Compensating effects of a system with a known transfer function
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.