Neural Network Control of a Class of Nonlinear Discrete Time Systems with Asymptotic Stability Guarantees 1

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1 9 Aeia Cotol Cofeee Hyatt Regey Riefot St Louis MO USA Jue - 9 h Neual Netwo Cotol of a Class of Noliea Disete ie Systes with Asyptoti Stability Guaatees alaje huati ad S Jagaatha Abstat I this pape a sigle ad ulti-laye eual etwo (NN) otolles ae deeloped fo a lass of oliea disete tie systes Ude a ild assuptio o the syste uetaities whih ilude uodeled dyais ad bouded distubaes by usig oel weight update laws ad a obust te loal asyptoti stability of the losed-loop syste is guaateed i otast with all othe NN otolles whee a uifo ultiate boudedess is oally show Siulatio esults ae peseted to show the effetieess of the otolle desig S I INRODUCION igifat eseah has bee pefoed i the past deade i the aea of eual etwo (NN) otol fo oliea syste he NNs beae popula [ due to thei futio appoxiatio apabilities whih ae utilized to lea uetaities Due to a futioal eostutio eos with a NN [ typially the otolle desigs ede a uifoly ultiately boudedess esult sie this eostutio eo is assued to be uppe bouded by a ow ostat [- he NN otolle desigs wee fist itodued fo otiuous-tie systes [- ad late exteded to otol oliea disete-tie systes [-5 Deelopet of stable otolles fo disete-tie systes is athe diffiult sie the fist diffeee of a Lyapuo futio adidate is ot liea with espet to its states i otast to a fist deiatie of a Lyapuo adidate fo otiuous-tie systes All these otolles elax the pesistee of exitatio oditio o the iput sigals he NN otolle desigs wee the exteded to a oe geeal lass of oliea systes with state ad output feedba [ ad fo oliea disete-tie systes [5 Reetly a obust itegal of the sig of the eo (RISE) feedba ethod is used i ojutio with a NN to show sei global asyptoti taig of otiuous-tie oliea systes [6 o esue asyptoti pefoae of the NN otolle a attept has bee ade i [7 fo a lass of otiuous ad disete-tie oliea systes by usig a seto boudedess assuptio o the uetaities [7 A sigle laye NN is utilized i the otolle desig y otast i this pape howee we deelop a suite of alaje huati (e-ail: btt7@stedu) ad S Jagaatha (eail: saagap@stedu) ae with the Depatet of Eletial ad Copute Egieeig Missoui Uiesity of Siee ad ehology (foely Uiesity of Missoui Rolla) Rolla MO 659 USA Reseah suppoted i pat by NSF I/UCRC o Itelliget Maiteae Systes Awad ad Itelliget Systes Cete NN otolles fo a lass of oliea disete-tie systes that guaatee loal asyptoti stability ude a ild assuptio o the uetaities [6-9 Note ulie othe NN otolle desigs [ -5 the poposed desig is guaateed to ede loal asyptoti stability he poposed otolles utilize the filte taig eo otio ad a obust te his ew obust te is a futio of the NN weights Iitially a liealy paaeteized NN is utilized i the otolle desig ad late exteded to ultilaye NNs he stability is show usig the Lyapuo theoy Fially a siulatio exaple is utilized to illustate the pefoae of the poposed NN otolles A Neual Netwos II ACKGROUND A geeal oliea otiuous futio f ( x) C ( s) whih aps f : S R whee S is a siply-oeted set of R ad C ( s) is the spae whee f is otiuous a be witte as = () f ( x) σ( V x) whee V ad epeset iput-to the hidde laye ad hidde-to-the output laye weights espetiely ad ( ) is a eual et futioal eostutio eo eto suh that N fo all x R Additioally the atiatio futios σ () ae bouded easuable o-deeasig futios fo the eal ubes oto [ whih ilude fo istae sigoid et e defie the output of a NN as y = ϕ ( x) () whee ( ) is the atual weight atix ad ϕ ( x( )) is the atiatio futio whih is seleted as a basis futio [ i ode to guaatee the futio appoxiatio Apat fo the sigle laye a gie otiuous futio f () ould be witte usig a thee laye NN as [ ( f ( x ) = ϕ ϕ ( ϕ ( x ( )))) ( ) () whee ad ae the ideal weights Additioally the ideal weights ae osideed to be bouded ad ad ϕ () ϕ () ad ϕ () ae the atiatio futios of the fist seod ad thid laye of the NN espetiely Next we defie the output of a thee-laye NN as /9/$5 9 AACC 9

2 y = ϕ ( ϕ ( ϕ ( x))) () whee ae the atual NN weights of the thid seod ad fist laye espetiely ad ϕ ( x( )) epeset the atiatio futio eto of the iput laye he ϕ ( ϕ ( x)) ϕ ( ϕ ( ϕ ( x))) deote the hidde laye ad output layes atiatio futio espetiely at the th istat Fo a ultilaye futio appoxiatio the atiatio futio eto eed ot be a basis futio [ 6 Next the lass of oliea disetetie syste to be osideed i this pape is itodued Dyais of the th-ode MIMO Syste Coside a th-ode ulti-iput-ulti-output (MIMO) disete tie oliea syste gie by x ( ) = x x ( ) = x x ( ) = f ( x) u d whee [ x = x x with x R i = i u( ) R is the iput eto ad d ( ) deotes a distubae eto at th istat with d d a ow ostat M Gie a desied tajetoy x ( ) ad its delayed alues we d defie the taig eo as e = x x d Defie the filteed taig eo ( ) R [ as = e e e (6) λ λ whee e e ae the delayed alues of the eo e ad λ λ ae ostat aties seleted so that z λ z λ is withi a uit dis Subsequetly (6) a be witte as ( ) = e λ e λ e (7) y substitutig (5) i (7) we get ( ) = f ( x ( )) x ( ) λ e ( ) λ e ( ) u ( ) d ( ) (8) d u as Now defie the otol iput u = x ( ) f ( x) ( ) d e e (5) λ λ (9) whee is a use seletable diagoal atix is the obust te eto whih is defied late ad f ( x) is a estiate of f ( x( )) he losed-loop eo syste beoes ( ) ( ) d f x = % () whee f % ( x ) = f( x ) f ( x ) is the futioal estiatio eo I the ext setio we popose the NN weight update law ad the obust te Additioally the stability of the poposed NN otolle will be deostated III NN CONROLLER DESIGN I this setio we popose a sigle ad a thee-laye NN based otolles by usig a oel weight update law by elaxig the pesistey of exitatio oditio ad etaity equialee piiple [ Iitially we oside a sigle laye NN the a ultilaye NN A Sigle laye etwo y osideig ostat ideal weights the oliea futio i (5) ould be witte as f ( x) = ϕ( x) whee the taget weight atix is assued bouded suh that Defie the NN futioal estiate as f ( x ) = ϕ ( x) () ad the weight estiatio eo as % = () hus the otol iput (9) is gie by = ( ) ϕ ( ) λ λ d u x x e e Substitutig the aboe equatio i (8) esults i the followig losed-loop filteed eo dyais as ( ) = Ψ d () ϕ x whee Ψ = % ( ) Defie the obust te as ( ) l = whee R is a ostat eto ad > is a ostat he pupose of the obust te is to ipoe the stability of the otolle as explaied late i this pape Substitutio of the obust te ito () edes w ( ) ( ) = Ψ d () whee = Addig ad subtatig w C i () whee C R is a ostat w eto he filteed taig eo dyais beoes ( ) = Ψ Ψ ( w C) (5) ( w% C ) whee Ψ ( ) = Next the followig lea o the odelig uetaity ad bouded distubaes is itodued befoe poeedig futhe Lea : he te ( ) opisig of the appoxiatio eos ad bouded distubae d( ) is assued to be uppe bouded by a sooth oliea futio of filte 95

3 taig eo ad the NN weights [6-9 as d d d w ( 5 ϕ ϕ) % d w% (6) whee d d d ad d ae oputable positie ostats Poof: Usig soe stadad o iequalities the fat that ϕ () eto is bouded by ostats fo RF sigoid ad tah it is easy to show that the eostutio eo is a futio of the filteed taig ad weight estiatio eos Rea : Siila elatioship is stated by a ube of eseahes [6-9 i otiuous ad disete-tie his assuptio is ild i opaiso with the assuptio that the futioal eostutio eo ad distubaes ae bouded aboe by a ow ostat Next i the theoe it will be show that the poposed otol law edes a asyptotially stable syste heoe : Let x ( ) be the desied tajetoy ad the d iitial oditios be bouded i a opat set S Coside bouded uetaities ad the otol law (9) be applied to the syste Let the NN weight update law be poided by w ( ) = w ϕ( ) ( ) (7) whee > is the leaig ate he the filte taig eo ( ) ad the NN weight estiatio eos w % ae loally asyptotially stable Poof: Coside a Lyapuo futio adidate as [ w w V t = % % he fist diffeee is gie by Δ V = ( ) ( ) tw [ % ( ) w% ( ) w% () w% () (8) Substitute the filte taig eo (5) i Δ V of (8) ad afte pefoig soe atheatial aipulatios we get Δ V = Ψ Ψ C Ψ C Ψ Ψ w w Ψ ( w C ) Ψ Ψ Ψ Ψ Ψ ( w C ) ( w C ) ( w C ) Ψ ( ) ( ) (9) Next the weight update law (7) is substituted i the seod te of (8) ad usig () to get Δ Δ Δ V = [( ) ( ) ( ) t w % w % w % w % w % w % = t[ Δw% Δ w % Δw% w % = = t ( ( ) ϕ )( ϕ ( )) ( ) ϕ % t [ w [ Ψ Ψ ( ) ( w C ) ϕ( Ψ Ψ ( ) ) ( w C ) w% ( Ψ Ψ ) ( w C ) ( ) ϕ ϕ Applyig Cauhy-Shwaz iequality ( a a a ) ( a a a ) ( ( a a a a a a ) ) fo the fist te i the aboe equatio ad applyig the tae opeato (gie a eto x R t( xx ) = x x ) to obtai 5 ϕ ϕ( ) Ψ Ψ Ψ Ψ ( w C ) ( w C ) 5ϕ ϕ ( ) ( w C Ψ e Ψ Ψ ) () Next the oeall fist diffeee of the Lyapuo futio adidate = a be obtaied fo (9) ad () as Ψ Ψ ( w C ) Ψ ( w C ) Ψ Ψ w w Ψ ( w C ) Ψ Ψ Ψ Ψ Ψ ( w C ) ( w C ) ( w C ) Ψ ( ) 5 ϕ ϕ 5ϕ ϕψ Ψ 5 ϕ ϕψ Ψ w C w C 5 ϕ ϕ ( ) Ψ Ψ Ψ Ψ Ψ 5 ϕ ϕ Ψ ( )( w C ) Ψ Afte soe atheatial aipulatio ad usig Lea the fist diffeee is ewitte as 96

4 Ψ Ψ 5 ϕ ϕ 5ϕ ϕψ Ψ d d d w% d % ( 5ϕ ϕ % % d w )( w w C C ) ( 5ϕ ϕ )( ww wc C C ) ( d /( 5ϕ )) w C C = i w i i aig d / = d ad applyig the Fobeius o the fist diffeee a be expessed i opat fo as ( 5ϕ d d ) ( ϕ 5ϕ 5ϕ d d ) w i % () Hee Δ V < poided the gais ae tae as d d ( 5 ϕ ) ϕ 5ϕ d d i = ( 5ϕ ) ad < << As log as the gais ae seleted aboe Δ V < i () whih shows stability i the sese of Lyapuo Hee ( ) ad w % ae bouded poided if ( ) ad w% ( ) ae bouded i the opat set S Suig both sides of () to show that both ad oeges w % appoahes to zeo asyptotially Next we exted the aboe esults fo a thee-laye NN otolle hee-laye NN otolle Hee oside a thee-laye NN by usig () the NN output of a oliea futio i (5) ould be witte as f ( x) ϕ ( ϕ ( ϕ ( x))) = () Defie the weight estiatio eos as = = ( ) = ( ) Next the followig fat a be stated Fat : he atiatio futios ae bouded by ow positie alues so that ϕ ( ) ϕ ϕ ( ) ϕ ϕ ( ) ϕ Defie atiatio futio eto eo as ad ϕ = ϕ ϕ % ϕ = ϕ ϕ % = % ad ϕ ϕ ϕ hus by usig () i the otol iput (9) we get ϕ ϕ ( ϕ u = x ( ) ( ( x))) d λ e λ e () he the losed-loop filteed eo dyais beoe = Ψ d % ϕ ( x ( )) () whee Ψ = % ϕ ( x) ad the obust te fo this otol desig is gie by = ( ) whee > is a ostat ad is a appopiate diesioed ostat eto to be defied late Hee () would be odified to ( ) = Ψ d % ϕ ( x) ( ) Next by addig ad subtatig ( ) C i the aboe equatio whee C is a appopiate diesioed ostat eto to get ( C ) ( ) = Ψ Ψ (5) ( ) whee % ( ) C Ψ ( ) = = d % φ ( x) ( ) he followig theoe guaatees asyptoti stability of the losed-loop syste usig the poposed otol law i () heoe : Let x ( ) be the desied tajetoy ad the d iitial oditios be bouded i a opat set S Cosideig bouded uetaities ad the otol law poposed i () whee the thee laye NN is tued olie usig the followig weight update laws fo the iput ad hidde layes as ( ) = ϕ [ y (6) ( ) = ϕ [ y (7) with y = ϕ i i i ad κ i i i = ae the weight update law fo the output-laye as w = w ϕ ( ) (8) whee i > i = deotes the leaig ate o adaptatio gais he the filte taig eo ( ) is loally asyptotially stable while the NN weight estiatio eos w% w% ad w % ae bouded Poof: Coside a Lyapuo adidate as % % % % t[ w% w% V = t[ w w t[ w w he fist diffeee is gie by Δ = w% ( ) w% ( ) w% w% ( ) ( ) V t[ t [ w% ( ) w% ( ) w% w% 97

5 t[ w ( ) w ( ) w w % % % % (9) Substitutig (5) to (8) i (9) olletig tes togethe ad opletig squae yields Ψ Ψ C Ψ C Ψ Ψ Ψ ( C ) ( ) ( ) Ψ Ψ Ψ ( ) Ψ ( ) Ψ ( ) ( C ) ( C ) ( C ) Ψ ( ) ( ) ( ϕ ϕ ) ( ϕ ϕ ) % ϕ ( ϕ ϕ) ( ϕ ) ϕ ( ) ( ϕ ( ) ) κ κϕ ( ϕ ϕ ) ( ϕ ϕ ) ( ϕ ϕ ) ( ϕ ϕ ) % ϕ ( ϕ ϕ ) ( ϕ ) κ ϕ ( ) ( ϕ ( ) ) κ ϕ ( ϕ ϕ ) ( ϕ ϕ ) Ψ ( ) Ψ Ψ Ψ Ψ Ψ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ϕ ϕ Ψ Ψ Ψ ( C ) 5 5 ϕ ϕ Ψ Ψ ( ) ( ) 5 5 ϕ ϕ ϕ ϕ ( C ) ( C ) ϕ 5ϕ () ( ) Next the followig lea is itodued Lea : Usig Lea the te ( ) ad the ideal weights of the NN ae assued to be bouded aboe by a sooth oliea futio of filte taig eo ad the NN weights [6-9 as ( ϕ κ ) ϕ ( ) i i i i i= ( ϕ ) i i (5 ϕ ϕ ) β β β β w% w% () whee β β β ad β ae oputable positie ostats Poof: Siila to Lea usig soe stadad o iequalities the fat that ϕ () ϕ () ad ϕ () etos ae bouded by ostats fo RF sigoid ad tah; ad the eostutio eo is a futio of the filteed taig eo ad the weight estiatio eos Usig Lea taig the Fobeious o of () ad taig C i C ( β /( 5 ϕ ϕ )) = i i γ ( β β ) we get ( ϕ ϕ ) ( ϕ ϕ ) % ϕ ( ϕ ϕ ) ( ϕ ) ( ϕ ϕ ) ( ϕ ϕ ) % ϕ ( ϕ ϕ ) ( ϕ ) ϕ β β ϕ ( 5 ) w% () whee β / = β ad γ = i= κ ϕ κ i i i 5 ϕ ( ) ( ϕ ) i i i () poided the followig gais ae seleted γ β β β β ϕ ( ) ϕ ( ) (8 ) he β β 5 ϕ ( ) = = ad he the fist diffeee i () whih shows stability i the sese of Lyapuo poided the gais ae seleted aboe Hee ( ) w % w % ad w% ae bouded poided if ( ) w% ( ) w% ( ) ad w% ( ) ae bouded i the opat set S Additioally by usig [5 we ould show that the taig eo as Hee ( ) oeges asyptotially I the ext setio siulatio esults ae itodued IV SIMULAION RESULS Coside the followig oliea disete-tie syste [5 X ( ) = Δ t( X ) X X ( ) = Δ t( F( X X )) U X D () 98

6 whee U [ u u X = [ x x X = [ x x F = ad the oliea futio X X gie by F( X X ) = [ M( X ) G( X X ) whee is Eo Asyptoti otolle pefoae [ M ( X ) = ad ( b b) a ba baa ( x ) ba baa ( x ) os os ba baa os( x ) ba G( X ) G ( ) X = si 98 os baax ( x ) ba ( x x ) with si 98 os G = b a a ( x x x ) ( x ) ( b b ) a ( x ) 98ba os( x x ) Also D( ) is the distubae eto whih is gie by D = fo < Δt se else D () is sapled at Δ t = se π Δt gie as si 5 = he syste i he desied tajetoies ae π Δt ad os 5 Additioally the paaetes of the oliea syste ae tae as a a b = ad b = he iitial oditios of the oliea = = syste ae hose to be X = [5 ad X = [ 5 Also the otolle gais ae hose to be = diag{ } ad λ = diag{9 9} Eo Eo Asyptoti otolle pefoae ie (Se) ouded otolle pefoae ie (Se) Figue : aig eo fo the efeee tajetoy by the asyptoti ad the bouded NN otolles he taig eos fo both the efeee tajetoies ae show i Figs ad he poposed asyptoti otolle (NN te obust te) is opaed agaist aothe bouded NN otolle peseted i [ without a obust te Fo the Figs ad it is eidet that the poposed asyptoti otolle ahiees a highly satisfatoy taig pefoae ee i the pesee of distubae whe opaed to a NN otolle that edes uifoly ultiately bouded esult O the othe had a stadad PD otolle edes ustable esults due to the size of the distubae ad theefoe it is ot suitable Eo ie (Se) ouded otolle pefoae ie (Se) Figue : aig eo fo the efeee tajetoy by the asyptoti ad the bouded NN otolles V CONCLUSIONS I this pape a suite of NN otolles wee deeloped fo oliea disete tie systes y usig a oel obust te ad based o ild assuptio o the NN appoxiatio eos ad i the pesee of bouded distubaes the asyptoti taig is deostated though Lyapuo aalysis Siulatio studies eify the theoetial ojetues REFERENCES [ S Jagaatha ad F L Lewis Disete-tie eual et otolle fo a lass of oliea dyaial systes IEEE as o Autoati Cotol ol o pp [ F L Lewis A Yesildie ad K Liu Multilaye eual-et obot otolle with guaateed taig pefoae IEEE as o Neual Netwos ol 7 pp Ap 996 [ S S Ge C C Hag ad Zhag Adaptie eual etwo otol of oliea systes by state ad output feedba IEEE as o Syste Ma Cybeetis- Pat ol 9 o 6 pp [ S Jagaatha ad F L Lewis Multilaye disete-tie eual-et otolle with guaateed pefoae IEEE as o Neual Netwos ol 7 o pp [5 S Jagaatha Neual etwo otol of oliea disete-tie systes CRC Pess Apil 6 [6 P M Pate MaKuis K Kaise ad E Dixo Asyptoti taig fo uetai dyais systes ia a ultilaye NN feed fowad ad RISE feedba otol stutue Po of the Aeia Cotol ofeee New Yo City NY USA pp [7 Xia D M Dawso M S de Queioz ad J Che A otiuous asyptoti taig otol stategy fo uetai oliea systes IEEE as o Autoati Cotol ol 9 o 7 pp 6- [8 Hayaawa M Haddad ad N Hoaiya Neual etwo adaptie otol fo a lass of oliea uetai dyaial systes with asyptoti stability guaatees IEEE as o Neual Netwos ol 9 o pp [9 C M Kwa D M Dawso ad F L Lewis Robust adaptie otol of obots usig eual etwo: global taig stability Po of the th ofeee o Deisio ad Cotol New Oleas LA pp