let me get this straight: Your ps2 is hooked up to a router, and your computer is also hooked up to that same router.
If this is the case then your computer's network config should be set to obtain IP automatically because the router will assign it an IP.
Your ps2 however, can be tricky. I believe you have to configure your router to give your ps2 a static route. In other words, opposed to your router handing out some random IP address to your computer and your computer accepting it because it's set (or should be set to) obtain IP automatically, for the ps2 however the router needs to give it a SPECIFIC IP address.
Most routers assign IP addresses in the range of 192.168.0.X, you should configure your router to assign your ps2 an address such as 192.168.0.10 if this is the case. This way your ps2 will always have the same adress when it's turned on and whenever you run radhost or an ftp client on your computer to connect to your ps2 the IP to connect to it should always be the same.
Just make sure that your ps2 is configured with the same IP address for all the different programs such as Ulaunch and SMS, etc as was noted before.
(note: The IP you configure for the ps2 on your router and the IP you tell the ps2 to use in programs like ulaunch and shit should be identical of course)
Ver. 3 NTSC independence exploited ps2, 120 gig WD HDD, HDLoader .8c, Ulaunchelf 4.04, Blackrhino linux on WD 40 gig HDD