CONTROL SYSTEMS. Chapter 3 : Time Response Analysis

Transcription

1 CONTROL SYSTEMS Chper 3 : Time Repoe Alyi GATE Objecive & Numericl Type Soluio Queio 4 [Prcice Book] [GATE EC 99 IIT-Mdr : Mrk] A uiy feedbck corol yem h he ope loop rfer fucio. 4( ) G () ( ) If he ipu o he yem i ui rmp, he edy e error will be (A) (B).5 (C) (D) A. (A) Sol. 4 Give : G, d r () u () Tkig Lplce rform of r(), we ge R () Sedy e error i give by, R() e lim E( ) lim GH ( ) ( ) e lim lim 4( ) 4( ) ( ) ( ) Hece, he correc opio i (A). Alerively, Velociy error coefficie i give by, 4( ) v lim G() H () lim ( ) Sedy e error for rmp ipu i give by, e v For ype - yem edy e error due o rmp ipu will be zero. Queio 6 [Prcice Book] [GATE EE 99 IIT-Mdr : Mrk] A fir order yem d i repoe o ui ep ipu re how i figure below, he yem prmeer d re repecively c (). r () c (). (A) 5, (B), 5 (C), (D), (ec)

2 A. (A) Sol. C () For fir order yem loop rfer fucio i d i repoe o ui ep ipu re how R () i figure below. c () c e / () ( ) For he give rfer fucio c () C () R () d i repoe. (ec). From he figure, ime co i. ec. O comprig wih bove drd equio, we ge, 5. Uig fil vlue heorem edy e oupu c be wrie, c lim c( ) lim C( ) From figure, c c. Hece, he correc opio i (A). Queio 7 [Prcice Book] (ec) lim R( ) lim r () c () Block digrm model of poiio corol yem i how i figure. R () Amplifier A [GATE EC 99 IIT-Delhi : 8 Mrk] Moor (.5) Y() () I bece of derivive feedbck ( ), deermie dmpig rio of he yem for mplifier gi A 5. Alo fid he edy e error o ui rmp ipu. (b) Fid uible vlue of he prmeer A d o h dmpig rio of he yem i icreed o.7 wihou ffecig he edy-e error obied i pr ().

3 Sol. Give : The give block digrm of poiiol corol yem i how i figure below. R () Amplifier A Moor (.5) Y() () Give : 5 G () d H () (.5) ( ) Cloed-loop rfer fucio for egive feedbck i give by, G () T() ( ).. (i) GH ( ) ( ) ( ) Trfer fucio for ecod-order yem wih ui ep ipu i give by, C ()..(ii) R () where, dmpig rio, url gulr frequecy O comprig equio(i) d (ii), we ge rd/ec. d Sedy e error i give by, R() e lim E( ) lim GH ( ) ( ) e lim lim. A. ( ) ( ) (b) The ope loop rfer fucio c be wrie, A A OLTF [ ( )] The cloed loop rfer fucio c be wrie, Y () A CLTF R () ( ) Trfer fucio for ecod-order yem wih ui ep ipu i give by, C () R () where, dmpig rio, url gulr frequecy A..(i).. (ii) 3

4 Velociy error coefficie i give by,.. (iii).9899 A A.. (iv) From equio (iii) d equio (iv), we ge 5 d 4. A. Queio 8 [Prcice Book] [GATE IN 99 IIT-Delhi : Mrk] For wh vlue of doe he yem how i figure hve zero edy e error [i.e., lim E( ) ] for ep ipu? v lim G( ) H( ) A A v lim [ ( )] Sedy e error i give by, e v. A.A O comprig equio (i) d (ii), we ge A.7 A A A E() 5 A. Sol. 4 (A) = (B) = (C) 4 (D) No vlue of (A) The give yem i how i figure. E() 5 DC gi of feedbck H() i H H H lim ( ) lim

5 Cloed-loop rfer fucio i give by, M() GH ( ) ( ) G () 5 ( ) ( 5 )( 4) ( )( 4) ( )( 4) M() ( )( 4) M() 3 9 ( ) (4) Sedy e error i give by, e lim E( ) lim R( ) M( ) H ( )( 4) e lim 4 R( ) 3 9 ( ) (4) For ui ep ipu R () ( )( 4) e lim ( ) (4) Hece, he correc opio i (A). Queio 9 [Prcice Book] A. Sol. 3 ( 5 )( 4)( ) [GATE IN 993 IIT-Bomby : Mrk] A ui ep i pplied o fir order yem wihou ime dely. The repoe h vlue of.64 ui mi d ui edy e. The rfer fucio of he yem i (A) (B) (C) (D) (A) The fir order yem ep repoe i give by, c () ke A, k e k A mi,.64 k e mi 6ec. Fir order rfer fucio i give by, k G () 6 A. 5

6 Queio [Prcice Book] [GATE IN 994 IIT-hrgpur : Mrk] 8 A uiy feedbck cloed loop ecod order yem h rfer fucio d i excied by.69 ep ipu of ui. The edy e error of he oupu i (A) (B). (C). (D). A. (B) Sol Give :.6 G( ) T().69 9 G( ).6..(i) Cloed-loop rfer fucio for egive feedbck i give by, G () T() G ( )..(ii) O comprig equio (i) d (ii), we ge 9 G ().6 Sedy e error i give by, R() e lim E( ) lim G ( ) 9.6 Hece, he correc opio i (B). Aigme [Work Book] [GATE EC 997 IIT-Mdr : 5 Mrk] The figure how he block digrm repreeio of corol yem. The yem i block A h impule repoe ha() e u(). The yem i block B h impule repoe hb () e u(). The block k i mplifier by fcor k. For he overll yem he ipu i x() d oupu y (). x () A B y () k Y() () Fid he rfer fucio whe k. X() (b) Fid he impule repoe whe k. (c) Fid he vlue of k for which he yem become uble. Sol. Give : Impule repoe ha() e u() Tkig Lplce rform, we ge HA() Impule repoe hb () e u() Tkig Lplce rform, we ge HB() 6

7 () k, G() HA() HB() Cloed-loop rfer fucio for egive feedbck i give by, Y() G() T() X () G() H() ( )( ) T() A. 3 3 ( )( ) (b) k Y () A B H () X() ( )( ) A d B H() Tkig ivere Lplce rform, we ge Impule repoe h () ( e e ) u () A. (c) Vlue of k for uble yem k T() 3k Chrceriic equio 3k Rouh Tbulio : + k 3 + k For ibiliy k k yem i uble. A. Queio 8 [Work Book] [GATE EC 999 IIT-Bomby : Mrk] A. If he cloed-loop rfer fucio T() of uiy egive feedbck yem i give by T... he he edy e error for ui rmp ipu i (A) (B) (C) (D) zero (D) T d r() = u()... Sol. Give :... T() (i) 7

8 Cloed-loop rfer fucio for egive feedbck i give by, G () T() GH ( ) ( ) O comprig equio (i) d (ii), we ge G ()... Tkig Lplce rform of r(), we ge R () The edy e error due o ui rmp ipu i give by, e v Velociy error coefficie i give by, ( ) v lim G( ) H( )... So, h e Hece, he correc opio i (D). Queio 3 [Prcice Book] v The block digrm of feedbck yem i how i he figure. Ipu G ( 3)... (ii) [GATE EC IIT-hrgpur : 5 Mrk] Oupu G V (). Sep repoe T Sol. () Fid he cloed loop rfer fucio. (b) Fid he miimum vlue of G for which he ep repoe of he yem would exhibi overhoo, how i figure. (c) For G equl o wice he miimum vlue, fid he ime period T idiced i he figure. () The give block digrm of feedbck yem i how below. Ipu G ( 3) Oupu G 8

9 Cloed-loop rfer fucio for egive feedbck i give by, G G () ( 3) T() G ( ) G ( 3) G G T () ( 3) G 3G (b) Give : Mximum pek overhoo i give by, MPO e. Dmpig rio i.6 For he give repoe rfer fucio i G T()..(i) 3G Trfer fucio for ecod-order yem i give by, C ()..(ii) R () where, dmpig rio, url gulr frequecy O comprig equio (i) d equio (ii), we ge 3 d G 3 G 3 G.6 G 6.5 A. (c) G' G G' 3.44 Dmped frequecy of ocillio i give by, d 3.53 (.44) 3.97 T T.96 ec. A. Queio 37 [Prcice Book] A. The forwrd ph rfer fucio of uiy feedbck yem i give by, 5 G 8 If e i he error o ui impule ipu he vlue of he performce idex 9 A. [GATE IN IISc-Bglore : Mrk] J e d (A) zero (B) ifiiy (C) (D).5 (D) i equl o

10 Sol. E() R () G () C () B () H() LT E () R () B () where e () E () E () R () CH () () E () R () EGH () () () E() G() H() R() R () E () GH ( ) ( ) Give : r () (), R (), H () Therefore, E () (5 ) ( ) ( ) ( ) (8 ) ( ) ( )(8 )( ) E () ( ) ( 8 ) ( ) ( 5 ) ( ) ( ) For cul e (): Give J e() d E() e() e d J E() J Queio [Work Book] ( )(8 )( ) lim ( ) ( 8 ) ( ) ( 5 ) ( ) ( ) J.5 () () C () 4 A ecod-order yem h he rfer fucio R () 4 4 he repoe c () of he yem i repreeed by (A).5 Sep Repoe (B) A. [GATE EC 3 IIT-Mdr : Mrk]. wih r () he ui-ep fucio, Sep Repoe Ampliude.5 Ampliude Time (ec) Time (ec)

11 (C). Sep Repoe (D). Sep Repoe Ampliude Time (ec) A. (B) Sol. Give : C () 4 R () 44 Trfer fucio for ecod-order yem i give by, C () R () where, dmpig rio, url gulr frequecy..(i)..(ii) 4 rd/ec 4 Sice, yem i criiclly dmped. The fil vlue c be clculed uig fil vlue heorem, 4 R( ) c lim c lim 44 For % olerce bd elig ime i give by, 4 4 T ec Thi me h he repoe c () will be ele o i fil vlue fer ec. Hece, he correc opio i (B). Queio 3 [Work Book] [GATE IN 4 IIT-Delhi : Mrk] A ceri yem exhibied overhoo of 6% whe ubjeced o ipu of u ( ), where u () i ep ipu. The dmpig rio d decy rio repecively re (A)(.8,.8) (B) (.5,.56) (C) (.,.66) (D)(.,.89) A. (B) Sol. Give : MPO = 6 % [By deful pek overhoo] Percege MPO i give by, % MPO e 6 e l(.6) Ampliude.5 5 Time (ec)

12 Dmpig rio ( ).5 Cocep of decy rio : c () c mx pek overhoo d pek overhoo p Decy rio d pek overhoo pek overhoo The d pek overhoo i give by, The decy rio i give by, 3 d %MPO( ) e Decy rio Hece, he correc opio i (B). Queio 5 [Prcice Book] (.5) e e (.5) e e [GATE EE 7 IIT-pur : Mrk] RLC circui how i figure. For ep ipu e i, he overhoo i he oupu e will be R L mh e C F e i A. (A) %, ice he yem i o uder dmped (B) 5 % (C) 6 % (D) 48 % (C)

13 Sol. The give RLC circui i how below. R L mh e C F e i Trform domi : R L E () E () i C E() Ei () C RL C Ei () E () RC LC E() E () R i LC L LC Chrceriic equio i give by, R [By VDR] L LC (i) Sdrd chrceriic equio for ecod order yem i give by, (ii) O comprig equio (i) d equio (ii), R d LC L R R LC R C L L L Mximum pek overhoo i give by, MPO e e.5 (.5) MPO.63 or 6.3% 6% Hece, he correc opio i (C). 3

14 Queio 5 [Prcice Book] [GATE EE 7 IIT-pur : Mrk] A. Sol. If he bove ep repoe i o be oberved o o-orge CRO, he i would be be hve he e i (A) ep fucio (B) qure wve of 5 Hz (C)qure wve of 3 Hz (D) qure wve of khz (C) The reoce frequecy i give by, 3 6 LC 4 rd/ec Selig ime for % olerce bd i give by, mec. 4.5 To oberve he rie repoe, ipu mu be pplied le upo.8mec. Pek ime i give by, p.36 mec 4 (.5) To oberve pek overhoo, he ipu mu be pplied le upo Ce : If ep ipu i pplied. e i p.36 mec. e e = p =.36 m =.8 m The give CRO i o-orge which me i c record he repoe. So i i o poible o red he rie repoe prciclly i will pper oly for oe i.e.,.8 mec. Afer.8 mec CRO oly diply co vlue. Therefore, he co ipu e hould be pplied gi d gi (i.e., qure wve) o h rie repoe pper lwy o CRO cree. Ce : If qure wve ipu i pplied. e i 4

15 For obiig rie repoe, T ( p d ) Opio (B) : For f 5 Hz T m 5 m.8 m The oupu of he CRO i co i bewee.8 m o m how below, e e.8 (m) Opio (C) : For f 3 Hz T.67 m 3 The oupu of he CRO i co i bewee.8 m o.67 m which i very le how below, e e.8.67 (m) Opio (D) : For f3 khz T3.5 m 3 Sice,.5 m i very much le h.8 m hece, he oupu wveform will o pper o he CRO cree. T Hece 3 Hz frequecy qure wve i he mo uible.67 mec becue oly for hi frequecy rie repoe will lwy pper o CRO cree. Therefore, i would be be o hve e i qure wve of 3 Hz. Hece, he correc opio i (C). Queio 53 [Prcice Book] [GATE EE 7 IIT-pur : Mrk] Coider he feedbck yem how below which i ubjeced o ui ep ipu. The yem i ble d followig prmeer 4,, 5 rd/ec d.7. The edy e vlue of Z i p i 5

16 i Z p A. (A) (B).5 (C). (D) (A) Sol. Give : R () The give feedbck yem i how below. R () E() i Z p C () G() The equivle repreeio c be drw how i figure below. R() E () p i G() C () Cloed-loop rfer fucio for egive uiy feedbck i give by, C () G'() T () R () G'() i p G() C () R () i p G( ) From he figure, he error igl c be wrie, i p G() E () R() C() R () i p G( ) E () i p G( ) From he figure, () i i Z E() ( p i) G( ) 6

17 i Z() ( p i) i Z() ( p i) Sedy e vlue c be clculed, z lim z( ) lim Z( ) i i Hece, he correc opio i (A). Queio 63 [Prcice Book] [GATE EE IIT-Mdr : Mrk] A wo-loop poiio corol yem i how below. R () Moor ( ) Y() k Tcho-geeror A. Sol. The gi k of he Tcho-geeror ifluece mily by (A) pek overhoo. (B) url frequecy of ocillio. (C) phe hif of he cloed loop rfer fucio very low frequecy. (D) phe hif of he cloed loop rfer fucio very high frequecy. (A) Give : A wo-loop poiio corol yem i how below. R () Moor ( ) Y() k Tcho-geeror O olvig he ier loop, we ge ( ) k ( k) ( ) Now, he overll rfer fucio c be wrie, Y() R() ( k )..(i) 7

18 Trfer fucio for ecod-order yem wih ui ep ipu i give by, C () R () where, dmpig rio, url gulr frequecy O comprig equio (i) d equio (ii), we ge ω d ξω k k So ξ Mximum pek overhoo i give by, MPO e MPO,..(ii)..(iii) Pek overhoo deped o dmpig fcor ξ d ξ i proporiol o gi from equio (iii). So, gi k of he Tcho-geeror ifluece mily by pek overhoo. Hece, he correc opio i (A). Queio [Work Book] [GATE EC/EE/IN 3 IIT-Bomby : Mrk] () The ope-loop rfer fucio of dc moor i give. Whe coeced i feedbck V () how below, he pproxime vlue of h will reduce he ime co of he cloed loop yem by oe hudred ime compred o h of he ope-loop yem i V () R () + () (A) (B) 5 (C) (D) A. (C) Sol. () Give : cloed loop ope loop where repree ime co d V () V () R () + () C () () For fir order yem loop rfer fucio i comprig wih R () ope loop V () Cloed-loop rfer fucio for egive uiy feedbck i give by, G () T() G ( ) Here G () () R () ( ) 8

19 Dividig umeror d deomior by () R () For fir order yem loop rfer fucio i cloed loop We hve cloed loop ope loop Hece, he correc opio i (C). Queio 67 [Prcice Book] C (). O comprig wih R () () R() we ge [GATE EC 5 (Se-3) IIT-pur : Mrk] The poiio corol of DC ervo-moor i give i he figure. The vlue of he prmeer re N-m/A, R, T L.H, J 5kg-m, B N-m/(rd/ec) d V/(rd/ec). The edy-e poiio repoe (i rdi) due o ui impule diurbce orque Td i. Td () b V () T R L J B () b A..5 Sol. Give : T N-m/A, R, L.H, J 5 kg-m, B N-m/(rd/ec), Td () V/(rd/ec) b V () T R L J B () b For ui impule Td () X() J B T () D b T ( J B ) ( R L ) 9

20 X() TD () ( JB)( R L) bt TD () () ( JB)( R L) bt () ( JB)( R L) bt Sedy e repoe c be clculed uig fil vlue heorem. Applyig fil vlue heorem, () lim ( ) lim ( JB )( R L ) T () D ().5 BR bt Hece, he correc wer i.5. Queio 68 [Prcice Book] [GATE EC 6 (Se-) IISc-Bglore : Mrk] I he feedbck yem how below G (). ( ) b T The ep repoe of he cloed -loop yem hould hve miimum elig ime d hve o overhoo. r G () y The required vlue of gi o chieve hi i. A. Sol. Give : G (), G'( ) G( ) The ecod order cloed loop rfer fucio wih egive uiy feedbck i give by, Y () G'() R() G'() Miimum elig ime d o overhoo From equio (i), Ad. (i)

21 IES Objecive Soluio Queio 4 [Prcice Book] [IES EE 99] Dmpig fcor d u-dmped url frequecy for he poiio corol yem i give by (A) J, J repecively (B), repecively fj J A. Sol. f J (C), repecively (D), J J f J repecively (C) Chrceriic equio of poiio corol yem i give by, f J J.(i) Sdrd form of ecod order chrceriic equio i give by,.(ii) O comprig equio (i) d (ii), we ge f J J J f J rd/ec J f J Hece, he correc opio i (C). Queio 4 [Prcice Book] [IES EE 995] A. Coider yem how i he give figure. If he yem i diribued o h c(), he c () for ui ep ipu will be (A) (B) (C) (D) (C) Sol. Give : C () U () C () U () C () Tkig ivere Lplce rform, we ge c () L c () c() Hece, he correc opio i (C).

22 Queio [Work Book] [IES EC 3] A. The ui impule repoe of yem hvig rfer fucio (A) (D) Sol. Give : (B) C () d R () R () C () Tkig ivere Lplce rform, we ge c () e Time co Time co i he ime which c () e.37 So, Hece, he correc opio i (D). (C) i how bove. The vlue of i: (D) Queio 54 [Prcice Book] [IES EE 5] Which oe of he followig i he edy e error of corol yem wih ep error, rmp error d prbolic error co p, v d repecively for he ipu ( )3 u( )? (A) (B) (C) (D) A. (D) Sol. Give : p () 3( ) () r u Tkig Lplce rform, we ge 3 6 R () 3 p Ui Impule Repoe p p

23 Sedy e error i give by, R() e lim lim G ( ) G ( ) e e lim 3 6 lim G ( ) G ( ) 3 6 lim G ( ) lim G ( ) 3 6 e p Hece, he correc opio i (D). Queio 7 [Prcice Book] [IES EE 7] For uiy feedbck corol yem wih forwrd ph rfer fucio G (), wh i error 5 rfer fucio We () ued for deermiio of error coefficie? 5 ( 5) (A) (B) (C) (D) A. (C) Sol. Give : G (), H () 5 W R () () e C () 5 We () R() G() We () R () 5 We () 5 R() 5 Hece, he correc opio i (C). Queio 77 [Prcice Book] [IES EE 8] A. Sol. I he ime domi lyi of feedbck corol yem which oe pir of he followig i o correcly mched? (A) Uder dmped : Miimize he effec of olieriie (B) Domi pole : Trie die ou more rpidly (C) Fr wy pole o he lef hlf of ple : Trie die ou more rpidly (D) A pole er o he lef of domi complex pole d er zero : Mgiude of rie i mll (B) Domi Pole : The pole h re cloe o he imgiry xi i he lef-hlf -ple give rie o rie repoe h will decy relively lowly, where he pole h re fr wy from he xi (relive o he domi pole) correpod o f-decyig ime repoe. Hece, he correc opio i (B). 3

24 Queio 89 [Prcice Book] [IES EE 9] I fluid flow yem wo fluid re mixed i pproprie proporio. The cocerio he mixig poi i y() d i i reproduced wihou chge, T d ecod ler he moiorig poi b(). Wh i he rfer fucio bewee b() d y()? (Where S i dice bewee moiorig poi d mixig poi) Td (A) e T d (B) e T d (C) e d (D) e T A. (C) Sol. Give : y() b( T d ) Tkig Lplce rform, we ge d Y () e T B () Y () T d e B () Hece, he correc opio i (C). Queio 97 [Prcice Book] [IES EC, ] A. Sol. Whe wo ideicl fir order yem hve bee ccded o-iercively he ui ep repoe o he yem will be (A) Over-dmped (B) Uder-dmped (C) U-dmped (D) Criiclly-dmped (D) Ccdig of wo fir order yem wih o-iercively R R Amplifier wih gi = Vi C C V Vi () RC RC V () V () T() Vi () ( RC) The pole-zero digrm of bove rfer fucio j pole /RC Two repeed pole. So, yem i criicl dmped ( = ). Hece, he correc opio i (D). Queio 8 [Prcice Book] [IES EC ] Aerio (A) : A ecod order yem ubjeced o ui impule ocille i url frequecy. Reo (R) : Impule ipu coi frequecie from o. Code : (A) Boh A d R re idividully rue d R i he correc explio of A. 4

25 (B) Boh A d R re idividully rue bu R i o he correc explio of A. (C) A i rue bu R i fle. (D) A i fle bu R i rue. A. (D) Sol. For, ecod order yem ubjeced o ui impule ocille i url frequecy bu i i o lwy rue. U-dmped c () Mrgil ble Imgiry j The impule repoe coi ll he frequecy compoe hvig frequecy repoe L () Hece, he correc opio i (D). Queio 3 [Prcice Book] [IES EE 4] The domi pole of ervo-yem re loced ( j). The dmpig rio of he yem i (A) (B).8 (C).77 (D).6 A. (C) Sol. Give : Pole re loced j. (i) Pole of ecod-order rfer fucio i give by, j. (ii) O comprig equio (i) d (ii), we ge d ( ) 4 4. (iii) 4. (iv) From equio (iii) d (iv), we ge Hece, he correc opio i (C). 5 f

26 Queio 35 [Prcice Book] [IES EE 4] A uiy feedbck ecod order corol yem i chrcerized by he ope loop rfer fucio G (), H () J ( B) J = mome of ieri, B = dmpig co d = yem gi The rie repoe pecificio which i o ffeced by yem gi vriio i (A) Pek overhoo (B) Rie ime (C) Selig ime (D) Time o pek overhoo A. (C) Sol. Give : G () d H () J ( B) Chrceriic equio i give by, GH ( ) ( ) B J B J J. (i) Chrceriic equio for drd ecod-order yem i give by,. (ii) O comprig equio (i) d (ii), we ge d J B J Pek overhoo i give by, M p e Sice deped o gi o pek overhoo ffeced by. Rie ime i give by, r d B J Sie d deped o gi o rie ime ffeced by, elig ime i give by, 4 4 8J B B J Selig ime i idepede of gi o will o be ffeced by. Hece, he correc opio i (C). 6