CONTROL SYSTEMS. Chapter 10 : State Space Response

Size: px
Start display at page:

Download "CONTROL SYSTEMS. Chapter 10 : State Space Response"

Transcription

1 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).

2 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).

3 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 ()

4 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

5 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 () Sol. The given ignal flow graph i hown below / U () Y ()

6 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 ] [ 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 () 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

7 () ( ) () () 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

8 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

9 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

10 (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)

11 [ 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 () () ()

12 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.

13 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() ( )

14 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

15 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] Adj[ I A] [ I A] I A 5 Adj[ I A] 6 I A ( 5) [ I A] ( 5) [ I A] B CI [ A] B[ C] 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

16 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

17 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) 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.

18 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.

19 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

20 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

21 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)

22 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

23 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()

24 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

25 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

26 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

27 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 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.

28 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

29 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

30 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

31 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 [ ]

32 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)

33 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

34 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

Sample Final Exam (finals03) Covering Chapters 1-9 of Fundamentals of Signals & Systems

Sample Final Exam (finals03) Covering Chapters 1-9 of Fundamentals of Signals & Systems Sample Final Exam Covering Chaper 9 (final04) Sample Final Exam (final03) Covering Chaper 9 of Fundamenal of Signal & Syem Problem (0 mar) Conider he caual opamp circui iniially a re depiced below. I LI

More information

10. State Space Methods

10. State Space Methods . Sae Space Mehods. Inroducion Sae space modelling was briefly inroduced in chaper. Here more coverage is provided of sae space mehods before some of heir uses in conrol sysem design are covered in he

More information

u(t) Figure 1. Open loop control system

u(t) Figure 1. Open loop control system Open loop conrol v cloed loop feedbac conrol The nex wo figure preen he rucure of open loop and feedbac conrol yem Figure how an open loop conrol yem whoe funcion i o caue he oupu y o follow he reference

More information

Chapter 7: Inverse-Response Systems

Chapter 7: Inverse-Response Systems Chaper 7: Invere-Repone Syem Normal Syem Invere-Repone Syem Baic Sar ou in he wrong direcion End up in he original eady-ae gain value Two or more yem wih differen magniude and cale in parallel Main yem

More information

CHAPTER 7: SECOND-ORDER CIRCUITS

CHAPTER 7: SECOND-ORDER CIRCUITS EEE5: CI RCUI T THEORY CHAPTER 7: SECOND-ORDER CIRCUITS 7. Inroducion Thi chaper conider circui wih wo orage elemen. Known a econd-order circui becaue heir repone are decribed by differenial equaion ha

More information

, the. L and the L. x x. max. i n. It is easy to show that these two norms satisfy the following relation: x x n x = (17.3) max

, the. L and the L. x x. max. i n. It is easy to show that these two norms satisfy the following relation: x x n x = (17.3) max ecure 8 7. Sabiliy Analyi For an n dimenional vecor R n, he and he vecor norm are defined a: = T = i n i (7.) I i eay o how ha hee wo norm aify he following relaion: n (7.) If a vecor i ime-dependen, hen

More information

EE202 Circuit Theory II

EE202 Circuit Theory II EE202 Circui Theory II 2017-2018, Spring Dr. Yılmaz KALKAN I. Inroducion & eview of Fir Order Circui (Chaper 7 of Nilon - 3 Hr. Inroducion, C and L Circui, Naural and Sep epone of Serie and Parallel L/C

More information

Chapter 6. Laplace Transforms

Chapter 6. Laplace Transforms 6- Chaper 6. Laplace Tranform 6.4 Shor Impule. Dirac Dela Funcion. Parial Fracion 6.5 Convoluion. Inegral Equaion 6.6 Differeniaion and Inegraion of Tranform 6.7 Syem of ODE 6.4 Shor Impule. Dirac Dela

More information

To become more mathematically correct, Circuit equations are Algebraic Differential equations. from KVL, KCL from the constitutive relationship

To become more mathematically correct, Circuit equations are Algebraic Differential equations. from KVL, KCL from the constitutive relationship Laplace Tranform (Lin & DeCarlo: Ch 3) ENSC30 Elecric Circui II The Laplace ranform i an inegral ranformaion. I ranform: f ( ) F( ) ime variable complex variable From Euler > Lagrange > Laplace. Hence,

More information

EECE 301 Signals & Systems Prof. Mark Fowler

EECE 301 Signals & Systems Prof. Mark Fowler EECE 31 Signal & Syem Prof. Mark Fowler Noe Se #27 C-T Syem: Laplace Tranform Power Tool for yem analyi Reading Aignmen: Secion 6.1 6.3 of Kamen and Heck 1/18 Coure Flow Diagram The arrow here how concepual

More information

CHAPTER. Forced Equations and Systems { } ( ) ( ) 8.1 The Laplace Transform and Its Inverse. Transforms from the Definition.

CHAPTER. Forced Equations and Systems { } ( ) ( ) 8.1 The Laplace Transform and Its Inverse. Transforms from the Definition. CHAPTER 8 Forced Equaion and Syem 8 The aplace Tranform and I Invere Tranform from he Definiion 5 5 = b b {} 5 = 5e d = lim5 e = ( ) b {} = e d = lim e + e d b = (inegraion by par) = = = = b b ( ) ( )

More information

13.1 Circuit Elements in the s Domain Circuit Analysis in the s Domain The Transfer Function and Natural Response 13.

13.1 Circuit Elements in the s Domain Circuit Analysis in the s Domain The Transfer Function and Natural Response 13. Chaper 3 The Laplace Tranform in Circui Analyi 3. Circui Elemen in he Domain 3.-3 Circui Analyi in he Domain 3.4-5 The Tranfer Funcion and Naural Repone 3.6 The Tranfer Funcion and he Convoluion Inegral

More information

CHAPTER 3 SIGNALS & SYSTEMS. z -transform in the z -plane will be (A) 1 (B) 1 (D) (C) . The unilateral Laplace transform of tf() (A) s (B) + + (D) (C)

CHAPTER 3 SIGNALS & SYSTEMS. z -transform in the z -plane will be (A) 1 (B) 1 (D) (C) . The unilateral Laplace transform of tf() (A) s (B) + + (D) (C) CHAPER SIGNALS & SYSEMS YEAR ONE MARK n n MCQ. If xn [ ] (/) (/) un [ ], hen he region of convergence (ROC) of i z ranform in he z plane will be (A) < z < (B) < z < (C) < z < (D) < z MCQ. he unilaeral

More information

6.8 Laplace Transform: General Formulas

6.8 Laplace Transform: General Formulas 48 HAP. 6 Laplace Tranform 6.8 Laplace Tranform: General Formula Formula Name, ommen Sec. F() l{ f ()} e f () d f () l {F()} Definiion of Tranform Invere Tranform 6. l{af () bg()} al{f ()} bl{g()} Lineariy

More information

Lecture 13 RC/RL Circuits, Time Dependent Op Amp Circuits

Lecture 13 RC/RL Circuits, Time Dependent Op Amp Circuits Lecure 13 RC/RL Circuis, Time Dependen Op Amp Circuis RL Circuis The seps involved in solving simple circuis conaining dc sources, resisances, and one energy-sorage elemen (inducance or capaciance) are:

More information

18.03SC Unit 3 Practice Exam and Solutions

18.03SC Unit 3 Practice Exam and Solutions Sudy Guide on Sep, Dela, Convoluion, Laplace You can hink of he ep funcion u() a any nice mooh funcion which i for < a and for > a, where a i a poiive number which i much maller han any ime cale we care

More information

Chapter 6. Laplace Transforms

Chapter 6. Laplace Transforms Chaper 6. Laplace Tranform Kreyzig by YHLee;45; 6- An ODE i reduced o an algebraic problem by operaional calculu. The equaion i olved by algebraic manipulaion. The reul i ranformed back for he oluion of

More information

6.302 Feedback Systems Recitation : Phase-locked Loops Prof. Joel L. Dawson

6.302 Feedback Systems Recitation : Phase-locked Loops Prof. Joel L. Dawson 6.32 Feedback Syem Phae-locked loop are a foundaional building block for analog circui deign, paricularly for communicaion circui. They provide a good example yem for hi cla becaue hey are an excellen

More information

More on ODEs by Laplace Transforms October 30, 2017

More on ODEs by Laplace Transforms October 30, 2017 More on OE b Laplace Tranfor Ocober, 7 More on Ordinar ifferenial Equaion wih Laplace Tranfor Larr areo Mechanical Engineering 5 Seinar in Engineering nali Ocober, 7 Ouline Review la cla efiniion of Laplace

More information

8. Basic RL and RC Circuits

8. Basic RL and RC Circuits 8. Basic L and C Circuis This chaper deals wih he soluions of he responses of L and C circuis The analysis of C and L circuis leads o a linear differenial equaion This chaper covers he following opics

More information

CONTROL SYSTEMS. Chapter 3 Mathematical Modelling of Physical Systems-Laplace Transforms. Prof.Dr. Fatih Mehmet Botsalı

CONTROL SYSTEMS. Chapter 3 Mathematical Modelling of Physical Systems-Laplace Transforms. Prof.Dr. Fatih Mehmet Botsalı CONTROL SYSTEMS Chaper Mahemaical Modelling of Phyical Syem-Laplace Tranform Prof.Dr. Faih Mehme Boalı Definiion Tranform -- a mahemaical converion from one way of hinking o anoher o make a problem eaier

More information

Instrumentation & Process Control

Instrumentation & Process Control Chemical Engineering (GTE & PSU) Poal Correpondence GTE & Public Secor Inrumenaion & Proce Conrol To Buy Poal Correpondence Package call a -999657855 Poal Coure ( GTE & PSU) 5 ENGINEERS INSTITUTE OF INDI.

More information

Laplace Transform. Inverse Laplace Transform. e st f(t)dt. (2)

Laplace Transform. Inverse Laplace Transform. e st f(t)dt. (2) Laplace Tranform Maoud Malek The Laplace ranform i an inegral ranform named in honor of mahemaician and aronomer Pierre-Simon Laplace, who ued he ranform in hi work on probabiliy heory. I i a powerful

More information

EE Control Systems LECTURE 2

EE Control Systems LECTURE 2 Copyrigh F.L. Lewi 999 All righ reerved EE 434 - Conrol Syem LECTURE REVIEW OF LAPLACE TRANSFORM LAPLACE TRANSFORM The Laplace ranform i very ueful in analyi and deign for yem ha are linear and ime-invarian

More information

NODIA AND COMPANY. GATE SOLVED PAPER Electrical Engineering SIGNALS & SYSTEMS. Copyright By NODIA & COMPANY

NODIA AND COMPANY. GATE SOLVED PAPER Electrical Engineering SIGNALS & SYSTEMS. Copyright By NODIA & COMPANY No par of hi publicaion may be reproduced or diribued in any form or any mean, elecronic, mechanical, phoocopying, or oherie ihou he prior permiion of he auhor. GAE SOLVED PAPER Elecrical Engineering SIGNALS

More information

Chapter 9 - The Laplace Transform

Chapter 9 - The Laplace Transform Chaper 9 - The Laplace Tranform Selece Soluion. Skech he pole-zero plo an region of convergence (if i exi) for hee ignal. ω [] () 8 (a) x e u = 8 ROC σ ( ) 3 (b) x e co π u ω [] ( ) () (c) x e u e u ROC

More information

Additional Methods for Solving DSGE Models

Additional Methods for Solving DSGE Models Addiional Mehod for Solving DSGE Model Karel Meren, Cornell Univeriy Reference King, R. G., Ploer, C. I. & Rebelo, S. T. (1988), Producion, growh and buine cycle: I. he baic neoclaical model, Journal of

More information

EECE 301 Signals & Systems Prof. Mark Fowler

EECE 301 Signals & Systems Prof. Mark Fowler EECE 30 Signal & Syem Prof. ark Fowler oe Se #34 C-T Tranfer Funcion and Frequency Repone /4 Finding he Tranfer Funcion from Differenial Eq. Recall: we found a DT yem Tranfer Funcion Hz y aking he ZT of

More information

2. VECTORS. R Vectors are denoted by bold-face characters such as R, V, etc. The magnitude of a vector, such as R, is denoted as R, R, V

2. VECTORS. R Vectors are denoted by bold-face characters such as R, V, etc. The magnitude of a vector, such as R, is denoted as R, R, V ME 352 VETS 2. VETS Vecor algebra form he mahemaical foundaion for kinemaic and dnamic. Geomer of moion i a he hear of boh he kinemaic and dnamic of mechanical em. Vecor anali i he imehonored ool for decribing

More information

Let. x y. denote a bivariate time series with zero mean.

Let. x y. denote a bivariate time series with zero mean. Linear Filer Le x y : T denoe a bivariae ime erie wih zero mean. Suppoe ha he ime erie {y : T} i conruced a follow: y a x The ime erie {y : T} i aid o be conruced from {x : T} by mean of a Linear Filer.

More information

Problem Set If all directed edges in a network have distinct capacities, then there is a unique maximum flow.

Problem Set If all directed edges in a network have distinct capacities, then there is a unique maximum flow. CSE 202: Deign and Analyi of Algorihm Winer 2013 Problem Se 3 Inrucor: Kamalika Chaudhuri Due on: Tue. Feb 26, 2013 Inrucion For your proof, you may ue any lower bound, algorihm or daa rucure from he ex

More information

1 1 + x 2 dx. tan 1 (2) = ] ] x 3. Solution: Recall that the given integral is improper because. x 3. 1 x 3. dx = lim dx.

1 1 + x 2 dx. tan 1 (2) = ] ] x 3. Solution: Recall that the given integral is improper because. x 3. 1 x 3. dx = lim dx. . Use Simpson s rule wih n 4 o esimae an () +. Soluion: Since we are using 4 seps, 4 Thus we have [ ( ) f() + 4f + f() + 4f 3 [ + 4 4 6 5 + + 4 4 3 + ] 5 [ + 6 6 5 + + 6 3 + ]. 5. Our funcion is f() +.

More information

Control Systems -- Final Exam (Spring 2006)

Control Systems -- Final Exam (Spring 2006) 6.5 Conrol Syem -- Final Eam (Spring 6 There are 5 prolem (inluding onu prolem oal poin. (p Given wo marie: (6 Compue A A e e. (6 For he differenial equaion [ ] ; y u A wih ( u( wha i y( for >? (8 For

More information

Solutions of Sample Problems for Third In-Class Exam Math 246, Spring 2011, Professor David Levermore

Solutions of Sample Problems for Third In-Class Exam Math 246, Spring 2011, Professor David Levermore Soluions of Sample Problems for Third In-Class Exam Mah 6, Spring, Professor David Levermore Compue he Laplace ransform of f e from is definiion Soluion The definiion of he Laplace ransform gives L[f]s

More information

DEPARTMENT OF ELECTRICAL AND ELECTRONIC ENGINEERING EXAMINATIONS 2008

DEPARTMENT OF ELECTRICAL AND ELECTRONIC ENGINEERING EXAMINATIONS 2008 [E5] IMPERIAL COLLEGE LONDON DEPARTMENT OF ELECTRICAL AND ELECTRONIC ENGINEERING EXAMINATIONS 008 EEE/ISE PART II MEng BEng and ACGI SIGNALS AND LINEAR SYSTEMS Time allowed: :00 hours There are FOUR quesions

More information

Algorithmic Discrete Mathematics 6. Exercise Sheet

Algorithmic Discrete Mathematics 6. Exercise Sheet Algorihmic Dicree Mahemaic. Exercie Shee Deparmen of Mahemaic SS 0 PD Dr. Ulf Lorenz 7. and 8. Juni 0 Dipl.-Mah. David Meffer Verion of June, 0 Groupwork Exercie G (Heap-Sor) Ue Heap-Sor wih a min-heap

More information

Discussion Session 2 Constant Acceleration/Relative Motion Week 03

Discussion Session 2 Constant Acceleration/Relative Motion Week 03 PHYS 100 Dicuion Seion Conan Acceleraion/Relaive Moion Week 03 The Plan Today you will work wih your group explore he idea of reference frame (i.e. relaive moion) and moion wih conan acceleraion. You ll

More information

Math 2142 Exam 1 Review Problems. x 2 + f (0) 3! for the 3rd Taylor polynomial at x = 0. To calculate the various quantities:

Math 2142 Exam 1 Review Problems. x 2 + f (0) 3! for the 3rd Taylor polynomial at x = 0. To calculate the various quantities: Mah 4 Eam Review Problems Problem. Calculae he 3rd Taylor polynomial for arcsin a =. Soluion. Le f() = arcsin. For his problem, we use he formula f() + f () + f ()! + f () 3! for he 3rd Taylor polynomial

More information

Physics 240: Worksheet 16 Name

Physics 240: Worksheet 16 Name Phyic 4: Workhee 16 Nae Non-unifor circular oion Each of hee proble involve non-unifor circular oion wih a conan α. (1) Obain each of he equaion of oion for non-unifor circular oion under a conan acceleraion,

More information

Design of Controller for Robot Position Control

Design of Controller for Robot Position Control eign of Conroller for Robo oiion Conrol Two imporan goal of conrol: 1. Reference inpu racking: The oupu mu follow he reference inpu rajecory a quickly a poible. Se-poin racking: Tracking when he reference

More information

Exponential Sawtooth

Exponential Sawtooth ECPE 36 HOMEWORK 3: PROPERTIES OF THE FOURIER TRANSFORM SOLUTION. Exponenial Sawooh: The eaie way o do hi problem i o look a he Fourier ranform of a ingle exponenial funcion, () = exp( )u(). From he able

More information

System of Linear Differential Equations

System of Linear Differential Equations Sysem of Linear Differenial Equaions In "Ordinary Differenial Equaions" we've learned how o solve a differenial equaion for a variable, such as: y'k5$e K2$x =0 solve DE yx = K 5 2 ek2 x C_C1 2$y''C7$y

More information

d 1 = c 1 b 2 - b 1 c 2 d 2 = c 1 b 3 - b 1 c 3

d 1 = c 1 b 2 - b 1 c 2 d 2 = c 1 b 3 - b 1 c 3 and d = c b - b c c d = c b - b c c This process is coninued unil he nh row has been compleed. The complee array of coefficiens is riangular. Noe ha in developing he array an enire row may be divided or

More information

23.2. Representing Periodic Functions by Fourier Series. Introduction. Prerequisites. Learning Outcomes

23.2. Representing Periodic Functions by Fourier Series. Introduction. Prerequisites. Learning Outcomes Represening Periodic Funcions by Fourier Series 3. Inroducion In his Secion we show how a periodic funcion can be expressed as a series of sines and cosines. We begin by obaining some sandard inegrals

More information

Frequency Response. We now know how to analyze and design ccts via s- domain methods which yield dynamical information

Frequency Response. We now know how to analyze and design ccts via s- domain methods which yield dynamical information Frequency Repone We now now how o analyze and deign cc via - domain mehod which yield dynamical informaion Zero-ae repone Zero-inpu repone Naural repone Forced repone The repone are decribed by he exponenial

More information

Linear Control System EE 711. Design. Lecture 8 Dr. Mostafa Abdel-geliel

Linear Control System EE 711. Design. Lecture 8 Dr. Mostafa Abdel-geliel Linear Conrol Sysem EE 7 MIMO Sae Space Analysis and Design Lecure 8 Dr. Mosafa Abdel-geliel Course Conens Review Sae Space SS modeling and analysis Sae feed back design Oupu feedback design Observer design

More information

Math 333 Problem Set #2 Solution 14 February 2003

Math 333 Problem Set #2 Solution 14 February 2003 Mah 333 Problem Se #2 Soluion 14 February 2003 A1. Solve he iniial value problem dy dx = x2 + e 3x ; 2y 4 y(0) = 1. Soluion: This is separable; we wrie 2y 4 dy = x 2 + e x dx and inegrae o ge The iniial

More information

ODEs II, Lecture 1: Homogeneous Linear Systems - I. Mike Raugh 1. March 8, 2004

ODEs II, Lecture 1: Homogeneous Linear Systems - I. Mike Raugh 1. March 8, 2004 ODEs II, Lecure : Homogeneous Linear Sysems - I Mike Raugh March 8, 4 Inroducion. In he firs lecure we discussed a sysem of linear ODEs for modeling he excreion of lead from he human body, saw how o ransform

More information

Voltage/current relationship Stored Energy. RL / RC circuits Steady State / Transient response Natural / Step response

Voltage/current relationship Stored Energy. RL / RC circuits Steady State / Transient response Natural / Step response Review Capaciors/Inducors Volage/curren relaionship Sored Energy s Order Circuis RL / RC circuis Seady Sae / Transien response Naural / Sep response EE4 Summer 5: Lecure 5 Insrucor: Ocavian Florescu Lecure

More information

Signal and System (Chapter 3. Continuous-Time Systems)

Signal and System (Chapter 3. Continuous-Time Systems) Signal and Sysem (Chaper 3. Coninuous-Time Sysems) Prof. Kwang-Chun Ho kwangho@hansung.ac.kr Tel: 0-760-453 Fax:0-760-4435 1 Dep. Elecronics and Informaion Eng. 1 Nodes, Branches, Loops A nework wih b

More information

Theory of! Partial Differential Equations-I!

Theory of! Partial Differential Equations-I! hp://users.wpi.edu/~grear/me61.hml! Ouline! Theory o! Parial Dierenial Equaions-I! Gréar Tryggvason! Spring 010! Basic Properies o PDE!! Quasi-linear Firs Order Equaions! - Characerisics! - Linear and

More information

Some Basic Information about M-S-D Systems

Some Basic Information about M-S-D Systems Some Basic Informaion abou M-S-D Sysems 1 Inroducion We wan o give some summary of he facs concerning unforced (homogeneous) and forced (non-homogeneous) models for linear oscillaors governed by second-order,

More information

Mon Apr 2: Laplace transform and initial value problems like we studied in Chapter 5

Mon Apr 2: Laplace transform and initial value problems like we studied in Chapter 5 Mah 225-4 Week 2 April 2-6 coninue.-.3; alo cover par of.4-.5, EP 7.6 Mon Apr 2:.-.3 Laplace ranform and iniial value problem like we udied in Chaper 5 Announcemen: Warm-up Exercie: Recall, The Laplace

More information

1 Solutions to selected problems

1 Solutions to selected problems 1 Soluions o seleced problems 1. Le A B R n. Show ha in A in B bu in general bd A bd B. Soluion. Le x in A. Then here is ɛ > 0 such ha B ɛ (x) A B. This shows x in B. If A = [0, 1] and B = [0, 2], hen

More information

Theory of! Partial Differential Equations!

Theory of! Partial Differential Equations! hp://www.nd.edu/~gryggva/cfd-course/! Ouline! Theory o! Parial Dierenial Equaions! Gréar Tryggvason! Spring 011! Basic Properies o PDE!! Quasi-linear Firs Order Equaions! - Characerisics! - Linear and

More information

MA 214 Calculus IV (Spring 2016) Section 2. Homework Assignment 1 Solutions

MA 214 Calculus IV (Spring 2016) Section 2. Homework Assignment 1 Solutions MA 14 Calculus IV (Spring 016) Secion Homework Assignmen 1 Soluions 1 Boyce and DiPrima, p 40, Problem 10 (c) Soluion: In sandard form he given firs-order linear ODE is: An inegraing facor is given by

More information

8.1. a) For step response, M input is u ( t) Taking inverse Laplace transform. as α 0. Ideal response, K c. = Kc Mτ D + For ramp response, 8-1

8.1. a) For step response, M input is u ( t) Taking inverse Laplace transform. as α 0. Ideal response, K c. = Kc Mτ D + For ramp response, 8-1 8. a For ep repone, inpu i u, U Y a U α α Y a α α Taking invere Laplae ranform a α e e / α / α A α 0 a δ 0 e / α a δ deal repone, α d Y i Gi U i δ Hene a α 0 a i For ramp repone, inpu i u, U Soluion anual

More information

Challenge Problems. DIS 203 and 210. March 6, (e 2) k. k(k + 2). k=1. f(x) = k(k + 2) = 1 x k

Challenge Problems. DIS 203 and 210. March 6, (e 2) k. k(k + 2). k=1. f(x) = k(k + 2) = 1 x k Challenge Problems DIS 03 and 0 March 6, 05 Choose one of he following problems, and work on i in your group. Your goal is o convince me ha your answer is correc. Even if your answer isn compleely correc,

More information

s-domain Circuit Analysis

s-domain Circuit Analysis Domain ircui Analyi Operae direcly in he domain wih capacior, inducor and reior Key feaure lineariy i preerved c decribed by ODE and heir I Order equal number of plu number of Elemenbyelemen and ource

More information

Lecture 20: Riccati Equations and Least Squares Feedback Control

Lecture 20: Riccati Equations and Least Squares Feedback Control 34-5 LINEAR SYSTEMS Lecure : Riccai Equaions and Leas Squares Feedback Conrol 5.6.4 Sae Feedback via Riccai Equaions A recursive approach in generaing he marix-valued funcion W ( ) equaion for i for he

More information

6 December 2013 H. T. Hoang - www4.hcmut.edu.vn/~hthoang/ 1

6 December 2013 H. T. Hoang - www4.hcmut.edu.vn/~hthoang/ 1 Lecure Noe Fundamenal of Conrol Syem Inrucor: Aoc. Prof. Dr. Huynh Thai Hoang Deparmen of Auomaic Conrol Faculy of Elecrical & Elecronic Engineering Ho Chi Minh Ciy Univeriy of Technology Email: hhoang@hcmu.edu.vn

More information

Chapter 8 The Complete Response of RL and RC Circuits

Chapter 8 The Complete Response of RL and RC Circuits Chaper 8 The Complee Response of RL and RC Circuis Seoul Naional Universiy Deparmen of Elecrical and Compuer Engineering Wha is Firs Order Circuis? Circuis ha conain only one inducor or only one capacior

More information

Chapter 2. First Order Scalar Equations

Chapter 2. First Order Scalar Equations Chaper. Firs Order Scalar Equaions We sar our sudy of differenial equaions in he same way he pioneers in his field did. We show paricular echniques o solve paricular ypes of firs order differenial equaions.

More information

PSAT/NMSQT PRACTICE ANSWER SHEET SECTION 3 EXAMPLES OF INCOMPLETE MARKS COMPLETE MARK B C D B C D B C D B C D B C D 13 A B C D B C D 11 A B C D B C D

PSAT/NMSQT PRACTICE ANSWER SHEET SECTION 3 EXAMPLES OF INCOMPLETE MARKS COMPLETE MARK B C D B C D B C D B C D B C D 13 A B C D B C D 11 A B C D B C D PSTNMSQT PRCTICE NSWER SHEET COMPLETE MRK EXMPLES OF INCOMPLETE MRKS I i recommended a you ue a No pencil I i very imporan a you fill in e enire circle darkly and compleely If you cange your repone, erae

More information

( ) ( ) ( ) () () Signals And Systems Exam#1. 1. Given x(t) and y(t) below: x(t) y(t) (A) Give the expression of x(t) in terms of step functions.

( ) ( ) ( ) () () Signals And Systems Exam#1. 1. Given x(t) and y(t) below: x(t) y(t) (A) Give the expression of x(t) in terms of step functions. Signals And Sysems Exam#. Given x() and y() below: x() y() 4 4 (A) Give he expression of x() in erms of sep funcions. (%) x () = q() q( ) + q( 4) (B) Plo x(.5). (%) x() g() = x( ) h() = g(. 5) = x(. 5)

More information

Chapter #1 EEE8013 EEE3001. Linear Controller Design and State Space Analysis

Chapter #1 EEE8013 EEE3001. Linear Controller Design and State Space Analysis Chaper EEE83 EEE3 Chaper # EEE83 EEE3 Linear Conroller Design and Sae Space Analysis Ordinary Differenial Equaions.... Inroducion.... Firs Order ODEs... 3. Second Order ODEs... 7 3. General Maerial...

More information

Section 3.5 Nonhomogeneous Equations; Method of Undetermined Coefficients

Section 3.5 Nonhomogeneous Equations; Method of Undetermined Coefficients Secion 3.5 Nonhomogeneous Equaions; Mehod of Undeermined Coefficiens Key Terms/Ideas: Linear Differenial operaor Nonlinear operaor Second order homogeneous DE Second order nonhomogeneous DE Soluion o homogeneous

More information

LAPLACE TRANSFORM AND TRANSFER FUNCTION

LAPLACE TRANSFORM AND TRANSFER FUNCTION CHBE320 LECTURE V LAPLACE TRANSFORM AND TRANSFER FUNCTION Professor Dae Ryook Yang Spring 2018 Dep. of Chemical and Biological Engineering 5-1 Road Map of he Lecure V Laplace Transform and Transfer funcions

More information

CHAPTER HIGHER-ORDER SYSTEMS: SECOND-ORDER AND TRANSPORTATION LAG. 7.1 SECOND-ORDER SYSTEM Transfer Function

CHAPTER HIGHER-ORDER SYSTEMS: SECOND-ORDER AND TRANSPORTATION LAG. 7.1 SECOND-ORDER SYSTEM Transfer Function CHAPTER 7 HIGHER-ORDER SYSTEMS: SECOND-ORDER AND TRANSPORTATION LAG 7. SECOND-ORDER SYSTEM Tranfer Funcion Thi ecion inroduce a baic yem called a econd-order yem or a quadraic lag. Second-order yem are

More information

Chapter 7: Solving Trig Equations

Chapter 7: Solving Trig Equations Haberman MTH Secion I: The Trigonomeric Funcions Chaper 7: Solving Trig Equaions Le s sar by solving a couple of equaions ha involve he sine funcion EXAMPLE a: Solve he equaion sin( ) The inverse funcions

More information

AN ANALYTICAL METHOD OF SOLUTION FOR SYSTEMS OF BOOLEAN EQUATIONS

AN ANALYTICAL METHOD OF SOLUTION FOR SYSTEMS OF BOOLEAN EQUATIONS CHAPTER 5 AN ANALYTICAL METHOD OF SOLUTION FOR SYSTEMS OF BOOLEAN EQUATIONS 51 APPLICATIONS OF DE MORGAN S LAWS A we have een in Secion 44 of Chaer 4, any Boolean Equaion of ye (1), (2) or (3) could be

More information

Chapter 7 Response of First-order RL and RC Circuits

Chapter 7 Response of First-order RL and RC Circuits Chaper 7 Response of Firs-order RL and RC Circuis 7.- The Naural Response of RL and RC Circuis 7.3 The Sep Response of RL and RC Circuis 7.4 A General Soluion for Sep and Naural Responses 7.5 Sequenial

More information

Homework-8(1) P8.3-1, 3, 8, 10, 17, 21, 24, 28,29 P8.4-1, 2, 5

Homework-8(1) P8.3-1, 3, 8, 10, 17, 21, 24, 28,29 P8.4-1, 2, 5 Homework-8() P8.3-, 3, 8, 0, 7, 2, 24, 28,29 P8.4-, 2, 5 Secion 8.3: The Response of a Firs Order Circui o a Consan Inpu P 8.3- The circui shown in Figure P 8.3- is a seady sae before he swich closes a

More information

Spring Ammar Abu-Hudrouss Islamic University Gaza

Spring Ammar Abu-Hudrouss Islamic University Gaza Chaper 7 Reed-Solomon Code Spring 9 Ammar Abu-Hudrouss Islamic Universiy Gaza ١ Inroducion A Reed Solomon code is a special case of a BCH code in which he lengh of he code is one less han he size of he

More information

Lectures 29 and 30 BIQUADRATICS AND STATE SPACE OP AMP REALIZATIONS. I. Introduction

Lectures 29 and 30 BIQUADRATICS AND STATE SPACE OP AMP REALIZATIONS. I. Introduction EE-202/445, 3/18/18 9-1 R. A. DeCarlo Lecures 29 and 30 BIQUADRATICS AND STATE SPACE OP AMP REALIZATIONS I. Inroducion 1. The biquadraic ransfer funcion has boh a 2nd order numeraor and a 2nd order denominaor:

More information

Solutions - Midterm Exam

Solutions - Midterm Exam DEPARTMENT OF ELECTRICAL AND COMPUTER ENGINEERING, THE UNIVERITY OF NEW MEXICO ECE-34: ignals and ysems ummer 203 PROBLEM (5 PT) Given he following LTI sysem: oluions - Miderm Exam a) kech he impulse response

More information

Serial : 4LS1_A_EC_Signal & Systems_230918

Serial : 4LS1_A_EC_Signal & Systems_230918 Serial : LS_A_EC_Signal & Syem_8 CLASS TEST (GATE) Delhi oida Bhopal Hyderabad Jaipur Lucknow Indore Pune Bhubanewar Kolkaa Pana Web: E-mail: info@madeeay.in Ph: -56 CLASS TEST 8- ELECTROICS EGIEERIG Subjec

More information

HOMEWORK # 2: MATH 211, SPRING Note: This is the last solution set where I will describe the MATLAB I used to make my pictures.

HOMEWORK # 2: MATH 211, SPRING Note: This is the last solution set where I will describe the MATLAB I used to make my pictures. HOMEWORK # 2: MATH 2, SPRING 25 TJ HITCHMAN Noe: This is he las soluion se where I will describe he MATLAB I used o make my picures.. Exercises from he ex.. Chaper 2.. Problem 6. We are o show ha y() =

More information

Module 2 F c i k c s la l w a s o s f dif di fusi s o i n

Module 2 F c i k c s la l w a s o s f dif di fusi s o i n Module Fick s laws of diffusion Fick s laws of diffusion and hin film soluion Adolf Fick (1855) proposed: d J α d d d J (mole/m s) flu (m /s) diffusion coefficien and (mole/m 3 ) concenraion of ions, aoms

More information

Problem Set #1. i z. the complex propagation constant. For the characteristic impedance:

Problem Set #1. i z. the complex propagation constant. For the characteristic impedance: Problem Se # Problem : a) Using phasor noaion, calculae he volage and curren waves on a ransmission line by solving he wave equaion Assume ha R, L,, G are all non-zero and independen of frequency From

More information

(b) (a) (d) (c) (e) Figure 10-N1. (f) Solution:

(b) (a) (d) (c) (e) Figure 10-N1. (f) Solution: Example: The inpu o each of he circuis shown in Figure 10-N1 is he volage source volage. The oupu of each circui is he curren i( ). Deermine he oupu of each of he circuis. (a) (b) (c) (d) (e) Figure 10-N1

More information

Y 0.4Y 0.45Y Y to a proper ARMA specification.

Y 0.4Y 0.45Y Y to a proper ARMA specification. HG Jan 04 ECON 50 Exercises II - 0 Feb 04 (wih answers Exercise. Read secion 8 in lecure noes 3 (LN3 on he common facor problem in ARMA-processes. Consider he following process Y 0.4Y 0.45Y 0.5 ( where

More information

5.2 GRAPHICAL VELOCITY ANALYSIS Polygon Method

5.2 GRAPHICAL VELOCITY ANALYSIS Polygon Method ME 352 GRHICL VELCITY NLYSIS 52 GRHICL VELCITY NLYSIS olygon Mehod Velociy analyi form he hear of kinemaic and dynamic of mechanical yem Velociy analyi i uually performed following a poiion analyi; ie,

More information

ME 391 Mechanical Engineering Analysis

ME 391 Mechanical Engineering Analysis Fall 04 ME 39 Mechanical Engineering Analsis Eam # Soluions Direcions: Open noes (including course web posings). No books, compuers, or phones. An calculaor is fair game. Problem Deermine he posiion of

More information

Mon Apr 9 EP 7.6 Convolutions and Laplace transforms. Announcements: Warm-up Exercise:

Mon Apr 9 EP 7.6 Convolutions and Laplace transforms. Announcements: Warm-up Exercise: Mah 225-4 Week 3 April 9-3 EP 7.6 - convoluions; 6.-6.2 - eigenvalues, eigenvecors and diagonalizabiliy; 7. - sysems of differenial equaions. Mon Apr 9 EP 7.6 Convoluions and Laplace ransforms. Announcemens:

More information

t + t sin t t cos t sin t. t cos t sin t dt t 2 = exp 2 log t log(t cos t sin t) = Multiplying by this factor and then integrating, we conclude that

t + t sin t t cos t sin t. t cos t sin t dt t 2 = exp 2 log t log(t cos t sin t) = Multiplying by this factor and then integrating, we conclude that ODEs, Homework #4 Soluions. Check ha y ( = is a soluion of he second-order ODE ( cos sin y + y sin y sin = 0 and hen use his fac o find all soluions of he ODE. When y =, we have y = and also y = 0, so

More information

!!"#"$%&#'()!"#&'(*%)+,&',-)./0)1-*23)

!!#$%&#'()!#&'(*%)+,&',-)./0)1-*23) "#"$%&#'()"#&'(*%)+,&',-)./)1-*) #$%&'()*+,&',-.%,/)*+,-&1*#$)()5*6$+$%*,7&*-'-&1*(,-&*6&,7.$%$+*&%'(*8$&',-,%'-&1*(,-&*6&,79*(&,%: ;..,*&1$&$.$%&'()*1$$.,'&',-9*(&,%)?%*,('&5

More information

Introduction to Congestion Games

Introduction to Congestion Games Algorihmic Game Theory, Summer 2017 Inroducion o Congeion Game Lecure 1 (5 page) Inrucor: Thoma Keelheim In hi lecure, we ge o know congeion game, which will be our running example for many concep in game

More information

Electrical Circuits. 1. Circuit Laws. Tools Used in Lab 13 Series Circuits Damped Vibrations: Energy Van der Pol Circuit

Electrical Circuits. 1. Circuit Laws. Tools Used in Lab 13 Series Circuits Damped Vibrations: Energy Van der Pol Circuit V() R L C 513 Elecrical Circuis Tools Used in Lab 13 Series Circuis Damped Vibraions: Energy Van der Pol Circui A series circui wih an inducor, resisor, and capacior can be represened by Lq + Rq + 1, a

More information

6.003 Homework #8 Solutions

6.003 Homework #8 Solutions 6.003 Homework #8 Soluions Problems. Fourier Series Deermine he Fourier series coefficiens a k for x () shown below. x ()= x ( + 0) 0 a 0 = 0 a k = e /0 sin(/0) for k 0 a k = π x()e k d = 0 0 π e 0 k d

More information

Chapter 15: Phenomena. Chapter 15 Chemical Kinetics. Reaction Rates. Reaction Rates R P. Reaction Rates. Rate Laws

Chapter 15: Phenomena. Chapter 15 Chemical Kinetics. Reaction Rates. Reaction Rates R P. Reaction Rates. Rate Laws Chaper 5: Phenomena Phenomena: The reacion (aq) + B(aq) C(aq) was sudied a wo differen emperaures (98 K and 35 K). For each emperaure he reacion was sared by puing differen concenraions of he 3 species

More information

EXERCISES FOR SECTION 1.5

EXERCISES FOR SECTION 1.5 1.5 Exisence and Uniqueness of Soluions 43 20. 1 v c 21. 1 v c 1 2 4 6 8 10 1 2 2 4 6 8 10 Graph of approximae soluion obained using Euler s mehod wih = 0.1. Graph of approximae soluion obained using Euler

More information

INDEX. Transient analysis 1 Initial Conditions 1

INDEX. Transient analysis 1 Initial Conditions 1 INDEX Secion Page Transien analysis 1 Iniial Condiions 1 Please inform me of your opinion of he relaive emphasis of he review maerial by simply making commens on his page and sending i o me a: Frank Mera

More information

Macroeconomics 1. Ali Shourideh. Final Exam

Macroeconomics 1. Ali Shourideh. Final Exam 4780 - Macroeconomic 1 Ali Shourideh Final Exam Problem 1. A Model of On-he-Job Search Conider he following verion of he McCall earch model ha allow for on-he-job-earch. In paricular, uppoe ha ime i coninuou

More information

FLAT CYCLOTOMIC POLYNOMIALS OF ORDER FOUR AND HIGHER

FLAT CYCLOTOMIC POLYNOMIALS OF ORDER FOUR AND HIGHER #A30 INTEGERS 10 (010), 357-363 FLAT CYCLOTOMIC POLYNOMIALS OF ORDER FOUR AND HIGHER Nahan Kaplan Deparmen of Mahemaic, Harvard Univeriy, Cambridge, MA nkaplan@mah.harvard.edu Received: 7/15/09, Revied:

More information

Chapter 6. Systems of First Order Linear Differential Equations

Chapter 6. Systems of First Order Linear Differential Equations Chaper 6 Sysems of Firs Order Linear Differenial Equaions We will only discuss firs order sysems However higher order sysems may be made ino firs order sysems by a rick shown below We will have a sligh

More information

EE 301 Lab 2 Convolution

EE 301 Lab 2 Convolution EE 301 Lab 2 Convoluion 1 Inroducion In his lab we will gain some more experience wih he convoluion inegral and creae a scrip ha shows he graphical mehod of convoluion. 2 Wha you will learn This lab will

More information

Math 106: Review for Final Exam, Part II. (x x 0 ) 2 = !

Math 106: Review for Final Exam, Part II. (x x 0 ) 2 = ! Mah 6: Review for Final Exam, Par II. Use a second-degree Taylor polynomial o esimae 8. We choose f(x) x and x 7 because 7 is he perfec cube closes o 8. f(x) x / f(7) f (x) x / f (7) x / 7 / 7 f (x) 9

More information

t is a basis for the solution space to this system, then the matrix having these solutions as columns, t x 1 t, x 2 t,... x n t x 2 t...

t is a basis for the solution space to this system, then the matrix having these solutions as columns, t x 1 t, x 2 t,... x n t x 2 t... Mah 228- Fri Mar 24 5.6 Marix exponenials and linear sysems: The analogy beween firs order sysems of linear differenial equaions (Chaper 5) and scalar linear differenial equaions (Chaper ) is much sronger

More information

e 2t u(t) e 2t u(t) =?

e 2t u(t) e 2t u(t) =? EE : Signals, Sysems, and Transforms Fall 7. Skech he convoluion of he following wo signals. Tes No noes, closed book. f() Show your work. Simplify your answers. g(). Using he convoluion inegral, find

More information