ME/ECE 851 Fall 2010
Homework # 7
Due Wednesday, November 17, 2010
1. Consider the LQR problem for system
x1 = x2 ,
x2 = x1 + u
and the performance index
[x2 (t) + u2 (t)]dt,
1
J=
>0
0
(a) Find the optimal stabilizing control in terms of .
(b) Find the cl
ME/ECE 851 Fall 2010
Homework # 8
Due Wednesday, December 1, 2010
1. Consider the system
x=
01
0 1
0
1
+
u,
1 0 x.
y=
We want to design an output feedback controller such that (i) all closed-loop eigenvalues
satisfy Re [] 2; (ii) The output y is asymptoti
ME/ECE 851 Fall 2010
Homework # 9
Due Friday, December 10, 2010
1. Consider the continuous-time system
x=
01
0 1
0
1
+
u,
y=
1 0 x.
We want to design a discrete-time output feedback controller such that (i) all discretetime system closed-loop eigenvalues
ME/ECE 851 Fall 2010
Homework # 6
Due Friday, November 5, 2010
1. Transform the pair
A=
0
3
1
1
01
0
0 3 1
1 4 1
0 1 0
,
B=
0
1
0
0
0
0
1
0
into controller form Ac = P AP 1, Bc = P B . What is the similarity transform matrix
in this case? What are the con
ME/ECE 851 Fall 2010
Homework # 2
Due Friday, September 24, 2010
1. Exercise 3.7
2. Exercise 3.17
3. Exercise 3.19
4. Consider the matrix A given by:
410
A := 0 4 3 .
004
(a) Find the eigenvalues of A and their corresponding eigenvectors.
(b) Transform A
ME/ECE 851 Fall 2010
Homework # 3
Due Wednesday, October 6, 2010
1. Exercise 3.36.
2. Exercise 3.38. Add the following part to the problem.
(c) Investigate asymptotic stability and BIBO stability of the system when the output is the
automobile vertical ac
ME/ECE 851 Fall 2010
Homework # 5
Due Wednesday, October 27, 2010
1. Ex 5.11
2. Ex 6.2
3. Consider the system
x = Ax + Bu, y = Cx + Du.
(a) Show that only controllable modes appear in eAt B , and therefore in the zero-state
response of the state.
(b) Show
ME/ECE 851 Fall 2010
Homework # 1
Due Friday, September 10, 2010
1. Exercise 1.10.
2. Find the state model of the circuit of Figure 1.3 when the state variables are taken
as the voltages across the capacitors and the current through the inductor. View the