PID Controller Design Based on Second Order Model Approximation by Using Stability Boundary Locus Fitting

Transcription

1 PID Controller Desgn Based on Second Order Model Aroxmaton by Usng Stablty Boundary Locus Fttng Furkan Nur Denz, Bars Baykant Alagoz and Nusret Tan Inonu Unversty, Deartment of Electrcal and Electroncs Engneerng, Malatya, Turkey Abstract Ths study resents a model reducton method based on stablty boundary locus (SBL fttng for PID controller desgn roblems. SBL analyss was commonly aled for controller stablzaton roblems. However, we use SBL analyss for the reducton of hgh order lnear tme nvarant system models to second-order aroxmate models to facltate analytcal desgn of closed loo PID control systems. The PID desgn s mlemented by a multle ole lacement strategy whch enforces the control system had real oles wth a desred tme constant secfcaton. Illustratve desgn examles are resented for the analytcal PID desgn of hgh-order lant models by means of second-order SBL model aroxmatons.. Introducton Effectve model order reducton strateges are manly needed for the develoment of analytcal methods for the solutons of desgn and analyss roblems of control engneerng ractce due to fact that real control roblems may ntroduce hgh-order models. Model order reducton ams to obtan a low-order aroxmaton of hgh-order systems whch exhbts smlar tme and frequency resonse as much as ossble under the oeraton ranges of control systems. In the control onts of vew, t s very desrable to have a good aroxmaton for tme and frequency doman roertes of hgh-order systems, artcularly wthn the low frequency regon. Some mortant studes on model order reducton can be found n [-8]. In the current aer, we roosed a model dentfcaton aroach that fts SBL of the hgh-order LTI model of orgnal systems to SBL of a lower order aroxmate LTI model n the closed loo PI controller desgn lane. Here, we consder SBL lnes of orgnal system as the fnger rnts of orgnal system model and hence SBL of the order reduced system model s ftted the samled SBL onts of the orgnal LTI model. In the lterature, SBL analyses were wdely utlzed for stablzaton of PID control systems [9-]. SBL analyses are based on the soluton of characterstc equaton of unty feedback closed loo control system for controller arameters. SBL s manly lotted n ( k, k arameter lane to llustrate vsually the stablty regon of PI controller coeffcents. Essentally, the fttng of SBL lnes of orgnal model and aroxmate models rovde an aroxmaton of characterstc equatons of ther transfer functons at the samled frequency onts and the man advantage of SBL fttng n model aroxmaton s that t ams to match stablty regons of orgnal and aroxmate models. Also, n the aer, we emloy the model order reducton based on SBL fttng n the analytcal desgn roblem of closed loo PID control systems. Frstly, we obtan the second order model reducton of hgh order lant functon by the fttng the SBL n a redefned frequency range and exress the coeffcents of the second order aroxmate lant model by solvng lnear equaton set. Secondly, we used these coeffcents for desgn of PID controller that formulated based on ole lacement on real axs for a desred tme constant secfcaton of control systems. These two tasks rovde a straghtforward soluton for the analytcal PID tunng roblem for hgh-order lant models accordng to second-order aroxmate models. Snce the second order models have a hase lmt of -8 o, the roosed method s very effectve for the hgh-order lants wth a hase lmtaton of -8 o. Numercal examles llustrate alcaton of the roosed method for analytcal controller tunng and model dentfcaton roblems.. Methodology.. Theoretcal Background Ths secton summarzes calculaton of SBL for hgh order lant transfer functons accordng to closed loo PI control system gven n Fgure [9,]. SBL analyss was emloyed to fgure out the ranges of controller arameters stablzng control systems. In ths study, we used SBL curvatures for model aroxmaton rooses. By consderng transfer functon of PI controller as C( k + ks and transfer functon of the hgh-order lant model as G ( N( / D( n Fgure, one can wrte the characterstc equaton of the system as follows, Δ( + ( k + ks G( sd( + ( k s + k N( ( In order to obtan SBL of G (, characterstc equaton gven by equaton ( s solved for s jω n frequency doman. When ths comlex characterstc equaton, exressed as Δ( j ω, s solved real and magnary arts of the comlex characterstc equaton yeld the followng equaton set: k NR( ω k ω NI ( ω ω DI ( ω ( k N I ( ω + k ω N R ( ω ω DR ( ω ( where, N R (ω and N I (ω are real and magnary arts of the numerator olynomal. Smlarly, the olynomals D R (ω and D I (ω are real and magnary arts of denomnator olynomal. By solvng k (ω and k (ω from equatons ( and (, we obtan SBL lne denoted by S ( k ( ω, k ( ω n ( k, k lane as follows. N I ( ω D I ( ω + N R ( ω D R ( ω k ( ω (4 N I ( ω + N R ( ω ω N R ( ω D I ( ω ω N I ( ω D R ( ω k ( ω (5 N ( ω + N ( ω I R 87

2 The SBL of lant G ( can be exressed as a set of onts n ( k, k lane as S {( k ( ω, k ( ω : Δ( k, k, ω ω (, } (6 r(t k N( + C( k G ( + s D( Fg.. The unty feedback PI control system for SBL calculaton of hgh order lant model G (... Second Order Model Reducton by SBL Fttng Lets exress the second-order aroxmate system n a general form as follows, a o G ( (7 b s + bs + b The characterstc olynomal of unty feedback closed loo PI control system for G ( s can be exressed as, Δ ( s + C ( s G ( s (8 In order to fnd out SBL of G ( s as exlaned n revous secton, the characterstc equaton of system s wrtten n the frequency doman as, Δ ( jω k a bω ( b + k a b ω (9 To fnd the coeffcents of G ( s, whch are fttng the samled SBL onts {( S ( k ( ω, k( ω, S ( k ( ω, k ( ω } for angular frequency samlng ω [ ω mn, ωmax ], one can wrte the followng equaton sets by consderng equaton (9, Δ ( jω k ( ω a bω ( ( b + k ( ω a b ω ω bω Δ ( jω k ( ω a b ( ( b + k ( ω a Then, four homogenous lnear equatons to obtan four lant functon coeffcents { a, b, b, b } are wrtten for two real art equatons as, k ( ω a bω & k ( ω a bω ( and two magnary art equatons as follows, ω b + k ( ω ωa bω b ω + k ( ω aω bω ( The soluton of these homogenous equatons can be deendent on an arbtrary a as follows, k ( ω k ( ω b ( a ω k ω (4 ω ω b a ( ω ω (5 k / k ( ω k ( a ω ω ω b (6 When these equatons are used n second-order aroxmate system model (equaton (7, a coeffcents can be reduced because of a factorzaton n numerator and denomnator olynomals and one obtans the reduced model wth resect to samled SBL onts as, (7 G ( k ( ω k ( ω ( s + ( k ( / ω ω s ω ω k ( ( + ω k ω ω ( k ω ω ω Accordng to equaton (7, whle calculatng numercally b, b and b by equatons 4-6, one can select value of the coeffcent a equal to one for the sake of smlcty. Here, SBL samlng onts from hgh-order model G (, whch are k ( ω, k ( ω, k ( ω and k ( ω, can be calculated by consderng equatons (4 and (5 for a desred frequency samlng ω [ ω mn, ωmax ]. The G ( jω ermts the secondorder dynamcs contanng squared of frequency ( ω n frequency doman and rovdes SBL (equaton (4 and (5 wrtten as, b + b ω b k ( ω and ω k ( ω (8 a a (a Magntude (db Phase (deg (b k Bode Dagram -8 - Frequency (rad/sec G( G ( wth SBL G ( by Xue et al. 8 Samlng Ponts 6 4 Fg.. The Bode dagrams and SBLs of the second order reduced models obtaned by SBL model reducton method and Xue s otmal model reducton method. We used the hase lmtaton of the second order transfer functons to determne whether the hgh-order model can be reduced to the second order model aroxmately. The hase of the second order model can be exressed as, 5 G( G ( w th SBL G ( by Xue et al k 88

3 Arg ( G ( jω tan ( bω /( b bω (9 The hase of G ( jω s lmted to Arg ( G ( jω < π for < ω <. When the hase of the hgh-order model s n the range of Arg ( G( jω < π, ths system can be reduced to the second order aroxmate model for PID control system desgn. Ths condton s referred to hase lmtaton condton for second order model reducton. Fgure shows the bode dagrams and SBL lnes of the second order reduced models of a hgh-order lant model, 4 s + s + 5s + 5s + 4 G ( ( s + s + 74s + 75s + 64s + 764s + 7 Ths lant transfer functon comles wth hase lmtaton condton of second order model reducton and second order aroxmate model well reresents ths lant model. Fgure (a demonstrates that frequency resonses of the roosed SBL model reducton method and Xue s otmal model reducton method [7] are well agreement wth the frequency resonse of hgh-order model. Fgure (b clearly shows that SBL model reducton method can rovde better fttng to SBL of hgh order model. Because, SBL model reducton method rovdes a soluton enforcng to match SBL lnes at the samlng onts... Analytcal PID Control System Desgn for Second Order Plant Models In order to tune PID controller wth resect to a desred tme constant secfcaton of the closed loo control system, we lace all oles of the closed loo PID control system on the real axs for equal tme constants. The closed loo PID control system wth the second-order lant model roduces three system oles. Lets denote them by P, P and P. Our ole lacement constrant for the equal real oles s exressed as follows, P P P ( τ where, arameter τ s the domnatng tme constant of control system. For enforcement of the multle ole lacement constrant n PID desgn, the characterstc olynomal of control system establshed as follows, Δ ( s + τ s ( The characterstc olynomal of the closed loo PID control system for the second-order aroxmate model, defned by equaton (7, s wrtten as, Δ ( b s + ( b + akd s + ( b + ak s + ak ( By equatng equaton ( and (, one obtans the PID coeffcents regardng to the ole lacement constrant gven n equaton ( as, / b τ b b k ; / τ k ; b / τ b k d (4 a a a The man advantages of ths PID desgn strategy can be summarzed for the control of second-order lants as follows, ( It enables to desgn PID controller by sngle arameter secfcaton (tme constant secfcaton and t also ensures the stablty of the control system va the real system ole lacement. ( Due to rovdng hgh gan control by settng a low tme constant, the roosed PID desgn can exhbt good dsturbance rejecton erformance for second-order lants. Fgure shows PID desgn of the second order unstable lant G ( / ( s s + for varous tme constant. As seen n fgure, settlng tme of the second order system s about T s τ. Ths enables to desgn PID controller wth resect to settlng tme secfcaton. Fgure 4 shows unt ste and unt dsturbance erformance of these PID desgns for G ( / ( s + s + []. Haer method [] rovdes a smooth ste resonse wth low dsturbance rejecton. The method of [] rovdes faster ste resonse wth hgh overshoot. However, the roosed PID desgn method wth τ. can rovde better ste resonse and dsturbance rejecton erformance for the second order system because of the hgh gan control and negatve reel ole lacement Ts T 4 6 τ. τ.4 τ Fg.. PID desgn for varous tme constant and corresondng settlng tmes Proosed Method τ. Haer s method Fg. 4. Unt ste and unt dsturbance resonses comarson of the roosed PID desgn method wth other PID desgn methods for second order system model. By usng the coeffcents of second-order aroxmate models a, b, b and b n the equaton (4, one obtans PID desgn rule for SBL method as follows, ( k ( ω k ( ω k τ ( ω ω (5 k ( ω k ( ω ω + k ( ω ω ω s Proosed Method τ. Haer s method T s Proosed Method τ. Haer s method

4 ( k ( ω k ( ω k (6 τ ( ω ω ( k ( ω k ( ω kd ( k ( ω / ω (7 τ ( ω ω The arameter τ /Ts s set to secfy a desred settlng tme of ste resonse. ω and ω are lower and uer oeratng frequency bounds of control system. The control system works n low frequency regon. Consderng equatons (5-(7, one should select ω > ω > to obtan ostve fnte real controller coeffcents.. Numercal Examles Examle : In ths examle, we desgn PID controller wth the tme constant τ. sec for the hgh order lant functon gven n equaton ( by usng second-order reduced model obtaned by SBL fttng. (a Magntude (db Phase (deg (b k Bode Dagram G( G ( -8 - Frequency (rad/sec G( G ( SP k Fg. 5. (a Bode lots of the hgh-order system ( G ( and the second-order aroxmate model ( G ( s ; (b SBL lnes of G ( and G ( s. Bode dagram of G ( shown n Fgure 5(a ndcates that hase resonse of G ( s lmted to -8 o and the hase lmtaton condton of second order model reducton s satsfed. We used two samlng onts for the angular frequency of ω. 4 rad/sec and ω rad/sec. The second-order aroxmate model was found as, G ( (8.994s + s + PID desgn was obtaned as k 68., k 994. and k d 8.8 for the tme constant τ. sec by equatons (5- (7. Fgure 5(a reveals a satsfactory overlang of Bode hase and magntude lots. Ths demonstrates that a good frequency resonse matchng was obtaned for the both system n ths examle. Fgure 5(b shows SBL lnes of the hgh-order system ( G ( and the second-order aroxmate model ( G ( s. Almost exact overlang of SBL lnes was obtaned due to fact that SBL of G ( system resents a SBL curvature that sutable to mlement by the second-order system dynamcs. The desgned PID control system exhbts almost the same ste resonse for the both orgnal and aroxmate systems as shown n Fgure 6. Ths ndcates that model order reducton s very effectve and PID control system desgn for the hgh-order lant G ( s very successful G ( Fg. 6. Ste resonses of closed loo PID control system for G ( and G ( s. Examle : Ths examle shows second-order SBL model dentfcaton of DC motor by stmulatng closed loo PI control system for oscllaton. Then, we desgn PID control system for the angular velocty of DC motor shaft ( wth settlng n.5 sec. DC motor control s gettng sgnfcance for the rotor seed control of unmanned electrcal aeral vehcle (UAV [] and smle DC motors are well characterzed modeled by second order transfer functons. In ths test, we used smle DC motor model n Smulnk/MATLAB [4]. ( The smle DC motor model n Smulnk/MATLAB was connected to a closed loo PI control system. ( By adjustng PI coeffcents, two PI confguratons oscllatng the control system was found out as llustrated n Fgure 7(a. Two SBL onts ( k, k (-4.98,. at ω.847 and ( k, k (-4.9,. at ω. 67 cause oscllaton of the closed loo control system. ( The second-order model of the DC motor model was obtaned by usng roosed SBL fttng as, G ( (9.95 s +.94 s In order to obtan a settlng tme roughly.5 sec, the tme constant should be taken as τ.5/5. sec. The PID desgn for G ( s was obtaned for ths model as k. 96, G( 8

5 k 9.5, k d. 4. Fgure 7(b comares the results of the PID controller for the both orgnal and dentfed models. (a (b ω.847 ω G ( DC Motor Fg. 7. (a Oscllaton of the closed loo PI control system wth smle DC motor model, (b Ste resonses of closed loo PID control systems for G ( s and DC motor model. 4. Conclusons The aer ntroduced a SBL based model aroxmaton aroach for the alcaton of model order reducton and demonstrated a soluton for analytcal PID control system desgn examles for hgh-order systems by usng model reducton. We observed that SBL model aroxmaton method can be helful for dealng wth stablty reservaton roblem n a desred frequency regon. Ths roerty s mortant for controller desgn methods emloyng the model reducton. One of the man drawbacks of the roosed analytcal PID desgn method s that t allows PID desgns for the lant comlyng wth the hase lmtaton condton for second order model reducton. Snce SBL onts yeld oscllatng system behavor, SBL aroxmaton can be used for dentfcaton of the real lants by fndng low frequency oscllaton onts of closed loo PI control systems. Ths modelng strategy can be useful for onlne auto-tunng of PID controllers for second order systems by detectng a coule of oscllatng PI controller onts. [] H. Pade, Sur la reresentaton aroachee d'une functon ar des fractons rato-nalles, Annales Scentques de P'Ecole Normale Suereure, Sere. (Sul. Vol. 9,. -9, 89. [] Y. Shamash, Model reducton usng the Routh stablty crteron and the Pade aroxmaton technque, Int. J. Control, Vol., , 975. [4] M.F. Hutton and B. Fredland, Routh aroxmatons for reducng order of lnear tme-nvarant system, IEEE Transacton Automatc Control, Vol.,. 9-7, 975. [5] C.F. Chen and L.S. Sheh, A novel aroach to lnear model smlfcaton, Int. J. Control, Vol.8, , 968. [6] O. Ismal, On multont Pade aroxmaton for dscrete nterval systems, Proceedngs of the Twenty Eghth Southeastern Symosum on System Theory, Washngton DC, USA: IEEE Comuter Socety, , 996. [7] D. Xue and Y.Q. Chen, Sub-otmun H ratonal aroxmatons to fractonal order lnear systems, Engneerng Techncal Conferences & Comuters and Informaton n Engneerng Conference DETC5-8474, Calforna, USA,. -, 5. [8] N. Tan, I. Kaya and D.P. Atherton, Comutaton of stablzng PI and PID controllers, In: Proc of the IEEE Int. Conf. on The Control Alcaton (CCA, ,. [9] N. Tan, I. Kaya, C. Yeroglu and D.P. Atherton, Comutaton of stablzng PI and PID controllers usng the stablty boundary locus, Energy Converson and Management Vol. 47, , 6. [] F.N. Denz, N. Tan, S.E. Hamamc and I. Kaya, Stablty regon analyss n Smth redctor confguratons usng a PI controller, Transactons of the Insttute of Measurement and Control, Vol. 7(5, , 5. [] M. Haer, Tunng rules for PID controller usng a DMC strategy, Asan Journal of Control, Vol.4,. 4-47,. [] J.C. Shen and H.K. Chang, PID Tunng Rules for Second Order Systems, 4 5th Asan Control Conference, , 4. [] R. Mansour, M. Bettaye and S. Djennoune, Aroxmaton of hgh order nteger systems by fractonal order reduced-arameters models, Mathematcal and Comuter Modellng, Vol.5,. 5-6,. [4] R. Akbar-Hasanjan, S. Javad and R. Sabbagh- Nadooshan, DC motor seed control by self-tunng fuzzy PID algorthm, Transactons of the Insttute of Measurement and Control, Vol.7, , 5. References [] E.J. Davson, A method for smlfyng lnear dynamc systems, IEEE Trans. Auto.Contr. Vol. AC-,. 9-,