Article A Hybrid Adaptive Unscented Kalman Filter Algorithm

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1 Preprnts ( NO PEER-REVIEWED Posted: 7 March 27 do:.2944/preprnts v Artcle A Hybrd Adaptve Unscented Kalman Flter Algorthm Jun He, Qnghua Zhang, Qn Hu and Guox Sun * School of Computer and Electronc Informaton, Guangdong Unversty of Petrochemcal echnology, Maomng 525, Chna; hejun_723@26.com (J.H.); fenglangren@vp.tom.com (Q.Z.); qn.hu@gdupt.edu.cn (Q.H.) * Correspondence: guox.sun@gdupt.edu.cn; el.: Abstract: In order to overcome the lmtaton of the tradtonal adaptve Unscented Kalman Flterng (UKF) algorthm n nose covarance estmaton for statement and measurement, we propose a hybrd adaptve UKF algorthm based on combnng Maxmum a posteror (MAP) crteron and Maxmum lelhood (ML) crteron, n ths paper. Frst, to prevent the actual nose covarance devatng from the true value whch can lead to the state estmaton error and arouse the flterng dvergence, a real-tme covarance matrces estmaton algorthm based on hybrd MAP and ML s proposed for obtanng the statement and measurement noses covarance, respectvely; and then, a balance equaton the two nds of covarance matrx s structured n ths proposed to mnmze the statement estmaton error. Compared wth the UFK based MAP and based ML, the proposed algorthm provdes better convergence and stablty. Keywords: hybrd adaptve; unscented alman flterng; maxmum a posteror; maxmum lelhood crteron. Introducton In the state estmaton technques for lnear stochastc systems, the Kalman flter has receved many attenton ndustral areas wdely and acheved some progress[]. However, n real applcaton, t s dffcult to obtan exact unnown nose statstcs nformaton whch may nduce large state estmaton errors and flter dvergence. o mae up for ths defect, a number of modfed flter methods have been publshed to weaen the mpact of uncertanly nose nformaton, and so the adaptaton of nose covarance matrces has become a mportant drecton for developng stablty and convergence Kalman fter[2]. Knds of extend Kalman flter (EKF) have been proposed for solvng the nonlnear problem n stochastc systems. EKF s the frst extend of the KF n hstorcally, whch approxmates the state parameter for nonlnear system by ts frst order lnearzaton [3-4]. Although t s smple and easy to use for nonlnear state estmaton, but ts promnent defects, the non-gaussan of nonlnear system state dstrbutons, have not been taen nto count and the devaton of estmaton mean and covarance for state parameter have been shamefully neglected. Moreover, as the sample number growng the estmaton errors resultng by the lnearzaton of Kalman flter can accumulate [5-6]. In order to reduce the constrants of the frst-order aylor expanson of EKF naturally, as reported n[7-8],a novel adaptve robust flter method whch can approxmate a probablty dstrbuton than to approxmate an arbtrary nonlnear transformaton, named unscented Kalman flter(ukf), has been proposed for state estmaton of nonlnear stochastc systems. he natural advantage of UKF s the thrd-order aylor expanson approxmate used to obtan the condtonal mean and

2 Preprnts ( NO PEER-REVIEWED Posted: 7 March 27 do:.2944/preprnts v 2 of 3 varance for Gaussan noses accurately, whch does not rely on any lnearzaton processng for state predctons and covarance. Consequently, compared wth the robust EKF, the UKF has been used n numerous ndustral applcatons, such as state estmaton and remanng useful lfe predct for rotary machnes [9-], battery health management [2-3], target tracng and satellte-aded navgaton [4-5], among others. However, n real ndustral applcaton, the unnown process nose s frequency encountered, and the accumulated state estmaton errors, n the case of the process nose covarance s not obtaned accurately, become more and more serous. o overcome these unfavorable factors, many researchers have pad ther attentons to two mportant drectons to mprove the stablty and convergence of UKF, and some wors have publshed n the two manly purposes, the one s how to mprove the matchng degree of nose dstrbuton obtaned by UKF and from a real stochastc system; the other s how to adjust the estmaton wndow for the UKF whch the covarance matrx of uncertanly nose s estmated at the present tme. o solve the problem that the naccuracy and dvergence of UKF when the pror nose statstc s unnown, an adaptve UKF algorthm based on maxmzng a posteror estmaton and exponental weghtng was proposed by authors [6]. In order to solve the state estmaton problem of nonlnear systems wthout nowng pror nose statstcal characterstcs, an adaptve UKF based on the maxmum lelhood prncple and expectaton maxmzaton algorthm was proposed[7]. Deng.et al n [8]proposed an adaptve unscented Kalman flter (AUKF) to estmate the tme-varyng parameters and states of a nd of nonlnear hgh-speed objects, whch a strong tracng flter and wavelet transform were employed to mprove the robustness of unscented Kalman flter (UKF) under the process nose s naccuracy, and the estmate accuracy by the varance of measurement nose, respectvely. In [9], an adaptve unscented Kalman flter algorthm wth dynamc thresholds of covarance used to update measurement nose covarance matrx real-tme was developed for satellte fault detecton and dagnoss by the authors. As reported n [2], an adaptve unscented Kalman flter method wth usng adaptve error covarance matchng technology to mprove the state estmaton accuracy was employed to estmate the battery energy and power capablty by authors. A robust Masrelez-Martn UKF whch can provde relable state estmates n the presence of both unnown process nose and measurement nose covarance matrces was presented [2]. In the estmaton process, an adaptve adjustment of the nose covarance usng covarance matchng n the UKF was mplemented by authors [2], whch was used to get a more accurate and robust state of charge estmaton. Meanwhle, Estmaton wndow, s one of the mportant factor whch can mae the stablty and accuracy of UKF be effected, has been concerned by the academc communty, recently [22-23]. hese wndowng approaches were adopted to estmate the unnown nose covarance matrx based on a set of nnovaton sequences n the wndow, nstead of predctng the nose covarance matrx by the hstorcal nnovaton sequences. From the revew of the related wor mentoned above, t can be found that, though there are some reports focusng on robust of adaptve UKF method or wndowng approaches adopted to estmate the unnown nose covarance matrx, separately, few wors concentrate on the adjustment of matchng degree between nose estmaton covarance obtaned by UKF and the truth value from real stochastc system, adaptvely. As stated before, a hybrd adaptve method helps to mae the unnown nose estmaton covarance computed by UKF match the truth value of real stochastc system as much as possble. he hybrd adaptve UKF method based on MAP and ML crteron for nonlnear stochastc system state estmaton contrbutes to mprove the stablty and convergence of

3 Preprnts ( NO PEER-REVIEWED Posted: 7 March 27 do:.2944/preprnts v 3 of 3 state estmaton based on UKF. Accordngly, n ths paper, there are three major contrbutons. Frst, we propose a hybrd adaptve UKF method for uncertanly nose covarance estmaton of a nonlnear stochastc system, and utlze the MAP method to obtan a suboptmal and unbased estmaton for slacng down the mpact of the mendacous and devaton for state estmaton. Second the maxmum lelhood prncple s used to obtan unnown nose estmaton covarance for real nonlnear stochastc system, no matter the MAP method and ML method, there les devaton of estmaton covarance and truth value. Last, the unnown nose covarance s estmated by the two algorthms (MAP and ML), respectvely, and then the two covarance are performed the expectaton of ther sum, whch looed as the new unnown nose estmaton covarance, and the new stablty dscrmnator rule of UKF algorthm s employed. 2. Unscented Kalman Flter Gven a nonlnear stochastc system whch descrbed as follows[24]: x = f (x ) + q () y = h(x ) + r (2) y R represents the measurement parameter, q n m where x R s the state parameter, and r ndcate the process nose and measurement nose, whch can meet the condton of zero mean and covarance beng defned as follows, respectvely. E[ q q ] = Q, E[ r r ] = R (3) All the UKF varants, as n the EKF, eep the structure of the Kalman flter for lnear system. here are two steps for compute the UKF, whch ncludng one predcton (or a pror estmaton) and one correcton (or update) les KF. A) Predct: Gven a set of + sgma ponts whch derved from estmated state, descrbed as blows: = ˆx, λ = ˆx + ( ( n+ ) P ), =,, n = ˆx ( ( n+ λ) P ), = n+,, where ( ( n+ λ) P ) s the th row/column of the matrx square root of ( + ) for n λ P the th tme step. hen the predcted sgma ponts propagated through the transton functon f ( ), and the pror of these predcted sgma ponts namely mean, and covarance are computed by: where m = f( ), =,, = wm = ˆ ˆ = wc( x )( x ) + Q = ˆx, P w represents the weght of th sgma pont state mean, estmaton, whch gven as follows, respectvely. (4) (5) w c s the weght of th covarance

4 Preprnts ( NO PEER-REVIEWED Posted: 7 March 27 do:.2944/preprnts v Scalng factor λ s defned as w w w m = n + λ = w + ( α + β) = wm =, =,, 2( n + λ) 2 c m c 4 of 3 λ = α 2 ( n+ κ) n (7) where α and κ were the scalng factor used to control the spread of the sgma ponts, β descrbes the related dstrbuton of x. B) Update: As well as n the predct step above, gven a set of + sgma ponts, whch used to compute the fltered mean and covarance. = ˆx, λ = ˆx + ( ( n+ ) P ), =,, n = ˆx ( ( n+ λ) P ), = n+,, And then, the observaton functon was used to project these sgma ponts, whch gven as follows: Y = h( ), =,, (9) hen, the predcted measurement mean and covarance were obtaned by recombned the weghted sgma ponts, and the state measurement cross-covarance matrx, all that can be wrtten as follows: wmy = ( ˆ )( ˆ zz = wc ) + = ( ˆ ˆ xz = wc x )( ) = yˆ = () P Y y Y y R P Y y (2) Equaton () and Equaton (2) were used to compute the gan of UKF, descrbed as follows: = (3) G P P xz zz And then, the updated state and updated covarance can be computed wth these formulate descrbed as: 3. Adaptve UKF (6) (8) () xˆ = x ˆ + G ( y yˆ ) (4) P = P G P G (5) zz In ths secton, a hybrd adaptve UKF wll be ntroduced, whch ncorporates the UKF based on MAP and the UKF based on ML, respectvely. Followng, we wll ntroduce the UKF based on MAP, the UKF based on ML crteron and hybrd adaptve UKF, respectvely. 3.. MAP Crteron here are two manly factors whch can lead to the flter dvergence, one s the unreasonable model for a stochastc system, and the other one s naccurate nose statstcal characterstcs, whch

5 Preprnts ( NO PEER-REVIEWED Posted: 7 March 27 do:.2944/preprnts v 5 of 3 have attracted many attentons n how to solve two problems adaptve. MAP, a classcal adaptve statstcal method, employed to develop a suboptmal nose estmator by Sage and Husa n 96s[25]. he estmators ˆQ and to maxmum the a-posteror densty functon, whch descrbed as. ˆR based MAP for Q and R at the step wll be obtaned such as p[ y( ) x( ), Q, R] p[x( ), Q, R] p[x( ), Q, R y( )] = (6) py [ ( )] Because of y ( ) s ndependent of the parameters x( ), Q, R. Furthermore, the state parameter and the covarance of measurement nose are assumed to be ndependent mutually, so that, the estmator based on MAP may be obtaned by maxmzaton the uncondtonal densty functon, and then the Equaton (6) rewrtten as follows. J ( ) = py [ ( ) x( ), QR, ] p[x( )] pqpr [ ] [ ] (7) he pror statstcs for Q and R were ntroduced detaled by Sage and Husa. So, the densty functons p[x( )] and py [ ( ) x( ), QR, ] were: p[x( )] = p[x()] p[x( ) x( ), Q] = 2 2 = p[ ] Q n+ ( 2 π ) exp x( ) f(x( )) x() x( ˆ ) 2 = [ ( ) x( ),, ] = [ ( ) x( ), ] p y Q R p y R = 2 2 Q P ( ) 2 = exp y ( ) h(x( )) m 2 (2 π ) R = R where n and m are the dmensons of state parameter and measurement parameter, respectvely. And then, combnng the densty functon gven n Equaton (7), Equaton (8) and Equaton (9), under the maxmzaton of J( ), the ˆQ and ˆR gven as: ˆ Q = x ˆ f( xˆ ) x ˆ f( xˆ ) (8) (9) (2) = ˆ R (x ˆ ˆ = y h ) y h( x ) (2) Although Sage and Hussas method has many advantages compared wth conventonal flter methods, such as real-tme, smple, et, ts stablty drop sharp under the unnown process nose and measurement nose. Amng at ths problem, adaptve UKF Flterng algorthm based on maxmum a posteror estmaton and exponental weghtng was publshed[6], n whch the fadng factor was ntroduced to substtute the weghtng coeffcent, therefore, the nose statstcs estmator based on fadng factor defned as: qˆ ( ) ˆ ˆ = d q + d x wmf ( ) = rˆ ( ) ˆ ˆ = d r + d y wmh( ) = ˆ MAP ˆ MAP R ( ) ( ˆ )( ˆ = d R + d εε wc y y ) = (22) (23) (24)

6 Preprnts ( NO PEER-REVIEWED Posted: 7 March 27 do:.2944/preprnts v where Qˆ = ( d ) Qˆ + d [ G εε G + P MAP MAP w ( xˆ ) ( xˆ = c ) ] 6 of 3 ε s the resdual of measurement parameter, whch can be computed by ε ˆ = y y, and =,,. he coeffcent d was wrtten as below β = db, =,, n b (26) d = b where b was the fadng factor, whch satsfed < b <, and β was the th weghtng factor, whch defned as β = β b and satsfed β =. Real tme estmaton and correcton for the unnown tme-varyng nose statstcs carred out wth the nose statstcs estmator, whch may overcome the shortcomngs of the tradtonal UKF n the nose statstcs of unnown tme-varyng flter. However, once the ampltude of real tme estmaton and correcton was not controlled well, whch may result n devaton of nose estmaton and ts theoretcal value. In order to mae up for ths defects, the ML prncple and EM crteron were used to employ an adaptve UKF Algorthm [7] ML Crteron ML crteron, a classcal parameter estmaton tool, s used for state parameter estmaton wdely. he estmaton based on ML crteron of state nose covarance Q and measurement nose covarance R was turned nto a new measurement resdual covarance adjustment parameter α problem. Assumng that the new measurement resdual covarance P ε defned: P ε = L+ = (25) P ε and an adaptve = εε (27) L where L s the sze of estmate wndow. Hence, the objectve functon used to approxmate the state nose estmaton covarance Q and the measurement nose covarance R based on ML crteron defned as follows: = ε + ε = L+ = L+ (28) O( α ) P ε P ε where represents the solvng determnant operatng, hen, carryng out dervatve for objectve functon O( α ),whch gven as follows: O( α ) = α P P ε ε = tr Pε ε Pε P ε ε = L+ α α where tr( ) s the trace of matrx operatng. o obtan the real tme measurement nose covarance R, we assumed that the state nose covarance Q s nown, and then the Equaton (29) can be rewrtten as: (29) tr( Pε ( P ) ) ε εε Pε = (3) = L+

7 Preprnts ( NO PEER-REVIEWED Posted: 7 March 27 do:.2944/preprnts v 7 of 3 due to the new measurement resdual covarance P ε,whch s the postve defnte matrx, n all the estmaton wndow, the Equaton (3) always set up. So the real tme state nose estmaton covarance and measurement nose covarance can be obtaned by followng Equatons, respectvely. 3.3 Hybrd Adaptve UKF ˆ R h(x ˆ ) h(x ˆ ) ML = εε wc y y (3) L = L+ = ˆ Q = ( + ( G ) ) ( ˆ )( ˆ ε Gε P w x x ) (32) ML c L = L+ = Wth the pror statstcal characterstcs of the nose were unnown, the estmaton covarance of the nose may be devate from the real value that leads to the ncrease of the state estmaton error, whch s the fundamental reason resultng n dvergence of the flterng. UKF method uses the Kalman flterng framewor. However, Kalman flterng s a growth of memory flterng. At the ntal stage of the flter, the updated covarance P s larger relatvely, hence, t s mportant to tae full use of the measurement nformaton whch has a stronger correcton ablty to mae the estmaton close to truth value fast. Unfortunately, wth the advance of the flterng process, P s gradually reduced, and the gan of the flterng process s gettng smaller and smaller, so that the correcton ablty of the new measurement data for the real state become more and more lttle, whch may be arouse the dvergence of the flterng. Whether the UKF based MAP or based ML crteron, the fnally goal s to obtan the nose state covarance Q and the measurement nose covarance R, whch closes to or converges to the real value. A suboptmal unbased estmator based on MAP, whch sometmes maybe arouse fluctuatons under the correcton ampltude controlled pershng. Nevertheless, the UKF based on ML, whch tunes the nose estmaton nto solvng the expectaton maxmzaton of a log lelhood functon, whch may fall nto local optmum. In order to mprove the stablty of UKF, wth unnown whch the nose state covarance Q and the measurement nose covarance R from the method based on MAP and ML, respectvely, s more close to the truth value. Assumng that the state nose covarance estmaton and the measurement nose estmaton obtaned by MAP and ML, defned as Q ˆ MAP, Rˆ MAP and Q ˆ ML, Rˆ ML respectvely. In order to reduce the nfluence for the flter on the stablty, whch arouse from estmaton based MAP and ML. he mean value of Q ˆ MAP, Qˆ ML and R ˆ MAP, R ˆ ML can obtan by the followng equatons: ˆ MAP ˆ ML ˆ New Q + Q Q 2 ˆMAP ˆML ˆ New R + R R 2 = (33) = (34) When the followng condton Equatons were satsfed, whch defned as follows, we loos the flter stablty and convergence. ˆ New ( ˆ New Q Q ) <Δζ (35) = ˆNew ( ˆ New R R ) <Δζ (36) = where Δ ζ s a pre-gven very small postve nteger. And then, the whole algorthm of ths proposed s ntroduced detaled as below. Algorthm of ths proposed Step: Equaton (4) to Equaton (5) used to calculate the conventon UKF method;

8 Preprnts ( NO PEER-REVIEWED Posted: 7 March 27 do:.2944/preprnts v 8 of 3 Step2:Equaton (22) and Equaton (23) employed to obtan the state nose mean and measurement nose mean, respectvely; Step3:Equaton (25) and Equaton (24) used to compute the state nose covarance Q ˆ MAP and measure covarance R ˆ MAP, Equaton (32) and Equaton (3) are used for gettng the state nose covarance Q ˆ ML and measure covarance R ˆ ML ; Step4: Equaton (33) and Equaton (34), are used to compute the state nose covarance Q ˆ New and measure covarance R ˆ New, at the same tme, the Equaton (35) and Equaton(36) employed to judge the stablty of the flter, f the flter s not convergence then go to Step2; Step5: Fnally, the estmaton results from Step2, Step3 and Step4 are sent to the Step for obtanng the correspondng UKF. 4. Smulaton and Case Study In ths secton, we provde a practcal case study to compare the performance wth the method presented n[6-7]. Subsequently, the results from smulaton are analyzed. A strongly nonlnear Gaussan system model, whch descrbed as follows, employed to verfy the effectveness of ths proposed method..2 x x = x + + q x (37) 2 y = x+ x + r 2 where q and r are the whte Gaussan nose, and the statstcal propertes of constant values defned as follows. q=.2, r =. Q =.6, R =.8 Assumed that the ntal value of the nonlnear system gven as ˆx =., (39) P =. ˆx, q and r are not related to each other. And the state nose statstcs characterstcs are unnown, the ntal state nose estmaton mean and covarance presents as below qˆ =.2, Qˆ =.5 In order to compare the performance of all nds of UKF convenently, the method ntroduced n [6] named Ln s UKF and the method reported n[7] descrbed as Wang s UKF, respectvely. And then, the state nose estmaton covarance ˆQ can be obtaned by usng these methods mentoned afore, the results of each method used to get the state nose estmaton covarance gven as followng Fgure s detaled. (38) (4)

9 Preprnts ( NO PEER-REVIEWED Posted: 7 March 27 do:.2944/preprnts v 9 of 3 Fgure. State estmaton covarance of Ln's UKF Fgure 2. State estmaton covarance of our UKF Fgure 3. State estmaton covarance of Wang s UKF Illustrated n Fgure, Fgure 2 and Fgure 3, the convergence speed of the state nose estmaton covarance of Ln s UKF s slower than that of the other, whch has a defect s when convergence to the optmal value t begns to dverge. he man reason whch leads to ths problem s the napproprate fadng factor adjustment maybe enhance or slacng down the effect for the maxmum a posteror based UKF method n error. Fgure 3 shows the Wang s UKF can qucly converge, however, as tme goes on, there s a suddenly dverge appearance and then slowly converge. he ML and EM based UKF can convergence fast, but t exsts over fttng defect, whch leads to burst dverge. he performance of our UKF for state nose estmaton covarance shows n Fgure 2,

10 Preprnts ( NO PEER-REVIEWED Posted: 7 March 27 do:.2944/preprnts v of 3 the convergence speed of our UKF s between Ln s UKF and Wang s UKF. Nevertheless, n all the computng process, the state nose estmaton covarance can converge to the theoretcal value stablty. Due to n the estmaton process, the mean of state nose estmaton covarance of Ln s and Wang s s computed, whch can avod the defects from a sngle method. On the bass of the above model, assumng that the state nose q and measurement nose r are Gaussan nose and obey Gauss dstrbuton, the state nose statstcs characterstc defne as Equaton (4), however, the measurement nose statstcs characterstc are unnown, but the ntal value of measurement nose statstcs characterstc represented as follows. rˆ =.3 Rˆ =.6 he smulaton results llustrate as Fgure 4, Fgure 5 and Fgure 6. (4) Fgure 4. Measurement nose covarance of Ln s UKF Fgure 5. Measurement nose covarance of our UKF

11 Preprnts ( NO PEER-REVIEWED Posted: 7 March 27 do:.2944/preprnts v of 3 Fgure 6. Measurement nose covarance of Wang s UKF Illustrated n Fgure 4, Fgure 5 and Fgure 6, at ntal phase, due to lac of pror nowledge and the fadng factor adjustng, the measurement nose covarance estmated by Ln s UKF has large fluctuatons, and then t can convergence to the theoretcal value. Descrbed as Fgure 6, n the ntal phase, the estmaton of measurement nose covarance has smaller fluctuatons than that of the other s. As the flter deepen, estmaton of measurement nose covarance can convergence to the theoretcal value accurately. Owng to n the flter process, our UKF maes good useful of the advantages of the Ln s UKF and Wang s UKF whch the mean of measurement nose covarance from Ln s UKF and Wang s UKF used to a new UKF teraton. Fgure 7 Performance of Ln s UKF Fgure 8. Performance of our UKF Fgure 9. Performance of Wang s UKF

12 Preprnts ( NO PEER-REVIEWED Posted: 7 March 27 do:.2944/preprnts v 2 of 3 From the Fgure 7, Fgure 8 and Fgure 9, t s very obvously to now that the performance of our UKF s better than the other two methods. In all the flter process, the state of nonlnear system, whch can approxmaton to theoretcal value. 5. Conclusons From the study of the adaptve unscented Kalman flter, t s very mportant to do best on mprovng the estmaton covarance of state nose and measurement nose precson. In ths paper, we ntroduce a hybrd adaptve UKF method for nonlnear system state estmaton, where the UKF based MAP and ML used to compute the estmaton covarance of state nose and measurement nose, respectvely, whle the hybrd adaptve UKF maes the estmaton covarance of state nose and measurement nose of the nnovaton as small as possble. he basc dea behnd our UKF method s to buld an balance operaton to obtan a closer optmum estmaton. And then employ the convergence strategy, to accomplsh the flter stablty decde. Convergence analyss shows that the proposed method s better than the Ln s and Wang s method, respectvely. Acnowledgments: hs wor was supported by the Natonal Natural Scence Foundaton of Chna (Grant: 633,66734),Guangdong Natural Scence Foundaton (Grant:26A333823). Author Contrbutons: Jun He proposed the deal of ths paper, and wrote the manuscrpt, Qnghua Zhang analyzed the data, and Qn Hu desgned the experments, and Guox Sun put forward amendments to ths manuscrpt. Conflcts of Interest: he authors declare no conflct of nterest. References. F. Auger, et al., "Industral Applcatons of the Kalman Flter: A Revew," Industral Electroncs, IEEE ransactons on, vol. 6, pp , C. D. Zuluaga, et al., "Short-term wnd speed predcton based on robust Kalman flterng: An expermental comparson," Appled Energy, vol. 56, pp , L. Salvatore, et al., "A new EKF-based algorthm for flux estmaton n nducton machnes," Industral Electroncs, IEEE ransactons on, vol. 4, pp , K. Jang, et al., "An extended Kalman flter for nput estmatons n desel-engne selectve catalytc reducton applcatons," Neurocomputng, vol. 7, pp , F. Meurer, et al., "Nonlnear state estmaton for the Czochrals process based on the weghng sgnal usng an extended Kalman flter," Journal of Crystal Growth, vol. 49, pp , J. A. Delgado-Aguñaga, et al., "Mult-lea dagnoss n ppelnes based on Extended Kalman Flter," Control Engneerng Practce. 7. S. J. Juler, et al., "A new approach for flterng nonlnear systems," n Amercan Control Conference, Proceedngs of the 995, 995, pp vol S. Juler, et al., "A new method for the nonlnear transformaton of means and covarances n flters and estmators," Automatc Control, IEEE ransactons on, vol. 45, pp , Q. Song and J.-D. Han, "An Adaptve UKF Algorthm for the State and Parameter Estmatons of a Moble Robot," Acta Automatca Snca, vol. 34, pp , 28.. M.. Sabet, et al., "Extended and Unscented Kalman flters for parameter estmaton of an autonomous underwater vehcle," Ocean Engneerng, vol. 9, pp , 24.

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