Old spec 5.4 example, closed-loop stability from the Bode diagram (ENGR5001)

Old spec 5.4 example, closed-loop stability from the Bode diagramTutorialOld spec ENGR2022:115 min

A unity feedback control system has the open-loop Transfer Function $KG(s) = \frac{K}{s(s + 1)^2}$

Formulas you may need

Unity feedback around $KG(s)$ : closed-loop characteristic equation $1 + KG(s) = 0$ ; marginal stability when $|KG(j\omega)| = 1$ at the frequency where $\mathrm{Arg}\,KG(j\omega) = -180^\circ$ (learn this)
Gain margin $= 1/|KG(j\omega_{pc})|$ at the phase crossover; phase margin $= 180^\circ + \mathrm{Arg}\,KG(j\omega_{gc})$ at the gain crossover (learn this)

(a)
Find the closed-loop Transfer Function and characteristic equation. Show that for $K = 2$ the closed loop is marginally stable, giving the closed-loop poles.
(b)
Using the open-loop frequency response, find the phase crossover frequency and $|KG|$ there, and hence the value of $K$ at the stability limit. Classify $K = 0.1$ , 2 and 10.
phase crossover frequency =rad/s
critical K =
(c)
For $K = 0.1$ , find the gain margin (in dB) and the phase margin.
gain margin =dB
phase margin =deg