Robust Centralized Fusion Kalman Filters with Uncertain Noise Variances

Transcription

1 ELKOMNIKA Indonean Jounal of Eleal Engneeng Vol., No.6, June 04, pp ~ 476 DOI: 0.59/elkomnka.v Robu Cenalzed Fuon Kalman Fle wh Unean Noe Vaane Wen-juan Q, Peng Zhang, Z-l Deng* Depamen of Auomaon, Helongjang Unvey Habn, Chan, *Coepongdng auho, e-mal: dzl@hlju.edu.n Aba h pape ude he poblem of he degnng he obu loal and enalzed fuon Kalman fle fo muleno yem wh unean noe vaane. Ung he mnmax obu emaon pnple, he enalzed fuon obu me-vayng Kalman fle ae peened baed on he wo-ae onevave yem wh he onevave uppe bound of noe vaane. A Lyapunov appoah popoed fo he obune analy and he obu auay elaon ae poved. I poved ha he obu auay of obu enalzed fue hghe han hoe of obu loal Kalman fle. Speally, he oepondng eady-ae obu loal and enalzed fuon Kalman fle ae alo popoed and he onvegene n a ealzaon beween me-vayng and eady-ae Kalman fle poved by he dynam eo yem analy (DESA) mehod and dynam vaane eo yem analy (DVESA) mehod. A Mone-Calo mulaon example how he obune and auay elaon. Keywod: muleno nfomaon fuon, enalzed fuon, unean noe vaane, mnmax obu Kalman fle Copygh 04 Inue of Advaned Engneeng and Sene. All gh eeved.. Inoduon he am of he muleno nfomaon fuon how o ombne he loal emao o loal meauemen o oban he fued emao, whoe auay hghe han ha of eah loal emao []. Fo he enalzed fuon opmal Kalman fle, all he loal meauemen daa ae aed o he fuon ene o oban a globally opmal fued ae emaon []. he dawbak of he Kalman fle ha only uable o handle he ae emaon poblme fo yem wh exa model paamee and noe vaane. Howeve, n many applaon poblem, hee ex uneane of he model paamee and/o noe vaane. Unde hee uneane he pefomane of he Kalman fle wll degade [3], and an nexa model may aue he fle o dvege. h ha movaed he degnng of he obu Kalman fle, whh guaane o have a mnmal uppe bound of he aual fleng eo vaane fo all admble uneane. In ode o degn he obu Kalman fle fo he yem wh he model paamee uneane, wo mpoan appoahe ae he Ra equaon appoah [4-6] and he lnea max nequaly (LMI) appoah [7-9]. he dadvanage of hee wo appoahe ha only model paamee ae unean whle he noe vaane ae aumed o be exaly known. he obu Kalman fleng poblem fo yem wh unean noe vaane ae eldom ondeed [0, ], and he obu nfomaon fuon Kalman fle ae alo eldom eeahed [, 3]. In h pape, ung he mnmax obu emaon pnple, he loal and enalzed fuon obu me-vayng and eady-ae Kalman fle ae peened baed on he woae onevave yem wh he onevave uppe bound of noe vaane. he onvegene n a ealzaon beween he me-vayng and eady-ae Kalman fle goouly poved by he dynam eo yem analy (DESA) mehod [4] and dynam vaane eo yem analy (DVESA) mehod [5]. Fuhemoe, a Lyapunov equaon appoah peened fo he obune analy, whh dffeen fom he Ra equaon appoah and he LMI appoah. he onep of he obu auay gven and he obu auay elaon ae poved, poved ha he obu auay of he enalzed fue hghe han ha of he loal obu Kalman fle. Reeved Deembe 9, 03; Reved Mah 8, 04; Aeped Mah, 04

2 4706 ISSN: he emande of h pape oganzed a follow. Seon gve he poblem fomulaon. he obu enalzed fuon me-vayng Kalman fle ae peened n Seon 3. he obu loal and enalzed fuon eady-ae Kalman fle ae peened n Seon 4. he obu auay analy gven n Seon 5. he mulaon example gven n Seon 6. he onluon popoed n Seon 7.. Poblem Fomulaon Conde he muleno lnea dee me-vayng yem wh unean noe vaane. x x w (),,, y H x L () n m Whee epeen he dee me, x R he ae, y R ubyem, w R he npu noe, m ubyem,, and he meauemen of he h he ommon dubane noe, R he meauemen noe of he h H ae known me-vayng mae wh appopae dmenon. L he numbe of eno. Aumpon. w, and ae unoelaed whe noe wh zeo mean and unknown unean aual vaane Q, R and R a me, epevely, Q, R and R Q, R R, afyng: ae known onevave uppe bound of,, and Q Q R R R R,,, L, (3) Aumpon. he nal ae x 0 ndependen of w, mean value and unknown unean aual vaane P 0 0whh afe: P 0 0 P0 0 and v and ha (4) Whee P0 0 a known onevave uppe bound of 0 0 P. Aumpon 3. he yem () and () unfomly ompleely obevable and ompleely onollable. Defnng:,,, v L (5) Whee v ae whe noe wh zeo mean and he onevave and aual vaane ae gven a:, R R R R R R v v j, R R R R v j Fom (3), we have: v v,,, L (6) v, j (7) R R,,, L, (8) ELKOMNIKA Vol., No. 6, June 04:

3 ELKOMNIKA ISSN: Robu Cenalzed Fuon me-vayng Kalman Fle Inodue he enalzed fuon meauemen equaon: y H x v (9) Wh he defnon: And, H H,, H, v v v y y,, yl v ha he onevave and aual vaane mae R and,, L (0) R a: Rv R R R R Rv R R R Rv, Rv R R R R Rv R R R R v () heefoe fom (3) and (8), aodng o he Lemma and Lemma n Appendx, we oban: R R () Baed on he wo-ae onevave yem () and (9) wh Aumpon -3 and onevave uppe bound Q and R, he globally opmal enalzed fued me-vayng obu Kalman fle ae gven a: ˆ xˆ x K y (3) = I K H (4) n = K P H H P H R (5) P P Q (6) he fued onevave fleng eo vaane P gven a: P In K H P (7) I an be ewen a he Lyapunov equaon: P P n H Q I K H K R K n (8) max. Wh he nal value xˆ 0 0, and P 0 0 P0 0 he aual pedon and fleng eo ae obaned a:, whee I n he n ndeny ˆ ˆ x x x x w (9) x x x In K H x K v (0) Robu Cenalzed Fuon Kalman Fle wh Unean Noe Vaane (Wen-juan Q)

4 4708 ISSN: Subung (9) no (0) yeld: x x In K H w K v () Τ he aual fued fleng eo vaane P Ε x x have:, aodng o (), we P P n H Q I K H K R K n (). heoem. Fo muleno unean yem () and (9) wh Aumpon -3, he aual enalzed fuon me-vayng Kalman fle wh he onevave uppe bound Q, Wh he nal value P 0 0 P0 0 R and P 0 0ae obu n he ene ha fo all admble aual vaane, 0 0afyng (3), (4) and (), fo abay me, we have: P P P Q R and (3) And P he mnmal uppe bound of P fo all admble uneane of noe vaane. We all he aual fued Kalman fle a he obu enalzed fuon Kalman fle. Poof. Defnng P P P, ubang () fom (8) yeld he Lyapunov equaon. P P U (4) U n Q Q In K H K R R K (5) Applyng (3), () and (5) yeld hau 0, and fom (4) we have: P 0 0 P 0 0 P 0 0 P 0 0 P (6) Hene fom (4), we have P 0 P 0, fo all me,.e. he nequaly (3) hold. akng Q Q, R R P0 0 P0 0, hen ompang (8) wh (), we have P P * * uppe bound P, we have P P P whh yeld ha uppe bound of P. he poof ompleed.. Applyng he mahemaal nduon mehod yeld and. Fo abay ohe P he mnmal Coollay. Fo unean muleno yem () and () wh Aumpon -3 and onevave uppe bound Q and Rv, mla o he obu enalzed fuon me-vayng Kalman fle, he obu loal me-vayng Kalman fle ae gven by: ˆ,,, L = In K H, K= P H R xˆ x K y (7) (8) ELKOMNIKA Vol., No. 6, June 04:

5 ELKOMNIKA ISSN: equaon []. R H P H R v (9) P P Q (30) P In K H P (3) he onevave loal fleng eo vaane P an be ewen a he Lyapunov P P n Q In K H K Rv K (3) Wh he nal value P0 0 P0 0 Lyapunov equaon.. And he aual fleng eo vaane ae gven by he P P n Q In K H K Rv K (33) Smlaly, he loal me-vayng Kalman fle ae alo obu,.e., P P,,, L (34) 4. Robu Loal and Cenalzed Fuon Seady-ae Kalman Fle heoem. Fo muleno unean me-nvaan yem () and (9) wh Aumpon and 3, whee,, H H, Q QR, R, R R,and Q Q, R R, R R ae all he onan mae, hen he aual enalzed fuon eady-ae Kalman fle ae gven by: ˆ xˆ x K y (35) =I K H, n K= H HH R (36) Q, P= In KH (37) he pedon eo vaane afe he eady-ae Ra equaon: = H H H R H Q (38) Whee he upep denoe eady-ae, he fued onevave fleng eo vaane P gven a: P P H Q I K H K R K (39) n n he fued aual fleng eo vaane P gven a: Robu Cenalzed Fuon Kalman Fle wh Unean Noe Vaane (Wen-juan Q)

6 470 ISSN: P P H Q I K H K R K (40) n n he aual enalzed fuon eady-ae Kalman fle (35) ae obu n he ene ha fo all admble uneane of noe vaane Q and R afyng (3) and (8), we have: v P P (4) And P he mnmal uppe bound of P. Poof. A, akng he lm opeaon fo (3)-(8), () and (3), we oban (35)- (4). akng Q Q, R R, fom (39) and (40), we have P P. If P abay ohe uppe bound of P fo all admble Q and R afyng Q Q, R R, hen we have P P P, whh yeld ha P mnmal uppe bound of P. he poof ompleed. Smlaly, he aual loal eady-ae Kalman fle ae gven by: ˆ xˆ x K y,,, L (4) =I K H, K = H H H R P I K H (43) n v, n he pedon eo vaane afe he eady-ae Ra equaon. = H H H R H Q v (44) he onevave and aual loal fleng eo vaane afy he eady-ae Lyapunov equaon. P P Q I KH KR K (45) n n v P P Q I KH KR K (46) n n v he aual loal eady-ae Kalman fle (4) ae obu,.e., P P,,, L (47) And P he mnmal uppe bound of P. heoem 3. Unde he ondon of heoem, and aume ha he meauemen y,,, L xˆ and ˆ ae bounded, hen he obu me-vayng and eady-ae Kalman fle x, xˆ and xˆ gven by (7) and (4), (3) and (35) have eah ohe he onvegene n a ealzaon, uh ha: ˆ x ˆ x 0, a,.a. (48) ˆ x ˆ x 0, a,.a. (49) Whee he noaon.a. denoe he onvegene n a ealzaon [5], and we have he onvegene of vaane. P P P P, a,,, L (50), ELKOMNIKA Vol., No. 6, June 04:

7 ELKOMNIKA ISSN: P P, P P, a (5) Poof. Aodng o he omplee obevably and omplee onollably of eah ubyem, he me-vayng loal Kalman fle (7) have he onvegene ha [6]: P, a,,, L (5) Fom (8) and (3), we have:, K K, P P, a,,, L (53) Seng, K K K n (7), applyng (53) yeld 0 K 0, a. Subang (4) fom (7), and defnng xˆ xˆ, we have: u (54) Wh u ˆ x K y. Nong ha aympoally able [7], and K y boundedne of xˆ. Hene we have 0 a able max, o alo unfomly aympoally able, hene 0 unfomly bounded, applyng Lemma 4 o (7) yeld he u. Applyng Lemma 4 o (54), nong ha,.e. he onvegene (48) hold. he onvegene of (49) an be poved mlaly. P P yeld he Lyapunov equaon. Fom (33) and (46), defnng U (55) n Q In KH KR v K P P U Q I K H K R K n n v (56) Fom (33), nong ha unfomly aympoally able, applyng K K, 0 and Lemma 3 yeld P bounded. Fom (56) yeld ha 0 Applyng Lemma 3 o (55) yeld 0, a,.e., P P pove (5) hold. he poof ompleed. U. hold. Smlaly, we an 5. he Auay Analy Defnon. he ae P of he uppe bound P of he aual fleng eo vaane P fo all admble uneane alled he obu auay o global auay of a obu Kalman fle, and P alled a aual auay. Fom h defnon, he malle P o P mean he hghe obu auay o aual auay. he obu auay gve he lowe bound of all poble aual auae yelded fom he uneane of noe vaane. heoem 4. Fo muleno unean yem () and () wh Aumpon -3, he auay ompaon of he loal and fued obu Kalman fle gven by: Robu Cenalzed Fuon Kalman Fle wh Unean Noe Vaane (Wen-juan Q)

8 47 ISSN: P P,,, L (57) P P P,,, L (58) P, P P P P,,, L (59) P P, P P P,,, L, (60) P P,,, L, P P P (6) Poof. Aodng o he obune (3) and (34), we have (57) and he f nequaly of (58). he eond nequaly of (58) ha been poven n [8]. akng he ae opeaon fo (57) and (58) yeld he nequale (59). A, akng he lm opeaon fo (57), (58) and (59) yeld (60) and (6). he poof ompleed. Fom he nequale (59), we an ee ha all admble aual ae P and P ae globally onolled by he uppe bound P and P, epevely, and he obu auay of he enalzed obu fue hghe han ha of eah loal obu Kalman fle. 6. Smlaon Example Conde a hee-eno me-nvaan akng yem wh unean noe vaane., y Hx,,,3 x x w (6) , 0, H I 0 (63) Whee x x, x he ampled peod, he ae, x and x ae he poon and veloy of age a me 0. w, ae ndependen Gauon whe noe wh zeo mean and unknown unean aual vaane Q, R and R epevely. In he mulaon, we ake Q, Q 0.8, R dag(.5,.5), R dag(,), R dag(3.6,.5), and R dag(3,.8), R dag(8,0.36), R dag(6,0.5), R dag(0.5,.8), R dag(0.38,), he nal value x 0 0 0, 0, P0 0 dag(.,.), P0 0 I 3. he ompaon of he fleng eo vaane mae and he ae of he obu eady-ae loal and enalzed fuon Kalman fle ae hown n able and able. hee mae and he ae vefy he auay elaon (60)-(6). he ae of he onevave and aual obu fleng eo vaane ae ompaed n Fgue. We ee ha he ae of he loal and fued obu me-vayng Kalman fle qukly onvege o hee of he oepondng eady-ae Kalman fle, whh how he obu auay elaon (59) and (6) hold. 3 able. he Conevave and Aual Auay Compaon of P and P,, 3, P P P 3 P ELKOMNIKA Vol., No. 6, June 04:

9 ELKOMNIKA ISSN: P P P P able. he Conevave and Aual Auay Compaon of P, P,,, 3, P, P P, P P, P 3 3 P, P.998, , , , P P P P P 3 P 3 P P /ep Fgue. he ae of he Conevave and Aual Loal and Fued Kalman Fle In ode o vefy he above heoeal auay elaon, akng 00 Mone Calo mulaon un, he mean quae eo (MSE) value a me of loal o fued obu Kalman fle ae defned a: Whee j x Τ j j j j ˆ ˆ j,,, 3, MSE x x x x j o xˆ denoe he h Aodng o he egody [9], we have: MSE P, a, j ealzaon of x o xˆ,,, 3, (64). (65) he MSE uve of he loal and fued me-vayng obu Kalman fle ae hown n Fgue, whh vefy he auay elaon (59) and (6), and vefy he egody (65). Robu Cenalzed Fuon Kalman Fle wh Unean Noe Vaane (Wen-juan Q)

10 474 ISSN: P MSE. P P P /ep MSE MSE MSE3 MSE Fgue. he Compaon of MSE and P,,, 3, P 3 P P 3 P 7. Conluon Fo muleno yem wh unean noe vaane, ung he mnmax obu emaon pnple, he loal and enalzed fuon obu Kalman me-vayng Kalman fle ae peened. Baed on he Lyapunov equaon appoah, he obune ae poved and he obu auay elaon ae alo poved. I poved ha he obu auae of he enalzed fuon Kalman fle ae hghe han hoe of he loal obu Kalman fle. he onvegene poblem of he obu loal and enalzed fuon me-vayng and eady-ae Kalman fle poved by he dynam eo yem analy (DESA) mehod and he dynam vaane eo yem analy (DVESA) mehod. h exenon of h pape o yem wh unean noe vaane and model paamee unde udy. Aknowledgemen h wok uppoed by he Naual Sene Foundaon of Chna unde gan NSFC , he Innovaon and Senf Reeah Foundaon of gaduae uden of Helongjang Povne unde gan YJSCX0-63HLJ. Refeene [] Hall DL, Lna JL. An noduon o muleno daa fuon, Poeedng of IEEE.997; 85(): 6-3. [] L XR, Zhu YM, Han CZ. Opmal lnea emaon fuon, Pa I: Unfed fuon ule, IEEE anaon on Infomaon heoy, 003; 49(9): [3] Syananda H. A mple mehod fo he onol of dvegene n Kalman fle algohm, Inenaonal Jounal of Conol. 97; 6(6): [4] Lew FL, Xe LX, Popa D. Opmal and obu emaon. Seond Edon. CRC Pe, New Yok [5] Lu X, Zhang H, Wang W. Robu Kalman fleng fo dee-me yem wh meauemen delayed. IEEE anaon on Cu and Syem-II: Expe Bef. 007; 54(6): [6] Xong K, We CL, Lu LD. Robu Kalman fleng fo dee-me nonlnea yem wh paamee uneane, Aeopae Sene and ehnology, 0; 8(): 5-4. [7] Jn XB, Bao J, Zhang JL. Cenalzed fuon emaon fo unean muleno yem baed on LMI mehod, Poeedng of he IEEE onfeene on Mehaon and Auomaon. 009; [8] Yang F, L Y. Robu e-membehp fleng fo yem wh mng meauemen: a lnea max nequaly appoah, IE Sgnal Poe, 0; 6(4): [9] Qu XM, Zhou J. he opmal obu fne-hozon Kalman fleng fo mulple eno wh dffeen oha falue ae, Appled Mahema Lee, 03; 6(): ELKOMNIKA Vol., No. 6, June 04:

11 ELKOMNIKA ISSN: [0] X HS. he guaaneed emaon pefomane fle fo dee-me depo yem wh unean noe, Inenaonal Jounal of Syem Sene. 997; 8(): 3-. [] Qu XM, Zhou J, Song EB, Zhu YM. Mnmax obu opmal emaon fuon n dbued muleno yem wh uneane, IEEE Sgnal Poeng Lee. 00; (9): [] Feng JX, Wang ZD, Zeng M. Dbued weghed obu Kalman fle fuon fo unean yem wh auo-oelaed and o-oelaed noe. Infomaon Fuon. 03; 4(): [3] Ahmad A, Gan M, Yang FW. Deenalzed obu Kalman fleng fo unean oha yem ove heeogeneou eno newok. Sgnal Poeng. 008; 88(8): [4] Ran CJ, ao GL, Lu JF, Deng ZL. Self-unng deoupled fuon Kalman pedo and onvegene analy. IEEE Seno Jounal. 009; 9(): [5] Deng ZL, Gao Y, L CB, Hao G. Self-unng deoupled nfomaon fuon Wene ae omponen fle and he onvegene. Auomaa. 008; 44(3): [6] Andeon BDO, Mooe JB. Opmal fleng, Pene Hall, Englewood Clff, NJ, 979. [7] Kamen EW, Su JK. Inoduon o opmal emaon, Spnge Velag, London Beln Hedelbeg, 999. [8] Deng ZL, Zhang P, Q WJ, Gao Y, Lu JF. he auay ompaon of muleno ovaane neeon fue and hee weghng fue. Infomaon Fuon. 03; 4(): [9] Ljung L. Syem denfaon. heoy fo he Ue, Seond Edon. Pae Hall PR Appendx Lemma. Le be he pove em-defne max,.e. 0, hen he followng L L max alo pove em-defne,.e., L L 0 (A.) Poof. Conde he haae polynomal of. I L I I I (A.) Addng all he ohe olumn o he f olumn yeld: I L L I L I I L I I (A.3) yeld: Subang he f ow fom eah ow ang off wh he eond ow o he Lh ow L I 0 I 0 I L I I L L 0 0 I (A.4) Whh yeld he haae equaon: L L I LI I 0 (A.5) Robu Cenalzed Fuon Kalman Fle wh Unean Noe Vaane (Wen-juan Q)

12 476 ISSN: I all egenvalue ae deemned by: LI 0, I 0 (A.6) Sne 0, hen L 0, o ha L ha all he egenvalue 0,,, whh ae L alo he egenvalue of. he ohe egenvalue of ae deemned by I 0,.e. L 0, whh yeld all he ohe egenvalue of ae 0,,, L. heefoe all egenvalue of ae non-negave,.e., 0. he poof ompleed. Lemma. Le R be he m m pove em-defne max,.e. R 0, he followng m mblok-dagonal max R alo pove em-defne,.e., R R R L dag, 0 (A.7) Wh m m ml. Lemma 3. [4] Conde he me-vayng Lyapunov equaon. P F P F U (A.8) Whee 0, he oupu P and he npuu ae he n nmae, and he n n mae F and F ae unfomly aympoally able,.e., hee ex onan 0 and j 0 uh ha: F,, 0, j, (A.9) j j j Whee he noaon denoe he nom of max, Fj Fj Fj Fj n. IfU bounded, hen P bounded. IfU 0, hen P 0 U alled o be bounded, f U (onan), fo abay 0. Fj, I Noe ha Lemma 4. [5] Conde a dynam eo yem. F u j,,, a. (A.0) Whee 0 bounded, hen n n, R, u R, and bounded. If u 0, hen 0 F unfomly aympoally able. If, a. u ELKOMNIKA Vol., No. 6, June 04: