It has been suggested that the use of closed-loop feedback in ventilators can highly reduce mortality and health problems in patients in need of respiratory treatments (Hahn, 2012). A good knowledge of the transfer functions involved is necessary for the design of an appropriate controller. In a study with 18 patients it was found that the open-loop transfer function from minute ventilation (MV) to end-tidal carbon dioxide partial pressure (P_ETCO₂) can be nominally modeled as:

a. Make a sketch of the root locus of the system indicating the breakaway points and the value k_c takes in each of them.

b. In the design of ventilators it is very important to have negligible overshoot with the fastest possible settling time. It has been suggested that a value of k_c = 5.35 will achieve these specifications. Mark the position of the closed-loop poles for this value of k_c and explain why this is a reasonable gain choice.

0.415k (s+ 0.092)(s+ 0.25) G(s) = s(s + 0.007)(s+0.207)