The answer to Books 1-3
The first three books of the Psalter have Royal Psalms at their seams (Pss. 2, 41, 72 & 89) indicating that these should be understood as a collection. Not that they stood as a collection independently from Books 4 and 5 but that they stand as a unity in and of themselves within the Psalter. Their focus is the Davidic covenant and the (apparent) failure of this covenant vocalised in Ps. 89:
Lord, where is your unfailing love?
You promised it to David with a faithful pledge.
Consider, Lord, how your servants are disgraced!
I carry in my heart the insults of so many people.
Your enemies have mocked me, O Lord;
they mock your anointed king wherever he goes.
This questioning is met by the assertion that Yahweh is king in Pss. 93, 95-99 and Book 4 is replete with assertions that Yahweh is the protector of Israel concluding with a Psalm on creation and two on the exodus, the two great events whereby Yahweh displays his sovereignty and love for Israel.
Book 5 then opens, following upon two exodus psalms, by saying:
Give thanks to the Lord, for he is good!
His faithful love endures forever.
Has the Lord redeemed you? Then speak out!
Tell others he has redeemed you from your enemies.
For he has gathered the exiles from many lands,
from east and west,
from north and south.
This answers the question that had been raised by the Babylonian captivity and Psalm 107 is characteristly Deutero-Isaianic.
Books 5 proceeds to unpack this by showing the hope of the Davidic king to be ‘renewed’ and so we find Pss. 110 and 132. The whole tenor of Book 5 seems to be eschatological so linking up with Deutero-Isaiah and Deutero-Zechariah culminating in Ps. 150 where the whole creation unites to praise Yahweh.