EECS 560 - Solutions to HW #1
Prof. Grizzle
1.
i (t)
is a solution
d2 i
di
+ a1
+ a0 i = bui
dt
dt2
and i (0) =
di (0)
= 0.
dt
(1)
Consider 3 (t) = 1 1 (t) + 2 2 (t). Then,
d (t)
d (t)
i) d 3 (t) = 1 1
+ 2 2
dt
dt
dt
2
2
2
d
(t)
d
2 (t)
1
+ 2
ii) d 2 3 (t
%
%
%
%
%
%
%
%
Example of using ODE45
to simluate a second order system
x1_dot = -3x1-2x2+sin(t)
x2_dot =
x1
initial condition x1(0)=2; x2(0)=8;
input u = sin(t)
function myscript
global a
a = 0.5;
t0 = 0; tart time
tf = 20; %end time
x10 = 0.1; %initial
% % % % %
Linear Systems, Homework 2 Date: 11/13/06 File: HW2_2.m Note: bicycle-rider system Name: Feng-Li Lian, [email protected]
c close all; clear all; % specifications g h V mvg mv mhg mh hv hh Iv Ih L L Lr h V B1 B2 l w2 fu f0 F = = = = = = =
% % % %
Linear Systems, Homework 2 Date: 11/13/06 File: HW2_1b.m Name: Feng-Li Lian, [email protected]
c close all; clear all; % %syms R1 R2 L C s t tau % 1a R1 R2 L C = = = = 2; 10; 0.5; 1;
Ap = [ -R1/L -1/L; 1/C -1/(C*R2)]; Bp = [ 1/L ; 0 ]; Cp
Project, Linear Systems, Fall 2008
Page 1
Your report should include WORD document, Matlab codes, , etc. Assigned: 9/17/08, Due on 1/20/09, 5pm, by e-mail to [email protected]
Project, Linear Systems, Fall 2008
Page 2
Project, Linear Systems,
EECS 560 - HW #7
Prof. Grizzle
Due November 1, 2013
At beginning of class
Reading Assignment: Either Chapters 3 and 4 of Williams and Lawrence or the time-invariant parts of
Chapter 6 of Chen. This is nice material.
1. For the control system
0 0 2
1
x =
EECS 560 - HW #9
Prof. Grizzle
Due November 22, 2013
At beginning of class
1. For which values of IR is the system x =
1
1
x
1
(a) stable in the sense of Lyapunov?
(b) asymptotically stable in the sense of Lyapunov?
(c) unstable?
2. Let A be a real n n ma
EECS 560 - Solutions to HW #8
Prof. Grizzle
1.
x1
x2
Let x =
x3 Note that x 3 = 1 x3 , which is clearly uncontrollable. Hence, we eliminate the corre
x
4
x5
sponding column & row to obtain
x 1
1
x 2 0
=
x 0
4
0
x 5
1
0
1
0
0
2
0
0
h
y= 0
EECS 560 - Solutions to HW #10
Prof. Grizzle
"
1.
(a)
det(I A) = det
1
1
#
= 2 1
= 1
unstable.
(b) First, check if the system is controllable:
Ab =
" #
1
0
Q=
"
0
b Ab =
1
#
1
0
dim Q = 2
c.c.
System can be stabilized.
(c) Desired char. poly. is
4()
= (
PLACE NAME OR INITIALS HERE:
1
Exam Number:
SOLUTIONS
EECS 560 Midterm Exam
HONOR PLEDGE: Copy (NOW) and SIGN (after the exam is completed): I have neither
given nor received aid on this exam, nor have I observed a violation of the Engineering Honor Code.
PLACE NAME OR INITIALS HERE:
1
Exam Number:
EECS 560 Midterm Exam
Wednesday, Nov. 3, 2010, 4:40-6:30 PM
By LAST Name
A to R in 1013 DOW
S to Z in 1005 DOW
HONOR PLEDGE: Copy (NOW) and SIGN (after the exam is completed): I have neither
given nor received a
PLACE NAME OR INITIALS HERE:
1
Exam Number:
EECS 560 Final Exam
Friday, December 13, 2013, 4:10 PM to 6:00 PM
A through J in Room 1017 DOW
K through L in Room 1690 BBB
M through Z in Room 1670 BBB
HONOR PLEDGE: Copy (NOW) and SIGN (after the exam is compl
EECS 560 - Solutions to HW #3
Prof. Grizzle
1. X =
("
a1
a2
#
)
ai IR . In principle, we should check each and every axiom, in order to show
a3 a4
that (X, IR) forms a vector space, but
it. "We simply
("we do# not
" do #
# " observer
#) that it is true.
EECS 560 - Solutions to HW #5
Prof. Grizzle
1. (a) eAt =
X
1 k k
t A
k!
k=0
"
0
A = I,
A4k+0 = I,
A=
0
#
1
1
0
A4k+1 = A,
,
2
A =
A4k+2 = A2 ,
"
1
0
0
1
"
#
3
,
A =
A4k+3 = A3
3 3
4 4
2 2
eAt = I + At + A t + A t + A t +
2!
3!
4!
2
4
= 1 + 0 t2 + 0 + t +
EECS 560 Exam I, Wed. Nov. 6, 2013
Starts at 4:40 PM sharp and ends at 6:30 PM sharp
Exam Topics:
1. The concept of state.
2. State variable models form I/O and transfer matrix representations.
3. Minimal state variable representations of SISO transfer fu
EECS 560 - HW #4
Prof. Grizzle
Due October 4, 2013
At beginning of class
Remark: I am assuming that you have had some experience with matrix algebra, and thus you know about
e-values and e-vectors. The main thing to learn in EECS 560 is how to use these q
EECS 560 - HW #2
Prof. Grizzle
Due September 20, 2013
At beginning of class
1. Construct state variable representations for the transfer matrices
3
s+2
U (s) (Two outputs and one input)
Y (s) =
s+1
s2 + s + 1
i
h
s+1
3
U (s) (One output and two inputs)
