Optimal Multistage Kalman Estimators

Transcription

1 2182 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 45, NO 11, NOVEMBER 2000 noting eem to be the fomulation of initial value fo the geneal diffeential-intego-algebaic epeentation A a matte of a fact, tanfe equivalence initial condition ae not conideed, ha been invetigated in [20] fo ational equation in Roenboc fom ACKNOWLEDGMENT The autho ae indebted to the anonymou eviewe fo helpful and inteeting comment The effot of the Aociate Edito in helping the pepaation of the final veion of the pape ae alo vey much appeciated REFERENCES [1] J C Willem, Fom time-eie to linea ytem Pat I: Finite dimenional linea time invaiant ytem, Automatica, vol 22, pp , 1986 [2], Fom time-eie to linea ytem Pat II: Exact modelling, Automatica, vol 22, pp , 1986 [3], Fom time-eie to linea ytem Pat III: Appoximate modelling, Automatica, vol 23, pp , 1987 [4], Paadigm and puzzle in the theoy of dynamical ytem, IEEE Tan Automat Cont, vol 36, pp , 1991 [5] H H Roenboc, State Space and Multivaiable Theoy London, UK: Nelon-Wiley, 1970 [6], The tanfomation of tict ytem equivalence, Int J Cont, vol 25, pp 11 19, 1977 [7] A C Pugh, G E Hayton, and A B Wale, Sytem matix chaacteization of input-output equivalence, Int J Cont, vol 51, pp , 1990 [8] J H Schumache, Tanfomation of linea ytem unde extenal equivalence, Linea Algeba Appl, vol 102, pp 1 33, 1988 [9] T Geet and J M Schumache, Impulive-mooth behavio in multimode ytem Pat I: State-pace and polynomial epeentation, Automatica, vol 32, pp , 1996 [10], Impulive-mooth behavio in multimode ytem Pat II: Minimality and equivalence, Automatica, vol 32, pp , 1996 [11] M A AI-Gwaiz, Theoy of Ditibution New Yo: Macel Dee, 1992 [12] M L J Hautu and L M Silveman, Sytem tuctue and ingula contol, Linea Algeba Appl, vol 50, pp , 1983 [13] T Geet, Invaiant ubpace and invetibility popetie fo ingula ytem: The geneal cae, Linea Algeba Appl, vol 183, pp 61 88, 1993 [14] G Doetch, Guide to the Application of Laplace Tanfomation, R Oldenboug, Ed Munich, 1961 [15] G C Veghee, B C Lévy, and T Kailath, A genealized tate-pace fo ingula ytem, IEEE Tan Automat Cont, vol AC-26, pp , 1981 [16] V Kucea and P Zagala, Contant olution of polynomial equation, Int J Cont, vol 53, pp , 1991 [17] S L Campbell and C D Meye, Genealized Invee of Linea Tanfomation New Yo: Dove, 1979 [18] P Van Dooen, The genealized eigentuctue poblem in linea ytem theoy, IEEE Tan Automat Cont, vol AC-26, pp , 1981 [19] S Weiland and A Stoovogel, Rational epeentation of behavio: Inteconnectability and tabilizability, Math Cont Sig Sy, vol 10, pp , 1997 [20] M Hou, A C Pugh, and G E Hayton, Genealized tanfe function and input output equivalence, Int J Cont, vol 68, pp , 1997 Optimal Multitage Kalman Etimato Fu-Chuang Chen and Chien-Shu Hieh Abtact An optimal multitage Kalman etimato (OMSKE) i popoed a a genealization of the optimal two-tage Kalman etimato fo the eduction of the computational buden of the Kalman etimato (KE) fo dicete-time linea time-vaying ytem with tiangula tanition matice Thi new filte i obtained by applying a multitage tanfomation to decouple the covaiance of the KE It i hown analytically that the computational complexity of the OMSKE i le than that of the KE and i minimum when the ytem tanition matix ha the maximum tage numbe Index Tem Augmented tate Kalman etimato, multitage Kalman etimato, optimal filte, two-tage Kalman etimato I INTRODUCTION Conide the poblem of etimating the tate of a dynamic ytem in which the ytem tanition matix ha an uppe tiangula fom A pecial cae of thi poblem i illutated by the etimation poblem of a dynamical ytem ubject to an unnown bia [1] [4] In thi pecific poblem, the ytem tate i augmented to include the bia tate, and the ytem i efomulated a an augmented ytem in which the ytem tanition matix i in an uppe tiangula fom It i common to ue the augmented tate Kalman etimato (ASKE) to etimate the augmented tate Howeve, to educe the computational cot and numeical eo, a two-tage Kalman etimato (TSKE) popoed by Fiedland [1] can be ued to obtain the Kalman etimate Unfotunately, Fiedland filte i optimal only fo contant biae Ignagni [2] conideed the tochatic cae fo applying Fiedland filte Howeve, the eult he obtained i uboptimal Alouani et al [3] popoed a ufficient condition fo Fiedland filte to be optimal when it i applied to tochatic ytem Howeve, thi ufficient condition i eldom atified in pactical ytem Anothe illutation of the conideing poblem i the tate etimation in a maneuveing taget tacing application, the tate i compoed of poition, velocity, and acceleation Recently [4], we genealized Fiedland filte by accounting fo the bia noie effect to obtain the optimal two-tage Kalman etimato (OTSKE), which i optimal in the ene that it can geneate the minimum-mean-quae-eo (MMSE) etimate of the ytem tate without any containt The OTSKE i deived by applying a two-tage U 0 V tanfomation to decouple the tate and the covaiance equation of the ASKE The main eaon that the OTSKE can educe the computational complexity of the ASKE i due to ode eduction It i moe expenive to implement an n1 +n2 -ode Kalman filte than two Kalman filte with ode n1 and n2 Hence, it i expected that a multitage filte, when applicable, can do bette than a two-tage filte in tem of computation The objective of thi note i to popoe a genealization of the OTSKE fo the poblem at hand to futhe implify the computational complexity of the OTSKE, and the obtained filte will be denoted a the optimal multitage Kalman etimato (OMSKE) Thi new filte i deived by applying a multitage U 0 V tanfomation, which i a genealization of the conventional two-tage U 0 V tanfomation, to /00$ IEEE Manucipt eceived Septembe 30, 1997; evied May 26, 1998, Novembe 25, 1998, Apil 30, 1999, Octobe 11, 1999, and May 10, 2000 Recommended by Aociate Edito, J Spall The autho ae with the Depatment of Electical and Contol Engineeing, National Chiao Tung Univeity, Hinchu, Taiwan, ROC ( fcchen@ccnctuedutw) Publihe Item Identifie S (00)

2 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 45, NO 11, NOVEMBER decouple the covaiance matice of the Kalman etimato (KE), and i compoed of covaiance-decoupled ubfilte It i hown analytically that the computational complexity of the OMSKE i le than that of the KE and i minimum when the ytem tanition matix ha the maximum tage numbe Thi pape i oganized a follow In Section II, we tate the poblem of inteet In Section III, the OMSKE i deived, and an algoithm fo implementing it i povided In Section IV, the computational complexity of the OMSKE i analyzed analytically, and it computational advantage ove the KE i alo hown Section V i the concluion Detailed poof ae povided in the Appendix II STATEMENT OF THE PROBLEM Conide the following dynamical ytem: X +1 A X + w (1) Y H X + (2) X 2 R n i the ytem tate, Y 2 R m i the meauement vecto, A i an -by- bloc uppe tiangula matix in which 3 n, and the matix H ha an appopiate dimenion The vecto w and ae zeo-mean white noie equence govened by Efw wlg 0 Q l, Ef lg 0 R l and Efw lg 0 0, 0 denote tanpoe and l denote the Konece delta function The initial tate X 0 i aumed to be a andom vaiable with EfX 0 g X 0 and CovfX 0g P 0 and i uncoelated with the white noie equence w and Note that the aumption of the uppe tiangula bloc fom in the ytem tanition matix i not a limitation ince it can be obtained by uing Houeholde tanfomation o uing the modifying equation peented in Rema B of the next ection On the othe hand, the ytem (1), (2) may tand fo an augmented-tate ytem and i bet illutated by the maneuveing taget tacing application; futhemoe, if 2, then the teated poblem in thi pape i educed to the conventional two-tage poblem in [1] [4] To obtain an optimal tate etimate fo the ytem (1), (2), the conventional KE may be ued, with the initial etimate X 0j0 X 0 and P 0j0 P 0 Howeve, the computational cot and the etimation eo of the KE inceae datically with the tate dimenion Hence, the KE model may be impactical to implement One method fo olving the above-mentioned poblem of the KE i to apply covaiance decoupled algoithm To thi end, the appoach of the two-tage U 0 V tanfomation in deiving the OTSKE [4] may be ued Howeve, the OTSKE applie only fo 2 In the next ection, we popoe a genealization of the two-tage appoach to futhe educe the complexity of the OTSKE fo >2, and the obtained new etimato will be denoted a the optimal multitage Kalman etimato (OMSKE) III OPTIMAL MULTISTAGE KALMAN ESTIMATORS The deivation of a multitage Kalman etimato i though a bloc-diagonalizing of the covaiance matice of the KE, ie, P and P j Thi can effectively educe the complexity of calculating thee covaiance, and hence the oveall Kalman filte algoithm The poblem i to find out ome U and V matice that atify the following: P U P U 0 and P j V P j V, 0 P (1) diagfp 1 (1); ; P (1)g We define the tuctue of the U and V matice a follow: U I n U U 0 I n 111 U (3) V I n V V 0 I n 111 V : (4) Thee, V ij and P i (1) tem ae to be detemined, 1 i< j To facilitate the deivation, we ued the following notation: M ij to denote the ij element of matix M A a fit tep, we define the following multitage U 0 V tanfomation, which i to be applied to the KE expeed by fx; K; P g, a X U X ; P U P U 0 (5) X j V X j ; K V K ; P j V P j V 0 (6) X, K, and P denote the tanfomed tate, gain, and covaiance, epectively Then, denote the invee tanfomation of U and V a U 01 ~ U V 01 ~ V Thu, (5) and (6) become I n U ~ U ~ 0 I n 111 U ~ I n V ~ V ~ 0 I n 111 V ~ (7) : (8) X ~ U X ; P ~ U P ~ U 0 (9) X j ~ V X j ; K ~ V K ; P j ~ V P j ~ V 0 : (10) Next, baed on (5), (6), (9), and (10), the tanfomed filte can be calculated ecuively via the two-tep ecuive ubtitution method [4] a X ~ U A 01 V 01 X 01 (11) X j ~ V U X + K (Y 0 H U X ) (12) P ~ U (A 01 V 01 P 01 (A 01 V 01 ) 0 + Q 01 ) ~ U 0 (13) K ~ V U P (H U ) 0 2fH U P (H U ) 0 + R g 01 (14) P j ( ~ V U 0 K H U )P ( ~ V U ) 0 : (15) To expe (11) (15) in ubfilte fom, we define the following notation: A 01 V 01 U 11 U U 0 U U U (16) H U [S S ] (17) Aij 01 + u(j0) li S i H i + u (i02) A il 01V lj 01 (18) H l U li (19)

3 2184 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 45, NO 11, NOVEMBER 2000 Fig 1 Bloc diagam of the optimal multitage Kalman etimato ( 3) in which u () denote the unit-tep function and 1 i j Uing (8) and (16) (19), and expanding (11) (15), we obtain the following covaiance-decoupled ubfilte i (i 1; ; ): X i A ii 01X i 01 + u i 01 (20) X i j X i + K i (Y i 0 S i X i ) (21) P i A ii 01P i 01(A ii 01) 0 + Q i 01 (22) K i P i (S i ) 0 fs i P i (S i ) 0 + R i g 01 (23) P i j (I 0 K i S i )P i (24) u i 01 u (0) Y i Y 0 u (i02) Q i 01 Q ii 01 + u (0) 2 R i X l 01 0 X l (25) S X ; Y 0 Y (26) 2 ( P l 01) 0 0 ( P l ) 0 (27) R + u (i02) S P (S ) 0 ; R 0 R : (28) The blending matice U and V ae then calculated by P j 01 (Ajj 01 )0 + Q ij 01 (P j )01 (29) V ij 0 K i + u (j0i02) K l S j (30) Q ij 01 Qij 01 + u(0) 2 lj+1 2 (U jl P l 01) 0 0 (U jl P l ) 0 (31) and 1 i<j Equation (20) (24), (29), and (30) ae deived in the Appendix Note that in the above algoithm, the meauement vecto and the meauement noie covaiance of ubfilte i ae eplaced by the innovation vecto and the innovation covaiance of ubfilte A a lat tep, the etimate of the KE can be obtained by the following OMSKE: ^X j V X j (32) with the following initial condition: X i 0j0 X i 0 0 u (0) P i 0j0 P ii 0 0 u (0) ^P j V P jv 0 (33) 0 X l 0j0 (34) 0 P l 0j0( 0 ) 0 (35) V ji 0 P ji 0 (P i 0j0) 01 (36) i ; ; 1 and j 1; ; The tuctue of the popoed OMSKE with 3i hown in Fig 1 Rema A: The OMSKE i mathematically equivalent to the KE Thi i deduced fom the inductive eaoning imbedded in the popoed two-tep iteative ubtitution method [4], and can be veified by uing inductive eaoning a in [3] In fact, it can be checed that the KE and the OTSKE can be thought of a pecial cae of the OMSKE with

4 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 45, NO 11, NOVEMBER and 2, epectively In ode fo (29) to be applicable, it i aumed that P j i noningula Thi i the mot often encounteed ituation in pactical application ince the KE i applied mainly to tochatic ytem the poce noie covaiance Q i poitive Howeve, thi etiction can alo be emoved and i decibed a follow If P j ij 0, then the matice U, 1 i<j, no influence on the obtained filte Thu, one can imply eplace (29) by 0 On the othe hand, if P i i ingula (6 0), then one only need to eplace (29) by U 1i U ;i [M ] ij mn U 1i P i U ;i P i M in 111 M ij : M mn 111 M mj (P i ) 01 (43) P j 01 (Ajj 01 )0 + Q ij 01 P j + (37) M + i the Mooe Penoe peudoinvee of M [11] Rema B: A mentioned in the peceding ection, the popoed multitage algoithm can be modified to be applied to geneal ytem Thi i done by modifying (18), (25), (27), and (31) by ij : U + u(i02) u i 01 : u i 01 + u (i02) Q i 01 : Q i 01 + u (i02) A il 01V lj 01 (38) X l 01 (39) ( P l 01) 0 (40) Q ij 01 : Qij 01 + (U jl P l 01) 0 (41) the ymbol : mean i eplaced by Baically, in the above modification, only the time update equation need to be evied Rema C: If the ytem [(1) and (2)] epeent a paallelly connected ytem, ie, A ij 0, Qij 0, and Hj 0fo 1 i<j, then the blending matice U and V become identity matice, and hence the OMSKE will be equivalent to a decoupled Kalman filte Rema D: The OMSKE can be thought of a a geneal tate-decentalized filteing algoithm in which the tate pace i decoupled while the output pace emain coupled Thi i diffeent fom conventional output-decentalized filteing algoithm, eg, [5] [8], the complexity eduction i mainly due to the decoupling of the meauement pace Rema E: The filteed eo covaiance given by (33) i not needed in the calculation of the ecuive algoithm, eg, (20) (24) Hence, it may be calculated only when filteed covaiance data ae needed On the othe hand, the popoed algoithm can be thought of a an altenative to the well-nown U 0 D covaiance factoization algoithm [9] unde the aumption that the ytem tanition matix i tiangulaized It i nown [10] that numeical tability of the covaiance factoization algoithm wa due to thei lac of enitivity to the choice of a pioi vaiance and poce noie level Thu, it i expected that the popoed OMSKE ha the advantage of both numeical accuacy and computational efficiency In thi pape, we focu ou attention on the computational efficiency of the OMSKE Rema F: To implement the OMSKE, (22), (27), and (29) ae calculated a follow: U 1i P i U ;i P i P i Q 1i 01 Q ii 01 +[U ] ii 0 [U ] i; i+1 2 (U ii P i 01 )0 (U i P 01 )0 i; i+1 (U P i )0 (U i P )0 (42) Finally, we give the pocedue fo implementing the OMSKE in each cycle a follow Step 1) Calculate, 1 i<j, by (18) Step 2) Calculate the time update equation fo ubfilte j (j ; ; 1) though: 21) calculate u j and 01 Xj by (25) and (20), epectively 22) calculate P j ij and U, i 1; ;, by (42) and (43), epectively Step 3) Calculate the new meauement matix S, i i 1; ;, by (19) Step 4) Calculate the meauement update equation fo ubfilte i (i 1; ;), by (21), (23), and (24) Step 5) Calculate the blending matix V ij, 1 i<j, by (30) Step 6) Calculate the Kalman etimate by (32) and (33) IV COMPUTATION EVALUATIONS To demontate the computational advantage of the OMSKE ove the KE, we ue floating-point opeation o flop in Matlab a a meaue of the computational complexity Each multiplication and each addition contibute one flop count To implify the dicuion of the computational load, we conide the pecial cae that ubfilte the ame dimenion (n i n n) A a fit tep, we lit the flop count of a geneal KE (KE g ), which ha tate dimenion n and meauement dimenion m, a follow: flop(ke g )6n 3 +(4m +3)n 2 +(4m 2 +4m +2)n +2m 3 + m 2 + m: (44) Note that if the ymmetic popety of the covaiance matice ha been ued, then the above flop count i educed to flop(ke )4n 3 +(4m +4:5)n 2 +(3m 2 +5m +2:5)n +2m 3 +0:5m 2 +1:5m (45) we ued KE to epeent the above implified KE Futhemoe, if the uppe tiangula popety of the ytem tanition matix, which ha bloc numbe, i ued, the flop count in (45) i futhe educed to flop(ke t ) 2+ 1:5 + 0:5 n m +3+ 1:5 2 n 2 +(3m 2 +5m +2:5)n +2m 3 +0:5m 2 +1:5m (46) and the obtained algoithm i efeenced to by the name KE t Then, the complexity of the OMSKE i evaluated a follow Fom (45), it i clea that the flop count of the covaiance-decoupled ubfilte i flop(subfilte) 4 4m +4:5 2 n3 + n 2 +(3m 2 +5m +2:5)n + (2m 3 +0:5m 2 +1:5m): (47)

5 2186 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 45, NO 11, NOVEMBER 2000 The flop count of the auxiliay calculation pecifically needed by the OMSKE, eg, the quantitie calculated in Step 1), 21), 23), 25), 3), 5), and 6) of the peceding ection, i given a follow: flop(auxiliay) n 3 +(3m +6)2 n 2 0 5m +6 2 n 2 + 0:5m( 0 1) n: (48) Uing (47) and (48), the flop aving, denoting by 4flop, of the OMSKE a compaed to the KE g, the KE, and the KE t ae given, epectively, by 4flop OMSKE KE n 3 + m 0 3+ m +1:5 2 n 2 + m 2 0 0:5m( +1)0 3:5 + 2 n 0 ( 0 1)(2m 3 +0:5m 2 +1:5m) +0:5m(m0 1) (49) 4flop OMSKE KE n 3 + m 0 1:5 + m +1:5 2 n 2 0 0:5m( 0 1) n 0 ( 0 1)(2m 3 +0:5m 2 +1:5m) (50) 4flop OMSKE KE : n 3 + m 0 3+ m +3 2 n 2 0 0:5m( 0 1) n 0 ( 0 1)(2m 3 +0:5m 2 +1:5m): (51) Note that the above pefomance i atified fo < n Howeve, if n, then the pefomance in (51) will inceae to 4flop OMSKE KE 2 3 n3 +(m03:5)n 2 +(0:5m 2 +2:5m )n 0 (n 0 1)(2m 3 +0:5m 2 +1:5m): (52) Thi additionally gained efficiency i mainly due to the fact that matix multiplication ae eplaced by cala multiplication To how the computational advantage of the OMSKE, we ue the elative impovement atio (RIR) of the OMSKE, which i defined by RIR OMSKE KE lim n!1 4flop OMSKE KE flop(ke i ) (53) a the pefomance index The RIR pefomance i intended fo the ituation the numbe of the meauement i negligible a compaed to the numbe of the ytem tate Thi i aumed fo applying the OMSKE and i uually the cae encounteed in pactical ytem Howeve, if thi i not the cae, a can occu in lage-cale o multieno ytem, then it i alway poible in pactice to poce the meauement equentially [8] o a to atify the above mentioned ituation Thu, ubject to n m, it i clea fom (44) (46), (49) (51), and (53) that the RIR of the OMSKE a compaed to the KE g, the KE, and the KE t ae given, epectively, by RIRKE OMSKE (54) RIRKE OMSKE (55) RIR OMSKE KE : (56) +3 Note that if i ufficiently lage, then the above RIR value will appoach 78%, 67%, and 33%, epectively Seeing fom the above eult, although the conventional KE, ie, KE g, can be implified by applying the ymmetic and tiangula popetie of the KE, the popoed OMSKE can futhe educe the complexity of thi implified KE, ie, KE t, fo the cae that the tate numbe i much lage than the meauement numbe Finally, fom the RIR pefomance [(54) (56)], we claim the following ey concept about the complexity iue of the multitage algoithm: the computational complexity of the multitage Kalman etimato i le if the lage i choen and ha the minimum value when the ytem tanition matix ha the maximum tage numbe V CONCLUSION In thi pape, the OMSKE i popoed The OMSKE i a genealization of the OTSKE [4] and i ued intead of the OTSKE when the uppe tiangula bloc numbe of the ytem tanition matix i geate than two It i hown by analytical eult that the computational complexity of the OMSKE i le than that of the implified KE (KE t ), which can be obtained by applying the ymmetic and tiangula popetie of the KE, and i minimum when the ytem tanition matix ha the maximum tage numbe Ou eult ugget that the OMSKE may eve a an altenative to the KE fo etimating the ytem tate of linea dynamical ytem ubject to uppe tiangula ytem tanition matice when the numbe of the ytem tate i lage and i much lage than the numbe of the meauement A inpied by the wo of Bieman [10], the numeical eliability of the OMSKE may be bette than that of the conventional KE The poblem of exploing the numeical tability of the OMSKE i unde invetigation APPENDIX 1) Uing (3) and (5), we X i X i + u (0) X l (57) 1 i Compaing (9) with (11) and uing (16), we X i A ii 01X 01 i + u (0) 2 X l 01: (58) Subtituting (58) into (57) and uing (25), we obtain u i 01 X i 0 A ii 01X i 01 u (0) X l 01 0 X l : (59)

6 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 45, NO 11, NOVEMBER ) Uing (3), (5), and the diagonalizing tuctue of P,we P ii P i + u (0) P l ( ) 0 (60) 1 i Compaing (9) with (13) and uing (16), we P ii A ii 01P i 01(A ii 01) 0 + Q ii 01 + u (0) P l 01( ) 0 : (61) Subtituting (61) into (60) and uing (27), we obtain Q i 01 P i 0 A ii 01P i 01(A ii 01) 0 Q ii 01 + u (0) 2f ( P l 01) 0 0 ( P l ) 0 g: (62) 3) Uing (15), (17), and the diagonalizing tuctue of P j,we P j with 2 fg2 11 fg fg2 fg2 21 fg fg2 fg2 1 fg fg2 I n (( V ~ U ) 12 ) 0 I n (( ~ V U ) ) 0 (( ~ V U ) ) I n fg ii 2 (I 0 K i S i )P i P i j (63) fg ij 2 ((~ V U ) ij 0 K S i j )P j 0 (64) 1 i and i<j Fom (64) and the fact that P j i noningula, we 4) Subtituting (65) into (12), we X i j X i + u (0) ( ~ V U ) ij K i S j : (65) + K i Y 0 j1 X i + K i Y 0 5) Fom (14) and (65), we obtain K i j1 S j P j (Sj )0 + R P i (S i ) 0 + u (0) ji+1 ji+1 S j X j i j1 K i S j X j S j X j : (66) K i S j P j (Sj )0 : (67) Canceling the ame tem on both ide of (67) and olving fo K i, we obtain K i P i (S i ) 0 i j1 S j P j (Sj )0 + R 6) Uing (3), (5), (9), and (13), we P ij P j + u(0) lj+1 P j 01 (Ajj 01 )0 + Q ij 01 + u (0) lj+1 01 : (68) P l (U jl )0 P l 01(U jl )0 (69) 1 i<j Auming that P j > 0 and olving (69) fo, we obtain P j 01 (Ajj 01 )0 + Q ij 01 (P j )01 (70) Q ij 01 i defined in (31) 7) Solving (8) fo V ~ ij, we obtain ~V ij ij 0V 0 u(j0i02) Uing (65), (8), (4), and (71), we K i S j (~ V U ) ij 0 V ij Solving (72) fo V ij V ij fg + u(j0i02) + u(j0i02) ( V ~ il U lj ~ V lj : (71) ~ U lj + V ~ ij 0 V il and uing (71) and (72), we obtain 0 K i S j + u(j0i02) 2 V lj 0 U lj (V lt ~ V tj + u(j0l02) 0 K i + u (j0i02) tl+1 j02 tl+1 j02 tl+1 0: tl+1 0 V ~ lt U tj ) + fg V ~ lt U tj 0 l01 ti+1 V ~ lt U tj 0 V ~ lt U tj 0 j02 V it li+2 ti+1 V ~ lj ): (72) K l S j (73) l01 ti+1 lt+1 V ~ tl U lj V it V it V ~ tl U lj V ~ tl U lj Fom (59), (66), (62), (68), (63), (70), and (73), we obtain (20) (24), (29), and (30)

7 2188 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 45, NO 11, NOVEMBER 2000 ACKNOWLEDGMENT The autho would lie to than the anonymou efeee fo thei inightful comment REFERENCES [1] B Fiedland, Teatment of bia in ecuive filteing, IEEE Tan Automat Cont, vol AC-14, pp , 1969 [2] M B Ignagni, Sepaate-bia Kalman etimato with bia tate noie, IEEE Tan Automat Cont, vol 35, pp , 1990 [3] A T Alouani, P Xia, T R Rice, and W D Blai, On the optimality of two-tage tate etimation in the peence of andom bia, IEEE Tan Automat Cont, vol 38, pp , 1993 [4] C S Hieh and F C Chen, Optimal olution of the two-tage Kalman etimato, IEEE Tan Automat Cont, vol 44, pp , 1999 [5] R A Singe and R G Sea, Inceaing the computational efficiency of dicete Kalman filte, IEEE Tan Automat Cont, vol AC-16, pp , 1971 [6] M F Haan, G Salut, M G Singh, and A Titli, A decentalized computational algoithm fo the global Kalman filte, IEEE Tan Automat Cont, vol AC-23, pp , 1978 [7] H R Hahemipou, S Roy, and A J Laub, Decentalized tuctue fo paallel Kalman filteing, IEEE Tan Automat Cont, vol 33, pp 88 94, 1988 [8] A Baong-Onana, M Daouach, and M Zaadzini, Computationally efficient optimal output decentalized etimation, Int J Cont, vol 58, pp , 1993 [9] C L Thonton and G J Bieman, Filteing and eo analyi via the UDU covaiance factoization, IEEE Tan Automat Cont, vol AC-23, pp , 1978 [10] G J Bieman and C L Thonton, Numeical compaion of Kalman filte algoithm: Obit detemination cae tudy, Automatica, vol 13, pp 23 35, 1977 [11] G H Golub and C F Van Loan, Matix Computation Baltimoe, MD: John Hopin Univ Pe, 1983 the failue of contol component and etain the deied contol ytem popetie, ee; [3], [5], [7], [8], and [10] [14] In [10], Veillette et al peented a methodology fo the deign of eliable linea contol ytem uch that the eulting deign guaanteed cloed-loop tability and H1 pefomance not only when all contol component ae opeational, but alo in the cae of ome admiible contol component outage Veillette [10] developed a pocedue fo the deign of H 2 tate feedbac contolle, the eulting deign could toleate the outage within a elected ubet of actuato while etaining the tability and the nown quadatic pefomance bound In the nonlinea cae, Yang et al [13] peented a eliable H1 contolle deign fo nonlinea ytem, which i a nonlinea veion of the eult given in [10] Howeve, the above eliable contolle deign method ae all baed on a baic aumption that contol component failue ae modeled a outage, which neglect the cae of patial degadation In thi pape, a moe geneal failue model i adopted fo actuato failue, which cove the cae of nomal opeation, patial degadation and outage Specifically, thi pape i concened with the poblem of eliable guaanteed cot contol fo uncetain nonlinea ytem The pape i oganized a follow The model of actuato failue, poblem, and ome peliminaie ae given in Section II In Section III, a deign method fo eliable guaanteed cot contol i peented, the eulting deign guaantee the obut tability and an H 2 pefomance bound againt plant uncetainty and actuato failue A numeical example i given to illutate the deign method in Section IV Section V conclude the pape II PROBLEM STATEMENT AND PRELIMINARIES Conide a cla of uncetain nonlinea ytem decibed by equation of the fom _x(t) f (x)+f 11 (x)1(x; )f 22 (x) Reliable Guaanteed Cot Contol fo Uncetain Nonlinea Sytem Guang-Hong Yang, Jian Liang Wang, and Yeng Chai Soh Abtact Thi pape i concened with the eliable contol deign poblem fo uncetain nonlinea ytem A moe pactical model of actuato failue than outage i adopted Baed on the Hamilton Jacobi inequality (HJI) appoach fom nonlinea contol theoy, a method fo deigning eliable tate feedbac contolle i peented The eulting contol ytem ae obutly table and with an pefomance bound againt plant uncetainty and actuato failue Index Tem Actuato failue, contol, Hamilton Jacobi inequalitie, nonlinea ytem, eliable contol, uncetain ytem I INTRODUCTION In the aea of eliable contol ytem deign, eveal deign method been developed fo the eulting cloed-loop ytem to toleate Manucipt eceived Octobe 5, 1998; evied Septembe 27, 1999 Recommended by Aociate Edito, M Ktic Thi wo wa uppoted by the Academic Reeach Fund of the Minity of Education, Singapoe, unde Gant MID-ARC 3/97 The autho ae with the School of Electical and Electonic Engineeing, Nanyang Technological Univeity, Nanyang Avenue, Singapoe Singapoe ( ejlwang@ntuedug) Publihe Item Identifie S (00) [g(x) +f 11 (x)1(x; )g 22 (x)]u(t); x(0) x 0 (1) x(t) 2 R n i the tate, u(t) 2 R m i the contol input, f (x), g(x), f 11 (x), f 22 (x), and g 22 (x) ae nown mooth mapping with f (0) 0, f 22 (0) 0, and 1(x; ) atifie 1 T (x; )1(x; ) I (2) with being an uncetain paamete vecto The cot function aociated with thi ytem i J 0 1 [Q(x)+u T (t)r 2u(t)] dt (3) Q(x) i a mooth poitive definite function with Q(0) 0, and R 2 > 0 i a ymmetic contant matix Fo the contol input u i, i 1; 2; 111;m, let u F i denote the ignal fom the actuato that ha failed In thi pape the following actuato failue model will be adopted: u F i i u i + i (u i ); i1; 2; 111;m (4) i > 0, and the uncetain function i (u i ) atifie 2 i (u i ) 2 i u 2 i ; i1; 2; 111;m (5) with i i 0 Rema 21: In the above model of actuato failue, if i 1, i 0, then it coepond to the nomal cae u F i u i When i i,it cove the outage cae If i > i, it coepond to the patial failue cae and alo can be egaded a patial degadation of the actuato /00$ IEEE