Question

In a unity-gain feedback control system, the plant \[ P(s) = \frac{0.001}{s(2s + 1)(0.01s + 1)} \] is controlled by a lag compensator \[ C(s) = \frac{s + 10}{s + 0.1} \] The slope (in dB/decade) of the asymptotic Bode magnitude plot of the loop gain at \( \omega = 3 \, \text{rad/s} \) is _________ (in integer).

Answer 1

Correct Answer: -60

To find the slope of the asymptotic Bode magnitude plot, we first identify the poles and zeros of the system. The plant has poles at \( s = 0 \), \( s = -0.5 \), and \( s = -100 \), while the compensator has a zero at \( s = -10 \) and a pole at \( s = -0.1 \). At \( \omega = 3 \, \text{rad/s} \), the slope of the Bode magnitude plot is dominated by the number of poles and zeros.
The net slope is: \[ +20 \, \text{dB/decade} \, (\text{zero}) - 20 \, \text{dB/decade} \, (\text{pole}) = -60 \, \text{dB/decade} \] Thus, the slope is \( \boxed{-60} \, \text{dB/decade} \).