Distributed Fusion Filter for Asynchronous Multi-rate Multi-sensor Non-uniform Sampling Systems

Transcription

1 Dstrbuted Fuso Flter for Asychroous Mult-rate Mult-sesor No-ufor Saplg Systes Jg Ma, Hogle L School of Matheatcs scece Helogjag Uversty Harb, Helogjag Provce, Cha ajg47@gal.co, lhogle8@63.co Shul Su Departet of Autoato Helogjag Uversty Harb, Helogjag Provce, Cha susl@hlju.edu.c Abstract hs paper s cocered th the dstrbuted fuso flterg proble for a class of asychroous ult-rate ultsesor o-ufor saplg dscrete stochastc systes, here the state s updated at the hghest saplg rate ad dfferet sesors ay have dfferet loer easureet saplg rates. Furtherore, the state s updated uforly ad the easureet s sapled o-uforly. he o-augeted state odels at each sesor are establshed by cosderg the syste oses. he local flters at easureet saplg pots of each sesor are desged based o the establshed state space odels by a ovato aalyss approach. Further, the local flters at state update pots are proposed. he correspodg flterg error covarace atrces are derved. Usg the covarace tersecto fuso algorth, the dstrbuted fuso flter s gve based o the local flters ad the local flterg error covarace atrces. he proposed algorth ca sgfcatly prove the estato accuracy copared to the prevous odelg ethod hch gores the syste ose. he sulato research verfes the effectveess of the proposed algorth. Keyords-ult-rate, ult-sesor, o-ufor saplg, covarace tersecto, dstrbuted fuso flter I. INRODUCION Recetly, the state estato proble for ult-sesor ult-rate systes has attracted lots of atteto due to the possblty to saple all the physcal sgals at oe sgle rate ay coplcated practcal systes. Geerally speakg, there are to cases for t. he frst case s the estato proble for sgle sesor syste [-], hch cludes to rates or four rates as follos: the state updatg rate, the easureet saplg rate, the estato updatg rate ad the estato output rate. he dffculty s ho to obta the state estato values at o easureet tes. he lear u varace optal state flter s preseted by state augetato approach []. Hoever, the proposed flter has expesve coputatoal cost due to the hgh deso augeted state. he other case s the estato proble for ult-sesor syste [3-6], here the state s updated at the fastest rate ad dfferet sesors have dfferet easureet saplg rates. Further, there s at least oe sesor th the saplg rate sae th updatg rate. he dffculty s ho to trasfor the fuso estato proble for ult-rate systes to the equvalet fuso estato proble for a sgle rate syste. Geerally, there are to ethods: ultscale theory [3-7] ad Kala flter [8-6]. Based o ultscale theory, usg avelet trasforato, Hog [3-4], Zhag [5] ad We [6-7] et al propose soe fuso strateges for dfferet sesors th the rato of the saplg rates beg a poer of to. Hoever, the hte ose s coverted to colored ose the state decoposto [3-4], hch results the suboptal state estators. he state augetato ethod s used [5-7], hch results the hgh deso coputatoal cost. Based o Kala flter, Ge [8], Ya ad Zhou [9-3] propose soe fuso algorths for dfferet sesors th the rato of the saplg rates beg ay postve ratoal. Hoever, the flters [8-9] have expesve coputatoal cost, the flter [] s suboptal sce soe avalable sesor forato s gored at soe stats ad the flters [-3] are also suboptal sce the syste ose s gored. Moreover, the cetralzed ad dstrbuted asychroous fuso algorths are preseted for cotuous stochastc syste [4]. he H flterg proble s also vestgated for ult-rate syste [5-6]. Motvated by the above dscusso, e cosder the dstrbuted fuso flterg proble for a asychroous ultrate ult-sesor o-ufor saplg dscrete stochastc syste. Frstly, the o-augeted state odels at each sesor are establshed by cosderg the syste oses. he, based o the establshed state space odels, the local flters at the easureet saplg pots ad at the state update pots are obtaed, respectvely. Further, the dstrbuted fuso flter s gve by the covarace tersecto fuso algorth [7]. he local flterg error covarace atrces are derved to copute the fuso eghts. II. PROBLEM FORMULAION Cosder the follog asychroous ult-rate ultsesor o-ufor saplg dscrete te varat lear stochastc syste th L sesors xt ( + ) Φxt ( ) + Γt ( ) () y H x + v,,,, L () hs ork as supported by Natoal Natural Scece Foudato of Cha, NSFC-67439, ad Progra for Ne Cetury Excellet alets Uversty uder Grat NCE

2 here xt () R s the syste state at t te, s the saplg perod, y R,,,, L are the kth easureets. x ( k ),,,, L are the states easured by r y ( k ). t () R ad v R are the syste ose ad easureet ose. Φ, Γ ad H are costat atrces th sutable desos. he syste state xt ( ) s updated at the hghest rate S x, the easureets are sapled at the loer rates S y, ad satsfes S : x Sy here,,,, L are the ko postve tegers. he easureets are sapled o-uforly. he subscrpt deotes the th sesor ad L s the uber of sesors. Assupto : t ( ) ad v ( k ) are ucorrelated hte oses th zero eas ad covarace atrces E[ t ( ) ( t)] Q, E[ v v ] Qv, here E deotes the atheatcal expectato, the superscrpt deotes the traspose. Assupto : he tal state x () s ucorrelated th t () ad v ( k ),,,, L ad satsfes that { x } μ E (), E ( x() μ )( x() μ ) P (3) { } Our a s to fd the dstrbuted fuso flter o ( t k ) based o the receved easureets ( y,, y()),,,, L easured by dfferet sesors th asychroous o-ufor saplg rates. A exaple of te versus sesor ap s depcted Fg., here the state updates uforly at the hghest rate ad sesor ad ay saple o-uforly at the loer saplg rates. he ratos of saplg rates betee state ad sesors are S : S ad S : S 3. x y he state data ca be dvded to blocks ad the legth of per block N (,,, L ) s the least coo ultple of,,,, L. he sesor should saple p N / tes per block. hs eas that the saplg of dfferet sesors ca be asychroous. We refer the readers to [-3] for ore detals. x sesor sesor data block Fgure. Saplg vs. sesor x y data block... Saplg pots III. DISRIBUED FUSION FILER A. Syste odelg I the follog, e ll establsh the o-augeted state odels at each sesor by cosderg the syste oses. heore : Uder Assuptos ad, the state space odel at sesor ca be set up as here x ( k + ) Φ x + (4) y H x + v,,,, L (5) Φ Φ (6) x ( k ) (/ )[ x ( ( k ) + + )] Φ Γ r ( k ) (/ ) ( k + r ) (7) (8) Also e have the follog ose statstc forato E[ ( ) ( )] (/ ) ( ) Φ Γ Γ Φ r + r + ( / ) r[ Φ ΓQΓ ( Φ ) r + r Φ ΓQ Γ ( Φ ) ] Q k k Q + (9) Q E[ ( k )] r + r Φ Γ r Γ Φ ( / ) r Q ( ) () Proof: Fro (7), e have x ( k + ) (/ ) x( k + + ) ( / )( x( k + ) + x( k + ) x( k + )) (/ )[ Φ x( ( k ) + ) + Φ Γ( k ) +... x k Φ Γk + Φ ( ( ) + ) + ( + )] Φ x + hereφ, are defed by (6) ad (8). Reark : Dfferetly fro referece [] here the state augetato approach s used ad the coputatoal cost becoes ore expesve as the legth of per block N creases, here the o-augeted state space odels are establshed ad the coputatoal cost s obvously reduced. 646

3 Dfferetly fro refereces [-3] here the establshed odels are coarse sce the syste ose s gored ad the errors of establshed odels becoe larger as the covarace atrx of the syste ose Q creases, here the oaugetato state space odels are establshed by takg the syste ose to accout. Reark : Fro (8)-(), e see that the syste ose ( k ) s colored ose hch s correlated at the sae ad adjacet te stats, hch ples x ( k ) s also correlated th ( k ). Hece, the tradtoal Kala flter s o loger applcable. B. Local flters at the easureet saplg pots I the follog, based o syste (4)-(5), e ll derve the local flters ( k k ) at the easureet saplg pots ad correspodg flterg error covarace atrces P ( k kby ) the ovato aalyss approach. heore : Uder Assuptos ad, the local flter at the easureet saplg pots of the th sesor subsyste of syste (4) (5) s coputed by ( k k) ( k k ) + K ε () ( k + k) Φ ( k k ) + L ε () ε y H ( k k ) (3) K k P k k H Q k (4) () ( ) ε () predctor, P ( k k ) ad P ( k k ) are the flterg ad predcto error covarace atrces. he tal values are ( ) μ ad P ( ) P. Proof: By projecto theory [8], e readly have () (3), here the flterg ga atrx s defed as K k x k k Q k (9) ( ) E[ ( ) ε ( )] ε ( ) Substtutg (5) to (3), the ovato ε ca be rertte as ε H x ( k k ) + v () Usg (), (9) ad x v, here the sybol deotes orthogoalty, (4) s obtaed. Fro (), the ovato varace atrx (6) s obtaed. he predcto ga atrx L s defed as Fro (4) ad (), e have L k x k+ k Q k () ( ) E[ ( ) ε ( )] ε ( ) + Φ Pk k H kx k k H () E[ x( k ) ε ] ( ) +E[ ( ) ( )] ΦPk ( k ) H +E[ kx ( ) ] H Usg (4) ad (), e have E[ x ] E[ ( k )] Q (3) L [ Φ P( k k ) + Q ] H Q (5) ε By substtutg (3) ad () to (), (5) follos. Fro (4), (), () ad (), the flterg ad predcto error equatos are obtaed as Qε () k HP( k k ) H + Q (6) v P k k P k k K k Q k K k (7) ( ) ( ) ( ) ε ( ) ( ) Pk ( + k) ( Φ L( kh ) ) Pk ( k )( Φ L( kh ) ) + ( Φ L H)( Q ) + Q ( Φ ( ) ) L k H + Q + L Q L (8) v here ε s the ovato sequece th covarace atrx Qε, K ( ) k ad L are the flterg ad predcto ga atrces, ( k k ) ad ( k k ) are the flter ad x ( k k) x ( k k) x ( k k ) K ε (4) x ( ) ( ) ˆ k + k x k + x( k+ k) ( Φ L ( kh ) ) x ( k k ) + k ( ) L( kv ) (5) Usg Pk ( k) E[ x ( k kx ) ( k k)] ad x ( k k) ε, e have the flterg error covarace atrx (7). Fro (5), x ( k k ) v ad ( k k ), the predcto error covarace atrx s coputed by Pk ( k) E[ x( k kx ) ( k k)] ( Φ L H) P( k k )( Φ L H) ( ) ( ) v + Φ E[ x ]( Φ L H) + Q + L k Q L k ( L ( kh ) )E[ x( k ) ] + (6) 647

4 By substtutg (3) to (6), (8) follos. C. Local flters at the state update pots Next, based o the local flter ( k k ) at the easureet saplg pots of syste (4) (5), e ll derve the local flter ( t k )( ( k ) + t k ) at the state update pots of syste () (). heore 3: Uder Assuptos ad, the local flter at the state update pots of the th sesor subsyste of syste () () s coputed by ( q k) Ψ ( k k) r r +, q ( k ) Ψ Φ Γ ˆ ( q r k ) + (7) l q ˆ ( ) ˆ ( ) l q x ˆ l k Φ x q k + Φ Γ ( ) l k, ( k ) + < l k (8) ˆ ( q+ r k) M ( q+ r k) ε, r,,, (9) r Γ Φ ε M ( q+ r k) (/ ) Q ( ) H Q (3) here Ψ ( Φ ), ( q k) s the flter of the state xq ( ) ( q ( k ) + ), ( l k ) s the flter of the state xl () ( ( k ) + < l k ), ˆ ( q+ r k), r,,, s the put hte ose flter, M ( q+ r k) s the put hte ose flterg ga atrx. he tal value ( k k) coputed by heore. Proof: Fro (7), e have xq ( ) x ( ( k ) + ) Ψ x r ( ) r, q ( ) k Ψ Φ Γ q + r s + (3) akg projecto of both sdes of (3) oto the lear space spaed by Ly (, y( k ),, y()) yelds (7). () by terato yelds xl () l q l q Φ xq ( ) + Φ Γl ( ) ( k ) + < l k (3) akg projecto of both sdes of (3) oto the lear space spaed by Ly (, y( k ),, y()), (8) s obtaed. Applyg the projecto theory [8], e have the put hte ose flter ˆ ( ) ˆ q+ r k ( q+ r k ) + M ( q+ r k) ε ( ) k q ( k ) +, r,,, (33) Usg q ( + r) L( y( k ), y( k ),, y()), e have ˆ ( q+ r k ). he put hte ose flterg ga M ( q+ r k) s defed as M q+ r k q+ r k Q k (34) ( ) E[ ( ) ε ( )] ( ) ε Fro () ad q ( + r) ( k k ), e have E[ q ( + r) ε ] E{ q ( + r)[ H( x ( k k )) + v] } )] ) r ( ) Γ Φ E[ q ( + r) x ( k H ( / Q H (35) By substtutg (35) to (34), (9) ad (3) are obtaed. o apply the covarace tersecto fuso estato algorth [7], e eed the coputato of covarace atrces of local flters, hch s stated belo. heore 4: Uder Assuptos ad, the local flterg error covarace atrx at the state update pots of the th sesor subsyste of syste () () s coputed by Pqk ( ) Ψ Pk ( k) Ψ r ( ( ) )( r s ) I ( ) r K k H Q s { } r r s { Q ( ) ( ( ) ) r I s K k H } r r [ ( ) P ( ) r q r k ( ) ] j Ψ ( Φ ) ΓM ( q+ j k) Q ε Ψ Φ Γ Γ Φ Ψ Ψ Φ Γ Γ Φ Ψ + Ψ Φ Γ + Γ jr,, j r M ( q r k r ) Γ ( Φ ) Ψ Φ Ψ + (36) l q l q Pl ( k) Φ Pq ( k)( Φ ) l q l q + q l+ s {[( I K H)( ) Q s } ( + q l+ j ) ( )] ( Φ Γ P ) j l k Γ Φ l q + q l+ s Φ Γ[ Q ( ) ( ( ) ) Γ Φ I s K k H + Φ Ψ Φ Γ { + + q l+ j l q j l q Φ Γ P ( ) ( ) l k Γ Φ l q Φ ΓM ( ) ( ) p,, p l k Q k ε p M ( l p k) Γ ( Φ ) P ( l k ) Γ ( Φ ) ] Ψ ( Φ ) + (37) P q+ r k Q M q+ r k Q k M q+ r k (38) ( ) ( ) ( ) ( ) ε } 648

5 here P ( q k ), q ( k ) + ad Pl ( k ), ( k ) + < l k are the flterg error covarace atrces, P ( q+ r k), r,,, are the estato error covarace atrces of the put hte ose q ( + r). he tal value P ( k k ) s coputed by heore. Proof: Subtractg (7) fro xq ( ), e have r Ψ Ψ Φ Γ r x ( q k ) x ( k k ) ( ) ( q + r k ) (39) Fro (39), the flterg error covarace atrx P ( q k) s coputed by Pqk x qk x qk ( ) E[( ( ))( ( )) ] Ψ P( k k) Ψ { E[ ( ) ( )] ( r r ) } j { ( Φ ) ΓE[ ( ) ( j )]} Ψ x k k q + r k Γ Φ Ψ Ψ q+ j k x k k j jr, + Ψ E[ ( Φ ) Γ( q+ j k) ( q r k ) Γ ( r Φ + ) ] Ψ (4) Subtractg (9) fro q ( + r), r,,,, e have the follog put hte ose flterg error equato ( q+ r k) ( q+ r) M ( q+ r k) ε (4) he, (38) ca be obtaed readly by coputg P ( q+ r k) E[ ( q+ r) ( q+ r)]. Usg (4), (), (4), x ( k k) ε ad xkk ˆ ( ) q ( + r), e have E[ x ( k k) ( q+ r k) ] E[ x ( k k) ( q+ r) ]( I K H )E[ x ( q+ r) ] r s (/ )( I K H) ΦΓQ s Ψ (4) O the other had, usg Assupto, (34) ad (4), e have E[ ( q+ j k) ( q+ r k)] P ( q+ j k), j r M ( q j k) Q M ( ), + ε q r k j r + Substtutg (4), (43) to (4), (36) s obtaed. Subtractg (8) fro (3), e have l q l q x ( q k (43) x ( l k) Φ ) + Φ Γ ( l k) (44) Fro (44), e have Pl ( k) E[ x( l kx ) ( l k)] l q ( )( l Φ P q q k Φ ) l q l q { E[ ( ) ( )] x ( ) q k l k } l q { E[ ( ) ( )]} ( l Φ Γ q ) l k x q k Φ l q Φ Γ ) p, p l k l p k Γ Φ + Φ Γ Φ + + E[ ( ) ( ( ) ] Fro (39) ad (4), e have E[ x( q k) ( l k)] Ψ E[ x ( k k) ( l ) ] r Φ Γ r + (45) Ψ E[ ( q r k) ( l k) ] (46) Usg (4), (43), (45) ad (46), (37) s obtaed. D. Dstrbuted fuso flter Based o the local flters heore 3 ad the flterg error covarace atrces heore 4, e have the follog dstrbuted fuso flter by applyg covarace tersecto fuso algorth [7]. heore 5: Uder Assuptos ad, the dstrbuted eghted fuso flter o ( t k ) at the state update pots s gve by L ˆ o ( t k) A ( t) x ( t k), ( k ) + t k (47) here the local flters of the th sesor ( ) ˆ t k x( q k) ( q ( k ) + ) ad ( ) ˆ t k x( l k) ( ( k ) + < l k ) are coputed by heore 3. he fuso eghts A ( t ) are coputed by L ()( ω () ) ( ) P A() t ω t t P ( t k) t k (48) here ω ( t) are set as follos tr( P ( t k)) L P L ω () t, ω ( t), () t tr( ( t k)) ω (49) here Pt ( k) Pq ( k), ( q ( k ) + ) ad Pt ( k) Pl ( k), ( ( k ) + < l k ) are coputed by heore 4. Based o heores -4, the coputatoal procedures of the dstrbuted fuso flter at the state update pots are as follos: ) Usg (), e ca obta the local flters ( k k ),,,, L at easureet update pots of syste (4) (5). ) Usg (7) ad (8), e ca obta the local flters ( q k) of the state xq ( ) ( q ( k ) + ) ad ( l k ) of the 649

6 state xl ( ) ( ( k ) + < l k ) at state update pots of syste () (). 3) Usg (36) ad (37), e ca obta the local flterg error covarace atrces P ( q k ) ad Pl ( k ). 4) Usg (48) ad (47), e ca obta the fuso eghts A () t ad the dstrbuted fuso flter ( t k ). Above, all steps are pleeted at each te step. IV. SIMULAION RESEARCH Cosder a trackg syste th 3 sesors xt ( + ) xt ( ) + t ( ).5 [ ] o (5) y x + v,,,3. (5) here s the state update perod. he state xt () [ st () st ()] Τ, here s( t ) ad s ( t) are the posto ad velocty of the target at te t. he easureet oses v ( k ),,,3 are depedet Gaussa oses th zero eas ad varaces Q ad are ucorrelated th k ( ). Our a s to fd the v dstrbuted fuso flter o ( t k ) by covarace tersecto algorth. We take 5 saplg data. I sulato, e set.s, Q.5, Q 3.5, Q.5, Q 3,, 3, 3 v, () [ ] v x ad P.I. Fg. gves the trackg perforaces of the dstrbuted fuso flter, here the sold curves deote the true values ad the dashed oes deote the flters. Fg. 3 gves the coparso curves of MSEs sulated by -tes Mote Carlo tests for all the local flters ad the dstrbuted fuso flter th the te ~5. Fro Fg.3, e see that the accuracy of the fuso flter s better tha that of all local flters. rue value ad Flter - v3 - rue value Fuso Flter -3 5 t/step 5 (a) Posto ad ts flter rue value ad Flter MSEs MSEs rue value Fuso Flter -6 5 t/step 5 (b) Velocty ad ts flter Fgure. Dstrbuted fuso flter Local Flter Local Flter Local Flter 3 Fuso Flter 3 t/step 4 5 (a) Posto Local Flter Local Flter Local Flter 3 Fuso Flter t/step (b) Velocty Fgure 3. Coparso of MSEs of local flters ad the dstrbuted fuso flter Next, e ake the coparso th referece [3], here the data receved rate α. Fg. 4 shos that the coparso of the MSEs curves of the to flters based o -tes Mote Carlo tests. Fro Fg. 4, e see that the accuracy of our flter s sgfcat better tha that of [3]. Fg. 5 shos the coparso of the accuulated ea square errors of 5 saplg data over a average of rus of Mote Carlo ethod (.e., 5 ( ) () l () l (/ ) ( ( ) ( )) t l t k x t, the superscrpt l deotes the lth sulato test). It ca be observed that the accuracy of our fuso flter becoe better as Q creases sce the state odels [3] gore the syste ose. he sulato results verfy the effectveess of the proposed algorth. 65

7 MSEs Our Flter Flter [3] 5 t/step 5 (a) Posto Our Flter Flter [3] V. CONCLUSION I ths paper, the dstrbuted fuso flterg proble for ult-rate ult-sesor o-ufor asychroous saplg systes s vestgated. he state s updated at the hghest ufor rate ad dfferet sesors have dfferet loer oufor easureet update rates. By cosderg the syste oses, the o-augeted state odels at each sesor are establshed. Based o the establshed state space odels, local flters at the easureet saplg pots ad the local flters at the state update pots are obtaed by applyg projecto theory, respectvely. Further, the dstrbuted fuso flter s gve by applyg the covarace tersecto fuso algorth. he flterg error covarace atrces are derved to copute the fuso eghts. Copared to referece [], the coputatoal cost s reduced sce the o-augetato approach s used. Copared to refereces [-3], the estato accuracy ca be proved sgfcatly sce the syste ose s cosdered. MSEs.6.4. ACKNOWLEDGMEN he authors are debt to the edtor ad reveers for valuable coets ad suggestos. Fgure 4. Coparso of MSEs of the flter ths paper ad the flter [3] AMSEs AMSEs 5 t/step (b) Velocty Our Flter Flter [3] Our Flter Flter [3] Fgure 5. Coparso of AMSEs of the flter ths paper ad the flter [3] Q (a) Posto Q (b) Velocty REFERENCES [] J. Sheg,. W. Che, S. L. Shah. Optal Flterg for Multrate Systes. IEEE ras. Crcus ad Systes II: Express Brefs, vol. 5, o. 4, pp: 8-3, 5. [] Y. Lag,. W. Che, ad Q. Pa, Mult-rate optal state estato, Iteratoal Joural of Cotrol, 9, vol. 8, o., pp , 9. [3] L. Hog. Mult-resolutoal Flterg Usg Wavelet rasfor. IEEE ras. Aerospace ad Electroc Systes, vol. 9, o. 4, pp: 44-5, 993. [4] L. Hog. Mult-resolutoal Dstrbuted Flterg. IEEE ras. Autoatc Cotrol, vol. 39, o. 4, pp: , 994. [5] L. Zhag, X. L. Wu, Q. Pa, H. C. Zag. Mult-resoluto Modelg ad Estato of Multsesor Data. IEEE ras. Sgal Processg, vol. 5, o., pp: 37-38, 4. [6] C. L. We., D.H. Zhou. Multscalae syste theory ad applcato. Bejg: sghua Uversty Press,, [7] C. L. We., Z. G. Che, L. P. Ya, D.H. Zhou. he ultscale recursve fuso estato based o dyac systes of ultrate sesors. Joural of Electrocs & Iforato echology, vol. 5,o. 3, pp: 36-3, 3. [8] Q. B. Ge, G. A. Wag,. H. ag, ad C. L. We. he research o a asychroous data fuso algorth based o saplg of ratoal uber tes. Acta electroca sca, vol. 34, o.3, pp: , 6. [9] L. P. Ya, B. S. Lu, ad D.H. Zhou, he odelg ad estato of asychroous ultrate ultsesor dyac systes, Aerospace Scece ad echology, vol., o., pp. 63-7, 6. [] L.P. Ya, B. S. Lu, D. H. Zhou, ad C. L. We, A class of state fuso estato algorth for ltrate ltsesor systes, Joural of Electrocs ad Iforato echology, vol. 9, o., pp , 7. [] L. P. Ya, B. S. Lu, ad M. Y. Zhou, Asychroous ultrate ultsesor forato fuso algorth, IEEE ras. Aerospace ad Electroc Systes, vol. 43, o. 3, pp , 7. [] L. P. Ya, C. Zhu, Y. Q. Xa ad M. Y. Fu, Optal state estato for a class of asychroous ultate ultsesor dyac systes, Proc. of 9th CCC, ,. 65

8 [3] L. P. Ya, D. H. Zhou, M. Y. Fu, ad Y. Q. Xa, State estato for asychroous ultrate ultsesor dyac systes th ssg easureets, IE Sgal Processg, vol. 4, o. 6, pp ,. [4] Y. Y. Hu, Z. S. Dua, ad D. H. Zhou, Estato fuso th geeral asychroous ult-rate sesors, IEEE ras. Aerospace ad Electroc Systes, vol. 46, o. 4, pp. 9-,. [5] Y. Lag,. W Che, Q. Pa. H Flterg of a class of ult-rate ult-sesor fuso systes. Proc. of Jot 48th IEEE CDC ad 8th CCC, , 9. [6] Y. Lag,. W, Che, ad Q. Pa, Mult-rate stochastc H flterg for etorked ult-sesor fuso, Autoatca, vol. 46, o., pp ,. [7] L. J. Che, P. O. Arabel, R. K. Mehra, Estato Uder Uko Correlato: Covarace Itersecto Revsted, IEEE ras. Autoatc Cotrol, vol.47, o., pp ,. [8] B. D. O. Aderso, ad J. B. Moore, Optal flterg, Egleood Clffs, NJ: Pretce-Hall,