Old spec 5.1 example, PI control stability and steady state (ENGR5001)

Old spec 5.1 example, PI control stability and steady stateTutorialOld spec ENGR2022:112 min

ENGR202 (old spec) Control 5.1 PI Control, closed loop, Hurwitz and steady state (slides 7-12)

PI control $C(s) = k_p + k_I/s$ is applied, with unity negative feedback, to the plant $X = \dfrac{\omega_n^2}{s^2 + 2\zeta\omega_n s + \omega_n^2}U$ .

Formulas you may need

PI control law $u(t) = k_pe(t) + k_I\displaystyle\int e\,dt$ , i.e. $C(s) = k_p + \dfrac{k_I}{s} = \dfrac{k_I + k_ps}{s}$ (learn this)
Third order Hurwitz: $a_0s^3 + a_1s^2 + a_2s + a_3 = 0$ stable if all $a_i > 0$ and $a_1a_2 > a_0a_3$ (learn this)

(a)
Find the closed-loop Transfer Function and characteristic equation.
(b)
Use the Hurwitz criterion to find the stability condition on the gains. For $\zeta = 0.5$ , $\omega_n = 1$ rad/s and $k_p = 1$ , what is the largest $k_I$ for stability?
maximum kI =
(c)
Find the steady state output for a unit step in the set point.
steady state output =