Lecture10-FromAnalogToDigitalControllers.pdf
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Lecture 10:
A Digital Control System
From Analog to Digital Controllers
( )
y t
(
)
u t
·
The sampling theorem
t
t
·
Aliasing and presampling lters
(
)
( )
u t
y t
Process
Unsampling signals
·
Hold
Sampler
y
k
u
k
·
Discrete time approximation of continuous time controller
u
k
D-A
Computer
A-D
y
k
Frequency domain
State space
t
t
The Sampling Theorem (Shannon 1949)
Proof idea
A continuous-time signal with Fourier transform
F
D
w
E
0
for
F
(
)
F
s
(
)
w
w
h
0
, is uniquely dened by its sampled values, provided
the sampling frequency
w
h
w
w
2
p
h
satises
w
2
w
0
.
s
s
-
2
w
0
2
w
-
2
w
- w
0
w
2
w
- w
w
N
N
N
N
N
N
N
N
The signal can be reconstructed by the interpolation formula
Let
F
D
w
E be the Fourier transform of
and expand the peri-
f
X
[
f
D
kh
E
sin
w
D
t
kh
E
2
odic function
F
s
D
w
E as a Fourier series
s
f
D
t
E
w
D
t
kh
E
2
s
X
[
X
[
k
[
E
1
h
C
k
e
ikh
w
F
s
D
F
D
k
E
w
w
w
s
k
[
k
[
It is straightforward to verify that
C
k
f
D
kh
E
k
0
1
2
,
,
, . . .
The frequency
w
2
is denoted
w
N
and called the
Nyquist
s
frequency
.
Hence M
f
D
kh
EN0
F
s
D
w
E 0
F
D
w
E 0
f
D
t
E
&
Dept. of Automatic Control, Lund
1
Aliasing and frequency folding
1
0
Aliasing and presampling lters
0
5
10
Time
F
(w )
F
s
(w )
-2w
N
- w
N
0
w
N
2w
N
-2w
N
- w
N
0
w
N
2w
N
w
w
2
Nyquist frequency
N
s
New frequencies since the system is NOT time-invariant
w
w
n
w
n
0
,
1
,
2
, . . .
sampl ed
s
Mini-problem
Example — Feedwater heating in a ship boiler
A wheel, rotating 90 turns per second, is lmed using a
camera with sampling time 10ms. What will the wheel rotation
look like on the lm?
38 min
Pressure
Steam
2 min
Valve
Feed
water
To boiler
Pump
Condensed
water
Temperature
2.11 min
Time
&
Dept. of Automatic Control, Lund
2
Pre- and postsampling lters
Example – Preltering
(a)
(b)
Typical control problem:
1
1
"Decrease the inuence of a low-frequency process distur-
bance despite high-frequency measurement noise."
0
0
·
Frequency separation
0
10
20
30
0
10
20
30
·
Prelter
w
N
(c)
(d)
1
1
Bessel, Butterworth, ITAE lters
Postsampling lters
·
0
0
Avoid exciting mechanical resonances
Higher order hold
0
10
20
30
0
10
20
30
Time
Time
w
0
.
9
,
w
0
.
5
,
w
0
.
1
(6th order Bessel lter)
d
N
alias
Consequence of using prelter
Unsampling signals
Pre- and postlter should be included in the process model
Exception: Fast sampling
·
Shannon
X
[
f
D
kh
E
sin
D
w
D
t
kh
E
2
E
s
f
D
t
E
A Bessellter can be approximated with a delay:
w
D
t
kh
E2
s
k
[
1
·
Zero order hold
0.01
·
First order hold
0.1
1
10
·
Predictive rst order hold
0
0.1
1
10
Frequency, rad/s
6th order Bessel (solid line), time delay (dashed line)
&
Dept. of Automatic Control, Lund
3
Shannon reconstruction
Zero and rst order hold
X
[
f
D
kh
E
sin
D
w
D
t
kh
E
2
E
s
f
D
t
E
w
D
t
kh
E2
s
k
[
The exact formula is not causal !
Approximation with information from
d
time steps ahead:
t
0
t
2
t
6
t
1
t
3
t
4
t
5
Time
E
n
d
X
w
t
2
E
sin
f
D
nh
s
t
f
D
kh
E
h
D
nh
t
kh
E
h
D
t
w
t
2
s
k
[
1
p
sin
t
p
t
0
t
0
t
2
t
6
Time
t
1
t
3
t
4
t
5
0
10
Time
Predictive rst order hold
Sinusoidal signal with
h
1
and
h
0
.
5
h=1
h=0.5
·
Use forward difference instead of backward difference
1
1
t
h
u
D
kh
t
E
u
D
kh
E
D
u
D
kh
h
E
u
D
kh
EE
0
0
0
5
10
0
5
10
·
Use model of the controller.
1
1
0
0
0
5
10
0
5
10
1
1
0
0
t
0
t
1
t
2
t
3
t
4
t
5
t
6
Time
0
5
10
0
5
10
Time
Time
&
Dept. of Automatic Control, Lund
4
Implementing a controller using a computer
H ( z)
»
G ( s)
u(t)
u kh
(
)
y kh
(
)
y (t)
{
}
{
}
A-D
Algorithm
D-A
Discrete time approximation of
continuous time controller
Clock
— Frequency domain
Want to get
A/D + Algorithm + D/A
G
D
s
E
Methods:
·
Approximate
s
i.e
G
D
s
E 0
H
D
z
E
·
Pole-zero matching
·
Use sample/hold to get
H
D
z
E from
G
D
s
E
Matlab
Approximation methods
Forward difference (Euler’s method)
SYSD = C2D(SYSC,TS,METHOD) converts the continuous
system SYSC to a discrete-time system SYSD with
sample time TS. The string METHOD selects the
discretization method among the following:
'zoh'
dx
D
t
E
dt
x
D
t
h
E
x
D
t
E
h
q
1
h
x
D
t
E
Backward difference
dx
D
t
E
dt
Zero-order hold on the inputs.
'foh'
Linear interpolation of inputs
(triangle appx.)
1
q
1
h
x
D
t
E
x
D
t
h
E
h
x
D
t
E
'tustin'
Bilinear (Tustin) approximation.
Trapezoidal method (Tustin, bilinear)
'prewarp'
Tustin approximation with frequency
prewarping.
The critical frequency Wc is specified
last as in C2D(SysC,Ts,'prewarp',Wc)
x
D
t
h
E
x
D
t
E
2
x
D
t
h
E
x
D
t
E
h
'matched'
Matched pole-zero method
(for SISO systems only).
&
Dept. of Automatic Control, Lund
5
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