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Signals and Systems Fall 2003 Lecture #20 20 November 2003 1. Feedback Systems 2. Applications of Feedback Systems
Signals and SystemsFall 2003
Lecture #2020 November 2003
1. Feedback Systems
2. Applications of Feedback Systems
A Typical Feedback System
Why use Feedback?
1. Reducing Effects of Nonidealities2. Reducing Sensitivity to Uncertainties and Variability3. Stabilizing Unstable Systems4. Reducing Effects of Disturbances5. Tracking6. Shaping System Response Characteristics (bandwidth/speed)
One Motivating Example
Open-Loop System Closed-Loop Feedback System
Analysis of (Causal!) LTI Feedback Systems: Black’s Formula
CT System
)()(1
)(
)(
)()(
sHsG
sH
sX
sYsQ
Black’s formula (1920’s)
Closed - loop system function
Forward gain — total gain along the forward path from the input to the outputLoop gain — total gain around the closed loop
gain loop-1
gain forward
Applications of Black’s FormulaExample
:
BCA
BA
sX
sY
'1
'
gain loop1
gain Forward
)(
)(
1)
2)
ABCA
BA
sX
sY
A
AA
1
'
)(
)(
1'
ABCA
AB
BCA
B
sD
sY
1
)1(
'1gain loop1
gain Forward
)(
)(
The Use of Feedback to Compensate for Nonidealities
Assume KP(jω) is very large over the frequency range of interest. In fact, assume
)(
1
)()(1
)(
)(
)()(
jGjGjKP
jKP
jX
jYjQ
— Independent of P(s)!!
1|)()(| jGjKP
Example of Reduced Sensitivity
1)The use of operational amplifiers2)Decreasing amplifier gain sensitivity
Example:(a) Suppose
(b) Suppose
10)099.0)(1000(1
1000
0.099) , 1000)(
1
11
)Q(j
G(jωjKP
)changegain %1( 9.9)099.0)(500(1
500)(
)changegain %50(
099.0)( , 500)(
2
22
jQ
jGjKP
Fine, but why doesn’t G(jω) fluctuate ?
Note:
For amplification, G(jω) must attenuate, and it is much easier to build attenuators (e.g. resistors) with desired characteristics
There is a price:
Needs a large loop gain to produce a steady (and linear) gain for the whole system.⇒ Consequence of the negative (degenerative) feedback.
)(
1)(
jGjQ
|)(|
1|)(|1|)(|
jGjKPjKPG
Example: Operational Amplifiers
If the amplitude of the loop gain
|KG(s)| >> 1 — usually the case, unless the battery is totally dead.
The closed-loop gain only depends on the passive components(R1 & R2), independent of the gain of the open-loop amplifier K.
Then Steady State 1
21
)(
1
)(
)(
R
RR
sGsX
sY
The Same Idea Works for the Compensation for Nonlinearities
Example and Demo:Amplifier with a Deadzone
The second system in the forward path has a nonlinear input-output relation (a deadzone for small input), which will cause distortion if it is used as an amplifier. However, as long as the amplitude of the “loop gain” is large enough, the input-output response ≅ 1/K2
───Much broader bandwidth, also
Improving the Dynamics of Systems
Example: Operational Amplifier 741The open-loop gain has a very large value at dc but very limited bandwidth
Not very useful on its own40
10)(
7
ssH
)(1040
10
)()(1
)()(
7
7
sGssHsG
sHsQ
GQ /1)0(
With feedback
Stabilization of Unstable Systems
P(s) — unstable Design C(s), G(s) so that the closed-loop system
is stable⇒ poles of Q(s) = roots of 1+C(s)P(s)G(s) in LHP
)()()(1
)()()(
sGsPsC
sPsCsQ
Try ： proportional feedback
Example #1: First-order unstable systems
1)(2
1)(
sGs
sP
KsC )(
Ks
K
s
Ks
K
sQ
2
21
2)(
Stable as long as K>2
Example #2: Second-order unstable systems
— Unstable for all values of K
— Physically, need damping — a term proportional to s ⇔ d/dt
1)(2
1)(
2
sGs
sP
Attempt #1： Proportional Feedback C(s)=K
Ks
K
s
Ks
K
sQ
2
21
2)(2
2
2
Example #2 (continued):
Attempt #2: Try Proportional-Plus-Derivative (PD) Feedback
— Stable as long as K2 > 0 (sufficient damping)and K1 > 4 (sufficient gain).
sKKsC 21)(
)4(
41
4 )(
122
21
221
221
KsKs
sKKs
sKKs
sKK
sQ
Example #2 (one more time):
Why didn’t we stabilize by canceling the unstable poles?
There are at least two reasons why this is a really bad idea:
a) In real physical systems, we can never know the precise values of the poles, it could be 2±∆.
b) Disturbance between the two systems will cause instability.
Demo: Magnetic Levitation
io = current needed to balance the weight W at the rest height yo
Force balance
Linearize about equilibrium with specific values for parameters
))((
))((
0
20
2
2
tyy
tiiW
dt
yd
g
W
─── Second-order unstable system)(4
10)(
)(10)(4
2
2
2
sIs
sY
titydt
dy
Magnetic Levitation (Continued):
—Stable!
)104(10
10)(
122 KsKs
KsQ
6.0 ,3.1 ,1 .. 21 KgE ：
2)3(
10)(
s
sQ
