1. ABE 463 Electro-hydraulic systems Laplace transform Tony Grift
Dept. of Agricultural & Biological Engineering University of Illinois

2. Pierre-Simon Laplace “The French Newton” (1749-1827)
Why do we need a Laplace Transform?
Definition Laplace Transform
Laplace Transform of functions
Unit step function
Ramp function
Exponential function
Cosine/Sine
Impulse function (dirac delta)
Laplace Transform of operations
Convolution

3. The Laplace transform can be used to transform a differential equation into an algebraic equation that can be solved. After transforming back to the time domain we obtain a solution of the differential equation in time.
Time domain:
Differential equation
Solution in
time domain
Inverse Laplace Transform
s-domain: algebraic equation
Solution in s-domain

4. The Laplace transform is a linear operation
Red frame: Important result

5. Laplace transform of unit step function
Definition Laplace Transform
The variable s is a constant under integration with respect to t

6. Laplace transform of a ramp function
Integration by parts
Blue frame: You should know this already

7. Laplace transform of an exponential function

8. Laplace transform of cosine function

9. Laplace transform of sine function

10. Laplace transform of an impulse ‘function’ (Dirac delta distribution)e
1/e
Writing as a McLaurin series
Writing as a McLaurin series

11. Laplace transform of impulse (Dirac delta distribution)

12. Check Laplace Transform of differentiation operation
Example
Is this correct?

13. Laplace transform of operations

14. Laplace transform of differentiation operation
Product rule:
Integration by parts

15. Laplace Transform of a function shifted in time

16. Laplace transforms of common functions and operations

17. Initial value and final value theorems

18. Initial value theorem proof
Flip over the limits

19. Final value theorem proof (simplified)

20. Final value theorem proof
Interchange the limits

21. Convolution

22. Convolution example: Moving average filter
1st iteration
2nd iteration
General
Continuous case

23. Convolve this vector
MatLab: conv(a,[3 -2 1])

24. Correct answer

25. Convolution in time domain = Multiplication in Laplace domain

26. The time equivalent of multiplication in the Laplace (and also Fourier) domain is called convolution
Impulse response of the system
at t - u
The total response is in fact the sum of all impulse responses over time weighted (multiplied) by the input signal

27. ABE 463 Electro-hydraulic systems Laplace transform The End
Dept. of Agricultural & Biological Engineering University of Illinois