Particle Filter Approach to Fault Detection andisolation in Nonlinear Systems

Transcription

1 Amercan Journal of Sgnal Processng 22,2(3): 46-5 DOI:.5923/j.ajsp Partcle Flter Approach to Fault Detecton andisolaton n Nonlnear Systems F. Soubgu *, F. BenHmda, A. Chaar Department of Electrcal Engneerng, Hgher School of Scences and echnques of uns, Unversty of uns, 5 Street aha Hussen, B.P uns, unsa Abstract hs paper ntroduces the partcle-flterng (PF) based framewor for fault dagnoss n non-lnear systems and nose and dsturbances beng Gaussan. In ths paper, we use the sequental Monte Carlo flterng approach where the complete posteror dstrbutons of the estmates are represented through samples or partcles as opposed to the mean and covarance of an approxmated Gaussan dstrbuton. We compare the fault detecton performance wth that usng the extended Kalman flterng and nvestgate the solaton performance on a nonlnear system. Keywords ecursve Bayesan approach, Extended Kalman flter, partcle flter, Parameter estmaton, fault detecton. Introducton Problem of fault detecton and solaton (FDI) n dynamc systems has receved consderable attenton for many years due to creasngly complex systems and relablty[2]. Dfferent types of approaches appearng n lterature, as can be seen from a large number of survey papers[-6]. he problem of fault detecton can be roughly dvded nto two major categores: Frst, we need to estmate the unnown and un measurable state varable of model and generate resduals on the bass of the avalable observatons and a model of the system. Secondly, we need to decde on the occurrence of a fault based on the resduals generated[2]. For the stochastc systems, much of development n fault detecton schemes has reled on the system beng lnear and dsturbances beng Gaussan. In such cases, the Kalman flter (KF) s nown to be optmal and employed for state estmaton. he nnovatons errors from the KF are used as the resduals, based on whch statstcal hypothess tests are carred out for fault detecton (see e.g. wllsy,[4] and Darouach et al.,[5]). However, n comparson wth lnear systems, the lterature addressng fault detecton (FD) for nonlnear stochastc systems s not smple, the man reason beng that the estmaton of the state vector of a nonlnear stochastc system s not easy. Suboptmal solutons use some form of approxmaton for non lnear systems wll addtve Gaussan nose and dsturbance by employng lnerasaton technques[9]. In ths case, Kalman flter s usually replaced by the extended Kalman flter (EKF) as a resdual generator. However, EKF * Correspondng author: faycalsoubgu@yahoo.fr (F. Soubgu) Publshed onlne at Copyrght 22 Scentfc & Academc Publshng. All ghts eserved s only an approxmaton method for non lnear flterng; there are no general results to guarantee that such approxmaton wll wor n most case. When the nonlnearty system s strong and non-gaussan dstrbutons, the performance of EKF wll descend or even become dvergent[9,,3]. For that the fault detecton for nonlnear stochastc systems s nown as a dffcult problem and very few results are avalable[8,5]. General soluton of the state estmaton problem s descrbed by the Bayesan recursve relatons. he closed form soluton of the Bayesan recursve relatons s avalable for a few specal cases (Gaussan or non- Gaussan). Durng the99s, the partcle flter (PF) whch domnated n recursve nonlnear state estmaton, has attracted much attenton and has been wdely appled n many felds (see e.g (Gordon and al.[7], Bolven and al.[8], Doucet and al.[], Benhmda and al.[9]). he PF solves the Bayesan recursve relatons usng Sequental Monte Carlo (SMC) methods. hese methods allow for a complete representaton of the posteror probablty densty functon of the states, so that any statstcal estmates, such as the Mnmum Mean Squared Error estmate (MMSE) and the Maxmum a Posteror Probabltes (MAP) can by easly computed. In year 2 Kadramanathan and al.[2] ntroduced Sequental Monte Carlo methods nto feld of fault detecton and solaton(fdi). Dfferent types of approaches appearng n lterature of FDI problem[,3] for solvng general nonlnear systems wth nown sets of possble faults. In ths latter, the partcle flter s combned wth the nnovaton based fault detecton technques to develop a fault detecton and solaton scheme. he paper s organzed as follows: secton 2 states the problem of nterest. In secton 3 we treat the ecursve Bayesan approach. An Innovaton-based fault detecton of the stochastc system and detected to update step of the Extended Kalman Flter n secton4. he partcle flter

2 Amercan Journal of Sgnal Processng 22,2(3): based detecton and solaton schemes are descrbed n sectons5, 6 and7. Fnally, we llustrate n secton 8 the smulaton results on a hghly non lnear system wtch demonstrate the effectveness of the partcle Flter. 2. Problem Formulaton he problem of fault detecton and solaton (FDI) consst of mang the decson on the presence or absence of faults n the motored system. In ths paper, the dynamcs of the system consdered s assumed nown and gven by dscrete tme nonlnear stochastc system gven by the state equaton () and the measurement equaton (2): x f( x, u, w) () y h( x, v) (2) nx nu nw nx where f : s a possbly nonlnear functon defnng the state at tme from the prevous state at tme, and u s the nown nput. he functon h : nx nv n y s a possbly nonlnear functon defnng the relaton between state, parameter and the measurement at tme. w and v the uncorrelated whte nose sequences of zero-mean and covarance matrces Q and >, respectvely. he ntal state s uncorrelated wth the whte noses processes w and v and x s a Gaussan random varable wth ε [ x] xˆ and x ε ( x xˆ )( x ˆ x) P ε denotes the expectaton operator. We denote by nx, nu, nw, ny and nv respectvely, the dmensons of the state, the nput vectors, process nose vectors, the measurement and measurement nose vectors. where [.] Let ( y ) {,,..., } D u denote the avalable measurement nformaton at the tme : he type of faults of nterest here are the falure type where the system parameter values to n new value reflected n a change n the state transton functon f (.) at tme and the measurement functon h (.) at tme such fault can be detected usng the state observer approach. 3. ecursve Bayesan Approach hs secton gves a bref nformal ntroducton to the basc recursve Bayesan approach. ecursve Bayesan state estmaton of dscrete-tme nonlnear stochastc systems has been the subject of a consderable research nterest over the last three decades. wo good surveys on nonlnear recursve estmaton are provded by (Sorenson, 988[]) and (Kulhavsy, 996[2]). Another good reference for nonlnear recursve and non recursve estmaton s (Jazwns, 97[3]). he dea of the Bayesan approach to state estmaton problems nvolve the constructon of the probablty densty functon of the current state x, based on the nput output data D observed up to nstant, more precsely, to estmate the condtonal probablty densty functon px ( D ). In general, no accurate tow estmator exsts, the Mnmum Mean Squared Error estmate (MMSE) and the Maxmum a Posteror Probabltes (MAP) estmate for nonlnear stochastc systems, even f the noses are assumed to be Gaussan or non-gaussan as follows x MMSE ε [ x D ] nx x p ( x D ) dx (3) x MAP arg max p ( x D ) (4) x From a Bayesan perspectve, the propagaton of the posteror px ( D ) of x based on the observaton sequence D, can be bascally acheved by performng three recursve steps. It can be expressed as follows Step (). Intalzaton: p( x D) p( x) (5) Step (2). Chapman-Kolmogorov equaton: ( ) ( ) ( ) p x D nx p x x p x D dx (6) Step (3). Bayes theorem update: Accordng to the Bayes theorem the posteror probablty densty follows from the relaton p( y x ) p( y D) p( x D ) p( y D) (7) p( y x ) ( ) ( ) nx p x x p x D dx p( y D) where the p( y D) the normalzng constant p ( y D) ( ) ( nx p y x p x D) dx (8) from the equatons (6) and (8), we obtan the equaton (7) p( y x ) nx p( x x) p( x D) dx p( x D ) (9) p y x p x D dx nx Step (4)., go to Step (2). In steps (2) and (3) px ( x) s defned by the state functon, Eq. and p( y x ) s the condtonal densty of observaton, gven the state at tme, Eq. 2. Smlarly the posteror probablty densty for smoothng px ( λ x) λ > and flterng, px ( λ x), λ can be defned. However ths paper s only concerned wth predcton, and thus the only stuaton of nterest s where λ >. he posteror probablty densty gven by the above steps s exact, but n general, t can be vewed as an nfnte dmensonal system, thus not mplementable. here s however a severe problem n Baysan state estmaton for nonlnear systems that maes s dffcult to readly the equatons (6) and (7) because they nvolve hgh dmensonal ntegratons. he most mportant specal case s when the system s lnear,.e, f (.) and h (.) are lnear and one assumes that the nose and the ntal state are Gaussan,.e., w ~ (,Q) v ~ (,). he soluton s provded by the Kalman flter. More detals and bacground for ths flter can be found n[3]. We also descrbe how, when the analytc soluton s ntractable, Extended Kalman flter, and Partcle flter approxmate the optmal Bayesan soluton.

3 48 F. Soubgu et al.: Partcle Flter Approach to Fault Detecton andisolaton n Nonlnear Systems 4. Innovaton-Based FDI Desgn One of the man dffcultes n fault detecton of the stochastc system descrbed by() and(2) s due to the presence of unnown and unmeasured state varables x. he dea s to generate estmates of the states and the predcted outputs. he resduals or nnovaton from the output predcton are used n a measure whch changes sgnfcantly under a falure type fault. For nonlnear Gaussan system, the states are estmated usng a extended Kalman flter[3], an approxmate sub-optmal estmate probablty densty functons descrbed n secton3, obtaned by lnearsaton, s recursvely gven accordng to be Gaussan,.e. p( x D) ( ˆ x,p) p( x D) ( ˆ x,p ) ( ) ( ˆ ) () () p x D x,p (2) where the states and matrx covarance are estmated accordng to the followng equatons: Predcton: ˆx ˆ f x,u (3) Correcton: P Φ P Φ Q (4) ( ) ( ) ˆ x ˆ x K y h ˆ x (5) K P P Ψ Ψ Φ 6) P Φ P K Ψ P (7) Where f ( x,u) h( x ) Φ x ˆx,u x Ψ x { ˆx { x he resdual or the nnovaton s then, r y ˆy (8) where the predcted output based on the EKF state estmate s gven by ( ˆx ) y h (9) ˆ It s well-nown that, under fault free or normal operaton, the nnovatons are zero mean Gaussan wth covarance Q r Ψ P Ψ Q (2) Any faults or changes n system dynamcs can therefore be detected by a change n the Weghted Squared esdual (WS) measure r l r Q r (2) hs however can lead to false alarms occurrng at a partcular nstant due to dsturbances and nose and a more robust decson functon for fault detecton s the weghted sum squared resdual (WSS) (see e.g.[4]) defned as follows: r j j j j j W j W (22) d l r Q r where W s the length of the sldng wndow wthn whch the resdual measure s summed. he wndow length W should be chosen n accordance wth the requrement for detecton tme and the fault alarm s set at tme when the condton[4] d > ε (22) s satsfed, ε beng the threshold. When usng extended alman flter to estmate the states and hence r. he r Q measure L wll thus consst of fluctuatons whch can n turn lead to hgher false alarm rates and also to faults not beng detected. 5. Partcle flterng (PF) We have so far presented two flterng methods that rely on Gaussan approxmaton. In ths secton, we shall present the partcle flter (PF), based on a Monte-Carlo technque, whch was frst proposed by Gordon (Gordon et al., 993). hereafter, a number of alternatve PF algorthms have been proposed. he PF uses sequental Monte-Carlo methods to approxmate the optmal flterng by representng the probablty densty functon (pdf) wth a swarm of partcles. Because the PF s able to handle any functonal nonlnearty and system or measurement nose of any probablty dstrbuton, t has attracted much attenton n the nonlnear non-gaussan state estmaton feld (Bolven et al.,[8]; Doucet et al.,[]). he objectve s to recursvely construct the posteror pdf p( x D ) of the state, gven the measured output D and assumng condtonal ndependence of the measurement sequence, gven the states. he PF wors n two stages: ) he predcton stage uses the state-transton model n () to predct the state pdf one step ahead. he pdf obtaned s called the pror. 2) he update stage uses the latest measurement to correct the pror va the Bayes rule. Partcle flters represent the pdf by N random samples (partcles) x wth ther assocated weghts w, normalzed N so that w. At tme nstant, the pror pdf p( x D) s represented by N samples x and the correspondng weghts w. o approxmate the posteror p( x D ) x and weghts w Samples x densty functon q( x x, y ), new samples are generated. are drawn from a (chosen) mportance usng the current measurement y w w and normalzed, and the weghts are updated, ( y ) ( ) q( x x, y ) p x p x x (23)

4 Amercan Journal of Sgnal Processng 22,2(3): w w N w he posteror pdf s represented by the set of weght samples, conventonally denoted by: N ( ) p x D w x x (24) N Here, we choose the mportance densty q( x x, y ) equal to the state-transton pdf p( x x). he weght update equaton (23) then becomes: ( w wp x ) y (25) A common problem of PF s the partcle degeneracy: after several teratons, all but one partcle wll have neglgble weghts. herefore, partcles must be resampled. A standard measure of the degeneracy s the effectve sample sze: Neff N 2 w hus, the degeneracy phenomenon can be detected when Neff < N thr where N thr s the threshold value N thr [, N]. When such condton s encountered, the resamplng algorthm s just appled. For more detals on partcle flters, refer to[3,,5]. 6. Partcle Flter for Fault Detecton For the purpose of fault detecton, the method proposed n ths paper s to desgn several partcle flters, each assumng a dfferent subset of the possble faults formulated n system descrbed by () and (2). he advantages of usng the complete probablty densty functon of the system state n fault detecton s bound to be superor to one whch uses approxmatons, such as n the extended alman flter (EKF). Our approach s precsely to replace the EKF based estmaton scheme by the partcle flter, and the weghted squared resdual (WSS) measure by an approprate nnovatons lelhood measure as the fault detecton crtera[,3]. he followng algorthm descrbes the complete partcle flter based fault detecton scheme: Partcle flter based fault detecton Algorthm Step : State predcton Samples { x :,...,N} are generated as partcle flter predcton step. Step 2: Output predcton he output predcton samples { y :,...,N} are generated usng the measurement equaton n (2), where, y h x Step 3: esdual generaton Sample mean of the predcted measurements s computed y y as, r Step 4: Fault detecton Compute the lelhood s gven by, N N p r D w he wndowed lelhood s λ p r D j W Or equvalently the negatve log lelhood s ν j W ( ) ln p( r D ) s computed and the condton ν ( ) presence of fault. > ε s tested for the 7. Augmented States Model for Fault Isolaton In the prevous secton s a fault detecton scheme whch cannot readly be extended to fault solaton. he dea s to vew the parameters as addtonal states. Usng the state augmentaton technque, a new state vector can be defned: x z θ Hereθ s the unnown fault parameter vector to be estmated. Because { θ } s not ergodc, θ cannot be traced n the PF algorthm. herefore, n order to trac the dynamcs of θ an artfcal dynamc nose vector s added to the model of the unnown parameterθ : θ w θ θ (26) where w θ s the parameter nose. hen the augmented system and measurement functons are respectvely defned as e z f ( z,u) w (27) e y h z v (28) where w w w θ. Because w θ s artfcal, ts statstcal propertes need to be determned. In ths paper, we as- sume that w θ s a zero-mean Gaussan whte nose process that s ntroduced for parameter evoluton to allow the exploraton of the parameter space. Gven the above system representaton (27) and (28), the partcle flter outlned n ths seton5, can be used to obtan the sample-based jont probablty functon densty of the state x and parameter vectorθ such an estmate based on partcle flter algorthm s gven by, N MMSE ẑ w z (29) j he estmate s essentally a weghted average of the partcles representng the underlyng dstrbuton[6,8]. he parameter estmate ˆθ can be compared to the nomnal values

5 5 F. Soubgu et al.: Partcle Flter Approach to Fault Detecton andisolaton n Nonlnear Systems θ θ ˆ θ. θ as a means for fault detecton and ts devaton 8. Numercal Example An example s presented n ths secton to llustrate the operaton of the partcle flter based fault detecton and solaton proposed n ths paper. he consdered system s descrbed by followng the dynamcal equatons, 6 θx, x2, θ2x2, u 25 f ( x,,x 2,, θ, θ2,u ) (3) x, 2 u 2 8. ( x ) 3, h x,x x x (3), 2,, 2, he ntal jont state vector, z [,,.,], P I 4 4. he measurement and process nose sgnals are both set to be Gaussan wth covarance matrces, Q, he nput and observaton shown n Fg.. he tme ndex s ncremented as,.,2. he nomnal values of the parameters are θ 5. and θ In the smulaton, the length of the data wndow for WSS calculaton s N 3 and the number of partcles n the partcle flter s N 6. he fault s smulated to occur at tme at whch tme the parameter θ 2 jumps from a value of θ 2 to 5. θ2 whle θ remans unchanged. he Weghted Sum Squared esdual (WSS) results for the EKF s shown n Fgure 2 and the negatve log lelhood for the partcle flter n Fgure 3. he EKF based approach fals to detect the occurrence of the fault around as evdenced by Fgure 2 where there s no sgnfcant change n WSS. he detecton performance s unacceptable. On the other hand, the partcle flter detects the fault at 4 a threshold value of ε 2.. he change n log lelhood s qute pronounced followng the onset of the fault. he combned fault detecton solaton (FDI) scheme proposed n secton7was also appled to the above example. Input YK Fgure. he nput u and observatony θ θ 2 WSS Fgure 2. (a) Parameterθ θ 2 estmates from the EKF and true values, (b) weght sum squared resdual (WSS) -EKF θ θ 2 Fgure 3. (a) Parameterθ θ 2 estmates from the PartcleFlter and true values, (b )Negatve log lelhood - Partcle Flter Fgure 5 shows that the change n the parameter θ 2 s traced followng an ntal transent and the estmate for the other parameterθ hovers around the nomnal parameter. he PF successfully tracs the system parameter, whle ths s not seen for EKF (Fgure3). hus we can safely conclude that the fault n the system s due to the change n the parameter θ 2 9. Conclusons Estmate θ Estmate θ Estmate rue WSS by EKF-approch Estmate θ Estmate θ Estmate rue novaton lelood by PF-approch 3 2 threshold hs paper, consders the Partcle flter based approach to fault detecton and solaton scheme developed. hs approach s applcable to general non-lnear systems wth Gaussan or non Gaussan dsturbance nose. he fault detecton performance compared wth that usng the EKF. he results from smulaton show that the detectablty of the partcle flterng approach s superor to the EKF based scheme, especally n the case where the system model s hghly nonlnear. he fault solaton scheme s also shown to dentfy the parameter assocated wth the fault and the level

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