*This is the first in a series of posts that are an attempt by me to understand why my *Industrial Measurement and Control *students need to study the Laplace Transform**.*

Consider a black box model that takes as an input signal a function and produces an output signal . For reasons that are as of yet not clear to me, we can take all of the derivatives of (and ) to vanish for .

### Definition

The *transfer function *of a black box model is defined as

,

where and are the Laplace Transforms of and .

Note we have the Laplace transform of a function is a function defined by

Poles of are complex numbers such that . For example, for an input signal , the transfer function has a pole at as

.

If the coefficients of are real, then in general the poles of the transfer are real or come in conjugate root pairs.

### Derivation of Formula for Transfer Function — Linear ODE Case

Consider a linear input/output system described by the differential equation:

,

where we can take .

If we assume that the input signal is of the form then because the system is linear we must also have the output signal as . Inserting these signals into the ODE yields:

,

and so the response of the system can be described by the two polynomials

(with )

and

.

The polynomial is nothing but the characteristic/auxiliary equation of the associated homogenous system. If it is not everywhere zero then we have

.

Therefore, the transfer function is

.

### Example: Complex Harmonic Signal

Suppose we have an input signal

,

and an output signal

.

Then we have

and

.

This is the *gain. *

### Example: Closed Loop Transfer Function

In this model, the output is measured and compared with the reference (ideal output) value . The controller then takes the error, , and changes the input of the system under control, , using some controller. This is known as a *closed-loop controller *or a *feedback controller.*

If we assume that the controller, system and sensor are linear and time-invariant (their transfer functions , and do not depend on time), we can use the Laplace transform to analyse the system. We have

,

and

.

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