1. Using Partial Fraction Expansion
Professor Walter W. Olson
Department of Mechanical, Industrial and Manufacturing Engineering
University of Toledo
Using Partial Fraction Expansion

2. Outline of Today’s Lecture
Review
Laplace Transform
Inverse Laplace Transform
Properties of the Laplace Transform
Final Value Theorem

3. Laplace Transform
Traditionally, Feedback Control Theory was initiated by using the Laplace Transform of the differential equations to develop the Transfer Function
The was one caveat: the initial conditions were assumed to be zero.
For most systems a simple coordinate change could effect this
If not, then a more complicated form using the derivative property of Laplace transforms had to be used which could lead to intractable forms
While we derived the transfer function, G(s), using the convolution equation and the state space relationships, the transfer function so derived is a Laplace Transform under zero initial conditions

4. Laplace Transform CAUTION: Some Mathematics is necessary!
The Laplace transform is defined as
Fortunately, we rarely have to use these integrals as there are other methods

5. Some Common Laplace Transforms
The Laplace Transform of the Impulse Function
The Laplace Transform of the Step Function
The Laplace Transform of a Unit Ramp:
The Laplace Transform of the 2nd power of t:
The Place Transform of the nth power of t:

6. Some Common Laplace Transforms
Laplace Trans Form of the exponentials:
Laplace Transforms of trigonometric functions:

7. Important Inverse Transforms

8. Properties of the Laplace Transform
Laplace Transforms have several very import properties which are useful in Controls
Now, you should see the advantage
of having zero initial conditions

9. Final Value Theorem
If f(t) and its derivative satisfy the conditions for Laplace Transforms, then
This theorem is very useful in determining the steady state gain of a stable system transfer function
Do not apply this to an unstable system as the wrong conclusions will be reached!

10. The Transfer Function

11. Partial Fraction Expansion
When using Partial Fraction Expansion, our objective is to turn the Transfer Function into a sum of fractions where the denominators are the factors of the denominator of the Transfer Function: Then we use the linear property of Laplace Transforms and the relatively easy form to make the Inverse Transform.

12. Partial Fraction Expansion
There are three cases that we need to consider in expanding the transfer function:
Case 1: All of the roots are real and distinct
Case 2: Complex Conjugate Roots
Case 3: Repeated roots

13. Case 1: Real and Distinct Roots

14. Case 1: Real and Distinct Roots Example

15. Case 1: Real and Distinct Roots An Alternative Method (usually difficult)

16. Case 1: Real and Distinct Roots Example

17. Case 2: Complex Conjugate Roots

18. Case 2: Complex Conjugate Roots Example

19. Case 3: Repeated Roots

20. Case 3: Repeated Roots Example

21. Heaviside Expansion

22. State Space Formulation
Example: The Nose Wheel

23. Example Now, we have a good view of the system structure
so that we can choose and adjust values, if needed

24. Example

25. Summary
Next Class: Nyquist Stability Criteria