Mech523 l19 Fmrlc Casestudy1

8/10/2019 Mech523 l19 Fmrlc Casestudy1

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Fall 2008 MECH523 : Intelligent Control 3

Learning mechanism (review)Learning mechanism (review)Tuning of ruleTuning of rule --base of fuzzy controllerbase of fuzzy controller s.ts.t . the. theC.L. system behaves like reference model.C.L. system behaves like reference model.

The learning mechanism consists of:The learning mechanism consists of:Fuzzy inverse modelFuzzy inverse model : Function to map: Function to map yyee (kT(kT ) to) tochanges in the plant inputchanges in the plant input p(kTp(kT ) that are necessary to) that are necessary to

forceforce yyee (kT(kT ) to be zero, or small. (today) to be zero, or small. (today ss lecture)lecture) Similar to fuzzy controller designSimilar to fuzzy controller design

KnowledgeKnowledge --base modifier base modifier : Function to modify fuzzy: Function to modify fuzzycontroller controller s rules rule --base to affect the needed changes inbase to affect the needed changes inthe plant inputs. (last lecture)the plant inputs. (last lecture)


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KnowledgeKnowledge --base modifier (review)base modifier (review)

If fuzzy controllerIf fuzzy controller hadhad generated control inputgenerated control input

thenthen yyee (kT(kT )) would have beenwould have been zero.zero.Next time when we have similar value of e andNext time when we have similar value of e andc, plant input will bec, plant input will be u(kTu(kT --T)+p(kTT)+p(kT ).).

FuzzyFuzzycontroller controller Plant

Plant


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Modification of outputModification of output MFsMFs (review)(review)Example: Assume fuzzy inverse model producedExample: Assume fuzzy inverse model producedp(kTp(kT )=0.5, for)=0.5, for e(kTe(kT --T)=0.75,T)=0.75, c(kTc(kT --T)=T)= --0.2.0.2.

FuzzyFuzzycontroller controller

Two rulesTwo rules ((RR 11 && RR 22 ))

are active in this case.are active in this case.

Move the correspondingMove the correspondingoutputoutput MFsMFs byby p(kTp(kT ).).

Output MFOutput MFfor Rfor R 11

Output MFOutput MFfor Rfor R 22


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Case study: Ship head direction controlCase study: Ship head direction control

Control ship head direction by actuating rudder angle.Control ship head direction by actuating rudder angle.u=5 (u=5 ( m/sm/s ): velocity in x): velocity in x --direction (constant)direction (constant)

An increase in the rudder angle will generally result in a An increase in the rudder angle will generally result in adecrease in the ship heading angle.decrease in the ship heading angle.

InputInputRudder angleRudder angle

OutputOutputShip headingShip heading


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FMRLC for direction controlFMRLC for direction control

ShipShip

Fuzzy controller Fuzzy controller

ReferenceReferencemodelmodel

Learning mechanismLearning mechanism

Fuzzy inverse modelFuzzy inverse model

KnowledgeKnowledge --basebasemodifier modifier


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Scaling effect onScaling effect on MFsMFs (review)(review)Scaling gains are same as expanding orScaling gains are same as expanding orcontracting input/output membership functions.contracting input/output membership functions.

Scaled FSScaled FS1/31/3 55

FSFS


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Input and outputInput and output MFsMFs of FCof FC

(121(121 MFsMFs areareOverlapping.)Overlapping.)


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Design of scaling gains of FCDesign of scaling gains of FCgg ee = 1/= 1/ (since the error(since the error e(kTe(kT ) can never be over) can never be over180 deg.)180 deg.)

gg cc = 100= 100 (since ship does not move much faster(since ship does not move much fasterthan 0.01than 0.01 radrad /sec (by simulations).)/sec (by simulations).)gg ee = 8= 8 /18 (since we want to limit/18 (since we want to limit (kT(kT ) between) between80 deg.)80 deg.)

Design of reference modelDesign of reference model


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Fuzzy inverse modelFuzzy inverse modelMFsMFs : Same as in the fuzzy controller : Same as in the fuzzy controller Scaling gains: By using Design procedure 1 inScaling gains: By using Design procedure 1 in

Appendix, Appendix,


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RuleRule --base of inverse fuzzy modelbase of inverse fuzzy modelCenters of output membership functionsCenters of output membership functions


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Rule interpretationsRule interpretationsi=0, j=0i=0, j=0

IfIf ee =0 and=0 and cc=0, then y is tracking=0, then y is tracking yymm perfectly, soperfectly, so

you should not update the fuzzy controller. Therefore,you should not update the fuzzy controller. Therefore,the output of the fuzzy inverse model will be zero.the output of the fuzzy inverse model will be zero.

i=1, j=2i=1, j=2

IfIf ee is positive andis positive and cc is positive, then change theis positive, then change theinput to the fuzzy controller that is generated toinput to the fuzzy controller that is generated toproduce these values ofproduce these values of ee andand cc by decreasing it.by decreasing it.

We want to increaseWe want to increase , and thus we want to decrease, and thus we want to decrease to achieve this.to achieve this.


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Simulation result: GradientSimulation result: Gradient --basedbased

Model Reference Adaptive ControlModel Reference Adaptive Control

H e a

d d i r e c

t i o n

( d e g )

H e a

d d i r e c

t i o n

( d e g )

Time (sec)Time (sec)

Slow convergence!Slow convergence!


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Simulation result:Simulation result: LyapunovLyapunov --basedbased

Model Reference Adaptive ControlModel Reference Adaptive Control

H e a

d d i r e c

t i o n

( d e g )

H e a

d d i r e c

t i o n

( d e g )

Time (sec)Time (sec)

Slow convergence!Slow convergence!


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Disturbance rejectionDisturbance rejection

Disturbance at rudder (deg)Disturbance at rudder (deg)

Head direction (deg)Head direction (deg)(FMRLC)(FMRLC)

Head direction (deg)Head direction (deg)(Gradient(Gradient --based MRAC)based MRAC)

Head direction (deg)Head direction (deg)((LyapunovLyapunov --based MRAC)based MRAC)


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DiscussionDiscussionIn this example, FMRLC gives:In this example, FMRLC gives:

Fast convergence compared toFast convergence compared to MRACsMRACs

Good disturbance rejection compared toGood disturbance rejection compared to MRACsMRACsHowever, in general:However, in general:

No guarantee that FMRLC is always better thanNo guarantee that FMRLC is always better thanMRACsMRACsNo guarantee of closedNo guarantee of closed --loop stabilityloop stability


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SummarySummaryFuzzy Model Reference Learning ControlFuzzy Model Reference Learning ControlHow to select fuzzy inverse modelHow to select fuzzy inverse model

Scaling gains (Two design procedures in Appendix)Scaling gains (Two design procedures in Appendix)Fuzzy system (Application dependent)Fuzzy system (Application dependent)

Ship direction control by FMRLCShip direction control by FMRLCEx: Read until p.346.Ex: Read until p.346.


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Fuzzy inverse model designFuzzy inverse model designDesign procedure 1Design procedure 1 for scaling gainsfor scaling gains

(Assume normalized I/O universe of discourse.)(Assume normalized I/O universe of discourse.)

1.1. Select as follows.Select as follows.

2.2. Apply step Apply step r(kTr(kT ) with typical magnitude. See) with typical magnitude. Seea)a) If there is unacceptable oscillations inIf there is unacceptable oscillations in y(kTy(kT ))

aroundaround yymm (kT(kT ), increase derivative gain), increase derivative gainb)b) IfIf y(kTy(kT ) is unable to) is unable to keep upkeep up withwith yymm (kT(kT ),),

decrease derivative gaindecrease derivative gainc)c) IfIf y(kTy(kT ) is acceptable) is acceptable w.r.tw.r.t .. yymm (kT(kT ), done!), done!


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Fuzzy inverse model designFuzzy inverse model designDesign procedure 2Design procedure 2 for scaling gainsfor scaling gains

(Assume normalized I/O universe of discourse.)(Assume normalized I/O universe of discourse.)

SetSet gg pp =0. (Begin without learning mechanism.)=0. (Begin without learning mechanism.)If it works fine, FC is already wellIf it works fine, FC is already well --designed.designed.

2.2. Choose gains by maximal magnitudes.Choose gains by maximal magnitudes.

3.3. Turn on learning mechanism by increasingTurn on learning mechanism by increasing gg pp slightly.slightly.(Slow update of output(Slow update of output MFsMFs in fuzzy controller.) Tunein fuzzy controller.) Tunethe inverse model if necessary.the inverse model if necessary.

4.4. Continue to increaseContinue to increase gg pp , and tune the inverse model if, and tune the inverse model ifnecessary. (Increased adaptation speed, possibility ofnecessary. (Increased adaptation speed, possibility ofoscillations and instability.)oscillations and instability.)

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Mech523 l19 Fmrlc Casestudy1