(Solved): Consider again the generic feedback system shown in Figure 5 , where \( G(s) \) is the plant's tra ...

Consider again the generic feedback system shown in Figure 5 , where \( G(s) \) is the plant's transfer function assumed to be stable and its Bode diagram is given in Figure 5 and \( C(s) \) is the controller's transfer function. 1. Let \( C(s)=k>0 \) be a proportional controller. By inspecting the Bode plot provided in Figure 5 , deduce: i) the crossover frequency and the phase margin of the loop transfer function obtained for \( k=100 \), [6 marks] ii) the maximum value of \( k \) for which the closed-loop system has phase margin at least \( 18^{\circ} \). [4 marks] 2. Let \( C(s)=k H(s) \), where \( k \) is a static gain, and \( H(s) \) is a phase advance compensator in the form \[ H(s)=\sqrt{\frac{\beta}{\alpha}} \frac{s+\alpha}{s+\beta} . \] Find the values of \( k, \alpha \) and \( \beta \) such that the crossover frequency and the phase margin of the resulting feedback system are \( 3 \mathrm{rad} / \mathrm{s} \) and \( 45^{\circ} \), respectively. [15 marks] [Hint: use the following tuning rules to design the phase advance compensator. In order to tune \( \mathrm{H}(\mathrm{s}) \) to attain desired \( \omega_{h}\left(0 \mathrm{~dB}\right. \) frequency) and \( \phi_{h} \) (maximum phase advance) choose: \[ \alpha=\left(\sqrt{\frac{1-\sin \left(\phi_{h}\right)}{1+\sin \left(\phi_{h}\right)}}\right) \omega_{h}, \quad \beta=\left(\sqrt{\frac{1+\sin \left(\phi_{h}\right)}{1-\sin \left(\phi_{h}\right)}}\right) \omega_{h} \] 3. draw the Bode diagram of the so-obtained controller \( \mathcal{C}(s) \). You can either use Matlab to have the exact Bode plot or you can draw the line approximation of the Bode diagram. [5 marks]