PI Servo with State-D Feedback for DC Motor Control with Consideration of Thermal Effect

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1 PI Servo wth Stte-D Feedbck for DC Motor Control wth Consderton of herml Effect *Surch Wongfooket ) nd hntch Kulworwnchpong ) ), ) Power System Reserch Unt, School of Electrcl Engneerng Insttute of Engneerng, Surnree Unversty of echnology, Nkhon Rtchsm, hlnd ) d54439@g.sut.c.th; ) hntch@gml.com ABSRAC hs pper presents PI-servo wth stte-d feedbck for DC motor control. he effect of wndng temperture s ncluded n the DC motor model. hs temperture cuses slow chnge n dynmc responses of the DC motor. he response of the proposed control nd the clsscl PI control re compred nd nvestgted. Wth the proposed control method, the effect of wndng temperture to the DC motor control cn be reduced. he control result shows tht the DC motor wth the proposed control gve stsfctory response nd provded good stblty.. INRODUCION DC motors cn be used extensvely n ndustres such s utomtc control system, robotc, electrcl vehcle, etc. Control of DC motor cn be used ether PI controllers or other. But the ncrese n motor s resstnce wth wndng temperture vrton cuses the motor s therml tme constnt to be ncresed whle ts electrcl tme constnt s reduced. In cse of permnent mgnet motors there re the reversble therml demgnetzton tht reduces the motor constnt presented by Rchrd H. Welch Jr. nd George W. Younkn (). hs therml effect cn degrde controller performnce. PI servo wth stte-d feedbck for LI systems hs recently been reported by Srwut, S., Wtchupong, W. nd Surch, W. (). It s notced tht most of these results re med for dsturbnce rejecton by stte-d nd commnd followng control. An dvntge over the conventonl stte feedbck s tht t results n smller dervtve gns. hs pper proposes PI-servo system wth stte-d feedbck control DC motor wth consderton of therml effect by usng the pole-plcement pproch. Secton of the pper explns the DC Motor wth therml effect. Our desgn gn PI nd stte D controller for DC motor re presented n Secton 3. Secton 4 provdes results for DC motor control, whle conclusons follow n Secton 5. ) Grdute Student ) Professor 4

2 Fg. Dgrm representng DC motor model. DC MOOR MODEL WIH HERMAL EFFEC DC motor model cn be represents by the dgrm n Fg.. he dynmc system cn be descrbed by stte Eq. (). b K t J J L Ke R Vt L L J () Copper wndng's resstnce t temperture chnge, s gven by Eq. () nd cn be found n Rchrd H. W. Jr. nd George W. Y.( )..393 R R () In permnent mgnet motors Ke nd Kt depend on ech specfc mgnet mterl nd temperture. It descrbed by Eq. (3) nd Eq. (4) respectvely s gven n Rchrd H.W. Jr. nd George W.Y. (). where L Ke Ke B (3) Kt Kt B (4) R = rmture resstnce ( ) t specfed temperture ( C) = rmture nductnce (H) 4

3 b = coeffcent of vscous frcton (Kg-m /s) J = rmture nert (Kg-m ) e K = motor voltge constnt (V-s/rd) t specfed temperture ( C ) K = motor torque constnt (Nm/A) t specfed temperture ( C ) t V t = ngulr speed of the motor shft (rd/s) = rmture current (A). = DC voltge fed to rmture crcut (V). = lod torque (Nm). = specfed temperture ( C ) = wndng s temperture ( C ) mb = mbent temperture ( C ) B = coeffcent for ech mgnet (Alnco =., SmCo =.35, NdFeB =., Ferrte =.)(/ C ) R th = therml resstnce ( C /W) = therml tme constnt (s) he temperture ncrese n DC motor cn be descrbed by Eq. (5). Eq. (6) llustrtes the ODE of chnged temperture s gven n Nst, K. De. nd Prsnt K. Sen (6). Power dsspton cn be expressed by Eq. (5). Dss P R (5) P R t P R P R e (6) Dss th Amb Dss th Amb d Dss th Amb (7) dt he followng prmeters of the motor re used: R.87, L.6 H, 6 b kg-m /s, J kg-m, Ke.6 V-s/rd, Kt.6 Nm/A, R th 6. C /W, B Alnco. / C,.565 Nm, V 5 V, 5 C, mb s, 3 C. Substtute these prmeters nto Eq. (). Eq. (8) cn be formed. t t, 78.3, , Vt (8) Bsed on Eq. (7) temperture rse n the DC motor s shown n Fg.. emperture strted rsng from 3 C to be sturte t 66.3 C wthn hours. Armture resstnce R, K e nd K t re chnged ccordngly by Eq. (), Eqs. (3) nd (4). Fg. 3 llustrtes the rmture resstnce chnged by the temperture. It strted from.87 to.355 t the sme hour perod. Fg. 4 exhbts the motor voltge nd torque 43

4 constnt K e nd K t. hey re slghtly reduced from.6 to emperture (celcous) Fg. emperture rse n the DC motor.6.4. R (ohm) Fg. 3 R wth the therml effect.6.4. Ke,Kt Fg. 4 K e nd K t wth the therml effect 44

5 Fg. 5 descrbed the speed of the DC motor wthout controller followng wth step nput Vt 5 V, nd the lod torque of.565 Nm. he response curves n Fg. 5() ndcte tht the response wth the therml effect cused the speed reducton down to 64.6 rd/s. In terms of the rse-tme, the response obtned wth the therml effect ws slower. Fg. 6 llustrtes the current of the DC motor. he response curves n Fg. 6() ndcte tht the current response wth the therml effect ws ncresed from 3.5 A to 3.56 A. In terms of the rse-tme, both responses wth nd wthout the therml effect re smlr Speed (rd/s) wth therml effect wthout therml effect Speed (rd/s) wth therml effect wthout therml effect () (b) Fg. 5 Speed of the DC motor wthout controller me (s) wth therml effect wthout therml effect 4 wth therml effect wthout therml effect (A).5.5 (A) () (b) Fg. 6 Current of the DC motor wthout controller me (s) 45

6 3. PI SERVO WIH SAE D FEEDBACK CONROL A DC motor system hvng sngle nput nd sngle output cn be descrbed by x Ax Bu (9) y Cx () he system sttes re fed bck through the gn mtrx K d, nd the error sgnl, the dfference between the reference nput (r) nd the output (y), s fed forwrd to the proportonl-ntegrl (PI) controller. hs error sgnl s denoted s. he block dgrm cn be shown n Fg. 7 to represent the control system. herefore, Eq. () nd Eq. () express the control sgnl nd the error sgnl, respectvely. u Kx k k () d p r y rcx () where s the output of the ntegrtor, k p nd k re controller prmeters. he desgn problem s to fnd the gns k p, k nd the mtrx K d. x Recll the DC motor stte mtrx s n Eq. (8). Let sttes form of the stte equton s descrbed n Eq. (9) cn be wrtten n detl s. he generl, 78.3 x 5.968, 44. x, 6.9 u (3) Snce, the speed of the motor s defned s the sngle output, Eq. (4) gve the output equton. y x (4) Fg. 7 Block dgrm representng the PI-servo nd stte-d feedbck control 46

7 Employng the Frobenus cnoncl form desgnted by AF nd - F x, x F BF cn be smply obtned from Eq. (5). where AF A, F B B (5) q qa qa (6) q h M (7) q s ( n) vector. M s the controllblty mtrx expressed by M B AB (8) One cn obtn the Frobenus form of the system, Eq. (3) s rnsformton of the system results n the Frobenus form of F A FB u. F F F 59,6, 43. F u (9) he control sgnl for the stte-d feedbck cn be wrtten s u KF () In whch K F kf kf nd Kd KF. herefore, F AFFBFKFF represents the system wth the nner feedbck loop. It s chrcterzed by n n d sdet sibfkfa F, whch s desred to be s ns n s s. Assgn the desred poles locted t 653, 47.. he desred chrcterstc polynoml cn be expressed s F s.4347s 438.4s 59,6 () For the nner loop control system, the chrcterstc polynoml s gven by d s s s, d f f For comprson, d s k s, 43. k s 59,6. () s s, where. Clculte the gn mtrx K d by Kd 47

8 Kd K d We cn wrte the equton descrbng the error dynmc s e A e B (3) u e where sgnl A Aˆ C, Bˆ ˆ B, A I BK A, d Bˆ IBKd B nd the control ue K p e (4) where K kp k p C. he error dynmc represented by Eq. (3) s chrcterzed by m m the chrcterstc polynoml s s s s, n whch. We cn wrte the error dynmc by o m m, 78.3 e 6.66,8 6,64.7 e ue. (5) Assgn the desred poles t.6 j 9.99,,595.7 for the PI-servo control prt, nd o s s s s s the chrcterstc polynoml. Clculte the gn mtrx Kp by K B A B A B A p m m For the error dynmc, s s s s represents the desred o m m chrcterstc polynoml. From the Cyley-Hmlton theorem, we cn wrte 3 A A A A I (6) 3 Resultng n k.739 nd k p m 4. SIMULAION RESULS o be compred wth the PI controller desgned by Zegler-Nchol s method to cheve the sme pole loctons s gven bove, one cn obtn the k.68 nd k 9.7. he step reference nput t 74 rd/s, nd the torque dsturbnce of.565 p 48

9 Nm re ppled t s. Fg. 8 llustrtes the current of the DC motor. he response curves n Fg. 8() ndctes tht the response of the PI controller nd the PI servo wth stte-d feedbck gve the sme result. he response of the therml effect cse s ncresed from 3.5 A to 3.56 A smlrly to tht of the uncontrolled cse. In terms of the rse-tme, the response of the PI controller hs overshoot lrger thn tht of the PI servo wth stte-d feedbck PI-D wth therml effect PI-D wthout therml effect PI wth therml effect PI wthout therml effect PI-D wth therml effect PI-D wthout therml effect PI wth therml effect PI wthout therml effect (A).5 (A) me (s) () (b) Fg. 8 Current response of the DC motor wth controllers 8 9 Speed (rd/s) PI-D wth therml effect PI-D wthout therml effect PI wth therml effect PI wthout therml effect Speed (rd/s) PI-D wth therml effect PI-D wthout therml effect PI wth therml effect PI wthout therml effect () (b) Fg. 9 Speed of the DC motor wth controllers me (s) Fg. 9 llustrtes the speed of the DC motor. he response curves n Fg. 9() ndcte tht the response wth the therml effect, wthout the therml effect, the PI controller nd the PI servo wth stte-d feedbck gve the sme result. In terms of the rse-tme, the speed response wth the PI controller experences overshoot lrger thn 49

10 tht of the PI servo wth stte-d feedbck. he overshoot of the responses s lso smlr. On the other hnd, the rse-tme of the therml effect cse s longer thn tht wthout the therml effect. For PI servo wth stte-d feedbck, the rse-tme wthout the therml effect s fster whle the overshoot of the response wth the therml effect cn be observed. 5. CONCLUSIONS An ntegrtor n the PI controller exhbts no stedy-stte errors n the response to the step nput. he stte-d feedbck hs n dvntge over the conventonl stte feedbck. It cn reduce the zero effect. hs pper hs presented control desgn method v the pole-plcement pproch for the DC motor speed control system wth consderton of the therml effect. he PI-servo wth stte-d feedbck control nd the PI control to cheve control objectves re smlr. Furthermore, the controller desgn by usng the proposed procedures s very smple nd effectve. REFERENCES Srwut, S., Wtchupong, W. nd Surch, W. (), PI-Servo wth Stte-D Feedbck Control for LI Systems, Reserch Journl of Appled Scences Engneerng nd echnology, Mxwell Scentfc Orgnzton, 4(7), Nst, K. De. nd Prsnt K. Sen. (6), Electrc drves, Prentce Hll of Ind Prvte Lmted, New Delh. Rchrd H., Welch Jr. nd Younkn, G.W. (), How emperture Affects Servomotor's Electrcl nd Mechncl me Constnts, Industry Applctons Conference, IEEE, 37(), Avlble; ture_when_specfyng_motors.pdf Avlble; 4