ME2142/TM3142 Introduction to Feedback Control Systems First half: Professor POO Aun Neow Second half: Professor V Subramaniam

Transcription

1 ME4/TM34 Itroductio to Feedbck Cotrol Syte Firt hlf: Profeor POO Au Neow Secod hlf: Profeor V Subri Textbook: Textbook: Moder Moder Cotrol Cotrol Syte Syte by by Richrd Richrd Doff Doff d d Robert Robert Bihop. Bihop. Pero Pero Pretice Pretice Hll Hll Aother Aother RefereceText: RefereceText: Moder Moder Cotrol Cotrol Egieerig Egieerig by by Ogt. Ogt. Pretice Pretice Hll Hll Webite Webite for for thi thi portio: portio:

2 Itroductio d Bic Cocept

3 Wht i i cotrol Syte? A cotrol yte i i itercoectio of of copoet tht will will provide deired yte repoe or or output repoe. The The tudy of of cotrol yte i i the the tudy of of dyic yte. A ttic yte eed o o cotrol. Exple of of cotrolled output: teperture, huidity, poitio, peed, preure, directio, liquid level, ltitude. Ad lo: ugr level i i hu, ifltio, iteret rte. 3

4 Wht i i cotrol Syte? I I order for for yte to to be be cotrollble, there ut be be cue-effect reltiohip for for it it copoet, i.e. i.e. there ut be be oe iput tht c c cue chge to to the the output preter to to be be cotrolled. (Source of eergy) Cotrollig/ctutig iput, Proce or Plt Output (cheicl proce, chie, idutril proce, ecooic proce) 4

5 Ope-loop cotrol Syte Deired output repoe Cotroller Plt or Proce Output I I ope-loop cotrol cotrol yte, o o feedbck fro fro the the output output i i ued ued to to cotrol cotrol the the yte. Bed Bed o o how how the the output output i i required, or or deired, to to repod, the the cotroller djut djut the the iput iput to to the the plt plt to to chieve thi. thi. Cotrol Cotrol will will oly oly be be ccurte if if plt plt i i highly highly predictble d dthere i i o o iterl iterl or or exterl diturbce. Geerlly ued ued oly oly whe whe good good cotrol cotrol perforce i i ot ot required. Exple: A A electric electric bred bred toter. Teperture cotrol cotrol of of iple iple wter wter heter heter for for the the hower. 5

6 Cloed-loop feedbck cotrol Syte R - E Cotroller U Plt Y Seor Feedbck R Set-poit or or Referece Iput Iput E Error Error U Plt Plt iput iput Y Cotrolled Vrible The The eor eor eure the the ctul ctul vlue vlue of of the the output, output, Y, Y, copre thi thi with with the the deired deired vlue, vlue, R, R, d d copute the the error, error, E. E. Bed Bed o o thi thi error error E, E, the the cotroller geerte the the iput, iput, U, U, to to the the plt plt o o to to brig brig Y to to the the deired deired vlue vlue R. R. 6

7 Cloed-loop feedbck cotrol Syte R - E Cotroller U Plt Y Seor Feedbck Geerlly ued ued whe whe good good cotrol cotrol perforce i i required. Accurte cotrol cotrol c c be be chieved eve eve i i the the preece of of plt plt vritio, d d iterl iterl or or exterl diturbce becue uch uch diturbce will will ffect ffect the the output output Y, Y, reflected i i the the error error E, E, t t thu thu will will cue cue the the plt plt iput iput U to to chge chge o o to to correct correct for for thee thee diturbce. C C becoe utble. Stbility becoe iportt coidertio. 7

8 Soe exple of of cotrol Syte? 8

9 Exple: Ope-loop v vcloed Loop Proce of of Wlkig: Deired output: poit where you you wt to to be. be. Cotroller: the the bri Plt or or proce: the the leg leg Ope-loop cotrol: Wlkig with your eye cloed. A wlkig Cloed-loop feedbck cotrol: Wlkig with your eye ope. The The eye eed the the ctul output, where you you re re d d where you you re re hedig, copute the the error i i poitio d d i i directio, d d iue cod to to the the plt, eig the the leg, to to ove i i uch wy wy o o to to reduce the the error. 9

10 Exple: Ope-loop v vcloed Loop Photo courtey U.S. Air Force Droppig Bob: Objective of of droppig bob fro height i i to to hit hit trget below. Deired output: Trget below Plt or or proce: the the bob with it it cotrol fi fi Ope-loop Cotrol or or dub bob The The cotroller, eig the the pilot or or bobrdier, eed to to etite hi hi ow ow height, velocity, ditce to to trget, wid coditio, d d chrcteritic of of bob to to decide whe d d where to to relee the the bob. Ofte, hudred of of bob re re eeded to to hit hit pecific trget. 0

11 Exple: Ope-loop v vcloed Loop Droppig Bob: Objective of of droppig bob fro height i i to to hit hit trget below. Deired output: Trget below Plt or or proce: the the bob with it it cotrol fi fi Cloed-loop Cotrol or or rt bob Seor re re icorported ito ito the the bob to to give give feedbck o o it it ctul poitio reltive to to the the trget. The The error ifortio i i the ued to to teer the the bob, uig it it cotrol fi, to to the the trget. Reult: oe oe trget oly eed oe oe bob. Seor: TV, TV, Ifrred, ler guided, or or GPS. See See lo: lo:

12 Exple: Ope-loop v vcloed Loop Siilr to to wlkig, here the the plt i i the the cr. cr. The The cotroller i i the the d d ctutio i i through the the ccelertor pedl (for (for peed cotrol) or or teerig wheel (for (for directio or or poitio-o-rod cotrol). Ope-loop cotrol: Drivig with eye cloed. Cloed-loop feedbck cotrol: Eye re re the the eor d d give give feedbck o o poitio, d d directio, of of cr cr o o rod, d d lo lo peed of of cr cr vi vi viul feedbck fro urroudig or or fro the the peedoeter.

13 Study of of cotrol Syte Study Study of of cotrol cotrol yte i i the the tudy tudy of of the the dyic of of the the yte. The The repoe of of the the cotrolled vrible Y to to y y iput iput R deped upo upo the the dyic of of the the Plt, Plt, Cotroller, d d the the Seor Seor or or Feedbck. R - E U Cotroller Seor Feedbck Plt Y Give Give cotrol cotrol yte, y f(r,t) f(r,t) eig tht tht y i i ot ot oly oly fuctio of of r, r, but but lo lo vrie vrie with with tie tie t. t. If If y f(r) f(r) the the the the yte yte i i ot ot dyic yte yte but but i i ttic. ttic. To To theticlly decribe the the dyic behvior of of the the cotrol cotrol yte yte d d it it copoet, differetil equtio re re ued. ued. 3

14 Lier d No-lier Syte A yte i i lier if if it it tify the the propertie of of uperpoitio d d hoogeeity/clig. A yte i i o-lier if if it it i i ot ot lier. Coider yte which h h the the repoe to to y y two two rbitrry iput u (t) (t) d d u (t) (t) y (t) (t) f(u f(u (t) (t) d d y (t) (t) f(u f(u (t) (t) Property of of Superpoitio i i tified if if the the output for for cobied iput of of u (t) (t) d d u (t) (t) i i y 3 3 f(u f(u (t) (t) u (t)) (t)) y (t) (t) y (t) (t) Property of of hoogeeity i i tified if if y 3 3 f(u f(u (t)) (t)) y y (t) (t) 4

15 Lier d No-lier Syte Coider yte which h h the the repoe to to y y two two rbitrry iput u (t) (t) d d u (t) (t) y (t) (t) f(u f(u (t) (t) d d y (t) (t) f(u f(u (t) (t) A yte i i lier if if the the propertie of of uperpoitio d d hoogeeity re re tified. The The bove yte will will be be lier if if the the followig i i tified y 3 3 f( f( u (t) (t) u (t)) (t)) y (t) (t) y (t) (t) I I geerl, rel rel phyicl yte re re o-lier if if the the opertig rge i i very very lrge, However, if if opertio i i coidered oly oly bout oe opertig poit, d d the the rge of of opertio i i ufficietly ll, ot yte c c be be coidered to to be be lier. 5

16 y f ( x ) x Lier d No-lier Syte Exple: Which of of the the followig yte re re lier? (i) (i) F x x (ii) (ii) y x (iii) (iii) y x x b For For y y cott A d d B d d y y two two iput x d d x,, (i) (i) F f(x f(x ) ) x x d d F f(x f(x ) ) x x Alo, F 3 3 f(ax f(ax Bx Bx ) ) (Ax Bx Bx ) ) Ax Ax Bx AF AF BF BF Thu propertie of of uperpoitio d d hoogeeity i i et. et. Thu lier. y f( x) x (ii) (ii) Ad Ad f( Ax) ( Ax) Ay Alo, f( x x ) ( x x ) f( x ) f( x ) x x Thu yte i i ot ot lier or oro-lier. y f( x) x b (ot (ot hoogeou) (uperpoitio violted) (iii) (iii) Ad Ad f( Ax) ( Ax) b Af( x) A( x b) (ot (ot hoogeou) Syte i i ot ot lier. C C be be how tht tht uperpoitio lo lo violted. 6

17 Lier Approxitio of of Syte Ay Ay o-lier yte c c be be lieried bout oe opertig poit d d c c be be coidered to to be be lier withi ll opertig regio bout tht tht poit For For y y fuctio y f( x) with with We We c c ue ue the the Tylor Serie expio bout oe opertig poit, x 0, 0, d d hve df ( x x ) d f ( x x ) y f( x ) dx x x! dx! x x 0 0 f( x ) 0 0 For For ll vritio bout the the opertig poit, ecod d d higher-order ter i i ( x x0 ) c c be be eglected. The y y0 ( x x0) or or y x which i i lier. y 7

18 Lier Approxitio of of Syte Exple: θ For For the the pedulu how i i the the figure, the the retorig torque due due to to grvity i i give by by T MgLiθ Derive the the lieried equtio bout the the opertig poit θ 0.. Solutio: d iθ T T0 MgL ( θ θ0) MgLco(0)( θ θ0) d θ 0 T 0 0 Sice with with,, θ θ 0 T MgLθ 8

19 Ed 9

20 ME4/TM34 Itroductio to to Feedbck Cotrol Syte Lplce Trfor

21 Mtheticl preliirie Coplex vrible σ jω j I Rel prt Igiry prt ω σ Re -ple

22 Coplex fuctio Coplex fuctio G ( ) G x jg y I Rel prt Igiry prt G θ t G x G y G G G ( ) G e y x jθ G G() y G θ G x Coplex cojugte G( ) G x jg y G()-ple 3

23 Euler Theore Euler Theore θ e j coθ j iθ It Coplex cojugte θ e j coθ j iθ Soe ueful forul: coθ iθ j ( jθ jθ e e ) ( jθ jθ e e ) 4

24 Lplce Trfor A theticl tool tool tht tht trfor difficult difficult differetil equtio ito ito iple iple lgebr proble where where olutio c c be be eily eily obtied. Defiitio: f ( t) 0 for t < 0 Ivere Trfor: πj c j c j F( ) e t d Norlly Tble of Lplce Trfor pir re ued for tkig the Lplce Trfro d the Ivere Trfor. 5

25 {} t Propertie of of Lplce Trfor Lierity: Exple Fro Fro Tble Tble L {} t d d L { i( t) } The The L { 3t 5i(t) } 4 7

26 3 6e {( t ) } 4 Propertie of of Lplce Trfor Trltio: If, the Exple Fro Fro Tble Tble L 3 6 { t } { } { } e The The L ( t ) d d L e t t ( ) Vrible trfor: 8

27 Propertie of of Lplce Trfor Derivtive: 9

28 Propertie of of Lplce Trfor Fil-Vlue Theore: li t f ( t) li 0 F( ) Iitil-Vlue Theore: f ( 0) li F( ) 0

29 Fidig the Ivere.. Uig L - - c j { F ( ) } F( ) e t d πj c j.. Uig the Tble of of Trfor Pir Uig propertie Uig Prtil Frctio Expio (We (We do do ot ot orlly ue ue Approch.. We We ue ue cobitio of of to to 4.) 4.)

30 Uig Tble d Propertie Exple Fid the ivere of 5 F( ) ( 3) 8 6 Solutio: Fro Tble L - t Lookup fro Tble ( 5 3) L - 5L L - L - ( 3) 3t 5te 4 i(4t) 4 Lookup fro Tble or or Ue Trltio property Lookup fro Tble Thu L - 3 { F( ) } t 5te t i(4t)

31 Uig Tble d Propertie Exple Fid the ivere of F( ) 8 e 5 Solutio: We write F( ) 8 e 5 e 8t ( ) Fro Tble L - i( ) t t Thu L - e i(t) ( ) Uig trltio property Alo Property L - { e F( ) } f ( t ) for t d 0 otherwie Thu L - { F( ) } e 0 ( t 8) i ( t 8) for for t t 8 < 8 3

32 4 Uig Prtil Frctio Uig Prtil Frctio Frequetly N() d D() beig polyoil i Exple or Eq(-) re the zero d pole of F() repectively. They c either be rel or coplex. If they re coplex, they lwy occur i cojugte pir. ) ( ) ( ) ( D N F p p p z z z,,, d,,, b b b b F > with ) ( 0 0 L L ) ( ) )( ( ) ( ) )( ( ) ( p p p z z z F L L

33 5 Uig Prtil Frctio Uig Prtil Frctio If Eq(-) h ditict pole, the F() c lwy be expded ito u of prtil frctio: Eq(-) where k re cott. To deterie the vlue of k, ultiply both ide of Eq(-) by (p k ) d let -p k. Eq(-3) p p p D N F L ) ( ) ( ) ( p k k k D N p ) ( ) ( ) (

34 Uig Prtil Frctio Exple Fid the ivere of F( ) 3 ( )( ) Solutio: We let 3 F( ) ( )( ) the ( 3) ( ) ( )( ( 3) ( ) ( )( Thu F( ) d t t f ( t) e e for t 0 6

35 Uig Prtil Frctio If Eq(-) h ultiple pole, ech ultiple pole p r of order q will be equivlet to prtil frctio of the for: b ( p r ) b ( p r ) L b ( q p r ) q where b k re cott. If Eq(-) h deoitor of the for ( b) ech of thee will be equivlet to prtil frctio of the for: N( ) c d F( ) L L( b) L b L where, b, c d d re cott. 7

36 Uig Prtil Frctio Give 3 F( ), fid f(t). ( ) Exple 3 Solutio: Let F( ) 3 3 ( ) b ( ) b ( ) b3 ( ) 3 Multiplyig both ide by () 3 Lettig 3 b ( ) b ( ) b3 b 3 we hve Coprig ter i, we hve b Coprig cott ter, we hve 3 b b b b 0 3 givig F( ) ( ) Thu d 3 f ( t) e t t e t 8

37 Solvig differetil equtio.. Tke the Lplce trfor of of the differetil equtio to to covert it it ito lgebric equtio. The Lplce trfor of of the depedet vrible i i the obtied uig iple Algebr... The tie fuctio of of the depedet vrible i i the obtied by by tkig the ivere of of the Lplce trfor. 9

38 Solvig differetil equtio Exple Solve for y(t) give && y y& 5y 3, y(0) 0, y& (0) 0 Solutio: Trforig or givig [ Y ( ) y(0) y& (0)] [ Y ( ) y(0) ] Y ( ) Y ( ) 5Y ( ) ( Y ( ) ( 3 5) Multiplyig both ide by d lettig 0, we get 5) Y ( ) b c 5 5Y ( )

39 Exple Solvig differetil equtio We hve Y ( ) ( 3 5) b c 5 Multiplyig both ide by d lettig 0, we get 3 5 Multiplyig both ide by the deoitor ( 5) we hve 3 ( 5) ( b c) Coprig ter i, 0 6 / 5 c c 6 / 5 Coprig ter i, Thu 3 3 Y ( ) /5 b b 3/ ( ) 3 0 ( ) Ad 3 3 t 3 t y( t) e co t e i t for t

40 Lplce Trfor END

41 Tble of Lplce Trfor Pir ME4/TM34 Mrch 005 Deprtet of Mechicl Egieerig & Bchelor of Techology Progre Ntiol Uiverity of Sigpore

42 Lplce Trfor Pir f (t) F () Uit ipule δ (t) Uit tep ( t ) 3 t 4 t ( )!, 5 t, 6 t e 7 t 8 t ( )! 9 t t e, e t (,, 3, ) (,, 3, )! te ( ), (,, 3, ) (,, 3, ) ( )! ( ) ω 0 i ωt ω co ωt ω ω ih ωt ω 3 coh ωt ω t 4 ( e ) t bt 5 ( e e ) b bt t 6 ( be e ) 7 b ( be b b t bt e ) ( ) ( )( b) ( )( b) ( )( b)

43 t t 8 ( e te ) t 9 ( t e ) ( ) ( ) 0 e t iωt e t coωt ω ζωt e iω ζ t ζ, ζ < ζωt 3 e i( ω ζ t φ) ζ φ t ζ ζ, ζ < ζω t 4 e i( ω ζ t φ) ζ φ t ζ ζ 5 coωt 6 ωt i ωt, ζ < ω ( ) ω ( ) ω ( ω ζω ω ζω ω ω ζω ω ) ω ( ω ) 3 ω ( ω ) 7 i ωt 3 ω ωt coωt ( ω ) 8 t ωt ω i ( ω 9 t co ωt ω ( ω ) ) 30 ( co ω co ω ) ω ω t t, ( ω ) ( ω )( ω ) ω 3 ( i ωt ωt coωt) ω ( ω )

44 Modellig of Phyicl Syte

45 Differetil Equtio U Plt Y I the plt how, the iput u ffect the repoe of the output y. I geerl, the dyic of thi repoe c be decribed by differetil equtio of the for d d dy d u dt d u dt L 0 b b L b b0 dt y dt y dt du dt Differetil equtio i i lier lierif if coefficiet re re cott or or fuctio oly oly of of tie tie t. t. Lier Lier tie-ivrit yte: if if coefficiet re re cott. Lier Lier tie-vryig yte: if if coefficiet re re fuctio of of tie. tie.

46 Modellig of of Phyicl Dyic Syte Mechicl Syte Fudetl Lw Lw Mechicl Syte Trltiol Syte Newto Lw Lw x f f x & ff i i pplied pplied force, force, i i i i g g x i i diplceet i i.. 3

47 Modellig of of Phyicl Dyic Syte Mechicl Syte Toriol Syte T J θ ω T J & θ J & ω T i i pplied pplied torque, torque, - - J i i oet of of ierti ierti i i g- g- i i diplceet i i rdi rdi i i the the gulr peed peed i i rd/ rd/ θ ω 4

48 Modellig of of Phyicl Dyic Syte Mechicl Syte --prig Trltiol: x x f i i teile force i i prig, i i prig cott, / f f Iportt: Note directio d ig f ( x x ) Rottiol: θ θ T re exterl torque pplied o the toriol prig, - G i toriol prig cott, -/rd G i toriol prig cott, -/rd T G θ ) ( θ 5

49 Modellig of of Phyicl Dyic Syte Mechicl Syte dper or or dhpot Trltiol:. x. x f f f i i teile force i i dhpot, b i i coefficiet of of dpig, -/ f b b( x& x& ) Rottiol: θ & T i i torque i i toriol dper, - b i i coefficiet of of toriol dpig, --/rd T b ( & θ & ) θ θ & 6

50 Modellig of of Phyicl Dyic Syte Exple Derive the the differetil equtio reltig the the output diplceet y to to the the iput diplceet x. x. b A y x Free-body digr t poit A, f d A f Note: Directio of of ff d ff d how ue they re teile. Sice 0, f give f f d 0 f d Sice d by& ( x y) f Thu Or ( x y) by& 0 b y& y x 7

51 The Trfer Fuctio The trfer fuctio of lier tie ivrit yte i defied the rtio of the Lplce trfor of the output (repoe) to the Lplce trfor of the iput (ctutig igl), uder the uptio tht ll iitil coditio re zero. Previou Exple b y& y x Auig zero zero coditio d d tkig tkig Lplce trfor of of both both ide ide we we hve hve by ( ) Y ( ) X ( ) Trfer Fuctio G ( ) Y ( ) X ( ) b Thi Thi i i firt-order yte. 8

52 Modellig of of Phyicl Dyic Syte Exple For For the the prig--dper yte yte how how o o the the right, right, derive derive the the trfer fuctio betwee the the output output x o d o d the the iput iput x i. i. x o Free-Body digr x o Note: f d f d ued to be teile. b f f d f give f f d & x o x i Thu ( x i x ) b( x& x& ) o i o & x o Or && x bx& x bx& o o o i x i Ad X o ( ) bx o ( ) X o ( ) bx i ( ) X i ( ) Trfer Fuctio G( ) X X i ( ) o ( ) b b. Thi i ecod-order yte. 9

53 Modellig of of Phyicl Dyic Syte Electricl Eleet Cpcitce e i C Q 0 q C e q Ce dq de i C dt dt t Q0 e i dt C 0 C i i iitil iitil chrge chrge (Coulob) Uit Uit of of C: C: Frd Frd (F) (F) Or Or E IX c I C Coplex ipedce X c /( C) Iductce Uit Uit of of L: L: Hery Hery (H) (H) Or Or Reitce e i R Uit Uit of of R: R: oh oh (( )) e i L i E IX L e e ir e i R Ω L L di dt t 0 I(L) e dt 0

54 Modellig of of Phyicl Dyic Syte Electricl Circuit- irchhoff Lw Lw Curret Lw: Lw: The The u u of of curret eterig ode ode i i equl equl to to tht tht levig levig it. it. i 0 Voltge Lw: Lw: The The u u lgebric u u of of voltge voltge drop drop roud roud cloed cloed loop loop i i zero. zero. e 0

55 Modellig of of Phyicl Dyic Syte Electricl Circuit- Exple RC RC circuit: circuit: Derive Derive the the trfer fuctio for for the the circuit circuit how, how, E IR i IX c R d d givig givig E o IX c E E o i X c R X c RC /( C) R /( C) e i i C e o Thi Thi i i firt-order trfer fuctio.

56 Modellig of of Phyicl Dyic Syte Electricl Circuit- Exple RLC RLC circuit: circuit: R L d d E IR IX i E o IX c L IX c e i i C e o givig givig E E o i R LC X X L c X c RC /( C) R L /( C) Thi Thi i i ecod-order trfer fuctio. 3

57 Modellig of of Phyicl Dyic Syte Opertiol Aplifier Propertie of of idel Op Ap v o A( v ) v Gi A i i orlly very lrge o tht copred with other vlue, ( v ) v i iued ll, equl to to zero. The iput ipedce of of the Op Ap i i uully very high (ued ifiity) o tht the curret i d i re very ll, ued zero. Two bic equtio goverig the opertio of of the Op Ap ( v v v v ) 0 or d i, i 0 0 4

58 Modellig of of Phyicl Dyic Syte Opertiol Aplifier Exple Z f i f i i v i i 0 Z i S - v o For the Op Ap, ue i 0 d v v 0. vi vo i i i f 0 0 Z Z The or or i f Therefore v o Z Z f i 5

59 Peret Mget DC DC Motor Drivig Lod R L e i eω T ω J b For For the the dc dc otor, otor, the the bck bck ef efi i proportiol to to peed peed d d i i give give by by e ω where where e i i the the voltge voltge cott. The The torque torque produced i i proportiol to to rture curret curret d d i i give give by by T i where where i i the the torque torque cott. Relevt equtio: T t t di e R i L dt e ω dω ti T J b ω dt Note: Note: By By coiderig power power i i power power out, out, c c how how tht tht e e t t 6

60 Ed 7

61 Trfer Fuctio d Block Digr Algebr

62 Block Digr Repreettio A block block digr i i grphicl tool toolc help help u u to to viulize the the odel odelof of yte yte d d evlute the the theticl reltiohip betwee their their eleet, uig uig their their trfer fuctio. The Trfer Fuctio Block Iput R() G() Syte C() Output G ( ) C( ) R( ) The trfer fuctio G() i defied oly for lier tie-ivrit yte d ot for olier yte. idepedet of the iput to the yte. Couttive G G GG Aocitive G G G G

63 Block Digr Eleet The Suig Poit Siged iput Y X X Y - Z - Z output Ay uber of iput. Oly oe output 3

64 Block Digr Algebr Whe Whe ipultig block block digr, the the origil origil reltiohip, or or equtio, reltig reltig the the vriou vriou vrible ut ut rei rei the the e. e. Block i erie or ccded block X Y Z G G X G G Z Whe block re coected i erie, there ut be o lodig effect. 4

65 Block Digr Algebr Block i prllel X G G Y X G G Y 5

66 6 G Z X Y G Z X Y G X Y X G X Y X X G Y Z X G Y Z Block Digr Algebr Block Digr Algebr G G Z X Y G X Y /G X Z Y X G /G

67 Cloed-Loop Feedbck Syte R E - G C B H R i clled the referece iput C i the output or cotrolled vrible B i the feedbck E (R B) i the error C E B E G GH i clled the feedforwrd trfer fuctio i clled the ope-loop trfer fuctio 7

68 Cloed-Loop Feedbck Syte R E - B G H C C GE G(R B) G(R HC) C( GH) GR C R G GH C R i the cloed-loop trfer fuctio C Alo E d G E R G C R GH E R i clled the error trfer fuctio 8

69 Cloed-Loop Cotrol Feedbck Syte D R E - M G c G p C B H G c i the cotroller trfer fuctio G p i the plt trfer fuctio M i the ipulted vrible D i the exterl diturbce C E G c G p i the feedforwrd trfer fuctio B E G G c p H i the ope-loop trfer fuctio 9

70 Cloed-Loop Cotrol Feedbck Syte D R E - M G c G p C B H G c i the cotroller trfer fuctio G p i the plt trfer fuctio M i the ipulted vrible D i the exterl diturbce C E G c G p i the feedforwrd trfer fuctio B E G G c p H i the ope-loop trfer fuctio 0

71 Cloed-Loop Cotrol Feedbck Syte D R E - M G c G p C B H Auig R 0, we c re-drw C R G GH G G c G G c p p H D - G p G c H C C D G GH G G p p G c H

72 Block Digr Mipultio Block Digr Mipultio Exple Exple R G - H F - I E D C D b Eb R G - H F - I E D/F C (D/F) Gb R - H FG - I E/G C D/F Gb Eb R - H FG - I E/G C D/F

73 Exple Block Digr Mipultio R - D/F - FG H E/G C R - D/F FG FGH E/G C I I R - D F FG FGH I E G C R D FG E F FGH G D FG E I F FGH G C 3

74 Ed 4

75 Syte Repoe Suppleetry redig:

76 - t Syte repoe The yte repoe coprie two prt, triet d tedy-tte. Triet Repoe ff - Stedy-Stte Repoe Output 6 6 Triet repoe tedy-tte repoe Stedy-Stte Error? 6 e tedy-tte error t The gitude of the triet repoe decree with tie d ultitely vihe levig oly the tedy-tte repoe. It i lwy ocited with the copoet e t with > 0 -

77 Syte Chrcteritic Equtio Coider the yte with the cloed-loop trfer fuctio, G c () how Iput R() G c () C() Output Syte with C( ) N Gc ( ) R( ) D ( ) ( ) where N c () d D c () re polyoil of. c c The yte chrcteritic equtio i give by D c ( ) Note tht the chrcteritic equtio i property of the yte d i ot depedet o the iput. 0 3

78 Syte Chrcteritic Equtio Exple Sprig--dper (Slide (Slide 8: 8: Modellig Modellig of of Phyicl Phyicl Syte) Syte) Trfer Trfer Fuctio Fuctio Chrcteritic Eq: Eq: G( ) X ( ) ( ) o X i b b b 0 R-C R-C circuit circuit (Slide (Slide : : Modellig Modellig of of Phyicl Phyicl Syte) Syte) E o Trfer Trfer Fuctio Fuctio E RC i Chrcteritic Eq: Eq: RC 0 Cloed-loop feedbck yte yte (Slide (Slide 8: 8: Block Block Digr Digr Algebr) Algebr) C G Trfer Trfer Fuctio Fuctio R GH Chrcteritic Eq: Eq: GH 0 4

79 Syte Chrcteritic Equtio C( ) R( ) Gc ( ) N D c c ( ) ( ) Chrcteritic equtio D c ( ) 0 The root of thi equtio re the cloed-loop pole d they deterie the triet repoe of the yte. Ech root, p, of thi equtio will cotribute ter i the tie repoe of the yte. Or c ( t) pt L Ae L Note tht if ll the root, p r, re egtive, the the triet repoe will evetully die wy t icree. But if y of the root i poitive, the the triet repoe will grow without boud tie icree. The yte i the id to be utble. e pt 5

80 Syte Repoe Give Give dyic yte: How do we pecify the chrcteritic of the repoe required? How do we copre it with other yte? How do we kow whether it repoe will dequtely eet our eed? How will we kow how it will repod to differet iput? We ue Stdrd tet iput to excite yte d oberve repoe Clify yte with iilr chrcteritic d idetify their perforce chrcteritic with yte preter. 6

81 Syte Repoe Tet igl ) ) Step Step iput iput r( t) A 0 A R ( ) 0 < 0 Whe A, we hve uit tep iput. Ued to tudy repoe to udde chge i iput. t t r(t) A t 0 t ) ) Rp Rp iput iput r( t) At 0 R ( ) A t 0 t < 0 Whe A, we hve uit rp iput. Ued to tudy repoe to grdul chge i iput. r(t) t 0 At t 7

82 Syte Repoe Tet igl 3) 3) Ipule iput iput r ( t) Aδ (0) i i the the uit-ipule fuctio or or Dirc Dirc delt deltfuctio R ( ) A Whe A, we hve uit ipule iput. r(t) A t 0 Ued to tudy repoe to udde hock or ipct. t 4) 4) Siuoidl iput iput r( t) Ai ωt 0 t 0 t < 0 Ued for frequecy repoe lyi. Iportt ethod. Will be dicu gi lter. r(t) t 0 t Tet igl () to (3) re ofte kow tie repoe or triet repoe lyi while tet igl (4) i kow frequecy repoe. 8

83 Syte Repoe Firt-order yte A firt-order yte yte c c be be writte writte i i the the tdrd for for C( ) R( ) T T i i kow kow the the tie tie cott d d deterie the the peed peed of of repoe. Exple Sprig-dper yte (Slide (Slide 7 of of Modellig of of Phyicl Syte) Y ( ) with T b X ( ) b. T. RC RC circuit circuit (Slide (Slide of of Modellig of of Phyicl Syte) E o with T RC E RC T. i. If the trfer fuctio re the e, the the repoe y(t) d e o (t) will be the e for the e iput i x(t) d e i (t).. 9

84 Syte Repoe Firt-order yte Repoe to to uit uit tep tep iput C( ) R( ) T.. Thu C( ) R( ) T with with R( ) / T ( / T ) A B / T Multiplyig both ide by d lettig 0 give A Multiplyig both ide by (/T) d lettig -/T give B - Therefore C( ) / T Uig tble c( t) t / T t / T e ( e ) for t 0 0

85 Syte Repoe Firt-order yte Repoe to to uit uit tep tep iput For c( t) ( e t / T ) Note: The ller the tie cott T, the fter the repoe. The hpe i lwy the e.

86 Syte Repoe Firt-order yte Repoe to to uit uit rp iput C( ) R( ) T.. Thu C( ) R( ) T with with R ( ) / T ( / T ) T T ( / T ) Uig tble c( t) For, c( t) ( t T t T ( e with the error e(t) r(t) c(t) T ( e Te t / T t / T ) t / T ) ) for t 0 r(t) c(t) t 0 r(t) e T t

87 Syte Repoe Firt-order yte Repoe to to uit uit ipule iput C( ) R( ) T with with R( ).. Thu C( ) T / T / T Or c( t) T e t / T For, r(t) c( t) T e t / T t 0 t 3

88 Syte Repoe Lier tie-ivrit yte Propertie Repoe to to Uit Ipule c ( t) T e t / T C( ) R( ) T c ( t) Uit Step ( e Chrcteritic Equtio T 0 T t / T ) c 3 ( t) Uit Rp ( t T Te t / T ) t T The The triet repoe ll ll coti the the ter ter e / which which i ideteried by by the the root root of of the the chrcteritic equtio d d the the preter T. T. Note tht the uit tep i the derivtive of the uit rp, d the uit ipule i the derivtive of the uit tep. Note tht iilrly, c (t) i the derivtive of c 3 (t) d c (t) i the derivtive of c (t). For lier tie-ivrit yte, the repoe to the derivtive of iput c be obtied by tkig the derivtive of the repoe to the iput. 4

89 Peret Mget DC DC Motor Goverig equtio di e Ri L eω dt E e Ω ( R L) I T ti T t I dω T J bω T ( J b) Ω dt e R i L eω The The Peret Mget DC DC otor. otor. T ω J b Block Block Digr E - L R I t T Ω J b e Ω e E - t ( L R )( J b) Ω e Ω e 5

90 Peret Mget DC DC Motor E - e Ω t ( L R )( J b) e The The Peret Mget DC DC otor. otor. Block Block digr the the becoe E - t / R J b e Ω Ω with Cooly L R << L c the be eglected Ω G E GH t / R ( J b) t e / R ( J b) τ t b t / R e / R J b J b τ b t / R t t J e e 6 / R / R

91 Speed Cotrol of of the the DC DC Motor E τ Ω τ b t J e / R The repoe to uit tep iput i firt order with tie cott of τ Ω t 0 t With With peed peed feedbck V - Error c Cotroller E τ Ω Ω V with c τ c ' c τ c τ ' τ τ c τ ' ' c c The reultt yte i till firt-order but the tie cott i ow uch ller, thu uch fter repoe. 7

92 Syte Repoe Secod-order yte A ecod-order yte will will be be of of the the for for Y ( ) X ( ) Exple RLC RLC circuit circuit (ee (ee Modellig of of Phyicl Syte) E o ( ).. E ( ) LC RC d Sprig--dper i e b c X X o i ( ) ( ) with with,, b, b, c, c, d d d e beig beig cott. cott. b b Stdrd For: For:.. C( ) R( ) ζ ω ω ( ) ω ζω ω 8

93 Cloed-Loop Poitio Feedbck Syte (Servoechi) R - E G c cotroller V τ Ω θ With G c beig proportiol gi p R p θ I tdrd fort - Θ R E Θ R ( τ ) G GH τ ω ζω ω p p with ω ω ζ p τ turl frequecy dpig rtio ζ p τ ζω τ 9

94 Tie Repoe Secod-order yte Coider Θ R ω ζω ω The root of the chrcteritic equtio re p, ζω ± ω ζ For 0 < ζ <, the root re pir of coplex cojugte p, ζω ± jω d where ω ω ζ i clled the dped turl frequecy d d the repoe i uderdped. For For ζ,, the the root re re equl p, ω d d the the yte i i id id to to be be criticlly dped. For For ζ >,, the the root root re re both both rel rel d d uequl p, ζω ± ω ζ d d the the yte yte i i id id to to be be overdped. 0

95 Step Repoe Secod-order yte Θ R ω ζω ω with with R( ) Therefore Θ ( ω ζω ω ) Uderdped Repoe 0 < ζ < Fro Fro tble tble (Etry (Etry 4 4 i i Dr Dr Che Che tble), tble), we we hve hve c( t) e ζωt ζ t 0 i( ω t d φ) 0 < φ < π ω d φ ω t ζ ζ ζ Thi Thi repreet decyig ocilltory repoe depedig upo upo with with frequecy of of ocilltio of of ω d ζ

96 Step Repoe Secod-order yte Criticlly dped Repoe ζ We We hve hve Θ ( ω ζω ω ) ω ( ω ) ω Fro Fro tble tble (Etry (Etry 8 8 i i Dr Dr Che Che tble tble with with ), ), we we hve hve t t c( t) ( e te ) ( ) c ωt ωt ( t) e ω te t 0 givig for Thi Thi repreet o-ocilltory repoe with with expoetilly decyig triet copoet d d zero zero tedy-tte error. error. The The peed peed of of decy decy of of the the triet copoet deped upo upo the the preter ω..

97 Step Repoe Secod-order yte Overdped Repoe ζ > Θ ( ( ω ζω ω ) ω )( b) with ( ζω ω ζ ω )( ζω ω ζ ) ζω ω ζ d b ζω ω ζ We We ue ue Etry Etry 7 7 i i Dr Dr Che Che tble. tble. ω t bt o tht c( t) ( be e b b t 0 ) for, C d C beig cott. We hve b ω t Ce C e bt The The repoe i i o-ocilltory, trt trt iitilly iitilly with with c( 0) 0 d d expoetilly rie rie to to c( ).. >> b If If ζ >>,, the the d d the the firt firt expoetil ter ter will will decy decy uch uch fter fter th th the the ecod. The The pole pole ( ) c c the the be be eglected d d the the yte yte behve like like firt-order yte. 3

98 Step Repoe Secod-order yte Norlized repoe curve For ft repoe, ζ 0.7 i uully deirble. If o overhoot i required, ζ > i uully ued. 4

99 Triet Repoe Specifictio Five Five eure of of triet perforce bed bed o o d d -order -order uderdped repoe Mxiu (percet) overhoot: c( t p ) c( ) M p 00% c( ) Dely tie Rie tie: 0% - 90%, or 5% - 95%, or 0% - 00% Pek tie Settlig tie: tie to rech d ty withi pecified liit, uully % or 5%. 5

100 Meure of of triet perforce We We hve hve c ( t) e ζωt ζ i( ω t d φ) ω d ω ζ φ t ζ ζ Rie Rie Tie Tie tr c( ) givig i( ω φ) 0 t r d t r or ω d t r φ 0 Thu ω t d r φ t ζ ζ t ζ ζ givig t r π β ω d 6

101 Meure of of triet perforce We We hve hve c ( t) e ζωt ζ i( ω t d φ) ω d ω ζ φ t ζ ζ Pek Pek Tie Tie t p dc( t) dt t t p (iω t d p ) ω ζ e ζω t p 0 givig i 0 ω or dt ω t 0, π, π, 3π, p d p t p π ω Therefore for the firt pek. d 7

102 8 We hve We hve ) i( ) ( φ ω ζ ζω t e t c d t ζ ζ φ t Mxiu Overhoot Mxiu Overhoot ζ ω ω d M p ) ( p p t c M ] ) / ( i[ ) / ( φ ω π ω ζ ω π ζω d d d e ) i( ) / ( φ π ζ π ζ ζ e A ) i( ζ φ π Therefore π ζ ζ ) / ( e M p Meure of triet perforce Meure of triet perforce

103 Meure of of triet perforce We We hve hve c ( t) e ζωt ζ i( ω t d φ) ω d ω ζ φ t ζ ζ Settlig Settlig Tie Tie t ζω The curve ± ( e t / ζ ) give the evelope curve of the triet repoe. t i foud to be pproxitely t 4T t 3T (% criterio) (5% criterio) where tie cott T ζω 9

104 Ed 30

105 Higher-Order Syte Repoe Cocept of Syte Stbility Suppleetry redig:

106 Higher-Order Syte repoe Higher-Order Syte repoe Coider the cloed loop trfer fuctio Coider the cloed loop trfer fuctio ) ( ) ( ) ( ) ( ) ( D N G R C c c c b b b b > with 0 0 L L The yte i id to be higher-order yte for >. There will be pole of G c (), or root of the deoitor D c (). Thee root c be either rel or coplex. If they re coplex, they will occur i coplex cojugte pir. G c () c thu be writte ) ( ) ( ) ( R C G c ) ( ) )( )( ( ) )( ( ) ( ) )( ( r r r q p p p z z z ω ω ζ ω ω ζ ω ω ζ L L L ) ( r q where

107 3 Higher-Order Syte repoe Higher-Order Syte repoe For uit tep iput, we c re-write C() i ter of prtil frctio q j r k k k k k k k k k j j c b p C ) ( ) ( ω ω ζ ζ ω ζω i which we ue tht ll the pole re ditict, i.e. ot repeted. The tie repoe will the be, by uig the Ivere Lplce Trfor r k k k k j t t p j t e e t c k k j ) i( ) ( φ ω ζ ζ ω ζ where k k k ζ ζ φ t For tble repoe, the pole ut ll hve egtive rel prt. The repoe of tble higher-order yte thu coprie u of uber of decyig expoetil curve d decyig dped iuoidl curve.

108 Higher-Order Syte repoe Plot of of pole o o the the -ple Rel Rel Pole Pole d d their their effect effect o o the the repoe I Ech rel pole will cotribute ter ito the repoe. j e p j t p p Re Rel pole o the S-Ple. The ore egtive the pole, or the frther wy fro the Igiry xi it i, the ore rpidly the expoetil ter decy to zero. I geerl, if two pole re uch tht p > 5 p, the the repoe cued by p i doit d tht for p c be eglected without lo of ccurcy. 4

109 Higher-Order Syte repoe Plot of of pole o o the the -ple Coplex cojugte pole pole d d their their effect effect o o the the repoe I ζ k ω k Ech coplex pir cotribute decyig dped iuoidl ter to the repoe. Lie of cott ζ ζω k φ The ore egtive the rel prt ζω, k k or the frther wy the pole re fro Re the Igiry xi, the ore rpidly the ter decy to zero. Coplex cojugte pole o the S-Ple. φ k The gle the pole ke with the Rel Axi deterie the dpig rtio, the greter the gle, the le the dpig rtio. φ t k ζ k ζ k 5

110 Soe typicl repoe Stble Stble yte 6

111 Soe typicl repoe Stble Stble yte 7

112 Soe typicl repoe A A Utble yte 8

113 Higher-Order Syte repoe Doit Pole Coplex cojugte pole pole d d their their effect effect o o the the repoe Cloed-loop pole o the S-Ple. The reltive doice of cloedloop pole i deteried by how fr they re fro the Igiry Axi, uig tht there re o zero erby. (Zero ffect the reltive gitude of the cott ter ocited with the pole, the cloer they re the ore the effect.) Uully the repoe will be djuted uch tht oe pir of coplex cojugte pole will be cloer to the Igiry Axi reltive to ll the other pole d the repoe cued by thi pir doite the overll repoe. Thi pir i clled the doit cloed-loop pole. 9

114 Cocept of of Syte Stbility A yte (lier or o-lier) i id to be BIBO (bouded iput, bouded output) tble if, for every bouded iput, the output i bouded for ll tie. A LTI (Lier Tie-Ivrit) yte ut hve ll pole i the left-hlf of the -ple (egtive rel prt) for it to be tble. I other word, the root of the chrcteritic equtio ut ll hve egtive rel prt. If pole, or pole, lie o the igiry xi, the yte i criticlly, or liitedly, tble. If lier yte i utble, eve i the bece of y iput, the output will grow without boud d becoe ifiitely lrge tie goe to ifiity. 0

115 Routh Stbility Criterio A yte i tble if ll the root of the yte chrcteritic equtio hve egtive rel prt. G( ) H ( ) 0 The proble i if the chrcteritic equtio i of order higher th two, it i ot ey to fid the root. (Of coure, there re coputer progr, e.g. MATLAB or OCTAVE, tht help with thi.) Fortutely, there i iple criterio, kow Routh Stbility Criterio (oetie lo kow the Routh-Hurwitz Stbility Criterio), which eble u to fid out the uber of root of the chrcteritic equtio tht lie o the right-hlf of the -ple, i.e. hve poitive rel prt, without hvig to fctor the chrcteritic polyoil.

116 Routh Stbility Criterio Procedure ) For the chrcteritic equtio 0 S S S S 0 > 0 We ue tht 0 ; i.e. y zero root hve bee reoved Exple: or ( ) 0 ue the equtio ) If y of the coefficiet i egtive or zero, the yte i ot tble. 3) If ll the coefficiet re poitive, there i till o gurtee tht ll the root hve egtive rel prt. We the for the Routh Arry d ue the Routh Criterio to deterie the uber of root with poitive rel prt.

117 f e e c c c b b b b L L L Routh Arry Routh Arry Routh Stbility Criterio Routh Stbility Criterio Chrcteritic equtio > S S S S > S S S S b 3 0 b b

118 f e e c c c b b b b L L L I developig the rry, etire row c be ultiplied by poitive uber to iplify the proce without ffectig the reult. I developig the rry, etire row c be ultiplied by poitive uber to iplify the proce without ffectig the reult. Routh Stbility Criterio Routh Stbility Criterio Siilrly Siilrly Routh Arry Routh Arry 3 b b b c 3 5 b b b c b b b c Routh Criterio tte tht the uber of root with poitive rel prt i equl to the uber of chge i ig of the coefficiet i the firt colu of the rry. Routh Criterio tte tht the uber of root with poitive rel prt i equl to the uber of chge i ig of the coefficiet i the firt colu of the rry. e e d d e f

119 Routh Stbility Criterio Exple Deterie the coditio for the followig equtio to to hve oly root with egtive rel prt Routh RouthArry If If 03 > 0, the there i i o ig chge d there i i o root with poitive rel prt. If If 03 < 0, the there re two ig chge. Therefore there re two root with poitive rel prt. 5

120 Specil Specil ce ce Routh Stbility Criterio A zero occur i the firt colu of y row while the reiig ter re ot zero, or there i o reiig ter. Solutio: The zero ter i replced by ll poitive uber d the rry i proceed ccordigly. Exple Routh RouthArry ε 3 3 If If the the ig ig of of the the coefficiet i i the the row row bove i i the the e tht tht below ( ( i i thi thi ce), the the there re re pir pir of of igiry root. If the ig of the coefficiet i the row bove i differet fro tht below, there i oe ig chge idictig oe root with poitive rel prt. 6

121 Routh Stbility Criterio Specil Specil ce ce If ll the coefficiet, or the oly oe coefficiet, i derived row re zero, it e tht there re root of equl gitude locted yetriclly bout the origi. Exple: The chrcteritic equtio hve fctor uch ( σ )( σ ) or ( jω)( jω). I Re For uch ce, for uxiliry polyoil with the coefficiet of the row bove the ll-zero row d uig the coefficiet of the derivtive of thi polyoil to replce the ll-zero row. 7

122 Specil Specil ce ce Exple Routh Stbility Criterio Note tht becue ot ll the coefficiet re poitive, thi idicte tht there i t let oe root with poitive rel prt. Routh RouthArry Ue thi uxiliry polyoil 4 P( ) 48 P& ( ) New New Routh RouthArry There i oe chge i ig i the firt colu oe root with ve rel prt. 8

123 Ed 9

124 Syte Type d Stedy-Stte Error Suppleetry redig:

125 Syte Type Coider the uity-feedbck yte R E - G C ( T )( Tb ) L( T ) With G( ) (-) N ( T )( T ) L( T ) p The preter N ocited with the ter S N i the deoitor repreet the Type of the yte. Exple: Type 0 if N0, Type if N d o o. I geerl, the higher the type uber, the better the tedy-tte ccurcy of the cloed-loop cotrol yte. However, the higher the yte type, the greter the proble with yte tbility.

126 Stedy-Stte Error Sttic Error Cott The e uity-feedbck yte R E - G C Error Trfer fuctio E R C C R R R G ( G) G G G Thu E( ) R( ) G( ) E R G d the tedy-tte error i e lie( t) e t li E( ) 0 R( ) li 0 G( ) 3

127 Stedy-Stte Error Sttic Error Cott For For uit-tep iput iput e R( ) li 0 G( ) R( ) d e li G( ) 0 G(0) Sttic Poitio Error Cott, p i defied lig( ) G(0) d 0 p e p ( T )( Tb ) L( T ) with G( ) N, ( T )( T ) L( T ) p p for Type 0 yte e for Type or higher yte 0 p e 4

128 Stedy-Stte Error Sttic Error Cott For For uit-rp iput iput e R( ) li 0 G( ) R ( ) d e li li 0 G( ) 0 G( ) Sttic Velocity Error Cott, v i defied v li G( ) 0 d ( T )( Tb ) L( T ) with G( ) N, ( T )( T ) L( T ) v v v 0 p for Type 0 yte for Type yte for Type or higher yte e v e e 0 e 5

129 Stedy-Stte Error Sttic Error Cott For For uit-ccelertio iput iput t r( t) for t 0, R ( ) 3 e d R( ) li 0 G( ) Sttic Accelertio Error Cott, i defied d ( T )( Tb ) L( T ) with G( ) N, ( T )( T ) L( T ) 0 li G( ) 0 p for Type 0 d Type yte for Type yte for Type 3 or higher yte e li G( ) e G( ) e e 0 e li 6

130 Stedy-Stte Error Sury of of tedy- Stte Stte error error Type 0 yte Step Iput r Type yte 0 Rp Iput r t Type yte 0 0 Accel. Iput r t / Type 0 yte hve fiite tedy-tte error for tep iput d cot follow rp iput. Type yte hve zero tedy-tte error for tep iput, fiite error for rp iput, d cot follow ccelertio iput. Type yte re eeded to follow rp iput with zero tedy-tte error. I geerl, the higher the ttic gi of the ope-loop trfer fuctio, G(), the ller the tedy-tte error. However, higher gi orlly led to tbility proble. 7

131 Cooly Ued Cotrol Actio D R E - M G c cotroller G p plt C Proportiol (P) (P) Cotrol pe G c ( ) p p i proportiol gi Thi i the iplet for of cotrol ctio d i ued o it ow or i cojuctio with other cotrol ctio. Itegrl (I) (I) Cotrol i i e dt Gc ( ) i i itegrl gi Itegrl cotrol reduce, or eliite, tedy-tte error. It lo icree the order of the yte d ke it ore proe to itbility. 8

132 Cooly Ued Cotrol Actio Derivtive (D) (D) Cotrol e G ( ) d & c Proportiol-plu-Itegrl (PI) (PI) Cotrol d d i derivtive gi Derivtive cotrol ctio i ki to ticiptory cotrol ctio. It ted to icree the dpig i, d tbility of, the yte. Although i doe ot directly ffect the tedy-tte error, it llow higher proportiol gi to be ued, thereby reducig tedy-tte error. It i ever ued o it ow but lwy with proportiol cotrol. e e dt p i Gc ( ) p Additio of itegrl cotrol ctio to proportiol cotrol reduce tedy-tte error but lo ted to ke the yte ore ocilltory. i 9

133 Cooly Ued Cotrol Actio Proportiol-plu-Derivtive (PD) (PD) Cotrol e e& G ( ) p d c p Additio of derivtive cotrol ctio dd ticiptory ctio d ted to icree yte dpig d tbility. d Proportiol-plu-Itegrl-plu Derivtive (PID) (PID) Cotrol i p e i e dt d e& Gc ( ) p d Becue three gi re ivolved, tuig, or djutig, of the gi i ot ey tk. Soetie G c () i writte i the for G ( ) Ti p Td c T i i Itegrl Tie T d i Derivtive Tie 0

134 Exple R Effect of of Cotrol Actio dc dc otor uder peed feedbck cotrol E - T G c D T C C i output peed R i referece iput T i otor torque D i diturbce/lod torque Proportiol (P) (P) Cotrol with with G c () c p p C R p T p T T p T p with p T The reultig yte i till firt-order d will lwy be tble. The reultig tie cott, T, i uch ller, thu givig uch fter peed of repoe. Stedy-tte error to uit tep i reduced fro /( ) to /( p ) but i till o-zero error ice the yte i Type 0. T p p

135 Effect of of Cotrol Actio dc dc otor uder peed feedbck cotrol Proportiol (P) (P) Cotrol Effect Effect of of Diturbce iput iput R E - T G c D T C D With R0, the block digr becoe - -M T p C C D T p T T For uit tep diturbce C( ) T p c p D ( ) / li C( ) 0 p A tep diturbce thu cue the output c(t) to chge. Therefore peed regultio i ffected.

136 3 Effect of Cotrol Actio dc otor uder peed feedbck cotrol Effect of Cotrol Actio dc otor uder peed feedbck cotrol D R C - E G c T T PI Cotrol PI Cotrol G i p c ) ( With PI cotrol, the ope loop trfer fuctio becoe i p Becoe Type yte: tedy-tte error to tep iput will be zero.. ) ( ) ( T i p i p i p T R C ) ( ) ( ) ( Alo, i p i p T T T / ) ( ω ζω ω p T i T ω i p T ζ with Icreig p icree dpig. Icreig i icree turl frequecy d reduce dpig.

137 Effect of of Cotrol Actio dc dc otor uder peed feedbck cotrol PI PI Cotrol effect effect o o diturbce R E - T G c D T C Gc ( ) p p i i D - - M T G c C C D With C( ) ( T /( T ) ( p ( T ) D ( ) / ) ( p i ) ( T ) ( p i ) i ) c li C( ) 0 0 4

138 Effect of of Cotrol Actio dc dc otor uder poitio feedbck cotrol R E - G c T ( T ) C C i output peed R i referece iput T i otor torque E i poitio error Proportiol (P) (P) Cotrol with with G c () c p p The reultig yte i Type ecod-order yte. Stedy-tte poitio error to tep iput i zero. Chrcteritic Equtio p 0 ( T ) T p 0 ζω ω 0 p With ω d T Icreig p icree ω d thu peed of repoe but reduce dpig. ζ p T 5

139 Effect of of Cotrol Actio dc dc otor uder poitio feedbck cotrol R E - G c T ( T ) C PD PD Cotrol with with Gc ( ) p d Ope-loop trfer i G OL ( ) ( ( T p d ) ) Cloed-loop yte rei Type yte with zero tedytte error to tep iput. Chrcteritic Equtio G OL ( ) 0 ( T ) ( ) 0 p Thu ω d T d p p d 0 T T ζω T d ζ p d T Additio of derivtive cotrol llow dpig to be icreed without ffectig yte type d turl frequecy. 6

140 Ed 7

141 Root Locu Alyi Suppleetry redig:

142 Root Locu Alyi E Coider the cloed-loop yte R G - B H C The triet repoe, d tbility, of the cloed-loop yte i deteried by the vlue of the root of the chrcteritic equtio G( ) H ( ) 0 or, i other word, the loctio of the cloed-loop pole o the ple. The ope-loop trfer fuctio c be writte i the for G( ) H ( ) where i djutble gi, the z d p re the zero d pole of the ope-loop trfer fuctio. A the gi chge, the vlue of the cloed-loop pole will chge d thu the triet repoe, d tbility. The root locu plot i plot of the loci of the cloed-loop pole o the -ple the gi vrie fro 0 to ifiity. ( z ( p )( )( z p ) L( ) L( z p ) )

143 Root Locu Alyi Exple Coider the yte with G( ) H ( ) ( )( ) Chrcteritic Eq: ( )( ) 0 Root Locu Plot Pole Pole v v P P P j j j j j j j j j j j j j j pole, thu 3 loci

144 Plottig the the Root Loci The The root loci loci re re plotted either Mully, or or Uig coputer progr uch MATLAB (ey (ey if if you you hve hve the the progr d d kow kow how how to to ue ue it.) it.) 4

145 Mul plottig Root Locu Cocept The The Chrcteritic Equtio i i writte i i the the for for G( ) H( ) F( ) where i i cott gi. gi. The root of the chrcteritic equtio re the vlue of which tify the equtio F( ) 80 for 0 ±, ± 3, ± 5, Or whe F( ) 80 ±, ± 3, ± 5, The gitude coditio i tified by hvig F( ) givig F( ) 5

146 Mul plottig Root Locu Cocept Deteriig the the phe phe gle gle for for F() F() Coider G( ) H ( ) F( ) ( p )( ( z p )( ) p 3 )( p 4 ) The where F Note tht θ θ ± 360 ( 3 4 ) φ θ θ θ θ 80 φ θ θ θ θ z ( ) p ( ) p ( ) 3 p ( ) 3 4 p ( 4 ),,3, (z ) ±, ± 3, ± 5, (p ) φ -z -p θ I 0 Re 6

147 Mul plottig Procedure d d Guidelie ) ) Locte the the pole pole d d zero zero of of the the ope-loop trfer, G()H(), fuctio o o the the ple. ) ) There There re re y y loci, loci, or or brche, pole pole of of the the G()H(). 3) 3) Ech Ech brch trt trt fro fro pole pole of of G()H() d d ed ed i i zero. zero. If If there there re re o o zero zero i i the the fiite fiite regio, the the the the zero zero re re t t ifiity. N( ) Reo: G( ) H ( ) 0 D( ) N( ) 0 D( ) D( ) 0 Whe 0, d root re root of D(). Whe >>, N ( ) 0 d root re root of N(). 7

148 Mul plottig Procedure d d Guidelie 4) 4) The The loci loci exit exit o o the the rel rel xi xi oly oly to to the the left left of of odd odd uber of of pole pole d/or zero. Coplex pole pole d/or zero zero hve hve o o effect effect becue, for for poit poit o o the the rel rel xi, xi, the the gle ivolved re re equl equl d d oppoite. Coider tet poit,, o I the rel xi how. Every Every pir pir of of coplex cojugte pole pole (or (or zero) will will cotribute pir pir of of gle, θ d d θ uch uch tht tht θ θ 360 They They c c thu thu be be igored. -z -3 θ -p -p - θ - θ 3 -p 3 j j 0 -j -j Re 8

149 Mul plottig Procedure d d Guidelie 4) 4) The The loci loci exit exit o o the the rel rel xi xi oly oly to to the the left left of of odd odd uber of of pole pole d/or zero. Coplex pole pole d/or zero zero hve hve o o effect effect becue, for for poit poit o o the the rel rel xi, xi, the the gle ivolved re re equl equl d d oppoite. Coider tet poit,, o I the rel xi how. Ech Ech rel rel pole pole or or zero zero to to the the left left of of poit poit doe doe ot ot cotribute to to the the gle gle u u d d thu thu c c be be igored. Ech pole/zero to the right of poit cotribute gle of ±80. A odd uber of the will thu cotribute totl of where ±, ± 3, ± 80 5, -z -3 θ -p -p - θ - θ 3 -p 3 j j 0 -j -j Re 9

150 Mul plottig Procedure d d Guidelie 5) 5) Becue coplex root root ut ut occur occur i i cojugte pir, pir, i.e. i.e. yetricl bout bout the the rel rel xi, xi, the the root-locu plot plot i i yetricl bout bout the the rel rel xi. xi. 6) Loci which terite t ifiity pproch yptote i doig o. For lrge vlue of, ech pole/zero cotribute equl phe gle. Thu ( Z P) θ 80 for ±, ± 3, ± 5, -p θ -z -p θ θ I 0 Re Z/PNo. of zero/pole θ θ 80 Z P -p 3 0

151 Mul plottig Procedure d d Guidelie 7) All the yptote trt fro poit o the rel xi with coordite I σ pi P Z Exple: For z i j j G( ) H ( ) ( )( ) Re The θ 60, 80, 300 Z P 0 3 -j -j σ 3 0 3

152 Mul plottig Procedure d d Guidelie 8) Brek-i d brekwy poit (o rel xi) At brek-i poit, vlue of icree the loci ove oto the rel xi d wy fro the brek-i poit. At brekwy poit, vlue of icree log the rel xi fro both ide d rech xiu t the brekwy poit. Alog the rel xi,. σ Thu, t the brek-i or brekwy poit, d d σ provided >0 d exit o the root loci. σ 0 Brek-i pt. Brekwy pt.

153 Mul plottig Procedure d d Guidelie 9) Igiry Axi Croig Two pproche: ) Ue Routh Criteri to deterie the vlue of t which the yte i criticlly tble. Thi i idicted by there zero i the firt colu but with o ig chge i the firt colu of the Routh Arry. b) Sice the root re o the igiry xi, by lettig jω i the chrcteritic equtio d olvig for ω d. Thi i doe by equtig both the rel d igiry prt of the chrcteritic equtio to zero. I Axi Croig 3

154 Mul plottig Procedure d d Guidelie 0) Agle of Deprture fro coplex pole d Agle of Arrivl fro coplex zero. Agle of Deprture Thi i doe by tkig tet poit very cloe to the coplex pole, or zero, d pplyig the gulr criteri. 4

155 Mul plottig Procedure d d Guidelie 0) Agle of Deprture fro coplex pole d Agle of Arrivl fro coplex zero. θ -p I Thi i doe by tkig tet poit very cloe to the coplex pole, or zero, d pplyig the gulr criteri. Exple, for digr o right, -z φ 0 Re φ θ ) ± 80 ( θ θ -p 5

156 Exple Root Locu Plot MATLAB Progr >> ghtf([],[ 0]) Trfer fuctio: ^3 ^ >> rlocu(gh) Pole t 0, ± j 6

157 Exple Root Locu Plot MATLAB Progr >> hzpk([-3-4],[0 -],[]) Zero/pole/gi: (3) (4) () >> rlocu(h) >> Pole t 0, - Zero t -3, -4 7

158 Exple Root Locu Plot MATLAB Progr >> gtf([],[ 5]) Trfer fuctio: ^ 5 >> gzpk([-],[0 -],[]) Zero/pole/gi: () () >> ghg*g Zero/pole/gi: () () (^ 5) >> rlocu(gh) Pole Pole t t Zero Zero t t 0,, ± j 8

159 Exple Root Locu Plot MATLAB Progr >> gtf([],[ 5]) Trfer fuctio: ^ 5 >> gzpk([],[0-4],[]) Zero/pole/gi: (4) >> ghg*g Zero/pole/gi: (4) (^ 5) >> rlocu(gh) Pole Pole t t 0, 4, ± j 9

160 Ed 0

161 Root Locu Alyi Exple

162 Root Locu Plottig Coider yte - ( )( ) ) ) Locte Locte pole pole d d zero zero I ) ) 3 pole pole give give 3 brche j 3) 3) Ech Ech brch trt trt fro fro pole pole d d ed ed t t zero. zero. 3 zero zero t t ifiity. (0 )(0 )(0 ) ( )( ) ( )( ) j 0 -j Re -j

163 Root Locu Plottig Coider yte - ( )( ) 4) 4) O O the the rel rel xi, xi, loci loci exit exit oly oly to to the the left left of of odd odd uber of of pole/zero. 5) 5) Root-locu plot plot will will be be yetricl bout bout the the rel rel xi. xi. I j j 6) 6) Ayptote re re t t gle gle 80 ± 60 or ± 80 or ± ) 7) Ayptote trt trt fro fro poit poit o o the the rel rel xi xi with with 3 0 σ j -j Re 3

164 Root Locu Plottig Coider yte - ( )( ) 8) 8) Brekqwy poit: poit: We hve 0 ( )( ) ( )( ) 0 3 ( 3 ) d d givig 0.46 or Brekwy pt. S I j j 0 Re Sice there i o loci t (correpodig to -8.3<0), the brekwy poit i t j -j 4

165 Root Locu Plottig Coider yte - ( )( ) 9) 9) Igiry Axi Axi Croig (): (): We hve givig Auxiliry Polyoil Croig t S I j j 0 Re 3 0 ± j -j -j 5

166 Root Locu Plottig Coider yte - ( )( ) 9) 9) Igiry Axi Axi Croig (b): (b): We hve Lettig jω 3 we hve ( jω) 3( jω) jω 0 3 or ( 3ω ) j(ω ω ) 0 Croig t S.44 I j j Thu ω ω 3 0 givig ± j Re d 3ω 0 givig 6 -j -j 6

167 Root Locu Plottig Coider yte - ( )( ) MATLAB Progr cler cler ll ll hzpk([],[0 - --],[]) rlocu(h) ed ed 7

168 Ed 8

169 Grphic Approch to deteriig the vlue of fro the Root Locu Coider the chrcteritic equtio G ( ) H ( ) F( ) 0 () If F() i writte i fctor of zero d pole, the we hve ( z)( z ) L( z ) 0 ( p )( p ) L( p ) Fro Equtio (), we hve F() / 80 eig gitude of d phe gle of 80 Equtio (3) i equivlet to the followig two coditio beig et ice h o phe gle. i) Phe gle of F() i -80 ii) Mgitude of i give by F( ) (4) The root locu coti ll the root of the chrcteritic equtio the preter vrie fro 0 to ifiity. Thi e tht y poit o the root locu will hve vlue of which tify Equtio (), (), (3) d (4). We c, therefore, ke ue of Equtio (4) to deterie the vlue of if we c fid the vlue of () F for y poit o the root locu plot. For thi, we c write () (3) F( ) z p z p L z L p For y poit o the -ple, the vector, r, where r i zero or pole, i vector trtig fro the poit r to the poit o the -ple. Refer to Slide 6 of the lecture preettio o Root Locu Alyi for illutrtio. Oce the vector i kow o the -ple, it legth c be deteried geoetriclly.

170 Exple (P7. i Dorf d Bihop): 0 ( 0) Coider the chrcteritic equtio give by v 0 (5) ( )( 00) The root locu plot i how i the figure o the ext pge. Here there i oe zero t -0 d three pole t 0, -, d -00. Thee re poited to by blue rrow i the figure. Suppoe we wih to fid vlue of v which ke the dpig rtio for the coplex pir of pole be 0.6. A lie for ζ 0. 6 i drw d where thi cut the root locu plot, t three poit, will be the three olutio eetig thi coditio. To deterie the vlue of v, tke for exple the olutio t the poit P. If ccurte plot i vilble, we c red the vlue of 7.5 ± j0. The vector for the igle zero d the three pole, 00,, d 00 re how d poited to by red rrow. (Refer to Slide 6 of Lecture Preettio for illutrtio of the vector.) The vlue of v t the poit P c the be obtied fro obtiig the legth of thee vector d will be give by 00 v d will work out to be v Note tht there re three olutio give by the coditio tht the dpig rtio for the coplex pir of pole beig 0.6. Thee three olutio re i) v.8; 99.9, 0.54 ± , 3 j ii) v 35; 85.9,, ± j0 iii) v 648;.7,, ± j59. 5 Note tht oly for ce (i) d (ii) bove will the pir of coplex pole be doit ice the pole t i the uch further to the left of the igiry xi. For ce (iii), the pole t will doite the repoe ice it i coprtively uch cloer to the igiry xi th the pir of coplex pole. The reultig repoe will the ot be ocilltory but will be like firt order repoe.

171 40 Root Locu 30 ζ Igiry Axi Rel Axi

172 Itllig OCTAVE i WINDOWS Ad Uig Octve for Cotrol Syte Alyi Prepred by: Xi Xuecheg Poo Au Neow Februry, 006 Deprtet of Mechicl Egieerig/Bchelor of Techology Progre Ntiol Uiverity of Sigpore

173 . Itroductio to GNU Octve GNU Octve i high-level lguge otly coptible with MATLAB. It i pririly iteded for uericl coputtio, for olvig coo uericl lier lgebr proble, ipultig polyoil, d itegrtig ordiry differetil d differetil-lgebric equtio. It lo h exteive tool for vriou fuctio icludig cotrol yte toolbox. The relly ice thig bout GNU Octve i tht it i freely reditributble oftwre. You y reditribute it d/or odify it uder the ter of the GNU Geerl Public Licee (GPL) publihed by the Free Softwre Foudtio. O-lie docuettio o the uge of Octve c be foud t Octve i pririly deiged for Liux opertig yte d the ltet verio of Octve lwy pper firt for Liux. Although there re oe verio of Octve for Widow which re directly itllble, their verio lg behid their Liux couterprt. I the Widow eviroet, Octve i ru uder Cygwi, Liux-like eviroet for Widow. To itll Octve, we firt dowlod the itller for Cygwi, which, whe executed, will itll Cygwi together with vriou pckge pecified, icludig the Octve pckge.. Step for itllig Cygwi with Octve uder Widow. ) Dowlod the Cygwi itller fro by clickig o the Itll or Updte ow ico. You c either ve it o dik firt or ru (ope) it. If it i ved o dik, reeber where it i ved, go to the directory d ru the cygwi etup.exe progr. ) Follow the tep (how below i creehot) to itll the Cygwi eviroet together with Octve pckge. Deprtet of Mechicl Egieerig/Bchelor of Techology Progre Ntiol Uiverity of Sigpore

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