Project 2. PID Controller Design with Root Locus

Transcription

1 ECSE 4440 Control Engneerng Sytem Projet PD Controller Degn wth Root Lou TA. Abtrat. ntrouton. PD Controller Degn wth Root Lou 4. Exerment 5. Conluon

2 . Abtrat The uroe of th rojet to learn to egn a ontroller wth hel of the root lou. Root Lou a ery ueful tool for ontroller egn. t make t oble for a ontrol egner to keth the trae of the ytem ole an to gra unertanng of the haratert of the ytem. Root lou an be ue for ontnuou ytem -lane an rete ytem z-lane.. ntrouton n the frt rojet, we hae learne that the loaton an trae of ole an zero ee the erformane of a ontroller regarng the tablty an ee of reone an trakng nut. Root lou a ery ueful tool an eay to ue n ontrol engneerng. Root lou metho enable a ontrol egner to raw a keth of trae of ytem ole by han or by oftware tool. The objet ytem to be ontrolle a ngle enulum ame a n rojet one. PD an PD ontroller wll be egne wth the hel of root lou. An the equalent amle ata ytem wll be egne. Thoe ontroller wll be tue for ther tablty an for ther erformane. The ontroller wll be mlemente an erfe wth mulnk mulaton an exerment.. PD Controller egn wth Root Lou. Root Lou for a P Controlle SytemTak The gen ytem.. θ θ. θ u gn. Root Lo for the ytem wth roortonal feebaktak f the ytem ontrolle by a mle roortonal feebak ytem e.g., u θ θe, the haratert funton of the ytem 0 The haratert funton an be alo exree a or the root lou of the P ontrolle ytem, only wll be onere a a tanar form. So, k ha ole at 0, an no zero. or 80 l lou an, l, large >,

3 gure A fgure how, the root lo annot go further beyon wth hangng. n other wor, olely annot make the ytem rtal amng an hae roer tranent to meet the efaton multaneouly. A the te reone of the nonlnear ytem hown n gure -, maller 0 an not ahee the fat reone tme an 00 or 400 ntroue the bg oerhoot, ollaton an teay tate error. gure -. Root Lou for PD ontrolle SytemTak or the PD ontrolle ytem wth a wahout flter, the haratert equaton z z 4

4 where ontant z. z z ha ole at, z. or large > 080 lou,the aymtote are z 80 l60 an, l, z z α. 0, an zero at gure Comare to P ontroller, ole for PD ontrolle ytem ha more freeom to loate at a ere oton. The root lo ha two oble form hown aboe. The loaton of the leftmot ole ee the hae of the lo. f the leftmot ole nearer to the zero than the rght ole n the rght lo, the zero oe not affet the lo of the rght ole. On the other han, the rght ole are nearer to zero n the left lo, the lo of the rght ole are attrate by the zero. A the root moe aroun the rle an aroahng to the ertal aymote n the rght lo, the amng haratert rtal amng or n a reaonable range whh erable. To ajut gan for the efaton, ang an ajut one ole of the wahout flter an one zero loate at z z to the root lou of the lant G untl the root lou of the erable hae. The root lou of the lant n gure an the ere lou of the PD ontrolle ytem n gure 4. 4

5 gure gure 4 or the ytem to be rtal amng, the root lo houl moe along the real ax n the root lou. Ung gen z4.6 an 00, then the houl be n the range of 0 <.45or for the lo to moe along real ax. or the nonlnear ytem n gure 5, ontroller erformane lmte by the aturator. n real ytem, the lant annot trak a abrut ontrol gnaln our ae, torque o that the PD gan houl be hoen n maller regon. or the gen efaton, z4.6, 0.8, 00 reult n the te reone n gure 6. The effet of the aturator an be omenate by the ntegral ant-wnu. An the lnear ytem oen t hae the Coulomb frton term o that the erformane an be wore than that of the nonlnear ytem. 5

6 gure 5. Root Lou for PD Controlle Sytem tak gure 6 The haratert equaton of PD ontrolle ytem 0 5 6

7 7 an the tanar form of the. z z 4 ha four ole nlung 0 an a zero at - where z. or large 80 0 lou >, the aymtote are, 4, m n an,, l l. The three aymtote hae the orgn aroun an angle of them are 40,00 80 an. The root lo an be kethe a gure 7 One of the ole attrate by the zero. The other ole gather at an erge along the three aymtote. A the keth of the root lo hown n fgure 7, the aymtote heang to the rght, ro the magnary ax whh mean the ytem untable. Therefore the range of the gan to kee the ytem table ery rtal an neee to be alulate. One of the ommon metho to tet the tablty of a ytem Routh Crteron tet ung the haratert olynomal. The haratert olynomal of the gen PD ontroller 0 4 z z 6 f the arameter are ubttute wth the numeral alue of rojet an gan alue ere n reou tak, the equaton beome

8 4 : : : : : et et et or the ytem to be table, eery element n frt olumn houl hae ame gn. rom the r term, <.05e5 an from the 4 th term, the numerator houl be negate ne the enomnator et to be ote n r term. nally, the two nequalte turn out <.05e5 an e6 < < < 70. The ntereton of two nequalte 990 < < 70. So the maxmum of 70. n the Matlab mulaton, the real alue of ytem ole are hown. The lat two are almot zero o that the maxmum 70 aetable. m*n^m*lg^; /; num[ ]; en[ z** z** 0]; ytfnum,en; rltooly; 70; lfeebak*y,; realolel an An the root lo wth thoe gan alo how the alue the maxmum for the ytem to be table n gure 8. 8

9 gure 8 Beaue the an z aule ue before, too mall for the ytem to meet the efaton of the rng tme, bgger an z mut be hoen uh a 0 an z.. The te reone wth the new arameter et omare wth that of the reou ytem n fgure 9. gure 9.4 Samle Data ontroller tak 4 The amle ata Smulnk hown n fgure 0. Drete equalent ontroller an be ere ung the formula G z z G Z. The retze wahout flter 9

10 W z z Z An retze ontroller z Z z z tz z z Z z z z z z z ex t tz z z z ex t 9 8 z*wz*z Gz gure 0 The ontrolle ytem an be mlfe a aboe. Ung Matlab, the z of the retze ytem an be alulate a z.0945z z.90z 5.706z z.904 e.. G_ztf,[ 0],t_; W_z*tf[ -],[ -ex-t_*],t_; *W_z; t_*tf[ 0],[ -],t_; feebakg_z,; _z*; rltool_z; [theta T]te/*feebak70*_z,,0; >>abole_z an Three ole are loate ne of unt rle an one ole on unt rle. So, the z margnally table. n t root lou n fgure, there are three zero an four ole. the ytem table a long a <.e4. 0

11 An the te reone of the ytem n gure. gure gure

12 gure The te reone of the nonlnear moel hown n fgure 4. gure 4 n the ae of the nonlnear ytem, the aetable range of muh maller a you an ee n the fgure 4. The aturator ntroue ome error at the tranent an the ntegrator n PD ontroller um u the error. Therefore f bg, t take longer for the tranent error to fae away. or the maxmum t_ keeng the ytem table, the ytem tranfer funton tete for t_ nreang by m e wth fxe gan0,400,400 an 00. One, the loe loo ytem ha ole oute of unt rle, then the ytem untable. Therefore the eon lat t_ houl be maxmum amlng tme. Th mlemente n the below Matlab oure... mn_t_e-; t_mn_t_:mn_t_:.5; 400; %rltool_z; %for :lengtht_

13 for G_ztf,[ 0],t_; W_z*tf[ -],[ -ex-t_*],t_; *W_z; t_*tf[ 0],[ -],t_; feebakg_z,; _z*; l_zfeebak*_z,; frt_0; f maxabolel_z> & frt_0 ['Untable ytem at ',numtr]; t_ break en en >> t_t Untable ytem at 4 an The te reone of the loe loo ytem for t_.0 e an t_.04 e are hown n fgure 4_. or t_.0e, the reone table but for t_.04 e, untable. gure 4_.5 Tme elay tak 5 n real ytem, tme elay haen to be unaoable. So, a ytem egner ha to oner the tme elay from the frt egnng tage. Tme elay tort the tranfer funton by multlyng ex T elay whh annot be exree by a olynomal exatly. The eay tool to aroxmate the funton the Pae

14 metho. Wth the Pae metho, the tranfer funton of tme elay an be aroxmate aorng to the orer, whh a ytem egner or the ytem nee. Wth the gan ere reouly 8,. an 400, the n orer ae aroxmaton ge the maxmum tme elay.09e an t orer.0. After n orer, the maxmum elay tme onere to /; 8; *tf[ 0],[ ]*tf,[ 0]; Gtf,[ 0]; Te.00:.00:.; frt_0; for :lengthte [num,en]aete,;htfnum,en; maxolemaxrealolefeebakg**h,; f maxole>0&frt_0; frt_; ['untable ',numtr]; Te en en untable 9 an n the nonlnear mulate ytem wth tme elay blok n fgure 5, the te reone for arou tme elay are hown n fgure 6. The fgure 7 how the error of the t orer aroxmaton t_elay.0. gure 5 4

15 gure 6 gure 7 4. Exermenttak 6 4. P Controlle Sytem A t tue n., the gan olely ahee the ere erformane. Small 0 make the ytem too low an bg 00 make the reone ollate or untable n fgure 8 an 9. 5

16 gure 8 gure 9 4. PD Controlle Sytem Ang D ontroller to P ontrolle ytem ahee better erformane than P ontrolle ytem uner ame gan. The reaon that a egner ha a atonal zero an a ole to hooe an make the erformane to meet the efaton. Een though the te reone of PD ontrolle ytem ha better haratert n re tme an oerhoot, the reone how ubtantal teay tate error n fgure 0. 6

17 gure 0 4. PD Controlle Sytem Proerly hoen ontroller an remoe the teay tate error ntroue by PD ontroller. But gan may woren the other erformane an tablty n fgure an fgure. gure 7

18 gure 5. Conluon PD ontroller egn ha been tue wth root lou. rom the haratert equaton of a ytem, root lo hae been rawn. The root lo hel a egner elet the arorate alue of a ytem arameter uh a gan an unertan the effet of the arameter araton. Root lou an be rawn ealy by han or oftware. Root lo hae been ue n ontnuou an rete tme ytem. Thoe egne ytem hae been mulate an erfe by mulnk mulaton an exerment. 8