or equivalently, (2) (4) Then we can state the following criterion for the stability of a closed-loop system:

Transcription

1 Istrumettio & Proess Cotrol Stility of Closed-loo Systems. Itrodutio A feedk otrol system must e stle s rerequisite for stisftory otrol. Cosequetly, it is of osiderle rtil imorte to e le to determie uder whih oditios otrol system eomes ustle. For exmle, wht vlues of the PID otroller rmeters K, τ I, d τ D kee the otroller roess stle? Defiitio of stility Before we roeed, we itrodue the followig defiitio for uostried lier systems. Notie tht the term uostried is used to refer to the idel situtio where there re o hysil limits o the outut vrile. Defiitio of stility. A uostried lier system is sid to e stle if the outut resose is ouded for ll ouded iuts. Otherwise, it is sid to e ustle. Chrteristi equtio Cosider the geerl lok digrm, whih is disussed i the revious hter. Usig lok digrm lger tht ws develoed i the revious hter, we oti GGvG y + G G G G m v y s GL + + G G G G m v d or equivletly, C G R G L () SP + Lod The stility hrteristis of the losed-loo resose will e determied y the oles of the trsfer futios G SP d G Lod. These oles re ommo for oth trsfer futios (euse they hve ommo deomitor) d re give y the solutio of the equtio G G G G 0 () + m v Equtio () is lled the hrteristi equtio for the geerlized feedk system. Let,,., e the roots of the hrteristi Equtio (): m v + G G G G ( s )( s ) K ( s ) (4) The we stte the followig riterio for the stility of losed-loo system: () A feedk otrol system is stle if ll the roots of its hrteristi equtio hve egtive rel rts (i.e. re to the left of the imgiry xis). Chter 6: Stility of Closed-loo Systems

2 Istrumettio & Proess Cotrol If y root of the hrteristi equtio is o or to the right of the imgiry xis (i.e. it hs rel rt zero or ositive), the feedk system is ustle. Figure rovides grhil iterrettio of this stility riterio. The qulittive effets of these roots o the trsiet resose of the losed-loo system re show i Figure. The left ortio of eh rt of this figure shows reresettive root lotios i the omlex le. The orresodig figure o the right shows the otriutios these oles mke to the losedloo resose to ste hge i the set oit. Similr resoses would our for ste hge i lod. Figure Stility regios i the omlex le for roots of the hrteristi equtio. The root lotios lso rovide iditio of how rid the trsiet resose will e. A rel root t s orresods to losed-loo time ostt of τ /. Thus, rel roots lose to the imgiry xis result i slow resoses. Similrly, omlex roots er the imgiry xis orresod to slow resose modes. The further the omlex roots re wy from the rel xis, the more osilltory the trsiet resose will e. Remrks The rodutg G G G G is lled oe-loo trsfer futio euse it reltes OL m v the mesuremet iditio y m to the set oit if the feedk loo is roke just efore the omrtor y G y (5) m OL s Chter 6: Stility of Closed-loo Systems

3 Istrumettio & Proess Cotrol Note tht the sme hrteristi equtio ours for oth lod d set-oit hges sie the term, +G OL, ers i the deomitor of oth terms i Equtio (). Thus, if the losed-loo system is stle for lod distures, it will lso e stle for set-oit hges. Figure Cotriutios of hrteristi equtio roots to losed-loo resose. Chter 6: Stility of Closed-loo Systems

4 Istrumettio & Proess Cotrol Exmle Cosider roess with the followig trsfer futios: 0 G, G f Gm, G K s Determie the rge of K vlues tht result i stle losed-loo system. Solutio The orresodig hrteristi equtio is + G G f m G G whih hs the root 0 + K 0 s s + 0K 0 s 0K The system is stle if <0 (i.e. K > /0). Exmle Cosider roess with the followig trsfer futios: G +, G f Gm, G 00 s + s + 0.s Determie wether the PI otroller stilize the system. Solutio The orresodig hrteristi equtio is 0 + G f GmGG + 0 K + s + s + s The equtio ove yields s + s + 0s with roots -7.85, j d.59.5j. The losed-loo system is ustle euse two roots of the hrteristi equtio hve ositive rel rts.. Routh Stility Criterio The riterio of stility for losed-loo systems does ot require lultio of the tul vlues of the roots of the hrteristi olyomil. It oly requires tht we kow if y root is to the right of the imgiry xis. Routh Stility Criterio is lytil tehique for determiig whether y roots of the olyomil hve ositive rel rts. This roh e lied oly to systems whose hrteristi equtios re Chter 6: Stility of Closed-loo Systems 4

5 Istrumettio & Proess Cotrol Chter 6: Stility of Closed-loo Systems 5 olyomils i s. Thus, the Routh Stility Criterio is ot diretly lile to systems otiig time delys, sie e -θs term ers i the hrteristi equtio. The Routh Stility Criterio is sed o hrteristi equtio tht hs the form o s s s L (6) We ritrrily ssume tht > 0. If < 0, we multily Equtio 6 y - to geerte ew equtio tht stisfies this oditio. First test. A eessry (ut ot suffiiet) oditio for stility is tht ll of the oeffiiets ( o,,,, K ) i the hrteristi equtio must e ositive. If y oeffiiet is egtive or zero, the t lest oe root of the hrteristi equtio lies o the right of, or o, the imgiry xis, d the system is ustle. Seod test. If ll of the oeffiiets re ositive, we ostrut the followig Routh rry: M M M K K K K where z Row + Routh Stility Criterio. A eessry d suffiiet oditio for ll roots of the hrteristi equtio to hve egtive rel rts is tht ll of the elemets i the left olum of the Routh rry re ositive.

6 Istrumettio & Proess Cotrol Remrks The umer of sig hges i the elemets of the first olum is equl to the umer of roots to the right of the imgiry xis. Exmle Determie the stility of system tht hs the hrteristi equtio s 4 + 5s + s + 0 Solutio Sie the s term is missig, its oeffiiet is zero. Thus, the system is ustle (First test). Exmle 4 Fid the vlues of otroller gi K tht mke the followig feedk otrol system stle. G G m GL, 5s +, s + G G f s + K Solutio The hrteristi equtio is 0s + 7s + 8s + + K 0 (7) All oeffiiets re ositive rovided tht +K > 0 or K > -. The Routh rry is K where 7(8) 0( + K 7 0 ) K ( + K ) 7(0) ( + K ) Chter 6: Stility of Closed-loo Systems 6

7 Istrumettio & Proess Cotrol To hve stle system, eh elemet i the left olum of the Routh rry must e ositive. Elemet will e ositive if K > 7.4/ Similrly, will e ositive if K > -. Thus, we olude tht the system will e stle if - < K <.6 This exmle illustrtes tht stility limits for otroller rmeters e derived lytilly usig the Routh rry; tht is, it is ot eessry to omute the roots of the hrteristi equtio or seify umeril vlue for K efore erformig the stility lysis.. Diret sustitutio method Diret sustitutio method is oveiet method for determiig the rge of otroller rmeters for whih the losed-loo resose is stle. The method is sed o the ft tht tht the roots of the hrteristi equtio vry otiuously with the loo rmeters. Cosequetly, t the oit of istility, t lest oe of the roots must lie o the imgiry xis of the omlex le s they ross from the left-hlf le to the right. This mes tht the roots re ure imgiry umers t the verge of istility. At this oit the loo is sid e mrgilly stle. This mes tht, t this oit, the hrteristi equtio must hve ir of ure imgiry roots t s, ± jωu. The frequey ω u with whih the loo osilltes is the ultimte frequey. The otroller gi t whih this oit of mrgil istility is rehed is lled the ultimte gi, K u. At gi just elow the ultimte, the loo osilltes with deyig mlitude, while t gi just ove the ultimte gi, the mlitude of osilltios ireses with time. At the oit of mrgil stility, the mlitude of osilltio remis ostt with time (Figure ). The ultimte eriod of osilltio T u is relted to the ultimte frequey, ω u, i rd/s, y Tu π ω (8) u The method of diret sustitutio osists of sustitutig s jωu i the hrteristi equtio. This results i omlex equtio tht e overted ito two simulteous equtios Rel rt 0 Imgiry rt 0 From these we solve for two ukows: oe is the ultimte frequey ω u, the other is y of the rmeters of the loo, usully the otroller gi t the oit of mrgil stility or ultimte gi. Geerlly, the losed-loo resose is ustle whe the otroller gi is greter th the ultimte gi. Chter 6: Stility of Closed-loo Systems 7

8 Istrumettio & Proess Cotrol Figure Resoses of losedloo with the otroller gi less th (), equl to (), d grter th () the ultimte gi. Exmle 5 Use the diret sustitutio method to determie K u for the system desried i Exmle 4. Solutio Sustitute s jω d K K u ito Equtio (7): 0 jω 7ω + 8 jω + + K or ( + K u u 0 7ω ) + j(8ω 0ω ) 0 Solve oth equtios + K 7ω 0 8ω 0ω ω(8 0ω ) 0 ω 0.8 ω ± K u u.6 u Thus, we olude tht K <.6 for stility. Equtio (9) idites tht t the stility limit, K K u.6, sustied osilltio ours tht hs frequey of ω rd/mi., if the time ostts hve uits of miutes. The orresodig eriod 7.0 mi. (9) Chter 6: Stility of Closed-loo Systems 8

9 Istrumettio & Proess Cotrol Performe riteri for losed-loo systems Itrodutio The futio of feedk otrol system is to esure tht the losed-loo system hs desirle dymi d stedy stte resose hrteristis. Idelly, we would like the losed-loo system to stisfy the followig erforme riteri:. The losed-loo system must e stle.. The effets of distures re miimized.. Rid, smooth resoses to set oit hges re otied. 4. Offset is elimited. 5. Exessive otrol tio is voided. 6. The otrol system is roust. I tyil otrol rolems, it is ot ossile to hieve ll of these gols sie they ivolve iheret oflits d trde-offs. For exmle, PID otroller settigs tht miimize the effets of distures ted to rodue lrge overshoots for set oit hges. O the other hd, if the otroller is djusted to rovide rid, smooth resose to set oit hge, it usully results i sluggish otrol for distures. Thus, the trde-off is required i seletig otroller settigs tht re stisftory for oth lod d set oit hges. Desig reltios for PID otrollers I this setio we osider some well-kow otroller desig reltios tht re sed o seifi model, mely the first-order lus time-dely model, Equtio (0). θs K e G( s) τ s + Desig reltios sed o itegrl error riteri A otroller desig reltios, whih is sed o erforme idex tht osiders the etire losed-loo resose, re develoed. Three erforme idies will e osidered i this ourse: Itegrl of the solute vlue of the error (IAE) 0 IAE e( t) dt () (0) Chter 6: Stility of Closed-loo Systems 9

10 Istrumettio & Proess Cotrol where the error sigl e(t) is the differee etwee the set oit d the mesuremet. Notie tht for P otroller, where offset ours, the itegrl give y Equtio does ot overge. I these ses, oe use modified itegrd, whih reles the error y y( ) y( t), sie this term does roh zero s t goes to ifiity. Itegrl of squred error (ISE) [ e t) ] ISE ( dt () 0 Itegrl of time-weighted solute error 0 ITAE t e( t) dt () A grhil iterrettio of the IAE erforme idex is show i Figure 4. Figure 4 Grhil iterrettio of IAE. The ISE will elize the resose tht hs lrge errors, whih usully our t the egiig of resose euse the error is squred. The ITAE will elize resose whih hs errors tht ersist for log time. The IAE will tret ll errors i uiform mer; thus, it llows lrger devitio th ISE. I geerl, ITAE is the referred itegrl error riterio sie it results i the most oservtive otroller settigs. Chter 6: Stility of Closed-loo Systems 0

11 Istrumettio & Proess Cotrol Desig reltios tht miimize the ITAE erforme idex re show i Tle. These reltios re sed o the first-order lus time-dely model, i.e. Equtio (0), d the idel PID otroller. Note tht the otiml otroller settigs re differet deedig o whether ste resoses to lod or set oit re osidered. For lod hges, the lod d roess trsfer futios re ssumed idetil. Tle. Cotroller desig reltios sed o ITAE erforme idex d firstorder lus time dely model. Exmle 6 For the roess model,.5s 4e G( s) 7s + Comre PI d PID otroller settigs sed o ITAE tuig reltios for oth lod d set oit hges. Solutio.5 KK K τ τ I τ 6.48 I The ITAE settigs re show elow: Chter 6: Stility of Closed-loo Systems

12 Istrumettio & Proess Cotrol Cotroller/Desig method K τ I τ D PI / lod PI / set oit PID / lod PID / set oit Figure 5 Comriso of otrollers desig usig ITAE riteri for () set oit d () lod hges. Figure 5 omres the ITAE otrollers. Desig for lod hges results i lrge overshoots for set-oit hges, while set-oit desig rodues sluggish resoses to lod distures. If set-oit hges d lod distures re oth likely to our, the omromise i the otroller settigs should e emloyed. Chter 6: Stility of Closed-loo Systems

13 Istrumettio & Proess Cotrol This exmle hs demostrted tht, i geerl, itegrl error riteri for set oit hges results i more oservtive otroller settigs th for lod hges. Cotroller tuig After otroller istlltio, the otroller settigs must usully e djusted util otrol erforme is osidered stisftory. This tivity is referred to s otroller tuig or field tuig of the otroller. I order to sve time d effort, it is desirle to hve relimiry estimtes of stisftory otroller settigs. A good first guess my e ville from exeriee with similr otrol loos. Altertively, if roess model is ville, itegrl error methods e emloyed to lulte otroller settigs. Field tuig, my still required to fie tue the otroller, eseilly if the ville iformtio is iomlete or ot very urte.. Cotiuous ylig method This roh hs lso ee referred to s loo tuig or the ultimte gi method. This method is desried s losed-loo method euse the otroller remis i the loo s tive otroller i utomti mode. A tyil exerimetl roh for PID otrollers e summrized s follows:. After the roess rehes stedy stte, remove the itegrl d derivtive modes of the otroller, levig oly roortiol otrol. O some PID otrollers, this requires tht the itegrl time (τ I ) e set to its mximum vlue d the derivtive time (τ D ) to its miimum vlue.. Selet vlue of roortiol gi (K ), distur the system, d oserve the trsiet resose. If the resose deys, selet higher vlue of K d gi oserve the resose of the system. Cotiue iresig the gi i smll stes util the resose first exhiits sustied osilltio, Figure 6. The vlue of gi d the eriod of osilltio tht orresod to the sustied osilltio re the ultimte gi (K u ) d the ultimte eriod (P u ). I erformig the exerimetl test, it is imortt tht the otroller outut does ot sturte. If sturtio does our, the sustied osilltio result eve thought K > K u. Tyil results re show i Figure 7. Defiitio. The ultimte gi K u is the lrgest vlue of the otroller gi K tht results i losed-loo stility whe roortiol oly otroller is used. Defiitio. The ultimte eriod P u is defied s the eriod of sustied ylig tht would our if roortiol otroller with gi K u were used. Chter 6: Stility of Closed-loo Systems

14 Istrumettio & Proess Cotrol. From the vlues of K u d P u foud i the revious ste, use the Ziegler-Nihols rules give i Tle 9. to determie otroller settigs (K, τ I, τ D ). These tuig reltios were emirilly develoed to rovide ¼ dey rtio, Figure 8. Figure 6 Resose of the loo with the otroller gi set equl to the ultimte gi K u. T u is the ultimte eriod. Figure 7 Determitio of K u usig Cotiuous Cylig Method. Tle. Ziegler-Nihols otroller settigs sed o the Cotiuous Cylig Method. otroller K P 0.5 K CU - - PI 0.45 K CU P U /. - PID 0.6 K CU P U / P U /8 τ I τ D Chter 6: Stility of Closed-loo Systems 4

15 Istrumettio & Proess Cotrol Figure 8 Qurter dey rtio resose to disture iut d to hge i set oit. Remrks. K u d P u e determied y the diret sustitutio method if the trsfer futios of ll of the omoets of the loo re kow qulittively.. The qurter dey rtio resose is very desirle for disture iuts euse it revets lrge iitil devitio from the set oit without eig too osilltory. However, it is ot s desirle for ste hges i set oit, euse it uses 50 % overshoot. This is euse the mximum devitio from the ew set oit i eh diretio is oe-hlf the reedig mximum devitio i the oosite diretio, Figure 8. This diffiulty esily e orreted y reduig the roortiol gi from the vlue redited y the formuls of Tle. I ft, the dey rtio is diret futio of the otroller gi, d e djusted t y time y simly hgig the gi. I other words, if for give roess the qurter dey rtio resose is too osilltory, redutio of the gi will smooth out the resose.. Shortomigs:.. It my e ojetiole euse the roess is ushed to stility limit. Cosequetly, if exterl distures or hge i the roess ours durig tuig, ustle oertio or hzrdous situtio ould our... This tuig roedure is ot lile to roesses tht re oe-loo ustle euse suh roesses re ustle t oth high d low vlues of K, ut stle for itermedite rge of vlues... Some roesses do ot hve ultimte gi; for exmle, first-order d seod-order roesses without time dely..4. The set of tuig rmeters eessry to oti the qurter dey rtio resose is ot uique, exet for the se of P otroller. Chter 6: Stility of Closed-loo Systems 5

16 Istrumettio & Proess Cotrol.5. Cotroller settigs i Tle should e regrded s first estimtes. Susequet fie tuig vi tril d error is ofte required.. Proess retio urve method This is oe-loo method; it is used s ltertive to Z-N method. Figure 9 shows tyil otrol loo i whih the otrol tio is removed d the loo oeed for the urose of itrodutio ste hge (M/s) to the vlve. The ste resose is reorded t the outut of the mesurig elemet. The ste hge to the vlve is oveietly rovided y the outut from the otroller, whih is i mul mode. The resose of the system (iludig the vlve, roess, d mesurig elemet) is lled the roess retio urve. Two differet tyes of roess retio urves re show i Figure 0 for ste hge ourrig t t 0. Figure 9 Blok digrm of otrol loo for mesuremet of the roess retio urve. This method is summrized i the followig stes:. After the roess rehes stedy stte t the orml level of oertio, swith the otroller to mul.. With the otroller i mul, itrodue smll ste hge i the otroller outut tht goes to the vlve d reord the trsiet, whih is the roess retio urve (Figure 0).. Drw stright lie tget to the urve t the oit of ifletio, s show i Figure 0. The itersetio of the tget lie with the time xis is the ret trsort lg (θ); the ret first-order time ostt (τ) is otied from: τ B u S (4) where Bu is the stedy stte ultimte vlue of B d S is the sloe of the tget lie. The stedy stte gi tht reltes B to M i Figure 9 is give y: Chter 6: Stility of Closed-loo Systems 6

17 Istrumettio & Proess Cotrol Figure 0 Tyil roess retio urves: () o-self regultig roess, () selfregultig roess. K Bu M (5) Note tht if the roess retio urve hs the tyil sigmoidl she show i Cse of Figure 0, the followig model usully rovides stisftory fit: θs B( s) K e GvG Gm (6) M ( s) τ s + 4. Usig the vlues of K, τ d θ from ste, the otroller settigs re foud from the reltios give i Tle. Notie tht the settigs give i this tle were develoed to rovide losed-loo resoses with dey rtio of ¼. Chter 6: Stility of Closed-loo Systems 7

18 Istrumettio & Proess Cotrol Tle Cohe d Coo Cotroller desig reltios. Remrks The dvtges of PRC method:. Oly sigle exerimetl test is eessry.. It does ot require tril d error.. The otroller settigs re esily lulted. However, it hs severl disdvtges:. The exerimetl test is erformed uder oe-loo oditios. Thus, if sigifit lod hge ours durig the test, o orretive tio is tke d the test results my e sigifitly distorted.. It my e diffiult to determie the sloe t the ifletio oit urtely, eseilly if the mesuremet is oisy d smll reorder hrt is used.. The reommeded settigs i Tles ted to result i osilltory resoses sie they were develoed to rovide /4 dey rtio. 4. The method is ot reommeded for roesses tht hve osilltory oe-loo resoses sie the roess model i Equtio (6) will e quite iurte. Chter 6: Stility of Closed-loo Systems 8

19 Istrumettio & Proess Cotrol Exmle 7 Chter 6: Stility of Closed-loo Systems 9