NARMA-L2 NEUROCONTROLLER FOR NONLINEAR SYSTEM

Mr. Kittisuk Srakaew

A Thesis Submitted in Partial Fulfillment of the Requirements

for the Degree of Master of Engineering Program in Mechanical Engineering

Department of Mechanical Engineering

Faculty of Engineering

Chulalongkorn University

Academic Year 2008

Copyright of Chulalongkorn University

~ A G ~ u f131iif% : ~?n?uqu~?~5u i~~ i - i i ~n~~ i~?u~ruu 'b~ i%9 idu . (NARMA-L2

NEUROCONTROLLER FOR NONLINEAR SYSTEM)

o . d ~ n ~ i ~ n u i i i n u i : 3n.ns.iaGu iYuniieity, n& 91.

# # 487064642 1 : MAJOR MECHANICAL ENGINEERING

KEYWORDS : NARMA / NEURAL NETWORK / NEUROCONTROLLER

KITTISUK SRAKAEW : NARMA-L2 NEUROCONTROLLER FOR NONLINEAR

SYSTEM. ADVISOR : ASSOC. PROF. RATCHATIN CHANCHAROEN, 91 pp.

This thesis designed and implemented the NARMA-L2 neurocontroller to control

nonlinear systems including water tank system and nonlinear pendulum system. The NARMA-L2

neurocontroller, first, learns and models the nonlinear system, then is recanfigured to be a

controller that eliminates both the nonlinearity and dynamic behavior of the system. The

NARMA-L2 neurocontroller computes the control effort based on reference position and the

actual position and its past value. Once the system eliminates the nonlinearity and dynamic

behavior, the closed loop system becomes implicit algebraic relation between the reference

position and the actual position. This means that the actual position do follow the reference

position in real time. Normally, there is a time delay between the control effort and the reference

position in the calculation, i.e., the current control effort controls the actual position to match the

reference position in the future time step.

In the first experiment, the NARMA-L2 neurocontroller is used to control the water tank

system that its cross section varies. The NARMA-L2 neurocontroller cannot eliminate the

dynamic efficiently in this case. The remedy is that the predefine dynamics is installed back to the

system such that the closed loop dynamics is as defined The NARMA-L2 neurocontrolller

combined with predefined dynamic is able to stabilize the system and also control the system

follow a desire trajectory.

In the second experiment, the NARMA-L2 neurocontroller is used to control the

pendulum system. In this case, the NARMA-L2 neurocontroller is able to eliminate nonlinearity

and dynamic of the system, and thus, able to perfectly control the system to follow a smooth

reference trajectory that is generated in real time using input device.

Department:. . .. . . -. . . M K M . ~ . . E~&.mhgg g g g -Student's

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[121 IA i l i i ~u~ l~mrn~no~nauqurz~uozJi~anauquu'afr NARMA-12 idnclaonauqu

iiurzuuAiinaiuuRnii~n'u i d o c ~ n a i u r n u i ~ n t u n ~ r ~ d u ~ a n a u ~ u ~ ~ n n (un*er~al

controller) uotdiuciuu'ain rzuu~ozrii/iuciuu"a?n~dnauqu~oiwuc~ir'u rmrrzuu

nauqu.rzciuuoamna Tfiuaznctnouna1umu1cnuo~6a~auqo~ l u r n r i i k n a i u l ~ i d u ~ ~ ~

iduuo.arzuu n a i u ~ ~ u i ~ n l u n ~ ~ ~ ~ l M ' ~ z ~ ~ u i i ~ ~ i i i u ~ ~ ~ w (Stabilization) imznaiumlnrolu

m ~ ~ a u ~ ~ ~ ~ 3 z u ' U R ' f i R i ~ i i u a ~ ~ (Trajectory Following Control)

Neural Network Input Hidden Layer Output Layer

Active 1 fhah 1

3.2 hu~i~iia~~i~uuid~iioyd~ao~~i%liuEod

iiu~iuu'ainuuuidor"Lo11up9sounniudu (Multilayer Percepm) ~ u l i ( i n h ~

dr:nou#?uiiarounniu~a I # ~ ~ ~ ~ ~ Y u ? u ~ ~ L ~ u u ~ L L ~ u ~ ~ I ~ B J I (Supervised learning) ~rn-I4 ~p9on iad16ounu (Backpropagation) diniums$orluiiuqiu buds:nouAiuno~

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dou6o (Weight) o:-llndiu~d~uu~~aom~60~n'ungnisi~~~~"wm~n~m (Error-correction) &fi n " ~ P l ~ P 9 ' 1 ~ ~ ~ 9 ~ n ~ ~ ~ ~ ~ ~ ~ ~ l ~ 9 1 ~ (Actual response) ~ ' ¶ J w P ~ ~ ~ F J Y ~ ~ u ~ ~ L ~ I M ~ I ~ (Target

response) ' l # 8 w q l ~ ~ b i ~ ~ n ~ ~ (Error signal) ~ t ~ ~ ~ i t u i i m n n 1 n ~ : ~ n d ~ 6 o ~ n n " L I i 4 1 ~ la t i idinrinmsnou~~~:~n~iu i

ouns~k~prnp9ounue.riiu~iuiGi1n8wnp9ounuo~idinoJiu

Neural Network Input Hidden Layer Output Layer

Tar* Ouput

3.2.1 I ~ U Y ~ I B ~ ~ ~ ~ ~ B U ~ U ~ ~ ~ Z U ~ [is]

~ ~ Y U ~ I ~ Q P G ? ~ ~ U Z U ~ B ~ L ~ U ~ (Single-Input Neuron) L s U O ~ ~ ~ ~ : ~ ~ Q ~ J & ~ I U U B ~

rjiu.r~uiiain ; I iiarouiin:: i guqm r~nm~qdd3.4 ninmio'uqw p (scalar input) a q n Q a

&:?u nmrniiimh w (Scalar weight) ~~8aQ1~1~~uiiUd1lui~on b (Bias) I#P(OI~U~U~B~(III? Y

n ~1nriua~~nri.r~d1ua~~1~1%1'ifl~1~6~161"~ (Scalar output) li?~~.r6<un1:sju (Activation

function)

Inputs General Neuron n m

n ' i ~ l i n u n : r i i l u ~ i o ~ d u z ~ ~ n u ' ~ d ~ u ~ ~ n i o ~ m ~ u ~ m v ~ ~ ~ ~ ~ ~ ~ n ' ~ u ~ n o i ~ ~ ( G ~

liaseu 1un~~iln<~uc~u~l&fi~adu~~n11~"uido1.1;a~i~l~q~oonu11ndt~u~~utdinului 4 w 4 ~oemruind~m ci?uY~6~unr~~uu:~ntn"~nt~u~~0nt~uu~1u.j1uua aafiiunmilull

d w d m ~.rla6~ut3c~tvuni01u'i~~~du~o~~ai~IIt n 1~u~enaunuaul46uu1n~ol.sn'h i%n3n d w d drl t

U O U ~ (Log-Sigmoid M i o n ) ~ ~ ~ . ~ T R I I U ~ I J ~ ~ . S YwYu~o~-J~uouRI V Z L ~ ~ ~ I Z U ~ ~ U R ~ uanuinqlfli(Gil& 1 n i o o u u ~ n ~ l X i i % ~ % ~ p ~ u n ~ ~ 3.3

1 a=- 1 + e-" (3.3)

3.2.2 I I U U ~ ~ ~ B ~ G ~ ~ B U W ~ ~ ~ ~ S M ~ B I [IS]

T ~ U $ ~ W L I X ~ ii~50u 1 s i?n i~ i~o~~O^~~]m1Xainn i1 I <a ilq~ p, . p, ..., pR LL~O::

ri?arQntiadivGnhwii w,,,, w,, . . . , w,,, 6 6 ~ r n i ~ i r o c m m ~ ~ ~ ~ ~ ~ u ~ d ~ 8 ~ ~ ~ ~ ~ ~ 1 ~ n d ~ u q n

p ( lput Matrix) I 6 n L W c i l ? n ~ ~ l j ; H ~ ~ W(Weight Matrix) ~4~6ffF141~~dd 3.6

Inputs Multiple-Input Neuron n m

1~d3.6 iinounrnuiluqcl [If]

~ 5 ~ ~ i n 5 u ~ n a . ' n ~ ~ a ~ n o ' a ~ l i ' 1 v G f ~ 0 ~ ~ n o i ~ 1 d ~ ~ ~ n ' ~ ~ ~ 1 ~ 1 ~ m ~ ~ m d ~ s i n i ? u i i ~

3aaf%n'l!Ym68ff b b%uq~pi (Net input) n luph~84

u w 2 < 2 l o V l ~ l G m (Neuron output) ~ u I J ~ ~ u u ~ ~ ~ ~ R ~ ~ A A ~ u

a = f (n) = f (Wp + b) (3.6)

Inputs Laycr of S Ncurons n m

1d43.7 4iaiu&i~a-~ [is]

Inputs First Layer Second Layer Third Layer n---

Y A Y t d B a s dauGuouqosGunii+u$ou (Hidden layer) un~aoiia6h%u~~~.tu8uqm p UUIRR 2

1: d ioiiiq6~ 2 uum s aunno~szii5uqmZo iio a'lmziirii~oii-qcr a' uum 9

Input Log-Sigmoid Layer Linear Layer n- -

f2(n) = n

h~u~ri i l~ninri~mof eii.jq6pii6qif

w;,, =lo, w ; ~ = 10, b: = -10, bi = 10, = 1, W& = 10, b2 = 0

I4'Suqm iidioir:nii.r [-221 r:lhnaounuo.rao.riia.r1u~.~~dd3.i0 imod * A w n m o u n u o a ~ : i ~ u ~ ~ ~ i t ~ 4 u ~ ~ ~ ~ ' u a.rSuwnu~ain~.rn"+uns~~uu~~iia~ou~uR~~~n

~m:enu~mdiu~dii~uo.r w ~ ~ ~ Q ~ ~ ~ u ~ Q ~ I u ~ ~ u ~ B ~ ' Q ~ ~ ~ I ~ ~ ~ ~ ~ I w I J I ~ ~ B I ~ ~ ~ I ~ ~ i ~ ~ d d u u ~ i ~ ~ r i ~ i m ~ % ' ~ ~ ~ ~ ~ w ' w n m ~ u ~ ~ o ~ ~ o ~ ~ ~ u . r ~ ~ t d ~ ~ ~ ~ ~ d a . r m ~ u ~ d d . i i $4

6 4 d 1im~wnroumro~ao~i1u~1u~u~unir~da~uri1~1~13~m~fw"n~~a~.rif

-IS<, i l , - 1 ~ 4 ~ 51, ~ i b i i20 , - 1 ~ b ~ ~ l (3.14)

I I

m~Q~u'~w~uu'ainldlflu~~uda:uanPtnBi~~qu'u r:#o~dirn~diud~~n'iJ~nu'n

imzn'~luiionuosiiuq~uLidnziu1~inu~:nui~ufiu t~szdil~~:uunilnanfi~nu'~d~fiu ri?d~:u1mmn'1n1nnl#oi1~~ioju61 ~ m d n u ' ~ u ~ ~ u n n ~ u u (Training Multilayer Networks) d s a scr nuuunoasm~~~uf~~uui~013~1 (Supervised learning) ~ n u s r ~ ~ ~ ~ ~ f i i ~ u o ~ ~ u q n i ~ n ~ n ' ~

idinumdnonnXo.rriu ~ u ~ n d 1 c i u r i ~ n o : ~ ~ o i ~ 1 d u " ~ ~ 1 u ~ 1 ~ ~ d ~ ~ 1 ~ a m i o 1 i q ~ 9 ~ ~ ~ u ' 1 u ~ 1 u

u8a ~1u1i~uuduun'u~o1~prfiid1~u1udno~~1do~n'u~~prnfi1~u1i~ni1~na1~iian~i1~n'~

oii .r l3 ein~uirrfiimnl~uu'iu~~u~doln"~~diio~iprnu'iu~iufi~~udo~ddoonoJiijn'ia~~ m u d 9 4 94 muriiidinlnuu~nd~m ~ ~ ~ ~ 7 d i u i 1 ~ ~ 1 ~ a ~ n ~ ~ u 1 4 n ' u n o a ~ n 1 ~ n ~ : ~ 1 u Q i o u n . i r ' u

4 (Backpropagation) %~0zd~znou~dhun1~$i1ua~n1n'1i~1oJ"~fl~1n'5:uuu'1u~1u fl12li1uaun'1

na-tui i~nm~ mrdiuaudirn~n~: a i u & o u n . i r ' u u o ~ c i i n a ~ ~ ~ i i ~ ~ n ~ ~ L~D:RII Jiu

~ ~ n o ? ~ o ~ ~ l Q 4 n 1 5 6 6 d f l 5 ~ ~ 1 ~ ~ 6 ~ ~ & @ U n a " ~ 1 (Backpropagation algorithm) d l n h Y '4

u ' l u 4 l ~ H n l u ~ ~ ~ ~ % % n i 3 n m r r ~ ~ 1 8 1 l u t i ~ a i n ~ ~ ~ ~ ~ i (Gradient Descent Method) iM~n°l lw'61

a * 9 A m~r~iiwnrauuo~waiui3fi~n1fii1n'a~1'o~uw1~unqm (Least mean square m) IRUQII$

Goqnn~r?Jn (Training Set) ~ ~ d r ~ ~ ~ n 0 ~ ~ ~ U ~ U ~ 9 9 i ~ ~ ~ b ~ l H ~ l U d 9 ~ ~ d ~ ~ P I ~ ~ 9 ~

lnus pq floiu~?n~~uq~uoau'1~41u unr tq R'orn~inko~~~if l inuiu (Target

adput) io1~nroau' iu~iu~~~nGiui i f iuui~u11n"~~oi iy~1~i l iu1u ain~uiriiarriimrd?u

c i 1 ~ i ~ 1 i i t c l o ~ u o a u ' i w i u ~ # m r a u ~ 1 a 1 u ~ ~ ~ n i m ~ i ~ ' ~ a o a ~ ~ i ~ o ~ ~ ~

Q Q F(X) = C e: =C (t, - a,)l

33.2 nimfilaz8~1ni~la~a~na1StlMei [is] 9cl ?sm3n~.rrriu811aniuatn~~~uui (Gradient descent) t ~ u ~ n i s ~ ~ i u i l ~ u n i r d i ~

dirn3iiiuoidiuaiu lmoarn'inird?ud~uuciili'inu"niinz~u~ioalu~"ami~~iinaiu~u r~ d q4qm wiohfl~n"~i jdln~ni lu1ni~m Cinror1~a6h F(* ~~iflufl~naun~~srn~wid1 x d

as ; d rill# F(# uclimnym mlnmw161 r iihlialnmrjiih

d, = -VF(x)

a, dociihsimsdeuf ( M ~ Rate)

d d a s 4 d, R'ot?n~closnunmwdilM'eii FN iiciiaanq .a~vmlk in%rnaa~3r~~1~11u~tu~

y w 2 msti;uuH~uriuwasa~naiu"W~o"(~~imr~~ia'~fio fiiuisnriiu~udi~iv~oLLnzluttofilma4u

n i u 1 . r o n 1 ~ 1 l f i n u ' n ~ ~ ' i l z l ~ ~ ~ o ~ ~ ~ n d s ' ~ k ; a ~ ~ ~ ~ ~ z 6 ~ f i 1 u ~ ~ u a ~ n ~ 1 ~ ~ ~ f l b i ' 6 ~ ~ m In-1 wC; (k + 1) = wr, (k) - as, a, (33 1)

by(k + 1) = b,"'(k) - as,!" (3.32)

f i d ~ ~ ~ 1 u ~ d r n ~ n $ k 8 ~ ~ U d

W m (k + 1) = W m ( k ) - asrn(am-l)T (3.33)

brn(k+l) = brn(k)-asm (3.34)

Finish Training A

ainnaiumuisouot(li1u;11~u"a5'nd~1~1i~oAnd~lM't4u~ad'izu1tllpi1n1nn~~~~ii

n~rr i i ( l i iu~iui ia5 'ou1d~~~n~"$~u~1unau~~ ~noiilmunrruu'iuciu~a5'n~niu~dttuu~l~

lurn3nfi.rttuuBinoaunzoonauusi?nauqu a d t n o ~ i l ~ u n r ~ u t t ) / a z t t u u ~ ~ b o ~ b ~ ~ ~ u Y

irslndiah 8in%~-ntn~wu~~u~#8~ni~~~~1~~1wu'a?n~uni~fia~1qu~zuu!~t?~tdu~# 4 C1 iiwnmsuaoqrzuulBiui~o~n13 a.rmonnoiilmunsvuiiu.ji~CaJnttuu NARMA-u dii

naianiln~olun1~B1no.rrruu'lit5~tdu~ki't~uoti1~~ lu~umouarnuzdinirdszuiol

tiuuB~noar~uuAiud~u~~~~as'nttuu NARMA-LZ a i n ~ u r i i ~ i u ~ i ~ d ~ ~ u i n f i ~ ~ a n a u ~ u

u"J1.l NARMA-L2 L ~ O I ~ U ~ I ~ ( ~ ? U ~ U ~ L Y U 1~~Bi?fl?~q~<?h NARMA-L2 %fl%IaZfl l3

rii~iun&un&riusi?na11~ud~una"udi~~t~ut8~idu (~aaback linearization control)

Dynamics)

cinrmmld (3.35) - (3.38) t~u!8~imnuuuhn8~~~81~11n81s"~e,~szuufliui~nduu

liuslXo~lu~dauuaou'~11iiuu0zniu1~noom~uu6~nauqut8oi1lM'rzuu1it~~td~

nmut8urzuui~~idulRmu9'1u n d m n s z ~ u 8 6 ~ f l i z a z n a ~ ~ % 1 ~ ~ l ~ ~ 1 u i ~ ~ o ' ~ l . n ' o ~ l u ~ d

aouwiliiuu ~ieoiooz~naiu~uG~~a~li~1u1znwi~tuu~1no.~dt~iu6it~u~~o d9wn'Zfla.i

n i u 1 s n o o n ~ u u 6 a n a u ~ u ~ o ~ n d u ~ 1 ~ ~ t ~ u ~ ~ ~ ~ d ~ d ~ d ~ z ~ ~ ~ n i w ~ k i ~unrfid~imuim 914 l s ~1uuuiinoa~~iufi~c~~ti~61IA~~siw1n~n~n1~~q~1t~nztoi~qm~o~rzuu~tan1i1.j~ fi

~ ~ u ~ m ~ f i ~ ~ t u u i ~ n o . j l ~ ~ r ~ ~ t a n ~ ~ i i o ~ d o ~ ~ a u ~ n ~ r n i ~ n ~ ~ ~ f i ~ l ~ ~ ~ n ~ u t a n ~ ( ~ i ~ ~

series) ~~u0i11oto~nruta111dtnu1znu,t8n1~i1n~~rz'~uiiaaiu~~1~o~ttnr~iu~~n1~

li1!d~#flf1.r~anau~ud~wn~'u~~~8~~~~1d~ douuuhno~oqnrul?m NARMA-~2

3.4.1 u~uuiiaaaaynbo~laai NARMA-LZ: [12,17]

6 6 l J ~ iln84 NARMA-L2 w'991~11~1 Oln L t l l ~ $11109 NARMA (Nonlinear Autoregressive B a d

wing AV-) ~~t3ut~uui1a~~1~rzuuua111l~sio~~o~1fi~i1t~16~m~~~~zu~nta~1lfiq

rr~naiu8uriuhtuu~~6~~tdun'u~i~~~b~iFs"qfi 6tnzo'urlauo.r~ruu~u~a~~anid~iuui a

ttuudiao~ NARMA aztumni~~uint~uui~no~lu~d~tuu~tmm mrqn NARMA ~rMl4iimnfi

lunirtmcl~r-uu uriazl4i3u~luddu~5wq~-utnzioi4qa~n'iU:uluni~tt~fi~5ruu

IRA x(k) E R', ~(k) E R liar y(k) E R {Q tiinlfifi 3uqcilttnz~oifim d W dd

f : R" X R + R", h : R" -+ R ttnz f, h E cW ( t j 3 u ~ ~ n ~ n u n a i u ~ ~ u ~ o ~ ) ~ ~ n r

my %I f (0, 0) = 0 unr h(0) = o (~rm~n$pfihvii~) nnrzuufiiurmm5 3.39 luu?taal8nd~.rriupfimqn ~ruuiirzi7u~unaiu~uw"ui

(Relative Degree) Ll'!l~¶J d f l l ~ l % Q ~ 6 ~ f i 9 l ~ ~ ~ 6 ~ ~ ~ 6 t ~ ~ f i 5 4 (Exact R-tion of The

systems) Ilibauttuuiiaoa NARMA RIUflUfrIT 3.40 ( ~ I U I ~ O ~ ~ W ~ I U ~ Z L ~ U ~ ~ L ~ U L ; U I # $

ninpnranaanr~~ntonfli~d~~~~ [121)

thl+iYanauqumwmmr 3.41 fiUt:uu.i~num5 3.49 r:~(~uirnnauqulN'toies'~m 4 w 9

uotszuu~iumiuiXuwitd~otnis~~~iu~o~ni~ iiciwiilunitd~u~nisnfit5anauqum1u

nunis 3.41 Zi?uiiutiuiiaininaiuuin t m r ~ o t ~ + n i s ( i i u a f u ~ ~ u ~ o u +tlinz~anlunir 4 d A* 1 nfithnauqu rtumrriauiauuiinop ~ ~ ~ ~ ~ - ~ ~ n u r u u ~ i u u i s i n m ~ d s r u i f u

a uuuhnoa NARMA #auoY nrutn6tn0~ ~nolXnniuisn Jiuinfita'anauqulN'

y(k + d ) = yr(k + d ) l f l ~ M . 1 ~

rlnnunlr 3.40 $1 F uiuuiu#auoynsurn6tno$ ( ~ a ~ l o r series expansion) 3QuIR

[(y(k), y(k - 1 ) ,..., y(k - n + I), u(k) = 0, u(k -l),u(k - 2) ,..., u(k - n + I)] 9 r I # ttuuqinot NARMA-LZ . IU~OIS 3.42 ( m u ~ s n i l n ~ i s i u n r t ~ u ~ t ~ u t ~ u ~ ~ d ~ ~ n w u a n t ~ n z

~ininnni~81454 [121)

aiou (Hidden Lyer) ~mr&boiiqos (output Layer) 1uiuiouarij~?roumiuit9idfiinufi

lunir dinu~o'uymvo~(riiu4iu 41ur14 u lm: y AlamlfifiiPduirrrfiinufi6aupii 4 dki A

1?nld5:?4 (Time delayed numbers) r 4 n ~ ~ f l l ~ ~ i a 4 R * ( Y ~ l ~ o ' u ~ ~ L ~ ~ ~ ~ ~ 1 i ~ ~ ~ ~ 4 3 ~ ~ ~ ~ ~ ~

d o u d u o ' u q m l X n ' u i ~ u a ~ ~ ~ w " ~ 1 3 ~ n ~ ~ d ~ ~ ~ i ~ d i ~ ~ n " ~ u f un: g ansi?odittdu

~inufil~icanilm~u~~Buqmu~~r:uufio 1 ~ i i a r n ' R m h o a ~ e 1 ~ ~ ~ 9 u ~ ~ ~ r u u ~ ~ 2 trl6

~ u y c l u o a G ~ w ~ u l u m r d s r ~ ~ ~ ~ ~ Y ~ n ' ~ f 66nr g ~ ( k ) , y(k) 6ln:: y(k -1)

Ncural Nctwork Approximation of g()

Neural Network Approximation of $0

A (Structure Model) d ~ Y o a m l n ~ v u ~ u w v n ~ ' l a ~ . ~ ~ R I ~ I J ~ I # I ~ ~ ~ R J I ~ ~ : : I I I I ~ ~ J I Y hm.i

thnu~hu?uu"arouuo:iiu?unrnAmulR' u n s i ? o r i ~ ~ ~ d u m u ~ a ' ~ ~ ~ ~ ~ ~ ~ w n w " u d ~ ~ ~ ~ ~ i ~ n o ~

n G ~ r n a s l ~ m i u ~ ~ ~ n i ~ 3.49

y = au - by - csin(y) (3.49)

L L ~ R J P N A I ~ 3.49 I $ Q ~ I ~ ~ : : U U L ? ~ I I O ~ ~ Q L U " Q ~ (Discrete time) %ilL?nld~ (Sampling

time) L i l n ' n r T o::l#Hunlr 3.50

y(k +2)=(2-bT) y(k + l ) - ( 1 - b T ) y ( k )

- cT2 sin ( y ( k ) ) + aT2u(k)

sinrmmr 3.53 srtiulffji ~ ( k + 1) du~an'huss y(k) nn:: y(k - 1) arizarifu

nin41uauomlmci*oa o unr y lXt8u 2 tm::~ sliuriiciu r::l#5uqmaa~ Jiaciut91wr'v

11~:lnmn'iRan'h fun:: g n'o y(k) un:: y(k -1) ~nann&arillrrnni~ 3.50 daulunir Y Y

fi1wum4iuauiia~ouluh(nioulX~~0~1::mcs'~nird~::0~imn'1v~au'i~ai~u'u niuimnmooa d u d I~~i'nnniQnu'io~iu kunisfiiwuniia~oud~~~~ncs'i~n'u 1~~a~R'onI44iuauiia~~unu~mq~

~ ~ I I X ~ ' I ~ ~ ~ ~ I R ~ U ~ I S A ~ ~ ' I U ~ I U ~ ~ ~ I Q ~ I U V Q I I L % ~ ~ ~ Q U ~ ' I I I ~ ~ ' 1crri::ni~on41uau~arsu

duin~&ldo1ra: : lu 'hulXd~~~n3ni~1~n1rd~: :u1mn' iu~~Ji~~1~~n'1~~1 lnrirrji~lX 4 W d A' d

u'iu~iuuasnu~aiu&4~uuin~u r ~ n : : B 1 ~ 8 u # ~ ~ l ~ s ' w u 1 n ~ l ~ n i ~ G i u a m ~ 1 n ~ ~ ~ 1 m $amas

Neural Network Approximation of g() f \

Neural Network Approximation off()

Edi3. 14 ~ ? ~ ~ R ? v ~ Y . J ~ ? ~ J NARMA-L2 [16]

I I

Plant T

L -

I'

3.4.23 niafiiw~m'hr(a~a4az'~~1Ym

lummauqu~ruuli?u~anauqu<als NARMA-LZ ri?nauqutrjiim~~~fiadqqifu

i d o f i ~ b n ~ i u ~ ~ ~ u i ~ ~ ~ d u i i n r w n a ' ~ ~ o ~ ~ z u u ri i lX~ruuil~o~lu~drnni~i'~~ntu'~~ (Micit Ycr 1 4

Algebraic Model) BIluffunl3 3.46 rh~ir'fflusnfil~uRioies'~99V043zu~ ~wun la r~ lu~qq ia

& a t 4 w4o~fiann5~voa-rruulX~d1~11ud~0~n1~ Lu%nis~fianni~ro~~ru~niur0nji1

1ki'Xau%nirfiinu~twnu0a~zuuil~

T Reference

Model .

i ~ n i s ~ r l n s n i d n i ~ ~ o ~ n i s 3.56 Q:!&

Y (s) = R(s)

~fid ~~R"o8qqirndito'a (reference signal)

riwutlums 3.55 lutlnrms 3.54 k

Y ( s ) = - s2 +As

(Yr ('1 - ' ( ~ 1 )

Y (s) = k

?+iks+k K(s>

01n~um3 3.57 Q Z ~ ~ U ~ I ~ ~ U ~ T U I ) ~ I ~ ~ ~ ~ ~ I U L ~ ~ ~ ~ U ~ I ~ L ~ ~ L ~ U ~ ~ ~ ~ ~ U Q (s-d

order System) ~ ~ n ~ ~ l 1 ~ 1 3 n 6 1 ~ ~ ~ ~ ~ n ~ ~ s ~ z u u ~ m ~ n 1 3 ~ d o n n ' l k unz A

nonunis 3.56 uuuu 0 1 a ~ a ~ ~ u u j l ~ o ~ ~ ~ ~ d ~ ~ ~ ~ ~ ~ i l ~ 6 1 ~ s i z d i A i ~ u m ~ W ' f,

dih f umr g, dih g si?n?uqu~~~iuisn~ia'mwa~81%645zuu!~~ri14fluu~& udlums

l4.nuoitsi?n~uclu~~o!iniuisnn~~~8fllw~anauQu~w"oii~knnin~~~szuu~~~ti79

nuuJai ~~d~~mlnlr jmu1sn. i i1n1snau~ulX~o1A~~1~~~szuu~~~1iud~~ia~i t5t I8~11u

h4rn.1 ~ u n ~ ~ ~ n i r i i i ~ u ~ b n u ~ t ~ ~ ~ ~ ~ ~ ~ ~ ~ a d ~ n " h f i ~ t l ~ ~ ~ m ~ u ~ u n i s 3.58 o-dlurlilW' ' 2 sruuiladn?iuntnu~w"uuu (~obustness) ~ ~ s ~ z ~ ~ u n i s A ~ ~ u a ~ n i ~ ~ l v i l ~ n " u s z u u lau

rrni~lni$fiinu~sz#o~d5n5wn~~~owna"muriivo~sruud~nt~na"~~~ urrrniaisnm

rtaflloni~ounriuaotszuu~m (Inverse Dynamics System) 6~~n l~qq l lB l~ lQ~41~681~~~9

-1.4 r v ~ ~ s r u u ~ m m i u ~ ~ u ~ a ~ w m o ~ n i s ~ ~ uoncin6~4i~~ioloud1flunisfiivum1wnv8aszuv

~ a & 4 ~ u j i i m s a ~ ~ a a ~ ~ ~ 1 ~ 8 i t 5 t ~ ~ G 1 ~ 8 a n a u ~ u u " a I s udoriinun18dqqlifu~it6~t~u d w ,Y 4 Atnauuu nnoi?roah (Steady state) W A I Q ~ ~ U O ~ U Q ~ ~ ~ ~ ~ Q ~ S ~ ~ % ; ~ ~ ~ B ~ ~ ~ W U ~ %~iawi!#

oiuuniscio~dd

rrvu61oumr 3.61 - 3.62 lununis 3.55 oz16 y(k + d ) = e,r(k + d ) +e,

fiinud8 e, IKIZ e, r~uciiaai c~dajiini~a~dniunis 3.63

Y (s ) = e, R(s)+e,

rimwerums 3.58 nsluaunis 3.64

Y (s) = Yr (s) + s2 +As

s2+h+keB s 2 + h + b , =f

Yr Yr (s) = - (3.68) S

e, = lim $

w v

4.1.2 Xanaq~5aTg NARMA-U (hnA~m~~fiJiwM'#m!u'n~~

nnnsnird 4.5 6 1 n u f i ~ ~ ~ u ~ ~ h u a i u ~ u n 1 ~ d ~ z u i ( u ~ 1 ~ ~ ~ h f lmr: g n'o

I%fiGoy nlurnAn$iuaiu ioo,ooo 4 i ? ~ a n i ~ l u n n ~ A Y Goy nin'in'u 0.01 iuiq hnnfitlu

p~d4.2 ~ m z ~ 1 n ~ ~ i i a i o ~ u o a ~ ~ i ~ ~ d 6 1 u a ~ d 1 a ~ I ~ Q ~ ~ ~ ~ I U ? U ~ ' ? J Q U ~ ~ I I ~ ~ ~ ~ H U I ~ O I U

Squared Error)

(fl) 8uqmuo4sz¶Ju

c i ~ . i ~an1~d1ae~ayu!w'~eleiqmre~~:~~amm~u#qq1~~d1~34

Jiinirnauqu~:uuhu~ana~quiial.r NARMA-~2 i 4 n n m ~ 1 ~ ~ ~ d 4 . 3 1 n ~ f i 1 ~ ~ m 1 ~

ifcyqioldia~t~5u#qqif~o11111u'~"ad (sim swkp) dnaiuzf 0.001-0.005 ~iirfia' niulu 3000 9 P a auin uuuimoti~znii4 0-40

~ in~dd4 .4 ~ ~ x u ~ I ~ ~ I ~ Q J ~ u I ~ K ~ ~ I ~ u ~ I . ~ ~ ~ L ~ ~ ~ u ~ 20 n.'loi6qfiro.rrzuu

I:ufiui+i o ido~arniiiuId 30 i u i ~ toi~fiuo~rzuu~~11iu1~0~mfiiu11ua~ii81~~41k; ~tnr i j

Piiainnaiu~~nnifim~ufi1d~n0~u0~nit~fifi~udqq1fu~1~8~o~d 0.92 tm11Xtiuiih

A PI

4.1.33 ~ m i ~ d ~ a o ~ n a ~ q ~ 1 ~ o ( r l a ~ 1 a ~ 1 q u g n ~ ~ d ~ ~ ~ ~ 1 i i ~ i ~ l ~ d n i ~ f i i s " ~ ~ a ' j . ~ 1

fiimrbinowtauqur:uu Tnufiinufilfl.l?nauquu"aIr NARMA-LZ Bd3:ilniniw

luni.iriiknnimnfins bi?umru8ti'iirj;n~nluIu~oi~~fivo~u'iu~iuu"as"n fhblurnrdr:uiol

diilq6+u g!#i~ e, im: e, miunumr 3.61-3.62 d61d3:lnolO.Z lm: 0.8 flludl~u im:

fiinufi1X8qqicud1~5~iSu0111uu"~d miuoir6inotnauquh?bD 4.1.3.1 W~nmrOino~

~ 1 U ~ ~ f l l ~ ~ d d 4 . 7

P c n ~ ~ n o . r n a u q u ~ X i o i c i ~ ~ r ~ u u ~ f i ~ ~ u i ~ u a i ~ b ~ a ~ c i d ~ i a n a u q u ~ n r i i ~ ~ ~

dr:Bnirnwlumrfiikwnimnfinq m1u~dd4.7 ~ : i ~ u I C ; j i ~ o i ~ ~ ~ o ~ r : u u ~ i n i u i ~ n

~ ~ R A I o J ~ Y ~ ~ I ~ u ~ I P ~ ~ ~ X Q ~ ~ I ~ I # d n ' i n n n a i u i i f i w r n f i i ~ d u f i i n ' ~ ~ ~ ~ ~ ~ ~ . ~ 7.35 idocsinri?

n a u q u ~ i ~ i l n r n f i i ~ f i n a i ~ ~ ~ n i i ~ u ~ ~ ~ i ~ u ~ m : w n i ~ u o c r : u u ~ ~ o t i i q n u u ~ ~ ~unri id

rnuirndiud~.rdr:iS~~niwluni~~auqu #aumrl#Ganau~uu'atr NARMA-LZ i a u h

Cls6htiiulou im:dimriiiuacuwnnnms"Bounn"U~w"ona1u~ulXr:uun1u1~n.P~~91u~iua sad ann#osm.i'l#i' ~fiu~wnd~~~ufil~iiur:u~~~n~on~rA~wufi~na"m~~ui~;~~dl#n'ur:u'~

R(s) = , kp (pr ( s ) - Y ( s ) ) s + kvs

(s2 + kvs + k p ) f (s) = Yr

kP

(s2 + k , , ~ + kp)E(s) = 0 (4.10)

E(s) = Y ( s ) - Y, ( s ) (4.1 1) sd d sr AinumIw' k p , kv m'iriu 0.25 un:: 0.7 mu 81th 1 ~ ~ ~ l 6 r r u a ~ o n f i o ~ n i ~ t 6 i o i i ~ f i

Yr Yr

INVERSE DYNAMICS

n-I.inmno4nauqu3zuuo'aI-Ino40'4 G U ~ A ~ ~ ~ : ~ B - I ~ ~ . U ~ ~ ~ : U P I R O d?u-ItuiLdlnn

1~16auu lmz3:ciudlun'44iPR-Iud-I6'u luma41no.mauqua:l4~anauquu'al.i NARMA-

LZ ~oqiia nauquIuiin~~lzmnrn (Cascade control) ~mur:i - Iao~m~nauqulX~~-I iy~vo~

~ : u u ~ m R i u ~ ~ ~ i f w 8 1 ~ 5 ~ $ $ ~ o ~ n - I ~

h , 8o naiuU~uoc~zciu~in'4~u~~mn'~riun'4

h , iio fiaiua4uoarz<u~i84di4flmn'iJ8un'4 d d ~ ( 4 ) ~ o m u n n f i ~ ~ u o t t k u u t9uW4dhuaanaiuq4yz&u6i

a d d 9 ~(~)nowunnui~~uo~n' . jd i4 t~uWsn"huosnaiu~4Tz6'udi

Q h ~ l l i P l i i f f 1 H n t ~ i ~ 8 4 Y U

Q2 ~ o d ? u w ~ i ~ ~ n n ~ ~ i ~ n ' ~ d i ~

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[I] McCulloch, W. and Pitts, W. 1943. A Logical Calculus of Ideas Immanent in Nervous

Activity, Bulletin of Mathematical Biophysics 5 : 11 5-1 13

[2] Hebb, D. 1949. The organization of behavior. New York : Wiley.

[3] Rosenblatt, F. 1958. The Perceptron: A Probabilistic Model for Information Storage and

Organization in the Brain, Psvchological Review 65 : 386-408

[4] Widrow, B. and Hoff, M. 1960. Adaptive Switching Circuits, 1960 IRE WESCON

Convention Record : 96-1 04

[5] Marvin Minsky and Seymour Papert. 1969. Perceptrons. Cambridge : IvlIT Press

[6] Paul Werbos. 1974. The Roots of Backpropagation. Doctoral's Thesis : Harvard University

[7] David Rumelheart and James McClelland. 1985. Parallel Distributed Processing. Cambridge :

MIT Press

181 Kumpati Narendra and Kannan Parthasarathy. 1990. Identification and Control of Dynamical

Systems Using Neural Networks, IEEE Transactions on Neural Networks 1 : 4-27

[9] Julio Tanomaru and Sigeru Omatu. 1991. Towards Effective Neuromorphic Controllers,

Proceedings of IECON International Conference on Control and Instrumentation : 1395-

1400

[lo] Greene, M. and Tan, E. 1991. Indirect adaptive control of a two-link robot arm using

regularization neural networks, Proceedings of IEEE Industrial Electronics Society : 952-

956.

[I 11 Nordgren, N. E. and Meck, P.H. 1993. An analytical comparison of a neural network and

a model-based adaptive controller, IEEE Transactions on Neural Networks 4 : 685-694

[12] Narendra, K. S. and Mukhopadhyay, s. 1997. Adaptive control using neural networks and

approximate models, IEEE Transactions on Neural Networks 8 : 475-485

[I31 Habibiyan, H.; Setayashi, S. and Alibieak, H. 2004. A Fuzzy-Gain-Scheduled Neural

Controller for Nuclear Steam Generators, Annals of Nuclear Enera 3 1 : 1765-1 78 1

[I41 Alexandru Floares. 2005. Genetic Programming and Neural Networks Feedback

Linearization for Modeling and Controlling Complex Pharmacogenomic Systems,

Proceedings of IEEE World Congress on Computational Intelligence : 75 10-75 17

[15] Martin T. Hagan and Howard Demuth. 1999. Neural Networks for Control, Proceedings

of American Control Conference 3 : 1642- 1656

[16] Howard Demuth and Mark Beale. 2000. Neural Network Toolbox for Use with Matlab,

Matlab User's Guide Version 4 : Mathworks Inc

[I71 Adetona, 0.; Sathananthan, S. and Keel, L. H. 2004. Approximation of the NARMA

Model of Non-Affine Plants, Proceeding of the American Control Conference 6 : 5502-

5507

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x = 0, ti,,-, = 0 ~ l f l ~ ~ ~ ~ ~ . I ~ ~ ~ ~ ~ ~ d ~ 0 1 8 (Implicit Function I'heory) AlU1IOLIAR.I x(k) Y

%u~duosnums~a'LdG

~ n u d ~ : R " x R " - ' -+R" i3u~-rA.a'uioiudo-rd~nin'aii;asn'u~~nu~aaLRn -4

x=o, Un-, = O Tnau ' a iu r i i a ~ n a x(k+n) o : u u A u t B e - r n ' i n ~ ~ ~ x(k)iln: i i 6 u Y

IJn = ~ ( k ) , u(k + I), ... u(k + n - 1) o:l6auflisaio!dG

x(k + n) = g[Yn(k),Un(k)l = g[y(k), y(k + I), ...y( k + n - I), u(k), u(k + I), .. .u(k + n - I)]

A d v d ~ n u d g : + w i~u~c6~ui~iuos~~~Innim~-rn"upanoJqn~1imm imza inauml

n.1 y(k + n) = h [x(k + n)] sz lk~uuhnoa NARMA ~ - r a u m ~ d o l d d

mnrzuumuauflir n.1 d r z r i u ~ u n ? i u ~ i u ~ u i (~elative Degree) i d i h d % ~ ~ n d

n a i k l ~ ~ u : ~ i t w a i o i o 1 6 ~ m ~ i a n i d eiom ~m:mtnroun~-rr:uu~~~~uiiniumr-r ( E X ~ C ~

Representation of The Systems) ~ # ~ ? u L L Y u ~ ~ D ~ NARMA kPk~ldd