Exercise 4.0

Exercise 4.0#

Control design ∙ Laplace/Frequency domain

We are given a mechanical system represented by the transfer function \(G(s)\). While an explicit expression for \(G(s)\) is not available, we have obtained its frequency response through a lot of hammer tests. The resulting frequency response is depicted in Fig. 19 below.

../_images/ex0-bode.svg

Fig. 19 Bode plot of the mechanical system \(G(s)\)#

  1. Design a proportional controller \(F(s)=K\) such that the following specifications are satisfied:

    • The closed-loop is stable.

    • The bandwidth is greater than \(5\) rad/s, but it does not exceed \(15\) rad/s.

    • The phase margin \(\varphi_m\) is larger than \(45^\circ\).

  2. What will be the worst-case resonance peak of the closed-loop transfer function \(G_c(s)\)?

Hint

Let the open-loop transfer function be \(G_o(s) = F(s) G(s)\). Then, its modulus and phase read

(6)#\[ \lvert G_o(i \omega) \lvert_{\text{dB}} = \lvert F(i \omega) \lvert_{\text{dB}} + \lvert G(i \omega) \lvert_{\text{dB}} \]
(7)#\[ \angle G_o(i \omega) = \angle F(i \omega) + \angle G(i \omega) \]