Given a system with Transfer Function and its desired output. Is it possible to find the required Input?
The transfer function is given as ...


Your transfer functions seems wrong. It's not "proper", i.e. there ought to be a $1$ in the denominator. You can try to rewrite it this way, but it would make it non-causal. Not sure if that's a typo or an error of your System Identification


  However, the inversed is not proper


No, your original transfer function is improper, the inverse is just fine. 


  If it is possible, could you kindly give me some broad instruction 



Make sure your original transfer function is proper, fix this first
Check if your corrected transfer function is causal, stable &amp; minimum phase, i.e. all poles and zeros inside the unit circle. If that's true, you can simply invert it by exchanging poles and zeros. 
If you have zeros outside of the unit circle, you can still invert this, but the result will be non-causal. You can still approximate an inverse by either adding bulk delay (if your application can tolerate that) or by least square error time domain solution.