All of these "successes" are the result of Sony's competition shooting themselves in the foot and a single third party game that drives sales.
PS1 - Nintendo goes full retard with expensive cartridges for the N64 as well as treating third parties like shit. Sega is also acting like tards by rapid firing new consoles at a ridiculous rate, eroding consumer confidence in them. This causes a shift in third part support towards Sony, which lets Final Fantasy VII save the PS1 from becoming the next Turbografix or Jaguar.
PS2 - Sega goes tits up after yet another console release. The DVD in the PS2 makes it the cheapest DVD player on the market at the time, and Sony continues to play well with third parties, so games like FFX and DMC take the PS2 from meh to commanding 80% of the console market. By the time the xbox and Gamecube were on the scene, the PS2 was too well established for it to be a thread.
PS3 - Nintendo doesn't bother competing and just panders to people who don't play games. MS creates an easy to develop for console and releases a year earlier than Sony. Sony offers an absurdly priced console with overly complicated architecture for increasingly incompetent devs. With competitors not being idiots, Sony falls flat on its ass.
PS4 - Xbox offers always online DRM, Kinect, SPORTS N' TV in its reveal, pissing off the xbox fanbase so bad that most of them went to the PS4. WiiU is a joke, and Nintendo is too late with the Switch to stop Sony from becoming dominant again.