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Chapter – Feedback
outputinput----------------- f D =
The general form
xF--- 4 D+
D 2 4D 16+ +---------------------------------=
An example
Transfer Functions
General Formsx·· 2nx·
n2x+ + f
M-----=
xD2 2nxD
n2x+ + f
M-----=
x D2 2nD n2+ + f
M-----=
xf---
1M-----
D2 2nD n2+ +
--------------------------------------------=particular
homogeneous
x· 1---x+ f=
xD 1---x+ f=
x D 1---+
f=
xf--- 1
D 1---+
-------------=
Feedback Systems
INPUT(e.g. gas) SYSTEM
(e.g. a car)
OUTPUT(e.g. velocity)
Control variable
vdesire d verror
+_
control car vac tualgasfunction
Controllers
PID
Rules (Heuristics)
1. If verror is not zero, and has been positive/negative for a while, increase/decrease gas2. If verror is very big/small increase/decrease gas3. If verror is near zero, keep gas the same4. If verror suddenly becomes bigger/smaller, then increase/decrease gas.5. etc.
u K p e K i e d t K dd ed t------
+ +=
Example
V V
+
-
amp motor
+
++
proportional
integral
derivative
Ki e
Kp e
Kdddt-----e
PID function
ue
+V
-V
Other Controller Types
g as K p v e r r o r K i v e r r o rd t+=
g a s K p v er ro r=
g a s K p v er ro r K d
d v er ro rd t
---------------- +=
Fo r a PI C o n t ro lle r
Fo r a P C o n t ro lle r
Fo r a PD C o n tro l le r
Block Diagram Manipulation
is equal to
c c rr+
-G D
H D
G D 1 G D H D +-------------------------------------
r G D e=
r G D c H D r– =
r 1 G D H D + G D c=
r G D 1 G D H D +-------------------------------------
c=
is equal to
c c rr+
+G D
H D
G D 1 G D H D –-------------------------------------
r G D e=
r G D c H D r+ =
r 1 G D H D – G D c=
r G D 1 G D H D –-------------------------------------
c=
is equal toc r crG D 1
G D -------------
r G D c=
c r 1G D -------------
=
is equal toc r
G D
r
c r
r
G D
G D
is equal toc c bbG D H D G D H D
x G D c=b H D x=b H D G D c=
x
is equal to
c rG D
c
c r
c
1G D -------------
G D
is equal toc
G D
b
ecG D
c 1G D -------------
b
e
e G D c b=
e G D c bG D -----------------
=
is equal to
c rG D
d
c rG D
G D dr G D c d =
r cG D dG D =
Kp
KiD----- KdD+ + 10
Fgas vactualvdesired verror
+-
1MD----------
Example – Simplify the diagram below
+5V -5V5K pote ntiom ete r
12V dc motor
shafts are couple d
-+
LM 675op-am p
2.2K
1K
PCI-1
200
data
acq
uisit
ion
card
from
Nat
iona
l Ins
trum
ents
Compute r Running La bview
-+
desired pos it ionvolta ge V d
gain K p
X
Example System
The Block Diagram
desiredpositionvoltage +
-
potentiometer
gain Kp op-amp motor shaftVd
System Error
ce r
b
G
H
+-
System error,
Feedback error,
S c r–=
e c b–=
Example – Find the system and feedback errors
G D KpD------= H D 1=
Given,
Type Transfer Function
Proportional (P)
Proportional-Integral (PI)
Proportional-Derivative (PD)
Proportional-Integral-Derivative (PID)
Lead
Lag
Lead-Lag
Gc K=
Gc K 1 1D-------+
=
Gc K 1 D+ =
Gc K 1 1D------- D+ +
=
Gc K1
1D+1
1D+----------------------- 1
2D+1
2D+-----------------------
=
1
Gc K 1 D+1 D+--------------------
=
Gc K 1 D+1 D+
-------------------- =
1
1
1
2
Other Controllers
ddt-----X AX BU+=
Y CX D+=
ddt-----X Y
DA
1D---- CB
U
+
+
+
+ X
State Variable Controller
Feedforward Controllers
process
feedforwardfunction
feedbackfunction
Gc1 Gc2 Gp1 Gp2
Cascade Controller
