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)
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
From Block Diagram Algebra… CLTF: Negative feedback control loop for FOPDT system with P-only controller
With Pade’s Approximation… Pade’s Approximation: CLTF:
This Reduces to… Therefore… CLTF:
Final Value Theorem In Laplace Domain 0 0 0
Now for C(t)… First we must simplify the denominator… for partial fraction decomposition CE:
Substitute for Easier Math… mpq CE:
Now Complete the Square
Simplify the last term Distribute the left side Factor the right
Solve for C
Solve for A
Solve for B
Substitute for m, p, q mpq
Separate the terms
Manipulate… to Fit Page 15 Formulas…
Inverse LaPlace Transform
Substitute A, B, C, m, p, q back into C(t)
Final Value Theorem In Time Domain 0 0
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