Lecture 2

steamgloomyElectronics - Devices

Nov 15, 2013 (3 years and 8 months ago)

64 views

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


Reading: FPE 8.2
-
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
, Addison
-
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
Frequency (rad/sec)
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
Frequency (rad/sec)
1
T

T

Nyquist

Frequency

1

0

phase

magnitude

Dominant effect is phase lag
Dec. 8, 2011

Feedback Control Systems (I) © Douglas Looze

29


Pade approximation for delay





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

order Pade


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)

Frequency (rad/sec)
ZOH
1st order Pade
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

6.4 rad/s 60
g M
 
  




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

 
 
   
 
 
189.6
  
add 70 phase lead at 6.4 rad/s
 
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)
Frequency (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
Frequency (rad/sec)
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
Frequency (rad/sec)
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