The proportional control system is more complex than an on-off control system like a thermostat, but simpler than a proportional-integral-derivative (PID) control system used in something like an automobile cruise control.
An on-off control is like driving a car by applying either full power or no power and varying the duty cycle, to control speed. The power would be on until the target speed is reached, and then the power would be removed, so the car reduces speed. When the speed falls below the target, with a certain hysteresis, full power would again be applied. It can be seen that this looks like pulse-width modulation, but would result in poor control.
Proportional control is how most drivers control the speed of a car. If the car is at target speed and the speed increases slightly, the power is reduced slightly, or in proportion to the error (the actual versus target speed), so that the car reduces speed gradually and reaches the target point with very little, if any, "overshoot", so the result is much smoother control than on-off control.
Further refinements like PID control would help compensate for additional variables like hills, where the amount of power needed for a given speed change would vary, which would be accounted for by the integral function of the PID control.
This can be mathematically expressed as
US Patent Issued to Air Power Systems on Oct. 30 for "Proportional Control Pneumatic Cylinder" (Oklahoma Inventor)
Oct 31, 2012; ALEXANDRIA, Va., Oct. 31 -- United States Patent no. 8,297,586, issued on Oct. 30, was assigned to Air Power Systems Co. Inc....