Information Fusion Kalman Smoother for Time-Varying Systems

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1 Infoaon Fuon alan oohe fo Te-Vayng ye Xao-Jun un Z- Deng Abac-- Fo he lnea dcee e-ayng ochac conol ye wh uleno coloed eaueen noe hee dbued opal fuon alan oohe ae peened baed on he opal nfoaon fuon ule weghed by ace dagonal ace cala n he lnea nu aance ene. Copaed wh he cenalzed fue hey ae locally opal bu ae globally ubopal. The accuacy of he fue wh a wegh hghe han ha of he fue wh cala wegh he accuacy of he fue wh dagonal a wegh beween boh of he. The accuacy of all he hee fue hghe han ha of each local alan oohe. Fuhe he coepondng hee eady-ae fuon fed-lag alan oohe ae alo gen fo he lnea dcee e-naan ochac conol ye whch can educe he on-lne copuaonal buden. In ode o copue opal wegh he new foula of copung he co-coaance aong local oohng eo ae peened. Two Mone Calo ulaon eaple fo acng ye how he pefoance of he popoed fue. Inde Te Te-ayng ye coloed eaueen noe dbued fuon alan oohe uleno nfoaon fuon T I. ITODUCTIO nfoaon fuon o daa fuon fo eaon ha wdepead applcaon nce any paccal poble nole daa fo ulple ouce. In ecen yea he nfoaon fuon alan fleng heoy ha been fuhe uded wdely appled n any feld uch a gudance defence oboc negaed nagaon age acng G poonng councaon gnal poceng conol ec[]. Fo alan fleng-baed fuon wo bac fuon ehod ae cenalzed decenalzed o dbued fuon ehod dependng on whehe aw daa ae ued decly fo fuon o no [2]. The cenalzed fuon ehod can ge he globally opal ae eaon by decly cobnng local eaueen daa. Bu dadanage ae ha ay eque a lage copuaonal buden hgh daa ae fo councaon. The dbued fuon ehod can ge he globally opal o ubopal ae eaon by cobnng Th wo uppoed by aonal aual cence Foundaon of Chna unde Gan FC X. J. un wh Depaen of Auoaon elongang Uney 58 Z.. Deng wh Depaen of Auoaca elongang Uney 58 abn Chna coepondng auho phone: e-al: dzl@hlu.edu.cn. he local ae eao. Th ehod ha condeable adanage: can faclae faul deecon olaon oe conenenly can nceae he npu daa ae gnfcanly. Thee ae wo appoache o dbued fuon alan fleng whch ae he nfoaon a appoach [34] he weghed coaance appoach [5-9]. The dbued fuon alan fle wh feedbac whou feedbac by he nfoaon a appoach ae equalen o he cenalzed fuon alan fle ge he globally opal ae eaon [4] bu hey all wll eque eene calculaon of local global nee coaance. The weghed coaance appoach a coaance auo-coaance co-coaance-baed weghed fuon appoach wh aou weghed fuon ule. I can elnae epene copuaonal equeen bu geneally ge a globally ubopal ae eaon wh a lgh lo of accuacy. o fa hee hae been eeal opal weghed fuon ule whch ae he fuon ule weghed by ace n he lnea nu aance ene [9] he au lelhood M fuon ule wh an aupon of a noal deny funcon [] he fuon ule weghed by cala n he lnea nu aance ene [8] he weghed lea quae W fuon ule [59]. oe ha he opal fuon eae elaed o he pefoance nde of opzaon whn a eced local lnea pace. All hee fuon ule ge locally opal eao whch globally ubopal copaed wh he cenalzed fuon eao. o fa he uleno nfoaon fuon eaon anly focued on he fleng fuon [457] bu he oohng fuon eldo epoed [9] he eaueen noe of eno uually aued o be whe noe bu he fuon eaon fo uleno ye wh coloed eaueen noe eldo epoed [2]. ecenly a nfoaon fuon eady-ae alan eao wa peened [9] whch can hle he fued fleng pedcon oohng poble bu only uable fo he e-naan ye wh whe eaueen noe. Fo he e-ayng uleno ye wh whe eaueen noe un [] peened a dbued fuon fed-lag alan oohe wh he coponen fuon weghed by cala whch equalen o he fue weghed by dagonal ace [9] bu no uable fo uleno ye wh coloed eaueen noe he fued alan oohe weghed by ace cala wee no peened. A dbued opal fuon 655

2 eady-ae alan fle wh a wegh wa peened fo ye wh coloed eaueen noe [2] bu he oohng fuon poble wa no oled n [2]. In ode o oecoe he aboe dawbac laon baed on he opal fuon ule weghed by ace dagonal ace cala n he lnea nu aance ene hee opal nfoaon fuon alan oohe ae peened fo he dcee e-aan lnea ochac conol ye wh uleno coloed eaueen noe n h pape. In ode o copue he opal wegh he foula copung he local eaon eo co-coaance ae peened whch ae dffeen fo he foula popoed by un []. Condeng he e-naan cae he hee nfoaon fuon eady-ae fed-lag alan oohe he foula copung he co coaance aong local eaon eo ae alo gen. Two Mone Calo ulaon eaple how he accuacy elaon of he popoed oohng fue. The ee of h pape oganzed a follow: The poble foulaon gen n econ 2. The hee dbued fuon alan oohe ae peened n econ 3. econ 4 ge he hee eady-ae fuon alan oohe. Two Mone Calo ulaon eaple ae gen n econ 5. The concluon peened n econ 6. II. OBM FOMUATIO Conde he dcee e-ayng lnea ochac conol ye wh eno Φ B u w z η 2 η A η ξ 3 whee dcee e ae he eaueen n he ae z p u he nown conol npu w ξ ae he whe noe η ae he coloed eaueen noe Φ ae e-ayng ace wh A copable denon. Aupon. w ξ ae ndependen whe noe wh zeo ean aance a ξ epecely. Aupon 2. The nal ae wh ean µ eo aance a ndependen of w ξ. The opal nfoaon fuon alan oohe poble o fnd he opal lnea nu aance fuon alan oohe ˆ > weghed by ace dagonal ace cala baed on he local alan oohe ˆ epecely. Inoducng new eaueen y z A z B u 4 ubung -3 no 4 we hae y Φ A w ξ 5 eng Φ A 6 w ξ 7 cobnng 5 we hae Φ B u w 8 y 9 The new ye 8 9 ae he lnea ochac conol ye wh coelaed whe noe w coelaed eaueen noe ha w [ w ] δ [ ] δ whee Ε he epecaon he upecp denoe he anpoe δ he onece dela funcon δ δ. III. DITIBUTD FUIO AMA MOOT ea [3]. Fo he uleno e-ayng ye -3 wh he aupon 2 he h eno ubye ha he local alan pedco ˆ ˆ B u y ξ p y ˆ 2 p f p Φ 3 p [ Φ ] 4 p 5 6 whee f ae he pedcon fleng p gan ace epecely. The pedcon eo aance a afe he cca equaon Φ Φ [ Φ ][ ] [ Φ ] 7 656

3 wh nal alue µ ˆ. oof. The poof of ea gen n [3] whch oed. Theoe. Fo he uleno e-ayng ye -3 wh he aupon 2 he co-coaance ace aong local pedcon eo ae gen a ] [ p p p p 8 o p p p p p p 9 wh he nal alue. oof. Fo [3] we hae he pedcon eo equaon w p p 2 whee w ndependen of. Ung 2 we hae 8. ea 2 [3]. Fo he uleno e-ayng ye -3 wh he aupon 2 he h eno ubye ha he local opal alan oohe ˆ ˆ 2 whee he oohng gan a gen a } { p 22 f 23 whee p f ˆ obaned fo ea he oohng eo aance a gen a 24 oof. The poof of ea 2 gen n [3] whch oed. Theoe 2. Fo he uleno e-ayng ye wh he aupon 2 he co-coaance ace aong local oohng eo a p p 25 whee > we defne p p p I n p 26 ] [ Ε 27 When n > we hae he followng equaon p p n p ] [ p p p δ 28 When n we hae p ] [ p p p ] [ p p 29 oof. Fo [3] we hae w p p 3 whch yeld p p ] [ w p 3 o we hae p p [ w p 32 whch yeld 28 ung 3 we oban 29. Theoe 3. Fo he uleno e-ayng ye -3 wh he aupon 2 hee dbued opal nfoaon fuon fed-lag alan oohe ae gen a Ω ˆ ˆ 33 Fo he fue wh a wegh we hae 657

4 [ Ω Ω ] e e e n n whee e I n I ] he fued eo aance a [ n gen a [ e e] 36 Fo he ue wh cala wegh Ω ω we hae [ ω ω ] e e e whee e [ ] denoe he ace of a he fue eo aance a gen a ω ω 39 Fo he fue wh dagonal a wegh we hae Ω dag ω ω 4 [ ω ω ] e e e 4 42 whee e [ ] ae he h dagonal eleen of. The ace of he fued eo aance a d gen a d [ e e] 43 Denong he cenalzed fuon eo aance a a c we hae he accuacy he elaon c 44 oof. Applyng he hee opal fuon foula weghed by ace dagonal ace cala n [9] we decely oban Theoe 3. IV. TADY-TAT FUIO AMA MOOT Fo he e-aan ye -3 wh conan ace Φ Φ B B A A ξ ξ we hae. If eey local eno ubye ha eady-ae alan eao we can oban he nfoaon fuon eady-ae fed-lag alan oohe whch can educe he on-lne couaon buden. Theoe 4. Fo uleno e-aan ye -3 wh aupon 2 he local eady-ae alan pedco gen a d whee ˆ ˆ Bu y 45 p y ˆ 46 p p Φ 47 [ p ΦΣ ] 48 f Σ 49 Σ p Σ afe he eady-ae cca equaon Σ Φ Σ Φ [ ΦΣ ][ Σ ] ] [ ΦΣ 53 he local eady-ae fed-lag alan oohe gen a ˆ ˆ 54 Whee he oohng gan a hown a follow Σ p 55 The eady-ae oohng eo aance a l gen a Σ 56 The eady-ae oohng eo co-coaance a Σ pσ Σ When n > we hae n p p 57 p [ p ] δ Σ p When n we hae p p Σ 58 p Σ p [ Σ p [ p ] p 58 oof. Fo he eady-ae alan fleng we hae ha p p p p Σ Σ a. Tang ea p ] 658

5 3 Theoe -3 we decly oban Theoe 4. Theoe 5. Fo uleno e-aan ye -3 wh aupon 3 he hee dbued fuon eady-ae fed-lag oohe gen by ˆ Ω ˆ 59 Whee ˆ ae copued a Theoe 4. The wegh Ω ae copued a Theoe 3 whee Ω ω ω ae eplaced by Ω ω ω elaon epecely. We hae he eady-ae accuacy d c 6 whee d denoe he eady-ae eo aance ace fo fue wh a wegh dagonal a cala wegh epecely denoe he eady-ae eo aance a of cenalzed fue. oof. Tang n Theoe 3 we aghfowad oban Theoe 5. c V. IMUATIO XAM aple. Conde he 3-eno dcee e-ayng acng ye wh coloed eaueen noe Φ B u w 6 z η 62 η A η ξ T.5T 2.5T 2.5T Φ T B T T..9co 2π.2.n.5 2π ξ. ξ 2 2π.2.8co..9n 2π ξ 3 c ρ n 2π. A.2 c. ρ Whee T he apled peod we ae T. 5 [ ] he ae. ae 2 3 he poon elocy acceleaon of age a e T 2 3 epecely. η ae he coloed eaueen whe noe w 2 2 w 2 d 2 c w ξ ae ndependen whe noe wh zeo ean aance ace ξ epecely. The conol u nown we [25] [575] aeu. The poble o copae [265] [76] he accuacy of local alan oohe ˆ 2 fued alan oohe ˆ θ 2 θ d cenalzed fue ˆ 2. The ulaon eul ae hown n c Fg.-Fg.3 Table. In Fg. Table we can ee ha he accuacy of fuon oohe hghe han ha of each local oohe he heoecal accuacy elaon 44 hold. 5 Mone Calo un ae caed ou he ean quae eo M cue hown n he Fg.2 Fg.3 whee he M alue a defned a M whee 2 ˆ 2 23 c ˆ 2 he h aple of he ochac poce ˆ 2 3 he apled nube. Fo Fg.2 Fg.3 we ee ha he accuacy of he fue hghe han ha of each local oohe he accuacy of he cenalzed fuon hghe han ha of hee weghed fue no obou becaue he M cue ae a ndnguhable /ep Fg. Copaon of ace 2 θ 2 θ d c of local fued oohng eo aance ace eno eno2 eno3 fuon weghed by cala fuon weghed by dagonal ace fuon weghed by ace cenalzed fuon aple 2. Conde he e-naan acng ye wh 3-eno coloed eaueen noe Φ w

6 z η 67 η a η ξ Φ.2. [ ] [] 3 [ ] M M /ep Fg. 2 The ean quae eo M cue od local fued oohe n 5 Mone Calo un eno eno2 eno3 fuon weghed by ace fuon weghed by cala fuon weghed by dagonal ace cenalzed fuon /ep Fg. 4 The ean quae eo M cue of local fued oohe n 3 Mone Calo un eno eno2 eno3 fuon weghed by ace fuon weghed by cala fuon weghed by dagonal ace cenalzed fuon M /ep Fg. 3 The ean quae eo M cue of he hee dbued fued cenalzed fued oohe n 5 Mone Calo un fuon weghed by ace fuon weghed by cala fuon weghed by dagonal ace cenalzed fuon M /ep Fg. 5 The ean quae eo M cue of he hee dbued fued cenalzed fued oohe n 3 Mone Calo un whee T he apled peod [ 2 3 ] he ae ae he poon elocy 2 fuon weghed by ace fuon weghed by cala fuon weghed by dagonal ace cenalzed fuon acceleaon of age a e T epecely. w ξ ndependen Gauan whe noe wh zeo ean 3 66

7 aance ace σ. 36 σ σ 2 2 w ξ ξ 2 σ ξ 3 7 epecely a. a 2. 3 a 3. 6.The poble o copue he accuacy of local fued eady-ae alan oohe. 5 Mone Calo un ae caed ou he ean quae eo M cue hown n he Fg.4 Fg.5 Table 2 whee we can oban he ae concluon a he eaple. Table. Copaon of local ace 2 fued ace 2 θ d c d c θ Table 2. Copaon of local ace 2 fued ace 2 θ d c ace d 2 2 c 2 alue I. OCUIO Fo he dcee e-ayng lnea ochac conol ye wh uleno wh coloed eaueen noe hee opal nfoaon fuon alan oohe hae been peened baed on he opal fuon ule weghed by ace dagonal ace cala n he lnea nu aance ene he coepondng hee eady-ae fuon alan oohe hae been peened fo he naan ye. In ode o copue he opal wegh he new foula of copung he co-coaance aong local oohng eo hae been peened. They ae locally opal ae globally ubopal. Two Mone Calo ulaon eaple hown he accuacy dncon of hee alan fue no obou o ha eployng he alan fue wh cala wegh o dagonal ace uable fo eal e applcaon. The popoed eul oecoe he laon dawbac n oe efeence. ACOWDGMT Th wo wa uppoed by aonal aual cence Foundaon of Chna unde Gan FC The auho wh o han he eewe fo he conuce coen. θ [4] Y. M. Zhu Z.. You J. Zhao.. Zhang X.. The opaly fo he dbued alan fleng fuon wh feedbac Auoaca ol. 37 pp [5]. A. Calon Fedeaed quae oo fle fo decenalzed paallel poce I Tanacon on Aeopace leconc ye ol. 34 pp [6]. X. aha. C. Chang ffcen algoh fo uleno ac fuon I Tanacon on Aeopace leconc ye ol. 34 pp [7].. un Z.. Deng Mul-eno opal nfoaon fuon alan fle Auoaca ol. 4 pp [8].. un Muleno nfoaon fuon whe noe fle weghed by cala baed alan pedco Auoaca ol. 4 pp [9] Z.. Deng Y. Gao. Mao Y. G. ao ew appoach o nfoaon fuon eady-ae alan fleng Auoaca ol. 4 no. pp [].. Deelopen of ac o ac fuon algoh oceedng of Aecan conol confeence Mayl June pp [].. un Dbued opal coponen fuon weghed by cala fo fed-lag alan oohe Auoaca ol. 4 pp [2].. un Z.. Deng Dbued opal fuon eady-ae alan fle fo ye wh coloed eaueen noe Inenaonal Joual of ye cence ol. 36 pp [3] Z.. Deng Opal aon Theoy wh Applcaon- Mondelng Fleng Infoaon Fuon aon abn: abn Inue of Technology e pp FC [] Y. Ba halon X.. Muleno Tacng: ncple Techngue. o CT: YB ublhng 995. [2] X.. Y. M. Zhu J. Wang C. Z. an Opal lnea eaon fuon-pa :Unfed fuon ule I Tanacon on Infoaon Theoy ol. 49 pp [3]. C. Chang T. Zh.. aha efoance ealuaon of ac fuon wh nfoaon a fle I Tanacon. Aeopace leconc ye ol. 38 no. 2 pp