581 lecture 36c
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May 9, 2007 Feedback Control Systems (II) © Douglas Looze 1
Lecture 36
ECE 581
Feedback Control Systems (II)
Doug Looze
May 9, 2007 Feedback Control Systems (II) © Douglas Looze 2
AnnounceProblem Set 6 available
– Due Tuesday, May 15Final exam
– Friday, May 18– 1:30 3:30 PM– Marston 211– 2005 exam on course site
Reading: FPE 8.3
May 9, 2007 Feedback Control Systems (II) © Douglas Looze 3
Last Time: Design by EmulationIdea
– Use continuous-time design model and objectives
– Design continuous-time controller
– Approximate continuous-time controller in discrete-time
– Analyze• Bode• Nyquist• Root locations• Simulation
May 9, 2007 Feedback Control Systems (II) © Douglas Looze 4
Today
Discrete-time design by emulation (cont.)– Matched pole-zero– Bilinear
• Tustin
• Trapezoidal
May 9, 2007 Feedback Control Systems (II) © Douglas Looze 5
Matched Pole-Zero Emulation
Exploit sTz e
– If pole is at in continuous-time, thenp
pole is at in discrete-timepTe
– Match zeros also
May 9, 2007 Feedback Control Systems (II) © Douglas Looze 6
Suppose
c
n sK s
d s
Polynomials
1
1
11
11
m ll
ciicp n k
k
cii
s sz
K
s sp
– Poles– Zeros
1n
ci ip 1
mci iz
– In general n m• If n > m
can augment zeros
zeros at n m
May 9, 2007 Feedback Control Systems (II) © Douglas Looze 7
Matched pole-zero method– Poles
1ci
np T
ie
– Zeros 1
cinz T
ie
zeros at 1n m
– Pick Kdp
Without0
Integrators/Differentiators
limd k l
c
K zT
K s
“DC Gain” unchanged
May 9, 2007 Feedback Control Systems (II) © Douglas Looze 8
where
1 1
1
1 1
1
1 1
1 1
ci
ci
m l lz T
id dp n k kp T
i
z e z
K z K
z e z
Without integrator/differentiators
May 9, 2007 Feedback Control Systems (II) © Douglas Looze 9
– What is Kdp?• Each term (non-zero pole or non-zero, finite zero) has factor in both• Zero (non-zero, finite)
• Pole (non-zero)
0
11
11
ci
ci
ci p Tp TsT
s
sp
ee e
May 9, 2007 Feedback Control Systems (II) © Douglas Looze 10
• Overall discrete-time proportional gain
1
1
1
1
ci
ci
n kp T
ik ldp cp m l
z T
i
e
K K T
e
May 9, 2007 Feedback Control Systems (II) © Douglas Looze 11
Modified matched pole-zero emulation– Used by Matlab c2d
Matched pole-zero emulation– Infinite zeros
• Relative degree n m• Add zeros at 1 in discrete-time for each infinite zero
– No effect if #poles = # zeros
May 9, 2007 Feedback Control Systems (II) © Douglas Looze 12
Comparison
5
5cs
K ss
0.1 sT
– 1 pole at origin
– 1 zero
– Gain
1 1n k
5 1 0ciz m l
1 00.5
11 0.1
1dpK
e
0.25
– Controller
0.610.25
1dz
K zz
0.5 0.61ciz Te e
May 9, 2007 Feedback Control Systems (II) © Douglas Looze 13
10-1
100
101
Mag
nitu
de (
abs)
10-1
100
101
-90
-60
-30
0
Pha
se (
deg)
Bode Diagram
Frequency (rad/sec)
Continuous-Time
Forw ard Diff
Backw ard Diff
Matched PZ
0.2 1c
sK s
s
0.1 sT
May 9, 2007 Feedback Control Systems (II) © Douglas Looze 14
Bilinear TransformationIntegral method
– Trapezoidal integration1
1
2 1
1
zs
T z
– Direct substitution
2 1
1d cz
K z KT z
2 1
1
z
T z
May 9, 2007 Feedback Control Systems (II) © Douglas Looze 15
– Assume continuous-time controller is rational
1
0 11
1
m mm
c n nn
b s b s bK s
s a s a
– Then
2 1 2 1
1 1
2 1 2 1
1 1
10 1
11
z z
T z T z
z z
T z T z
m mm
d n nn
b b bK z
a a
1
1
n
n
z
z
rational
May 9, 2007 Feedback Control Systems (II) © Douglas Looze 16
Mapping 2 1
1
zs
T z
2 2
sz s zT T
2 2s z s
T T
1 12 2
sT sTz
12
12
sT
zsT
2 1
1
zs
T z
Bilinear transform pair
May 9, 2007 Feedback Control Systems (II) © Douglas Looze 17
Poles and zeros–
Suppose has a pole at : Re 0c c cK s s p p (OLHP)
12
12
c
dc
p T
pp T
12
12
c
dc
p T
pp T
2
2
c
c
pT
pT
2
2
jT
jT
cp j
May 9, 2007 Feedback Control Systems (II) © Douglas Looze 18
2 22 2
T T
2 22 22 2
T T
• Note
22
22
2
2d
Tp
T
1
2 2
T T
Inside unit circle
– OLHP gets mapped to unit disk
May 9, 2007 Feedback Control Systems (II) © Douglas Looze 19
s-plane z-plane
StableStable
-j
-1 1
j
12
12
sT
zsT
2 1
1
zs
T z
May 9, 2007 Feedback Control Systems (II) © Douglas Looze 20
10-1
100
101
Mag
nitu
de (
abs)
10-1
100
101
-90
-60
-30
0
Pha
se (
deg)
Bode Diagram
Frequency (rad/sec)
Continuous-Time
Matched PZ
Bilinear
May 9, 2007 Feedback Control Systems (II) © Douglas Looze 21
Summary
Bilinear emulation
May 9, 2007 Feedback Control Systems (II) © Douglas Looze 22
Next Time
Example