Designing an Inverted F-Antenna
This have I seen done in practice by trimming the length "L" of the antenna, but this would change the resonant frequency and the impedance simultaneously, or maybe I'm missing something.


The impedance is determined by the length from the feed pin to the short post. Trimming the "L" dimension of your PIFA will accomplish this but in a detrimental way that will change your resonance frequency. Moving the feed closer to the short post will decrease impedance and moving it further will increase impedance.


  ...the input impedance of the antenna is dependent on the distance of the feed point from the grounded end. The portion of the antenna between the feedpoint and the ground plane is essentially behaving as a short-circuit stub. Thus, the designer can match the antenna to the system impedance by setting the position of the feed point...
  Wikipedia.com - Inverted F-Circuit


Impedance is done by a calculation of measurements, as can be seen in the diagram below:



source PIFA – Planar Inverted F Antenna Iulian Rosu, YO3DAC / VA3IUL 

I recommend this whole .PDF file as it explores the subject of PIFA circuits in clear language. 

As you can see, the resonance of the antenna is based off of the length and width of the plate (L1 and L2 here respectfully). 

I have yet to find a reputable source on determining the location of the feed but will update this post when I do.

Update: the feed location (aka transmission line) is highly characteristic to your substrate and dielectrics. I am still trying to find a way to soundly calculate it. I have found some software which may help you calculate the transmission line characteristics under the microstrip section. I don't have a Windows computer so I can't say for sure.