Transcript Basic Control Actions - Government College of Engineering

By: Dr. S. B Chikalthankar [email protected]
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Dr. Sanjay Chikalthankar
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 The word control is usually taken to mean regulate, direct, or
command
 In modern usage the word system has many meanings.
 A system is an arrangement, set, or collection of things connected or
related in such a manner as to form an entirety or whole.
 A system is an arrangement of physical components connected or
related in such a manner as to form and/or act as an entire unit.
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 “A control system is an arrangement of physical
components connected or related in such a manner as to
command, direct, or regulate itself or another system.”
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 In engineering and science we usually restrict the meaning of control
systems to apply to those systems whose major function is to
dynamically or actively command, direct, or regulate.
 a control system, and a control system may be part of a larger
system, in which case it is called subsystem or control subsystem
 we define two terms: input and output in next slide which help in
identifying, delineating, or defining a control system
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Feedback control
 It is the most important and widely used control strategy
 It is a closed-loop control strategy
Block diagram
disturbance
y
manipulated
comparator
sp
+
set-point
–
error
controller
variable
y
process
controlled
variable
transmitter
 The input is the stimulus, excitation or command applied to a control
system, typically from an external energy source, usually in order to
produce a specified response from the control system.
 The output is the actual response obtained from a control system. It
may or may not be equal to the specified response implied by the
input.
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 The purpose of the control system usually identifies or defines the
output and input.
 If the output and input are given, it is possible to identify,
delineate, or define the nature of the system components.
 Control systems may have more than one input or output.
 Often all inputs and outputs are well defined by the system
description.
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 A thermostatically controlled heater or furnace automatically regulating the
temperature of a room or enclosure is a control system.The input to this
system is a reference temperature, usually specified by appropriately setting a
thermostat. The output is the actual temperature of the room or
enclosure.
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 When the thermostat detects that the output is less than the
input, the furnace provides heat until the temperature of the
enclosure becomes equal to the reference input. Then the
furnace is automatically turned off. When the temperature
falls somewhat below the reference temperature, the furnace
is turned on again.
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• Control objective ; To keep the tank temperature atTthe desired value
by adjusting
Q
TR the rate of heat input from the heater.
Schematic diagram for a stirred-tank control system.
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Basic components in the feedback control loop
 Process being controlled(stirred tank).
 Sensor and transmitter.
 Controller.
 SCR and final control element(electrical heater)  Actuator.
Dr.
Chikalthankar lines(electrical cables) between the various instruments.
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 Sanjay
Transmission
Transmission
line
D/A converter
A/D converter
u(t)
serial port
y(t)
Personal
Computer
Controller
influent
effluent
DO sensor
Actuator
Process
power
blower
Sensor
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Typical equipment for process control using computer.
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250 B.C ; Greeks, water level controller
Their mode of operation was very similar to that of the level
regulator in the modern flush toilet.
1788 ; James Watt, fly-ball governor
It played a key role in the development of stream power.
1930s ; PID controller became commercially available
The first theoretical papers on process control were published.
1940s ; Pneumatic PID controller
1950s ; Electronic PID controller
Late 1950 ~ 1960s ; The first computer control applications
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 Control systems are classified into two general categories: open-loop
and closed-loop systems.The distinction is determined by the control
action, that quantity responsible for activating the system to produce
the output.
 The term control action is classical in the control systems literature, but the
word action in this expression does not always directly imply change,
motion, or activity.
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 Open Loop =
An open-loop control system is one in which the control action is
independent of the output.
 Close Loop=
A closed-loop control system is one in which the control action is
somehow dependent on the output.
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 Two outstanding features of open-loop control systems are:
1. Their ability to perform accurately is determined by their
calibration. To calibrate means to establish or reestablish the inputoutput relation to obtain a desired system accuracy.
2. They are not usually troubled with problems of instability, a concept to
be subsequently discussed in detail.
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 Closed-loop control systems are more commonly called feedback
control systems
 Feedback is that characteristic of closed-loop control systems which
distinguishes them from open-loop systems.
 More generally, feedback is said to exist in a system when a closed
sequence of cause-and-effect relations exists between system variables.
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 Feedback is that property of a closed-loop system which permits the
output (or some other controlled variable) to be compared with the
input to the system (or an input to some other internally situated
component or subsystem) so that the appropriate control action may
be formed as some function of the output and input.
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 Control systems engineering consists of analysis and design of control
systems configurations.
 Analysis is the investigation of the properties of an existing system.
The design problem is the two methods exist for design:
1. Design by analysis
2. Design by synthesis
 Design by analysis is accomplished by modifying the characteristics
of an existing or standard system configuration, and design by
synthesis by defining the form of the system directly from its
specifications.
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 To solve a control systems problem, we must put the specifications or description of system
configuration and its components into form amenable to analysis or design.
 Three basic representations of components and systems are used extensively in the study of
control systems:
1.
Mathematical models, in the form of differential equations, difference equations,
and/or other mathematical relations, for example, Laplace- and z-transforms
2.
Block diagrams
3.
Signal flow graphs
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 The following five definitions are examples of control laws,
or control algorithms.
1. An on-off controller (two-position, binary controller) has
only two possible values at its output U, depending on the input
e to the controller.
2. A proportional (P) controller has an output U proportional to
its input e , that is, U = Kpe, where K , is a proportionality constant.
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3. A derivative (D) controller has an output proportional to the derivative of
its input e , that is, U = KD de/dt, where KD is a proportionality constant.
4. An integral (I) controller has an output U proportional to the integral of its
input e, that is, U = K , / e ( t ) dt, where K , is a proportionality constant.
5. PD, PI, DI, and PID controllers are combinations of proportional (P),
derivative (D), and integral (I) controllers.
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block diagram of an industrial
control system
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Classifications of Industrial
Controllers
 1. Two-position or on-off controllers
 2. Proportional controllers
 3. Integral controllers
 4. Proportional-plus-integral controllers
 5. Proportional-plus-derivative controllers
 6. Proportional-plus-integral-plus-derivative controllers
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 A PID controller is a simple three-term controller. The letters P, I
and D stand for:
1. P – Proportional
2. I – Integral
3. D - Derivative
 The transfer function of the most basic form of PID controller is
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we assume the controller is used in a closed-loop unity feedback
system. The variable e denotes the tracking error, which is sent to the
PID controller. The control signal u from the controller to the plant is
equal to the proportional gain (KP) times the magnitude of the error
plus the integral gain (KI ) times the integral of the error plus the
derivative gain (KD) times the derivative of the error.
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 We are most interested in four major characteristics of the closed-
loop step response. They are
1. Rise Time: the time it takes for the plant output y to rise beyond
90% of the desired level for the first time.
2. Overshoot: how much the peak level is higher than the steady state,
normalized against the steady state.
3. Settling Time: the time it takes for the system to converge to its
steady state.
4. Steady-state Error: the difference between the steady-state output
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and
the desired output.
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 Typical steps for designing a PID controller are
1. Determine what characteristics of the system needs to be
improved.
2. Use KP to decrease the rise time.
3. Use KD to reduce the overshoot and settling time.
4. Use KI to eliminate the steady-state error.
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 Ziegler and Nichols conducted numerous experiments and
proposed rules for determining values of KP, KI and KD
based on the transient step response of a plant. They
proposed more than one methods, but we will limit
ourselves to what’s known as the first method of ZieglerNichols here. It applies to plants with neither integrators nor
dominant complex-conjugate poles, whose unit-step
response resemble an S-shaped curve with no overshoot. This
S-shaped curve is called the reaction curve.
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PID controller is the controller that has the three basic control modes of
Proportional(P), Integral(I) , and Derivative(D).
– PID controllers are still used widely in industry because of their
simplicity, robustness, and successful practical applications.
– In spite of the development of many advanced control algorithms, nearly
80% of the controllers in the industrial field are PID controller.
Flow control system.
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Schematic diagram of a feedback controller.
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u P (t )  kc ( ys (t )  y(t ))
Proportional(P) part:
Integral(I) part:
u I (t ) 
Derivative(D) part:
kc
 ( y (t*)  y(t*)) dt *
(8.2)
d ( ys (t )  y(t ))
dt
(8.3)
t
I
(8.1)
s
0
uD (t )  kc D
Where y s (t )and y (tdenote
the set point(the desired process output) and
)
process out put. Constants
kcare
, I ,called
 D proportional gain, integral
time and derivative time, respectively.
•
PID controller is sum of the above three part as follows.
u PID (t )  u P (t )  u I (t )  u D (t )
 kc ( ys (t )  y(t )) 
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kc
 ( y (t*)  y(t*)) dt *
t
I 0
d ( ys (t )  y(t ))
 kc D
Dr. Sanjay Chikalthankar
dt
s
(8.7/18/2015
4)
Steady-state error.
For usual process(i.e., open-loop stable processes), the control output
should be nonzero to keep the process output in a nonzero set point.
• Consider the following PD controller( the following derivation is
applicable to P controller case).
d ( ys (t )  y(t ))
uPD (t )  kc ( ys (t )  y(t ))  kc D
(8.6)
dt
PD(or P) controller output uPD (tis)always zero at steady-state if the error
is zero(i.e.,
).y (t )  y s (t )
 PD(or P) controller cannot be nonzero constant when the error is zero
at steady-state. So, the PD(or P) controller cannot keep the process output
in a nonzero set point for open-loop stable processes.
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• Offset can be calculated as follows. Here denotes
ss steady state and is
the static
k gain(or DC gain) of the process.
yss (t )  k  uss (t )  k  kc ( ys  yss (t ))
(8.7)
ys
: y s  y ss (t ) 
(8.8)
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1  k  kc
Integral part(= reset or floating control part) : Since the
integral part is not necessarily zero even though the error at steadystate is zero, it plays an important role in rejecting the offset.
uPID, ss (t ) 
kc
I
 ( y (t*)  y(t*)) dt*  nonzero constant
t
0
1. Transfer function.
s
U I (s) kc 1
 
E (s)  I s
(8.9)
(8.10)
2. Disadvantages
 Not immediate corrective action.
 Practically PI controller is used.
 Oscillatory response.
 Reduce the stability of the system.
Solution ; proper tuning of the controller or including
derivative control action which tends to counteract the destabilizing effects.
 Reset windup( or integral windup).
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Reset windup( or Integral windup).
• Sustained error  Large integral term  Saturation of controller output
 Further buildup of the integral term while the controller is saturated is
referred to as reset windup or integral windup.
• Reset windup occurs when a PI or PID controller encounters a sustained
error, for example, during the start-up of a batch process or after a large setpoint change.
Figure 8.5. Reset windup during a set-point change.
t  t1
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•Derivative part(= rate action,pre-act or anticipatory control part) : Since
this
part represents approximately the increment of the error after time from
the present time , it plays a role in rejecting
the future error in advance
by
d
t
increasing the control output proportional to the future incremental error.
e(t)
 d de(t ) / dt
d
e(t)
present (t) future t(   d )
1. Transfer function.
U D ( s)
 kc D  s
(8.11)
E (s)
2. Advantage : This part enhance the
robustness of the PID controller
by considering abrupt change of the
error.
Figure 8.6. Extrapolation using the
derivative of the error
3. Disadvantage : If the process measurement is noisy, this term will change widely and amplify the noise
unless the measurement is filtered.
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8.3.2.3 Ideal PID Controller.
1. Transfer function.


U PID ( s)
1
 kc 1 
  D s
E ( s)
 Is

(8.12)
Electronic or pneumatic device that provides ideal derivative
action cannot be built(is physically unrealizable). Commercial controllers
approximate the ideal behavior as follows.
 I s  1    D s  1 
U PID ( s)
 kc 
(8.13)


E ( s)
  I s   D s  1
where  is a small number, typically between 0.05 and 0.2.
2. Disadvantages.
 Derivative kick
 Proportional kick
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Example of PID Control
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PID Control Dynamics
Assume inlet process temperature
[Ti(t)] decreases; this leads to a
decrease in outlet temperature, T(t)
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PID Control Dynamics
At ta,
•the amount of error is positive and small -> PI
control offers small correction
•But, the derivative (slope) of e(t) is large and
positive
•Indicates process is heading in the wrong
direction
•Derivative action kicks in and does most
of the control
At tb,
•the amount of error is positive and large -> PI
control offers more correction
•But, the derivative (slope) of e(t) negative
•Indicates process is heading in the “right”
direction (error decreasing)
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•Derivative action subtracts from
PI mode
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PID Control- Advantages
 Reduces overshoot
 Decreases oscillations around the set point
 Commonly used in slow processes (such as temperature loops)
 Not good for fast and/or “noisy” processes (overly responsive)
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PID Controls-Advantages
 Most common controller in the CPI.
 Came into use in 1930’s with the introduction of pneumatic
controllers.
 Extremely flexible and powerful control algorithm when
applied properly.
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On-Off Controllers
–The controller output of ideal on-off controller.
u max
uon off (t )  
u min
where u maxand u mindenote the on and off values, respectively.
On-off controller can be considered to be a special case of P
controller with a very high controller gain
– Advantage : Simple and inexpensive controllers.
– Disadvantage
 Not versatile and ineffective.
 Continuous cycling of the controlled variable and excess
wear on the final control element.
– Usage :Thermostats in heating system.
Domestic refrigerator.
Noncritical industrial applications.
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Typical Response of Feedback Control Systems
Figure




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Typical process response with feedback control.
C is the deviation from the initial steady-state.
No feedback control make the process slowly reach a new steady-state.
Proportional control speeds up the process response and reduces the
offset.
Integral control eliminates offset but tends to make the response
oscillatory.
Derivative control reduces both the degree of oscillation and response
time.
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Effect of controller gain
Figure
.
kc
. Process response with proportional control.
 Increasing the controller gain.
 less sluggish process response.
 Too large controller gain.
 undesirable degree of oscillation or even unstable response.
 An intermediate value of the controller gain
 best control result.
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Effect of integral time
Figure
.
I
. PI control: (a) effect of integral time (b) effect of controller gain.
 Increasing the integral time.
 more conservative(sluggish) process response.
 Too large integral time.
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 too long time to reach to the set point after load upset or set-point change
occurs.
 Theoretically, offset will be eliminated for all values of
.
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
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I
Effect of derivative time
Figure
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D
. PID control: effect of derivative time.
• Increasing the derivative time.
 improved response by reducing the maximum deviation, response
time and the degree of oscillation.
• Too large derivative time.
 measurement noise tends to be amplified and the response may be
oscillatory.
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• Intermediate value of
isdesirable.
D
Tuning a control loop is the adjustment of its control
parameters (gain/proportional band, integral gain/reset,
derivative gain/rate) to the optimum values for the desired
control response. Stability (bounded oscillation) is a basic
requirement, but beyond that, different systems have
different behavior, different applications have different
requirements, and requirements may conflict with one
another.
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Integral Control of Systems.
Integral control of the system eliminates the
steady-state error in the response to the step
input..
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a) Magnitude and phase response (b) Nyquist plot of the
frequency response of a PD element
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The form
of the PD controller most often encountered in industry, and the one
most relevant to tuning algorithms is the standard form. In this form
the Kp gain is applied to the Iout, and Dout terms, yielding:
where
Ti is the integral time
Td is the derivative time
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In this standard form, the parameters have a clear physical meaning. In
particular, the inner summation produces a new single error value which is
compensated for future and past errors.
The addition of the proportional and derivative components effectively predicts
the error value at Td seconds (or samples) in the future, assuming that the loop
control remains unchanged. The integral component adjusts the error value to
compensate for the sum of all past errors, with the intention of completely
eliminating them in Ti seconds (or samples). The resulting compensated single error
value is scaled by the single gain Kp.
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Properties of Proportional, Integral,
Derivative Control
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Properties of Proportional Action
c(t )  c0  K c e(t )
Gc ( s )  K c
 Closed loop transfer function
Kc K p
Y (s)

Ysp ( s )
Kc K p  1
p
Kc K p  1
s 1
base on a P-only controller
applied to a first order process.
 Properties of P control
◦ Does not change order of process
◦ Closed loop time constant is
smaller than open loop p
◦ Does not eliminate offset.
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Offset Resulting from P-only Control
Offset
Setpoint
1.0
3
2
0
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1
Time
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Proportional Action for the Response of
a PI Controller
ysp
ys
cprop
Time
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Proportional Action
 The primary benefit of proportional action is that it speedup
the response of the process.
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Properties of Integral Action
c(t )  c0 
Y (s)

Ysp ( s )
 I p
Kc K p
 p 
Kc
I

t
0
e(t ) dt
 Based on applying an I-only
1
s2 
I
Kc K p
s  1
 I p
Kc K p
1
I
 
2  p Kc K p
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controller to a first order
process
 Properties of I control
◦ Offset is eliminated
◦ Increases the order by 1
◦ As integral action is increased,
the process becomes faster, but
at the expense of more
sustained oscillations
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Integral Action for the Response of a PI
Controller
ysp
ys
cint
Time
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Integral Action
 The primary benefit of integral action is that it removes
offset from setpoint.
 In addition, for a PI controller all the steady-state change in
the controller output results from integral action.
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Properties of Derivative Action
de(t )
c(t )  c0  K c D
dt
K c K p D s
Y (s)
 2 2
Ysp ( s)  p s  2 p  K c K p D s  1
 Closed loop transfer function for derivative-only control
applied to a second order process.
 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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Derivative Action for the
Response of a PID Controller
ysp
ys
cder
Time
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Derivative Action
 The primary benefit of derivative action is that it reduces the
oscillatory nature of the closed-loop response.
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Response to Torque Disturbances
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Proportional Control
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Proportional-Plus-Integral Control
 To eliminate offset due to torque disturbance, the
proportional controller may be replaced by a proportionalplus-integral controller.
 If integral control action is added to the controller, then, as
long as there is an error signal, a torque is developed by the
controller to reduce this error, provided the control system is
a stable one.
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It is important to point out that if the controller was an integral controller,
then the system always becomes unstable because the characteristic
equation
will have roots with positive real parts. Such an unstable system cannot be
used in practice.
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Proportional-Plus-Derivative Control
. Thus derivative control introduces a damping effect. A typical
response curve c ( t ) to a unit step input is shown in
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Sometimes it is useful to write the PD regulator in laplace
transform:
Having the PID controller written in Laplace form
and having the transfer function of the controlled system
makes it easy to determine the closed-loop transfer
function of the system.
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Dr. Sanjay Chikalthankar
7/18/2015
.
The gain parameters are related to the parameters of the standard
form through
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Dr. Sanjay Chikalthankar
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 Effect of Proportional,
Integral & Derivative Gains on the
Dynamic Response
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Dr. Sanjay Chikalthankar
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Change in gain in P controller
• Increase in gain:
 Upgrade both steadystate and transient
responses
 Reduce steady-state
error
 Reduce stability!
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Dr. Sanjay Chikalthankar
7/18/2015
P Controller with high gain
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Dr. Sanjay Chikalthankar
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Integral Controller
 Integral of error with a constant gain
 increase the system type by 1
eliminate steady-state error for
a unit step input
 amplify overshoot and oscillations
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Dr. Sanjay Chikalthankar
7/18/2015
Change in gain for PI controller
• Increase in gain:
 Do not upgrade steadystate responses
 Increase slightly
settling time
 Increase oscillations
and overshoot!
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Dr. Sanjay Chikalthankar
7/18/2015
Derivative Controller
 Differentiation of error with a constant gain
 detect rapid change in output
 reduce overshoot and oscillation
 do not affect the steady-state response
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Dr. Sanjay Chikalthankar
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Effect of change for gain PD controller
• Increase in gain:
 Upgrade transient
response
 Decrease the peak and
rise time
 Increase overshoot
and settling time!
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Dr. Sanjay Chikalthankar
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Changes in gains for PID Controller
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Dr. Sanjay Chikalthankar
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 These rules are used to determine Kp, Ti and Td for PID
controllers
 First Method: The response is obtained experimentally to a
unit step input. The plant involves neither integrators nor
differentiators
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Second Method
 Set Ti= inf and Td=0, increase Kp from 0 t a critical value
Kcr where the output exhibits sustained oscillations.
 Use Kcr , Pcr and Table 10-2 to determine the parameters of
the controller
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Proportional-Integral Control
 Proportional (P-only) control algorithm leaves permanent,
steady-state error (offset; shown last time)
 Some processes require precise control at the prescribed set
point
 Need additional intelligence/functionality for the controller
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Proportional-Integral Control
 Add integral (or reset) action to the P-controller
Kc
 I = integral (or reset)
dtseconds)
mt  time
m (usually
K c et in minutes
etor

 I controllers (+/-)
 Same convention for reverse/direct acting
 Two parameters to tune (Kc and I)
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For 1% increase in c(t) [e(t) = r – c; e(t) = -1%]
mt   m  K c et  
I
mt   50%  K c  1 
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Dr. Sanjay Chikalthankar
 et dt
Kc
Kc
I
mt   50%  K c 
 1dt
Kc
I
t
7/18/2015
mt   50%  K c 
t  50%  K c
I
controller response governed by P-control only K
c


m
t

50
%

K

t
c
 As t increases (t ≠ 0),
I
controller response governed by PI-control
 At t = 0
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Dr. Sanjay Chikalthankar
Kc
7/18/2015
Proportional-Integral Control
 At t = I
mt   50%  K c   I  50%  K c  K c
 I taken by proportional mode
integral mode repeats action
 As I decreases, controller integrates faster (integral action
increases)
 Want to emphasize I-action more? => Small I
 Want to emphasize P-action more? => Large I
Kc
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Dr. Sanjay Chikalthankar
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PI Control – Equation
mt   m  K c et  
Kc
I
 et dt
 As long as error present, controller integrates error and changes




output
Once error = 0, controller output doesn’t change anymore
At tf, Error = 0 does NOT mean
At tf, Error = 0 means
et dt  0

The constant removes the offset
 et dt  constant
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Example of PI Control
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r
c
m
Assumptions:
• qi = qo = 200 gpm (steady-state values)
• h = 8 ft (steady-state value)
• Valve is linear
• Closed => no flow => 0 gpm
• Wide open => maximum flow => 500 gpm
Dr. Sanjay Chikalthankar
95
• Tank height (equipment, not liquid level) = 127/18/2015
ft
PI Control: Tank Level
 Set point,
8 ft
r
100 66.7%
12 ft
 Bias,
200 gpm
m
100  40%
500 gpm
 For 
direct acting, PI-controller

Kc
Kc
m  m  Kce 
 et dt  m  Kc r  c 
I
m  40%  K c 66.7%  c  
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Dr. Sanjay Chikalthankar
I
Kc
I
 r  cdt
 66.7%  cdt
7/18/2015
PI Control: Tank Level
 Suppose Kc = 2
 Suppose I = 0.1
 Suppose qi increases to 300 gpm
 If no control => tank overflows
 If flow control compensates, flow out (qo) becomes 300 gpm and
m (valve position) must change
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Dr. Sanjay Chikalthankar
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PI Control: Tank Level
 Now
300 gpm
New steady state valve
m
100  60%
position
500 gpm
 Once at new steady-state (from before; still applies after tuning)
At SS, r = c, e = 0
Kc
m  40%  K c 66.7%  c  

I
60%  40%  10 
98
Kc
I
 66.7%  cdt
 0dt  40%  
Kc
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Dr. Sanjay Chikalthankar
Integral
is (-1) because of direction of error (c > r) at last time interval
7/18/2015
approaching steady state
PI Control: Tank Level




99
2
60%  40% 
 40%  20%  60% !
0.1
Controlled variable (c) and height (h) remain unchanged!
No steady-state error, or offset – removed by the integral action
85% of all controllers are PI
Need to “tune” two parameters (Kc and I)
Dr. Sanjay Chikalthankar
7/18/2015
PI Control Dynamics
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Dr. Sanjay Chikalthankar
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Reset Rate
 Some controllers use the term “Reset Rate”
parameter (instead of reset time, I)
1
R
I 
( ) as a tuning
 IR
I
 Control algorithm with Proportional Band and Reset Rate
100
100 IR
mt   m 
et  
et dt

PB
PB
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Dr. Sanjay Chikalthankar
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In most commercial control systems, derivative action is
based on PV rather than error. This is because the digitised version
of the algorithm produces a large unwanted spike when the SP is
changed. If the SP is constant then changes in PV will be the same
as changes in error.
Therefore this modification makes no difference to the way the
controller responds to process disturbances.
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Dr. Sanjay Chikalthankar
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Most commercial control systems offer the option of also basing the
proportional action on PV. This means that only the integral action responds to
changes in SP.While at first this might seem to adversely affect the time that the
process will take to respond to the change, the controller
may be retuned to give almost the same response - largely
By increasing Kp.
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Dr. Sanjay Chikalthankar
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The modification to the algorithm does not affect the way the controller
responds to process disturbances, but the change in tuning has a beneficial effect.
Often the magnitude and duration of the disturbance will be more than halved.
Since most controllers have to deal frequently with process disturbances and
relatively rarely with SP changes, properly tuned the Modified algorithm can
dramatically improve process performance
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Dr. Sanjay Chikalthankar
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Having the PID controller written in Laplace form and
having the transfer action of the controlled system makes it easy
to determine the closed-loop transfer function of the system.
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Dr. Sanjay Chikalthankar
7/18/2015
 SCHAUM’S OUTLINE OF FEEDBACK and
CONTROL SYSTEMS, 2E
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THANK YOU
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