International Journal
on Marine Navigation
and Safety of Sea Transportation
Volume 3
Number 4
December 2009
431
1 INTRODUCTION
The process of the ship movement steering can be
divided into several control subsystems, e.g. the
ship’s course and/or speed stabilization, damping of
roll angle, dynamic ship positioning (DSP), guidance
along trajectory etc. One of them is the control sys-
tem for precise steering of the ship moving with the
low and very low speed. Such kind of the vessel mo-
tion is also known as a crab movement. This regula-
tion process means the full control of velocities dur-
ing translation of the ship with any drift angle, e.g.
motion ahead, astern and askew or rotation in place.
No other help (tugs, anchors etc.) is required for this
process.
In the beginning, the precise steering systems
were installed as
extensions
of DSP units on research
ships, drilling vessels, cable and pipe laying ships
and similar ones. Nowadays these systems are mount-
ed on ferries, passenger ships, shuttle tankers, FSO
and dredging vessels (Fossen 2002).
The exemplary manoeuvres under such a steer-
ing are presented in Fig.1. It gives, among others, the
following advantages:
− the increasing safety of the vessel, especially
on constrained water with intensive traffic (har-
bours, navigation channels, closed or inner roads
etc.), owing to ability to perform e.g. a fast anti-
collision manoeuvre on very small area,
− the possibility of resignation of tugs cooperation
for e.g. berthing or mooring manoeuvres,
Figure 1: The exemplary situations when precise manoeuvres
during berthing are needed and expected.
− the ability to pass along very shallow and tortu-
ous navigation channels, inaccessible for ships
with conventional drivers e.g. near attractive
touristic places (islands, gulfs, fiords, etc.).
The H
2
and Robust H
inf
Regulators Applied to
Multivariable Ship Steering
W. Gierusz
Gdynia Maritime University, Gdynia, Poland
ABSTRACT: The main goal of this task was a calculation of the two multivariable regulators for precise
steering of a real, floating, training ship. The first one minimized the H
2
norm of the closed-loop system. The
second one was related to the H
inf
norm. The robust control approach was applied in this controller with the
usage of the structured singular value concept. Both controllers are described in the first part of the paper. De-
tails of the training vessel and its simulation model then are presented. The state model of the control object
obtained via identification process is described in the next section. This model with matrices weighting func-
tions was the base for creation of ’the augmented state model’ for the open-loop system. The calculation re-
sults of the multivariable controllers is also shown in this section. Several simulations were performed in or-
der to verify the control quality of both regulators. Exemplary results are presented at the end of this paper
together with final remarks.
432
For this purpose the ship has to be equipped with
at least a few driving devices like: main propellers,
tunnel thrusters, jet-pump thrusters, or azipods (a
blade rudder is useless in such operations). They al-
low to steer the ship in the manual manner, but it rare-
ly leads to satisfying results - therefore the multivari-
able controller seems to be a reasonable solution.
Figure 2: The block diagram of the multivariable ship control
system.
The regulation of three ship’s velocities:
surge, sway and yaw often needs the ’usage’ of only
one velocity at a time (see Fig. 1), therefore the con-
trol system should ensure complete or almost com-
plete decoupling steering of the ship.
The whole described system (see Fig. 2) consists
of three elements:
− the measuring subsystem,
− the multivariable regulator,
− the thrust allocation unit.
As it was pointed out the precise steering of the
vessel is performed with very slow velocities. The
standard navigation devices for measuring of mo-
tion parameters have poor accuracy in these work
conditions. Therefore ship’s velocities have to be
estimated (reconstructed) from position coordinates
and a value of the heading. The Kalman filters are
commonly used for this purpose (Anderson & Moore
2005).
The ship as a control object has very disadvanta-
geous features:
− the characteristics of the ship strongly and in
the nonlinear manner depend on operating condi-
tions e.g. the ship’s velocity, the direction of the
motion, ship load, water depth, proximity of other
ships, wharfs, etc.
− the allowance for all these factors in the model
is very difficult and even after it has been done it
leads to a badly complicated structure useless for
synthesis,
− the linearization of the model in many working
points gives a family of the models and the family
of regulators. Next it generates another problem
with the process of proper controllers shockingless
switching.
A control system designer has two main ways to
overcome these problems. One of them is matching
regulator to the real plant during the control process
i.e. adaptation of the control system - see for exam-
ple Astrom and Wittenmark books or (Niederlinski,
Moscinski & Ogonowski 1995). The second way is
evaluation of the bounds of the plant (ship) changes
and including them into the regulator synthesis pro-
cess (Skogestad & Postlethwaite 2003), (Zhou 1998).
The last approach is often named H
inf
robust con-
trol and requires a minimization of a process matrix
norm called H
inf
(Doyle, Glover, Khargonekar &
Francis 1989).
The matrix norms are very convenient ways for
formulation of performance criterions, especially in
multivariable systems. One can use two norms: H
inf
and H
2
. Controllers related to each norm are com-
monly named ’H
inf
regulator’ and ’H
2
regulator’.
The synthesis of both controllers for a ship is the ob-
jective of this paper.
2 THE H
INF
AND H
2
REGULATORS
2.1 Problem formulation
The feedback controller design can be formulated for
the general configuration
of the MIMO system shown
in Fig.3 (note opposite directions of signals - from
right to left hand side, more convenient for matrix op-
erations used in multivariable systems).
Figure 3: The block diagram of the closed-loop system with
weighting functions for selected signals. The meaning of the
particular signals is as follows: ˜r = references vector, ˜p - vec-
tor of disturbances, n˜ - noises vector, ˜ey - weighted control
errors, ˜eu - weighted control signals.
The concept of weighting functions is a conven-
ient way of introducing different signal specifications
into a MIMO process:
− the signals scaling operation is easy to perform by
means of this functions,
− one can distinguish between more and less im-
portant components of the signals vectors (e.g. in
errors vector) by proper gain coefficients, intro-
duced into these functions,
− the designer requirements related to the particular
signals can be formulated for specified frequency
ranges in a natural way.
433
Note the different sense of functions W
u
, W
s
on
the one hand and W
p
, W
n
, W
z
on the other one. Func-
tions matrices W
s
and W
u
define designer require-
ments for steering quality in the system while func-
tions matrices W
p
, W
n
and W
z
form input signals in
frequency domain. One can write the following
equations based on the Fig.3:
GuWpWWrWWe
spszSy
−−=
~~
(1)
uWe
uu
=
(2)
GuWnWpWrWv
snpz
−−−=
~~~
(3)
Kvu =
(4)
Above equations can be rewritten in more com-
pact form:
×=
u
n
p
r
P
v
e
e
u
y
~
~
~
(5)
where matrix P has the form:
−−−
−−
=
GWWW
W000
GW0WWWW
P
npz
u
spszs
(6)
Matrix P is called the augmented plant (model
plant) due to weighting functions vectors included in
it. Introducing the input vector
[ ]
T
nprd
~~~
=
and
the weighting error vector
[ ]
T
uy
eee =
one can
write:
×=
u
d
P
v
e
(7)
vKu ×=
(8)
The last equations
enable to build the generalized
configuration exposed in Fig.4.
Figure 4: The generalized closed-loop system configuration.
Now the weighting error vector can be expressed
in the form:
( )
dKP,Te
ed
×=
(9)
where matrix T
ed
can be obtained by means of the
Lower Linear Fractional Transformation (Redhef-
fer 1960).
The control system design can be treated as a
process of calculating a controller K such which
maintain small certain weighted signals (e.g. con-
trol errors). One of the possible way to define the
’smallness’ of signals (or transfer matrices) are ma-
trix norms H
inf
and H
2
(Skogestad & Postlethwaite
2003) expressed by the following equations:
( ) ( ) ( )
[ ]
( ) ( )
[ ]
ωσ
ωωω
π
ω
jTsT
jTjTsT
eded
ededed
∞∈
∞
+∞
∞−
=
×=
∫
,0
2
max
2
1
dtr
(10)
2.2 The H
2
regulator
The H
2
optimal control problem is to find a control-
ler K which stabilizes the closed-loop system (pre-
sented in Fig. 4) and minimizes the H
2
norm of this
system. The minimization of the H
2
norm is per-
formable only for strictly proper systems. When the
plant P is written in state model form:
×
=
u
d
x
DDC
DDC
BBA
v
e
x
22212
12111
21
(11)
the part D11 and D22 must be a matrices of zeros for
such a system.
The well-known LQG controller can be treated as
a special case of the H
2
regulator, when a weighting
factor in LQG performance criterion is included into
weighting function W
u
(Zhou 1998).
The regulator which minimizes the H
2
norm of
the system ensures the proper steering quality repre-
sented by the matrix weighting functions W
s
and/or
W
u
(see Fig.3), but under assumption that the plant
model is adequate and accurate.
2.3 The H
inf
regulator
The goal of H
inf
regulator is similar to that of the H
2
one, but now one wants to minimize the H
inf
norm
with the condition:
( )
ℜ∈><
∞
γγγ
,0 ,KP,T
ed
The value
γ
has the sense of the energy ratio be-
tween error vector e and exogenous input vector d.
When the
γ
tends to its minimal value the above
formulation is often named the optimal H
∞
control
problem
(Skogestad & Postlethwaite 2003).
The regulator which minimizes the H
inf
norm of
the system ensures similar quality of the steering for
434
any combinations of exogenous input signals
formed by matrix weighting functions W
p
, W
n
and
W
z
(note that this is not warranted by H
2
regulator).
2.4 The robust regulator
However this steering quality is only achieved under
the same assumption that the plant model is accu-
rate. If the real plant differs (e.g. due to operating
conditions) from the model used during controller
synthesis this quality can be significantly poor. The
differences between the object and the model are
usually named the system uncertainties (Doyle
1982).
There are several sources of uncertainties which
can be introduced into the ship model:
− changes the physical parameters of the vessel due
to different work conditions (e.g. load, trim, depth
of water, etc.),
− errors in estimation process for model coeffi-
cients values,
− neglected nonlinearities inside the object (e.g. re-
lated to hydrodynamics phenomena),
− measurement and filtration process errors (e.g.
biases),
− unmodelled dynamics, especially in the high fre-
quency range,
accepted (chosen) limitation of the model order.
All uncertainties can be divided into two classes:
parametric ones, related to the particular model coef-
ficients and others - nonparametric ones. Introduc-
tion of the concept of uncertainties into the model-
ling process means that one considers not only the
one nominal model of the object G
n
(jω), but a fami-
ly of models G
D
spread around this nominal model.
The uncertainties can be introduced into the sys-
tem model in different ways, depending on their
types and locations, but all of them are represented
by means of two components:
− the first one is the ”pure” uncertainty Δ, bounded
in the H
inf
norm sense i.e.
1≤∆
∞
− the second one it is the weighting function model-
ing the magnitude and shape of the uncertainty in
the frequency domain.
Consequently, any closed-loop system with un-
certainties contains three basic components: the gen-
eralized (augmented) plant P, the controller K that
has to be obtained and the set of ”pure” uncertainties
Δ, collected in the matrix form (see Fig.5).
Figure 5: The generalized closed-loop system configuration
with uncertainties.
The augmented plant P consists of the nominal
object model G
n
and of all matrices of weighting
functions (modeling the performance requirements,
forming input signals and describing the uncertain-
ties). Note that the augmented plant P for H
inf
con-
troller synthesis slightly different from this plant for
H
2
one.
2.5 The ship subsystems as a control object
The control object denoted Gn (see Fig.3) in the
considered system consists of four elements: the al-
location unit, thrusters set, the ship and the filters
system (Gierusz 2006). It has three inputs: two de-
manded forces
x
τ
and
y
τ
for longitudinal and lateral
directions of movement and one moment
p
τ
for
turning (in the ship-fixed frame) and three outputs:
estimated values of velocities surge
u
ˆ
, sway
v
ˆ
and
yaw
r
ˆ
(see Fig.6).
Figure 6: The block diagram of control object.
3 CASE STUDY
3.1 The training ship
The H
2
and H
inf
robust controllers was applied to
steer a floating training ship. The vessel named
’Blue Lady’ is used by the Foundation for Safety of
Navigation and Environment Protection at the Silm
lake near Ilawa in Poland for training of navigators.
It is one of the series of 7 various training ships ex-
ploited on the lake.
The ship ’Blue Lady’ is an isomorphous model of
a VLCC tanker, built of the epoxide resin laminate
in 1:24 scale. It is equipped with battery-fed electric
drives and the two persons control steering post at
435
the stern T˙ he silhouette of the ship is presented in
Fig.7.
The main parameters of the ship are as follows:
Length over all LOA = 13,78[m]
Beam B = 2,38[m]
Draft (average) - load condition Tl = 0,86[m]
Displacement - load condition Δl = 22,83[t]
Speed V = 3,10[kn]
The high-fidelity, fully coupled, nonlinear simu-
lation model of this ship was built for controllers
synthesis. Special attention was paid to the proper
modeling of the ship’s behaviour during movement
with any drift angle (e.g. astern or askew). The block
diagram of the model is presented in Fig. 8 (see
(Gierusz 2001) for detailed description of this mod-
el).
Figure 7: The outline of the training ship ”Blue Lady”
Figure 8: The block diagram of the ’Blue Lady’ simulation model. Input signals for the model are as follows (from top to bottom):
mean wind velocity - Vw , mean wind direction - γw , revolutions of the main propeller - ngc, blade rudder angle - δc, relative
thrust of the bow (stern) tunnel thruster - sstdc (sstrc), relative thrust of the bow pump thruster - ssodc, turn angle of the bow pump
thruster - αdc, relative thrust of the stern pump thruster - ssorc, turn angle of the stern pump thruster - αrc. The output signals of the
model are: surge - u, sway - v, yaw - r, position coordinates - x,y and the heading - Ψ.
436
3.2 The linear model identification
The synthesis processes of both controllers de-
scribed in this paper need a linear model of the ob-
ject. There are two ways to create it: a linearization
of a nonlinear (e.g. simulation) model of the vessel
dynamics or identification way. The second ap-
proach was used in presented work.
Every identification experiment was performed as
a simulation run in Simulink environment. More
than one hundred of experiments were performed for
this purpose (Gierusz 2006).
During identification process, it turned out, that
three subsystems demonstrated weak correlation be-
tween output and input signals
uu
pyx
→→→
ττυτ
,,
, therefore these subsystems
were canceled from the whole model (see Fig. 9).
Figure 9: Control object paths to be identified.
Finally, the third order state model was obtained.
The average values of coefficients, obtained in all
identification experiments were chosen as the values
of parameters of the nominal model Gn. Note values
of coefficients equal 0 in the channels cancelled dur-
ing identification process (see Fig. 9).
×
+
+
×
=
3
2
1
3
2
1
3
2
1
0
00
0
00
τ
τ
τ
υ
υυυ
υ
υυυ
rrrru
r
uu
rrrru
r
uu
bbb
bb
b
x
x
x
aaa
aa
a
x
x
x
(13)
=
3
2
1
100
010
001
x
x
x
x
r
u
υ
(14)
The resultant model is state controllable and ob-
servable - see (Gierusz & Tomera 2006) for details.
This model was used for H
2
controller synthesis.
For synthesis of the robust regulator five paramet-
ric uncertainties (denoted δ
i
, i = 1, ..., 5) were intro-
duced into the state model due to the wide range of
variations of parameter values acquired in various
experiments. This model had the form:
×
+
+
+
+
×
+
+
+
=
3
2
1
15
14
3
2
1
13
12
11
3
2
1
0
00
0
00
τ
τ
τ
δ
δ
δ
δ
δ
υ
υυυ
υ
υυυ
rrrru
r
uu
rrrru
r
uu
bbb
bb
b
x
x
x
aaa
aa
a
x
x
x
(15)
=
3
2
1
100
010
001
x
x
x
x
r
u
υ
(16)
The coefficients values of the state model of
the ship dynamics with values of uncertainties are
collected in the table 1 below.
Table 1: The values of model coefficients.
___________________________________________________
Wsp. Nominal Real Relative
value uncertainty value uncertainty value [%]
___________________________________________________
a
uu
-3.36*10
-3
2.64*10
-3
78
a
vv
-9.00*10
-3
5.00*10
-3
64
a
vr
-2.00*10
-4
a
ru
-3.00*10
-3
a
rv
-1.00*10
-3
a
rr
-7.75*10
-3
4.05*10
-3
52
b
uu
+3.62*10
-3
1.51*10
-3
42
b
vv
+2.06*10
-3
b
vr
+1.61*10
-5
2.89*10
-5
179
b
ru
+3.00*10
-5
b
rv
+1.15*10
-5
b
rr
+8.00*10
-3
___________________________________________________
3.3 The controllers synthesis
3.3.1 H
2
regulator
The state model, presented via equations (13) and
(14), could be arranged into ’augmented state model
of the open-loop process’ (Balas, Doyle, Glover,
Packard & Smith 2001), which was necessary to
compute the multivariable controller which mini-
mized H
2
norm.
The three tracking velocity errors e
u
, e
v
and e
r
were chosen as a performance criterion. It was as-
sumed that these expected errors would depend on
frequency of the reference signals. These require-
ments were transferred into the matrix of the
weighting functions W
s
for each velocity. The ma-
trix of the weighting function W
zad
was introduced
instead, to moderate the reference signals rate and
consequently to constrain the possibly large ampli-
tude of the steering signals.
The block diagram of model for this process is
presented in Fig. 10.
437
Figure 10: The block diagram of the augmented open- loop
process for H2 controller synthesis. Symbols denote: BL3 nom
- state model of the control object; Dop - adaptation matrix;
Wzad - filters for reference signals; Ws - weighting functions
for control perfor- mance. Numbers in parentheses denote sizes
of the signal vectors.
The synthesis of the regulator was made by
means of the algorithm named ’h2syn’ from ’µ
Analysis and Synthesis Toolbox’(see (Balas et al.
2001) for more details).
The computed regulator is of order 15:
( ) ( ) ( )
tvBtxAtx
15x3
r
15x15
r
×+×=
(17a)
( ) ( )
tvtx
x3
r
x15
r
×+×=
33
DC
c
τ
(17b)
The value of the closed-loop system H
2
norm was
12.14 and the value of the H
∞
norm was between
23.9365 and 23.9604. This last value means that the
H
2
controller is not a robust one for the described
sys- tem.
3.3.2 Hinf regulator
Apart from uncertainties related to changing
proper- ties of the plant, (see equations (15) and
(16)) two multiplicative, nonparametric uncertainties
were introduced to the presented ship control sys-
tem. The first one modelled inaccuracy in input sig-
nals (related to transmission errors) with the matrix
of weighting function W
wyk
, and the second one
modelled measuring and filtering errors in the output
plant with the matrix of weighting function W
pom
.
The state model of the control object with all
weighting functions was rebuilt into ’augmented
state model of the open-loop process’ much more
complicated then one presented in Fig.10:
Figure 11: The block diagram of the augmented open- loop
process. Symbols denote: Δ
1
- structured uncertainties block;
Δ
2
- input uncertainty with weighting functions Wwyk; Δ
3
-
measuring and filtering uncertainty with weighting functions
Wpom; BL3_nom- state model of the control object; Dop -
adaptation matrix; Wzad - filters for reference signals; Ws -
weighting functions for robust performance. Numbers in paren-
theses denote sizes of the signal vectors.
The algorithm named ’D-K iteration’ from men-
tioned Matlab toolbox was used to compute the ro-
bust Hinf controller for the system presented in
Fig. 11. The obtained regulator in state model form
was of high order equal to 41 - the same as the open-
loop system (with the scaling matrices D - see (Balas
et al. 2001) for the meaning of such matrices).
The value of H
inf
norm was 0.56 < 1 which en-
sures the robust property of the controller.
Therefore the order reduction procedures were
performed. Finally the controller of the order 21 was
obtained.
The regulator order seems to be quite high, but it
is worth to remember what the introduction of para-
metric uncertainties to the plant model is. It means
that the obtained controller should steer properly
(in weighing functions sense) the object which can
change its characteristic in a very wide range. There-
fore, the controller for such object should not be so
simple.
4 RESULTS ANALYSIS AND FINAL
REMARKS
The examination of both control systems was per-
formed during simulation runs with the ship’s non-
linear simulation model.
Every Figure is divided into two parts. The left-
hand side presents the results of the steering with the
H
2
controller and the right-hand side presents the
same trials performed with the robust regulator.
This example is illustrated by means of 3 Figures:
− the trajectory, drawn by ship’s silhouettes eve-
ry 60[s],
438
− ship’s velocities (reference signals and real val-
ues), supplemented by wind velocity runs (pre-
sented in Beaufort scale)
− command signals from the regulators.
The results were recalculated to start both trajec-
tories from point (0,0) and the initial heading was
chosen as 0 [deg].
One can compare the tracking errors for all veloc-
ities in all presented examples . The following for-
mula was used for this purpose:
( ) ( )( ) { }
∑
=
=−=
T
i
cq
ruqiqiq
T
J
1
2
,, ,
1
υ
(18)
where
c
q
reference signal for particular velocity,
q
ˆ
estimated value from Kalman filter,
T
= 1000, 1400, 2800 successively for first, second
and third example.
Figure 12: The trajectory of the ship in the first exam- ple
drawn by silhouettes every 60[s]. Initial heading ψ0 = 0[deg],
the trial period t = 1000[s]. An arrow indicate the average wind
direction.
Figure 13: The velocities of the ship in the first example - from
the top: surge, sway and yaw. The bottom figures present the
wind speed in Beaufort scale (recalculated in the ship model
scale 1:24). Solid lines denote real values, dashed lines - com-
mands.
Figure 14: The commands from controllers - from the top: for
surge - τ
x
, for sway - τ
y
and for yaw - τ
p
.
Figure 15: The trajectory of the ship in the second example
drawn by silhouettes every 60[s]. Initial heading ψ
0
= 0[deg],
the trial period t = 1400[s]. An arrow indicate the average wind
direction.
Figure 16: The velocities of the ship in the second example -
from the top: surge, sway and yaw. The bottom figures present
the wind speed in Beaufort scale (recalculated in the ship mod-
el scale 1:24). Solid lines denote real values, dashed lines -
commands.
439
Figure 17: The commands from controllers - from the top: for
surge - τ
x
, for sway - τ
y
and for yaw - τ
p
.
Figure 18: The trajectory of the ship in the third example
drawn by silhouettes every 60[s]. Initial heading ψ
0
= 0[deg],
the trial period t = 2800[s]. An arrow indicate the average wind
direction.
Figure 19: The velocities of the ship in the third example -
from the top: surge, sway and yaw. The bottom figures present
the wind speed in Beaufort scale (recalculated in the ship mod-
el scale 1:24). Solid lines denote real values, dashed lines -
commands.
Figure 20: The commands from controllers - from the top: for
surge - τ
x
, for sway - τ
y
and for yaw - τ
p
.
The comparisons are presented in the tables (val-
ues x 10
6
):
Example 1
Controller J
u
J
v
J
r
H
2
35 127 191
H
inf
1 73 29
Example 2
Controller J
u
J
v
J
r
H
2
369 3 660
H
inf
37 1 230
Example 3
Controller J
u
J
v
J
r
H
2
2650 205 3280
H
inf
920 109 2270
The similar calculations one can perform for con-
trol effort for both regulators using the formula:
( )( ) { }
∑
=
==
T
i
ss
pyxsi
T
J
1
2
,, ,
1
τ
τ
where
s
τ
control signal from regulator in the particular
channel,
T = 1000, 1400, 2800 successively for first, second
and third example.
The results are presented in the tables:
Example 1
Controller
x
J
τ
y
J
τ
p
J
τ
H
2
30 805 34
H
inf
1 728 1
440
Example 2
Controller
x
J
τ
y
J
τ
p
J
τ
H
2
393 7 225
H
inf
180 5 132
Example 3
Controller
x
J
τ
y
J
τ
p
J
τ
H
2
7720 723 1104
H
inf
5120 444 633
5 REMARKS
− The fully coupled, simulation model of the ship
with acceptable accuracy gives possibilities to
perform the identification trials instead of costs
and time consuming full-scale experiments. One
can the build the multidimensional linear model
and estimate the system uncertainties: their rang-
es and sources, based on the results from simula-
tion runs.
− The introduction of parametric uncertainties into
the plant model enables to cover the changes of
object characteristics (even nonlinear) in the all
range of assumed work conditions. On the other
hand it causes the increasing difficulty in the con-
troller synthesis.
− Very important advantage (or attribute) of both
regulators is its fixed structure and constant val-
ues of coefficients. It means that navigators do
not need to adjust any coefficients of these con-
trollers.
− The H
2
controller works worse than the robust
one. One can compare tables with results for con-
trol quality and steering effort. One of the main
reasons for such a steering can be the lack of the
robust properties of the regulator (see the H
inf
norm of this regulator).
− Both systems were tested in the presence of a
medium level of wind, in spite of fact that exter-
nal disturbances were not taken into account dur-
ing controllers synthesis processes. The robust
regulator still seems to be a better one in such
work conditions. The external disturbances one
can try to introduce into the controller synthesis
process but often no enough adequate regulator is
obtained (eg. without robust properties).
− As one can see in Fig. 12 - Fig.19, the steering is
almost de-coupling despite the full matrices B, C
and D in the controllers.
− The both closed-loop systems are stable under all
tested work conditions.
− The most important problems are related to yaw
steering (especially for H
2
controller). One of the
possible sources was the gyrocompass (with its
accuracy 0.2[deg]) and one was the fact that the
training ship is high weatherly.
− In general regulator calculated for one ship can
not be transferable to another one due to linear
object model specified for particular ship. It is a
similar situation like with PID controllers in
many industrial processes. But the possibility of
using a simulation model of the ship’s dynamics
instead a real ship for experiments for H
2
or H
inf
robust controller synthesis seems to be a great
advantage of described approach.
REFERENCES
Anderson, B. & Moore, J. (2005), Optimal Filtering, Dover
Publications, UK.
Balas, G., Doyle, J., Glover, K., Packard, A. & Smith, R.
(2001), μ-Analysis and Sythesis Toolbox. Ver.4, The
Mathworks Inc., Natick, USA.
Doyle, J. (1982), ‘Analysis of feedback systems with struc-
tural uncertainties’, IEE Proc., Part D 129(6), 242–250.
Doyle, J., Glover, K., Khargonekar, P. & Francis, B. (1989),
‘State-space solutions to standard H2 and H∞ control prob-
lems’, IEEE Trans. on Automatic Control 34(8), 831–847.
Fossen, T. (2002), Marine Control Systems, Marine Cybernet-
ics, Trondheim, Norway.
Gierusz, W. (2001), Simulation model of the ship handling
training boat ,,Blue Lady”, in ‘Int. IFAC Conference Con-
trol Applications in Marine Systems CAMS’01’, Glasgow,
Scotland.
Gierusz, W. (2006), ‘The steering of the ship motion - a μ-
synthesis approach’, Archives of Control Sciences 16/1, 5–
27.
Gierusz, W. & Tomera, M. (2006), ‘Logic thrust allocation ap-
plied to multivariable control of the training ship’, Control
Engineering Practice 14, 511–524.
Niederlinski, A., Moscinski, J. & Ogonowski, Z. (1995), Adap-
tive control (in polish), PWN, Warszawa, Poland.
Redheffer, R. (1960), ‘On a certain linear fractional transfor-
mation’, Journal of Mathematical Physics 39, 269–286.
Skogestad, S. & Postlethwaite, I. (2003), Multivariable Feed-
back Control - Analysis and Design, John Wiley and Sons,
Chichester, UK.
Zhou, K. (1998), Essentials of Robust Control, Prentice Hall,
Upper Saddle River, USA.
