Chapter 2 Robust Covariance Intersection Fusion Steady-State Kalman Filter with Uncertain Parameters

Transcription

1 Chapter 2 Robust Covarance Intersecton Fuson Steady-State Kalman Flter wth Uncertan Parameters Wenjuan Q, Xueme Wang, Wenqang Lu and Zl Deng Abstract For the lnear dscrete tme-nvarant system wth uncertan parameters and known nose varances, a robust covarance ntersecton (CI) fuson steady-state Kalman flter s presented by the new approach of compensatng the parameter uncertantes by a fcttous nose. Based on the Lyapunov equaton approach, t s proved that for the prescrbed upper bound of the fcttous nose varances, there exsts a suffcently small regon of uncertan parameters; such that ts actual flterng error varances are guaranteed to have a less-conservatve upper bound. Ths regon s called the robust regon. By the searchng method, the robust regon can be found. Its robust accuracy s hgher than that of each local robust Kalman flter. A Monte-Carlo smulaton example shows ts effectveness and the good performance. Keywords Covarance ntersecton fuson Robust Kalman flter Uncertan parameters Fcttous nose approach Robust regon 2.1 Introducton Multsensor nformaton fuson Kalman flterng has been appled to many felds, such as sgnal processng, data fuson, and target trackng. For Kalman flterng fuson, there are two basc fuson methods: The centralzed and dstrbuted fuson methods. For the dstrbuted fuson method, the three-weghted state fuson approaches weghted by matrces, dagonal matrces, and scalars have been presented. In order to compute the weghts, the cross-covarances among the local flterng errors are requred. However, n many practcal applcatons, the computaton of the cross-covarance s very dffcult [1]. In order to overcome ths lmtaton, the covarance ntersecton fuson algorthm has been presented [2]. W. Q X. Wang W. Lu Z. Deng (&) Department of Automaton, Helongjang Unversty, Harbn, Chna e-mal: dzl@hlju.edu.cn Sprnger-Verlag Berln Hedelberg 2015 Z. Deng and H. L (eds.), Proceedngs of the 2015 Chnese Intellgent Automaton Conference, Lecture Notes n Electrcal Engneerng 336, DOI / _2 13

2 14 W. Q et al. In ths paper, a robust CI fuson steady-state Kalman flter s presented for system wth uncertan parameters and known nose varances. Two mportant approaches used to develop the robust Kalman flter are the Rccat equaton approach [3] and the lnear matrx nequalty (LMI) approach [4]. More research references on ths topc are usng these two approaches; however, n ths paper, a new approach s presented by compensatng the uncertan parameters by a fcttous nose whch converts the system wth uncertan parameters nto the system wth nose varance uncertantes [5]. Ths paper extends the robust CI fuson Kalman flter wth uncertan nose varances [5] to the robust CI fuson Kalman flter wth uncertan parameters. Compared wth the suboptmal Kalman flter wthout fcttous nose, the proposed robust Kalman flter can sgnfcantly mprove the flterng performance, and ts robust accuracy s hgher than that of each local robust Kalman flter. 2.2 Local Robust Steady-State Kalman Flter Consder the multsensor uncertan system wth model parameters uncertantes xtþ ð 1 Þ ¼ ðu e þ DUÞxt ðþþcwðþ t y ðþ¼h t xt ðþþv ðþ; t ¼ 1;...; L ð2:1þ ð2:2þ where t s the dscrete tme, xðtþ 2R n s the state to be estmated, y ðtþ 2R m s the measurement of the th subsystem, wðtþ 2R r ; v ðtþ 2R m are uncorrelated whte noses wth zero means and known varances Q and R, respectvely. Φ e, Γ, and H are known constant matrces wth approprate dmensons. L s the number of sensors. U ¼ U e þ DU s uncertan transton matrx, ΔΦ s the uncertan parameter dsturbance. Assume that Φ and Φ e are stable matrces. ξ(t) s a uncertan fcttous whte nose wth zeros mean and upper-bound varance Δ ξ > 0, whch s used to compensate the uncertan model parameter error term ΔΦx(t) n(2.1), so that the systems (2.1) and (2.2) wth uncertan model parameters can be converted nto the followng worst-case conservatve system wth known model parameters and nose varances Q, R, and Δ ξ. x e ðt þ 1Þ ¼ U e x e ðþþw t e ðþ; t w e ðþ¼cw t ðþþn t ðþ t y e ðþ¼h t x e ðþþv t ðþ; t ¼ 1;...; L ð2:3þ ð2:4þ Assume that each conservatve subsystem s completely observable and completely controllable. The conservatve local steady-state optmal Kalman flters are gven as

3 2 Robust Covarance Intersecton Fuson Steady-State Kalman Flter 15 ^x e ðtjtþ ¼ W ^x e ðt 1jt 1ÞþK y e ðþ t ð2:5þ W ¼ ½I n K H ŠU e ; K ¼R H T H R H T 1; þ R P ¼½I n K H ŠR ð2:6þ where I n s an n n dentty matrx, Ψ s a stable matrx, and Σ satsfes the steadystate Rccat equaton h R ¼ U e R R H T ðh R H T þ R Þ 1 H R U T e þ CQCT þ D n ð2:7þ where the symbol T denotes the transpose. Defne ~x e ðtjtþ ¼ x e ðþ ^x t e ðtjtþ, applyng (2.3) and (2.5), we have ~x e ðtjtþ ¼ W ~x e ðt 1jt 1Þþ½I n K H ŠCwðt 1Þ þ ½I n K H Šnðt 1Þ K v ðþ t ð2:8þ Applyng (2.8) yelds that the conservatve local flterng error varances P and cross-covarance P j satsfy the conservatve Laypunov equaton P j ¼ W P j W T j þ ½I n K H ŠCQC T T I n K j H j TþK þ ½I n K H ŠD n I n K j H j R j Kj T d j; ; j ¼ 1;...; L ð2:9þ where δ j s the Kronecker δ functon, δ =1,δ j =0( j). Remark 2.1 Notce that n (2.5), the conservatve measurements y e (t) are unavalable, only the actual measurements y (t) are known. Therefore, replacng the conservatve measurements y e (t) wth the known actual measurements y ðþ,we t obtan the actual local Kalman flters as ^x ðtjtþ ¼ W ^x ðt 1jt 1ÞþK y ðþ t ð2:10þ From (2.10) and (2.11) we have ~x ðtjtþ ¼ W ~x ðt 1jt 1Þþ½I n K H ŠDUxðt 1Þ þ ½I n K H ŠCwðt 1Þ K v ðþ t ð2:11þ So the actual local flterng error varances and cross-covarance are gven as P j ¼ W P j W T j þ ½I n K H ŠDUXDU T T I n K j H j þ ½I n K H ŠCQC T Tþ I n K j H j ½ In K H ŠDUC j W T j þ W C T Tþ DUT I n K j H j K R j Kj T d j ð2:12þ

4 16 W. Q et al. where X ¼ E xt ðþx T ðþ t ; C ¼ E xt ðþ~x T ðtjtþ. From (2.1), X satsfes the followng Lyapunov equaton X ¼ UXU T þ CQC T ð2:13þ Applyng (2.1) and (2.11), we have the Lyapunov equaton C ¼ UC W T þ UXDU T ½I n K H Š T þcqc T ½I n K H Š T ð2:14þ Lemma 2.1 [6] Consder the Lyapunov equaton wth U to be a symmetrc matrx P ¼ FPF T þ U ð2:15þ If the matrx F s stable (all ts egenvalues are nsde the unt crcle) and U s postve (sem)defnte, then the soluton P s unque, symmetrc, and postve (sem-)defnte. Theorem 2.1 For uncertan systems (2.1) and (2.2) wth uncertan parameters, the actual local steady-state Kalman flter (2.10) s robust n the sense that there exsts a regon < ðþ DU, such that for all admssble uncertan model parameter DU 2<ðÞ DU, the correspondng actual flterng varances P have the upper-bound P,.e., P \P ð2:16þ and < ðþ DU s called the robust regon of the local robust Kalman flter (2.10). Proof Defne DP ¼ P P, subtractng (2.12) from (2.9) yelds DP ¼ W DP W T þ U ðduþ ð2:17þ U ðduþ ¼½I n K H ŠD n ½I n K H Š T ½I n K H ŠDUXDU T ½I n K H ½I n K H ŠDUC W T W C T DUT ½I n K H Š T ð2:18þ From (2.6) yelds I n K H ¼ P R 1, so we have det½i n K H Š ¼ det P det R 1 6¼ 0; ½I n K H Š s nvertble. Snce D n [ 0, then U 0 ¼ ½I n K H ŠD n ½I n K H Š T [ 0. Accordng to the property of the contnuous functon, as ΔΦ 0, we have U ðduþ!u 0 [ 0. Hence there exsts a suffcently small regon < ðþ DU, such that for all admssble DU 2< ðþ DU,wehaveU ðduþ [ 0. Applyng Lemma 2.1 yelds DP [ 0,.e., (2.16) holds, and < ðþ DU s called the robust regon of uncertan parameters for the local robust Kalman flter (2.10). The proof s completed. Š T

5 2 Robust Covarance Intersecton Fuson Steady-State Kalman Flter Robust CI Fuson Steady-State Kalman Flter For multsensor uncertan tme-nvarant systems (2.1) and (2.2), the robust steadystate CI-fused Kalman flter s presented as X L ^x CI ðtjtþ ¼ P CI ¼1 x P 1 ^x ðtjtþ ð2:19þ P CI ¼ " # 1 XL ; XL ¼1 x P 1 ¼1 x ¼ 1; x 0 ð2:20þ The optmal weghtng coeffcentsω are obtaned by mnmzng the performance ndex 8 " # 1 9 < X L = J ¼ mn trp CI ¼ mn tr x P 1 x ð2:21þ x 2 ½0; 1Š : ; ¼1 x 1 þþx L ¼ 1 The actual error varances are gven as P CI ¼ P CI " # X L ¼1 X L j¼1 x P 1 P j P 1 j x j P CI ð2:22þ It s proved that [7] the local robustness (2.16) yelds the robustness of the CI fuser for all DU 2< CI DU ¼ TL ¼1 < ðþ DU P CI P CI ð2:23þ where the symbol denotes the ntersecton of sets. Theorem 2.2 [8] The local and CI fuson robust Kalman flters have the followng robust accuracy relatons trp \trp ; ¼ 1;...; L trp CI trp CI trp ; ¼ 1;...; L ð2:24þ ð2:25þ Remark 2.2 Takng the trace operaton for (2.16), we have trp \trp ; ¼ 1;...; L. The trace trp s called robust accuracy or global accuracy of a robust Kalman flter, the trace trp s called ts actual accuracy. Theorem 2.1 shows that the robust accuracy of the CI fuson Kalman flter s hgher than that of each local robust Kalman flter. The actual accuracy of the local or CI fuser s hgher that ts robust accuracy.

6 18 W. Q et al. Fg. 2.1 The robust regon of the local robust Kalman flter ^x 1 ðtjtþ detu δ detu t/step 2.4 Smulaton Example Consder a 2-sensor tme-nvarant system wth uncertan model parameters xðt þ 1Þ ¼ðU e þ DUÞxðtÞþCwðtÞ ð2:26þ y ðtþ ¼H xðtþþv ðtþ; ¼ 1; 2 ð2:27þ 0:43 0:32 In the smulaton, we take U e ¼ ; DU ¼ 0 0 ; C ¼ 1 ; H 0: d 0 1 ¼ ½1 0Š; H 2 ¼ I 2 ; Q ¼ 1; R 1 ¼ 1; R 2 ¼ dagð6; 0:36Þ; d s the uncertan parameter. The smulaton results are shown n the followng. From Fgs. 2.1 and 2.2, the necessary and suffcent condton of U ðdþ [ 0 s that det U ðdþ [ 0, so we can Fg. 2.2 The robust regon of the local robust Kalman flter ^x 2 ðtjtþ detu2 detu δ t/step

7 2 Robust Covarance Intersecton Fuson Steady-State Kalman Flter 19 Table 2.1 The robust and actual accuracy comparson of trp and trp, =1,2,CI trp 1 ðtrp 1 Þ trp 2 ðtrp 2 Þ trp CI ðtrp CI Þ (0.7412) (1.1673) (0.6299) Fg. 2.3 The covarance ellpses of the local and CI fuson robust Kalman flters P 1 P P CI P CI P 2 P obtan that the robust regon of ^x 1 ðtjtþ s < ð1þ d : 0:72\d\0:43, the robust regon of ^x 2 ðtjtþ s < ð2þ d : 0:73\d\0:48, so the robust regon of CI fuser s < CI d : 0:72\d\0:43. When δ = 0.2 n the robust regon, the traces comparsons of the conservatve and actual flterng error varances are gven n Table 2.1, whch verfy the accuracy relatons (2.24) and (2.25). In order to gve a geometrc nterpretaton of the matrx accuracy relatons, the ¼< ð1þ d \< ð2þ d covarance ellpse of varance Ps defned as the locus of ponts x : x T P 1 x ¼ c, where P s n n the varance matrx and x R n and cs a constant. Generally, we select c = 1 wthout loss of generalty. It has been proved n [8] that P 1 P 2 s equvalent to that the covarance ellpse of P 1 s enclosed n that of P 2. The matrx accuracy relatons are gven based on the covarance ellpses as shown n Fg From Fg. 2.3, we see that the ellpse of P s enclosed n that of P, the ellpse of P CI s enclosed n that of P CI. These verfy that the accuracy relatons (2.16) and (2.23) hold. In order to verfy the above theoretcal results for the accuracy relaton, takng the Monte-Carlo smulaton wth 1,000 runs, the mean-square error (MSE) curves of the local and CI-fused Kalman flters are shown n Fg. 2.4; we see that the values of the MSE ðtþ, =1,2,CI are close to the correspondng trp and the accuracy relatons ( ) hold.

8 20 W. Q et al trp 2 1 MSE trp 1 trp CI t / step MSE1 MSE2 MSECI Fg. 2.4 The MSE curves of the local and CI fuson robust Kalman flters 2.5 Concluson For multsensor systems wth uncertan parameters and known nose varances, the local and CI-fused steady-state Kalman flters are presented by the new approach of compensatng the parameters uncertantes by a fcttous nose. It s proved that the local and CI-fused Kalman flters are robust for all admssble uncertan parameters n the robust regon, ths s, the actual flterng error varances have a less-conservatve upper bound, and the robust accuracy of the CI fuser s hgher than those of the local robust Kalman flters. When the fcttous nose varance s prescrbed, by the searchng method, the robust regon can be found. Acknowledgments Ths work s supported by the Natonal Natural Scence Foundaton of Chna under Grant NSFC and NSFC References 1. Sun XJ, Gao Y, Deng ZL, L C, Wang JW (2010) Mult-model nformaton fuson Kalman flterng and whte nose deconvoluton. Inf Fuson 11: Juler SJ, Uhlmann JK (2007) Usng covarance ntersecton for SLAM. Robot Auton Syst 55 (1): Qu XM, Zhou J (2013) The optmal robust fnte-horzon Kalman flterng for multple sensors wth dfferent stochastc falure rates. Appl Math Lett 26(1): Yang F, L Y (2012) Robust set-membershp flterng for systems wth mssng measurement: a lnear matrx nequalty approach. IET Sgnal Process 6(4): Q WJ, Zhang P, Feng WQ, Deng ZL (2013) Covarance ntersecton fuson robust steady-state Kalman flter for multsensor systems wth unknown nose varances. Lect Notes Electr Eng 254:

9 2 Robust Covarance Intersecton Fuson Steady-State Kalman Flter Kalath T, Sayed AH, Hassb B (2000) Lnear Estmaton. Prentce Hall, New York 7. Deng ZL, Zhang P, Q WJ, Gao Y, Lu JF (2013) The accuracy comparson of multsensor covarance ntersecton fuser and three weghtng fusers. Informaton Fuson 14: Q WJ, Zhang P, Deng ZL (2014) Robust weghted fuson Kalman flters for multsensor tmevaryng systems wth uncertan nose varances. Sgnal Process 99:

10