Any papers or sources to learn the methods to compute the smoothed shocks?

[jpfeifer]

See

Variance Decomposition

Dynare uses standard time series techniques for linear state space models. For forecast error variance decompositions, see Lütkepohl (2005): New Introduction to Multiple Time Series Analysis, Chapter 2.3.3. For the shock decomposition, A Kalman Filter/Smoother Approach is used. See Durbin/Koopman (2012): Time Series Analysis by State Space Methods or Hamilton (1994)

But that will not work for nonlinear models. The starting point in this case may be https://www.stats.ox.ac.uk/~doucet/doucet_johansen_tutorialPF2011.pdf