CONTROL SYSTEMS Chaper : Sae Space Repone GATE Objecive & Numerical Type Soluion Queion 5 [GATE EE 99 IIT-Bombay : Mark] Conider a econd order yem whoe ae pace repreenaion i of he form A Bu. If () (), hen yem i (A) Conrollable (B) Unconrollable (C) Obervable (D) Unable An. (B) Sol. Le a b () e u () c d () f. (i) Sae equaion i given by, A Bu. (ii) On comparing equaion (i) and (ii), we ge A b, B e c d f a be ae bf AB c d f ce df Since () () and () () e f and a b c d The conrollabiliy mari i defined a, n [ B AB A B... A B] Where n number of ae variable If, hen he yem i conrollable. If, hen he yem i no conrollable. Q C e ae bf f ce df Q ce def aef bf C ce de ae be [ e f] e cd ab e abab abcd Thu, he yem i no compleely conrollable. Hence, he correc opion i (B).
Queion 7 The mari of any ae-pace equaion for he ranfer funcion i R () [GATE EE 994 IIT-Kharagpur : Mark] C () of he yem hown below in figure R () C () An. Sol. (A) (B) (C) (D) [] (C) Tranfer funcion of given yem T ( ). C () R () C() C() R() dc() c () () r d Taking c () So, dc() () r c () d () r [ ] [ ][ ] [ r( )] The ae pace mari i [ ]. Hence, he correc opion i (C). Alernaively : Signal flow graph for given block diagram i hown below. u() y c () () r r( ).(i) The ae pace repreenaion of yem i given by, A Bu.(ii) On comparing equaion (i) and (ii), we ge A Hence, he correc opion i (C).
Queion 8 [GATE IN 994 IIT-Kharagpur : 5 Mark] A fir order mari differenial equaion of a yem i given a u, y [ ] (i) Find he ranfer funcion of he yem. (ii) Find he oluion of he ae and he oupu when he inpu i a uni ep and he iniial condiion of he ae i. Sol. Given : u, [ ] y. (i) Sae equaion i given by, A Bu.(ii) Oupu ae equaion i given by, y C Du. (iii) On comparing equaion (i), (ii) and (iii), we ge A, B, C [ ] (i) Tranfer funcion i, T() () (ii) Given : (), u () uni ep, U () Sae equaion for a uni-ep inpu () ZIR ZSR.(iv) X() () I A I A BU () X() () Calculaion of ZIR : () ZIR I A () ZIR ZIR Calculaion of ZSR : ZSR I A BU ()
Queion ZSR ZSR / ZSR From equaion (iv), we ge X() ( )( ) ( )( ) X() ( )( ) ( )( ) Applying parial ranform, we ge X() ( ) ( ) ( ) ( ) X() ( ) ( ) ( ) ( ) X() X() Taking invere Laplace ranform, we ge () u() e u(), () e u() y () () () y() u() An. [GATE EC 996 IISc-Bangalore : 5 Mark] Obain a ae pace repreenaion in diagonal form for he following yem dy dy dy 6 6y6 u( ) d d d dy dy dy Sol. Given : 6 6y6 u( ) d d d Taking Laplace ranform, we ge Y() 6 Y() Y() 6 Y() 6 U() Y () 6 T () U () 66 Y() 6 b b b T() U() ( )( )( ) b, b 6, b 4
b b b Y U U U Y () X() X() X() () () () () Le y () () () ().(i) b Where X () U() b X () U() b X () U() X () X () bu() Taking invere Laplace, we ge bu() ().(ii) Similarly, bu () ().(iii) bu() ().(iv) From (i), (ii), (iii) and (iv), we ge b b u b Y( ) [ ] An. Queion [GATE EC 996 IISc-Bangalore : 5 Mark] From he ignal flow graph hown in figure. Obain he ae pace model wih,, and 4 a ae variable and wrie he ranfer funcion direcly from he ae pace model. 6 4 / U () Y () 4 5 7 9 Sol. The given ignal flow graph i hown below. 6 4 4 / U () Y () 4 5 7 9 5
From ignal flow graph, 4 4 u4579 y 4 64 From he above equaion he ae pace repreenaion in vecor mari noaion i given below. [ u ] 4 9 7 5 4 [ y] [ 4 6] 4 (i) Noice ha he la row of A conain he negaive value of he coefficien of he homogeneou par of he differenial equaion in acending order, ecep for he coefficien of he highe-order erm, which i uniy. (ii) B i a column mari wih he la row equal o and he re of he elemen are all zero. Thi mari i known a a phae variable canonical form. I i alo referred a conrollable canonical form (CCF). (iii) C i a row mari wih coefficien of numeraor in acending order of ranfer funcion. The ranfer funcion can be wrien a, Y ( ) 64 TF An. U 4 () 5 7 9 Queion [GATE EC 997 IIT-Madra : Mark] A cerain linear ime invarian yem having he ae and he oupu equaion given below. u y dy If (), (), u(), hen i d (A) (B) (C) (D) None of hee An. (A) Sol. Given : u y () () () 6
() ( ) () () y dy d dy () () d Hence, he correc opion i (A). Queion 4 [GATE EC 997 IIT-Madra : 5 Mark] For he circui hown in he figure choe ae variable a,, o be i (), L v (), c i () L 4 4 e () H F V c H i L i L Sol. Wrie he ae equaion A B[()] e The given circui diagram i hown below. i i i 4 4 e () H F V c H i L i L Applying KVL in loop, we ge di L e () i e () ( il i) e () il i.(i) d dil Vc d i.(ii) 4 Uing (ii) in (i), we ge dil Vc di L d e () il d 4 dil dil e () il Vc d d dil 4 e () il Vc.(iii) d dvc i il.(iv) d 7
Uing (ii) in (iv), we ge dil Vc dv C d il d 4 dvc dil Vc il.(v) d 4 d 4 Uing (iii) in (v), we ge dvc 4 e () il Vc Vc il d 4 4 dvc e () il Vc il.(vi) d 6 6 dil vc 4iL.(vii) d From equaion (iii), (vi), (vii) he ae equaion in vecor mari noaion i 4 il V C e( ) 6 An. 6 i L 4 Queion 6 [GATE EE 998 IIT-Delhi : 5 Mark] The ae pace repreenaion of a yem i given by, 5 6 Find he Laplace ranform of he ae raniion mari. Find alo he value of a if () (). Sol. 5 Given : 6. (i) () Sae equaion i given by, A Bu.(ii) On comparing equaion (i) and (ii), we ge 5 A 6 The ae raniion mari i given by, () L I A 5 5 [ I A] 6 6 Adj[ I A] [ I A] I A Adj[ I A] 6 5 8
I A [( 5) 6] 5 6 ( )( ) () I A ( )( ) ( )( ) An. 6 5 ( )( ) ( )( ) Taking invere Laplace ranform, we ge e e e e () 6( e e ) e e Calculaion of ZIR : ZIR () () e e e e ZIR 6( e e ) e e () e e ZIR = () 6( e e ) () e e. An. Queion [GATE EE IIT-Kharagpur : 5 Mark] A e e e Conider he equaion () A() given : e e e e (i) Find a e of ae () and () uch ha (). (ii) Show ha I A (), ( ) (iii)from (), find he mari A. A e e e Sol. Given : e and e e e () (i) Soluion of homogenou equaion i given by, () () () () e e e () () e e e () () ( e e ) () e ()..(i) () ( e ) () ( e e ) ()..(ii) Subiuing and in equaion (i), () ( e e ) () e () ().74 ().7 () () ( e e ) () e () ().46 ().7 () () e () ( e e ) ().7 () Sum of ae () ().745 ().7 ().7 () An. 9
(ii) Laplace ranform of ae raniion mari can be wrien a, L[()] [ I A] ( ) ( ) L[ ( )] ( ) ( ) () [ IA] Adj[ I A] Adj[ I A] ( ) I A..(iii) where A A () e () L [ e ] L [ ( )] Hence proved. (iii) Laplace ranform of ae raniion mari can be wrien a, [ I A] ( ) From equaion (iii), Adj[ I A] [ I A] a b I A c d a b ( ) A c d ( ) A An. Queion [GATE EE IIT-Kharagpur : Mark] Given he homogeneou ae-pace equaion The eady ae value of lim ( ), (A) (B) An. (A) Sol. Given : Calculaion of ZIR : () () () given he iniial ae value of (C) [ I A] () L [ I A] Adj[ I A] I A ( )( ) T (), i (D)
[ I A] () () ( )( ) ( )( ) ( )( ) L ( )( ) L Applying parial fracion, we ge ( ) ( ) () L e e e () e Seady ae value can be calculaed a, e lim Hence, he correc opion i (A). Queion [GATE EC IISc-Bangalore : 5 Mark] The block diagram of a linear ime invarian yem i given in he figure i X () U () Y () Sol. (a) Wrie down he ae variable equaion for he yem in mari form auming he ae vecor o be T [ ( ) ( )]. (b) Find ou he ae raniion mari. (c) Deermine y (),, when he iniial value of he ae a ime are (), and (). (a) From he figure given, we have () u() () () u() () () y () () ()
From he above equaion, () () u () () (). (i) () y () [ ] () Sae equaion i given by, A Bu. (i) An. (b) The ae raniion mari i given by, () L [ I A] On comparing equaion (i) and (ii), we ge A, B [ I A] Adj[ I A] I A ( )( ) ( )( ) ( )( ) [ I A] ( )( ) ( ) ( )( ) () L ( ) e e e () An. e (c) From equaion, y () () ().. (i) Soluion of ae equaion i given by, () ZIR ZSR () A A( ) () ( ) e e Bu d Aume he inpu i zero a no given in he queion. If inpu i zero hen we conider zero inpu repone. A e e e () e e e () e () e () e () e e e e e () e From equaion (i), we ge y () e e e e An.
Queion 7 [GATE EE IISc-Bangalore : 5 Mark] Obain a ae variable repreenaion of he yem governed by he differenial equaion d y dy dy y u( ) e, wih he choice of ae variable a y, ye. Alo find (), d d d given ha u () i a uni ep funcion and (). d y dy Sol. Given : y u( ) e d d y..(i) dy ye..(ii) d Differeniaing he above equaion (i), we ge y..(iii) From equaion (i), (ii) and (iii), y e y e..(iv) Differeniaing he above equaion (ii), we ge ye ye ye ye..(v) yu() e y y..(vi) Subiue equaion (vi) in (v), we ge ( u( ) e y y) e ye ye ye u() ye ye ye u () ye ye..(vii) Subiuing equaion (i) and equaion (iv) in (vii), we ge u () [ e ] e ye u () e e u ()..(viii) From equaion (iv), (viii) he ae equaion in vecor mari noaion. e u () From equaion (i) he oupu equaion i y [ ] () calculaion : Inpu i ep and () From equaion (viii), u () Taking Laplace ranform, we ge X () () X () Given () X() ( )
Applying parial fracion, we ge X() Taking invere Laplace ranform, we ge () e Queion An. [GATE EE IIT-Madra : Mark] The following equaion define a eparaely ecied dc moor in he form of a differenial equaion d B d K K V a d J d LJ LJ The above equaion may be organized in he ae-pace form a follow : d d d P d QVa d d Where he P mari i given by B K K B (A) J LJ (B) L J J (C) K B L J J (A) Sol. Given : Le, d d d P d QV d d a d.. (i) d d d d K B d K V a d LJ J d LJ B K K J LJ LJ K B K V a V a (D) B K J LJ LJ J LJ.. (ii) From equaion (i) and (ii), we ge B K A J LJ Hence, he correc opion i (A). 4
Queion 4 [GATE IN 4 IIT-Delhi : Mark] Y () 6 A ae pace repreenaion for he ranfer funcion U() 5 6 i A Bu, y C. Where A, B 6 5. The value of C will be (A) (B) 6 (C) (D) 6 An. (B) Sol. Given : Y () 6 U() 56, 6 5 B Tranfer funcion i given by, T () CI [ A] B D.. (i) [ I A] 6 5 6 5 Adj[ I A] [ I A] I A 5 Adj[ I A] 6 I A ( 5) 6 5 5 [ I A] ( 5) 6 6 56 6 5 [ I A] B 56 6 5 6 6 CI [ A] B[ C] 56 56 C 6 C 6 Only poible opion for C i Hence, he correc opion i (B). Alernaively : In direc decompoiion form, (i) The la row of A conain he negaive value of he coefficien of he homogeneou par of he differenial equaion in acending order, ecep for he coefficien of he highe-order erm, which i uniy. (ii) B i a column mari wih he la row equal o and he re of he elemen are all zero. (iii) C i a row mari wih coefficien of numeraor in acending order of ranfer funcion. Noe : Numeraor erm of ranfer funcion give he C mari. 5
Queion 6 [GATE EC 4 IIT-Delhi : Mark] The ae variable equaion of a yem are u,, y u. The yem i (A) Conrollable bu no obervable. (B) Obervable bu no conrollable. (C) Neiher conrollable nor obervable. (D) Conrollable and obervable. An. (D) Sol. Given : u,, y u. (i) Sae equaion i given by, A Bu.(ii) Oupu ae equaion i given by, y C Du.(iii) On comparing equaion (i) and (ii), we ge A, B The conrollabiliy mari i defined a, n [ B AB A B... A B] Where n number of ae variable If, hen he yem i conrollable. If, hen he yem i no conrollable. AB Q C Q C Thu, he yem i compleely obervable. The obervabiliy mari i defined a, QO C A C... ( A ) C Where n number of ae variable T T T n T T If QO, hen he yem i obervable. If QO, hen he yem i no obervable. T T AC Q O Q Hence, he correc opion i (D). O and C 6
Queion 4 7 [GATE EE 6 IIT-Kharagpur : Mark] ( ) For a yem wih he ranfer funcion H (), he mari A in he ae pace form 4 A Bu i equal o (A) (B) (C) (D) 4 4 4 4 An. (B) Sol. Given : ( ) 6 H() 4 4 Le Y () X () H () X () U() where X() U () 4 (i) Y() 6 U()...(ii) Uing equaion (i), we have ( 4 ) X( ) U( ) X () 4 X () X () X () U () Taking invere Laplace ranform, d d d 4 u d d d (iii) (iv) d d (v) d (vi) d From equaion (iii), (iv), (v) and (vi), we have d 4u (vii) d Uing equaion (v), (vi) and (vii) we have, u 4 AX Bu (viii) When A and B 4 ( ) Noe : In original Gae queion H(). Here order of characeriic equaion i wo. Order 4 of characeriic equaion give number of ae variable, o accordingly here mu be only wo ae variable. Bu from given opion i i oberved ha order of yem mari i hree which a number of ae variable. So given funcion H( ) ha been correced by aking characeriic equaion a 4 inead of 4.
Queion 4 8 [GATE IN 6 IIT-Kharagpur : Mark] The ae variable repreenaion of a plan i given by A Bu, y C, where i he ae variable, u i inpu and y i he oupu. Auming zero iniial condiion, he impule repone of he plan i given by, (A) ep( A ) (B) ep[ A( )] Bu( ) d (C) Cep( A) B (D) C ep[ A( )] Bu( ) d An. (D) Sol. Given : A Bu, y C () Soluion of above ae equaion in ime domain i given by, () ZIR ZSR ( ) ep( A) () ep A( ) Bu( ) d In cae of zero iniial condiion we conider zero ae repone. () ZSR () ep A( ) Bu( ) d Oupu y C C ep A( ) Bu( ) d Hence, he correc opion i (D).. Saemen For Linked Anwer Queion 44 & 45. Conider a linear yem whoe ae pace repreenaion i () A(). If he iniial ae vecor of he e yem i (), hen he yem repone i (). If he iniial ae vecor of he yem e e change o () hen he yem repone become (). e Queion 44 [GATE EC 7 IIT-Kanpur : Mark] The Eigen value and Eigen vecor pair ( i, vi) for he yem are (A), and, (B), and, (C), and, (D), and, An. (A) Sol. Given : () A() (i) e If he iniial ae vecor of he yem i (), hen he yem repone i (). If he e e iniial ae vecor of he yem change o () hen he yem repone become () e.
Soluion of ae equaion of homogenou yem repreened by above equaion i given a, () () () Le () 4 () () (ii) 4 e When () ; ( ) e Puing above epreion of () and () in equaion (ii), we have, e e 4 e (iii) e 4 (iv) e When () ; ( ) e Puing above epreion of () and () again in equaion (ii), we have, e e 4 e (v) e 4 (vi) From equaion (iii) and (v), we have e e e e (vii) e e (viii) From equaion (iv) and (vi) we have e e 4 e e From equaion (vii), (viii), (i) and () we have, e e e e () e e e e Taking Laplace ranform, ( )( ) ( )( ) () (i) ( )( ) ( )( ) Taking Laplace ranform of equaion (i), we have X () () AX () X () [ I A] () X () ()() 9
where () [ I A] Adj[ I A] () (ii) [ I A] From (i) and (ii), we have, Adj[ I A] I A ( )( ) where I A ( )( ) (iii) Adj[ I A] Characeriic equaion of yem i given by, I A (iv) ( )( ), Roo of characeriic equaion are pole of he yem which alo repreen eigenvalue of yem mari A. So eigenvalue of A,, Le be eigen vecor of A, hen, [ I A] For, eigen vecor For
Queion 48 If hen For, eigen vecor Hence, he correc opion i (A).. Saemen For Linked Anwer Queion 48 & 49. The ae pace equaion of a yem i decribed by A Bu y C. [GATE EE 8 IISc-Bangalore : Mark] Where i ae vecor, u i inpu, y i oupu and A, B, C The ranfer funcion G() of hi yem will be (A) (B) An. (D) Sol. Given : A Bu and y C Where A, B, C and D [ I A] Adj[ I A] I A ( ) Adj[ I A] [ I A] I A [ I A] ( ) The ranfer funcion i given by, T() = CI [ A] B T() ( ) ( ) ( ) ( ) T() ( ) ( ) Hence, he correc opion i (D). (C) (D)
Queion 49 [GATE EE 8 IISc-Bangalore : Mark] A uniy feedback i provided o he above yem G() o make i a cloed loop yem a hown in figure. r () G () y() For a uni ep inpu r(), he eady ae error in he oupu will be (A) (B) (C) (D) An. (A) Sol. Given : G () ( ) and H( ) r () u () Taking Laplace ranform, we ge R () For ep inpu eady ae error i given by, e K p where K p lim ( ) e Hence, he correc opion i (A). Queion 5 [GATE EC 9 IIT-Roorkee : Mark] d p Conider he yem A Bu wih A and B d q where p and q are arbirary real number. Which of he following aemen abou he conrollabiliy of he yem i rue? (A) The yem i compleely ae conrollable for any nonzero value of p and q. (B) Only p = and q = reul in conrollabiliy. (C) The yem i unconrollable for all value of p and q. (D) We canno conclude abou conrollabiliy from he given daa. An. (C) p Sol. Given : A, B q The conrollabiliy mari i defined a, n [ B AB A B... A B] Where n number of ae variable If, hen he yem i conrollable. If, hen he yem i no conrollable. p p p AB q q q
Queion 55 An. Sol. p p So q q Thu, he yem i no conrollable for all value of p and q. Hence, he correc opion i (C). Noe : (i) Mari A i in diagonal canonical form bu no diinc eigenvalue hence we can no apply direc concep o e conrollabiliy. (ii) Mari A i no even in Jordan canonical form even here are repeaed roo in diagonal elemen becaue in JCF here hould be above he repeaed roo like A Thi concep canno be ued here. If A i in diagonal canonical form (DCF) or Jordan canonical form (JCF) he pair (A, B) i compleely conrollable if all he elemen in he row of B ha correpond o he la row of each Jordan block are nonzero.. Common Daa For Queion 55 & 56. The ignal flow graph of a yem i hown below. The ae variable repreenaion of he yem can be (A) u y.5 (C) u y.5.5 (B) The given ignal flow graph i hown below. From given SFG, y.5.5 y.5( ).5.5 u / /.5 U() Y() [GATE EC IIT-Guwahai : Mark] (B) u y.5 (D) u y.5.5 / /.5 U() Y()
u y.5 Hence, he correc opion i (B). Queion 6 [GATE EC/EE/IN IIT-Delhi : Mark] The ae variable decripion of an LTI yem i given by, a a u a y Where y i he oupu and u i he inpu. The yem i conrollable for (A) a, a, a (B) a, a, a (C) a, a, a (D) a, a, a An. (D) Sol. Given : a a u a.. (i) y.. (ii) Sae equaion i given by, A Bu. (iii) On comparing equaion (i) and (iii), we ge a A a and B a The conrollabiliy mari i defined a, n [ B AB A B... A B] Where n number of ae variable If, hen he yem i conrollable. If, hen he yem i no conrollable. a AB a a a 4
a aa aa a ( aa ) A B AAB a a a Q C a and a Hence, he correc opion i (D). Alernaively : Given ae pace model i in Jordan canonical form i.e. repeaed roo in diagonal elemen.. If A i in diagonal canonical form (DCF) or Jordan canonical form (JCF) he pair (A, B) i compleely conrollable if all he elemen in he row of B ha correpond o he la row of each Jordan block are nonzero.. All he elemen below he main diagonal are zero, o a.. Some of he elemen immediaely above he muliple order eigenvalue on he main diagonal are, o a a. Queion 75 [GATE EE 5 (Se-) IIT-Kanpur : Mark] For he yem governed by he e of equaion d u d d u d y Y () The ranfer funcion i given by U () ( ) ( ) ( ) ( ) (A) (B) (C) (D) ( ) ( ) ( ) ( ) An. (A) Sol. Given : d u d d u d y u y.. (i).. (ii) Sae equaion i given by, A Bu. (iii) 5
Oupu ae equaion i given by, y C Du. (iv) On comparing equaion (i), (ii), (iii) and (iv), we ge A, B and C Tranfer funcion i given by, T() C[ I A] B. (v) [ I A] Adj I A I A I A Adj[ I A] I A ( ) [ I A] From equaion (v), we ge T () T() 4 ( ) T() Hence, he correc opion i (A). Queion 76 [GATE IN 5 IIT-Kanpur : Mark] A yem i repreened in ae-pace a A Bu where and. The value of for 6 B which he yem i no conrollable i. An. ( ) Sol. Given : A, 6 B AB 6 6 The conrollabiliy mari i defined a, n [ B AB A B... A B] Where n number of ae variable 6
If, hen he yem i conrollable. If, hen he yem i no conrollable. For a yem o be unconrollable, i conrollabiliy deerminan hould be equal o zero. Q C α 6 6 6 Hence, he correc anwer i. Queion 78 7 [GATE EC 6 (Se-) IISc-Bangalore : Mark] A econd-order linear ime-invarian yem i decribed by he following ae equaion d () () () d d () () u () d Where () () are he wo ae variable and () c () (), hen he yem i (A) conrollable bu no obervable (B) obervable bu no conrollable (C) boh conrollable and obervable (D) neiher conrollable nor obervable An. (A) Sol. Given : () () u() And () () u() () () u() () () u() u c () (), A B, C For conrollabiliy 6 B AB Hence yem i conrollable. For obervabiliy C Q CA Q Hence yem i no obervable.
IES Objecive Soluion Queion 5 [IES EE 994] Conider he cloed-loop yem hown in he figure. U () Y () The ae model of he yem i (A) u, y (B) u, y (C) u, y (D) u, y An. (A) Sol. From he given figure, he oupu equaion i, The ae equaion are, Hence, he correc opion i (A). Queion 8 [IES EE 995] Conider he yem Y() X () y [ ] u u b u () () () b c () d d () The condiion for complee ae conrollabiliy and complee obervabiliy i (A) d, b, b and d can be anyhing (B) d, d, b and b can be anyhing (C) b, b, d and d can be anyhing (D) b, b, b and d can be anyhing An. (A) 8
Sol. Given : b u () () () b c () [ d d] (). (i). (ii) Sae equaion i given by, A Bu. (iii) Oupu ae equaion i given by, y C Du. (iv) On comparing equaion (i), (ii), (iii) and (iv), we ge A b,, and D = B b C d d The conrollabiliy mari i defined a, n [ B AB A B... A B] Where n number of ae variable If, hen he yem i conrollable. If, hen he yem i no conrollable. B AB b b b AB b b b b b bb bb b b b b If b, If b = non-zero erm, he yem i ae conrollable. The obervabiliy mari i defined a, T T T n T T Q O C A C... ( A ) C Where n number of ae variable If QO, hen he yem i obervable. If QO, hen he yem i no obervable. If T T T Q [ C A C ] O d, Q T A, T d C d d d T T AC d d d d d QO d dd dd d d d d O 9
If d non zero erm, he yem i ae obervable. Hence, he correc opion i (A). Alernaively : b () () u() b La row of Jordan block Fir row of Jordan block b mu be non-zero for conrollabiliy () () Hence, he correc opion i (A). c () [ d d ] () d mu be non-zero for obervabiliy Queion 6 [IES EE 997] The ae equaion of a dynamic yem i given by, () A() A 4 4 The Eigen value of he yem would be (A) real non-repeaed only (B) real non-repeaed and comple (C) real repeaed (D) real repeaed and comple An. (D) Sol. Given : () A() A 4 4 Eigen value can be obained a, AI AI 4 4
AI 4 4 AI ( )[( )( )( )( ) (6) AI ( )[9 6] AI [ ][ 6 5] 6 6 AI [ ] 6 j8 AI [ ] The Eigen value of he yem are,, and j4. Hence, he correc opion i (D). Queion 7 [IES EE 997] Conider he following ae equaion for a dicree yem ( k) ( k) ( k) uk ( ), ( k ) ( k) yk ( ) [ ] 4 uk ( ) ( k) 4 4 The yem given above i (A) conrollable and obervable. (B) unconrollable and unobervable. (C) unconrollable and obervable. (D) conrollable and unobervable. An. (B) Sol. Given : ( k) ( k) uk ( ) ( k ) ( k) 4 4.(i) yk ( ) [ ( k) ] 4 uk ( ) ( k).(ii) Sae equaion i given by, A Bu. (iii) Oupu ae equaion i given by, y CDu. (iv) On comparing equaion (i), (ii), (iii) and (iv), we ge A, 4 4 B and C [ ]
The conrollabiliy mari i defined a, n [ B AB A B... A B] Where n number of ae variable If, hen he yem i conrollable. If, hen he yem i no conrollable. For he given ae equaion, conrollabiliy e mari i, Q B AB C Thu, Therefore, he yem i unconrollable. The obervabiliy mari i defined a, QO C A C... ( A ) C Where n number of ae variable AB 4 4 Q C T T T n T T If QO, hen he yem i obervable. If QO, hen he yem i no obervable. For he given ae equaion, obervabiliy e mari i, T T T Q [ C A C ] O T A T T AC Q O 4, 4 T C Thu, Q O Therefore, he yem i unobervable. Hence, he correc opion i (B). 4 4 4 4 4 4 4 4
Queion [IES EE ] The conrol yem hown in he given figure i repreened by he equaion y u Mari G y u u y u y (A) (B) (C) (D) An. (C) Sol. From he given figure, y ( u y y) y u and y u y y Hence, he correc opion i (C). Queion 7 [IES EE ] The yem mari of a dicree yem i given by A 5 The characeriic equaion i given by (A) z 5z (B) z (C) z z5 (D) z An. (A) Sol. Given : A 5 z z zi A z 5 z5 z5 z
Characeriic equaion i given by, zi A zz ( 5) z 5z Hence he correc opion i (A). Queion [IES EE ] Le and u y b ] where b i an unknown conan. Thi yem i (A) Obervable for all value of b. (B) Unobervable for all value of b. (C) Obervable for all non-zero value of b. (D) Unobervable for all non-zero value of b. An. (C) Sol. Given : and u y b ] From he given ae equaion : A, B and C b The obervabiliy mari i defined a, T T T n T T Q O C A C... ( A ) C Where n number of ae variable If QO, hen he yem i obervable. If QO, hen he yem i no obervable. A T b, C T For he given ae equaion, obervabiliy e mari i, T T T QO C A C T T b b AC b b b QO b b If b: Q O If b.5: Q O If b: Q O Thu, he yem i obervable for all non-zero value of b. Hence, he correc opion i (C). 4