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Introduction to Control Systems
Control systems are an integral part of modern society and numerous applications are all around Us
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the difference in the difference between the actual response and the desired response
is the error of control system
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We will understand
all these reasons with the help of a practical example
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Then in that case we will use a rotating knob, and we will
rotate this knob
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We 'll discuss the basics of control systems in which the very first lecture will be the
system response characteristics
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we will discuss the concept of feedback and the types of
feedback
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After
completing this chapter, we will move on to the next chapter, which is the root locus diagram
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after that
we will discuss the solution of state equations and we will also have the discussion on the concept of
controllability and observability, along with the theory discussion
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Along with this this, this course is also beneficial for college goers who have the
control systems as a subject
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The output differs from the input due to two
factors
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For instance, when we
enter inside the elevator and push the fourth-floor button, the input is instantaneous, but the output of
any control system is gradual as the system has to work to generate the output
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The accuracy of the system at steady state is one of the factors that can cause
the output to differ from the input
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In the upcoming lecture, we will delve into system configurations with two different types; open-loop
systems and closed-loop systems
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an example of a toaster
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The advantages of using an open loop system are simple in construction and design because it
does not have a complex mechanism
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the major disadvantage of using any open lovers is it is poorly

equipped to handle disturbance and as it can't handle disturbance
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in order to remove
the disadvantages of open loop system
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Closed Loop Systems
Open loop systems consist of two distinct sections:
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The controller section, which regulates the necessary input to ensure the desired output is
achieved
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Feedback can help compensate for disturbances and improve the accuracy of the system
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If the temperature drops below a certain
point, the air conditioner will kick in to maintain the temperature
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Open loop systems have low accuracy because they cannot handle disturbances, whereas closed loop
systems have high accuracy because they are capable of handling disturbances
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It is a type of integral transform that allows us to convert a signal from its standard time
domain representation to its frequency domain representation
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This function is equal to 1 when t is
greater than 0 and 0 when t is less than 0
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We can
also find the inverse Laplace transforms of these functions using the method of partial fractions
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Review of Laplace Transform
Laplace transform plays a significant role in determining the Laplace domains of different time domain
functions
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Linearity property: This property is a combination of homogeneity and the superposition principle
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For example, let us assume we have Laplace transform of the following three functions:
Unit step function: u(t)
Exponential function: e-3t
Sine function: sin(2t)
We can easily find out the Laplace transform of f(t) by using the linearity property
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We will discuss some more properties in the next lecture
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See you in the next
one
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Using the time scaling property, we can determine that shifting in the time domain is equivalent to
exponential multiplication in the frequency domain with the same sign
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Thus, we have covered the Laplace
transform property
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Shifting a function to the right
side by a factor of 2 in the time domain corresponds to an exponential multiplication with the same sign
in the frequency domain if f(t) is a time domain function with a Laplace transform of f(s-s0)
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We
still have two more properties to review - the differentiation property and the convolution property
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Review of Laplace Transform (Part 4)
The fifth property is the time differentiation property, also known as differentiation in the time domain
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The initial value of the original function is the
initial condition of the function f(t) when t is equal to 0 minus
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By using Laplace
transforms, we can convert a complicated differential equation into a simple algebraic equation
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Given that y(0) = 4 and y'(0) = 6, find the Laplace transform of y''(t) + 5y'(t) + 4y(t) =
8t
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We will discuss the convolution property in the next lecture
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Review of Laplace Transform (Part 5)
The impulse response of an LTI system is defined as the output of the system when the input signal is an
impulse
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In the upcoming sections of this course, we will discuss additional aspects of LTI
systems
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The convolution property of Laplace
transform is a multiplication operation in the frequency domain
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Mathematically, the output
signal y(t) is equal to the convolution of the input signal x(t) with the impulse response signal h(t):

y(t) = x(t) * h(t)
Review of Laplace Transform (Part 6)
The inverse Laplace transform is a method used to find the time domain function from a given frequency
domain function
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However, if we split these two factors into two different fractions, we can easily
calculate the inverse Laplace transform
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In this case we have (1/1)
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In the previous lecture, we discussed examples of inverse Laplace transform
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We will solve this by splitting the two terms as two different fractions
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When we solve this equation, we get 2b+c=-6
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We can easily calculate the inverse Laplace transform using the frequency shifting
property we discussed in the example
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I learned this
lecture here
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We have calculated this using partial fractions
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Let's cover some important points about LTI systems:
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Time-converged LTI systems follow the superposition principle of additivity and homogeneity in
the Laplace transform
The solution to an LTI system's response to an input follows the system's model
If we time shift the input by a factor of 1, the response is a time-shifted version of the original
response
The response of an LTI system equals the sum of the system's response to the input and the
response to a time-shifted input
The state response of a system is given by the system's response to the zero-state input
response
All initial conditions of an LTI system must be 0 to produce output
The output of a linear system is equal in terms of the initial condition and the response to the
input
The properties of LTI systems follow both linear and time-invariant systems
If we are given input x1(t) with a rectangular waveform from 0 to 2, we can find x2(t) by shifting
x1(t) by 1
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If we introduce a time shift to the input or response, the output response will be a time-shifted
version of the initial response
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One
of these requirements is that the initial conditions must be zero
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Meanwhile, the 0 input response is the zero state response where no
external input is applied to the system
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The electric circuit must be charged for
the system to function, with the charges stored in the capacitor, which acts as the power source
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The system must meet certain criteria to be considered linear time-invariant (LTI)
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The initial conditions must be zero

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The 0 input response is the zero state response with no external input applied to the system
The total output must be equal to the zero state response (Zsr)
The electric circuit is charged with the charges stored in the capacitor, which acts as the power
source
The charges present in the capacitor power the system, which in turn charges the electrical
system
A non-linear system will not meet the requirements of an LTI system
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