Genesis 18:19

Because his holiness (piety, chasidutha) is manifest before Me, (and) that he will instruct his sons, and the men of his house after him, to keep the ways that are right before the Lord, to do justice and judgment, that the Lord may bring upon Abraham that which He hath spoken concerning him.

FOR I HAVE KNOWN HIM (‘YEDATIV’), TO THE END (‘LEMA’AN’) THAT HE MAY COMMAND HIS CHILDREN. Rashi comments: “For I have known him, as the Targum takes it, is an expression denoting affection, just as A kinsman (‘moda’) of her husband’s; And I know thee. Still the main connotation of all these expressions is that of knowing, for he who holds a person in affection and draws him to himself knows him well and is familiar with him. But if you explain it as the Targum does — i.e., “I know that he will command his children” — then the word lema’an (to the end) does not fit into the sense.”
It is possible that the word yedativ means “I have raised him and elevated him so that he shall command his children after him to do that which is right before Me, and therefore I will make him a great and mighty nation so that he should serve Me. In a similar sense are the verses: I know thee (‘yedaticha’) by name; What is man, that Thou knowest htm? Or the verse may be stating, I know that he will command, does not fit the context. Ramban proceeds to show from Exodus 23:12 that the words lema’an yanuach mean she’yanuach (that he may rest); here likewise, lema’an asher yetzaveh means she’yetzaveh (that he will command). Thus, the word lema’an is seen to fit into the context. and in a similar sense is the verse, So that thine ox and thine ass may have rest (lema’an yanuach), meaning that he may have rest.
The correct interpretation appears to me to be that the word yedativ literally means “knowing.” He is thus alluding that G-d’s knowledge, which is synonymous with His Providence in the lower world, is to guard the species, and even the children of men are subject despite it to the circumstantial evil occurrences until the time of their visitation comes. But as regards His pious, He directs His Providence to know each one individually so that His watch constantly attaches to him, His knowledge and remembrance of him never departs, as it says: He withdraweth not His eyes from the righteous. There are many verses on this theme, as it is written, Behold, the eye of the Eternal is toward them that fear Him, and other verses besides.

כי ידעתיו, as a symbol of righteousness.

למען אשר יצוה את בניו, G’d did all of the foregoing in order that Avraham would instruct his sons to emulate the ways of G’d having personal experience of G’d’s great love for mankind, seeing how His love extended even to the wicked.

ושמרו...לעשות צדקה ומשפט למען הביא ה' על אברהם את אשר דבר אליו. The ultimate purpose of G’d in revealing to Avraham His manner of meting out justice was to ensure that He would have an opportunity to fulfill all the promises He had made to Avraham. [promises which were largely conditional on Avraham’s offspring being true to his teachings. Ed.] (compare 17,7)

את ביתו ואת בניו, both the members of his household who were not biologically related to his as well as his family, in order to teach us that a man is obligated to monitor the conduct of all members of his household and is responsible for their misconduct if he did not use his authority to put a stop to it. David spells out this responsibility of the head of the household in Psalms 101,6-7 when he said: עיני בנאמני ארץ לשבת עמדי הולך בדרך תמים הוא ישרתני. לא ישב בקרב ביתי עושה רמיה, “my eyes are on the trusty men of the land, to have them at my side. He who follows the way of the blameless shall be in my service.” He who deals deceitfully shall not live in my house;” Concerning the wicked Solomon says in Proverbs 29,12: “a ruler who listens to lies, all his servants become wicked.”

על אברהם, meaning on his descendants after him.