I am given the following diagram:
Now, I am being asked to find the $p, a, b, k$ variables respectively with given steady-error function $e(t) = \dfrac{1}{2}(e^{-2t}+e^{-8t})$ with $U(s) = \dfrac{1}{s}$. Currently, I found the transfer functions as follows:
$$ \begin{align*} E(s) & = R(s) - Y(s) \\ Y(s) & = G(s)[E(s)H(s)] \\ E(s) & = \dfrac{1}{1+G(s)H(s)} = \dfrac{s^p(bs+1)}{s^p(bs+1) + (s+a)} \end{align*} $$
$$ \begin{align*} G_{yr}(s) = \dfrac{k(s+a)}{s^p(bs+1) + (s+a)} \\ G_{er}(s) = \dfrac{s^p(bs+1)}{s^p(bs+1) + (s+a)} \\ \end{align*} $$
Based on the steady-error function, I wrote the E(s) like this:
$$ E(s) = \mathscr{L}[e(t)] = \dfrac{1}{2}(\dfrac{1}{s+2} + \dfrac{1}{s+8}) = \dfrac{s + 5}{s^2+10s+16} \equiv G_{er}(s) $$
Therefore, I can get $p = 1, a = \dfrac{16}{5}, b = \dfrac{1}{5}$. From the right graph, I can tell that the steady-state error should be $0$. So, I attempted to use the final-value theorem,
$$ e = 0 = \lim_{s\to0}sE(s) = \lim_{s\to0}s{\dfrac{s+5}{s^2+10s+16}\dfrac{1}{s}} = \dfrac{5}{16} $$
However, I am not sure where to get the specific value $k$ that fits into the equation. To me, I know that the $k$ which is usually called a proportional gain in PID control system has the use to reduce steady-state error, but how can I establish a well equation to find the value $k$?
Thank you for your time and help.
From the $y(t)$ graphics and considering $p=1$
$$ y_{ss}=1=\lim_{s\to 0}sR(s)G(s) = \lim_{s\to 0}s\frac 1s\frac{k(s+a)}{s(bs+1)+s+a}=k $$
hence $k=1$