We wish to design an automatic speed control for an automobile. Assume that

1) the car has a mass m of 1000 kg,

2) the accelerator is the control U and supplies a force on the automobile of 10 N per degree of accelerator motion, and

3) air drag provides a friction force proportional to velocity of 10 N · sec/m.

a) Obtain the transfer function from the control input U to the velocity fo the automobile

b) Assume the velocity changes are given by

V (s) =1/(s+0.002) U(s) + 0.05/(s+0.02) W(s)

Where V is given in meters per second, U is in degrees, and W is the percent grade of the road. Design a proportional control law U = -Kp V that will maintain a velocity error of less than 1 m/sec in the presence of a constant 2% grade.

c) Discuss what advantage (if any) integral control would have for this problem.

d) Assuming that pure integral control (that is, no proportional term) is advantageous; select the feedback gain so that the roots have critical damping (? = 1).