KALMAN FILTER BASED METHOD FOR FAULT DIAGNOSIS OF ANALOG CIRCUITS

Transcription

1 Metrol. Meas. Syst., Vol. XX (03), No., pp METROLOGY AND MEASUREMENT SYSTEMS Idex , ISSN KALMAN FILTER BASED METHOD FOR FAULT DIAGNOSIS OF ANALOG CIRCUITS Xifeg Li, Yogle Xie, Dogjie Bi, Yogcai Ao School of Automatio Egieerig, Uiversity of Electroic Sciece ad Techology of Chia, Chegdu, 673, Chia (* , Abstract This paper presets a Kalma filter based method for diagosig both parametric ad catastrophic faults i aalog circuits. Two major iovatios are preseted, i.e., the Kalma filter based techique, which ca sigificatly improve the efficiecy of diagosig a fault through a iterative structure, ad the Shao etropy to mitigate the ifluece of compoet tolerace. Both these cocepts help to achieve higher performace ad lower testig cost while maitaiig the circuit s fuctioality. Our simulatios demostrate that usig the Kalma filter based techique leads to good results of fault detectio ad fault locatio of aalog circuits. Meawhile, the parasitics, as a result of ehacig accessibility by addig test poits, are reduced to miimum, that is, the data used for diagosis is directly obtaied from the system primary output pis i our method. The simulatios also show that decisio boudaries amog faulty circuits have small variatios over a wide rage of oiseimmuity requiremets. I additio, experimetal results show that the proposed method is superior to the test method based o the subbad decompositio combied with coherece fuctio, arise recetly. Keywords: Aalog fault diagosis, sigature extractio, Kalma filter, Shao etropy. 03 Polish Academy of Scieces. All rights reserved. Itroductio Testability of aalog circuits has become more importat recetly due to the rapid developmet i aalog VLSI chips, mixedsigal systems, ad systemochip (SoC) products [][3]. Sice complete testig of a aalog circuit specificatio ca be costly, sometimes impossible, aalyzig the output respose of the circuit uder test (CUT), which is called sigature aalysis, is the key to successful applicatio of defectedorieted testig for high fault ad yield coverage [4]. Sigature aalysis i aalog circuits is complicated because the ucertaity comes from the poor iput sigal ad toleraces i the compoet values of the aalog circuit [4]. Further, the imprecise measuremet of the aalog circuit ca also cause the respose of the circuit to be differet from the ideal respose. These make it madatory to compress the output respose i order to produce a sigature of the output for fault diagosis. Ufortuately, the sigature of the faulty respose may correspod to a faultfree sigature, which is usually called aliasig. The aliasig results i a failure of fault diagosis. Further, aliasig may also happe if the sigature of the faulty respose is ot robust. Our motivatio for this research is to miimize the aliasig probability by icorporatig oise iformatio i geeratig the robust sigature.. Prelimiaries ad Cotributios A sigificat amout of methods dealig with sigature aalysis i aalog circuits has bee proposed, ad oly some of them are reviewed here. First, several importat schemes are Article history: received o Ja., 03; accepted o May. 0, 03; available olie o Ju. 03, 03; DOI: 0.478/mms03007.

2 X.F. Li, Y.L. Xie: KALMAN FILTER BASED METHOD FOR FAULT DIAGNOSIS OF ANALOG CIRCUITS briefly reviewed, ad the the oisebased sigature extractio scheme, o which our techique is based, is discussed... Previous work Papakostas ad Hatzopoulos [5] proposed a correlatiobased compariso of the aalog sigatures scheme for improvig the fault discrimiatio capacity, which itroduced a ew criterio by the use of a weighted factor. Faults i the CUT that lead to similar aalog sigatures ca be detected ad idetified. This approach is appealig sice the probability of aliasig is sharply reduced. Subbad filterig testig usig a itegrator scheme for aalog respose sigature compactig was proposed i [4]. I this scheme, a subbad filter takes the test respose, ad the decomposes the sigal for each frequecy bad. The decomposed sigal for each frequecy bad is the fed ito its respective itegrator, which was used for sigature compressio. This scheme has good performace o parameter faults ad catastrophic faults. Catelai ad Fort [6] proposed a efficiet scheme for aalyzig sigatures of aalog circuits usig a radial basis fuctio etwork classifier. Their mai idea is to use eural etworks to perform fault diagosis of aalog circuits. The etwork is traied by meas of a fault dictioary, which was previously costructed ad cotaied examples of fault sigatures. A idex of the ovelty of the fault is obtaied as a output of the etwork, ad this is used as a output of sigature compressio. It is assumed that the eural etworks already exist as a part of the test overhead for the fault diagosis. Coefficietbased testig employig the bouds of circuit trasferfuctio coefficiets as the sigature is proposed by Guo ad Savir [7], [8]. I their scheme, the values of the trasferfuctio coefficiets of the good circuit are supposed to lie i their idividual hypercubes. Wheever oe or more of the CUT s coefficiet values slip outside their hypercube, a differet trasfer fuctio that reflects the existece of a detectable fault is obtaied. Sice this scheme tries to get the correspodig trasferfuctio, sigificat computatioal overhead is eeded. I [9], Yag et al. combied the slopebased model ad the testpoit selectio algorithm for aalog circuits test. I this scheme, the slope is used for sigature compressio. Although the slope sigature ca diagose both parametric ad catastrophic faults, it leads to a sigificat computatioal cost. Deg et al. [0] propose a ew approach based o the subbad decompositio combied with coherece fuctios. Their mai idea is that the Volterra kerel is decomposed i subbad, ad the decomposed respose sigals are processed by a coherece fuctio to extract the fault sigature. Sice this scheme requires computig the Volterra series ad coherece fuctio, the sigificat computatioal cost is also cosiderable... Work i this paper From above illustratios of the previous work, it is easily see that traditioal methods for fault diagosis of aalog circuits are maily focused o the fault locatio at the cost of augmetig the algorithm complexity or addig the test odes for data acquisitio. A robust method, which could explore the iformatio cotaied i the measuremet data of the CUT as much as possible without addig additioal test odes, is appealig. Meawhile, the method should idetify the ew iformatio from the ew measured data without acquirig more complexity of the algorithm, with aid of this iformatio, a robust ad sesitive sigature ca be obtaied so effectively that the fault locatio becomes more reliable ad robust.

3 Metrol. Meas. Syst., Vol. XX (03), No., pp Thaks to the Kalma filter techique, the objective that we wat to fulfil the above task ca be easily achieved. To the best kowledge of the authors, the Kalma filter based method for fault diagosis of aalog circuits has ot bee reported. Especially, the oisebased diagosis method preseted here is also ew. I this paper, the Kalma filter is used to acquire the variace of the Gaussia distributio, which is uique for a cocrete fault. Based o the Gaussia distributio, the Shao etropy is used as the sigature parameter to idetify the fault. This paper focuses o locatig the fault ad miimizig the aliasig probability effectively whe a parametric or catastrophic fault occurred i the aalog circuit. Simulatio ad experimetal results show the robustess of the proposed approach compared with the subbad decompositio ad coherece fuctio based method [0]. The remaider of this paper is orgaized as follows: I sectio III, a fault model is established, the Kalma filter is reviewed, the Kalma filter based method for fault diagosis of aalog circuits is illustrated. Two simulatios described i Sectio IV demostrate the approach. Sectio V provides a experimet validatio of the usefuless of the proposed method for fault diagosis i a actual aalog circuit. Coclusios are give i Sectio VI. 3. Diagosis method 3.. Diagosis model I order to fid the relatio betwee faultfree ad faulty output sigals, a ew fault model is proposed at first. Due to the iheret ucertaity comig from the measuremet ad system disturbace, the output of the aalog circuit may vary withi a acceptable rage aroud its omial value ad this will ot be treated as a fault. Therefore, it is reasoable to assume that the ideal faultfree output is costat, i.e., ideal faultfree output is used as the omial value. The actual faultfree respose deviates from the ideal faultfree output peacefully. Otherwise, the circuit is supposed to be faulty. So, the measured output of the aalog circuit ca be cosidered as the realizatio of a radom process, i which the ideal faultfree output ca be viewed as its mea value or omial value. With this i mid, the aalog circuit testig ca be viewed to address the problem of tryig to extract the sigature of the output sigal xî R of the CUT that is govered by the liear stochastic differece equatio x = x w () k k k With a measuremet y Î that is R yk = xk vk () vk The radom variable wk ad represet the process ad measuremet oise, respectively. They are assumed to be a ormal probability distributio: wk ~ N(0, Q), vk ~ N(0, R), where Q ad R represet the process oise variace ad measuremet oise variace, respectively. The subscript k i () ad () deotes the discrete time.

4 X.F. Li, Y.L. Xie: KALMAN FILTER BASED METHOD FOR FAULT DIAGNOSIS OF ANALOG CIRCUITS Remark : For aalog circuits with a catastrophic fault, we are assumig that the measuremet ucertaity vk is fixed, that is, the variace of v k stays costat durig the process of measuremet. xk i (), havig a large deviatio from the faultfree output, correspods to wk with a large variace. Remark : For a parametric fault, if it is supposed that the measuremet ucertaity v k i () is fixed as explaied i Remark, xk i () shows a small deviatio from the faultfree output, which correspods to wk with a small variace. Remark3: There is o such thig as a perfect measuremet. Measuremet of the variable are iflueced by a umber of elemetal error sources, such as errors caused by variatios i ambiet temperature, humidity, pressure, vibratios, electromagetic iflueces; usteadiess i the steadystate pheomeo beig measured, ad others. So, here, the measuremet ucertaity is quatitatively described by v k, which idicates the ucertaity itroduced by the imperfect measuremet. 3.. Noiseehaced testig scheme The method of usig suitable oise to ehace performace has bee proposed as a powerful sigal processig scheme [5], [6], [7]. However, to the best of the authors kowledge, oiseaided fault diagosis of aalog circuits is rarely reported. A detailed discussio of the oiseehaced fault diagosis scheme will be give i this sectio, ad the scheme proposed below is based o Kalma filterig. I our scheme, recallig (), the faultfree respose ad the white oise with a certai mea ad variace make the output of the faulty circuit. Furthermore, because of the uique role of each compoet i aalog circuits, it is reasoable to assume that the white oise correspods oetooe with a cocrete compoet fault i the circuits. So, it is atural to use the white oise as the objective of the fault sigature aalysis. The proposed scheme has a Kalma filter as the preprocessig block show i Fig.. The preprocessig block ca be represeted as the geeral Kalma filter with its filter characteristics depedig o the applicatio ad specificatio of the CUT. The Kalma filter takes the respose measured by the test equipmet, the geerates the result of oise estimatio for the curret state of the CUT. The oise is the fed ito the sigature geeratio blocks. The sigature has its ow faultfree value ad uique error tolerace. At last, the classifier is used as a comparator to detect ad locate faults i the CUT. respose Kalma filter (oise estimatio) sigature geeratio classifier pass/fail Fig.. Sigature aalysis scheme usig Kalma filterig.

5 Metrol. Meas. Syst., Vol. XX (03), No., pp Kalma filterig Sice the Kalma filter is used as the mai part of our proposed respose compressio scheme, a outlied discussio of the Kalma filter will be give i this sectio. I 960, R. E. Kalma published his famous paper describig a recursive solutio to the discretedata liear filterig problem []. Sice that time, due i large part to advaces i digital computatio, the Kalma filter has bee the subject of extesive research ad applicatio, particularly i the area of autoomous or assisted avigatio. Theoretically, the Kalma filter is a estimator for what is called the liearquadratic problem, which is the problem of estimatig the istataeous state of a liear dyamic system perturbed by white oise by usig measuremets liearly related to the state but corrupted by white oise. The resultig estimator is statistically optimal with respect to ay quadratic fuctio of estimatio error. Practically, the Kalma filter provides a meas for iferrig the missig iformatio from idirect ad oisy measuremet ad is also used for predictig the likely future courses of dyamic systems that people are ot likely to cotrol []. The Kalma filter addresses the geeral problem of tryig to estimate the state xî R of a discretetime cotrolled process that is govered by the liear stochastic differece equatio x = Ax Bu w (3) k k k k with a measuremet z m Î that is R The radom variables k z = Hx v (4) k k k w ad v k represet the process ad measuremet oise, respectively. They are assumed to be idepedet of each other, white, ad with ormal probability distributios: w ~ N(0, Q) (5) k v ~ N(0, R ) (6) k where Q ad R represet process oise covariace ad measuremet oise covariace, respectively. I practice, the process oise covariace Q ad measuremet oise covariace R matrices might slightly chage with each time step or measuremet, however here we assume they are costat. The matrix A i the differece equatio (3) relates the state at the previous time step k to the state at the curret step k. The l matrix B relates the l optioal cotrol iput uî R to the state x k. The m matrix H i the measuremet equatio (4) relates the state to the measuremet z k. We defie xˆ Î R (ote the super mius ) to be a prior state estimatio at step k give k kowledge of the process prior to step k, ad xˆ kî R to be a posterior state estimate at step k give measuremet z k. We the defie a priori ad a posteriori state estimate errors as e = x xˆ (7) k k k

6 X.F. Li, Y.L. Xie: KALMAN FILTER BASED METHOD FOR FAULT DIAGNOSIS OF ANALOG CIRCUITS e = x xˆ (8) k k k The a priori estimate error covariace is the P T = E[ e e ] (9) k k k Ad the a posteriori estimate error covariace is T P = E[ e e ] (0) k k k The Kalma filter estimates a process by usig a form of feedback cotrol: the filter estimates the process state at some time ad the obtais feedback i the form of (oise) measuremet. As such, the equatios for the Kalma filter fall ito two groups: time update equatios ad measuremet update equatios. The time update equatios are resposible for projectig forward (i time) the curret state ad error covariace estimates to obtai a priori estimates for the ext time step. The measuremet update equatios are resposible for the feedback, i.e. for icorporatig a ew measuremet ito the a priori estimate to obtai a improved a posteriori estimate [3]. The specific equatios for the time ad measuremet updates are preseted below. Discrete Kalma filter time update equatios are xˆ = Axˆ Bu () k k k T Pk APk A Q = () Discrete Kalma filter measuremet update equatios are K = P H ( HP H R) (3) T T k k k xˆ = xˆ K ( z Hxˆ ) (4) k k k k k P = ( I K H) P (5) k k k After each time ad measuremet update pair, the process is repeated with the previous a posteriori estimates used to project or practical implemetatios much more feasible tha (for example) a implemetatio of a Wieer filter [4] which is desiged to operate o all of the data directly for each estimate Noise estimatio based o Kalma filterig Step : Mote Carlo procedure for oise estimatio of ideal good circuit Due to the tolerace of aalog compoets, the faultfree output voltage is ot exactly equal to the respose derived from the trasfer fuctio of the CUT. So, i order to get the

7 Metrol. Meas. Syst., Vol. XX (03), No., pp actual respose of the CUT, it is idispesable to apply the Mote Carlo simulatio to estimatio of the mea ad variace of the output sigal, respectively. Step : The Kalma filter procedure for oise estimatio of a actual good circuit Owig to the evirometal oise impact, the curret distributio is ot equal to the desity distributio calculated i Step. Thus, it is reasoable to use the ideal faultfree respose as xk i eq.() ad the output comig from the actual circuit as yk of eq.(). Usig the Kalma filter method, we adjust the variace of wk to make the Euclidea distace betwee the ideal faultfree respose ad actual faultfree respose miimal, at that momet, the variace of wk is supposed to be the variace of the distributio of the actual faultfree respose. Step 3: The Kalma filter procedure for oise estimatio of a faulty circuit Because each fault correspods to oe cocrete Gaussia distributio, the fault ca be located if the distributio is idetified. The cocrete implemetatio is as follows. a) The mea ad variace are set to zero ad a small positive real umbers 0, respectively. The oe specific Gaussia is obtaied. b) Usig the Kalma filter to estimate the curret state k x, where xk is the actual faultfree output i eq.(), ad the faulty output is viewed as the measured sigal of curret state y k i eq.(). c) Calculate the Euclidea distace betwee xkad the measured faulty output y k. If the Euclidea distace meets the certai miimum requiremet, stop the process ad the correspodig variace i eq.() is what we wat. Else icrease s i to s i ad set s i as the ew variace of Gaussia distributio, ad the go to step b). The Euclidea distace is defied as where,,, / ù å x j h j ú (6) j= é d( x, y) = ê ( ) ë û x= ( x, x..., x ), y= ( hh..., h ) represet data sequece Etropy aalysis for sigature extractio Thaks to the Kalma filter based method i Sectio 3.4, the distributio correspodig to a cocrete fault is obtaied. The we use Shao s etropy to get the fault sigature Give pi ( i=,..., ) represets the probability of evet i happeig, Shao s etropy is defied: H( p, p, p ) =å p log (7) p i i= i Here, we use the base for the logarithm fuctio i Shao s etropy. The cotiuous versio of Shao s etropy is H( p) =ò p( x)log dx (8) p( x)

8 X.F. Li, Y.L. Xie: KALMAN FILTER BASED METHOD FOR FAULT DIAGNOSIS OF ANALOG CIRCUITS where p( x ) deotes the probability distributio of the system. Here, based o our assumptio, p( x) is the Gaussia distributio, that is, p( x) has the form: x p( x) = exp( ) ps s (9) where s represets the variace of Gaussia distributio 4 H Sigma Fig.. Sigature extractio usig Shao s etropy. Xaxis deotes the variace of Gaussia distributio, Yaxis deotes Shao s etropy From Fig., it is easily see that the Shao etropy is a mootoe fuctio related to the variace of Gaussia distributio. So Shao s etropy is uique for a defiite Gaussia distributio with a fixed variace. Therefore, Shao s etropy could be used as a tool for sigature compressio, that is to say, if the variace of Gaussia distributio is differet, the Shao etropy is also differet. Because differet faults result i differet variace of Gaussia distributio, the Shao etropy ca be used as the uique sigature for a cocrete fault of CUT. I fact, the physical explaatio of Shao s etropy is the iformatio used to be captured from imprecise kowledge that the oise cotais. The higher the Shao etropy, the worse the system quality is Classifier We use the miimum variace priciple as the criteria to match the fault(s), that is, we calculate the Euclidea distace betwee the predefied fault sigature ad the oe obtaied from the proposed method based o the measured output of the CUT. Oce the miimum is foud, the correspodig fault is regarded as the fault which had happeed i the actual aalog circuit Algorithm of the sigature extractio of the proposed diagosis scheme To make the above descriptio of the oise estimatio ad sigature extractio more clear, the flow diagram is used to help arrive at our purpose, that is, Fig.3 ad Fig.4, which are employed to exhibit the process of producig the faultfree sigature ad the faulty sigature of the aalog circuit uder test (CUT) via Kalma filterig ad Shao s etropy techiques, respectively.

9 Metrol. Meas. Syst., Vol. XX (03), No., pp It is oted that the geeratio of the sigature of the faultfree aalog circuit eeds Mote Carlo simulatio before usig the Kalma filterig techique, that is because for actual testig sceario, the actual faultfree oise ad its sigature are also eeded to estimate effectively whe there is o effective referece respose for calculatio. It is also worth otig that the iitial value of the variace s i (ors j ) ca be a rather big umber, ad decrease with a fixed step util the orm requiremet is satisfied, i fact, the iitial value of the variace ca also be set to be a small umber ad icrease with a fixed step util the correspodig orm coditio is fulfilled. I our practice, we set the iitial value of s ) as 0.0 deviated from its omial value of variace ad the icreased scheme is s i (or j employed to execute the iterative process. Vary each circuit compoets withi its tolerace limit ad apply Mote Carlo simulatio for faultfree CUT to obtai respose deoted{ } s i i= Sample the respose of the actual faultfree CUT to obtai respose deoted { } s i i= Sample the respose of the actual faultfree CUT deoted { } s i i= Sample the respose of the actual faulty CUT deoted{ f } i i s = Set Q= R= s i Eq.(5) ad (6) 0 Set Q= R= s i Eq.(5) ad (6) 0 Let x = k si, apply Kalma filter described i Sectio 3. to get the estimated respose of the actual faultfree CUT deoted { ˆ } s i i= Let x = k { si}, apply Kalma filter described i Sectio 3. to get the estimated respose of the actual faulty CUT deoted { ˆ } s i i= Use d( x, y) defied i Eq.(6) to calculate d( sˆ, s ) i i Use d( x, y) defied i Eq.(6) to f calculate d( sˆ, s ) i i No d( sˆ i, si ) < e Set Q= R= s j i Eq.(5) ad (6) No f d( sˆ i, si ) < e Set Q= R= s j i Eq.(5) ad (6) Yes Yes s j is the desired variace ad via Eq.(9), the Gaussia desity p( x) is obtaied s j is the desired faulty variace ad via Eq.(9), the Gaussia desity p( x) is obtaied Substitute p( x) ito Eq.(8), the Shao etropy is obtaied Substitute p( x) ito Eq.(8), the Shao etropy associated with fault is obtaied Fig. 3. Algorithm for faultfree sigature extractio. Fig. 4. Algorithm for fault sigature extractio. 4. Simulatios Here, simulatios will be carried out to illustrate the idea of this paper. All simulatio work is fiished i a persoal computer with a 3GHz processor ad GB radom access memory. The programs used i the simulatios are developed by the authors i MATLAB.

10 X.F. Li, Y.L. Xie: KALMAN FILTER BASED METHOD FOR FAULT DIAGNOSIS OF ANALOG CIRCUITS 4.. Simulatio Leapfrog filter circuit Results of applyig the proposed method to a bechmark circuit are explaied i this sectio. The leapfrog filter circuit i Fig.5 is from [8]. All the resistors are 0 kw, ad capacitor C ad C4 are 0.0 m F, ad capacitors C adc 3 are 0.0m F. The leapfrog filter is a kid of low pass filter, which is based o MITEL semicoductor s complemetary metaloxidesemicoductor (CMOS) techology. Several faults are iserted ito the leapfrog filter, ad the filter is simulated with a siusoidal iput of khz frequecy ad ±.5V amplitude. The simulatio is doe usig PSPICE. The samplig frequecy is 00 khz ad a total of 048 samples were gathered i 0ms. The simulatio results are give i Table I. I the table, % error meas the error betwee a faulty circuit ad the faultfree circuit i percetage ad is the biggest by calculatio. ˆ s deotes estimated variace of the Gaussia distributio. C R 3 V i 0 R R 4 R4 5 5 R R 6 C 0 R 7 R 3 C3 R 4 C 3 R 8 4 R 9 6 R 0 7 R 9 V o Fig. 5. Leapfrog filter. Fault No. TABLE I RESULTS OF SIGNATURE ANALYSIS USING KALMAN FILTER METHOD FOR FAULT LOCATION IN THE LEAPFROG FILTER Fault Faulty value ˆ s value (%error) Shao etropy value H Node 0&4 short 00W (.0).078 Node 0&4 short 00W (9.8) Node 0&0 short 00W (59.9).77 4 Node 5& short 00W (79.9) Node 9&0 short 00W (3.0).06 6 Node 6&3 short 00W (.0) Node 6&7 short 00W (7.).84 8 Node 9&3 short 00W (3.).09 9 R ope MW (46.4) R3 ope R5 ope R9 ope R0 ope MW MW MW MW (9.9) (5.5) (36.) (4.3) R variatio (0k>8k) 8kW (5.).0 5 R3 variatio (0k>8k) 8kW (7.).47 6 R7 variatio (0k>7.5k) 7.5kW (8.3).88 7 R variatio (0k>8k) 8kW (6.).63 8 C3 variatio (0.0u>0.03u) 0.03u (47.7) Actual faultfree *.008 (0).050

11 Metrol. Meas. Syst., Vol. XX (03), No., pp For a catastrophic fault, the proposed method gives good results for fault locatio. Because of the legth limitatio ad the emphasis of this paper, oly the radomly selected four catastrophic faults are aalyzed. I Table I, for example, for cases 4, all the values of Shao s etropy are differet from the faultfree case, clearly a 00% detectable rate. It is also oted that the Shao etropy i case, case, case 3, case 4 are.078, 3.86,.77,.897, respectively, ad.050 for faultfree circuit, which is uique for differet faults. So the differet fault cases ca be separated easily usig Shao s etropy as the sigature. The parametric faults are more complicated sice variatios of faulty elemets are smaller tha those of the catastrophic faults. The proposed method ca also give good results for locatig the parametric fault. For example, the Shao etropy for case 4 i Table I is.0, whereas the value of case 5 is.47. The sigatures for case 4 ad case 5 are differet, so these two faults ca be separated defiitely. The same coclusio ca also be draw for other fault cases i Table I. Data Bak No. TABLE II ROBUSTNESS COMPARISON BETWEEN COHERENCE FUNCTION RESULTS AND SHANNON S ENTROPY RESULTS FOR FAULTS 4 AND 5 IN TABLE Fault Coherece Fuctio (subbad No.) Shao s etropy value H R variatio (0k>8k) 3.(5).97 R variatio (0k>8k).8(5).0 3 R variatio (0k>8k).57(5) R variatio (0k>8k) R variatio (0k>8k) R variatio (0k>8k) R3 variatio (0k>8k) R3 variatio (0k>8k) R3 variatio (0k>8k) R3 variatio (0k>8k) R3 variatio (0k>8k) R3 variatio (0k>8k) 3.3(5).97(5).68(5) 0.0(7) 0.005(7) 0.05(7) 0.009(7) 0.04(7) 0.08(7) Because a robust fault sigature ca dramatically reduce the probability of aliasig, here a aalysis will be give to illustrate the robustess of the sigature produced by the proposed approach. The method preseted i [0] is employed for compariso. The parametric faults for aalysis are fault cases 4 ad 5 i Table I. The results of compariso are show i Table II. From Table II, we see that the method proposed i [0] gives good discrimiatio for differet parametric faults, but its sigature, that is, the value of coherece fuctio is ot robust. For example, the maximal value of the coherece fuctio of case 4 is 3., ad the miimal value is.57 (for a egative value, we take its absolute value). The biggest relative error is 04.5% for the sigature used i the method [0]. Whereas the biggest relative error is oly 0.06% whe the Shao etropy is used as the sigature based o the Kalma filter method. For fault case 5, the same coclusio ca also easily be draw from Table II. Sice sigature with the property of big variatio is proe to aliasig, which makes the fault locatio impossible evetually, a robust sigature could greatly reduce the aliasig probability ad make the fault detectio ad locatio more easily. Thus, the sigature produced by usig the proposed method is much more efficiet tha that of usig the method i [0] for fault diagosis of aalog circuits. Fig.6 shows the origial fault respose ad the faultfree output. The fault respose estimated by usig the proposed method is depicted i Fig.7

12 X.F. Li, Y.L. Xie: KALMAN FILTER BASED METHOD FOR FAULT DIAGNOSIS OF ANALOG CIRCUITS.5 Leapfrog filter simulatio Faultfree R Fault 0.5 Amplitude(V) Time(s) x 0 3 Fig. 6. Leapfrog filter: Faultfree respose (blue dotted lie) ad R parameter fault respose (red solid lie). Xaxis deotes the time (s), Yaxis deotes the amplitude (v)..5 Noise Iserteded Kalma Estimate R Fault 0.5 Amplitude(V) Time(s) x 0 3 Fig. 7. Leapfrog Filter: The Kalma filter based method for realizatio of the R parametric fault respose (red dotted lie) by usig the faultfree respose plus Gaussia white oise with a proper variace (blue solid lie). Xaxis deotes the time (s), Yaxis deotes the amplitude (v). 4.. Simulatio State variable filter circuit The state variable filter [8] i Fig.8 is used as the secod bechmark circuit for the simulatio. All the capacitors are 0.0mF ad all the resistors are 0 kw except R, R =3 kw. The iput to system is a khz siusoidal sigal ad the output of the system is measured at the bad pass output pi. The samplig frequecy is 5 khz ad samples are gathered for 0.4ms.

13 Metrol. Meas. Syst., Vol. XX (03), No., pp V i R R HPO R 3 R 5 C C 3 4 BPO R 4 LPO R 7 R 6 Fig. 8. State variable filter. Fiftee fault cases icludig six catastrophic faults ad ie parametric faults are cosidered. All of the fault cases are radomly give. The testig results are show i Table III, where % error meas the error betwee a faulty circuit ad the faultfree circuit i percetage ad is the biggest by calculatio. ˆ s deotes the estimated variace of Gaussia distributio. From Table III it is easily see that the proposed method ca locate catastrophic faults effectively, as is the parametric faults. That is, there is a oetooe mappig betwee the fault ad the Shao etropy, which is used for sigature i our scheme. For example, the Shao etropy of fault case is.8 ad that of case 3 is.95, they are all differet from the Shao etropy of the faultfree case, ad sice they are ot equal, these two fault cases ca be discrimiated easily. The same coclusio ca also bee draw for ay other fault cases i Table III. Further, because of the robustess of the Shao etropy, the sigature we derived here is early a costat, so the declaratio of the fault for the CUT is more reliable. Fault No. TABLE III RESULTS OF SIGNATURE ANALYSIS USING KALMAN FILTER METHOD FOR FAULTS LOCATION IN THE STATE VARIABLE FILTER Fault Faulty value ˆ s value (%error) Shao etropy value H Node &3 short 00W (3.3).8063 Node 3&4 short 00W (.9) Node &4 short 00W (36.43) Node &3 ope Node &4 ope Node 4&3 ope MW MW MW (.9) (.60) (.3) R variatio (3k>.4k).4kW (4.57) R3 variatio (0k>8k) 8kW (0.40) R variatio (0k>7.5k) 7.5kW (6.3) R4 variatio (0k>8k) R5 variatio (0k>8k) R6 variatio (0k>8k) R7 variatio (0k>8k) C variatio (0.0u>0.03u) C variatio (0.0u>0.03u) 8kW 8kW 8kW 8kW 0.03u 0.03u (5.5) (5.54) (6.99) (0.73) (.89) (3.35) Actual faultfree * (0) The Experimet Other tha the simulatios, a experimet is carried out o the actual circuit to verify the proposed method.

14 X.F. Li, Y.L. Xie: KALMAN FILTER BASED METHOD FOR FAULT DIAGNOSIS OF ANALOG CIRCUITS The circuit uder test is TowThomas filter circuit. The structure ad parameters of this circuit [8] are illustrated i Figure 9, that is, R = R = R3 = R4 = 6kW, R5 = R6 = 0kW, C = C = F. The actual circuit is show i Figure 0, ad the amplifiers i the circuit are TL084 produced by TI. The circuit is stimulated by a V, 0 khz siusoidal sigal, ad the output of the faultfree ad faulty circuit are measured by a oscilloscope produced by Tektroix. The sample rate of the oscilloscope is GS/s ad its bad width is 00M. Eight fault cases are radomly chose, ad the test results are listed i Table IV. R3 Vi R C R4 _ R5 R6 _ Vout R C _ Fig. 9. TowThomas filter. Fig. 0. The actual TowThomas filter. I Table IV, fault ad fault represet two types of parametric faults of oe compoet, that is, for fault, the value of compoet C is smaller tha its omial value, whereas for fault, the value is bigger tha the omial value of C. The same is the case with fault 3 ad fault 4, fault 5 ad fault 6, respectively. Fault 7 is a multiple parametric fault caused by the variatios of the C ad R6 ad R together. Fault 6 is the same case as fault 7. Table IV is also a illustratio of the good discrimiatio effects for differet parametric faults i a aalog circuit by usig the proposed method. All the results show that the faults ca be precisely located without aliasig. For example, for fault case, the Shao etropy is 0.853, whereas its value is for fault case. They are differet values ad all of them are ot equal to the value of the faultfree circuit, so fault case ad fault case ca easily be separated. The same coclusio ca be draw for ay other cases as well. Fault No. TABLE IV RESULTS OF SIGNATURE ANALYSIS USING KALMAN FILTER METHOD FOR FAULT LOCATION IN THE TOWTHOMAS FILTER Fault Faulty value ˆ s value (%error) Shao etropy value H C Variatio (F>0.8F) 0.8F (3.80) C variatio (F>.F).F (.8) R6 variatio (0k>9.k) 9.kW (4.56) R6 variatio (0k>k) kw (3.04) R variatio (6k>4.7k) 4.7kW (.66) R variatio (6k>6.9k) 6.9kW (5.09) Fault Fault 3 Fault 5 0.8F 9.k 4.7 k (5.3) Fault Fault 4 Fault (0.38) F k 6.9k 0 Fault free * (0) 0.879

15 Metrol. Meas. Syst., Vol. XX (03), No., pp No. TABLE V ROBUSTNESS OF SIGNATURE BY USING KALMAN FILTER BASED METHOD FOR DIAGNOSIS IN THE TOWTHOMAS FILTER Fault Faulty value ˆ s value Shao etropy value H C Variatio (F>0.8F) 0.8F C Variatio (F>0.8F) 0.8F C Variatio (F>0.8F) 0.8F C Variatio (F>0.8F) C Variatio (F>0.8F) C Variatio (F>0.8F) C Variatio (F>0.8F) C Variatio (F>0.8F) C Variatio (F>0.8F) C Variatio (F>0.8F) C Variatio (F>0.8F) C Variatio (F>0.8F) 0.8F 0.8F 0.8F 0.8F 0.8F 0.8F 0.8F 0.8F 0.8F Furthermore, the multiple parametric faults ca also be located by usig the proposed method. For example, for fault 7, its sigature the Shao etropy is , while for fault 8, its sigature value is , which is differet from the value of fault case 7, thus these two multiple parametric faults ca both be located. Especially, the sigature obtaied by usig the proposed method also shows strog robustess for a actual circuit. For example, show i Table V, the fault case i Table IV is cosidered, that is, the C dow drift parametric fault, the differece betwee its maximal sigature value ad miimal value is oly So, the probability of aliasig is greatly reduced by usig the proposed method for the fault diagosis of actual aalog circuits. 6. Coclusios A sigature with robust discrimiatio capacity is crucial for fault locatio i aalog circuits. The oiseaid scheme provides a way to produce the robust sigatures. Whe faults occur, oe special oise correspodig to the faults arises. To use the iformatio cotaied i the oise for sigature extractio, a ew fault model ad a oise estimatio method for fault locatio i aalog circuits have bee proposed i this paper. The approach combies Kalma filterig with Shao s etropy to aalyze the sigature i aalog circuits. Simulatios ad experimetal results show that the approach provides a robust sigature both for catastrophic faults ad parametric faults. Based o these sigatures, the parametric faults ad the catastrophic faults i aalog circuits ca be located effectively ad reliably. Compared with the existig method (the subbad method employig coherece fuctio [0]), the approach is this paper ca produce a more stable sigature i locatig parametric compoets i aalog circuits. So the proposed method improves the discrimiatio capacity strogly. The simulatio examples ad experimetal results show the advatages. Further, the proposed method oly requires the measuremet of the output sigal from the system primary output pis so that both the parasitics ad the testig cost are reduced. Sice the Kalma filter ca be built with a small hardware cost, the proposed approach i this paper ca be used for the olie testig sceario. Ufortuately, the proposed method ca oly be applied to the diagosis of liear aalog circuits due to the limitatio itroduced by the Kalma filter. However, with the help of oliear Kalma filter methods, such as the exteded Kalma filter (EKF), the approach i this paper ca also be exteded to diagose the fault i oliear aalog circuits, ad this will be our future work. The reader is ivited to attack this problem.

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