Old spec 4.4 and 4.5 examples, proportional and integral action on a second order plant (ENGR5001)

Old spec 4.4 and 4.5 examples, proportional and integral action on a second order plantTutorialOld spec ENGR2022:215 min

ENGR202 (old spec) Control 4.4 Proportional Action (slides 4-14) and 4.5 Integral Action (slides 4-8)

The plant is the generalised second order system with unity steady state gain, $X = \frac{\omega_n^2}{s^2 + 2\zeta\omega_n s + \omega_n^2}U$ in a unity negative feedback loop with set point $D$ , error $E = D - X$ and controller $U = C(s)E$ .

Formulas you may need

Series rule and negative feedback rule $\dfrac{G_1}{1 + G_1G_2}$ (learn this)
Final Value Theorem $\lim_{t\to\infty}x(t) = \lim_{s\to 0}sX(s)$ (learn this)

(a)
Proportional action $C(s) = k_p$ . Find the closed-loop Transfer Function, the closed-loop characteristic equation, and the steady state error for a unit step in $D$ . Evaluate the error for $k_p = 0.5$ and $k_p = 2$ .
error, kp = 0.5 =
error, kp = 2 =
(b)
Integral action $C(s) = k_I/s$ . Find the closed-loop Transfer Function, characteristic equation and steady state value for a unit step. What is the price paid?
steady state output (I action) =