24-10-2016, 11:18 AM
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INTRODUCTION
1.1 LINEAR PROCESS
Linear models describe a continuous response variable as a function of
one or more predictor variables. They can help you understand and predict the
behavior of complex systems or analyze experimental, financial, and biological
data.
A linear series, involving measurement in one dimension only; pertaining
to lengthlinear measure. Of or relating to the characteristics of a work of art in
which forms and rhythms are defined chiefly in terms of line.
“Linear development” refers to those developments that are constructed
in a linear fashion across the landscape; i.e., power lines, roads, railways,
pipelines (gas, oil), telecommunications infrastructure and man-made
waterways.
Linear relationships can be expressed in a graphical format where the
variable and the constant are connected via a straight line or in a mathematical
format where the independent variable is multiplied by the slope coefficient,
added by a constant, which determines the dependent variable.
1.2 NON LINEAR PROCESS
Nonlinear control is one of the most challenging topics in modern control
theory. Although linear control system theory has been well developed, it is the
nonlinear control problems that present the most headaches. The main reason
that a nonlinear process is difficult to control is because there could be so many
variations in process nonlinear behavior. Therefore, it is difficult to develop a
single controller to deal with the various nonlinear processes.
Many common process control problems exhibit nonlinear behavior, in
that the relationship between the controlled and manipulated variables depends
on the operating conditions. For example, if the dynamic behavior of a nonlinear process is approximated by a linear model such as a first order
transfer function, the model parameters (e.g.steady-state gain, time constant,
time delay)
Depends on nominal operating conditions. If the process is only mildly
nonlinear or remains in the vicinity of a nominal steady state, then the effects of
the non-linearity may not to serve. In these situations, conventional feedback
control strategies can provide adequate performance.
But many important industrial processes including high purity distillation
columns, highly exothermic chemical reactions, Ph neutralizations, and batch
systems can exhibit higher nonlinear behavior. These processes may be required
to operate over a wide range of conditions due to large process upsets or set
point changes. When conventional PID controllers are used to control highly
nonlinear processes, the controllers must be tuned very conservatively in order
to provide stable behavior over the entire range of operating conditions.
But conservative controller turning can result in serious degradation of
control system performance. There are other situations where conventional PID
control is inadequate, for example, when the process gain changes sign (e.g.
some reactor control problems).
Process control research has largely emphasized the analysis of linear
systems (via transfer function and state-space models) and the design of linear
controllers. In a similar vein, industrial practice has traditionally relied on
linear control laws, the ubiquitous PI and PID control algorithms. But within the
last 15 years, model-based control strategies such as model predictive control
(MPC) have become the preferred control technique for difficult multivariable
control problems in oil refineries and petrochemical plants. Because the current
generations of MPC systems are largely based on linear dynamic models such
as step response and impulse response models, the resulting linear controllers
must be conservatively turned for highly nonlinear problems.
In view of the shortcomings of linear controllers for highly nonlinear
processes, there are considerable incentives for developing more effective
control strategies that incorporate knowledge of the nonlinear characteristics.
During the past decade, there has been a resurgence of interest in developing
nonlinear control strategies that are appropriate for process control.
DEFNITIONS
1.3.1 Set point
The set point is a value for a process variable that is desired to be
maintained. For example, if a process temperature needs to keep within 5 °c of 100
°c, then the set point is 100 °c. A temperature sensor can be used to help maintain
the temperature at set point.
1.3.2 Process variable
A process variable is a condition of the process fluid (a liquid or gas)
That can change the manufacturing process in some way.
1.3.3 Measured variable
The measured variable is the condition of the process fluid that must be kept
at the designated set point.Sometimes the measured variable is not the same as the
process variable. For example, a manufacturer may measure flow into and out of a storage tank to determine tank level. In this scenario, flow is the measured
variable, and the process fluid level is the process variable.
1.3.4 Manipulated variable.
The factor that is changed to keep the measured variable at set point is called
the manipulated variable.
1.3.5 Error
Error is the difference between the measured variable and the set point and
can be either positive or negative. In the temperaturecontrol loop example, the error
is the difference between the 110 °cmeasured variable and the 100 °c set point-that
is, the error is +10°c.The objective of any control scheme is to minimize or eliminate
error. Therefore, it is imperative that error be well understood. Any error can be seen
as having three major components. These three components are shown in the figure
on the following page.
1.3.6 Magnitude
The magnitude of the error is simply the deviation between the values of the
set point and the process variable. The magnitude of error at any Point in time
compared to the previous error provides the basis for determining the change in
error. The change in error is also an important value.
1.3.7 Offset
Offset is a sustained deviation of the process variable from the Set point. In
the temperature control loop example, if the control System held the process fluid
at 100.5 °c consistently, even though theset point is 100 °c, then an offset of 0.5 °c
exists.
1.3.8 Load disturbance
A load disturbance is an undesired change in one of the factors that can
affect the process variable. In the temperature control loop Example, adding cold
process fluid to the vessel would be a load Disturbance because it would lower the
temperature of the process Fluid.
1.3.9 Delay Time (td):
It is the time taken for response to reach 50% of the final value, for the
very first time.
1.3.10 Rise Time (tr):
It is the time taken for response to raise from 0 to 100% for the very first
time. For under damped system, the rise time is calculated from 0 to100%.but
the over damped system it is the time taken by the response to raise from 0 to
90%.for critically damped system, it is the time taken for response to raise from
5% to 95%.
1.3.11 Peak Time (tp);
It is the time taken for the response to reach the peak value the very first
time or it is the time taken for the response to reach the peak overshoot, in
control (MP).
1.4 MATHEMATICAL MODEL.
A mathematical model is a description of a process using mathematical
concepts. The process of developing a mathematical model is termed as
mathematical modeling. Mathematical modeling is used to explain the identified
system and to study the effects of different components, and to make
predictions about the process behavior.
1.4.1 Transfer Function
The block diagram is to represent a control system in diagram form. In
other words practical representation of a control system is its block diagram. It
is not always convenient to derive the entire transfer function of a complex
control system in a single function. It is easier and better to derive transfer
function of control element connected to the system, separately. The transfer
function of each element is then represented by a block and they are then
connected together with the path of signal flow. For simplifying a complex
control system, block diagrams are used. In the figure below, there are two
elements with transfer function Gone(s) and Gtwo(s). Where Gone(s) is the transfer
function of first element and Gtwo(s) is the transfer function of second element
of the system.
In addition to that, the diagram also shows there is a feedback path
through which output signal C(s) is fed back and compared with the input R(s)
and the difference between input and output E(s) = R(s) – C(s) is acting as
actuating signal or error signal
Gain Scheduler
A gain scheduler runs in the controller’s microprocessor and monitors the
process variable to determine when the process has entered a new operating
range. It then updates the controller with a predetermined set of tuning
parameters designed to optimize the closed-loop performance in that range.
Gain scheduling is particularly appropriate for processes that speed up or
slow down as the process variable rises and falls. It also works if the process
becomes more or less sensitive to the controller’s efforts as the process variable
changes.
A gain scheduler provides the best of both worlds. It allows the controller
to be tuned for any number of operating ranges so that an optimal set of tuning
parameters can be downloaded into the controller depending on the current
value of the process variable.
Unfortunately, that’s a lot of work. The control engineer implementing
the gain schedule must first determine how the full span of the process variable
should be partitioned into distinct operating ranges that adequately represent all
the possible variations in the process’s behaviour.
1.5.2 Working of Gain Scheduling
Gain scheduling facilitates process control when the gains and the time
constants vary with the current value of the process variable. Gain scheduling is
particularly appropriate for processes that speed up or slow down as the process
variable rises and falls.
1.5.3 Gain Scheduler Advantages
A gain scheduler provides the best of both worlds. It allows the controller
to be tuned for any number of operating ranges so that an optimal set of tuning
parameters can be downloaded into the controller depending on the current
value of the process variable.
Unfortunately, that’s a lot of work. The control engineer implementing
the gain schedule must first determine how the full span of the process variable
should be partitioned into distinct operating ranges that adequately represent all
the possible variations in the process’s behavior. In the water tank example,
“nearly empty,” “half full,” and “almost full” would be obvious choices, but
there could also be several or several dozen ranges in between, depending on
how severely the process’s behavior varies as the water level changes.
The implementer would then have to operate the process within each
range and tune the controller for optimal closed-loop performance each time.
She would then load the resulting sets of tuning parameters into the gain
schedule to be retrieved by the controller whenever the process variable enters
the operating range that corresponds to each set. In the tank example, that would
require the implementer to fill the tank a little, tune the controller to maintain
the level there, record the resulting tuning parameters in the gain schedule fill
the tank a little more, and repeat until the entire range of tank levels had been
covered.
1.6 TYPES OF CONTROL SYSTEMS
There are various types of control system but all of them are created to
control outputs. The system used for controlling the position, velocity,
acceleration, temperature, pressure, voltage and current etc. Are examples of
control systems? Let us take an example of simple temperature controller of the
room, to clear the concept. Suppose there is a simple heating element, which is
heated up as long as the electric power supply is switched on. As long as the
power supply switch of the heater is on the temperature of the room rises and
after achieving the desired temperature of the room, the power supply is
switched off. Again due to ambient temperature, the room temperature falls and
then manually the heater element is switched on to achieve the desired room
temperature again. In this way one can manually control the room temperature
10
at desired level. This is an example ofmanual control system. This system can
further be improved by using timer switching arrangement of the power supply
where the supply to the heating element is switched on and off in a
predetermined interval to achieve desired temperature level of the room. There
is another improved way of controlling the temperature of the room. Here one
sensor measures the difference between actual temperature and desired
temperature. If there is any difference between them, the heating element
functions to reduce the difference and when the difference becomes lower than
a predetermined level, the heating elements stop functioning. Both forms of the
system are automatic control system. In former one the input of the system is
entirely independent of the output of the system. Temperature of the room
(output) increases as long as the power supply switch is kept on. That means
heating element produces heat as long as the power supply is kept on and final
room temperature does not have any control to the input power supply of the
system. This system is referred as open loop control system. But in the later
case, the heating elements of the system function, depending upon the
difference between, actual temperature and desired temperature. This difference
is called error of the system. This error signal is fed back to the system to
control the input. As the input to output path and the error feedback path create
a closed loop.
Hence, there are two main types of control system. They are as follow
1. Open loop control system
2. Closed loop controlsystem
1.6.1 OPEN LOOP CONTROL SYSTEM
A control system in which the control action is totally independent of
output of the system then it is called open loop control system. Manual control
system is also an open loop controlsystem.Fig1.5shows the block diagram of
11
open loop control system in which process output is totally independent of
controller action.
CONTROLLERS
1. Controllers improve steady state accuracy by decreasing the steady state
errors.
2. As the steady state accuracy improves, the stability also improves.
3. They also help in reducing the offsets produced in the system.
4. Maximum overshoot of the system can be controlled using these
controllers.
5. They also help in reducing the noise signals produced in the system.
6. Slow response of the over damped system can be made faster with the
help of these controllers.
7. Now what are controllers? A controller is one which compares controlled
values with the desired values and has a function to correct the deviation
produced.
1.8.1 Types of Controllers
Let us classify the controllers. There are mainly two types of
controllersand they are written below: Continuous Controllers: The main feature
of continuous controllers is that the controlled variable (also known as the
manipulated variable) can have any value within the range of controller’s
output. Now in the continuous controller’s theory, there are three basic modes
on which the whole control action takes place and these modes are written
below. We will use the combination of these modes in order to have a desired
and accurate output.
1. Proportional controllers.
2. Integral controllers.
3. Derivative controllers.
Combinations of these three controllers are written below:
1. Proportional and integral controllers.
2. Proportional and derivative controllers.
Now we will discuss each of these modes in detail.
1.8.2 Proportional Controllers
We cannot use types of controllers at anywhere, with each type
controller; there are certain conditions that must be fulfilled. With proportional
controllers there are two conditions and these are written below:
1. Deviation should not be large; it means there should be less deviation
between the input and output.
2. Deviation should not be sudden.
We are in a condition to discuss proportional controllers, as the name
suggests in a proportional controller the output (also called the actuating signal)
is directly proportional to the error signal. Now let us analyze proportional
controller mathematically.
A(t)=Kp*e(t)
Where Kp is proportional constant also known as controller gain. It is
recommended that Kp should be kept greater than unity. If the value of Kp is
greater than unity, then it will amplify the error signal and thus the amplified
error signal can be detected easily.
1.8.3 Advantages of Proportional Controller
Now let us discuss some advantages of proportional controller.
1. Proportional controller helps in reducing the steady state error, thus
makes the system more stable.
2. Slow response of the over damped system can be made faster with the
help of these controllers.
Disadvantages of Proportional Controller
Now there are some serious disadvantages of these controllers and these
are written as follows:
1. Due to presence of these controllers we some offsets in the system.
2. Proportional controllers also increase the maximum overshoot of the
system.
1.9 PERFORMANCE INDICES
1.9.1 Accuracy
Accuracy is the measurement tolerance of the instrument and defines the
limits of the errors made when the instrument is used in normal operating
conditions. Accuracy can be improved by using feedback elements. To increase
accuracy of any control system error detector should be present in control
system.
1.9.2 Sensitivity
The parameters of control system are always changing with change in
surrounding conditions, internal disturbance or any other parameters. This
change can be expressed in terms of sensitivity. Any control system should be
insensitive to such parameters but sensitive to input signals only.
1.9.3 Noise
An undesired input signal is known as noise. A good control system
should be able to reduce the noise effect for better performance.
1.9.4 Stability
It is an important characteristic of control system. For the bounded input
signal, the output must be bounded and if input is zero then output must be zero
then such a control system is said to be stablesystem.
INTRODUCTION
1.1 LINEAR PROCESS
Linear models describe a continuous response variable as a function of
one or more predictor variables. They can help you understand and predict the
behavior of complex systems or analyze experimental, financial, and biological
data.
A linear series, involving measurement in one dimension only; pertaining
to lengthlinear measure. Of or relating to the characteristics of a work of art in
which forms and rhythms are defined chiefly in terms of line.
“Linear development” refers to those developments that are constructed
in a linear fashion across the landscape; i.e., power lines, roads, railways,
pipelines (gas, oil), telecommunications infrastructure and man-made
waterways.
Linear relationships can be expressed in a graphical format where the
variable and the constant are connected via a straight line or in a mathematical
format where the independent variable is multiplied by the slope coefficient,
added by a constant, which determines the dependent variable.
1.2 NON LINEAR PROCESS
Nonlinear control is one of the most challenging topics in modern control
theory. Although linear control system theory has been well developed, it is the
nonlinear control problems that present the most headaches. The main reason
that a nonlinear process is difficult to control is because there could be so many
variations in process nonlinear behavior. Therefore, it is difficult to develop a
single controller to deal with the various nonlinear processes.
Many common process control problems exhibit nonlinear behavior, in
that the relationship between the controlled and manipulated variables depends
on the operating conditions. For example, if the dynamic behavior of a nonlinear process is approximated by a linear model such as a first order
transfer function, the model parameters (e.g.steady-state gain, time constant,
time delay)
Depends on nominal operating conditions. If the process is only mildly
nonlinear or remains in the vicinity of a nominal steady state, then the effects of
the non-linearity may not to serve. In these situations, conventional feedback
control strategies can provide adequate performance.
But many important industrial processes including high purity distillation
columns, highly exothermic chemical reactions, Ph neutralizations, and batch
systems can exhibit higher nonlinear behavior. These processes may be required
to operate over a wide range of conditions due to large process upsets or set
point changes. When conventional PID controllers are used to control highly
nonlinear processes, the controllers must be tuned very conservatively in order
to provide stable behavior over the entire range of operating conditions.
But conservative controller turning can result in serious degradation of
control system performance. There are other situations where conventional PID
control is inadequate, for example, when the process gain changes sign (e.g.
some reactor control problems).
Process control research has largely emphasized the analysis of linear
systems (via transfer function and state-space models) and the design of linear
controllers. In a similar vein, industrial practice has traditionally relied on
linear control laws, the ubiquitous PI and PID control algorithms. But within the
last 15 years, model-based control strategies such as model predictive control
(MPC) have become the preferred control technique for difficult multivariable
control problems in oil refineries and petrochemical plants. Because the current
generations of MPC systems are largely based on linear dynamic models such
as step response and impulse response models, the resulting linear controllers
must be conservatively turned for highly nonlinear problems.
In view of the shortcomings of linear controllers for highly nonlinear
processes, there are considerable incentives for developing more effective
control strategies that incorporate knowledge of the nonlinear characteristics.
During the past decade, there has been a resurgence of interest in developing
nonlinear control strategies that are appropriate for process control.
DEFNITIONS
1.3.1 Set point
The set point is a value for a process variable that is desired to be
maintained. For example, if a process temperature needs to keep within 5 °c of 100
°c, then the set point is 100 °c. A temperature sensor can be used to help maintain
the temperature at set point.
1.3.2 Process variable
A process variable is a condition of the process fluid (a liquid or gas)
That can change the manufacturing process in some way.
1.3.3 Measured variable
The measured variable is the condition of the process fluid that must be kept
at the designated set point.Sometimes the measured variable is not the same as the
process variable. For example, a manufacturer may measure flow into and out of a storage tank to determine tank level. In this scenario, flow is the measured
variable, and the process fluid level is the process variable.
1.3.4 Manipulated variable.
The factor that is changed to keep the measured variable at set point is called
the manipulated variable.
1.3.5 Error
Error is the difference between the measured variable and the set point and
can be either positive or negative. In the temperaturecontrol loop example, the error
is the difference between the 110 °cmeasured variable and the 100 °c set point-that
is, the error is +10°c.The objective of any control scheme is to minimize or eliminate
error. Therefore, it is imperative that error be well understood. Any error can be seen
as having three major components. These three components are shown in the figure
on the following page.
1.3.6 Magnitude
The magnitude of the error is simply the deviation between the values of the
set point and the process variable. The magnitude of error at any Point in time
compared to the previous error provides the basis for determining the change in
error. The change in error is also an important value.
1.3.7 Offset
Offset is a sustained deviation of the process variable from the Set point. In
the temperature control loop example, if the control System held the process fluid
at 100.5 °c consistently, even though theset point is 100 °c, then an offset of 0.5 °c
exists.
1.3.8 Load disturbance
A load disturbance is an undesired change in one of the factors that can
affect the process variable. In the temperature control loop Example, adding cold
process fluid to the vessel would be a load Disturbance because it would lower the
temperature of the process Fluid.
1.3.9 Delay Time (td):
It is the time taken for response to reach 50% of the final value, for the
very first time.
1.3.10 Rise Time (tr):
It is the time taken for response to raise from 0 to 100% for the very first
time. For under damped system, the rise time is calculated from 0 to100%.but
the over damped system it is the time taken by the response to raise from 0 to
90%.for critically damped system, it is the time taken for response to raise from
5% to 95%.
1.3.11 Peak Time (tp);
It is the time taken for the response to reach the peak value the very first
time or it is the time taken for the response to reach the peak overshoot, in
control (MP).
1.4 MATHEMATICAL MODEL.
A mathematical model is a description of a process using mathematical
concepts. The process of developing a mathematical model is termed as
mathematical modeling. Mathematical modeling is used to explain the identified
system and to study the effects of different components, and to make
predictions about the process behavior.
1.4.1 Transfer Function
The block diagram is to represent a control system in diagram form. In
other words practical representation of a control system is its block diagram. It
is not always convenient to derive the entire transfer function of a complex
control system in a single function. It is easier and better to derive transfer
function of control element connected to the system, separately. The transfer
function of each element is then represented by a block and they are then
connected together with the path of signal flow. For simplifying a complex
control system, block diagrams are used. In the figure below, there are two
elements with transfer function Gone(s) and Gtwo(s). Where Gone(s) is the transfer
function of first element and Gtwo(s) is the transfer function of second element
of the system.
In addition to that, the diagram also shows there is a feedback path
through which output signal C(s) is fed back and compared with the input R(s)
and the difference between input and output E(s) = R(s) – C(s) is acting as
actuating signal or error signal
Gain Scheduler
A gain scheduler runs in the controller’s microprocessor and monitors the
process variable to determine when the process has entered a new operating
range. It then updates the controller with a predetermined set of tuning
parameters designed to optimize the closed-loop performance in that range.
Gain scheduling is particularly appropriate for processes that speed up or
slow down as the process variable rises and falls. It also works if the process
becomes more or less sensitive to the controller’s efforts as the process variable
changes.
A gain scheduler provides the best of both worlds. It allows the controller
to be tuned for any number of operating ranges so that an optimal set of tuning
parameters can be downloaded into the controller depending on the current
value of the process variable.
Unfortunately, that’s a lot of work. The control engineer implementing
the gain schedule must first determine how the full span of the process variable
should be partitioned into distinct operating ranges that adequately represent all
the possible variations in the process’s behaviour.
1.5.2 Working of Gain Scheduling
Gain scheduling facilitates process control when the gains and the time
constants vary with the current value of the process variable. Gain scheduling is
particularly appropriate for processes that speed up or slow down as the process
variable rises and falls.
1.5.3 Gain Scheduler Advantages
A gain scheduler provides the best of both worlds. It allows the controller
to be tuned for any number of operating ranges so that an optimal set of tuning
parameters can be downloaded into the controller depending on the current
value of the process variable.
Unfortunately, that’s a lot of work. The control engineer implementing
the gain schedule must first determine how the full span of the process variable
should be partitioned into distinct operating ranges that adequately represent all
the possible variations in the process’s behavior. In the water tank example,
“nearly empty,” “half full,” and “almost full” would be obvious choices, but
there could also be several or several dozen ranges in between, depending on
how severely the process’s behavior varies as the water level changes.
The implementer would then have to operate the process within each
range and tune the controller for optimal closed-loop performance each time.
She would then load the resulting sets of tuning parameters into the gain
schedule to be retrieved by the controller whenever the process variable enters
the operating range that corresponds to each set. In the tank example, that would
require the implementer to fill the tank a little, tune the controller to maintain
the level there, record the resulting tuning parameters in the gain schedule fill
the tank a little more, and repeat until the entire range of tank levels had been
covered.
1.6 TYPES OF CONTROL SYSTEMS
There are various types of control system but all of them are created to
control outputs. The system used for controlling the position, velocity,
acceleration, temperature, pressure, voltage and current etc. Are examples of
control systems? Let us take an example of simple temperature controller of the
room, to clear the concept. Suppose there is a simple heating element, which is
heated up as long as the electric power supply is switched on. As long as the
power supply switch of the heater is on the temperature of the room rises and
after achieving the desired temperature of the room, the power supply is
switched off. Again due to ambient temperature, the room temperature falls and
then manually the heater element is switched on to achieve the desired room
temperature again. In this way one can manually control the room temperature
10
at desired level. This is an example ofmanual control system. This system can
further be improved by using timer switching arrangement of the power supply
where the supply to the heating element is switched on and off in a
predetermined interval to achieve desired temperature level of the room. There
is another improved way of controlling the temperature of the room. Here one
sensor measures the difference between actual temperature and desired
temperature. If there is any difference between them, the heating element
functions to reduce the difference and when the difference becomes lower than
a predetermined level, the heating elements stop functioning. Both forms of the
system are automatic control system. In former one the input of the system is
entirely independent of the output of the system. Temperature of the room
(output) increases as long as the power supply switch is kept on. That means
heating element produces heat as long as the power supply is kept on and final
room temperature does not have any control to the input power supply of the
system. This system is referred as open loop control system. But in the later
case, the heating elements of the system function, depending upon the
difference between, actual temperature and desired temperature. This difference
is called error of the system. This error signal is fed back to the system to
control the input. As the input to output path and the error feedback path create
a closed loop.
Hence, there are two main types of control system. They are as follow
1. Open loop control system
2. Closed loop controlsystem
1.6.1 OPEN LOOP CONTROL SYSTEM
A control system in which the control action is totally independent of
output of the system then it is called open loop control system. Manual control
system is also an open loop controlsystem.Fig1.5shows the block diagram of
11
open loop control system in which process output is totally independent of
controller action.
CONTROLLERS
1. Controllers improve steady state accuracy by decreasing the steady state
errors.
2. As the steady state accuracy improves, the stability also improves.
3. They also help in reducing the offsets produced in the system.
4. Maximum overshoot of the system can be controlled using these
controllers.
5. They also help in reducing the noise signals produced in the system.
6. Slow response of the over damped system can be made faster with the
help of these controllers.
7. Now what are controllers? A controller is one which compares controlled
values with the desired values and has a function to correct the deviation
produced.
1.8.1 Types of Controllers
Let us classify the controllers. There are mainly two types of
controllersand they are written below: Continuous Controllers: The main feature
of continuous controllers is that the controlled variable (also known as the
manipulated variable) can have any value within the range of controller’s
output. Now in the continuous controller’s theory, there are three basic modes
on which the whole control action takes place and these modes are written
below. We will use the combination of these modes in order to have a desired
and accurate output.
1. Proportional controllers.
2. Integral controllers.
3. Derivative controllers.
Combinations of these three controllers are written below:
1. Proportional and integral controllers.
2. Proportional and derivative controllers.
Now we will discuss each of these modes in detail.
1.8.2 Proportional Controllers
We cannot use types of controllers at anywhere, with each type
controller; there are certain conditions that must be fulfilled. With proportional
controllers there are two conditions and these are written below:
1. Deviation should not be large; it means there should be less deviation
between the input and output.
2. Deviation should not be sudden.
We are in a condition to discuss proportional controllers, as the name
suggests in a proportional controller the output (also called the actuating signal)
is directly proportional to the error signal. Now let us analyze proportional
controller mathematically.
A(t)=Kp*e(t)
Where Kp is proportional constant also known as controller gain. It is
recommended that Kp should be kept greater than unity. If the value of Kp is
greater than unity, then it will amplify the error signal and thus the amplified
error signal can be detected easily.
1.8.3 Advantages of Proportional Controller
Now let us discuss some advantages of proportional controller.
1. Proportional controller helps in reducing the steady state error, thus
makes the system more stable.
2. Slow response of the over damped system can be made faster with the
help of these controllers.
Disadvantages of Proportional Controller
Now there are some serious disadvantages of these controllers and these
are written as follows:
1. Due to presence of these controllers we some offsets in the system.
2. Proportional controllers also increase the maximum overshoot of the
system.
1.9 PERFORMANCE INDICES
1.9.1 Accuracy
Accuracy is the measurement tolerance of the instrument and defines the
limits of the errors made when the instrument is used in normal operating
conditions. Accuracy can be improved by using feedback elements. To increase
accuracy of any control system error detector should be present in control
system.
1.9.2 Sensitivity
The parameters of control system are always changing with change in
surrounding conditions, internal disturbance or any other parameters. This
change can be expressed in terms of sensitivity. Any control system should be
insensitive to such parameters but sensitive to input signals only.
1.9.3 Noise
An undesired input signal is known as noise. A good control system
should be able to reduce the noise effect for better performance.
1.9.4 Stability
It is an important characteristic of control system. For the bounded input
signal, the output must be bounded and if input is zero then output must be zero
then such a control system is said to be stablesystem.