ENGR202 2019 Q B1 (ENGR5001)

ENGR202 2019 Q B1Past paperOld spec ENGR2022:125 marks30 min

In this question, the differential equation of an uncontrolled system is given by the following mathematical model:

$\frac{d^2x}{dt^2} - 0.2\frac{dx}{dt} + 0.05x = 2.3u(t)$

where $x(t)$ and $u(t)$ are the output and input, respectively.

Formulas you may need

Laplace with zero initial conditions: $\dot x \to sX$ , $\ddot x \to s^2X$ (learn this)
PD controller $U(s) = (K_P + K_D s)(V(s) - X(s))$ (given in the question)
Second order characteristic equation $s^2 + 2\zeta\omega_n s + \omega_n^2 = 0$ (given in the question)
Unity feedback: $\dfrac{CG}{1 + CG}$ (learn this)

(a)
Determine the Transfer Function form of the uncontrolled system and hence show that the characteristic equation is given by $s^2 - 0.2s + 0.05 = 0$ .
[3]
(b)
Using the characteristic equation stated in part (a), determine the poles of the uncontrolled system and the stability condition, and sketch the general form of the time response.
pole real part =
pole imaginary part (+/-) =
[3]
(c)
Draw the block diagram of a proportional-derivative (PD) control system, as applied to the Transfer Function determined in part (a). Hint: expressed as a Transfer Function, the PD controller takes the following standard form: $U(s) = \left(K_P + K_D s\right)\left(V(s) - X(s)\right)$ where $V(s)$ represents the set point.
[3]
(d)
For the control system in part (c), design a PD controller (i.e. determine values of $K_D$ and $K_P$ ) such that the closed-loop damping ratio $\zeta = 1.1$ and the natural frequency $\omega_n = 0.3$ rad/s. Hint: the general form of the characteristic equation for a second order system is: $s^2 + 2\zeta\omega_n s + \omega_n^2 = 0$ . What is the significance of designing this controller such that $\zeta \ge 1$ ?
K_D =
K_P =
[12]
(e)
What are the potential disadvantages of the PD control system selected above? Suggest alternatives, explaining the reasons for your answer.
[4]