Smother crops like sorgam-sudan, or buckwheat, at least a year in advance of planting will help control weeds. I would be careful about tilling it in though. Tillage can disturb the weed seeds back to the surface causing them to sprout and undo all the work you did suppressing the weeds.
Asparagus loves to be mulched, so once the cover crop has winter killed let the residue remain on the ground as a mulch and add more in the form of manure or straw. That should suppress the weeds and give your crop a kick start. Good luck and post your crop when it comes in.