ME2142/ME2142E Feedback Control Systems First half: Professor POO Aun Neow Second half: Professor V Subramaniam

Transcription

1 ME4/ME4E Feedbck Cotrol Sytem Firt hlf: Profeor POO Au Neow Secod hlf: Profeor V Subrmim Referece Text Cotrol Sytem Egieerig 5 th Editio by Nom S Nie. Joh Wiley & So RefereceText: Moder Cotrol Egieerig by Ogt. Pretice Hll Webite for thi portio: ME4/TM34 Feedbck Cotrol Sytem

2 Itroductio d Bic Cocept ME4/TM34 Feedbck Cotrol Sytem

3 Wht i cotrol Sytem? A cotrol ytem i itercoectio of compoet tht will provide deired ytem repoe or output repoe. The tudy of cotrol ytem i the tudy of dymic ytem. A ttic ytem eed o cotrol. Exmple of cotrolled output: temperture, humidity, poitio, peed, preure, directio, liquid level, ltitude. Ad lo: ugr level i hum, ifltio, iteret rte. ME4/TM34 Feedbck Cotrol Sytem 3

4 Wht i cotrol Sytem? I order for ytem to be cotrollble, there mut be cue-effect reltiohip for it compoet, i.e. there mut be ome iput tht c cue chge to the output prmeter to be cotrolled. (Source of eergy) Cotrollig/ctutig iput, Proce or Plt Output (chemicl proce, mchie, idutril proce, ecoomic proce) ME4/TM34 Feedbck Cotrol Sytem 4

5 Ope-loop cotrol Sytem Deired output repoe Cotroller Plt or Proce Output I ope-loop cotrol ytem, o feedbck from the output i ued to cotrol the ytem. Bed o how the output i required, or deired, to repod, the cotroller djut the iput to the plt to chieve thi. Cotrol will oly be ccurte if plt i highly predictble d there i o iterl or exterl diturbce. Geerlly ued oly whe good cotrol performce i ot required. Exmple: A electric bred toter. Temperture cotrol of imple wter heter for the hower. ME4/TM34 Feedbck Cotrol Sytem 5

6 Cloed-loop feedbck cotrol Sytem R - E Cotroller U Plt Y Seor Feedbck R Set-poit or Referece Iput U Plt iput E Error Y Cotrolled Vrible The eor meure the ctul vlue of the output, Y, compre thi with the deired vlue, R, d compute the error, E. Bed o thi error E, the cotroller geerte the iput, U, to the plt o to brig Y to the deired vlue R. ME4/TM34 Feedbck Cotrol Sytem 6

7 Cloed-loop feedbck cotrol Sytem R - E Cotroller U Plt Y Seor Feedbck Geerlly ued whe good cotrol performce i required. Accurte cotrol c be chieved eve i the preece of plt vritio, d iterl or exterl diturbce becue uch diturbce will ffect the output Y, reflected i the error E, t thu will cue the plt iput U to chge o to correct for thee diturbce. C become utble. Stbility become importt coidertio. ME4/TM34 Feedbck Cotrol Sytem 7

8 Some exmple of cotrol Sytem? ME4/TM34 Feedbck Cotrol Sytem 8

9 Exmple: Ope-loop v Cloed Loop Proce of Wlkig: Deired output: poit where you wt to be. Cotroller: the bri Plt or proce: the leg Ope-loop cotrol: Wlkig with your eye cloed. A wlkig m Cloed-loop feedbck cotrol: Wlkig with your eye ope. The eye eed the ctul output, where you re d where you re hedig, compute the error i poitio d i directio, d iue commd to the plt, meig the leg, to move i uch wy o to reduce the error. ME4/TM34 Feedbck Cotrol Sytem 9

10 Exmple: Ope-loop v Cloed Loop Photo courtey U.S. Air Force Droppig Bomb: Objective of droppig bomb from height i to hit trget below. Deired output: Trget below Plt or proce: the bomb with it cotrol fi Ope-loop Cotrol or dumb bomb The cotroller, meig the pilot or bombrdier, eed to etimte hi ow height, velocity, ditce to trget, wid coditio, d chrcteritic of bomb to decide whe d where to relee the bomb. Ofte, hudred of bomb re eeded to hit pecific trget. ME4/TM34 Feedbck Cotrol Sytem 0

11 Exmple: Ope-loop v Cloed Loop Droppig Bomb: Objective of droppig bomb from height i to hit trget below. Deired output: Trget below Plt or proce: the bomb with it cotrol fi Cloed-loop Cotrol or mrt bomb Seor re icorported ito the bomb to give feedbck o it ctul poitio reltive to the trget. The error iformtio i the ued to teer the bomb, uig it cotrol fi, to the trget. Reult: oe trget oly eed oe bomb. Seor: TV, Ifrred, ler guided, or GPS. See lo: ME4/TM34 Feedbck Cotrol Sytem

12 Study of cotrol Sytem Study of cotrol ytem i the tudy of the dymic of the ytem. The repoe of the cotrolled vrible Y to y iput R deped upo the dymic of the Plt, Cotroller, d the Seor or Feedbck. R - E U Cotroller Seor Feedbck Plt Y Give cotrol ytem, y f(r,t) meig tht y i ot oly fuctio of r, but lo vrie with time t. If y f(r) the the ytem i ot dymic ytem but i ttic. To mthemticlly decribe the dymic behvior of the cotrol ytem d it compoet, differetil equtio re ued. ME4/TM34 Feedbck Cotrol Sytem

13 Lier d No-lier Sytem A ytem i lier if it tify the propertie of uperpoitio d homogeeity/clig. A ytem i o-lier if it i ot lier. Coider ytem which h the repoe to y two rbitrry iput u (t) d u (t) y (t) f(u (t)) d y (t) f(u (t)) Property of Superpoitio i tified if the output for combied iput of u (t) d u (t) i y 3 f(u (t) u (t)) y (t) y (t) Property of homogeeity i tified if y 3 f(u (t)) y (t) ME4/TM34 Feedbck Cotrol Sytem 3

14 Lier d No-lier Sytem Coider ytem which h the repoe to y two rbitrry iput u (t) d u (t) y (t) f(u (t))) d y (t) f(u (t)) A ytem i lier if the propertie of uperpoitio d homogeeity re tified. The bove ytem will be lier if the followig i tified y 3 f( u (t) u (t)) y (t) y (t) I geerl, rel phyicl ytem re o-lier if the opertig rge i very lrge, However, if opertio i coidered oly bout ome opertig poit, d the rge of opertio i ufficietly mll, mot ytem c be coidered to be lier. ME4/TM34 Feedbck Cotrol Sytem 4

15 Lier d No-lier Sytem Exmple: Which of the followig ytem re lier? (i) F x (ii) y x (iii) y mx b For y cott A d B d y two iput x d x, (i) F f(x ) x d F f(x ) x Alo, F 3 f(ax Bx ) (Ax Bx ) Ax Bx AF BF Thu propertie of uperpoitio d homogeeity i met. Thu lier. y f( x) x (ii) Ad f( Ax) ( Ax) Ay Alo, f( x x ) ( x x ) f( x ) f( x ) x x Thu ytem i ot lier or o-lier. y f( x) mx b ME4/TM34 Feedbck Cotrol Sytem (ot homogeou) (uperpoitio violted) (iii) Ad f ( Ax) max ( ) b Af( x) Amx ( b) (ot homogeou) Sytem i ot lier. C be how tht uperpoitio lo violted. 5

16 Lier Approximtio of Sytem Ay o-lier ytem c be lieried bout ome opertig poit d c be coidered to be lier withi mll opertig regio bout tht poit For y fuctio y f( x) with We c ue the Tylor Serie expio bout ome opertig poit, x 0, d hve df ( x x ) d f ( x x ) y f( x ) dx x x! dx! x x 0 0 For mll vritio bout the opertig poit, ecod d higher-order term i ( x x ) c be eglected. The y m x y y mx ( x ) 0 0 or 0 y f( x ) 0 0 which i lier. ME4/TM34 Feedbck Cotrol Sytem 6

17 Lier Approximtio of Sytem Exmple: θ For the pedulum how i the figure, the retorig torque due to grvity i give by T MgL iθ Derive the lieried equtio bout the opertig poit. Solutio: Sice with θ 0, 0 θ 0 d iθ T T MgL ( θ θ ) MgL co(0)( θ θ ) dθ θ 0 T T 0 0 MgLθ ME4/TM34 Feedbck Cotrol Sytem 7

18 Ed ME4/TM34 Feedbck Cotrol Sytem 8

19 ME4/ME4E Feedbck Cotrol Sytem Itllig OCTAVE i WINDOWS Ad Uig OCTAVE for Cotrol Sytem Alyi Jury 009 Deprtmet of Mechicl Egieerig, d Bchelor of Techology Progrmme Ntiol Uiverity of Sigpore

20 . Itroductio to GNU Octve GNU Octve ( i high-level lguge motly comptible with MATLAB ( It i primrily iteded for umericl computtio, for olvig commo umericl lier lgebr problem, mipultig polyomil, d itegrtig ordiry differetil d differetil-lgebric equtio. It lo h fcilitie for diplyig the reult of computtio i vriou form of grph which mke it relly ice. The relly ice thig bout GNU Octve i tht it i freely reditributble oftwre. You my reditribute it d/or modify it uder the term of the GNU Geerl Public Licee (GPL publihed by the Free Softwre Foudtio ( O-lie documettio o the uge of Octve c be foud t OCTAVE for Cotrol Sytem Study Similr to MATLAB, OCTAVE lo h exteive tool for vriou fuctio icludig cotrol ytem toolbox d it i thee tht we will primrily be itereted i i thi firt itroductory coure o cotrol ytem. 3. Dowlodig, Itllig d Uig OCTAVE Iformtio o dowlodig, itllig d uig OCTAVE c ll be foud from OCTAVE webite t A copy of the ltet pre-built itller for OCTAVE for both Widow d Mc OS X c be foud vi the mi OCTAVE ite or t For thoe uig Widow, do the followig to itll copy of Octve 3.0.3:. dowlod d ru the itller file octve etup.exe. Dowlodig could tke few to te of miute depedig upo your coectio peed.. Durig itlltio, click ext t the followig cree

21 3

22 For the ext cree, chooe Guplot d click ext. Click ext o the ubequet cree d fiih t the lt cree to complete the itlltio. 3. Tet tht the itlltio i ruig properly by luchig OCTAVE START>All Progrm>GNU Octve>Octve to ope Octve widow. Note tht you c lo view Octve documettio. 4. I the Octve widow, t the commd lie, type (uderlied below) Octve exe:> ytf(,[,,4]); Octve exe:> tep(y); The figure below, which w copied to clipbord d pted here, hould the pper of tep repoe for uderdmped d-order ytem. 5. Cogrtultio, you hve uccefully itlled Octve Verio

23 6. There re few demotrtio progrm for cotrol i the Octve uite: DEMOCotrol: i demo/tutoril progrm for vriou cotrol toolbox fuctio. AlDemo: Stte-Spce Alyi demo. FRDemo: Frequecy Repoe demo. ModDemo: Model Mipultio demo. RLDemo: Root Locu demo. You c ru thee by typig the demo/tutoril me t the commd lie. 4 Some Note o Uig Octve 4. Cotrol Sytem Specifictio: The cotrol ytem uder tudy c be pecified i everl wy. () By uig vector to repreet the coefficiet of the umertor d deomitor polyomil of the cloed-loop trfer fuctio. For exmple, to pecify the ytem with the cloed-loop trfer fuctio?????????? The followig commd c be ued: Octve exe:3> um[ 3]; Octve exe:4> de[ 4]; Octve exe:5> ytf(um,de); or, ltertively, the followig: Octve exe:6> ytf([ 3],[ 4]); i which [ 3] pecifie the polyomil????? d [ 4] pecifie the polyomil?????????. To diply the dt tructure of the ytem, the yout commd c be ued : Octve exe:7> yout(y); Octve exe:8> yout(y); (b) By uig vector to pecify the zero d pole of the cloed-loop ytem: For exmple, to defie the ytem with the cloed-loop trfer fuctio????????????????? we c ue the commd: octve exe:9> um[-3]; octve exe:0> de[0 - -6]; octve exe:> k5; octve exe:> y3zp(um,de,k); or,ltertively, the followig: octve exe:3> y4zp([-3],[0 - -6],5); To diply the dt tructure of the ytem, the yout commd c be ued : octve exe:4> yout(y3); octve exe:5> yout(y4); 4. Triet Repoe: The uit impule repoe d the uit tep repoe of ytem c be eily obtied by uig the impule fuctio d the tep fuctio repectively. Thee fuctio produce plot of the triet repoe. For exmple: octve exe:6> impule(y); 5

24 octve exe:7> tep(y4); 4.3 Root Locu Plot: Root locu plot c eily be obtied uig the fuctio rlocu(y,[ic, mi_k, mx_k]), or where y i the ope-loop trfer for which the root locu plot i deired, ic i the icremet i the gi prmeter, d mi_k d mx_k re the miimum d mximum vlue of the gi prmeter for the plot. For exmple, if the root locu plot i deired of the ytem med y4 pecified i Sectio 4. bove, the followig commd c be ued: octve exe:8> rlocu(y4); octve exe:9> rlocu(y4,0.0,0,5); 6

25 ME4/ME4E Feedbck Cotrol Sytem Review of Lplce Trform ME4/TM34 Feedbck Cotrol Sytem

26 Mthemticl prelimirie Complex vrible σ jω j Im Rel prt Imgiry prt ω σ Re -ple ME4/TM34 Feedbck Cotrol Sytem

27 Complex fuctio Complex fuctio G ( ) G x jg y Im Rel prt Imgiry prt G θ t G x G y G G G ( ) G e y x jθ G G() y G θ G x Complex cojugte G()-ple G( ) G x jg y ME4/TM34 Feedbck Cotrol Sytem 3

28 Euler Theorem Euler Theorem θ e j coθ j i θ It Complex cojugte θ e j coθ j i θ Some ueful formul: coθ i θ j ( jθ jθ e e ) ( jθ jθ e e ) ME4/TM34 Feedbck Cotrol Sytem 4

29 Lplce Trform A mthemticl tool tht trform difficult differetil equtio ito imple lgebr problem where olutio c be eily obtied. Defiitio: Ivere Trform: L f { } f t) F( ) e t ( dt ( t) 0 for t < Normlly Tble of Lplce Trform pir re ued for tkig the Lplce Trfrom d the Ivere Trform. 0 L - { ( ) } f ( t) 0 c j F F( ) e t d πj c j ME4/TM34 Feedbck Cotrol Sytem 5

30 Lplce Trform Ed of L ME4/TM34 Feedbck Cotrol Sytem 6

31 Propertie of Lplce Trform Lierity: L { f ( t) b g( t) } L { f ( t) } b L { g(t) } F( ) b G( ) Exmple From Tble L The L - F( ) b G( ) { F ) } b L - L { } L - t d L { i( t) } t 5i(t) 4 { } { } 3 ( { G() } f ( t) b g( t) ME4/TM34 Feedbck Cotrol Sytem 7

32 3 6e {( t ) } 4 Propertie of Lplce Trform Trltio: If f ( t ) t g( t), the { } L g( t) e F( ) 0 t < L { e t f ( t) } F( ) Exmple From Tble L 3 6 { t } 4 3 6e The L ( t ) d L 3 6 { } { e } t t 4 4 ( ) Vrible trform: ME4/TM34 Feedbck Cotrol Sytem L{ f ( t) } F ( ) 8

33 Propertie of Lplce Trform Derivtive: L { f ( t) } ' L { f ( t) } f (0) F( ) f (0 ) { } '' ' ' L f ( t) L{ f ( t) } f (0) f (0) F( ) f (0) f (0) L... { f ( t) } f ( ) f (0) f (0) L ' ( ) f (0) L - L - L - { ' F ( ) } tf ( t) { '' } F ( ) t f ( t) ME4/TM34 Feedbck Cotrol Sytem... { } F ( ) ( ) t f ( t) 9

34 Propertie of Lplce Trform Fil-Vlue Theorem: lim t f ( t) lim 0 F( ) Iitil-Vlue Theorem: f ( 0) lim F ( ) ME4/TM34 Feedbck Cotrol Sytem 0

35 Fidig the Ivere. Uig L - c j F F( ) e t d πj c j { ( ) }. Uig the Tble of Trform Pir 3. Uig propertie 4. Uig Prtil Frctio Expio (We do ot ormlly ue Approch. We ue combitio of to 4.) ME4/TM34 Feedbck Cotrol Sytem

36 Uig Tble d Propertie Exmple Fid the ivere of 5 F ( ) ( 3) 8 6 Solutio: From Tble L - t Lookup from Tble L - 5L - ( 5 3) 8 6 L - L - ( 3) 3t 5te 4 i(4t) 4 Lookup from Tble or Ue Trltio property Lookup from Tble Thu L - 3t { F( ) } t 5te i(4t) ME4/TM34 Feedbck Cotrol Sytem

37 Uig Tble d Propertie Exmple Fid the ivere of F( ) 8 e 5 Solutio: We write F( ) 8 e 5 e 8t ( ) From 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 ME4/TM34 Feedbck Cotrol Sytem 3

38 ME4/TM34 Feedbck Cotrol Sytem 4 Uig Prtil Frctio Uig Prtil Frctio Frequetly N() d D() beig polyomil i Exmple or Eq(-) re the zero d pole of F() repectively. They c either be rel or complex. If they re complex, they lwy occur i cojugte pir. ) ( ) ( ) ( D N F m p p p z z z,,, d,,, m b b b b F m m m m > with ) ( 0 0 L L ) ( ) )( ( ) ( ) )( ( ) ( m p p p z z z F L L

39 ME4/TM34 Feedbck Cotrol Sytem 5 Uig Prtil Frctio Uig Prtil Frctio If Eq(-) h ditict pole, the F() c lwy be expded ito um of prtil frctio: Eq(-) where k re cott. To determie the vlue of k, multiply 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 ) ( ) ( ) (

40 Uig Prtil Frctio Exmple 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 ME4/TM34 Feedbck Cotrol Sytem 6

41 Uig Prtil Frctio If Eq(-) h multiple pole, ech multiple pole p r of order q will be equivlet to prtil frctio of the form: b ( p r ) b ( p r ) L b ( q p r ) q where b k re cott. If Eq(-) h deomitor of the form ( b) ech of thee will be equivlet to prtil frctio of the form: N( ) c d F( ) L L( b) L b L where, b, c d d re cott. ME4/TM34 Feedbck Cotrol Sytem 7

42 Uig Prtil Frctio Give 3 F( ), fid f(t). ( ) Exmple 3 Solutio: Let F( ) 3 3 ( ) b ( ) b ( ) b3 ( ) 3 Multiplyig both ide by () 3 Lettig 3 b ( ) b ( ) b3 we hve b 3 b Comprig term i, we hve Comprig cott term, we hve 3 b b b b 0 3 givig Thu F ( ) ( ) 3 d f ( t) e t t e t ME4/TM34 Feedbck Cotrol Sytem 8

43 Solvig differetil equtio ME4/TM34 Feedbck Cotrol Sytem 9

44 Solvig differetil equtio Exmple Solve for y(t) give && y y& 5y 3, y(0) 0, y& (0) 0 Solutio: Trformig 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 ( ) ME4/TM34 Feedbck Cotrol Sytem 0

45 Solvig differetil equtio Exmple We hve Y( ) ( 3 5) b c 5 Multiplyig both ide by d lettig 0, we get 3 5 Multiplyig both ide by the deomitor ( 5) we hve 3 ( 5) ( b c) Comprig term i, 0 6/ 5 c c 6/ 5 Comprig term i, Thu 3 3 Y ( ) /5 b b 3/ ( ) 3 0 ( ) Ad 3 3 t 3 t y( t) e cot e i t for t ME4/TM34 Feedbck Cotrol Sytem

46 Lplce Trform END ME4/TM34 Feedbck Cotrol Sytem

47 Tble of Lplce Trform Pir ME4/ME4E Jury 008 Deprtmet of Mechicl Egieerig & Bchelor of Techology Progrmme Ntiol Uiverity of Sigpore

48 Lplce Trform Pir f (t) F () Uit impule δ (t) 4 Uit tep u (t) 3 t t ( )!, 5 t, (,, 3, ) 6 t e 7 t 8 t ( )! 9 t t e, e t (,, 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)

49 t t 8 ( e te ) t 9 ( t e ) ( ) ( ) 0 e t i ωt e t co ωt ω ζω t e i ω ζ t ζ, ζ < ζωt 3 e i ( ω ζ t φ) ζ φ t ζ ζ, ζ < ζω t e i ( ω ζ t φ) 4 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 ( coω coω ) ω ω t t 3 ( i ωt ωt coωt) ω, ( ω ) ( ω )( ω ) ω ( ω )

50 ME4/ME4E Feedbck Cotrol Sytem Modellig of Phyicl Sytem The Trfer Fuctio ME4/TM34 Feedbck Cotrol Sytem

51 Differetil Equtio U Plt Y I the plt how, the iput u ffect the repoe of the output y. I geerl, the dymic of thi repoe c be decribed by differetil equtio of the form d y dt m m d y dy d u d u du L 0 y bm bm L b b dt dt dt dt dt 0 u Differetil equtio i lier if coefficiet re cott or fuctio oly of time t. Lier time-ivrit ytem: if coefficiet re cott. Lier time-vryig ytem: if coefficiet re fuctio of time. ME4/TM34 Feedbck Cotrol Sytem

52 Modellig of Phyicl Dymic Sytem Mechicl Sytem Fudmetl Lw Mechicl Sytem Trltiol Sytem Newto Lw x f m f m mx & f i pplied force, m i m i g x i diplcemet i m. ME4/TM34 Feedbck Cotrol Sytem 3

53 Modellig of Phyicl Dymic Sytem Mechicl Sytem Toriol Sytem T J θ ω T J θ & Jω& T i pplied torque, -m J i momet of ierti i g-m θ ω i diplcemet i rdi i the gulr peed i rd/ ME4/TM34 Feedbck Cotrol Sytem 4

54 Modellig of Phyicl Dymic Sytem Mechicl Sytem - prig Trltiol: x x f i teile force i prig, i prig cott, /m f f Importt: Note directio d ig f ( x x ) Rottiol: T re exterl torque pplied o the toriol prig, -m G i toriol prig cott, -m/rd θ θ T G( θ ) θ ME4/TM34 Feedbck Cotrol Sytem 5

55 Modellig of Phyicl Dymic Sytem Mechicl Sytem dmper or dhpot Trltiol:. x. x f f f i teile force i dhpot, b b i coefficiet of dmpig, -/m f b( x& ) x& Rottiol: θ & T i torque i toriol dmper, -m b i coefficiet of toriol dmpig, -m-/rd T b ( θ & & ) θ θ & ME4/TM34 Feedbck Cotrol Sytem 6

56 Modellig of Phyicl Dymic Sytem Exmple Derive the differetil equtio reltig the output diplcemet y to the iput diplcemet x. b A y x Free-body digrm t poit A, f d A f Note: Directio of f d f d how ume they re teile. Sice m 0, f m give f f d 0 Sice Thu Or f d & by d ( x y) f ( x y) by& 0 b y& y x ME4/TM34 Feedbck Cotrol Sytem 7

57 The Trfer Fuctio The trfer fuctio of lier time ivrit ytem i defied the rtio of the Lplce trform of the output (repoe) to the Lplce trform of the iput (ctutig igl), uder the umptio tht ll iitil coditio re zero. Previou Exmple b y& y x Aumig zero coditio d tkig Lplce trform of both ide we hve by ( ) Y ( ) X ( ) Trfer Fuctio G( ) Y ( ) X ( ) b Thi i firt-order ytem. ME4/TM34 Feedbck Cotrol Sytem 8

58 Modellig of Phyicl Dymic Sytem Exmple For the prig-m-dmper ytem how o the right, derive the trfer fuctio betwee the output x o d the iput x i. m x o Free-Body digrm m x o Note: f d f d umed to be teile. b f f d f m give f f d mx& o x i Thu ( x i x ) b( x& x& ) o i o m& x o Or m && x bx& x bx& o o o i x i Ad m X o ( ) bx o( ) X o ( ) bx i ( ) X i ( ) Trfer Fuctio G( ) X X i ( ) o ( ) m b b. Thi i ecod-order ytem. ME4/TM34 Feedbck Cotrol Sytem 9

59 Modellig of Phyicl Dymic Sytem Electricl Elemet Cpcitce e i C Or Complex impedce q C e q Ce dq i dt E I C C I C(E) IX c de dt X c /( C) Reitce e i R Uit of R: ohm ( ) Iductce e i L Uit of L: Hery (H) i e e ir e i R Ω L L di dt t 0 e dt Or E IX L I(L) ME4/TM34 Feedbck Cotrol Sytem 0

60 Modellig of Phyicl Dymic Sytem Electricl Circuit- irchhoff Lw Curret Lw: The um of curret eterig ode i equl to tht levig it. i 0 Voltge Lw: The um lgebric um of voltge drop roud cloed loop i zero. e 0 ME4/TM34 Feedbck Cotrol Sytem

61 Modellig of Phyicl Dymic Sytem Electricl Circuit- Exmple RC circuit: Derive the trfer fuctio for the circuit how, E IR i IX c R d E o IX c e i i C e o givig E E o i X c R X c RC /( C) R /( C) Thi i firt-order trfer fuctio. ME4/TM34 Feedbck Cotrol Sytem

62 Modellig of Phyicl Dymic Sytem Electricl Circuit- Exmple RLC circuit: R L d E IR IX IX i E o IX c L c e i i C e o givig E E o i X R X LC L c X c RC /( C) R L /( C) Thi i ecod-order trfer fuctio. ME4/TM34 Feedbck Cotrol Sytem 3

63 Modellig of Phyicl Dymic Sytem Opertiol Amplifier Propertie of idel Op Amp v o A( v v) Gi A i ormlly very lrge o tht compred with other vlue, ( v ) i umed mll, equl to zero. v The iput impedce of the Op Amp i uully very high (umed ifiity) o tht the curret i d i re very mll, umed zero. Two bic equtio goverig the opertio of the Op Amp ( v v v v) 0 or d i, i 0 0 ME4/TM34 Feedbck Cotrol Sytem 4

64 Modellig of Phyicl Dymic Sytem Opertiol Amplifier Exmple Z f i f i i v i i 0 Z i S - v o For the Op Amp, ume i 0 d v v 0. The i o i i 0 or 0 i f v Z i v Z f Therefore v o Z Z f i v i V o i ( ) V ( ) Z Z f i ME4/TM34 Feedbck Cotrol Sytem 5

65 Modellig of Phyicl Dymic Sytem Opertiol Amplifier Exmple Z f i f i i v i i 0 Z i S - v o For the followig v o Z Z f i v i v v Z Z f R R R f C o f f i R i i R C i p i ME4/TM34 Feedbck Cotrol Sytem 6

66 Permet Mget DC Motor Drivig Lod R L e i ω e T ω J b For the dc motor, the bck emf i proportiol to peed d i give by where e i the voltge cott. The torque produced i proportiol to e ω rmture curret d i give by where i the torque cott. Relevt equtio: T T i t di e R i L dt t dω ti T J b ω dt e ω Note: By coiderig power i power out, c how tht e t ME4/TM34 Feedbck Cotrol Sytem 7

67 Ed ME4/TM34 Feedbck Cotrol Sytem 8

68 ME4/ME4E Feedbck Cotrol Sytem Sytem Triet/Time Repoe ME4/TM34 Feedbck Cotrol Sytem

69 Sytem repoe The ytem repoe comprie two prt, triet d tedy-tte. Triet Repoe Stedy-Stte Repoe Stedy-Stte Error The mgitude of the triet repoe decree with time d ultimtely vihe levig oly the tedy-tte repoe. It i lwy ocited with the compoet e t with > 0 ME4/TM34 Feedbck Cotrol Sytem

70 Sytem Chrcteritic Equtio Coider the ytem with the cloed-loop trfer fuctio, G c () how Iput R() G c () C() Output Sytem with C ( ) N G c ( ) R( ) D ( ) ( ) where N c () d D c () re polyomil of. c c The ytem chrcteritic equtio i give by D c ( ) Note tht the chrcteritic equtio i property of the ytem d i ot depedet o the iput. 0 ME4/TM34 Feedbck Cotrol Sytem 3

71 Sytem Chrcteritic Equtio Exmple Sprig-m-dmper (Slide 9: Modellig of Phyicl Sytem) Trfer Fuctio Chrcteritic Eq: G( ) X ( ) o X i ( ) m m b b b R-C circuit (Slide : Modellig of Phyicl Sytem) E o Trfer Fuctio E RC i 0 Chrcteritic Eq: RC 0 Cloed-loop feedbck ytem (Slide 8: Block Digrm Algebr) C G Trfer Fuctio R GH Chrcteritic Eq: GH 0 ME4/TM34 Feedbck Cotrol Sytem 4

72 Sytem 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 determie the triet repoe of the ytem. Ech root, p, of thi equtio will cotribute term time repoe of the ytem. 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 time icree. The ytem i the id to be utble. e pt i the ME4/TM34 Feedbck Cotrol Sytem 5

73 Sytem Repoe Give dymic ytem: How do we pecify the chrcteritic of the repoe required? How do we compre it with other ytem? How do we kow whether it repoe will dequtely meet our eed? How will we kow how it will repod to differet iput? We ue Stdrd tet iput to excite ytem d oberve repoe Clify ytem with imilr chrcteritic d idetify their performce chrcteritic with ytem prmeter. ME4/TM34 Feedbck Cotrol Sytem 6

74 Sytem Repoe Tet igl ) Step iput r( t) A 0 A R ( ) < 0 Whe A, we hve uit tep iput. Ued to tudy repoe to udde chge i iput. t t 0 r(t) A t 0 t ) Rmp iput r( t) At 0 A R ( ) Whe A, we hve uit rmp iput. Ued to tudy repoe to grdul chge i iput. t t < 0 0 r(t) t 0 At t ME4/TM34 Feedbck Cotrol Sytem 7

75 Sytem Repoe Tet igl 3) Impule iput r ( t) Aδ (0) i the uit-impule fuctio or Dirc delt fuctio R ( ) A Whe A, we hve uit impule iput. r(t) A t 0 Ued to tudy repoe to udde hock or impct. t 4) Siuoidl iput r( t) Ai ωt Ued for frequecy repoe lyi. 0 Importt method. Will be dicu i the ecod hlf of coure. t t < 0 0 r(t) t 0 t Uig tet igl () to (3) re ofte kow time repoe or triet repoe lyi while uig tet igl (4) i kow frequecy repoe. ME4/TM34 Feedbck Cotrol Sytem 8

76 Sytem Repoe Firt-order ytem A firt-order ytem c be writte i the tdrd form C( ) R( ) T T i kow the time cott d determie the peed of repoe. Exmple Sprig-dmper ytem (Slide 7 of Modellig of Phyicl Sytem) Y ( ) with T b X ( ) b. T RC circuit (Slide of Modellig of Phyicl Sytem). E E o i with RC T If the trfer fuctio re the me, the the repoe y(t) d e o (t) will be the me for the me iput i x(t) d e i (t).. T RC ME4/TM34 Feedbck Cotrol Sytem 9

77 Sytem Repoe Firt-order ytem Repoe to uit tep iput. C( ) R( ) T with Thu C( ) R( ) T 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 ME4/TM34 Feedbck Cotrol Sytem 0

78 Sytem Repoe Firt-order ytem Repoe to uit tep iput For c( t) ( e t / T ) Note: The mller the time cott T, the fter the repoe. The hpe i lwy the me. ME4/TM34 Feedbck Cotrol Sytem

79 Sytem Repoe Firt-order ytem Repoe to uit rmp iput. C( ) R( ) T with Thu C( ) R( ) T ( 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) t 0 c(t) r(t) e T t ME4/TM34 Feedbck Cotrol Sytem

80 Sytem Repoe Firt-order ytem Repoe to uit impule iput C( ) R( ) T 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 ME4/TM34 Feedbck Cotrol Sytem 3

81 Sytem Repoe Lier time-ivrit ytem Propertie Repoe to Uit Impule 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 Rmp ( t T Te t / T ) t T The triet repoe ll coti the term e / which i determied by the root of the chrcteritic equtio d the prmeter T. Note tht the uit tep i the derivtive of the uit rmp, d the uit impule i the derivtive of the uit tep. Note tht imilrly, c (t) i the derivtive of c 3 (t) d c (t) i the derivtive of c (t). For lier time-ivrit ytem, the repoe to the derivtive of iput c be obtied by tkig the derivtive of the repoe to the iput. ME4/TM34 Feedbck Cotrol Sytem 4

82 Permet Mget DC Motor Goverig equtio di e R ω i L e 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 Permet Mget DC motor. T ω J b Block Digrm E - L R I t T Ω J b e Ω e E - ( L R )( J b) t Ω e Ω e ME4/TM34 Feedbck Cotrol Sytem 5

83 Permet Mget DC Motor E - e Ω ( L R )( J b ) e The Permet Mget DC motor. Ω ME4/TM34 Feedbck Cotrol Sytem Block digrm the become E - t t / R J b e Ω with Commoly L << R J b L c the be eglected Ω G E GH / R ( J b) t e / R ( J b) τ t b t t / R e / R J b τ t / R b t t J e e 6 / R / R

84 Speed Cotrol of the DC Motor E τ Ω τ b t J e / R The repoe to uit tep iput i firt order with time cott of τ Ω t 0 t With peed feedbck V - Error c Cotroller τ ME4/TM34 Feedbck Cotrol Sytem E Ω c Ω τ c ' V c τ c τ ' τ τ c with τ ' ' c c The reultt ytem i till firt-order but the time cott i ow much mller, thu much fter repoe. 7

85 Sytem Repoe Secod-order ytem A ecod-order ytem will be of the form Y( ) X ( ) d e b c with, b, c, d d e beig cott. Exmple RLC circuit (ee Modellig of Phyicl Sytem) E o ( ). E ( ) LC RC Sprig-m-dmper i X X o i ( ) ( ) m b b Stdrd Form:. C( ) R( ) ω ME4/TM34 Feedbck Cotrol Sytem ζ ω ( ) ω ζω ω 8

86 Cloed-Loop Poitio Feedbck Sytem (Servomechim) R - E G c cotroller V τ Ω θ With G c beig proportiol gi p θ R - E p ( τ ) I tdrd formt Θ R Θ R G GH τ ω ζω ω p p with ω ω ζ p τ ζ turl frequecy dmpig rtio p ζω τ τ ME4/TM34 Feedbck Cotrol Sytem 9

87 Sytem Repoe Secod-order ytem Exmple: Determie the vlue of gi for the cloed-loop ytem to hve udmped turl frequecy of 4. Wht will the be the dmpig fctor? 5 8 5() 8 Chrcteritic equtio: GH ςω ω ω ( 5) ςω 4 ς ME4/TM34 Feedbck Cotrol Sytem 0

88 Time Repoe Secod-order ytem Θ ω Coider R ζω ω The root of the chrcteritic equtio re p, ζω ± ω ζ For 0 < ζ <, the root re pir of complex cojugte p, ζω ± jω d where ω d ω ζ i clled the dmped turl frequecy d the repoe i uderdmped. For ζ, the root re equl p d the ytem i id to be, ω criticlly dmped. For ζ >, the root re both rel d uequl p, ζω ± ω ζ d the ytem i id to be overdmped. ME4/TM34 Feedbck Cotrol Sytem

89 Step Repoe Secod-order ytem Θ R ω ζω ω with R( ) Therefore Θ ( ω ζω ω ) Uderdmped Repoe 0 < ζ < From tble (Etry 4 i Tble), we hve c( t) e ζωt ζ t 0 i( ω d t φ) 0 < φ < π ω d φ t ω ζ ζ ζ Thi repreet decyig ocilltory repoe depedig upo frequecy of ocilltio of ω d ζ with ME4/TM34 Feedbck Cotrol Sytem

90 Step Repoe Secod-order ytem Criticlly dmped Repoe ζ We hve Θ ( ω ζω ω ) ω ( ω ) From tble (Etry 8 i Dr Che tble with givig ( ) c c( t) ( e t ω te t ωt ωt ( t) e ω te for t 0 ), we hve Thi repreet o-ocilltory repoe with expoetilly decyig triet compoet d zero tedy-tte error. The peed of decy of the triet compoet deped upo the prmeter. ) ω ME4/TM34 Feedbck Cotrol Sytem 3

91 Step Repoe Secod-order ytem Overdmped 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 ue Etry 7 i Dr Che tble. o tht for t c( t) ω b ( be b e t bt ) 0, C d C beig cott. We hve t Ce C The repoe i o-ocilltory, trt iitilly with expoetilly rie to c( ). b ω e c( 0) bt 0 d >> b If ζ >>, the d the firt expoetil term will decy much fter th the ecod. The pole ( ) c the be eglected d the ytem behve like firt-order ytem. ME4/TM34 Feedbck Cotrol Sytem 4

92 Step Repoe Secod-order ytem Normlized repoe curve For ft repoe, ζ 0.7 i uully deirble. If o overhoot i required, ζ > i uully ued. ME4/TM34 Feedbck Cotrol Sytem 5

93 Triet Repoe Specifictio Five meure of triet performce bed o d -order uderdmped repoe Mximum (percet) overhoot: c( t p ) c( ) M p 00% c( ) Dely time Rie time: 0% - 90%, or 5% - 95%, or 0% - 00% Pek time ME4/TM34 Feedbck Cotrol Sytem Settlig time: time to rech d ty withi pecified limit, uully % or 5%. 6

94 Meure of triet performce We hve c( t) e ζωt ζ i( ω d t φ) ω d ω ζ φ t ζ ζ Rie Time 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 ME4/TM34 Feedbck Cotrol Sytem 7

95 Meure of triet performce We hve c( t) e ζωt ζ i( ω d t φ) ω d ω ζ φ t ζ ζ Pek Time t p dc( t) dt t t p (iω d t p ) ω ζ e ζω t p 0 givig i 0 ω or ω dt p 0, π, π, 3π, t d p Therefore t p π ω d for the firt pek. ME4/TM34 Feedbck Cotrol Sytem 8

96 ME4/TM34 Feedbck Cotrol Sytem 9 We hve We hve ) i( ) ( φ ω ζ ζω t e t c d t ζ ζ φ t Mximum Overhoot Mximum 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 performce Meure of triet performce

97 Meure of triet performce We hve c( t) e ζωt ζ i( ω d t φ) ω d ω ζ φ t ζ ζ Settlig Time t ζω The curve ± ( e t / ζ ) give the evelope curve of the triet repoe. t i foud to be pproximtely t 4T t 3T (% criterio) (5% criterio) where time cott ME4/TM34 Feedbck Cotrol Sytem T ζω 30

98 Ed ME4/TM34 Feedbck Cotrol Sytem 3

99 A Note o eglectig L O Slide 6 i the preettio o Sytem Repoe, it w metioed tht the iductce L c be eglected. Below re ome ote o thi. Coider the trfer fuctio ω V ( L ' R )( J We c re-write thi ω V ( τ )( τ ) b) where ', L / R R b τ d τ J / b. Coider the repoe to uit tep iput, V() /. The Ω( ) ( τ )( τ ) From the Lplce Trform Tble (Che Tble etry 7), we write the ivere trform /( τ τ ) Ω ( ) ( / τ )( / τ ) /( τ τ ) ϖ t ) ( (/ τ )(/ τ ) / τ / τ τ e τ t / τ t / τ ( e τ t / τ t / τ ( τ e τ e τ For implicity, coider the ce whe. Coider lo tht τ d with τ << τ with τ τ / 0. / t /0 The ω ( t) ( e 0e t ) () 9 If, becue τ << τ, we eglect τ, the we hve ) )

100 From Etry 4 i Dr Che Tble, we hve /0 Ω( ) ( τ ) (0 ) ( /0) /0 /0 t /0 t /0 ω ( t) ( e ) e () The two repoe repreeted by Equtio () d () re plotted i the figure below () () Time, ec The two curve how bove, plotted uig Equtio () d () re pproximtely imilr, meig tht we c eglect τ d the repoe will till be reobly ccurte with the impler trfer fuctio.

101 A Note o Derivtio of Pek Time, T p (Slide 7 i Sytem Repoe) The output repoe i c( t) e ζωt ζ i( ω t φ) d () with ω d ω ζ d Differetitig Eq () by prt, we hve d c( t) ( ζωe dt ζ ζωt ) i( ω t φ) d ζ φ t. ζ e ζωt ζ ( co( ω t φ) ω ) d d e ζωt ζ e ζωt ζ [ ζω i( ω t φ) ω co( ω t φ) ] d d d [ ζω ( i( ω t)coφ co( ω t)iφ) ω ( co( ω t)coφ i( ω t)iφ) ] d d d d d Notig tht co φ ζ d iφ ζ, we hve c& ( t) e e ζωt ζ ζωt ζ [( ζω coφ ω iφ)i( ω t) ( ζω iφ ω coφ)co( ω t) ] d d [( ζ ω ω ( ζ )) i( ωdt) ( ζω ζ ζ ωζ )co( ωdt) ] d d e ζωt ζ [ ω i( ω t) ] d Lettig c& ( t) 0 t t t p, we thu hve e ζω t p ζ [ ω i( ω t )] 0 d p. p Sice e ζω 0 for t <, t p we hve i( ω dt p ) 0 givig i( ω dt p ) 0, π,π,3π, Chooig the firt pek give t p π ω d

102 ME4/ME4E Feedbck Cotrol Sytem Stedy-Stte Chrcteritic ME4/TM34 Feedbck Cotrol Sytem

103 Sytem Type Coider the uity-feedbck ytem R E - G C ( T )( Tb ) L( Tm ) With G( ) (-) N ( T )( T ) L( T ) p The prmeter N ocited with the term S N i the deomitor repreet the Type of the ytem. Exmple: Type 0 if N0, Type if N d o o. A free term i the deomitor repreet itegrtio. The higher the type umber, the better the tedy-tte ccurcy of the cloed-loop cotrol ytem. However, the higher the ytem type, the greter the problem with ytem tbility. ME4/TM34 Feedbck Cotrol Sytem

104 Stedy-Stte Error Sttic Error Cott The me uity-feedbck ytem R E - G C Error Trfer fuctio E R R C C R R G ( G) G G G Thu E( ) R( ) G( ) E R G d the tedy-tte error i e lim e( t) e t lim E( ) 0 R( ) lim 0 G( ) ME4/TM34 Feedbck Cotrol Sytem 3

105 Stedy-Stte Error Sttic Error Cott e R( ) lim 0 G( ) For uit-tep iput R( ) d e lim G( ) 0 G(0) Sttic Poitio Error Cott, p i defied lim G( ) G(0) d 0 p e p ( T )( Tb ) L( Tm ) with G( ), N ( T )( T ) L( T ) p p for Type 0 ytem e for Type or higher ytem 0 p e ME4/TM34 Feedbck Cotrol Sytem 4

106 Stedy-Stte Error Sttic Error Cott For uit-rmp iput e R( ) lim 0 G( ) Sttic Velocity Error Cott, v i defied v lim G( ) 0 R ( ) d e lim lim 0 G( ) 0 G( ) d ( T )( Tb ) L( Tm ) with G( ), N ( T )( T ) L( T ) v v v 0 p for Type 0 ytem for Type ytem for Type or higher ytem e v e e e 0 ME4/TM34 Feedbck Cotrol Sytem 5

107 Stedy-Stte Error Sttic Error Cott For uit-ccelertio iput t r( t) for t 0, R ( ) ME4/TM34 Feedbck Cotrol Sytem e d R( ) lim 0 G( ) Sttic Accelertio Error Cott, i defied d ( T )( Tb ) L( Tm ) with G( ), N ( T )( T ) L( T ) 0 lim G( ) 0 p for Type 0 d Type ytem for Type ytem for Type 3 or higher ytem 3 e e lim G( ) G( ) e e e lim 0 6

108 Stedy-Stte Error Summry of tedy- Stte error Type 0 ytem Step Iput r Type ytem 0 Rmp Iput r t Type ytem 0 0 Accel. Iput r t / Type 0 ytem hve fiite tedy-tte error for tep iput d cot follow rmp iput. Type ytem hve zero tedy-tte error for tep iput, fiite error for rmp iput, d cot follow ccelertio iput. Type ytem re eeded to follow rmp iput with zero tedytte error. I geerl, the higher the ttic gi of the ope-loop trfer fuctio, G(), the mller the tedy-tte error. However, higher gi ormlly led to tbility problem. ME4/TM34 Feedbck Cotrol Sytem 7

109 Ed ME4/TM34 Feedbck Cotrol Sytem 8

110 ME4/ME4E Feedbck Cotrol Sytem Block Digrm Algebr ME4/TM34 Feedbck Cotrol Sytem

111 Block Digrm Repreettio A block digrm i grphicl tool c help u to viulize the model of ytem d evlute the mthemticl reltiohip betwee their elemet, uig their trfer fuctio. The Trfer Fuctio Block Iput R() G() Sytem C() Output G ( ) C( ) R( ) The trfer fuctio G() i defied oly for lier time-ivrit ytem d ot for olier ytem. I property of the ytem d i idepedet of the iput to the ytem. Commuttive G G GG Aocitive G G G G ME4/TM34 Feedbck Cotrol Sytem

112 Block Digrm Elemet The Summig Poit Siged iput Y X X Y - Z - Z output Ay umber of iput. Oly oe output ME4/TM34 Feedbck Cotrol Sytem 3

113 Block Digrm Algebr Whe mipultig block digrm, the origil reltiohip, or equtio, reltig the vriou vrible mut remi the me. Block i erie or ccded block X Y Z G G X G G Z Whe block re coected i erie, there mut be o lodig effect. ME4/TM34 Feedbck Cotrol Sytem 4

114 Block Digrm Algebr Block i prllel X G G Y X G G Y ME4/TM34 Feedbck Cotrol Sytem 5

115 ME4/TM34 Feedbck Cotrol Sytem 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 Digrm Algebr Block Digrm Algebr G G Z X Y G X Y /G X Z Y X G /G

116 Cloed-Loop Feedbck Sytem 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 ME4/TM34 Feedbck Cotrol Sytem 7

117 Cloed-Loop Feedbck Sytem 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 Alo E C G d E R G C R GH E R i clled the error trfer fuctio ME4/TM34 Feedbck Cotrol Sytem 8

118 Cloed-Loop Cotrol Feedbck Sytem D R E - M G c G p C B H G c G p M D i the cotroller trfer fuctio i the plt trfer fuctio i the mipulted vrible 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 ME4/TM34 Feedbck Cotrol Sytem 9

119 Cloed-Loop Cotrol Feedbck Sytem D R E M G c G p C - B H C R G GH G G c G G c p p H Aumig R 0, we c re-drw D C - G p G c H C D G GH G G p p G c H ME4/TM34 Feedbck Cotrol Sytem 0

120 Block Digrm Mipultio Exmple: Determie C()/R() D D E R - - F G C H I Whe mipultig block, mut eure C() doe ot chge, o tht C()/R() remi me. ME4/TM34 Feedbck Cotrol Sytem

121 Block Digrm Mipultio Exmple: Determie C()/R() We wih to move thi To move igl b to igl to before block F fter Block G Aume me of igl how b F-FcD b F-Fc(D/F)F ME4/TM34 Feedbck Cotrol Sytem

122 Block Digrm Mipultio Exmple: Determie C()/R() R - - D F D G E C R - D/F (D/F) - F b G E Eb C H H Gb I I E/G Eb C Gb Eb R - D/F - FG Gb C R - D/F - FG E/G C H H I C Gb (E/G)Gb I ME4/TM34 Feedbck Cotrol Sytem 3

123 Block Digrm Mipultio Exmple G GH R - D/F - FG H E/G C R - D/F FG FGH E/G C I I R - D FG F FGH I E G C R D FG E F FGH G D FG E I F FGH G C ME4/TM34 Feedbck Cotrol Sytem 4

124 Ed ME4/TM34 Feedbck Cotrol Sytem 5

125 ME4/ME4E Feedbck Cotrol Sytem Cocept of Sytem Stbility ME4/TM34 Feedbck Cotrol Sytem

126 ME4/TM34 Feedbck Cotrol Sytem Higher-Order Sytem repoe Higher-Order Sytem repoe Coider the cloed loop trfer fuctio Coider the cloed loop trfer fuctio ) ( ) ( ) ( ) ( ) ( D N G R C c c c m b b b b m m m m > with 0 0 L L The ytem i id to be higher-order ytem for >. There will be pole of G c (), or root of the deomitor D c (). G c () c thu be writte ) ( ) ( ) ( R C G c ) ( ) )( )( ( ) )( ( ) ( ) )( ( r r r q m p p p z z z ω ω ζ ω ω ζ ω ω ζ L L L ) ( r q where Rel Root Rel Root Complex root Complex root q j r k k k k k k k k k j j c b p ) ( ω ω ζ ζ ω ζω

127 ME4/TM34 Feedbck Cotrol Sytem 3 Higher-Order Sytem repoe Higher-Order Sytem repoe For uit tep iput, we c re-write C() i term 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 ume tht ll the pole re ditict, i.e. ot repeted. The time repoe will the be, by uig the Ivere Lplce Trform 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 mut ll hve egtive rel prt. The repoe of tble higher-order ytem thu comprie um of umber of decyig expoetil curve d decyig dmped iuoidl curve.

128 Higher-Order Sytem repoe Plot of pole o the -ple Rel Pole d their effect o the repoe Im p p Re Rel pole o the S-Ple. Ech rel pole will cotribute term ito the repoe. The more egtive the pole, or the frther wy to the left from the Imgiry xi it i, the more rpidly the expoetil term decy to zero. I geerl, if two pole re uch tht p > 5 p, the the repoe cued by p i domit d tht for p c be eglected without lo of ccurcy. j e p t j ME4/TM34 Feedbck Cotrol Sytem 4

129 Higher-Order Sytem repoe Plot of pole o the -ple Complex cojugte pole d their effect o the repoe Im ζ k ω k Ech complex pir cotribute decyig dmped iuoidl term to the repoe. Lie of cott ζ ζω k φ The more egtive the rel prt ζω, k k or the frther wy to the left the pole Re re from the Imgiry xi, the more rpidly the term decy to zero. Complex cojugte pole o the S-Ple. φ k The gle the pole mke with the Rel Axi determie the dmpig rtio, the greter the gle, the le the dmpig rtio. φ k t ζ ζ k k ME4/TM34 Feedbck Cotrol Sytem 5

130 Some typicl repoe Stble ytem ME4/TM34 Feedbck Cotrol Sytem 6

131 Some typicl repoe Stble ytem ME4/TM34 Feedbck Cotrol Sytem 7

132 Some typicl repoe A Utble ytem ME4/TM34 Feedbck Cotrol Sytem 8

133 Higher-Order Sytem repoe Domit Pole Complex cojugte pole d their effect o the repoe Cloed-loop pole o the S-Ple. The reltive domice of cloedloop pole i determied by how fr they re from the Imgiry Axi, umig tht there re o zero erby. (Zero ffect the reltive mgitude of the cott term ocited with the pole, the cloer they re the more the effect.) Uully the repoe will be djuted uch tht oe pir of complex cojugte pole will be cloer to the Imgiry Axi reltive to ll the other pole d the repoe cued by thi pir domite the overll repoe. Thi pir i clled the domit cloed-loop pole. ME4/TM34 Feedbck Cotrol Sytem 9

134 Cocept of Sytem Stbility A ytem (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 time. A LTI (Lier Time-Ivrit) ytem mut 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 mut ll hve egtive rel prt. If pole, or pole, lie o the imgiry xi, the ytem i criticlly, or limitedly, tble. If lier ytem i utble, eve i the bece of y iput, the output will grow without boud d become ifiitely lrge time goe to ifiity. ME4/TM34 Feedbck Cotrol Sytem 0

135 Routh Stbility Criterio A ytem i tble if ll the root of the ytem chrcteritic equtio hve egtive rel prt. G( ) H( ) 0 The problem i if the chrcteritic equtio i of order higher th two, it i ot ey to fid the root. (Of coure, there re computer progrm, e.g. MATLAB or OCTAVE, tht help with thi.) Fortutely, there i imple criterio, kow Routh Stbility Criterio (ometime lo kow the Routh-Hurwitz Stbility Criterio), which eble u to fid out the umber 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 polyomil. ME4/TM34 Feedbck Cotrol Sytem

136 Routh Stbility Criterio Procedure ) Form the chrcteritic equtio 0 S S S S 0 > 0 0 We ume tht 0; i.e. y zero root hve bee removed. Exmple: or ( ) 0 ue the equtio ) If y of the coefficiet i egtive or zero, the ytem 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 form the Routh Arry d ue the Routh Criterio to determie the umber of root with poitive rel prt. ME4/TM34 Feedbck Cotrol Sytem

137 ME4/TM34 Feedbck Cotrol Sytem 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 b 3 0 b b b

138 ME4/TM34 Feedbck Cotrol Sytem f e e c c c b b b b L L L I developig the rry, etire row c be multiplied by poitive umber to implify the proce without ffectig the reult. I developig the rry, etire row c be multiplied by poitive umber to implify the proce without ffectig the reult. Routh Stbility Criterio Routh Stbility Criterio Similrly Similrly Routh Arry Routh Arry 3 b b b c 3 b b b c 3 5 b b b c 3 5 b b b c b b b c b b b c Routh Criterio tte tht the umber of root with poitive rel prt i equl to the umber of chge i ig of the coefficiet i the firt colum of the rry. Routh Criterio tte tht the umber of root with poitive rel prt i equl to the umber of chge i ig of the coefficiet i the firt colum of the rry. e e d d e f e e d d e f

139 Routh Stbility Criterio Exmple Determie the coditio for the followig equtio to hve oly root with egtive rel prt Routh Arry If 03 > 0, the there i o ig chge d there i o root with poitive rel prt. If, the there re two ig chge. Therefore 0 3 < 0 there re two root with poitive rel prt. ME4/TM34 Feedbck Cotrol Sytem 5

140 Routh Stbility Criterio Specil ce A zero occur i the firt colum of y row while the remiig term re ot zero, or there i o remiig term. Solutio: The zero term i replced by mll poitive umber d the rry i proceed ccordigly. Exmple Routh Arry ε 3 3 If the ig of the coefficiet i the row bove i the me tht below ( i thi ce), the there re pir of imgiry root. If the ig of the coefficiet i the row bove i differet from tht below, there i oe ig chge idictig oe root with poitive rel prt. ME4/TM34 Feedbck Cotrol Sytem 6

141 Routh Stbility Criterio Specil ce If ll the coefficiet, or the oly oe coefficiet, i derived row re zero, it me tht there re root of equl mgitude locted ymmetriclly bout the origi. Exmple: The chrcteritic equtio hve fctor uch ( σ )( σ ) or ( jω)( jω ). For uch ce, form uxiliry polyomil with the coefficiet of the row bove the ll-zero row d uig the coefficiet of the derivtive of thi polyomil to replce the ll-zero row. ME4/TM34 Feedbck Cotrol Sytem 7

142 Routh Stbility Criterio Specil ce Exmple Note tht becue ot ll the coefficiet re poitive, thi idicte tht there i t let oe root with poitive rel prt. Routh Arry Ue thi uxiliry polyomil 4 P( ) 48 P& ( ) New Routh Arry ME4/TM34 Feedbck Cotrol Sytem There i oe chge i ig i the firt colum oe root with ve rel prt. 8

143 Ed ME4/TM34 Feedbck Cotrol Sytem 9

144 ME4/ME4E Feedbck Cotrol Sytem Root Locu Alyi ME4/TM34 Feedbck Cotrol Sytem

145 Root Locu Alyi Coider the cloed-loop ytem R E - G C B H The triet repoe, d tbility, of the cloed-loop ytem i determied 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 form 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 from 0 to ifiity. ME4/TM34 Feedbck Cotrol Sytem ( z )( z ( p )( p ) L( z ) L( m p ) )

146 Root Locu Alyi Exmple Coider the ytem with G( ) H ( ) ( )( ) Chrcteritic Eq: ( )( ) 0 Root Locu Plot Pole v P P P j j j j j j j j j j j j j j pole, thu 3 loci ME4/TM34 Feedbck Cotrol Sytem 3

147 Plottig the Root Loci The root loci re plotted either Mully, or Uig computer progrm uch OCTAVE (ey if you hve the progrm d kow how to ue it.) ME4/TM34 Feedbck Cotrol Sytem 4

148 Mul plottig Root Locu Cocept The Chrcteritic Equtio i firt writte i the form G( ) H( ) F( ) where i cott gi. 0 The root of the chrcteritic equtio re the vlue of which tify the equtio F( ) 80 for ±, ± 3, ± 5, Or whe F( ) 80 ±, ± 3, ± 5, The mgitude coditio i tified by hvig F( ) givig F( ) ME4/TM34 Feedbck Cotrol Sytem 5

149 Mul plottig Root Locu Cocept Determiig the phe gle for 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 ( p3,,3, ) ( 4 ) 4 p (z ) ±, ± 3, ± 5, (p ) φ -z -p θ Im 0 Re ME4/TM34 Feedbck Cotrol Sytem 6

150 Mul plottig Procedure d Guidelie ) Locte the pole d zero of the ope-loop trfer, G()H(), fuctio o the ple. ) There re my loci, or brche, pole of the G()H(). 3) Ech brch trt from pole of G()H() d ed i zero. If there re o zero i the fiite regio, the the zero re t ifiity. N( ) Reo: G( ) H ( ) 0 D( ) N( ) 0 D( ) Whe 0, D( ) 0 d root re root of D(). Whe >>, N ( ) 0 d root re root of N(). ME4/TM34 Feedbck Cotrol Sytem 7

151 Mul plottig Procedure d Guidelie 4) The loci exit o the rel xi oly to the left of odd umber of pole d/or zero. Complex pole d/or zero hve o effect becue, for poit o the rel xi, the gle ivolved re equl d oppoite. Coider tet poit,, o the rel xi how. Every pir of complex cojugte pole (or zero) will cotribute pir of gle, d uch tht θ θ θ θ 360 They c thu be igored. -z -3 θ -p - θ - θ 3 -p 3 Im j j 0 -j Re -p -j ME4/TM34 Feedbck Cotrol Sytem 8

152 Mul plottig Procedure d Guidelie 4) The loci exit o the rel xi oly to the left of odd umber of pole d/or zero. Complex pole d/or zero hve o effect becue, for poit o the rel xi, the gle ivolved re equl d oppoite. Coider tet poit,, o the rel xi how. Ech rel pole or zero to the left of poit doe ot cotribute to the gle um d thu c be igored. Ech pole/zero to the right of poit cotribute gle of ± 80. A odd umber of them will thu cotribute totl of 80 where ±, ± 3, ± 5, -z -3 θ -p -p - θ - θ 3 -p 3 Im j j 0 -j -j Re ME4/TM34 Feedbck Cotrol Sytem 9

153 Locu exit to left of odd No. of zero/pole Exmple: Coider the chrcteritic equtio GH ( 3) ( j)( j) ( z) ( p )( p ) F( ) Coider tet poit,, o the Oe rel zero t -z-3; xi to the left of the pole t 0. Oe zero t 0; -p Im j j Complex cojugte pole t -p--j, d -p-j -z p 0 0 Re -j -p -j ME4/TM34 Feedbck Cotrol Sytem 0

154 Locu exit to left of odd No. of zero/pole Exmple: Coider the chrcteritic equtio GH ( 3) ( j)( j) ( z) ( p )( p ) F( ) Coider tet poit, S, o the rel xi to the left of ONE pole t 0. -p Im j Thi tet poit will be prt of the root locu if F( ) 80 p θ j Or φ θ θ θ 80 3 Note tht, d θ ϑ θ 3 0 φ 0 80 Poit S form prt of the root locu. -z φ -3 z p - θ 3 -p - θp 0 0 -j -j Re ME4/TM34 Feedbck Cotrol Sytem

155 Locu exit to left of odd No. of zero/pole Exmple: Coider the chrcteritic equtio GH ( 3) ( z) F( ) ( j)( j) ( p )( p ) Coider tet poit, S, o the rel xi to the left of the zero t -3. Note tht, d Thu ϑ θ 3 0 φ θ 80 F() 0 -p -p p -z φ -3-3 θ Im Im j j j j pθ 0 00 Re Re Poit S i ot prt of of the root locu. Note S i to the left of eve No. of zero/pole z p -p -p θ 3 -j -j -j -j ME4/TM34 Feedbck Cotrol Sytem

156 Mul plottig Procedure d Guidelie 5) Becue complex root mut occur i cojugte pir, i.e. ymmetricl bout the rel xi, the root-locu plot i ymmetricl bout the rel xi. 6) Loci which termite t ifiity pproch ymptote i doig o. For tet poit S t ifiity, ech pole/zero cotribute equl phe gle. Thu for ( Z P) θ 80 ±, ± 3, ± 5, -p θ -z -p θ θ Im 0 Re Z/PNo. of zero/pole θ θ 80 Z P -p 3 ME4/TM34 Feedbck Cotrol Sytem 3

157 Mul plottig Procedure d Guidelie 7) All the ymptote trt from poit o the rel xi with coordite Im σ pi P Z Exmple: For m z i j j G( ) H ( ) ( )( ) Re The θ 60, 80, 300 Z P 0 3 -j -j σ ME4/TM34 Feedbck Cotrol Sytem 4

158 Mul plottig Procedure d Guidelie 8) Brek-i d brekwy poit (o rel xi) At brek-i poit, vlue of icree the loci move oto the rel xi d wy from the brek-i poit. At brekwy poit, vlue of icree log the rel xi from both ide d rech mximum t the brekwy poit. Alog the rel xi,. σ Thu, t the brek-i or brekwy poit, d d σ provided >0 d root loci. ME4/TM34 Feedbck Cotrol Sytem 0 σ exit o the Brek-i pt. Brekwy pt. 5

159 Mul plottig Procedure d Guidelie 9) Imgiry Axi Croig Two pproche: ) Ue Routh Criteri to determie the vlue of t which the ytem i criticlly tble. Thi i idicted by vlue of zero i the firt colum but with o ig chge i the firt colum of the Routh Arry. b) Sice the root re o the imgiry xi, by lettig jω i the chrcteritic equtio d olve for ω d. Thi i doe by equtig both the rel d imgiry prt of the chrcteritic equtio to zero. Im Axi Croig ME4/TM34 Feedbck Cotrol Sytem 6

160 Mul plottig Procedure d Guidelie 0) Agle of Deprture from complex pole d Agle of Arrivl from complex zero. Agle of Deprture Thi i doe by tkig tet poit very cloe to the complex pole, or zero, d pplyig the gulr criteri. ME4/TM34 Feedbck Cotrol Sytem 7

161 Mul plottig Procedure d Guidelie 0) Agle of Deprture from complex pole d Agle of Arrivl from complex zero. θ Im -p Thi i doe by tkig tet poit very cloe to the complex pole, or zero, d pplyig the gulr criteri. Exmple, for digrm o right, -z φ 0 Re φ ( θ θ ) ± 80 θ -p ME4/TM34 Feedbck Cotrol Sytem 8

162 Exmple Root Locu Plot MATLAB Progrm >> ghtf([],[ 0]) Trfer fuctio: ^3 ^ >> rlocu(gh) Pole t 0, ± j ME4/TM34 Feedbck Cotrol Sytem 9

163 Exmple Root Locu Plot MATLAB Progrm >> hzpk([-3-4],[0 -],[]) Zero/pole/gi: (3) (4) () >> rlocu(h) >> Pole t 0, - Zero t -3, -4 ME4/TM34 Feedbck Cotrol Sytem 0

164 Exmple Root Locu Plot MATLAB Progrm >> gtf([],[ 5]) Trfer fuctio: ^ 5 >> gzpk([-],[0 -],[]) Zero/pole/gi: () () >> ghg*g Zero/pole/gi: () () (^ 5) >> rlocu(gh) Pole t Zero t 0,, ± j ME4/TM34 Feedbck Cotrol Sytem

165 Exmple Root Locu Plot MATLAB Progrm >> gtf([],[ 5]) Trfer fuctio: ^ 5 >> gzpk([],[0-4],[]) Zero/pole/gi: (4) >> ghg*g Zero/pole/gi: (4) (^ 5) >> rlocu(gh) Pole t 0, 4, ± j ME4/TM34 Feedbck Cotrol Sytem