Chromosome Representation - Multiobjective Optimization Methodology: A Jumping Gene Approach - page 230

Appendix D: Design of
the Fuzzy PID Controller

The.structure.of.the.fuzzy.proportional-integral-derivative.(PID).controller.
given. in. Figure 5.5. is. similar. to. the. conventional. PID. controller,. but. self-
tuned.control.gains.are.used..These.gains.are.derived.from.nonlinear.func-
tions.of.the.input.signals,.while.its.fuzzy.control.laws.are.derived.from.the.
classical. discrete. PID. controller.. In. the. design,. only. simple. membership.
functions. and. minimal. fuzzy. logic. if-then. rules. are. used.. Moreover,. the.
normal. processes. in. fuzzy. controlled. laws. (e.g.,. the. fuzzification,. control.
rule.execution,.and.defuzzification.steps).are.all.embedded.in.the.final.for-
mulation. of. the. fuzzy. control. laws,. which. are. in. an. explicit. conventional.
form.. Thus,. the. resultant. controller. simply. works. just. like. a. conventional.
PID.controller.
The. stability. of. this. fuzzy. PID. controller. has. been. analyzed. [1-3]. and. is.
guaranteed..Its.superior.performance.has.also.been.demonstrated.in.many.
simulations.and.practical.examples.[1].
Referring. to. Figure 5.3,. the. fuzzy. PI. + D. control. law. can. be. derived. as.
follows

u nT = u nT T − + K ∆ u nT

( ) ( ) ( )

PID PI uPI PI

. (D.1)

+ u nT T − − K ∆ u nT

( ) ( )

.

D uD D

D ( ∆ . are. the. incremental. control. outputs. of. the.
fuzzy.PI.controller.and.the.fuzzy.D.controller,.respectively;.and. T .is.the.sam-
pling.time..The.fuzzy.PI.and.D.controllers.are.then.designed.by.the.standard.
procedure.of.fuzzy.controller.design,.which.consists.of.fuzzification,.control.
rule.base.establishment,.and.defuzzification.[4,5]..A.fuzzification.unit.is.used.
to. transform. the. numerical. input. signal. into. some. fuzzy. values,. while. the.
defuzzification.step.is.used.to.transform.the.final.fuzzy.value.into.an.output.
signal.from.the.controller..These.two.steps.require.heuristic.rules.and.mem-
bership.functions.to.encode.the.desired.system.response.characteristics.and.
controller.dynamics.
In.this.design,.simple.membership.functions.of.the.input.and.output.of.the.
fuzzy.PI.and.fuzzy.D.controllers.are.used,.as.shown.in. Figures.D.1 .a nd . D.2,.
respectively..Based.on.a.simple.control.rule.and.the.centroid.defuzzification.

PI (

)

where. u ∆ nT . and. u nT

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