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Inversion of Transfer Function created with tfest

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Hey guys,
I used the tfest function to create the transfer function for a real electronic circuit. The estimation is pretty decent (about 99%). The transfer function has 3 zeros and 3 poles. The vector which was used as the output was the result of a measurement on the system. The input is based on a theoretic calculation. It should be a mono-exponential. In the picture below you see the actual measured output together with the result of the lsim command on the estimated transfer function and the input.
Obviously it worked quite well. For my task it is however important to reconstruct the input signal for a given output signal. My idea was to create the inverse transfer function and then use lsim(inv(tf),output) in order to obtain the input signal. The result is however very poor (see below):
Could you suggest a solution for a better reconstruction of the input signal?
Thanks a lot! Regards T

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