Design of Optimum Controllers for Gas Turbine Engines

Transcription

1 Desgn of Optmum Controllers for Gas Turbne Engnes Junxa Mu 1, Dav Rees 1, Cer Evans 1 an Neophytos Chras 1 School of Electroncs, Unversty of Glamorgan Pontypr, CF37 1DL, Wales, UK Phone: +44() Fax: +44() E-mal: jmu@glam.ac.uk or rees@glam.ac.uk Praxs Crtcal Systems Lmte, Manvers Street, Bath, BA1 1PX, UK Tel: +44() Fax: +44() E-mal: neophytos.chras@praxs-cs.co.uk Abstract Ths paper presents controllers base on lnear an nonlnear moels of an arcraft gas turbne engne. These moels along wth the performance nexes,, an are use to estmate the parameters of a PID controller. The parameters an hence the qualty of the controllers are very much epenent on the accuracy of the moels. It s shown that the Nonlnear AutoRegressve Movng Average wth exogenous (NARMAX) representaton proves a comprehensve benchmark for controller esgn, snce t moels the global ynamcs of the engne. However, the lnear moels prove an accurate representaton for small sgnal nputs an gve a relable metho for estmatng the controller parameters for a lmte operatng range. A comparson between controllers usng lnear an NARMAX moels s mae. 1 Introucton Gas turbnes are now extensvely use n aerospace, marne an nustral applcatons. Wth ths ncreasng use n a verse range of applcatons, esgnng of controllers for optmum performance s an mportant factor n the context of mprove effcency an envronmental frenlness. One of the essental requrements of any optmum controller esgn s the avalablty of accurate moels of the ynamcs of the system beng controlle. The work presente n ths paper s base on controller esgns carre out for a Rolls Royce Spey Mk arcraft gas turbne. Although t s no longer n servce, the Spey possesses the same characterstcs, for control purposes as a moern engne such that of the EJ ftte to the Eurofghter [1]. Ths work presents controllers usng moels base on work carre out by Evans et al. [, 3] an Chras et al. [4, 5, 6] n the last sx years. Gas turbne moelng eals wth the ynamc relatonshp between the fuel flow an the shaft spee. Evans et al. [, 3] concentrate on testng the engne usng small-ampltue multsne sgnals an frequency-oman technques to obtan lnear moels of hgh accuracy at fferent operatng ponts. Chras et al. [4, 5, 6] prouce several global nonlnear moels of the engne, usng block-structure (Wener, Hammersten, etc) moels, NARMAX representatons an neural networks. In ths work the proportonal, ntegral an ervatve (PID) controller s use, where the parameters are estmate usng the prevously estmate lnear an nonlnear moels. PID controllers are stll the most popular approach to control nustral processes n spte of contnual avances n control theory [7]. The fact that more than 9% of all control loops are PID controllers s not only ue to ther smple structure but also ue to ther robustness for most nustral applcatons. However for a varety of reasons most process controllers are surprsngly poorly tune. Therefore, tunng of PID controllers s an mportant ssue. A varety of PID controller tunng methos have been evelope n the last 5 years, such as Zegler-Nchols rule [8], Zegler-Nchols complementary rule [9], symmetrc optmum rule [1], some overshoot rule an no overshoot rule [11], refne Zegler-Nchols rule [1] an so on. These methos are apple rectly snce they prove smple tunng rules to etermne the PID parameters. However, snce they rely on a mnmum amount of ynamc nformaton, the realze close loop response s less than optmum. The paper wll nvestgate the tunng of PID controllers for the control of a gas turbne, usng lnear an nonlnear moels. The nonlnear moel s base on the NARMAX representaton. Three performance nexes wll be use, Integral of Error Square () [13], Integral of Absolute Error () an Integral of Tme Absolute Error () [14]. The Gas Turbne A gas turbne s mae of three basc components: a compressor, a combuston chamber an a turbne. Fgure 1 shows a smplfe schematc of the Rolls-Royce Spey engne. Ar s rawn nto the engne by the compressor, whch compresses t an elvers t to the combuston chamber. Wthn the combuston chamber the ar s mxe wth fuel an the mxture s gnte, proucng a rse n temperature an hence an expanson of gases. These gases are exhauste through the engne nozzle but frst pass through the turbne, whch s esgne to extract suffcent energy from them to keep the compressor rotatng. Both the compressor an the turbne are splt nto low pressure

2 (LP) an hgh pressure (HP) stages. The HP turbne rves the HP compressor an the LP turbne rves the LP compressor. They are connecte by concentrc shafts, whch rotate at fferent spees, enote N H an N L. These shaft spees are the prmary outputs of a gas turbne, from whch the nternal pressures an thrust can be calculate. estmate at each operatng pont usng a moel selecton an valaton proceure escrbe n etal n [, 3]. It must be stresse here that estmaton of lnear moels nvolves the removal of the means of the ata recors. In orer to compensate for ths, a constant term s ae to the moels gven below. 55%N H moel.63( s +.997).3939s e () ( s +.897)( s +.7) 65%N H moel.71( s +.518).17s e (3) ( s +.43)( s +.485) 75%N H moel Fgure 1. Smplfe schematc of the Rolls Royce Spey engne 3 Lnear Moels Work conucte by Jackson [15] showe that for a gven statonary pont the hgher orer nonlnear thermoynamc moels erve from the engne physcs can be reuce to lnear moels of the same orer as the number of engne shafts. More recently, Hll [16] apple tme oman technques to evelop screte moels. Although lnear screte moels wth goo nput an output propertes were estmate, problems wth the applcaton of tme oman technques were reporte [17]. More recently, Evans et al. [, 3] use multfrequency sgnals an frequency oman technques to obtan lnear moels of the engne. Multsne an Inverse Repeat Maxmum Length Bnary Sequences (IRMLBS) were use at ampltues of up to ±1% of the steay state fuel flow. The errors ue to nose an non-lnearty were assesse an foun to be small for these perturbatons. The use of multfrequency sgnals an frequency oman technques were foun to be well sute to ths problem snce they allowe the accurate estmaton of lnear moels at fferent operatng ponts, by mnmzng the effects of nonlnear storton an by accurately moelng the pure tme elay. Parametrc entfcaton n the frequency oman nvolves selectng the parameters of an s-oman moel wth pure tme elay T an s efne as: na nb b + b1 s +... bnbs H( s) a + a s +... a s 1 na e st Equatons (), (3), (4) an (5) show parametrc moels (1).69( s +.75).143s e (4) ( s +.616)( s +.44) 85%N H moel.653( s +.58).455s e (5) ( s +.97)( s +.418) 4 Nonlnear NARMAX Moel In orer to entfy a global moel capable of representng the engne ynamcs at all operatng ponts, Chras et al. [4, 5, 6] use tme- an frequency-oman nonparametrc analyss an a forwar-regresson algorthm to estmate NARMAX moels for the engne. The avantages of usng the NARMAX representaton are that they are lnear n the parameters, there s no restrcton n the nature of the exctaton, an the NARMAX representaton nclues a famly of other nonlnear representatons such as blockstructure moels. The moel s efne as: k) F{ k 1),..., k ny), u( k 1),..., u( k n ), e( k 1),..., e( k n )} + e( k) u F s a nonlnear functon; k), u(k) an e(k) represent the engne output, nput an nose sgnals respectvely; n y, n u an n e are the assocate maxmum lags. The selecton of the NARMAX structure s base on the error reucton rato (ERR) an s efne as: ERR N g w k 1 N k 1 y ( k) e ( k) (6) (7)

3 where g are the coeffcents; w (k) are the terms of an auxlary moel constructe n such a way that the terms w (k) are orthogonal over the ata recors. A forwarregresson algorthm s employe to select the term wth the hghest ERR at each step. The proceure s usually stoppe by usng an nformaton crteron such as the AIC, whch s efne as σ ε θ p e θ p + AIC N log ( σ ε ( )) kp (8) ( ) beng the varance of the resuals assocate wth a p-term moel an k s a penalzng factor. A concatenate ata set of small sgnal tests was use for structure selecton an parameter estmaton of the NARMAX moel as shown n Fgure. The moel shown n equaton (9) was then estmate usng the estmaton an valaton proceure escrbe n etal n [4, 5, 6]. It was shown that the moel s capable of moellng the statc an ynamc behavor of the engne for small an large sgnals, coverng a we operatng range efne by the ata use shown n Fgure. Ths moel wll be use n the remaner of ths paper to esgn controllers for the engne. Amp (cc/s) 4 3 (a) operatng set pont requre, an the controller nput s the error sgnals generate between the operatng set pont an the shaft spee. In orer to prevent engne surge, the rate lmter s neee. The rsng slew rate of the rate lmter s 4cc/s an the fallng slew rate s 4cc/s. Furthermore ue to the lmt on fuel fee, saturaton occurs at 44cc/s. In ths paper, three fferent ntegral performance nexes are use to select the PID parameters. To mplement optmum control effectvely, gan scheulng wll have to be use at fferent operatng ponts. Engne Throttle r( + Measure Engne - Varables e( PID Controller Fuel Injecton System Turbne Engne u 1 ( Fgure 3. Generalze Control Arrangement Fuel Flow Rate Lmter u ( Saturaton u 3 ( The control requrements are that there s mnmum overshoot an a fast response tme. It must be stresse here that the engne thrust an power are relate to the shaft spee, thus oscllaton n the engne output wll cause the engne power to oscllate an create ffculty for the plot Tme (s) (b) 9 Shaft Spee (%NH) Tme (s) Fgure. Concatenate ata set use for estmaton (a) measure fuel flow (b) measure HP shaft spee n ) u( 3.31e.7.56e 4 5 u( n )... n ) 5 Gas Turbne Control The control arrangement for the gas turbne to control the HP shaft spee s shown n Fgure 3. The performance of the gas turbne s prmarly epenent on the shaft spee of the engne, an ths s etermne by the fuel fee, whch s controlle. The engne throttle gves the (9) 6 Optmum PID Controller Usng Lnear Moels Because of the rate lmter an saturaton n the forwar path, the analyss of the system response usng lnear s- oman moels s not easly carre out. The optmum PID parameters are obtane by usng the moels gven n equatons ( - 5) an by teratng the system (Fgure 3) to obtan the mnmum performance nexes through equatons {(11), (13), (14), (16), (17), (19)}. The tunng of the controller an the evaluaton an mnmzaton of the performance nexes were one usng a nonlnear mnmzaton routne n MATLAB. The parameter space for K p, K was n the range between an 1 an for K n the range between an. The eal transfer functon of a PID controller expresse n the s-oman s shown as u1 ( s) K e K + s p + K s (1) where K p s the proportonal gan, K s the ntegral gan an K s the ervatve gan. The performance nexes were expresse as follows:

4 Integral of Error Square () For the contnuous tme case 4 J1( K p, K, K ) ( r( ). t (11) For the screte tme case t 8 J 1 ( K p, K, K ) T ( r( ) (1) n where r(, r( are the esre set ponts an, are the HP shaft spees (%N H ), T s the samplng tme (.5s). Thus the optmum PID controller esgn can be state as mn J1( K K p, K, K, K, K ) p (13) resulte n zero. The optmum PID parameters are shown n Table 1. Set pont Inex K p K I Mn. Inex (%N H ) (%N H ) (%N H ) (%N H ) Table 1. The optmum PID parameters at fferent operatng ponts for the lnear moels (K ) Integral of Absolute Error () 8 step response 8.5 step response For the contnuous tme case 8 4 J ( K p, K, K ) r(. t (14) For the screte tme case t 8 ( K p, K, K ) T r( n J (15) The optmum PID controller esgn s shown as mn J ( K K p, K, K, K, K ) p Integral of Tme Absolute Error () For the contnuous tme case 4 (16) J 3 ( K p, K, K ) t r(. t (17) For the screte tme case t 8 3 ( K p, K, K ) T nt r( n J (18) The optmum PID controller esgn s shown as mn J 3 ( K K p, K, K, K, K ) p (19) The optmum search for the ervatve parameters K (a) Tme(s) (b) Tme(s) Fgure 4. The step responses for the 7-8%N H range usng the, an (a) Large range (b) Small range Fgure 4 shows the step responses for a range between 7%N H an 8%N H wth controllers estmate usng fferent nexes. It can be seen that the response of the system s acceptable whchever performance nex was use although the nex gave controller parameters that resulte n a lttle more overshoot. All the performance nexes gave smlar controller parameters an consequently smlar tme response profles. The responses of the fuel fee for a seres of steps from 5%N H to 9%N H s shown n Fgure 5 usng the nex. Fgure 5 shows that the percentage ncrease n fuel fee ncreases as the set pont s ncrease. Ths emonstrates the nonlnear characterstc of the fuel fee system. The results ncate that gan scheulng s essental for the optmum control of the gas turbne. The step responses also show that the PI controller s aequate for the gas turbne system.

5 45 4 step response f u ( 44 > then u ( 44 (3) 3 Fuel flow (cc/s) else u u ( ) +.66u 3 ( n The NARMAX moel usng the controller s efne by n ) e ( 3.31e 4 5 u ( n )... 3 n ) (4) Tme(s) Fgure 5. The responses of the fuel fee for the 5-9%N H range usng the 7 Optmum PID Controller Usng the NARMAX Moel The parameters of the nonlnear NARMAX moel are lnear, an the recursve methoology s use to select the PID parameters. The control arrangement s shown n Fgure 3 by usng screte tme nstea of usng contnuous tme. The samplng tme of the NARMAX moel s.5s. u 1 () represents the ntal output of the controller, an u 1 ( represents the samplng pont n output of the controller. The equatons use to evaluate the screte representaton of the PID controller are e( r( () 8 u1( K pe( +.5K e(... n (1) e( e( + K + u1().5 Snce the rsng slew rate of the rate lmter s 4cc/s an the fallng slew rate s 4cc/s, the output of the rate lmter (u ) s represente by u f 1( u ( > 4.5 then u.5*4 + u ( n 1) ( u f 1( u ( < 4.5 then u.5*( 4) + u ( n 1) ( u1( u ( f 4 < < 4.5 then u u ( ) ( 1 n For the saturaton, ts output (u 3 ) s shown as () Evaluatng equatons () through to (4) recursvely, the performance nexes are calculate through equatons {(1), (13), (15), (16), (18), (19)}, an the optmum PID parameters are obtane. The optmum search for the ervatve parameters K resulte n zero. The optmum PID parameters for the 1%N H step response tests from 5%N H to 9%N H are shown n Table. Set pont Inex K p K Mn. Inex (%N H ) (%N H ) (%N H ) (%N H ) Table. The optmum PID parameters at fferent operatng ponts for the NARMAX moel (K ) Step response (a) Tme(s) Step response (b) Tme(s) Fgure 6. The step responses for the 7-8%N H range usng the, an (a) Large range (b) Small range

6 An example of the system step response s shown n Fgure 6 for the 7-8%N H step, an s gven for the three performance nexes. The response of the fuel fee from 7%N H to 8%N H usng the nex s shown n Fgure 7. by the ncrease n the nex, whch s shown n Table 3. Agan gan scheulng s essental n orer to mplement the optmum controller effectvely. 8 Step response 36 Step response 8 Fuel fee(cc/s) Tme(s) Fgure 7. The response of the fuel fee for the 7-8%N H range usng the From Fgure 6 t can be seen that the response of the system s acceptable whchever performance nex was use. The rse tme s fast an there s mnmal overshoot, although the nex gave controller parameters that resulte n a small overshoot. Furthermore, the PI structure s aequate for controllng the gas turbne. Wth the ncrease n the operatng pont, both K p an K are ncrease, confrmng the non-lnearty of the engne. 8 Comparson of the Controllers Usng the Lnear an Nonlnear NARMAX Moels The results obtane show that the controllers erve usng lnear an nonlnear moels gve comparable an acceptable responses. The values of the parameters obtane for the respectve operatng range are very smlar for the ntegral acton term, but n general the proportonal gan term s somewhat hgher usng the lnear moels compare wth the NARMAX moel. The one excepton s the 5-6%N H range. Fgure 8 compares the fference between the performance of the controllers esgne usng the lnear an nonlnear moels. It presents the step response for the 7-8%N H range usng the parameters obtane from the nex for both moels. For the sake of comparson, the controller settngs obtane for the 5-6%N H range are also nclue. It can be seen that the controller settngs usng the lnear moel gve a slght overshoot, compare wth the controller settngs usng the NARMAX moel, whch s crtcally ampe. The response profle usng the parameters obtane for the 5-6%N H range s very much overampe. The less than optmum settngs are ncate Tme(s) Fgure 8. The step responses usng the NARMAX moel for the 7-8%N H range by usng NARMAX (sol) an lnear (asho moel controller parameters for the same range, NARMAX (otte) an lnear (ashe) moel controller parameters for the 5-6%N H range Moel Controllers Inex 5-6%N H lnear moel controller %N H NARMAX moel controller %N H lnear moel controller %N H NARMAX moel controller 7.81 Table 3. The nex usng the controllers erve from 5-6%N H an 7-8%N H lnear an NARMAX moels apple to the NARMAX moel for the 7-8%N H range. 9 Conclusons The paper has presente lnear an nonlnear NARMAX moels of a gas turbne engne to esgn a controller wth optmum settngs. The parameters of the controllers are very much epenent on the accuracy of the moels. The NARMAX representaton s a global moel an covers both small an large sgnal nputs, an therefore proves a comprehensve benchmark for controller esgn. However, the lnear moels prove an accurate representaton for small sgnal nputs an gve a relable metho for estmatng the controller parameters for ths operatng regme. The performance nexes,, an show that both moels gve comparable an acceptable responses although there s a lttle more overshoot usng the compare wth the an, an all three nexes gve parameters that are smlar for the respectve operatng range. The values of ntegral terms usng the lnear an nonlnear NARMAX moels are very smlar,

7 but the proportonal gan terms are somewhat hgher usng the lnear moels except for the 5-6%N H range. The PI controller s aequate for the gas turbne system. Because of the non-lnearty of the gas turbne, gan scheulng s essental n orer to obtan the optmum control of the engne. Acknowlegments Ths work was conucte on ata gathere at the Defence Evaluaton an Research Agency (DERA) at Pyestock wth the support of Rolls Royce plc. The authors woul lke to thank all staff nvolve, especally Dr D. C. Hll of Rolls Royce. References [1] Da, G. J., A. E. Sutton an A. W. M. Greg. Multvarable control of mltary engnes, AGARD Conference Proceengs No. 57 Avance Aero- Engne Concepts an Controls, 8, pp.1-1, [] Evans, C., D. Rees an D. Hll. Frequency oman entfcaton of gas turbne ynamcs, IEEE Transactons on Control Systems Technology, 6, pp , [3] Evans C, P J Flemng, D.C. Hll, J. P. Norton, I Pratt, D. Rees an K. Rorguez-Vaswquez. Applcaton of system entfcaton technques to arcraft gas turbne engnes, Control Engneerng Practce, 9, pp ,. [4] Chras N. Lnear an nonlnear moellng of gas turbne engnes, Ph.D. Thess, Unversty of Glamorgan,. [5] Chras N, C. Evans, D. Rees. Nonlnear moellng an valaton of an arcraft gas turbne, IFAC Symposum Nonlnear Control System, St. Petersburg, 1. [6] Chras N, C. Evans, D. Rees. Global nonlnear moellng of gas turbne ynamcs usng NARMAX structure, ASME Journal of Engneerng an Power, Accepte for publcaton,. [7] K. J. Åström, T. Hägglun. The future of PID control, Control Engneerng Practce, 9, pp , 1. [8] Zegler, J. G., & Nchols, N. B. Optmum settngs for automatc controllers, Transatons of ASME, 64, pp , 194. [9] Mantz, R. J., & Taccon, E. J. Complementary rules to Zgler an Nchols rules for a regulatng an trackng controller, Internatonal Journal of Control, 49(5), pp , [1] Voa, A., & Lanau, I. D. A metho for the autocalbraton of PID controller, Automatca, 31, pp.41-53, [11] Seborg, D. E., T. F., & Mellchamp, D. A. Process Dynamcs an Control, Wley, New York, [1] Hang, C. C., Astrom, K. J., & Ho, W. K. Refnements of the Zegler-Nchols tunng formula, IEE Proceengs-D, 138 (), pp , [13] Zhuang, M., & Atherton, D. P. Automatc tunng of optmum PID controllers, IEE Proceengs-D, 14 (3), pp.16-4, [14] Pessen, D. W. A new look at PID-controller tunng, Transactons of the Amercan Socety of Mechancal Engneers, Journal of Dynamc Systems, Measurement an Control, 116, pp , [15] Jackson D. Investgatng of state space archtectures for engne moels, Rolls Royce plc, Report TDR 9331, [16] Hll, D.C. System entfcaton of gas turbne engne, Ph.D. Thess, Unversty of Brmngham, School of Electronc an Electrcal Engneerng, [17] Hll, D.C. Ientfcaton of gas turbne ynamcs: tme-oman estmaton problems, ASME Gas Turbne Conference, paper 97-GT-31, pp.1-7, [18] Grace, A. Optmzaton Toolbox for Use wth MATLAB, The Mathworks Inc, 1996.