You will need:
Isopropyl alcohol 90%+ (rubbing alcohol)
Petroleum jelly (optional)
Disassembled PSP (can be found elsewhere)
Nerves of steel.
Step 1: Analog joystick removal
Prestep 2: Flip the faceplate over and remove the two silver screws that retain the analog joystick. Remove the joystick from the faceplate. Be careful not to lose the cap.
Step 1: Flip the joystick upside down, and gently pry the silver metal plate tabs on the side. You will remove it to find the PCB as in the picture.