Outlier-Tolerant Kalman Filter of State Vectors in Linear Stochastic System

Transcription

1 (IJCS Internatonal Journal of dvanced Computer Scence and pplcatons, Vol., No., Outler-Tolerant Kalman Flter of State Vectors n Lnear Stochastc System HU Shaoln State Key Laboratory of stronautcs an, 74, P.O.Box 55-6, Chna; Huajang Ouyang School of Engneerng, The Unversty of Lverpool Lverpool, L69 GH, UK; Karl Mene CSC School, Royal Insttute of Technology SE- 44, Stocholm, Sweden SUN Guoj System Engneerng Insttute, an Jaotong Unversty an, 74, Chna bstract The Kalman flter s wdely used n many dfferent felds. Many practcal applcatons and theoretcal results show that the Kalman flter s very senstve to outlers n a measurement process. In ths paper some reasons why the Kalman Flter s senstve to outlers are analyzed and a seres of outler-tolerant algorthms are desgned to be used as substtutes of the Kalman Flter. These outler-tolerant flters are hghly capable of preventng adverse effects from outlers smlar wth the Kalman Flter n complexty degree and very outler-tolerant n the case there are some outlers arsen n samplng data set of lnear stochastc systems. Smulaton results show that these modfed algorthms are safe and applcable. Keywords- Kalman flter; Outler-tolerant; Outler; Lnear stochastc system. I. INTRODUCTION The Kalman flter s not only a wdely used tool to estmate or to reconstruct states of a dynamc system n modern control but also famous powerful tool to extract useful nformaton from nosy sgnals n sgnals processng. The Kalman flterng algorthms have many advantages: t s optmal n lnear estmator set and sutable for onlne processng because of ts recursve relatonshp; t can be used n statonary system and non-statonary system; and t can be used n mult-dmensonal processes, etc. Snce Kalman and Bucy put forward ths lnear optmal teratve flterng algorthm wth the development of state-space theory n the early 96s, ts applcatons have become more and more wde-spread n many dfferent engneerng felds, such as process control, stochastc control and navgaton, etc. It s also very useful n fault dagnoss of dynamc systems. lthough the Kalman flter possesses many advantages stated above, recent research has revealed that the Kalman flterng algorthm s not robust (see [,] aganst perturbaton n a model or observed/measured data. Practcal experence n usng Kalman flter to process sgnals also ndcates that outlers n the measured data would degrade the performance of Kalman flter. How to mporve the Kalman flterng algorthm s an open queston n the cases when there are outlers n measurement data sequence because outlers are unavodable and could lead to a consderable devaton of the estmated target from the true system status when usng Kalman-flter based algorthms. Durovc, et al. (999 dscussed robust estmaton wth unnown nose statstcs; Nhal et al. (99 and Chan et al. (5 bult a new robust Kalman flter algorthm wth outlers respectvely; Tng et al. (7 revewed the Kalman flter and suggested a nd of robust Kalman flterng wth Bayesan weghts so as to overcome negatve effects from outlers. Ths paper analyzes systematcally the adverse effect of outlers on Kalman flter and establshes a seres of outlertolerant flterng algorthms. Numercal results of smulated examples show the valdty of the outler-tolerant algorthms. II. EFFECTS OF OUTLIERS ON KLMN FILTER The Kalman flter s used to estmate the state vector of the followng lnear stochastc dynamc-measurement system Y n ( R, Y R m H If both the mult-dmensonal dynamc nose process { and dynamc measurement error seres { possess the followng propertes: (a E{ E (b cov(, (c R ( cov(,, R ( cov(, then the optmal estmator of the state vector n model ( can be expressed by the followng recursve relatonshps: K ( ( H ( ( I K ( { H K H E H ( R R ( ( ( ( 7 P a g e

2 (IJCS Internatonal Journal of dvanced Computer Scence and pplcatons, Vol., No., These recursve relatonshps can ntutvely express that the best estmator of the state ( at tme t s composed of two parts: the best estmator of the state at tme t and the samplng nnovaton as follows E ( Y H ( ( Obvously, f the addtonal samples Y are normal ( values, the samplng nnovaton E wll mae some correct ( modfcaton to predctor wth rato K to get the next ( optmal estmator. On the other hand, f the addtonal ( samples Y are outlers, the resultant nnovaton E wll be abnormal and the abnormal nformaton wll result n an ( erroneous modfcaton to predcton x by the same rato K, whch lead to the flterng estmators devatng from normal states. In secton V of ths paper, an example s gven to substantate the negatve nfluence of outlers on the Kalman flterng estmators of state vectors. Fgure plots the smulaton results, whch ndcates that the Kalman flterng of state vectors s far from normal states when there are outlers n samplng data seres. For a practcal measurement system or devce, outlers are nevtable n samplng data because of faulty operatons or recordng errors. In some cases, there may be complcated abnormal measurement data exstng n samplng processes, such as step-type jumps or patchy outlers, etc. For example, when the flght of a spacecraft was traced by an mpulse radar, (occasonally as hgh as 5 of measurement data dsplay serous devatons from the trend formed by most other samples. So t s very mportant to modfy the Kalman flterng algorthms or reduce the negatve effects of outlers on Kalman flterng estmators of state vectors. III. TCTICS TO DEL WITH OUTLIERS IN SMPLE Consderng that t s very dffcult to dagnose outlers n a large quantty of data and that the Kalman flterng s stll wdely used, a best effort s made n ths paper to mprove fault-tolerance of the lnear optmal recursve flterng algorthm. The mproved flters should possess the followng propertes: The algorthm should be recursve and easy to use; When there are a few abnormal samples n the measured data seres, the flterng algorthm must have a strong ablty to overcome negatve effects from abnormal data, or must restrct the negatve effects wthn prescrbed bounds; When there are no abnormal samples, the algorthm s capable of mang full use of useful nformaton to acheve hgh flterng accuracy. Consderng these three restrctve condtons stated above, a set of algorthms whch are smlar to the Kalman flter are establshed as follows r ( ( E ( G E K ( ( r E ( where the functon seres { ( are segment-wse smooth and bounded, K s the gan and G s a weghtng matrx. When and G I (dentty matrx are selected, ( n equaton ( s reduced to the conventonal Kalman flter. It s found that the man reason for outler-tolerance of conventonal Kalman flter s that the functon sequence { potentally treat all of the nnovatons (normal and abnormal equally. Ths s the root cause why Kalman flter s unable to deal wth outlers. In order to endow a flter antoutlers capablty, a sensble tactc s to select a sutable whch must decrease or dmnsh to zero when r ncreases. IV. OPTIML SELECTION OF In order to reduce the negatve effects of outlers n state flterng, some analyses of formulae ( must be done as follows: ( ( ( (4 K ( ( ( ( G / ( where ( = G E ( s one-step predcted weghted resdual. For threshold constant seres { c used to control the dfference between the fltered values and the predcted values of state vectors, should be sutably chosen so as to mae sure that the followng nequalty holds ( / ( { ( ( ( ( (5 c where s the maxmum egenvalue of matrx K G K. It s easy to see that there are qute many dfferent nds of functon seres { satsfyng nequalty (5.ll of the functon seres are denoted as a set S: S {{ { c :, r( r, r [,, N (6 The way to select functon seres { from the set S s examned below. Theorem :For a lnear stochastc system, when the nose seres { x,,, ;,, are stochastc sequences that possess a normal dstrbuton and are mutually ndependent, f 8 P a g e

3 the weghtng matrx sequence are G H H R (, then the functon seres { S whch lead to the least errors of flterng estmaton are as follows: where C, r ( r C C, r r C s the sutably chosen threshold constant. Proof:It follows from equatons ( and ( that the flterng error can be expressed as follows: E ( E ( E ( E( ( ( ( ( ( E ( ( ( Usng the basc propertes on condtonal probablty dstrbuton, the followng results may be deduced E( ( ( ( ( { ( E E ( ( ( ( ( E {( ( ( E ( E( { E( ( ( ( Y,, Y Y,, Y Y,, Y Note that the project property of Kalman flterng has been used n the last equaton of the above expresson. So the followng result can be obtaned: E ( E ( ( E ( In order to nvestgate how to mnmze the error n equaton (8, equatons ( and ( are used to deduce the followng expresson: ( ( ( (IJCS Internatonal Journal of dvanced Computer Scence and pplcatons, Vol., No., E E{[ ( r ] K E (9 dstrbutons of sequence {, x,, x ; It s obvous that functon seres { S whch were prescrbed n the equaton (7 mnmze the equaton (9. In the followng part of ths secton, we wll dscuss how to calculate wegh matrx sequence { G and threshold constants C (,,. Hypothess. Suppose that { and { are two ndependent nose sequences and they satsfy followng relatonshps: (7 (8 a. E, E, Ex x, Ex x R xx ; b. E R, E R ( ( ( ( ; c. E, Ex, Ex Hypothess : Suppose that { and { are two ndependent nose sequences and they satsfy followng dstrbutons: x N( x, R ; N(, R ( ; N(, R ( Lemma : If a lnear stochastc system defned by equaton ( satsfes hypothess and hypothess, then stochastc sequence { x,,, ;,, obeys the law of a multvarate normal dstrbuton and ther jont covarance matrxc s as follows: C a Ca ( Ca( Ca( ( Ca( a( where operator represents the bloc-dagonal matrx formed by the two bloc matrces on ether sde of the operator; C a ( s the covarance matrx of the measured nose sequence, whch can be expressed as the nverse matrx of the tr-dagonal matrx = ( a, whch s defned as follows j a a a a a a,,,,,, j R R R R ( ( R R ( ( ( ( R ( j,, j,..., ( In formula ( matrx C a ( s the covarance matrx of the measured nose sequence { : C ( dag{ R (,, R ( a Lemma : If the lnear stochastc system defned by equaton ( satsfes hypothess and hypothess, the jont x y,, y are normal and ther covarance matrx s gven by the followng relatonshps Ca Cb a HC CaH H, H HCaH Ca where the superscrpt denotes transpose of a matrx. nalyzng these two lemmas descrbed above, t s found that the jont dstrbuton of sequence { y,, y s also normal. H 9 P a g e

4 (IJCS Internatonal Journal of dvanced Computer Scence and pplcatons, Vol., No., If the lnear stochastc system defned by equaton ( satsfes those two hypotheses, denotng the jont covarance matrx of { y,, y by D ( and the covarance matrx of E ( by,t s easy to verfy the propertes of Kalman flterng algorthms that d D d ( D ( D d ( ( where d cov( y, y, the auto-varance matrx of stochastc vector y ; matrx D d ( s the covarance matrx of { y,, y and y. Weghtng matrx G and weghted resdual ( can be calculated as follows G ( d / E D ( d ( D ( D d ( (4 It can be proven that ( follows the standard multvarate normal dstrbuton. Supposng that the measured nformaton s an m-dmensonal vector, the weghted resdual ( s an m-dmensonal standard normal varable, the norm of whch s equal to r : r E ( E ( (5 ( m where ( m denotes the dstrbuton of degrees-offreedom m. Snce the random varable r satsfes the dstrbuton ( m, the threshold constant c (m satsfes P ( r c ( m Generally parameter s taen as.5 or.5. If r c ( m, t s beleved wth (- % confdence that the addtonal nformaton from the measured data y s reasonable; otherwse, when c (, t s also beleved r m wth % confdence that the addtonal nformaton from the measured data y s unreasonable and hence ths addtonal nformaton must be dscarded. From the above analyss and consderng equaton (6, t s nown that C can be gven by / C ( c ( m (6 where s the maxmum egenvalue of matrx K K. In specfc applcatons n engneerng, C may be chosen wth our experence. V. NUMERICL SIMULTION Supposng that the coeffcent matrces and covarance matrx of errors defned by equaton ( are as follows..9.., H....7 and the covarance of the model errors are respectvely equal to R R ( ( dag{.5,.5,.5 dag{.,. Usng Monte Carlo method and selectng the ntal states of the system as the followng x x x ( ( ( (. one hundred peces of smulaton data are generated and denoted by the set S. Let devatons of the 5 th and 75 th peces of data be y (, y ( 5 ( 5,75 The new set wth these two outlers s denoted by S*. The estmates on S* made by equatons ( and ( are shown n fgures and (a Flterng Estmaton of (b Flterng Estmaton of 4 P a g e

5 (IJCS Internatonal Journal of dvanced Computer Scence and pplcatons, Vol., No., (c Flterng Estmaton of Fgure Kalman Flters of State Varables R (a Flterng Estmaton of (b Flterng Estmaton of (c Flterng Estmaton of Fgure Fault-tolerant Flters of State Varables R It can be seen clearly that the outlers have a very negatve effect on conventonal Kalman flterng and the outlerstolerant modfcaton method proposed n the paper s capable of overcomng ths negatve effect and s relable. CKNOWLEDGEMENTS The research was supported by the Sweden Insttute Grant (SI 548/5-, the Natonal Natural Scence Foundaton of Chna (No and the Tanyuan Fund of Natonal Natural Scence Foundaton of Chna (No.64. The frst author would le to than Dean Ingrd Melnder of the Computer Scence and Communcaton (CSC of the Royal Insttute of Technology (KTH for her frendly help when he vsted the CSC of KTH of Sweden. REFERENCES [] Hu Shaoln, Fan Jngcheng. Bounded Influence Flter of the Dynamc- Measurement System. Control Theory and pplcatons,99.(: 645 [] Han Congzhao, Wang Yuejuan, Wan Bawu. Stochastc System Theory. Press of the an Jaotong Unversty,987 [] Martn D, Mntz. Robust Flterng and Predcton for Lnear System wth Uncertan Dynamcs a game theoretc approach.ieee Trans. utomatc Control,98, C-8 [4] Martn D. Robust Method for Tme Seres,cademc Press Inc, 98 [5] Hu Shaoln,Wang aofeng,karl Mene,Huajang Ouyang.Outlertolerant fttng and onlne dagnoss of outlers n dynamc process samplng data seres.n rtfcal Intellgence andcomputatonal Intellgence, pp.95-4, LNI 74(Deng Mao,Le Wang,eds Sprnger Press, [6] F Carlos,S lcala, J Qn. Unfed nalyss of Dagnoss Methods for Process Montorng. In: Proc of the 7 th IFC Symp on Fault Detecton, Supervson and Safety of Techncal Processes, Span,9,7- [7] Nhal Y, Bovas, MacGregor J. alman flter n the presence of outlers Communcatons n Statstcs - Theory and Methods Volume, Issue 5-6, 99 [8] Durovc, Z.M., Kovacevc, B.D.: Robust estmaton wth unnown nose statstcs. IEEE Transactons on utomatc Control 44,9 96,999 [9] Chan, S.C., Zhang, Z.G., Tse, K.W.: new robust Kalman flter algorthm under outlers and system uncertantes. In: IEEE Internatonal Symposum on Crcuts and Systems. IEEE,47 4,5 [] Jo-nne Tng, Evangelos Theodorou, Stefan Schaal. Kalman Flter for Robust Outler Detecton. Computatonal Learnng & Motor Control Lab Unversty of Southern Calforna,IROS,7 4 P a g e