EE571
Solution to HW#3
0. Have you picked a lab partner to work with you? Your first lab is coming soon! 1a) The analogous circuit for the first problem is: Original Circuit Analogous Circuit
w=input Torque F1 J1 2 1 F2 Na: 1 J2
1/F2=1 Na:1 2:1
K
1/F1=1 w
Due Monday, February 8
1. a)
EE572
HW #6
$
$
$
Use the fact that the solution to, x k +1 = Ax k is x k = A k x 0 to find the Ztransform of A k (hint: take the Z$
$
$
transform of both x k +1 = Ax k and x k = A k x 0 and solve for the Ztransform of A k )
EE572  Solution to HW #6
1. a)
$
$
$
Use the fact that the solution to, Zcfw_x k +1 = Ax k = zX ( z ) zx 0 = AX ( z ) or X ( z ) = [ zI A]1 zx 0 to find the Z$
$
$
transform of A k (hint: take the Ztransform of both x k +1 = Ax k and x k = A k x 0 and
EE572
HW #7
Due Wednesday, February 10
^
^
1. a)
If x k+1 = A x k + B w k , find the eigenvalues and eigenvectors then use the similarity transformation, xk
= Pzk, to determine which of the eigenvalues are controllable (you may use Matlab if you wish):
1
EE572  Solution to HW #7 1. a) i) Solution: Find eigenvalues from det[sIA] = 0 = (s3)(s1). Therefore, eigenvalues are s1= 3 and s2= 1. The eigenvectors are found from [siIA]Pi=0. The eigenvectors are P1=[1 1]T and P2=[1 1]T . Thus, a similarity tr
Due Wednesday, February 17
1.
EE572
HW#9
Consider the following discretetime multiinput state variable model:
 2 2
2  1
x k+1 =
xk +
wk 0
0  2
 1 1
a)
Find the eigenvalues of the openloop system.
b)
Is the system stable?
c)
Is the system con
EE572  Solution to HW #8
1. a)
Solution: Consider the following discretetime state variable model (don't use Matlab):
4 4
1
x k+1 =
x k + w k 1
4 4
2
0 1
0
$
$
The characteristic equation is s2+8s+0=0. Thus, A pv =
and b pv = 1 . Thus, the sim
Solution
1.
EE572
HW#9
Consider the following discretetime multiinput state variable model:
 2 2
2  1
x k+1 =
xk +
wk 0
0  2
 1 1
a)
Find the eigenvalues of the openloop system. Solution: the eigenvalues are cfw_2, 2
b)
Is the system stabl
Due Monday, March 22nd (Happy Spring Break)
1.a)
EE572
HW #15
Sketch the root locus of the following splane openloop pole zero configurations (Use Matlab's rlocus() command to check your
answers) :
j
i)
j
ii)
j
iii)
j4
(two poles at the origin)
j4
(two
EE572
0.
1.a)
Solution to HW #15
Check your scores under the grades option and make sure all records are correct!
Sketch the root locus of the following splane openloop pole zero. Solution:
j
i)
60
j
ii)
j
iii)
j4
j4
(two poles at the origin)
(two poles
Due Monday, March 29
EE572
HW #17
1. Keep working on your project!
2. Consider the system from HW#16:
T
s
W(s) +
G c (z)
G zoh(z)
10
s( s + 8)
Y(s)

Recall that we have already designed a lead compensator, Gc(z), to meet the following specifications:
ts
Solution
2.
EE572
HW #16
Given the system:
Ts
W(s) +
G lead(z)
G zoh(z)
10
s( s + 8)
Y(s)

a) Find an sdomain model for the openloop system including the ZOH if Ts = 10 msec.
Solution: From class we learned that the approximate transfer function for GZ
Due Wednesday, April 7
EE572
HW #18
Note: PreLab 4 is also due next Monday. You should be able to do it, now.
1.a)
Determine the type number of the following openloop Zdomain transfer functions:
i ) G( z ) =
10( z + 1) 2
z ( z 1)
ii ) G ( z ) =
10( z +
EE572 Solution to HW #17
1. Consider the system from HW#16:
T
s
W(s) +
G zoh(z)
G c (z)
10
s( s + 8)
Y(s)

Recall that we have already designed a lead compensator, Gc(z), to meet the following specifications:
ts 0.4 sec and Mp2%
(see solution to HW#16 on
EE572  Solution to HW #18 1.a) Determine the type number of the following openloop Zdomain transfer functions:
i ) G( z ) =
10( z + 1) 2 z ( z 1)
ii ) G( z ) =
10( z + 1)3 z 2 ( z 1)
iii ) G ( z ) =
10( z + 1)3 z ( z 1) 2
Solution: i) type 1 ii) type 1
EE572
Soln to RISHW #5*
1 0
1
5
x=
x + 2 w, x(0) = 6 , and
Given the continuous state variable model,
0 3
y = [3 4]x
the corresponding discrete nextstate approximation to this model,
x k +1 = Ax k + Bwk , x0
with sampling period T.
y = Cx
k
k
a
EE572
RISHW #5*
Due Wednesday, February 2nd (Happy Ground Hogs Day)
0. Please complete Lab 1 that we took data for after class today. It's due on Wednesday, too! On
days when you have a lab due, I usually give you a short, oneproblem HW just to keep you
EE572
HW#2
Due Monday, January 24
0.
Pick a Lab partner and form a group of 45 people for the project. It is helpful if at least one of your
group members has some experience in assembly language programming.
1. a) Represent the number 3.625 in 8 bit fi
EE571 Solution to HW#1 1. a) Find the state variable model of the form
& x = Ax + Bw, x( 0+ ) for the following electrical network:
1 ohm
+
iL
1 0 u( t)+ w ( t) u( t)
Amp s
Vc 
1 /3 F
2 o hm s 1 /2 H
(Hint: Let x =
v C ) iL
(make sure to include your
EE571  Solution to HW#2 1. a) Find the state variable model of the form
& x = Ax + Bw, x ( 0 + ) y = Cx + Dw
for the following electrical network:
1/3 F
1 ohm + Y2 1/2 H 10u(t)+w2(t)u(t) Y1
10u(t)+w1(t)u(t)
+ 
2 ohm
(make sure to include your initial
Due Wednesday, September 11
0.
EE571
HW#3
Pick a lab partner to work with. Your first lab is coming soon.
1. a) Find an analogous circuit then a state variable model for the rotational system shown
below:
w=input Torque
Na:1
1
F1
K
F2
J1
J2
2
Let F1=F2=1
Due Monday, Sept. 9th
1. a)
EE571
Find the state variable model of the form
HW#2
x Ax Bw, x( 0 )
for the following electrical network:
y Cx Dw
1/3 F
1 ohm
+ Y2 
1/2 H
2 ohm
10u(t)+w1(t)u(t) +

10u(t)+w2(t)u(t)
Y1
(make sure to include your initial con
EE571
Solution to HW#4
1a) The solution to the state equation is x(t)=eAt x(0). To evaluate, first lets find the eigenvalues of A:
s 2 1
2 1
2
A=
sI A = 1 s 2 = s 4 s + 3 = ( s 1)( s 3) = 0 s1 = 1 and s 2 = 3
1 2
Next, lets find the eigenvectors of A:
1
Due Monday, September 1
EE571
HW#4
0.
Go back and finish HW3 Problem 2 now that you know how to do it.
1. a)
Recall the first matrix from HW#3, problem 2. Since you have already found the
eigenvectors/eigenvalues for this matrix, for the system given belo
Due Monday, September 19
EE571
Easy 1 Problem Assignment! HW#7
0.
Lab 1 (the in lab portion) is due Monday. Each person should email his/her own lab.
1.
Given the following state variable model (see HW#4):
4 2
0
1
x =
x 1 w, x ( 3) 0
2 4
y 1 0x
EE571
Solution to HW#7
t
1a) The solution is x(t) = e
A(t t 0 )
e
x( t 0 ) +
A(t )
Bw( )d . To evaluate eAt, first lets find the eigenvalues of A:
t0
s + 4 2
4 2
A=
sI A =
= s2 + 8s + 12 = ( s + 2)( s + 6) = 0 s1 = 2 and s2 = 6
2 4
2 s + 4
2c1
1 0
Due Wednesday, September 21
EE571
HW#8
Since it is football season, you have decided to invest in a satellite dish. Rather than pay for a DirectTV technician to come and
align your dish antenna, you decide to build your own automatic tracking systems that
EE571
Solution to HW#8
1a) The block diagram for the satellite tracking system is:
1
2
comm
+

5
1+s/(40
)
V in
Ka
+

1/Ra
Va +
60

Motor Ckt
Amplifier
Receiver
Ia
1
Vb
(J L/Na +J )s
m
Kt
T
0.5
142.45/s
m
Torque
1/Na
1/12,000
Gear
out
out
1/s
Integrato