PID ControlOverall Course Objectives
Tuesday, May 5, 2015
PID Control
•
Develop the skills necessary to function as an
industrial process control engineer.
– Skills
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Tuning loops
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Control loop design
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Control loop troubleshooting
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Command of the terminology
– Fundamental
understanding
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Process dynamics
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Feedback control
PID Controls
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Most common
controller in the CPI.
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Came into use in
1930’s with the introduction of pneumatic controllers.
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From the general feedback control loop and using
the properties of transfer functions, the following expressions can be derived:
Characteristic Equation
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Since setpoint tracking and disturbance
rejection have the same denominator for their closed loop transfer functions,
this indicates that both setpoint tracking and disturbance rejection have the
same general dynamic behavior.
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The roots of the denominator determine the
dynamic characteristics of the closed loop process.
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The characteristic equation is given by:
Feedback Control Analysis
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The loop gain (KcKaKpKs)
should be positive for stable feedback control.
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An open-loop
unstable process can be made stable by applying the proper level of feedback
control.
Characteristic Equation Example
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Consider the dynamic behavior of a P-only
controller applied to a CST thermal mixer (Kp=1; tp=60
sec) where the temperature sensor has a ts=20 sec and ta
is assumed small. Note that Gc(s)=Kc.
Example Continued- Analysis of the Closed Loop Poles
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When Kc
=0, poles are -0.05 and -0.0167 which correspond to the inverse of tp and ts.
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As Kc
is increased from zero, the values of the poles begin to approach one another.
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Critically damped
behavior occurs when the poles are equal.
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Underdamped
behavior results when Kc is increased further due to the
imaginary components in the poles.
In-Class Exercise
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Determine the
dynamic behavior of a P-only controller with Kc equal to 1 applied
to a first-order process in which the process gain is equal to 2 and the time
constant is equal to 22. Assume that Gs(s)
is equal to one and Ga(s) behaves as a first-order process with a
time constant of 5.
PID Control Algorithm
Definition of Terms
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e(t)- the
error from setpoint [e(t)=ysp-ys].
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Kc- the controller gain is a tuning parameter and
largely determines the controller aggressiveness.
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tI- the
reset time is a tuning parameter and determines the amount of integral action.
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tD- the
derivative time is a tuning parameter and determines the amount of derivative
action.
Transfer Function for a PID Controller
Example for a First Order Process with a PI Controller
Example of a PI Controller Applied to a Second Order
Process
Properties of Proportional Action
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Closed loop transfer function base on a P-only
controller applied to a first order process.
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Properties of P control
– Does
not change order of process
–
Closed loop time constant is smaller
than open loop tp
–
Does not eliminate offset.
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Based on applying an I-only controller to a
first order process
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Properties of I control
–
Offset is eliminated
– Increases
the order by 1
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The primary
benefit of integral action is that it removes offset from setpoint.
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In addition, for
a PI controller all the steady-state change in the controller output results
from integral action.
Properties of Derivative Action
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Closed loop transfer function for
derivative-only control applied to a second order process.
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Properties of derivative control:
–
Does not change
the order of the process
–
Does not
eliminate offset
–
Reduces the
oscillatory nature of the feedback response
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Another way to express the controller gain.
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Kc in this formula is
dimensionless. That is, the controller
output is scaled 0-100% and the error from setpoint is scaled 0-100%.
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Proportional band is equal to 200%.
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The range of the error from setpoint is 200 psi.
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The controller output range is 0 to 100%.
Conversion from Kc to PB
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Controller gain is equal to 15 %/ºF
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The range of the error from setpoint is 25 ºF.
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The trapezoidal approximation of the integral.
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Note the difference in proportional, integral,
and derivative terms from the position form.
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Velocity form is the form implemented on DCSs.
Correction for Derivative Kick
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Derivative kick occurs when a setpoint change is
applied that causes a spike in the derivative of the error from setpoint.
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Derivative kick can be eliminated by replacing
the approximation of the derivative based on the error from setpoint with the
negative of the approximation of the derivative based on the measured value of
the controlled variable, i.e.,
Correction for Aggressive Setpoint Tracking
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For certain process, tuning the controller for
good disturbance rejection performance results in excessively aggressive action
for setpoint changes.
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This problem can be corrected by removing the
setpoint from the proportional term.
Then setpoint tracking is accomplished by integral action only.
The Three Versions of the PID Algorithm Offered on
DCS’s
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(1) The original
form in which the proportional, integral, and derivative terms are based on the
error from setpoint
Guidelines for Selecting Direct and Reverse Acting
PID’s
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Consider a direct acting final control element
to be positive and reverse to be negative.
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If the sign of the product of the final control
element and the process gain is positive, use the reverse acting PID algorithm.
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If the sign of the product is negative, use the
direct acting PID algorithm
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If control signal goes to a control valve with a
valve positioner, the actuator is considered direct acting.
Level Control Example
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Process gain is negative because when flow out
is increased, the level decreases.
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If the final control element is direct acting,
use direct acting PID.
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For reverse acting final control element, use
reverse acting PID.
In-Class Exercise
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Write the position
form of the PID algorithm for Example 3.4, and assume that the control valve on
the feed line to the mixer has an air-to-close actuator. Use the form that is not susceptible to
derivative kick. Specify whether the
controller is a direct-acting or reverse-acting controller.
In-Class Exercise
•
Write the
velocity form of the PID algorithm for Example 3.1, and assume that the control
valve on the feed line to the mixer has an air-to-open actuator. Use the form that is not susceptible to
derivative kick or proportional kick. Specify whether the controller is a
direct-acting or reverse-acting controller.
Filtering the Process Measurement
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Filtering reduces
the effect of sensor noise by approximating a running average.
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Filtering adds
lag when the filtered measurement is used for control.
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Normally, use the
minimum amount of filtering necessary.
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f- filter factor (0-1)
Feedback Loop with Sensor Filtering
Effect of Filtering on Closed Loop Dynamics
Analysis of Example
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tf is equal to
Dt (1/f-1) as f
becomes small, tf becomes
large.
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As tf is increased, tp’ will increase.
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Critical issue is
relative magnitude of tf compare to tp.
Effect of the Amount of Filtering on the Open Loop
Response
Effect of a Noisy Sensor on Controlled Variable
without Filtering
Effect of a Noisy Sensor on Controlled Variable with
Filtering
An Example of Too Much and Too Little Filtering
Relationship between Filter Factor (f), the
Resulting Repeatability Reduction Ratio (R) and the Filter Time Constant
(tf)
Key Issues for Sensor Filtering
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To reduce the
effect of noise (i.e., R is increased), f must be reduced, which
increases the value of tf. Filtering
slows the closed-loop response significantly as tf becomes larger than 10% of tp.
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The effect of
filtering on the closed-loop response can be reduced by increasing the
frequency with which the filter is applied, i.e., reducing Dtf.
PID Controller Design Issues
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Over 90% of
control loops use PI controller.
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P-only: used for fast responding processes that do
not require offset free operation (e.g., certain level and pressure
controllers)
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PI: used for fast
responding processes that require offset free operation (e.g., certain flow,
level, pressure, temperature, and composition controllers)
Integrating Processes
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For integrating
processes, P-only control provides offset-free operation. In fact, if as integral action is added to
such a case, the control performance degrades.
•
Therefore, for
integrating processes, P-only control is all that is usually required.
PID Controller Design Issues
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PID: use for sluggish processes (i.e., a process
with large deadtime to time constant ratios) or processes that exhibit severe
ringing for PI controllers. PID controllers
are applied to certain temperature and composition control loops. Use derivative action when:
Analysis of Several Commonly Encountered Control Loops
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Flow control
loops
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Level control
loops
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Pressure control
loops
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Temperature
control loops
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Composition
control loops
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DO control loop
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Biomass
controller
Flow Control Loop
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Since the flow sensor and the process (changes
in flow rate for a change in the valve position) are so fast, the dynamics of
the flow control loop is controlled by the dynamics of the control valve.
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Almost always use PI controller.
Deadband of a Control Valve
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Deadband of industrial valves is between
±10%-±25%.
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As a result, small changes in the air pressure
applied to the valve do not change the flow rate.
Deadband of Flow Control Loop
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A control valve (deadband of ±10-25%) in a flow
control loop or with a positioner typically has a deadband for the average
flow rate of less than ±0.5% due to the high frequency opening and closing
of the valve around the specified flow rate.
Level Control Loop
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Dynamics of the sensor and actuator are fast
compared to the process.
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Use P-only controller if it is an integrating
process.
Pressure Control Process
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The sensor is generally faster than the
actuator, which is faster than the process.
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Use P-only controller if it is an integrating
process otherwise use a PI controller.
Temperature Control Loop
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The dynamics of the process and sensor are
usually slower than the actuator.
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Use a PI controller unless the process is
sufficiently sluggish to warrant a PID controller.
Analysis of PI Controller Applied to Typical
Temperature Loop
Further Analysis of Dynamic of a Typical Temperature
Control Loop
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Note that as the
controller gain is increased, i.e., KcKp increase, the
closed loop time constant becomes smaller.
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Also, note that
as the controller gain is increased, the value of z decreases.
Composition Control Loop
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The process is usually the slowest element
followed by the sensor with the actuator being the fastest.
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Use a PI controller unless the process is
sufficiently sluggish to warrant a PID controller.
DO Control Loop
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The process and the sensor have approximately
the same dynamic response.
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This is a fast responding process for which
offset-free operation is desired.
Therefore, PI controller should be used.
Biomass Controller
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The process for this system is the slowest
element.
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Because the process is a high-order sluggish
process, a PID controller is required.
Overview
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The
characteristic equation determines the dynamic behavior of a closed loop
system
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Proportional, integral, and derivative action
each have unique characteristics.
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There are a number of different ways to apply a
PID controller.
•
Use a PI controller unless offset is not
important or if the process is sluggish.
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When analyzing the dynamics of a loop, consider
the dynamics of the actuator, the process, and the sensor separately.
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