Covariance Intersection Fusion Kalman Estimator for the Two-sensor Time-delayed System

Transcription

1 Covarance Interecton Fuon Kalman Etmator for the Two-enor Tme-elaye Sytem Jnfang Lu Department of Automaton Helongang Unverty Harbn Chna Zl Deng Yuan Gao Department of Automaton Helongang Unverty Harbn Chna Abtract For a two-enor lnear crete tme-nvarant tochatc ytem wth tme-elaye meaurement an equvalent ytem wthout meaurement elay obtane by applyng the meaurement tranformaton metho. Ung the moern tme ere analy metho bae on the ARMA nnovaton moel the local teay-tate optmal Kalman etmator are obtane. Then everal fuon Kalman etmator are preente nclung the Kalman fuer weghte by matrce agonal matrce calar an covarance nterecton repectvely. Compare wth other three fuer the covarance nterecton ) fuon Kalman etmator can avo a large computatonal buren an hanle the fuon problem for the ytem wth unknown crocovarance. The accuracy comparon of three weghtng fuer wth the fuer preente. It prove that the accuracy of the fuer hgher than that of each local etmator an lower than that of optmal Kalman fuer weghte by matrce. A Monte-Carlo mulaton example how the accuracy relaton an ncate that the actual accuracy of the fuer may be hgher than thoe of the Kalman fuer weghte by agonal matrce an calar an cloe to that of the Kalman fuer weghte by matrce o t ha goo performance. Keywor- nformaton fuon Kalman etmator; tme-elaye meaurement; covarance nterecton fuon; unknown crocovarance; contency. I. INTRODUCTION Multenor nformaton fuon ha been apple to many fel for t obvou avantage of obtanng more epenable etmator []. Several optmal weghte fuon metho have been preente: the nformaton fuon Kalman etmator weghte by matrce [3] agonal matrce [4] an calar [5] repectvely. All of them nee to compute the crocovarance among local Kalman etmator o a large computatonal buren an complex computaton can t be avoe [67]. But ometme the cro-covarance are unknown or ffcult to compute. So a covarance nterecton ) fuon etmator wth contency an robutne wa preente an evelope n [8-] whch can olve the fue flterng problem wth unknown cro-covarance. It avo the computaton of cro-covarance an gve a upper boun of t actual etmaton error varance whch nepenent of unknown cro-covarance o that t ha the contency an robutne. For the ytem wth unknown cro-covarance an tmeelaye meaurement a two-enor fuon Kalman flter Th work upporte by the Natonal Natural Scence Founaton of Chna ) Automatc Control Key Laboratory of Helongang Unverty Support Program for Young Profeonal n Regular Hgher Eucaton Inttuton of Helongang Provnce 55G3). 586 preente n []. Bae on the moern tme ere metho the two-enor fuon Kalman etmator prector flter an moother) for the ytem wth unknown cro-covarance are preente n [3]. An the accuracy relaton among the local the matrce weghte an the fuon Kalman etmator rgorouly prove n [3]. Then a fat equental covarance nterecton S) Kalman flterng algorthm [4] preente whch how that the mult-enor Kalman fuer can approxmately be cloe to everal two-enor Kalman fuer o the reearch of two-enor Kalman fuer wthout lo of generalty. In th paper for a two-enor ytem wth tme-elaye meaurement whch much more complcate to compute the cro-covarance than general ytem a meaurement tranformaton approach preente to convert the ytem nto an equvalent one wthout meaurement elay [67]. Then the fuon metho apple to obtan the teay-tate Kalman fuer whch can hanle the ytem wth unknown crocovarance but tll wth hgher accuracy comparng wth local etmator. It avo the computatonal of the cro-covarance o the computatonal buren can be reuce gnfcantly. The avantage of the fuer that t ha contency.e. the actual varance for fuer ha a theoretcal upper boun an the accuracy of fuer hgher than that of each local etmator an lower than that of optmal fuer weghte by matrce wth known cro-covarance. II. PROBLEM FORMULATION Coner a two-enor lnear crete tme-nvarant tochatc ytem wth tme-elaye meaurement xt + ) = Φxt ) + Γwt ) ) ) t = H x t τ ) + ξ ) t = ) where t the crete tme τ > the tme elay tep n m of the th enor xt ) R the tate ) t R the h m meaurement wt ) R an ξ t) R are uncorrelate whte noe wth ero mean an varance Q w an Q ξ repectvely. Φ Γ an H are contant matrce wth compatble menon Φ H ) a completely obervable

2 par wth the obervable nex β an Φ Γ ) the completely controllable par. The problem to fn the local teay-tate optmal Kalman etmator = the fue etmator m N < N = N > ) weghte by matrce agonal matrce calar an covarance nterecton repectvely an then to compare the accurace among thee etmator. III. THE LOCAL STEADY-STATE OPTIMAL KALMAN ESTIMATOR Introucng the new meaurement y t ) an meaurement noe v t ) y ) t = t+ τ ) 3) v ) t = ξ t+ τ ) 4) From ) the new meaurement equaton wthout meaurement elay obtane by y ) t = H x) t + v ) t = 5) where the new meaurement noe v t ) whte noe wth ero mean an varance Qv = Q ξ. Denotng the lnear pace panne by the tochatc varable t+ t+ N ) ) a L t+ t+ N) ) an the lnear pace panne by the tochatc varable y t+ N τ) y t+ N τ ) ) a Ly t+ N τ) y t+ Nτ ) ) then the relaton between the two lnear meaurement pace L t+ t+ N ) ) = Ly t+ N τ) y t+ Nτ ) ) 6) So t yel that = N τ ) 7) where the lnear mnmum varance etmator of xt ) bae on L t+ t+ N) ) an N τ ) that of xt ) bae on y t+ N τ ) y t+ N τ ) ). From 7) t obvou that the problem of fnng the local teay-tate optmal Kalman etmator bae on the meaurement t+ t+ N ) ) equvalent to that of fnng the local teay-tate optmal Kalman etmator N τ ) bae on the meaurement y t+ N τ) y t+ N τ ) ). For N > τ N = τ an N < τ the etmator calle the moother flter an prector repectvely. Wth the efnton x ) ) ˆ t t+ N = x t x an x = x t) we have the relaton a x = x N τ ) = 8) Defnng the teay-tate etmator error varance an cro-covarance a P =Ε [ x x ] an P =Ε [ x x ] repectvely. So from 8) we have P = P N τ N τ ) = 9) wth P N τ N τ ) =Ε [ x N τ ) x N τ )]. Defne from 7) an 9) we have k = N τ k = N τ ) = k ) ) P = P k k ) ) wth efnng P k k ) =Ε [ x t t+ k ) x k )]. Hence the problem of fnng the local teay-tate optmal Kalman etmator an the cro-covarance P N ) for the tme-elaye ytem ) an ) equvalent to that of fnng the local teay-tate optmal Kalman etmator k) an the cro-covarance P k k ) for the ytem ) an 5) wthout tme-elay. From ) 3) 4) an 5) we have y t = H I q Φ Γ q w t + t+ 3) ) n ) ) ξ τ ) where q the backwar hft operator. Introucng the leftcoprme factoraton H I q q = A q B q 4) n Φ) Γ ) ) where A q ) an B q ) are polynomal matrce wth the nx form X q ) = X + X q + + Xn q X x n x X = > nx ) A = Im B =. So the local ARMA nnovaton moel obtane a A q y t = D q t = 5) ) ) ) ε) n where D q ) = D + Dq + + Dn q table.e. all ero of et D x ) le oute the unt crcle) D = Im an m the nnovaton proce ε t) R whte noe wth ero mean an varance matrx Q ε D q t = B q w t + A q v t 6) ) ε ) ) ) ) ) 587

3 where D q ) an Q ε can be obtane by Gever-Wouter teratve algorthm [5]. Lemma [63]. For the two-enor ytem ) an 5) the local teay-tate optmal Kalman prector t+ t) gven by where t+ t) = Ψ t t ) + K y t) = 7) p p Ψ = Φ K H 8) p p + H M H M Φ K p = 9) H β M Φ β where Ψ p a table matrx K p the prector gan the T T peuo-nvere of matrx X efne a X + = X X) X. M can recurvely be compute a M =A M A M + D ) n n wth M = < ) M = Im. The precton error varance Σ an the precton error cro-varance Σ atfy the Lyapunov equaton Σ = Ψ ΣΨ + ΓQ Γ + K Q K = ) T T T p p w p v p Σ = Ψ Σ Ψ + ΓQ Γ = ) p p w The k tep local teay-tate optmal Kalman prector k ) gven by t t+ k = t+ k + t+ k k 3) k ) Φ ) The Kalman precton error varance an cro-covarance are gven by k k k ) k k ) ) Φ ΣΦ Φ ΓQwΓ Φ = Pk k = + 4) max k k ) k k ) ) Φ ΣΦ Φ ΓQwΓ Φ = = + 5) wth the efnton ) =Ε [ x k) x k)] an P k k ) =Ε [ x k ) x k )] k k. The local teay-tate optmal Kalman flter t t ) gven by t t) = Ψ t t ) + K y t) = 6) f f + Im QvQ ε Φ M H H K f = 8) β HΦ M β The correponng teay-tate Kalman flterng error varance P =Ε[ x t t) x t t)] an cro-covarance P =Ε[ x t t) x t t)] are gven by P = Ψ PΨ + I K H ) ΓQ Γ I K H ) + K Q K T f f n f w n f f v f P = Ψ PΨ + I K H ) ΓQ Γ I K H ) f f n f w n f 9) = 3) The local teay-tate optmal Kalman moother k ) k > ) gven by where k ) = t t ) + K r) ε t+ r) = 3) r = K r r ) ΣΨ p k = H Q ε 3) ε ) t = y ) t H t t ) 33) The Kalman moothng error varance ) =Ε [ x k) x k)] an cro-covarance P k k ) =Ε [ x t t+ k ) x k )] are gven by k T ) = Σ ) Σ + v) ) > r = Pk k K r H H Q K r k 34) mn k k ) T ) = Σ ) Σ ) > r = K r H H K r k k 35) Lemma [6]. For the two-enor ytem ) an 5) the teay-tate Kalman etmaton error varance ) =Ε [ x k) x k)] an cro-covarance P k k ) =Ε [ x t t+ k ) x k )] are gven by cae. when k < k < Pk k) =Σ k= 36) P k k ) =Σ k = k = 37) k k k) Φ ΣΦ Φ Γ wγ Φ = Pk k) = + Q k 38) where Ψ = I K H ) Φ 7) f n f 588

4 k k ) kk k k k+ r+ r ) = Φ Ψp ΣΦ + Φ Ψp ΓQwΓ Φ r=k k r r + Φ ΓQwΓ Φ k k 39) r = k ) ) kk k ) = p Φ Σ Ψ Φ k r k+ r+ k) ΦΓQwΓΨp Φ r=k + k r r + ΦΓQwΓΦ k k 4) r = k k k ) r r ) = Ψp ΣΦ + ΨpΓQwΓ Φ r = k = k 4) k k k) r r ) = Φ ΣΨp + Φ ΓQwΓ Ψp r = cae. when k k k k = 4) P) = P P ) = P 43) k = Σ Σ + v r = Pk k) K r)[ H H Q] K r) 44) k k r = n r p n p r= = P k k ) [ I δ K r) HΨ ] Σ [ I δ K ) HΨ ] mn k k) r p w p 45) r= u= u u + K ) rhψ ΓQΓ Ψ H K ) r cae 3. when k < k k k k) k k ) ) = Φ ΣΨ p Φ ΣΨ p ) = H K k k k r r r + r) ΦΓ wγψp ΦΓ wγψp r= = r= 46) + Q + Q H K ) cae 4. when k k < k k k ) ) ) k k ) = Ψp ΣΦ K ) HΨp ΣΦ = k k k r r + r r ΨpΓQwΓ Φ K ) H Ψp ΓQwΓ Φ r= = r= 47) + + IV. THE COMPARISON OF THE SEVERAL FUSION METHODS A. Optmal fuon Kalman etmator weghte by matrce For the two-enor ytem ) an 5) the optmal fuon Kalman etmator weghte by matrce [3] gven by t t+ = Ω + Ω m = Ω k ) + Ω k ) 48) where the weghte matrx gven by = 49) [ Ω Ω ] [ e P ) e] e P ) P = P ) = P k k )) = 5) where e = [ I I ] P ) m N gven a n n n n n n an the fue error varance matrx = 5) P [ e P ) e] m B. Optmal fuon Kalman etmator weghte by agonal matrce The optmal fuon Kalman etmator weghte by agonal matrce [4] gven by t t+ = A + A = A k ) + A k ) 5) where the weghte agonal matrce A N ) are gven by A = ag a a ) = 53) n t t+ an can be gven n component form ˆ t t+ x t t+ ) ˆ t t+ N = x t t+ = 54) ) ˆ n t t+ N xn t t+ So the r th component of can be expree by ˆ r r r = = a x r = n 55) where the optmal weghte coeffcent vector rr e P ) ar [ a r ar ] rr e P ) e 56) where [ ] e = an efnng rr) rr) rr P P P = rr) rr) 57) P P where P rr ) N ) the rrth ) agonal component of P k k ) an the optmal component fue error varance 589

5 Pr N ) an the optmal fue error varance P N ) are gven by rr r = = = = P [ e P ) e ] P A P k k ) A C. Optmal fuon Kalman etmator weghte by calar 58) The optmal fuon Kalman etmator weghte by calar [5] gven by = b + b = b k ) + b k ) 59) the optmal weghte coeffcent bn ) [ b b ] gven a where [ ] e = an tr tr Ne ) = ep bn ) = ep 6) P N tr P tr P N tr ) = )) = tr P tr P where = k k ) =. 6) The correponng optmal fue etmaton error varance P N ) gven by = = P = b b P 6) D. fuon Kalman etmator From the extng reult t well known that the computaton of the cro-covarance between local flterng very complex an the compute buren very large. So a fuon metho [8-] wa preente to avo th hortcomng. Coner a two-enor ytem ) ) an 5) the fuon Kalman etmator only epene on the varance P N ) an P N ) can be gven a = P [ ω P ) + ω ) P ) ] 63) where the fuon error varance matrx P N ) gven by P = [ ω P ) + ω ) P ) ] 64) wth P = ) =. The weghte coeffcent ω [] obtane by mnmng the performance nex J=mntr P = mn tr{[ ω P ) ω ω [] + 65) ω ) P ) ] } where the notaton tr enote the trace of matrx an the optmal weghtng coeffcent ω can be olve by the gol ecton metho or Fbonacc metho []. Theorem. The actual fuon error varance matrx N ) gven by P P = P [ ω P ) + ω ω ) P ) P P ) + ω ω ) P ) P P ) + ω ) P ) ] P 66) Proof. From 64) we have t yel P [ ω P ) + ω ) P ) ] = I 67) xt ) = P [ ω P ) xt ) + ω ) P ) xt )] 68) Subtractng 63) from 68) we get the actual fuon error x t t+ = P [ ω P ) x t t+ + ω ) P ) x ] 69) Then ubttutng 69) nto the efnton P =Ε [ x x ] yel the actual fuon error varance.e. 66) hol. The proof complete. Theorem. For local an fue Kalman etmator we have the accuracy relaton Pm P Pm P Pm P Pm P = 7) P P 7) tr P tr P tr P tr P = 7) m tr P tr P tr P tr P = 73) m Proof. From 48) m the unbae lnear mnmum varance ULMV) etmate of xt ) bae on the lnear pace L m panne by ˆ x ). From 5) 59) 63) we have θ Lm θ = whch yel 7). It ha been proven P P an tr Pm tr P tr P tr P = n the reference [386] o we have 7) an 7) hol. In the performance nex 65) takng ω = we have J = tr P an takng ω = we have J = tr P. Hence the optmal weghtng coeffcent ω [] yel tr P tr P tr P tr P.e. n 59

6 tr P tr P. Takng trace operaton for 7) an 7) we have 73) hol. The proof complete. V. SIMULATION EXAMPLE Coner the two-enor trackng ytem wth tme-elaye meaurement T.5T xt + ) = xt ) + wt ) T 74) ) t = H x t τ ) + ξ ) t = 75) H = [ ] H = τ = τ = 76) where xt ) R the tate t ) the meaurement wth elay for enor wt ) an ξ t) are whte noe wth ero mean an varance Q w an Q ξ repectvely. Accorng to the correponng tranformaton we obtan the meaurement y ) t wthout tme elay a y t) = t+ τ ) = Hx t) + v t) where v t) = ξ t+ τ ) the tranforme meaurement whte noe wth ero mean an varance Qv = Q ξ. Thu the problem of fnng the local teay-tate optmal Kalman etmator N = ) bae on the meaurement t ) converte nto the problem of fnng the local teay-tate optmal Kalman etmator N τ ) bae on the new meaurement y t ) repectvely. Further applyng the Kalman flterng metho yel the teay-tate optmal fuon Kalman etmator m weghte by matrce agonal matrce an calar an fue Kalman etmator. For N = N = an N = the local an fue etmator are calle the correponng prector flter an moother repectvely. In mulaton we take T =. Q w =.8 Q ξ =.5 Q ξ =.4. In orer to gve a powerful geometrc nterpretaton wth repect to accuracy relaton of local an fuer the covarance ellpe for a covarance matrx P efne a the T locu of pont { x: x P x = c} where c a contant. In the equel c = wll be aume wthout lo of generalty. It wa prove [] that P a P b equvalent to that the ellpe for P a encloe n the ellpe for P b.the mulaton reult are hown n Fg Fg 3. From Fg - Fg 3 applyng 7) an 73) t clearly hown that the covarance ellpe for Pm N ) le wthn the nterecton of ellpe for P N ) an P ) N the ellpe for fue theoretcal covarance P N ) encloe the nterecton regon of ellpe for P N ) an P ) N an pae through the four pont of nterecton of ellpe for ) P N an P ) N. The ellpe for fue actual covarance P N ) le wthn the ellpe for theoretcal covarance P N ) t/tep Fgure. The accuracy comparon of local an fue prector P ) P ) t/tep P ) Fgure. The accuracy comparon of local an fue flter ) P P ) P ) P m ) P P m ) ) P ) P ) m P ) ) P P ) ) P P ) P ) P ) P ) P ) P ) t/tep Fgure 3. The accuracy comparon of local an fue moother 59

7 ) ) ) ) ) ) ) m ) ) ) ) ) ) m ) Fgure 4. Comparon of MSE curve for local an fue Kalman prector ) Fgure 6. Comparon of MSE curve for local an fue Kalman moother In orer to verfy the above theoretcal reult for accuracy relaton Monte-Carlo run are performe for t = 3 the reult are hown n Table I - Table III an Fg 4- Fg 6. TABLE I. ACCURACY OF LOCAL AND FUSED KALMAN PREDICTORS ) ) m ) ) ) ) ) ) ) ) ) ) m ) TABLE II. ACCURACY OF LOCAL AND FUSED KALMAN FILTERS ) ) m) ) ) ) ) Fgure 5. Comparon of MSE curve for local an fue Kalman flter TABLE III. ACCURACY OF LOCAL AND FUSED KALMAN SMOOTHERS ) ) m) ) ) ) ) where the traght lne enote N ) = ) f N ) f = m ) an N ) the curve enote the correponng mean quare error MSE t ). From Fg - Fg 6 an Table I - Table III we ee that the fuer ha goo performance whoe accuracy hgher than that of each local etmator an cloe to the accuracy of the optmal etmator weghte by matrce becaue MSE t ) cloe to MSE ) m t. Remark: In th mulaton example we ee the ellpe of P N ) P N ) an P N ) have nterecton whch mean 59

8 that thee ellpe o not have neceary encloe relaton. An we ee that the accuracy N ) of the fuer hgher than the accurace N ) an N ). However n ome mulaton example the accurace N ) an N ) are hgher than the accuracy N ). VI. CONCLUSION For the two-enor ytem wth tme-elaye meaurement an wth unknown cro-covarance a fuon Kalman etmator preente whch can avo the complex computaton of the cro-covarance between local etmator. It prove that t accuracy hgher than that of each local etmator an lower than that of optmal fuer weghte by matrce. A two-enor ytem Monte-Carlo mulaton example how the accuracy relaton among the optmal fuon Kalman etmator weghte by matrce agonal matrce calar an the fuon Kalman etmator.e. the accuracy of Kalman fuer hgher than that of each local Kalman etmator an lower than but cloe to that of the Kalman fuer weghte by matrce. ACKNOWLEDGMENT The author thank to the upport from Natonal Natural Scence Founaton of Chna ) Automatc Control Key Laboratory of Helongang Unverty Support Program for Young Profeonal n Regular Hgher Eucaton Inttuton of Helongang Provnce 55G3). REFERENCES [] D. L. Hall J. Llna An ntroucton to multenor ata fuon Proceeg of the IEEE vol. 85 p [] Y. Bar-Shalom X. R. L Etmaton wth applcaton to trackng an navgaton Thagalngam Krubaraan John Wley & Son Inc. [3] S. L. Sun Z. L. Deng Mult-enor optmal nformaton Kalman flter Automatca vol. 4 pp [4] Z. L. Deng Y. Gao C. B. L G. Hao Self-tunng ecouple nformaton fuon Wener tate component flter an ther convergence Automatca vol. 44 pp [5] S. L. Sun Mult-enor nformaton fuon whte noe flter weghte by calar bae on Kalman prector Automatca vol. 4 pp [6] X. J. Sun Z. L. Deng Informaton fuon Wener flter for the multenor multchannel ARMA gnal wth tme-elaye meaurement IET Sgnal Proceng vol. 3 no. 5 pp [7] X. J. Sun Y. Gao Z. L. Deng C. L J. W. Wang Mult-moel nformaton fuon Kalman flterng an whte noe econvoluton Informaton Fuon vol. pp [8] J. Juler J. K. Uhlman Non-vergent etmaton algorthm n the preence of unknown correlaton Proceeng of the IEEE Amercan Control Conference Albuquerque NM USA p [9] P. O. Arambel C. Rago R. K. Mehra Covarance nterecton algorthm for trbute pacecraft tate etmaton Proceeng of the Amercan Control Conference Arngton VA p [] S. J. Juler J. K. Uhlman General ecentrale ata fuon wth unknown cro covarance Proceeng of the SPIE Aeroence Conference SPIE p [] S. J. Juler J. K. Uhlman General ecentrale ata fuon wth covarance nterecton n: M. E. D.L. Lggn Hall J. Llna E.) Hanbook of Multenor Data Fuon Theory an Practce Secon Eton CRC Pre 9 pp [] W. J. Q Y. Gao Z. L. Deng Covarance Interecton Fuon Kalman Flter for Sytem wth Delaye Meaurement Scence Technology an Engneerng vol. no. 3 pp [3] Y. Gao Z. L. Deng Covarance Interecton Fuon Kalman Etmator Apple Mechanc an Materal vol. -6 pp [4] Z. L. Deng P. Zhang W. J. Q J. F. Lu Y. Gao Sequental covarance nterecton fuon Kalman flter Informaton Scence vol. 89 pp [5] M. Gever W. R. E. Wouter An nnovaton approach to the cretetme tochatc realaton problem Quarterly Journal on Automatc vol. 9 pp [6] L. J. Chen P. O. Arambel R. K. Mehra Etmaton uner unknown correlaton: Covarance nterecton revte IEEE Tran Automatc Control vol. 47 pp