Steady state error using final value theorem

Question

From the continuous time control system above, i need to find the steady state error using the final value theorem in response to a unit ramp input signal. How do I begin solving this kind of question. I've looked at many examples but couldn't figure out how to start with this one.

Answer 1

As proposed by @Zeeklees you need to find the transfer function from the reference $r(s)$ to the error $e(s)$

$$H(s)=\dfrac{e(s)}{r(s)}=\dfrac{1}{1+G_c(s)G(s)}$$

In order to get this result look at the summation point here, we have

$$e(s) = r(s)-G_c(s)G(s)e(s).$$

Solve this for $e(s)/r(s)$ to get the previous result.

The final value theorem states that (you have to check the conditions under which you can apply the theorem!)

$$\lim_{t\to \infty}e(t) =\lim_{s\to 0^+}se(s)=\lim_{s\to 0^+}sH(s)r(s)$$

for a unit step response we have $r(s)=1/s$. Hence we obtain

$$\lim_{t\to \infty}e(t) =\lim_{s\to 0^+}H(s)=\lim_{s\to 0^+}\dfrac{1}{1+G_c(s)G(s)}$$