# Transfer function | definition, theory, properties of transfer function in control system

In this article we will discuss about the transfer function and also about the various application of transfer function in control system. The control system is associated with all our daily works. Even when you are opening this website and reading this article this task is also a control system, which involves your brain and hand to synchronize in a coordinated system. In any type of control system proper representation is very vital, because the further analysis of that system is dependent on that representation. Transfer function is such a mathematical model, which is used to represent the characteristics of a physical system. If an input is provided to that function, it will transfer the corresponding output value. So we can tell that, a transfer function represents a linear relationship in between the input and output of any system.

This is the very basic and easiest definition of transfer function, but a transfer function usually consists of several complex variable. Actually, in each and every control system, we provide some informations such as the input, and after the processing on that input, suitable output is generated. The output of the system actually shows the response of the system on that particular input. The transfer function model of a particular system is generated by combining various output or responses for various inputs to the system. Suppose, in a machine factory, 50 kg of copper is needed to make 60 motors. Therefore, if we provide 50 kg of copper ( as input) into the assembly chain, then after the processing, 60 motors (as output) are manufactured. So, the transfer function of that system is 60 / 50 ( that is Output / Input).

## Concept Of Transfer function

The prime objective of the transfer function is to provide answer to the question ” What will the system’s response for a particular input ? “. More concisely, this model tells us about the output of the system for any input. It is actually a simplified representation of any linear time invarient (LTI) system. Processing with very high order system is quite difficult, but using transfer function we can easily describe a high order LTI system.The concept of transfer function is based on the theory of Laplace transformation. The Laplace transform is a widely used mathematical tool which is used to analyze the behaviour of a linear time invariant system. Actually, near about all physical systems are non-LTI type, but in some specific region of operation these systems approach to linearity. The advantage of using Laplace transform is that it can convert any complex higher order differential equation into a simple algebraic equation which is much more easier to deal with. By definition, Laplace transform of any system function f(t) is:

In control system, the transfer function of any linear time invariant system is defined as the ratio of laplace transform of the system output to the laplace transform of the system input, considering its initial conditions to be null (zero). Suppose, we provide an input function c(t) to a LTI system and we get the output function y(t). Then the transfer funstion of the system will be,
G(s) = Laplace transform of output function / Laplace transform of input function.
∴G(s) = L{y(t)} / L{c(t)}
G(s) = Y(s) / C(s)

The another way to define the transfer function is described as follows. We can also tell that, transfer function of a system is the Laplace transform of the impulse respose of that system. Now, we should know the concept of impulse respose of a system. Consider a linear time invariant system whose input is r(t) and output is y(t), and this system can be described in the following manner when the system is initially in relaxed condition.Here, the function g(t) is termed as the impulse response of the system.
Hence, the Laplace transform of the impulse response i.e Laplace transform of the function g(t), represents the transfer function G(s) of the LTI system. We can see that, transfer function of a system is often represented in the frequency domain but the impulse response description is said to be in time domain.

## Properties Of Transfer function

Some properties of transfer functions are listed as follows:

• The transfer function is only defined for Linear Time Invarient System (LTI System), so this model is not defined for non linear or time varient system.
• To determine the transfer function model, all initial conditions are set to null value.
• Suppose, there is a set of input variables and a set of output variables for a LTI system. Then the transfer function of that system will be the ratio of laplace tarnsform of the input to the laplace tarnsform of the output.
• The transfer function of an LTI system is independent of the input of that system.

## Procedure to construct the transfer function model

There are two ways by which we can construct the transfer function model of a system. They are described as follows.

• Generally in a lumped LTI system, we can build mathematical model of that system based on the physical laws and these models are nothing but a set of differential equations. So, by applying the Laplace transform on those differential equations we can easily construct the transfer function of that system.
• Transfer function model of any system can also be determined from the set of input and output data. This procedure is termed as curve fitting method.

Let’s check out the method of determining the transfer function.

Suppose, a linear system is represented by the following differential equation:

d2y/dt2 + a1(dy/dt) + a2y = u

∴ s2Y(s) + asY(s) + a2Y(s) = U(s)

Taking Laplace transform on both sides; (s2 + a1s + a2)Y(s) = U(s)

The transfer function is G(s) = Y(s) / U(s) = 1 / ( s2 + a1s + a2 )

See the following example:

See the another example.

This equation Q(s) / V(s) represents the transfer function of the RLC circuit system.

## Application or Use of transfer function:

Actually, transfer function is a very useful tool or model in control system. This model represents the behaviour of the system output as a function of system input. Transfer function also represent the frequency response of the system and it is also helpful to determine the stability of the system.