1. Chapter 4
Laplace Transforms

2. Overall Course Objectives
Develop the skills necessary to function as an industrial process control engineer.
Skills
Tuning loops
Control loop design
Control loop troubleshooting
Command of the terminology
Fundamental understanding
Process dynamics
Feedback control

3. Laplace Transforms
Provide valuable insight into process dynamics and the dynamics of feedback systems.
Provide a major portion of the terminology of the process control profession.

4. Laplace Transforms Useful for solving linear differential equations.
Approach is to apply Laplace transform to differential equation. Then algebraically solve for Y(s). Finally, apply inverse Laplace transform to directly determine y(t).
Tables of Laplace transforms are available.

5. Method for Solving Linear ODE’s using Laplace Transforms

6. Some Commonly Used Laplace Transforms

7. Final Value Theorem
Allows one to use the Laplace transform of a function to determine the steady-state resting value of the function.
A good consistency check.

8. Initial-Value Theorem
Allows one to use the Laplace transform of a function to determine the initial conditions of the function.
A good consistency check

9. Apply Initial- and Final-Value Theorems to this Example
Laplace transform of the function.
Apply final-value theorem
Apply initial-value theorem

10. Partial Fraction Expansions
Expand into a term for each factor in the denominator.
Recombine RHS
Equate terms in s and constant terms. Solve.
Each term is in a form so that inverse Laplace transforms can be applied.

11. Heaviside Method Individual Poles

12. Heaviside Method Individual Poles

13. Heaviside Method Repeated Poles

14. Heaviside Method Example with Repeated Poles

15. Example of Solution of an ODE
ODE w/initial conditions
Apply Laplace transform to each term
Solve for Y(s)
Apply partial fraction expansions w/Heaviside
Apply inverse Laplace transform to each term

16. Overview
Laplace transforms are an effective way to solve linear ODEs.