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Unformatted text preview: Name l MAE 340: Test f February 11', 2003 Number 120 are 4 points each; Questions 21 and 22 are 10 points each Answer Questions 1—10 based on the following problem statement: A system is modeled by the differential equations 55, =30;2 —)'cl)+l,1(x2 —x,)+2F1(t)+5F2(t)
5&2 =—4(ic2 —J'c1)—25(x2 —x1)+5F2(t)+F,(t) where Fla) and ‘F2 (1) are inputs. The outputs for this system are defined as
y1=6(x2 —x1)+2(jc2 —;isl) , y2 = 3x1 —x1 +4F;(t)+2F2,(t) Write the state—space and output equations for this system in the standard form ‘q'qu+Bu , y=Cq+Du' i. The size of the A matrix is: ~
(a) 4 x 2 (b) 2 x 4 (c) 2 x 2 (e) Other 2. The size a the B matrix is:
m (b) 4 x 4 (c) 2 x 2 (d) 2 x 4 (e) Other 3. The size of the C matrix is: (a) 4 x 2 (c) 2 x 2 (d) 4 x 4 (e) Other
4. The size of the D matrix is: v
(a) 4 x 4 (b) 4 x 2 @ (d) 2 x 4 (e) Other ‘ , x "X‘
5. Write a state vector for this system: i * LXZ
,_ r , r r. _, W 1:1 or 1 9 '
9‘ ‘N X 6. Write the input vector for this system: . 7. Write the A matrix for this system: \. *Q/ 0 0 l O W!
£inng A; o 0 0 l ' 5Q” I 00 L m ll —3 3 iin ~3 u '3
5“ ‘25” as 4 “Ar "‘0 0 0’
maxim  l 2’ as" 8_. Write theBmatrix for this system: 4 5/, +4 Owl)ng
{E
\i
MW
«gerx’lg
Ere/Vie
{~sz ‘0 o : 2
[g I 9. Write the C matrix for this system; , . ~~ I x (0 ~ 1 a D
C : ‘ " {0 (o '2 2 C : l r 1 i
10. Write the D matrix for this system: _
if” 0 O F O O
(3 I > 5
z i ’2,
L. 4 Z L 4
A system’s characteristic equation is given by: s2 +23 +5 = 0 5 A “if 4 ‘20
‘ ‘IL ‘ 2.
ll. Writethe general expression for the homogeneous solution:  1( 1L 2 .
0 ~ 4 «H (t): Ce?“ (2* + (D) 2. e time constant of the homogeneous solution is:'
(a) 1 sec \ (b) 71 sec (C) V2 sec (d) Other (e) Not enough information 13. The settling time of the homogeneous solution is: a) 1 sec (b) 2 sec 14. The per' a  e homogeneous solution is:
a) 1 sec @ (c) V2 sec '(d) Other 15. The value of the solution at t = 1 sec is: a) 0.37 (b) 0.02 (c) 4/3 sec (e) Not enough information (e) Not enough information 1mm ~_“’\.
(e) Not enough information ‘ (c) 0.63 (d) Other ‘ 16. Assume t  . ,tem is modeled by 55+3i+10x=4sin 3t . The system is:
(a) Unstable (b) Stable (c) Marginally stable (cl) Not enough information to know 1 7. Assume that a syste
(a) Unstable \(b) Stable . sume that a syste
(a) Unstable (b) Stable m is modele  '+18x=0. The system is:
(c) Marginally stable (d) Not enough information to know 21 m is modeled by X—Zic—leze . The system is: (c) Marginally stable (d) Not enough information to know 19. A system is modeled by 56+7X+12x =The settli ‘ the system is: 5 : "Thy/H43
(a) 3 secs (b) 1 sec (c) 1/3 sec {5 x .(d) 4 sec“ (e) Other '1; ’1
. . .. . . V — ‘3‘ 4C ‘ *3 
20. The particular solution for x+ 2x+x = 21 +5 :5: ~ ; r 7, “ / 9 (a) xp = 6t+7 (c) xp : 2t+5 I (d) Not enough info .(e) None oftheabove xP»; ell«LC; {75, "11} Cl 27‘ “(ﬁzcu V (7,:5’4:l
21. For the system shown, find the ordinary differential equation model. gxgagl‘l’i‘m 1114:2553 "Y, W : {laﬁi‘l'i‘w a"; i‘hé‘éﬁééé mm":
) L  l 22. Sketch the solution of 4i+8jc+104x= 8 for the time period t=0 to the {i “7% =
settling time, with x(0) =10 and jc(0) = 0. a 'Z. ,— ,, M ' f ’— =1 7L 1 v s \ _
all 3M?“ mm 1 7 ’1 % xm' Ce *5‘m(37t+p5 + 3 v + ,. g w Name MAE 340: Test I February 11,2003 Number 120 are 4 points each; Questions 21 and 22 are 10 points each Answer questions i—l 0 based on the following problem statement: A system is modeled by the differential equations 551 =11(2'c2 —2'c,)+3($c2 x1)+5F1(t)+2F2(t)
X2 = _25(i2 *x1)_4(x2 ‘x1)+F2(t)+5Fl(t) where F10) and F2 (t) "are inputs. The outputs for this system are defined as
y1 = 2(Jc2 —x])+6(J'r2 —ic,) , y2 = 4x,—x,+21«;(r)+3F2(t) Write the state—space and output equations for this system in the standard form 1. The size of the A matri ‘ 
(a)2x4 _ (02x2 (d)4x2 (e)0ther
2. The size of the B matri '
‘ (a)2x4 « (c)2x2 (d)4x4 (e)Other
3. The size of the C matrix is: '
(a) 4 x 2 (b) 2 x 2 (d) 4 x 4 (e) Other
4. The size 0 :2 D matrix is:
> '(b) _2 x 4 (c) 4 x 2 (d) 4 x 4 (e) Other
' (fig X,
5. Write a state vector for this system’s; ; 7;! E V a: ,7?
l1 / “XL 7! r
6. Write the input vector for this system: 7gL ) . H at {in
a , L s
L 7. Write the A matrix for this system: h d“ _ 0? l 0 ' 0 0 i "0
“yr ‘9 x“ r _ r3 ~il 3 ll . l o O 0 i
lﬁémﬂ‘ A‘0 O, 0, l A ~$ 3 *ll ll
4 23 J} ~23 4 ‘4 ZS. _23
8. Write the B matrix for this system: 0 O 0 0 § 7» s 0 0 l3: 0 o {3 ' ’ 1 S i i s 9. Write the C matrix for this system: I ‘ ~l ’(0 1 '(L/f‘TZ '2 ‘é C ’ ~ 4* 0 0 ‘L ~ 1 0 4 o i
10. Write the D matrix for this system: f0 0
2 ivy—L 3 A system’s characteristic equation is given by: s2 +4s+13 = 0 . (I ’1 ll.Write the general expression for the homogeneous solution: , 1'
~2* \ ‘ ‘
xJﬂzge 5m8itﬂ 2. The time constant of the h a oeneous solution is: ‘ .
(a) 1 sec (b) 72 sec (d) Other (e) Not enough information
13. The settlin ime of the homogeneous solution is:
a) 1 sec (c) 4/3 sec (d) Other (e) Not enough information 14.1The period of the homogeneous solutio ' ' »
a) 3 sec (b) 7: sec (c) V2 sec (cl) Other ’ (e) Not enough information 15. The value of the solution at t = 1 sec is: y
a) 0.37 (b) 0.02 (c) 0.63 (d) Other e) Not enough information ‘2'. The system is: A sume that a system is modeled by 56+3x—10x : e
(a) Unstable (b) Stable (c) Marginally stable (d) Not enough information to know 17. Assumet — . stem is modeled by 36+ 6x+18x : 43in 3t. The system is: (a) Unstable (b) Stable (c) Marginally stable (d) Not enough information to know
18. Assume that a system is modeled b ' 12x : 0. The system is: (a) Unstable (b) Stable (c) Marginally stable (d) Not enough information to know 9. A system is modeled by 56+5jc+ 6x = 0. The settling time of the system is: S,+ {2 3,14
@ (b) 1 sec (c) 1/2 sec (d) 4 sec (e) Other 5. ‘— _;__/~ [)2. f ,2,
. Nb 1‘ 9 r articular solution for 56+x+x=6t+13 is: l 7 ~ “2/ 3
m (b) xp =2t+l (c) xp = 6t+13 (d) Not enough info (e) Noneoftheabove :xfzclﬁl(Lg C, :C:
":6. (34340:? (X
21. For the system shovfn, find the dinary differential equation model. i ~+ '3' : friars?wa a“? ééiﬁé (“is”)
._ in . .  A‘xﬁflgﬂwwﬁ .X\ {X ; ngg‘lﬁm asgfwtﬁfﬁ WM) m lﬂ‘ .3
5“ » 22. Sketch the solution of 456+16X+80x = 8 for the time period t =0 to the
settling time, with x(0) = 5 and J'c(0) = 0.
1 275+ 472 Haﬁz a ,7;
_, f4j: ~ ‘ 1A 2 if“; .  ’ \ t, \ ,1 /
Si]; “J’s:— 1~ 4" (moi : 2:? :1 5,,ng Name MAE 340: Test I February 11, 2003 Number 120 are 4 points each; Questions 21 and 22 are 10 points each Answer Questions 1—10 based on the following problem statement: A system is modeled by thevdifferential equations I 56} = ~25(ic2 — 5c.) — 4(x2 — xl ) + F10) + 5F2 (1‘)
x2 =11u2 ‘x1)+3(x2 Tx1)+5F2(t)+2F1(t) where EU) and F2 (1‘) are inputs. The outputs for this system are defined as
yl ='4(x2 — x1) — (x2 — xl)+ 2F; (t)+3F2 (t) , y2 = Zi‘f 6xl Write the state—space and output equations for this system in the standard form q=Aq +Bu , y=Cq +Du l. The‘ size of e A matrix is:
m (b) 2 x 4 (c) 2 x 2 (d) 4 x 2 (e) Other
2. The size of the B matrix is:
(a) 2 x 4 (b) 2 x 2 (d) 4 x 4 (e) Other
3. The siz o e C matrix is: _ 2
w” (b) 4 x 4 (c) 4 x 2 (d) 2 x 2 (e) Other
4. The size of the D matrix is: ,
(a) 4 x 4 (b) 2 x 4 (c) 4 x 2 (e) Other 6. Write the input vector for this system: XL 7. Write the A matrix for this system: ' 79  7i
5. Write a state vector for this system: % j or : ﬁr
O , . O 0
0 I h g o 0 I A 4 2 '4
“Q” A“ 4 ~4 2f—2§" "" O 0 O
gdzi‘hm ' vs 3 ~Il \l 23 e“ ‘3 Lb. Write the B matrix for this system: 0 O f O O — r : / 5' _ '0 g j O O
2 f 5 1 5f 9. Write e C matrix for this system: i
C ;  Ar ‘i l "l C : c 4 I 4 s I 10. Write the D matrix for this system: ’23
(3'00 A system’s characteristic equation is given by: s2 + 6s +10 2 0 I li.Write the general expression for the homogeneous solution:
' a ﬁt A / /‘ CIﬁ Siakxh T 12. The time constant of the homogeneo "on is: ' .
(a) 1 sec (b) 71 sec (c) V2 sec (d) Other ‘ (e) Not enough information 13. The settling time of the u o n o eneous solution is: ' " r
a) 1 sec (b) 2 sec (d) Other (e) Not enough information
14. The period of the horn   a  us solution is:
a) 3 sec (b) 7: sec (d) Other (e) Not enough information 15. The value of the solution at t = 1 sec is: V a) 0.37 (b) 0.02 ’ (c) 0.63 (d) Other (e) Not enough information
: J Han (a) Unstable (b) Stable ‘ c) Marginally stable (d) Not enough information to know V . ‘ \me that a system is modeled by ii—jc + 12x 2 {2’ . The system is:
(a) Unstable (b) Stable (c) Marginally stable (d) Not enough information to know 18. Assume th  system is modeled by 56+2ic+12x= 6sin 2t. The system is:
(a) Unstable (b) Stable (c) Marginally stable (d) Not enough information to know 19. A system is modeled by ji+4jc+3x= he settling time of the system is: ’41: ’10”, V
(a) 2 secs y (b) 1 sec (c) 3 sec '(d) 4 sec (e) Other '5‘ 2 2 ‘_\H_/ //1
' ‘ . t
20. The particular solution for X+3x+x=2t+13 is: w :L'}: ‘i ,7)
(a) xp = 6t+7 (b) xp =2t+1 (c) xp = 2t+13 (d) Not enough info 1 I ‘:C‘ ct:i§*é:3r 7(
21. For the system shoan, find the or inary differential equation model. 7"; 75‘ 9 11+ 73:19 i} :iisegl’t‘a‘ah of: u '4 s
{XI} 5' , g, ’ f ' . i £1 _ 
. , v , ., IX :3» : Mahala s f‘rkédﬁm mi 2%,.
W 2, k, '7 ‘1:“ ‘3” _ I a
a t. ! {Aﬂﬁ ~53: ’Yzﬂ'g‘rﬁ ’3“ O my; M: ,‘ .5 5
s 1 tuftAri ¥ is {Afﬁdavit Ni? 22. Sketch the solution of 3x+12x+ 120x 2 6 for the time period t = O to the
settling time, with x(0) = 20 and 2&(0) = 0. ’36 +471+40x11 "’5 (X‘s: ‘/10 nil/t ‘2 ice
5“ —, ’4ixlm"('o \ Sb, 6 M S 1,; . 1 “2%.; Wm ...
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