Old spec 5.2 example, PID control stability (ENGR5001)

Old spec 5.2 example, PID control stabilityTutorialOld spec ENGR2022:112 min

ENGR202 (old spec) Control 5.2 PD, PV and PID Control, PID example (slides 10-12)

PID control $u = k_pe + k_I\int e\,dt + k_d\,de/dt$ 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

PID: $C(s) = k_p + \dfrac{k_I}{s} + k_ds$ (learn this)
Third order Hurwitz: stable if all coefficients positive and $a_1a_2 > a_0a_3$ (learn this)

(a)
Find the closed-loop Transfer Function and characteristic equation, and check the steady state gain.
steady state gain =
(b)
Use the Hurwitz determinant to find the stability condition relating the gains. For $\zeta = 0.5$ , $\omega_n = 1$ rad/s, $k_p = 1$ and $k_d = 1$ , find the largest $k_I$ for stability.
maximum kI =