Nonlinear System Modelling: How to Estimate the. Highest Significant Order

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1 IEEE Insrumenion nd Mesuremen Technology Conference nchorge,, US, - My Nonliner Sysem Modelling: ow o Esime he ighes Significn Order Neophyos Chirs, Ceri Evns nd Dvid Rees, Michel Solomou School of Elecronics, Universiy of Glmorgn Ponypridd, Rhondd Cynon Tf $,CF7DL,Wles,U Phone: , Fx: E-mil: drees@glm.c.u bsrc new mehod for esiming he highes significn order of nonlineriy of Volerr ype sysems is presened. The mehod is bsed on he use of mulisine signls nd he possibiliy of esing he sysem hree differen mpliudes. The performnce of he proposed mehod is demonsred in simulion nd i is shown h i is possible o esime he highes order of nonlineriy of Volerr ype sysems very ccurely. The mehod cn be used o provide essenil prior nowledge bou he nonlineriy nd hus id he ccure represenion of he sysem under es. eywords nonliner sysems, Volerr series, NRMX models, order of nonlineriy, frequency domin, frequency response, mulisine signls. I. INTRODUCTION The Volerr funcionl series represenion consiues useful wy of represening nonliner sysem since i cn be seen s nurl exension of liner sysem heory. The Volerr ernels hve direc physicl significnce nd cn ofen be given physicl inerpreion, or be reled o he sysem s consiuen elemens []. Efficien mehods for mesuring hese Volerr ernels hve been previously developed, for exmple [-5], nd he success of hese mehods depends on he ssumpion h he highes significn order of nonlineriy of he sysem under es is nown. In [6] prcicl lgorihms for deermining he highes significn order of nonliner sysems were presened. These lgorihms re bsed on number of mesuremens equl o n o deermine if he nh-order nonliner erm is significn or no for given signl mpliude. The number of mesuremens hus increses wih he order of nonlineriy nd cn become unresonbly lrge for higher order nonlineriy. In his pper simple frequency-domin mehod is developed o esime he highes significn order of nonlineriy bsed on only hree ess differen mpliudes. The proposed echnique is bsed on he use of mulisine signls nd he biliy o es he sysem hree differen mpliudes. complee nlysis of he echnique is presened followed by compuer simulions nd n experimenl illusrion on prcicl sysem. II. II.. Volerr series PROBLEM FORMULTION n nlyic response funcion cn be represened by n infinie series clled he Volerr series. This is generlision of he impulse response funcion of liner sysems nd is composed of he convoluion inegrl y h τ u τ dτ nd sic nonlineriy represened by Tylor series. u u n u... n u n y The Volerr series is hen given by n y h... hn τ τ τ n u τ i d,,... n i τ which represens sum of oupus of prllel sub-sysems clled Volerr funcionls illusred grphiclly in he schemic digrm in Figure. Nonliner sysem idenificion bsed on he Volerr represenion requires he mesuremen of he ernels hn τ, τ,... τ n. Severl pproches exis in he lierure for he esimion of hese prmeers, he mos common bsed on he exension of correlion mehods for liner sysems i

2 nd he use of whie Gussin signls. The complexiy of his model depends on he highes significn order of nonlineriy n, which is dependen on he dynmic rnge of he inpu signl. For exmple, for smll signl mpliudes higher-order responses cn be negleced nd only he lower-order responses re considered s dominn. For sufficienly lrge mpliudes however, he conribuion of he responses genered by higher-order nonlineriies is more significn nd hey should be en ino ccoun. I is hus necessry o deermine he highes significn order of nonlineriy relive o he rnge of inpu signl mpliudes h will be pplied o he sysem. The implicions o he ccurcy of he esimion of he Volerr ernels is lso mor fcor for he correc esimion of he highes significn order of nonlineriy s poined ou in [7]. If he order esime is oo smll, he resuling Volerr ernels will be highly inccure. On he oher hnd if he order is overesimed n excessive number of mesuremens will be required which will me he whole process ime-consuming. h h h : h Figure. Grphicl represenion of he Volerr series. II.. Nrmx Models The Nonliner uoregressive Moving verge wih exogenous inpus NRMX pproch ws inroduced by Leonriis nd Billings [8, 9] nd Chen nd Billings [] s mens of describing he inpu-oupu relionship of nonliner sysem. The model represens he exension of he wellnown RMX model o he nonliner cse, nd is defined s y,..., y n y, u,..., u nu,... y F e 4 e,..., e ne repre- where F is nonliner funcion; y, u nd e sen he oupu, inpu nd noise signls respecively; nd n y, n u, nd ne re heir ssocie mximum lgs. The NRMX represenion consiues powerful ool for nonliner modelling nd i includes fmily of oher nonliner represenions such s bloc-srucured models nd Volerr series []. The mos involved s in NRMX modelling is o selec he pproprie regressors i.e. model erms o build he model srucure. The number of cndide regressors l for given order of nonlineriy n is given by equion 5, where i cn be seen h i increses rpidly wih he increse of he order of nonlineriy nd mximum inpu, oupu nd noise lgs. n l li n y nu ne i] / i i [ wih l n n n 5 y u e ving in mind h he corresponding number of models l o choose from is given by M, he ssessmen of ech possible model is no prcicl, reinforcing he need o use srucure selecion echnique o selec he mos pproprie regressors for inclusion in he model. ll srucure selecion echniques exmine lrge number of model erms nd uilise cerin crieri for heir inclusion or removl from he model. The whole procedure cn be considerbly simplified if priorinowledge is uilised, such s nowledge bou he highes significn order n. This mens h he serch spce of cndide models cn be significnly nrrowed, nd srucure selecion lgorihms cn be fcilied wih greer confidence, since he possibiliy of hving spurious componens enering ino he model srucure will be minimised. Therefore, i is ppren h nowledge of n provides significn dvnges since he idenificion procedure cn be grely simplified, he ccurcy of he esimed model cn be preserved nd hus he esimion ime is reduced. This provided he moivion for he developmen of n nlyicl echnique o esime he highes significn order of he nonlineriy for Volerr models. The proposed echnique cn lso be pplied in NRMX modelling since he Volerr represenion belongs o he NRMX fmily. III. ORDER TEST LGORITM The proposed mehod is bsed on Frequency Response Funcion FRF mesuremens hree differen mpliudes. The use of periodic signls is essenil in his cse since sysemic errors rising from FFT lege problems cn be voided nd he signl-o-noise rios of he d records cn be improved by verging over number of periods []. The use of periodic signls lso llows he direc esimion of he FRF s he rio of he men vlues of he oupu nd inpu coefficiens, he discree es frequencies M m M ˆ m 6 M U U m M m

3 where M is he number of mesured periods. The periodic signl used hroughou his sudy is he mulisine signl which is sum of n rbirry ensemble of hrmoniclly reled cosines F i u cos 7 where is vecor of mpliudes i vecor of hrmonic numbers nd he signl funmenl, Φ vecor of phses nd F he number of cosines in he signl. The relive phses of he hrmonics mus be crefully seleced in order o minimise he signl cres funcor. The lowes cres fcor chieved o de is by he l mehod proposed in []. n exmple of en hrmonic mulisine signl wih fundmenl frequency of.5z nd hrmonic vecor i,,, is shown in Figure Time s mp Frequency z mp b Figure. Mulisine signl in ime domin nd b frequency domin. In order o presen he proposed mehodology in cler nd undersndble wy he lgorihm will be derived using wo-one consecuive mulisine given by cos cos u. 8 This is used o excie sic nonliner sysem conining single qudric nonlineriy s shown in Figure. u u u y Figure. Sic qudric sysem. The oupu y of he sysem is given by: cos4... cos cos... cos cos... cos cos... cos y 9 From he bove equion i is cler h he complex mpliudes he wo es frequencies re given by: If he sysem is esed using n inpu signl u where is consn hen he complex mpliudes he wo oupu frequencies will be given by: Finlly, he sysem is esed using signl u where is noher consn. I follows h, The corresponding vlues of he FRFs he es frequencies re hus given by:

4 The following index cn hen be clculed r r r 4 Ifhesmeprocedureisfollowedndr is clculed for sysemwihsingle rd order nonlineriy hen he previous index is given by: r 5 nd if r is clculed for sysem wih single 4 h order nonlineriy hen he index is given by: r 6 generl formul reling his index wih he mximum order of nonlineriy n cn hus be derived s n n r 7 n nd n cn be clculed since i is he only unnown. I mus be sressed here h he bove equion is sricly vlid for sysems wih single nonlineriy. In prcice hough his is no usully he cse since sysems will conin oher orders of nonlineriy s shown by he generlised sic nonliner sysem in Figure 4. u u u u... n u n Figure 4. Generl sic nonliner sysem. y In his cse equion 7 is no sricly vlid since he simplificions h occurred in 4 do no e plce. The new formul for he index r is funcion of he frequency mpliudes, nd he coefficiens n f n n n n fn f f n n... n fn f... r 8 From he bove equion i cn be seen h he dominn erm is he erm which corresponds o he highes order of nonlineriy n. This of course depends on he vlue of he coefficien nd he choice of nd. Neverheless i cn hus be esily seen h equion 8 is very close pproximion of equion 7 especilly for high vlues of n. I mus be sressed here h he mehod is no dependen on wo-one mulisine s he one used o derive equions 9 o 8. The use of brodbnd mulisine is recommended since wo one mulisine could fil in cses where he inpu o he sysem is shped by he dynmics of he sysem before enering he nonliner elemens s in he Wiener cse. The proposed lgorihm is hus summrised s follows: Sep : Excie he sysem under es wih mulisine signl hree mpliudes defined by u, u nd u. Sep : Clcule he FRFs, nd he hree mpliudes using equion 6. Sep : Clcule he index r Sep 4: Obin n esime of he mximum order of nonlineriy using equion 7. IV. EXPERIMENTL ILLUSTRTIONS IV.. Simuled exmples To illusre he effeciveness of he proposed lgorihm he simple mmersein nd Wiener models shown in Figure 5 were excied using consecuive mulisine conining hrmonics wih fundmenl frequency of.5 z nd pe mpliude of. The liner pr of he models is he sme s liner model idenified for he igh Pressure P shf of Rolls-Royce gs urbine engine by Evns e l. [4]. The d

5 records were corruped by ddiive Gussin noise of zero men nd uni vrince. u u.6976 s.99 s.597 s.768 5u.5u -.u -.8u 4.u 5 y y b 5y.5y -.y -.8y 4.y s.99 s.597 s.768 Figure 5. mmersein model b Wiener model. The resuls of he es lgorihm for he mximum nonlineriy order for boh sysems re shown in Tble for hree differen combinions of nd. I cn be seen h he proposed lgorihm gives good indicion for he mximum order of nonlineriy, even hough i is cler h he resuls obined for he Wiener model re no s ccure s he resuls for he mmersein model. This ws of course expeced since in he Wiener cse he inpu is filered by he liner pr of he model before pssing hrough he nonlineriy. The resuls re more encourging if higher-order erm, i.e. he 9 h -order erm.y 9 is dded o he sic polynomils of he wo models. I cn be seen from Tble h even hough he coefficien of he 9 h -order erm is very smll compred o he oher coefficiens, he lgorihm deecs he conribuion of his erm very well. Tble. Mximum Order of Nonlineriy for mmersein nd Wiener models n5 Model.5,.5.5,.5, 4 mmersein z z Sysems wih differen sic polynomils were esed nd i ws concluded h he lgorihm is cpble of deecing he mximum order of nonlineriy of sysems lie Wiener, mmersein nd Wiener-mmersein nd Volerr series. I ws lso observed h he ccurcy of he lgorihm is improved for high orders of nonlineriy s is clerly suggesed by he resul in equion 8. I is lso cler h furher invesigion is required on he prcicl issues concerning he proposed echnique such s he opimum selecion of he mpliude levels nd. IV.. nonliner elecricl circui nonliner mechnicl resoning sysem mss, viscous dmping, nonliner spring is simuled wih n elecricl circui. The displcemen y is reled o he force u by he following nonliner, second-order differenil equion. d y dy m d y by u 9 d d s Schouens e l. [5] noed, he cul relized circui is no in perfec greemen wih 9, since smll qudric erm ws deeced in he mesuremens. I ws lso noed h for smll exciions he spring becomes lmos liner so h he underlying liner sysem consiss of second-order resonnce sysem. Two differen signls, consecuive mulisine f f,,,,n, N 6 nd f.98 z nd n odd mulisine f f,,,5,n-, N nd f.98z whereusedoeshesysemhreedifferen mpliudes. Figure 6 shows he mpliude responses of he sysem hree differen mpliudes, obined using he consecuive mulisine. The exisence of nonlineriy in he sysem cn be esily visulized s he evoluion of he sysem dynmics wih growing inpu signl mpliude 5 5 Wiener Tble. Mximum Order of Nonlineriy for mmersein nd Wiener models n9 Model.5,.5.5,.5, 4 Gin db mmersein Wiener Frequency z Figure 6. Evoluion of sysem dynmics growing exciion levels:,, 7 mv RMS. Tble shows he resuls of he proposed lgorihm when pplied o he mesured d using he wo mulisines. I cn be seen h he proposed echnique gives good pproximion of he mximum order of nonlineriy in he sysem even hough i under predics he exc order. This is o be ex-

6 peced since nonliner sysems which chnge dynmics wih inpu exciion level, belong o he fmily of he Wiener-lie srucures nd more specificlly hose srucures where he inpu is filered by he liner pr of he model before pssing hrough he nonlineriy. s previously sed, he ccurcy of he lgorihm improves s he order of he nonlineriy increses. I is cler h in his cse he order of he nonlineriy is quie low, which ffecs he ccurcy of he lgorihm. Neverheless he proposed lgorihm gives good indicion of he vlue of he mximum order of nonlineriy. Tble. Mximum Order of Nonlineriy for he nonliner mechnicl resoning sysem Consecuive mulisine Exciion levels:,, 7 mv RMS. Odd mulisine Exciion levels:, 4, 7 mv RMS... V. CONCLUSIONS mehodology hs been presened o esime he highes order of nonlineriy of Volerr ype sysems. The proposed lgorihm is bsed on he use of mulisine signls nd he clculion of he FRFs of he sysem under es hree mpliudes. The use of mulisine signls llows he direc esimion of he FRFs, he sysemic errors rising from FFT lege problems cn be voided nd he signl-o-noise rios of he d records cn be improved by verging over number of periods. The performnce of he proposed lgorihm ws demosred in simulion using simple bloc srucure models which belong o he Volerr series fmily. The proposed echnique ws lso illusred on nonliner mechnicl resoning sysem which is nonliner circui wih mximum rd order nonlineriy. I ws shown h he proposed lgorihm provides good pproximion of he mximum order of nonlineriy in sysem. The lgorihm is suible o be used for sysems h belong o he Volerr series fmily s well s lrge number of NRMX srucures. hn ll he sff involved. REFERENCES [] G.. ung nd L. W. Sr, The inerpreion of ernels n overview, nnls of Biomedicl Engineering, vol. 9, pp , 99. [] L. O. Chu nd. Lio, Mesuring Volerr ernels II, In. Journl of Circui Theory nd pplicions, vol. 7, pp. 5-9, 989. []. W. Lee nd M. Schezen, Mesuremen of he Wiener ernels of non-liner sysem by cross-correlion, In. Journl of Conrol, vol., pp. 7-54, 965. [4].. Brer nd R. W. Dvy, Mesuremen of second-order Volerr ernels using pseudorndom ernry signls, In. Journl of Conrol, vol. 7, no., pp. 77-9, 978. [5] M. Schezen, The Volerr nd Wiener Theories of Nonliner Sysems, Wiley Inerscience, New or, 98. [6] L. O. Chu nd. Lio, Mesuring Volerr ernels III: ow o esime he highes significn order, In. Journl of Circui Theory nd pplicions, vol. 9, pp. 89-9, 99. [7]Q.Zhng,B.Sui,D.T.Weswicnd.R.Luchen, Fcorsffecing Volerr ernel esimion: Emphsis on lung issue viscoelsiciy, nnls of Biomedicl Engineering, vol. 6, pp. -6, 998. [8] I. J. Leonriis nd S.. Billings, Represenions of non-liner sysems: he NRMX model, In. J. of Conrol, vol. 49, no., pp. -, 985. [9] I. J. Leonriis nd S.. Billings, Inpu-oupu prmeric models for non-liner sysems. Pr II: sochsic non-liner sysems, In. J. Conrol, vol. 4, no., pp. 9-44, 985. [] S. Chen nd S.. Billings, Represenions of non-liner sysems: he NRMX model, In. J. of Conrol, vol. 49, no., pp. -, 989. [].P. Liu, Idenificion of Nonliner Sysems: The NRMX Polynomil Model pproch, Ph.D. disserion, Universiy of Sheffield, Deprmen of uomic Conrol & Sysems Engineering, U, 988. [] C. Evns, D. Rees nd. Borrell, Idenificion of ircrf gs urbine dynmics using frequency-domin echniques, Conrol Engineering Prcice, vol. 8, no. 4, pp ,. [] P. Guillume, Idenificion of muli-inpu muli-oupu sysems using frequency-domin mehods, Ph.D. disserion, Vrie Universiei Brussel, Deprmen ELEC, Belgium, 99. [4] C. Evns, N. Chirs, Pric Guillume nd D. Rees, Mulivrible modelling of gs urbine dynmics, SME Turbo Expo', Congress, New Orlens, GT-8,. [5] J. Schouens, R. Pinelon,. Rolin nd T. Dobrowieci, Frequency response funcions mesuremens in he presence of nonliner disorions, uomic, vol. 7, no. 6, pp ,. This pper illusres how frequency-domin echniques nd mulisine signls cn be used in order o provide essenil priori nowledge bou he nonlineriy in sysem nd hus id he idenificion procedure. VI. CNOWLEDGEMENTS Tesing of he nonliner mechnicl resoning sysem ws conduced in he Elecricl Mesuremen Deprmen ELEC of he Vrie Universiei Brussel VUB wih he permission of Professor John Schouens. The uhors would lie o