Right now, unless someone has a custom Dreamcast-to-PS3 converter, at bare minimum you would need two converters –
A) a Dreamcast to PS2 converter (easy to find); and
B) a PS2-to-PS3 converter (harder to find, but they’re out there).
**** And before you ask, YES, you use both converters at the same time. A lot of people have plugged one converter into another and found them to work fine with little to no lag. No-lag isn’t guaranteed… ***
The only other option is to replace the original PCB with either a PS3 controller PCB or a multi-console PCB like the MC Cthulu. In the long run, a multi-console PCB is a better move unless your heart is set on keeping the original controller in stock condition. Very few of these things become that desirable/collectible and the Dreamcast peripheral market is pretty much a cheap zone except for the Virtual-On Twin Stick for the Dreamcast… That has some desirability/collectibility. Agetec is also a bit more expensive for reasons explained below.
The rest of the Dreamcast arcade sticks pretty much get gutted/converted for new consoles and refitted with arcade parts if they fit… The Agetec is generally the Dreamcast refit stick base of choice because of the adequate interior space for Japanese arcade levers (JLF and most Seimitsu series levers) and space area for both 30mm pushbuttons and replacement PCB’s like the Cthulu family. Stock Agetec is a decent if not spectacularly competent joystick and most experienced modders end up refitting Agetec cases with better parts down the line unless they’re collectors. People that use these sticks as primary arcade sticks tend to “retune” them with superior parts.
MC Cthulu is the replacement PCB I installed in my modded Agetecs but then again I’m weird in that I kept my old consoles (PS2, Saturn, Dreamcast) for the fighting games and other console-era-specific games not ported to the newer consoles. (There are tons of “girl fighters” and anime fighters that never got ported to newer consoles. Face it – most of these games were awful to begin with but if you have a favorite nonetheless–!) The MC Cthulu is the most versatile PCB on the market. The only reason it isn’t bought by more people is because it’s a pain for many individuals to try and solder in 360 support with this PCB.
****** REPEAT after me – SOLDERING is the least-liked aspect of modding in general! When they can, people like to avoid having to solder as much as possible…! ******
The Paewang PCB is the only current in-production PCB (besides the similar PCB in the Qanba and Eightarc joysticks) that has dual 360/PS3 support built-in.