6.241 Spring 2011
Midterm Exam
March 27, 2011
Problem 1
Let A Cnn , and B Cmm . Show that X (t) = eAt X (0)eBt is the solution to X =
AX + X B .
1
Solution Recalling the denition of matrix exponential, eAt = i! (At)i , it is
i=0
clear that, for any matrix
6.241 Spring 2011
Final Exam
5/16/2011, 9:00am 12:00pm
The test is open books/notes, but no collaboration is allowed: i.e., you should not
discuss this exam or solution approaches with anybody, except for the teaching sta.
Problem 1
Let (A, b, c, 0) be a
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:
01
0
x = Ax + bu , A =
, b=
00
1
The soluti
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 1
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 7 Solutions
Exercise 15.1 (a) The system is causal if the impulse response is right-sided. Consider a sequence
eat u
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Department of Electrical Engineering and Computer Science
6.241: Dynamic SystemsFall 2008
Homework 6 Solutions
Exercise 13.1 (a) The system described by
x=z
z = 4x3 + 2x
has an equilibrium point at (0, 0) for any valu
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(et e2t )
for t 0
One
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., u2 . Since yn is given as
2
yn =
n
hi un1
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 Cm | x Cn such that y = Ax
R (A) = cfw_z Cm | y z
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