Divide the ground beef into four portions, and form patties around pineapple rings so that none of the pineapple is showing. In a small saucepan, mix together the ketchup, brown sugar, and mustard. Heat until sugar is dissolved. Set aside.
Place burgers on the grill, and cook for about 5 minutes per side, or until well done. Spoon some of the brown sugar sauce over the burgers before serving.