1. ISAT 412 -Dynamic Control of Energy Systems (Fall 2005)
The Laplace Transform
ISAT 412 -Dynamic Control of Energy Systems
(Fall 2005)

2. Second Order O.D.E. Solution
General solution to the second order, linear, constant coefficient, homogeneous O.D.E.
Assume

3. Characteristic Equation
Assuming this form gives us the characteristic equation for the 2nd order equation
Solve for the roots of the characteristic equation (factor or quadratic equation)

4. General 2nd Order, Homogeneous Solution
The general homogeneous solution to the second order equation is given by
We can express this result in three different ways depending upon the roots of the characteristic equation

5. Three cases Distinct real roots Repeated real roots
Complex conjugate roots

6. Particular Solution
The non-homogeneous solution is found in the same manner as for the 1st order equation
Assume that the particular solution takes the form of the non-homogeneous term (forcing function)

7. Laplace Transform
The Laplace transform solution method provides a systematic and general method for solving linear O.D.E.s

8. Laplace Transform
The Laplace transform takes the function x from the time domain into the Laplace (or “s”) domain
It transforms an O.D.E. in the time domain into an algebraic equation in the Laplace domain

9. Laplace Transform Notation
Denote by using the capitalized letter corresponding to the lower-case symbol of the transformed function

10. Solving an O.D.E. using the Laplace Transform
Transform the O.D.E. into the Laplace domain
Simplify the resulting algebraic equation and solve for the transformed variable of interest
Use the inverse Laplace transform to convert back to the time domain

11. Laplace Transform Pairs
Relate common function in the time domain to their Laplace transform in the “s” domain
Page 115 (and front cover) of your textbook

12. Properties of the Laplace Transform
Linearity
Note – this DOES NOT apply to products of functions!

13. Properties of the Laplace Transform
Consult page 117 of the textbook for additional properties of the Laplace transform
In particular, note the Laplace transform of first and second derivatives in the time domain
Properties 2, 3, and 4