$8\mathrm{.}7.$ a$)$ Calculate the roots of the characteristic equations and relate them to the

dynamic behaviors of the closed$-$loop systems in Figure $8\mathrm{.}4$a through d$.$

FIGURE $8\mathrm{.}4$

Disturbance responses for three$-$tank mixing process under proportional control subject to

a disturbance in feed composition $(x_A{)}_B$ of $0\mathrm{.}8\%$A and open$/(\%$A$)]$: $(a)$ without

control; $($b$)$ with proportional control, $K_c=10$;$(c)$ with proportional control, $K_c=100$;

$($d$)$ with proportional control, $K_c=220.$