Chapter 4: The Laplace Transform
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In control system, mathematical modelling plays an important role. With the presence of mathematical equation, differential equation is often present in it. And we must solve for its solution, and here comes the Laplace transform. If you want to learn about the Laplace transform you can google it. You still need to manually solve for Laplace transform because as an engineer, it’s a must. We have the most powerful tool, MATLAB®, but manual solving is an engineer’s must skill. But nevertheless, Matlab will surely help us solve for the solution.
So to start,
I want you to pick up your pen/pencil and solve for the Laplace transform of
this equation:
t3u(t) <--> ??
Did you get it? So let me help you,
Solution:
t4u(t) = ℒ{t4}
tn
= t4 = f(t)
F(s)
= ℒ{f(t) = t4}
= n!/sn + 1 = 24/s5
Let’s check in MATLAB. In order to do it in
MATLAB, we need to have a variable (just like in the equation we need to use
the letters ‘s’ and ‘t’). A variable which has no value, but just a symbol for
our computation. To do this, we type:
>>syms s t w x
Then we type our function:
>>Fs = laplace(t^3) %use capital Fs to denote it is in
laplace form
Fs =
24/s^5
So we are correct in our manual computation. What
if we are given the Laplacetransorm and we need to get the function. It’s easy,
again we type:
>>syms s t w x %with space each letters and from syms
>>ft = ilaplace(24/s^5) %use small ft to denote the function
ft =
t4
another example:
>>ft = ilaplace((s+1)/(s^3+5*s+6))
ft =
2*exp(-3*t) – exp(-2*t) %exp
means the letter ‘e’ function in our calculator or the euler’s constant
So that’s it the basic of laplace
transform. You could try more challenging laplace transform, or other function.
Just try also to solve it manually and to be not too much dependent on the software.
Thanks.
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