Asymptotic Regional Boundary Observer in Distributed Parameter Systems via Sensors Structures

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1 Sesors,, 37-5 sesors ISSN 44-8 by MDPI hp:// Asympoc Regoal Boudary Observer Dsrbued Parameer Sysems va Sesors Srucures Raheam Al-Saphory Sysems Theory Laboraory, Uversy of Perpga, 5, aveue de vlleeuve, 6686 Perpga, Frace. E-mal : saphory@uv-perp.fr Receved: 6 March / Acceped: 4 Aprl / Publshed: 6 Aprl Absrac: The purpose of hs paper s o sudy he cocep of asympoc regoal observer coeco wh he characerzaos of sesors. We gve varous resuls relaed o dffere ypes of measuremes, of domas ad boudary codos. Furhermore, we show ha he measuremes srucures allow he exsece of regoal observer ad we gve a suffce codo for such observer. We also show ha, here exss a dyamcal sysem for he cosdered sysem s o observer he usual sese, bu may be regoal boudary observer. Key words: Sesors, -deecably, -observers, Dffuso sysem.. Iroduco The observer heory was roduced by Lueberger [] ad s geeralzed o fe dmesoal corol sysems descrbed by sem-group operaors []. The aalyss of dsrbued parameer sysems has receved much aeo he leraures [3, 4, 5]. The sudy of hs oo, hrough he srucure of he sesors ad he acuaors, was developed by El Ja ad Prchard [6, 7]. The oo of regoal aalyss was exeded by El Ja ad Zerrk e al. [8, 9]. Ths sudy movaed by cera cocree-real problems, hermc, mechac, evrome [, ]. If a sysem defed o a doma s represeed as he Fg., he we are eresed he regoal sae o or Г of he doma. The cocep of asympoc regoal aalyss was roduced recely by Al-Saphory ad El Ja [, 3, 4], cosss sudyg he behavor of sysems o all he doma bu oly o parcular regos or Г of he doma. I hs pape we are oly eresed he asympoc regoal boudary sae o a rego Г of he boudary. The purpose of hs paper s o gve some resuls relaed o he

2 Sesors, 38 lk bewee he Г-observer ad he umber of sesors, her locaos, he geomercal domas ad boudary codos. measuremes Fgure. Doma, regos, Г ad locaos of measuremes. The paper s orgazed as follows. Seco cocers he formulao problem ad prelmares. Seco 3, devoes o he roduco of Г-deecably problem. I seco 4, we gve some defos ad characerzaos cocers Г-observer ad sraegc sesors. We show ha here exss a couer-example of he case Г-observer whch s o observe he whole domae. I he las seco, we llusrae applcaos wh varous suaos of sesor locaos ad we characerze such a regoal boudary observer o a dffuso sysem.. Problem Saeme Le be a regula bouded ad ope se of R, wh smooh boudary ad be a oempy gve subrego of, wh posve measureme. We deoe = ], [ ad = ], [. We cosder he sysem descrbed by he followg parabolc equao z z z η, = Az = = z wh he oupu fuco y., = Cz.,. ad we assume ha Z, U, O be separable lber spaces where Z s he sae space, U he corol space ad O he observao space. Usually we cosder Z =, U = L,, R p ad O = L,, R q where p ad q hold for he umber of acuaors ad sesors. A s a secod order lear dffereal operaor whch geeraes a srogly couous sem-group S A o Z ad s selfadjo wh compac resolve. The operaors B L R p, Z ad C L Z, R q deped o he srucure of acuaors ad sesors [7]. Uder he gve assumpos, he sysem. has a uque soluo gve by.

3 Sesors, 39 z., = SA z. + SA τ Bu τ dτ.3 The measuremes ca be obaed by he use of zoe, powse or les sesors whch may be locaed or [7]. Le us recall ha a sesor by ay couple D, f where D deoe closed subses of, whch s spaal suppor of sesor ad f L D defes he spaal dsrbuo of measureme o D. Accordg o he choce of he parameers D ad f, we have varous ypes of sesors. A sesors may be a zoe ypes whe D. The oupu fuco. ca be wre he form y = Cz., = z f d.4 D A sesor may also be a po whe D = {b} ad f = δ. - b where δ s he Drac mass coceraed b. The, he oupu fuco. may be gve by he form y = Cz., = z δb b d.5 I he case of boudary zoe seso we cosder D = wh oupu fuco. ca he be wre he form ad f L. The y = Cz., = z η, f η dη.6 The operaor C s ubouded ad some precauos mus be ake [7, 5]. The operaor K defed by K L R z CS A z q :,, ad he case of eral zoe sesors s, lear ad bouded wh a adjo * q K : L,, R * * * * A τ y S C y dτ The race operaor of order zero / γ : * s lea surjecve ad couous wh adjo deoed by γ Cosder a subdoma of ad le χ be he fuco defed by χ / / : z χ z = z * where z s he resrco of he sae z o. We deoe by χ he adjo of χ. Le χ be he fuco defed by χ where z χ z = z : z s he resrco of he sae z o.

4 Sesors, 4 The operaor T : / s gve by T = χ γ T Recallg some defos cocer he oo of -observably as [9, 6] : The auoomous sysem assocaed o.-. s sad o be exacly respecvely weakly -observable f : * / * Im χ γ K = respecvely Im χ / γ K = The sue D, f q of sesors s sad o be -sraegc f he sysem. ogeher wh he oupu fuco. s weakly -observable. For he dual resuls cocerg he acuaors srucures [7]. 3. Regoal boudary deecably The ma reaso for roducg -deecably s, he possbly o cosruc a -esmaor for he curre sae of he orgal sysem. Ths cocep s exeded by Al-Saphory ad El Ja [3]. For hs objecve we recall some defos cocer hs cocep : The sysem. s sad o be -sable, f he operaor A geeraes a sem-group whch s sable o he space /. I s easy o see ha he sysem. s -sable, f ad oly f, for some posve cosas M ad α, we have γ S α. / Me, A If S A s sable sem-group /, he for all z, he soluo of he assocaed auoomous sysem sasfes lm γ z., = lm γ S z. = 3. / / A The sysem. ogeher wh he oupu. s sad o be -deecable, f here exss a q operaor : R / such ha A C geeraes a srogly couous sem-group S whch s sable o /. If a sysem s orgal sysem. If we cosder he sysem x = Ax + x = x x η, = -deecable, he s possble o cosruc a asympoc -observer for he y., Cy he x, esmaes asympocally he sae z, because he error e, = z -x, sasfes e = A C e wh e, = z -x,. The, f he sysem s -deecable, s possble o choose whch realzes lm e., / =. 3.

5 Sesors, 4 Remark 3.. I hs pape we oly eed he relao 3. o be rue o a gve subrego of he rego lm χγ z., = lm χγ S z. = 3.3 / / A we may refer o hs as -sably. The sysem. s sad o be -sable, f he operaor A geeraes a sem-group whch s sable o /. The sysem.-. s sad o be -deecable f here exss a operaor q : R / such ha A C geeraes a srogly couous sem-group S whch s sable o /. For more deal [8]. oweve oe ca easly have he followg resuls : Corollary 3.. If he sysem. ogeher wh oupu fuco. s exacly -observable, he s -deecable. Ths resul leads o: γ > χ γ = 3.4 such ha S z. CS z., z / A / A L,, O Thus, he oo of -deecably s a weaker propery ha he exac -observably as Ref. [6]. 4. Asympoc -observer ad sraegc sesors I hs seco, we propose a approach whch allows o deermae a regoal asympoc esmaor of z o, based o he eral asympoc -observer. Ths mehod eed some defos cocer regoal observer as []. 4.. Defos ad characerzaos Cosder he sysem. -. ogeher wh he dyamcal sysem x = F x = x x η, = x + G u + y 4. where F geeraes a srogly couous sem-group S F whch s sable o X =,.e. : M, > such ha α F F χ S F F α F. M e 4. p ad, q G L R ad L R,. The sysem 4. defes a asympoc - esmaor for χ T where z, s a soluo of he sysem. -. f

6 Sesors, 4 χ lm x., T., = ad χ T maps DA o D F where x, s he soluo of he sysem 4.. The sysem 4. specfes a -observer for he sysem. -. f he followg codos hold: q. There exss M L R, L ad N L L such ha M C + N χt = I.. χ TA + F χt = G C ad = χtb. 3. The sysem 4. defes a asympoc -esmaor for χ T. The sysem 4. s sad o be a dey -observer for he sysem. -. f ad Z = X. The sysem 4. s sad o be a reduced-order -observer for he sysem. -. f Z = O X. E r sesors Fgure : The cosdered doma ad he subrego r. The boudary regoal observer may be see as eral regoal observer f we cosder he followg rasformaos. Le R be he couous lear exeso operaor [9] R : / such ha γ R = 4.3 / h, h,, h Le r> s a arbrary ad suffcely small real ad le he ses E = B z, r ad = E z r where Bz, r s he ball of radus r ceered z, ad where s a par of r Fg.. Proposo 4.. If he sysem. -. s exacly respecvely weakly r -observable, he s exacly respecvely weakly -observable see [9]. From hs resul, we ca deduce he followg proposo : Proposo 4.. The dyamcal sysem 4. s Lueberger. -., he s Lueberger -observer. r -observer for he sysem Proof: Le x / ad x, s exeso o / usg 4.3 ad race heorem, here exss R x, wh a bouded suppor such ha

7 Sesors, 43 γ R x = x 4.4 Sce he sysem 4. s regoal observer o r so we ca deduce ha : The sysem 4. s regoal observer o, here exss a dyamcal sysem wh ha r x X such χ T z = χ R x he we have * χ γ χ χt z = x 4.5 The equaos. ad 4.5 allow y x C = χ γ χ * z χt ad here exs wo lear bouded operaors R ad S sasfy he relao RC+ χ γ χ χ T = I * There exss a operaor F s regoally sable o r, he s regoally sable o [8]. Fally he sysem 4. s a regoal boudary observer o. r 4. Suffce codo for -observer As Refs. [7,8], we develop a characerzed resul ha lks he -observer ad sraegc sesors ad we gve a suffce codo for -observer. For ha purpose, we assume ha here exss a complee se of egefucos ϕ of A, assocaed o he egevalues λ wh a mulplcy s ad suppose ha he fucos ψ defed by ψ = χ γ ϕ, s a complee se /. If he sysem. has J usable modes, he we have he followg heorem. Theorem 4.3. Suppose ha here are q zoe sesors D ad he specrum of A coas J, f q egevalues wh o-egave real pars. If he followg codos are sasfed :. q > s. rak G = r,, =,..., J wh G = G where sup j < ψ., f. > j = ψ b j < ψ f >.,. j L D L zoe sesor case s = s < ad j =,..., s. The he dyamcal sysem powse sesor case boudary zoe sesor case

8 Sesors, 44 x = Ax C x z., x = x x η, = 4.6 s -observer for he sysem. -.,.e. lm x., T z., / =. Proof : The proof s lmed o he case of zoe sesors he followg sapes : Sep. Uder he assumpos of seco, he sysem. ca be decomposed by he projecos P ad I - P o wo pars, usable ad sable. The sae vecor may be gve by where z, s he sae compoe of he usable par of he sysem., may be wre he form z z z η, = Az = z = ad z, s he compoe sae of he sable par of he sysem. gve by z = Az z = z z η, = The operaor A s represeed by a marx of order s, s gve r r r A = dag λ,..., λ,..., λ,..., λ,..., λ,..., λ ] ad PB = G, G,..., G ] [ J J J = J = [ Sep. The codo of hs heorem, allows ha he sue D, f q of sesors s - sraegc, he subsysem 4.7 s weakly -observable [6] ad sce s fe dmesoal, he s exacly -observable. Therefore s -deecable, ad hece here exss a operaor such ha A whch s sasfed he followg : c M, > such ha A α e M e α ad he we have / J z α., / M e Pz. / Sce he sem-group geeraed by he operaor A s sable o /, he here exs M, α > such ha

9 Sesors, 45 α α τ / + / / z, / whe. Fally, he sysem -. z., M e I P z. M e I P z. u τ dτ ad herefore. s -deecable. Sep 3. Le e = z x where x, s he soluo of he sysem 4.6. Dervg he above equao ad usg he equaos. ad 4.6, we oba e z x = = A z Ax z., x = A C e q Sce he sysem. -. s -deecable, here exss a operaor, / L R, such ha he operaor A C, geeraes a sable, srogly couous sem-group S o he space / whch s sasfed he followg relaos : M, α > such ha Fally, we have χ γ S. / α M e e., / χ γ S. e. M e e. / wh e = z x ad herefore lm e., / = α Thus, he dyamcal sysem 4.6 a -observer for he sysem. -.. Remark 4.4. From heorem 4.3., we ca deduce he followg saemes :. A dyamcal sysem whch s a -observer s -observer.. If a sysem s -observe he s observer every subse of, bu he coverse s o rue. Ths may be prove he followg example: Example 4.5. Cosder a wo-dmesoal sysem descrbed by he dffuso equao z, = z z, = z z η, η, = where =],[ ],[. The operaor A = geeraes a srogly couous sem-group S o A he lber space gve by 4.9 A, m= λm S z = e < z, ϕ > ϕ m m

10 Sesors, 46 where λ = + m π, ϕ, a cos π cos mπ ad m m = m Cosder he dyamcal sysem x, = x, C x z, x, = x x η, η, = / am = λ m. 4. q q where L R, Z, Z s a lber space ad C : R s a lear operaor. Cosder he boudary sesor, f defed by = {} ], [ ad f η, η = cosπη. Thus, he oupu fuco ca be wre by y = z η,η, f η,η dη d η 4. If he sae z s defed by z, = cos πcosπ, he he sysem s o weakly observable,.e. he sesor, f s o sraegc ad herefore he sysem s o deecable. Thus, he dyamcal sysem 4. s o observer for he sysem see [6]. ere, we cosder he rego = ],[ {} Fg. 3 ad he dyamcal sysem Fgure 3. Doma, rego ad locao of zoe sesor. x, = x, x = x x η, η, = C x,, z,, 4. q where L R, /. I hs case, he sysem s weakly observable ad he sesor, f s -sraegc [9]. Thus, he sysem s -deecable [3]. Fally he dyamcal sysem 4.9 s -observer for he sysem [8].

11 Sesors, Applcao o sesors srucures I hs seco, we cosder he dsrbued dffuso sysems defed o =, a [ ], [. We ] a explore varous resuls relaed o dffere ypes of measuremes, domas ad boudary codos. 5. Case of a zoe sesor We sudy he followg cases : 5.. Recagular doma I hs case, he sysem. -. s gve by z z, z η, η, υ y = z,, = z = =, z η, η f η, η dη dη Σ 5. where Σ = ], [, D s he locao of zoe sesor ad above sysem represe he heacoduco problem see Ref. []. Le = { a } ], a[ be a rego of o ], a [ ], a[ The egefucos ad he egevalues are gve by ad ψ = m cos π cos mπ 5. aa a a m λm = + π 5.3 a a If a /a, he he mulplcy of s = m ad oe sesor ca be guaraeed -sraegc sesor see []. The dyamcal sysem 4.6 may be gve by x = x x = x x η, η, = C x,, z,, Le he measureme suppor s recagular wh D = [ l, + l] [ l, + l]. If f s symmerc abou = ad f s symmerc wh respec o =, he we have he followg resul : Corollary 5.. The dyamcal sysem 5.4 s -observer for he sysem 5. f / a ad m a N for every, m =,..., J. / 5.4

12 Sesors, 48 I he case where ad f L, he sesor D, f may be locaed o he boudary = η l, η + ] { a } he we have : [ l a D a a η - - a η a a - η Fgure 4. Doma, rego ad locaos D,, of zoe sesor. Corollary 5... Oe sde case : Suppose ha he sesor D, f s locaed o = [ η l, η + l] { a } ad f s symmerc wh respec o η = η, he he dyamcal sysem 5.4 s - observer for he sysem 5. f η / a N for every, m =,..., J.. Two sde case : Suppose ha he sesor D, f s locaed o =, η + ] {} {}, η + ] [ l ad f s symmerc wh respec o [ l η = η ad he fuco f s symmerc wh respec o η = η, he he dyamcal sysem 5.4 s -observer for he sysem 5. f η a ad mη a N for every, m =,..., J. / / 5.. Dsk doma The sysem 5. may be gve by he followg form z θ, z θ, z a, θ, υ y = z θ, = z = = θ z r, θ, f r, θ dr dθ 5.5 Σ where < θ < π, = D, a, r = a > ad = θ [,π ], > are defed as Fg. 5. Le he egefucos ad egevalues cocerg he rego = D a, θ of wh θ [,π ] are defed by q λ =,, m m β m 5.6 where β m are he zeros of he Bessel fucos J ad

13 Sesors, 49 ϕ m θ = J β m r m ϕ θ = J β r cos θ, m 5.7 m m ϕ, = s, m m r θ J r β m θ p p p θ θ a p θ θ a Fgure 5. Doma, rego ad locaos p, p of eral boudary zoe sesors. wh mulplcy s m = for all m ad s m = for all m =. I hs case, he -sraegc sesor s requred a leas wo zoe sesors D, f q where D = r, θ q see [6]. The dyamcal sysem 5.4 ca be wre by x θ, = x θ, + x θ, = x θ x a, θ, = υ If f ad D are symmerc wh respec o C z θ, x θ, θ =, for all q, he we have : θ 5.8 Corollary 5.3. The dyamcal sysem 5.8 s for every, =,..., J. observer for he sysem 5.5 f θ θ /π N o Whe he sesors are locaed o ad he fuco, f q θ = θ, q as Fg. 5. So, we have. f s symmerc wh respec Corollary 5.4. The dyamcal sysem 5.8 s for every, =,..., J. observer for he sysem 5.5 f θ θ / π N 5. Case a powse sesor I hs subseco, we cosder he followg cases :

14 Sesors, The doma =, a [ ], [ ] a Now he sysem 5.5 s gve by he form z z, z η, η, υ y If = z,, = z = =, z η, η f η, η dη dη b = b, b he, we have : Σ 5.9 b σ b a a a b b b b a b a a Fgure 6. Recagular doma, rego ad locaos b, σ of powse sesors. Corollary Ieral case : If b / a ad mb / a N for every, m=,..., J, he he dyamcal sysem 5.4 s -observer for he sysem Flame case : Suppose ha he observao s gve by he flame sesor where σ = Im γ s symmerc wh respec o he le b = b, b, f b / a ad mb / a N for every, m=,..., J, he dyamcal sysem 5.4 s -observer for he sysem Boudary case : If mb / a N for every m =,..., J, he he dyamcal sysem 5.4 s - observer for he sysem The doma = D, a The sysem 5.5 may be gve by he followg form z θ, = z θ, z θ, = z θ z a, θ, = υ y = z θ, δ p r r, θ θ dr dθ Σ 5.

15 Sesors, 5 where θ π. The sesors may be locaed p = r, ad p = r, or p = a, θ Fg. 7. θ θ D D θ θ a θ θ a Fgure 7. Dsc doma, rego ad locaos p, p of eral boudary powse sesors. Corollary If he sesors p, δ p q are locaed p = r, θ ad θ θ / π N for every, =,..., J, he he dyamcal sysem 5.8 s -observer for he sysem 5... If he sesors p, δ p are locaed p = a, θ q ad θ θ / π N for every, =,..., J, he he dyamcal sysem 5.8 s -observer for he sysem Cocluso The cocep suded hs paper s relaed o he case of dey -observer coeco wh he sesors srucure. For hs class of parabolc dsrbued parameer sysems, may eresg resuls cocerg he choce of he sesor are explored ad llusraed specfc suaos of he doma. I smlar way, we ca characerze he geeral case reduced-order -observer by sesors srucure. A mpora exeso of hese resuls, relaed o he problem of regoal boudary grade observe coeco wh he srucure of sesors; s uder cosderao. Refereces. Lueberge D. Observers for mul-varable sysems. IEEE Trasacos o Auomacs Corol 966,, Gressag, R.; Lamo, G. B. Observers for sysems characerzed by sem-groups. IEEE Trasacos o Auomacs Corol 975,, Fuj, N.; ra, M. A fe-dmesoal asympoc observer for a class of dsrbued parameer sysems. Ieraoal Joural of Corol 98, 3, ou, M.; Mulle P.C. Desg of observers for lear sysems wh ukow pus, IEEE Trasacos Auomac Corol 99, 37, Cura, R. F.; Zwa. J. Iroduco o fe dmesoal lear heory sysems; Sprger- Verlag : New York, 995.

16 Sesors, 5 6. El Ja, A.; Prchard, A. J. Sesors ad acuaors dsrbued parameer sysems Ieraoal Joural of Corol 987, 46, El Ja, A.; Prchard, A. J. Sesors ad corols he aalyss of dsrbued sysems; Ells orwood seres Mahemacs ad s Applcaos, Wley : New York, El Ja, A.; Smo, M. C.; Zerrk, E. Regoal observably ad sesor srucures. Sesors ad Acuaors 993, 39, Zerrk, E.; Badraou, L.; El Ja, A. Sesors ad regoal boudary sae recosruco of parabolc sysems. Sesors ad Acuaors 999, 75, -7.. El Ja, A.; Zerrk, E.; Smo, M. C.; Amouroux, M. Regoal observably of a hermal process. IEEE Trasacos o Auomac Corol 995, 4, Al-Saphory, R.; El Ja, A. Sesors ad asympoc -observer for dsrbued dffuso sysems. Sesors,, Al-Saphory, R.; El Ja, A. Sesors srucures ad regoal deecably of parabolc dsrbued sysems. Sesors ad Acuaors, 9, Al-Saphory, R.; El Ja, A. Sesors characerzaos for regoal boudary deecably of dsrbued parameer sysems. Sesors ad Acuaors, 94, Al-Saphory, R.; El Ja, A. Asympoc regoal sae recosruco. Ieraoal Joural of Sysems Scece o appear. 5. Cura, R. F. Fe dmesoal compesaors for parabolc dsrbued sysems wh ubouded corol ad observao. SIAM J. Corol ad Opmsao 984,, Zerrk, E.; Badraou, L. Sesors characerzao for regoal boudary observably. Ieraoal Joural of Appled Mahemacs ad Compuer Scece,, Zerrk, E.; Bououlou, L.; El Ja, A. Acuaors ad regoal boudary corollably of parabolc sysems. Ieraoal Joural of Sysems Scece, 3, Al-Saphory, R. Asympoc regoal aalyss for a class of dsrbued parameer sysemes, Ph.D. hess, Uversy of Perpga, Frace,. 9. Dauray R.; Los, J.L. Aalyse mahémaque e calcul umérque pour les sceces e les echques; sére scefque 8, Masso : Pars, Kamura, S.; Saka S.; Nshmura, M. Observer for dsrbued-parameer dffuso sysems. Elecrcal egeerg Japa 97, 9, El Ja, A.; El Yacoub, S. O he umber of acuaors parabolc sysem, Ieraoal Joural of Appled Mahemacs ad Compuer Scece 993, 3, Sample Avalably: Avalable from he auhors. by MDPI hp:// Reproduco s permed for ocommercal purposes.