I at first thought that since the rest of the chords were in a cycle of Tonic -> Subdominant -> Dominant -> Tonic, that Db was some kind of Dominant. But the dominant of what, Gb? And how does that explain its resolution to Eb? So then I thought "Well, maybe it's prolonging the subdominant, giving me a plagal cadence" but the cycle of the rest of the chords and the fact that it fits neatly into the progression seems to contradict that. Neither Secondary Dominant, nor Secondary Subdominant seem to fit the bill. Neopolitan doesn't fit the bill either because C minor doesn't occur until the next chord cycle.
So, what is the function of this Db chord? Prolonging the Subdominant? Some kind of Deceptive resolution? A dominant of some sort? Simply being there for the sake of Chromaticism?