CONTROL SYSTEMS. Chapter 3 : Time Response Analysis. [GATE EE 1991 IIT-Madras : 2 Mark]

Transcription

1 CONTROL SYSTEMS Chper 3 : Time Repoe lyi GTE Objecive & Numericl Type Soluio Queio 4 [GTE EC 99 IIT-Mdr : Mrk] 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 () (B).5 (C) (D). () Sol. 4 Give : G, d r () u () Queio 6 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 (). lerively, 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. [GTE EE 99 IIT-Mdr : Mrk] fir order yem d i repoe o ui ep ipu re how i figure below, he yem prmeer d re repecively c (). r () c (). () 5, (B), 5 (C), (D), (ec)

2 . () 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 (). Queio 7 [GTE EC 99 IIT-Delhi : 8 Mrk] Block digrm model of poiio corol yem i how i figure. (ec) lim R( ) lim R () mplifier r () c () Moor (.5) Y() () I bece of derivive feedbck ( ), deermie dmpig rio of he yem for mplifier gi 5. lo fid he edy e error o ui rmp ipu. (b) Fid uible vlue of he prmeer 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 () mplifier Moor (.5) Y() () Give : 5 G () d H() (.5) ( ) Cloed-loop rfer fucio for egive feedbck i give by, G () ( ).. (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.. ( ) ( ) (b) The ope loop rfer fucio c be wrie, OLTF [ ( )] The cloed loop rfer fucio c be wrie, Y () CLTF R () ( ) Trfer fucio for ecod-order yem wih ui ep ipu i give by, C () R () where, dmpig rio, url gulr frequecy..(i).. (ii) 3

4 Velociy error coefficie i give by, lim G( ) H( ).. (iii) (iv) From equio (iii) d equio (iv), we ge 5 d 4.. Queio 8 [GTE 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 v lim [ ( )] Sedy e error i give by, e v.. O comprig equio (i) d (ii), we ge.7 E() 5. Sol. 4 () = (B) = (C) 4 (D) No vlue of () 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, G () M() 5 GH ( ) ( ) ( ) ( 5 )( 4) ( )( 4) ( )( 4) M() 3 ( 5 )( 4)( ) ( )( 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 (). Queio 9 [GTE IN 993 IIT-Bomby : Mrk]. Sol. 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 () (B) (C) (D) () The fir order yem ep repoe i give by, c () ke, k e k mi,.64 k e mi 6ec. Fir order rfer fucio i give by, k G () 6. 5

6 Queio [GTE IN 994 IIT-hrgpur : Mrk] 8 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 () (B). (C). (D).. (B) Sol Give :.6 G( ) T ().69 9 G( ).6..(i) Cloed-loop rfer fucio for egive feedbck i give by, G () 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). Queio 4 [GTE EC 997 IIT-Mdr : 5 Mrk] The figure how he block digrm repreeio of corol yem. The yem i block h impule repoe h() 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 () 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 h() e u() Tkig Lplce rform, we ge H() Impule repoe hb () e u() Tkig Lplce rform, we ge HB() 6

7 () k, G() H() HB() Cloed-loop rfer fucio for egive feedbck i give by, Y() G() X () G() H() ( )( ). 3 3 ( )( ) (b) k Y () B H () X() ( )( ) d B H() Tkig ivere Lplce rform, we ge Impule repoe h () ( e e ) u (). (c) Vlue of k for uble yem k 3k Chrceriic equio 3k Rouh Tbulio : + k 3 + k Queio 33. For ibiliy k k yem i uble.. [GTE EC 999 IIT-Bomby : Mrk] If he cloed-loop rfer fucio of uiy egive feedbck yem i give by T... he he edy e error for ui rmp ipu i () (B) (C) (D) zero (D) T d r() = u()... Sol. Give : (i) 7

8 Cloed-loop rfer fucio for egive feedbck i give by, G () 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 36 v The block digrm of feedbck yem i how i he figure. Ipu G ( 3)... (ii) [GTE 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 Queio 46 Cloed-loop rfer fucio for egive feedbck i give by, G G () ( 3) 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..(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. (c) G' G G' Dmped frequecy of ocillio i give by, d 3.53 (.44) 3.97 T T.96 ec.. C () 4 ecod-order yem h he rfer fucio R () 4 4 he repoe c () of he yem i repreeed by ().5 Sep Repoe (B). [GTE EC 3 IIT-Mdr : Mrk]. wih r () he ui-ep fucio, Sep Repoe mpliude.5 mpliude Time (ec) Time (ec) 9

10 (C). Sep Repoe (D). Sep Repoe mpliude Time (ec). (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 6 [GTE IN 4 IIT-Delhi : Mrk] 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 ()(.8,.8) (B) (.5,.56) (C) (.,.66) (D)(.,.89). (B) Sol. Give : MPO = 6 % [By deful pek overhoo] Percege MPO i give by, % MPO e 6 e l(.6) mpliude.5 5 Time (ec)

11 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 (.5) e e (.5) e e Hece, he correc opio i (B). Queio 67 [GTE 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 i Z p. () (B).5 (C). (D) () Sol. Give : R ()

12 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( ) 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 ().

13 Queio 83 wo-loop poiio corol yem i how below. R () Moor ( ) [GTE EE IIT-Mdr : Mrk] Y() k Tcho-geeror. Sol. The gi k of he Tcho-geeror ifluece mily by () 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 () Give : 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) 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,..(i)..(ii)..(iii) MPO e MPO, 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 (). 3

14 Queio 85 [GTE 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 () + () () (B) 5 (C) (D). (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 ope loop R () V () Cloed-loop rfer fucio for egive uiy feedbck i give by, G () G ( ) Here G () () R () ( ) Dividig umeror d deomior by () R () C For fir order yem loop rfer fucio i (). O comprig wih () we ge R () R() cloed loop We hve cloed loop ope loop Hece, he correc opio i (C). 4

15 Queio 88 [GTE EC 4 (Se-) IIT-hrgpur : Mrk] For he followig feedbck yemg (). The % elig ime of he ep repoe i ( )( ) required o be le h ecod. r C () G () y. Which oe of he followig compeor C () chieve hi? () 3 5 (C) Sol. Give :.3 (B) 5 G () ( )( ) (C) ( 4) (D) Cloed-loop rfer fucio for egive uiy feedbck i give by, G () ( )( ) T () G ( ) ( )( ) ( )( ) 33 Trfer fucio for ecod-order yem i give by, C () R () (i)..(ii) where, dmpig rio, url gulr frequecy O comprig equio (i) d (ii), we ge 3.5 For % olerce bd elig ime i give by, ec.5 Thu, i order o mke elig ime le h ec. Proporiol derivive (PD) coroller hould be ued. From opio (C) where C() = ( + 4). New rfer fucio T'( ) c be wrie, ( 4) CG () () ( )( ) T'( ) CG ( ) ( ) ( 4) ( )( ) ( 4) ( 4) T'( ) 38 5 O comprig equio (i) d (iii), we ge 5.5..(iii) 5

16 For % olerce bd elig ime i give by, 4 4.6ec <.5 Hece, he correc opio i (C). Queio 94 [GTE 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/, 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..5 Sol. Give : T N-m/, R, L.H, J 5kg-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 ) 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. pplyig fil vlue heorem, () lim ( ) lim ( JB )( R L ) bt ().5 BR bt Hece, he correc wer i.5. T () D 6

17 Queio 97 I he feedbck yem how below G (). ( ) [GTE EC 6 (Se-) IISc-Bglore : Mrk] 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.. 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), d. (i) IES Objecive Soluio Queio 4 [IES EE 99] Dmpig fcor d u-dmped url frequecy for he poiio corol yem i give by () J, J repecively (B), repecively fj J. Sol. f (C), J J J repecively (D), f 7 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)

18 O comprig equio (i) d (ii), we ge f J J f J J rd/ec J f J Hece, he correc opio i (C). Queio 4 [IES EE 995] Coider yem how i he give figure. If he yem i diribued o h c(), he c () for ui ep ipu will be () (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). Queio 4 [IES EC 3]. The ui impule repoe of yem hvig rfer fucio () (D) Sol. Give : (B) C () d R () R () C () Ui Impule Repoe 8 (C) i how bove. The vlue of i: (D)

19 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). Queio 56 [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( )? () (B) (C) (D) p p p p. (D) Sol. Give : r () 3( ) u () Tkig Lplce rform, we ge 3 6 R () 3 Sedy e error i give by, R() e lim lim G ( ) G ( ) e e 3 6 lim lim G G 3 6 lim G ( ) lim G ( ) ( ) ( ) 3 6 e p Hece, he correc opio i (D). Queio 76 [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) () (B) (C) (D) (C) Sol. Give : G (), H () 5 9

20 W R () () e C () 5 We () R() G() We () R () 5 We () 5 R() 5 Hece, he correc opio i (C). Queio 8 [IES EE 8] I he ime domi lyi of feedbck corol yem which oe pir of he followig i o correcly mched? () 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) pole er o he lef of domi complex pole d er zero : Mgiude of rie i mll. (B) Sol. 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). Queio 93 [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 () e T d (B) e T d (C) e d (D) e T. (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 [IES EC, ] Whe wo ideicl fir order yem hve bee ccded o-iercively he ui ep repoe o he yem will be () Over-dmped (B) Uder-dmped (C) U-dmped (D) Criiclly-dmped. (D) Sol. Ccdig of wo fir order yem wih o-iercively

21 R R mplifier wih gi = Vi C C V Vi () RC RC V () V () Vi () ( RC) The pole-zero digrm of bove rfer fucio j Hece, he correc opio i (D). Queio [IES EC ] erio () : ecod order yem ubjeced o ui impule ocille i url frequecy. Reo (R) : Impule ipu coi frequecie from o. Code : () Boh d R re idividully rue d R i he correc explio of. (B) Boh d R re idividully rue bu R i o he correc explio of. (C) i rue bu R i fle. (D) i fle bu R i rue.. (D) Sol. For, ecod order yem ubjeced o ui impule ocille i url frequecy bu i i o lwy rue. U-dmped pole /RC Two repeed pole. So, yem i criicl dmped ( = ). c () Mrgil ble Imgiry j The impule repoe coi ll he frequecy compoe hvig frequecy repoe L() Hece, he correc opio i (D). f

22 Queio 36 [IES EE 3] forcig fucio fucio i ( ) u( ) i pplied o lier yem. The Lplce rform of he forcig () 3 e (B) e (C) e e (D). (D) Sol. Give : f() ( ) u( ) ( ) u( ) f () ( ) u( ) u( ) Tkig Lplce rform, we ge e e F () e 3 3 e 3 Hece, he correc opio i (D). Queio 39 [IES EE 4] The domi pole of ervo-yem re loced ( j). The dmpig rio of he yem i () (B).8 (C).77 (D).6. (C) Sol. Give : Pole re loced j.(i) Pole of ecod-order rfer fucio i give by, O comprig equio (i) d (ii), we ge j.(ii) ( ) 4 4. (iii) d 4.(iv) From equio (iii) d (iv), we ge Hece, he correc opio i (C). Queio 43 [IES EE 4] 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 () Pek overhoo (B) Rie ime (C) Selig ime (D) Time o pek overhoo (C)

23 Sol. Give : G () d H( ) J ( B) Chrceriic equio i give by, GH ( ) ( ) J B B J J.(i) Chrceriic equio for drd ecod-order yem i give by,.(ii) O comprig equio (i) d (ii), we ge J d 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). 3