Poodles, schnauzers, first gen doodles etc, all should be non shedding. However make sure, the trigger for your asthma is hair, as more often than not it's dander (the flakey bits coming off the skin) that trgger, so dogs like labradors etc can be a much bigger problem.
I have 2 chows very hairy and blow binliners of coat, my husband was chronically allergic to all animals before we rescued our frst, but after 4 weeks was better with her than me!!!