This is an example of a joint state-space model combining a decomposition of unemployment into NAWRU+gap and a Phillips curve. This is from the the 2nd Edition of the State-Space/DSGE e-course. The most complicated part of this is choosing the proper decomposition and restrictions for unemployment, which is the subject of two other examples from the course. The addition of second observable (inflation) creates a more complex state-space model, but the key to good results is a good unemployment decomposition (which might require a different model for a different country).
A similar model is Laubach and Williams(2003) while Matheson and Stavrev(2013) is a similar model but with time-varying coefficients in the Phillips curve.