It is known that mammals have hormonal regulation mechanisms that help maintain almost constant calcium plasma levels (0.08–0.1 g/L in dairy cows). This control is necessary to maintain healthy functions, as calcium is responsible for diverse physiological functions, such as bone formation, intracellular communications, and blood clotting. It has been postulated that the mechanism of calcium control resembles that of a PI (proportional-plus integral) controller. PI controllers are placed in cascade with the plant and used to improve steady-state error. Assume that the PI controller has the form

where K_{P} and K_{I} are constants. Also assume that the mammal’s system accumulates calcium in an integrator-like fashion, namely P(s) = 1/V_{s}, where V is the plasma volume. The closed-loop model is similar to that of Figure P8.3, where G(s) = G_{c}(s)P(s) (Khammash, 2004).

a. Sketch the system’s root locus as a function of K_{P}, assuming K_{I} > 0 is constant.

b. Sketch the system’s root locus as a function of K_{I} , assuming K_{P} > 0 is constant.