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.
(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 (d = 1).
(e) For the integral feedback you found in part (c), make a bode plot of the ratio of closed-loop response to road grade divided by the open-loop response to road grade. At what frequency is the ratio greater than 1 (0dB)? If the car is cruising at 25 m/s (about 60mph), what is the minimum distance between peaks of rolling hills for which the integral feedback controller will reduce the speed variations, compared to a simple "constant throttle" controller?