Transfer Functions | Control Systems 1.3

In the previous tutorial, we saw how we can model physical systems. In this tutorial, we shall move forward to learn about transfer functions. Before that, why do we need a transfer function? As we saw in the previous tutorial, a mathematical model of a system is simply an ordinary differential equation and to obtain the response of the system, we would have to solve that differential equation which is tedious. When we transform this equation to the s - domain using Laplace transforms, it reduces to simple algebraic equations that are relatively easy to solve. This transformed model of the system in the s-domain is called a transfer function.

Now let’s look at a more formal definition of a transfer function.

Mathematically speaking, if C(s) is the Laplace transform of the output function and R(s) is the Laplace transform of the input function, then the transfer function G(s) is given by:

Let’s take a look at this quick example.

A unit step signal is given as input and the output is obtained as shown above.

Let’s refer to this table and take the Laplace transform.

The input applied to the system is:

Which, then taking the Laplace transform, you get,

The output, on the other hand, is,

Taking the Laplace transform of this, we get,

With this input and output relation, we find the transfer function is simply:

Now, let’s consider a different scenario. What if R(s) (in the Laplace, or s-domain) is equal to 1? In that case, G(s) = C(s). This means the transfer function is just the Laplace transform of the output when the Laplace transform of the input is 1. This now gets us thinking, what is the input whose Laplace transform is 1? It’s the impulse signal or impulse function.

An impulse signal "δ(t)" is a tall and narrow signal which is obtained as shown.

As a becomes very small (almost zero), 1/a becomes infinitely tall. This means that the width of the above shown rectangle becomes essentially 0 and the height becomes infinite yet their area remains equal to 1.

Hence the impulse signal or the impulse function takes zero value for all values of time except for t = 0.

As the area remains equal to 1, we can say

When we apply this impulse signal as the input to a system, the output thus obtained is called the impulse response. Practically speaking, an impulse signal is like a sudden short-lived disturbance to the system and impulse response is just how the system would react naturally to this quick disturbance. Let’s use an example to be more clear. Consider a pendulum.

When you flick the pendulum with your fingers, that acts as an impulse. And the manner in which the pendulum reacts is its impulse response.

As we discussed earlier, if impulse δ(t) is the considered the input, then its Laplace transform equivalent,

And as per the definition of transfer function,

So, the transfer function is the same as the output in the case of an impulse input.

With these two examples, you can see how the input and output are related through the transfer function. Hence, the transfer function is just the Laplace transform of the output that is obtained when an LTI system is excited with an impulse signal.

We shall discuss steps to be followed to determine the transfer function of a system with the help of the simple RLC circuit which we modelled in the previous tutorial.

Step 1: Determine the mathematical model equations of the given system.

Applying Kirchhoff’s voltage law to the loop shown above,

Step 2: Identify the system’s input and output variables.

Here v_i(t) is the input and v_o(t) is the output.

Step 3: Transform the input and output equations into s-domain using Laplace transforms assuming the initial conditions to be zero.

In this example, we assume the initial current through the inductor to be zero and the initial voltage across the capacitor to be zero.

Now, let’s take the Laplace transform of the obtained input and output equations. As these are fairly simple, we can do a straight conversion using the table.

Step 4: Obtain the ratio of the Laplace transform of the output to the Laplace transform of the input.

In summary, we first found the equations in the time domain, and then took the Laplace transform of those equations assuming initial conditions to be zero. We then take the ratio of the output to input, giving us the transfer function. This general procedure can be followed to find the transfer function of most of the systems.

It is worth noting some important points regarding transfer functions here:

• Transfer functions can be used to study the response of the system to various inputs and understand the nature of the system which we shall be seeing in the coming tutorials.
• Transfer functions are independent of the nature and magnitude of the input.
• Transfer functions do not tell anything regarding the composition of the system. This means that it is possible for different systems to have the same transfer function.

Transfer functions in general are represented as shown.

Here,

The poles are just the roots of the denominator polynomial of the transfer function. Similarly, the zeros are the roots of the numerator polynomial of the transfer function. In other words, the poles are those values of s for which the transfer function becomes infinite and zeros are those values of s for which the transfer function becomes zero. The following example will make this more clear.

Let’s take a transfer function,

We shall now try and determine the poles and zeros of this transfer function.

Zeros - Roots of numerator polynomial.

Poles - Roots of denominator polynomial.

Poles and zeros can be purely real numbers or can occur in complex conjugate pairs. We shall be extensively using these terms in this tutorial series.

At the end of the previous tutorial, we modeled a DC motor where we obtained the mathematical model equation:

Where,

For simplicity, let’s consider all the constants to be equal to one.

Taking Laplace transform, we get

Let the speed of the motor be considered as the output, so we know:

Taking the Laplace transform of the output,

Now, per the definition of transfer function,

Hence, we have obtained a transfer function of a DC motor. Now with this, we shall simulate this transfer function with Scilab XCOS.

To get started with XCOS, refer the below tutorial:

And as we run the simulation, we obtain

All we need to infer from simulating this model is that as we apply a certain voltage to a motor, the speed of the motor rises gradually from 0 and settles at a constant value and hence the transfer function can be used to analyze the behavior of the system under various inputs. Let’s not worry much about the response now as we shall do a detailed analysis of system response in the coming tutorials.

In this tutorial, we started with defining a transfer function and then we obtained the transfer function for a series RLC circuit by taking the Laplace transform of the voltage input and output the RLC circuit, using the Laplace transform table as a reference. We then looked at some properties of transfer functions and learnt about poles and zeros. At the end, we obtained the transfer function of the DC motor we modelled in the previous tutorial and then had a look at its simulated response.

In the next tutorial, we shall be learning about the block diagrams and techniques to reduce block diagrams. Until then, try obtaining the transfer function of the parallel RLC circuit and the mechanical systems that we had modelled in the previous tutorial as that will greatly help with retention as we move forward.