tests carried
outinMATLAB/Simulinkforvariouscontrolsignals:
o
and
o
n .Therangeofchangesofthesesignalsis
asfollows:
15 15
o
degwiththeresolutionof
1degand
5 160
o
n rpmwiththeresolutionof5
rpm,whichresultsinasetof992operatingpoints.
Eachcombinationof the control signals and their
corresponding parameters of the ship movements:
n
u ,
n
v ,
n
r and
n
determines the nominal
operating point of the ship. The resulting functions
,
n
un
,
,
n
vn
,
,
n
rn,
,
n
n areshownin
Figures2,3,4and5,respectively.
Figure 2. The surge velocity in the nominal operating
points.
Figure3.Theswayvelocityinthenominaloperatingpoints.
Figure4.Theyawrateinthenominaloperatingpoints.
Figure5.Therollangleinthenominaloperatingpoints.
As a result of the linearization performed in the
whole range of the nominal control signals linear
state‐spacemodelsofthecontainershipareobtained:
22222
22 2
() [ () ] [ () ]
() [ () ],
nn
nn
tt t
tt
xAxxBuu
yyCx x
(18)
where:
22
22
11 12 14 15
21 22 23 24 25
T
31 32 33 34 35
T
2
41 42 43 44 45
64 65
T
11 21 31 41
T
2
12 22 32 42
00
0
0
(,) ,
0
001000
000 0
00
(,) ,
00
n
n
n
n
T
aa aa
aaaaa
aaaaa
aaaaa
aa
bbbb
bbbb
xx
uu
xx
uu
Afxu
x
Bfxu
u
C
22
T
T
2
100000
(,) ,
000001
n
n
xx
uu
gxu
x
45
with the entries
ij
a
and
ij
b
depending on the
values of surge velocity
n
u , sway velocity
n
v , yaw
angular velocity
n
r , roll angle
n
and control
signals
T
noo
n
u in the nominal operating
points of the container vessel. Now, the full state
vector
tx of the vessel can be taken as:
12
T
tt
xx .Therefore,thestatevectoroftheship
isasfollows:
.
T
t nuv prx (19)
Finally,thefulllinearizedmodelofthe container
vesselisdescribedbythematrices:
1
1
2
22
,, .
A0
B
A
BC0C
BA
0
(20)
The obtained linear models (20) with known
parametersarethe starting point forapplying many
known methods for linear multivariable control
system design. When the linear MIMO systems are
consideredmultivariablemodal(or possibly optimal
LQG/LQR)controllersareusuallydesigned.
In the case of non‐measurable state variables,
modal controllers used in the proposed control
system structure are mult
ivariable dynamic systems
withparametersdefinedintimedomainby:
() () ()
() () (),
rrrr
rr r
ttt
ttt
xAxBe
uCx De
(21)
where:
, , , .
rrrr
A
ABF LCB LC FD 0 (22)
Here,
F is the state feedback matrix related to
thestatevectorcomponentsoftheplantmodels,and
L
is the gain matrix of full‐order Luenberger
observers, which reconstruct the state vector of the
plant linear models (20). Synthesis of modal
controllers is based on the use of the various
techniquesofpoleplacementinstableregionsofthe
s‐plane. As it was shown in (Bańka et al. 2013) the
designed moda
l controllers may be calculated using
four methods: Eigenvalues Method (EM),
Eigenvectors Method (EVM), Polynomial Method
(PM) and Polynomial Matrix Equations Method
(PME) which in case of MIMO pla nt yield different
results for the same data taken for calculations. The
method we finally choose should depend on the
numerical condit
ions for the plant given, its local
linearmodelsaswellastheresultofcalculationswe
need.IntheEM,EVM,andPMmethodsthesynthesis
of MIMO modal controllers base on separately
finding the matrices
F
and
L
for which,
according to (22), their “standard” state‐space
equationshavebeenformulated.InthePMEmethod,
insteadofseparatelycalculatingthematrices
F
and
L
,thecontrollertransferfunction matrix is directly
obtained at one go by solving the Diophantine left
polynomial matrix equation. More particular details
on this subject may be found in (Bańka et al. 2013).
The controller presented in the paper has been
synthetizedwiththeuseofEVMmethod.
If strictly causal moda
l controllers based on the
full‐order Luenberger observers are selected then
designing performed directly in the time domain as
well as in s‐domain (without solving polynomial
matrix equations) leads to calculating the feedback
matrix
F
which places the closed‐loop system
matrixeigenvaluesin the desired locations on the s‐
plane and the weight matrix
L
of the full‐order
Luenberger observer for appropriately desired
observer poles. In the case of measurable state
variables it is sufficient to determine the state
feedback gain matrix
F in order to synthesize
modal control system in time domain. If the plant
model is described by matrices (20) the vector of
commandedcontrolsignalsisasfollows:
() () ,
cref o
ttuFxxu (23)
which shifts the poles of a linear plant model to
desired locations, which in our case are as follows:
[‐0.11,‐0.12,‐0.13,‐0.14,‐0.15,‐0.16,‐0.17,‐0.18].These
experimentally assumed values allow us to achieve
sufficiently fast dynamics of control system and
reduceoutputsignalsovershootsandcontrolsignals
sa
turation.Thereferencestatevector
ref
x
isdefined
as:
0 0000 .
T
ref ref ref ref
nux (24)
Here
ref
n is the reference shaft speed
corresponding to the reference surge velocity of the
shipin a steady‐state for
0
. The resultingset of
992 local controllers has been used to create
multivariable adaptive controller with stepwise
varyingparameters.Thiscontrolleristunedwithfour
measuredauxiliarysignalsincluding:surgeandsway
speedcomponentsoftheshipwithrespecttowateras
wellasyawrateandrollangleoftheship,whichare
shownin Figures2,3,4and5.The current nominal
operating point is determined by minimiz
ation of a
quadraticfunctional
:
n
J
2222
max max max max
,
n
uvr
uvr
J
(25)
where
u ,
v ,
r ,
are auxiliary signals
deviations from thevalues in the nominal operating
points, whereas
max
u ,
max
v ,
max
r ,
max
are the
maximum values of auxiliary signals in the whole
rangeofnominaloperatingpoints.
Theblock diagram of theproposed multivariable
adaptivecontrolsystemisshowninFig.6.Itconsists
of the state feedback matrix
F whose entries are
switched in a stepwise manner according to the
currentoperatingpointoftheship.
46
Figure 6. Block diagram of the proposed control system
structure.
If the state vector of the ship model (1) is not
measurable the state feedback matrix should be
replaced by an adaptive modal controller(21) based
on the Luenberger observer or the Kalman filter
(Bańkaetal.2013).
The stability of the above described closed loop
system with modal (gain
‐scheduled) controller has
beenprovedbytheuseofthestabilitytheoryof the
nonsmooth system given in (Shevitz and Paden,
1994),usedsuccessfullye.g.in(Lee
et al.,2001).
4 SIMULATIONTESTS
The usability of the propose control system is
illustrated with a multivariable adaptive control
systemforthenonlinearMIMOmodelofacontainer
vessel (1). The goal of the presented control system
was a simultaneous control of the course angle and
forward speed of the container
ship. Results of
simulations carried out in the MATLAB/Simulink
environmentarepresentedinFig.7and8.Theinitial
statevectoroftheshipwas:
0 0408.1400000 .
T
x (26)
which means that the ship goes forward with the
speedof8.14knots.Thefirstmaneuveratt=100swas
the change of the desired forward speed to 25.44
knots.Thenafter200sthedesiredcourseanglewas
changedto20
0
with keepingtheshipforward speed
at 25.44 knots. Both changes have been done
according to the assumed ship dynamics and all
maneuvershavebeendonewithacceptablevaluesof
the control signals: rudder angle and shaft speed,
presentedinFig.8.
Figure 9 presents values of indices i and
j which
denotethe current operating point.Changes of their
valuesshowmomentsinwhichthefeedbackmatrixF
entriesaremodified(switched).
5 CONCLUSIONS
In the paper an adaptive control system for the
nonlinearMIMOplantwasproposedandtested.The
utilized adaptive gain scheduling modal controller
allows one to
control a strongly nonlinear process,
herethemodelofacontainervessel.Thesynthesisof
the controller is based on the linearization of a
nonlinear ship model in operating points
corresponding to the set of 992 typical operating
regimes.Theadaptivecontrollerparametersvaryina
stepwise way on the
basis of auxiliary signals
measured during ship operation. The presented
example of multivariable control of the ship, shows
efficiencyof this methodandthe appropriateness of
its use to the direct control or as a part of more
complex control systems, e.g. a model loop in the
MFCcontrolstructure(Dworak
etal.2012b).
Figure7.Thecourseangleandspeed oftheship.
Figure8.Rudderangleandshaftspeed.
Figure9.MomentsofswitchingofthefeedbackmatrixF.
47
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