What Is the Steady State Value of U with Both R and V as Unit Steps?

[Pavoo]

Homework Statement

Consider this control system below:

R = set point
E = remaining error
V = interference

My question is, if both R and V are unit step $\frac{a}{s}$, what will the value of U be when time t$\rightarrow$$\infty$ ?

Homework Equations

This question is based on the final value theorem of Laplace transform such as:

Other relevant transfer functions:

$\frac{U}{V}=\frac{-GR*GP}{1+GR*GP}$

$\frac{U}{R}=\frac{GR}{1+GR*GP}$

The Attempt at a Solution

$\lim_{s\rightarrow0}\frac{U}{V}+\lim_{s\rightarrow0}\frac{U}{V}=s*\frac{a}{s}*\frac{-GR*GP}{1+GR*GP}+s*\frac{a}{s}*\frac{GR}{1+GR*GP}$

Is that an ok solution?
The question is how do I use the final value theorem if BOTH signals are step? I know how to do it with one signal, that's easy, but how do I calculate when two signals are step?
I have no answer to this question, but rather asking how I should approach these kind of problems.

 

[mighty2000]

you can approach this problem by using the final value theorem to solve for the steady state value of U. The final value theorem states that the steady state value of a signal is equal to the limit of the signal as s approaches zero. In this case, both R and V are unit step functions, so their limits as s approaches zero will be equal to 1.

Using the transfer functions given, you can solve for the steady state value of U by substituting 1 for both R and V. This will give you the following equations:

\frac{U}{V}=\frac{-GR*GP}{1+GR*GP}=\frac{-GR*GP}{1+GR*GP}=1

\frac{U}{R}=\frac{GR}{1+GR*GP}=\frac{GR}{1+GR*GP}=1

Solving for U, we get:

U=\frac{-GR*GP}{1+GR*GP}=\frac{GR}{1+GR*GP}

Since both R and V are unit step functions, their values will remain constant at 1 as time approaches infinity. This means that the value of U will also approach a constant value as time approaches infinity.

In summary, the value of U when t approaches infinity will be equal to \frac{GR}{1+GR*GP}.

it is important to understand and use mathematical tools such as the final value theorem to solve problems in control systems. It is also important to approach problems systematically and to use relevant equations and transfer functions to solve for unknown variables.