An automotive company has hired you to design a cruise control system to achieve a desired constant speed for one of their new line of vehicles. The cruise control system needs to be designed for a vehicle with a mass of 1000 kg and a viscous damping constant of 100 N-s/m which models the drag. If the standard wheels for the vehicle have a radius of 0.65 m and it is possible to apply an accurate motor torque, design a proportional (P) controller that only takes the vehicle 10 seconds to increase the speed by 5 m/s without overshooting to avoid causing the car to exceed the speed limit. To receive full credit for this problem you must include the following as justification:
(a) Write the differential equation and derive the plant transfer function relating vehicle speed (V) to motor torque (T). This transfer function should be G(s)=1/(650s +65).
(b) A block diagram representation of the closed-loop system with desired speed (Va), actual speed (V), controller (D), plant (G), and any possible sources of disturbances.
(c) Calculation of the location of the desired closed-loop poles (s*) in the Laplace domain.
(d) A root locus diagram of the proportional controlled system showing the location of the desired closed-loop poles (s*).
(e) Calculation of the proportional gain.