Lecture 2

Ηλεκτρονική - Συσκευές

15 Νοε 2013 (πριν από 4 χρόνια και 8 μήνες)

81 εμφανίσεις

Dec. 8, 2011

Feedback Control Systems (I) © Douglas Looze

1

Lecture
24

ECE 580

Feedback Control Systems (I)

MIE 444

Automatic Controls

Doug Looze

Dec. 8, 2011

Feedback Control Systems (I) © Douglas Looze

2

Announce

PS 8 Due

Final Exam

Monday, December
12

10:30 AM

ELAB 323

Open book, notes

No electronic devices

Sample exam from
2009

Dec. 8, 2011

Feedback Control Systems (I) © Douglas Looze

3

Last time

Effect of sampling in control loop

Equivalent discrete
-
time control loop

Stability

Mapping of s
-
plane to z
-
plane

Nyquist

Bode

sT
z e

Dec. 8, 2011

Feedback Control Systems (I) © Douglas Looze

4

Bode plots

10 10
log vs. log
j T
L e

Magnitude plot

Phase plot

10
vs. log
j T
L e

Nyquist plots

Im vs. Re
j T j T
L e L e
 
Dec. 8, 2011

Feedback Control Systems (I) © Douglas Looze

5

Today

Design by emulation

Matched pole
-
zero

Tustin

Example

-
8.3

Dec. 8, 2011

Feedback Control Systems (I) © Douglas Looze

6

Design by Emulation

Idea

Use continuous
-
time design model and
objectives

Design continuous
-
time controller

Approximate continuous
-
time controller in
discrete
-
time

Analyze

Bode

Nyquist

Root locations

Simulation

Dec. 8, 2011

Feedback Control Systems (I) © Douglas Looze

7

Emulation

dk
u

e t

d
D z
k
e
ZOH

d
u t

u t

e t

c
D s

Want

d
u t u t

Or

dk k
u u

Dec. 8, 2011

Feedback Control Systems (I) © Douglas Looze

8

Example

c
a
D s
s a

u t au t ae t
  

Approximate differential equation by difference
equation

Dec. 8, 2011

Feedback Control Systems (I) © Douglas Looze

9

Methods

Backward difference

1
1
z
s
T

Forward difference

1
z
s
T

Bilinear (trapezoidal)

1
1
2 1
1
z
s
T
z

Tustin

transformation

Can use

Pre
-
warping

Dec. 8, 2011

Feedback Control Systems (I) © Douglas Looze

10

Methods not based on differential operation

Step invariant

1
1
c
d
D s
D z z
s

 
 
 
 
 
 
Z

Impulse invariant

d c
D z D s

Z

Matched pole
-
zero

sT
z e

Dec. 8, 2011

Feedback Control Systems (I) © Douglas Looze

11

Matched Pole
-
Zero Emulation

Exploit

sT
z e

If pole is at in continuous-time, then
p
pole is at in discrete-time
pT
e

Match zeros also

Dec. 8, 2011

Feedback Control Systems (I) © Douglas Looze

12

Suppose

c
n s
D s
d s

Polynomials

1
1
1
1
1
1
m l
l
ci
i
cp
n k
k
ci
i
s s
z
K
s s
p

 

 
 

 

 
 

Poles

Zeros

1
n k
ci
i
p

1
m l
ci
i
z

In general
n

m

If
n

>
m

can augment zeros

zeros at
n m
 
Dec. 8, 2011

Feedback Control Systems (I) © Douglas Looze

13

Matched pole
-
zero method

Poles

1
ci
n
p T
i
e

Zeros

1

ci
m
z T
i
e

zeros at 1
n m
 

Pick
K
dp

Without
0
Integrators/
Differentiators
lim
d
k l
c
D z
T
D s

 

 
 
 
“DC Gain” unchanged

Dec. 8, 2011

Feedback Control Systems (I) © Douglas Looze

14

where

1 1
1
1 1
1
1 1
1
1 1
ci
ci
m l
l
z T
n m
i
d dp
n k
k
p T
i
z e z
D z K z
z e z

 

 

 
 
 

Dec. 8, 2011

Feedback Control Systems (I) © Douglas Looze

15

Matched pole
-
zero emulation

Modified m
atched
pole
-
zero emulation

Used by Matlab c2d

Infinite zeros

Relative degree
n

m

Modified adds 1 less zero at

1 in discrete
-
time for
each infinite zero

See Franklin, Powell, Workman,
Digital Control of
Dynamic Systems
-
Wesley, 1990

Transfer function is strictly proper

However, can include computation time by using asynchronous
sample, ZOH

See Ogata,
Discrete
-
time Control Systems
, Prentice
-
Hall, 1987

No effect if #poles = # zeros

Dec. 8, 2011

Feedback Control Systems (I) © Douglas Looze

16

Bilinear Transformation

Integral method

Trapezoidal integration

1
1
2 1
1
z
s
T
z

Direct substitution

2 1
1
d c
z
D z D
T z

 

 

 
2 1
1
z
T z

Dec. 8, 2011

Feedback Control Systems (I) © Douglas Looze

17

Assume continuous
-
time controller is rational

1
0 1
1
1
m m
m
c
n n
n
b s b s b
D s
s a s a

  

  

Then

2 1 2 1
1 1
2 1 2 1
1 1
1
0 1
1
1
z z
T z T z
z z
T z T z
m m
m
d
n n
n
b b b
D z
a a
 
 
 
 

  

  

1
1
n
n
z
z

rational

Dec. 8, 2011

Feedback Control Systems (I) © Douglas Looze

18

Mapping

2 1
1
z
s
T z

2 2
sz s z
T T
  
2 2
s z s
T T
 
  
 
 
1 1
2 2
sT sT
z
 
  
 
 
1
2
1
2
sT
z
sT

2 1
1
z
s
T z

Bilinear transform pair

Dec. 8, 2011

Feedback Control Systems (I) © Douglas Looze

19

Poles and zeros

Suppose D has a pole at :Re 0
 
c c c
s s p p
1
2
1
2
c
d
c
p T
p
p T

1
2
1
2
c
d
c
p T
p
p T

2
2
c
c
p
T
p
T

2
2
j
T
j
T
 
 
 

 
c
p j
 
 
Dec. 8, 2011

Feedback Control Systems (I) © Douglas Looze

20

2 2
2 2
T T
 
   
  
   
   
2 2
2 2
2 2
T T
   
   
    
   
   

Note

2
2
2
2
2
2
d
T
p
T
 
 
 
 
 
 

 
 
 
 
1

2 2
T T
 
  

Inside unit circle

OLHP gets mapped to unit disk

Dec. 8, 2011

Feedback Control Systems (I) © Douglas Looze

21

s
-
plane

z
-
plane

Stable

Stable

-
j

-
1

1

j

1
2
1
2
sT
z
sT

2 1
1
z
s
T z

Dec. 8, 2011

Feedback Control Systems (I) © Douglas Looze

22

Comparison

5
c
s
D s
s

0.1 s
T

1 pole at origin

1 zero

Gain

1 1
n k
 
5 1 0
ci
z m l
   

1 0
0.5
1
5 0.1
1
dp
D
e

1.27

Controller (matched pole
-
zero)

0.61
1.27
1
d
z
D z
z

0.5
0.61
ci
z T
e e

 
Dec. 8, 2011

Feedback Control Systems (I) © Douglas Looze

23

Controller (bilinear)

1
1
2 1
0.1
1
d c
z
D z D
z

 

 

 
1
20 5
1
1
20
1
z
z
z
z

1.25 0.75
1
d
z
D z
z

Zero at 0.6
Dec. 8, 2011

Feedback Control Systems (I) © Douglas Looze

24

10
-1
10
0
10
1
Magnitude (abs)
10
-1
10
0
10
1
-90
-60
-30
0
Phase (deg)
Bode Diagram
Continuous-Time
Matched PZ
Bilinear
Dec. 8, 2011

Feedback Control Systems (I) © Douglas Looze

25

Design by Emulation

Idea

Use continuous
-
time design model and
objectives

Design continuous
-
time controller

Approximate continuous
-
time controller in
discrete
-
time

Analyze

Bode

Nyquist

Root locations

Simulation

Dec. 8, 2011

Feedback Control Systems (I) © Douglas Looze

26

Continuous
-
time design model

ZOH
G s

G s

des
G s

Continuous
-
time controller design

r
u
z

y
e

c
D s
d

des
G s
Dec. 8, 2011

Feedback Control Systems (I) © Douglas Looze

27

Continuous
-
time model

Could use

des
G s G s

ZOH
1
G s
 

Better approximation of ZOH in loop is

ZOH
1
sT
e
G s
sT

Not rational

Approximate

Dec. 8, 2011

Feedback Control Systems (I) © Douglas Looze

28

10
-2
10
-1
10
0
Magnitude (abs)
10
-1
10
0
10
1
-180
-135
-90
-45
0
Phase (deg)
Bode Diagram
1
T

T

Nyquist

Frequency

1

0

phase

magnitude

Dominant effect is phase lag
Dec. 8, 2011

Feedback Control Systems (I) © Douglas Looze

29

2
2
1
2 8
1
2 8
sT
Ts
Ts
e
Ts
Ts

  

  
1
2
1
2
sT
Ts
e
Ts

To first order in
Ts

Dec. 8, 2011

Feedback Control Systems (I) © Douglas Looze

30

ZOH
1
sT
e
G s
sT

1
1
2
1
1
2
Ts
Ts
sT
 

 
 
 
 

 
 
1 1
1
2 2
1
2
Ts Ts
Ts
sT
 
  
 

 
 

 
 

ZOH
1
1
2
G s
T
s

1
st

G
ZOH
(
s
) is 1
st

order lag

Dec. 8, 2011

Feedback Control Systems (I) © Douglas Looze

31

10
-1
10
0
Magnitude (abs)
10
-1
10
0
10
1
-90
-60
-30
0
Phase (deg)

ZOH
1
T

T

Nyquist

Frequency

Dec. 8, 2011

Feedback Control Systems (I) © Douglas Looze

32

Design by Emulation

Continuous
-
time objectives

Design model

des
2
1
1

T
G s G s
s

r
u
z

y
e

c
D s
d

des
G s

Emulate controller using MPZ or Tustin

Dec. 8, 2011

Feedback Control Systems (I) © Douglas Looze

33

Example

System

1
0.1 s
1
G s T
s s
 

Objectives

10% 0.2 s (2 samples)
p r
M t
 

Analysis

Peak overshoot

Rise time

0.6 60
M
 
  
1.8
9
0.2
n

 

4 2
9 1 4 0.6 2 0.6 6.4 rad/s
g

   
Dec. 8, 2011

Feedback Control Systems (I) © Douglas Looze

34

Account for ZOH

ZOH
1
sT
e
G s
sT

1
1
2
T
s

1
0.05 1
s

Design system

ZOH
des
G s G s G s

1
1 0.05 1
s s s

 
Dec. 8, 2011

Feedback Control Systems (I) © Douglas Looze

35

Continuous
-
time design

g M
 
  

1 1
6.4
90 tan 6.4 tan
20
des g
G j

 
 
   
 
 
189.6
  
 
Dec. 8, 2011

Feedback Control Systems (I) © Douglas Looze

36

Use symmetric optimum design

1 sin70
1 sin70

 

 
0.0311

1
z
g
T
 

1
6.4 0.0311

0.881

0.0274
z
T

0.881 1
0.0274 1
c cp
s
D s k
s

Dec. 8, 2011

Feedback Control Systems (I) © Douglas Looze

37

Gain to achieve crossover

1
des g c g
G j D j
 

0.0274 6.4 1
6.4 0.881 6.4 1
cp
des
j
k
G j j

7.5

0.881 1
7.5
0.0274 1
c
s
D s
s

Dec. 8, 2011

Feedback Control Systems (I) © Douglas Looze

38

10
-6
10
-4
10
-2
10
0
10
2
Magnitude (abs)
10
-1
10
0
10
1
10
2
10
3
-270
-225
-180
-135
-90
Phase (deg)
Bode Diagram
Gm = 8.46 (at 26.9 rad/sec) , Pm = 61.8 deg (at 6.24 rad/sec)
Continuous
-
time design

Dec. 8, 2011

Feedback Control Systems (I) © Douglas Looze

39

0
0.2
0.4
0.6
0.8
1
1.2
0
0.2
0.4
0.6
0.8
1
Step Response
Time (sec)
Amplitude
Dec. 8, 2011

Feedback Control Systems (I) © Douglas Looze

40

Emulate

Matched pole/zero

Continuous
-
time

Discrete
-
time

Pole

36.5

0.02600

Zero

1.14

0.8927

Gain

241

68.08

0.8927
68.08
0.026
dm
z
D z
z

Dec. 8, 2011

Feedback Control Systems (I) © Douglas Looze

41

Bilinear

2 1
1
d c
z
D z D
T z

 

 

 
Continuous
-
time

Discrete
-
time

Pole

36.5

0.2920

Zero

1.14

0.8926

Gain

241

90.21

0.8926
90.21
0.292
db
z
D z
z

0.8927
68.08
0.026
dm
z
D z
z

Dec. 8, 2011

Feedback Control Systems (I) © Douglas Looze

42

10
1
10
2
Magnitude (abs)
10
-1
10
0
10
1
0
30
60
Phase (deg)
Bode Diagram
Continuous-time
Matched Pole-Zero
Bilinear
Controllers

Dec. 8, 2011

Feedback Control Systems (I) © Douglas Looze

43

10
0
10
1
-180
-150
-120
Phase (deg)
Bode Diagram
10
-1
10
0
10
1
Magnitude (abs)
Continuous-time
Matched Pole-Zero
Bilinear
Loop TFs

Dec. 8, 2011

Feedback Control Systems (I) © Douglas Looze

44

Margins

Phase Margin

Crossover

C
-
Time

61.8
°

6.24

Matched PZ

51.8
°

6.37

Bilinear

59.6
°

6.63

Dec. 8, 2011

Feedback Control Systems (I) © Douglas Looze

45

Step Response
Time (sec)
Amplitude
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
0
0.2
0.4
0.6
0.8
1
Continuous-time
Matched Pole-Zero
Bilinear
Step Response