MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Department of Electrical Engineering and Computer Science
6 241: Dynamic SystemsFall 2008
Homework
Solutions
Exercise 13 1 (a) The system described by
x = z
z =
4x3 + 2 x
has an equilibrium point at (0, 0) for any val
6.241 Spring 2011
Midterm Exam
March 27, 2011
Problem 1
Let A 2 Cnn , and B 2 Cmm . Show that X(t) = eAt X(0)eBt is the solution to X =
AX + XB.
P
1
Solution Recalling the denition of matrix exponential, eAt = 1 i! (At)i , it is
i=0
clear that, for any ma
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Department of Electrical Engineering and Computer Science
6.241: Dynamic SystemsSpring 2011
Homework 8 Solutions
Exercise 17.4 1) First, in order for the closed loop system to be stable, the transfer function
from ( w
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Department of Electrical Engineering and Computer Science
6.241: Dynamic SystemsSpring 2011
Homework 1 Solutions
Exercise 1.1 a) Given square matrices A1 and A4 , we know that A is square as well:
A1 A2
A=
0 A4
=
Note
1
Solution to 6.262 Final Examination 5/21/2009
Solution to Question 1
a) We rst solve the steady state equations for the Markov process. As we have seen many
times, the pi , i 0 for a birth-death chain are related by pi+1 qi+1,i = pi qi,i+1 , which in th
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Department of Electrical Engineering and Computer Science
6.241: Dynamic SystemsSpring 2011
Homework 4 Solutions
Exercise 4.7 Given a complex square matrix A, the denition of the structured singular value
function is
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Department of Electrical Engineering and Computer Science
6.241: Dynamic SystemsFall 2007
Homework 3 Solutions
Exercise 3.2 i) We would like to minimize the 2-norm of u, i.e., kuk2 . Since yn is given as
2
yn =
n
X
hi
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Department of Electrical Engineering and Computer Science
6.241: Dynamic SystemsSpring 2011
Homework 7 Solutions
Exercise 15.1 (a) The system is causal if the impulse response is right-sided. Consider a sequence
e at
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Department of Electrical Engineering and Computer Science
6.241: Dynamic SystemsSpring 2011
Homework 10 Solutions
Exercise 23.1 a) We are given the single input LTI system:
0 1
0
x = Ax + bu , A =
, b=
0 0
1
The solut
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Department of Electrical Engineering and Computer Science
6.241: Dynamic SystemsSpring 2011
Homework 9 Solutions
Exercise 21.1 We can use additive perturbation model with matrices W and given in Figure 21.1.
W21
0
0
P
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Department of Electrical Engineering and Computer Science
6.241: Dynamic SystemsSpring 2011
Homework 5 Solutions
Exercise 7.2 a) Suppose c = 2. Then the impulse response of the system is
h(t) = 2(e
t
e
2t
)
for t
0
On
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Department of Electrical Engineering and Computer Science
6.241: Dynamic SystemsFall 2007
Homework 2 Solutions
Exercise 1.4 a) First dene all the spaces:
R(A) = cfw_y 2 Cm | 9x 2 Cn such that y = Ax
R? (A) = cfw_z 2 C