Question No$.01$

Continuous Stirred Tank Reactor

The above figure shows a continuous stirred tank reactor $($CSTR$)$ in which an aqueous solution of sodium acetate $(CH3\frac{C}{O}ONa)$ is neutralized with hydrochloric acid $(HCl)$ to maintain a particular $pH$ value. The amount of acid in the mix is controlled by varying the rotational speed of a feeding peristaltic pump. A nominal linearized transfer function between $HCl$ flowrate to $pH$ is given on the next page.

$G(s)=\frac{0\mathrm{.}9580{10}^{-4}s-0\mathrm{.}01197{10}^{-4}}{s^3+0\mathrm{.}5250s^2+0\mathrm{.}0126s+0\mathrm{.}000078}$

a$.$ Find the system in state$-$space phase$-$variable$-$form.

b$.$ Determine the controllability and observability of the CSTR$.$

c$.$ Use state$-$feedback methods to design a regulator that will yield an overdamped output $pH$ response with a settling time of $Ts=5$ minutes for a step input change in $pH.$