ANALOG FABRICATION OF PID CONTROLLER

ANALOG FABRICATION OF PID CONTROLLER
1
ANALOG FABRICATION OF PID
CONTROLLER
A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE
REQUIREMENTS FOR THE DEGREE OF
Bachelor of Technology in Electrical Engineering
By
Tapan Kumar Swain
Vaibhav Baid
Under supervision of
Prof. Sandip Ghosh
Department of Electrical Engineering
National Institute of Technology, Rourkela
May 2014
2
CERTIFICATE
This is to certify that the project entitled, “Analog fabrication of PID Controller”
submitted by Tapan Kumar Swain and Vaibhav Baid is an authentic work carried out
by them under my supervision and guidance for the partial fulfillment of the requirements
for the award of End semester thesis Submission in Electrical Engineering at National
Institute of Technology, Rourkela (Deemed University).
Prof. Sandip Ghosh
Rourkela Dept. of Electrical Engineering,
National Institute of Technology,
Rourkela,
Orissa-769008
3
ABSTRACT
The PID controller has been used and dominated the process control industries for
a long time as it provides the control action in terms of compensation based on
present error input(proportional control), on past error(integral control) and on
future error if recorded by earlier experience or some means(derivative control).
The PID controllers have excellent property of making the system response faster
and at the same time reduce the steady state error to zero or at least to a very small
tolerance limit. The work below starts with study of individual components of the
controller and their responses in a certain environment for different test signals
(say a step or sine wave input).The problem is to design a PID controller using
appropriate analog circuit as well as understand and utilize the advantages of all
the three terms. The below work is for the study of an analog PID controller using
operational amplifiers and fabricate the controller on hardware after testing the
individual terms:-proportional, integral and derivative.
4
ACKNOWLEDGEMENT
We express our gratitude and sincere thanks to our supervisor Prof. Sandip Ghosh,
Professor Department of Electrical Engineering for his constant motivation and
support during the course of our thesis. We truly appreciate and value his esteemed
guidance and encouragement from the beginning of our thesis. We are indebted to
him for having helped me shape the problem and providing insights towards the
solution.
We, Tapan Kumar Swain and Vaibhav Baid, are thankful to each other for cooperating and encouraging and being alongside for the literature review and
background study, the hardware implementation, experiments and the whole thesis
in the same project work.
We extend our gratitude to the researchers and scholars whose hours of toil have
produced the papers and thesis that we have utilized in our project.
110EE0233
110EE0583
5
LIST OF TABLES
 TABLE 1:- Transfer function of various controllers using Op Amps
17
 TABLE 2:-Effects of increasing a parameter independently
21
 TABLE 3:-Datasheet of LM741
23
 TABLE 4:-Components used in fabricating PID controller
31
LIST OF FIGURES
 FIGURE 01:-Block diagram of a PID controller
14
 FIGURE 02:-An inverter circuit
16
 FIGURE 03:-Circuit diagram of a PID controller
18
 FIGURE 04:- Pin configuration of 741 opamp
22
 FIGURE 05:- Circuit symbol of an opamp
23
 FIGURE 06:- signal buffer circuit
24
 FIGURE 07:- signal inverter circuit
25
 FIGURE 08:- signal summer circuit
26
 FIGURE 09:- signal differentiator circuit
26
 FIGURE 10:- proportional controller
27
 FIGURE 11:- derivative controller
28

FIGURE 12:-integral controller
29

FIGURE 13:- complete pid controller circuit
30
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TABLE OF CONTENTS
 CERTIFICATE
02
 ABSTRACT
03
 ACKNOWLEDGEMENT
04
 LIST OF TABLES
05
 LIST OF FIGURES
05
 TABLE OF CONTENTS
06
1. INTRODUCTION
07
2. BACKGROUND AND LITERATURE REVIEW
12


PID CONTROL
TRANSFER FUNCTION REPRESENTATION
3. ANALOG PID IMPLEMENTATION



EFFECT OF GAIN PARAMETERS ON PERFORMANCE
THE 741 OPAMP
OPAMP REALISATIONS
13
15
16
18
21
24
4. CHOICE OF CIRCUIT PARAMETERS
27
5. TEST RESULTS
32
6. CONCLUSION
33
REFERENCES
33
7
INTRODUCTION
Looking back to the history of the PID controller, the PID controllers in the initial
days were all pneumatic. In fact, the experimentations were carried out all with
pneumatic controllers by Ziegler and Nichols. But the nature of pneumatic
controllers was slow. The electronic controllers started replacing the conventional
pneumatic controllers after the development of electronic devices and operational
amplifiers. However, with the advent of the microprocessors and microcontrollers,
the implementation with digital PID controllers has now become the main focus of
development. The topmost benefit of using digital PID controllers is that the
controllers’ parameters can be programmed effortlessly; consequently, without
changing any hardware, they can be changed. Furthermore, besides generating the
control action, the same digital computer can be used for a number of other
applications.
But here we are concerned with the analog PID controller design, and how they
can be implemented in actual practice. The design of automatic control systems is
perhaps the most important function that the control engineer carries out. We may
analyze and find out methods to do the design in certain cases while mostly we do
design based on trial and error basis. This requires that we should put some
restrictions and constraints along with pre-specified performance conditions in
order to get a better quality control in terms of performance. So, design requires
various factors to be taken care of.
8
Every control system designed for a specification or specific application has to
meet certain performance specifications. Some methods specifying the
performance of a control system are:1. By set of specifications in time domain and/or in frequency domain such as
peak overshoot, settling time, gain margin, phase margin, steady-state error
etc.
2. By optimality of a certain function, e.g., an integral function.
In addition to performance specification, some other constraints are also always
imposed on the control system design. Say for example, the tracking antenna
control system where an actuator is designed for movement of antenna. Depending
on required performance, power supply available, space and economic limitations
etc., it could be servomotor (ac or dc) or hydraulic motor. Size is determined by
inertia, velocity and acceleration ranges of antenna. Gear trains for higher speeds
may be required.
From this discussion, it is evident that the choice of plant components is dictated
not only by performance but also size, weight, available power supply, cost etc.
Therefore, the plant generally cannot meet the performance specifications. Though
the designer is free to choose alternative components, this is generally not done
because of cost, availability and other constraints.
However, some components of a plant, its replacement are not a big problem
because of low- cost and wide- range of availability of such amplifiers. Merely by
gain adjustments, it may be possible to meet the given specifications on
performance of simple control systems. In such cases, gain adjustment seems to be
9
the most direct and simple way of design. However, in most practical cases, the
gain adjustment does not provide the desired result. As it is usually the case,
increasing the gain reduces the steady-state error but results in oscillatory transient
response or even instability. Under such circumstances, it is necessary to introduce
some kind of corrective subsystems to force the chosen plant to meet the
specifications. These subsystems are known as controllers/compensators and their
job is to compensate for the deficiency in the performance of the plant.
There are basically two approaches to control system design problem:1. We select the configuration of the overall system by introducing controller
and then choose the performance parameters of the controller to meet the
given specifications on performance.
2. For a given plant, we find overall system that meets the given specification and
then compute the necessary controller.
The first approach will be used below in the work.
So, we find that plant components are determined considering various factors and
plant cannot meet these specifications. For this gain adjustment seems suitable, as
replacing by alternative components may be costly or impractical. This is because
the steady state error transfer function is inversely proportional to open loop gain
and is given by:-
=
Where
G(s) = open loop transfer function or gain
(1)
10
However gain adjustment using such Proportional gain (P) leads to oscillatory
transient response and may lead to instability, although it reduces steady state error
to some extent.
So, we use a PID controller which can have the advantage of making the system
response faster, reduce the steady state error to zero or within a desirable tolerance
limit. The use of PID controller, however, is avoided in some process industries
now-a-days and they prefer PI controller because the derivate control poses some
problems. Here we study each of the control parameters viz., proportional,
integral, derivative individually or with combination as PD or PI and then we can
fabricate a PID controller on hardware for an arbitrary plant using appropriate
tuning techniques and meanwhile understand the advantages that can be more
prominent and utilized for a particular specification. The fundamental difficulty
with PID control is that it is a feedback system, with constant parameters, and no
direct knowledge of the process, and thus overall performance is reactive and a
compromise. While PID control is the best controller in an observer without a
model of the process, better performance can be obtained by overtly modeling the
actor of the process without resorting to an observer.
PID controllers, when used alone, can give poor performance when the PID loop
gains must be reduced so that the control system does not overshoot, oscillate or
hunt about the control set point value. They also have difficulties in the presence
of non-linearities, may trade-off regulation versus response time, do not react to
changing process behavior, and have lag in responding to large disturbances.
11
The most significant improvement is to incorporate feed-forward control with
knowledge about the system, and using the PID only to control error.
Alternatively, PIDs can be modified in more minor ways, such as by changing the
parameters, improving measurement, or cascading multiple PID controllers.
Another problem faced with PID controllers is that they are linear, and in
particular symmetric. Thus, performance of PID controllers in non-linear systems
is variable. For example, in temperature control, a common use case is active
heating but passive cooling, so overshoot can only be corrected slowly – it cannot
be forced downward. In this case the PID should be tuned to be over damped, to
prevent or reduce overshoot, though this reduces performance (it increases settling
time).
A problem with the derivative term is that it amplifies higher frequency
measurement or process noise that can cause large amounts of change in the
output. It does this so much, that a physical controller cannot have a true derivative
term, but only an approximation with limited bandwidth. It is often helpful to filter
the measurements with a low-pass-filter in order to remove higher-frequency noise
components. As low-pass filtering and derivative control can cancel each other
out, the amount of filtering is limited. So low noise instrumentation can be
important. A nonlinear median filter may be used, which improves the filtering
efficiency and practical performance. In some cases, the differential band can be
turned off with little loss of control. This is equivalent to using the PID controller
as a PI controller.
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BACKGROUND AND LITERATURE REVIEW
A proportional-integral-derivative controller (PID controller) is one of the
widely used controllers in industries for controlling feedback systems. PID
controller calculates an error value also called actuating signal which is the
difference between a measured process variable or the output value and a desired
value or set point input. The error is controlled or reduced by manipulating
/adjusting the inputs that the PID controller receives and thus it produces a
command signal to the plant for error correction.
The error correction is done for a control system in 3 ways basically viz.,
proportional, integral and derivative. A controller can use either of these term or
their combinations, however, integral and derivative control are achieved along
with proportional control. PID controller is the one which has all the three terms in
it. So, three separate constant parameters are calculated and hence it is also called
a three-term control. In time domain this may be interpreted as: P depends on
the present error, I on the accumulation of past errors, and D is a prediction
of future errors, based on present rate of change. The weighted sum of these three
actions is used to adjust the process via a control element such as the position of
a control valve, a damper, or the power supplied to a heating element.
In the absence of knowledge of the underlying process, a PID controller has
historically been considered to be the best controller. By tuning the three
parameters in the PID controller, the controller can provide control action desired
for specific process requirements. The response of the controller can be described
13
in terms of the responsiveness of the controller to an error, the degree to which the
controller overshoots the set point, and the degree of system oscillation. However,
we should understand that the use of the PID algorithm for control does not
guarantee optimal control of the system or system stability.
Some applications may require using only one or two actions to provide the
appropriate system control. This is achieved by setting the other parameters to
zero. A PID controller will be called a PI, PD, P or I controller in the absence of
the respective control actions. PI controllers are fairly common, since derivative
action is sensitive to measurement noise, whereas the absence of an integral term
may prevent the system from reaching its target value due to the control action.
PID CONTROL
The PID control scheme is named after its three correcting terms, whose sum
constitutes the manipulated variable (MV). The proportional, integral, and
derivative terms are summed to calculate the output of the PID controller.
Defining U(t) as the controller output, the final form of the PID algorithm is given
by:
U(t) = MV(t) =
∫
Where
: Proportional gain, a tuning parameter
: Integral gain, a tuning parameter
: Derivative gain, a tuning parameter
(2)
14
: Error, (Set point- output value)
: Time or instantaneous time (the present)
: Variable of integration; takes on values from time 0 to the present
The general block diagram for a PID controller is shown below in fig1
r(t)
y(t) _ +
Plant/process
U(t)
P
+
∑
+
I
∫
+
D
Fig 1- block diagram of a PID controller
e(t)
∑
15
The above block diagram and equation shows the PID controller behavior in time
domain form. The time domain analysis is used for real-time results and to
determine various gain parameters like rise time, peak overshoot, steady-state error
etc. However, there is another form of representation that helps in determining the
performance parameters like stability, gain and phase margins etc. This form is
given below.
TRANSFER FUNCTION REPRESENTATION
Sometimes it is useful to write the PID control equation in Laplace transform form
which is given by:
G(s) =
=
(3)
are the proportional, derivative and integral gain
respectively. This transfer function can be realised using various RLC circuits,
opamp circuits etc. This function is in frequency domain thus, being used for
frequency domain analysis. As we can see from the transfer function, it has
one pole at s=0 i.e. origin and two zeros. The addition of a pole to the system
and that too on the imaginary axis makes the system sluggish. This form is
helpful in designing a controller based on stability criteria where we may be
having bode plot of the system, or its root locus diagram, by using various
tuning techniques.
16
ANALOG PID IMPLEMENTATION
For implementing the PID controller we can use both digital and/or analog
circuits. Digital PID is implemented using integrated circuits while we can use
various circuits using operational amplifiers in case of analog design of which one
is shown in fig 3.
From fig 3, we see that it is basically three different inverter circuits with different
values of impedances
and
. The inverter circuit is shown in fig 2.
𝑍𝑓 𝑠
𝑍𝑖 𝑠
Fig 2- an inverter circuit
The above inverter circuit has a closed loop gain given by:-
G(s) = -
(4)
17
For different values of
and
, we can get various control actions and
thus implement different types of controllers as shown in Table[1].
Table 1:-
controller
Transfer
G(s)
P
-
PI
-[
PD
-[
Function
]
]
PID
Transfer function of various controllers using Op Amps
So, the transfer functions using Op Amp for PID controller can be as in Table [1]
is
G(s) = -
(5)
Or,
The transfer function can take following shape as per the diagram
Shown in fig3 as follows:
[
]
(6)
18
This circuit contains a summer circuit that sums up command signal generated by
each of the control terms and finally an inverter is used for getting positive value
of transfer function.
Proportional
Integral
Error
Summer
inverter
Derivative
Fig 3-Circuit diagram of a PID controller
EFFECTS OF GAIN PARAMETERS ON PERFORMANCE
Let us consider a second order system. The overall transfer function for a closed
loop second order system can be written in standard form as:
19
(7)
The study of second order systems is important because it is simpler and higher
order systems can be approximated to a fair extent by second order systems and
thus, one can get fair idea about the dynamics of the system and steady state error.
The dynamics refers to the response of a system response to an abnormal condition
such as lightning, sudden rise of voltage, constantly increasing input etc., and such
systems are studied using test signals like impulse, step, ramp etc.
The dynamics can be analyzed by knowing the damping ( ) and undamped natural
frequency (
overshoot (
).This can give known from the system response viz., peak
), rise time (
), settling time (
), steady-state error (
).
For a step input, these values are given by following equations:-
1. Rise time (
) is the time required by response to rise from 10% to 90% of
final value for overdamped system and 0 to 100% for underdamped system.
√
√
2. Peak overshoot
(8)
), is normalized difference between peak of response
and steady state output normalized w.r.t. to steady output.
20
√
3. Settling time (
(9)
) is the time required for the response to reach and stay
within specified limit of its final value called tolerance band (2-5%).This
value is for 5% band,
(10)
4. Steady state error
) is the error between the actual output and desired
output as tends to infinity.
(11)
By introduction of PID controller we can control these above system dynamics
using tuning methods and thus determine various parameters. The effects of these
parameters on system response are shown in table below.
We can see that with increase in the value of
, we get better steady state
stability as it reduces the steady-state error. The integral control can nullify the
steady state error but cost paid is that it makes the system sluggish. While above
two lead to oscillatory response initially, the derivative control makes the
overshoot within limit and also improves the settling time.
21
Table 2:-
Parameter Rise
Overshoot Settling
time
Decrease
Increase
Steady-
Stability
time
state error
Small
Decrease
Degrade
Degrade
change
Decrease
Increase
Increase
Eliminate
Minor
Decrease
Decrease
No effect in Improve if
change
theory
is small
Effects of increasing control parameter independently
Table [2] shows how change in various gain parameters affects the response of the
system both transient and steady state.
THE 741 OPAMP
The OPAMP stands for operational amplifier. The opAmp is an amplifier with
some specific important characteristics. As the word amplifier suggests, the
function of an operational amplifier (op amp) is to amplify a voltage. However, the
operational amplifier does much more than that. It also functions as a buffer and as
a cascade which are two functions that enable simple circuits to be assembled into
complex circuits to create higher level functions which are called operations¾
hence the name operational amplifier.
22
Op amps have five terminals that are important. The voltage that is amplified is the
difference between the voltage at the ‘+’ terminal
terminal
and the voltage at the ‘-’
, as shown in figure below. The amplified voltage is the output voltage
. Unlike the resistor and capacitor, which are both “passive” (unpowered)
devices, the opamp is an “active” device. Indeed, the op amp needs a voltage
supply for the amplification. The
and the
terminals are the positive and
negative supply voltages, respectively. The op amp schematic and the chip that
we’ll use are shown in figure below. Generally it is available in integrated chips.
The pin configuration of 741 IC is as shown below.
1
8
Offset
unused
2
7
3
6
offset
4
5
Fig 4- Pin configuration of 741 opamp
23
Fig 5- Circuit symbol of an opamp
DATASHEET FOR LM741
Absolute maximum Ratings
TABLE 3:-
Supply Voltage
Power Dissipation
Differential
Voltage
LM741A
LM741
LM741C
±22V
±22V
±18V
500 mW
500 mW
500 mW
±30V
±30V
±15V
±15V
Continuous
Continuous
−55°C to +125°C
−55°C to +125°C
Input ±30V
Input Voltage
±15V
Output Short Circuit Continuous
Duration
Operating
−55°C to +125°C
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Temperature Range
Storage
Temperature Range
−65°C to +150°C
−65°C to +150°C
−65°C to +150°C
Datasheet for 741 IC
“Absolute Maximum Ratings “indicate the limits beyond which damage to the
device may occur. Operating Ratings indicate the conditions for which the device
is functional, but do not ensure specific performance limits. For operation at
elevated temperatures, these devices must be derated based on thermal resistance.
For supply voltages less than ±15V, the absolute maximum input voltage is equal
to supply voltage.
OPAMP REALISATIONS
SIGNAL BUFFER:-
It is a circuit configuration in which input equals output. The importance
of this circuit is that it isolates the input and output side. Since, the input current of
opAmp is 0, loading effect is 0. So, we can measure the actual input without error
due to loading. Its circuit diagram is shown below.
Fig 6- signal buffer circuit
25
SIGNAL INVERTER:-
This circuit changes the polarity of the input signal with amplification and
the gain value is,
(12)
Fig7- signal inverter circuit
SIGNAL ADDER/SUMMER:-
This circuit helps in summing up various signals .Here, output voltage is given by
(13)
26
R
R
R
R
Fig 8- signal summer circuit
SIGNAL SUBTRACTOR/DIFFERENTIATOR:-
This circuit gives the difference of the two inputs given to the opamp circuit,
provided all the resistances should have same value as shown in the circuit below.
Here output signal is given as
(14)
R
R
R
R
Fig 9- signal differentiator circuit
27
CHOICE OF CIRCUIT PARAMETERS
We need to initially determine the values of
,
and
for a certain PID
controller. Since our plant is unknown we assume our plant to be anything
arbitrary and thus our controller should be tunable one.
We need a PID controller for 0<=
<=100, 0<=
<=10 and an arbitrary
as per
requirement.
Firstly, we need to test each components of the controller viz., proportional,
integral and derivative terms separately and then integrate them together. So, we
assemble the components for the proportional controller. As 0<=
chose our
pot and
<=100, we
ohms. We chose a 741 opAmp for this
purpose. Initially, we set up the board as shown in circuit diagram below.
100k pot
Sine input
1kohm
Output
Fig 10- proportional controller
28
Then, we supplied a sinusoidal voltage wave from a function generator as input to
the controller circuit. The input and output waveforms were viewed in a CRO. The
results were noted and waveforms were traced in tracing paper. The experiment
was repeated by varying the values of
using potentiometer. Results were viewed
and traced.
Now, we needed to do the same test with the derivative controller. Here, we
needed to supply a ramp input and check the output. Since, ramp signal cannot be
generated due to saturation, so, we used a triangular wave input to the controller.
As we required 0<=
<=10, we use a 10 micro Farad capacitor, a 1K resistor and
a 1M pot for the purpose, as shown in below circuit diagram. Waveforms were
viewed in CRO and traced in tracing paper.
1M pot
Triangular input
1kohm
10µF
Output
Fig 11- derivative controller
Next, we repeated the test for integral controller with circuit diagram as shown
below. Components required were 1 micro Farad capacitor and 1M pot and a
29
small, resistance say 1kohm was put in series with the capacitor as shown in the
circuit diagram above.
Input given to the controller was a square wave. Output waveforms were viewed
and traced in a tracing paper. Results were obtained for different values of
by
varying the potentiometer. The same was repeated by replacing the 1 micro Farad
capacitor with a 10 micro Farad capacitor.
1µF/10µF
Square input
1M pot
Output
Fig 12- integral controller
Now, after performing these entire tests we move on to fabricate our PID
controller .The circuit diagram for the design is shown below. The components
used for whole process are shown in table below. The components are assembled
together and the connections were made as per circuit diagram on the bread board.
The supply voltages for the 741 opamps are not shown in the circuit diagram.
Supply voltage of ±15V was given to the IC’s. Then inputs were supplied using
function generator and the required waveforms were traced on the tracing paper.
Finally, after the testing the components were removed from the bread board and
fabrication was started. Thus, fabrication of PID controller was completed.
30
100k
100k pot
100k
Process
1k
100k
100k
output
10µF
Set point
1M pot
100k
100k
100k
1M pot
1k
100k
10µF
Fig 13- complete pid controller circuit
dedicated as input buffer, while one for output buffer. The process variable and set
point variable are given at the input and we get the same values of input at the
output terminals. Next, both the inputs are subtracted using another opamp IC
which uses four equal resistors of 100k each. This generates an error signal at its
output. The output of this is given to each of the individual controllers viz.,
proportional, integral and derivative. The controllers are nothing but three signal
inverters with two resistors in proportional control and one capacitor and one
resistor in both integral and derivative controls with their position exchanged in
each. The output of the three controllers is summed up using a summer circuit and
then passed through a buffer circuit. By using the buffer circuit, we are isolating
31
the whole control circuit from outside loads. The controls of the variables are
achieved using the three potentiometers as shown in figure. As we can see that
proportional term contains a 100 pot, derivative and integral terms contain 100M
pots, which is done in order to achieve required range of values of
and
.
In the derivative control, we find a small resistor of 1k. This is given in order to
save the capacitor from short circuiting because we now that uncharged capacitor
when connected to a voltage source acts like a short circuit initially. So, this
resistor limits the short circuit current.
TABLE 4:-
Sl. No
Components used
Quantity
1
OpAmps (741)
8
2
100k pot
1
3
1M pot
2
4
100k resistors
8
5
1k resistor
2
6
1microFarad capacitor
1
7
10microFarad capacitor
2
8
Soldering kit
-
9
Multimeter
-
10
CRO
-
11
±18v power supply
-
32
12
Connecting wires
As required
13
14
Bread
board
7815
2
15
Fan to fan connecter
As required
16
Berg strip
As required
board/proto -
Components used in fabricating PID controller
TEST RESULTS
The supply voltage given was ±18v through an adapter which was then converted
to ±15V using 7815. Then required test was performed. After performing the test
on proportional controller, it was found that the peak to peak value of sine
waveform got reduced with increase in the value of
, and the waveform
approached a steady dc value with almost no ripples.
The output of the derivative controller was a square wave corresponding to a
triangular input as expected. The variation of
had practically no effect on the
waveform except that there was a slight variation in duty cycle for variation of
from 0 to 10.
The output of integral controller showed both positive and negative peaks when a
1 micro Farad capacitor was used. When the capacitor was replaced by a 10 micro
Farad capacitor the output waveform was similar to input wave with a large rise
and decay time.
33
CONCLUSION
It was found that the proportional controller reduces the transients to an
appreciable extent and thus,
should have high value. The derivative controller
acts on the rate of change of input and thus converts the triangular wave to a
square wave. It is very sensitive to changes or variations in the input.
As far as integral controller was concerned, it had a slow rise and decay time
making system sluggish. The output waveforms were found almost as expected
and thus, the analog pid controller was fabricated finally.
REFERENCES
1] Nagrath, I.J. and Madan, Gopal, Control system engineering, 5th Edition, New
Age International Publisher, 2007
2] Bennett, Stuart (1993), A history of control engineering, (1930-1955), IET.
P.48. ISBN 978-0-86341-299-8
3] Ang, K.H., Chong, G.C.Y., and Li, Y. (2005). PID control system analysis,
design, and technology, IEEE Trans Control Systems Tech, 13(4), pp.559-57
4] King, Myke, Process Control: A Practical Approach, Wiley, 2010, p. 52
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