Suppose we have a system which we want to know the exact transient times. In ideal case, we can extract the transient times, but in practice it will be affected by another system with a known transfer function. We know the system with a transfer function changes the transient times and also the shape of output. How can we obviate its effects on the (main) system?
That is pretty broad question, as it depends on the system.
However, your problem is extremely common: Every communication system works because a receiver is able to estimate the original events (symbols sent by the transmitter) after the signal has gone through a system with transfer function (the channel).
So, receivers need to do exactly that: reverse the effect of the channel as good as possible. The thing that does that is usually called an equalizer; the equalizer you pick, and how much info about the system you need to make that equalizer work, depends on your "main system".
So, if you need to solve that: go ahead; define your main system as well as possible. Find stochastic descriptions for the parts of the system that you don't know with certainty. Then, find a channel model that fits that model, and look for equalizers used with such a channel.