Laplace Transforms

Who on earth was this Laplace and why did he invent Laplace Transform?

Well, the story goes like The life of Pierre-Simon, Marquis de Laplace is only tangentially related to the topic of Control systems.

Though after the development of Laplace transform it allowed Linear Differential Equations which approximate the behavior of the system. His method changed how we see control systems.

By doing Laplace transformations we could get a system into algebraic equations which is easier to solve. 

What does Laplace Transform do???

Let me tell you this in a very raw manner,

It solves very complex problems in a very simple manner hence it is a key element to control systems.

Block Diagram is very useful to be able to represent a system as a series of blocks with discrete inputs and outputs.

In order for a model of a system to work, we must be able to express the behavior of a system mathematically and to represent the output of a system as a function of its inputs.

While doing Linearization we know how a complex system could be represented by a simplified linear differential equation. This is also very useful, but these sorts of equations cannot be used to develop a block diagram because the input and output variables cannot easily be isolated. Take, for instance, the most general form of a linear differential equation:

For even a low-order differential equation of this form, it is prohibitively complex to isolate the input and output terms in order to intuit the behavior of the system. Ultimately it is the ability to simplify a system's behavior enough to be able to make informed design decisions. This is where the Laplace transform comes in. Because it allows for differential equations to be solved in a simple algebraic format, it also allows for the inputs and outputs of a system to be isolated and expressed in a much more intuitive form.

Significance of Laplace Transform

The Laplace transformation is a relation between Two functions: f(t) which is defined with the relation to the time variable and F(s) which is defined in the complex frequency variable

The easiest way to describe Laplace Transform is as a more general form of Fourier Transform. Where the Fourier transform takes a periodic signal and deconstructs it into a component of sine and cosine waves, The Laplace Transform but with the added capability to account for growth & Decay of the signal over time. the value of the ability to graph system response with respect to the real and complex components of s will be explored in the root locus module, but suffice to say that S-Plane is a powerful tool.

For now, all you need to understand is that the complex frequency variable allows for a practical way to represent the frequency response of a system and is a useful frame for doing algebraic analysis of a system response.

How does it work

The transform is defined by a relation

so if the integral on the right side of that equation converges, then we know that there exists a function F (s) which is also referred to as the Laplace Transform of the function f(t).

Conversely, given a frequency domain function F(s) the process can be reversed using inverse Laplace transform which looks like this:

First of all, u(t) is a step function centered at 0, so the resulting function f(t) is only defined for a time greater than zero. Also, you have to make an informed decision about the value of 𝛿 so that the resulting integral converges.

I know Maths is Hard. and that's why we don't use actual integration to all these transformations. 

We just need to remember the following table and you will be fine calculating pretty much many things.

There are also some rules that can be used to shift, modify, or combine known transforms to find the Laplace Transform of a new function. The first and most useful is that the Laplace transform is linear, meaning that L[k f(t) = K F(s) and L[g(t) +h(t)] = G(s) + H(s). This allows for simple scaling and a combination of known transforms. For example:

You can also shift a known transform in the time domain with L[F (t-T)]= e-sTF(s) or in the frequency domain with L[e-at f(t)] = F(s+a). Scaling a known transform follows the relation L[f(at)] = F(). So we can take a sine function and scale it like so:

You can also shift a known transformation in the domain with 

or in the frequency domain with 

Scaling a known transform follows the relation 

So we can take a sine function and scale it like so:

Finally, you can take the nth derivative of a time function using this formula:

where fn (0-) is the initial value of the nth derivative of the function f(t). This will become very important when we look at applications of Laplace transforms while learning Block Diagrams. 

Conversely, you can take the integral of a function using this formula :

See you further in the Next Article of Block Diagrams...