Question

Question 1 a) Determine the final value of $X(z) = \frac{1}{1 - z^{-1}} - \frac{1}{1 - e^{-aT}z^{-1}}$, $a > 0$ by using the Final Value Theorem [7 marks] b) Find the Nyquist rate for the following analogue signal $x(t) = 3 \cos(50\pi t) + 10 \sin(300\pi t) - \cos(100\pi t)$ [8 marks] c) A continuous time PD controller can be discretized as follows: $u(t) = K_p e(t) + K_d \frac{de(t)}{dt} \implies u(kT) = K_p e(kT) + K_d \left[\frac{e(kT) - e((k-1)T)}{T}\right]$ where k represents the discrete time instants and T is the sampling period. State the condition T, $K_p$ and $K_d$ should satisfy for this controller to be stable. [10 marks]

          Question 1
a)
Determine the final value of $X(z) = \frac{1}{1 - z^{-1}} - \frac{1}{1 - e^{-aT}z^{-1}}$, $a > 0$ by using the Final Value Theorem
[7 marks]
b)
Find the Nyquist rate for the following analogue signal
$x(t) = 3 \cos(50\pi t) + 10 \sin(300\pi t) - \cos(100\pi t)$
[8 marks]
c)
A continuous time PD controller can be discretized as follows:
$u(t) = K_p e(t) + K_d \frac{de(t)}{dt} \implies u(kT) = K_p e(kT) + K_d \left[\frac{e(kT) - e((k-1)T)}{T}\right]$
where k represents the discrete time instants and T is the sampling period.
State the condition T, $K_p$ and $K_d$ should satisfy for this controller to be stable.
[10 marks]

Question 1
a)
Determine the final value of X(z) = (1)/(1 - z^-1) - (1)/(1 - e^-aTz^-1), a > 0 by using the Final Value Theorem
[7 marks]
b)
Find the Nyquist rate for the following analogue signal
x(t) = 3 cos(50π t) + 10 sin(300π t) - cos(100π t)
[8 marks]
c)
A continuous time PD controller can be discretized as follows:
u(t) = Kp e(t) + Kd (de(t))/(dt) u(kT) = Kp e(kT) + Kd [(e(kT) - e((k-1)T))/(T)]
where k represents the discrete time instants and T is the sampling period.
State the condition T, Kp and Kd should satisfy for this controller to be stable.
[10 marks]

Added by Latoya B.

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