I want to install a pergola on my concrete patio, if this can be done, how can I secure the posts to the existing concrete?

The old way is to use steel pins and drill in the center of the post and concrete floor to set the post. Many old porches uses down weight as the (nailing) effect

The bottom of the post needs to be sealed well because of mostiure wicking into that bottom.

I would drill into the concrete and then set in a galvanized post box with and expanding anchor bolt. This will give side to side protection as well as lift protection from the wind. Something along this line Amazon.com: Steel 4" x 4" Mounting Plate: Patio, Lawn & Garden

Bsmom If the bracket Bal1 suggested is visually not good for looks (although for strength is great and will rust) , you could angle drill in the post and use concrete Tap Cons to attach it better for twist and lift

Tapcons- The Original Blue Concrete Screw || Blue Tapcon Concrete Screws

There are other "Plate" system but all show one way or another. Some of the others lift the post off the ground but visually its ugly. I have used them and then put a wood 1by wood shoe or "boot" as we call it to hide the plates

You can get ones that are hot dipped galvanized too Galvanized Steel Bracket for 2x4 Mounting to Flat Surface