ace 82.6 .. Fail 2005
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ECE 826 Fall 2003
Homework # 12
Due Wednesday, December 3, 2003
1. Consider the system
x(t) =
0
1
1 1
y (t) =
1 1 x(t)
x(t) +
0
1
u(t)
Design an output feedback controller such that all closed-loop eigenvalues satisfy
Re cfw_ 2 and the closed-loop transfe
ECE 826 Fall 2003
Solution of Homework # 5
1. (a)
1
31
2 0
W = [B AB A2 B ] = 0
0 4 0
rank(W ) = 2. Hence, the system is not controllable.
(b)
C
M = CA =
CA2
1
1
1
2 1
1
1
1
1
0 1 1
Rows number 1, 2, and 4 are linearly independent. Thus, rank(M ) = 3
ECE 826 Fall 2003
Homework # 11
Due Wednesday, November 26, 2003
1. Consider the system of Problem 3 in Homework # 9, together with the state feedback
control that was given in the solution of that homework.
(a) Assuming that you only measure x1 , design
ECE 826 Fall 2003
Homework # 10
Due Wednesday, November 19, 2003
k
F
f f f
E
m
f f f
p
Eq
1. The mass-spring system shown above is modeled by
mq + kq = F
Set 2 = k/m and F = ku to rewrite the model as
q + 2q = 2u
(a) Using x1 = q and x2 = q as th
ECE 826 Fall 2003
Homework # 1
Due Wednesday, September 3, 2003
1. Exercise 2.1 of the textbook.
2. Figure 1 shows an RC network connecting ampliers. The ampliers are assumed to
have linear inputoutput characteristics represented by vi = Ki ui , where ui
ECE 826 Fall 2003
Homework # 3
Due Wednesday, September 17, 2003
1. Find the transition matrix of the system
x=
1
x
1+t
for t 0.
2. Find the transition matrix of the system
e2t
e2t
1 + e2t
1 e2t
x=
3. Find the exponential matrix exp(At) for
001
A= 0 1 0
ECE 826 Fall 2003
Homework # 5
Due Wednesday, October 8, 2003
1. Consider the system
30
2
1
2 x(t) + 0 u(t)
x(t) = 2 1
4 0 3
0
1
11
2 1 1
y (t) =
x(t)
(a) Is the system controllable?
(b) Is it observable?
2. Consider the system
2t2
1+t2
1
x(t) =
1
2t2
x
ECE 826 Fall 2003
Homework # 6
Due Wednesday, October 15, 2003
1. Exercise 13.5.
2. The linearized state model of a satellite about a nominal trajectory is given by
x=
0
1
3 2
0
0
0
0 2/r
00
0 2r
01
00
x +
00
10
00
0 1/r
u
where and r are positive consta
ECE 826 Fall 2003
Homework # 2
Due Wednesday, September 10, 2003
1. Exercise 2.6 of the textbook.
2. Exercise 2.13 of the textbook.
3. Consider the matrix dierential equation
X (t) = A(t)X (t),
X (t0 ) = E
where A, X , and E are n n matrices and A(t) is c
ECE 826 Fall 2003
Homework # 4
Due Wednesday, September 24, 2003
1. Study internal stability of the system
1 1
x(t) = 0 1 x(t)
00
for all possible values of the real parameters and .
2. Exercise 6.3 of the textbook.
3. Exercise 6.4 of the textbook.
4. Co
ECE 826 Fall 2003
Homework # 7
Due Wednesday, October 22, 2003
1. Exercise 13.11.
2. Exercise 13.15
3. Consider two systems represented by the state models cfw_A1 , B1 , C1 , D1 and cfw_A2 , B2 , C2 , D2 .
Suppose that (A1 , B1 ) and (A2 , B2 ) are contr
ECE 826 Fall 2003
Homework # 8
Due Wednesday, October 29, 2003
1. Consider the system
2 1 1
10
1 x + 0 1 u
x = 1 1
10
3 1 2
y=
2 1 1
1 0
1
x
Is the realization minimal? If not, nd a minimal realization.
2. Find a minimal realization of the transfer functi
ECE 826 Fall 2003
Homework # 9
Due Wednesday, November 12, 2003
You can use Matlab in Problem 2 and 3, but not in Problem 1.
1. Consider the system
2 1 1
1
1 x + 0 u
x = 1 1
3 1 2
1
Design a state feedback control to assign all closed-loop eigenvalues at