How does feedback in power supply work considering output change delay?
This is where control systems come in to play.  I would recommend one of the most common control systems: a PID controller.  Actually, you probably only need a PI controller.  PID stands for Proportional Integral Derivative.  The idea is to adjust the 'control input' (in your case, duty cycle) based on some function of the 'error' (in your case, the difference between the desired voltage and the measured voltage).  The proportional component adjusts the control input based directly on the error (say, duty cycle = 10 * error voltage).  The integral component adjusts the control input based on the integral of the error (for example, duty cycle = 0.1 * running sum of error voltage).  The derivative component adjusts the control input based on the derivative of the error (say, duty cycle = 5 * difference between current measurement and last measurement).  It is quite simple to implement a PID algorithm in software.

The proportional component does most of the work, but it means that there will always be some 'error' required to get the desired control input.  This is where the integral component steps in: the gain builds up over long periods of time, eliminating offsets on long time scales.  The derivative term mostly helps with the step response (overshoot, ringing, response time, etc.).  

The building blocks required to construct a PID loop (gain/attenuation blocks/amplifiers, summers, differentiators, integrators) can all be built out of analog circuits (e.g. op amps) so it is possible to construct a purely analog PID loop.  I actually designed a board a while back with a simple analog feedback loop for a thermoelectric cooler controller.  The PID loop itself consisted of a single op amp with a number of resistors and capacitors.  A DAC chip provided the set point that was compared with the voltage across a thermistor.  The same can be done with a power supply, though you generally want much shorter time constants.