# Control Systems/Transfer Functions

## Transfer Functions

A **Transfer Function** is the ratio of the output of a system to the
input of a system, in the Laplace domain considering its initial
conditions and equilibrium point to be zero. This assumption is relaxed
for systems observing transience. If we have an input function of
*X(s)*, and an output function *Y(s)*, we define the transfer function
*H(s)* to be:

$$H(s) = {Y(s) \over X(s)}$$

Readers who have read the Circuit Theory
book will recognize the transfer function as being the impedance,
admittance, impedance ratio of a voltage divider or the admittance ratio
of a current divider.

![](Laplace_Block.svg "Laplace_Block.svg"){width="400"}

## Impulse Response

For comparison, we will consider the time-domain equivalent to the above
input/output relationship. In the time domain, we generally denote the
input to a system as *x(t)*, and the output of the system as *y(t)*. The
relationship between the input and the output is denoted as the
**impulse response**, *h(t)*.

We define the impulse response as being the relationship between the
system output to its input. We can use the following equation to define
the impulse response:

$$h(t) = \frac{y(t)}{x(t)}$$

### Impulse Function

It would be handy at this point to define precisely what an \"impulse\"
is. The **Impulse Function**, denoted with δ*(t)* is a special function
defined piece-wise as follows:

$$\delta(t) = \left\{
\begin{matrix} 
  0, & t < 0
\\
  \mbox{undefined}, & t = 0
\\
  0, & t > 0
\end{matrix}\right.$$

The impulse function is also known as the **delta function** because
it\'s denoted with the Greek lower-case letter δ. The delta function is
typically graphed as an arrow towards infinity, as shown below:

![](Delta_Function.svg "Delta_Function.svg")

It is drawn as an arrow because it is difficult to show a single point
at infinity in any other graphing method. Notice how the arrow only
exists at location 0, and does not exist for any other time *t*. The
delta function works with regular time shifts just like any other
function. For instance, we can graph the function δ*(t - N)* by shifting
the function δ*(t)* to the right, as such:

![](DeltaN_Function.svg "DeltaN_Function.svg")

An examination of the impulse function will show that it is related to
the unit-step function as follows:

$$\delta(t) = \frac{du(t)}{dt}$$

and

$$u(t) = \int \delta(t) dt$$

The impulse function is not defined at point *t = 0*, but the impulse
must always satisfy the following condition, or else it is not a true
impulse function:

$$\int_{-\infty}^\infty \delta(t)dt = 1$$

The response of a system to an impulse input is called the **impulse
response**. Now, to get the Laplace Transform of the impulse function,
we take the derivative of the unit step function, which means we
multiply the transform of the unit step function by s:

$$\mathcal{L}[u(t)] = U(s) = \frac{1}{s}$$

$$\mathcal{L}[\delta(t)] = sU(s) = \frac{s}{s} = 1$$

This result can be verified in the transform tables in **The
Appendix**.

### Step Response

Similar to the impulse response, the **step response** of a system is
the output of the system when a unit step function is used as the input.
The step response is a common analysis tool used to determine certain
metrics about a system. Typically, when a new system is designed, the
step response of the system is the first characteristic of the system to
be analyzed.

## Convolution

However, the impulse response cannot be used to find the system output
from the system input in the same manner as the transfer function. If we
have the system input and the impulse response of the system, we can
calculate the system output using the **convolution operation** as such:

$$y(t) = h(t) * x(t)$$

Where \" \* \" (asterisk) denotes the convolution operation. Convolution
is a complicated combination of multiplication, integration and
time-shifting. We can define the convolution between two functions,
*a(t)* and *b(t)* as the following:

$$(a*b)(t) = (b*a)(t) = \int_{-\infty}^\infty a(\tau)b(t - \tau)d\tau$$

(The variable τ (Greek tau) is a dummy variable for integration). This
operation can be difficult to perform. Therefore, many people prefer to
use the Laplace Transform (or another transform) to convert the
convolution operation into a multiplication operation, through the
**Convolution Theorem**.

### Time-Invariant System Response

If the system in question is time-invariant, then the general
description of the system can be replaced by a convolution integral of
the system\'s impulse response and the system input. We can call this
the **convolution description** of a system, and define it below:

$$y(t) = x(t) * h(t) = \int_{-\infty}^\infty x(\tau)h(t - \tau)d\tau$$

## Convolution Theorem

This method of solving for the output of a system is quite tedious, and
in fact it can waste a large amount of time if you want to solve a
system for a variety of input signals. Luckily, the Laplace transform
has a special property, called the **Convolution Theorem**, that makes
the operation of convolution easier: 

The Convolution Theorem can be expressed using the following equations:

$$\mathcal{L}[f(t) * g(t)] = F(s)G(s)$$

$$\mathcal{L}[f(t)g(t)] = F(s) * G(s)$$

This also serves as a good example of the property of **Duality**.

## Using the Transfer Function

The Transfer Function fully describes a control system. The Order, Type
and Frequency response can all be taken from this specific function.
Nyquist and Bode plots can be drawn from the open loop Transfer
Function. These plots show the stability of the system when the loop is
closed. Using the denominator of the transfer function, called the
characteristic equation, roots of the system can be derived.

For all these reasons and more, the Transfer function is an important
aspect of classical control systems. Let\'s start out with the
definition: 

If the complex Laplace variable is *s*, then we generally denote the
transfer function of a system as either *G(s)* or *H(s)*. If the system
input is *X(s)*, and the system output is *Y(s)*, then the transfer
function can be defined as such:

$$H(s) = \frac{Y(s)}{X(s)}$$

If we know the input to a given system, and we have the transfer
function of the system, we can solve for the system output by
multiplying:

$$Y(s) = H(s)X(s)$$

### Example: Impulse Response

### Example: Step Response

### Example: MATLAB Step Response

## Frequency Response

The **Frequency Response** is similar to the Transfer function, except
that it is the relationship between the system output and input in the
complex Fourier Domain, not the Laplace domain. We can obtain the
frequency response from the transfer function, by using the following
change of variables:

$$s = j\omega$$

![](Fourier_Block.svg "Fourier_Block.svg"){width="400"}

Because the frequency response and the transfer function are so closely
related, typically only one is ever calculated, and the other is gained
by simple variable substitution. However, despite the close relationship
between the two representations, they are both useful individually, and
are each used for different purposes.

