Chapter 4 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 initial-value 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 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.