Delay-Dependent State Estimation for Time Delay Systems

Transcription

1 WSEAS TRANSACTIONS o SYSTEMS ad CONTROL Mohammad Al Pakzad, Bja Moav Dlay-Dpd Sa Esmao for Tm Dlay Sysms MOHAMMAD ALI PAKZAD Dparm of Elcrcal Egrg Scc ad Rsarch Brach, Islamc Azad Uvrsy Thra IRAN m.pakzad@srbau.ac.r BIJAN MOAVENI Dparm of Ralway Egrg Uvrsy of Scc Tchology Thra IRAN b_moav@us.ac.r Absrac: - I hs arcl, a obsrvr wh dlay-dpd sably codos has b prsd for m dlay sysms wh ukow pus, whr h m dlay rms xs h sa ad oupu of h sysm. Th dsgg of hs obsrvr, whos prformac dx s h H orm, has b formulad as a olar opmzao problm. By usg h f characrzao of a Lyapuov fucoal quao, suffc codos hav b proposd o guara h xsc of a dsrabl H flr. Ths f characrzao ca b calculad by mas of a marx xpoal fuco. Svral umrcal xampls hav also b smulad o show h ffcvss ad smplcy of h proposd obsrvr. Ky-Words: - m dlay sysm, Lyapuov fucoal, sa dlay sysm, H flr, obsrvr, Kalma flr Iroduco A dyamc sysm whos sa varabls ar smaos of h sa varabls of aohr sysm s calld h obsrvr of ha sysm. Ths xprsso was frs roducd 963 o h hory of lar sysms by Lubrgr []. H showd ha for vry obsrvabl lar sysm, a obsrvr ca b dsgd whos smao rror (.. h dffrc bw h ral sa of h sysm ad h obsrvr sa) bcoms zro a vry cosdrd spd. I fac, a obsrvr s a dyamc sysm whos pus ar h procss pus ad oupus, ad whos oupus ar h smad sa varabls. I ca b sad ha a smaor of sa s a dspsbl mmbr of h corol sysms hory, ad has mpora applcaos fdback corol, sysm suprvso ad h faul dagoss of dyamc sysms. I h corol procss, s of assumd ha h ral sa vcors xs ad ar avalabl h masurm of h oupu; whl pracc, hs s o h cas, ad s cssary o dvs a obsrvr ordr o provd a smao of sa vcors. If h smao ad rcosruco of all h sa varabls s dd, h full-ordr obsrvrs, ad f h smao ad rcosruco of a umbr of sa varabls s dd, h rducd-ordr obsrvrs ar usd. Durg h las dcad, h hory of obsrvr dsg for m dlay sysms has b wdly complad [-8]. Th smao of sa varabls s a mpora dyamc modl, whch adds o our kowldg of dffr sysms ad hlps us aalyz ad dsg varous corollrs. Dffr approachs hav b usd for h dsgg of obsrvrs, cludg: h coorda chag approach [9], h LMI mhod [], rducg rasformao chqu [], facorzao approach [], polyomal approach [3], modal obsrvr [4], rducd-ordr obsrvr [5] ad h oupu jco basd obsrvr [6]. I [7], hrough a algbrac approach, a obsrvr wh dlay-dpd sably for sysms wh o oupu dlay has b prsd. I [8], a obsrvr has b proposd ha uss h H orm as h prformac dx. Th H flr has b cosdrd [9], [] by applyg h dlay dpd sably codos, whch h marx qualy has b usd. W also frquly cour h ssu of sa dlay corol problms ad physcal sysms. I rc yars, h sysms wh dlay sa hav aracd h ao of may rsarchrs, ad umrous E-ISSN: Issu, Volum 8, Jauary 3

2 WSEAS TRANSACTIONS o SYSTEMS ad CONTROL Mohammad Al Pakzad, Bja Moav approachs hav b proposd for h valuao of sably hs sysms (s [] ad [, 8] ad h rfrcs cd hm). Tm-dlayd sysms play sgfca rols horcal as wll as praccal flds; ad hs fluc ca b obsrvd umrous rsarch arcls wr o varous problms ha volv hs class of sysms [3-4]. Th goal of hs arcl s o dsg a obsrvr for m dlay sysms whch h m dlay rms xs h oupu ad h sa varabls, ad also h pus ar mxd wh os ad h sysm oupu accompas masurm oss. I h dsgg of hs obsrvr w hav usd h H orm as h prformac dx. Howvr, dsp h usfulss of h H orm, fw obsrvrs hav usd as h prformac dx. I [3] ad [4] a mhod has b proposd for h calculao of h H orm of m dlay sysms by mas of h dlay Lyapuov quao. I [5], a obsrvr has b offrd for m dlay sysms by applyg h dlay-dpd sably codos. I should b mod ha dlay-dpd approachs ar grally mor cosrvav ha dlay-dpd os. I hs arcl, for h smao of sysm sas, a H obsrvr has b proposd whos dsg uss h dlay-dpd sably codos. No ha wh hr ar o m dlay rms, h opmal H orm obsrvr s a sadard Kalma flr. Thrfor, h proposd flr ca b prsd as a Kalma flr for h m dlay sysms. Th opmal H orm obsrvr wll b dsgd by mployg h f characrzao of a Lyapuov fucoal quao as a marx xpoal fuco ad applyg h ucosrad olar opmzao algorhm. Fally, h proposd obsrvr hs arcl wll b usd o sma h curr sas basd o h m dlay sysm, whr h m dlay rms xs h sa ad h oupu of h sysm. Ths arcl has b orgazd h followg mar. I sco, for h dfo of h obsrvr, h cssary mahmacs has b prsd. I sco 3, h calculao of h H orm ad h Lyapuov fucoal quao has b offrd for h sa dlay sysm. I sco 4, h mhod of flr dsg has b dscrbd. I sco 5, ordr o s h praccal usfulss of h proposd chqu, has b appld for solvg h smao problm of svral lar sysms wh m dlay. Ad fally, h summary ad cocluso of h obad rsuls hav b prsd h las sco. Iroduco of h proposd obsrvr Cosdr lar m-vara sysms dscrbd by. x () = Ax() + Ax( h) + Bω() + Bu() y() = C x() + C x( h) + C ν () () p Whr x R s h sa, ω R s h procss q r os, u R s h pu, y R s h r masurm, ad ν R s h masurm os. Th h s cosa kow m dlay h sas ad h oupus. I s assumd ha ν ad ω ar ucorrlad wh Gaussa procsss, whch sasfy E{ ω( )} =, E{ ω( ) ω( s)} = Iδ( s) E{()} ν =, E{()()} ν ν s = Iδ( s) () Th objcv of hs papr s o drv a H flr for a m dlay sysm (), whr a flr has h followg form: x ˆ() = Ax ˆ ˆ () + Ax ( h) K C x() + C x( h) y () + B u() ( ˆ ˆ ) Dfg h smao rror () as () x () x ˆ() w oba (3) G : ( ) = A ( ) A( h) + Bξ ( ) (4) Whr A A KC, A A KC ω() B [ B, KC], ξ ( ) ν () Th H orm of h rror sysm s usd as h prformac dx sma T G = J( k, h) = lm E () () d T T (5) If hr ar o m dlay rms (.., A = ad C = ), h () bcoms E-ISSN: Issu, Volum 8, Jauary 3

3 WSEAS TRANSACTIONS o SYSTEMS ad CONTROL Mohammad Al Pakzad, Bja Moav x () = A x() + B ω() + B u() y() = C x() + C ν () ad h flr, mmzg h H orm (5) for hs o-dlayd sysm, s h sadard Kalma flr. Thus w ca call h proposd flr mmzg (5) a Kalma flr for m dlay sysms. 3 H Norm Compuauo Th H orm of G s xprssd rms of h marx fuco Ps ( ) h x horm. Thorm : If s sabl, h G = Tr( B P() B) (6) Whr P(), s s h s couously dffrabl ad sasfs P() = P () P ( s) = A P() + A P( h s), s h P () + P () + I = (7) Rmark : s rlad o h Lyapuov fucoal of sa dlay sysm (4). L V ( φ), φ C[ h,] b dfd by V ( φ) φ () P() φ() + φ () P( r) Aφ( h + r) dr h h + φ ( h + r) A P( r s) Aφ( h + r) dsdr h (8) Whr Ps () P ( s) f s <. Equao (7) s drvd from d V ( x ) = x ( ) x ( ) (9) d x ( r) x( + r), r h, Whr [ ] Rmark : If hr ar o m dlay rms, h rsul Thorm bcoms a sadard H orm compuao. S, for xampl, Thorm 3.3. [7]: h H orm of a sabl o-dlay sysm s gv by G = Tr( B PB) () Whr AP + PA + I= No ha codos (7) ar quval o hos () f h =, A = ad C =. Th proof of Thorm wll b gv usg Lmma ad. Lmma : If sysm G s sabl, h G = Tr( G ( jω) G ( jω)) dω () π Proof: Th rsul s sadard (s Chap 3.3 [7]). Lmma : If G s sabl ad P(), s s h sasfs (7), h P jω jω dω () () = ( ) ( ) π Whr j h ( jω) jωi A A ω (3) Proof: S [9]. (Proof of Thorm ) From Lmma, Tr( B P() B) = Tr B ( jω) ( jω) Bdω π = Tr{ B ( jω) ( jω) B} dω π, w hav Sc f ( jω) dω= f ( jω) dω Tr( B P() B) = Tr{ B ( jω) ( jω) B} dω π = Tr{ G ( jω) G( jω) } dω π Sc Tr( AB) = Tr( BA) whvr AB ad BA ar squar marcs, w hav Tr( B P() B) = Tr{ G ( jω) G ( jω) } dω= G π E-ISSN: Issu, Volum 8, Jauary 3

4 WSEAS TRANSACTIONS o SYSTEMS ad CONTROL Mohammad Al Pakzad, Bja Moav Th las qualy s from (). If G s sabl, h G ca b compud from P () Thorm. How o chck h sably of G wll b cosdrd lar Thorm ; frs w wll cosdr how o compu P () h x lmma. Noao: For a marx M M m m m m m m m m m = gv by M dos complx cojuga raspos of M h colum srg csm s dfd by [ csm m m m m m m m m m ] Lmma 3: If G s sabl, h P () ad P( h ) sasfyg (7) ar gv by ( I A ) + ( A I) ( I A ) T + ( A I) R R csp () csi = csp ( h) Whr T : = T T T, T k R Row vcor T, k s dfd by T, k k k T : =,, j ( ) + j ( j ) + (4) Whr k R, k s a row vcor whos k-h lm s ad all ohr lms ar. Ad * [ R R ] [ ] V Marcs ad V * ar from h sgular valu dcomposo of h followg I J Hh = U V ( xp( )) * (5) Whr U ad V ar uary marcs, ad R s a dagoal marx whos dagoal lms ar ozro sgular valus of ( I J xp( Hh) ). L T j do a marx wh (, j) -ry qual o ad all ohr rs qual o zro, ad l T R b h block marx T, T j (.., h (, j) -block of T s T j ). Marcs H ad J ar dfd by ( I A ) ( I A ) T I H, J = ( I A ) T ( I A ) I Proof: S []. No ha P () ca b compud from h marx xpoal (5) ad a smpl lar quao (4). Thus f G s sabl, h w ca asly compu H orm: s (6). Now h sably of G s cosdrd Thorm, whr a sably codo for rval dlay h s provdd. Thorm : Suppos G s sabl for h =. If H has magary gvalus { jω,, jωk } ad hr corrspodg gvcors ar gv by ν, νk, ν, ν k, ν =,, νk = ν ν, k, h G s sabl for h whr h s dfd by ν h m l k jω ν, l, l+ (6) whr ν, l, s ay ozro lm of ν l. Thorm s provd usg Lmma 4 ad 5. Lmma 4 s basd o h fac ha f G s sabl for E-ISSN: Issu, Volum 8, Jauary 3

5 WSEAS TRANSACTIONS o SYSTEMS ad CONTROL Mohammad Al Pakzad, Bja Moav h = ad G dos o hav ay magary pols for h, h G s sabl for h. Lmma 4: G s sabl for h f G s sabl for h = Th followg quao dos o hav ay roos for h, h : ) d( j h jωi A A ω ) = (7) Proof: S [3]. Sably of G for h = ca b asly chckd from gvalus of A + A. O h ohr had, chckg whhr (7) has ay roos for h s o asy: (7) should b chckd for all ω < ad h < h I h x lmma, s show ha a roo jω of (7) (f ay) s a gvalu of H. Lmma 5: If (7) has a rooω, h s a gvalu of H. Proof: Suppos (7) has a roo jω for h ; h hr xss x( C ) such ha j h x ( jωi A A ω ) = (8) Takg h raspos (o complx cojuga), w Oba jωh ( jωi A A ) x = L α C α α α = x α b dfd by jωh (9) whr, α, s a complx umbr. L ν b dfd by (u s h complx cojuga of u ) u v u () Whr αx α x u = C α x () Th horm s provd f w show ha hs ν ( v from h cosruco) sasfs ( jωi H) ν = : ha s, jω s a gvalu of H. From h dfo of H, w oba ( jωi H) ν jωi ( I A ) ( I A ) T = ν ( I A ) T jωi + ( I A ) ( jωi ( I A ) ) u ( I A ) Tu = ( jωi + ( I A ) ) u + ( I A ) Tu () Paro ( jωi H) ν o complx vcors ad l h -h block of ( jωi H) ν b dod by C. Th r, s gv by r r = ( jωi A ) α x A ( T α + T α + + T α ) x Nog h followg rlao ( T α + T α + + T α ) x = ( T α + T α + + T α ) α = jωh αα W oba jωh jωh ( ) jωh ( ω ) jωh jωh jωh = ( ω ) αα α α r j I A A = α jωi A A α = α j I A A x =, Th las qualy s from (8). Sc r+ = r, (s ()), w hav r =, +. Hc, ( jωi H) ν =, whr ν (sc x ). Proof of Thorm : From h proof of Lmma 5, f (7) has a roo ω for h ( k), h ω s a gvalu of H. Furhrmor, h corrspodg gvcor of H s of h form: E-ISSN: Issu, Volum 8, Jauary 3

6 WSEAS TRANSACTIONS o SYSTEMS ad CONTROL Mohammad Al Pakzad, Bja Moav v = x x x x x x x x jωh jωh jωh jωh xx jωh jωh xx Thus h ca b compud as follows: T whr α s a cosa ad s chos so ha pkh (, ) JKh (, ) wh h h( k). Wh hs paly fuco, a cosrad opmzao problm (3) ca b rplacd by h followg ucosrad opmzao problm: m J ( K, h) G ( k, h) + p( K, h) (4) K P h = v l, l j v ω, l+ Whr v, l, l s ay ozro lm of v. If h mmum valu of h ( k) s h h (7) dos o hav a roo for h. From Lmma 4, hs provs h horm. Rmark 3: Oc a flr ga K s drmd, w ca chck h sably of h rror sysm (4) (Thorm ) ad compu s H orm (Thorm). No ha f h < h( k) (.., G s sabl), h J P ( K, h) = J( K, h). Also o ha f h h( k),h J P ( K, h ) s domad by h paly fuco pkh (, ). Thus h paly fuco pkh (, ) prvs usabl rgo sarchg wh h H orm s bg mmzd. A al valu of K ca b chos by mmzg J( K,): h al valu corrspods o h Kalma flr ga for a o-dlayd sysm. Mmzao problm (4) ca b solvd, for xampl, usg a ucosrad olar opmzao fuco fmuc MATLAB opmzao oolbox. 4 Flr Dsg I hs sco, h syhss algorhm of a H flr (3) s proposd, whr h algorhm s formulad as a cosrad olar opmzao problm. Wh mmzg H orm of G ovr K usg Thorm, should b guarad ha G s sabl. If h flr ga K s gv, h sably of G ca b chckd usg Thorm, whch provds a uppr sably boud h( k ) (.., G ( K, h) s sabl as log as h < h ). Thus fdg a opmal K, whch sablzs G ad mmzs G ( K, h) ca b formulad as follows: m J( K, h) G ( K, h) k subjc o h < h ( k ) (3) (3) s a cosrad olar opmzao problm whos global soluo s dffcul o fd. A subopmal approach s proposd o compu K usg paly mhods [6]. A paly fuco s dfd by f h < h ( K ) pkh (, ) α h h f h h K ( ) ( ) 5 Numrcal Exampl I hs sco, h smulaos hav b prformd by mas of h MATLAB sofwar. Exampl : Cosdr h followg frs-ordr m dlay sysm: x () = x() x( h) +.5 ω() + u() y() = x() + x( h) +.5 ν () (5) whr ω () ad v () ar h vcors of h pu os ad masurm os, rspcvly. I s assumd ha hs oss ar Gaussa procsss wh a avrag of zro ad ha ω () ad v () ar ucorrlad ad hy sasfy rlao (). I hs xampl, h =.5. Th opmzao problm (4) s solvd by mas of h Malab opmzao oolbox, ad for hs purpos, h opmzao fuco fmuc Malab s usd. By usg h =, h al valu for flr ga K s obad. Th valu of α h paly fuco has b adjusd a. Th valus calculad for h =.5 ar as follows: K =.73, h =.9, G ( K, h) =.77 Usg h compud flr ga, sa smao smulao was do, whr a u sp sgal was E-ISSN: Issu, Volum 8, Jauary 3

7 WSEAS TRANSACTIONS o SYSTEMS ad CONTROL Mohammad Al Pakzad, Bja Moav appld o h corol pu u () a m s. Th smulao rsul s gv Fg. paly fuco s s o. Th compud valus ar as follows:.77,.639, (, ).4 K = h = G K h =.76 Usg h compud flr ga, sa smao smulao was do, whr a u sp sgal was appld o h corol pu u () a m s. Th smulao rsuls ar gv Fg. ad Fg.3: ca b s ha h proposd H flr smas sysm sas wll. Fg. : Smulao rsul : ru sa ad smad valu ca b s ha h proposd H flr smas sysm sas wll. To s how h m dlay affcs smao prformac, H flrs wr dsgd for dffr h valus. As s Tabl, compud H orm crass as m dlay h crass. Tabl. Tm dlay ffcs o smao prformac. h =. h =.3 h =.5 h =.8 G ( K, h ) Fg. : Smulao rsul: ru sa (h frs lm of sa x) ad smad valu Varac of acual smao rror Exampl : I hs problm, h H flr s dsgd for h scod-ordr sysm gv h followg rlao. x () = () ( ) x + x h. + ω() u(). + (6) y() = [ ] x() + [ ] x( h) +.5 ν () whr ω () ad v () ar zro-ma, ucorrlad wh Gaussa procsss sasfyg (). Th m dlay s s o b h =.3. Opmzao problm (4) was solvd usg Malab opmzao oolbox. Th al valu of h flr ga K s compud usg h =, ad α h Fg. 3: Smulao rsul: ru sa (h scod lm of sa x) ad smad valu To s how h m dlay affcs smao prformac, H flrs wr dsgd for dffr h valus. E-ISSN: Issu, Volum 8, Jauary 3

8 WSEAS TRANSACTIONS o SYSTEMS ad CONTROL Mohammad Al Pakzad, Bja Moav As s Tabl, compud H orm crass as m dlay h crass. Varac of acual smao rror, whch was compud from a smulao, also crass as m dlay h crass. Ths vrfs a commo blf ha h m dlay advrsly affcs o smao prformac. Tabl. Tm dlay ffcs o smao prformac. h =. h =.3 h =.5 h =.7 G ( K, h ) Varac of acual smao rror Fg. 4: Smulao rsul: ru sa (h frs lm of sa x) ad smad valu Exampl 3: Cosdr h followg hrd-ordr sysm wh dlayd oupu ad sa: (7) x () = 3 x() 5 + x( h) ω( ) + u( ). y( ) = x( ) + x( h) +.5 ν ( ) [ ] [ ] whr ω () ad v () ar h vcors of h pu os ad masurm os, rspcvly. I hs xampl h =.6. Th opmzao problm (4) s solvd by mas of h Malab opmzao oolbox, ad for hs purpos, h opmzao fuco fmsarch Malab s usd. By usg h =, h al valu for flr ga K s obad. Th valu of α h paly fuco has b adjusd a 5. Th valus calculad for h =.6 ar as follows: Fg. 5: Smulao rsul: ru sa (h scod lm of sa x) ad smad valu.5338 K =.5596, h =.64, G ( K, h) = Th smulao rsuls ar gv Fg.4, Fg.5 ad Fg.6: ca b s ha h proposd H flr smas sysm sas wll. Fg. 6: Smulao rsul: ru sa (h hrd lm of sa x) ad smad valu E-ISSN: Issu, Volum 8, Jauary 3

9 WSEAS TRANSACTIONS o SYSTEMS ad CONTROL Mohammad Al Pakzad, Bja Moav As s Tabl 3, compud H orm crass as m dlay h crass. Tabl 3. Tm dlay ffcs o smao prformac. h =. h =.3 h =.6 h =. G ( K, h ) Varac acual of smao rror As s obsrvd, h cras of m dlay has a oppos ffc o h smao prformac, ad wh h cras of m dlay, h smao rror varac also crass. 6 Cocluso I hs arcl, a mhod was proposd for h dsgg of H obsrvrs for lar sysms wh m dlay h oupu ad sa varabls. By usg h f characrzao of a Lyapuov fucoal quao, h xsc of suffc codos for achvg h rgh soluo ad guarag h propr covrgc ra of h smao rror was valuad. Ths obsrvr provdd sasfacory rsuls praccal applcaos. Fally, by dsgg obsrvrs for hr lar sysms wh m dlays, h ffcvss of h proposd approach was dmosrad. Rfrcs: [] D. G. Lubrgr, A Iroduco o Obsrvrs, IEEE Tras. Auoma. Cor., vol. Ac-6, No. 6, 97, pp [] Y. M. Fu, G. R Dua, ad S. M. Sog, Dsg of ukow pu obsrvr for lar mdlay sysms, Iraoal Joural of Corol, Auomao, Sysms, vol., o 4, 4, pp [3] K. Subbarao, ad P. Muraldhar, A Sa Obsrvr for LTI Sysms wh Dlayd Oupus: Tm-Varyg Dlay, Procdgs of h Amrca Corol Cofrc, (Washgo, U.S.A), 8, pp [4] C.-M. Zhag, G.-Y. Tag, M. Ba, Sa obsrvr dsg for lar sysms wh dlayd masurms, Cofrc Procdgs - IEEE Iraoal Cofrc o Sysms, Ma ad Cybrcs, 8, pp [5] A. Faouh, O. Sam, ad J. M. Do, A ukow pu obsrvr dsg for lar mdlay sysms, Procdgs of h 38s IEEE Corol ad Dcso Cofrc,(Arzoa, USA) 999., pp [6] Y. S. Suh, Kalma Flr for Oupu Dlay Sysms, Procdgs of h 4s IEEE Corol ad Dcso Cofrc,Vol. 3, (Las Vgas, U.S.A.),, pp [7] O. Sam, A. Faouh, ad J. M. Do, Furhr rsuls o ukow pu obsrvrs dsg m-dlay sysms, Procdgs of h 4s IEEE Corol ad Dcso Cofrc, Florda, USA,, pp [8] Y. S. Suh, H. J. Kag, Y. S. Ro, H Flr for Tm Dlay Sysms, Iraoal Joural of Corol, Auomao, Sysms, Vol. 4, No. 5, 6, pp [9] M. Hou, P. Zk, ad R. J. Pao, A obsrvr dsg for lar m-dlay sysms, IEEE Tras. o Auomac Corol, vol. 47, No.,, pp. -5. [] M. Darouch, Lar fucoal obsrvrs for sysms wh dlays sa varabls, IEEE Tras. o Auomac Corol, vol. 46, No. 3,, pp [] A. E. Parso ad Y. A. Fagbdz, A obsrvr for m lag sysms, IEEE Tras. o Auomac. Corol, vol. 34, No. 4, 989, pp [] Y. X. Yao ad Y. M. Zhag, Paramrzao of obsrvrs for m dlay sysms ad s applcao obsrvr dsg, IEE Proc. Corol Thory Appl., vol. 43, o. 3, 996, pp [3] O. Sam, Ukow pu robus obsrvr for m dlay sysm, Proc. of h 36h IEEE Cofrc o Dcso ad Corol, 997, pp [4] J. Lyva-Ramos ad A. E. Parso, A asympoc modal obsrvr for lar auoomous m lag sysms, IEEE Tras. Auoma. Cor., vol. 4, No. 7, 995, pp [5] M. Darouach, P. Prro, ad E. Rchard, Dsg of rducd-ordr obsrvrs whou ral dlays, IEEE Tras. Auoma. Cor., vol. 44, No. 9, 999, pp [6] M. Hou, P. Zk, ad R. J. Pao, A obsrvr dsg for lar m-dlay sysms, IEEE Tras. Auoma. Cor., vol. 47, No.,, pp. -5. [7] P. M. Na, R. Spah, A algbrac approach o dsg obsrvrs for dlay-dpd sably of sysms wh sgl oupu dlay, E-ISSN: Issu, Volum 8, Jauary 3

10 WSEAS TRANSACTIONS o SYSTEMS ad CONTROL Mohammad Al Pakzad, Bja Moav Procdgs of h Amrca Corol Cofrc, (Sa Fracsco),, pp [8] A. Faouh, O. Sam, ad J. M. Do, H obsrvr dsg for m-dlay sysms, Proc. of h 37h Cofrc o Dcso ad Corol, (Florda, USA), 998, pp [9] E. Frdma, U. Shakd, ad L. X, Robus H flrg of lar sysms wh m-varyg dlay, IEEE Tras. Auoma. Cor., vol. 48, No., 3, pp [] H. Gao ad C. Wag, Dlay-dpd robus H ad L /L flrg for a class of ucra olar m-dlay sysms, IEEE Tras. Auoma. Cor., vol. 48, No. 9, 3, pp [] S. Pakzad, ad M. A. Pakzad, Sably codo for dscr sysms wh mulpl sa dlays, WSEAS Trasacos o Sysms ad Corol, Vol.6, No.,, pp [] Y. S. Suh ad S. Sh, Sably of Sa Dlay Sysms Basd o F Characrzao of a Lyapuov Fucoal, Tras. of h Socy of Isrum ad Corol Egrs, Vol.35, No.9, 999, pp [3] M. A. Pakzad, ad S. Pakzad, Sably map of fracoal ordr m-dlay sysms, WSEAS Trasacos o Sysms, Vol., No.8,, accpd. [4] E. Jarlbrg, J. Vabrv, ad W. Mchls., Explc xprssos for h H orm of mdlay sysms basd o h dlay Lyapuov quao, Procdgs of h 49s IEEE Corol ad Dcso Cofrc, (Alaa, U.S.A),, pp [5] E. Jarlbrg, J. Vabrv, ad W. Mchls., Characrzg ad compug h H orm of m-dlay sysms basd o h dlay Lyapuov quao, IEEE Tras. Auoma. Cor., vol. 56, No. 4,, pp [6] M. S. Mahmoud, Robus Corol ad Flrg For Tm-Dlay Sysms, Marcl Dkkr, Nw York,. [7] D. Lubrgr, Opmzao by Vcor Spac Mhods, Wly, Nw York, 969. [8] M. A. Pakzad, Kalma flr dsg for m dlay sysms, WSEAS Trasacos o Sysms, Vol., No.8,, accpd. [9] M.Gr ad D.J.N.Lmbr, Lar Robus Corol, Prc-Hall, Eglwood Clffs, NJ, 995. [3] H. Wzhag, Gralzao of Lyapuov s horm a lar dlay sysm, J. Mah. Aa. & Appl., vol. 4, No., 989, pp [3] J. Chasso, A mhod for compug h rval of dlay valus for whch a dffral dlay sysm s sabl, IEEE Tras. Auoma. Cor., vol. AC-33, No., 988, pp [3] L. Pkar, Roo locus aalyss of a rardd quaspolyomal, WSEAS Trasacos o Sysms ad Corol, Vol.6, No.3,, pp [33] M. Bahador ad M. Baharak, Corol Dsg for Ucra Sgularly Prurbd Sysms wh Dscr Tm Dlay, WSEAS Trasacos o Sysms ad Corol, Vol.6, No.,, pp [34] Z. L, ad C. Schmd, A Gralsd Mmum Varac Corollr for Tm-Varyg MIMO Lar Sysms wh Mulpl Dlays, WSEAS Trasacos o Sysms ad Corol, Vol.7, No.5,, pp [35] T. Hashmoo, ad T. Ammya, Sablzao of Lar Tm-varyg Ucra Dlay Sysms wh Doubl Tragular Cofgurao, WSEAS Trasacos o Sysms ad Corol, Vol.4, No.9, 9, pp [36] K. Zakova, O Typ of Corollr Dsg for Dlayd Doubl Igraor Sysm, WSEAS Trasacos o Sysms ad Corol, Vol.3, No., 8, pp [37] Y. Chag, S. Ch, Sac Oupu-Fdback Smulaous Sablzao of Irval Tm- Dlay Sysms, WSEAS Trasacos o sysms, Vol. 7, No. 3, March 8, pp [38] R. Brca, A. Exbarra, Idusral PC-basd ral-m corollrs appld o scod-ordr ad frs-ordr plus m dlay procsss, WSEAS Trasacos o sysms, Vol. 7, No. 9, 8, pp [39] Y. K, C. Mao, Sably aalyss of BAM ural works wh ral rm ad m dlay, WSEAS Trasacos o sysms, Vol., No.,, pp [4] E. Gyurkovcs, T. Takacs, Oupu Fdback Guarag Cos Corol by Marx Iquals for Dscr-Tm Dlay Sysms, WSEAS Trasacos o sysms, Vol. 7, No. 7, 8, pp E-ISSN: Issu, Volum 8, Jauary 3