PID Loop Shaping Design Goals
• -20 dB/dec Slope in the Crossover
Region
• High Gain at Low Frequencies
• Continued Decrease in Magnitude after
the Crossover
– we will discuss this issue at a later date
• Open-loop Crossover Equals (almost)
Closed-Loop Bandwidth, ω BW
High Gain at Low Frequencies
• Allows output to closely follow (“track”)
the desired input at low frequencies
• The higher the gain is the lower the
error is, and therefore the tighter the
closed-loop transfer function (CLTF)
holds to a magnitude ratio of one (0 dB).
Slope in the Crossover Region
• If the slope is –20 dB/dec in the region near
the crossover, then the open-loop transfer
function phase should be around –90 deg.
– Gives ~ 90 degree phase margin, good stability
• Schinstock’s rule of thumb for safe designs
is
– “–20 dB/dec for two decades; one before the
crossover and one after.”
CLTF and Error TF
xc
+
e
xc
T ( s) =
xc
x
G (s)
G( s) = x / e = OLTF
_
x
G (s)
1 + G (s)
e
S ( s) =
1
1 + G( s )
S ( s) ≅
T ( s ) = x / xc = CLTF
S ( s) = e / x c = ErrorTF
1
for G ( s) large
G ( s)
T ( s) ≅ G ( s) for G( s) small
op
Lo
en
Op
slope in region near
crossover should
be -20 dB/dec
Freq
p
Loo
sed
Clo
Err
or
h
hig t low
a ’
in eq
ga fr
Magnitude Ratio
PID Loop Shaping Design Goals
continued
decrease
in mag’
after
crossover
Likely Design Steps
1.Choose desired bandwidth (crossover)
frequency.
2.Choose “compensation” (the poles and
zeros of the controller) to achieve the
desirable shape.
3.Choose the “gain” to obtain the correct
crossover.
bandwidth of
CLTF is near
crossover of OLTF
1
Precision ….????
Simplified Model
PWM
Amplifier
Voice coil is similar to a
DC motor - linear force is
proportional to the current
Shuttle
DSP &
Laser Cards
F=k f i
Voice
Coil
X-Slide Scale &
Spindle Encoder
m
k
Ic
s
Closed-Loop
Amplifier
Dynamics
PID
Controller
xc
- 20
Coil
Current
ms 2 + bs + k
Plant
x
kf
I
m
F
b
(desired x = R)
-
1
2
ms + bs
Freq
Significant
High Freq’
Dynamics
(e.g. Amplifier)
• High low-frequency gain of the openloop system (for good tracking)
• Closed-loop bandwidth ω BW of 50 Hz
• Phase margin of at least 70 degrees
(actual x = C)
F
dB
/d
ec
Design Goals
x
b = 0.6 lbf-sec/in
PID
z1 , z 2
k/ m
ω bw
Example Problem
m = 2 lbf-sec2/in
zeros
from
controller
- 20
ms 2 + bs + k
s
R(s) +
breakpoint
from plant
dB
/d
ec
x
kf
I
Coil
Current
k d s 2 + k p s + ki
Plant
ec
/d
dB
-60
k d s 2 + k p s + ki
Open-Loop Transfer Function
OLTF Magnitude
Current
Command
PID
Controller
b
Interferometer
System Transfer Functions
xc
x
C(s)
PID ⇒
k d s 2 + k p s + ki
s
k ( s + z1 )( s + z 2 )
= d
s
2
Desired Open-Loop TF
Desired Open-Loop TF
120
120
100
100
80
80
System pole ~ 0.3 rad/sec
-60 dB
decade
Magnitude, dB
Magnitude, dB
decade
60
-40 dB
40
decade
20
-20 dB
0
-60
1.0E-01
1.0E+00
1.0E+01
1.0E+02
1.0E+03
-40 dB
40
decade
20
Arbitrarily
set z 1 ~ 3 rad/sec
-20
ωBW ~ 314 rad/sec
-40
decade
0
Set z 2 ~ 31.4 rad/sec
-20
60
ωBW ~ 314 rad/sec
-40
-60
1.0E-01
1.0E+04
-20 dB
1.0E+00
1.0E+01
Frequency, rad/sec
1.0E+02
1.0E+03
1.0E+04
Frequency, rad/sec
Finish the PID Controller
Desired Open-Loop TF #2
120
-
1
k d (s + 3)(s + 31 .4) F
2
s
2s + 0.6 s
C(s)
System pole ~ 0.3 rad/sec
100
80
decade
Magnitude, dB
R(s) +
• Find the value of kd that gives the
desired bandwidth of ~50 Hz
• What is the phase margin for this
controller?
60
-40 dB
40
20
0
Arbitrarily
set z 1 ~ 0.3 rad/sec
decade
-20 dB
-20
ωBW ~ 314 rad/sec
-40
-60
1.0E-01
1.0E+00
1.0E+01
1.0E+02
1.0E+03
1.0E+04
Frequency, rad/sec
PID Controller #2
R(s) +
-
k d (s + 0. 3)(s + 31 .4 )
s
F
Questions
1
2s 2 + 0.6 s
• Find the value of kd that gives the
desired bandwidth of ~50 Hz
• What is the phase margin for this
controller?
C(s)
• If the “force amplifier” saturated at 100
lb, what is the largest step input that will
no cause saturation?
• Plot the response of the system to ±0.01
inch sine waves at
– 1 Hz (6.28 rad/sec)
– 5 Hz
– 10 Hz
– 25 Hz
What is the max tracking
error in each case?
3