Why Is the Box in Red the Transfer Function?

[influx]

Homework Statement

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

 

[FactChecker]

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.

 

[influx]

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

 

[influx]

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) ?

 

[FactChecker]

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.

 

[influx]

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