PFC controller - How does it work?
In general

PFC converters in general have to fulfill two criteria. Firstly, the input power equals the output power (neglecting converter losses), meaning that
\begin{align}
V_{in,rms} \cdot I_{in,rms} = V_{out} \cdot I_{out}
\end{align}
Secondly, the shape of the input current has to follow the shape of the input voltage to ensure high power factor and with as little harmonic content as possible to fulfill THD criteria. The consequence of these criteria is that the output power will have a fluctuation of double the grid frequency. The effect on the dc side is normally handled by including a large output capacitance. 

Control scheme

For explaining the details of a PFC circuit and controller, I like to base it on a boost converter operated in CCM with a simplified control scheme with a cascaded control. 

The outer control loop starts by comparing the measuring the output voltage, \$v_{meas}\$, to the reference voltage level, \$v_{ref}\$. The resulting error, \$e_{volt}\$, is fed to a voltage controller (PI\$_{v}\$). The controller output can be thought of as the needed average or RMS current needed to feed the output load. In the next step, this value is multiplied with a dimensionless signal that gives a instantaneous current that is the same shape as the grid voltage with zero phase shift, \$i_{ref}\$. Again, the reference current is compared to the measured value, \$i_{meas}\$, to produce an error signal, \$e_{current}\$, which is fed to the current controller (PI\$_i\$). The current controller output can be thought of as the instantaneous duty cycle, \$d\$, for the boost converter switch, Q. The duty cycle is transformed to a PWM signal by comparing it to a sawtooth signal. The duty cycle over a half period will start high, then decrease, and then go high again towards the end. This is the same as in your figure "c)", where the duty cycle follows the grey area. 

This circuit is fairly simple to setup in a simulator, making sure that the current controller has a bandwidth much higher than \$2 \cdot f_{grid}\$ and low bandwidth for the voltage controller. The measured output voltage also typically has a filter on it. 

Your referenced control system will function in a similar manner, just with a different implementation. Of course, more complex control schemes exist.