Unknown input extended Kalman filter-based fault diagnosis for satellite actuator

Transcription

1 Internatonal Conference on Computer and Automaton Engneerng (ICCAE ) IPCSI vol 44 () () IACSI Press, Sngapore DOI: 776/IPCSIV44 Unnown nput extended Kalman flter-based fault dagnoss for satellte actuator Wang Zh a, and Sh Jun a Computer Scence and Engneerng College, X an echnologcal Unversty, X an 7, Chna Abstract Unnown nput Kalman flter(uikf) s extended to nonlnear systems, and then appled to detect the early gradual faults of satellte flywheels to mprove the tmelness of fault detecton, to avod the occurrence of major accdents A set of structured resduals are constructed to acheve fault solaton, and Wald sequence detecton method s employed to process flter resduals Fnally, the fault dagnoss logc s gven Numercal smulaton results show that ths method s effectve and able to qucly detect faults Keywords: satellte actuator; fault dagnoss; UIKF; sequence detecton Introducton he atttude control system s one of the most complex sub-systems of satellte, and has very hgh probablty of falure Accordng to a survey n lterature [], the faults that occur n ths sub-system account for more than % of all satellte faults Snce the exstence of rotatng parts, the flywheels are the hghest ncdence of fault of atttude control components In recent years, some achevements have been made n the feld of fault dagnoss for satellte actuator In lterature [], the method of fault detecton and dagnoss for satellte reacton wheels s proposed, whch uses state space approxmaton to study neural networ fault recognton wth nonlnear parameter he lterature [] employs a set of detecton flters to detect the faults of satellte reacton wheels In addton, fault dagnoses based on expert systems have also been wdely studed Fault dagnoss of nonlnear systems and robust fault dagnoss are the hot and dffcult ssues n the current study As a class of smple constructon, hgh unversalty of the nonlnear state estmator, the extended Kalman flter (EKF) s subject to a wde range of attenton both n theory or practce For robust fault dagnoss, a well-nown method s the unnown nput observer (UIO), the basc dea of whch s to use the extra degrees of freedom n Luenberg observer desgn to mae the outputs and the unnown nputs (Includng dsturbances, modelng uncertantes, etc) decouplng [4,5] On the bass of lnear unnown nput observer, Ktands studed the problem of unbased mnmum varance estmaton of lnear stochastc system wth unnown dsturbances, and the proposed algorthm was called the unnown Input Kalman flter(uikf) [6] hrough the combnaton of EKF and UIKF, a flterng algorthm appled to nonlnear system wth unnown dsturbances s proposed, whch s called unnown nput extended Kalman flter(uiekf) n ths paper he UIEKF s used to detect the early gradual faults of satellte flywheels, whch can mprove the tmelness of fault detecton, and provde reconstructon nformaton for the atttude control system of satellte to avod the occurrence of major accdents Satellte Model and Fault Descrpton When the servo s three-orthogonal-flywheel, the nematcs and dynamc equaton of satellte are descrbed as follows Correspondng author el:567 E-mal address: zh_wang_xa@6com 7

2 q () t = ( q () t q () t E) ω () t q () t = q () t ω () t () I ω t ω t Iω t Jω t = u t () () ( () F ()) () Where q R, q Rdenote quaternon vector and q q q = ; q s the sew symmetrc matrx of q, E s unt matrx; I R R ( Kg m ) s the total moment of nerta matrx of satellte, ω R ( rad/s) s ts nertal angular velocty vector, u R ( Nm ) s the control torque actng on satellte, J R R ( Kg m ) s the moment of nerta matrx of flywheel, ωf R (rad/s) s ts angular velocty vector hen, the state equaton of satellte atttude control system can be denoted as follows () = ( ()) () x t f x t Bu t () As long contnuous mechancal movement, the flywheels are the hghest ncdence of fault of atttude control components Snce the frcton torque ncreases and other reasons, the flywheels may appear gradual faults Such falures n the early stages are dffcult to detect, as tme ncreases, the flywheel falures become clear, t wll affect the platform s normal posture herefore, ths type of falure should be dagnosed as early as possble, and then the control system s reconfgured to avod a greater mpact For smplcty, ths paper only consder the bas-type falure, that s where () t system can be expressed as () () ρ () u t u t t ρ s the bas-type fault ndcator, () t UIEKF Algorthm = =,, () ρ wll mean that the bas-type fault occurs hen, the fault () = ( (), ()) ρ () x t f x t u t B t (4) Consderng factors such as dsturbances and un-modeled dynamcs, the dscrete model of the satellte () s obtaned as follows x = f x u E x d w y = h x u v m q Where y R s the measurement vector; d R s the unnown nput vector whch denotes unnown dsturbances and modelng uncertantes; f ( x, u ), E( x ) and h( x, u ) are formed by smooth nonlnear functons; System nose w and measurement nose v are zero-mean Gaussan whte noses, the covarance matrx of them s respectvely denoted as Q, R Assumpton: he dstrbuton matrx E E( x) of the unnown nput s full column ran, and H = h x, E satsfes ran H E = ran E = q Based on the assumpton, the UIEKF algorthm for the system (5) s as follows xˆ f xˆ, u = P = F P F Q ( ) x ˆ = x ˆ L y h x ˆ, u (6) ( ) P = I K H P Γ ϒ Φ ϒ Γ (5) Where L = K Γ ϒ ; K P H = Φ ; Γ = I K H Eˆ ; ( ˆ H E ) ( H Eˆ ) ( H Eˆ ) ϒ = Φ Φ ; H P H R Φ =, 7

3 and the matrx F, H and E ˆ s respectvely determned by F = f ; H x ( xˆ, u ) = h x ( xˆ, u ) and Eˆ ( ˆ E x) = he dfference between the UIEKF and EKF s that, the gan matrx L and the fltered state error covarance matrx P n UIEKF are amended on the results of EKF, and the correcton terms are drectly related wth the dstrbuton matrx E of unnown nput It can be nown by the assumpton that L meets the constrant of dsturbance decouplng L ˆ ˆ H E = E (7) hs constrant ensures that the state estmaton obtaned by the flter (6) s unbased, and the fnal state error covarance matrx s large than the EKF 4 Fault Isolaton and Resdual reatment Based on the UIEKF algorthm, ths secton frst gves the robust fault detecton and solaton strateges, and then dscusses the treatment of resduals Consderng the equaton (5), the fault model of satellte s represented as follows ϑ x = f x u E x d x ρ w y = h x u v Fault solaton s realzed by constructng a set of structured resdual, each of whch s senstve to a subset of the faults and robust for the remanng faults One of the most commonly used structured resdual desgns s that, there are three UIEKF desgned, each of UIEKF s only decoupled for one dmensonal fault and all the dsturbances and senstve for the remanng two-dmensonal faults [7] hat s, for the followng system (9), there are followng three UIEKF desgned to acheve fault solaton ϑ x = f x u E x d x ρ w y = h x u v (8) =,, (9) Clearly, more degrees of freedom are requred to acheve fault solaton, that all E( x ), ( ϑ x) assumpton Flter resdual s respectvely defned as ( ˆ, ) γ y h x u =, =,, satsfy Next, the Wald sequence detecton method wll be employed to handle the flterng resduals and detect the flywheel faults Under the condton that the w and v are zero mean whte nose, f the flterng resdual r does not contan fault nformaton, t s also a zero mean whte nose sequence and obeys the normal dstrbuton N (, σ ) hs modal wll be expressed as H If the flterng resdual contans fault nformaton, ts Whte characterstcs wll be destroyed and t can be supposed to obey the normal dstrbuton N ( μ, σ ) hs modal wll be expressed as H () he optmal decson law of the Wald sequence detecton method can be gven by the followng lelhood rato functon Accordng to the ndependence between γ, law can be mproved as follows Formula () can be further smplfed as ( (),, ) (),, p r r n H λ ( n) = ln p r r n H λ ( ) j γ ( j) ( n) =,, () and the normal dstrbuton, the foregong decson ( γ ( j) μ) σ n e = ln () j= ( γ ( j) ) σ e μ λ ( n) = λ ( n ) γ ( n) μ σ () 74

4 Accordng to the gven false alarm rateα and mssng report rate β, two thresholds can be calculated as follows And then the decson rule s If λ L L < n < L, the testng wll contnue β = ln α L ( ): λ () : λ α = ln β H n L H n L Accordng to all the decsons made by three UIEKF, the result of fault solaton can be obtaned by the followng logc () H th flywheel fault H j, j 5 Numercal Smulaton and Conclusons he man physcal parameters of a satellte are gven as follows I = [ 54797, 5,5; 5, 55, 4;5, 4, 4448] wo covarance matrces are W = dag[,,] ; V = dag[,,] Samplng perod s seconds It s set that between the 5th samplng pont and th samplng pont, the th actuator appears slowly varyng gan type of fault, the fault value s Nm, < ρ 5 = sn ( 4 ) Nm, 5 he smulaton results are shown n Fgures through 6 In the smulaton, Fg and Fg 5 show that the output resduals of th UIEKF and th UIEKF contan fault nformaton, and the faults are qucly detected by the Wald sequence detecton method (Fg and Fg 6); the output resdual of th UIEKF doest not contan fault nformaton It can be nown from the fault dagnoss logc that the th satellte flywheel appears fault he smulaton results show that the proposed fault detecton and solaton strateges are effectve Fg Resdual Curve (th UIEKF) Fg wo hresholds and Decson Curve (th UIEKF) Fg Resdual Curve (th UIEKF) Fg 4 wo hresholds and Decson Curve (th UIEKF) 75

5 Fg 5 Resdual Curve (th UIEKF) 6 References Fg 6 wo hresholds and Decson Curve (th UIEKF) [] Ma afazol, A study of on-orbt spacecraft falures, Acta Astronautca, vol 64, 9, pp95-5 [] HA aleb, RV Patel, and K Khorasan, Fault detecton and solaton for uncertan nonlnear systems wth applcaton to a satellte reacton wheel actuator, Systems, Man, and Cybernetcs, 7, pp 4-45 [] N Mesn, K Khorasan, Fault detecton and solaton n a redundant reacton wheels confguraton of a satellte, Systems, Man, and Cybernetcs, 7, pp 5-58 [4] J Anzurez-Marn, N Ptalua-Daz, O Cuevas-Slva, and J Vllar-Garca, Unnown nput observers desgn for fault detecton n a two-tan hydraulc system, Robotcs and Automotve Mechancs Conference, CERMA '8 Electroncs, Sept 8-Oct 8, pp 7 78 [5] Stefen Hu, Stanslaw H Za, Low-Order Unnown Input Observers, 5 Amercan Control Conference, Portland, OR, USA, June 8-, 5, pp [6] Ktands P K, Unbased mnmum-varance lnear state estmaton, Automatca, vol, 987, pp [7] L Lng-la, Robust fault dagnoss of nonlnear systems, PhD Dssertaton, snghua Unversty, 6 76