The King/Queen is Here
Toward the end of Bameh Madlikin, we read about the blowing of the shofar to alert the populace that Shabbat is arriving. In addition to functioning as an alert to cease work and prepare for Shabbat, I think we can understand it in another way. As many believe, we are better able to feel God's presence on Shabbat. The shofar blasts on Shabbat can similarly be seen in this light. One understanding of the mitzvah of shofar on Rosh Hashanah is that they are used to signal God's coronation. The shofar blasts signal that the King/Queen is being crowned. In a similar vein, the shofar can also serve as an announcement that the King/Queen is approaching. Thus, the pre-Shabbat blasts can also serve as a call to everyone who hears that God's presence is arriving with the comming of Shabbat.
This reading would explain why these blasts are mentioned in the Mishnah in Sukkah (chapter 4 if I remember correctly) along with the other shofar sounds that were blown in the temple, even though the pre-Shabbat blasts appear at first glance to have no ritual function. But, in reality, these blasts are not just in order to prevent people from working but also to alert them to God immanent arrival for the Shabbat.