Linear Control Systems: Designed by Input/Output Methods
EE 4231

Fall 2010
S. Boyd
EE102
Table of Laplace Transforms
Remember that we consider all functions (signals) as dened only on t 0.
General
f (t ) f +g f ( R) df dt dk f dtk
t
F (s) = F +G F
0
f (t)est dt
sF (s) f (0) sk F (s) sk1 f (0) sk2 f ( ) d G(s) = F (s) s df dk1 f
Linear Control Systems: Designed by Input/Output Methods
EE 4231

Spring 2014
EE4231
Fall 2014
Solutions to Exam 1
Problem 1.
(a) The transfer function from R to Y is
Y (s)
=
R(s)
1
2+
s
1/s
1 + 3/s
=
2s + 1
.
s(s + 3)
(b) The transfer function from R to Y has a zero at 0.5 and two poles at 0, 3.
(c) The transfer function from W to
Linear Control Systems: Designed by Input/Output Methods
EE 4231

Spring 2014
EE 4231 Linear Control Systems
1.
2.
3.
Course information
Introduction to Automatic Control
History and the evolution of ideas
Examples of control systems
Feedback: technology, engineering &
systems science
course information
Instructor: Tom Posbergh
Of
Linear Control Systems: Designed by Input/Output Methods
EE 4231

Spring 2014
AEM4321/EE4231
Fall 2014
Solutions to Exam 2
Problem 1.
The loop gain for the unitary feedback system is
1
2
2kp
D(s)G(s) = kp
=
.
3
1 + Td s s(s + 2)
Td s + (2Td + 1)s2 + 2s
(a) For the closed loop system
T (s) =
D(s)G(s)
2kp
=
.
3
1 + D(s)G(s)
Td s + (
Linear Control Systems: Designed by Input/Output Methods
EE 4231

Spring 2014
swam/1351mm e E2 20!
amework Q (n/Mom
b/ It
(89
Imaginaiy
N:0,P=O=>Z=N+P=0
The closedloop system is stable for any K > O.
'ch
.  The Bode plot shows an initial phase of 0 hence the Nyquist starts
on the positive real axis at A. The Bode ends with a ph
Linear Control Systems: Designed by Input/Output Methods
EE 4231

Spring 2014
AIS/Mam [@4431 Jig/MW
Sea/Wm: 710 #omemr/K 5
ab/em I .
(a) With all the negatives, the problem statement might be confusing.
With the G(s) as given, MATLAB needs to plot the negative locus,
which is the regular positive locus for G. The locus is plotted b
Linear Control Systems: Designed by Input/Output Methods
EE 4231

Spring 2014
Q/C'zj/zo/M
Examgk 0995 cfw_New LT: SBSHLYM
[509 +75ch +51%) =uc+3 um = Input
\CB :OUdeT
mm unms gage) ,gcm
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let 30 435 k
W or): gen
Linear Control Systems: Designed by Input/Output Methods
EE 4231

Fall 2013
G( jw)
G( jw)
Im
w
Re(G( jwi )
G( jwi )
Re
G( jwi )
Im(G( jwi )
w0
Im
w
G( jwi )
90
Re
w0
w
Im
w0
G( jwi ) 90
Re
Im
w0
1
0.5
45
wT 1
Re
Im
w
w0
0.5
1
Re
Im
w0
w
0.5
1
Re
wn
wn
wn
0
Nyquist Diagrams
From: U(1)
6
4
To: Y(1)
Imaginary Axis
2
0
2
4
MATLAB c
Linear Control Systems: Designed by Input/Output Methods
EE 4231

Fall 2013
Lead Controller
Monday, November 23, 2009
12:21 AM
week11 Page 1
Sunday, November 29, 2009
10:54 PM
week11 Page 2
Nyquist of a lead controller
Monday, November 23, 2009
12:24 AM
week11 Page 3
Maximum phase of a lead term
Monday, November 23, 2009
12:25 AM
Linear Control Systems: Designed by Input/Output Methods
EE 4231

Fall 2013
Building Blocks
Tuesday, October 27, 2009
11:41 PM
week8 Page 1
Complex Roots
Tuesday, October 27, 2009
11:43 PM
week8 Page 2
Decomposition of realrational transfer functions
Tuesday, October 27, 2009
11:45 PM
week8 Page 3
First order System
Wednesday, O
Linear Control Systems: Designed by Input/Output Methods
EE 4231

Fall 2013
Frequency Response of LTI systems
Friday, October 23, 2009
9:43 AM
Week7 Page 1
Sinusoidal Response
Friday, October 23, 2009
9:52 AM
Week7 Page 2
SinusoidalResponse
Friday, October 23, 2009
9:57 AM
Week7 Page 3
Summary
Friday, October 23, 2009
10:00 AM
We
Linear Control Systems: Designed by Input/Output Methods
EE 4231

Spring 2014
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