A unity feedback system has an open-loop transfer function given by

\[
G(s)=\frac{250}{s[(s / 10)+1]}
\]

The following lag compensator added in series with the plant yields a phase margin of \(50^{\circ}\),

\[
D_{c}(s)=\frac{(s / 1.25)+1}{50 s+1} .
\]

(a) Using the matched pole-zero approximation, determine an equivalent digital realisation of this compensator.

(b) The following transfer function is a lead network designed to add about \(60^{\circ}\) of phase at \(\omega_{1}=3 \mathrm{rad} / \mathrm{sec}\),

\[
H(s)=\frac{s+1}{0.1 s+1}
\]

Assume a sampling period of \(T=0.25 \mathrm{sec}\), and compute and plot in the z-plane the pole and zero locations of the digital implementations of \(H(s)\) obtained using (i) Tustin's method and (ii) pole-zero mapping. For each case, compute the amount of phase lead provided by the network at \(z_{1}=e^{j \omega_{1} T}\).

(c) Using log-scale for the frequency range \(\omega=0.1\) to \(\omega=100 \mathrm{rad} / \mathrm{sec}\), plot the magnitude Bode plots for each of the equivalent digital systems found in part (a), and compare with \(H(s)\). (hint: Magnitude Bode plots are given by \(|H(z)|=\left|H\left(e^{j \omega T}ight)ight|\) )