Gain fusion algorithm for decentralised parallel Kalman filters

B.S.Paik and J.H.Oh

Abstract: A iiew gain fiision algorithm i s proposed for application to deccnlralised sensor systems. ‘I’hc proposed algorithm gives coinputer-efficient suboptimal cstiination rcsults, such that it reconsLructs tlic global estimate and covariancc kom local Kalinan filter gains and cstiinates without significant loss o f accuracy. Coinparcd to the conventionnl algorithm, thc snialler coinmunicntion rcquiremcnt and the removal o f thc calculutioii requircmcril of invcrsc covariances make the proposed algorithm murc suitablc for real titnc applications. A numerical exainple sliciws that the propnsed algorithm provides a convincing suboptimal decentraliscd algorithm. T i l addition, the proposcd gain fusion algorithm CBH bc easily cxteiided to accointnodatc local ICalmiln filters with rctluced (irder.

1 Introduction

In the decentralised real-time inuhisensor iiieasiircniciit systems, such as tracking, surveillance and navigation systems, thc optimal cstiinatioii problem is one o f the most critical concerns. Conventional Kalmnn filtcring methods can be applied to multisensor systems to provide optimal cstiination. But the typically large state diinension in tlic system model and the high d a h rntcs of thesc incthods huvc forced thc developinciit of thc decentralised parallel proccsslng mctbods [ i ] in recent ycars.

Deccntralised estimation offcrs considcrable advantages: it can provide tkc most logically fcasible processing schcincs in many circummnces, it can facilitate fault dctcction and isolation inore cotivcniently, it cat1 incrcase thc input data rates significantly, and it can improve thc throughput moderately 121.

Extensivc likratiirc has been published o n this subject. Early contributions to decentralised estimation theory by Speyer [3] and others [4, 53 provided lrsehl insights about diffcrcnt approaches to partitioned cstimntion. Based 011 these, Ken [6] proposed a dccciitralised liltering structure with fault-tolerant features. Bicrinan [7] developed an optimal and practical dcccnrralised filtering method for orbit estimation purposes. I-lashemipour et ul. [XI dcvcl- opcd decentraliscd Kalman filtering structures for niulti- scnsor networks with parnllcl processing capability. Carlson [SI develol~cd the fcdewteci filtering method based on information-sharing principlcs. In [IO], Liggitis et ai, provided a coniprehensivc survey of the rclated work.

All of the prcvious works result in the global estiinate of the state in a decentralised system, but tlicy all still require extensive cnlculalions of the local and global iiivcrse

c) IF.E, 2000 UX Pmccedirig online no. ZUOOO141 DOI: i 0,1049/ip-cla:20000 14 1 Pnper first received 10th Jmc and in reviscd fomm 30th Scptcmbcr I999 Thc authors arc with the Department of Mcclianical Enginecring, Korea Ativniiccd Iiistilulc of Sciencc and Technology (KAIST), 373- I, Kurong- dong, Yusoiig-gu, Tacjcon, Kepuhlic of Korea 305-701 E-mail: {pbsjunholi) @Iii..hb.knist.iic.kr

I.% Pm.-Cuuntroi Tiieoy &PI.. Yol. 147. No I , Jw1nr-y 2000

covariances. The gain fiision algoritlirn can eliminate this costly computational requiremciit: instead of tlic iiiversc covariances, ~1 global processor receives tlic inlbrmation in the form of II K a l n m gain transfer from local systems aiid fulmulates tlic global estimate. Examples of relevant work in this area include [ l I , 121. In [I 11, Oardiier proposcd the gain transfcr nlgorithni I’or decentralised estimation. This algorithm docs not usc inversc covariances, but, as it requires thal. the filtcr be updatcd sequentially, it is riot thc parallel processing stnic!ure in triic sence. In [ I 21, Kim et nl. developcd a federatcd Kalman liltcr using a gain fusion algorithm. Siticc the application is limited to only when all local sensors producc equivalent information, it cannot be considcrcd as the gciicral form of tlic gain fusion algorithm for thc decentlalised estimation.

A IICW gain fiisiun algorithm is introduced in this papcr with parallel proccvsing capability and with inore gencral form for deccntralised estimation. The proposcd gain fusion algorithm can bc easily cxtcnded to nccoininodatc local Kalman filters with reduced order [2 ] , i.c. each local inodcl has a statc diinensioii smaller than the global state vector. To demonstrstc the basic principles o f thc proposcd algoritlini inore clearly, each local inodel is assumed to have,jidl urdw stato vector in this paper. [t is assumed that, in every scan, all the local sensors acquire and process their measiircments axid then communicate with a global p111ceSS6r.

2 System description

Considcr a model of III inultiscnsor systems, whose local variables are subscripted (.Ii, where i = I , . , . , m [13]. They cach take the observations z,(kj of a global system stutc vector x(k) according LO the following observatioii formula:

(1) z,{k) = Hj(k)x(kj -I- vj(lt)

whcrc x(k) E Yl”, zi(k) E W’r and H i ( / ) E CJV f t . Thc incasureincnls are corrupted by noise, v,(k) “(0, R,(k)) with R,(k) E BVJiY~’r, modelled as an unco~relateil wliitc sequence.

17

" Thesc locd obscrvation variables are the partitions of the respective global counterparts z(k), H(k) and v(k):

qit) Iz;f(kj. I . 1 Z:(k>jT (24

H(k) = [H:(k), * . , HTJ(k)lT (2b)

~ ( k ) = IvT(k), I . . , (2ul

so there i s a global observation formula:

z(k) = H(k jx(k) 4- v(k) (3 I where z(k) E W , H(k) E 91' and v(k) - N(0, R(k)) with R(k) E 9 l p p. The dimcnsion, p of the global observation vector is defined as p = xrLl p p This partitioning was made cxplic,it in CS]. Assume further h a t the local noise pariitions are imcorre~rrtcd:

R[k) = E{v(k>vT(!)] = blackdiadlj,,Rl(k), . . . , Snlt,,,(k)] (41

where hk! = 1 when k= I and 6,

state transition formula:

0 otherwise. The staic dynamics are modelled RS the following lincar

~ ( k + 1) = F(k)~(k) + G(k)w(k) ( 5 )

where V(k) E 91""" is the state transition matrix fiorn k to IC+ I , G(k) c !UpInry is tkc noise modcl and w(k)-NN(O, Q(k)) is the associated procoss noisc which is modelled as an uncorrelated white scquence with E{ w(k)d (I)! =&Q(/.C) aiid Q{k) E %'?xq. Finally, assume that the process and measurement noises are uncorrelated:

SIV(~C)W?' + {ljli) = o (6 )

For all k? j , definc

i ( k l j ) E{x(k)\x(O. . . . , dj)} (7)

to bc the estimatc (cxpected value) of the state x(k) at a time step k based on the observations taken up to a time step j and

P(k1.j) = NXIN -i<kIj))(x(,t) - W l i ) ~ ~ + I Z ~ I ) , . . I , z ~ ) j (8)

to bc the mean squared error in this estimatc. For thc assumed modcl described above, thc centralised

and the conventional decentralised Kalman filter formulas arc: well dcscribed in (1-lOl. Eqns. 9 aird 10 me the fiisiorr equations to update a global filter of the conventional optimal dccenntralised algorithm, which will be used as the basic modcl ta be compared with the proposed algo- rithm later in this paper:

P-yk -t 1 Jk + I)i(k f I Ik -t I ) m

= P; + 1 ~k + 1 ] i i ( k + 1 ~k t. I j i=l

- ( m - l)P-'(k 4- l(k)%(k + Ilk) 19) 01

P - ' ( k $ - ~ j k + l ) = r ) P ; ' ( k - r l J k - f . l ) i= l

- (m - I)P-l(k 6 Ilk) (10)

where the information feedback from the global filter to the local filters is permitted. The following well known expres- sions are used for the fcedback configuration:

i , ( k + I lk ) = Ei(k 4- l(k) U 1)

3 Gain fusion algorithm

Ln this Section, thc gain fusion algorithm will be dcveloped for the application to the decenlraliscd sensor systcma with pmalIc1 processing capabilities. The proposed gain fusion aIgorithni for mu [tisensor integration involves the informa- tion feedback from the gIobaI filter to the iocai filters. To dcrive the gain fusion algorithm, first, consider the globally optimal ICalmon gain matrix, which can ba rowrittui as

K(k -t 1 ) = P(k f I/k)HT + (k + I)S-'(k t I j (13)

where S ( k + I)=EZ(k+ 1)P(k4 I)k)HT f ( k + 1)+R(k+ I ) and i s the cavntiancc o f the residual of the global model. From the recent study of Carlson [9] and Felter [141, the covariance of thc residual of the global model can be replaccd by its upper bound as

S(k + 1 ) 5 blockdiag[ylSI(k 4- I ) , . . . , y,,,S,(k 3. I ) ] (14)

whcre Si(k+ i j=Hi(k+ l)Pf(k+ 1 l k j ~ ; r ( k s l ) s ( ! ly , ) R,(k+ 1) and is the covariance of thc ~osidual of the ith local modcl. The detailed derivarion of eqn. 14 is given in the Appendix (Section 8.1). Although this is clearly suboptimal, thc assumptian of the block diagonal nature of s(k +- I) allows a decentraliscd gain fusion algorithm. Setting the covariance of the residual of the global modcl to its upper bound and using eqns. 22, and 13, the following result of the globally suboptimal Kalmatl gain matrix can bc obtaincd:

'The Kalnian gain of the ith [mal iiltcr can then be dcfined bY

1 Yi

Ki(k + 1) = ,Pi(k + 1lk)H; f ( k 3- l)S;'(k 5 I ) (16)

where Si&+- 1) is defined us in eqn, 14. Using eqns. 12, 15 and 16, the suboptimal Kalman gain matrix can be parti- tioned as

K(k -+ i) = [K,(/c + I ) ! . . . IK,(k -+ I)] (17)

Therefore, combining cqna. 2b and 17 results in the global error covariance as follows:

For the global crror covariance, the Joscph stabilised formula is adopted in eqn. 18, and, for the global cstitnate, it can be represented as

where z ( k f l) , H(k+ 1) and K(k+ I ) are given by cqns. 2a, 2b and 17. Froin the cxprcssion of the local Kalinan filtcr, cqn. 21 can be obtained as follows:

Consequently, combining eqns. I t , 20 and 21, it can be concluded that tbc global estimate is presented by

As a result, the gain fusion algorithm for miltisensor integration involves the following stcps, hcginning with initial conditions:

A . Ex’xlrpoliltion of glokaiiy integrated resid$:

B. Reset locadfn’iters: Local filters arc rcsct with

C. Ohssrvational updale of locnl guins and estimaies:

D. GloAalfi,~ian: The global fusion of m locally estimated results is

* I

i ( k $ t lk+ 1) = -pi@+ I l k + 1) - (m - l ) i ( k + I l k ) I= I

(30)

K j ( k f I)H;(k + 1) P(k + 1 Ik) i= I 1

I K,(k + l)Hj(lc + 1) i= I

+ 2 K,(k + l)Ri(k + 1)K:(k f 1) (3 I ) i= I

It can now be seen that the local filters can pass thcir. rcspcctivo %,(k+ i l k+ l) , &(k+ I), H,(k+ 11 and R,(k+ 1>, i = 1,. . , ,m, on to the global filter, which, in h m , can compute its global estimate.

It can be seen that this gain fusion algorithm eliminates the costly computational rcquirement of the calculations of the invcrsc covariances. The need for the calculation ofthc observational update of the local covariances to obtain thc global cstimate is no longer ncccssary, and the proposcd gain fusion algorithm can be easily extcnded to wcom- modate Local Kalinan filters with reduced order. Also, by feeding back the global CI priori estimate and error covar- iancc to the local filters, indircct mensiiremeiit sharing i s allowed bctween the Iocal filters. Thus, from eqn. 30, ifone of the local scnsors missed a measurcinent, then the global estimate is fused with thc global a priori cstimate instead nf thc observational updatc cstimate for that scnsor, and from cqn. 3 1 , the fusion or thc global error covariancc is dnnl: with the information from the rest of tlic sciisors, Thcreforc, the proposed algorithm can operate normally, even though a slight loss OP accuracy is expected. In addition, i t can be shown that, using thc approximation of eqn. 14, the global error covariance matrix of the proposed algorithm is positive semideflnitc, as described in the Appendix (Section 8.2). Thus, the proposed decen- tralised K h a n filter can be coiisidcred as a suboptimal decentralised Kalinan filter.

4 Computational aspects

To slzow the efficiency of the proposcd algorithm, the computational coinplcxitics of the selected conventional dccciitralised algorithm and the proposed algorithm are evaluatcd and compared. In this paper, the numbcr of flops per cycle is used to cxpress the computational complexities of the algorithms, since it gives a rough idea of the relative computational complcxitics of the algorithms.

Table 1 shows the amount o f operations rcquired at the global proccssor pet cycle to updatc its own global Kalman filter implemented by the two methods: the one with the conventional decentraliscd algorithm based on eqns. 9 and 10, and thc other with the proposcd gain fusion algorithm

Table 1: Operatfon count for global filter

Algorithm No. of flops Communication

Conventional (m+Z)($ +1.5d t0.5ni-k $ + n m(t+ + n) Proposed n3 4-M + Z O + Cj?!, (r+ p , - t np, +n,# mn+ c , ( 2 n p j +,$)

99

based on eqns. 30 and 31, Table 1 also shows ilic comparison of the required cominiinication per cycle between the local and thc global processor, in the ahscncc of thc infortnation feedback which i s idcritical in both algorithms.

The abovc rcsull sliows that the proposed algorithm requires less computation and communication per cyclc than thc conventional algorithm for thc application to m multisensor systcms. Also, it suggests that the proposed algorithtii is tnorc advantageous in the sittiation tlitlt the statc variable dimension n is mnch larger ttiati the observa- tion variable dimension p .

5 Illustrative example

To demonstrate the efficiency and accuracy of this new gain hision algorithm for multisensor integration, a simple nuincrical cxainple is presented. Cnnsidcr the following linear system, which is a inodifiad version of a tracking model in [15]:

0.80 0.20 0.00

with the initial condition io =0, Po =0.041,, wlicrc x1,x2 and x3 are the position, velocity and acceleration, rcspec- tivcly, of a flying object. Here, thc system noise sequeiicc {w(/c)} is apscudormidom sequence (i.c. uncouelated zero- incan Gaussian wliite noisc sequence) with {Q(k) ) = {O.OOII , ) . In this example, the position and the velocity of the flying objcct arc assumed to bc obscrvcd at each time instant, so that z,(/) = H,(k)x(k) + v,(k), i = 1 , 2 , with H I ( / ) = [ I 0 03, H,(k) = [0 1 01 c m be defined, where thc observation noisc sequences { v i (k ) ) arc also pseudoran- dom sequences with {X,(k)=O.I} and {R2(k)=0.01].

Based on the above tracking system, the perforniancc of the proposed algorithm is compared with thc conventional optimal algorithin. For the conventional optimal algoritlim, the expressions dcscribcd in Section 2 are uscd, and for the

-2 . . ... ... . .._.._.. . . . .. . . . .

.L 5 10 15 20 25 30 35 40 45 5b

normalised position error

f i g . 1 AW-maIi.red position w f w i

- convciitioiiol oprimal algorithm . . , ,

I00

proposed algorithm

c L.

e .-

.f YA

Fig. 2

5

3 I !

cunvenrinnnl optitnal algorithm . , . . . proposcd algorithm

proposed algcirithm, the expressions where yi is replaccd by m the number of thc local sensors, arc uscd. Computer simulation results sliiiwn in Figs. 1 and 2 are the normal- iscd cstitnation error's o f tlic state components X I and x2, whicli represent position anti velocity, rcspcctively, togcthcr with the 95%) probability region. Thc solid curvcs dciiute the simulation rcsults obtaiiictl by using the coriveritional optimal algorithm, a n d thc dotted curves give that of using tlic proposed algorithm. The dashed CUTVCS represent the 95Yn probability region which is within the intcrval [ - I .96, 1.961. l 'he absc.issa deiiotcs the observation points. Figs. 3 and 4 show the RMS cstimation ermix o f position and vclocity, respectively. All simulations are evaluatcd idetitically in terms of the ensemble averagc of the 50 Monte Carlo runs. The normal- iscd cstimation errors sliowii in Figs. 1 and 2 cxcced the 95% lcvcl only at a few obscrvation points. Thus, the proposed dcccntrdised fiIter appears to be almost consis- tciit. From the RMS cstimation errors shown in 1:igs. 3 and 4, it caii bc seen that, allhough some difterences arc visible, the accuracy ofthe proposed algorithm is very close to that o f the conventional optinial decentralised algorithm used as a hcnctimnrk. Thew rcsuIts confirm thal thc proposed algorithm i s a convincing suboptimal decentraliscd algo- rithm.

0.25

0,20

OIL--. , I I I I I I

5 i o 15 20 25 30 35 40 45 50 RMS posilion error

Fig. 3 RMSpusitiioii wmr .. ... ctiiir,entional oprinial nlgrrritlim , . . . . proposad nlgoritlini

lIX I'roc.-Uonlrd T h i y Appl.. Val. 147. Nu. I , .Iuaaiury 2000

0.26 I

Table 2: Operation count for numerlcal example -. - -.

Algorithm No. of flops Communication

Conventional 180 24 Proposed 81 12

The computational complexity of this examplc, where p m = 3 , m = 2 , p I = 1,p2=1, issunimarisedin‘l’ahlc2. The anlolint of coinniunication o f the proposcd algorithin in this examplc has been derivcd using nail -I- zyLi (rrpr), since thcw is no need fur cach local proccssor to transmit H i and Ri to the global processor in the timc invariant system. From I’ablc 2, it can bc seen that the proposcd algorithm has a considelahlc advantagc in terms of the requirements ol computation and communication.

6 Summary and conclusions

A new gain fusion algorithin for decciihlised parallel Kdman filters has hecn proposcd in this papcr. The proposcd algorithm gives computcr-efficient siiboptiin;il estimation results, such that it rcconstructS the global estimate und covariance froin the local Kalinari gains and estimatcs without significant loss uf accuracy. Compared tu other currently proposed methods that requirc the invei sion or potentially large inatriccs, the compiilational rcquire- nient is subslatitially reduced in the proposed algorithm, so that it bccornes more suitable for tiic rcal-time multisensor integration. In thc proposed algori~hm, the nccd for thc calculation of tiic observational update u f the local covar- iances to obtain thc global cstiiiiate is no longer necessary. Also, the proposed gain fusion a[goritbm can bc casiIy extendcd LO accommodate local Kalman filters with rcduccd order.

In a numoiical exainplc or the tracking system, simula- tion results show that the propoacd algorithm provides a convincing suboptimal decetitraliscd algorithm, aiid it c m be sccn that the proposed algorithm has a considcrahle advantage in terms of thc i’cquiirernents of‘ computation and comniiinicution.

7 References

1 MINKLER, G., nnrl MINKLLR, J.: ‘Thcory antl application of Knlimn filtcrirlg’ (Magcllaii Hook Coiupnny, 1093)

Ifi~’1: I’roc.-Cuiitrui Tlimry Appl.. C’oi. I4 7. ,%J. 1, Jnn,wnry 2000

2 ROY, S., HASHEMIPOUK, II.R., and LAUl3,A.J.: ‘Squiireroo1 paial lcl K a h n rillering iising rediicerl-ortlcr local filters’, !ERE fionr., 1991, AKS-27, (2), pp. 276-289

3 SPEYER, J.t,.: Lomputatiun Iind traiistiiissioii rrquiwmciits for I I dcccntnilized lincar-qundratic-~aussi.lii colitid Ipi-oublcni’, ~ h w . , 1979, AC-24, (2), pp. 266-269

4 WILLSKY, A.S., UULLO. M., CASTANON. D.A., I,F,VY, U,, iind VERGHESE, C.C.: ‘Cunihiiiing and updiiting o f local estimates and 1-cogional maps along sctfi of oiic diiiiensiorial tiacks’, I E I 3 ’ nuns., 1982,

5 CASTANON, U.A., niid TI~,NIXETZIS, D.: 'Distributed csiimalion algorithm for iionhicar systcnis’, /MI? Tww., 19B.5, AC-30, (3, pp. 418-425

6 KBKK, T.: ‘Dccciitnlizcti filtcring and rcdiindanLy iswiagcriicnt fur iniilti.sen~or navigation’, I I X E EYIIIS., 1987, AES-23, (I), pp. 83-1 19

7 riIERMAN, G.J.. iind BELZER, M.R.: ‘A dccciitrnlizcd square root iiiforinritiun fillcr/sinothcr’, Pruccctliiigs ofthc 24th IEFX Confcmiicr: 011 Dwisiun o d Conrrd, 1985, pp. 1902-1905

8 IIASHI~MIPOUR, H.R., ROY, S,, antl I,AUH, A.3.: ‘Dcccnlnlizcd slnictirirs for mallcl k h a n filtci-irie’. 1EKk‘ l i a n r . 1 OBH. AC-33.

AC-27, (4), pp , 799-SI3

U , . - . . , (11% pp, 8R-93 ’

9 CARLSON, N.A.: ‘Fcdciatcd sqww ruot lillcr for dcccntralirerl parallel PI’OECsscs’, /LE6 RWns., 1990, AES-26, (3), PI). 517-525

10 LIGCiINS II, M.L. CMONG, CY., KADAR, I., ALFOKU, M.G., - VANNtCOI,A, V., ant1 lIIOMOPOUl.OS, 5.: ‘Uistributctl liwion nrchi-

lcrliircs and nlgoritliins Tor largct hacking’, Rwc. JE/?E, 1997, 85, (1)- nt?. 95-107

11 EARDNRR, \ V L niid IEONDKS, C.’l’,: ‘Gain t i -ansh algorithm: an nlgonrhm for tlzecnlvalizcd Iiicrrri-chical cstiination’, lilt. 1 Conad, 1990, 52. (21. DD. 279-292 . - _ , . , ,

12 K M , J,K , JEP, RI., and LEE, I.G.: ‘A fcdcixtcd Krdnxin liltcr design using B gaiii fusiou al@~ithm’, IFAC Syiuposiiini on Atitoinatic Coiitrol in hcrospacc, 1998, ipp. 395 401

13 nRRG, ‘LM., and TIURRAN’I1WII\’TE, H J . : ‘On distributed ond dccciiirulized cstiination’, Pi-ucccdirigs of tlic A,iimicon riini~nl Cor$),-

14 PELTER, S.C.: ‘A furmalion flight relative navigation systcin’. F1i.D. diswtation, 1995, Statc Uii ivcrdy ofNcw York at Biiightuntoii

15 CIIEN, G . , nndCI.IUL C.K.: ‘A modified adaptive Kalmanfiltcrfor rcal- timc application’, IEEE Trcriis,, 1991, AES-27, (I], pp. 149-153

~ I ~ C C , 1992, pp. 3304-3305

8 Appendix

8.1 Proof of eqn. 12 ‘1’11~ covavinncc OF the inriavation process of thc global inodel is cxprcssed as

S(k + I ) = A(k 4- I ) + R(k + 1 ) (33) wherc R(k+ I ) F ‘ 3 V ’ ” Y and A(k+ I)=H(k-t 1) P ( k f I)I/c)H1(k+ I ) E WX‘’. H(k+ I ) is always a block diagonal matrix, wlicrcas A(k+ I ) is not, in general. In the subsequelit rlcvcloprnent ofthis proof, the tiinc index li will be droppcd in order to simplify the notation. In eqn. 33, the matrix A can be ieplaccd by its uppcr bound as follows:

A 5 bluckdiag[g,A,, , , , , Y, ,~A~, ] (34) with

I

I Yi E-= 1 (35)

wherc m is the iiutnlm of locul sciisors. Ai E W+ xPr is the ith term o f a consecutive scrics of a diagohally located submatrix of A such that each matrix may bc o r a differcnl size and tlicy do not overlap each other with C ;Ll p , = I J . Eqn. 34 is satisfied only when inatriccs A and Ai are positivc semidefinitc. Eqn. 34 tnay bc proved, as suggested by Felter [ 141, hy showing that the following incquality is satisfied for all z E W’:

. . .

XT [*‘l 1 : : +xT[yl;l ’ . _ 0 ( 3 6 )

U p ] “ . %I ... 7,,%,

wlwc ad is the elemctit of the matrix A. Eqn, 36 can be provcd by showing that it will bc satisfied

with yi, which minimises tlic right side of cqn. 36. Thc finding o f y i , among thc oncs constraincd by eqn. 35, becorncs the optimisation problem with constraitits, which

1 0 1

can bc solved by the following method of Lagrangc multi- pliers:

~ ( y l , . . . , y n J = z T [ I l l ::: 0 ]z+A(c;-l) i=l

" . YIflA,,

(37)

The value of 1. that minimises the exprcssioii for F can be obtaincd by taking the partial derivative with respcct to yi and setting it equal to zero:

l3F d - = z7A.z. I 1 - = 0, zi E W, z = I , . . I , In (38)

* V i Y i

and

(39)

Taking U slimmation on both sidcs of eqn. 39 with respcct to i and using eqn. 35 yioltl

i= I

Therefore, the value of yi that minimises the expression for F can be obtained as follows:

(41)

Using the definition of' a vector yi as eqn. 42, the vector z can be expressed as q n . 43:

yi = [O. , .$ . . . 0IT, zj f W' (42)

(43)

When zTAn = 0, cqn 36 is satisfied. When zTAz > 0, to prove cqn. 36, we first definc the n o m of the vector z as

llzll = (z'Az$ (44)

wherc z E DV' a i d z T h > 0. In lhis case, the n o m of the vector yi can be represented as

Substituting eqn. 45 into eqn. 41 results in

Using cqn. 46, the right side of eqn. 36 becomcs

102

From cqm. 43 and 44, the lefi side of eqn. 36 can be reprcscnted as

Therefore, by the thcorcrn of the triangular incqualily, eqn. 36 is proved, and so is oqn. 34. Consequently, eqn. 14 can be derived from cqns. 33 and 34.

8.2 Proof of positive semidefiniteness of suboptimal error covariance In this proof, the supci'script '*' will be used to denote the optiinal quantity. Prom eqns. 25 and 3 1, thc suboptimal error covariance of tho global model cun be expressed as

wherc

K(k+ I ) is suboptimal Kalmai~ gain and S ( k + 1) i s thc suboptimal covariance of thc rcsidual of the global model in the proposed algorithm. However, the optimal error covariance of the global inodel cm be writtcn as

@(k+ I l k + 1)

=f[(k), *(klk>] - K(k + l)H(k + lY[(k), @(kIk)l

- , f ' [ ( R ) , P(klk)]HT(k f I>KT(k -I- 1)

3- K(k + l ) S ( k 4- I)KT(k + 1) (50)

where

+ G(k f l)Q(k -t I)GT(k+ I )

A@+ 1) is optimal Kalman gain and $(k+ 1 ) is the optimal covariancc of the residual of the global model. It is assumcd that the suboptimal crror covariance is identical with th: optimal one at some instant such that P(k1k) = P(klk). Thcn tlie difference of the crror covariance bctwccn tlie suboptimal and optimal cases can be rcprc- sented as

P(k + I l k + 1) - P(k + Ilk $- I )

From ,eqn. 13, the optimal and the suboptimal Kalman gain, K(k+ 1) and K(R+ l) , in eqn. SI can be written as

K ( k + 1) = f [ ( k ) , P(klk)]H"'(k + l ) k ' ( k + I )

= f [ ( k ) , P(klk)]HT(k + l)S-'(k + 1) (52)

K(k -k I ) = f [ ( k ) , P(klk)]HT(k 4- I)S-'(k f 1) (53) Substituting eqns. 52 nnd 53 into eqn. 5 1, and considering the symmetric property of the error covariance matrix, eqn. 5 1 bccomcs

P(k+ I l k + l )-P(k+ l1k-t- I )

x H(k + I)S[(M, P(klk)l

- S-'(k + l)j['H(k -t l ] j - [ (k) , P(klk)]

=f[(k), P(klk)]HT(k -t l ) [ k ' ( k f 1) - S - ' ( k i- t)]

+ 2f[(k), P(klk)]HT(k + l ) [ k l ( k + 1>

(54)

From the approximation of eqn. 14, the following condi- tion is always satisfied:

P ( k Jr 1) - S - y k + I ) 'I: 0 ( 5 5 )

Therefore, from eqns. 54 and 5 5 , the suboptimal error covariancc o f tlic global modcl in thc proposed algorithm can be rcprcsontcd as

P(k + 1 lk + 1) 2 P(k + Ilk + 1) 0 (56)

This completes the proof.

LEE Pmc.-Conlrol Theory Appl., Yo!. 147, No. 1. Jonumy 200110 103