This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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 ' ...
View
Full Document
 Fall '11
 Staff

Click to edit the document details