The following Content. Objective. Introduction to Process Control. What have we talked in MM1-MM5? MM5? Software: Control Tutorials for Matlab

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1 Objecive Iroucio o Proce Corol Zheyu Yag Aalborg Uiveriy Ebjerg Semeer 5 of Fall 4, h://caueauck/~yag/cour e/4fall/roce4hml /8/4 Proce Corol To uera a gra he fuameal kowlege abou feeback corol heory: claical corol heory Moer corol heory To be able o aalyze, yheize a imulae corol yem uig iffere meho To be able o uera iurial corol yem /8/4 Proce Corol Wha have we alke i MM-MM5? MM5? Hiory of feeback corol Block iagram for yem moelig Time-omai ecificaio Traie erformace, eay-ae error, yem ye PID coroller Frequecy reoe aalyi a eig Boe lo, gai a hae margi, bawih, Deig of yamic comeaio Lea, lag comeaor BIBO Sabiliy (Rouh crierio) Nyqui abiliy /8/4 Proce Corol 3 /8/4 Proce Corol 4 The followig Coe Par oe: Ehace he Claical corol heory MM6: Moellig Iue a Corol Secificaio MM7: PID Corol a Examle MM8: Frequecy Reoe Meho - I MM9: Frequecy Reoe Meho - II MM:Roo locu meho Par wo: Moer corol heory MM: Iroucio o ae ace meho MM: Corol Deig for Full Sae Feeback MM3: Corol eig uig eimaor MM4: Iroucio of Referece Iu MM5: Iegral Corol a LQR corol /8/4 Proce Corol 5 Sofware: Corol Tuorial for Malab h://wwwegiumicheu/grou/ cm/homeexhml /8/4 Proce Corol 6

2 MM6 Moelig Iue & Corol Secificaio ) Iroucio ) Moelig iue 3) Secificaio 6 Iroucio The fuameal goal of CE i o fi echically, eviromeally, a ecoomically feaible way of acig o yem o corol heir ouu o eire level of erformace i he face of uceraiy of he roce a i he reece of ucorolable exeral iurbace acig o he roce - Graham C Goowi /8/4 Proce Corol 7 /8/4 Proce Corol 8 6 Wa fly-ball goveror Hiorical Perio of Corol Theory The iurial revoluio (86) The Seco Worl War (94-945) The uh io ace (6, 7) Ecoomic globalizaio (8) Shareholer-value hikig /8/4 Proce Corol 9 /8/4 Proce Corol 6 CE Alicaio Domai /8/4 Proce Corol 63 Baic Corol Syem Referece - iu Feeforwar comeaor Feeback comeaor Alicaio: Regulaor yem Servo or oiio yem Trackig yem acuaor Meaure oie eor iurbace Pla Targe: Cloe-loo abiliy Diurbace aeuaio Goo comma reoe Robue /8/4 Proce Corol

3 MM6 Moelig Iue & Corol Secificaio ) Iroucio ) Moelig iue LTI moelmalab Block iagrammalab 3 Digial corol yem 3) Secificaio /8/4 Proce Corol 3 6 Moelig Iue Moel Aribue mahemaical Coiuou-ime Iu-ouu Dyamic SISO Liear Parameric Time-ivaria Coraig aribue No-mahemaical Dicree-ime Sae ace aic MIMO oliear Noarameric Time-varyig /8/4 Proce Corol 4 6 Moelig Iue Moellig 6 Examle: Trai Syem Theoreical (hyical law) Whie-box ieificaio Srucure eermiaio Time-omai Recurive Direc Exerimeal Black-box ieificaio Parameer eimaio Frequecy-omai No-recurive Iirec Moelig Tuorial Problem: coier a oy rai coiig of a egie a a car Aumig ha he rai oly ravel i oe irecio, we wa o aly corol o he rai o ha i ha a mooh ar-u a o, alog wih a coa-ee rie The ma of he egie a he car will be rereee by M a M, reecively The wo are hel ogeher by a rig, which ha he iffe coefficie of k F reree he force alie by he egie, a he Greek leer, mu, reree he coefficie of rollig fricio Semeer /8/4 6: Moellig a Proce imulaio Corol 5 /8/4 Proce Corol 6 6 Examle: Trai Syem (co ) 6 Examle: Trai Syem (co ) M kg M 5 kg k N/ec F N u ec/m g 98 m/^ M; M5; k; F; u; g98; um[m M*u*g ]; e[m*m *M*M*u*g M*kM*M*u*u*g*gM*k M*k*u*gM*k*u*g]; yf(um,e) Liview(y) /8/4 Proce Corol 7 /8/4 Proce Corol 8 3

4 6 Coiuou C LTI Syem Moel Differeial equaio y ζ y y u Trafer fucio Num-e form m Y ( ) b b G ( ) U ( ) a a Zero-ole form m X FX Y HX GU JU m ( z ) Y ( ) i i G ( ) e g, G ( ) U ( ) ( )( ) ( i ) i, ζ ± ζ /8/4 Proce Corol 9 L b L b m, e g, G ( ) ζ 63 Moel Exreio i Malab Sae-ace form y(f,g,h,j) Num-e rafer fucio form yf(um,e) Zero-ole rafer fucio form yzk(z,p,) Mole exchage y(f(um,e)) or [F,G,H,J] (f(um,e)) [um,e] f(a,b,c,d,iu) [A,B,C,D]f(um,e) /8/4 Proce Corol 64 Coecio Block Diagram Referece iu - Forwar comeaor Feeback comeaor acuaor eor iurbace Pla 65 Coecio i Malab - I Serie coecio of wo LTI y erie(y,y) y erie(y,y,ouu,iu) w r - D() A(S) P() y y F() S() /8/4 Proce Corol /8/4 Proce Corol 65 Coecio i Malab - II 65 Coecio i Malab - III Parrallel coecio of wo LTI y arallel(y,y) y arallel(y,y,i,i,ou,ou) y y Feeback coecio of wo LTI y feeback(y,y) y feeback(y,y,ig) y feeback(y,y,feei,feeou,ig) /8/4 Proce Corol 3 /8/4 Proce Corol 4 4

5 r() 66 Digial Corol Syem - D/A a D (z) A(S) P() hol clock F (z) A/D a amler S() /8/4 Proce Corol 5 w 66 Zero-Orer Hol Equivale H h ( z) ( z D( ) ) Ζ{ } Malab imlemeaio y c(y,t,meho) 'zoh : Zero-orer hol The corol iu are aume iecewie coa over he amlig erio T 'foh : Triagle aroximaio (moifie fir-orer hol) The corol iu are aume iecewie liear over he amlig erio T 'ui : Biliear (Tui) aroximaio 'rewar : Tui aroximaio wih frequecy rewarig 'mache : Mache ole-zero meho /8/4 Proce Corol 6 Demom file ca be owloa from coure web Pla: G()/(), lea comeaor: D()7()/() NumG[]; DeG[ ]; NumD[7 4]; DeD[ ]; ygf(numg,deg); ydf(numd,ded); ycfeeback(yg*yd,); T5; ydlf([/t],[ /T]); yclfeeback(yg*yd*ydl,); Sublo(,,); imule(yc); hol; imule(ycl,'r'); Sublo(,,); zma(yc); gri; hol;zma(ycl,'r'); hol off figure boe(yc,'b',ycl,'r'); hol off yzohc(yc,t,'zoh'); ytusinc(yc,t,'ui'); yzpc(yc,t,'mache'); boe(yc,'b',yzoh,'r',ytusin,'g',yzp,'b') /8/4 Proce Corol 7 MM6 Moelig Iue & Corol Secificaio ) Iroucio ) Moelig iue 3) Corol Secificaio /8/4 Proce Corol 8 63 Corol Secificaio 3 Syem Sabiliy 63 Cloe-loo abiliy Defiiio: BIBO abiliy 63 Goo comma reoe 33 Diurbace aeuaio 34 Robue /8/4 Proce Corol 9 Deermiaio meho: o Imule reoe fucio/equece o Roo of characeriic equaio o Rouh abiliy crierio o Gai a hae margi o Nyqui abiliy crierio /8/4 Proce Corol 3 5

6 Sabiliy Imule Reoe BIBO abiliy (FC 3-4) The coiuou yem wih imue reoe h() i BIBO able if a oly if h() i aboluely iegrallable The icree yem wih imue reoe h[] i BIBO able if a oly if h[] i aboluely ummable Sabiliy Characeriic Roo Aymoic ieral abiliy Occur whe all ole (roo of he chraceriic equaio) of he coiuou yem are ricly i he LHP of he -lae Occur whe all ole (roo of he chraceriic equaio) of he icree yem are ricly iie he ui circle of he z-lae (Malab: roo(e)) /8/4 Proce Corol 3 /8/4 Proce Corol 3 Sabiliy Rouh Crierio Rouh abiliy crierio (FC 5-) For a able yem all he eleme i he fir colum of he Rouh array mu be oiive Examle (ee exbook) 3 Corol Secificaio 3 Cloe-loo abiliy 3 Goo comma reoe 33 Diurbace aeuaio 34 Robue /8/4 Proce Corol 33 /8/4 Proce Corol Syem Performace Syem ye: Coiuou corol yem Digial corol yem Aalyi omai: Time omai ecificaio Frequecy ecificaio Differe erio: Dyamic raie reoe Seay-ae reoe /8/4 Proce Corol 35 Coiuou Syem Time Domai (I) Dyamic reoe (raie reoe) Imule reoe: imule(y) H ( ) ζ ζ : amig raio Se reoe: e(y) EXAMPLE: y: Sy: h ( ) e : auralfre quecy i( )( ) σ ζ, ζ um[]; um[ ]; e[ ]; e[ 3]; imule(f(um,e),'r',f(um,e),'b') e(f(um,e),'r',f(um,e),'b') /8/4 Proce Corol 36 σ 6

7 Coiuou Syem Time Domai (II) goo raie reoe (See FC 6-3) Rie ime r Selig ime Overhoo M 8 r 4 6 ζ 4 6 σ πζ / ζ Peak ime M e, ζ π, ζ H( ) ζ ζ : amigraio : auralfrequecy /8/4 Proce Corol 37 Coiuou Syem Time Domai (III) goo eay-ae reoe (See FC -6) Lalace raform Fial value heorem Seay-ae error a yem ye (ye, ye I, ye II, ) uiy feeback y T ( ) e lim k lim G o ( ) e v lim G o ( ) e v Velociy coa Acceleraio coa a lim G o ( ) e a /8/4 Proce Corol 38 Poiio-error coa F ( ) f ( ) e lim f ( ) lim F ( ) Seay-ae error a yem ye Coiuou Syem Frequecy Domai Frequecy reoe (See FC ) u ( ) U y ( ) U A G ( ) θ < G ( ) i( ) A i( θ ) j j {Re[ G ( j )]} a Im[ G ( j )] Re[ G ( j )] {Im[ G ( j )]} /8/4 Proce Corol 39 /8/4 Proce Corol 4 Coiuou Syem Frequecy Domai Bawih a reoa eak Bawih:meaure he ee of reoe ( 3B frequecy) Boe lo echique (See FC 35) /8/4 Proce Corol 4 Coiuou Syem Frequecy Domai Boe lo boe(y); boe(y,w) boe(y,'plosyle',,yn,'plosylen') Comue gai a hae margi a aociae croover frequecie [Gm,Pm,Wcg,Wc] margi(y) margi(y) EXAMPLE: um[ ]; e[ 3 4] margi(um,e) /8/4 Proce Corol 4 7

8 Effec of Pole a Zero SGRID geerae a gri over a exiig coiuou -lae roo locu or ole-zero ma Lie of coa amig raio (zea) a aural frequecy (W) are raw H ( ) ζ : amig raio ζ h ( ) : auralfre quecy e ζ σ i( )( ) σ ζ, ζ zma(sy); /8/4 Proce Corol 43 ZGRID geerae a gri over a exiig icree z-lae roo locu or ole-zero ma Lie of coa amig facor (zea) a aural frequecy (W) are raw i wihi he ui Z-lae circle /8/4 Proce Corol 44 Effec of Aiioal Pole A aiioal ole i he LHP will icreae he riig ime igificaly if he exra ole i wihi a facor of 4 of he real ar of he comlex ole Examle: um[36] e[ 6 36] yf(um,e) roo(e) yy*f(,[ ]) yy*f(65,[ 65]) y3y*f(5,[ 5]) e(y,'b',y,'g',y,'r', y3,'--') lege('orig', 'y', 'y', 'y3') figure, boe(y,'b',y,'g',y,'r', y3,'--') lege('orig', 'y', 'y', 'y3') Dowloa aiioalolezerom /8/4 Proce Corol 45 Effec of Aiioal Zero A aiioal zero i he LHP will icreae he overhoo if he zero i wihi a facor of 4 of he real ar of he comlex ole A aiioal zero i he RHP will ere he overhoo (a may caue he e reoe o ar ou i he wrog irecio omiimum-hae yem Examle : um[36]; e[ 6 36]; yf(um,e); yy*f([ 3],3) yy*f([ 9],9) y3y*f([ 5],5) e(y,'b',y,'g',y,'r', y3,'--') figure, boe(y,'b',y,'g',y,'r', y3,'--') Dowloa aiioalolezerom Examle : um[36]; e[ 6 36]; yf(um,e); umz[-36 36] ez[ 6 36]; yzf(umz,ez) e(y,'b',yz,'r') /8/4 Proce Corol 46 Exercie Oe See he iribue aer /8/4 Proce Corol 47 8