Desensitized Cubature Kalman Filter with Uncertain Parameter

Transcription

1 Desestzed Cubature Kalma Flter wth Ucerta Parameter asha Lou School of Electrc ad Iformato Egeerg Zhegzhou Uversty of Lght Idustry Zhegzhou Cha (e-mal: Abstract: A robust desestzed cubature Kalma flterg (DCKF) for olear systems wth ucerta parameter s proposed. Sestvty matrces are defed as the tegral form ad desestzed cost fucto s desged by pealzg the posteror covarace trace by a sestvty-weghtg sum of the posteror sestvtes. he DCKF ga s obtaed by mmzg the desestzed cost fucto to amed the state estmato. he the sestvty propagato of the state estmate errors s descrbed ad the sestvty of the root square matrx s obtaed by solvg a specal equato. he effectveess of the proposed DCKF was demostrated by two umercal smulatos wth ucerta parameters. Keywords: Ucerta Parameters; Desestzed Kalma flter; Cubature Kalma Flter; Sestvty matrx; olear system.. INODUCION Nolear state estmato plays a mportat role a wde varety of applcatos such as target tracg avgato for the aerospace vehcle chemcal plat cotrol mult-sesor data fuso etc. May olear flters based o the Bayesa framewor have bee put forward over four decades. Oe evasble dffculty of the applcato of the Bayesa estmato to the practce problems s that the realstc dyamc systems are ofte olear [ 2]. hese olear flters ca be roughly classfed as three categores. he frst oe s the flters based o the aalytcal approxmato approach (also called fucto approxmato approach) such as the exteded Kalma flter (EKF) ad the secod-order EKF [3]. he secod oe s the flters based o the determstc samplg approach (also called sgma pot Kalma flter) whch cota the usceted Kalma flter (UKF) [4] dvded dfferece flter [4] ad Gauss-Hermte quadrature flter [6]. he thrd oe s the flters based o the Mote Carlo smulato such as sequetal Mote Carlo [7] ad partcle flter [8]. For the frst category calculatg Jacobas or Hessas are ofte umercally ustable ad computatoally tesve the flters. he olear flters the secod category suffer from the curse of dmesoalty or dvergece or both [2]. For the thrd category the flters may solve ay o-gaussa estmato problems but the computatoal load s ofte expesve []. ecetly a olear flter based o the Bayesa framewor whch has bee proposed for the olear Gaussa systems s the cubature Kalma flter (CKF) [9 0]. he core dea of the CKF s usg the cubature role to approxmate the multdmesoal tegrals wth determstc cubature pots where s the state vector dmeso. I the CKF algorthm the mea ad covarace of the state ad measuremet vectors are calculated by propagatg the cubature pots through the olear fucto. Comparso of the EKF ad the UKF the CKF s usually appled the hgh-dmesoal systems ad has more accuracy ad umercal stablty [9]. Subsequetly the CKF s used may applcatos such as avgato [] sesor data fuso [] etc. I these flters above there s a fudametal assumpto that the dyamc models ca be accurately modeled wthout ay uow statstcal propertes of the oses or ucerta parameters [3]. Whe the assumpto s satsfed the performace of these flters wll be hghly sestve to the dyamc model ucertates ad deterorate apprecably [3]. A desestzed optmal cotrol methodology s exteded by Karlgaard ad She [4] to the robust flter desg problem that the performace sestvty of the flters respected to model ucerta parameters ca be reduced. he desestzed Kalma flter s desged by pealzg the cost fucto cosstg of the posteror covarace trace usg a weghted orm of the state error sestvtes ad mmzg ths cost fucto to obta the desestzed state estmates [4]. Subsequetly the desestzed dvded dfferece flterg [4] ad the desestzed usceted Kalma flterg (DUKF) [6] are preseted. For the DUKF there s o cotuous propagato for the sestvtes of the sgma pots betwee the teratos because a ew set of sgma pots s always resampled at the ext terato. She ad Karlgaard [6] sllfully desged a uque way to propagate the sestvtes of the state estmate error ad the a pror/posteror covarace matrces for the DUKF. hs paper proposes a robust cubature Kalma flter for systems wth ucerta parameter. A desestzed cost fucto of the robust desestzed cubature Kalma flterg (DCKF) s desged by pealzg the a posteror covarace trace by a sestvty-weghtg sum of the a posteror sestvtes. he a ga matrx of the DCKF s obtaed by mmzg the ew cost fucto to amed the state estmato. Secto 2 brefly troduces the recursve Bayesa framewor ad the CKF algorthm. Secto 3 presets the propagato of the sestvtes ad the DCKF s proposed aturally. wo umercal smulatos about a

2 vertcally fallg body model ad a hoverg helcopter model (HHM) are aalyzed Secto 4. he coclusos are summarzed Secto CUBAUE KALMAN FILE Cosder the dscrete olear process ad measuremet models wth addtve oses gve by x f( x c u ) w () z hx ( cu ) v (2) where x s the state vector ad z s the m measuremet vector. f s the dyamc vector-valued fucto h s the olear measuremet vector-valued fucto. c s referred to as the ucerta parameter vector. u s the ow cotrol put vector. w ad v are depedet zero-mea Gaussa ose processes ad ther covarace are respectvely Q ad. hey satsfy [ j ] j [ j ] j [ j ] E ww Q E vv E wv 0 (3) where j s the Kroecer delta fucto ad Q s postve sem-defte ad s postve defte. 2. ecursve Bayesa flter uder Gaussa doma Uder the Gaussa doma the predctve desty p( x Z ) the flter lelhood desty p( z Z) ad the a posteror desty p( x Z) are both Gaussa. Whe the a posteror desty p( x ˆ ) N( Z x P) at tme t ad ther dstrbutos respectvely are p( x ˆ Z ) N( x P ) p( z ˆ Z ) N( z Pzz ) (4) p( x Z ) N( xˆ P ) where Z u z s the hstory of put measuremet pars up to tme t N() s the Gaussa desty symbol the superscrpts deote a pror ad deote a posteror. x ˆ x ˆ z ˆ are the state estmates ad the measuremet estmate respectvely; P P P zz are the correspodg error covarace matrces respectvely. he the fuctoal recurso of the Bayesa flter reduces to a algebrac recurso. Oly the meas ad covaraces of varous codtoal destes ecoutered the tme ad measuremet updates are eeded to calculate. he recursve Bayesa flter uder the Gaussa approxmato s summarzed as follows [9]: me update xˆ E[ x Z ] E[ f( x c u ) w Z ] E[ f( x c u ) Z ] (5) f( x c u ) p( x Z ) dx ˆ N( ) d f( x c u ) x P x P E[ xx z: ] f ( x c u ) f ( x c u ) N( xˆ P ) dx xˆ xˆ Q where x ˆ x x s the a pror estmato error ad c s the referece value of the parameter vector c. where Measuremet update (6) zˆ [ z Z ] h( x c u ) ( x P ) x (7) ˆ E N d P ˆ zz h( x ) ( ) ( ) c u h x c u N x P dx zz ˆ xz ( ) ( ) N d (8) P xh x cu x P x xz (9) x [ ˆ x K z z ] (0) P E[ x x ] P P xz K KPxz KPzz K () K Pxz ( P zz ) (2) x xˆ x s the a posteror estmato error. he Kalma optmal ga K s obtaed by mmzg the cost fucto J r( P ) whch r deotes the trace of the matrx ad the result s (2). 2.2 Cubature Kalma flter algorthm he cubature Kalma flter s a type of Bayesa flter uder Gaussa assumpto. he CKF approxmates the mea ad covarace of a radom varable propagated uder a olear fucto by the cubature rule. Uder the Gaussa assumpto the fuctoal recurso of the Bayesa flter reduces to a algebrac recurso whch the multdmesoal tegrals ca be approxmated by the cubature rule [9 7]. he cubature rule approxmates the -dmesoal Gaussa weghted tegral by usg the mea μ ad covarace P. he formulato s f( x) N( μ P) dx ( ) f μ Pξ (3) where P s a square-root factor of the covarace P whch satsfy P= P( P ) ; ξ s the th elemet of cubature pots ad ts value comes from the followg set e e2 e e e2 e (4) where e ( 2 ) s the ut vector whch the th elemet s oe ad others are zeros e =[0 00 0]. Based o the state estmate xˆ ad covarace at tme t the cubature pots are geerated as follows χ ˆ j Pξ j x j 2 (5) Each of the propagated cubature pots s computed through the olear fucto as χ j f ( χ j cu ) j 2 (6)

3 he the a pror state ad the correspodg error covarace (5) ad (6) are evaluated as xˆ χ j (7) j P χ χ x x Q (8) j j j edraw the cubature pots by usg x ˆ ad P χ ˆ j P ξ j x j 2 (9) he by usg (7) to (9) the predcted measuremet ad ts correspodg covaraces are Ζ h( χ cu ) (20) j j zˆ Ζ (2) j j zz j j j P Ζ Ζ zz (22) P χ Ζ xz (23) xz j j j Fally the a posteror estmate ad the assocated covarace are gve by x ( ˆ x K z z ) (24) P P Pxz K KPxz KPzz K (25) K Pxz ( P zz ) (26) where the CKF ga K s obtaed by mmzg ts cost fucto J r( P ). c 3. DESENSIIZED CUBAUE KALMAN FILE I ths secto the robust DCKF s preseted by usg the Sestvty propagato equato ad the CKF method secto 2. he dffculty for the DCKF compared wth the DEKF s the propagato of the sestvtes because the cubature pots are always redraw tme after tme the flterg ad ths maes that there s o cotuty about the cubature pots from oe terato to the ext. So the propagato of the sestvtes s troduced frstly as the lterature [6] ad the DCKF algorthm s summarzed table Sestvty propagato equato for the recursve Bayesa flter Uder the basc assumptos of the recursve Bayesa flter such as o model ad parameter ucertates the state estmate are ubased. It meas that the optmal estmato errors satsfy E[ x] 0 E[ x ] 0 (27) Whe the model parameter has ucertaty the basc assumptos of the recursve Bayesa flter caot be satsfed ad the state estmates may be based ad eve dvergece. So the state estmate error sestvtes of each parameter c could be defed as [4] x ˆ x s (28) c c x ˆ x s (29) c c where c deotes the th compoet of the parameter vector. Note that the sestvty of the true state s x c 0 (28) ad (29) because the true state does t vary wth the assumed value of the parameter vector c. he propagato equatos of the recursve Bayesa flter are defed as f( x c u ) E[ ] c s f( x c u ) N( xˆ P ) dx c (30) s s K γ (3) where the sestvty of the olear measuremet fucto s defed by ( ) ( ) E[ hx c u ] hx c γ u N( xˆ P ) dx (32) c c Note that the sestvty of ga matrx s assumed as K c 0 (3) because ay K c 0 meas that the soluto for the optmal ga s a fucto of the resdual whch volates the bass for the lear update equato gve () [4]. 3.2 Sestvty propagato of cubature pots he propagato of the sestvtes bases o the sestvty of cubature pots whch s obtaed by tag partal dervatve of the cubature pot geerated equato such as (5) ad (9). he propagato equatos of the sestvtes ca be obtaed by tag partal dervatve of the propagato equatos of the CKF such as (6)-(8) ad (20)-(25). he sestvty propagato algorthm of the cubature pots s summarzed able. able. Sestvty propagato algorthm of cubature pots me update () Compute the sestvtes of step cubature pots ( 2 j 2 2 ) χ P ξ (33) j j s c c (2) Propagate the sestvty cubature pots χ j f ( χ j cu ) χ j f ( χ j cu ) (34) c χ j c c (3) Evaluate the sestvtes of the pror state estmate xˆ j s χ (35) c j c (4) Evaluate the sestvtes of the pror covarace matrx

4 P j j χ χ j χ j χ c j c c s x xs (36) Measuremet update (5) Compute the sestvtes of the redraw cubature pots χ j P ξ s (37) c c (6) Evaluate the sestvty of the predcted measuremet cubature pots Ζ j h( χ j cu ) χ j h( χ j cu ) (38) c χ j c c (7) Evaluate the sestvtes of the predcted measuremet zˆ j γ Ζ (39) c c (8) Evaluate the sestvtes of the ovato covarace matrx P zz j j Ζ Ζ j Ζ j Ζ c j c c (40) γ zˆ zˆ γ j (9) Evaluate the sestvtes of the cross-covarace matrx P xz j j χ Ζ j χ j Ζ c j c c (4) s zˆ xˆ γ (0) Evaluate the sestvtes of the posteror state estmate s s K γ (42) () Evaluate the sestvtes of the posteror covarace matrx P P P xz Pxz K K c c c c (43) Pzz K K c he sestvty of the root square matrx P must be computed whe the sestvtes of the cubature pots ad redraw cubature pots are calculated. By tag partal dervatve of equato P= P( P ) whch s dfferet from the equato She ad Karlgaard [6] they satsfy equato P = P ( P) P ( P ) (44) c c c he soluto P c of (44) ca be obtaed as [8] P c 2 c P (45) where s a arbtrary sew symmetrc matrx whch satsfes ad are o-sgular matrx such that P I. 3.3 Desestzed cubature Kalma flter algorthm Uder the recursve Bayesa flter framewor the CKF s troduced the above secto. he DCKF s aturally obtaed based o the CKF by usg the sestvty propagato of cubature pots. Wth the sestvtes of the posteror state estmate able a desestzed cost fucto of the DCKF whch cossts of the posteror covarace ad a weghted orm of the posteror sestvty s pealzed by a sestvty-weghtg sum of the sestvtes s J d r( P ) s W s (46) where W s a symmetrc postve sem-defte weghtg matrx for the th sestvty. Substtutg (25) ad the sestvtes of the posteror state estmate able to (46) tag partal dervatve of J d wth respected to the ga matrx K ad settg the partal dervatve K 0 gves the soluto J d zz xz KP W Kγ γ P W s γ (47) Also the ga K s obtaed by algebracally solvg wth the lear equato (47). he DCKF algorthm s summarzed able 2. able 2. DCKF algorthm me update () Italze the state vector ˆx 0 the auxlary matrx P 0 ad the sestvty parameters s0 ad P0 c 2. (2) Factorze P = ( ) P P (48) (3) Evaluate the cubature pots χ j usg (5) ad the sestvtes χ c usg (45) ad (33). j (4) Evaluate the propagated cubature pots χ j usg (6) ad the sestvtes χ j c usg (34). (5) Estmate the predcted state x ˆ usg (7) ad the pror sestvty s usg (35). (6) Estmate the predcted covarace matrx P usg (8) ad the sestvty P c usg (36). Measuremet update (7) Factorze P = P ( P ) (49) (8) Evaluate the redraw cubature pots χ j usg (9) ad the sestvtes χ j c usg (45)

5 ad (37). (9) Evaluate the propagated cubature pots of measuremet equato Ζ j usg (20) ad the sestvtes Ζ j c usg (38). (0) Estmate the predcted measuremet z ˆ usg (2) ad the sestvty γ usg (39). () Estmate the ovato covarace matrx P zz usg (22) ad the sestvty P zz c usg (40). (2) Estmate the cross-covarace matrx P xz usg (23) ad the sestvty P xz c usg (4). (3) Obta the ga matrx K by solvg (47). (4) Estmate the posteror state x ˆ usg (24) ad the posteror sestvty s usg (42). (5) Estmate the posteror covarace matrx P usg (25) ad the sestvty P c usg (43). 4. NUMEICAL SIMULAION o verfy the effectveess of the proposed DCKF algorthm the prevous secto the vertcal fallg body model wth oe ucerta parameter [8] ad the HHM wth two ucerta parameters are cosdered [2]. he perfect CKF (perf. CKF) ad the mperfect CKF (mp. CKF) are employed to compare the performace of the proposed algorthm. he perfect meas that the true values of the parameters are all ow exactly perfect CKF; he mperfect meas that the mperfect CKF the true values of the ucerta parameters are ot ow ad oly the referece values comg from the prevous experece are ow. to be a zero-mea Gaussa ose wth covarace E[ v ] 0 ft. Here we assumed that there s ucertates the costat parameter c ad the true values of c subjects to a uform dstrbuto c U(3 4 c5 4 c ). he true state ad tal estmates are gve as x(0) 30 ft; x2(0) 2 0 ft/s; x3(0) 0 (52) x ˆ (0) 30 ft; x2(0) 2 0 ft/s; x3(0) 3 0 (53) ad the tal covarace s gve as P 4 0 dag 0ft40ft / s 0 (54) otal tme of the smulato s assumed to be 60s ad the fourth-order uge-kutta method s used to dscretzed the cotuous state equato wth a sample frequecy 0Hz. 200 Mote Carlos are ru to evaluate the performace of the three flters. I the mperfect CKF ad the DCKF the ucerta parameter c s set as the correspodg referece value 4 c 2 0. For the DCKF the sestvty-weghtg matrx s set to W dag (55) 4. Fallg body model wth oe ucerta parameter I ths example we use the perf. CKF the mp. CKF ad the proposed DCKF to estmate the alttude x () t velocty x 2 () t ad costat ballstc coeffcet x () 3 t of a vertcally fallg body wth a hgh velocty [ ]. A tracg radar devce s used to record the rage measuremets betwee the radar ad the fallg body. he system equatos are gve by x () t x2() t w() t 2 x () t x2() t x3()exp{ t x() t c} g w2() t (50) x () t w () t 3 3 where c s a costat whch s the relatoshp betwee the ar desty ad the alttude ad t s referece value s 4 2 c 2 0 g 32.2ft s s the gravtatoal accelerato. w () t s the process ose that affects the th equato wth 2 Ew [ ( t)] 0 ( 23). he dscrete-tme rage measuremet from the radar s gve by Fg.. Sestvtes of the mperfect CKF ad the DCKF for state x he state sestvtes respected to the ucerta parameter c wth logarthmc scales are show Fgs. to 3. Compared to the mperfect CKF the proposed DCKF almost has smaller sestvtes to the ucerta parameter. Fgures 4 to 5 show that the root mea squared errors (MSEs) of the above three flters. Whe the process model s dsturbed by the ucerta parameter the MSE for the DCKF s smaller tha that of the mperfect CKF. I a word the proposed DCKF ca reduce the sestvtes of the state estmate errors respect to the ucerta parameter compared wth the mperfect CKF ad has a better performace tha the mperfect CKF whe the process model parameter s mperfect. z M ( x H) v (5) where M 0 ft s the radar horzotal dstace from the 5 body s vertcal le of fall ad H 0 ft s the radar alttude above the groud level. v s the measuremet ose assumed

6 Fg.2. Sestvtes of the mperfect CKF ad the DCKF for state x 2 Fg.5. x2 MSE Fg.3. Sestvtes of the mperfect CKF ad the DCKF for state x 3 Fg.6. x3 MSE 4.2 Hoverg helcopter model wth two ucerta parameters I ths example the hoverg helcopter model s gve by [2] Fg.4. x MSE c c2 g x x Klqr x (56) z x v (57) where the state vector s x [ x x2 x3 x4] whch x s the horzotal velocty x 2 s the ptch agle of the fuselage ad ts dervatve x 3 ad x 4 s perturbato from a groud pot referece. g s the accelerato from gravty ad ts value s g K lqr s a costat vector gve by Klqr [ ]. v s the measuremet ose. he two ucerta model parameters c ad c 2 are assumed to be uform dstrbutos whch are respectvely c U( ) ad... he tal true state of the HHM s x 0 [ ] (58) ad the tal state of the flter s set as xˆ 0 x 0 wth the tal covarace P ˆ 0 I.

7 I smulato the total tme terval s assumed to be 4s ad the cotuous state equato has bee dscretzed usg the fourth-order uge-kutta method wth a sample frequecy 20Hz. For performace evaluato 200 Mote Carlo rus are performed for three flters. I the mperfect CKF ad the DCKF the two ucerta parameters are set as ther referece values whch are c 0. ad c2 0.. For the DCKF the sestvty-weghtg matrces are set to W W dag (59) he true state ad ts estmates of three flters MSE sestvtes respect to the ucerta parameters ad mea cost fuctos of the three flters are performed the followg. he true state of the HHM ad ts estmates of the mperfect CKF ad the DCKF oe Mote Carlo test are show Fg. 7. he true values of the two parameters ths smulato case are c ad c respectvely. Fgure 8 represets the state sestvtes respect to the ucerta parameter c the mperfect CKF ad the DCKF wth logarthmc scales because the state sestvtes to c 2 are smlar. From Fg. 8 t ca be see that the mperfect CKF almost has a larger sestvtes to the parameter ucertates tha the DCKF. he mea cost fuctos of three flters wth logarthmc scales are show Fg. 9. Here for the dfferet cost fuctos of the three flters the cost fucto of the perfect CKF s Jc r( P ) ad the cost fucto of the mperfect CKF ad the DCKF s computed usg (49). It ca be see that the mea cost fucto of the perfect CKF s the smallest; the mea cost fucto of the DCKF taes secod place ad the mea cost fucto of the mperfect CKF s the largest oe. So the DCKF reduces the sestvtes of the state estmate errors respect to the parameter ucertates ad mmzes the correspodg cost fucto compared wth the mperfect CKF. Fgure 0 shows that the MSEs of the three flter. he MSE for the DCKF s larger tha the perfect CKF ad smaller tha the mperfect CKF whe the process model has the ucerta parameters. hese results demostrate that the DCKF ca effectvely reduce the estmato errors the presece of the ucerta parameters. Fg.9. Mea cost fucto Fg.7. rue values of the HHM ad estmates of the mperfect CKF ad the DCKF

8 Fg.8. Sestvtes of the mperfect CKF ad the DCKF A flter cosstecy test based o the ormalzed mea error (NME) test descrbed referece s troduced to hghlght the effectveess of the covarace estmates accoutg for errors the state estmates. he results of the NME cosstecy test are show Fg.. It ca be see that all the state test statstcs of the proposed DCKF are well below Fg.0.MSE the requred threshold ad ths shows that the DCKF covarace estmate accurately predcts the state estmate errors. But the results of the mperfect CKF dcate that t has a poor cosstecy descrbg the states. hat s to say the DCKF cosstetly provdes excellet state estmates whe the process model parameter has ucertates.

9 Fg.. NME cosstecy test 5. CONCLUSIONS I ths paper the state estmato of olear systems wth ucerta parameter s studed based o the desestzed optmal cotrol methodology. he defto of the state estmate error sestvty to the ucerta parameter s troduced to the recursve Bayesa flter. he robust desestzed cubature Kalma flterg (DCKF) for olear systems wth ucerta parameter uder Bayesa framewor s proposed. he desestzed cost fucto of the DCKF s desged by pealzg the posteror covarace trace by a sestvty-weghtg sum of the a posteror sestvtes ad the ga of the DCKF s obtaed by mmzg the desestzed cost fucto to amed the state estmato. I the DCKF the sestvty propagato of the state estmate errors s descrbed ad a specal equato whch s dfferet from the equato of She ad Karlgaard s solved to obta the sestvty of the root square matrx. wo umercal smulatos are computed to verfy the effectveess of the proposed DCKF. EFEENCES efereces: Daum F.: 'Nolear flters: Beyod the Kalma flter' Aerospace ad Electroc Systems Magaze IEEE (8) pp Arasaratam I. Hay S. Ellott.J.: 'Dscretetme olear flterg algorthms usg Gauss- Hermte quadrature' P Ieee (5) pp Jazws A.H.:'Stochastc processes ad flterg theory'( Academc Press 970 ed.970) 4 Juler S. Uhlma J. Durrat-Whyte H.F.: 'A ew method for the olear trasformato of meas ad covaraces flters ad estmators' Automatc Cotrol IEEE rasactos o (3) pp Norgaard M. Poulse N.K. av O.: 'New developmets state estmato for olear systems' Automatca () pp Ito K. Xog K.: 'Gaussa flters for olear flterg problems' Automatc Cotrol IEEE rasactos o (5) pp Doucet A. Godsll S. Adreu C.: 'O sequetal Mote Carlo samplg methods for Bayesa flterg' Stat Comput (3) pp Gordo N.J. Salmod D.J. Smth A.F.M.: 'A Novel approach to olear/o-gaussa Bayesa state estmato' IEE Proceedgs F (adar ad Sgal Processg) (2) pp Arasaratam I. Hay S.: 'Cubature Kalma flters' Automatc Cotrol IEEE rasactos o (6) pp Ja B. X M. Cheg Y.: 'Hgh-degree cubature Kalma flter' Automatca (2) pp Su. X M.: 'Hypersoc etry vehcle state estmato usg hgh- degree cubature alma flter'. AIAA Atmospherc Flght Mechacs Coferece Atlata GA Fera dez-prades C. Vlà-Valls J. Bayesa olear flterg usg quadrature ad cubature rules appled to sesor data fuso for postog. Commucatos 200 IEEE Iteratoal Coferece o. 200: IEEE. 3 Gelb A.:'Appled optmal estmato'( MI press 974 ed.974) 4 Karlgaard C.D. She H.J.: 'Desestzed Kalma flterg' IE adar Soar & Navgato () pp Karlgaard C.D. She H.J.: 'obust state estmato

10 usg desestzed dvded dfferece flter' Isa (5) pp She H. Karlgaard C.D.: 'Sestvty reducto of usceted Kalma flter about parameter ucertates' IE adar Soar & Navgato (4) pp Ho Y. Lee.: 'A Bayesa approach to problems stochastc estmato ad cotrol' Automatc Cotrol IEEE rasactos o (4) pp Hodges J.H.: 'Some matrx equatos over a fte feld' Aal d Matematca Pura ed Applcata Seres () pp Crassds J.L. Jus J.L.:'Optmal estmato of dyamc systems'( Chapma & Hall\CC Press 202 d ed. ed.202) 20 Smo D.:'Optmal state estmato: Kalma H fty ad olear approaches'( Wley- Iterscece 2006 ed.2006) 2 Madaa. Sgla P. Sgh. Scott P.D.: 'Polyomal-Chaos-Based bayesa approach for state ad parameter estmatos' Joural of Gudace Cotrol ad Dyamcs (4) pp Athas M. Wsher. Bertol A.: 'Suboptmal state estmato for cotuous-tme olear systems from dscrete osy measuremets' Automatc Cotrol IEEE rasactos o (5) pp Lou. Fu H. Zhag Y. Wag Z.: 'Cosder uobservable ucerta parameters usg rado beaco avgato durg Mars etry' Adv Space es (4) pp