Approximation of Pure Time Delay Elements by Using Hankel Norm and Balanced Realizations

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1 БЪЛГАРСКА АКАДЕМИЯ НА НАУКИТЕ BULARIAN ACADEY OF SCIENCES ПРОБЛЕМИ НА ТЕХНИЧЕСКАТА КИБЕРНЕТИКА И РОБОТИКАТА 64 PROBLES OF ENINEERIN CYBERNETICS AND ROBOTICS 64 София Sofia Approxiaio of Pur Ti Dlay El by Uig al Nor ad Balad Ralizaio Ka Prv Sy ad Corol Dpar Thial Uivriy of Sofia 756 Sofia Eail: Abra: Th papr oidr h probl of pur i dlay l approxiaio by uig al or ad balad ralizaio. Th wighd dlay l ar rprd i r of Pad ad Kauz ri or by uig Lagurr Pad ad Kauz hif opraor dripio. Quaiaiv rror boud o dlay approxiaio ar giv. al or ad balad ralizaio ar ud o dri h ordr of ruaio. Th rdud ordr odl ar opard wih h full ordr approxiaio o dri h ffiiy of h prd hod. Kyword: pur i dlay Lagurr Pad ad Kauz hif opraor approxiaio balad ralizaio al or approxiaio.. Iroduio ay idurial pro xhibi i dlay i hir bhaviour. Evry aio of h orol ipu for uh pro will aff h aurd oupu afr rai aou of i. Th dlay l odl boh h xiig i dlay i raporig rgy arial ad iforaio a wll a h xi of highr ordr r by auulaio of i lag. Th pr of i dlay ipo rai liiaio o ahivabl fdba prfora. Th dlay l ha a rai dabilizig ff o h dyai y. Thi ff a b obrvd i 4

2 Bod diagra: h agiud rpo ha a oa valu whil h pha rpo ha a oa ra of hag ad ira wih frquy o ifiiy. Ti dlay ira h y pha lag ad alo giv ri o oraioal rafr fuio of h y aig h or diffiul o aalyz ad orol. Fro a ahaial poi of viw i dlay y ar ifii diioal aig ha hir a i a ifii diioal vor. Thr i a broad rag of hod ad algorih for u wih raioal rafr fuio odl whil rlaivly fw xi for pro wih irraioal rafr fuio uh a ho oaiig i dlay. A aural quio ari whhr i i poibl o approxia h dlay y odl by a fii diioal o li h raioal rafr fuio. Th raioal fuio approxiaio of h i dlay l i uually vry iaura [7 9 ]. Ay raioal rafr fuio a hav oly a fii pha lag whra h pha lag of h pur i dlay i uboudd. Th arbirary larg high frquy pha iah rul i approxiaio rror of a la hudrd pr a frqui whr h pha rror i ( +)π for o aural. Thrfor h raioal approxiaio of h dlay a oly ovr a fii frquy bad [4 9 ]. Poially br approxiaio ay b produd wh propri of o wighig fuio ( ) i a io aou i h approxiaio of h dlay l i.. h wighd dlay odl () i xplord. Wighd approxiaio i vry raoabl i i pro iduri fir ordr plu i dlay ad od ordr plu i dlay odl ar ooly ud o drib h y bhavior [8]. Thi papr oidr h probl of raioal rafr fuio approxiaio of pur i dlay l. Th i dlay i prd by Lagurr Kauz ad Pad powr ri or hif opraor dripio ruad by uig balad ad al or ralizaio. Crai frquy wigh ar addd o h i dlay l ad rror boud ar opud for h drivd approxiaio. Svral urial xapl ar prd viualizig h auray of h i dlay approxiaio.. Raioal fuio approxiaio of pur i dlay l Th ai ir for approxiaig a i dlay wih o raioal fuio li i h ap o dal wih a fii diioal y iad of h orrpodig ifii diioal o. Th i dlay l i uually approxiad by a allpa raioal fuio. Th al igular valu for all-pa fuio ar all uiy hrfor h rror bw a all-pa fuio ad i lowr ordr raioal fuio approxiaio will b grar or qual o o ad h h approxiaio will b vry iaura. Thi do o iply howvr wh a all-pa y i od wih a raioal wighig fuio. Th approxiaio hod ud o widly i prai ar bad o h praio of h dlay l a a raio of wo polyoial [7]: 5

3 6 ( ) () Q () е Q whr Q () i a abl polyoial of dgr. Th polyoial Q ( ) wh ud for h [ ] Pad ri approxiaio of h dlay l i giv i h for [7]: i i () () ( i)! i ( i)!! i Q = i i = ( ). =! i= ( )(! i)! i! So low ordr Q () polyoial ar: Q () = + Q () = + + Q() = If w do by P h [ ] Pad approxia of h fuio for = j w hav h followig rror boud [9]: () j P ( j) whr ψ = ( ). 44 ψ + ψ ψ. Pad hod giv a opial ovrg ra for = i r of p or. If h odiio ( j) i aifid for o > ad p h for + p h followig rror boud i valid [9]: raioal approxiaio of fuio of h for ( ) ( ) (4) () P ( ) () p ψ whr ψ = ( ). 44. A w approah for raioal approxiaio of dlay y i bad o hif opraor hiqu [5 6]. Th advaag of uig hif opraor o fro h fa ha ay ipora orhooral ba uh a Lagurr ad Kauz ba ar ow o b idud by h orrpodig hif opraor. I i alo ow ha h dlay opraor i a hif opraor. A ipl approxiaio hiqu u h laial rlaiohip of xpoial fuio l i prd i h for [] (5) = h hif opraor S L dfid by x ( ) ( + ) = li x = li +. If h i dlay (6) S f = f f L + i alld a ulipl Lagurr hif wih ulipliiy. Th rror bw h i dlay ad h Lagurr hif i boudd by h xprio []

4 7 (7) ( ) φ L i j S whr ( ) φ =. If a abl dlay y i prd by h rlaio () () = whr h raioal fuio ( ) ha a rlaiv dgr ad oiuou uppr boud alog h iagiary axi giv by (8) ( ) h l j whr low ad high oa l h > ar uh ha h l = h h rror bw h Lagurr hif approxiaio ad h dlay y i giv by h forula a []: (9) if β h = ; h l l L S β () if β = h h l L S β β β whr ( ) = β. Aohr ipora hif opraor i h Kauz hif K : S dfid by [5] () f f S = K f ad alld a ulipl Kauz hif of ulipliiy. Th rror boud bw h Kauz hif ad h dlay y i giv by h iqualiy [5] () ( ) K 48 C S whr C i a oa dfid by h xprio ( ) C j 48 C < ad = i h rlaiv dgr of ( ). If 48 C h rror boud i prd a

5 () I h a wh ( C ) (4) SK li ϕ( β ) S K. 4 h rror boud a b prd a follow [5]: 4 β whr ϕ( β ) = up ( j) ad 48( C ) β ( ) ( ). A xio of Kauz hif i h ulipl Pad hif of ulipliiy S P : alo ow a Pad- hif ad dfid by h xprio [6] + (5) S f = f f P. + + Th rror boud bw h ulipl Pad- hif ad h dlay y i giv by h iqualiy [6] C (6) SP whr = 4 ad C i a oa aifyig ( ) C < 44. For if 44( C ) ( ) 5 ( 4) C j ad alo ad for = 4 if 5 C 44 h followig rror boud i valid: (7) 5 ( C ) S P 4 7. Th a wh h rlaiv dgr i 5 h rror boud a b iplifid a follow: (8) 5 SP li ϕ5 ( β ) 4 7 β ϕ β = up j β. whr ( ) ( ) β Th pur i dlay l i odld by uig ri or hif opraor approxiaio. Th ordr of ruaio of ri rpraio or h ulipliiy ubr of hif opraor dripio ar drid by applyig h al or ad balad ralizaio. Th quaii ar drid by xplorig h approxiaio rror of h orrpodig ralizaio. 8

6 . Balad ruaio ad al or approxiaio of dlay l A popular hod for odl rduio of fii diioal y whih a b ud i o ifii diioal a i h balad rlizaio hod. A balad ralizaio i h o whr h orollabiliy ad obrvabiliy graia ar qual ad diagoal. Th diagoal ri of h graia ar alld al igular valu. By ruaig h a pa ari of a balad ralizaio w obai a rdud ordr odl wih good approxiaio propri. odl rduio rquir h liiaio of o of h a variabl fro h origial y rpraio. Th y i fir raford io a balad for ad h o of h a variabl ar ruad whil prrvig abiliy. Au a abl liar i ivaria y dribd by i a pa odl: (9) x &() = Ax( ) + Bu( ) y () = Cx( ) x ( ) = x. Coidr h orollabiliy opraor of h y dfid a: () L : L ( ) R A whr ( τ ) ( L u)( ) = Bu( τ d )τ ad h obrvabiliy opraor of h y dfid a () Lo : R L ( ) Lo x h h orollabiliy ad obrvabiliy graia whr ( )( ) A = C x () Aτ A τ A τ Aτ W ( ) = BB dτ ad W ( ) = C C dτ ar h arix rpraio of h ap L L ad L o L o whr wih h ar uprrip h adjoi opraor i aigd. If h liar y i abl orollabl ad obrvabl h h graia W = liw ( ) ad W = liw ar h uiqu oluio of Lyapuov quaio: o o ( ) () AW + W A + BB = ad A Wo + Wo A + C C =. Th all aou of rgy dd o ov h y fro zro o a x i giv by E = x W x whil h rgy obaid by obrvig h oupu of h y wih a iiial odiio x ad o ipu fuio i giv by Eo = x Wo x. Thrfor o way o rdu h ubr of a i o liia ho whih rquir a larg aou of ipu rgy E o b rahd ad yild a all aou of obrvaio rgy E o a h oupu. Th goal i o loo for a bai i h a pa whr orollabiliy ad obrvabiliy ar quival i o. Suh a bai xi if W = Wo = diag( σ K σ ) σ i = K ar h al igular i valu of h y. Approxiaio i hi bai a pla by ruaig h o 9

7 x = x x K x for <. Approxiaio by balad ruaio prrv abiliy ad h or of h rror bw h origial ad h ruad y i giv by h xprio Σ σ + K +. iiial a vor x = [ x x K x ] T o h rdud a vor [ ] T (4) ( ) Σ + σ A balaig raforaio i opud a P = S U R whr Wo = R R i a Choly dopoiio of h obrvabiliy graia ad RW R = US U i h igular valu dopoiio of h xprio i h lf id. Thi yp of balaig i ow a Lyapuov balaig bau i i bad o olvig Lyapuov quaio for h graia. Thi approah ay ur ou o b iffii pially for larg probl du o h rapid day of h al igular valu. To avoid h diffiuli o odifiaio of h origial balaig algorih ar propod o prv arix ivrio. O of h o popular odifiaio of h balaig algorih i h Squar Roo Algorih [ ]: If w pariio h ari W = [ W W ] ad V = [ V V ] ad W o = LL (Choly dopoiio) W = UU (Choly dopoiio) U L = WΣV (SVD dopoiio) P = Σ V L (iilariy raforaio) P =UWΣ (iilariy raforaio) Σ d Σ = Σ w obai h iilariy raforaio a follow: P = Σ V L R ad P = UWΣ R. Th rdud ordr y of diio obaid fro h origial o by balad ruaio i P AP PB Σ =. CP Fially w obai h rror bw h y dribd by () = () wih () abl raioal wighig fuio ad i balad ruaio of h dlay ri approxiaio. To obai h rror boud w u h riagl propry of or: (5) Σ = S + S Σ S + S Σ whr S do ay of h dlay l hif approxiaio prd i Sio. Th fir rror or i (5) i boudd by h iqualii (9) () () () (4) or (6) (7) (8) ad h od rror or i boudd by h r (4). Th al or approxiaio i bad o h or idud by h al opraor. I i doai h al opraor i dfid a

8 Γ : L ( ) L( ) whr Γ u = P+ ( h u) axi of h igal L ( ) i a projio o h poiiv i u ovolvd wih h ipul rpo h (). Th rul of hi ovoluio i obaid a (6) ( )( ) ( ) ( ) Γ = h τ u τ dτ u <. Th al opraor ha h irpraio of h y fuur oupu y() = Γu () bad o h pa ipu u ( ). Th al opraor a alo b prd a a opoiio of ap fro h pa ipu o h iiial a ad fro h iiial a o h fuur oupu. Thi opoiio of ap i prd by wo ohr opraor []: h orollabiliy opraor L L ( ) R whr L : Aτ u = Bu( τ ) dτ ad h obrvabiliy opraor Lo : R L ( ) A whr Lo x = C x. Thu h al opraor a b oidrd a a opoiio of h orollabiliy ad obrvabiliy ap i.. Γ = L o o L = LoL. Th -or of a abl y al opraor i giv a Γ = Γ Γ = ρ W W (7) ( ) whr ( ) W W o ρ pr h pral radiu of h graia produ ad i ow a h al or of h y Σ dod by Σ. Th al or i idud -or fro h pa ipu o fuur oupu. Th al or approxiaio hory i bad o h followig rul []: i) giv abl y Σ ad Σ of dgr ad whr > hr hold Σ Σ σ +( Σ) ; ii) h wo-or of ay L y Σ i o l ha h al or of i abl par Σ + i.. Σ Σ + ; iii) giv a abl y Σ hr xi a y Σ havig xaly abl pol ad Σ Σ = σ +( Σ). Furhror Σ Σ i all-pa. Th y Σ i alld all-pa dilaio of h y Σ ; iv) giv { K } a abl y Σ ad a poiiv ubr ε uh ha σ ( Σ ) > ε > σ +( Σ) hr xi a y Σ wih abl pol ad Σ Σ = ε. Furhror Σ Σ i all-pa ad Σ i alld ε all-pa dilaio of Σ. If Σ i ε all-pa dilaio of Σ ad σ + ( Σ) ε σ ( Σ) h Σ ha xaly abl pol ad σ ( Σ) Σ Σ < ε σ + Σ = h Σ Σ = σ +( Σ).. If + ( ) ε I a pa h opial al or approxiaio i obaid a follow []. Au ha σ i a al igular valu of ulipliiy r of h y Σ. Trafor h y io a balad for ad pariio h y ari a o

9 A A A = A A B B = ( ) B C = C C D W W = σir Wo Wo =. σir Th h approxiaio probl oluio i giv by h y ari: Aˆ = Γ ( σ A + WoAW σcub ) Bˆ = Γ ( WoB + σ CU ) Cˆ = CW + σ UB Dˆ = D σu Γ = W W σ I o whr U i a uiary arix aifyig B = CU U U = I igular valu driig h iz of h rror. 4. Exprial rul ad σ i h al W xai a pur i dlay l wih a abl raioal wighig fuio of fir ad od ordr. Thi yp of rafr fuio i qui popular i prai bau i fi h odl of a larg variy of idurial orol pro. Coidr h fir ordr odl wih a i dlay () =. Au ha = T + ad T a h valu. ad. W oidr fir h [ 4 4] Pad approxiaio of h dlay l. Th offii of h approxiaig polyoial ar obaid a follow: Q = [ ] Th al igular valu of h raioal approxiaio of ( ) dlay l i approxiad by [ 4 4] Pad ri ar how i Fig.. whr h Fig.. al igular valu of [4 4] Pad Fig.. al igular valu of 4h ordr approxiaio wih ordr wigh: Lagurr hif ordr wigh: T= (---); T =. ( ); T = (- -); T = (-.-) T = (---); T =. ( ); T = (- -); T = (-.-)

10 Th fir obrvaio fro Fig. i ha wh T = or h approxiaio i for h pur dlay l wihou a wighig fuio h all al igular valu of h raioal fuio ar qual o o ad h rror of approxiaio ao b rdud furhr. Thi rul how ha i i o poibl o approxia h dlay opraor arbirarily loly by raioal fuio bu h iuaio hag wh h approxiaio hod i applid o wighd dlay l. Th od obrvaio i ha h rror of approxiaio dra by iraig h i oa of h wighig fuio. Th al igular valu of a ulipl Lagurr hif wih ulipliiy approxiaio for h a valu of T ar how i Fig.. Th approxiaio polyoial ha h followig offii: Q = [ ]. Iigifia diffr for h al igular valu i h a of h fourh ordr Pad approxiaio i obrvd. For xapl for T = h [ 4 4] Pad ri approxiaio ha al igular valu S = { } whil h Lagurr hif 4 S = approxiaio ha { } Fig.. al igular valu of 4h ordr Fig. 4. al igular valu of 4h Lagurr hif: T =T = (---); T =.5 T = ( ); ordr: Pad (---); hif Lagurr-4 ( ); T =. T = (- -); T = T = (-.-.) hif Kauz- (- -); hif Pad- (-.-.) I i agai ha by lowig dow h filr dyai h approxiaio rror i rdud. Nx w oidr a i dlay l wih a od ordr wighig fuio of h for () =. W xplor four a for ( T + )( T + ) diffr filr i oa: T = T = ; T =.5 T = ; T =. T = ad T = T =. Th approxiaio odl ud i Lagurr hif wih ulipliiy = 4. Th rul ar how i Fig.. Siilarly o h a of h fir ordr wighig fuio wh T = T = h al igular valu ar qual o o. Wh o of h i oa i iraig h al igular valu hav highr ra of dli ad h approxiaio rror bo allr.

11 Wh boh i oa ar larg h dli ra i h fa o. Fig. 4 how h al igular valu for T =.5 ad T = for h approxiaio odl: [ 4 4] Pad ri Lagurr hif wih = 4 Pad ad Kauz hif wih =. I i obrvd ha all four approxiaio odl giv alo h a al igular valu wih h a ra of day. Thi rul how ha i h low pa frquy rag h approxiaio hod hav iilar approxiaio apabilii. W oidr a odl of i dlay wih a fir ordr wigh: () =. T + Au ha = 5 ad T =. Th al igular valu for [ 4 4] Pad ri approxiaio ar S = { L } ad for hif Lagurr-4 approxiaio ar S = { }. For odl rduio w apply h balad ruaio hod. Fig. 5 pr h ui p rpo for a wighd i dlay l whih i approxiad by [ 4 4] Pad ri. Th i rpo for h full ordr y h rdud fourh hird od ad fir ordr y ar how. For all approxiaio odl xp for h fir ordr approxiaio h diffr i h i rpo appar oly i h ady a valu. Siilar rul ar obrvd for h hif Lagurr-4 approxiaio odl i Fig. 6. I i obrvd ha h rdud fourh ad hird ordr odl loly approah h full ordr odl. Thrfor h balad ruaio hod a rliably b ud for wighd approxiaio of i dlay l. Nx w au a i dlay l wih a od ordr wighig fuio () = whr T = 5 T T + T =. ad =. ( )( ) + Fig. 5. Ui p rpo of [4 4] Pad Fig. 6. Ui p rpo of hif approxiaio wih ordr wigh: Lagurr-4 approxiaio ordr wigh: full ordr (---); rdud 4h ordr (- -); full ordr (---); rdud 4h ordr (- -); rdud rd ordr ( ); rdud d ordr (-. -. ); rdud rd ordr ( ); rdud d ordr (--); rdud ordr (xxx) rdud ordr (xxx) 4

12 W oidr h odl of ulipl Lagurr hif wih ulipliiy = 4 approxiaio. Th al igular valu of h wighd dlay l ( ) approxiad by h hif Lagurr-4 ar: S = { }. Th p rpo o h full ordr odl h rdud fifh fourh ad hird ordr odl by appplyig h al or approxiaio hiqu for odl rduio ar how o Fig. 7. Fig. 7. Ui p rpo of hif Lagurr-4 Fig. 8. Ui p rpo of hif Pad- approxiaio wih d ordr wigh fuio: approxiaio wih d ordr wigh full ordr (---); rdud 5h ordr (- -); rdud fuio: full ordr (---); rdud 5h ordr (- -); 4h ordr ( ); rdud rd ordr (-.-.) rdud 4h ordr ( ); rdud rd ordr (-.-.) Th al or approxiaio hod giv h y rror i r of h al or: h rror bw h full ordr ad rdud 5h ordr approxiaio of ( ) i.6 h rror bw h rdud 5h ordr ad h rdud 4h ordr hif Lagurr-4 approxiaio i.69 ad iilarly h rror bw h rdud 4h ordr ad h rdud rd ordr hif Lagurr-4 approxiaio i.48. I i obrvd fro Fig. 7 ha h ui p rpo diffr bw h full ordr ad rdud ordr y appar i h ady a valu ad i i vry all. Siilar rul ar obaid wh h hif Pad- approxiaio i applid o h wighd dlay y. Th al igular valu for h wighd i dlay l wih h a of parar valu ar obaid a follow: S = { }. Th ui p i rpo of h full ordr hif Pad- approxiaio ad h rdud 5h ordr 4h ordr ad rd ordr y ar how i Fig. 8. I i obrvd ha h rror bw h y rpo ira whih i du o h largr valu of h al igular valu of h hif Pad- approxiaio ad hrfor h largr al or diffr bw h rdud ordr approxiaio. 5. Coluio Th papr oidr h probl of h pur i dlay l approxiaio by applyig al or ad balad ralizaio. Th i dlay l i odld i r of Pad ad Kauz ri or Lagurr Kauz ad Pad hif 5

13 opraor rpraio. Th rror boud for h diffr approxiaio odl ar diud ad i i how ha i dlay a b approxiad ufully oly wh uilizig o wighig fuio. Th fir ordr ad od ordr wighig fuio ar ud ha orrpod o h prai o odl idurial pro o of by fir ordr or od ordr lag ad i dlay. Th al or ad balad ralizaio ar ud o rdu h ordr of h wighd i dlay approxiaio. Th balad ruaio hod i a br of h faily of Lyapuov yp approah for balad odl rduio. I i bad o a urially ffii algorih uilizig Choly dopoiio of boh graia. Th al or approxiaio u h adard hiqu of quially rduig h odl ordr by applyig balad ralizaio ad obaiig σ allpa dilaio of h rror y. Boh hod for odl rduio ar d wih urial xapl ad h orrpodig rror ar alulad i r of h al igular valu. Th rul obaid ofir h ffiiy of h hif opraor approxiaio of i dlay l i obiaio wih h balad ad al or odl rduio hiqu. R f r. A o u l a A. Approxiaio of Larg-Sal Dyaial Sy. Philadlphia SIA Publ. 5.. l o v r K. All al Nor Approxiaio of Liar ulivariabl Sy ad Thir L Error Boud. Iraioal Joural of Corol Vol No L a J. Aalyi o h Lagurr Forula for Approxiaig Dlay Sy. IEE Tra. o Auo. Cor. Vol No L i u Y. B. D. O. A d r o. odl Rduio wih Ti Dlay. IEE Pro. Vol. 4D 987 No ä i l a P. J. P a r i g o. Lagurr ad Kauz Shif Approxiaio of Dlay Sy. Iraioal Joural of Corol Vol No ä i l a P. J. P a r i g o. Shif Opraor Idud Approxiaio of Dlay Sy. SIA Joural o Corol ad Opiizaio Vol No i r i L. Z. P a l o r. Corol Iu i Sy wih Loop Dlay. I: adboo of Nword ad Ebddd Corol Sy. D. riu-varali ad W. Lvi Birhäur Ed N o r y-r i o J. E. C a a h o. Corol of Dad Ti Pro. Brli Sprigr Parigo J. So Frquy Doai Approah o h odl Rduio of Dlay Sy. Aual Rviw i Corol Vol P r v K. Approxiaio of Pur Ti Dlay by Balad Truaio. I: Pro. of h Aivrary Cofr 4 Yar DIA UCT Sofia Sofia R i h a r d J. Ti Dlay Sy: A Ovrviw of So R Adva ad Op Probl. Auoaia Vol Z h o u K. J. D o y l K. l o v r. Robu ad Opial Corol. Uppr Saddl Rivr Pri all

14 Аппроксимация элемента чистого запаздывания с использованием Ханкеловой нормы и балансовой реализации Камен Перев Кафедра Системы и управления Технический университет София 756 София Eail: (Р е з ю м е) В статье рассматривается задача аппроксимации элемента чистого запаздывания с использованием Ханкеловой нормы и балансовой реализации. Взвешанный элемент запаздывания представлен рядами Паде и Каутца или оператором перемещения Лагера Паде и Каутца. Заданы количественные оценки ошибки аппроксимации чистого запаздывания. Ханкловые нормы и балансовые реализации исспользуются для определения порядка перерыва аппроксимации. Редуцированная модель сравнивается с аппроксимацией польного порядка для определения эффективности представленных методов. 7