Why Is the Box in Red the Transfer Function?

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Discussion Overview

The discussion revolves around understanding the components of a transfer function in a control system context, specifically addressing the roles of different terms in the equation Y(s) = G(s)U(s) + additional terms. Participants explore the definitions of forced and free responses, the implications of initial conditions, and the stability of the system based on the given expressions.

Discussion Character

  • Homework-related
  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • Some participants propose that the box in red represents the transfer function because it relates how the input U is transferred to the output Y, specifically being the coefficient of U in the equation.
  • Others argue that the distinction between the free response and forced response is based on whether the response is influenced by the input U, with the free response not being driven by U.
  • Participants question how the terms are identified as free or forced responses and seek clarification on the reasoning behind these classifications.
  • There is a discussion about why the equation includes additional terms beyond the standard form Y(s) = G(s)U(s), with some noting that these terms arise from initial conditions of Y, Y', and Y'' that are independent of U.
  • Some participants mention that the expression 3e^(-t) - e^(-3t) indicates system stability, as the exponential terms diminish over time, suggesting that disturbances do not grow exponentially.

Areas of Agreement / Disagreement

Participants express varying levels of understanding regarding the classification of terms in the transfer function and the implications of initial conditions. While some points are clarified, the discussion remains unresolved on certain aspects, particularly the reasoning behind the classification of responses.

Contextual Notes

The discussion highlights the dependence on definitions of forced and free responses, as well as the role of initial conditions in determining the form of the transfer function. There are unresolved questions about the classification of terms and the implications for system behavior.

influx
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Homework Statement


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Homework Equations


N/A

The Attempt at a Solution


I understand how they got the answer and the calculations they did but I have 3 questions about this screenshot.

1) Why the box in red is the transfer function? Is there a way to tell this from the Y(s) = ... expression?
2) Why is the second term the free response (green box) and the first term the forced response (blue box)?
3) Why does the 3e^(-t) - e^(-3t) confirm that the system is stable?

Thanks
 
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influx said:
1) Why the box in red is the transfer function? Is there a way to tell this from the Y(s) = ... expression?
That is the part that relates how U is transferred to Y. It is the coefficient of U in the equation.
2) Why is the second term the free response (green box) and the first term the forced response (blue box)?
The free response is not "forced" by the input U. The "forced" response is forced by the input U.
3) Why does the 3e^(-t) - e^(-3t) confirm that the system is stable?
As time, t, increases in the positive direction, the exponentials disappear. If the exponents were positive, the any tiny disturbance would grow exponentially.
 
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FactChecker said:
The free response is not "forced" by the input U. The "forced" response is forced by the input U.

I understand that but how did this then lead to the conclusion which of the terms is which?

Thanks
 
FactChecker said:
That is the part that relates how U is transferred to Y. It is the coefficient of U in the equation.

Generally we've Y(s) = G(s)U(s) but in this case it's Y(s)=G(s)U(s) + another term. Is there a reason why we don't have the usual Y(s) = G(s)U(s) ?
 
influx said:
Generally we've Y(s) = G(s)U(s) but in this case it's Y(s)=G(s)U(s) + another term. Is there a reason why we don't have the usual Y(s) = G(s)U(s) ?
The other terms are coming from the initial conditions of Y, Y' and Y''. U is not involved in driving those. The current Y, Y' and Y'' are called state variables. Since some of them have nonzero initial values, their effect is independent of U and is added in.
 
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FactChecker said:
The other terms are coming from the initial conditions of Y, Y' and Y''. U is not involved in driving those. The current Y, Y' and Y'' are called state variables. Since some of them have nonzero initial values, their effect is independent of U and is added in.

That makes sense. Thanks
 

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