$($d$)$ Design a PID controller in the frequency domain for the phugoid dynamics to control transient response. In addition, the PID controller must provide an overshot of less than $50\%,$ transient time $20s,$ and raising time $10s.$

The following table provides the coeficicents and their derivatives

w$.$r$.$to $x$ at the trim point:

$\backslash$table$[[C_T=0\mathrm{.}12,\frac{del{C}_T}{\text{d}\text{e}\text{l}\text{V}}=0\mathrm{.}00005,\frac{del{C}_T}{\text{d}\text{e}\text{l}\text{V}}=0\mathrm{.}001,\frac{del{C}_T}{\text{d}\text{e}\text{l}\text{q}}=0\mathrm{.}001,\frac{del{C}_T}{\text{d}\text{e}\text{l}}=0\mathrm{.}001$