Classical PID Controllers
Eng R. L. Nkumbwa
School of Technology
Copperbelt University
2010
PID Controllers
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This note examines a particular control structure that has become
almost universally used in industrial control.
It is based on a particular fixed structure controller family, the socalled PID controller family.
These controllers have proven to be robust and extremely
beneficial in the control of many important applications.
PID stands for:
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P (Proportional)
I (Integral)
D (Derivative)
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General Control System Arrangement
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What is Classical Control?
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Classical control involves the choice of a
suitable controller in the transfer function
Gc(s) so that closed performance meets the
specifications as in figure on the previous
slide.
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Historical Note
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Early feedback control devices implicitly or explicitly
used the ideas of proportional, integral, and
derivative action in their structures. However, it was
probably not until Minorsky’s work on ship steering
published in 1922, that rigorous theoretical
consideration was given to PID control.
This was the first mathematical treatment of the type
of controller that is now used to control almost all
industrial processes.
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University, School of Technology
The Current Situation
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Despite the abundance of sophisticated
tools, including advanced controllers, the
Proportional, Integral, Derivative (PID
controller) is still the most widely used in
modern industry, controlling more that 95%
of closed-loop industrial processes.
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So, what is a Controller?
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A controller is a device that generates an
output signal based on the input signal it
receives.
The input signal is actually an error signal,
which is the difference between the
measured variable and the desired value, or
set point.
See figure below for the Controller.
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Heat Exchanger Control System
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Structure of a Controller
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Structure of a Controller
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This input error signal represents the amount of
deviation between where the process system is
actually operating and where the process system
is desired to be operating.
The controller provides an output signal to the
final control element, which adjusts the process
system to reduce this deviation.
The characteristic of this output signal is
dependent on the type, or mode, of the
controller.
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Modes of Controllers
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The mode of control is the manner in which a control
system makes corrections relative to an error that exists
between the desired value (set point) of a controlled
variable and its actual value.
The mode of control used for a specific application
depends on the characteristics of the process being
controlled. For example, some processes can be
operated over a wide band, while others must be
maintained very close to the set point.
Deviation is the difference between the set point of a
process variable and its actual value. This is a key term
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Nkumbwa, Copperbelt
used when discussing various modes
ofL. control.
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Four Modes of Controllers
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Four modes of control commonly used for
most applications are:
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Proportional (P)
Proportional plus Reset (PI)
Proportional plus Rate (PD)
Proportional plus Reset plus Rate (PID)
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Other Authors state it differently as follows
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Four Modes of Controllers
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Proportional Controllers (P)
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Each mode of control has characteristic
advantages and limitations.
The modes of control are discussed in this and
the next several sections of this module.
In the proportional (throttling) mode, there is a
continuous linear relation between value of the
controlled variable and position of the final
control element.
In other words, amount of valve movement is
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proportional to amount of deviation.
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Proportional Controllers
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Three terms commonly used to describe the
proportional mode of control are proportional band, gain
and offset.
Proportional band, (also called throttling range), is the
change in value of the controlled variable that causes full
travel of the final control element.
Gain, also called sensitivity, compares the ratio of
amount of change in the final control element to amount
of change in the controlled variable.
Mathematically, gain and sensitivity are reciprocal to
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Proportional Controllers
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Offset, also called droop, is deviation that
remains after a process has stabilized. Offset is
an inherent characteristic of the proportional
mode of control. In other words, the proportional
mode of control will not necessarily return a
controlled variable to its set point.
Proportional control is also referred to as
throttling control.
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Reset or Integral Controller (I)
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Integral control describes a controller in which
the output rate of change is dependent on the
magnitude of the input.
Specifically, a smaller amplitude input causes a
slower rate of change of the output.
This controller is called an integral controller
because it approximates the mathematical
function of integration.
The integral control method is also known as
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reset control.
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Definition of Integral Control
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A device that performs the mathematical function
of integration is called an integrator.
The mathematical result of integration is called
the integral.
The integrator provides a linear output with a rate
of change that is directly related to the amplitude
of the step change input and a constant that
specifies the function of integration.
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Example of an Integral Output for a
Fixed Input
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Integral Flow Rate Controller
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Integral Flow Rate Controller
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Properties of Integral Control
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The major advantage of integral controllers is that they
have the unique ability to return the controlled variable
back to the exact set point following a disturbance.
Disadvantages of the integral control mode are that it
responds relatively slowly to an error signal and that it
can initially allow a large deviation at the instant the error
is produced.
This can lead to system instability and cyclic operation.
For this reason, the integral control mode is not normally
used alone, but is combined with another control mode.
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Proportional Plus Reset Control (PI)
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Proportional plus reset control is a combination
of the proportional and integral control modes.
This type control is actually a combination of two
previously discussed control modes, proportional
and integral.
Combining the two modes results in gaining the
advantages and c o m p e n s a t i n g f o r t h e
disadvantages of the two individual modes.
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Characteristics of the PI
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The main advantage of the proportional control mode is
that an immediate proportional output is produced as
soon as an error signal exists at the controller as shown
in Figure below.
The proportional controller is considered a fast-acting
device.
This immediate output change enables the proportional
controller to reposition the final control element within a
relatively short period of time in response to the error.
See below figure for the response of the PI Control
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University, School of Technology
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Disadvantages of the
Proportional Control
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The main disadvantage of the proportional control mode
is that a residual offset error exists between the
measured variable and the set point for all but one set of
system conditions.
The main advantage of the integral control mode is that
the controller output continues to reposition the final
control element until the error is reduced to zero.
This results in the elimination of the residual offset error
allowed by the proportional mode.
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Disadvantages of the
Proportional Control
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The main disadvantage of the integral mode is
that the controller output does not immediately
direct the final control element to a new position
in response to an error signal.
The controller output changes at a defined rate
of change, and time is needed for the final
control element to be repositioned.
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PI Equations
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PI Characteristics
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Characteristics of the PI
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The combination of the two control modes is
called the proportional plus reset (PI) control
mode.
It combines the immediate output characteristics
of a proportional control mode with the zero
residual offset characteristics of the integral
mode.
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Example of the PI for the Plant
Heat Exchanger
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Effects of Disturbance on Reverse
Acting Controller
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Effects of Disturbance on Reverse
Acting Controller
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By adding the reset action to the proportional
action the controller produces a larger output for
the given error signal and causes a greater
adjustment of the control valve.
This causes the process to come back to the set
point more quickly. Additionally, the reset action
acts to eliminate the offset error after a period of
time.
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Reset Wind-Up
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Proportional plus reset controllers act to eliminate the offset error
found in proportional control by continuing to change the output after
the proportional action is completed and by returning the controlled
variable to the set point.
An inherent disadvantage to proportional plus reset controllers is the
possible adverse effects caused by large error signals.
The large error can be caused by a large demand deviation or when
initially starting up the system.
This is a problem because a large sustained error signal will
eventually cause the controller to drive to its limit, and the result is
called "reset windup."
Because of reset windup, this control mode is not well-suited for
processes that are frequently shut down and started up.
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University, School of Technology
Proportional plus Rate control (PD)
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Proportional plus rate control is a control mode in which a
derivative section is added to the proportional controller.
Proportional plus rate describes a control mode in which
a derivative section is added to a proportional controller.
This derivative section responds to the rate of change of
the error signal, not the amplitude; this derivative action
responds to the rate of change the instant it starts.
This causes the controller output to be initially larger in
direct relation with the error signal rate of change.
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PD Controller
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PD Controller
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Proportional plus Rate control (PD)
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The higher the error signal rate of change, the
sooner the final control element is positioned to
the desired value.
The added derivative action reduces initial
overshoot of the measured variable, and
therefore aids in stabilizing the process sooner.
This control mode is called proportional plus rate
(PD) control because the derivative section
responds to the rate of change of the error
signal.
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Definition of the Derivative Control
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A device that produces a derivative signal is
called a differentiator. Figure below shows the
input versus output relationship of a
differentiator.
The differentiator provides an output that is
directly related to the rate of change of the input
and a constant that specifies the function of
differentiation.
The derivative constant is expressed in units of
seconds and defines the differential
controller
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output.
Derivative Output for a Constant Rate
of change
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Rate Control Output
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Rate Control Output
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The differentiator acts to transform a changing
signal to a constant magnitude signal as shown
in Figure above.
As long as the input rate of change is constant,
the magnitude of the output is constant.
A new input rate of change would give a new
output magnitude.
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Rate Control Output
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Derivative cannot be used alone as a control
mode.
This is because a steady-state input produces a
zero output in a differentiator.
If the differentiator were used as a controller, the
input signal it would receive is the error signal.
As just described, a steady-state error signal
corresponds to any number of necessary output
signals for the positioning of the final control
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element.
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Rate Control Output
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Therefore, derivative action is combined with
proportional action in a manner such that the
proportional section output serves as the
derivative section input.
Proportional plus rate controllers take advantage
of both proportional and rate control modes.
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Proportional Action
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As seen in Figure below, proportional action
provides an output proportional to the error.
If the error is not a step change, but is slowly
changing, the proportional action is slow.
Rate action, when added, provides quick
response to the error.
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Example of the Proportional Plus rate
Control
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To illustrate proportional plus rate control, we will
use the same heat exchanger process that has
been analyzed in previous chapters (see Figure
below).
For this example, however, the temperature
controller used is a proportional plus rate
controller.
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Proportional Plus rate Control for the
Heat Exchanger
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Effects of Disturbance
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Industrial Applications
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Proportional plus rate control is normally used with large
capacity or slow-responding processes such as
temperature control.
The leading action of the controller output compensates
for the lagging characteristics of large capacity, slow
processes.
Rate action is not usually employed with fast responding
processes such as flow control or noisy processes
because derivative action responds to any rate of change
in the error signal, including the noise.
Proportional plus rate controllers are useful with
processes which are frequently started up and shut down
because it is not susceptible to reset
Engwindup.
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Proportional Plus Integral Plus
Derivative Control (PID)
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Proportional plus reset plus rate controllers combine
proportional control actions with integral and derivative
actions.
For processes that can operate with continuous cycling,
the relatively inexpensive two position controller is
adequate.
For processes that cannot tolerate continuous cycling, a
proportional controller is often employed.
For processes that can tolerate neither continuous
cycling nor offset error, a proportional plus reset
controller can be used.
For processes that need improved stability and can
Eng R.
L. Nkumbwa,
tolerate an offset error, a proportional
plus
rate Copperbelt
controller
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is employed.
Proportional Plus Integral Plus
Derivative Control (PID)
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However, there are some processes that cannot
tolerate offset error, yet need good stability.
The logical solution is to use a control mode that
combines the advantages of proportional, reset,
and rate action.
This chapter describes the mode identified as
proportional plus reset plus rate, commonly
called Proportional-Integral-Derivative (PID).
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Controller Action
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When an error is introduced to a PID controller,
the controller’s response is a combination of the
proportional, integral, and derivative actions, as
shown in Figure below.
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Controller Action
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Assume the error is due to a slowly increasing measured
variable. As the error increases, the proportional action of
the PID controller produces an output that is proportional
to the error signal.
The reset action of the controller produces an output
whose rate of change is determined by the magnitude of
the error.
In this case, as the error continues to increase at a
steady rate, the reset output continues to increase its rate
of change.
The rate action of the controller produces an output
whose magnitude is determined by the rate of change.
When combined, these actions produce
output
as
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shown in Figure above.
Controller Action
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As you can see from the combined action
curve, the output produced responds
immediately to the error with a signal that is
proportional to the magnitude of the error
and that will continue to increase as long as
the error remains increasing.
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PID Controller Response Curves
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PID Response Curves
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Figure above demonstrates the combined
controller response to a demand disturbance.
The proportional action of the controller stabilizes
the process.
The reset action combined with the proportional
action causes the measured variable to return to
the set point.
The rate action combined with the proportional
action reduces the initial overshoot and cyclic
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period.
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PID Controller
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PID Controllers Equations
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PID Controller Characteristics
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Design of PID Controllers
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Based on the knowledge of P, I and D
– trial and error
– manual tuning
– simulation
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University, School of Technology
Tuning of PID Controllers
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Because of their widespread use in practice, we present
below several methods for tuning PID controllers.
Actually these methods are quite old and date back to the
1950’s.
Nonetheless, they remain in widespread use today.
In particular, we will study
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Ziegler-Nichols Oscillation Method
Ziegler-Nichols Reaction Curve Method
Cohen-Coon Reaction Curve Method
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Ziegler-Nichols Method
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Ziegler-Nichols method
– based on a open-loop process
– based on a critical gain
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Ziegler-Nichols (Z-N)
Oscillation Method
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This procedure is only valid for open loop stable
plants and it is carried out through the following
steps:
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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.
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Ziegler-Nichols (Z-N)
Oscillation Method
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Record the controller critical gain Kp = Kc and the
oscillation period of the controller output, Pc.
Adjust the controller parameters according to next
slide; there is some controversy regarding the PID
parameterization for which the Z-N method was
developed, but the version described here is
applicable to the parameterization of standard
form PID.
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Ziegler-Nichols tuning using the
oscillation method
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Reaction Curve
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Reaction Curve Based Methods
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A linearized quantitative version of a simple plant
can be obtained with an open loop experiment,
using the following procedure:
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1. 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.
2. 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).
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Reaction Curve Based Methods
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3. Record the plant output until it settles to the
new operating point. Assume you obtain the curve
shown on the next slide. This curve is known as
the process reaction curve.
In the figure on next page, m.s.t. stands for
maximum slope tangent.
4. Compute the parameter model as follows
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Plant Response
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Plant Response
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Z-N Tuning Using Reaction Curve
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PI Z-N Tuned Reaction Curve
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Observation
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We see from the previous slide that the
Ziegler-Nichols reaction curve tuning method
is very sensitive to the ratio of delay to time
constant.
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Cohen-Coon Reaction
Curve Method
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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.
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Cohen-Coon Tuning Reaction Curve
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Lead-Lag Compensators
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Closely related to PID control is the idea of
lead-lag compensation. The transfer function
of these compensators is of the form:
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