Recent Developments on the Modelling of Aircraft Gas Turbines

Transcription

1 Prsntd in 3 rd Intrnational Confrnc on Control Thory and Applications, Prtoria, Dcmbr 1 Rcnt Dvlopmnts on th Modlling of Aircraft Gas Turbins Nophytos Chiras Cri Evans David Rs School of Elctronics, Univrsity of Glamorgan, CF37 1DL Wals, UK nchiras@glam.ac.uk Abstract In this papr th orthogonal stimation algorithm is usd to stimat a NARMAX modl for an aircraft gas turbin. Th prformanc of th modl is validatd using a rang of small and larg signal tsts and by xamining th modl s static and dynamic charactristics. Th NARMAX rprsntation allows th dirct mapping to th frquncy domain by computing th Highr Ordr Frquncy Rspons Functions (HOFRF thus allowing physical intrprtation to b mad. N L. Ths shaft spds ar th primary outputs of a gas turbin, from which th intrnal ngin prssurs and thrust can b calculatd. 1 Introduction Th modlling of gas turbin dynamics has bn th subjct of considrabl study sinc th arly days of jt propulsion. Gas turbins ar now xtnsivly usd in aro, marin and industrial applications. With such widsprad and incrasing applications, th accurat modlling of such ngins is an important issu. Modlling of gas turbins is rquird both in th dvlopmnt and oprational stags of an ngin s lif. Modls can b usd to prdict th ngin s prformanc, dsign and tst th prformanc of th ngin control systms and prdict normal ngin bhaviour for on-lin fault dtction purposs. Modls ar usually validatd against ral data and in som cass a physical intrprtation of th paramtrs can b mad. This allows initial assumptions about th ngin charactristics to b chckd. This papr dals with th nonlinar rlationship btwn th ful flow and shaft spd dynamics of a Rolls Royc Spy Mk aircraft gas turbin. Although no longr in srvic, th Spy possss th sam charactristics, for control purposs as a modrn ngin such that of th EJ fittd to th Eurofightr [1]. Th gas turbin A gas turbin is mad up of thr basic componnts: a comprssor, a combustion chambr and a turbin. Air is drawn into th ngin by th comprssor, which comprsss it and dlivrs it to th combustion chambr. Within th combustion chambr th air is mixd with ful and th mixtur is ignitd, producing a ris in tmpratur and hnc an xpansion of gass. Ths gass ar xhaustd through th ngin nozzl but first pass through th turbin, which is dsignd to xtract sufficint nrgy from thm to kp th comprssor rotating. A schmatic diagram of th Rolls-Royc Spy ngin tstd in this work is shown in Fig. 1. It can b sn that both th comprssor and th turbin ar split into low prssur (LP and high prssur (HP stags. Th HP turbin drivs th HP comprssor and th LP turbin drivs th LP comprssor. Ths ar connctd by concntric shafts, which rotat at diffrnt spds, dnotd N H and Fig. 1. Simplifid schmatic of a Rolls Royc Spy ngin. Work conductd by Jackson [] for Rolls Royc plc, showd that th highr-ordr nonlinar thrmodynamic modls drivd from th ngin physics could b rducd to linar modls of th sam ordr as th numbr of ngin shafts. Th modls wr first linarisd at a sris of oprating points, using small prturbations, and thn a modl rduction procdur was mployd. Work don by Hill [3] xamind th application of tim-domain mthods to stimat discrt z-domain ngin modls. Evn though linar discrt modls with good input-output proprtis wr stimatd, problms with th application of tim-domain tchniqus wr rportd [4]. Mor rcntly, tams at th univrsitis of Glamorgan, Birmingham and Shffild utilisd th sam data to invstigat various idntification tchniqus on a Rolls-Royc Spy ngin. Data wr gathrd undr sa-lvl static conditions at th Dfnc Evaluation & Rsarch Agncy (DERA at Pystock. Multisin and invrs rpat maximum lngth binary squncs (IRMLBS wr usd at amplituds of up to ±% of th stady stat ful flow (W f. In addition highr-amplitud signals, such as triangular wavs and thr-lvl priodic signals with input amplituds of up to ±4% W f, wr usd. Ths causd th HP shaft spd to vary btwn 6% and 8% of its maximum valu (%N H. Evans t al. [-7] from th Univrsity of Glamorgan, usd frquncy tchniqus on th small-amplitud multisin and IRMLBS data to stimat linar modls at diffrnt oprating points. Th rrors du to nois and nonlinaritis wr assssd and found to b small for ths small-signal modls. Th fact that th dc gains and th dynamics of ths modls chang with oprating point showd that th gas turbin is nonlinar, so th nd is apparnt for a mor complt gas turbin dscription using nonlinar modls. In Evans t al. [8] an xtndd last-squars algorithm with optimal smoothing was usd by Norton, to idntify tim-varying transfr function modls to rprsnt larg transint and nonquilibrium ffcts and provid a mor dtail insight into th slow

2 Prsntd in 3 rd Intrnational Confrnc on Control Thory and Applications, Prtoria, Dcmbr 1 thrmal dynamics of th ngin. In ordr to idntify a modl capabl of rprsnting th ngin at all oprating points, Rodriguz [9] usd a multiobjctiv gntic programming approach on th sam data and allocatd wights to various objctivs, to assss thir significanc in th structur slction of Nonlinar AutoRgrssiv Moving Avrag with Xognous inputs (NARMAX modls of th ngin. A simpl NARX modl was idntifid which was abl to rprsnt both th small and larg signal dynamics of th ngin. In this papr a NARMAX modl for th ngin is stimatd using a forward-rgrssion orthogonal stimation algorithm. Th modl is validatd using a rang of small and larg signal tsts and by comparing th modl s static and dynamic bhaviour with masurd static data and stimatd linar dynamics. Th modl is mappd to th frquncy domain by computing th HOFRFs and th nrgy mchanisms rvald by th intraction of frquncis ar xamind. 3 NARMAX Modlling Lontaritis and Billings [], introducd th NARMAX approach as a mans of dscribing th input-output rlationship of a nonlinar systm. Th modl rprsnts th xtnsion of th wllknown ARMAX modl to th nonlinar cas, and is dfind as k = F( k 1,..., k n, u( k 1,..., u( k n, ( k 1,..., ( k n + ( k whr F is a nonlinar function; y (k, u(k and (k rprsnt th output, input and nois signals rspctivly, and n,, and n u y ar thir associat maximum lags. Th NARMAX rprsntation constituts a powrful tool for nonlinar modlling sinc it includs a family of othr nonlinar rprsntations such as block-structurd modls and Voltrra sris [11]. In addition, th modl is linar-in-th-paramtrs so that linar last-squars paramtr stimation tchniqus can b asily applid. A wll stablishd procdur for structur slction of a NARMAX polynomial modl is basd on th rror rduction ratio (ERR dfind in [1] as N i wi k = 1 N g ( k ERR i = ( y ( k k = 1 whr g i ar th cofficints and w i (k ar th trms of an auxiliary modl constructd in such a way that th trms w i (k ar orthogonal ovr th data rcords. A forward-rgrssion algorithm [1] is mployd to slct at ach stp th trm with th highst ERR, in othr words th trm which contributs most to th rduction of th rsidual varianc. Th procdur is usually stoppd using an information critrion such as th AIC, dfind as ( ( θ kp AIC = N log (3 σ ε p + whr ( is th varianc of th rsiduals associatd with a p- σ ε θ p trm modl and k is a pnalising factor. This procdur was found to work vry wll whn idntifying polynomial NARMAX modls y nu (1 and it dos xclud th vast majority of insignificant candidat trms by working towards th rduction of th rsidual varianc. A concatnatd data st of small signal tsts is usd for structur slction and paramtr stimation. This is achivd by combining small-signal IRMLBS tsts at diffrnt oprating points to covr a rang from % to 8% N H as shown in Fig.. Th forwardrgrssion orthogonal stimation algorithm is applid to ths data and th slctd trms with thir associatd ERRs and th cofficints of th slctd modl ar shown in Tabl 1. Amp (cc/s (b (a Fig.. Concatnatd data st usd for stimation (a masurd ful flow (b HP shaft rspons. Tabl 1 Quadratic NARX Modl 1. Modl 1 trms ERR i θ i k u( k k k 1 * k constant Th modl quality can b assssd using highr ordr corrlation functions [13], but no dfinit conclusion can b drawn unlss cross-validation is mployd. Nonparamtric validation is mployd which consists of th simulation of th stimatd modl with diffrnt masurd inputs and comparison of th rsults with th masurd outputs. Fig. 3 shows a tim-domain comparison btwn th modl output and th masurd gas turbin output for a small signal multisin tst, whr it can b sn that a vry good match is achivd. Similar rsults ar obtaind whn th nonlinar modl is validatd against small signal tsts at diffrnt oprating points, suggsting that Modl 1 is suitabl to b usd in th plac of th family of linar modls prviously stimatd. Modl 1 is also validatd against high-amplitud data. Th nxt tst, shown in Fig. 4, consists of a thr-lvl squnc of priod s with an input amplitud of ±% W f. This tst is th most dmanding availabl ngin tst sinc it ssntially consists of a sris of positiv and ngativ larg-amplitud stp rsponss. It can b again sn that th modl shows a good rspons suggsting that th modl is capabl of rprsnting th ngin dynamics at high amplituds. An important tst to assss th modl prformanc is to xamin its static bhaviour at diffrnt oprating points. Th static bhaviour of th modl was thrfor drivd through simulation

3 Prsntd in 3 rd Intrnational Confrnc on Control Thory and Applications, Prtoria, Dcmbr 1 and was compard with th static bhaviour of th ngin drivd from masurd data. It can b sn from Fig. that th modl is capabl of modlling th static bhaviour of th ngin throughout th oprating rang. In ordr to study th dynamic bhaviour of th modl with oprating point th modl was simulatd with multisin signals at th four oprating points for which linar modls wr prviously idntifid. Tabl shows a comparison of th dominant mods idntifid from ngin data using small signal tsts and frquncy domain tchniqus, and th dominant pols idntifid whn simulation data was gnratd using th nonlinar modl. It is again clar from th tabl that th local linarisation of th nonlinar modl is vry clos to that of th ngin at th diffrnt oprating points (a (b Fig. 3. Multisin HP shaft tst. (a complt signal (b portion of th signal. Modl 1 output (dashd masurd output (solid Ful flow (cc/s Fig.. Static rlationship btwn shaft spd and ful flow (solid bars Modl (dashd. Tabl. Linar modls idntifid from ngin data Vs local linar modls obtaind from Modl 1. Oprating point (%N H Estimatd modls Locally linar modls Tabl 3 Quadratic NARX Modl. Modl trms θ i constant k k.6334 u( k u( k k 1 * k Fig. 4. Thr-lvl HP shaft tst. Modl 1 output (dashd masurd output (solid. Th structur slction procdur basd on th ERR yildd a nonlinar modl which is capabl of modlling th ful fd to shaft dynamics of th ngin at both small and larg amplituds. During this work othr diffrnt structurs wr also invstigatd which wr found to hav vry good input-output proprtis. On of ths modls is shown in Tabl 3. This is a similar modl as th on idntifid by Rodriguz [9] and by Chiras t al. [14]. Th lattr, usd nonparamtric analysis to limit th sarch spac of modls undr considration and thn an xhaustiv sarch to stimat th modl shown in Tabl 3. Th prformanc of this modl, rfrrd to th rmaindr of this txt as Modl was illustratd in [14] for both small and larg amplitud inputs. It was shown that Modl is capabl of rprsnting th ngin dynamics at small and larg amplituds Ful flow (cc/s Fig. 6. Static rlationship btwn shaft spd and ful flow. Modl 1 (circls Modl (dashdot. Th prformanc of Modl is vry similar to that of Modl 1 stimatd using th orthogonal stimation algorithm. This can b illustratd in Fig. 6 and Tabl 4 whr a comparison of th static and dynamic bhaviours of th two modls is illustratd. From

4 Prsntd in 3 rd Intrnational Confrnc on Control Thory and Applications, Prtoria, Dcmbr 1 Fig. 1 it can b sn that th static bhaviour of Modl 1 is almost idntical to that of Modl. At th sam tim th dominant mods of th local linar modls drivd by th linarisation of th two modls, ar similar as shown in Tabl 4. Tabl 4. Local linar modls obtaind from nonlinar Modl 1 Vs local linar modls obtaind from nonlinar Modl. Linar modls from Modl Oprating point (%N H 4 Frquncy-domain analysis Linar modls from Modl On way of analysing a discrt NARMAX modl is by dirct mapping to th frquncy domain [1]. To this nd, th probing mthod [1] was usd to driv th analytical xprssions for th tru first- and scond-ordr gnralisd frquncy rspons functions for both stimatd modls as shown in th Appndix. Th amplitud of th tru first-ordr (linar FRF for Modl 1 is shown in Fig. 7 whr it can b sn that th ngin is a low-pass systm with a gain at zro frquncy approximatly qual to 8dB. It should b notd hr that providd th stimatd modl is th tru modl of th systm th rspons shown in Fig. 7 is th tru first-ordr frquncy rspons function of th systm, which in gnral is diffrnt from th biasd stimat obtaind from traditional spctral analysis. Th amplitud of th scond-ordr HOFRF for Modl 1 is shown in Fig. 8. Clos inspction rvals th sam typ of low-pass charactristic as in th first-ordr cas. Th gain of th scondordr HOFRF is found to b at last 7dB down compard with th gain of th first-ordr FRF. This is of cours significantly lss that th first-ordr FRF hnc th ffct of th scond-ordr HOFRF at th systm output is not significant unlss th input amplitud is larg. It is clar from Fig. 8 that th ffct of th scond-ordr HOFRF is apparnt at th paks of th scond-ordr HOFRF whr th gain is at its maximum. This corrsponds to th lins of frquncy dfind by f 1 =, f = and f 1 +f = maning that th quadratic nonlinarity in th systm will only gnrat nonlinar contributions at th input frquncis and at dc. (m + 1 f1 + f = f1 + f ± m =,1,,... th numrator of H (f 1,f bcoms zro. If th prvious quation is rarrangd to giv (m + 1 f = f1 f = ± m =,1,,... ( thn th frquncis whos diffrnc conforms with th abov will caus a gorg. H1 (f (db Frquncy (Hz Fig. 7. First-ordr GFRF for Modl 1. H(f1,f (db 1 f1 (Hz f1 (Hz f (Hz f (Hz Fig. 8. Scond-ordr HOFRF for Modl 1. (4 Th paks on th scond-ordr HOFRF at th lins of frquncy dfind by f 1 = and f =, indicat that thr will b significant intrmodulation ffcts on th output rspons if th signal contains a significant dc bias as in th ngin cas. This phnomnon is not limitd to th dc cas sinc ths lins also rprsnt othr low frquncy componnts, which will intract and crat nw harmonics at th output of th systm. Th paks on th scond-ordr HOFRF at th lins of frquncy dfind by f 1 +f = indicat that thr is a significant dc shift in th output rspons. This ngin charactristic can b also obsrvd from th tim rcords in Fig.. H(f1,f (db 1 f1 (Hz f1 (Hz f (Hz An intrsting fatur, which is shown clarly in th plot, is th xistnc of two paralll dp gorgs. This is bcaus two xponntial phasors cancl occasionally. In this cas it can b asily sn from th numrator of H (f 1,f in th Appndix that whn f (Hz Fig. 9. Scond-ordr HOFRF for Modl.

5 Prsntd in 3 rd Intrnational Confrnc on Control Thory and Applications, Prtoria, Dcmbr 1 This gorg is a product of th k-1k- trm in Modl 1 sinc this trm dtrmins th numrator of H (f 1,f. Fig. 9 shows th amplitud of th scond-ordr HOFRF of Modl. It can b sn that th two modls possss idntical frquncy charactristics xcpt at th lins of frquncy dfind in quation (. Sinc th modls ar almost quivalnt in trms of thir static and dynamic bhaviour, and thir HOFRFs ar almost idntical apart from th singularitis in Modl 1, thn Modl is proposd as an appropriat modl to approximat th nonlinar bhaviour of th gas turbin. Conclusions In this papr a forward-rgrssion orthogonal stimation algorithm was usd to stimat a NARMAX modl for an aircraft gas turbin. Th prformanc of th modl was validatd using a rang of small and larg signal tsts and by xamining th modl s static and dynamic bhaviour. Th stimatd modl was also compard with a prviously stimatd modl. It was shown that th static and dynamic charactristics of th two modls ar almost idntical. Th NARMAX rprsntation allowd th dirct mapping to th frquncy domain by computing th HOFRFs. Th analysis of ths HOFRFs rvald th nrgy transfr mchanisms rsulting from th nonlinarity, and facilitatd th slction of Modl as th most appropriat nonlinar modl for th gas turbin. Acknowldgmnts Work was conductd on data gathrd at th Dfnc Evaluation & Rsarch Agncy at Pystock with th support of Rolls Royc plc. Th authors would lik to thank all th staff involvd. Rfrncs [1] G.J. Dadd, A. E. Sutton and A. W. M. Grig. Multivariabl control of military ngins, AGARD Confrnc Procdings No. 7 Advancd Aro-Engin Concpts and Controls, papr 8, pp. 1-1, [] D. Jackson. Invstigation of stat spac architcturs for ngin modls, Rolls Royc plc, Rport TDR 9331, [3] D.C. Hill. Systm idntification of gas turbin ngins, Ph.D. dissrtation, Univrsity of Birmingham, School of Elctronic and Elctrical Enginring, [4] D.C. Hill. Idntification of gas turbin dynamics: timdomain stimation problms, ASME Gas Turbin Conf., papr 97-GT-31, pp. 1-7, [] C. Evans, D. Rs and D. Hill. Frquncy-domain idntification of gas turbin dynamics, IEEE Transactions on Control Systms Tchnology, 6,, pp , [6] C. Evans, A. Borrll and D. Rs. Tsting and modlling gas turbins using multisin signals and frquncy-domain tchniqus, ASME Journal of Enginring for Gas Turbin and Powr, 11, pp , [7] C. Evans, D. Rs and A. Borrll. Idntification of aircraft gas turbin dynamics using frquncy-domain tchniqus, Control Enginring Practic, 8, 4, pp ,. [8] C. Evans, P.J. Flming, D.C. Hill, J.P. Norton, I. Pratt, D. Rs and K. Rodriguz-Vazquz. Application of systm idntification tchniqus to aircraft gas turbin ngins, Control Enginring Practic, 9, pp , 1. [9] K. Rodriquz-Vazquz. Multiobjctiv volutionary algorithms in non-linar systm idntification, Ph.D. Thsis, Univrsity of Shffild, Dpartmnt of Automatic Control & Systms Enginring, [] I. J. Lontaritis, and S.A. Billings. Input-output paramtric modls for nonlinar systms, Int. J. Control, 41,, pp , 198. [11] Y. P. Liu. Idntification of Nonlinar Systms: Th NARMAX Polynomial Modl Approach, Ph.D. dissrtation, Univrsity of Shffild, Dpartmnt of Automatic Control & Systms Enginring, UK,1988. [1] S.A. Billings, Chn, S. and M.J. Kornbrg. Idntification of MIMO nonlinar systms using a forward-rgrssion orthogonal stimator, Int. J. Control, 49, 6, pp , [13] S.A. Billings, and W.S.F. Voon. Structur slction and modl validity tsts in th idntification of nonlinar systms, IEE Procdings, 13, 4, pp , [14] N. Chiras, C. Evans, D. Rs. Nonlinar Gas Turbin Computr modlling using NARMAX structurs, Procdings of th 17 th IEEE Instrumntation and Masurmnt Tchnology Confrnc, Baltimor, 3, pp ,. [1] J.C. Pyton Jons, and S.A. Billings. Rcursiv algorithm for computing th frquncy rspons of a class of non-linar diffrnc quation modls, Int. J. Control,,, pp , APPENDIX πjf H 1( f = πjf πj f First-and scond-ordr transfr functions for Modl 1..3 H ( f H ( f πj ( f1 + f πj ( f1 + f ( + H ( f1, f = πj ( f1+ f πj ( f1 + f First-and scond-ordr transfr functions for Modl πjf πj f πj ( f H 1( f = 1 + f H ( πjf πj f f1, f = πj ( f1 + f πj( f1 + f 1.7 H ( f H ( f.6