How do you find the zero-input response in the frequency domain?

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SUMMARY

The discussion focuses on finding the zero-input response of a system in the frequency domain, specifically addressing the equation required for its calculation. The zero-input response is determined by incorporating initial conditions into the analysis, which is essential for accurate results. The participant initially struggled with this concept but ultimately clarified that setting the input to zero is necessary, alongside considering initial conditions. This understanding resolves the confusion surrounding the calculation of the zero-input response.

PREREQUISITES
  • Understanding of frequency domain analysis
  • Familiarity with system response concepts
  • Knowledge of initial conditions in control systems
  • Basic proficiency in Laplace transforms
NEXT STEPS
  • Study the derivation of zero-input response equations in control systems
  • Learn about the role of initial conditions in system dynamics
  • Explore Laplace transform applications in frequency domain analysis
  • Review examples of zero-input and zero-state responses in engineering textbooks
USEFUL FOR

Control system engineers, electrical engineers, and students studying system dynamics who need to understand the calculation of zero-input responses in the frequency domain.

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I'm working on a problem and one part of it is finding the zero-input response of a system in the frequency (or "s") domain.

The zero-state response can be easily found in the frequency domain, it's just:

Y_{zs} = X(S)H(S)

But I can't for the life of me find the equation for calculating the zero-input response. I've been looking through my textbook and online and simply can't find anything.. How is it done?
 
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Well, after considerable effort I finally figured it out. Zero-input means you set the input to zero :-)

Actually, there was more to it than that. You have to include initial conditions in the equation or it won't work, and that was the part that was tripping me up.

Case closed!
 

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