COMPARISON OF TWO PI METHODS
APPLIED TO FDI ON SHIPS DYNAMICS
Ramon Ferreiro Garc?a1, Manuel Haro Casado2, Francisco J. Velasco3
ABSTRACT
Most of nonlinear type one and type two control systems suffers from lack of
detectability when model based techniques are applied on fault detection and
isolation (FDI) tasks. This research is centred on frequency techniques applied
to identify ship?s model parameters (PI) including nonstructured or partially
known structured models using backpropagation neural networks as functional
approximators. The results of the comparison of two strategies based in frequen
cy techniques are presented. Such frequency techniques are:
? Mapping the frequency response associated to system parameters
when a closed loop controlled ship is excited by the wellknown harmonic bal
ance test (HBT).
? Mapping the frequency response associated to system parameters
when closed loop controlled ship is excited by a group of sinusoidal inputs added
to the manipulated variable (CLFRT).
With achieved frequency response mappings, system parameters are
associated by means of functional approximation techniques. In this case, Feed
forward neural networks trained with backpropagation conjugate gradient algo
rithm are massively used. Finally, PI results are used in FDI tasks, where nomi
nal plant parameters are matched against online estimated parameters on a par
ity space approach.
Keywords: Backpropagation, Conjugate gradient, Parameter identification,
Fault detection, Frequency response, Harmonic balance, Neural Networks.
JOURNAL OF MARITIME RESEARCH 21
1 Profesor de la E T Superior de N?utica y M?quinas, Universidad de La Coru?a, (ferreiro@udc.es).
2 Profesor de la Universidad de C?diz, (manuel.haro@uca.es).
3 Profesor de la E T Superior de N?utica, Universidad de Cantabria, (velasco@unican.es), .
Journal of Maritime Research, Vol. II. No. 3, pp. 2140, 2005
Copyright ? 2005. SEECMAR
Printed in Santander (Spain). All rights reserved
ISSN: 16974840
art. 2.qxp 18/02/2007 13:29 P?gina 21
INTRODUCTION
Safety in process industry can be strongly related to the detection and isola
tion of the features indicative of changes in the sensors actuators or process perform
ance. In using modelbased approaches, when the models describing the process are
accurate, the problem of fault detection may be solved by observertype filters. These
filters generate the socalled residuals computed from the inputs and outputs of the
process. The generation of these residual signals is the first stage in the problem of
fault detection and isolation (FDI). To be useful in the FDI task, the residuals must
be insensitive to modelling errors and highly sensitive to the faults under considera
tion. In that regard, the residuals are designed so that the effects of possible faults are
enhanced, which in turn increases their detectability. The residuals must also
respond quickly. The residuals are tested in order to detect the presence of faults.
Various FDI methods have been previously reported, such as the papers of Willsky,
A. S. (1976), Isermann, R. (1984), Frank, P. M. (1987a), Gertler, J. J. (1988), Patton,
R. J. and Chen, J. (1991). Among the classic books on the subject are those of Him
melblau, D. M. (1978), Pau, L. F. (1981), Basseville M. (1986) .
Model based fault detection methods
Fault detection methods based on process and signal models include actua
tors, processes and sensors for which inputs and output variables must be precisely
measured. Such methods deal mainly with parameter estimation, state observers and
parity equation methods. If measuring system fails, fault detection methods based
on the use of input/output measurements yields ambiguous and/or erroneous results.
A lot of research on model based fault detection methods has been carried out
during the last three decades. In this section a brief list on process model based fault
detection methods is given:
1.Fault detection with parameter estimation Gertler, J. J. (1988) Isermann, R.
(1992), Isermann, R. (1993), Mosler, O., Heller and R. Isermann (2001),
Newmann, D. (1991).
? Equation error methods
? Output error methods
? Frequency techniques
2. Fault detection with stateestimation.
(a) Dedicated observers for multioutput processes.
? State Observe, excited by one output Clark, R. N. (1978a):
? Kalman filter, excited by all outputs ] Mehra, R. K. and Peschon, J.
(1971):, Willsky, A. S. (1976),.
? Bank of state observers, excited by all outputs Willsky, A. S. (1976),
? Bank of state observers, excited by single outputs Frank, P. M. (1987a)
COMPARISON OF TWO PI METHODS APPLIED TO FDI ON SHIPS DYNAMICS
Volume II. Number 3. year 200522
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? Bank of state observers, excited by all outputs except one Frank, P. M.
(1987a).
(b) Fault detection filters for multioutput processes Beard, R. V. (1971).
3. Fault detection with parity equations Isermann, R. (1984), Gertler, J. J.
(1991):, Patton, R. J. and Chen, J. (1994):.
(a) Output error methods.
(b) Polynomial error methods.
4. Fault detection using analytical redundancy Ragot J., Maquin D., Kratz, F.,
2000.
(a) Static analytical redundancy.
(b) Dynamic analytical redundancy.
No general method exists for solving all FDI cases. Successful FDI applica
tions are based on a combination of several methods. Practical FDI systems apply
analytical redundancy using the socalled firstprinciples like actionreaction bal
ances such as mass flow rate balance, energy flow rate balance, force/torque/power
balances and commonly, the mathematical balance of any causeeffect equilibrium
condition.
As stated before, diagnosing techniques previously mentioned, when applied
to nonlinear type one and type two processes, suffers from lack of detectability.
With regard to residuals, they are the outcomes of consistency checks between the
plant observations and a mathematical model. The three main ways to generate
residuals are parameter estimation, observers and parity relations. For parameter
estimation, the residuals are the difference between the nominal model parameters
and the estimated model parameters. Derivations in the model parameters serve as
the basis for detecting and isolating faults.
In most practical cases the process parameters are partially not known or not
known at all. Such parameters can be determined with parameter estimation meth
ods by measuring input and output signals if the basic model structure is known.
There are two conventional approaches commonly used which are based on the
minimization of equation error and output error. The first one is linear in the param
eters and allows therefore direct estimation of the parameters (least squares) in non
recursive or recursive form. The second one needs numerical optimisation methods
and therefore iterative procedures, but may be more precise under the influence of
process disturbances. The symptoms are deviation of the process parameters. As the
process parameters depend on physically defined process coefficients, determination
of changes usually allows deeper insight and makes fault diagnosis easier Isermann,
R. (1984. These conventional methods of parameter estimation usually need a
process input excitation and are especially suitable for the detection of multiplicative
faults. Parameter estimation requires an input/output correct measuring system.
RAM?N FERREIRO GARC?A, MANUEL HARO CASADO, FRANCISCO J. VELASCO
JOURNAL OF MARITIME RESEARCH 23
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Some drawbacks of such methods are:
? the possibility of faulty measuring signals,
? an unknown model structure or
Goals to be achieved
Aforementioned diagnosing techniques, when applied to nonlinear type
one and type two processes, suffer from lack of detectability. For this reason the fol
lowing work will be oriented to the problem of fault detection, fault isolation, and
fault estimation by a novel parameter estimation method using residual generation
on the basis of parity space approach. The proposed parameter estimation method is
based on functional approximation techniques implemented with backpropagation
neural network (BPNN) even under a faulty measuring system.
The main tasks to be carried out are:
? Implementation of a fault tolerant data acquisition method by means of
frequency based techniques to achieve a consistent database
? Implementation of two PI methods based on the association of frequency
responses with functional approximation
? Comparison of both methods on a ship model steering process
Subsequent sections are devoted to the description of such technique, result
ing in an interesting complement or substitute to some conventional mentioned
techniques.
FREQUENCY BASED PARAMETER ESTIMATION TECHNIQUES
Introduction
This research work is focused on the problem of fault detection, fault isola
tion on the basis of parameter estimation by functional approximation implemented
with backpropagation neural networks associated to frequency techniques on non
linear type one and type two systems, for which serious problems with detectability
exist.
Conventional parameter estimation techniques are affective if measuring sys
tem operates free of faults. That means, measurement equipment operates without
drift errors. Consequently, when the possibilities of sensor drift errors exist, the
methods described below, in this section, are proposed.
Process Characteristics Based on HB Tests
The output of a nonlinear system under the effect of a disturbance or a step
input may go into a steadystate oscillation about an equilibrium point and still be
considered stable. Such an oscillation is called a limit cycle and is a periodic though
not sinusoidal oscillation whose amplitude and frequency are dependent only upon
COMPARISON OF TWO PI METHODS APPLIED TO FDI ON SHIPS DYNAMICS
Volume II. Number 3. year 200524
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the magnitude of the input and the characteristics of the system for non linear sys
tems and are dependent only upon the characteristics of the system for linear systems.
The ultimate amplitude and frequency are particular characteristics of all
transfer functions. HB is a procedure formerly used in process controllers autotun
ing which is an attractive technique for determining the real time ultimate frequency
and gain of a process. When stationary processes are under consideration, the tech
nique does provide a useful tool for the process parameter changes detection as
shown in this work. If a change in the values of ultimate frequency and ultimate
amplitude is observed, this means that some parameters of transfer function have
changed. The method requires a relay feedback or a closed loop controlled by a relay
around the setpoint as shown in figure 1.
Fig. 1. Structure of the HBT
With regard to figure 2, a relay of height h is inserted as a feedback controller.
The manipulated variable m is increased by h above the steadystate value. When the
controlled variable x crosses the setpoint, the relay reduces m to a value h below the
steadystate value. The system will respond to this ?bangbang? control by produc
ing a limit cycle, provided the system phase angle drops below 180?, which is true
for all real processes.
The period of the limit
cycle is the ultimate period
(Pu) for the transfer func
tion relating the controlled
variable x and the manipu
lated variable m. So the ulti
mate frequency is given as
(1)
A series Fourier expansion
of the relay output shows
GP
fu fx
u y x m
servo
xSP
RAM?N FERREIRO GARC?A, MANUEL HARO CASADO, FRANCISCO J. VELASCO
JOURNAL OF MARITIME RESEARCH 25
a
h
time
time
x2
xSP
x1
m2
m
Fig. 2. Input output signals of HBT
u
u P
?
?
2
=
art. 2.qxp 18/02/2007 13:30 P?gina 25
that the amplitude of first harmonic component is , and the error signal has the
amplitude
(2)
The condition for sustained oscillation (limit cycle) is that
(3)
Consequently the ultimate amplitude of the transfer function is given by
(4)
where
h = height of the relay,
a = amplitude of the primary harmonic of the output x.
It follows that if any change in system parameters takes place, then the ulti
mate frequency, ultimate amplitude, or both, will change also. Such concept can be
defined as a function of ultimate frequency and ultimate amplitude of primary har
monic of the output. This property is expressed as:
(5)
with Pi the system parameter set. So that, the condition to asseverate system param
eter invariance, which means to confirm that no parameter has changed is
(6)
where uN and auN are the nominal ultimate frequency and amplitude respectively
corresponding to the nominal parameters set PiN.
It should be noted that Eq. (5) and (6) give us approximate values for ?u and
au because the relay feedback introduces an additional nonlinearity into the system.
However, for most systems, the approximation is close enough for engineering pur
poses.
Nevertheless, when systems transfer functions are influenced by any auxiliary
or external variable, (variables different of the input/output of the transfer function),
uNuNiNu aPfa ,)(),( ?? ==
MiPfa iu L,1);(),( ==?
uK
h
a
?
4
=
?? ?=)(arg uiG
)(
1
u
u iG
K
?
=and
)(
4
uiG
h
a ?
?
=
COMPARISON OF TWO PI METHODS APPLIED TO FDI ON SHIPS DYNAMICS
Volume II. Number 3. year 200526
?
h4
art. 2.qxp 18/02/2007 13:30 P?gina 26
?
they should be taken into account. As consequence of the existence of such variables
(6) can be rearranged as follows:
(7)
If the relay of height h inserted as a feedback controller is externally forced to
change its eight to a new value, which means to change the manipulated variable,
then a different pair of ultimate period and amplitude is achieved. Such idea is
expressed as
(8)
Consequently, the application of (5) yields
(9)
Expression (8) states any pair of ultimate period and amplitude of the group
described by (9) is function of the complete set of plant parameters and related
external variables (coupling variables).
Process Characteristics Based on CLFRT
Frequency response is understood as the gain and phase response of a plant or
other unit under test at all frequencies of interest. Although the formal definition of
frequency response includes both the gain and phase, in common usage, the fre
quency response often only implies the magnitude (gain). In this study phase
response must be considered.
The frequency response H(f ) is defined as the inverse Fourier Transform of
the Impulse Response h(?) of a system.
(10)??
??
?
= ?? ?? dehfH fj2)()(
),(
),(
),(
),(
22
11
ii
iui
u
u
VPf
a
a
a
=
?
?
?
M
),(
),(
),(
2222
1111
iuiii
u
u
ahm
ahm
ahm
?
?
?
??
??
??
M
);,(),( iiu VPfa =?
RAM?N FERREIRO GARC?A, MANUEL HARO CASADO, FRANCISCO J. VELASCO
JOURNAL OF MARITIME RESEARCH 27
art. 2.qxp 18/02/2007 13:30 P?gina 27
Frequency response measurements require the excitation of the system with
energy at all relevant frequencies. The fastest way to perform the measurement is to
use a broadband excitation signal that excites all frequencies of interest simultane
ous, and use FFT techniques to measure at all of these frequencies at the same time.
Using random noise excitation best minimizes noise and nonlinearity, but short
impulses or rapid sweeps (chirps) may also be used. The selected excitation signal for
this study is of the type given as
(11)
with two or three relevant frequencies and same amplitude yielding for the case of
three relevant frequencies
(12)
Excitation function can be applied simultaneously or sequentially. Obviously when
simultaneously, the CLFRT is faster that sequentially but under noisy systems accu
racy is poorer.
When the desired resolution bandwidth of interest is less than about 100
kHz, the fastest way to measure the frequency response functions is to use FFT
based techniques as it is done in this work.
For proper measurement, it is also important to take into account the nature
of the type of signals that we are dealing with.
As a rule of thumb, if there is a given percent distortion or noise in the sys
tem, the error will be of the same order of magnitude. The output must be statisti
cally correlated to the input. This assumption is normally true in high fidelity analog
systems. However, in mechanical systems, as well as systems with complex transmis
sion mechanism and/or with digital encoding, echo cancelling, and other adaptive
techniques, this assumption may not be fulfilled. To account for all of the above, it
can be used digital signal processing techniques, including FFT and crossspectral
methods.
The output of a stable nonlinear system under the effect of a sinusoidal con
tinuous disturbance consists in a steadystate oscillation about an equilibrium point
and still be considered stable. Such an oscillation similar to a limit cycle is a periodic
though not sinusoidal oscillation whose amplitude G? and phase ?? is dependent
only upon the magnitude of the input and the characteristics of the system.
When stationary processes are under consideration, the technique does pro
vide a useful tool for the process parameter changes detection as shown in this work.
)()()()( 332211 tSinAtSinAtSinAtf ???? ++=
?
=
=
n
i
ii tSinAtf
1
)()( ??
COMPARISON OF TWO PI METHODS APPLIED TO FDI ON SHIPS DYNAMICS
Volume II. Number 3. year 200528
art. 2.qxp 18/02/2007 13:30 P?gina 28
If a change in the characteristic values of the frequency response (amplitude and
phase) is observed, this means that some parameters of transfer function has
changed. The method requires a sine generator added to a closed loop controller as
shown in figure 3.
Fig. 3. Structure of the CLFRT
It follows that if any change in system parameters takes place, then, the mag
nitude and phase will change also. This property is expressed as:
(13)
with Pi the system parameter set. So that, the condition to asseverate system parame
ter invariance, which means to confirm that no parameter has changed, is
(14)
where G?N and ??N are the nominal amplitude and phase respectively correspon
ding to the nominal parameters set PiN.
It should be noted that Eq. (13) and (14) give us approximate values for G?N
and ??N because the measuring system introduces an additional error into the system
which must not be relevant. However, for most systems, the approximation is close
enough for engineering purposes.
Nevertheless, when a system transfer function is influenced by any auxiliary
or external variable, (variables different of the input/output of the transfer function),
they should be taken into account. As consequence of the existence of such variables,
(13) can be rearranged as follows:
(15)),(),( ii VPfG =?? ?
NNi
GPfG ???? ?? ,)(),( ==
MiPfG i L,1);(),( ==?? ?
GP
fx
u y x
Controller
xS
( )? ? tSinA ?
fu
RAM?N FERREIRO GARC?A, MANUEL HARO CASADO, FRANCISCO J. VELASCO
JOURNAL OF MARITIME RESEARCH 29
art. 2.qxp 18/02/2007 13:30 P?gina 29
If a sinusoidal function of amplitude A inserted in parallel with a feedback
controller is forced to change its frequency to a new value, then a different pair of
amplitude and phase as frequency response is achieved. Such idea is expressed as
(16)
for identification purposes the amplitude of the excitation signal can be selected
such that A1 = A2 = ?.An yielding
Consequently, the application of (15) yields
(17)
Expression (17) states that any pair of amplitude and phase is function of the
complete set of plant parameters and related external variables (coupling variables).
Advantages of the methods
These methods has several distinct advantages over conventional parameter
estimation methods:
a) It doesn?t depend on the output measuring errors (drift of system output
sensors)
),(
),(
),
),(
22
11
ii
nn
VPf
G
G
G
=
??
??
??
?
?
?
M
),(,
),(,
),(,
222
111
nnn
GA
GA
GA
??
??
??
??
??
??
?
?
?
M
),(,
),(,
),(,
2222
1111
nnnn
GA
GA
GA
??
??
??
??
??
??
?
?
?
M
COMPARISON OF TWO PI METHODS APPLIED TO FDI ON SHIPS DYNAMICS
Volume II. Number 3. year 200530
art. 2.qxp 18/02/2007 13:30 P?gina 30
b) No a priori knowledge of the system parameters is needed. The method
automatically results in a sustained oscillation at the excitation frequency
of the process. The only parameter that has to be specified is the frequency
and amplitude of excitation signal or the relay eight for HBT.
c) They are closed loop tests, so the process will not drift away from the set
point. This is precisely why the methods works well on highly nonlinear
processes. The process is never pushed very far away from the steadystate
conditions
GENERAL PROCEDURE
To fulfil the requirements for training a feedforward backpropagation neural
network, a database for every PI method is needed. For the case of HBT, a database
relating ship parameters Pij and the associated pairs of (aUj, PUj) must be achieved.
Figure 4(a) shows the implementation of HBT to achieve the actual demanded data.
For the case of CLFRT, a database relating ship parameters with the magnitude and
phase responses are needed. Such pairs of ultimate values corresponding to actual
plant parameters are nominal ultimate values if, and only if, plant parameters are
nominal. In this situation it is assumed a fault free plant operation mode. Figure 4(b)
shows the implementation of CLFRT to achieve the actual demanded data
Fig .4. Tasks to achieve the necessary data to be used in NN training. (a), HBT. (b) CLFRT.
The simplest idea to identify only one parameter consists in applying a func
tional approximation technique based in the use of backpropagation neural networks
properly trained as shown in figure 5. Figure 5 illustrates the case of a plant with
known model structure and three (but could be any other quantity) accessible
(known by any means) parameters P1, P2 and P3. It shows a ship in which an HBT is
HBT PUN
aUN
PiN
Ship Database
CLFRTT G?
??
PiN
Ship & controller Database
? A.Sin(?t)
(a)
(b)
RAM?N FERREIRO GARC?A, MANUEL HARO CASADO, FRANCISCO J. VELASCO
JOURNAL OF MARITIME RESEARCH 31
art. 2.qxp 18/02/2007 13:31 P?gina 31
executed. As consequence of applied HBT, a database is filled with achieved data.
With the recent data contained into the database, a training session is performed
and a NN based model for the patter parameter is achieved. The same sequence can
also be performed under CLFRT in the same order that for HBT.
Figure 6 shows the scheme
adopted for comparison pur
poses on the basis of parame
ter estimation, which consists
in a group of trained neural
networks, ready to identify
one or more plant parame
ters, w hen real time or actual
data is applied to the inputs.
So that, the tasks necessary to
identify at least a plant
parameter, requires again the
on line HBT or CLFRT
tasks to obtain the actual pair
of ultimate gain and period or
alternatively, the frequency
response data. By introducing
such actual values, including
the rest of known parameters
to the neural network inputs,
it yields at the output, the
actual value of the plant
unknown parameter. The
accuracy in the value of the
estimated parameter is crucial
because it will be straight
away applied on the last
phase of the FDI task.
APPLICATION TO PI USING
A NOMOTO MODEL
In order to validate the PI
methods by comparison of
model parameters accuracy,
one of the simplest models of a ship is selected Ferreiro Garc?a R., Haro Casado
M.(2006),. The transfer function from rudder angle ? to heading ? is described for
our purposes under a Nomoto model as
COMPARISON OF TWO PI METHODS APPLIED TO FDI ON SHIPS DYNAMICS
Volume II. Number 3. year 200532
Database containing plant
parameters and results of HBT
NN
P2
P3
Pu
au P1
Target
data
Backpropagation
NN training phase
Pattern
data
P1 P2 P3 Pu au
HBT
Ship
CLFRT
Fig. 5 . Neces
sary tasks to
achieve a neural
network ship
parameter esti
mator.
Trained
NN.
P1
P2
P3
Pu
au
P?1
FDI
algorithm HBT
Vi
Trained
NN.
P1
P2
P3
G?
??
P?1
FDI
algorithm CLFRT
Ship
Vi
Comparison analysis
Fig. 6. Scheme of paramostic tasks and comparison of PI
methods.
art. 2.qxp 18/02/2007 13:31 P?gina 32
(18)
where according with ?strom and Wittenmark (1989), model parameters can be
approached as
(19)
with l the ships length in m, u the ship velocity in m/s, A the rudder area in m2, and
D, the ship?s displacement in m3. According described Nomoto model, the parame
ters a and b depends on the ship velocity u and ship displacement D. The rudder
servo operates with a speed of 4 degrees/s limited to ? 30 degrees. The ruder area is
assumed as 20 m2. Consequently, the simulation model necessary to validate both
the parameter identification procedures is of the type
(20)
where the servo model is shown in figure 7.
Fig. 7. Simplified rudderservo model
The closed loop controller is an adaptive PID adjusted by gain scheduling to
adapt its parameters as function of ship displacement under the criterion of mini
mum overshoot and response time. According such requirements, the PID parame
ters are shown in able I.
)(
.0
)(
0
2
l
u
ass
D
Al
l
u
sG
+
??
???
?
=
5
l
u
aa 0=
D
Al
l
u
b
2
.0 ??
???
?
? 5and,
)()(
)(
)(
ass
b
s
s
sG
+
==
?
?
RAM?N FERREIRO GARC?A, MANUEL HARO CASADO, FRANCISCO J. VELASCO
JOURNAL OF MARITIME RESEARCH 33
s
4
?SP ?

art. 2.qxp 18/02/2007 13:31 P?gina 33
Table I. PID parameters as function of displacement
Training procedure
Both PI methods are then applied by simulation on the closed loop model,
and consequently, two databases shown in tables II and III were achieved. Input data
to achieve the necessary database uses ship velocities from 4 to 10 m/s while ship
displacement varies from 4000 to 10000 tons on the container ship, where rudder
area is 20 m2.
Table II. Database for CLFRT method
D/103 4 6 8 10
PID
Kp 4 3.4 2.9 2.6
Ti 120 135 160 190
Td
COMPARISON OF TWO PI METHODS APPLIED TO FDI ON SHIPS DYNAMICS
Volume II. Number 3. year 200534
u D/103 G?1 ??1 G?2 ??2
4 4 0,0801045 68,0906 0,0416869 89,5841
4 6 0,0815297 73,8223 0,0415882 95,779
4 8 0,0820465 78,9383 0,0407358 103,086
4 10 0,0813522 83,8892 0,0394238 109,301
6 4 0,07857 63,2443 0,0413649 81,0538
6 6 0,0788952 65,1473 0,041328 84,9855
6 8 0,0793861 67,5302 0,0413907 87,3135
6 10 0,0797526 70,0747 0,041473 90,4957
8 4 0,0779755 61,4826 0,041303 79,5276
8 6 0,0781531 63,0128 0,0413122 80,7345
8 8 0,0784074 63,4854 0,0412631 81,2487
8 10 0,0786551 64,0515 0,0411353 84,4931
10 4 0,0778708 59,4287 0,0414477 76,4577
10 6 0,0777866 61,1995 0,0412255 79,3707
10 8 0,0778946 62,6008 0,0412056 80,3609
10 10 0,0779463 63,2211 0,0411875 80,8526
art. 2.qxp 18/02/2007 13:31 P?gina 34
The data of such consistent databases achieved on the basis of Nomoto model
is used in backpropagation neural network training phase. The training algorithm
selected is the conjugate gradient FletcherReeves implemented on the Neural Net
work toolbox of Matlab under offline training sessions. Consequently, the selected
neural network architecture is defined by means of the Matlab expression:
net = newff(minmax (p), [10, 10, 1], {?tansig?,?tansig?,?purelin?},?traincgf?)
which consists of a feedforward Backpropagation NN with two hidden layers and
ten neurons per layer, trained by means of ,?traincgf ? algorithm of Matlab.
After several training sessions with different data structures from the data
base, some of the training results are shown in table IV.
Mean Square Error (MSE) is an acceptable performance index to evaluate the
estimates accuracy of training results as shown in table I. By comparing the values of
rows corresponding to the index MSE, some differences are observed, which indi
cates the degree of accuracy that could
be expected when online parameter
estimation method is applied. For
CLFRT, MSE = 0.0989 and for HBT
MSE = 0.12088. Consequently,
CLFRT procedure is expected to be
more effective than HBT.
Some results
Figure 8 shows both PI meth
ods ready to be used on parameter
identification. The results of a PI ses
sion shows that the accuracy of
parameters achieved by using
CLFRT is better than using the
HBT, according with the MSE value
of table I. Consequently, with accept
able parameter identification values,
the residuals of the parity space
RAM?N FERREIRO GARC?A, MANUEL HARO CASADO, FRANCISCO J. VELASCO
JOURNAL OF MARITIME RESEARCH 35
u D/103 Pu a
4 4 88.2353 0.181148
4 6 93.75 0.163727
4 8 93.75 0.109534
4 10 88.2353 0.0852799
6 4 68.1818 0.338723
6 6 75 0.270966
6 8 75 0.18365
6 10 75 0.157505
8 4 62.5 0.399597
8 6 62.5 0.282363
8 8 62.5 0.233929
8 10 62.5 0.174988
10 4 55.5556 0.623681
10 6 55.5556 0.351874
10 8 55.5556 0.290567
10 10 55.5556 0.245539
Table III. Database for HBT method
Training algorithm Epochs MSE Gradient Test
Traincgfsrchcha 100/100 0.0989 5.426e6 CLFRT
Traincgfsrchcha 100/100 0.1209 2.979e6 HBT
Table IV. Training characteristics
art. 2.qxp 18/02/2007 13:31 P?gina 35
approach are achieved in the FDI tasks. FDI task is then carried out using simple
rules in a rule based task. Residuals necessary to implement the last phase of FDI
task, are achieved by comparing the actual ship velocity with the estimated one. As
consequence of such comparison, display 2 and display 3 of figure 8 shows respec
tively ?0.1147 and 0.09682. Under the assumption of a fault free process, then
CLFRT method is more accurate than HBT.
Fig. 8. Implementation of both PI methods for comparison purposes.
CONCLUSIONS
The aim of this research was to compare both PI strategies which consists in
develop and implement a method to detect and isolate faults on the basis of parame
ter variation detection on a ship even when under faulty measuring systems (output
sensor drift). Furthermore, the method is focused towards plants of type 1 and type
2 where conventional PI methods are not quite effective.
The most relevant advantages of these strategies are:
? PI method when applying HBT and CLFRT doesn?t depend on the qual
ity of measuring system in cases of sensor drift.
? The online test can be applied with the ship operating under nominal
setpoints, without disturbing or interrupting the ship course for instance.
COMPARISON OF TWO PI METHODS APPLIED TO FDI ON SHIPS DYNAMICS
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? The strategy is useful under partially known model structures.
As a result of comparison of both methods, CLFRT is more accurate than
HBT, and hence, FDI task is more reliable.
The most relevant disadvantages of both frequency based methods are due to:
? The time necessary to estimate the parameters increase with both, the
accuracy and number of parameters.
? The computational effort increase with the required accuracy and the
number of parameters to be estimated.
ACKNOWLEDGMENTS
The author wishes to acknowledge the financial support of the Spanish MICYT
and FEDER Funding at DPI200300512 project
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COMPARISON OF TWO PI METHODS APPLIED TO FDI ON SHIPS DYNAMICS
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COMPARACI?N DE DOS M?TODOS DE
IDENTIFICACI?N DE PAR?METROS APLICADOS
A LA DETECCI?N Y AISLAMIENTO DE FALLOS
EN LA DIN?MICA DE BUQUES
RESUMEN
La mayor parte de los sistemas no lineales de los tipos uno y dos sufren de escasez
de detectabilidad al aplicar m?todos de detecci?n y aislamiento de fallos basados
en modelos. Este trabajo est? centrado en t?cnicas frecuenciales aplicadas para
identificar par?metros del modelo de un buque incluyendo modelos no estruc
turados o parcialmente estructurados, utilizando aproximadoRes funcionales
basados en redes neuronales por propagaci?n hacia atr?s. Se presentan los resul
tados de la comparaci?n entre dos estrategias de identificaci?n basadas en t?cni
cas frecuenciales. Tales t?cnicas frecuenciales consisten en:
?Mapear la respuesta frecuencial asociada con los par?metros del buque
cuando al aplicar el m?todo del balance arm?nico en lazo cerrado (HBT).
?Mapear la respuesta frecuencial asociada con los par?metros del buque
cuando se a?ade una se?al de excitaci?n a la salida del controlador por reali
mentaci?n (CLFRT).
Las citadas respuestas frecuenciales acumuladas en una base de datos aso
ciadas a los par?metros del buque, proporcionan los aproximadores funcionales
para la identificaci?n de par?metros. Se utilizan masivamente redes neuronales
artificiales entrenadas por propagaci?n hacia atr?s mediante el algoritmo del
gradiente conjugado. Finalmente, los resultados de las tareas de identificaci?n de
fallos son utilizados en la detecci?n y aislamiento de fallos en l?nea y los resulta
dos de ambos m?todos de identificaci?n son comparados entre s? para establecer
baremo de calidad.
INTRODUCCI?N
Este trabajo comienza describiendo los antecedentes de los m?todos de detec
ci?n y aislamiento de fallos en base a la estimaci?n de par?metros. A continuaci?n se
describen dos m?todos frecuenciales (HBT y CLFRT) de estimaci?n de par?metros
del buque en base a la utilizaci?n de redes neuronales como aproximadotes funcionales.
Seguidamente, se realiza una aplicaci?n de la identificaci?n de par?metros utilizando
los m?todos frecuenciales propuestos con el modelo de Nomoto y posteriormente se
comparan los resultados de ambos m?todos para establecer un baremo de calidad.
RESULTADOS
La figura 8 muestra ambos m?todos de identificaci?n de par?metros listos
para identificaci?n. Los resultados de una sesi?n de identificaci?n muestran que la
RAM?N FERREIRO GARC?A, MANUEL HARO CASADO, FRANCISCO J. VELASCO
JOURNAL OF MARITIME RESEARCH 39
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precisi?n en la determinaci?n de los par?metros conseguida mediante el m?todo
CLFRT es mejor que la conseguida mediante HBT, de acuerdo con el criterio MSE
(mean square error) de la tabla I. En consecuencia, se aplican las t?cnicas de detec
ci?n y aislamiento de fallos o anomal?as con valores aceptables de identificaci?n de
par?metros que proporcionan los correspondientes residuos dentro del espacio de
paridad. La tarea de detecci?n y aislamiento es llevada a cabo por medio de razon
amiento basado en reglas. Los residuos necesarios para implementar la ?ltima fase de
aislamiento de anomal?as se llevan a cabo comparando la velocidad real del buque
con la velocidad estimada por el modelo dependiente de los par?metros estimados.
Como consecuencia de tal comparaci?n, se han obtenido resultados de los residuos
de 0.1147 para HBT y 0.09682 para CLFRT, constatando que es de mayor calidad
el m?todo de CLFRT frente a HBT, bajo la suposici?n de un ensayo en condiciones
nominales o libre de fallos.
CONCLUSIONES
El objetivo de este trabajo consist?a en comparar ambos m?todos de identifi
caci?n de par?metros posteriormente utilizados para la detecci?n y aislamiento de
anomal?as en base a modelos, a?n ante fallos de medida de los sensores de salida
debidos a desv?os de la misma (drift).
Adem?s, el m?todo est? enfocado hacia plantas de los tipos uno y dos donde
exhiben inherentemente falta de detectabilidad. Tal, es el caso de los buques.
Las ventajas m?s relevantes de tales estrategias son:
? Los m?todos de identificaci?n de par?metros basados en t?cnicas frecuen
ciales como HBT y CHFRT no dependen de la calidad de la medida
cuando ?sta est? afectada de desv?o constante.
? Las pruebas en l?nea pueden ser aplicadas con el buque operando en
condiciones normales y dentro del punto de consigna nominal sin pertur
bar sus tareas de operaci?n.
? Estas estrategias pueden ser utilizadas con estructuras de modelos parcial
mente conocidas
Como resultado de las comparaciones de calidad de ambos m?todos, se tiene
que CLFRT es m?s preciso que HBT y por consiguiente, las tareas de detecci?n y
aislamiento resultan mas eficaces con el m?todo CLFRT
Las desventajas mas relevantes de tales t?cnicas de identificaci?n se enumer
an a continuaci?n como:
? El tiempo necesario para estimar los par?metros del modelo aumenta con
la precisi?n requerida y el n?mero de par?metros a determinar
? El esfuerzo computacional aumenta con la precisi?n requerida y el
n?mero de par?metros a ser estimados
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