Proceedng of the European Control Conference 9 Budapet, Hungary, Augut 6, 9 WeA6. Centralzed PID Control by Decouplng of a BolerTurbne Unt Juan Garrdo, Fernando Morlla, and Francco Vázquez Abtract Th paper deal wth the control of a nonlnear bolerturbne unt, whch a x multvarable proce wth great nteracton, hard contrant and rate lmt mpoed on the actuator. A control by decouplng methodology appled to degn a multvarable PID controller for th unt. The PID controller obtaned a a reult of approxmatng an deal decoupler ncludng ntegral acton. Three addtonal proportonal gan are tuned to mprove the performance. Becaue of the nput contrant, a condtonng ant wndup trategy ha been ncorporated n the controller mplementaton n order to get a better repone. A good decoupled repone wth zero trackng error acheved n all mulaton. The reult have been compared wth other controller n lterature, howng a mlar or better repone. Keyword: bolerturbne unt, multvarable PID control, centralzed control by decouplng, ant wndup technque. A I. INTRODUCTION bolerturbne unt a x proce howng nonlnear dynamc under a wde range of operatng condton []. In order to acheve a good performance, the control of the unt mut be carred out by multvarable control tratege. In fact, the requrement to control multaneouly everal meaure wth trong couplng, jutfy the ue of any multvarable control. In recent year many reearcher have pad attenton to the control of bolerturbne unt ung dfferent control methodologe, uch a robut control, genetc algorthm (GA) baed control, fuzzy control, gancheduled approach, nonlnear control and o on. Robut multvarable controller ung H loophapng technque [], [] acheve good robutne and zero trackng error; however, H controller encounter dffculte n dealng wth the control nput contrant. Genetc algorthm baed method uch a GA/PI or GA/LQR control [4], [5] may caue large overhoot or teadytate trackng error. Nonlnear controller [6] how good Manucrpt receved October 5, 8. Th work wa upported by the Spanh CICYT under Grant DPI 7 65. Th upport very gratefully acknowledged. Moreover, J. Garrdo thank the FPU fellowhp (Ref. AP649) of the Spanh Mntry of Scence and Innovaton. J. Garrdo wth the Department of Informatc and Numerc Analy, Unverty of Córdoba, Span (correpondng author, phone: 4957879; fax: 4957879; emal: pgajuj@uco.e). F. Vázquez wth the Department of Informatc and Numerc Analy, Unverty of Córdoba, Span (emal: fvazquez@uco.e). F. Morlla wth the Department of Informatc and Automatc, UNED, Madrd, Span (emal: fmorlla@da.uned.e). decouplng, trackng and robutne properte n dfferent operatng pont, becaue they conder the nonlnear dynamc of the ytem n ther degn tage. Th paper llutrate the applcaton of a multvarable PID controller to the bolerturbne unt condered n []. Th PID controller a full matrx controller (centralzed control) degned by the methodology of decouplng control [79]. The centralzed control depcted n Fg. ha been choen ntead of decentralzed control (dagonal control) becaue the bolerturbne unt how gnfcant nteracton []. PID decouplng control degn methodology wthn the framework of a unty feedback control tructure ha been teted by the author for TITO (twonput twooutput) procee n [9], obtanng good reult. r () r () r () k () k () k () k () k () k () k () k () k () u () u () u () g () g () g () g () g () g () g () g () g () y () y ( ) y ( ) Fg.. x centralzed control ung a unty feedback control tructure Moreover, becaue of hard contrant mpoed on the actuator of the bolerturbne unt, an ant wndup trategy ued to mprove the repone of the ytem. In ecton II the nonlnear multnput/multoutput (MIMO) bolerturbne model preented. Secton III deal wth the PID centralzed control by decouplng of th ytem. In ecton ISBN 9789669 Copyrght EUCA 9 47
J. Garrdo et al.: Centralzed PID Control by Decouplng of a BolerTurbne Unt WeA6. IV the reult are evaluated n comparon wth other author. Fnally, ecton V preent the man concluon of th paper. II. THE BOILERTURBINE MODEL The bolerturbne model ued n th paper wa developed by Bell and Atröm []. The model a thrd order nonlnear multvarable ytem wth great nteracton, hard contrant and rate lmt mpoed on the actuator. The dynamc of the unt gven by 9/8 x.8ux.9u.5u 9/8 x.7u.6x.x x 4u.u.9 x/ 85 y x y x y.5.7x ac qe / 9 67.975 where tate varable x, x and x denote drum preure (kg/cm ), power output (MW) and flud denty (kg/m ), repectvely. The nput u, u and u are the valve poton for fuel flow, team control, and feedwater flow, repectvely. The output y the drum water level (m) regardng the operatng reference level, o t can take potve and negatve value. Varable a c and q e are team qualty and evaporaton rate (kg/), repectvely, and they are gven by c.58x.8x5.6 x.94.4x q.854u.47 x 45.59u.54u.96 e a Due to actuator lmtaton, the control nput are ubject to the followng contrant: u,, u.7 u. u.5 There are everal typcal operatng pont of the Bell and Atrom model (), but the lnear control degn for the unt found n lterature uually take the lnearzed model at the operatng pont x =[8 66.65 48] T, u =[.4.69.4] T and y =[8 66.65 ] T. The lnearzed model gven by the followng tranfer functon matrx G(). () () () G 98. 6 49.. 4. 578 58. 58. 7 44. 9655.. 48. 6 65. 466 9. 4. 49. 978. 5657. 59. 79 There a common pole =.59 n all t element, a well a a common pole =. n the econd row (y ), and a common ntegrator n the thrd row (y ). Th x matrx ued to degn the centralzed control by decouplng. For nteracton analy we obtan the Relatve Gan Array (RGA) of the model. Snce G() contan ntegrator element n the thrd row, the RGA calculated n the alternatve way decrbed n []. The expreon of the RGA.9.684.58 RGA.994.76.47.4.4 The hgh number of RGA value whch are far of the unt how a proce wth great nteracton, a t wa expected. Furthermore, t hgh condton number of 587 ndcatve of an llcondtoned plant. So, calng the proce hould be advable n order to reduce th number. Neverthele, calng not ued becaue t ntegrated n the propoed degn methodology. III. PID DECOUPLING CONTROL In [9], centralzed PID control by decouplng for TITO procee preented. In th ecton th degn methodology extended to x procee gven by T (4) (5) g() g () g() G () g() g () g() (6) g() g () g() where the proce controlled by a control law dependng on the error gnal, uch a t hown n Fg. Th, u() k() k() k() r() y() u() k() k() k() r() y() u() k() k() k() r() y() where K() the x fullcro coupled multvarable tranfer matrx of the controller. The paradgm of decouplng control [79] propoe to fnd a K() uch that the cloed loop tranfer matrx G() K() [I G() K()] decoupled over ome dered bandwdth. Th goal enured f the open loop tranfer (7) 48
Proceedng of the European Control Conference 9 Budapet, Hungary, Augut 6, 9 WeA6. matrx L()=G() K() dagonal. For th reaon, the technque ued n decouplng control are very mlar to the technque ued to degn decoupler. Then, aumng that the open loop tranfer matrx hould be dagonal l () L() l () l ( ) the followng expreon for the deal controller by decouplng obtaned K() G() L() G l G l G l g l gl gl G l G l G l g l gl gl G G l G l G l g l gl gl where the complex varable ha been omtted, where G j the cofactor correpondng to g j () n G(), and where the G() nne tranfer functon g()= j are the equvalent j G() procee for the nne decoupled SISO loop [7] controlled by k j () repectvely. It can be een n (9) that each column of K() are related to the ame dagonal element of L(). Therefore, pecfyng the three l () tranfer functon enough to determne the nne element k j () of the controller from expreon (9). A. How to pecfy the l () The degn problem n (9) wll have oluton f the pecfcaton of l () are well propoed, n other word, f they take nto account the dynamc of the three correpondng equvalent procee, the achevable performance pecfcaton of the correpondng SISO cloed loop ytem, and, not le mportant, that the controller mut be realzable. Snce the cloed loop mut be table and wthout teadytate error due to et pont or load change, the open loop tranfer functon l () mut contan an ntegrator. Then, the followng general expreon for l () propoed [9]: (8) (9) l() k () l () Parameter k become a tunng parameter n order to met degn pecfcaton and l () mut be a ratonal tranfer functon takng nto account the common not cancelable dynamc of the correpondng equvalent procee. Typcal not cancelable dynamc are the nonmnmum phae zero and untable pole. In the x ytem under revew (4), l ()= choen for l () and l (), nce ther correpondng equvalent procee do not have common nonmnmum phae zero and untable pole. In th cae, the cloed loop tranfer functon ha the typcal hape of a frt order ytem k h() k T () wth tme contant T =/k. Then, n order to determne k t enough to pecfy the tme contant of the cloed loop ytem. T =5 and T =.5 are elected, o k =.4 and k =.8. z On the other hand, l ()= choen for l () becaue the correpondng equvalent procee have a pole n =. Now, the cloed loop tranfer functon gven by the followng expreon, a econd order ytem wth a zero n =z. z k h() k z z k kz () It pole are characterzed by the undamped natural frequency and the dampng factor k n kz ; () 4z Partcularly, t uffcent to elect k =4z n order to acheve pole wth crtcal dampng (=) and n =z. In the controller degn z =. and k =.4 are tuned, o t obtaned a ytem wth crtcal dampng and n =. n the thrd loop. Thee adjutable parameter k have been choen n order to have the mlar ettlng tme of the ytem output repone wth other author method, and to have nto account the actuator contrant of the ytem. Increang tunng parameter k n the controller matrx, the correpondng th ytem output repone become fater, but the output energy of the th column controller of K() and ther correpondng actuator grow larger, tendng to exceed ther output capacte n practce. So, tunng parameter k a tradeoff between the achevable ytem repone performance and the actuator contrant. Conequently, after electng the three tranfer functon l (), the dagonal equvalent open loop proce L() the followng 49
J. Garrdo et al.: Centralzed PID Control by Decouplng of a BolerTurbne Unt WeA6..4.8 L () (4).4 Then, the nne element k j () of the multvarable centralzed controller by decouplng K() are obtaned by replacng (4) and (4) n (9). Neverthele, the reultng element do not have PID tructure. In order to get a centralzed PID control by decouplng, model reducton technque baed on the frequency repone are ued, jut a t decrbed n the next ubecton. B. Ung PID tructure If t ntended that controller become PID controller wth fltered dervatve, t neceary to force the followng tructure n all controller element TDj k j() kkpj TIj jtdj (5) where t appear the controller wth t four parameter: proportonal gan (K Pj ), ntegral tme contant (T Ij ), dervatve tme contant (T Dj ) and dervatve tme noe flter contant ( j ). Alo note that the ame tunng parameter k appear n k, k and k. For PI tructure dervatve tme contant forced to zero. After applyng th reducton to K(), a matrx K PI () wth eght PI controller (6) wa elected for the control of the bolerturbne unt. PID tructure ha alo been teted n ome element but t how a wore performance wth the nonlnear model. The reducton carred out n the frequence from 4 to.5 rad/. K PI(). 4. 6. 8. 4. 6. 8 76. 9. 9 95. 7 7 7 587. 56. 587. 56. (6)... 44 4. 9. 44. 49. 555. 8 88. 95. 8 The ngular value plot of the deal controller by decouplng and the reduced K PI () controller are hown n Fg.. It can be hown that they are cloe at the low frequence but dfferent at the hgh frequence. So the reduced controller wll have mlar performance a the deal controller at low frequence. Sngular Value (db) 6 4 4 6 Sngular Value 8 4 Frequency (rad/ec) Fg.. Sngular value plot of the deal controller (old lne) and the reduced K PI controller (dotted lne) C. Condtonng ant wndup trategy The bolerturbne unt ubject to hard contrant n control nput gnal (), therefore the controller need to be equpped wth ome protecton mechanm agant the wndup effect. Otherwe t performance could deterorate when the nput gnal contrant are exceeded. In th paper t ued a multvarable ant wndup technque that decrbed n [4] and called condtonng. To ue th trategy the controller element have to be bproper, that, the number of pole equal the number of zero. In the cae of PID tructure gven by (5) th condton fulflled. The condtonng ant wndup cheme hown n Fg., where K the hgh frequency gan matrx and K() the tranfer matrx that fulfl the followng equaton K() K K() (7) In addton, the Contrant block mut ncorporate a model of the control gnal contrant. E() K Contrant K() U at () Fg.. Condtonng ant wndup cheme K For the degned controller K PI (), K and K() matrce can be ealy dentfed from (6) where each element expreed a the um of a gan (K j ) and a ratonal tranfer functon K j (). K nonngular, whch t neceary to the control mplementaton. So, n order to mplement the 4
Proceedng of the European Control Conference 9 Budapet, Hungary, Augut 6, 9 WeA6. control depcted n Fg., we need the x gan matrx K and t nvere, the x ntegrator matrx K() and the actuator contrant model (). IV. SIMULATION RESULTS In th ecton, we analyze the performance of the developed controller for the nonlnear bolerturbne unt decrbed n ecton II through mulaton and comparon wth other author. Specfcally, we compare the propoed controller wth a robut controller of Tan n [], and wth a decentralzed multvarable nonlnear controller (MNC) n [6]. Although the controller of Tan wa degned va loophapng H approach, then t wa reduced to four PI element. The propoed NMC n [6] baed on tate pace repreentaton of the nonlnear ytem (). After defnng the dered cloed loop equaton for each output, controller parameter are obtaned n order to compenate nteracton a dturbance. To tet the performance of the propoed controller, degned n ecton III, three mulaton are gven. In the frt one we how the uperorty of the propoed controller over the Tan controller wth regard to decouplng when the bolerturbne model () ued wthout nput contrant. In the other mulaton we compare the propoed controller wth MNC and Tan control ung the nonlnear dynamc model () wth the nput contrant () and condtonng ant wndup trategy for the propoed and Tan controller. In the econd mulaton we change from the nomnal pont to another operatng pont that cloe. In the thrd one, a large operatng pont change mulated. y (kg/cm) y (MW) y (m) 5 5 8 4 5 6 6 4 5 6.4.. 4 5 6 Tempo () Fg. 4. Sytem output repone for two et pont change from nomnal operatng pont (Propoed control: old lne; Tan: dotted lne) In the frt mulaton, at t=, drum preure y ncreaed from 8 to kg/cm, at t=, power output y ncreaed from 66.65 to MW, and drum level y kept at m. In Fg. 4 the ytem output repone for the propoed and Tan controller are hown. The propoed control acheve a better decouplng than the Tan controller. In addton, t ha a moother repone and wthout overhoot n y and y, and a lower devaton n y. In the econd mulaton we ue the nonlnear model of the bolerturbne wth actuator contrant and we carry out the ame two et pont change of the frt mulaton, but both change occur at the ame tme t=. In Fg. 5 and Fg. 6 the ytem repone for the propoed and the two prevou controller are hown. Fgure 5 how the tme repone of output. The repone of the three dfferent controller are mlar; however, the propoed control acheve a moother repone and wthout overhoot n y, and a lower devaton n y. Alo, control gnal of th controller (Fg. 6) are le aggreve than control gnal of the other controller. y (kg/cm) y (MW) y (m) 5 8 5 5 5 5 4 45 5 6 5 5 5 5 4 45 5... 5 5 5 5 4 45 5 Tempo () Fg. 5. Output gnal of the ytem for a change from nomnal operatng pont to a near operatng pont (Propoed control: old lne; Tan: dotted lne; MNC: dahed lne) u u u.8.6.4. 5 5 5 5 4 45 5.9.8.7 5 5 5 5 4 45 5.5 5 5 5 5 4 45 5 Tempo () Fg. 6. Input gnal of the ytem for a change from nomnal operatng pont to a near operatng pont (Propoed control: old lne; Tan: dotted lne; MNC: dahed lne) 4
J. Garrdo et al.: Centralzed PID Control by Decouplng of a BolerTurbne Unt WeA6. To how that the propoed lnear controller can operate well n a wde operatng range, we conder a large operatng pont change at t= n the lat mulaton. Drum preure ncreae from 75.6 to 4 kg/cm, power output from 5. to 8 MW, and drum level from.97 to.98 m. The ytem repone for the degned controller are hown n Fg. 7 and Fg. 8 n comparon wth the other two control. The three controller have a very mlar performance, but the propoed control acheve a bt lower ettlng tme n output y. In addton, t control gnal are the moothet. y (kg/cm) y (MW) 8 6 4 8 4 5 6 7 8 9 5 y (m) u u u 5 4 5 6 7 8 9 4 5 6 7 8 9 Tempo () Fg. 7. Output gnal of the ytem for a large operatng pont change (Propoed control: old lne; Tan: dotted lne; MNC: dahed lne).5 4 5 6 7 8 9.8.6.4 4 5 6 7 8 9.5 4 5 6 7 8 9 Tempo () Fg. 8. Input gnal of the ytem for a large operatng pont change (Propoed control: old lne; Tan: dotted lne; MNC: dahed lne) Alo LQR and GA/LQR controller n [5] have been mulated and contrated wth the propoed controller; however they are not llutrated n th work becaue ther repone were wore than the repone of other two control that have been choen for comparon here. V. CONCLUSION In th paper, we decrbe the applcaton of a new degn methodology of multvarable PID control to a bolerturbne unt. The degn procedure cont of three tep: frt, an deal decoupler ncludng ntegral acton determned. Second, the decoupler approxmated wth PID controller. Thrd, three proportonal gan are tuned to acheve pecfcaton. Due to the hard nput contrant of the plant, the multvarable controller mplemented wth an ant wndup compenaton. Smulaton reult how that the controller ntroduced n th paper well done for the nonlnear bolerturbne ytem. Interacton are reduced, zero trackng error acheved and t can operate well n a wde operatng range. The reult have been contrated wth other controller n lterature and the propoed control how a mlar or better performance. REFERENCES [] R. D. Bell and K. J. Atröm, Dynamc model for bolerturbnealternator unt: data log and parameter etmaton for a 6 MW unt, Lund Inttute of Technology, Sweden, Report TFRT9, 987. [] W. Tan, Y. G. Nu, and J. Z. Lu, H control for a bolerturbne unt, IEEE Conference on Control Applcaton, 999, pp. 878. [] W. Tan, H. J. Marquez, T. Chen, and J. Lu, Analy and control of a nonlnear bolerturbne unt, Journal of Proce Control, vol. 5, no. 8, pp. 8889, 5. [4] R. M. Dmeo and K. Y. Lee, Genetcbaed control of a bolerturbne plant, Proceedng of the rd Conference on Decon and Control, 994, pp. 557. [5] R. M. Dmeo and K. Y. Lee, BolerTurbne control ytem degn ung a genetc algorthm, IEEE Tranacton on Energy Converon, Vol., Num. 4, 995, pp. 75759. [6] D. L, H. Zeng, Y. Xue, and X. Jang, Multvarable nonlnear control degn for bolerturbne unt, Proceedng of the 6th World Congre on Intellgent Control and Automaton, 6, pp. 758 75. [7] Q. G. Wang, Decouplng Control. Lecture Note n Control and Informaton Scence; 85. SprngerVerlag,. [8] T. Lu, W. Zhang, and F. Gao, Analytcal decouplng control trategy ung a unty feedback control tructure for MIMO procee wth tme delay, Journal of Proce Control, vol. 7, pp. 786, 7. [9] F. Morlla, F. Vázquez, and J. Garrdo, Centralzed PID Control by Decouplng for TITO Procee, Proceedng of th IEEE Internatonal Conference on Emergng Technologe and Factory Automaton, 8, pp. 85. [] F. Vázquez, F. Morlla, and S. Dormdo, An teratve method for tunng decentralzed PID controller, Proceedng of the 4th IFAC World Congre, 999, pp. 49496. [] F. Vázquez, Deño de controladore PID para tema MIMO con control decentralzado, Ph.D. The, UNED, Madrd,. [] F. Vázquez and F. Morlla, Tunng decentralzed PID controller for MIMO ytem wth decouplng, Proceedng of the 5th IFAC World Congre,, pp. 778. [] B. A. Ogunnake and W. H. Ray, Proce Dynamc, Modelng, and Control. Oxford Unverty Pre, 994. [4] G. C. Goodwn, S. F. Graebe, and M. E. Salgado, Control Sytem Degn. Prentce Hall,. 4