The purpose of a cruise control is to keep the velocity of a car constant. The driver drives the car at the desired speed, the cruise control system is activated by pushing a button and the system then keeps the speed constant. The major disturbance comes from changes of the slope of the road which generates forces on the car due to gravity. The cruise control system measures the difference between the desired and the actual velocity and generates a feedback signal which attempts to keep the error small in spite of changes in the slope of the road. The feedback signal is sent to an actuator which influences the throttle and thus the force generated by the engine.

It is assumed that the force developed by the engine is proportional to the control signal sent to the throttle. Introducing parameters for a particular car in fourth gear, the model is described by the following equation;
Dv/dt +0.02v = u-10θ
where
θ = road slope steepness in %.
u = 1 at full throttle (maximuminput value)
u = control signal sent to the throttle and has a value 0 v = car velocity

a) Derive the transfer function and draw a block diagram of the arrangement.

b) What is the maximum speed at full throttle on a horizontal road (Θ=0) and how long will it take to reach the maximumspeed?

c) Calculate the input signal u required in order to maintain a car speed of 100 kph with the road slope steepness is 4%.

d) If the car is cruising at 100 kph on a horizontal road, briefly explain what will happen if the slope steepness suddenly changes to 10%.