The closed-loop vehicle response in stopping a train depends on the train’s dynamics and the driver, who is an integral part of the feedback loop. In Figure P8.3, let the input be R(s) = vr the reference velocity, and the output C(s) = v, the actual vehicle velocity. (Yamazaki, 2008) shows that such dynamics can be modeled by G(s) = G_{d}(s)G_{t}(s) where

represents the driver dynamics with h, K, and L parameters particular to each individual driver. We assume here that h= 0.003 and L =1. The train dynamics are given by

where M = 8000 kg, the vehicle mass; k_{e} = 0.1 the inertial coefficient; k_{b} = 142.5, the brake gain; K_{p} = 47.5, the pressure gain; τ = 1.2 sec, a time constant; and f = 0.24, the normal friction coefficient.

a. Make a root locus plot of the system as a function of the driver parameter K.

b. Discuss why this model may not be an accurate description of a real driver-train situation.