I don't know what you are doing to attempt to kill the Perovskia. If all you are doing is giving it an occasional spray with glyphosate, it isn't going to get rid of it.  Spray the existing plant.  Then, revisit the plant several times a week and spray anything that is green and growing.  Do this for the rest of the growing season..  Or better yet, dig it up and get rid of roots.  This is faster and more permanent.  Everywhere a stem comes out of the ground, dig up that plant with roots.

Your unknown plant a tree that is considered a junk or trash tree... Ailanthus or tree of heaven. It grows rapidly from underground roots.  It pops up right next to houses in urban areas and is fine growing in the country, too. 

Again, if you are just spraying the tops of the plants and not going back for weeks and weeks, it just pops up again.  Or if you are cutting down either plant and then spraying the ground or the cut stem, this is going to do little to even annoy it.  Glyphosate needs to go on as much leaf surface as possible.

Or dig up as much of the root system with the top as possible.  You need to learn to recognise these problem plants are remove them when they are tiny.  It is much less work.