1 / 16

Mastering Laplace Transforms for Industrial Process Control Engineering

Develop essential skills in tuning loops, control loop design, and troubleshooting. Gain a fundamental understanding of process dynamics and feedback control using Laplace Transforms. Learn to solve linear ODEs, apply Initial- and Final-Value Theorems, and utilize methods like Partial Fraction Expansions and Heaviside Method.

carolynbridges
Download Presentation

Mastering Laplace Transforms for Industrial Process Control Engineering

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Chapter 3 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.

More Related