FAULT DETECTION FOR NONLINEAR SYSTEMS WITH MULTIPLE PERIODIC INPUTS. Z. Y. Yang and C. W. Chan

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1 FAULT DETECTION FOR NONLINEAR SYSTES WITH ULTIPLE PERIODIC INPUTS Z. Y. Yang and C. W. Chan Dep. of echanical Engineering, Univ. of Hong Kong, Pokfulam Road, Hong Kong Absrac: Faul diagnosis is imporan o improve he reliabiliy of engineering sysems. os exising faul diagnosis echniques consider only sysems wih non-periodic inpus, hough periodic signals are also common in conrol sysems. In his paper, faul diagnosis of nonlinear sysems subjec o period inpu consising of muliple fundamenal frequencies is presened. A faul diagnosis echnique is derived by he local asympoic approach, which reduces o a χ es for a given confidence level. The procedure o implemen his echnique is presened, and is performance illusraed by a simulaion example involving a nonlinear sysem. Copyrigh 006 UKACC Keywords: faul deecion, asympoic local approach, periodic signal, hypohesis es. INTRODUCTION The inense ineres in faul diagnosis is moivaed mainly by he requiremens for indusrial processes o operae more safely and reliably. In he pas en o weny years, many echniques have been developed (see e.g., Venkaasubramanian e al. (003); Basseville (988); Frank (990)). odel based echniques are popular in faul diagnosis. However, hey are derived assuming he availabiliy of accurae models of he sysems (Isermann, 005; Frank and Ding, 994). Furher, hese echniques are mainly devised assuming he inpu of he sysems is nonperiodic. As periodic exciaions are also common in pracice (Hsiao, 995), i is herefore imporan o develop faul diagnosis echniques for his class of sysems. A echnique for deecing fauls in nonlinear sysems subjec o inpus wih one fundamenal frequency is proposed by Yang and Chan (005) based on he asympoic local approach and he frequency esimaion mehod proposed by Quinn and Hannan (00), and a χ es is proposed o deec fauls for a given confidence level. Following hese resuls, faul diagnosis echniques for inpus ha have muliple fundamenal frequencies are developed in his paper. The proposed echnique is illusraed by a simulaion example involving a nonlinear sysem conaminaed by Gaussian whie noise. The paper is organized as follows. I is shown in Secion ha for sysems wih periodic inpus, he residuals generaed for faul diagnosis are likely o conain periodic componens. An analysis of he frequencies likely o appear in he oupu is presened in Secion 3. In Secion 4, a brief review of he asympoic local approach is presened, followed by he derivaion of he proposed faul diagnosis echnique. In Secion 5, a simulaion example is used o illusrae he proposed deecion echnique.. FAULTS IN SYSTES WITH PERIODIC INPUTS Consider a discree single-inpu-single-oupu nonlinear sysem, y () = g( y ( ),..., y ( ny )) () + f( u( ),..., u( nu )) + ε ( ) where u() and y() are respecively he inpu and he oupu, wih maximum lags n u and n y, ε() is he whie noise wih a disribuion: N(0, ), g(.) and f(.) are single-value smooh nonlinear funcions. Le y ( ) = { y ( ),, y ( n y )} and u ( ) = { u ( ),, u ( n u )}, hen rewrie () as, y () = g( y ( )) + f( u ( )) + ε () ()

2 Le u() be a periodic funcion consising of disinc frequencies {ω,, ω R }. Assuming he sampling inerval τ is chosen, such ha he period expressed in he number of samples is: T S = T L /τ, where T L is he leas common facor of he periods a hese frequencies. For convenience, i is assumed here ha sufficienly good approximaion of g(.) and f(.), denoed by g ˆ(.) and f ˆ(.), can be obained using modeling mehods, such as fuzzy logic, neural neworks, neurofuzzy neworks (Brown and Harris, 994), or he suppor vecor neurofuzzy neworks (Chan e al., 00). Similarly, f ˆ(.) is also assumed o be a single value funcion. Replacing g(.) and f(.) by heir esimaes g ˆ(.) and f ˆ(.), he esimaed oupu yˆ ( ) can be compued from () as, y ˆ() = gˆ( y ( )) + fˆ ( u ( )) (3) The residual r () is given by, r () = y () y ˆ() = g ( y ( )) + f ( u ( )) +ε() (4) where g (.) = g(.) gˆ (.) and f (.) = f(.) fˆ (.). If g ˆ(.) and f ˆ(.) are respecively sufficienly good esimaes of g(.) and f(.), hen boh g ~ (.) and f (.) approach zero, and hence r() approaches ε(). In his case, r() is also Gaussian disribued. However, if a faul occurs, eiher g(.) or f(.) would depar from heir nominal value, and ~ ~ g(.) or f (.) will no longer be zero. As a conrol sysem may subjec o periodic and/or non-periodic inpu, he analysis of periodic componens generaed by he non-lineariy of he sysem is discussed firs in he nex secion. 3. ANALYSIS OF FREQUENCY COPONENTS IN THE OUTPUT I can be readily shown ha periodic componens are inroduced ino he residuals. Since u ( + TS ) = u ( ), i follows ha f ( u ( + TS )) = fu ( ( )). Consequenly, ~ f (.) is almos periodic, and from (4), r() is also almos periodic (Chua and Ushida, 98). The frequency componens in he oupu of he nonlinear sysem () subjec o periodic inpu wih muli-frequency can be quie complex. Consider a muli-frequency periodic inpu (Lang and Billings, 000; Ushida and Chua, 984; Yue e al., 005), R u () = Ai cos( ωi+ ϕi) (5) i= where R is he number of inpu frequencies, A i, ω i and φ i are respecively he ampliude, frequency and phase of he i h inpu frequency componen. The frequency componens in he oupu of () are a combinaion of he inpu frequencies and heir negaives, denoed by Ω base, as given below (Lang and Billings, 000), Ω = [ ω,..., ω, ω,..., ω ] (6) base R R To convenienly represen all possible combinaions, define a frequency-mix vecor, v (Lang and Billings, 000): v = [ m R,..., m, m,..., mr ] (7) where {m i, i = -R,, R} are nonnegaive inegers. For a n h order nonlinear sysem, v saisfies he following consrain, m R m + m mr = n (8) where n is he order of sysem non-lineariy. Denoing he se of v s for he n h order sysem nonlineariy by V n, Vn = { v: m R m + m mr = n} (9) The frequencies in he oupu of he n h order nonlinear sysem, denoed by Ω n, is given by Ω { T n = Ωbasev : v Vn} (0) I follows ha for a nonlinear sysem wih a nonlinear order of N, he possible oupu frequency se Ω ou is, Ω = Ω ou N i () i= As an illusraion, consider he following example. Example Consider a 3 rd order nonlinear sysem. The maximum non-lineariy is herefore 3. Le he periodic inpu of he sysem conain wo frequencies, ω and ω. From (6) and (7), he inpu frequency base is: Ω base = [-ω, -ω, ω, ω ], and he frequency-mix vecor is: v = [m -, m -, m, m ]. From (8), for he sysem s firs order nonlineariy, v saisfies: m - +m - +m +m =, and from (9) he frequency-mix vecors in V are: [,0,0,0], [0,,0,0], [0,0,,0] and [0,0,0,]. Similarly, for he sysem s second order nonlineariy, he vecors in V are: [,,0,0], [0,,,0], [0,0,,], [,0,0,], [,0,,0], [0,,0,], [,0,0,0], [0,,0,0], [0,0,,0], [0,0,0,]; and for he hird order non-lineariy: [0,0,0,3], [0,0,,], [0,0,,], [0,0,3,0], [0,,0,], [0,,,], [0,,,0], [0,,0,], [0,,,0], [0,3,0,0], [,0,0,], [,0,,], [,0,,0], [,,0,], [,,,0], [,,0,0], [,0,0,], [,0,,0], [,,0,0], [3,0,0,0]. Therefore, he oupu frequency se Ω ou consiss of he following frequencies: ω, ω, ω +ω, ω -ω, ω, ω, 3ω, 3ω, ω +ω, ω +ω, ω -ω, ω -ω, and 0. I should be noed ha for a sampled sysem, he range of frequencies ha can be deeced is deermined by he sampling inerval and he number of daa available for analysis. The upper bound is given by he sampling heory, which requires he sampling frequency o be a leas wice he highes frequency in he signal, i.e., ω upper =π/τ. And he lower bound is ω lower =π/τ, where is he number of daa ha should cover a leas one complee period. Therefore, he deecable frequency se is given as, Ω =Ω ( ω, ω ) () d ou lower upper In pracice, he nonlinear order N is difficul o obain. Therefore, choosing an appropriae N is crucial. A small N is likely leading o he omission of high order nonlineariy, while a large N increases

3 unnecessary effors in deecing high frequencies ha are insignifican, as he energy in he frequency componens decreases geomerically as he order increases. 4. FAULT DETECTION As discussed in he previous secion, he faul deecion problem can now be reformulaed as one ha deecs wheher he residual r() conains any periodic and/or non-periodic componens in addiion o noise. For he periodic componens, i is necessary o examine every possible combining of he frequencies in he inpu conain in Ω d given by (). If any signal wih frequencies in Ω d is deeced, hen a faul has occurred. A novel mehod for deecing boh he non-periodic and he periodic componens is presened in his secion. 4. Asympoic Local Approach Consider he saisics R(), R () = rk ( ) (3) k= + where he residual r() is obained from (4). The saisics R() can be inerpreed as he normalized cumulaive sum of r() up o wihin a moving window, chosen sufficienly large o smooh ou he random elemens in r(). I is shown by Zhang e al. (998) ha if ε() in () is Gaussian disribued, hen R() is also Gaussian disribued: N(0, ). If here are fauls in he sysem, he mean of R() is no longer be zero (Basseville, 998; Wang e al., 00). Le R () ξ () = (4) The faul deecion can be formulaed as a hypohesis es. As ξ() is χ -disribued wih one degree of freedom (Basseville, 998), he hypohesis ha a faul has occurred holds, if ξ() exceeds he hreshold λ obained from he χ -disribuion able for a given confidence level. This echnique has been applied successfully o deec small and incipien fauls under noise (Wang e al, 00; Zhang e al., 998). However, is performance for residuals conaining periodic componens is no saisfacory. For example, if he widh of he moving window is idenical o he period of he oscillaion, hen clearly no fauls can be deeced, as he mean of he moving window is close o zero even hough here are large oscillaions wihin he window. 4. Deecion of Periodic Componens Before deriving he echnique for deecing he periodic componens in he residual r(), he following resuls are inroduced firs. Lemma For a sufficienly large ε()~n(0, ), hen ε( k)sin( ωk) N(0, ) k= +, and and ε( k)cos( ωk) N(0, ) k= + The proof is given in he Appendix. The moivaion of he proposed mehod can be illusraed by he following example. Consider he case when here is a faul, and he residual r() conains a periodic funcion, as follows, r () = asin( ω) + ε () where ε() is Gaussian noise defined in (). In his case, a is nonzero when a faul occurs, bu is zero when he sysem is operaing normally. As R() approaches zero irrespecive of he value of a, i would be difficul o deec from (3) wheher a = 0, and hence he hypohesis es derived from he asympoic local approach fails o deec fauls in a sysem wih periodic inpu. Since he energy of whie Gaussian noise is uniformly disribued over all frequencies, whils ha for a periodic signal is he highes near he frequencies in he signal (Quinn and Hannan, 00), similar o he resonance phenomenon. Consequenly, he energy compued a he frequencies in he periodic signal will be more significan han a any oher frequencies. Now, define anoher saisics, R, (Quinn and Hannan, 00) as follows, R'( ) = r( k)sin( ωk) k= + = asin( ωk) k= + (5) + ε ( k)sin( ωk) k= + The firs erm on he righ side of (5) is nonzero, and can be approximaed by a /. Following he resuls in Lemma, he second erm has a disribuion: N(0, ). I implies ha he effec of he sinusoidal signal is o inroduce a consan o he mean of R'( ). Then he exisence of a sinusoid signal can be reformulaed as deecing he change in he mean of R'( ), which can be readily achieved by he hypohesis es described above. This saisics can be applied o deec fauls in nonlinear sysems. However, i should be noed ha he proposed mehod can only deec one frequency a a ime. To deec all possible frequencies, i is necessary o repea he procedure o all frequencies in he deecable frequency se Ω d given by (). As e jω is a general represenaion of sinusoid funcions for approximaing periodic signals, (5) can be rewrien as follows wih R'( ) replaced by η(), jωk η() = ( ) r k e k= + (6) η() can be decomposed ino,

4 jωk η() = r( k) e k= + = ( )( sin( ) cos( )) rk j ωk + ωk (7) k= + = ( R + R ) where R = r( k)sin( ωk) and k= + R = r( k)cos( ωk). From Lemma, he k= + disribuion of R and R for sysems operaing normally are: R N(0, ) and R N(0, ) Then η() is χ disribued wih degree of freedom, η() = ( R + R) χ () (8) However, if r() conains a periodic signal wih a frequency of ω and he period in he number of samples is T=π/ωτ, hen where ER [ ] = E[ ( xk ( ) + ε( k))sin( ωk)] k= + = xk ( )sin( ωk) k= + + T = ( xk ( )sin( ωk ) T + k= + + mod(, T) k= + = ( S O) + T xk ( )sin( ωk)) + T T is S = x( k)sin( ωk) = x( k)sin( ωk) k= + k= he sum in a complee period, and [.] is he runcaion + mod(, T) operaor, and O = x( k)sin( ωk) is he sum k= + in he remainder ime, where mod[,t] is he remainder of he quoien of and T. I is assumed ha O is bounded and S 0. These condiions can be readily saisfied in pracical applicaions. If, he widh of he moving window, is sufficienly large, hen ER [ ] S 0 T Similar resuls can be obained for R. Since he mean of boh R and R are nonzero, i follows ha he mean of η() is also nonzero. As discussed previously, he es is repeaed for all deecable frequencies in Ω d. A oal of K + ess is required; K ess for he frequencies in Ω d, and he oher one for he non-periodic componen in r(). Denoe he corresponding saisics for he frequencies in Ω d by η (), η (),, η K (), in ascending order of frequency, and for non-periodic componen, ξ() as given by (4). I should be noed from (4) and (8) ha ξ()~χ () and η i ()~χ (), for i=,,,k. For a given false alarm rae, wo hresholds λ for χ () and λ for χ () can be obained from he χ disribuion able. The hypohesis ha he mean of hese saisics is zero holds, if ξ()<λ and η i ()<λ, for i=,,,k. Oherwise, a faul is deeced. 4.3 On-line faul deecion procedure The on-line faul deecion procedure is presened below. Sep Obain he approximaions g ˆ(.) and f ˆ(.) using modeling mehods such as fuzzy logic, neural neworks, or neurofuzzy neworks. Sep Selec a sufficienly large window for compuing he cumulaive sum of residual, and confidence level for he χ es. Obain λ, he hreshold for a given false alarm rae from he χ disribuion able. Sep 3 Compue he mean m 0 and sandard deviaion v 0 of he residual sequence from raining daa. This is necessary as he mean of he residual in pracice is generally nonzero, and is sandard deviaion unknown. Sep 4 Choose an appropriae N, he maximum order of sysem non-lineariy. Form he deecable frequency se Ω d from (), and compue K, he number of deecable frequencies. Sep 5 For each wih r() and replaced by r()- m 0 and v 0, on-line compue ξ() by using (4), and η i () for every frequency in Ω d given by () by using (8). If ξ()>λ or η i ()>λ, hen a faul occurs. 5. SIULATION EXAPLE Example Consider he following nonlinear sysem, ay( ) y( ) y () = y ( ) + y ( ) + y ( ) bu( ) + + cu( ) + ε ( ) u ( ) + where ε() is he whie Gaussian noise wih a disribuion: N(0,0. ). Nominal values of he parameers are: a=.5, b=.4, c=.. The inpu is u()=sin(8πτ)+0.8sin(0πτ), which consiss of wo frequencies. The sampling period τ is se o 0.05s, and he discree form is u()=sin(0.4π)+0.8sin(0.5π). The maximum order of sysem non-lineariy N is chosen o be 3. Then following he resul of Example in Secion 3, he oupu frequency se consiss of: 0, π, 6π, 8π, 0π, π, 6π, 8π, 0π, 4π, 6π, 8π and 30π. The widh of he moving window is se o 00, i.e., daa are colleced over a period of 5s. From discussions in Secion 3, he lower and upper bounds of he deecable frequencies are 0.4π and 0π, and hence he deecable frequency se given by () consiss of π, 6π, 8π, 0π, π, 6π, 8π. The number of deecable frequencies is K=7. For a given false alarm rae of %, he hresholds λ and λ

5 obained from he χ disribuion able are and 9.03 respecively. The proposed faul diagnosis echnique is used o deec and isolae he following fauls ha occurs a 30s: (a) componen faul wih a f =.3, acuaor fauls wih (b) b f =. and (c) c f =, and (d) sensor faul wih y f ()=0.9y(). The resuls of he simulaion are shown in Figs. o 5, illusraing ha all fauls are deeced successfully. 6. CONCLUSIONS In his paper, a faul deecion echnique for nonlinear sysems subjec o inpus wih muliple fundamenal frequencies is proposed. The echnique is derived by he local asympoic approach, and he faul diagnosis is decided by χ es for a given confidence limi. As only one frequency can be esed each ime, i is necessary o repea he es for all possible combinaion of he inpu frequencies. The procedure for implemening he proposed echnique is presened, and is performance illusraed by a simulaion example involving a nonlinear conaminaed by Gaussian whie noise. Fig. 3. Deecion resuls for faul (b). The X axis is ime and η 3 () and η 4 () are deeced. Fig. 4. Deecion resuls for faul (c). The X axis is ime and ξ(), η (), η 6 () and η 7 () are deeced. Fig.. The periodic inpu wih frequencies 8π and 0π. Fig. 5. Deecion resuls for faul (d). The X axis is ime and ξ(), η (), η 3 (), η 6 ()and η 7 () are deeced. REFERENCES Fig.. Deecion resuls for faul (a). The X axis is ime and ξ() is deeced. Basseville,. (988). Deecing changes in signals and sysems - A survey. Auomaica, Vol. 4, Basseville,. (998). On-board componen faul deecion and isolaion using he saisical local approach. Auomaica, Vol. 34,

6 Brown,. and Harris, C. J. (994). Neurofuzzy Adapive odelling and Conrol. Prenice-Hall, Englewood Cliffs, NJ. Chan, C. W., Jin, H., & Cheung, K. C. e al (00). Faul deecion of sysems wih redundan sensors using consrained Kohonen neworks. Auomaica, Vol. 37, Chua, L. O. and Ushida, A. (98). Algorihms for compuing almos periodic seady-sae response of nonlinear sysems o muliple inpu frequencies. IEEE Trans. Circuis and Sysems, Vol. CAS-8, Frank, P.. (990). Faul deecion in dynamical sysems using analyical and knowledge based redundancy a survey and new resuls. Auomaica, Vol. 6, Frank, P.., & Ding, X. C. (994). Frequencydomain approach o opimally robus residual generaion and evaluaion for model-based fauldiagnosis. Auomaica, Vol. 30, Hsiao, F. H. (995). Sabiliy analysis of nonlinear muliple ime-delay sysems wih a periodic inpu. Conrol Theory and Advanced Technology, Vol. 0, Isermann, R. (005). odel-based faul-deecion and diagnosis saus and applicaions. Annual Reviews in Conrol, Vol. 9, Lang, Z. Q. and Billings, S. A. (000). Evaluaion of oupu frequency responses of nonlinear sysems under muliple inpus. IEEE Trans. on Circuis and Sysems-II: Analog and Digial Signal Processing, Vol. 47, Quinn, B. G. & Hannan, E. J. (00). The esimaion and racking of frequency. Cambridge Universiy Press. Ushida, A., Chua, L. O. (984). Frequency-domain analysis of nonlinear circuis driven by muli-one signals. IEEE Trans. on Circuis and Sysems, Vol. 3, Venkaasubramanian, V., Rengaswamy, R., & Kavuri, S. e al (003). A review of process faul deecion and deecion. Compuers and Chemical Engineering, Vol. 7, Wang, Y., Chan, C. W., & Cheung, K. C. (00). Online faul diagnosis of nonlinear sysems based on neural neworks using he asympoic local approach. Asian Journal of Conrol, Vol. 3, Yang, Z. Y. and Chan, C. W. (005). Faul deecion of nonlinear sysems wih periodic inpu. Proc. of IEEE Inernaional Conference on Indusrial Technology, Yue, R., Billings, S. A. and Lang, Z. Q. (005). An invesigaion ino he characerisics of non-linear frequency response funcions. Inernaional Journal of Conrol, Vol. 78, , Zhang, Q., Basseville,., & Benvenise, A. (998). Faul deecion and isolaion in nonlinear dynamical sysems: A combined inpu-oupu and local approach. Auomaica, Vol. 34, APPENDIX: Proof of Lemma For convenience, le A = ε ( k)sin( ωk) k= + and A = ε ( k)cos( ωk). Since ε(k) is k= + normally disribued, he linear combinaion of A and A is also normally disribued. Taking expecaion of A gives EA [ ] = E[ ε( k)sin( ωk)] k= + = E[ ε( k)]sin( ωk) k= + = 0 and he variance of A is: V[ A] = E[( A E[ A]) ] = EA [ ] = [ ε ( k ) ε ( p )sin( ωk )sin( ωp )] = k= + p= + [ [ ( ) ]sin E ε k ( ωk ) k= + + k= + p= + k p E[ ε( k)] E[ ε( p)]sin( ωk)sin( ωp)] = sin( ωk) k= + cos( ωk) = k= + = cos( ωk) k= + As, hen V[ A ] =. Similar resul can also obain for A. Therefore, for a sufficienly large, A and A are disribued: N(0, ).