Proportional, Integral, and Derivative Controller Design Part 1

Transcription

1 A Suna onlin continuing ducation cour roportional, ntgral, and rivativ ontrollr ign art by tr J nndy

2 roportional, ntgral, and rivativ ontrollr ign art A Suna onlin continuing ducation cour. ntroduction: n thi cour, th dign and application of roportional -plu- ntgral -plu- rivativ controllr i dicud. control i a tchniqu ud xtnivly in fdback control yt. t origin dat back to th 9 th cntury, bing ud for govrnor pd control, and inc thn in nurou application with a wid varity of actuator and nor. h controllr i ipl tructur; bing th u of thr tr a th na ipli. t configuration i illutratd in Figur.. h tructur provid for a fairly-wid rang of tuning adjutnt in a fdback control loop, pcially for rlativly ipl proc. So failiarity with fdback control ay hlp in providing a bttr undrtanding of th cour atrial. Rfrnc [, ] provid xcllnt covrag of controllr dign a includd in thi cour. Ential apct of fdback control ar includd, but on ight alo rviw rfrnc on th ubjct [3, 4] or Sunca our 8. Figur. Functional Block iagra ontrollr controllr ar ud in any control application; poibly bing th ot coon for of fdback control copnation. With fdback control, th output tat of a phyical yt to b controlld rfrrd to a th plant, proc, or load i aurd by a nor. h aurd tat i fd back and copard to a dird tat. h trinology for th dird tat vari dpnding on th application; oftn trd t point in proc control, a rfrnc ignal in circuit dign, or a coand input fro an outr control loop. h controllr dtrin th diffrnc btwn th aurd and dird tat; th control loop rror, to gnrat a control ignal that rduc th rror. hi quat to a ngativ fdback control loop; cntral to fdback control thory. A u th rror, it intgral and drivativ to driv a control ignal driving th rror to a null tat. h controllr can b tructurd in any configuration; -only,,,, plu othr to b dicud. control i cntral to ot proc control yt; but can alo b found in nurou application othr than proc control ranging fro poitioning control loop to pointing, tracking and platfor tabilization control loop. h can alo b intgratd with opyright 7 tr J. nndy ag of 4

3 roportional, ntgral, and rivativ ontrollr ign art A Suna onlin continuing ducation cour highr lvl control tratgi uch a odl prdictiv control, adaptiv controllr and fuzzy logic control dcribd in art of th cour. h vratility of th ay rid i a fairly-ipl control tructur, ay to iplnt in oftwar or hardwar, offring loop gain adjutnt, an intgrator to rduc or null rvo rror, and a drivativ th provid pha lad to iprov loop tability or act a a prdictiv lnt. An xapl of a fdback control yt i an indutrial proc, hown in Figur., whr it i dird to aintain a working fluid at a contant tpratur. Figur. Fluid pratur ontrol roc Ful i input to a cobution chabr, which hat th fluid. A control valv i th actuator controlling how uch ful i input to th chabr; bing adjutd bad upon a control ignal fro th controllr. A tpratur nor i th fdback nor that aur th tpratur of th working fluid at th output of th cobution chabr. h aurd tpratur i fd back to th controllr and copar it to th dird tpratur or t point, calculating a diffrnc or rror. f th tpratur i l than th t point, th rror could b ud by a controllr to forc or hat into th roo. h proportional and intgral gain will t th rpon ti. A th t point i rachd, th intgrator will hlp null th rror and th drivativ tr rduc any ovrhoot hutting down th hating or cooling ourc to th roo. n thi cour, control dign will b prntd fro th prpctiv of claical control dign tchniqu. ontrol thodology i oti dividd into claical and odrn control thod; ffctivly diffrnt approach to olving a control probl, ach having thir own advantag and diadvantag. laical control dal dirctly with th diffrntial quation that dcrib th dynaic of a plant or proc, tranforing th into frquncy dpndnt tranfr function for analyi. h tranfr function i th ratio of two frquncy dpndnt polynoial who root opyright 7 tr J. nndy ag 3 of 4

4 roportional, ntgral, and rivativ ontrollr ign art A Suna onlin continuing ducation cour dcrib th rpon of th plant in a frquncy doain. h controllr or copnator, uch a th controllr dicud in thi cour, hap th clod fdback loop rpon for a givn plant rpon, to achiv th control prforanc objctiv. Modrn control thory u Stat Spac thod to valuat th rpon of a yt a wll a gnrat th control for it. hi rli havily on linar algbra and atrix thory. h approach i vry hlpful whn olving probl with yt that hav high ordr dynaic or ulti-input, ulti-output probl. ontrol yt dign and analyi rli havily on athatic. Bfor focuing on th controllr dign, th nxt thr ction will provid an ovrviw of gnral fdback control yt architctur, rlationhip that govrn any controllr dign, and th athatic of forulating of th control yt dynaic in both th ti and frquncy doain.. Fdback ontrol Block iagra: h fdback control yt block diagra provid both a viual a wll a a bai for th athatical rprntation of th fdback control yt. t i an iportant part of th control yt dign and analyi. n gnral, th fdback control block diagra will b coplx incorporating any lnt and vral fdback loop. o dcrib jut th baic lnt, a vry ipl block diagra i hown in Figur 3.. Figur 3. Baic Fdback ontrol Block iagra hi ipl loop includ a controllr, plant, and fdback nor. h input ar th coand input cd in, diturbanc d and nor noi n. h coand, which i th dird plant output, i th rfrnc variabl and oti dnotd a t-point in proc control application. t i applid to th input uing junction and copard with th aurd plant output. h rvo rror btwn th coand and aurd output i calculatd and fd to th controllr ; a i a pcific typ of controllr -> a dcribd in thi cour. h controllr gnrat th driv ignal to th plant that rduc th rror until th coand input qual th actual output or rain within a dird offt. n an actual control loop dign, th could b th priary haping function for th control loop, howvr, othr copnator uch a lad, lag, or lad-lag ay b rquird to obtain th tability argin dird; copnating for th lag of pcific control loop coponnt. opyright 7 tr J. nndy ag 4 of 4

5 roportional, ntgral, and rivativ ontrollr ign art A Suna onlin continuing ducation cour h dynaic of ach block can b xprd in th ti or frquncy doain. h ti doain i vry uful for iulating th loop rpon, and valuating prforanc a a function of ti. A t of diffrntial quation which can includ non-linar tr ar ud to dcrib th dynaic for ach block. h frquncy doain i xcllnt for linar analyi. h gain of ach lnt can b charactrizd a a function of frquncy; ffctivly aking ach lnt a frquncy dpndnt gain. h output rpon can b valuatd a a function of frquncy, and th controllr adjutd in th frquncy doain, to iprov th loop prforanc. 3. i and Frquncy oain Rprntation of Loop ynaic: h plant dynaic ar norally dcribd by a t of diffrntial quation and function for lnt that ar non-linar in ti or ulti-variabl dpndnci. f th plant i linar or can b linarizd about an oprating point, th frquncy doain rpon can b dcribd by uing th Laplac tranfor or a diffrntial oprator. h frquncy doain i iportant for analyi and dign; a ud in nuing ction. 3. Frquncy oain finition: h Laplac tranfor of a ti dpndnt variabl ft i dfind a: F t f t dt hi intgral xit for th coplx variabl σj with any ral part σ>. h variabl πf whr f i frquncy in hrtz. iffrntiating th right-hand id of thi xprion n ti with rpct to ti, rult in th iportant rlationhip: n n t n f t dt F k A diffrntiatd function in ti i algbraically rlatd to it tranfor a L[f n t]-> n F whr L dnot th Laplac tranfor. h uation tr on th right account for th initial condition of ach diffrntiatd tr. For frquncy analyi, initial condition ar oftn aud to b zro o that: f n t t dt n F for k initial f nk condition wo iportant oprator rlationhip, a litd in abl., can b drivd fro thi xprion. abl. Laplac ranfor rivativ and ntgral Oprator ranfor ranfor Notation Not rivativ f t t dt F L f t > F ntgral f t t dt F L f t > F f t f t dt opyright 7 tr J. nndy ag 5 of 4

6 roportional, ntgral, and rivativ ontrollr ign art A Suna onlin continuing ducation cour h intgral rlation i obtaind, noting th drivativ of th intgral i th function, o that: f t t t dt F f t dt F F F h ignificanc of th rlationhip i thy allow diffrntial quation to b convrtd to algbraic xprion iilar to th u of diffrntial oprator. h invr Laplac tranfor can b ud to convrt a frquncy doain function back to on in th ti doain. h invr tranfor i givn by: c j t f t F d π j c j hi coplx intgral i valuatd along th path cj in th coplx plan fro c-j to cj whr c i any ral nubr > σ for which th path cj li in th rgion of convrgnc of th tranfor F. Although a owhat coplicatd dfinition, ot control txtbook or nginring rfrnc hav vry thorough tabl of both th Laplac tranfor and it invr tranfor for nurou function o on nd not valuat th tranfor or it invr dirctly. t i alo iportant to not that although th dfinition of th Laplac variabl i σj, whn prforing frquncy doain analyi, th variabl will b quatd to j whr πf. h σ tr i aud zro inc for frquncy doain analyi a function i bing valuatd bad upon it rpon to priodic inuoidal ignal. h ral axi σ in th -doain rprnt an xponntial dcay or growth factor not rlvant for thi analyi. 3. Evaluation of F: h frquncy dpndnt function F i a coplx nubr that can b xprd in tr of a agnitud and pha or ral and iaginary part a: jθ F F R F j F ; j h coplx function i quantifid in tr of it agnitud and pha a dfind in abl.. abl. oplx Function finition Function Sybolic Exprion Quantitativ Exprion F jθ F F co θ j F inθ ral part R F F coθ iaginary part F F inθ agnitud F R F F pha Φ F a tan[ F/ RF ] For frquncy rpon analyi F could rprnt th clod loop tranfr function LF or opn loop tranfr function OLF of th control yt. Rfrring to Figur 3., th ar drivd in ction 4. and givn by: opyright 7 tr J. nndy ag 6 of 4

7 roportional, ntgral, and rivativ ontrollr ign art A Suna onlin continuing ducation cour opyright 7 tr J. nndy ag 7 of 4 Snor Fdback ontrollr lant of Lalac ranfor OLF F LF F,,,, A th tranfr function i th ratio of two frquncy dpndnt polynoial, th thr tranfr function can alo b writtn a th ratio of a nurator N and dnoinator polynoial: ; ; N N N h corrponding LF and OL ar thn givn by; ; N N N OLF N N N N N LF ypically, th plant for application i a low ordr low gain tabl tranfr function. h controllr provid ot of th gain for th loop a rquird pr th prforanc pcification. Gnrally, on dir unity gain fdback o that caling aociatd with th fdback nor i noralizd to on by an invr caling contant. f fdback i not unity gain, th quantity S will cal th clod loop gain a can b n in th prviou xprion. h LF and OLF ar ud to valuat th control loop rpon and tability, and ar dcribd in or dtail in th nxt ction. h yt rpon can b valuatd via Bod, Nyquit, or Nichol plot and analyi. Bod analyi plot th agnitud and pha of th OLF and LF to valuat rpon and i rlativly traightforward to undrtand. h OLF i plottd in tr of it opn loop gain OLG and th opn loop pha. Bod tability critria dfin two critical frqunci; th gain croovr frquncy, fg, which i th frquncy at which th OLG cro on and th pha croovr frquncy, f, which i th frquncy at which th OLF pha, Φ, cro -8. A tabl pha argin i th aount that th OLF pha angl i > -8 whn th OLG at th gain croovr frquncy. h gain argin i th agnitud of th OLG rlativ to unity gain whn th OLF pha go through -8 at th pha croovr frquncy. h quantiti ar uarizd in abl 3.. abl 3. Bod Stability ritria Stability Mtric Exprion Stability ritrion ha Margin M ] [ 8 G f Φ OLF 8 ] [ > > Φ M or f OLF Gain Margin GM f OLG > < GM or f OLG h gain argin and pha argin ar norally pcifid in th prforanc rquirnt for a control loop. h gain argin i oftn xprd in dcibl, or:

8 roportional, ntgral, and rivativ ontrollr ign art A Suna onlin continuing ducation cour GM db Log GM Log OLG f t can b notd that, givn th OLF atifi th Bod critrion, th loop i tabl if fg < f. A th Bod plot i ud oftn in th cour, an xapl i hown in Figur 4. for an OLF givn by:.6 z OLF p ξ r r whr : z π f z π.59 p π f p π.59 π f π.6366 ; ξ.5 r r h olid lin ar for th opn loop gain, which i cald in unit of db on th right-hand id of th plot; and th dottd lin ar for th opn loop pha cald in dgr on th lft-hand id of th plot. h plot of th OLG i in olid black with a lin at db in olid light blu. Each pol contribut - db pr dcad of lop to th agnitud plot and ach zro db pr dcad of lop. h quadratic tr contribut -4 db pr dcad of lop. h OLF pha i th dottd rd lin and a lin at th -8 rfrnc in dottd dark blu. Each pol contribut -9 of pha lag to th pha plot and ach zro 9 of pha lad. h quadratic tr contribut -8 of pha lag. A hown in th figur, th yt ha a gain croovr frquncy at.5 z, a M~3 and a GM~-3 db and i tabl bad upon th Bod tability critrion. hi i a low bandwidth yt and th rlativly low M indicating o ocillatory rpon can b xpctd. opyright 7 tr J. nndy ag 8 of 4

9 roportional, ntgral, and rivativ ontrollr ign art A Suna onlin continuing ducation cour Figur 4. Exapl Bod lot for Exapl 3.3 i to Frquncy oain onvrion: Uing th Laplac tranfor, th rlationhip btwn th ti and frquncy doain for a firt and cond ordr plant can b valuatd a hown in abl 4.. For both plant, to hortn notation xt qual th plant input and yt th output. h tabl dcrib th diffrntial quation, it frquncy doain convrion and th plant and output rpon. For th t ordr plant th tr i th plant ti contant whil S i th caling factor. abl 4. t and nd Ordr i and Frquncy oain Rprntation t Ordr lant nd Ordr lant ti y t y t x t y t ξ y t y t x frquncy Y X plant output Y S S S t ξ Y X S ξ X X S S X X ξ tranfor L y t Y ; L y t Y ; L x t X ; L y t Y For th nd ordr yt, ζ i th daping contant and th natural frquncy of th plant. hi gnral procdur can b applid to a diffrntial quation of any ordr. 4. y Fdback Loop Rlationhip: Each block in th fdback control loop block diagra, Figur 3., can b xprd in th frquncy doain uing th Laplac tranfor dcribd in th lat ction, auing all contraint ar t. h rlationhip ar coon to all linar control loop including on uing a controllr, dicud hrin. n th frquncy doain, block can b tratd algbraically with ach lnt of th loop in th block diagra tratd a a frquncy dpndnt function. Rfrring to th block diagra of Figur 3., an xprion for th control ignal, dnotd a u and th output can b obtaind algbraically a: ontrol:u cd in n output dfining rvo rror a: cd in output thn: u n ; h output i thn givn a: output [ d u ] d n or, output d cd in output n opyright 7 tr J. nndy ag 9 of 4

10 roportional, ntgral, and rivativ ontrollr ign art A Suna onlin continuing ducation cour opyright 7 tr J. nndy ag of 4 Solving for th output: n d in cd output h firt tr in brackt dcrib th rpon fro th coand input to th output and i trd th clod loop tranfr function LF, with a agnitud calld th clod loop gain LG or: ; LG LF h control copnator gain,, i norally vry high and th priary gain tr uch that agnitud of th product of th thr tr ** >> ovr th ffctiv oprating bandwidth of th loop. hi product i th opn loop tranfr function OLF and it agnitud calld th opn loop gain OLG: ; OLG OLF h fdback tr i oftn a unity gain or cald to provid unity gain in th fdback path a it i iply ning th plant output. hi bing th ca, th clod loop gain, LG i unity ovr th oprating bandwidth of th loop and roll off toward zro at high frquncy. f th fdback gain i not unity, it will chang th clod loop gain fro on to th invr of th fdback gain agnitud i.. /. h cond xprion account for th ipact of diturbanc on th rpon. h tranfr function in brackt ultiplying th diturbanc i oftn trd th coplianc a in how copliant th loop i to a diturbanc. oplianc hi tr attnuat th diturbanc, priarily du to th high gain control tr in th opn loop gain. hrfor, th highr th controllr gain, th or diturbanc rjction and th l ipact th diturbanc will hav on th output. owvr, thi gain will b liitd by th tability of th loop. Finally, a odld, th lat tr i nor noi which, a hown ffctivly, add dirctly to th output inc it i ultiplid by th clod loop gain. A it i baically input to th a uing junction a th coand input, thi would b xpctd. h rvo rror btwn th coand input and th output i givn by: n d in cd output in cd h rror i ffctivly th accuracy with which th output follow th coand input. h firt tr in brackt that i ultiplying th coand input i oftn calld th loop nitivity function.

11 roportional, ntgral, and rivativ ontrollr ign art A Suna onlin continuing ducation cour For unity fdback ~ thi tr i ffctivly on ovr th opn loop gain iilar to th coplianc function dicud prviouly, or: Snitivit y for h tranfr function for th plant can vary ignificantly bad upon th application. A will b dicud, th work bt with iplr firt and cond ordr plant. For proc control application, dlay or dad ti ut oftn b accountd for and a ipl thr paratr plant odl can oti b ud to dcrib it rpon a: L proc caling ; rpon ti con tant ; L dad or dlay ti opnation for th dlay i not alway poibl with a ipl controllr. f th dlay i all rlativ to th apling priod, a ipl firt ordr aylor Sri -L [-L] or ad approxiation -L [-L]/[L] ay b ufficint. owvr, if th dlay i ignificant rlativ to th apling priod, th would nd to b cobind with a or advancd control algorith uch a th Sith rdictor [, ] dicud in part of thi cour. h rror will b a function of th attnuation providd by thi tr a wll a th agnitud of th coand, diturbanc, and noi. On final rror rlationhip of iportanc i th tady tat rror which i dfind a: SS li [ ] li cd in t i a aur of th rror rpon or ffctivly th accuracy of a tabl yt a ti go to infinity. t i norally valuatd againt thr tandard input ignal: a tp, rap or rat, and parabolic input or acclration. h ignal hav th Laplac tranfor: tp rap parobolic tp ; rap ; parobolic 3 h tady tat rror i oftn ud a part of th control loop prforanc pcification. h rquird rpon to th input ignal will dtrin what i trd th loop yp ; which rfr to th opn loop pol at or th nubr of intgrator i.. / tr in th OLF. f w dcrib a gnral input to th yt a x/ n which can b any of th abov input dpnding on th valu of n thn th tady tat rror can b writtn a: x x ESS li [ Error ] li li n n n Fro thi xprion, abl 5. i gnratd dcribing th tady tat rror a a function of loop typ and input. abl 5. Stady Stat Error vru Loop yp and nput Stady Stat Error opyright 7 tr J. nndy ag of 4

12 nput Stp n Rap n roportional, ntgral, and rivativ ontrollr ign art A Suna onlin continuing ducation cour yp Syt no opn loop pol at tp yp Syt on opn loop pol at yp Syt two opn loop pol at Mor than two opn loop pol at rap arabolic n3 li [ ] V V parobolic li [ ] A Fro th tabl, it i obrvd that inc th or controllr includ an intgrator or pol at, uing it will rult in at lat a yp yt. hi i oftn dirabl inc th rror du to a tp input to a yp loop will nulld whra with yp loop rult in an rror offt. h offt i th diffrnc btwn th dird input and actual output which i a function of th loop gain. h yp loop cannot track a rap or parabolic input a th rror will continu to grow infinitly with ti for both input typ. A yp loop will null th rror to a tp input at a pd bad on th loop bandwidth. h yp loop will track a rap with an offt btwn th input and output which i a function of th loop gain. t cannot track a parabolic input a th rror will continu to grow with ti. h yp loop track both a tp and rap with zro rror and a parabolic input with an offt btwn th input and output which i a function of th loop gain. Loop yp gratr than two can track all thr input with zro rror, howvr a loop with thi 5. h ontrol Algorith: aving dcribd th gnral fdback control yt architctur and rlationhip that govrn th output rpon, th gnral thory and dign of th controllr which rlat to th gnral architctur a and hown in Figur 5. i dvlopd. A opyright 7 tr J. nndy ag of 4

13 roportional, ntgral, and rivativ ontrollr ign art A Suna onlin continuing ducation cour Figur 5. Fdback ontrol Loop with ontrollr h controllr tructur, paratrizd in tr of gain and rfrncd frquntly in th litratur, i: u t t d t t h control ignal variabl ut wa dfind ction 4. with th controllr bing th u of thr tr: A tr proportional to th rror with gain A tr proportional to th intgral of th rror with gain A tr proportional to th drivativ of th rror with gain h ti doain rprntation, paratrizd in tr of gain, can b convrtd to th frquncy doain uing th Laplac ranfor rlationhip dfind in ction 3, a: u hr ar vral vrion of th controllr that do not includ all thr tr; proportional plu intgral, proportional plu drivativ or hav a lightly diffrnt tructur uch a proportional plu intgral and proportional plu drivativ. h controllr vratility can b obrvd fro th ipact it ha on ky clod loop yt ti rpon charactritic that includ: Ri i: ti rquird for th plant output to rach 9% of th dird lvl for th firt ti Ovrhoot: aount th pak lvl xcd th tady tat valu; noralizd to thi valu Sttling i: ti rquird for yt to rach th tady tat opyright 7 tr J. nndy ag 3 of 4

14 roportional, ntgral, and rivativ ontrollr ign art A Suna onlin continuing ducation cour Stady-tat Error: diffrnc btwn th tady tat output and th dird output h gnral ffct that incraing ach controllr paratr ha on th rpon i uarizd in abl 6.. abl 6. Effct of ncraing ontrollr aratr on Syt Rpon aratr Ri i Ovrhoot Sttling i Stady Stat Error dcra incra indtrinat dcra dcra incra incra liinat for tp indtrinat dcra dcra indtrinat hrfor, an incra in will dcra ri ti and tady tat rror but incra ovrhoot and tndncy toward ocillation. h intgral tr will ak th yt at lat a yp loop with th tady tat rror a dcribd in abl 5.. n gnral, incraing rduc tability argin, incraing ovrhoot and ttling ti, but dcra ri ti. Finally, an incra in dcra both ovrhoot and ttling ti. owvr, it hould alo b notd that incraing th tr too uch can lad to intability a wll a aplification of noi. hi lat charactritic of th drivativ tr oftn doinat it prforanc o that in any application it i not ud. On final not rgarding th u of th : givn it only ha thr control paratr, it do hav liitation. t can b highly ffctiv controlling ipl tabl plant but, gnrally, not tho that hav highr ordr dynaic, high frquncy od ubjct to ocillation, or rquirnt with high prciion. h can uually b ud if th plant rpon to a tp input i iilar to that of a firt ordr yt. Anothr tandard tructur, which i actually th ntrnational Socity of Autoation SA tandard for, i paratrizd by ti aociatd with th intgral and drivativ. hi for i oftn ud in proc control application. h gain ar paratrizd rlativ to th ti priod aociatd with intgration and diffrntiation a: / whr i th intgral ti, and * whr i th drivativ ti. With thi paratrization, th prcding quation can b xprd a: t u t t d t h frquncy doain quivalnt i thn givn by: u h proportional, intgral, and drivativ control tr copnat for th pat, prnt and futur rror rpon. h intgral tr will alway try to null th tady tat rror and bco th noinal control tr a th rror convrg. h intgral ti can b conidrd a a filtr ti contant in a poitiv fdback loop a hown for a controllr in Figur 6. []. opyright 7 tr J. nndy ag 4 of 4

15 roportional, ntgral, and rivativ ontrollr ign art A Suna onlin continuing ducation cour opyright 7 tr J. nndy ag 5 of 4 Figur 6. ontrollr onfigurd a oitiv Fdback through a Filtr h control ignal i givn by: u u with Solving th quation for rult in th control ignal:. u h drivativ tr can provid a pha lad of up to ~5 and b conidrd a a prdictiv lnt can b n fro it dfinition: li t t t t t t u with thi tr ar that it aplifi noi, pcially at high frquncy and if it i too larg which can lad to ocillation and intability in a controllr. Noi can b attnuatd by adding a pol to roll-off th frquncy rpon or ffctivly filtr th ignal at high frquncy. Effctivly thi i th iplntation of th drivativ tr a a high pa filtr. h tructur with th roll off frquncy i: rolloff f a whr a a u π 5. i vru Frquncy oain Rprntation: h two rprntation provid diffrnt prpctiv of th controllr. n th ti doain, th controllr conit of intgral, proportional, and drivativ part corrponding to control tr that copnat for th pat, prnt and futur rror rpon; ach wightd by thir rpctiv gain. ntuitivly thi a raonabl approach, providing thr tat of th rror dynaic to u for loop rpon copnation. n th frquncy doain, tr cannot b tratd paratly and th bnfit ar not quit a obviou a in ti doain. Without th drivativ tr, th controllr frquncy rpon can aily b intrprtd with th rprntation: n th frquncy doain, thi controllr conit of a pol at and a zro at /. n th ti doain, thi zro will norally cau an ovrhoot in th yt tp rpon dpndnt upon

16 roportional, ntgral, and rivativ ontrollr ign art A Suna onlin continuing ducation cour th agnitud of th zro and othr lnt of th control loop. h frquncy rpon i hown for 6.8, and 59. hi corrpond to.6 and hnc ha a zro at / 9.4. h frquncy rpon i hown in Figur 7.. Figur 7. Frquncy Rpon h ffct of th intgrator pol at i to provid a high gain with a - db pr dcad lop to th agnitud of th controllr at low frquncy ayptotically convrging to th valu of th proportional gain. h inflction point i at th zro frquncy or 9.4/*π ~.5 z. h zro tr provid db pr dcad of gain cauing th agnitud to lvl out a n in th rpon plot. h on iu with th controllr i it provid no pha lad, with a 9 pha lag du to th intgrator at low frquncy; ayptotically approaching zro pha at highr frquncy. t will not work with yt which hav a pha lag of 8 or or and in gnral, it will hav th ffct of rducing th loop pha argin. h controllr frquncy rpon i lightly or difficult to intrprt. h controllr rprnt an ipropr tranfr function, du to th drivativ tr, with two zro in th nurator and on pol at zro in th dnoinator. h pol at zro will provid a - db pr dcad lop to th agnitud of th controllr at low frquncy and ach zro a db pr dcad lop at highr frqunci dpnding on th valu of ach zro. hi can b n in ithr tandard for of th tranfr function: opyright 7 tr J. nndy ag 6 of 4

17 roportional, ntgral, and rivativ ontrollr ign art A Suna onlin continuing ducation cour opyright 7 tr J. nndy ag 7 of 4 h agnitud *, pha Φ, and zro of both for of th tranfr function ar givn in abl 7.. abl 7. Standard For Magnitud, ha, Zro Standard Gain Standard i j j j j j Φ a tan a tan Zro, Z Z 4 4 t can b obrvd that if / > ¼ or * / > /4 th zro ar coplx root. h zro of th plant and copnator will alo b th zro of th clod loop yt. For a pair of zro, th frquncy rpon agnitud will dip or notch at ~in / /*. A an xapl, th frquncy rpon i hown for 6.8, 59, and. hi corrpond to.6 and.6 o /.5 and th zro ar coplx -3.±7j. h controllr frquncy rpon i hown in Figur 8. with a notch at ~. z. Figur 8. Frquncy Rpon

18 roportional, ntgral, and rivativ ontrollr ign art A Suna onlin continuing ducation cour h ovrall rpon with th drivativ i doinatd by thi tr at high frqunci, th gain incraing at db pr dcad. hi charactritic i on iu with a ipl iplntation of th controllr and th raon it i norally iplntd a a high pa filtr with a pol to roll off th rpon at high frquncy a oppod to th db pr dcad lop. Of cour, th ovrall bnfit of th will dpnd on th charactritic of th proc itlf. f it i alrady low frquncy or havily filtrd, th roll-off ay not b rquird. n gnral, howvr, it i good practic inc th drivativ will alway diffrntiat noi aplifying it pcially at high frquncy. 5. lod Loop Syt Siulation with a ontrollr: o provid a viualization of th ffct ach tr ha on th clod loop rpon, an xapl bad upon th block diagra in Figur. i providd with th tr dfind a: π f ; f z π f a a F noi and diturbanc a π f ; f 5 z rolloff h plant ha a pol at zro and z. h dird clod loop bandwidth i ~ z o xpctation would b th drivativ tr would b ncary. owvr, to valuat th contribution of ach tr, firt proportional only i ud with *π* 6.8. h output and rror rpon of th clod loop to a tp input i hown in Figur 9.. rolloff Figur 9. Output and Error i Rpon to a Stp for roportional Only ontrol opyright 7 tr J. nndy ag 8 of 4

19 roportional, ntgral, and rivativ ontrollr ign art A Suna onlin continuing ducation cour t i n that th rpon i ocillatory with an ffctiv bandwidth tiatd fro th priod of ~.6 z. h rror to th tp do go to zro, but i du to th plant having a pol at zro which ak it a yp loop abl 4. not th controllr. f th plant did not hav th pol at zro, thr would b a tady tat rror invrly proportional to th gain of th controllr. h frquncy rpon plot of th clod and opn loop gain ction 4., clod loop gain LG and opn loop gain OLG and opn loop pha ction 3. i hown in Figur.. h opn loop gain croovr frquncy i 3 z whn opn loop gain, clo to that obrvd but th pha angl i ~-6. hi tranlat to a low pha argin M of 8 M 8-6 which rult in th ocillation obrvd. Figur. Frquncy Rpon for roportional only With an intgral tr i59 addd to th controllr, th pha argin will b rducd furthr inc th intgrator induc a 9 pha lag. h rpon i hown in Figur. with th a and i untabl. h frquncy rpon of th controllr can b n in Figur.a. h intgral tr doinat with high gain at low frquncy rolling off at db pr dcad until it lvl nar th zro frquncy and i doinatd by th proportional gain. h pha rpon i opyright 7 tr J. nndy ag 9 of 4

20 roportional, ntgral, and rivativ ontrollr ign art A Suna onlin continuing ducation cour alo hown, adding 9 of lag at low frquncy that cau th intability. h yt frquncy rpon i hown in Figur.b. h opn loop gain croovr frquncy i till at 3 z but th pha i nvr gratr than -8 rulting in yt intability Figur. Output and Error i Rpon to a Stp for ontrol Figur.a ontrollr Frquncy Rpon Figur.b Syt Frquncy Rpon h contribution of th intgral tr incra with dcraing intgral ti, howvr, thi alo rduc tability argin. craing i i ffctivly an incra in intgral ti. Rducing both control gain, p and i, will tabiliz th loop, but thi alo rult in a lowr bandwidth or lowr rpon ti. With th p and i gain rducd, th rulting ti rpon hown in Figur 3. and th yt frquncy rpon in Figur 4.b. h gain croovr frquncy i opyright 7 tr J. nndy ag of 4

21 roportional, ntgral, and rivativ ontrollr ign art A Suna onlin continuing ducation cour l than z but th opn loop pha i ~ -6 o thr i a poitiv M indicating potntial tability, howvr, th rpon i till too low. Figur 3. Output and Error i Rpon to a Stp for ontrol Rducd Gain Figur 4.a Frquncy Rpon Rducd Gain Figur 4.b Syt Frquncy Rpon Finally, th drivativ tr i addd to th controllr, foring a tructur. h drivativ tr i iplntd a a high pa filtr with th roll-off frquncy at 5 z. For thi ca, th pha lad of th drivativ tr hlp conidrably by daping th ocillation. h ti rpon i hown in Figur 5.. Ocillation ha bn rducd ignificantly with only ~5% axiu ovrhoot. opyright 7 tr J. nndy ag of 4

22 roportional, ntgral, and rivativ ontrollr ign art A Suna onlin continuing ducation cour Figur 5. Output and Error i Rpon to a Stp for ontrol h frquncy rpon of th i hown in Figur 6.a illutrating th ffct of th intgral gain at low frquncy, th drivativ tr crating th notch in th id-frquncy rang z and th roll off at high frquncy. t i alo obrvd that th pha angl will add a poitiv pha to th loop nhancing tability not pha fold-ovr i du to plot rang. h yt frquncy rpon plot i hown in Figur 6.b. h gain croovr frquncy i now at z, a dird, with a pha argin > 8 --which i highr than ndd, but illutrat th poitiv ipact of th drivativ tr. Figur 6.a Frquncy Rpon Figur 6.b Syt Frquncy Rpon 5.3 iffrnt For of th ontrol Algorith: hr ar vral aniftation of th, and diffrnt for of th ay apply bttr to diffrnc application. For xapl, for proc control application tnd to b lowr and tporal rprntation ar or iportant for opyright 7 tr J. nndy ag of 4

23 roportional, ntgral, and rivativ ontrollr ign art A Suna onlin continuing ducation cour opyright 7 tr J. nndy ag 3 of 4 tuning prforanc. A highr bandwidth targt tracking application dign ay rly or on placnt and tuning for a dird frquncy rpon o that frquncy paratr ar or rlvant. hr ar two tandard algorith rprntation a dicud prviouly. On i paratrizd in tr of a gain, intgral ti, and drivativ ti a: u or rplacing th drivativ with a high pa filtr: a a u h othr tandard for i paratrizd in tr of abolut gain a: u Again, th drivativ can b rplacd with a high pa filtr a: a a u With th two tandard-for, th proportional gain ar quivalnt and th othr gain rlatd a: / whr i th intgral ti, and * whr i th drivativ ti. n th lattr rprntation, th paratr valu ar rlatd to abolut gain rathr than ti aociatd with intgration and th drivativ. A th paratr linarly wight th oprator of ach tr, it i uful in analytical calculation; it alo ha th advantag that it i poibl to obtain pur proportional, intgral, or drivativ action by finit valu of th paratr. A lightly diffrnt vrion of th controllr i th forward path product of a proportional intgral cacadd with a proportional drivativ controllr * a givn by: u hi vrion i uch air to work with in th frquncy doain a tr ar factord o that both zro, th pol at zro, and gain ar aily dlinatd. On intrprtation of thi for i a controllr oprating on th prdictd valu th rror ignal. h paratrization of thi tructur and th tandard tructur ar not quivalnt. h factord for can b quatd to th tandard for by ultiplying out th factor and quating lik cofficint: ; ; Equating th tandard for to a factord for i poibl only if th zro of th tandard for ar ral which rquir >4*. h factord for can alo u th high pa filtr for noi attnuation a prviouly dicud, or: a a u n thi aniftation, th tr can alo b intrprtd a a lad copnator. Expanding thi tr rult in:

24 roportional, ntgral, and rivativ ontrollr ign art A Suna onlin continuing ducation cour opyright 7 tr J. nndy ag 4 of 4 a a a a u h lad pol i at a and th zro at a/a. A final iplntation dicud u a proportional plu intgral in th forward path oprating on th rror with a proportional plu drivativ controllr in th output fdback path a illutratd in th block diagra hown in Figur 7.. Figur 7. ontrollr Architctur h ontrollr tranfr function ar: ; ' h controllr i thn givn by: output in cd u Subtituting th tranfr function quation rult in: ' output in cd output in cd u h paratr of thi controllr ar rlatd to th tandard for with abolut gain a: h controllr proportional gain ut b on for a zro tady tat rror. hi for ha advantag that th drivativ oprat on th output rathr than th rror which i oftn a oothr function o gnrat a l noi in rpon. 6. ontrol Loop rforanc Spcification: h pcification dfin th dird fdback loop prforanc. Fro a practical prpctiv, th control loop dignr ut undrtand th ur

25 roportional, ntgral, and rivativ ontrollr ign art A Suna onlin continuing ducation cour rquirnt to dvlop a atifactory pcification. y charactritic oftn pcifid by a ur ar accuracy and rpon ti for a dfind coand ignal or rang of ignal and th diturbanc and noi nvironnt. Fro thi inforation, th dignr ut dvlop th control rquirnt and pcification. A dicud prviouly, accuracy and diturbanc rjction ar iprovd by a high gain controllr, a i rpon ti. A gain i ffctivly proportional to bandwidth, thi ipli a highr bandwidth iprov prforanc. owvr, a th bandwidth incra, th loop tability argin bgin to dcra, or ovrhoot rult, control lnt can aturat, and th noi btwn loop lnt can b aplifid. So, th dign i a tradoff btwn th ur rquirnt and control loop pcification that t th rquirnt with ufficint tability argin. ontrol pcific pcification can b dfind in ithr th ti or frquncy doain and nd to b drivd fro yt prforanc pcification. n th ti doain thy ay rlat dirctly, a dicud in ction 5., with rpon pcifid in tr of: ri ti, ovrhoot, ttling ti, and tady tat rror all of which ar a function of bandwidth and th control copnator tructur. n th frquncy doain, thy ar oftn drivd fro diturbanc rjction and axiu allowabl rvo rror rquirnt; a thy rlat to yt prforanc. o achiv thi prforanc th dignr can pcify th ronant pak, pak frquncy, gain croovr frquncy, bandwidth which in o ca ar dirctly rlatd, and th iniu allowabl pha and gain argin. h ronant pak i a axiu of th gain at th pak frquncy. Anothr iportant pcification i th control loop typ which a dicud prviouly rfr to th nubr of intgrator within th loop; ithr du to th controllr or th plant dynaic. h loop typ or nubr of intgrator will dtrin th accuracy axiu rror with which th output follow an input coand or rjct a diturbanc in th long tr. h intgrator alo provid vry high gain at low frquncy which i good for diturbanc rjction and iprovd accuracy; howvr, ach pol at alo provid a -9 pha hift which can ipact tability. h robutn of th controllr dign ut alo b conidrd. f th plant paratr chang for whatvr raon, what will happn to th loop tability argin? h controllr dign ut account for th variation. hi can b accoplihd by digning for th wort-ca plant variation and nuring th yt t th tability argin ovr th full rang of plant variation; auring th plant rpon i.. plant idntification and changing control paratr to account for th plant chang ithr anually or on a chduld bai, or a uing an adaptiv controllr that autoatically adjut th control paratr a a function of th plant variation. 7. ontrollr ign: Mot gnral thod for control yt dign can b applid to control; within th contraint of th controllr tructur. controllr dign can b catgorizd a drivd fro anual tuning and aurnt proc and/or tho or analytically dtrind. A nubr of pcial thod ad pcifically for control hav alo bn dvlopd; oftn calld tuning thod. rrpctiv of th thod, howvr, th dign ut opyright 7 tr J. nndy ag 5 of 4

26 roportional, ntgral, and rivativ ontrollr ign art A Suna onlin continuing ducation cour alway account for charactritic of th load diturbanc, nor noi, proc uncrtainty rfrnc ignal o that a robut dign i achivd. 7. Manual uning Bad on roc Maurnt [, ]: h tuning algorith dvlopd by Ziglr and Nichol ar aong th ot wll-known thod rfrncd in litratur. hy ar bad on charactrization of proc dynaic by a fw controllr paratr and ipl quation rlating th to paratr. hr ar two balin thod trd th tp rpon and frquncy rpon thod along with any ubqunt odification drivd fro th thod to iprov prforanc 7.. h Ziglr and Nichol Stp Rpon Mthod [, ]: h tp rpon thod charactriz th aurd opn loop plant tp rpon with two paratr, a and L. hi thod i applicabl to plant with a boundd tp rpon; rult bing bt whn th rpon i clo to bing firt ordr. h dfinition of th paratr ar hown in Figur 8 with th full tp rpon to th lft and th xpandd rgion rlvant to th dign paratr a and L to th right. Figur 8 haractrization of a Stp Rpon in th Ziglr Nichol Stp Rpon Mthod. h tangnt to th tp rpon lop at it axiu valu i dtrind and a lin i drawn intrcting th ti and tat rpon ax. h intrction with th coordinat ax provid th two paratr a hown in th figur. h ti intrval intrctd on th ti axi i trd L and th tat dlta intrctd on th tat rpon axi a. h controllr paratr ar a function of th paratr a hown in abl 8. with p bing an tiat of th clod loop priod or rpon ti. abl 8. ontrollr aratr for Ziglr Nichol Stp Rpon Mthod ontrollr p /a 4L.9/a 3L 5.7L./a L L/ 3.4L opyright 7 tr J. nndy ag 6 of 4

27 roportional, ntgral, and rivativ ontrollr ign art A Suna onlin continuing ducation cour An iprovd tp rpon thod can b ud for tabl proc with dlay a wll a intgrating proc. hi thod charactriz th unit tp rpon by thr paratr, L and. h dfinition of th paratr ar iilar to tho for th balin tp rpon thod and ar hown in Figur 9.. hi paratrization atch plant odl with th tranfr function tiatd a: L L Figur 9. Stp Rpon haractrization 3 aratr Modl Again, th paratr and ar obtaind fro aurd tp rpon data. h tranfr function i a firt ordr yt with ti dlay. aratr L i dtrind fro th intrcpt of th tangnt with largt lop with th ti axi a wa dcribd for th balin tp rpon thod. aratr L i alo dtrind a th diffrnc btwn th dlay ti and th ti th tp rpon rach 63% of it tady tat valu. aratr i th tatic gain of th yt. aratr L i th tiatd ti dlay and paratr th tiatd ti contant. h ignificanc of th dlay can b valuatd by th rlativ dlay ratio: L η L t provid an tiat of th tporal rpon drivr; th rpon ti lag,, or th procing dlay, L. control oftn uffic for proc that ar dlay doinatd, i.. whn η i clo to on whil drivativ action i typically bnficial for proc with all rlativ dlay η. o obtain iprovd tuning rul, dcribd in [4], th author ud a dign thod to axiiz th intgral gain contraind for robutn o that th axiu nitivity i <.4. Uing thi procdur, on tuning rul for an iprovd controllr wa: opyright 7 tr J. nndy ag 7 of 4

28 roportional, ntgral, and rivativ ontrollr ign art A Suna onlin continuing ducation cour.3 for L < L.5 for < L 8L for L <..8 for. < L <.4L for < L 7.. h Ziglr Nichol Frquncy Rpon Mthod [, ]: A cond thod dvlopd by Ziglr and Nichol i a ipl charactrization of th frquncy rpon of th plant dynaic. h dign i bad on knowldg of th control loop frquncy whn plant pha i -8. On th Nyquit plot of th proc tranfr function ; thi i th point whr th Nyquit curv intrct th ngativ ral axi. On a Bod lot, it i quivalnt to a pha croovr frquncy for th opn loop plant rpon; dfining a boundary btwn a tabl and untabl rpon. h Ziglr Nichol Frquncy Rpon Mthod i charactrizd by two paratr, th frquncy 8 and th gain at that frquncy 8 i8. h paratr u /8 and u π / 8, trd th ultiat gain and th ultiat priod ar ud to driv th controllr gain. h paratr ar dtrind by connct a controllr to th proc; tting th paratr o that control action i proportional, i.., and and thn incraing th gain lowly until th proc ocillat. h gain at which thi occur i u and th ocillation priod i u. h paratr of th controllr ar thn givn by abl 9.. h frquncy rpon thod i an pirical tuning procdur whr th controllr paratr ar obtaind dirctly fro proc aurnt cobind with o ipl rul. For a proportional controllr, th rul i to iply incra th gain until th proc ocillat and thn rduc it by 5%. abl 9. ontrollr aratr for Ziglr Nichol frquncy rpon thod ontrollr p.5u u.4u.8u.4u.6u.5u.5u.85u An ant [] of th Ziglr Nichol Mthod i that th tuning rul wr dvlopd to provid clod loop yt with good attnuation of load diturbanc. h thod wr bad on xtniv iulation. h dign critrion wa quartr aplitud dcay ratio, which an that th aplitud of an ocillation hould b rducd by a factor of four ovr a whol priod. hi corrpond to clod loop pol with a rlativ daping of about ξ., which i too all. ontrollr dignd by th Ziglr Nichol rul inhrntly giv clod loop yt with poor robutn. owvr, thy ar ipl to u and do provid ball park tiat for th controllr paratr with th final tuning don by trial and rror. 7. Analytical ign Approach: Analytical approach odl atching critrion, again dicud in part, gnrally rfr to i trying to atch th yt rpon to that of a dird opyright 7 tr J. nndy ag 8 of 4

29 roportional, ntgral, and rivativ ontrollr ign art A Suna onlin continuing ducation cour rfrnc odl or ii ignal or trying to plac pol of th yt rpon at tho dird for a pcifid rpon. hr ar any control loop dign tchniqu; ot involv o for of loop haping and/or algbraic pol-zro placnt and cancllation. Loop haping can b accoplihd uing Bod or Nyquit plot analyi. Viually, th Bod plot i ay to intrprt and adjut th rpon pcially if iplntd in a athatical coputr aidd dign progra i.. MatLab or Mathad. h Bod plot can b ud with aurd plant gain and pha rpon data, th plant-pol zro tructur do not hav to b known xactly although th Bod dign till ubjct iniu pha critrion. Algbraic thod appar traightforward but rquir knowldg of th plant pol-zro tructur and car ut b takn to inur tabl dign. For xapl, pol and zro cancllation ut b don with o nitivity analyi bcau thy ar nvr known xactly and in o ca, can chang or drift. h i pcially tru if th pol and zro ar not tabl; dirct cancllation can rult in a pol/zro pair that ar untabl both root in right half -plan a oppod cancllation of th untabl root. h loop yp can b adjutd uing th or controllr. Whn uing any control tr with a pur intgrator, th output hould b liitd or othr aur takn to inur windup of th intgrator do not occur. A coupl of ipl dign xapl ar providd. 7.. Frquncy oain ign: Au for a unity fdback loop th dird gain croovr frquncy fg 4 z and th pha argin > 55. hi provid two critria to atify with th copnator: OLG fg M 8 Φ OLF fg 55 or Φ OLF fg 5 Fro th dfinition of th opn loop unity gain tranfr function in ction 3., thi can b xprd a: N jg N jg ; G π fg jg jg Φ jg jg 5 Φ N jg Φ N jg Φ jg Φ jg h plant i givn by: L h plant paratr L i th caling factor. For a doubl intgrator, th pha lag i -8 o it can b aud to obtain th dird rpon a proportional plu drivativ controllr iplntd with a roll-off filtr or baically a lad copnator of th for: opyright 7 tr J. nndy ag 9 of 4

30 roportional, ntgral, and rivativ ontrollr ign art A Suna onlin continuing ducation cour π 6 α π 6 π 6 π 6 π 6 π 6 whr α ; ; α h pol frquncy at 6 z i chon a th roll-off frquncy for th loop, ffctivly liit frquncy rpon and rduc highr frquncy noi. Rprnting th pol and zro in tr of a frquncy provid for bttr viualization of th dign proc. t i alo uful to cancl plant paratr with a caling factor o ffctivly th loop rpon i hapd only by th noralizd plant rpon and th copnator. h copnator i thn xprd a: α L π 6 h OLF i thn: OLF α π 6 hi will b a yp control loop. h valu for th copnator gain and zro ar obtaind by firt obtaining th valu of th zro fro th pha argin condition and olving for th gain fro th OLG condition. Fro thi xprion, th pha coponnt ar givn by: G G Φ N jg a tan ; Φ jg a tan α π 6 Φ j ; Φ j 8 N G Subtituting into th xprion for pha argin and olving for α: G G 5 a tan a tan 8 ; G π 4 α π 6 π 4 or 55 a tan 4 α π 4 97 olving for α : α 97 π f Z f Z 5.4z tan69 π h pha argin critrion i atifid with th copnator zro at 5.4 z. h agnitud critrion i now ud to dtrin th valu of that atifi th opn loop gain condition at th gain croovr frquncy. G opyright 7 tr J. nndy ag 3 of 4

31 roportional, ntgral, and rivativ ontrollr ign art A Suna onlin continuing ducation cour opyright 7 tr J. nndy ag 3 of π π π π π π π π π π π π π π π or j j j or f j j j G G G G G h copnator gain that atifi th opn loop gain croovr frquncy condition i thn: π π α 7.. opnator rivd fro a Modl LF: Anothr thod for digning th controllr i to pcify a clod loop yt odl bad upon th dird prforanc. For xapl, rpon ti or bandwidth, ovrhoot, tability argin could b pcifid. With th clod loop tranfr function odl dfind and th plant tranfr function known, th copnator can b dtrind. Auing unity gain fdback and uing th dfinition for th clod loop rpon: odl odl ; N N N or OLF LF N LF M M M OL OL M M Μ Μ Μ Μ Μ Μ Μ Μ h copnator cancl th plant and ubtitut an opn loop tranfr function that giv th dird clod loop tranfr function. hr ar o iportant contraint: f th plant ha a pol in th right half -plan pol with a poitiv ral part thn th odl M ut b chon uch th M-NM ha th a root. f th plant ha a zro in th right half -plan thn th dird M ut hav th a right half -plan zro o b ralizabl, th xc pol of th dird ut b qual or gratr than th xc pol of th plant. f th nurator polynoial of a tranfr function i of ordr nz and th dnoinator polynoial of ordr np thn th xc pol ar np-nz. Sipl xapl ar providd with a dird LF and plant of variou for. Au th dird clod loop rpon i a cond ordr odl with bandwidth ff and daping contant ς givn by:

32 roportional, ntgral, and rivativ ontrollr ign art A Suna onlin continuing ducation cour opyright 7 tr J. nndy ag 3 of 4 loop yp control a attnuation or loop rror for ; ; ζ ζ ζ ζ ζ ζ ζ ζ π ζ Μ << Μ Μ f OL OL abl. provid th controllr that provid th dird clod loop rpon for a givn plant tranfr function. h plant paratr ar dfind a: frquncy natural undapd plant ; daping contant plant ζ ; f π abl. ontrollr for ird lod Loop Rpon and lant lant ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ 7.3 ign u: o obtain good prforanc fro a controllr it i ncary to conidr iu that ipact or liit prforanc. A nubr of practical iu hav bn dicud. A ntiond, ipl controllr lik th and controllr ar not uitabl for all proc. h controllr i uitabl for proc with alot onoton tp rpon, iilar to that for a firt ordr plant, providd that th rquirnt ar not too tringnt. A coupl of iu ntiond prviouly wr rlatd to th intgral and drivativ tr, th and othr ar dicud in or dtail in thi ction. h ain iu, any of which ar coon to ot controllr, ar: rivativ r: hi iu ha bn dcribd and can ot aily b undrtood by ipl diffrntiation of a inuoid which rult in th output bing hiftd by 9 and aplifid by th radian frquncy *π*f; th highr th frquncy th gratr th aplification. hi i alo obviou fro th frquncy rpon of th diffrntiator agnitud which go to infinity a frquncy incra. h iplntation of th drivativ a a high pa filtr with a roll-off frquncy that uppr th high frquncy noi rduc thi probl. Effctiv noi can alo b gnratd within th rror ignal du to diturbanc or iply th functionality of th driv coponnt. n ction 5.3 iffrnt For of th ontrol Algorith, a tructur uing output fdback to iplnt th drivativ tr wa dicud that could itigat any potntial iu a th output i oftn a oothr function than th rror o gnrat a l noi in th rpon.

33 roportional, ntgral, and rivativ ontrollr ign art A Suna onlin continuing ducation cour ntgrator Windup: U of an intgrator in a phyical yt co with th poibility intgrator aturation. hi ffctivly liinat th intgrator a a control tr; convrting a linar yt rpon to a non-linar on. Windup rult whn th intgral action aturat. All actuator hav liit and for a control yt with a wid rang of oprating condition, th controllr can rach th liit. h fdback loop i thn brokn and th yt run opn loop bcau th actuator rain at it liit indpndnt of th proc output. f a controllr with intgrating action i ud, th rror will continu to b intgratd. hi an that th intgral tr ay bco vry larg or, in nc, wind up. t i thn rquird that th rror ha oppoit ign for a long priod bfor thing rturn to noral. h conqunc i that any controllr with intgral action ay giv larg tranint whn th actuator aturat. h iplt thod for addring windup ar liiting th intgral tr or input to th intgral. owvr, thi can alo liit prforanc and gnrally do not rally olv th probl. A bttr thod i to aur th intgrator output and a it approach th actuator aturation lvl and provid fdback that rduc th input; oti trd th tracking and back calculation approach. A block diagra of th thod i hown in Figur. []. Figur. racking and Back alculation Wind-up opnation St oint Wighting []: With th tandard control law, a tp chang in th rfrnc ignal rult in an ipul to th control ignal du to th drivativ tr. hi probl can b avoidd by filtring th rfrnc valu bfor bing input to th controllr or by uing proportional only on part of th rfrnc ignal. hi i calld t point wighting. A controllr givn by: u β cd in output σ cd in output opyright 7 tr J. nndy ag 33 of 4

34 roportional, ntgral, and rivativ ontrollr ign art A Suna onlin continuing ducation cour h intgral tr rain unchangd; bing bad on rror fdback to nur th dird tady tat. h contant β and σ ar additional paratr ud to wight th input coand aociatd with th proportional and drivativ tr. hi controllr can b xprd a: u β σ cd in output Effctivly th for ha two dgr of frdo bcau th ignal path fro output to u i diffrnt fro th cd in to u. h tranfr function fro cd in to u i: u β σ cd in h tranfr function fro output to u i: u output With thi controllr, th yt will rpond to load diturbanc and aurnt noi a with th tandard controllr. h rpon to rfrnc valu, howvr, can b odifid by th paratr β and σ. Ovrhoot for t point chang will b inial for β inc th rfrnc i only part of th intgral tr. h paratr σ i norally zro to avoid larg tranint in th control ignal du to uddn chang in th t point. h controllr with t point wighting i vry iilar in for to th controllr dcribd in ction 5.3. h controllr with β and σ i oftn calld an intgral proportional plu drivativ controllr. For β and σ it i th controllr fro ction 5.3. With thi tructur, tuning i air, inc, and can firt b dtrind to dal with load diturbanc, aurnt noi and proc uncrtainty. h t point rpon can thn b adjutd by chooing th paratr β and σ. 8. o ating Syt: hi xapl i a iplifid vrion of fdback control for hating a hou. h u of a controllr for thi application will b xaind. h phyical configuration of th control loop i illutratd in Figur.. h hou i idalizd a a box filld with air at a unifor tpratur. h wall of th hou ar conidrd a pur ritanc to hat tranfr with no nrgy torag capacity. h ovrall cofficint of hat tranfr i U and th hat tranfr ara i A. h outdoor nvironnt tpratur i, vari with ti thrby acting a a diturbanc to th control yt. h tpratur i aurd by a tpratur nor in a throtat or tpratur controllr ountd inid th hou. h dird tpratur can alo b t by thi dvic, or in today nvironnt ay vn intrfac to a art phon with an A that lt it b t rotly. t i aud that th tpratur i convrtd to a voltag or currnt with a caling contant V with unit Volt/ F and procd by lctronic or or likly in today nvironnt a icro-controllr. opyright 7 tr J. nndy ag 34 of 4

35 roportional, ntgral, and rivativ ontrollr ign art A Suna onlin continuing ducation cour Figur. oncpt Fdback ontrol onfiguration for ating a ou h controllr will tak th diffrnc btwn th dird and actual tpratur and gnrat a control ignal to th furnac. hi ignal control an actuator that incra or dcra th ga flow and ffctivly th aount of hat into th roo. For thi ipl xapl, th actuation and hat gnration ar odld a a linar proc; in rality, it i a coplx proc. n dvloping a odl of th yt, it i aud that initially th hou i in quilibriu with contant valu of and. h furnac will thn b upplying ufficint hat to balanc th hat lo to th nvironnt. Any diturbanc or chang in th dird tpratur will rult in an incra or dcra in hat input fro thi original valu. All variabl hould b conidrd dviation fro thir initial quilibriu condition. h ipl firt ordr thral dynaic odl for th hat balanc in th hou th control loop plant in th ti doain i: thral nrgy tord thral nrgy in thral nrgy laving th hou or M t QM t U A t t BU whr QM thral nrgy in hr M a of air in hou pcific hat of air at contant prur BU BU au U A 5 ; M 8 r F F onvrting to th frquncy doain and rarranging tr: opyright 7 tr J. nndy ag 35 of 4

36 roportional, ntgral, and rivativ ontrollr ign art A Suna onlin continuing ducation cour QM U A M whr.r 43 c U A hi rprnt th odl of th plant; in tr of prviou notation: F U A 5 43 BU / r h tpratur i thn givn by: QM U A A fdback control yt aur th hou tpratur and copar it to a dird tpratur S or th t point. h hat input i aud proportional to thi tpratur diffrnc, a iplification for th xapl. h control loop i hown in Figur.. A th plant rpon i dfind by a firt ordr rpon, a raonabl choic for th controllr,, i a typ iplntation. h xapl will analyz th rpon for a only and control. Figur. Block diagra of ho hating control rvo loop Fro th block diagra, QM i givn by: Q BU M VQ V S / Subtituting into th xprion for tpratur th loop dynaic can b xprd a: R opyright 7 tr J. nndy ag 36 of 4

37 roportional, ntgral, and rivativ ontrollr ign art A Suna onlin continuing ducation cour opyright 7 tr J. nndy ag 37 of 4 F volt volt R BU F R BU au A U V VQ S V VQ / /.67 ] [ Solving for : A U V VQ S V VQ V VQ h rror btwn th t point tpratur and th actual tpratur i givn by: A U V VQ S V VQ S A dicud prviouly, th rror i a function of th agnitud of th outid and t point tpratur and th rror attnuation providd by th controllr gain. h iplt controllr i a proportional gain or with unit of volt /volt. Subtituting into th prviou xprion and uing th plant dfinition: V VQ S V VQ S V VQ S A U A U whr A U A U A U A U λ λ λ λ Stability i not an iu, a th opn loop ha only on ral ngativ pol. h tr λ i baically th attnuation factor who agnitud i bad upon th valu of. h ti contant of th yt will alo b rducd by thi factor. A thi i a yp yt, thr will b a tady tat rror to a tp chang. For a tp chang in and S, th tady tat rror i givn by: S SS S S SS E Li Li E δ δ λ δ λ λ δ λ λ h tp chang in th input will ot likly co fro a nw t point, a th tpratur of th nvironnt would not b xpctd to chang intantanouly o: tp chang E S SS δ λ h gain i chon to provid an attnuation factor λ that i conitnt with th aount of allowabl offt that can b tolratd btwn th nw t point and actual tpratur. For xapl, with th gain wa chon a:

38 roportional, ntgral, and rivativ ontrollr ign art A Suna onlin continuing ducation cour U A VQ h rulting attnuation would b a factor of or -4 db. h achivabl gain will alo dpnd upon th actual driv charactritic of th furnac. h gain croovr can b dtrind by a Bod plot or tiatd dirctly fro th OLF a fg~.4 z. Givn th balin -only rult, a proportional plu intgral controllr dign i xaind nxt. With a proportional plu intgral controllr th yt i yp and th tady tat offt to a tp qual to zro. h copnator ha th for: α ; α 4. U A ; α.45 ;. c VQ h opn loop tranfr function i: V α [ ] VQ V VQ V U A lod loop yt dnoinator i now a quadratic polynoial. Sinc all cofficint ar poitiv, th root will hav ngativ ral part o that th yt will b tabl. h tability argin ar obtaind uing a Bod plot. h opn loop gain and pha ar hown in Figur 3.. V opyright 7 tr J. nndy ag 38 of 4

39 roportional, ntgral, and rivativ ontrollr ign art A Suna onlin continuing ducation cour opyright 7 tr J. nndy ag 39 of 4 pha argin Figur 3. Bod plot for Exapl with controllr h gain croovr frquncy i ~.67 z, th M~67. h gain argin i infinit inc th intgrator lag i offt by th zro o th nt lag i only that of th plant which bing firt ordr i 9. ar ut b takn in an actual dign to nur th highr bandwidth do not put too uch dand on th furnac driv. h xprion for th rror for thi copnator i: [ ] α α α λ λ α α V VQ V VQ V VQ V VQ r S r V VQ S V VQ S A U b A U b A U whr b b A U A U A U A U ; ; For a tp chang, th tady tat rror will b zro inc: b b Li Li E S r S SS δ δ λ For a rap or tadily incraing tpratur chang, th tady tat rror i: