Optimum controller settings : Evaluation criteria IAE, ISE,
ITAE and 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.
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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 K p, K I, K D
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.
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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.
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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 K p
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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.
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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.
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One Quarter Decay Ratio Criterion
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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.
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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 ratio criteria it uses
only isolated characteristics of the dynamic response. It is more
precise. In ratio criteria many combinations of controller settings
are possible; but in integral criteria only one combination is
possible which certainly reduces the error.
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Time Integral Performance Criteria 1.Integral of the Square
Error (ISE) ISE criteria give more weight to larger deviations
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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.
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Time Integral Performance Criteria 3.Integral of the Time
weighted Absolute Error (ITAE) ITAE criteria weights deviations
more heavily as time increases.
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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.
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Time Integral Performance Criteria Different criteria lead to
different controller designs. For the same time integral criterion,
different input changes lead to different designs.
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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 K P, K I and K D 3.Select
the controller which gives the best value for the performance
criterion.
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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 K P to produce faster
response, the system become oscillatory and may lead to
instability.
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Derivative Control a.Anticipates future errors and introduces
appropriate action. b.Introduces a stabilizing effect on the
closed-loop response of a process.
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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
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.
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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.
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PROCESS REACTION CURVE METHOD
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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 y m (t) is called
PROCESS REACTION CURVE.
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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.
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PROCESS REACTION CURVE METHOD
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Static Gain Time constant = slope of the sigmoidal response at
the point of inflection. Dead time t d = time elapsed until the
system responded Derive expressions for the best controller
settings
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PROCESS REACTION CURVE METHOD Proportional Controller
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PROCESS REACTION CURVE METHOD Proportional Integral
Controller
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PROCESS REACTION CURVE METHOD Proportional - Integral -
Derivative Controller
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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.
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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.
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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, 0 . Let M be the amplitude ratio of the systems
response at the cross over frequency.
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ZIEGLER-NICHOLS METHOD 4.Compute the following two quantities:
Ultimate Gain K U = 1/M Ultimate period of sustaining cycling, P U
= 2/ CO min/ cycle 5.Using the values of K U and P U compute
controller settings.
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ZIEGLER-NICHOLS METHOD Mode KPKP T I (minutes) T d (minutes)
Proportional K U /2-- Proportional-Integral K U /2.2P U /1.2-
Proportional-Integral-Derivative K U /1.7P U /2P U /8
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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 . The reset time and derivative time are
based on the period of oscillation, P, which is always greater than
the ultimate period P U.
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DAMPED OSCILLATION METHOD For PID control T D = P/6 T I =
P/1.5