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Impulse response and transfer function
Control system is a system which controls another system. Such system is always needed as the output of a system is desired to be comfortable with operation. As the civilization progressed the automation of
machinery became necessary and control system got upgraded more and more. Control system is usually implemented in two ways : one is the by open loop method and the other is closed loop method. In open loop method the output
of any electrical system depends only on variation of the input. No external factors will influence the input of the system. The output will not be fed back to system to modify the input or compare with it . A simple example
of the such system is remote controlling a Television or a timer controlling some machines. The old fashioned form of open loop system was a type of manual control. We know what a manual control is . Some human action is needed to
trigger something in a functioning system. More precisely , the output of such system is not
measured and reused in the system.
Before we try to fathom the control engineering, we need to know how a electrical system can be defined. A system is like a box which is connected with two bars on two opposite side. One may be thought as input and other may be thought as output. When something (signals in electronic system) passes through the input, it gets modified and driven out of the box through other bar. The box is the crucial part which perform operation on the input. The input in electrical system is a kind of electrical signal and output is also some form of current or voltage signal. The output may be called the response of the system. Suppose we have an inductor and a capacitor connected in series inside the box. Then the output of the system will be an oscillating signal. The box is thus turning an input signal into time varying signal. It may seem confusing at the moment because we still need to understand what the box is called when we want to analyze it mathematically. Mathematically speaking, the box is the transfer function of the system. More rigorously it is the response of the system when the input is unit impulse signal, asuming all other initial conditions are zero.
The transfer function is the ratio of output C(s) and input R(s). When R(s) = 1 , the input is impulse signal. An example of output function C(s) is given as Laplace transform of C(t). But it is mandatory that the transfer function is only applicable to linear-time-invariant system(LTI). In system and signal analysis impulse response fully characterize the system. Impulse response is the response of the system when it is excited by all the frequencies. Once we know the impulse response in frequency domain , we can multiply it with any input signal's spectrum to get the system response. The multiplication in frequency domain is the same thing as performing convolution of input signal with the impulse response. When we convolute input signal with impulse response, we get the output of a system. Convolution is a type of integration. Convolution shows how graph of a function changes when another graph is shifted over it. Convolution is done by finding the area under the common part of two graphs( functions). In control system the impulse response (in complex frequency[s = a + jw where j = squareroot(-1)] domain) is called the transfer function of the system.
Closed loop feedbackThe closed loop feedback control system shown in the figure has also a transfer function which characterizes the system.
Fig: Typical Control system and transfer function.
The feedback branch has the transfer factor H(s) while the upper branch has that of G(s).
Let us think of a simple resistive circuit with Resistance R and voltage source v is connected in series. We think of output is the voltage drop (IR) across the resistor and input is the voltage v. Then the transfer
function of the system is simply unity (considering ratio of output to input). But for many complicated circuits this is not the case. We can have higher order linear differential equation for modelling the system. Then the transfer function will be
some rational function containing numerator and denominator, which are Laplace transforms of other functions. When the input signal is a unit impulse then its Laplace transform is unity and the transfer function is the response
of that signal because transfer function is the ratio of output to input signal's Laplace transform. A unit impulse signal can be thought as a function which has infinitely tall spike at the origin. This is also known as dirac
Fig: dirac delta function.
This is not some ordinary function but designed to solve specific problems occurred in physics and engineering. The infinite spike can be approximated by a curve around the origin as the width decreases. The infinite spike must diverge so as to make the total area under the curve unity. This is the special property of delta function. When something happens for a very brief moment like a bat hitting a tennis ball , the force acting on the ball can be described by impulse function. The total change in momentum happens during that brief interval. Similar situation occurs in electrical engineering when impulse function describes phenomena that last only a very infinitesimally small period of time. Laplace transformation is an essential method to study transfer function because it gives us the freedom to characterize the system in complex plane. Laplace transform converts a time domain signal into complex frequency[s = a + jw where j = squareroot(-1)] domain signal. For more study you can follow This link.
Fig: Laplace transform.
The function f(t) is time domain signal whereas Y(s) is a complex domain signal. The method of transforming from time domain signal to complex domain signal is the integral equation as shown in the figure. It is a type of integral transform where there is usually a factor named kernel. The kernel in Laplace transform is exp(-st). Substituting other kernel would give different integral transform. In case of Fourier transform the kernel is exp(-iwt).
Fig: Transfer function of second order system.
As given in the figure above, the control system is of second order in complex frequency( s). The denominator is a second order or quadratic equation of s , which can be converted to time domain equation by applying
Inverse Laplace transformation. We can then find its time response by necessary algebraic and geometric manipulations. It is like solving differential equation for voltage or current as a function of time(t).
There are many methods of determining the system's stability. One of those is analyzing Nyquest-plot in the complex plane. Bode plot is another method of analyzing system stability. Nyquest plot can handle singularities
that may occur in the right half complex plane. The singularities is any point where any function or its derivative diverges. Simply, the value of such function is undefined at the singularity point. Singularity occurs
in many areas of both mathematics and physics. At singularity all the laws of physics go hay wire. As an example , big bang was a special kind of singularity which general relativity can not explain and quantum mechanical
laws becomes much more relevant. These are the topics that strictly fall under quantum gravity or string theory. We should focus on control system engineering here.