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Tom Penick
[email protected]
www.teicontrols.com/notes
AutomaticControls.pdf
5/10/2000
Page 1 of 10
INTRODUCTION TO AUTOMATIC CONTROLS
INDEX
adjoint.
...............................
6
block diagrams.
..................
4
closed loop system.
.......
5, 10
E
(
s
)....................................
6
e
(
t
).....................................
6
error
steady state tracking.
........
6
tracking.
...............................
6
e
ss
.....................................
6
glossary.
...........................
10
impulse function.
................
3
inputs.
................................
5
inverse Laplace transform1, 2
lag compensation.
...............
7
Laplace transform.
..............
1
lead compensation.
.............
7
Mason's gain rule.
...............
4
open loop system.
.........
5, 10
PD Controllers .
.................
8
phase lag compensation .
.....
7
phase lead compensation.
....7
PI Controllers.
...................
8
PID.
.................................
10
PID Controllers .
.................
8
pole of a function.
...........
2, 3
proportional derivative.
.......
8
proportional integral.
..........
8
proportional integral
derivative.
.....................
8
R
(
s
)....................................
5
r
(
t
).....................................
5
ramp .
.................................
5
residues.
.........................
2, 3
repeated roots.
.................
3
stability.
.............................
2
state vector model.
..........
5, 6
steady state tracking error .
..6
steadystate response.
.......
10
step.
...................................
5
tracking error.
.................
5, 6
LaPlace transform .
...........
6
transient response.
............
10
trig identities.
.....................
9
type 0 system .
....................
5
type 1 system .
....................
5
type 2 system .
....................
5
unit ramp.
...........................
5
unit step.
............................
5
unity feedback.
...................
4
zero of a function.
...........
2, 3
LAPLACE TRANSFORMS
We use
Laplace transforms
because we are dealing
with
linear dynamic systems
and it is easier than
solving differential equations.
We don't use
Fourier
transforms
because we are dealing with the
transient response and because a Fourier transform
won't handle a system that "blows up".
LAPLACE TRANSFORM
The Laplace transform is used to convert a function
f
(
t
)
in the
time domain
to a function
F
(
s
)
in the
s
domain
,
where
s
is a complex number:
( 29 ( 29
0
st
F
s
e ft dt
∞

=
∫
f
(
t
)
is 0 for
t
<0
.
f
(
t
)
can "blow up" or be piecewise.
We are
free to
pick
the value of
s
to make the integral converge;
however, once the calculation is made you can use the result
everywhere.
For example if
( 29
10
t
f
te
=
, then
s
must be 10 or
greater to do the integration.
But the result is
( 29 ( 29
1
/
10
F
ss
=
, in which
s
can be less than 10.
Misc:
sj
=σ+ϖ
,
1
jx
e
=
INVERSE LAPLACE TRANSFORM
The inverse Laplace transform is used to convert a
function
F
(
s
)
in the
s
domain
to a function
f
(
t
)
in the
time domain
, where
s
is a complex number:
( 29
( 29
1
2
Cj
st
f
t
Fse ds
j
+∞
∞
=
π
∫
In the conceptual view,
c
is
a real number defining a
line in the splane as
shown at right.
All poles of
F
(
s
)
must lie to the left of
this line.
Poles are always
symmetric about the real
axis.
cj
c+j
Real axis
c
Imaginary axis
∞
∞
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View Full Document Tom Penick
[email protected]
www.teicontrols.com/notes
AutomaticControls.pdf
5/10/2000
Page 2 of 10
INVERSE LAPLACE TRANSFORM
Second Order Conjugate Pair Example
( 29
(
29(
29
100
1
1
0
1
10
Fs
s
s j
sj
=
+
+
+
( 29
10
1
0
1
0
cos1
0
sin10
101
tt
f
t
e
t
et

=
A second order
conjugate pole pair in
the lefthand side of the
splane results in a
damped sinusoid in the
time domain.
SYSTEM STABILITY
Stable:
A system is stable is there are no roots in the right
hand plane and no repeated roots on the
j
ϖ
axis.
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This note was uploaded on 04/03/2012 for the course EE 366 taught by Professor Pore during the Spring '08 term at University of Texas at Austin.
 Spring '08
 Pore

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