Design controller to get this response

We have the following closed loop system where a is from 1 to 10 and p from 0 to 1(excluding 1).

We want to design controllers H(s) and C(s) so that the step response is inside this area This is a problem from an exam sheet so a computer can't be used. Anything I should do is by hand.

Ι though C(s) could be just a gain C(s)=k. Then I get the following closed loop transfer function. $$G(s)=\frac{ka}{s+1+p+ka}$$ Now I could pick the right value for k so that my function in time domain decays faster than e^-2t which is the case shown in the picture. However steady state error can't be zero because I don't know where p is. Final value theorem for step input gives for s=0 $$y(t=\infty)=H(0)\frac{ka}{1+p+ka}$$ In other problems without p in the denominator I would set this equal to 1 and find H(s) but here p is a moving pole. What do I do?

• What type of controller are you looking for? Saying "controller" is pretty vague. – KingDuken Jul 21 '17 at 21:42
• A 1st order system should do it, with appropriate constants for C(s) and H(s) – Chu Jul 21 '17 at 22:20
• @Chu I edited , can you check again what my problem is? – John Katsantas Sep 3 '17 at 18:54
• Try $C(s)=\frac{K}{s}$, $H(s)=1$. Envelope of underdamped 2nd order is $1\pm\frac{e^{-\zeta\omega_nt}}{\sqrt{1-\zeta^2}}$, which equates to $\small1\pm e^{-2t}$. For overdamped, just need the lower envelope. Do this for the four limiting conditions of a and p, and select the K that fits all! Haven't tried it - just a thought. Can't be more complicated than that for an exam question. – Chu Sep 4 '17 at 7:34
• I think $\zeta \omega_n>2$ does the job. – Chu Sep 4 '17 at 13:10