PID Control

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PID Control

• spring-mass-damper problem

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In an experiment, the following output was
obtained and the system parameters were found to
be ? c/(2(k1m1)1/2)0.1203, ?n (k1/m1)1/2
18.81 rad/sec m1 0.5773 kg, k1 204.25 N/m, c
2.6135 N sec/m x1(0) ? 0.025 m 2.5 cm, from
rest.
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• lets model this system using Simulink
• governing differential equation
• for our case, f(t) 0, (0) 0, x1(0) 0.025
  m
• thus

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new case ? 0.938, ?n 10.43 rad/sec 1.66
cycles/sec
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• suppose the mass is initially positioned at the
  free length and is at rest
• we want to move the mass to the position x 0.3
  m
• how do we determine f(t)?

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• proportional controller for the underdamped case
• f(t) (1 N/m) e(t)

large steady state error large settling
time never gets close to the desired position
x, m
Simulink
time, sec
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Kp 10,000 N/m
x, m
Kp 1000 N/m
Kp 100 N/m
Kp 10
time, sec
as proportional gain Kp increases steady state
error improves rise time improves settling time
worsens overshoot and frequency of oscillation
worsens
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free response
overdamped system new case ? 10.43, ?n
10.43 rad/sec 1.66 cycles/sec
was 4.5 N sec/m
underdamped case
this overdamped case
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• proportional controller for the overdamped case

Kp 1 N/m
underdamped case
x, m
Kp 1 N/m
this overdamped case
Simulink
time, sec
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Kp 10,000 N/m
Kp 1000 N/m
Kp 100 N/m
Kp 10
proportional controller looks pretty good for
overdamped case when gain value is high
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Kp 10,000,000 N/m
Kp 1,000,000 N/m
Kp 100,000 N/m
response can overshoot for overdamped case if
gain value Kp becomes too large
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• suppose we have a critically damped system (?1)

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critically damped case, Kp 1
Simulink
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critically damped case for various Kp
Kp 100,000 N/m
Kp 10,000 N/m
Kp 1000 N/m
Kp 100 N/m
Kp 10
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Kp 1 N/m for all 3 cases
c 4.5 N sec/m, ? 0.094
c 47.9583 N sec/m, ? 1
c 500 N sec/m, ? 10.43
all cases (underdamped, critically damped, and
overdamped) have the same steady state error
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• apparent rules of thumb for a 2nd order system
  using only a proportional controller
• for all three cases, increasing Kp reduces the
  steady state error
• if Kp gets too large you will getoscillations
• proportional controller can workpretty well,
  unless the system isunderdamped

underdamped
overdamped
critically damped
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underdamped
overdamped
critically damped
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• how can we do better?
• lets try to reduce the steady state error
• next try ? PI controller
• we will add a new term to f(t)
• this will be proportional to the time integral of
  the error

• what about the limits of the integration?
• do we integrate from tdinosaur age to current
  time?

• - usually only add together the last n error
  terms
• note that e(t) may be positive or negative
• if it stays positive or negative for a long time,
  the contribution to f(t) can become significant
• so, if there is a steady state error, eventually
  this error will add up and give a boost to f(t)

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• lets look at the underdamped system again

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underdamped system
Kp 1 N/m, Ki 1 N sec/m
Kp 1 N/m, Ki 0
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underdamped system
Kp 1 N/m, Ki 100 N sec/m
Kp 1 N/m, Ki 300 N sec/m
Kp 1 N/m, Ki 500 N sec/m
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underdamped system
Kp 0.1 N/m, Ki 250 N sec/m
Kp 5 N/m, Ki 250 N sec/m
Kp 50 N/m, Ki 250 N sec/m
Kp 500 N/m, Ki 250 N sec/m
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overdamped system
Kp 1 N/m, Ki 1 N sec/m
Kp 1 N/m, Ki 0
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overdamped system
Kp 1 N/m, Ki 100 N sec/m
Kp 1 N/m, Ki 300 N sec/m
Kp 1 N/m, Ki 500 N sec/m
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critically damped system
Kp 1 N/m, Ki 1 N sec/m
Kp 1 N/m, Ki 0
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critically damped system
Kp 1 N/m, Ki 100 N sec/m
Kp 1 N/m, Ki 300 N sec/m
Kp 1 N/m, Ki 500 N sec/m
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critically damped system
Kp 1 N/m, Ki 1000 N sec/m
Kp 1 N/m, Ki 2000 N sec/m
Kp 1 N/m, Ki 5000 N sec/m
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overdamped system
underdamped system
critically damped system
critically damped system w/ larger Ki values
Kp 1 N/m, Ki 100 N sec/m
Kp 1 N/m, Ki 300 N sec/m
Kp 1 N/m, Ki 500 N sec/m
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• Can we decrease the overshoot and the settling
  time?

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• lets look at the underdamped case againKp 100
  N/m

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underdamped case, Kp 100 N/m
x (position)
m
error
time, sec
how can we reduce the oscillations ?
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underdamped case, Kp 100 N/m
x (position)
x (position)
error
m
derivative of error
error
derivative of the error
time, sec
lets look at the derivative of the error if we
have a positive error, and is positive, then
the error is getting worse ! so, whenever we
have a positive error and is positive, lets
add more to f(t) lets add Kd ? , where Kd is
some constant
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e(t) xdesired - x
lets look at all the possible cases
I
II
III
IV
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underdamped case, Kp 100 N/m
xdesired 0.3 m
x (position)
m
derivative of error / 3
I
II
I
II
time, sec
for this example, the error is always positive,
but goes positive and negative lets look at
a proportional and derivative controller (PD)
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underdamped case, Kp 100 N/m
xdesired 0.3 m
Kd 0
Kd 10 N/m-sec
m
Kd 100
time, sec
the Kd term helped tremendously with the
oscillations, but didnt change the steady state
error
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In Summary
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In Summary
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PID Controller
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PID Controller
underdamped case, Kp 100 N/m, Ki 250 N sec/m
xdesired 0.3 m
Kd 0
m
Kd 10 N/m-sec
Kd 100
time, sec
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General Tips for Designing a PID Controller
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• Obtain an open-loop response and determine what
  needs to be improved
• Add a proportional control to improve the rise
  time
• Add a derivative control to improve the overshoot
  
• Add an integral control to eliminate the
  steady-state error
• Adjust each of Kp, Ki, and Kd until you obtain a
  desired overall response. (TUNING!)

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PI Control with Velocity Feedback
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underdamped system PI controller
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underdamped system PI controller with velocity
feedback
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underdamped system PI controller with velocity
feedback
Kp 5 Ki 250
Kv 0
Kv 1
m
Kv 5
Kv 10 N-sec/m
time, sec
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underdamped system
Kp 5 Ki 250
m
Kv 10 N-sec/m
time, sec