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2 PARTICLE SWARM
OPTIMIZATION
The PSO methodology has been recently introduced
in the fied of Evolutionary Computing by (Kennedy
and Eberhart, 1995). The main idea is to use
mechanisms found by studying the flight behaviour
of bird flocks (Heppner and Grenander, 1990). The
method may be used to solve optimization problems
using the next analogy. If a roosting area is set, then
the birds will form flocks and will fly towards this
area, “landing” when they arrived there. The
roosting area may be seen as an optimal or a near-
optimal solution in the search space. The birds may
represent points in the search space that will move in
time towards this solution. The search process is
guided by an objective function and each point is
able to evaluate the value of this function (the
fitness) for its current location. The movements of
the points during search will no longer resemble the
move of the birds in a flock, but rather the
movement of the particles in a swarm. There are two
mechanisms that are employed during this
exploration of the search space. First, each point in
the swarm memorizes the best location (in terms of
fitness) he ever passed through. Second, each point
is aware of the best location that the whole swarm
ever passed through, i.e. the global best location.
The new location of a particle is computed as
follows. Using the vector notation from Physics, the
direction vector of each particle is updated using the
vectors that point from the current location towards
the two previously mentioned locations. The search
process stops when all other points draw closer than
a very small given distance to one point. This point
is considered to be the solution of the optimization
problem.
The variant of PSO used in this paper starts from
a set of points around origin in the parameters m-
dimensional space mentioned in Subsection 2.1.3.
Fortunately, the probability to find points with large
fitness around origin is very high (see Table 1). This
means that it is very likely that the search process
starts with particles found very close to optimal
solutions. The exploration of the search space
follows the rules discussed above. The stop
condition is modified as follows. It was noticed that
if the global best location does not modify for a
relatively small number of iterations then this
location is a optimal solution. On the basis of this
observation, the search process will be stopped if the
global best location does not change for 3 iterations.
Using the analogy above, if the roosting is found
then the global best location will not further modify
and, therefore, there is no reason to wait until all the
birds landed.
3 CASE STUDY
The DAMADICS benchmark flow control valve was
chosen as the case study for this method. More
information on DAMADICS benchmark is available
via web, http://www.eng.hull.ac.uk/research/control
/damadics1.htm. The valve was extensively modeled
and a MATLAB/SIMULINK program was
developed for simulation purposes (Sa da Costa and
Louro, 2003). The data relative to the behavior of
the system while undergoing a fault was generated
using as inputs to the simulation real data, normal
behavior and some faulty conditions, collected at the
plant. This method provides more realistic
conditions for generating the behavior of the system
while undergoing a fault. It also makes the FDI task
more difficult because the real inputs cause the
system to feature the same noise conditions as those
in the real plant. However the resulting FDI systems
will deal better when applied to the real plant.
The system is affected by a total of 19 faults. In
this paper only the abrupt manifestation of the faults
has been considered. A complete description of the
faults and the way they affect the valve can be found
in (Louro, 2003).There are several sensors included
in the system that measure variables that influence
the system, namely the upstream and downstream
water pressures, the water temperature, the position
of the rod, and the flow through the valve. These
measurements are intended for controlling the
process but they can also be used for diagnosis
purposes, which means that the implementation of
this sort of system will not imply additional
hardware. Two of these sensors, the sensor that
measures the rod position (x) and the sensor that
measures the flow (F) provide variables that contain
information relative to the faults. The difference dP
between the upstream pressure (P1) sensor
measurement and the downstream pressure (P2)
sensor measurement is also considered (besides F
and x) as it permits to differentiate F17 from the
other faults. For the rest of the faults, the previous
difference has always negligible values (close to
zero).
The effects of six out of the 19 faults on this set
of sensor measurements are not distinguishable from
the normal behaviour, {F4, F5, F8, F9, F12, F14}.
So, in the following, these cases are not studied.
They can be dealt with if further sensors are added
to the system. Also, there can be distinguished three
groups of faults, {F3, F6}, {F7, F10}, and {F11,
F15, F16}, that share similar effects on the
measurements. Due to the large overlapping, a fault
member in one of the previous groups can be easily
mistaken with faults in the same group. This
problem is solved in recent studies by using a hybrid
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