AN APPLICATION OF THE TCRBF NEURAL NETWORK IN MULTI-NODE FAULT DIAGNOSIS METHOD

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1 XIX IMEKO World Congress Fundamental and Appled Metrology September 6 11, 2009, Lsbon, Portugal AN APPLICATION OF THE TCRBF NEURAL NETWORK IN MULTI-NODE FAULT DIAGNOSIS METHOD Zbgnew Czaja 1, Mchał Kowalewsk 2 1 Gdansk Unversty of Technology, Faculty of Electroncs, Telecommuncatons and Informatcs, Department of Optoelectroncs and Electronc Systems, Gdansk, Poland, zbczaja@pg.gda.pl 2 Gdansk Unversty of Technology, Faculty of Electroncs, Telecommuncatons and Informatcs, Department of Optoelectroncs and Electronc Systems, Gdansk, Poland, Mchal.Kowalewsk@et.pg.gda.pl Abstract Ths paper presents the new self-testng method for dagnoss of analog parts n mxed-sgnal embedded systems controlled by mcrocontrollers. The tested analog part s stmulated by a snus-wave suppled by the onboard generator and ts responses are sampled n selected nodes by mcrocontrollers ADC. The measurement space s represented by dfferences between values of selected node voltages. Fault detecton and localzaton s performed by a Two-Center Radal Bass Functon (TCRBF) Neural Network. The dagnoss procedure was mplemented and smulated n a PC. Keywords fault dagnoss, neural network, BIST 1. INTRODUCTION At present, embedded electronc systems whch are characterzed by an ntellgent unt, often based on a mcrocontroller, a dgtal sgnal processor (DSP) or a programmable devce (e.g. FPGA, CPLD), predomnate on the electronc market, because nformaton about the operatonal envronment and controlled objects are often obtaned va analog sensors. Analog sgnals are transmtted and ntally processed n analog parts, however analog-todgtal processng and data processng are realzed by dgtal parts. Therefore, the analog part has to work correctly, because an ncorrect measurement sgnal can lead to a wrong decson of the control unt, what can even result n damage of the controlled devce. Hence, the embedded system should be able to run self-testng of analog parts. Self-testng of analog crcuts bases on fault dagnoss methods. When elaboratng these methods we have to take nto consderaton contnuous values of crcut elements, the contnuous nature of stmulaton and response sgnals and the fact that these sgnals can assume the shape of any functon, the presence of element tolerances and crcut nonlneartes. Addtonally, especally for non-electrcal objects modelled by electrcal crcuts we have only poorly defned system models. Thus neural networks can be a very effcent soluton n fault dagnoss methods. Many types of neural networks are used as fault classfers: backpropagaton [1], probablstc [2], self-organzng [3] radal bass functon [4,5], neural networks based on TCRB Functons [6]. In ths paper a new self-testng procedure wll be presented. It bases on a new mult-port fault dagnoss method and the smplfed TCRBF neural classfer [7]. The dagnoss method s used to create a fault dctonary n the form of dspersed localzaton curves deally suted for the TCRBF neural classfer. Addtonally, measurements of parameters of the tested analog part are performed by the reconfgurable BIST created from perpheral devces of the control unt controllng the electronc embedded system. 2. FAULT DIAGNOSIS METHOD The method wll be llustrated on an exemplary analog part represented by the low-pass Tow-Tomas flter shown n Fg. 1. Fg. 1. Tested analog crcut low-pass Tow-Tomas flter, where R1 = R2 = R3 = R4 = R5 = R6 = 10 kω, C1 = 33 nf, C2 = 6.8 nf Generally, the self-testng procedure conssts of three stages. At frst, durng the pre-testng stage, the TCRBF neural classfer s constructed on the bass of a famly of dspersed localzaton curves. These curves descrbe the behavour of the tested crcut followng changes of ts elements values. The second stage s responsble for snuswave stmulaton of the tested crcut and for measurements of real and magnary values of voltages n accessble nodes. At the last stage, the TCRBF neural classfer detects and localzes a fault of the analog part Idea of the method Let X N be a set of N accessble measurement nodes X N = {X 1, X 2,..., X N }. Thus, we have the followng set of values of node voltages: u N = {v n,w n } n = 1,..., N, where

2 u n = v n + j w n. These voltages can drectly create a (2 N)- dmensonal measurement space n a way smlar to [9,10], but n ths case localzaton curves representng devatons of values of crcut elements often characterze wdely dfferent lengths, whch depends on crcut senstvty, and often they are not stuated optmally. Ths makes t dffcult to perform correct fault localzaton. Hence, t s proposed to create a new measurement space, where coordnates have the followng forms: {v n,n-1, w n,n-1 } n = 2,.., N, where v n,n-1, = v n - v n-1, w n,n-1, = w n - w n-1. That s we create the space represented by dfferences between values of node voltages. In ths case we mprove the dynamcs of length changes of localzaton curves. Lengths are more smlar between themselves and curves are more separated as shown n Fg. 2 and Fg. 3. It wll be explaned on the example of the crcut shown n Fg. 1. The denomnator for all ts node transfer functons s followng: ( R R R R X R R R + X X R ) d = R (1) R6 where X 1 = -1/(jω C 1 ), X 2 = -1/(jω C 2 ). It s dependent on all elements, especally on R 1. However, the numerators for next node transfer functons have the forms: Thus, we obtan the followng dfferences between numerators of transfer functons for respectve nodes: (2) n two three-dmensonal spaces (Fg. 2, 3). Localzaton curves n these fgures were drawn for ±50% changes of element values wth reference to ther nomnal values p j nom. Fg. 2. The map of localzaton curves for the tested crcut n the 3-dmensonal measurement space: Re(u 2,1 ), Im(u 2,1 ), Re(u 3,2 ). Localzaton curves are a graphcal descrpton of crcut propretes followng from changes of ts element values. All curves cross at a pont whch s the nomnal state of the crcut. Assumpton of element tolerances causes dsperson of localzaton curves. Thus we need a classfer wth good generalzaton capabltes to correctly localze faults. Ths requrement can be fulflled by the TCRBF classfer, whch s constructed on the bass of dspersed localzaton curves. It s seen from (2) and (3) that the dfference m 2 m 1 between the 1 st and the 2 nd node functon numerators s dependent on the same elements as the respectve numerators. The same stuaton s for the m 3 m 2 dfference. That s we decrease the sze of the measurement space what decreases the sze of the fault dctonary, smultaneously keepng the same level of the localzaton resoluton. But ths operaton ncreases the dynamc of measurement sgnals, because sgnals for neghbourng nodes are moved about 180 and they added to themselves (see Fg. 1). The new measurement space s descrbed by the followng transformaton: (3) T ( p ) N = ( n= 2 Re + Im ( u ( p ) u ( p )) n n 1 ( u ( p ) u ( p )) ) n n 1 2n 2 2n 3 where: n - s a coordnate vector along the n axs, u n (p ) complex value of voltage n the node n for a change of p value of the -th element, = 1,..., I, I the number of crcut elements, N the number of accessble measurement nodes. The transformaton (4) maps the changes of crcut element values {p } =1,,I nto a famly of localzaton curves placed n the (2 N 2)-dmensonal measurement space. For the crcut shown n Fg. 1 we have a 4- dmensonal space. Thus the transformaton (1) s presented (4) Fg. 3. The map of localzaton curves for the tested crcut n the 3-dmensonal measurement space: Re(u 2,1 ), Im(u 2,1 ), Im(u 3,2 ) The measurement procedure The measurement procedure s performed by mcrocontroller ATmega16 and ts nternal devces [11]: analog multplexer, 10-bt ADC and 16-bt Tmer 1 (Fg. 4). The tested analog part s stmulated by the snus-wave and ts responses are sampled n partcular nodes by the ADC n moments establshed by the Tmer.

3 vn = wn = ( un,1 un,1 + un,2 un,2 ) ( u u u u ) n,1 n,2 n,2 n,1 where the value χ = 1 / U n s constant and known, what consderably smplfes calculatons of these values, because we use only addton and multplcaton operatons. Next t calculates dfferences between these values: v n,n-1, = v n - v n-1, w n,n-1, = w n - w n-1. χ χ (5) Fg. 4. Example of the electronc embedded system n the selftestng confguraton Tmngs of the measurement procedure are shown n Fg. 5. Each sgnal s sampled three tmes, where tme dstances between samples are set to one fourth of the perod T. Tme dstances between samples for subsequent sgnals are equal to a half of the perod. Thanks to ths, t s allowed to sample all sgnals at the same moments n relaton to the start of samplng, because samplng of the next sgnal s shfted about one perod. Samplng of the nput sgnal u n s needed to establsh a random shft tme T α of the samplng seres. The thrd sample of each sgnal s used to elmnate the voltage offset. The frst u n,1 and second u n,2 samples of the node voltage sgnal u n are used to calculate ther real and magnary values. Fg. 6. The algorthm of the measurement procedure 3. USE OF THE TCRBF NEURAL NETWORK Fg. 5. Tmngs of the measured sgnals durng the self-testng procedure n the nput node and nodes X 1 and X 2. The algorthm of the measurement procedure (Fg. 6) conssts of a measurement part whch s responsble for control of the measurement. The man functon confgures the ADC, starts the tmer and wats for the end of samplng (t wats for the measurement of nne voltage samples). The nterrupt servce of the tmer starts the ADC converson and actualzes the countng tme between next samples. The nterrupt servce of the ADC converson complete saves the voltage samples and changes the channel of the analog multplexer to measure the next node voltage. Durng the calculaton part, the mcrocontroller computes the real v n and magnary w n parts of the n-th node voltage basng on: Detecton and localzaton of faults n the proposed method are performed by a neural network wth TCRB functons [6,7]. The usefulness of the TCRB functon n dctonary methods based on localzaton curves follows from the fact that the localzaton curve (Fg. 7a) can be transformed only wth a few TCRB functons (Fg. 7d). In advance, some experments wth Radal Bass Functon Neural Networks (RBFNN) for dagnoss purposes were performed [4,5]. Unfortunately an accurate transformaton of localzaton curves wth RBFNN requres a lot of Radal Bass (RB) functons n the archtecture (Fg. 7b). Hence t appears an dea to construct a new neuron for better transformaton of stretched clusters of dspersed localzaton curves. The TCRB functon radally maps the space around a lne segment wth endponts c (1) and c (2) (Fg. 7c) wth the followng equaton: n 1 2 y( x ) = exp ( ( )) ( ) x w x, (6) 2 s x = 1

4 where: s(x) s the scalng functon descrbng the localzaton curve dsperson changng form σ 1 to σ 2 and w (x) ( = 1, 2,, n) are center functons dependng on coordnates of centers c (1) and c (2) [6]. real values M y 1 v 2,1 w 2,1 M 2 y 2 v 3,2 f 1 f 2 f 0,1 - values w 3,2 Input layer 1 st hdden layer wth TCRB functons M C y C λ 2 nd hdden layer wth maxmum functons Output layer Indcaton faulty component f C Indcaton state of the crcut s Fg. 9. The TCRBF neural network structure. Fg. 7. Idea of localzaton curve transformaton wth RB and TCRB functons. Applyng some TCRB functons to each localzaton curve of the testng crcut gves the possblty to obtan nformaton about dstances between the measurement pont and localzaton curves. Ths nformaton s gven as a vector of real values from the range (0, 1]. If the measurement pont les near the localzaton curve then one of TCRB functons assgned to that curve has the value close to 1. Otherwse, f the measurement pont les far from the localzaton curve, all TCRB functons assgned to ths curve have values close to 0. The number of requred TCRB functons for each localzaton curve depends on ts curvature. For a curve smlar to a straght lne only one TCRBF s needed. The more bent the curve s the more TCRB functons are requred. A graphcal llustraton of actvaton regons of TCRB functons for exemplary dspersed localzaton curves s shown n Fg. 8. Fg. 8. Graphcal llustraton of actvaton regons of TCRB functons for exemplary dspersed localzaton curves Archtecture of TCRBF classfer The TCRBF classfer proposed n ths paper s a threelayer feed-forward network (Fg. 9). Its archtecture s based on other neural networks wth RB functons (RBFNN, Probablstc, Generalsed Regresson). TCRB functons assgned to localzaton curves are placed n the 1 st hdden layer. They play the role of radal mappng of dspersed localzaton curves to real values from the range (0, 1]. Dsperson s caused by crcut element tolerances. Neurons n the 2 nd hdden layer group TCRB functons n C classes, whch nclude sngletons and ambguty groups, and produce maxmum values (M) of TCRBF outputs. The output layer s a perceptron wth a hard lmt transfer functon, whch produces a vector f wth 0 and 1 values. Each neuron n the output layer s fed only wth two sgnals: maxmum value of TCRB functons assgned to a class and constant bas λ. The network output vector f combnes nformaton about the state of the testng crcut and moreover about the fault class. An addtonal summng functon sums perceptron output values and gves the value s = f 1 + f f C, whch presents the state of the testng crcut as a sngle nteger value from the range [0, C]. The TCRBF classfer s able to dstngush between three states of the crcut: fault-free state (s = C), sngle fault or ambguty group (1 s < C), multple fault (s = 0). A. Fault free state If the measurement pont s located near the nomnal pont then all TCRB functons located closest to the nomnal pont and assgned to dfferent classes have output values y 1,, y C greater than λ. Hence the sum of all values from the perceptron outputs gves s = C. It ndcates a fault-free state of the crcut. B. Sngle fault or ambguty group Else f the measurement pont s located near one of localzaton curves assgned to a fault class and far from others, then only one value y s grater than λ and the sum of all values from the perceptron outputs gves s = 1. In ths case a sngle one n the output vector f ndcates a fault class (sngle fault or ambguty group). We meet wth ambguty also when 1 < s < C. In ths case the measurement pont s located near more than one localzaton curves assgned to dfferent fault classes.

5 C. Multple fault Fnally f the measurement pont s located far from any of localzaton curves then all values y are lower than λ and the sum of all values from the perceptron outputs gves s = 0. In ths case we meet wth a multple fault and t s mpossble to ndcate whch element s faulty. A constant parameter λ (0, 1), appled to the perceptron layer, makes t possble to descrbe the szes of decson regons n the measurement space. For λ close to 1 decson regons are narrowed to spaces located near localzaton curves. It s best suted for crcuts wth small element tolerances. Otherwse, f λ s close to 0, decson regons are wde and crcuts wth greater element tolerances can be correctly dagnosed Constructon of TCRBF classfer In contrast to RBF or feed-forward back-propagaton (FFBP) neural networks, TCRBFNN does not requre tranng. Centers of TCRB functons are placed on localzaton curves of the testng crcut wth a known model. Scalng parameters of TCRB functons are obtaned on the bass of Monte Carlo analyss n selected centers. The center selecton method bases on lnear nterpolaton of localzaton curves. Startng from two centers placed on extreme ponts of the localzaton curve, n next steps ponts farthest to the nterpolaton curve are selected as new centers and also as new nterpolaton nodes. Ths procedure s repeated untl the maxmum dstances between localzaton and nterpolaton curves are lower then the assumed dstance d max. In the second step for all selected centers the Monte Carlo analyss s performed n order to obtan localzaton curves dsperson. Wth assumpton of normal dstrbuton of ponts n each cluster, standard devatons of multvarate normal dstrbuton are estmated and used as scalng parameters of TCRB functons. Fg. 10 presents nterpolaton curves and data clusters obtaned from the Monte Carlo analyss wth assumpton of 3% element tolerances for the crcut shown n Fg. 1. bnd dentfcaton curves wth fault classes. In case of lack of the crcut model, when only expermentally acqured measurements are avalable, TCRBF centers can be obtaned wth the fuzzy c-means clusterng algorthm [4]. The parameter λ of the output layer s selected expermentally n order to decrease the classfcaton error Smulaton results The proposed dagnoss procedure wth the TCRBF classfer was used for faults detecton and localzaton n the low-pass flter shown n Fg. 1. Three TCRBF classfers wth dfferent archtectures were constructed (Table 1) and ther generalzaton capabltes were compared. A dfferent number of the followng nput coordnates was used n each classfer: Re(u 2,1 ), Im(u 2,1 ), Re(u 3,2 ), Im(u 3,2 ). Input dmenson Table 1. TCRBF classfers parameters. Number of TCRB functons Network archtecture Number of TCRB functons per each element R 1 R 2 R 3 R 4 R 5 R 6 C 1 C 2 2D D D The number of TCRB functons and classes n each classfer depends on the dmenson of the nput space (except the curvature of localzaton curves). For more dmensons, localzaton curves are more dstant from each other and faults are more dstngushable. In ths case more TCRB functons are needed and more classes can be created. Localzaton curves of some elements cover each other and form ambguty groups, for example {R 4, C 2 }. Hence one needs to apply TCRB functons only for one element n the ambguty group. An ncrease n the number of nput dmensons from 2 to 3 caused separaton of localzaton curves n the ambguty group {R 3, R 5, R 6, C 1 }. For ths group two classes were formed: {R 3,C 1 } and {R 5, R 6 }. Graphcal llustratons of the 1 st hdden layer (wth TCRB functons) actvaton regons n 2- and 3-dmensonal nput spaces are shown n Fg. 11 and Fg. 12. We can see a very good ft to localzaton curves. Fg. 10. Interpolaton curves and data clusters obtaned for the crcut shown n Fg. 1 and used n TCRBF classfer constructon stage. Connectons between the 1 st and the 2 nd hdden layer are made easly, as we know the crcut model and know how to Fg. 11. Graphcal transformaton of dspersed localzaton curves by the TCRBF neural classfer n the 2-dmensonal measurement space wth coordnates: Re(u 2,1 ), Im(u 2,1 ).

6 4. CONCLUSIONS Fg. 12. Graphcal transformaton of dspersed localzaton curves by the TCRBF neural classfer n the 3-dmensonal measurement space wth coordnates: Re(u 2,1 ), Im(u 2,1 ), Re(u 3,2 ). The classfcaton error of the created TCRBF classfers dd not exceed 2% and was manly caused by mproperly classfed sgnatures near the nomnal state of the crcut. The TCRBF classfer constructed on a data set belongng to dspersed localzaton curves s able to dstngush between the nomnal state of the crcut, sngle faults and multple faults (Fg. 13). Ths feature can be acheved wth the RBF neural network, but as mentoned earler, a correct transformaton of localzaton curves requres a sgnfcant ncrease n the number of RB functons. Fg. 13. Decson regons of the TCRBF classfer wth the archtecture n the 2-dmensonal measurement space for λ = 0.5. In the case of the FFBP neural network wth the archtecture , traned on the same data set as the TCRBF classfer, the classfcaton error on the test set exceeds 30%. It follows from the fact that neurons n FFBP network dvde the pattern space usng hyperplanes. Hence decson regons of that network go beyond dspersed localzaton curves and extend to nfnty. To decrease ths effect the FFBP network requres addton of a multple faults data set n the tranng data set. Addtonally the number of samples of the tranng set assgned to dspersed localzaton curves must be ncreased. Ths consderably extends the classfer constructon procedure wth reference to the TCRBF neural network. The new self-testng method wth the TCRBF classfer presented n ths paper s an effcent way to dagnose analog parts n mxed-sgnal embedded systems controlled by mcrocontrollers. The novelty of the method les n creatng a new measurement space represented by dfferences between values of node voltages. The TCRBF classfer used to detect the state of the crcut, as opposed to other types of neural classfers (back-propagaton, radal bass, probablstc), has generalzaton capabltes deally suted for dctonary methods based on localzaton curves. Unlke FFBP and RBF networks, the TCRBFNN requres a lower sze of the data set used for constructon. Hence the constructon phase s shorter and less burdensome. Furthermore, on the bass of the lmted sze of the constructon data set, the TCRBF classfer gves the possblty to extend dagnostc nformaton about the testng crcut. It s possble to dstngush between a fault-free state of the crcut, a sngle fault, an ambguty group and a multple fault. REFERENCES [1] Czaja Z., Zelonko R. Fault dagnoss n electronc crcuts based on blnear transformaton n 3-D and 4-D spaces, Trans. on Instr. and Measurement, vol. 52, nº 1, pp , [2] Yang Z. R., Zwolnsk M. Applyng a robust heteroscedastc probablstc neural network to analog fault detecton and classfcaton, IEEE Trans. CAD IC&S, vol. 19, pp , [3] Collns P., Yu S., Eckersall K. R., Jervs B. W., Bell I. M., Taylor G. E. Applcaton of Kohonen and supervsed forced organzaton maps to fault dagnoss of electronc analog crcuts, Electronc Letters, vol. 30, nº 22, pp , [4] Catelan M., Fort A. Fault dagnoss of electronc analog crcuts usng a radal bass functon network classfer, Measurement, vol. 28, Issue 3, pp , [5] Toczek W., Kowalewsk M. A neural network based system for soft faults dagnoss n electronc crcuts, Metrology and Measurements Systems, vol. 12, nº 4, pp , [6] Kowalewsk M. Two-center radal bass functon network for classfcaton of soft faults n electronc analog crcuts, In proc. of the IMTC 2007 Conference, Warsaw, Poland, [7] Czaja Z., Kowalewsk M. A new method for dagnoss of analog parts n electronc embedded systems wth two center radal bass functon neural networks, 16 th IMEKO TC4 Symposum, pp , Italy, Florence, September, [8] Czaja Z., A dagnoss method of analog parts of mxed-sgnal systems controlled by mcrocontrollers, Measurement, vol. 40, nº 2, pp , [9] Czaja Z., Usng a square-wave sgnal for fault dagnoss of analog parts of mxed-sgnal electronc embedded systems, IEEE Transactons on Instrumentaton and Measurement, vol. 57, nº. 8, pp , August, [10] Czaja Z., A fault dagnoss algorthm of analog crcuts based on node-voltage relaton, 12th IMEKO TC1-TC7 Jont Symposum, pp , Annecy, France, September, [11] Atmel Corporaton, 8-bt AVR mcrocontroller wth 16k Bytes In-System Programmable Flash, ATmega16, ATmega16L, PDF fle, Avalable from: