Adaptive Control 2

8/9/2019 Adaptive Control 2

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ADAPTIVE CONTROL

SYSTEMS


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MRAC MODEL REFERENCE ADAPTIVE CONTROL

SYSTEM It can be considered as an ADAPTIVE SERVO

SYSTEM. It consists of two oo!s inner oo! "o#ter oo! It consists of a reference $ode in o#ter oo!. Ordinar% feed bac& is caed inner oo!.

Para$eters are ad'#sted on basis of feedbac&fro$ t(e error. Error is t(e difference between !rod#ced o#t!#t

" reference $ode )a#e.


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Adjustment of system parameters

in a MRAC can be obtained in twoways.

GRADIENT METHD !MIT R"#E$

#%&N' (TA)I#IT% THER%


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MODEL

CONTROLLER PLANT

ADJUSTMENT

MECHANISM

*C

YM

* Y

CONTROLLER PARAMETERS

)#C* DIAGRAM + MRAC


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MRAC IS COMPOSED OF

Pant containin+ #n&nown !ara$eters

Reference $ode Ad'#stabe !ara$eters containin+ contro

aw or oss aw

Ordinar% feed bac& oo!


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MIT R*LE

It is de)eo!ed b% INSTR*MANTATION LA,ORATORY

at MIT.

To consider MIT r#e we #se a cosed oo! res!onse inw(ic( controer (as one ad'#stabe !ara$eter -.

T(e desired cosed oo! res!onse is s!ecified b% a

$ode w(ose o#t!#t is %$.

T(en error e %$/ %. 0% is ori+ina o#t!#t1.


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Error criteria seected (ere is

,!-$/01 !e$1

And t(is oss f#nction is to be $ini$i2ed to $a&e t(e s%ste$ controed.To ac(ie)e t(is t(e !ara$eters are c(an+ed in t(e direction of ne+ati)e

+radient of 3.

d- 23 4, 2 3e 4e

dt d- 4-

T(is is caedGRADIENT orMIT ru5e.

3 adaptation 6ain

4e0 4- sensiti7ity deri7ati7e!informs 8ow error isinf5uenced by -


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T(e oss f#nction can aso be c(osen as

,!-$9e9

d- 2 3 4e si6n e :GRADIENT METHD;

dt 4-

first MRAC is imp5emented by t8is function

d- 2 3 si6n 4e si6n !e$

dt 4- : si6n2si6n a56orit8m;

used in te5ecommunication w8ere simp5e

imp5ementation < fast computin6 are re=uired


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+or mu5ti7ariab5e systems

- is considered as 7ector

4e0 4- is considered as 6radient of t8e

error wit8 respect to t8e parameter


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A&ICATIN( +R MIT

R"#EADAPTATION OF A FEED FOR4ARD 5AIN

MRAS FOR A FIRST ORDER SYSTEM


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ADAPTATION OF A FEED FOR4ARD

5AIN

PRO,LEM 6Ad'#st$ent of feed forward +ain

ASS*MPTIONS6 Process is inear wit( t(e transfer f#nction 750s1 50s1 is &nown 7 is #n&nown !ara$eterDESIRED CONDITION6Transfer f#nction 5$0s1 s(o#d be e8#a to 7o50s1

7ois +i)en constant


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9 :

9 750S1

;

?

- " "C

%m

MRAC +R AD,"(TMENT + +EED+R>ARD GAIN )% MIT R"#E


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T(e constant kfor t(is !ant is #n&nown. @owe)er a

reference $ode can be for$ed wit( a desired )a#e of k

and t(ro#+( ada!tation of a feedforward +ain t(e

res!onse of t(e !ant can be $ade to $atc( t(is $ode.

T(e reference $ode is t(erefore c(osen as t(e !ant

$#ti!ied b% a desired constant ko.

T(e cost f#nction c(osen (ere is


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T(e error is t(en restated in ter$s of t(e transfer f#nctions

$#ti!ied b% t(eir in!#ts.

As can be seen t(is eB!ression for t(e error

contains t(e !ara$eter thetaw(ic( is to be #!dated.

To deter$ine t(e #!date r#e t(e sensiti)it%

deri)ati)e is cac#ated and restated in ter$s of t(e

$ode o#!#t6


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Fina% t(e MIT r#e is a!!ied to +i)e an eB!ression for#!datin+ theta. T(e constants kand koare co$bined into

gamma.

To t#ne t(is s%ste$ t(e )a#es of koand gammacan be

)aried.


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NTE( N DE(IGN >ITH MIT

It is i$!ortant to note t(at t(e MIT r#e b% itsef does not

+#arantee con)er+ence or stabiit%.An MRAC desi+ned #sin+ t(e MIT r#e is )er% sensiti)e

to t(e a$!it#des of t(e si+nas.

As a +enera r#e t(e )a#e of gammais &e!t s$a.T#nin+ of gammais cr#cia to t(e ada!tation rate and

stabiit% of t(e controer.


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IF 50S1=S?

*C is sin#soida si+na wit( fre8#enc% rad=s

7

7

Var% )a#es of ARD GAIN CNTR# "(ING #%A&"N' R"#E


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T8e error e=uations are


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T8e 7a5ue of d70dt is :w8ic8 is 7e semidefinite;

As time deri7ati7e of #ypano7 function is ne6ati7e

semi definite.

)y usin6 5emma of 5ypano7 we can s8ow error

6oes to Fero.

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http://www.control.hut.fi/Kurssit/AS-74.185/luennot/lu5ep.pdfhttp://www.control.hut.fi/Kurssit/AS-74.185/luennot/lu6ep.pdfhttp://www.control.lth.se/~FRT050/Exercises/ex4sol.pdfhttp://www.control.lth.se/~FRT050/Exercises/ex5sol.pdfhttp://www.igi.tugraz.at/helmut/Presentations/AdaptiveControl.htmlhttp://www.igi.tugraz.at/helmut/Presentations/AdaptiveControl.htmlhttp://www.irs.ctrl.titech.ac.jp/~dkura/ic2004/lec405.pdfhttp://www.irs.ctrl.titech.ac.jp/~dkura/ic2004/lec406.pdfhttp://mchlab.ee.nus.edu.sg/Experiment/Manuals/EE5140/adap1.pdfhttp://mchlab.ee.nus.edu.sg/Experiment/Manuals/EE5140/adap1.pdfhttp://mchlab.ee.nus.edu.sg/Experiment/Manuals/EE5140/adap1.pdfhttp://mchlab.ee.nus.edu.sg/Experiment/Manuals/EE5140/adap1.pdfhttp://www.irs.ctrl.titech.ac.jp/~dkura/ic2004/lec406.pdfhttp://www.irs.ctrl.titech.ac.jp/~dkura/ic2004/lec405.pdfhttp://www.igi.tugraz.at/helmut/Presentations/AdaptiveControl.htmlhttp://www.igi.tugraz.at/helmut/Presentations/AdaptiveControl.htmlhttp://www.control.lth.se/~FRT050/Exercises/ex5sol.pdfhttp://www.control.lth.se/~FRT050/Exercises/ex4sol.pdfhttp://www.control.hut.fi/Kurssit/AS-74.185/luennot/lu6ep.pdfhttp://www.control.hut.fi/Kurssit/AS-74.185/luennot/lu5ep.pdf

Adaptive Control 2

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