Curve fitting with a Heaviside function

Hello, I am trying to fit a set of data to a delayed exponential rise with the following equation heaviside*(x-k)*A*(1-exp(-bx)). I'm using cftool to do this. However, the tool spits out answers that are wrong. I tried replacing the exponential data with a simple linear data set and it seems like the tool does not like the Heaviside function much. Has anyone else encountered this problem? I really don't want to trim the data set.

Thanks in advance.