A TERM PAPER ON TYPES OF CONTROLLERS,
AND THEIR APPLICATIONS
NAME: AGBONHESE JAMES
MATRIC NO: 12/ENG05/005
DEPARTMENT: MECHATRONICS
ENGINEERING.
COURSE CODE: EEE441.
COURSE TITLE: SERVOMECHANISM.
SUBMITTED TO: MR KOLAPO ALLI.
DATE: NOVEMBER 2015.
CONTROLLERS.
Most industrial processes require that certain variables such as flow, temperature level or
pressure should remain at or near some reference value called SET POINT. The device that
serves to maintain a process variable value at the set point is called a CONTROLLER.
A Controller is a device that receives data from a measurement instrument, compares that data to
a programmed set point, and if necessary, signals a control element to take corrective action.
Controllers may perform complex mathematical functions to compare activities of a set of data to
the set point or they may perform simple addition or subtraction functions to make comparisons.
Controllers always have an ability to receive input, to perform a mathematical function with the
input, and to produce an output signal.
Uses of Controllers:
1. Controllers improve steady state accuracy by decreasing the steady state errors.
2. As the steady state accuracy improves, the stability also improves, thus they improve
stability.
3. They also help in reducing the offsets produced in the system.
4. Maximum overshoot of the system can be controlled using 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 controllers.
Types of controllers:
Based on control mechanism function there are three types of controllers;



Discrete controllers.
Multistep controllers.
Continuous controllers.
DISCRETE CONTROLLERS.
Discrete controllers are controllers that have only two modes or positions: on and off. A common
example of a discrete controller is a home hot water heater. When the temperature of the water in
the tank falls below set point, the burner turns on. When the water in the tank reaches set point,
the burner turns off.
 On/Off Controllers:
An on-off controller is the simplest form of temperature control device. The output from the
device is either on or off, with no middle state. An on-off controller will switch the output only
when the temperature crosses the set point. For heating control, the output is on when the
temperature is below the set point, and off above set point. Since the temperature crosses the set
point to change the output state, the process temperature will be cycling continually, going from
below set point to above, and back below. In cases where this cycling occurs rapidly, and to
prevent damage to contactors and valves, an on-off differential, or “hysteresis,” is added to the
controller operations. This differential requires that the temperature exceed set point by a certain
amount before the output will turn off or on again. On-off differential prevents the output from
“chattering” or making fast, continual switches if the cycling above and below the set point
occurs very rapidly. On-off control is usually used where a precise control is not necessary, in
systems which cannot handle having the energy turned on and off frequently, where the mass of
the system is so great that temperatures change extremely slowly, or for a temperature alarm.
One special type of on-off control used for alarm is a limit controller. This controller uses a
latching relay, which must be manually reset, and is used to shut down a process when a certain
temperature is reached.
MULTISTEP CONTROLLERS.
Multistep controllers are controllers that have at least one other possible position in addition to
on and off.
CONTINUOUS CONTROLLERS.
Controllers automatically compare the value of the PV to the SP to determine if an error exists. If
there is an error, the controller adjusts its output according to the parameters that have been set in
the controller. But this needs a tuning and tuning parameters essentially determine: How much
correction should be made? The magnitude of the correction (change in controller output) is
determined by the proportional mode of the controller. How long the correction should be
applied? The duration of the adjustment to the controller output is determined by the integral
mode of the controller. How fast should the correction be applied? The speed at which a
correction is made is determined by the derivative mode of the controller.
 Proportional Controllers:
Proportional controls are designed to eliminate the cycling associated with on-off control. A
proportional controller decreases the average power supplied to the heater as the temperature
approaches set point. This has the effect of slowing down the heater so that it will not overshoot
the set point, but will approach the set point and maintain a stable temperature. This
proportioning action can be accomplished by turning the output on and off for short intervals.
This “time proportioning” varies the ratio of “on” time to “off” time to control the temperature.
The proportioning action occurs within a “proportional band” around the set point temperature.
Outside this band, the controller functions as an on-off unit, with the output either fully on
(below the band) or fully off (above the band). However, within the band, the output is turned on
and off in the ratio of the measurement difference from the set point. At the set point (the
midpoint of the proportional band), the output on: off ratio is 1:1; that is, the on-time and offtime are equal. If the temperature is further from the set point, the on- and off-times vary in
proportion to the temperature difference. If the temperature is below set point, the output will be
on longer; if the temperature is too high, the output will be off longer. The proportional band is
usually expressed as a percentage of full scale, or degrees. It may also be referred to as gain,
which is the reciprocal of the band.
Note that in time proportioning control, full power is applied to the heater, but cycled on and off,
so the average time is varied. In most units, the cycle time and/or proportional band are
adjustable, so that the controller may better match a particular process. In addition to
electromechanical and solid state relay outputs, proportional controllers are also available with
proportional analog outputs, such as 4 to 20 mA or 0 to 5 Vdc. With these outputs, the actual
output level is varied, rather than the on and off times, as with a relay output controller. One of
the advantages of proportional control is the simplicity of operation. It may require an operator to
make a small adjustment (manual reset) to bring the temperature to set point on initial startup, or
if the process conditions change significantly. Systems that are subject to wide temperature
cycling will also need proportional controllers.
Depending upon the process and the precision required, either a simple proportional control or
one with PID may be required. Processes with long time lags and large maximum rates of rise
(e.g., a heat exchanger), require wide proportional bands to eliminate oscillation. The wide band
can result in large offsets with changes in the load. To eliminate these offsets, automatic reset
(integral) can be used. Derivative (rate) action can be used on processes with long time delays, to
speed recovery after a process disturbance. In a proportional controller the output (also called the
actuating signal) is directly proportional to the error signal. Writing this mathematically we have,
A(t) α e(t)
Removing the sign of proportionality we have,
A(t) = k p × e(t)
Where Kp is the 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.
Figure 1.0: A Proportional controller.
The proportional controller gives an output value that is proportional to the error value with a
gain value of Kp. The proportional response can be adjusted by multiplying the error by a
constant Kp, called the proportional gain constant, the transfer function can be written as,
KK p
C(S)
1
+ TS
=
K
R(S)
1+
1 + TS
𝐶(𝑆)
=
𝑅(𝑆)
𝐾𝐾𝑝
1+𝐾𝐾𝑝
Where; T ′ =
.
1
1+ 𝑇 ′ .𝑆
T
1+KKp
Thus unit step response for Proportional Controller will be,
C(t) =
−ST
KK p
(1 − e T′ )
1 + KK p
Effect of adding Proportional Controller in the system:
On adding a proportional controller in system, time response of system improves by a factor of
. Also on adding proportional controller, steady state offset arises between desired
response and output response as shown in figure 1.1.
Figure 1.1: The System response with and without P-controller
Advantages of Proportional Controllers.
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 Controllers.
1. Due to presence of these controllers we some offsets in the system.
2. Proportional controllers also increase the maximum overshoot of the system.
Applications of P controller:
P controllers are suited to noncritical control applications which can tolerate steady-state error in
the event of disturbances: e.g. pressure, flow rate, level and temperature control. P control action
provides rapid response, although its dynamic properties can still be improved through additional
control components.
 Integral Controllers:
As the name suggests in integral controllers the output (also called the actuating signal) is
directly proportional to the integral of the error signal. Now let us analyze integral controller
mathematically. Writing this mathematically we have,
t
A(t) α ∫ e(t)dt.
0
Removing the sign of proportionality we have,
t
A(t) = K i × ∫ e(t) dt
0
Where Ki is the integral constant also known as an controller gain. An integral controller is also
known as a reset controller.
Figure 1.2: An I-Controller
In an integral controller, the manipulation equals the integral of the error over time, multiplied by
a gain KI Figure 1.2 shows the block diagram of Integral controller employed for a SISO system.
The closed loop transfer function can be written as,
K
C(S)
T1 S (1 + TS)
=
K
R(S)
1+
T1 S(1 + TS)
C(S)
K
=
R(S)
K + Ti S + TTi S 2
For a Step input, R(s) = 1/s, we get the steady state error as,
e(s) =
T1 S (1 + TS)
1
.
T1 S (1 + TS) + K S
ess = lims→0 s . e(s) = 0
Effect of adding Integral Controller in system:
The step response of this closed loop system with integral action is shown in figure 1.3. The
integral term enhances the movement of the process towards desired point. It also eliminates the
residual steady-state error that produces with a pure proportional controller. From the transfer
function, it is observed that use of integral controller leads to increasing order of closed loop
system which may cause instability, slow and oscillatory response. However the system has a
major advantage that integral controller produces zero steady state error.
The drawbacks of integral controller can be rectified if we use Proportional controller along with
Integral one.
Figure 1.3 Response using an Integral controller.
Advantages of Integral Controllers.
1. Due to their unique ability they can return the controlled variable back to the exact set
point following a disturbance that’s why these are known as reset controllers.
Disadvantages of Integral Controllers.
1. It tends to make the system unstable because it responds slowly towards the produced
error.
 Derivative Controllers:
We never use derivative controllers alone. It should be used in combinations with other modes
of controllers because of its few disadvantages which are written below:
1. It never improves the steady state error.
2. It produces saturation effects and also amplifies the noise signals produced in the system.
Now, as the name suggests in a derivative controller the output (also called the actuating signal)
is directly proportional to the derivative of the error signal. Writing this mathematically we
have,
A(t) α
de(t)
dt
Removing the sign of proportionality we have,
A(t) = K d ×
de(t)
dt
Where Kd is proportional constant also known as controller gain. Derivative controller is also
known as rate controller.
Fig 1.4: Block diagram of a Derivative Controller.
A derivative controller uses the derivative of the error instead of the integral. Figure 1.4 shows
the building blocks of a differential controller. In this closed loop system, the transfer function
can be written as,
Td SK
C(S)
(1 + TS)
=
Td SK
R(S)
1+
(1 + TS)
C(S)
Td SK
=
R(S)
1 + Ts + Td SK
e(s) =
Td SK
1
.
1 + Ts + Td SK S
For Step input R(s) = 1/s, the steady state error would be,
Effect of adding Differential Controller in system:
Derivative controller improves stability of the system and it also improves settling time.
Derivative of the error can be calculated by determining slope of the error over time and
multiplying this term with derivative gain τd .
Advantages of Derivative Controller.
The major advantage of derivative controller is that it improves the transient response of the
system.
 Proportional and Integral Controller:
As the name suggests it is a combination of proportional and an integral controller the output
(also called the actuating signal) is equal to the summation of proportional and integral of the
error signal. Writing this mathematically we have,
t
A(t) α ∫ e(t)dt + A(t) α e(t)
0
Removing the sign of proportionality we have,
t
A(t) = K i ∫ e(t) dt + K p e(t)
0
Where Ki and kp proportional constant and integral constant respectively.
The advantages and disadvantages are the combinations of the advantages and disadvantages of
proportional and integral controllers.
Applications of PI controller:
Control loops allowing no steady-state error.
Examples: pressure, temperature, ratio control, etc.
 Proportional and Derivative Controller:
As the name suggests it is a combination of proportional and a derivative controller the output
(also called the actuating signal) is equals to the summation of proportional and derivative of the
error signal. Writing this mathematically we have,
A(t) α de(t) + A(t) α e(t)
Removing the sign of proportionality we have,
A(t) = K d
de(t)
+ K p e(t)
dt
Where Kd and kp proportional constant and derivative constant respectively.
The advantages and disadvantages are the combinations of advantages and disadvantages of
proportional and derivative controller.
Applications of PD Controllers.
PD controllers are employed in all applications where P controllers are not sufficient. This
usually applies to controlled systems with greater lags, in which stronger oscillation of the
controlled variable ñ caused by a high K p value must be prevented.
 PID Controllers:
The proportional with integral and derivative control, or PID. This controller combines
proportional control with two additional adjustments, which helps the unit automatically
compensate for changes in the system. These adjustments, integral and derivative, are expressed
in time-based units; they are also referred to by their reciprocals, RESET and RATE,
respectively. The proportional, integral and derivative terms must be individually adjusted or
“tuned” to a particular system using trial and error. It provides the most accurate and stable
control of the three controller types, and is best used in systems which have a relatively small
mass, those which react quickly to changes in the energy added to the process. It is
recommended in systems where the load changes often and the controller is expected to
compensate automatically due to frequent changes in set point, the amount of energy available,
or the mass to be controlled.
There are also other features to consider when selecting a controller. These include auto- or selftuning, where the
instrument will automatically calculate the proper proportional band, rate and reset values for
precise control; serial communications, where the unit can “talk” to a host computer for data
storage, analysis, and tuning; alarms, that can be latching (manual reset) or non-latching
(automatic reset), set to trigger on high or low process temperatures or if a deviation from set
point is observed; timers/event indicators which can mark elapsed time or the end/beginning of
an event. In addition, relay or triac output units can be used with external switches, such as SSR
solid state relays or magnetic contactors, in order to switch large loads up to 75 A.
PID controller is commonly used for SISO systems. Figure 1.5 shows the basic blocks of a SISO
system. It has single input and single output. It has a controller which controls the operation of a
process based on the feedback received.
Figure 1.5 A SISO system.
For a PID controller, the output can be expressed in terms of input as follows:
U(t) = K p [e(t) + Td
de(t)
1 t
+ ∫ e(t)dt]
dt
Ti 0
And the transfer function of PID controller can be written as,
C(S) = K p [1 + Td S +
1
]
Ti S
Where
Kp → Proportional Gain.
τd → Derivative Time.
τi → Integral Time.
PID controller consists of Proportional, Integrator and Differentiator Controllers which can be
understood by considering a first order system SISO whose transfer function can be written as,
P(S) =
K
1 + TS
Types of PID controller:

Parallel PID Controller
Figure 3.4.10 shows the parallel configuration of PID controller. In general, this type of PID is
preferred over series one because in Parallel PID, output of one controller does not affect the
output of other. This allows freedom to control parameters independently.
Figure 1.6: Configuration of PID controller

Series PID Controller
Figure 3.4.11 shows the typical configuration of series PID controller. In this type of controller,
the output of a controller affects the output obtained from the other. Therefore the order of
controller must be taken into account during designing such configuration.
Figure 1.7: Series PID controller.
Applications of PID controllers.
Control loops with second- or higher-order systems that require rapid stabilization and do not
allow steady-state error. PID controllers have wide variety of applications in manufacturing
industry. Some of them are listed as follows.
1. PID control is used in automatic car steering when it is integrated with Fuzzy Logic.
2. In movement detection system of modern seismometer.
3. In water/oil level monitoring in tanks.
4. Head positioning of a disk drive.
5. Automated inspection and quality control.
6. Manufacturing process control: CNC machine tools.
7. Chemical process control: flow control, temperature control.
8. Automatic control of material handling equipments.
9. Automatic packaging and dispatch.
10. To ensure safety during manufacturing operations.
Conclusions:
1. Proportional Controller improves system response time. It provides high proportional
gain which results into very low rise time and thus improves the response system.
2. Integral Controller makes the system steady with error approaches zero. But Integral
controller may increase instability of a system and may cause oscillations. However in
Proportional system provides very low value of Integral gain resulting in very low
amount of oscillations.
3. Derivative controller improves system settling time and also improves stability.
Figure 1.8: Comparison of different control systems.
A comparison of systems with no controller, only proportional controller, only Integral
controller and both proportional and integral controller can be seen in Figure 1.8. It can be seen
that the response curve produced by PI controller is better in comparison with that obtained by
only P, only I and without any controller. PI controller has the advantages of both the P as well
as I controllers. Therefore in general, it is recommended not to employ integral and derivative
controllers on their own. They are always to be used in conjunction with a proportional
controller.
Other Types of Controllers Include;

Switching Controllers:
As opposed to a continuous-action controller, this type of controller does not have a continuous
output signal. The output signal can only be switched on or off. But this can also be used for
controlling purposes.
Types of modulation:
Pulse width modulation (PWM): ASCO Numatics uses pulse width modulation in its electronic
control units.
Pulse amplitude modulation
Pulse frequency modulation
In pulse width modulation (PWM), the 24 V DC supply voltage is transformed into rectangular
pulses with different width. the output signal is no longer a constant signal but a sequence of
pulses which is repeated at a certain time interval, or period.

Bang-Bang Controllers:
Despite the low-brow sounding name of the Bang-Bang controller, it is a very useful tool that is
only really available using digital methods. A better name perhaps for a bang-bang controller is
an on/off controller, where a digital system makes decisions based on target and threshold
values, and decides whether to turn the controller on and off. Bang-bang controllers are a nonlinear style of control that this book might consider in more detail in future chapters.
Consider the example of a household furnace. The oil in a furnace burns at a specific
temperature -- it can't burn hotter or cooler. To control the temperature in your house then, the
thermostat control unit decides when to turn the furnace on, and when to turn the furnace off.
This on/off control scheme is a bang-bang controller.

Programmable Automation Controller:
A programmable automation controller (PAC) can be described as a "mashup" between a PC
and a PLC in that it typically offers the benefits of both in a single package. Therefore, it's
becoming more common that PLC vendors position their higher-end controllers as PACs —
largely because their higher-end products incorporate more connectivity options and broader
control capabilities than their PLC lines. In contrast to PC-based control, more often than not, a
PAC will have lower running and maintenance costs.
One advantage of a PC-based controller is faster computing speed and greater data storage area,
but not necessarily faster I/O access. Will your machine accommodate a larger PC-based
controller? Is the environment around the machine harsh in any way? PACs are usually smaller
and more robust.
Fig: 1.7 b: Diagrams for PAC.
How are PACS Different?
PACs differ from the hardware you're probably using now in several ways. In effect, PACs
expand the capabilities of hardware you're using now by merging features of more traditional
PLC , DCS, and RTU systems, plus adding some capabilities from PCs.
PLC. Traditional PLC (programmable logic controller) systems provided discrete-logic-based
control of input/output (I/O) signals, using ladder logic programming.
PLC: Traditional PLC (programmable logic controller) systems provided discrete-logic-based
control of input/output (I/O) signals, using ladder logic programming.
DCS: DCS (Distributed control system) technology traditionally provided process control—
batch control where product variations are made according to recipes, or continuous process
control.
RTU: The traditional remote terminal unit (RTU—also called a remote telemetry unit) provided
multiple communication options for monitoring remote assets, such as radio towers or pipelines.
PC: PC-based control traditionally linked an adapter card on a computer to I/O, with custom
applications written for control and communication.
 Limit Controller:
Limit Controller Functions When the measured temperature (PV) exceeds the limit SP, the limit
output relay turns OFF and the OUT1 operation indicator turns ON. If the limit output relay
turns OFF (limit alarm is ON), the limit output relay will remain off until the operator manually
resets the Limit Controller.
Selecting Upper/Lower Limit: The upper/lower limit selection setting enables switching
between upper limit and lower limit operation. The default setting is for upper limit operation.
Select either upper limit or lower limit: Resetting Limit Outputs Limit outputs can be reset by
pressing the Level Key/Reset Key for 1 second min. while in the operation level. The limit
output reset operation can be used to reset limit outputs and annunciates outputs. When the limit
status is OFF, the limit output is cleared and the limit output relay turns ON. When the limit
status is ON (limit over status), limit outputs will not be reset. The annunciator output turns OFF
regardless of the limit status. Event inputs used as reset inputs (for resetting limit outputs) can be
received while in the operation level, adjustment level, or protect level.

The Step Motor and Stepper Motor Controller:
With the help of a stepper motor controller, step motors convert electrical energy into precise
mechanical motion. The stepper motor rotates a specific incremental distance per each step. The
number of steps that are executed controls the degree of rotation of the motor’s shaft. This
characteristic makes step motors excellent for positioning applications. For example, a 1.8°
stepper motor executing 100 steps will rotate exactly 180° with some small amount of noncumulative error. The speed of step execution controls the rate of motor rotation. A 1.8° step
motor executing steps at a speed of 200 steps per second will rotate at exactly 1 revolution per
second.
The stepper motor controller can very accurately control how far and how fast the stepper motor
will rotate. The number of steps the motor executes is equal to the number of pulse commands it
is given by the controller. A stepper will rotate a distance and at a rate that is proportional to the
number and frequency of its pulse commands.
For a typical step motor based system. The stepper motor controller, step motor driver and motor
must all be present in one form or another. Each component’s performance will have an effect
on the others.
First is the pulse generator, also known as a stepper motor controller or indexer. The pulse
generator will output command pulses that the motor will follow. By altering the frequency of
the pulse train, the pulse generator can instruct the motor to accelerate, run at a speed, decelerate
or stop. A pulse generator must be present, otherwise the motor will not move.
Next is the motor driver. The stepper driver’s function is to control the magnitude and direction
of current flow into the motor windings. The driver takes the pulses from the pulse generator and
determines how and when the windings should be energized. The windings must be energized in
a specific sequence to generate motion.
Finally there is the step motor itself.
A step motor has two primary parts; the rotor, the moving piece, and the stator, the stationary
piece. The stator contains coils of wire called windings. The rotor spins on bearings or bushings
inside the stator. All step motors operate through the principle of the rotor following a rotating
magnetic field created by sequencing the flow of current through the stator windings. Each
NMB stepper has two phases, which are groups of electrically connected windings. As current is
passed through each phase, the motor take “steps,” or small movements to keep in synchronism
with the magnetic field. The degree of rotation per step depends on the style of driver used and
the construction of the motor.
Are there different types of stepper motor controllers?
In fact, not every stepper motor controller is the same. For one thing, a stepper motor controller
can be either open loop or closed loop. The difference between the two is that an open loop
system sends a consistent rate of power to the motor, assuming that the rotating field that the
rotor follows is consistent. A closed loop system uses feedback to adjust power based on the
kind of load the motor is bearing. In other words, in a closed loop system, information is looping
back to the controller, which then makes the necessary adjustments. In an open loop system, no
feedback is provided.
Most motor applications work with an open loop system, because it is simpler and less
expensive. Since these applications generally call for a motor that will behave consistently, no
feedback is needed, so it would be wasteful to opt for a closed loop controller. However, if the
motor behavior will need to vary for maximum effectiveness, a closed loop system will be
necessary.
Step Motor Advantages
Step motors have several advantages over other types of motors. One of the most impressive is
their ability to position very accurately. NMB’s standard step motors have a step angle accuracy
of +/-5%. The error does not accumulate from step to step. This means that a standard stepper
can take a single step and travel 1.8° +/- 0.09°. Then it can take one million steps and travel
1,800,000° +/-0.09°. This characteristic gives a step motor almost perfect repeatability. In motor
terms, repeatability is the ability to return to a previously held position. A step motor can
achieve the same target position, revolution after revolution.
Breakdown of Step Motor Benefits:
1. Accuracy & Repeatability – Ability to position accurately.
2. Responsiveness & Quick Acceleration – Step motors have low rotor inertia, allowing them
to get up to speed quickly. This makes step motors an excellent choice for short, quick
moves.
3. Excellent torque for their size – Step motors have the highest torque per cubic inch of any
motor.
4. Positioning Stability – Unlike other types of motors, step motors can be held completely
motionless in their stopped position.
5. Open Loop Control – Open loop control is simpler, more reliable and less expensive than
feedback based (closed loop) control. In closed loop systems, encoders are used to count the
number of steps taken by the motor. The number of steps taken is compared to the number
of step commands given. This feedback is used to make position corrections or initiate
alarm signals. Encoders and their associated electronics add additional cost to a motion
control system. Assuming that a step motor is properly sized for its load, it should never
miss a step, making an encoder unnecessary.
6. Cost and Reliability – Step motor technology is reliable and proven. It is the most cost
effective method of precision position control.

Digital Controllers:
Digital control is a branch of control theory that uses digital computers to act as system
controllers. Depending on the requirements, a digital control system can take the form of a
microcontroller to an ASIC to a standard desktop computer. Since a digital computer is a
discrete system, the Laplace transform is replaced with the Z-transform. Also since a digital
computer has finite precision (See quantization), extra care is needed to ensure the error in
coefficients, A/D conversion, D/A conversion, etc. are not producing undesired or unplanned
effects.
The application of digital control can readily be understood in the use of feedback. Since the
creation of the first digital computer in the early 1940s the price of digital computers has
dropped considerably, which has made them key pieces to control systems for several reasons:

Inexpensive.

Flexible: easy to configure and reconfigure through software

Scalable: programs can scale to the limits of the memory or storage space without extra
cost

Adaptable: parameters of the program can change with time

Static operation: digital computers are much less prone to environmental conditions than
capacitors, inductors, etc.
Digital Controller Implementation;
A digital controller is usually cascaded with the plant in a feedback system. The rest of the
system can either be digital or analog.
Typically, a digital controller requires:

A/D conversion to convert analog inputs to machine readable (digital) format

D/A conversion to convert digital outputs to a form that can be input to a plant (analog)

A program that relates the outputs to the inputs
Output Program

Outputs from the digital controller are functions of current and past input samples, as
well as past output samples - this can be implemented by storing relevant values of input
and output in registers. The output can then be formed by a weighted sum of these stored
values.
The programs can take numerous forms and perform many functions

A digital filter for low-pass filtering

A state space model of a system to act as a state observer

A telemetry system
Stability
Although a controller may be stable when implemented as an analog controller, it could be
unstable when implemented as a digital controller due to a large sampling interval. During
sampling the aliasing modifies the cutoff parameters. Thus the sample rate characterizes the
transient response and stability of the compensated system, and must update the values at the
controller input often enough so as to not cause instability.
When substituting the frequency into the z operator, regular stability criteria still apply to
discrete control systems. Nyquist criteria apply to z-domain transfer functions as well as being
general for complex valued functions. Bode stability criteria apply similarly. Jury criterion
determines the discrete system stability about its characteristic polynomial.
Design of digital controller in s-domain
The digital controller can also be designed in the s-domain (continuous). The Tustin
transformation can transform the continuous compensator to the respective digital compensator.
The digital compensator will achieve an output which approaches the output of its respective
analog controller as the sampling interval is decreased.
Tustin transformation deduction
Tustin is the Padé(1,1) approximation of the exponential function
:
And its inverse
We must never forget that the digital control theory is the technique to design strategies in
discrete time, (and/or) quantized amplitude (and/or) in (binary) coded form to be implemented in
computer systems (microcontrollers, microprocessors) that will control the analog (continuous in
time and amplitude) dynamics of analog systems.
Fig 1.8: A digital controller.
Fig 1.9: A digital controller block diagram.

Process Controllers:
Process control is an engineering discipline that deals with architectures, mechanisms and
algorithms for maintaining the output of a specific process within a desired range. For instance,
the temperature of a chemical reactor may be controlled to maintain a consistent product output.
Process control is extensively used in industry and enables mass production of consistent
products from continuously operated processes such as oil refining, paper manufacturing,
chemicals, power plants and many others. Process control enables automation, by which a small
staff of operating personnel can operate a complex process from a central control room.
Process control may either use feedback or it may be open loop. Control may also be continuous
(automobile cruise control) or cause a sequence of discrete events, such as a timer on a lawn
sprinkler (on/off) or controls on an elevator (logical sequence).
A thermostat on a heater is an example of control that is on or off. A temperature sensor turns
the heat source on if the temperature falls below the set point and turns the heat source off when
the set point is reached. There is no measurement of the difference between the set point and the
measured temperature (e.g. no error measurement) and no adjustment to the rate at which heat is
added other than all or none.
A familiar example of feedback control is cruise control on an automobile. Here speed is the
measured variable. The operator (driver) adjusts the desired speed set point (e.g. 100 km/hr) and
the controller monitors the speed sensor and compares the measured speed to the set point. Any
deviations, such as changes in grade, drag, wind speed or even using a different grade of fuel
(for example an ethanol blend) are corrected by the controller making a compensating
adjustment to the fuel valve open position, which is the manipulated variable. The controller
makes adjustments having information only about the error (magnitude, rate of change or
cumulative error) although settings known as tuning are used to achieve stable control. The
operation of such controllers is the subject of control theory.
A commonly used control device called a programmable logic controller, or a PLC, is used to
read a set of digital and analog inputs, apply a set of logic statements, and generate a set of
analog and digital outputs.
For example, if an adjustable valve were used to hold level in a tank the logical statements
would compare the equivalent pressure at depth setpoint to the pressure reading of a sensor
below the normal low liquid level and determine whether more or less valve opening was
necessary to keep the level constant. A PLC output would then calculate an incremental amount
of change in the valve position. Larger more complex systems can be controlled by a Distributed
Control System (DCS) or SCADA system.
Fig 2.0: Diagrams for process controllers.
Types of processes using process control
Processes can be characterized as one or more of the following forms:

Discrete – Found in many manufacturing, motion and packaging applications. Robotic
assembly, such as that found in automotive production, can be characterized as discrete
process control. Most discrete manufacturing involves the production of discrete pieces
of product, such as metal stamping.

Batch – Some applications require that specific quantities of raw materials be combined
in specific ways for particular durations to produce an intermediate or end result. One
example is the production of adhesives and glues, which normally require the mixing of
raw materials in a heated vessel for a period of time to form a quantity of end product.
Other important examples are the production of food, beverages and medicine. Batch
processes are generally used to produce a relatively low to intermediate quantity of
product per year (a few pounds to millions of pounds).

Continuous – Often, a physical system is represented through variables that are smooth
and uninterrupted in time. The control of the water temperature in a heating jacket, for
example, is an example of continuous process control. Some important continuous
processes are the production of fuels, chemicals and plastics. Continuous processes in
manufacturing are used to produce very large quantities of product per year (millions to
billions of pounds).
Applications having elements of discrete, batch and continuous process control are often called
hybrid applications.
Examples

An anti-lock braking system (ABS) is a complex example, consisting of multiple inputs,
conditions and outputs.

Aircraft stability control is a highly complex example using multiple inputs and outputs.

Programmable logic controller:
A programmable logic controller, PLC, or programmable controller is a digital computer used
for automation of typically industrial electromechanical processes, such as control of machinery
on factory assembly lines, amusement rides, or light fixtures. PLCs are used in many machines,
in many industries. PLCs are designed for multiple arrangements of digital and analog inputs
and outputs, extended temperature ranges, immunity to electrical noise, and resistance to
vibration and impact. Programs to control machine operation are typically stored in batterybacked-up or non-volatile memory. A PLC is an example of a "hard" real-time system since
output results must be produced in response to input conditions within a limited time, otherwise
unintended operation will result.
Before the PLC, control, sequencing, and safety interlock logic for manufacturing automobiles
was mainly composed of relays, cam timers, drum sequencers, and dedicated closed-loop
controllers. Since these could number in the hundreds or even thousands, the process for
updating such facilities for the yearly model change-over was very time consuming and
expensive, as electricians needed to individually rewire the relays to change their operational
characteristics.
The functionality of the PLC has evolved over the years to include sequential relay control,
motion control, process control, distributed control systems, and networking. The data handling,
storage, processing power, and communication capabilities of some modern PLCs are
approximately equivalent to desktop computers. PLC-like programming combined with remote
I/O hardware, allow a general-purpose desktop computer to overlap some PLCs in certain
applications. Desktop computer controllers have not been generally accepted in heavy industry
because the desktop computers run on less stable operating systems than do PLCs, and because
the desktop computer hardware is typically not designed to the same levels of tolerance to
temperature, humidity, vibration, and longevity as the processors used in PLCs. Operating
systems such as Windows do not lend themselves to deterministic logic execution, with the
result that the controller may not always respond to changes of input status with the consistency
in timing expected from PLCs. Desktop logic applications find use in less critical situations,
such as laboratory automation and use in small facilities where the application is less demanding
and critical, because they are generally much less expensive than PLCs.
Fig 2.1: Diagrams for PLC.
As an example, say a facility needs to store water in a tank. The water is drawn from the tank by
another system, as needed, and our example system must manage the water level in the tank by
controlling the valve that refills the tank. Shown is a "ladder diagram" which shows the control
system. A ladder diagram is a method of drawing control circuits which pre-dates PLCs. The
ladder diagram resembles the schematic diagram of a system built with electromechanical
relays. Shown are:

Two inputs (from the low and high level switches) represented by contacts of the float
switches

An output to the fill valve, labelled as the fill valve which it controls

An "internal" contact, representing the output signal to the fill valve which is created in
the program.

A logical control scheme created by the interconnection of these items in software
In ladder diagram, the contact symbols represent the state of bits in processor memory, which
corresponds to the state of physical inputs to the system. If a discrete input is energized, the
memory bit is a 1, and a "normally open" contact controlled by that bit will pass a logic "true"
signal on to the next element of the ladder. Therefore, the contacts in the PLC program that
"read" or look at the physical switch contacts in this case must be "opposite" or open in order to
return a TRUE for the closed physical switches. Internal status bits, corresponding to the state of
discrete outputs, are also available to the program.
In the example, the physical state of the float switch contacts must be considered when choosing
"normally open" or "normally closed" symbols in the ladder diagram. The PLC has two discrete
inputs from float switches (Low Level and High Level). Both float switches (normally closed)
open their contacts when the water level in the tank is above the physical location of the switch.
When the water level is below both switches, the float switch physical contacts are both closed,
and a true (logic 1) value is passed to the Fill Valve output. Water begins to fill the tank. The
internal "Fill Valve" contact latches the circuit so that even when the "Low Level" contact opens
(as the water passes the lower switch), the fill valve remains on. Since the High Level is also
normally closed, water continues to flow as the water level remains between the two switch
levels. Once the water level rises enough so that the "High Level" switch is off (opened), the
PLC will shut the inlet to stop the water from overflowing; this is an example of seal-in
(latching) logic. The output is sealed in until a high level condition breaks the circuit. After that
the fill valve remains off until the level drops so low that the Low Level switch is activated, and
the process repeats again.
| (N.C. physical
(N.C. physical
|
|
Switch)
Switch)
|
|
Low Level
High Level
Fill Valve
|
|------[ ]------|------[ ]----------------------(OUT)---------|
|
|
|
|
|
|
|
|
|
|
Fill Valve |
|
|------[ ]------|
|
|
|
|
|
A complete program may contain thousands of rungs, evaluated in sequence. Typically the PLC
processor will alternately scan all its inputs and update outputs, then evaluate the ladder logic;
input changes during a program scan will not be effective until the next I/O update. A complete
program scan may take only a few milliseconds, much faster than changes in the controlled
process.
Programmable controllers vary in their capabilities for a "rung" of a ladder diagram. Some only
allow a single output bit. There are typically limits to the number of series contacts in line, and
the number of branches that can be used. Each element of the rung is evaluated sequentially. If
elements change their state during evaluation of a rung, hard-to-diagnose faults can be
generated, although sometimes (as above) the technique is useful. Some implementations forced
evaluation from left-to-right as displayed and did not allow reverse flow of a logic signal (in
multi-branched rungs) to affect the output.
Cascade controllers:
The simplest cascade control scheme involves two control loops that use two measurement
signals to control one primary variable. In such a control system, the output of the primary
controller determines the set point for the secondary controller. The output of the secondary
controller is used to adjust the control variable. Generally, the secondary controller changes
quickly while the primary controller changes slowly. Once cascade control is implemented,
disturbances from rapid changes of the secondary controller will not affect the primary
controller. To illustrate how cascade control works and why it is used, a typical control system
will be analyzed. This control system is one that is used to adjust the amount of steam used to
heat up a fluid stream in a heat exchanger. Then an alternative cascade control system for the
same process will be developed and compared to the typical single loop control. The figure
below shows the performance of cascade control vs. single-loop control in CST heater
Fig 2.2: Diagrams for cascade controllers.
Cascade control gives a much better performance because the disturbance in the flow is quickly
corrected
Example of Cascade Control
Figure 2.2. Single loop control for a heat exchanger
In the above process, the fluid is to be heated up to a certain temperature by the steam. This
process is controlled by a temperature controller (TC1) which measures the temperature of the
exiting fluid and then adjusts the valve (V1) to correct the amount of steam needed by the heat
exchanger to maintain the specified temperature. Figure 2.3 shows the flow of information to
and from the temperature controller.
Figure 2.3: Flow of information when single loop feedback control is used for a heat exchanger
Initially, this process seems sufficient. However, the above control system works on the
assumption that a constant flow of steam is available and that the steam to the heat exchanger is
solely dependent on opening the valve to varying degrees. If the flow rate of the steam supply
changes (i.e. pipeline leakage, clogging, drop in boiler power), the controller will not be aware
of it. The controller opens the valve to the same degree expecting to get a certain flow rate of
steam but will in fact be getting less than expected. The single loop control system will be
unable
to
effectively
maintain
the
fluid
at
the
required
temperature.
Implementing cascade control will allow us to correct for fluctuations in the flow rate of the
steam going into the heat exchanger as an inner part of a grander scheme to control the
temperature of the process fluid coming out of the heat exchanger. A basic cascade control uses
two control loops; in the case presented below (see Figure 2.4), one loop (the outer loop, or
master loop, or primary loop) consists of TC1 reading the fluid out temperature, comparing it to
TC1set (which will not change in this example) and changing FC1set accordingly. The other loop
(the inner loop, or slave loop, or secondary loop) consists of FC1 reading the steam flow,
comparing it to FC1set (which is controlled by the outer loop as explained above), and changing
the valve opening as necessary.
Figure 2.4: Cascade control for a heat exchanger
The main reason to use cascade control in this system is that the temperature has to be
maintained at a specific value. The valve position does not directly affect the temperature
(consider an upset in the stream input; the flow rate will be lower at the same valve setting).
Thus, the steam flow rate is the variable that is required to maintain the process temperature.
The inner loop is chosen to be the inner loop because it is prone to higher frequency variation.
The rationale behind this example is that the steam in flow can fluctuate, and if this happens, the
flow measured by FC1 will change faster than the temperature measured by TC1, since it will
take a finite amount of time for heat transfer to occur through the heat exchanger. Since the
steam flow measured by FC1 changes at higher frequency, we chose this to be the inner loop.
This way, FC1 can control the fluctuations in flow by opening and closing the valve, and TC1
can control the fluctuations in temperature by increasing or decreasing FC1set .
Thus, the cascade control uses two inputs to control the valve and allows the system to adjust to
both variable fluid flow and steam flow rates. The flow of information is shown in figure 2.5.
Figure 2.5: Flow of information when cascade control is used for a heat exchanger
In order to accomplish this, relationships between the primary and secondary loops (see
definitions below) must be defined. Generally, the primary loop is a function of the secondary
loop.
A possible example of such relations is:
Primary and Secondary Loops
In Figure 3, there are two separate loops. Loop 1 is known as the primary loop, outer loop, or the
master, whereas loop 2 is known as the secondary loop, inner loop, or the slave. To identify the
primary and secondary loops, one must identify the control variable and the manipulated
variable. In this case, the control variable is the temperature and the reference variable is the
steam flow rate. Hence, the primary loop (loop 1) involves the control variable and the
secondary loop (loop 2) involves the reference variable. The information flow for a two loop
cascade control system will typically be as shown in Figure 2.6. Please note that the user sets the
set point for loop 1 while the primary controller sets the set point for loop 2.
Figure 2.6: Information flow of a two loop cascade control
In addition to this common architecture, cascade control can have multiple secondary loops;
however, there is still one primary loop and a main controlled variable. Unfortunately, with
multiple inner loops, tuning the PID becomes even more challenging, making this type of
cascade less common. The secondary loops can be either independent of each other, or
dependent on each other, in which case each secondary loop affects the set point of the other
secondary loop. When tuning such controller, the inner most loop should be tuned first. The loop
that manipulates the set point of the inner-most loop should be tuned next and so for. The figure
below shows an example of using two secondary loops, independent of each other, in a fuel
combustion plant. In this combustion furnace, the master controller controls the temperature in
the furnace by changing the set point for the flow of fuels A and B. The secondary loops
correspond to the change in the set point for the flow, by opening or closing the valves for each
fuel.
Fig 2.7: Diagrams for cascade controller.
Cascade control is generally useful when

A system error affects the primary control variable only after a long period of time as it
propagates through dead time and lag time.

A system has long dead times and long lag times.

Multiple measurements with only one control variable are required for better response to
a disturbance of a system.

Variance occurs in multiple streams.
General Cascade Control Schematic
The reactor below needs to be cooled during continuous-feed operation of an exothermic
reaction. The reactor has been equipped with a cooling water jacket with the water flow rate
being controlled by cold water valve. This valve is controlled by two separate temperature
controllers. An “inner-loop” or “slave” (highlighted in orange) temperature transmitter
communicates to the slave controller the measurement of the temperature of the jacket. The
“outer-loop” or “master” (green) temperature controller uses a master temperature transmitter to
measure the temperature of the product within the reactor. The output from the slave controller
is fed into the master controller and used to adjust the cold water valve accordingly.
Fig 2.8: Diagrams for cascade controller.
Fig 2.9:
for cascade
Diagrams
controller.
Step 1: Write down
equations for each
control loop
all the
stage of the
Master Loop
Slave Loop
Gp1 and Gp2 are the process operators and are usually of the form:
Where n is a natural number.
Gc1 and Gc2 are the control operators and depend on the type of controller used. For PID
controllers, they would be:
Gm1 and Gm2 are the measurement operators and usually are just equal to 1. Note that there are
no equations for the "intersections" A and B shown on the diagram.
Step 2: Simplify the equations for the slave loop
Solve for Y2(t)
Step 3: Simplify the equations for the master loop
Solve for Y1(t)