System Control Query - Find y(t) from Y(s)

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SUMMARY

The discussion focuses on finding the inverse Laplace transform to derive y(t) from the given Y(s) function in control systems. The provided function is Y(s) = (1/s) * (k / (s(s+1)+k)), which simplifies to Y(s) = (1/s) * (k / ((s+1/2)^2 + (k-1/4))). The inverse Laplace transform yields y(t) = (k / √(k-1/4)) * ∫(0 to t) e^(-t/2) * sin(√(k-1/4) * t) dt. This formula is crucial for analyzing system responses in control theory.

PREREQUISITES
  • Understanding of Laplace transforms and their properties
  • Familiarity with control systems and system response analysis
  • Knowledge of integral calculus, specifically definite integrals
  • Experience with mathematical modeling in engineering contexts
NEXT STEPS
  • Study the properties of the Laplace transform in detail
  • Learn about the application of inverse Laplace transforms in control systems
  • Explore numerical methods for evaluating integrals, particularly in engineering
  • Investigate the stability analysis of control systems using the derived y(t)
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Control engineers, system analysts, and students studying control theory who need to understand the application of Laplace transforms in system response analysis.

DrMath
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Dear PF Friends,

Given the system to be controlled.
I'm stuck in finding the inverse laplace (to find y(t) from Y(S)) - any advice?
 

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Must forgive me if it not correct. I'm trying

Y(s)=\frac{1}{s} ( \frac{k}{s(s+1)+k}) = \frac{1}{s} ( \frac{k}{(s+1/2)^2+ (k-1/4)})

Invert
y(t)=\frac{k}{\sqrt{k-1/4}} \int_0^t e^{-t/2} \sin{(\sqrt{k-1/4})t } dt
 

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