Biomedical Engineering Reference
In-Depth Information
For.a.general.control.problem,.it.is.desirable.to.obtain.the.optimal.results.
for.different.system.performance..Consider.a.step.input.
r t
)
(
.and.the.output.
response.
y t
)
(
;.the.task.is.to.optimize.the.following.objectives:
.
1.. Minimize.the.maximum.overshoot.of.the.output
Obj
=
max ( )
y t
1
t
.
2.. Minimize.the.settling.time.of.the.output
( )
Obj
2
= ,.such.that.
0.98
t
s
r
≤
y t
≤
1.02 ,
r
∀ ≥
t
t
s
.
.
3.. Minimize.the.rising.time.of.the.output
(
1
)
(
2
)
Obj
=
t
=
t
−
t
,.such.that.
y t
=
0.1
r
.and.
y t
=
0.9
r
.
r
3
2
1
.
5.7.3.2 Results and Comparisons
Referring.to.the.simulation.cases.given.in.Section.4.5.4,.the.primary.sche-
mata.competition.(PSC).sets.cannot.reach.their.equilibria.in.a.population.
with. limited. size.. Instead,. the. proportions. of. schemata. fluctuate. around.
the. equilibrium. point,. and. it. is. known. that. the. deviation. caused. by. the.
cut-and-paste.operation.is.smaller.than.that.caused.by.the.copy-and-paste.
operation.
MOGAs. with. and. without. JG. were. compared;. 500. iterations. were. run.
for. both. algorithms,. and. the. population. size. was. ixed. at. 300.. The. chro-
mosome. length. was.
L
=. 96;. the. number. of. order.
n
. PSC. sets. was.
( )
. for.
L
n
= 1, 2, ⋯,
L
.
Since. it. is. impossible. to. trace. the. high-order. sets. due. to. their. huge. size,.
only.order.1.and.order.2.PSC.sets.were.analyzed..Figures 5.6.and.5.7.show.
the.mean.absolute.errors.from.the.equilibria.of.all.order.1.and.order.2.PSC.
sets,. respectively.. Their. means. and. variances. over. the. last. 100. generations.
were.calculated.and.are.shown.in
.
Table 5.16
..
It.can.be.easily.noted.that.the.
deviation.from.equilibrium.for.the.cut-and-paste.operation.was.the.smallest,.
which.agrees.with.the.results.presented.in.Section.4.5.4;.that.is,.a.more.even.
distribution.of.schemata.was.promoted.using.the.JG.in.a.GA.
Figure 5.8
.
shows. the. obtained. set. of. all. rank. 1. solutions. for. the. control.
problem;.
Figure 5.9.
gives.the.projections.on.different.two-dimensional.(2D).
planes.. Whether. the. JG. is. embedded. in. an. MOGA. or. not,. the. MOGA. can.
always. ind. some. optimal. solutions,. which. demonstrates. the. great. search-
ing. ability. of. the. MOGA.. However,. the. effect. of. using. the. JG. is. obvious..
Additional.different.optimal.solutions.can.be.located.by.embedding.the.JG.
into.the.MOGA,.which.gives.a.rich.set.of.solutions,.thus.enhancing.the.diver-
sity.of.the.solutions.
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