Iellge Corol ad Auoao 3 78-89 hp://dxdoorg/436/ca3 Publshed Ole February (hp://wwwscrporg/oural/ca) Sesors ad Regoal Grade Observably of Hyperbolc Syses Sar Behadd Soraya Reab El Hassae Zerr Maheacs Depare Faculy of Exac Sceces Uversy Meour Cosae Algera MACS ea Depare of Maheacs Faculy of Sceces Uversy Moulay Isal Mees Morocco Eal: {hebah reabsoraya err3}@yahoofr Receved Augus ; revsed Noveber 3 ; acceped Noveber ABSRAC hs paper preses a ehod o deal wh a exeso of regoal grade observably developed for parabolc syse [] o hyperbolc oe hs cocers he recosruco of he sae grade oly o a subrego of he syse doa he ecessary codos for sesors srucure are esablshed order o oba regoal grade observably A approach s developed whch allows he recosruco of he syse sae grade o a gve subrego he obaed resuls are llusraed by uercal exaples ad sulaos Keywords: Dsrbued Syses; Hyperbolc Syses; Observably; Regoal Grade Observably; Sesors; Grade Recosruco Iroduco For a dsrbued paraeer syse evolvg o a spaal doa IR he oo of regoal observably cocers he recosruco of he al sae o a subrego of Characerao resuls ad approaches for he recosruco of regoal sae are gve [34] Slar resuls were developed for he sae grade of parabolc syses [] hs led o he so-called regoal grade observably ad cocers he possbly o recosruc he grade o a subrego whou he owledge of he syse sae he sudy of grade observably s ovaed by real applcaos he case of sulao probles also here exs syses for whch he sae s o observable bu he sae grade s observable exaple s gve [] I hs paper we prese a exeso of he above resuls o regoal grade observably o hyperbolc syses evolvg o a spaal doa ha s o say oe ay be cocered wh he observably of he sae grade oly a crcal subrego of More precsely le (S) be a lear hyperbolc syse wh suable sae space ad suppose ha he al sae y ad s grade y are uow ad ha easurees are gve by eas of oupu fucos (depedg o he uber ad srucure of he sesors) he proble cocers he recosruco of he sae grade o he subrego of he syse doa whou ag o accou he resduel par o \ Here we cosder he proble of regoal grade observably of hyperbolc syses ad we esablsh codo ha allows he recosruco of he al grade o such a subrego Ad he paper s orgaed as follows he secod seco s devoed o defos ad characeraos of hs oo for hyperbolc syses I he hrd seco we esablsh a relao bewee regoal grade observably ad sesors srucure he fourh seco s focused o regoal recosruco of he al grade I he las seco we gve a uercal approach exedg he Hlber Uueess Mehod developed by J os [5] ad llusraos wh effce sulaos Regoal Grade Observably e be a ope bouded subse of IR wh a re- gular boudary Fx ad le deoe by Q ad Cosder he syse descrbed by he hyperbolc euao y x A y x yx yx y y y Q () where A s he secod order ellpc lear operaor wh regular coeffces Copyrgh ScRes
S BENHADID E A 79 Euao () has a uue soluo ; ; y C H C [6] Suppose ha easurees o syse () are gve by a oupu fuco: C y () where C: H IR s a lear operaor depedg o he srucure of sesors e us recall ha a sesor s defed by a couple D f where D s he locao of he sesor ad f D s he spaal dsrbuo of easurees o D I he case of a powse sesor D b ad f b s he Drac ass coceraed b see [7] e y y y ad CyCy he he syse ay be wre he for y A (3) y y y I wh A A A has a copac resolve ad geeraes a srogly couous se-group S o a subspace of he gve by Hlber sae space r y w cos y y w s w S y r s y w y w w cos w H orhooral s a bass of egefucos of A ad he assocaed egevalues wh ulplcy r he (3) ads a u- ue soluo y Sy e us defe he observably operaor K H H : ; h CS h IR whch s lear ad bouded wh s ado deoed by K ad le be he operaor where : H H y y y y y y : H y y y yy x x x ad respec- whle her ados are deoed by vely Defo he syse () ogeher wh he oupu () s sad o be exacly (resp approxaely) grade observable f K I K resp Ker Such a syse wll be sad exacly (resp approxaely) G-observable For a posve ebesgue easure subse of we also cosder he operaors where ad : : y y y y y y : y y whle her ados deoed by ad respecvely ad gve by where y y y y : : y o y y o \ ad : y o y y o \ We fally roduce he operaor K : ; IR Copyrgh ScRes
8 S BENHADID E A Defo ) he syse () ogeher wh he oupu Euao () s sad o be exacly regoally grade observable or exacly G-observable o f I ) he syse () ogeher wh he oupu euao () s sad o be approxaely regoally grade observable or approxaely G-observable o f Ker he oo of regoal G-observably o ay be characered by he followg resuls 3 Proposo 3 ) he syse () ogeher wh he oupu Euao () s exacly G-observable o f ad oly f oe of he followg proposos s holds a) For all here exss c such ha c K ; IR b) Ker I K ) he syse () ogeher wh he oupu Euao () s approxaely G-observable o f ad oly f he operaor N s posve defe 4 Proof ) a) le us cosder he operaor h Id ad g K Sce he syse s exacly G-observable o we have I h I g ad by he geeral resul gve [8] hs s euvale o c such ha h ; IR c g b) e y he y sce he syse () s exacly G-observable o here exss ; IR such ha yk e pu y y y where y K ad K he y y Ker ad y I K Coversely le y he y here exs y ad y I K such ha y y y ad y y y y y ; Ker K here exss ; IR Sce y I such ha y K hus gves y K I y K whch ) e such ha N So ; IR whch eas ha Ker ad sce () s approxaely G-obser- vable he ha s N s posve defe Coversely le such ha he N here for ha s he syse s approxaely G-observable o 5 Rear 4 ) If a syse s exacly (resp approxaely) G-observable o s exacly (resp approxaely) G-observable o ) here exs syses whch are o G-observable o he whole doa bu ay be G-observable o soe subrego 6 Exaple 5 e we cosder he wo-desoal syse descrbed by he hyperbolc syse he operaor y x x yx x Q yx x y x x yx x y x x y A x x whch he egevalues are π assocaed o he egefucos w x x s π x s π x Measurees are gve by he oupu fuco y x x f x x dx dx D where D s he sesor suppor ad f x x sπ x s he fuco easure e he subrego 6 3 ad we cosder he al sae Copyrgh ScRes
S BENHADID E A 8 y x x sπ x s π x y x x cos π x cos3π x he he al sae grade o be observed s g g x x x x We have he resul 7 Proposo 6 π cos π x s π x πs π x cos π x πs π x cos 3π x x s 3π x 3π cosπ he grade g s o approxaely G-observable o he whole doa however s approxaely G-observable o he subrego 8 Proof o prove ha g s o approxaely G-observable o we us show ha gker K We have Sce K g g g w cos we have IN ad g w s w for w π D s for π g w π cos πx cos x dx s π x s π x dx 4 π s π x s π x dx g w π s π cos π x cos π x dx cos 3π x x cos π x dx s π x dx 6 π cos π x s π x dx s 3π x cos π x dx hs gves K g g ad he he syse s o approxaely G-observable oce O he oher had g ay be approxaely G-observable o K g g he Ideed suppose ha Sce for g w cos g w s w s cos large eough he se fors a coplee orhooral se of bu for we have g w w D g w w f D ; ad IN we have π w s D wch gves IN g w g w Bu for we have hus π g w 3 3 4 3 K g g 3 Grade Sraegc Sesors he purpose of hs seco s o esablsh a l bewee regoal grade observably ad he sesors srucure e us cosder he syse () observed by sesors D f whch ay be powse or oe 3 Defo 3 A sesor D f (or a seuece of sesors) s sad o be grade sraegc o f he observed syse s G-observable o such a sesor wll be sad G-sraegc o We assue ha he operaor A s of cosa coeffces ad has a coplee se of egefucos deoed by w orhooral H Copyrgh ScRes
8 S BENHADID E A assocaed o he egevalues of ulplcy r Assue also ha r sup r s fe he we have he followg resul 3 Proposo 3 If he seuece of sesors D f s G-sraegc o he ad ra M r where ad r ad M r r w x w x b po wse case oe case D s he row vecor he elees of whch are wh w w dx ; for ; r 33 Proof he proof s developed he case oe sesors he seuece of sesors D f s G-sraegc o f ad oly f u u ; IR Suppose ha he seuece of sesors D f s G- sraegc o ad here exss wh M r ra he here exss Z such ha (4) ad M (5) r e verfyg w r w r (6) e verfyg w r w r ad le ad he u K u assue ha he ad he Ku v v H H u v v Iegrag o we oba v v x v v x u u K bu we have ad r x x w w r w w Usg he fac ha rl lp w wlp l p (7) Copyrgh ScRes
S BENHADID E A 83 ad rl lp w w lp l p he we oba ad r rl lp x l p x rl l p r fro (5) (6) ad (7) we oba r l p lp w w lp lp w x w x r l lp wl cos p rl l w p wl s p l p x hus hs gves u K x x u ; IR ad whch coradcs he fac ha he seuece of sesors s G-sraegc 34 Rear 33 ) he above proposo ples ha he reured uber of sesors s greaer ha or eual o he larges ulplcy of egevalues ) By fesally deforg of he doa he ulplcy of he egevalues ca be reduced o oe [9] Coseuely he regoal G-observably o he subrego ay be possble oly by oe sesor 4 Regoal Grade Recosruco I hs seco we gve a approach whch allows he recosruco of he al sae grade o of he syse () hs approach exeds he Hlber Uueess Mehod developed for corollably by os [6] ad do ae o accou wha us be he resdual al grade sae o he subrego \ Cosder he se F o\ HH H H where for he syse x A x Q x x x x (8) has a uue soluo ; ; C H C We cosder he oe sesor case where he syse () s observed by he oupu fuco y x x D D s he sesor suppor he fuco of easure ad we cosder a se-or o F defed by F d x D where x s he soluo of (8) he reverse syse gve by x A x D f Q x x x has a uue soluo ; ; C H C [5] We deoe he soluo x x by x e cosder he operaor where P F by P x ad (9) () () ad cosder he rerograde syse whch has a uue soluo Copyrgh ScRes
84 S BENHADID E A Z x y A Z x D f Q x Zx Zx Z Z x by Z x ZC ; H C ; [5] We deoe he soluo ad Z x by Z x he he regoal grade observably urs up o solve he euao PZ Z where Z Z Z Z ad Z Z Z Z 4 Proposo 4 () (3) If he sesor D f s G-sraegc o he he euao (3) has a uue soluo whch s he grade of he al sae o be observed o 4 Proof ) e us show frs ha f he syse () s G-observable he () defes a or o F Cosder a bass w of he egefucos of A whou loss of geeraly we suppose ha he ulplcy of he egevalues are sple he F x o whch s euvale o w cos w s he se s cos orhogoal se of D w x fors a coplee w w f x w w ad sce he sesor we have he we oba D f x D D f s regoally G-sraegc o he w x D w w Coseuely ad hus Coversely c ad (cosas) sce ; ; C H C ad fro o () s a or ) e deoe by F copleo of F by he or () ad F be s dual We show ha s a soorphs fro F o F Ideed le ˆ ˆ F ad ˆ he correspodg soluo for he proble (8) ulply he frs euao of he syse () by ˆ x ad egrae o Q we oba x ˆ ˆ A x x Q Q ˆ x f D x x l l D for he frs er we oba ˆ ˆ x x Q ˆ A ˆ x x Q Usg Gree forula for he secod er we oba ˆ ˆ x x x A f D l Q l D Q ˆ A x Q ˆ ( ) ˆ ( ) ( ) ( ) d A x x A ˆ( ) x f D dxd x x l l D ad wh he boudary codos we oba ˆ ˆ ˆ d x x l l Usg Cauchy-Schwar eualy we have c Copyrgh ScRes
ˆ ˆ F ˆ ˆ ˆ ˆ F F S BENHADID E A 85 M s a order of rucao Hece whch proves ha s a soorphs ad coseuely he Euao (3) has a uue soluo whch correspods o he sae grade o be observed o he subrego 43 Rear 4 he prevous approach ca be esablshed wh slar echues whe he oupu s defed by eas of eral or boudary powse sesors 5 Nuercal Approach I hs seco we gve a uercal approach whch leads o explc forulas for y y o We cosder he case where he syse () s observed by he oupu euao y x 5 Proposo 5 ( D ) If he sesor D f s G-sraegc o he he al grades y ad y ay be approached by ˆ y ad ˆ y respecvely M M w w l ˆ x y (4) y w cos d w w x xl o o \ M M w w l ˆ x y (5) y s w d w w x x l o o \ Fˆ F where 5 Proof I he prevous seco has bee see ha he regoal recosruco of he al sae grade o urs up o solve he Euao (3) For ha cosder he fucoal P Z Z d Z Z x Ad solvg Euao (3) urs up o e wh respec o Afer develope ad whe we oba ad l d x ( D) w w w 4 x For large eough we have ( D) d x w w w 4 x O he oher had we have sce x w w x x w w x he w w w o (6) x w x x x ad w w w o (7) x w x x x we oba w x Z w Z Copyrgh ScRes
86 S BENHADID E A ad Z w Z w x he ao of (3) s euvale o solve he wo followg probles ad If w 4 w Z x If w w x ( D) w w 4 x ( D) w w Z x whch soluos are ad w w Z w Z w x ( D) w w x ( D) Now le x Z of he syse () wh (8) (9) w x be he soluo w f y s Z s s d s x ad hus Z Z x s w sds w Z x w he we oba cos y s x x sds w y s x x ad Z w y s w ( D) l w s sd s w x xl Z w w f ( D l y s w cos s d s w x xl Wh hese developes accordg o (8) ad (9) we oba ad w w f w l x y s w cos sd s w l x x w w w l x y s w s d x x We replace ha he relao (6) ad (7) we oba f s s w l ad x w y w l x s w cos d o f s s w w x xl x w w l x ys w s d o f s s w w x xl We cosder a rucao up o order M M IN he we oba he relao (4) ad (5) We defe a fal error ˆ ˆ y y y y he good choce of M wll be such ha Copyrgh ScRes
S BENHADID E A 87 ad we have he followg algorh: Algorh Sep : Daa: he rego he sesor locao D ad Sep : Choose a low rucao order M Sep 3: Copuao of ˆ y ad ˆ y by he forulae (4) ad (5) Sep 4: If he sop oherwse Sep 5: M M ad reur o sep 3 53 Rear 5 f y ad y are regular eough we have a regular syse sae so easurees ay be ae wh powse sesor I hs case we oba slar forulaes as he prevous proposo gve by M M w b w l b ˆ x y () yb w cos d w w x xl o o \ M M w b w l b ˆ x y () yb w s d w w x xl o o \ 6 Sulaos 6 Exaple I hs seco we develop a uercal exaple ha leads o resuls relaed o he choce of he subrego he sesor locao ad he al sae grade O we cosder he oe desoal syse yx yx x yx y x () yx y x y y Measurees are gve by he oupu fuco ' y b (3) he prevous syse s G-observable o [7] f ad oly f bs IN We deoe ha uercally a rraoal uber does o exs bu ca be cosdered as rraoal f rucao uber exceeds he desred precso e 5 ad he sesor s locaed a b 4855 he al grade o be recosruced s gve by ad y A x s 3π y A s 4πx 4πxcos 4πx he coeffces A A are chose such ha he uercal schee be sable ad order o oba a reaso- able aplude of y ad y le us ae A 5 ad A Applyg he prevous algorh usg he forulae () () we respecvely oba he Fgures ad for 46 66 ad respecvely he Fgures 3 ad 4 for he esaed grade s obaed wh error 4 8 o 46 66 For he grade s recosruced wh error 449 o 6 Sulag Coecures Now we show uercally how he error grows wh respec o he subrego area I eas ha he larger he rego s he greaer he error s he obaed resuls are preseed able Fgure Ihebeps: al sae grade y (couous le) ad esae al sae grade bar ()y (dashed le) Copyrgh ScRes
88 S BENHADID E A able Evoluo error wh respec o he area of he subrego 449 494 35 3 484 436 4 3474 5683 5 4666 8634 able Evoluo error wh respec o he al sae grade aplude Fgure Nouhaeps: al speed grade y (couous le) ad esae al speed grade bar ()y (dashed le) A 5 8 4 5 568 385 3 985 536 4 4 8634 5 Also how boh he error decreases wh respec o he aplude A of he al sae grade For hs le ae he subrego 46 66 ad A We oe ha he recosruco error depeds o he a- plude of al sae grade I eas ha he greaer he aplude s he greaer he error s he obaed resuls are preseed able Fgure 3 Ihebeps: al sae grade y (couo us le) ad esae al sae grade bar ()y (dashed le) Fgure 4 Nouhaeps: al speed grade y (couou s le) ad esae al speed grade bar ()y (dashed le) 7 Cocluso Grade Observably o a subrego eror o he spaal evoluo doa of hyperbolc syse s cosdered A relao bewee hs oo ad he sesors srucure s esablshed ad uercal approach for s recosruco s gve hs allows he copuao of he al sae grade whou he owledge of he syse sae Illusraos by uercal sulaos show he effcecy of he approach Ieresg uesos rea ope he case where he subrego s par of he boudary of he syse doa hs ueso s uder cosderao REFERENCES [] E Zerr H Bourray ad A El Ja Regoal Flux Recosruco for Parabolc Syses Ieraoal Joural of Syses Scece Vol 34 No -3 3 pp 64-65 do:8/77368 [] E Zerr ad H Bourray Regoal Grade Observably for Parabolc Syses Ieraoal Joural of Appled Maheacs ad Copuer Scece Vol 3 3 pp 39-5 Copyrgh ScRes
S BENHADID E A 89 [3] A El Ja M C So ad E Zerr Regoal Ob- servably ad Sesor Srucures Ieraoal Joural o Sesors ad Acuaors Vol 39 No 993 pp 95- do:6/94-447(93)84- [4] E Zerr Regoal Aalyss of Dsrbued Paraeer Syses PhD hess Uversy Med V Raba 993 [5] J os ad E Magees Problèe aux es o Hoogèes e Applcaos Duod Pars 968 [6] J os Corôlablé Exace Perurbao e Sa- blsao des Sysèes Dsrbués Duod Pars 988 [7] A El Ja ad A J Prchard Sesors ad Acuaors he Aalyss of Dsrbued Syses I: E Horwood Ed Seres Appled Maheacs Joh Wley Hoboe 988 [8] V reogue Aalyse Focoelle Edo MIR Moscou 985 [9] A M Mchele Perurbaoe dello Sopero d u Ellpc d po Varaoale Relaoe ad ua Varaoe del Capo ( Iala) Rcerche d Maeaca Vol XXV Fasc II 976 [] A El Ja ad S El Yacoub O he Nuber of Acuaors Dsrbued Syse Ieraoal Joural of Appled Maheacs ad Copuer Scece Vol 3 No 4 993 pp 673-686 Copyrgh ScRes