Transfer Function

Intro

In control systems, a transfer function helps in describing the characteristics of a system. A transfer function is a ratio between the output and input of a Linear Time Invariant system. The transfer function is the $\mathcal{L}aplace$ transformed function of the impulse response. A system is Linear Time invariant, if the following two conditions hold;

• Linear: The relation between the output and input is a linear mapping i.e. a system injected with an input $\alpha$x(t) has an output $\alpha$y(t)
• Time Invariant: The output will not change based on when the input is injected i.e. it does not matter if the input is injected now or an amount T later, the same input will lead to the same input.

Block Schematics

In the figure below, a block scheme is shown which visualizes the general form of a transfer function. In this scheme, the input to the system is denoted by X(s), the system itself by H(s), and the output by Y(s).

With this block scheme, the relation between the output in terms of the input and the system can be derived as follows;

This relation in words essentially means that the arriving input at H(s) is multiplied by the system. The final transfer function is expressed as;

The general form of a transfer function is now known, in the next section there is an elaboration on computing the tf of a mass-spring-damper system.

TF of a mass-spring-damper system

Now we will investigate the transfer function of a mass-spring-damper system. The dynamics of such a system can be described by the following equation;

Now by taking the $\mathcal{L}aplace$ transform of this equation the transfer function can be found. Since this is an ordinary differential equation the $\mathcal{L}aplace$ transform can easily be computed by replacing the $n^{th}$ derivative with $s^{n}X(s)$.

In general, the $\mathcal{L}aplace$ transform is a more complex operation to take than there is shown now. However, a more in-depth theory on the $\mathcal{L}aplace$ can be found in Laplace. Also, in the next articles, there will be elaborated on the stability and control of various systems.