Exercise 5.1

Analysis ∙ Robustness of feedback controllers

Consider the control system in Fig. 39, where \(v(t)\) is a sinusoidal disturbance \(v(t) = \sin(\omega t)\).

Fig. 39 Block diagram of the control system

The proportional controller has a positive gain, that is, \(K > 0\).

1. Compute, as a function of \(K\), the absolute value of the sensitivity function at \(\omega = 1\) rad/s.

2. Determine for what values of \(K\) the disturbance \(v(t)\) gets attenuated[1] by a factor of \(9\) at the frequency \(\omega= 1\) rad/s.

If necessary, use the asymptotic Bode plot of \(G(s)\) shown in Fig. 40.

Fig. 40 Magnitude Bode plot of \(G(s)\)

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[1]

That is, the effect of \(v(t)\) on the output \(y(t)\) is \(\frac{1}{9}\) of the maximum amplitude of \(v(t)\), in this case \(1\). In this case, we say that the closed-loop achieves an attenuation factor of \(9\) for disturbances at frequency \(\omega = 1\).