Loop Shaping. Professor Walter W. Olson Department of Mechanical, Industrial and Manufacturing Engineering University of Toledo. Outline of Today’s Lecture. Review PID Theory Integrator Windup Noise Improvement Static Error Constants (Review) Loop Shaping Loop Shaping with the Bode Plot
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Professor Walter W. Olson
Department of Mechanical, Industrial and Manufacturing Engineering
University of Toledo
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Without PID
With PID
Kv(dB)
Error
signal
E(s)
Output
y(s)
Input
r(s)
Controller
C(s)
Plant
P(s)
Open Loop
Signal
B(s)
Sensor
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3 db
Roll off Rate dB/dec
Resonant peak gain, dB
Bandwidth rps
Resonant peak frequency rps
Gain cross over
frequency rps
Increase of gain
also increases
bandwidth and
resonant gain
Break frequency
corresponds to the
component pole or zero
Poles bend the magnitude and phase down
Zeros bend the magnitude and the phase up
Mechanical
Lead
Compensator
b1
xi
b2
x0
k
y
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Y
R
C(s)
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The root locus from the complex poles has very little damping and causes the vibration seen in the response. There is a pole at 0.01 on the real axis that is dominant and causes the sluggish behavior.
Strategy: Use a lead compensator to bend the curves to the left and into the 0.6 damping region. The zero of the compensator should counteract the vibrational mode
The initial design with the pole at 5 and the zero at 1 had the desired effectof bending the root locus to the left and removing most of the vibration. However
the pole is still too close to the origin such that 0.6 damping can not be achieved.
We achieved the specifications once the pole of the compensator was moved
out to 9 and we adjusted the gain for the 0.6 damping.
Mechanical
Lead
Compensator
Note:
the lead compensator opensup the high frequency regionwhich could cause noise problems
The Lead compensator addsphase
b1
xi
b2
x0
k
y
f
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Aircraft Pitch
Rate Dynamics
Compensator
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Y
R
C(s)
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Current System:
Reading the plot:
When design a lead
compensator first adjust
the gain to meet the static
error condition
In this case the gain needs
the be increase by 180 or
20Log10180= 45.1 dB
added
33dB
Initial design: Phase good but Kvnot
Gain needs to be increased by about 20 dB
Final design:
Mechanical
Lag
Compensator
xi
b2
k
a
b
b1
Note that the lag compensator causes
a drop in the magnitude and phase
This could be useful in reducing bandwidth, and improving gain margin; however it might reduce phase margin
x0
Linear Motor
Rate Dynamics
Compensator
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Y
R
C(s)
1
Current System:
Phase margin is
low and the static
velocity error constant
must be improved.
Start by correcting the
static velocity error constant
4.9dB
Gain Adjusted:
Gain and Phase
Margin problems
try placing a pole
at 0.01 rps and
adjust the zero
Need to shape
the curve like this
Need to move PM to here
Final Design
Next: Sensitivity Analysis