Chelsea released a statement on Wednesday confirming an agreement had been reached to sign left-back Luis from the La Liga champions.
After four years in Madrid and a 2013-14 UEFA Champions League runners-up medal, Luis held a farewell media conference following the announcement and the 28-year-old said the tough decision was based on his dream of playing in the Premier League.
"I can say that it has been the most difficult decision I've ever made in my life," Luis told reporters.
"When the first offer from Chelsea came, the first thing I did was talk to [Jose Luis] Caminero and the club and explain my reasons.
"The truth is that as a child I always dreamed of playing in the Premier League, and decided to communicate it to the club so they could see that that was my goal, a change of scenery. I think that my time here was up at this stage."
Luis added: "The reason for my leaving is neither economical nor because I'm at odds with Atletico de Madrid.
"On the contrary, I'm still an Atletico [man] and will be for the rest of my life. I feel very identified with this club and at 28, I very much wanted to try a different league."
