Decks, Patios, Porches and Docks - unstable patio cover

I have built a free standing 8X12 patio cover over my concrete patio. I used 4X4 posts at the corners attached to the patio using mounting brackets. The cover is open with 2X6s spaced about 15 inches apart. They are attached to double 12 foot 2X6s sandwiching the posts. I am going to use 1X2s or 2X2s in the opposite direction on top to provide shade. My problem is that my posts do not seem to be mounted to the concrete securely enough. The cover is just off the back of my house and I could use some kind of decorative brace to tie two of the posts to the house but I am afraid that it might look tacky. Someone has suggested that I pour concrete footings on top of the existing patio and attach them to the patio by drilling holes in the patio and running the rebar down into them. The cover as a whole seems tied together well it just has some wobble when you push on a post.