Problem $4$: $R_1=0\mathrm{.}5mi\frac{n}{f}{t}^2,R_2=2mi\frac{n}{f}{t}^2,A_1=10f{t}^2,K_v=2\mathrm{.}5f\frac{t^3}{m}i\frac{n}{m}A,A_2=0\mathrm{.}8f{t}^2,K_m=1\mathrm{.}5m\frac{A}{m}in,{}_m=0\mathrm{.}5$min.$q_1=\frac{1}{R_1}{h}_1,q_2=\frac{1}{R_2}{h}_2..[14\mathrm{.}22]$

$($a$)G(s)=\frac{H_1^{\text{'}}(s)}{Q_i^{\text{'}}(s)}.$

$($b$)$ ultimate gain $K_{cu}{}_c.$

$($c$){}_c=1\mathrm{.}369ra\frac{d}{m}in,K_{cu}.$

$($d$)$ Z$-$N $$ PI $?$

$($e$)$ PI ${}_c.$

$($f$){}_c=1\mathrm{.}086ra\frac{d}{m}in$ gain margin$.$

$($g$)$ Phase margin$,{}_g=0\mathrm{.}925ra\frac{d}{m}in.$

$($h$)$ Phase margin$.$

Figure $1$: Storage tank system