New Guaranteed H Performance State Estimation for Delayed Neural Networks

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1 Ieraoal Joural of Iformao ad Elecrocs Egeerg Vol. o. 6 ovember ew Guaraeed H Performace ae Esmao for Delayed eural eworks Wo Il Lee ad PooGyeo Park Absrac I hs paper a ew guaraeed performace sae esmao problem for sac eural eworks wh mevaryg delay s vesgaed. A ew Lyapuov-Krasovsk fucoal s roduced o mprove he performace. Moreover wh he help of lower boud lemma a upper-boud of a lear combao of posve fucos weghed by he verses of covex parameers s obaed. wo smulao examples are gve o prove he effecveess of he proposed heorem. Idex erms ae esmao sac eural eworks H-fe performace recprocally covex approach mevaryg delay. I. IRODUCIO eural eworks have araced cosderable aeo from academc research ad dusral applcaos durg he pas decades. Varous successful applcaos have bee fouded may felds such as paer recogo mage processg opmzao problems ad adapve corol. me delay s wdely exss may praccal sysems such as chemcal or process corol sysems ad eworked corol sysems []. Also me delay may exs eural eworks because of her fe swchg speeds ad commucao me. ce hese me-delay may duce sysem sably ad performace degradao he sably aalyss of delayed eural eworks has become a mpora ssue ad may resuls have bee repored he leraure []-[4]. ae esmao problem of eural ework s very praccal ad heorecally mpora ssue whch has bee suded rece years [5]-[7]. I may praccal applcaos he euro saes are o always measurable he eural eworks oupus sce may be very dffcul ad expesve o acqure all he sae formao of he euro saes large-scale eural eworks. Bu he sae formao may be ceraly ecessary for some applcaos such as sysem modelg ad sae feedback corol. herefore hs case he euro saes should be esmaed by measuremes Mauscrp receved Augus ; revsed epember. hs research was suppored by he MKE(he Msry of Kowledge Ecoomy) Korea uder he IRC (Iformao echology Research Ceer) suppor program supervsed by he IPA(aoal I Idusry Promoo Agecy) (IPA- -H--) & IPA--(H--)). hs research was suppored by World Class Uversy program fuded by he Msry of Educao cece ad echology hrough he aoal Research Foudao of Korea (R-). Wo Il Lee s wh he Deparme of Elecrcal Egeerg Pohag Uversy of cece ad echology (POECH) Pohag Republc of Korea (e-mal: wlee@ posech.ac.kr). PooGyeo Park s wh he Dvso of ICE ad Deparme of Elecrcal Egeerg Pohag Uversy of cece ad echology (POECH) Pohag Republc of Korea (e-mal: ppg@posech.ac.kr). proves he mporace of he sae esmao problem for eural eworks. Recely [5] proposed a guaraeed performace sae esmaor for sac eural eworks wh mevaryg delay. Bu he process of dervg lower bouds of oe egral erm [5] roduced a approxmao leadg o a lle coservaveess. I hs paper we propose a ew guaraeed performace sae esmaor for delayed eural eworks based o a ew Lyapuov-Krasovsk fucoal. By applyg [8] s lower boud lemma a mproved performace s obaed. hs paper s orgazed as follows. he sae esmao problem s formulaed eco. eco proposes a ew guaraeed H-fy performace sae esmaor for delayed sac eural eworks. I eco 4 wo smulao examples are gve o prove he effecveess of he proposed heorem. II. PROBLEM FORMULAIO Cosder he delayed sac eural ework subjec o ose dsurbaces: : Σ x () = Ax () + f( Wx ( d( ) ) + J) + Bw() () y () = Cx () + Dx ( d ()) + Bw () () z () = Hx () () xs () = φ() s s [ ] (4) where x () = [ x() x () x()] R s he euro sae m vecor; y () R s he ework oupu measureme; p z () R s a lear combao of he saes o be q esmaed; w () R s a ose dsurbace belogg o L [ ) ; A = dag{ a a a } s a dagoal marx wh posve eres a > ; W s he ercoeco marces represeg he weghg coeffces of he euros; f ( x( )) = [ f ( x ( ) f ( x ( )) f ( ( ))] x s he euro acvao fuco; J = [ J J J ] s a exeral pu vecor; φ () s s a al codo o [ ] ad d () s a me-varyg delay of he sysem sasfyg d () d () μ. (5) I hs paper we choose a sae esmaor for he eural ework ( Σ ) as DOI:.776/IJIEE..V.4 9

2 Ieraoal Joural of Iformao ad Elecrocs Egeerg Vol. o. 6 ovember Σ : x ˆ() = Ax ˆ() + f( Wx ˆ( d ()) + J) + K[ y () Cx ˆ() Dxˆ( d( ))] (6) z ˆ( ) = Hx ˆ() (7) xs ˆ( ) = s [ ] (8) p where x ˆ() R z ˆ() R ad K s he sae esmaor ga marx o be deermed. Defe he errors o be e () = x () xˆ( ) ad z() = z() z ˆ( ). he he error-sae sysem s represeed by Σ : e () = ( A+ KCe ) () KDe ( d ()) + ψ ( We ( d () ) E xˆ( d ( ))) + ( B KB ) w( ) (9) z() = He() () where ψ ( We() xˆ()) = f ( Wx() + J ) f ( Wxˆ() + J ). Defo : he error sysem ( Σ ) s sad o be globally sable wh H performace γ f for some scalar γ > here exss a proper sae esmaor ( Σ ) such ha he equlbrum po of he resulg error sysem (9) wh w () s globally asympocally sable ad E z() < γ w() () R > (j=) Z ad wo dagoal marces Λ= dag( λ j λ λ ) > Γ= dag( γ γ γ ) > such ha he followg LMIs hold + R + R > () Ω Ω R W LΛ P Ω Ω 9 * Ω R R W LΓ D G * * Q R R * * * Q * * * * Ω 55 * * * * * R R * * * * * * Λ+ Z * * * * * * * Ω P * * * * * * * * γ I Ω9 * * * * * * * * * P + X < (4) Ω Ω R W LΛ P Ω Ω 9 * Ω R R W LΓ D G * * Q R R * * * Q * * * * R R * * * * * Ω 66 * * * * * * Λ+ Z * * * * * * * Ω P * * * * * * * * γ I Ω9 * * * * * * * * * P + X where < (5) Uder zero-al codos for all ozero w () L [ ) where η() = η () η() d. Assumpo. he euro acvao fucos () f ( ) sasfy he followg Lpschz codo f ( x) f ( y) l x y R (= ) () x y wh L = dag( l l l ) >. III. GUARAEED H PERFORMACE AE EIMAOR hs seco s dedcaed o he desg of a guaraeed H performace sae esmaor for he delayed sac eural ework. A delay depede LMI based codo wll be esablshed. heorem. Uder Assumpo gve posve scalars μ ad prescrbed cosa γ > sae esmao problem of he delayed sac eural ework ( Σ ) wh guaraeed H performace s solvable f here exs appropraely dmesoed marces P > Q > (=) > j Ω = + + PA A P GC C G Q Q 9 + H H + R Ω = GD + Ω = PB GB Ω = APCG Ω = ( μ) Q + + R R Ω =Ω =R R Ω =( μ) Z Γ Ω = BPBG 9. I hs case a desred he sae esmaor ga marx K s gve as K = P G. Proof. Choose a Lyapuov-Krasovsk fucoal caddae as V(()) e = e () Pe() + e () s Qe() + e () s Q e() s d( ) ( s) Q ( s) ds e ( s) e( s) dsd + + η e( sesdsd ) ( ) + e( sresdsd ) ( ) dη + + η + d( ) () sresdsd () dη ψ ( We( s) xˆ( s)) Zψ( We( s) xˆ( s)) ds. (6) Calculag he me-dervave of V(()) e alog he rajec- 94

3 Ieraoal Joural of Iformao ad Elecrocs Egeerg Vol. o. 6 ovember ores of sysem () ad og ha d () sasfes (5) yelds V (()) e e ()[ P( A+ KC) ( A+ KC) P+ Q + Q + ]() e e ( ) PKDe( d( )) + e ( ) Pψ ( We( d ( )) xˆ ( d()) + e () PB ( KB) w () ( μ) e ( d ()) Qe ( d ()) + ψ ( We( ) xˆ( )) Zψ( We( ) x ˆ( )) ( μψ ) ( We( d( )) x ˆ( d ( ))) Zψ ( We( d ( )) x ˆ( d ( ))) e( ) Qe( ) ( ) Q ( ) + ( )[ + Q + ( R + R )] ( ) e () sesds () s() + e sre + () () d sr(). Usg Jese s equaly [9] oe ca oba e () s e() d( ) d( ) = e () s e() e () s e() (7) d( ) d( ) ( d ( ))[ e( sds ) ] [ esds ( ) ] d () d () d ( )[ e( sds ) ] [ esds ( ) ] d( ) d( ) d () d () (8) () s () d( ) d( ) = () s () () s () [ e ( d()) e ( )] [ e( d()) e( )] d () [ e ( ) e ( d( ))] [ e( ) e( d( )] d () (9) () s R() + d( ) d( ) d( ) + + d( ) () s R() d( ) = () s R() d () s R() [ e () e ( s) ds] R[ e() e( s) ds] d( ) d( ) d () d () d( ) [ e ( d()) e ( s) ds] R[ e( d()) d () d( ) e( s) ds] + [ e ( ) e ( d( ))] R[ e( ) d () e( d( ))] [ e ( ) e ( d( ))] R[ e( ) e( d( ))] d () () + () s R () + d( ) + d( ) d( ) d( ) () s R () d( ) = () s R () d () s R () [ e ( s) dse ( d( ))] R [ e( s) ds d( ) d( ) d () d () d( ) e ( d ( ))] [ e ( sds ) e ( )] d () d( ) R [ e( s) ds e( )] d () + [ e ( d()) e ( )] R [ e( d()) e( )] [ e ( d( )) e ( )] R [ e( d( )) e( )]. d () () Le us defe = d ()/ ad λ() = col{ (() e e ( d () ) (( ed ()) e ( ))}. he gaherg he posve fucos weghed by he verses of he covex parameers { } or equvalely from (9)-() ad applyg lower { / d ( ) /( d ( ))} boud lemma [8] for sasfyg () we have λ () + R + R λ() whch produces a upper-boud as [ e ( ) e ( d( ))]( + R )[ e( ) e( d( ))] d () [ e ( d ( )) e ( )]( + R)[ e ( d ( )) e ( )] d () e () e ( d ()) + R e () e ( d ()) ( ( )) ( ) ( ( )) ( ). ed e + R ed e () By assumpo oe ca oba ha for ay We ψ ( We xˆ) f ( Wx+ J ) f ( Wxˆ+ J ) = l We WxWxˆ where W = [ w ] w w s he -h row vecor of W. he he followg equales hold ψ ( We() xˆ()) Λψ( We() xˆ()) + ψ ( We( ) xˆ ( )) ΛLWe( ) () 95

4 Ieraoal Joural of Iformao ad Elecrocs Egeerg Vol. o. 6 ovember ψ ( We( d( )) xˆ( d ( ))) Γψ( We ( d ( )) x ˆ( d ( ))) + ψ ( We( d ()) x ˆ( d ( ))) ΓLWe ( d ()) (4) For ay dagoal marces Λ= dag( λ λ λ ) > ad Γ= dag( γ γ γ ) >. Defe J () = [ z () sz( s) γ w() sws ()] ds for >. he for ay o-zero w () L [ ) J () () s w() sws ()] ds+ Ve ()) Ve (()) z [ z z( s) γ = () γ () () [ z s z( s) w s w s + V ( e( s))] ds. (5) From he codo () ad (7)- (4) ca be see ha () z() γ w () w() + V( e() ξ ()[ Ω + Ω XΩ d () Ω Ω ( d ()) Ω Ω ] ξ() where 4 4 ) (6) ξ() = [ e () e ( d()) e ( ) ( ) d( ) esd () s esds () ψ ( We( ) x ˆ( )) d( ) d () d () ψ ( We( d( ) ) x ˆ( d( ))) w( ) ] (7) Ω Ω R W LΛ P Ω 9 * Ω R R W LΓ * * Q R R * * * Q Ω = * * * * R R * * * * * R R * * * * * * Λ+ Z * * * * * * * Ω * * * * * * * * γ I Ω = PA A P PKC C K P Q Q H H + R Ω = PKD + Ω = PB PKB 9 [ A KC KD I B KB ] Ω = ( + ) X = + Q + R + R Ω = Ω =. [ I ] [ I ] 4 ce () d ( d ()) Ω 4 4 (8) Ω Ω Ω s a covex combao of he marces Ω Ω ad Ω Ω o d () 4 4 ca be o-coservavely hadled by wo boudary LMIs: oe for d () = ad he oher for d () =. Pre- ad pos- mulplyg wo LMIs by dag{ I I I I I I I I I PX }. Usg followg equaly for ay real P > ad X > PX P P X P X X P X + = ( ) ( ) PX P P + X (9) Ad applyg he chage of varable such ha K = P G oe ca deduce he LMIs ()-(5) mply z () s z () s γ w () s w(s) + V (()) e s < for w (). herefore J () < from (5) for > hus () holds. Globally asympocally sably of he equlbrum po of he error sysem (9) wh w () s acheved f V (()) e < holds. Oe ca easly prove ha he codo V (()) e < s guaraeed by he LMIs ()-(5). We skp he specfc proof due o space lmao. IV. IMULAIO EXAMPLE wo smulao examples are gve hs seco o llusrae he effecveess of he developed approach. Example. Cosder a delayed sac eural ework ( Σ ) wh he followg parameers: A = B =... D = [.5 ] W = B = [ ] H = C = J =. he acvao fuco he me varyg delay ad he ose dsurbace are ake as f ( x) = ah( x) wh L = I d ( ) = cos(.4 ) wh = ad μ =. ad w () = /(.8+.) for > respecvely. he solvg heorem by resorg o he LMI solver he Malab LMI Corol oolbox he sae esmaor ga marx ca be foud as K = Wh he opmal H performace dex γ =.75. I m s easy o oce ha hs resul s a mproved resul ha γ =.6 [5]. Fg. represes he error e () for m radom al values. Example. Cosder a delayed sac eural ework ( Σ ) wh he followg parameers: A = W =

5 Ieraoal Joural of Iformao ad Elecrocs Egeerg Vol. o. 6 ovember. B =..4. B = [ ] C = smulao examples proved he mproveme of he proposed heorem compared o exsg oe. I s worh ocg ha he proposed heorem ca be wdely applcable corol felds such as sae feedback corol problems ad large scale eural eworks ad so o. Fg.. he error e ( ) for radom al value. ABLE I: COMPARIO OF HE OPIMAL H PERFORMACE IDEX γ ( μ ) (.9.) (.9.5) (.8.7) [5] heorem D = [ ] H =.5 L = I. Usg heorem he opmal H performace dex γ ca be derved a dffere ( μ ) ad hese are m summarzed able. I s easy o see ha our mehod ca oba much beer γ ha hose [5]. m V. COCLUIO I hs paper he guaraeed H-fe performace sae esmao problems s suded for delayed sac eural eworks. Based o a ew Lyapuov-Krasovsk fucoal we solved he guaraeed H-fe performace sae esmao problem. Moreover wh he help of [8] s lower boud lemma we could oba a mproved H-fe performace resuls for delayed sac eural eworks. I s show ha he guaraeed H-fe performace sae esmaor ga marx ca be foud by solvg LMIs. wo REFERECE [] J. P. Rchard me-delay sysems: A overvew of some rece advaces ad ope problems Auomaca vol. 9 o. pp [] Y. He G. P. Lu D. Rees ad M. Wu ably aalyss for eural eworks wh me-varyg delay IEEE ras. eural eworks vol. 8 o. pp [] X M. Zhag ad Q. L. Ha ew Lyapuov-Krasovsk fucoals for global asympoc sably of delayed eural eworks IEEE ras. eural eworks vol. o. pp [4] Y. Lu Z. Wag ad X. Lu Asympoc sably for eural eworks wh mxed me-delays: he dscree-me case eural eworks vol. o. pp [5] H. Huag G. Feg ad J. Cao Guaraeed performace sae esmao of sac eural eworks wh me-varyg delay eurocompug vol. 74 o. 4 pp [6] H. Huag ad G. Feg Robus sae esmao for ucera eural eworks wh me-varyg delay IEEE ras. eural eworks vol. 9 o. 8 pp [7] Y. He Q. G. Wag M. Wu ad C. L Delay-depede sae esmao for delayed eural eworks IEEE ras. o eural eworks vol. 7 o. 4 pp [8] P. Park J. W. Ko ad C. Jeog Recprocally covex approach o sably of sysems wh me-varyg delays Auomaca vol. 47 o. pp [9]. Boyd L. E. Ghaou E. Fero V. Balakrsha Lear Marx Iequales ysem ad Corol heory Phladelpha PA: IAM 994 pp. -. Wo Il Lee receved hs B.. degree elecroc ad elecrcal egeerg from Kyugpook aoal Uversy. He s currely sudyg oward hs Ph.D. a Pohag Uversy of cece ad echology (POECH). Hs curre research eress clude robus Lear Parameer Varyg (LPV) delayed sysems ad eural eworks. PooGyeo Park receved hs B.. ad M.. degrees Corol ad Isrumeao Egeerg from eoul aoal Uversy Korea 9 ad 99 respecvely ad he Ph.D. degree Elecrcal Egeerg from aford Uversy U..A ce 996 he has bee afflaed wh he Dvso of Elecrcal ad Compuer Egeerg a Pohag Uversy of cece ad echology where he s currely a Professor. Hs curre research eress clude robus Lear Parameer Varyg (LPV) Recedg Horzo Corol (RHC) ellge ad ework-relaed corol heores sgal processg ad wreless commucaos for persoal area ework (PA). 97