There were some cool analog controllers for the PS1, but no consistency in implementation or compatibility.
People forget that the system came out in 1995, but the dual analog pad wasn't released until 1997 and the dual rumble pad wasn't released until 1998.
So hardly any of the games from 95-97 support analog control.
There are a few games that were made for use with special controllers, but the crazy part is that not all will work with analog controllers other than the specific one they were designed for.
The Namco NeGcon was released in 1995 at system launch for use with Ridge Racer.
You twist the controller to steer.

Sony Analog Joystick - April 1996

Sony Dual Analog Controller - August 1997
(you know what they look like)
Sony Dual Shock Controller - May 1998 (rumble)
(you know what they look like)
Namco JoGcon - 1998 Packaged with Ridge Racer Type 4 - Featured full directional force feedback
Interesting....the sucker was actually optical:
http://nfgworld.com/mb/thread/382-Namco-JogCon
I'd never seen the one chopperthedog posted.
Guess I need to add it to my list and find out some history on it.

chopperthedog, I thought RR Type 4 was the only PS1 game to have real directional ffb.
Does it work with other games, or just rumble?