Introduction to control systems and basics

Introduction to control systems
Dr. Ankita Malhotra,
SVKM'S DJ Sanghvi College of Engineering
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Unit -3

Introduction
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 Control system means any quantity of interest in a machine or
mechanism is altered in accordance with a desired manner.
Basic/open loop control system
Manually controlled closed-loop system

Basic Definitions
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 Controlled Variable -The controlled variable is the quantity or
condition that is measured and controlled. The control signal or
manipulated variable is the quantity or condition that is varied by
the controller so as to affect the value of the controlled variable.
Normally, the controlled variable is the output of the system.
control signal s given to the system to correct or limit deviation
of the measured value from a desired value.
 System- It is a combination of components that work together to
perform desired task.
 Disturbance- A disturbance is a signal that tends to adversely
affect the value of the output of a system. If a disturbance is
generated within the system, it is called internal, while an
external disturbance is generated outside the system and is an
input.

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 Feedback Control. Feedback control refers to an operation that, in
the presence of disturbances, tends to reduce the difference between
the output of a system and some reference input and does so on the
basis of this difference. Here only unpredictable disturbances are so
specified, since predictable or known disturbances can always be
compensated for within the system.
 Open –loop system- In an open-loop system the control action is
independent of the output of the system, OR when the output
quantity of the system is not fed back to the system.

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 Closed loop system-In a closed loop system the control action is
dependent on the output of the system, where the output quantity is
considerably controlled by sending a command signal to the input.
 Feedback- Normally feedback signal has opposite polarity to input
signal. This is called negative feedback. The advantage is the
resultant signal taken form the comparator output is smaller in
magnitude and can be easily handle by the system. This is called
actuating signal. This actuating signal becomes zero when the
desired output is reached.
* Note- It should be emphasized that for systems in which the inputs
are known ahead of time and in which there are no disturbances it is
advisable to use open-loop control. Closed-loop control systems have
advantages only when unpredictable disturbances and/or
unpredictable variations in system components are present.

Advantages and Disadvantages on open-loop system
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The major advantages of open-loop control systems are as follows:
 1. Simple construction and ease of maintenance.
 2. Less expensive than a corresponding closed-loop system.
 3. There is no stability problem.
 4. Convenient when output is hard to measure or measuring the
output precisely is economically not feasible. (For example, in the
washer system, it would be quite expensive to provide a device to
measure the quality of the washer’s output, cleanliness of the
clothes.)
The major disadvantages of open-loop control systems are as follows:
 1. Disturbances and changes in calibration cause errors, and the
output may be different from what is desired.
 2. To maintain the required quality in the output, recalibration is
necessary from time to time.

Transfer function
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 The transfer function of a control system is defined as the ratio of the
Laplace transform of the output variable to Laplace transform of the
input variable assuming all initial conditions to be zero.
* Laplace transform is an integral transform that converts a
function of a real variable often time) to a function of a complex
variable (complex frequency). The transform has many applications in
science and engineering because it is a tool for solving differential
equations. In particular, it transforms differential equations into
algebraic equations and convolution into multiplication.
The Laplace transform of a function f(t), defined for all real numbers t ≥
0, is the function F(s), which is a unilateral transform defined by
F(s)=

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 The transfer function of a linear, time-invariant, differential equation
system is defined as the ratio of the Laplace transform of the output
(response function) to the Laplace transform of the input (driving
function) under the assumption that all initial conditions are zero.
Consider the linear time-invariant system defined by the following
differential equation:
where y is the output of the system and x is the input. The transfer
function of this system is the ratio of the Laplace transformed output
to the Laplace transformed input when all initial conditions are zero,
or

Effects of feedback
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 Let the open loop gainof the system is G(s),feed back loop gain
is H(s), output C(s), inputR(s), Then feedback signal Bs(s) as
given:

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 Overall gain-As given by equation 1, the gain of open-loop system
is reduced by factor in the feedback system. If the
feedback gain is positive, overall gain will reduce, however if the
feedback gain is negative overall gain will increase.
 Stability: If the system is able to follow the input command signal,
the system is stable. If not, then system becomes unstable.
As can be seen from equation 1, If G(s).H(s)= -1, then the output of
the system becomes infinite for finite input. Thus the system
becomes unstable.
 Sensitivity: It depends on the system parameters. For a good control
system it is desirable that the system is insensitive to parameter
changes.
Sensitivity-
It can reduced by increasing G(s).H(s), i.e. by properly selecting
feedback signal.

Comparison of open loop and closed loop systems
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Classification of control systems
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Control systems can be classified based upon:
 a) Method of analysis and design, as linear and non-linear systems-
In linear systems principle of superposition can be applied, while it
cannot be applied on non-linear systems.
 b) The type of signal as time varying, time-invariant, continuous or
discrete data systems.- If the parameters of the system are unaffected
by time , then the system is time-invariant. In time-varying system,
parameters of the system change with time. Further if the signal not
varying continuously with time, but is in the form of pulses, it is
discrete data systems, while for continuously varying signal makes a
continuous data system
 c)Type of system components like electro-mechanical, hydraulic,
thermal etc.
 d) The main purpose such as position control, velocity control etc.

Examples of control systems
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 Speed Control System. The amount of fuel admitted to the engine is adjusted
according to the difference between the desired and the actual engine
speeds. Watt’s speed governor is used to control speed. If the actual speed
drops below the desired value due to disturbance, then the decrease in the
centrifugal force of the speed governor causes the control valve to move
downward, supplying more fuel, and the speed of the engine increases until
the desired value is reached. On the other hand, if the speed of the engine
increases above the desired value, then the increase in the centrifugal force
of the governor causes the control valve to move upward. This decreases
the supply of fuel, and the speed of the engine decreases until the desired
value is reached.
 In this speed control system, the plant (controlled system) is the engine and
the controlled variable is the speed of the engine. The difference between
the desired speed and the actual speed is the error signal. The control signal
(the amount of fuel)to be applied to the plant (engine) is the actuating
signal. The external input to disturb the controlled variable is the
disturbance.

Speed control system
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 Temperature Control System. The temperature in the electric
furnace is measured by a thermometer, which is an analog device.
The analog temperature is converted to a digital temperature by an
A/D converter. The digital temperature is fed to a controller through
an interface. This digital temperature is compared with the
programmed input temperature, and if there is any discrepancy
(error), the controller sends out a signal to the heater, through an
interface, amplifier, and relay, to bring the furnace temperature to a
desired value.

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 Business Systems. A business system is a closed-loop system. A
good design will reduce the managerial control required. Note that
disturbances in this system are the lack of personnel or materials,
interruption of communication, human errors, and the like. The
establishment of a well-founded estimating system based on statistics
is mandatory to proper management.
Engineering operational control system

Mathematical Modelling of
Electrical systems
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 Basic laws governing electrical circuits are Kirchhoff’s current law
and voltage law. Kirchhoff’s current law (node law) states that the
algebraic sum of all currents entering and leaving a node is zero.
Kirchhoff’s voltage law (loop law) states that at any given instant the
algebraic sum of the voltages around any loop in an electrical circuit
is zero.

Transfer function
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 LRC Circuit. Consider the electrical circuit shown in Figure. The
circuit consists of an inductance L (henry), a resistance R (ohm), and
a capacitance C (farad). Applying Kirchhoff’s voltage law to the
system, we obtain the following equations:
(1)
(2)
Equations (1) and (2) give a mathematical model of the circuit.

Transfer Function Model
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 A transfer-function model of the circuit can also be obtained as
follows:
Taking the Laplace transforms of Equations (1) and (2),
assuming zero initial conditions, we obtain:
 If ei is assumed to be the input and eo the output, then the
transfer function of this system is found to be:

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 Transfer Functions of Cascaded Elements:
Assume that ei is the input and eo is the output. The capacitances C1
and C2 are not charged initially.
(3)
(4)
(5)

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 Taking the Laplace transforms of Equations (3) to (5), respectively,
using zero initial conditions, we obtain:
 Eliminating I1(s) from Equations (6) and (7) and writing Ei(s) in
terms of I2(s), we find the transfer function between Eo(s) and Ei(s)
to be:
(6)
(7)
(8)

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 The term R1C2s in the denominator of the transfer function
represents the interaction of two simple RC circuits. When the
second circuit is connected to the output of the first, a certain amount
of power is withdrawn, The degree of the loading effect determines
the amount of modification of the transfer function.

Electrical circuit involving an operational amplifier
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Since e’ =0
Taking Laplace transform, assuming initial conditions to be zero

Multiple input, Multiple Output Systems
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 The response to the reference input can be obtained by assuming
U(s)=0 and the block diagram becomes:

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 Similarly, the response to U(s) is obtained by assuming R(s)=0, the
block diagram now becomes:

Block diagram reduction Technique
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 A complex block diagram configuration can be simplified by
certain rearrangements of block diagram using the rules of block
diagram algebra.

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 Laplace transformed network:
 Applying KVL:
 Vi(s)= I(s)[R+1/Cs]+Vo(s)
 I(s)=I1(s) +I2(s)
 = Vo(s)/R + Vo(s).Cs
 Vi(s)= [Vo(s)/R +Vo(s).Cs][R+1/Cs] + Vo(s)

Reduce the given block diagram
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Step 1 – Eliminating feedback paths present in the forward path

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Step 2 – Combining the two blocks in series
Step 3 – Eliminating the feedback path

Reduce the given block diagram
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Step 1 –Shifting the take-off beyond G3

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Step 2 – Combining G2 & G3 and eliminating feedback loop (H3):
Step 3 - Eliminating feedback path H1/G3

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Step 4 - Combining all the three blocks

Signal Flow Graphs
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Definitions
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Example of signal flow graph
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Construction of signal flow graphs
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Let:
Step 1:
Step 2 : Nodes are connected to x2 according to first equation

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 Step 3 : Similarly the signal flow graphs of each
equation are constructed

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 Step 4: Combining all the signal flow graphs we get
final signal flow graph of the circuit:

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 For the constructed signal flow graph in the previous
example, the following calculations can be done:
Gain Calculation using Mason’s gain
formula

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Therefore:

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Thus , the overall system gain is given by

Practice problems on signal flow graphs
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Solution:
Step 1- Shifting the take off point:

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 Solution using signal flow graph-

Obtain the TF of given block diagram
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Shifting H2 towards right of G3

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TF using signal flow graphs
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Introduction to control systems and basics

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