Hi Debbie,
Centaurea macrocephala 'Lemon Fluff' can be cut back after bloom to about half size. In fall, after a hard frost, you can cut it down near the ground. It's a tall plant that should be placed at the back of the border. It's considered an invasive in Washington state.
http://www.perennials.com/seeplant.html?item=1.125.100]
http://www.diggingdog.com/pages2/plantpages.php/P-0834
I've not grown this variety of Centaurea, but have found that dead heading will often result in a second flush of bloom, though not quite as lush.
Newt
When weeding, the best way to make sure you are removing a weed and not a valuable plant is to pull on it. If it comes out of the ground easily, it is a valuable plant.