Embed Size (px)
Citation preview
© 2004 HellersteinFeedback Control of Computing Systems
Feedback Control of Computing SystemsM2: Signals and Z-Transforms
Joseph L. HellersteinIBM Thomas J Watson Research Center, [email protected]
September 21, 2004
2 © 2004 HellersteinFeedback Control of Computing Systems: M2 - Signals
Motivating Example
( )r k ( )y k( )e kController
NotesServer
NotesSensor
( )u k
( 1) ( ) ( 1)Iu k u k K e k ( 1) (0.43) ( ) (0.47) ( )w k w k u k ( 1) 0.8 ( ) 0.72 ( ) 0.66 ( 1)y k y k w k w k
( )w k
( ) ( ) ( )e k r k y k
The problem
Want to find y(k) in terms of KI so can design control system that is stable, accurate, settles quickly, and has small overshoot.
But this is difficult to do with ARX models.
The SolutionUse a different representation
This module focuses on signals: time varying data
3 © 2004 HellersteinFeedback Control of Computing Systems: M2 - Signals
M2:Lecture
4 © 2004 HellersteinFeedback Control of Computing Systems: M2 - Signals
Agenda
Z-Transform representation of signals Z-Transforms of common signals Properties of z-transforms of signals Poles Effect of poles Final value theorem
Reference: “Feedback Control of Computer Systems”, Chapter 3.
5 © 2004 HellersteinFeedback Control of Computing Systems: M2 - Signals
Z-Transform of a Signal
0 1 2 3 40
1
2
3
4
5
6
Time domain representationu(0)=1u(1)=3u(2)=2u(3)=5u(4)=6
u(k)
z domain representation1z0 +3z-1 +2z-2 +5z-3 +6z-4
0
If { ( )} (0), (1),... is a signal, then its -Transform is
( ) ( ) kk
u k u u z
U z u k z
0
1
2
1: 0 (current time)
: 1 (one time unit in the future)
: 2 (two time units in the future)
z k
z k
z k
z is time shift; z-1 is time delay
k
6 © 2004 HellersteinFeedback Control of Computing Systems: M2 - Signals
Write the Z-Transforms or Plots for the Following Finite Signals
0 1 2 3 4 5-2
-1
0
1
2
3
4
5
k
u(k
)
4321 4542)( zzzzzU
(Drop exponents >0.)
0 1 2 3 4-1
0
1
2
3
4
5
k
v(k
)
32154)()( zzzzzUzV
7 © 2004 HellersteinFeedback Control of Computing Systems: M2 - Signals
0 1 2 3
1
time (k)
Impulse 0,0)(;1)0( kkyyy(k)
1
...001)( 210
zzzzY
Common Signals: Impulse
5)3(5 :Example 3 uz
8 © 2004 HellersteinFeedback Control of Computing Systems: M2 - Signals
0 1 2 3
1Step
time (k)
y(k)
0 1 2
1
( ) 1 1 1 ...
1
1
1
Y z z z z
zz
z
0,1)( kky
Common Signals: Step
9 © 2004 HellersteinFeedback Control of Computing Systems: M2 - Signals
Properties of z-Transforms of Signals
0 1 2
0 1 2
1 0 1
0 1
1 2 3
Signals: ( ) (0) (1) (2) ...
( ) (0) (1) (2) ...
Shift: ( ) (0) (1) (2) ...
(1) (2) ...
Delay: ( ) / (0) (1) (2) ..
U z u z u z u z
V z v z v z v z
zU z u z u z u z
u z u z
U z z u z u z u z
0 1 2
0 1 2 0 1 2
.
Scaling: ( ) (0) (1) (2) ...
-Transform of { ( )}
Sum of signals: (0) (1) (2) ... (0) (1) (2) ...
aU z au z au z au z
z au k
u z u z u z v z v z v z
0 1 2 ( (0) (0)) ( (1) (1)) ( (2) (2)) ...
( ) ( )
u v z u v z u v z
U z V z
10 © 2004 HellersteinFeedback Control of Computing Systems: M2 - Signals
Infinite Length Signals
0 5 10 15 200
2
4
6
8
10
1
5
...555)( 21
z
z
zzzU
11 © 2004 HellersteinFeedback Control of Computing Systems: M2 - Signals
Construct a Z-Transform for the Following Infinite Length Signal
0 5 10 15 200
2
4
6
8
10
( )Y z
0 5 10 15 200
2
4
6
8
10
5 ( ), ( ) is the unit stepW z W z
0 5 10 15 200
2
4
6
8
10
103z
0 5 10 15 200
2
4
6
8
10
115 ( )z W z
)(53)(5)( 1110 zWzzzWzY
12 © 2004 HellersteinFeedback Control of Computing Systems: M2 - Signals
Common Signals: Geometric
az
z
zaazzY
...1)( 22
kaky )( :Geometric
0 5 10 15 200
0.2
0.4
0.6
0.8
1
a=0.8
8.0
...64.08.01)( 21
z
z
zzzY
13 © 2004 HellersteinFeedback Control of Computing Systems: M2 - Signals
Poles of a Z-Transform
Definition: Values of z for which the denominator is 0
Easy to find the poles of a geometric:az
zzV
)(
5 2( )
1 0.8
z zY z
z z
Quick exercises: What are the poles of the following Z-Transforms?
Easy if sum of geometrics
Harder if expanded polynomial
Poles determine key behaviors of signals
Pole is a.
06.05.0
3)(
2
zz
zzV
14 © 2004 HellersteinFeedback Control of Computing Systems: M2 - Signals
Effect of Pole on the Signal
-5
0
5a=0.4
-5
0
5a=0.9
-5
0
5a=1.2
0 5 10-5
0
5a=-0.4
0 5 10-5
0
5a=-0.9
0 5 10-5
0
5a=-1.2
az
zaky k
)(
What happens when|a| is larger?|a|>1?a<0?
Larger |a|Slower convergence
|a|>1Does not converge
a<0Oscillates
,...),,1(...1
Why?
222 aazaazaz
z
15 © 2004 HellersteinFeedback Control of Computing Systems: M2 - Signals
Final Value Theorem
Provides an easy way to determine the steady state value of a signal
Limit as k becomes large
)()1(lim)( 1 zVzv z
(if V(z) has all of its poles inside the unit circle)
circleunit theinside is if ,0)1(lim
01)1(lim
11
)1(lim
1
1
1
aaz
zz
zz
zz
z
z
z
Final value of the unit step is 1.
Final value of the impulse is 0.
16 © 2004 HellersteinFeedback Control of Computing Systems: M2 - Signals
Applying the Final Value Theorem
)3.0)(4.1(
2.0)(
:signal for the valuefinal theisWhat
42.5)|()1(
)3.0)(1(
38.0)(
1
zz
zzV
zWz
zz
zzW
z
17 © 2004 HellersteinFeedback Control of Computing Systems: M2 - Signals
Describe the following Signals
Components of descriptionDoes the signal convergeFast or slow convergenceOscillations, if any
5 2( )
1 0.8
z zY z
z z
7 3 10( )
0.6 1 0.2
zU z
z z z
Step of 5 Geometric at .8
5 (2)(0.8)k
13 (7)( 0.6) (10)(.02) , 0
7, 0
k k k
k
0 2 4 6 8 10-10
-5
0
5
10
15
20
3)()()1(lim 1 uzUzz
(1) 3 (7)( .6) 10 17.2u Delayed geometric
Delayed step
18 © 2004 HellersteinFeedback Control of Computing Systems: M2 - Signals
M2:Labs
19 © 2004 HellersteinFeedback Control of Computing Systems: M2 - Signals
Plot The Signals
4.03.1
3.12)( .3
)1(
410
1
2)( 2.
/)34(2)( 1.
2
2
42
53
zz
zzzY
zzz
z
zzY
zzzzY
20 © 2004 HellersteinFeedback Control of Computing Systems: M2 - Signals
Find the Final Values
12
72)(
)2.02.1(
16
09.
3)(
2
2
zz
zzY
zz
z
z
zzY
21 © 2004 HellersteinFeedback Control of Computing Systems: M2 - Signals
Describe the Following Signals
Which converges more quickly? Which of the following oscillate? Do any of the following fail to converge?
24.0
2.1
32.02.1
2
2
2
zz
z
zz
z
