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30 Sep 2022
Skill-Lync
In Control System Engineering, the response of a system is analyzed for designing a suitable controller. The controller is needed to control the system behaviour as per requirements. As technology is moving toward automation, the role of sensors and actuators play a crucial role in any engineering field.
Sensors sense the system output and send the signal to the controller. The controller gives the command to the actuator so that the manipulated input to the system will be varied. In order to design a suitable controller for a system, the characteristics of the system need to be analyzed first. Hence, the system response will be studied either in the time domain or frequency domain.
The time domain graph depicts how the signal varies with respect to time. In this time-domain graph, the magnitude of the signal is represented at each instant time. If it is a continuous system, the signal will be represented as continuous. If it is a discrete system, the magnitude of the signal will be represented in distinct time intervals. The Cathode Ray Oscilloscope(CRO) device is the common device used to analyse the signal in a time domain.
The frequency domain graph shows how much of a signal lies within each frequency domain. Here in the frequency domain, the signal is represented by the sum of sinusoidal waves of different frequencies and each with a certain amplitude. Frequency domain analysis is very common in Control system Engineering. Thus, this refers to analyzing the signals with respect to the frequency of each signal.
In the time domain, the amplitude of the signal is plotted against the time and the frequency of the signal is untold. In the frequency domain, the amplitude of the signal is plotted against the frequency whereas how the signal varies with time is not given. Hence, each domain of signal representation provides us with different kinds of information about the same signal. Based on the objective, the analysis in one domain is advantageous over the other. The following methods are used to transform from the time domain to the frequency domain.
Signals, given in one domain, can be transformed to another domain using a mathematical transformation method called the Fourier formula (Fourier transform and Fourier series). Fourier stated that any signal in the time domain can be represented as a summation of sinusoidal waveforms of different frequencies. Sinusoidal waveforms are used because they do not change their shape when they pass through an LTI (Linear Time Invariant) system. x(t) is the signal in the time domain that can be transformed to the frequency domain by Fourier transform. It is given by,
Laplace transform decomposes the signals in the time domain into a domain of both sine and exponential functions. It is otherwise called an s-domain. The sine wave frequency wave is described by ω, and the amplitude is described by 𝛔. Therefore,
x(t) is the signal in the time domain that can be transformed to the frequency domain by Laplace transform and it is given by,
Unlike Fourier transform, Laplace transform gives information about the magnitude gain of a signal. Hence, the Laplace transform of the signal is used in control system analysis. The application of the Laplace transform makes a differential equation into an algebraic equation that is much easier to manipulate and to design a suitable compensator for systems. z-Transform is used for discrete systems. It is considered a discrete-time equivalent of the Laplace transform.
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Navin Baskar
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