Consider a unity-gain negative feedback system consisting of the plant G(s) (given below) and a proportional-integral controller. Let the proportional gain and integral gain be 3 and 1, respectively. For a unit step reference input, the final values of the controller output and the plant output, respectively, are

$\mathrm{G}(\mathrm{s})=\frac{1}{\mathrm{s}-1}$

Given system can be drawn as shown below,

Here,

X(s) → Controller output

C(s) → Plant output

$C(s)=R(s)\frac{(3+\frac{1}{s})\left(\frac{1}{s-1}\right)}{1+(3+\frac{1}{s})\frac{1}{s-1}}$ = $R(s)\frac{(3s+1)}{s(s-1)+3s+1}$ = $\frac{(3s+1)}{s(s-1)+3s+1}\times \frac{1}{s}$

∴ c(∞) = $\underset{s\to 0}{lim}sC(s)=1$

$X(s)=[R(s)-C(s)](3+\frac{1}{s})$

$X(s)=[R(s)-C(s)]\left(\frac{3s+1}{s}\right)$

$\mathrm{x}(\mathrm{\infty })=lim{\mathrm{s}}_{\mathrm{s}\to 0}\mathrm{X}(\mathrm{s})=\underset{\mathrm{s}\to 0}{lim}[\mathrm{R}(\mathrm{s})-\mathrm{C}(\mathrm{s})](3\mathrm{s}+1)$

$x(\mathrm{\infty })=\underset{s\to 0}{lim}(\frac{1}{s}-\frac{1}{s}\frac{(3s+1)}{(s^2+2s+1)})(3s+1)$

$=\underset{s\to 0}{lim}\frac{1}{s}\left[\frac{s^2+2s+1-3s-1}{s^2+2s+1}\right](3s+1)$

$=\underset{s\to 0}{lim}\frac{1}{s}\left[\frac{s^2-s}{s^2+2s+1}\right](3s+1)$ = $\underset{s\to 0}{lim}\frac{(s-1)(3s+1)}{s^2+2s+1}=-1$