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Unformatted text preview: Exam #2 ;
EEL 5173 (Spring 2005) i Name: Bill) Lang;
SS#: Please show all work for partial credit. (100 points total) Problem 1. (25 points) Consider a mechanical system shown in the ﬁgure: k1 n k2 k1 u2__. 1 Its differential equation is given by d2 t ml (:1; ) +(kl +k2)y1(t)‘k2.V2 =u1
d2 t ' m2 y2()—k2y,(t)+(k,+k2)y2=u2 dt2
Express the system in the state space representation, considering y1(t) and y2(t) to be the
output and u1(t) and u2(t) to be the input (note that this is a twoinput twooutput system). 9W Let x: 2' Wm y; n] y' U1
Y1 / I YA”
VL’
[kl 2 \
X‘:X2,XL:—k;’ yll’é/l‘l—i’l—i
'   KL kn‘kz
stx‘t, Xt m V.  m Vanni
. O ' O O 00
>0 K+t< Xl 40
\;._i 0 £ 0 ml u
“H “M 00
k0 0 o I
W O _l<I‘H<L 0 0J— Problem 2. (25 points) Suppose x is a column vector 4:]— Then with respect to the standard basis (e1,ez,e3) the representation is just x = [7 2 4]T.
Suppose we take as a basis the vectors p1 ii p: {it a i What is the representation of x in the new basis? Gown; X=Q1T, +04% M3 7’5 I kt ‘
2 :Q, Z +% f +a3[g
4 3 '4 8 37> (JULM @2318 Q
l / ;:‘7 901 (m NWﬂKu ofy [Hit in: M} ii
,16 \2
/" I Problem 3. (25 points) For the following system 561 0 O 0 xl
562 = l 0 2 x2
3'63 0 1 1 x3 ﬁnd its eigenvalues, eigenvectors, and then diagonalize it. 3 b
Show v. 0 0 O
A = 1 0 2
0 1 1
Its characteristic polynomial is
A 0 0
A(A) = det(AI — A) = det —1 A —2
0 —1 A — 1 = A[A(A — 1) — 2] = (A — 2)(A‘+ 1)A Thus A has eigenvalues 2, —1, and 0. The eignevector associated with A = 2 is any nonzero solution of i —2 0 0 1
(A—21)q1= 1 —2 2 ‘q1=0
o 1 —1 Thus q1 = [O 1 l]’ is an eigenvector associated with A = 2. Note that the eigenvector is not
unique, [0 a a]’ for any nonzero real (1 can also be chosen as an eigenvector. The eigenvector associated with A = ——1 is any nonzero solution of
1 0 0
(A  (—1)I)q2 = 1 1 2 (12 = 0
0 1 2 which yields q; = [0 — 2 l]’. Similarly, the eigenvector associated with A = 0 can be
computed as q3 = [2 1 —— 1]’. Thus the representation of A with respect to {q1, q2, q3} is 2 o 0
A: 0 —1 0 0 0 0 It is a diagonal matrix with eigenvalues on the diagonal. This matrix can also be obtained by
computing A=Q‘1AQ
with
0 o 2
Q=[q1q2q3]= 1 —2 1 (3,55,; Problem 4. (25 points) For the following system EH? illilHil“
QOLAQWK Cmdvnllalm‘lan‘ha moth“)? “I
F: [e m : [I‘ 4] WM) :{ is it controllable? No’t ' ...
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This note was uploaded on 01/14/2012 for the course EEL 5173 taught by Professor Staff during the Fall '11 term at University of Central Florida.
 Fall '11
 Staff

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