Module 2
Module 2
 Optimum controller settings : Evaluation criteria –
IAE, ISE, ITAE and ¼ decay ratio – determination
of optimum settings for mathematically described
processes using time response and frequency
response – tuning – Process Reaction Curve
method – Ziegler Nichols method – Damped
oscillation method.
Evaluation Criteria
 To define What is a GOOD control, which may differ from
process to process.
 How to Select the type of feedback controller – P, PI, PD or PID
 How to adjust parameters – Kp, KI, KD
We can:
1. Keep the maximum deviation (error) as small as possible.
2. Achieve short settling times.
3. Minimize the Integral of errors until the process has settled
to its desired set point.
Performance Criteria
If the criterion is to ‘return to desired value as soon as possible’,
then we select closed loop response ‘A’.
If the criterion is to ‘keep the maximum deviation as small as
possible’ we select response ‘B’.
Performance Criteria
 Steady State Performance Criteria
 Dynamic Response Performance Criteria
Steady State Performance Criteria
- ‘Zero error at steady state’


Proportional controller cannot achieve
zero error due to offset, but PI mode can.
For a Proportional control steady state error
tends to zero when Kp →∞
Performance Criteria
Dynamic Response Performance Criteria
based on two criteria:
 Simple Performance Criteria:
o uses only a few points of the response.
o They are simpler, but only approximate.
 Time Integral Performance Criteria:
o uses entire closed loop response from t=0 to t= very
large.
o They are precise, but difficult to use.
Simple Performance Criteria
 Different parameters like Overshoot, rise time, settling
time, decay ratio and frequency of oscillation of the
transient are considered.
Most popular is DECAY RATIO criterion.
One Quarter Decay Ratio Criterion
One Quarter Decay Ratio Criterion
The measure of decay ratio is found by adjusting the
control loop until the deviation from the disturbance is
such that each deviation peak is down by one quarter
from the preceding peak.
Time Integral Performance Criteria
The shape of complete closed loop response from
time t=0 until steady state reached is used.
It uses the entire response, but in the case of ¼ ratio
criteria it uses only isolated characteristics of the
dynamic response.
It is more precise.
In ¼ ratio criteria many combinations of controller
settings are possible; but in integral criteria only one
combination is possible which certainly reduces the
error.
Time Integral Performance Criteria
1. Integral of the Square Error (ISE)
ISE criteria give more weight to larger deviations
Time Integral Performance Criteria
2. Integral of the Absolute value of Error (IAE)
It seems the best criterion for process control,
since the penalty for control is generally a linear
function of the error.
Time Integral Performance Criteria
3. Integral of the Time weighted Absolute Error (ITAE)
ITAE criteria weights deviations more heavily as
time increases.
Time Integral Performance Criteria
 If we want to suppress large errors, ISE is better
than IAE, because the errors are squared and
thus contribute more to the value of integral.
 For the suppression of small errors, IAE is better
than ISE because when we square small errors
they become even smaller.
 To suppress errors that persists for longer times,
ITAE criterion is better because of the presence of
‘t’ in the integral term.
Time Integral Performance Criteria
 Different criteria lead to different controller designs.
 For the same time integral criterion, different input
changes lead to different designs.
Selection of Feedback Controller
1. Define an appropriate performance criterion (ISE, IAE
or ITAE)
2. Compute the value of the performance criterion using
a P or PI or PID controller with the best settings for the
adjusted parameters KP , KI and KD
3. Select the controller which gives the ‘best’ value for
the performance criterion.
Proportional Control
a. Accelerates the response of a controlled process.
b. Produces an offset.
Integral Control
a. Eliminates any offset.
b. Higher maximum deviation.
c. Produces sluggish, long oscillating response.
d. If we increase KP to produce faster response, the
system become oscillatory and may lead to instability.
Derivative Control
a. Anticipates future errors and introduces appropriate
action.
b. Introduces a stabilizing effect on the closed-loop
response of a process.
CONTROLLER TUNING
Process of deciding what values to be used for its
adjusted parameters for a feed back controller.
3 General approaches for controller tuning:
1. Use simple criteria such as ¼ decay ratio, minimum settling
time, minimum largest error and so on. (Since it provides
multiple solutions, additional specifications needed to be
considered to reach a single solution and new value to
parameters)
2. Use time integral performance criteria (IAE, ISE, ITAE). (it is
cumbersome and relies heavily on mathematical model. If
applied experimentally on actual process it is time
consuming.)
3. Use Semi empirical rules which have been proven in
practice.
PROCESS REACTION CURVE METHOD
(Cohen and Coon Method)
 Also called OPEN LOOP TRANSIENT RESPONSE
METHOD.
 Opening the control loop by disconnecting the
controller output from the final control element.
 Can be used only for systems with ‘self
regulation’.
PROCESS REACTION CURVE METHOD
PROCESS REACTION CURVE METHOD
 Open the control loop by disconnecting the
controller output from the final control element.
 Introduce a step change of magnitude ‘A’ in the
variable ‘c’ which actuates FCE.
 Record the value of output with respect to time.
This curve ym(t) is called ‘PROCESS REACTION
CURVE’.
PROCESS REACTION CURVE METHOD
 PRC is affected by the dynamics of process,
sensor and FCE.
 Cohen and Coon observed a ‘sigmoidal shape’
which can be approximated to a first order
system with a dead time.
PROCESS REACTION CURVE METHOD
PROCESS REACTION CURVE METHOD
Static Gain
Time constant
= slope of the sigmoidal response
at the point of inflection.
Dead time td = time elapsed until the system responded
 Derive expressions for the best controller settings
PROCESS REACTION CURVE METHOD
Proportional Controller
PROCESS REACTION CURVE METHOD
Proportional – Integral Controller
PROCESS REACTION CURVE METHOD
Proportional - Integral - Derivative Controller
PROCESS REACTION CURVE METHOD
Observations:
 Gain of PI controller is less than P controller
because integral mode makes the system more
sensitive (oscillatory).
 The stabilizing effect of derivative control mode
allows the use of higher gains in PID controller
compared to P and PI controllers.
ZIEGLER-NICHOLS METHOD
(Ultimate Cycling Method)
 Also called CLOSED LOOP TUNING METHOD.
 This method based on frequency response
analysis.
 Adjusting closed loop until steady oscillations
occur, controller settings are then based on the
conditions that generate the cycling.
ZIEGLER-NICHOLS METHOD
1. Bring the system to desired operational level (Design
condition).
2. Reduce any Integral and derivative action to their
minimum effect.
3. Using proportional control only and with feedback loop
closed, introduce a set point change and vary
proportional gain until the system oscillates
continuously. The frequency of continuous oscillations
is the cross over frequency, ‘ω0’. Let ‘M’ be the
amplitude ratio of the system’s response at the cross
over frequency.
ZIEGLER-NICHOLS METHOD
4. Compute the following two quantities:
Ultimate Gain
KU= 1/M
Ultimate period of sustaining cycling,
PU = 2π/ω
COmin/
cycle
5. Using the values of KU and PU compute controller
settings.
ZIEGLER-NICHOLS METHOD
Mode
KP
Proportional
KU/2
Proportional-Integral
Proportional-Integral-Derivative
TI
(minutes) (minutes)
-
KU/2.2 PU/1.2
KU/1.7
Td
PU/2
-
PU/8
DAMPED OSCILLATION METHOD
 Sustained oscillations for testing purpose are not
allowable in many plants. So Ziegler- Nichols Method
cannot be used.
 More accurate than Closed loop method
 By using only proportional action and starting with a
low gain, the gain is adjusted until the transient
response of the closed loop shows a decay ratio of ¼.
 The reset time and derivative time are based on the
period of oscillation, P, which is always greater than the
ultimate period PU.
DAMPED OSCILLATION METHOD
For PID control
TD = P/6
TI = P/1.5