velocity forms calculate the change in controller output,

1
26 PID Algorithms

• Ref Sections 6.3, 6.4

2
PID Algorithms

• Available in various forms with various pros and
  cons
• Five versions will be presented here
• Three Position Form versions
• Two Velocity Form Versions

3
Position and Velocity Forms

• Position Forms ? directly calculate the new value
  of controller output, c(t)
• Velocity Forms ? calculate the change in
  controller output, ?c(t).
• The actual controller output is then calculated
  from the previous value c(t-?t) and ?c(t).

4
(1) PID, derivative of error, integral

• Position form, calculates c(t)
• c0 is the controller bias, the value of c(t) when
  the controller is activated (t0).
• PRO basic version shows terms clearly
• CON derivative of error subject to derivative
  kick
• CON includes integral not compatible with
  digital computers

5
Derivative Kick

• Error is Setpoint minus Process measurement, e(t)
  ysp(t) ys(t)
• Setpoint can change instantly in a step change
  this causes e(t) to have a step change whenever
  setpoint is changed
• The sudden step change in e(t) causes the
  derivative of e(t) to be infinite this causes
  the derivative term to be huge whenever the
  setpoint is changed (derivative kick)

6
Process measurement is steady
ys(t)
Setpoint changed suddenly
ysp(t)
Error generated by setpoint change
e(t)
8
Derivative of vertical step is 8 this causes
huge change in c(t), called derivative kick.
de/dt
7
(1) PI control equation (version 1)

• Reminder Any of the versions of the PID
  algorithm can be adapted to PID, PI or P-only
  control.

8
(2) PID, derivative of measurement, integral
Note the minus sign
Derivative term is now based on ys(t) rather than
e(t).

• Position form, calculates c(t)
• PRO eliminates derivative kick
• CON includes integral not compatible with
  digital computers

9
Under normal operating conditions, when setpoint
is not changing
Note the minus sign
10
Process measurement is steady
ys(t)
Setpoint changed suddenly
ysp(t)
Error generated by setpoint change
e(t)
Since the step does not appear in ys(t) there is
no kick in dys/dt.
dys/dt
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(3) PID, derivative of measurement, summation
Integral replaced by summation
Derivative replaced by finite difference

• Position form, calculates c(t)
• PRO eliminates derivative kick
• PRO includes summation compatible with digital
  computers
• CON Summations can cause overflow errors
  (crashes) on digital computers
• CAUTION ?t is the time between data samples, ?t
  must be small compared to process response time

12
Velocity Form Development

• Evaluate position form at t to get c(t)
• Evaluate position form at t-?t to get c(t-?t)
• Subtract to get ?c(t) c(t) - c(t-?t)

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Evaluate position form (version (3)) at t
Evaluate position form (version (3)) at t - ?t
Subtract to get velocity form
P
I
D
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(4) PID, velocity form
Summation is gone

• PRO c0 is gone
• PRO Integral is gone (no overflow errors)
• CON Calculates ?c(t), not c(t) must calculate
  c(t) separately
• CON Derivative term is more complicated (10 of
  controllers use D term)

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Proportional Kick

• The P term is still based on error, so a step
  change in setpoint causes a step change in error,
  and the P term responds strongly to that.
• Proportional kick is much less severe than
  derivative kick but for some systems even
  proportional kick is considered too aggressive.
• The P term can also be based on process
  measurement (next slide) to eliminate the
  proportional kick.
• This is not common.

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(5) PID, another velocity form
e(t) replaced by ys(t) note the sign change

• PROs and CONs Same as version (4), plus
• PRO Eliminates proportional kick
• CON Eliminates proportional kick controller
  only responds to SP change through I term (slow)

Version (4) is the most commonly used algorithm.
17
PID Algorithm, Laplace Space
P
I
D
This is version (1) in Laplace space note how
E(s) appears in each term, and has been factored
out.
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Example 6.2

• Determine the dynamic behavior (i.e., check the
  poles) of a PI controller Kc 1, tI 1
  applied to a 2nd order process Kp 1, tp 5, ?
  2. Assume Ga(s) 1, Gs(s) 1.

PI controller
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Ex 6.2, contd Characteristic Equation
Gc(s) Ga(s) Gp(s) Gs(s) 1 0
Substitute values for every parameter
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Ex 6.2, contd Check Poles
Solve for values of s (poles)
s -0.764 s -0.018 0.228 i
Dominant root is closest to origin
Real part is negative, so stable
Imaginary part, so oscillates