CONTROL SYSTEMS. Chapter 2 : Block Diagram & Signal Flow Graphs GATE Objective & Numerical Type Questions

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1 ONTOL SYSTEMS hapter : Bloc Diagram & Signal Flow Graph GATE Objective & Numerical Type Quetion Quetion 6 [Practice Boo] [GATE E 994 IIT-Kharagpur : 5 Mar] educe the ignal flow graph hown in figure below, to obtain another graph which doe not contain the node e 5. (Alo remove any elf-loop from the reulting graph). t 4 e t e t 3 e 3 t 34 e 4 t 46 e 6 t 5 t 5 t 35 t 54 By eliminating the node e 5, the ignal flow graph i hown below. e 5 t 4 e e t 3 e 3 t 34 e 4 t 46 e 6 t tt tt tt The node equation e et e t t et t e t e ( t t ) et et t e t t t t e e e e tt t5t5 t5t5 t5t5 The redrawn ignal flow graph i hown below. t4 t t t e t5t5 e t 3 e 3 t 34 e 4 t 46 e 6 t 5 5 t35t5 t t 5 5 tt Quetion 0 [Practice Boo] Draw the ignal flow graph for the following et of algebraic equation y ay gy3 y3 ey cy4 y4 by dy4 y Hence, find the gain y. The ignal flow graph i drawn below. tt 5 54 An. [GATE E 998 IIT-Delhi : 5 Mar]

2 GATE AADEMY - Bloc Diagram & Signal Flow Graph y a g y e y3 c y 4 d Forward path : P a Individual loop : L eg, L d, L3 bcg Two non-touching loop : LL egd Determinant : ( LL L3) LL eg d bcg egd Path factor : L d Uing Maon gain formula tranfer function can be written a, () P () So, the tranfer function i y P a( d) P y egd bcgegd b An. Quetion 3 [Practice Boo] [GATE E 000 IIT-Kharagpur : 8 Mar] For the linear, time-invariant ytem whoe bloc diagram i hown in figure with input xt () and yt (), xt () yt () 4 xt () 3 t (a) Find the tranfer function. (b) Find the tep repone of the ytem [i.e. find yt () when xt () i a unit tep function and the initial condition are zero]. (c) Find yt (), if xt () i a hown in figure and initial condition are zero. (a) The given bloc diagram i hown below with input xt () and yt (). xt () yt () 4 xt () 3 t Tranfer function of the integrator i Integrator

3 GATE AADEMY - 3 Bloc Diagram & Signal Flow Graph The ignal flow graph for the given bloc diagram can be drawn a, X() Y() 4 Forward path : P 4 3 Individual loop : L, L 4 3 Determinant : ( LL) Path factor : Forward path touche all the loop. 0 Uing Maon gain formula tranfer function can be written a, () P () So, the tranfer function i Y () P P X() Y() X() (b) Given : xt () ut () Taing the Laplace tranform of x(t), we get X() Y () Tranfer function i X() 43 Y () 3 6 ( 43) ( )( 3) 3 Taing invere Laplace tranform of Y(), we get t 3t yt () e e 3 6 (c) Given : xt () ut ( ) ut ( ) Taing the Laplace tranform of x(t), we get X() ( e e ) Y () Tranfer function i X() 43 Y() ( e e ) ( 43) Y() e e ( )( 3) e e An. An.

4 GATE AADEMY - 4 Bloc Diagram & Signal Flow Graph Taing invere Laplace tranform of Y(), we get t nd yt ( ) L [ Y( )] [ Term] [ Term] t ( t) 3( t) Where, [ Term] ut ( ) e ut ( ) e ut ( ) 3 6 nd ( t) 3( t) [ Term] ut ( ) e ut ( ) e ut ( ) 3 6 Quetion 6 [Practice Boo] [GATE EE 00 IIT-Kanpur : Mar] An electrical ytem and it ignal-flow graph repreentation are hown in the figure repectively. The value of G and H, repectively, are An. Vi () Z () Z () Z () Z () 3 4 I () I () H 3 3 (A) Z () (), Z (B) Z 3() 3(), Z Z() Z3() Z4() Z() Z3() Z() Z3() Z4() Z() Z3() 3 3 () Z () (), Z Z 3() 3(), Z Z() Z3() Z4() Z() Z3() Z() Z3() Z4() Z() Z3() () An electrical ytem and it ignal-flow graph repreentation are hown in the figure repectively. Vi () Z () Z () Z () Z () 3 4 I () I () H From firt figure, Vi ( Z( ) Z3( )) I Z3( ) I (i) ( Z( ) Z3( ) Z4( )) I Z3( ) I 0 (ii) From econd figure, I() GI() (iii) I() GV i () HI() (iv) omparing eq. (ii) and (iii), we get Z3() G Z() Z3() Z4() omparing eq. (i) and (iv), we get H Z3() Z () Z 3() Hence, the correct option i (). Quetion 7 [Practice Boo] [GATE EE 003 IIT-Madra : Mar] The bloc diagram of a control ytem i a hown in figure. The tranfer function G () Y ()/ U () of the ytem i 9 V () 0 V () 0 V () i V () i I () I () G G G3 I () I () G G G3 V () 0 V () 0 u() t Integrator Integrator yt () 3

5 GATE AADEMY - 5 Bloc Diagram & Signal Flow Graph (A) (B) () An. (B) The given bloc diagram i hown below u() t Integrator Integrator yt () 3 Signal flow graph for the given bloc diagram can be drawn a, 3 Forward Path : P, Individual loop : 3 L 8 3, L, L Two Non-touching loop : LL Determinant : ( LL L3) ( LL ) L Path factor : All the loop touch forward path. 0 Uing Maon gain formula tranfer function can be written a, () P () So, the tranfer function i Y() U() Y ( ) U ( ) P P ( 6) ( 9) Y () U() Hence, the correct option i (B). 9 U () Y ()

6 GATE AADEMY - 6 Bloc Diagram & Signal Flow Graph Quetion 0 [Practice Boo] [GATE EE IIT-Kanpur : Mar] The ytem hown in figure below. b 0 c 0 b c P a 0 a can be reduced to the form X Y P An. with (A) X c0 c, Y, Z b 0 b ( a0a) ( b b0) () X c c0, Y, Z ( aa ) 0 The bloc-diagram can be redrawn a, c (3) Z ( c 0 c) (B) X, Y, Zb 0b ( a a ) 0 X c c0, Y, Z b b0 ( a a0) () b () () c 0 (4) (5) P (6) b 0 a0 a The ignal flow graph of the given bloc diagram can be drawn a, () c b () () c 0 (3) (4) a (5) P a 0 cp 0 cp Forward path : P c0 P, P c P a Individual loop : L a, L 0 a a 0 b 0

7 GATE AADEMY - 7 Bloc Diagram & Signal Flow Graph bp 0 bp L3 Pb0, L 4bP 0 0 Determinant : ( L L L3L4) a a b P bp Path factor : All the loop touch forward path. 0, 0 Uing Maon gain formula tranfer function can be written a, () P () So, the tranfer function i cp 0 cp () PP P () a a0 b0p bp ( c0 c ) P cp 0 cp ( a a 0) ( abp ) ( a0 b0p) ( b0 b ) P ( a a ) X Y P 0 (i) xyp () yzp omparing equation (i) and (ii), we get c0 c xy a a0 b0 b yz a a0 Hence option i correct. (ii) Quetion [Practice Boo] [GATE IN IIT-ooree : Mar] A filter i repreented by the ignal flow graph hown in the figure. It input i x(t) and output i y(t). The tranfer function of the filter i An. X () Y () ( ) (A) (B) ( ) (A) The given ignal flow graph i hown in figure below. X () Y () ( ) () Forward path : P, P Z ( )

8 GATE AADEMY - 8 Bloc Diagram & Signal Flow Graph Individual loop : L Determinant : L Path factor : All the loop touch forward path. 0, 0 Uing Maon gain formula tranfer function can be written a, () P () So, the tranfer function i () P P P U() L Hence, the correct option i (A). Quetion 8 [Wor Boo] [GATE EE 04 (Set - 03) IIT-Kharagpur : Mar] The bloc diagram of a ytem i hown in the figure. () G() If the deired tranfer function of the ytem i () () then Gi () (A) (B) () / 3 An. (B) () Given : () The bloc diagram can be converted into ignal flow graph a hown in below. () G() Forward path : P G () G () G () Individual loop : LG(), LG(), L3 G () Determinant : ( L L L3) G() G() G () G ( ) G ( ) Path factor : All the loop touch forward path. 0

9 GATE AADEMY - 9 Bloc Diagram & Signal Flow Graph Uing Maon gain formula tranfer function can be written a, () P () So, the tranfer function i () P G () P () G () G ( ) G ( ) Let G () () () Hence, the correct option i (B). Quetion 6 [Practice Boo] [GATE IN 04 IIT-Kharagpur : Mar] onider the control ytem hown in figure with feed forward action for rejection of a meaurable diturbance d(t). The value of K, for which the diturbance repone at the output y(t) i zero mean, i K dt () r() t 50 yt () (A) (B) () An. Given : y( ) 0 The given diagram i hown below. dt () K r() t 50 yt () onider rt () 0then repone due to dt () can be written a, Y () 0 Y ()50 KD () D () 50 K Y() D() Y() K D () 50 onider tep diturbance, D () K Y() 50 Zero mean = Zero Average = Zero contant = Zero teady tate Steady tate mean final value i.e. to get teady tate value we can apply final value theorem. y( ) lim y( t) lim Y( ) t 0

10 GATE AADEMY - 0 Bloc Diagram & Signal Flow Graph K K lim K 0 K 50 Hence, the correct option i. Quetion 8 [Practice Boo] [GATE E 05 (Set - 0) IIT-Kanpur : Mar] By performing cacading and/or umming/differencing operation uing tranfer function bloc G () and G (), one ANNOT realize a tranfer function of the form G () (A) G() G() (B) G () () G() G() G () An. (B) Since we have only G () and G (). For option (A), For option (), G G GG G G() G() G () G G G G For option, G G GG G Hence, the correct option i (B). Quetion 9 [Wor Boo] [GATE EE 05 (Set - 0) IIT-Kanpur : Mar] For the ignal-flow graph hown in the figure, which one of the following expreion i equal to the Y () tranfer function? X () X ( ) 0 X () X () G G G G Y() An. G (A) G ( G ) G (B) G ( G ) (B) Given ignal flow graph i hown below. G () GG G GG

11 GATE AADEMY - Bloc Diagram & Signal Flow Graph X () 0 X () X () G G Y() G G Y() Forward Path : P G Individual Loop : L GG, L G Determinant : GG G G( G) Path Factor : All the loop touch forward path. Uing Maon gain formula tranfer function can be written a, P G T() G( G) Hence, the correct option i (B). Quetion 9 [Practice Boo] Find the tranfer function Y () X() of the ytem given below. G [GATE EE 05 (Set - 0) IIT-Kanpur : Mar] G X() H Y() Y() An. G G G G (A) (B) HG HG HG HG () From the bloc diagram Y G( X HY) G( X HY) Y X( GG) HY( G G) Y H( GG) X( G G) Y G G X H( G G) Hence, the correct option i (). ESE Objective Type Quetion G G () HG ( G) G G HG ( G) Quetion [Practice Boo] [IES E 99] From the ignal flow graph hown in the figure, the value of x 6 i x a b G d e x x 4 x 6 5 x c x 3

12 GATE AADEMY - Bloc Diagram & Signal Flow Graph An. (A) de ax bx cx3 () 3 ( ) (B) ( abc)( xx x3)( d e) ( ax bx cx )( d e) abc de( xx x3) (A) From the given ignal flow graph x4 axbx cx3 x6 ex5, x5 dx4 x6 dex4 x de( ax bx cx ) 6 3 Hence, the correct option i (A). Quetion [Practice Boo] [IES EE 994] In the feedbac ytem hown in given figure, the noie component of output i given by (aume high loop gain at frequencie of interet) G() () H () H () An. N() N() N () N() (A) (B) () H() H() H() H() (B) Given : The ignal flow graph of a given ytem i hown in figure below. G() () N () H () H () H () H () With () 0ignal flow graph, N() H () G () N() H () Forward path : P ( H( )) G( ) G( ) H( ) Individual loop : L GH () () H() Determinant : L GH () () H() Path factor : 0 Uing Maon gain formula tranfer function can be written a, n () GH () () P N() G() H() H() () GH () () N () GH () H()

13 GATE AADEMY - 3 Bloc Diagram & Signal Flow Graph Aume high loop gain at frequencie of interet, GH () () H() () GH () () N() G() H() H() H() () N () () N () H() H() Hence, the correct option i (B). Quetion 0 [Wor Boo] [IES EE 999] In the ytem hown in the given figure, to eliminate the effect of diturbance D() on, the tranfer function G () hould be d D () Gd () () ( 5) An. (A) ( 0) ( 0) 0 (B) () (B) Uing Maon gain formula tranfer function can be written a, 5 0 Gd ( )( ) () 0 ( 5) D () ( 5) For the noie to become zero, 5 0 Gd ( ) 0 0 ( 5) 0 ( 5) ( 0) Gd () Hence, the correct option i (B). 0 ( 0) Quetion 9 [Practice Boo] [IES EE 000] n () A cloed-loop ytem i hown in the given figure. The noie tranfer function [ n() = output N() correponding to noie input N()] i approximately () G () H () H () for G ( ) H ( ) H ( ) (B) for G( ) H( ) H( ) H() for G ( ) H ( ) H ( ) H () H () for G ( ) H ( ) H ( ) G () H () H () (A) G() H() () N()

14 GATE AADEMY - 4 Bloc Diagram & Signal Flow Graph An. (B) Given : The ignal flow graph of a given ytem i hown in figure below. () G () H () H () With () 0ignal flow graph, N() H () G () N() Forward path : P ( H( )) G( ) G( ) H( ) Individual loop : L G() H() H() Determinant : L G() H() H() Path factor : 0 Uing Maon gain formula tranfer function can be written a, n () P G() H() P N() G() H() H() Aume high loop gain at frequencie of interet, G() H() H() () G() H() N() G() H() H() H() Hence, the correct option i (B). Quetion 37 [Practice Boo] [IES EE 004] onider the following three bloc diagram A, B and hown below. H () Bloc Diagram A Bloc Diagram B

15 GATE AADEMY - 5 Bloc Diagram & Signal Flow Graph An. Bloc Diagram Which one of the following tatement i correct in repect of the above bloc diagram? (A) Only A and B are equivalent (B) Only A and are equivalent () Only B and are equivalent A, B and are equivalent For bloc diagram Eliminating feedbac loop and erie bloc we get, Shifting tae off point after, we get, Eliminating feedbac loop and reducing erie bloc diagram we get, Thu, bloc diagram A and B are equivalent.

16 GATE AADEMY - 6 Bloc Diagram & Signal Flow Graph For bloc diagram B : Eliminating the feedbac loop and reducing erie bloc we get, Hence, the correct option i. Quetion 40 [Practice Boo] [IES EE 005] G G An. G3 H What i the overall tranfer function of the bloc diagram given above? GG GG 3 GG GG 3 GG GG (A) (B) () GG GG 3 GH G3H GG3H (A) Given : The bloc diagram of the given ytem i hown below. G G 3 3 G3 H Fig. (a) earrange the above bloc diagram a how in below figure (b) G G G 3 Fig. (b) Eliminating the feedbac loop and forward path fig. (b) we get, G G G3 GH H Fig. (c) By olving the cacade combination of figure (c) we get,

17 GATE AADEMY - 7 Bloc Diagram & Signal Flow Graph ( G G ) G GH 3 Fig. (d) The overall tranfer function of the ytem i given by, ( G G3) G GH GG GG3 GH Hence, the correct option i (A). Alternatively : The ignal flow graph of a given bloc diagram i given below. G G Forward path: P G G GG G3 H P G3G GG3 Individual loop : L GH Determinant : ( L ) ( GH) GH Path factor : 0, 0 Uing Maon gain formula tranfer function can be written a, n P P T.F. P GG GG 3 T.F. GH GG GG 3 T.F. GH Hence, the correct option i (A). Quetion 44 [Practice Boo] [IES EE 007] The bloc diagram for a particular control ytem i hown in the figure. What i the tranfer function () for thi ytem? () () An. a b (A) a (B) a () b b b a Given : The bloc diagram of the given ytem i hown below. () b a a b Fig. (a)

18 GATE AADEMY - 8 Bloc Diagram & Signal Flow Graph earrange the above bloc diagram a hown in figure below. () Eliminating the feedbac loop and forward path, we get () b a By olving the two cacade combination of two bloc, we get () b a The overall tranfer function i given by, b () b () a a () b () a Hence, the correct option i. Alternatively : The ignal flow graph of a given bloc diagram i hown below. () a Forward path : P P b b Individual loop : L a a Determinant : ( L ) ( a ) a Path factor : 0, 0 Uing Maon gain formula tranfer function can be written a, n P P T.F. PKK K b b T.F. a a ( a)/ b T.F. ( b)/ a Hence, the correct option i. a Quetion 47 [Practice Boo] [IES EE 009] b b () () K

19 GATE AADEMY - 9 Bloc Diagram & Signal Flow Graph An. For what value of K, are the two bloc diagram a hown above equivalent? (A) (B) () ( ) Given : The bloc diagram of the given two ytem i hown below. () () K Fig. (a) Fig. (b) For the option A, B and condition for both the ytem equivalent are not atified. Therefore, we tae K. From the figure (a) the tranfer function i given by, () () earrange the bloc diagram of figure (b) with () K, we get, educing the erie bloc and forward path, we get () () ( )( ) () [( )( )] () ( )( ) () () Hence, the correct option i.

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