This document discusses various empirical techniques for tuning PID controllers, including the Ziegler-Nichols and Cohen-Coon reaction curve methods. The Ziegler-Nichols method determines controller parameters by forcing sustained oscillations and measuring the ultimate gain and period. The Cohen-Coon method analyzes the open-loop step response curve to estimate parameters. Simulation tools can also assist in empirically tuning controllers to achieve desired closed-loop responses.

1. Tuning PID Controllers
Eng. Mahmoud Hussein
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Gc(s)
Controller
+
+ n
sensor
noise
+
w load disturbance
+
Gp(s)
Plant
u
control
y
output
+
-
r
reference
input, or
set-point
e
sensed
error

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3. Analytical Vs. Empirical Techniques
• Analytical Tuning Techniques
▫ Frequency response analysis
• Empirical Tuning Techniques
▫ On-line tuning after the control system is installed
▫ Computer simulation
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4. Empirical Tuning Techniques
• Two Steps:
▫ Determination of the dynamic characteristics of
the control loop
▫ Estimation of the controller tuning parameters
that produce a desired response
• Tuning Techniques
▫ Ziegler-Nichols Oscillation Method
▫ Ziegler-Nichols Reaction Curve Method
▫ Cohen-Coon Reaction Curve Method
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5. Z-N Closed Loop Method
Z-N Continuous Cycling Method
By Ziegler and Nichols (1942)
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6. Quarter Decay Ratio Response By
Ultimate Gain
• Use the proportional controller to force
sustained oscillations.
• Goal is to determine ultimate gain and ultimate
period of oscillation of the control loop
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7. Quarter Decay Ratio Response By
Ultimate Gain
• Ziegler-Nichols method specifies a decay ratio of
one-fourth (1/4)
• Decay ratio = ratio of the amplitudes of two
successive oscillations
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8. Tuning Procedure
• Set the true plant under proportional control,
with a very small gain
• Increase the gain until the loop starts oscillating
▫ Note that linear oscillation is required and that it
should be detected at the controller output.
• Record the controller critical gain (ultimate
gain) Kp = Kc (Ku) and the oscillation period of
the controller output, Pc (Pu).
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9. Tuning Procedure
• Calculate the parameters according to the
following formulas
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Controlle
r Type
Proportional
Gain (Kp)
Integral
Time (tI)
Derivative
Time (tD)
P-only Ku/2 - -
PI Ku/2.2 Pu/1.2 -
PID Ku/1.7 Pu/2 Pu/8

10. Example
• Consider the given plant
▫ Find the parameters of a PID controller using the
Z-N oscillation method.
▫ Obtain a graph of the response to a unit step input
reference.
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 3
1
1
)(
+

s
sGp

11. Example Solution
• Applying the procedure we find
▫ Kc = 8 and ωc = 3
• Hence, from Table we have
▫ Kp = 4.8
▫ Tr = 1.81
▫ Td = 0.45
• Try the same procedure for the second order
system
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 2
1
1
)(
+

s
sGp

12. Drawbacks
• Time-consuming
▫ If the process dynamics are slow
• The process is pushed to the stability limits
▫ Risk of plant damage
• For first-order and second-order model without
time delays the ultimate gain does not exist
▫ Because the closed-loop system is stable for all
values of Kp. However, in practice, it is unusual
for a control loop not to have an ultimate gain.
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14. Tuning Procedure
• Same procedure as Z-N closed loop method
• TLC tuning values tend to reduce oscillatory
effects and improves robustness.
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15. Z-N Open-loop Method
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16. Process Reaction Curve
• From the open loop step response, the system
dynamics can be determined
• The controller settings are based on the open-
loop step response, or in other words, the
process reaction curve
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17. Tuning Procedure
• With the plant in open loop, take the plant
manually to a normal operating point
▫ Say that the plant output settles at y(t) = y0 for a
constant plant input u(t) = u0.
• At an initial time, t0, apply a step change to the
plant input, from u0 to u
▫ This should be in the range of 10 to 20% of full
scale
• Record the plant output until it settles to the
new operating point. Assume you obtain the
curve shown on the next slide.
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18. Tuning Procedure
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• Compute the parameter model as follows
m.s.t. stands for maximum slope
tangent

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20. Tuning Procedure
• Cohen and Coon carried out further studies to
find controller settings which, based on the same
model, lead to a weaker dependence on the ratio
of delay to time constant. Their suggested
controller settings are shown below
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21. Simulink Control Design 3.0
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22. PID Controller Block
▫ PID_Controller_Automated_Tuning.mdl
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23. Initial PID Design
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24. Adjusting Response Time To Tune
Parameters
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