PID Tuning Sigurd Skogestad NTNU, Trondheim, Norway

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PID Tuning Sigurd SkogestadNTNU, Trondheim,
Norway
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Tuning of PID controllers

• SIMC tuning rules (Skogestad IMC)()
• Main message Can usually do much better by
  taking a systematic approach
• Key Look at initial part of step response
• Initial slope k k/?1
• One tuning rule! Easily memorized
• Reference S. Skogestad, Simple analytic rules
  for model reduction and PID controller design,
  J.Proc.Control, Vol. 13, 291-309, 2003
• () Probably the best simple PID tuning rules in
  the world

?c 0 desired closed-loop response time (tuning
parameter) For robustness select ?c ?
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Need a model for tuning

• Model Dynamic effect of change in input u (MV)
  on output y (CV)
• First-order delay model for PI-control
• Second-order model for PID-control

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Step response experiment

• Make step change in one u (MV) at a time
• Record the output (s) y (CV)

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First-order plus delay process
? Delay - Time where output does not change ?1
Time constant - Additional time to reach 63 of
final change k steady-state gain ? y(1)/? u
k slope after response takes off k/?1
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Model reduction of more complicated model

• Start with complicated stable model on the form
• Want to get a simplified model on the form
• Most important parameter is usually the
  effective delay ?

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Example
Half rule
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half rule
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original
1st-orderdelay
2nd-orderdelay
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Approximation of zeros
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Derivation of SIMC-PID tuning rules

• PI-controller (based on first-order model)
• For second-order model add D-action.
• For our purposes it becomes simplest with the
  series (cascade) PID-form

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Basis Direct synthesis (IMC)
Closed-loop response to setpoint change
Idea Specify desired response and from this
get the controller. Algebra
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IMC Tuning Direct Synthesis
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Integral time

• Found Integral time dominant time constant (?I
  ?1)
• Works well for setpoint changes
• Needs to be modified (reduced) for integrating
  disturbances
• Example. Almost-integrating process with
  disturbance at input
• G(s) e-s/(30s1)
• Original integral time ?I 30 gives poor
  disturbance response
• Try reducing it!

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Integral Time
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Integral time

• Want to reduce the integral time for
  integrating processes, but to avoid slow
  oscillations we must require
• Derivation

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Conclusion SIMC-PID Tuning Rules
One tuning parameter ?c
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Some insights from tuning rules

• The effective delay ? (which limits the
  achievable closed-loop time constant t2/2 ) is
  independent of the dominant process time constant
  t1
• It depends on t2/2 (PI) or t3/2 (PID)
• Use (close to) P-control for integrating process
• Beware of large I-action (small tI) for level
  control
• Use (close to) I-control for time delay process

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Some special cases
One tuning parameter ?c
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Another special case IPZ process

• IPZ-process may represent response from steam
  flow to pressure
• Rule T2
• SIMC-tunings

These tunings turn out to be almost identical
to the tunings given on page 104-106 in the Ph.D.
thesis by O. Slatteke, Lund Univ., 2006 and K.
Forsman, "Reglerteknik for processindustrien",
Studentlitteratur, 2005.
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Note Derivative action is commonly used for
temperature control loops. Select ?D equal to ?2
time constant of temperature sensor
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Selection of tuning parameter ?c

• Two main cases
• TIGHT CONTROL Want fastest possible
  control subject to having good robustness
• Want tight control of active constraints
  (squeeze and shift)
• SMOOTH CONTROL Want slowest possible control
  subject to acceptable disturbance rejection
• Want smooth control if fast setpoint tracking is
  not required, for example, levels and
  unconstrained (self-optimizing) variables
• THERE ARE ALSO OTHER ISSUES Input saturation
  etc.

TIGHT CONTROL
SMOOTH CONTROL
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TIGHT CONTROL
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TIGHT CONTROL
Typical closed-loop SIMC responses with the
choice ?c?
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TIGHT CONTROL
Example. Integrating process with delay1. G(s)
e-s/s. Model k1, ?1, ?11
SIMC-tunings with ?c with ?1
IMC has ?I1
Ziegler-Nichols is usually a bit aggressive
Setpoint change at t0
Input disturbance at t20
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TIGHT CONTROL

• Approximate as first-order model with k1, ?1
  10.11.1, ?0.10.040.008 0.148
• Get SIMC PI-tunings (?c?) Kc 1 1.1/(2
  0.148) 3.71, ?Imin(1.1,8 0.148) 1.1

2. Approximate as second-order model with k1,
?1 1, ?20.20.020.22, ?0.020.008
0.028 Get SIMC PID-tunings (?c?) Kc 1
1/(2 0.028) 17.9, ?Imin(1,8 0.028) 0.224,
?D0.22
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TIGHT CONTROL
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Tuning for smooth control
SMOOTH CONTROL

• Tuning parameter ?c desired closed-loop
  response time
• Selecting ?c? (tight control) is reasonable
  for cases with a relatively large effective delay
  ?
• Other cases Select ?c gt ? for
• slower control
• smoother input usage
• less disturbing effect on rest of the plant
• less sensitivity to measurement noise
• better robustness
• Question Given that we require some disturbance
  rejection.
• What is the largest possible value for ?c ?
• Or equivalently The smallest possible value for
  Kc?

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Closed-loop disturbance rejection
SMOOTH CONTROL
d0
-d0
ymax
-ymax
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SMOOTH CONTROL
Minimum controller gain for PI-and
PID-control Kc Kc,min u0/ymax u0
Input magnitude required for disturbance
rejection ymax Allowed output deviation
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SMOOTH CONTROL

• Minimum controller gain
• Industrial practice Variables (instrument
  ranges) often scaled such that
• Minimum controller gain is then

(span)
Minimum gain for smooth control ) Common default
factory setting Kc1 is reasonable !
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Example
SMOOTH CONTROL
?c is much larger than ?0.25
Does not quite reach 1 because d is step
disturbance (not not sinusoid)
Response to step disturbance 1 at input
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Application of smooth control
SMOOTH CONTROL
LEVEL CONTROL

• Averaging level control

If you insist on integral action then this value
avoids cycling
Reason for having tank is to smoothen
disturbances in concentration and flow. Tight
level control is not desired gives no
smoothening of flow disturbances. Let u0
? q0 expected flow change m3/s (input
disturbance) ymax ?Vmax -
largest allowed variation in level m3 Minimum
controller gain for acceptable disturbance
rejection Kc Kc,min u0/ymax From the
material balance (dV/dt q qout), the model is
g(s)k/s with k1. Select KcKc,min.
SIMC-Integral time for integrating process ?I
4 / (k Kc) 4 ?Vmax / ? q0 4
residence time provided tank is nominally half
full and ?q0 is equal to the nominal flow.
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More on level control
LEVEL CONTROL

• Level control often causes problems
• Typical story
• Level loop starts oscillating
• Operator detunes by decreasing controller gain
• Level loop oscillates even more
• ......
• ???
• Explanation Level is by itself unstable and
  requires control.

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Integrating process Level control
LEVEL CONTROL
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How avoid oscillating levels?
LEVEL CONTROL
0.1 ¼ 1/?2
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Case study oscillating level
LEVEL CONTROL

• We were called upon to solve a problem with
  oscillations in a distillation column
• Closer analysis Problem was oscillating reboiler
  level in upstream column
• Use of Sigurds rule solved the problem

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LEVEL CONTROL
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Rule Kc u0/ymax 1 (in scaled variables)
SMOOTH CONTROL

• Exception to rule Can have Kc lt 1 if
  disturbances are handled by the integral action.
• Disturbances must occur at a frequency lower than
  1/?I
• Applies to Process with short time constant (?1
  is small) and no delay (? ¼ 0).
• Then ?I ?1 is small so integral action is
  large
• For example, flow control

Kc Assume variables are scaled with respect to
their span
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Summary Tuning of easy loops
SMOOTH CONTROL

• Easy loops Small effective delay (? ¼ 0), so
  closed-loop response time ?c (gtgt ?) is selected
  for smooth control
• ASSUME VARIABLES HAVE BEEN SCALED WITH RESPECT TO
  THEIR SPAN SO THAT u0/ymax 1 (approx.).
• Flow control Kc0.2, ?I ?1 time constant
  valve (typically, 2 to 10s)
• Level control Kc2 (and no integral action)
• Other easy loops (e.g. pressure control) Kc 2,
  ?I min(4?c, ?1)
• Note Often want a tight pressure control loop
  (so may have Kc10 or larger)

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Selection of ?c Other issues

• Input saturation.
• Problem. Input may overshoot if we speedup
  the response too much (here speedup ?/?c).
• Solution To avoid input saturation, we must obey
  max speedup

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A little more on obtaining the model from step
response experiments
?1 ¼ 200 (may be neglected for ?c lt 40)

• Factor 5 rule Only dynamics within a factor 5
  from control time scale (?c) are important
• Integrating process (?1 1)
• Time constant ?1 is not important if it is much
  larger than the desired response time ?c. More
  precisely, may use
• ?1 1 for ?1 gt 5 ?c
• Delay-free process (?0)
• Delay ? is not important if it is much smaller
  than the desired response time ?c. More
  precisely, may use
• ? ¼ 0 for ? lt ?c/5

time
? ¼ 1 (may be neglected for ?c gt 5)
?c desired response time
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Step response experiment How long do we need to
wait?

• RULE May stop at about 10 times effective delay
• FAST TUNING DESIRED (tight control, ?c ?)
• NORMALLY NO NEED TO RUN THE STEP EXPERIMENT FOR
  LONGER THAN ABOUT 10 TIMES THE EFFECTIVE DELAY
  (?)
• EXCEPTION LET IT RUN A LITTLE LONGER IF YOU SEE
  THAT IT IS ALMOST SETTLING (TO GET ?1 RIGHT)
• SIMC RULE ?I min (?1, 4(?c?)) with ?c ?
  for tight control
• SLOW TUNING DESIRED (smooth control, ?c gt ?)
• HERE YOU MAY WANT TO WAIT LONGER TO GET ?1 RIGHT
  BECAUSE IT MAY AFFECT THE INTEGRAL TIME
• BUT THEN ON THE OTHER HAND, GETTING THE RIGHT
  INTEGRAL TIME IS NOT ESSENTIAL FOR SLOW TUNING
• SO ALSO HERE YOU MAY STOP AT 10 TIMES THE
  EFFECTIVE DELAY (?)

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• Integrating process (?c lt 0.2 ?1)
• Need only two parameters k and ?
• From step response

Response on stage 70 to step in L
Example. Step change in u ?u 0.1 Initial
value for y y(0) 2.19 Observed delay ?
2.5 min At T10 min y(T)2.62 Initial
slope
y(t)
2.62-2.19
7.5 min
?2.5
t min
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Conclusion PID tuning
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Cascade control
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Tuning of cascade controllers
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Cascade control serial process
d6
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Cascade control serial process
d6
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Tuning cascade control serial process

• Inner fast (secondary) loop
• P or PI-control
• Local disturbance rejection
• Much smaller effective delay (0.2 s)
• Outer slower primary loop
• Reduced effective delay (2 s instead of 6 s)
• Time scale separation
• Inner loop can be modelled as gain1
  2effective delay (0.4s)
• Very effective for control of large-scale systems

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CONTROLLABILITY
Controllability

• (Input-Output) Controllability is the ability
  to achieve acceptable control performance (with
  any controller)
• Controllability is a property of the process
  itself
• Analyze controllability by looking at model G(s)
• What limits controllability?

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Controllability
CONTROLLABILITY

• Recall SIMC tuning rules
• 1. Tight control Select ?c? corresponding to
• 2. Smooth control. Select Kc
• Must require Kc,max gt Kc.min for controllability
• )

max. output deviation
initial effect of input disturbance
y reaches k d0 t after time t y reaches
ymax after t ymax/ k d0
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Controllability
CONTROLLABILITY
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Example Distillation column
CONTROLLABILITY
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Example Distillation column
CONTROLLABILITY
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Conclusion controllability

• If the plant is not controllable then improved
  tuning will not help
• Alternatives
• Change the process design to make it more
  controllable
• Better self-regulation with respect to
  disturbances, e.g. insulate your house to make
  yTin less sensitive to dTout.
• Give up some of your performance requirements