Chapter 3 Laplace Transforms Overall Course Objectives Develop

Chapter 3 Laplace Transforms

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

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.

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.

Method for Solving Linear ODE’s using Laplace Transforms

Some Commonly Used Laplace Transforms

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.

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

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

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.

Heaviside Method Individual Poles

Heaviside Method Repeated Poles

Heaviside Method Example with Repeated Poles

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

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