1. Transforming C(s) into c(t): Negative Feedback Control with Proportional Only Controller Jigsaw Team Estrogen Stephanie Wilson Amanda Newman Jessica Raymond (We Laplace Transforms)

2. Goal: Advantages and disadvantages of different t 0 (dead time) values System: Any FOPDT system in a negative feedback loop with a Proportional-Only Controller

3. From Block Diagram Algebra… CLTF: Negative feedback control loop for FOPDT system with P-only controller

4. With Pade’s Approximation… Pade’s Approximation: CLTF:

5. This Reduces to… Therefore… CLTF:

6. Final Value Theorem In Laplace Domain 0 0 0

7. Now for C(t)… First we must simplify the denominator… for partial fraction decomposition CE:

8. Substitute for Easier Math… mpq CE:

9. Now Complete the Square

11. Simplify the last term Distribute the left side Factor the right

12. Solve for C

13. Solve for A

14. Solve for B

15. Substitute for m, p, q mpq

16. Separate the terms

17. Manipulate… to Fit Page 15 Formulas…

18. Inverse LaPlace Transform

19. Substitute A, B, C, m, p, q back into C(t)

20. Final Value Theorem In Time Domain 0 0

21. CONCLUSION Advantages of Increased t 0  Reaches steady state faster  Larger values of stable operating gain  Bigger gain is better Disadvantages of Increased t 0  Increase in t 0 results in a decrease in the ultimate 0 0gain of the loop (Kcu)  Poor performance  Limited values of stable operating gain