Controller Design (to determine controller settings for P, PI or PID controllers) Based on Transient Response Criteria

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Controller Design (to determine controller
settings for P, PI or PID controllers) Based on
Transient Response Criteria
Chapter 12
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• Desirable Controller Features
• The closed-loop system must be stable.
• The effects of disturbances are minimized, i.e.,
  good disturbance rejection.
• Quick and smooth responses to the set-point
  changes are guaranteed, i.e., good set-point
  tracking.
• Off-set is eliminated.
• Excessive controller action is avoided.
• The control system is robust, i.e., it is
  insensitive to changes in operating conditions
  and to inaccuracies in process model and/or
  measurements.

Chapter 12
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Chapter 12
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Simplified Block Diagram
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Simplified Block Diagram
D(s)
B(s)
P(s)
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Example
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Chapter 12
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Example 12.1
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Chapter 12
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Chapter 12
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• Alternatives for Controller Design
• Direct synthesis (DS) method
• Internal model control (IMC) method
• Controller tuning relations
• Frequency response techniques
• Computer simulation
• On-line tuning after the control system is
  installed.

Chapter 12
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Direct Synthesis
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Direct Synthesis Steps

1. Specify desired closed-loop response (transfer
   function)
2. Assume process model
3. Solve for controller transfer function

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Direct Synthesis to Achieve Perfect Control
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Direct Synthesis to Achieve Finite Settling Time
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Example
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Example
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Direct Synthesis for Time-Delayed Systems
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Taylor Series Approximation
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Example 1
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Example 2
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Pade Approximation
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Example
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Example 12.1
Use the DS design method to calculate PID
controller settings for the process
Chapter 12
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Consider three values of the desired closed-loop
time constant .
Evaluate the controllers for unit step changes in
both the set point and the disturbance, assuming
that Gd G. Repeat the evaluation for two cases

1. The process model is perfect ( G).
2. The model gain is 0.9, instead of the
   actual value, K 2. Thus,

Chapter 12
The controller settings for this example are

3.75 1.88 0.682
8.33 4.17 1.51
15 15 15
3.33 3.33 3.33
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Chapter 12
Figure 12.3 Simulation results for Example 12.1
(a) correct model gain.
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Simulation results for Example 21.1(b) incorrect
model gain.
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Chapter 12
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PID vs. IMC
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PID Controller Design Procedure Based on IMC
Method Step 1 factor process model
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PID Controller Design Procedure Based on IMC
Method Step 2 derive IMC transfer function
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PID Controller Design Procedure Based on IMC
Method Step 3 derive PID transfer function
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Chapter 12
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Example
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Controller Synthesis Criteria in Time Domain

• Time-domain techniques can be classified into two
  groups
• (a) Criteria based on a few points in the
  response
• (b) Criteria based on the entire response, or
  integral criteria

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Approach (a)

• Based on settling time, overshoot, rise time,
  decay ratio (Fig. 5.10 can be viewed as
  closed-loop response).
• Several methods based on 1/4 decay ratio have
  been proposed, e.g., Cohen-Coon and
  Ziegler-Nichols.

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Chapter 12
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Chapter 12
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Approach (b) - Criteria

• Integral of absolute value of error (IAE)
• Integral of square error (ISE)
• Time-weighted IAE (ITAE)

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Approach (b) - Remarks

• Pick controller parameters to minimize integral.
• IAE allows larger overall deviation than ISE
  (with smaller overshoots).
• ISE needs longer settling time
• ITAE weights errors occurring later more heavily
• Approximate optimum tuning parameters are
  correlated with K, ?, ? (Table 12.3).

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Chapter 12
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Chapter 12
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Example 1
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Example 1

ITAE 0.169 1.85
IAE 0.195 2.02
ISE 0.245 2.44
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Example 2
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Chapter 12
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Summary of Tuning Relationships 1. KC is
inversely proportional to KPKVKM . 2. KC
decreases as ?/? increases. 3. ?I and ?D
increase as ?/? increases (typically ?D 0.25 ?I
). 4. Reduce Kc, when adding more integral
action increase Kc, when adding derivative
action 5. To reduce oscillation, decrease KC
and increase ?I .
Chapter 12
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Disadvantages of Tuning Correlations 1.
Interactions are ignored (decreased stability
limits). 2. Derivative action equipment
specific. 3. First order time delay model can
be inaccurate. 4. Kp, t can vary. 5.
Resolution, measurement errors decrease stability
margins. 6. ¼ decay ratio not conservative
standard (too oscillatory).
Chapter 12
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Example 12.4
Consider a lag-dominant model with
Chapter 12
Design four PI controllers

1. IMC
2. IMC based on the integrator
   approximation in Eq. 12-33
3. IMC with Skogestads modification
   (Eq. 12-34)
4. Direct Synthesis method for disturbance rejection
   (Chen and Seborg, 2002) The controller settings
   are Kc 0.551 and

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Evaluate the four controllers by comparing their
performance for unit step changes in both set
point and disturbance. Assume that the model is
perfect and that Gd(s) G(s).
Solution
The PI controller settings are
Chapter 12
Controller Kc
IMC 0.5 100
(b) Integrator approximation 0.556 5
(c) Skogestad 0.5 8
(d) DS-d 0.551 4.91
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Figure 12.8. Comparison of set-point responses
(top) and disturbance responses (bottom) for
Example 12.4. The responses for the Chen and
Seborg and integrator approximation methods are
essentially identical.
Chapter 12
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Chapter 12
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On-Line Controller Tuning

1. Controller tuning inevitably involves a tradeoff
   between performance and robustness.
2. Controller settings do not have to be precisely
   determined. In general, a small change in a
   controller setting from its best value (for
   example, 10) has little effect on closed-loop
   responses.
3. For most plants, it is not feasible to manually
   tune each controller. Tuning is usually done by a
   control specialist (engineer or technician) or by
   a plant operator. Because each person is
   typically responsible for 300 to 1000 control
   loops, it is not feasible to tune every
   controller.
4. Diagnostic techniques for monitoring control
   system performance are available.

Chapter 12
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Controller Tuning and Troubleshooting Control
Loops
Chapter 12
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Ziegler-Nichols Rules These well-known tuning
rules were published by Z-N in 1942
controller Kc tI tD
P PI PID 0.5 KCU 0.45 KCU 0.6 KCU - PU/1.2 PU/2 - - PU/8
Chapter 12
Z-N controller settings are widely considered to
be an "industry standard". Z-N settings were
developed to provide 1/4 decay ratio -- too
oscillatory?
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Modified Z-N settings for PID control
controller Kc tI tD
original Some overshoot No overshoot 0.6 KCU 0.33 KCU 0.2 KCU PU/2 PU/2 PU/3 PU/8 PU/3 PU/2
Chapter 12
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Chapter 12
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Chapter 12
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Chapter 12
Figure 12.15 Typical process reaction curves (a)
non-self-regulating process, (b) self-regulating
process.
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Chapter 12
Figure 12.16 Process reaction curve for Example
12.8.
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Chapter 12
Figure 12.17 Block diagram for Example 12.8.
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Chapter 12