  We introduce controlled $KK$-theory groups associated to a pair $A$, $B$ of
separable $C^*$-algebras. Roughly, these consist of elements of the usual
$K$-theory group $K_0(B)$ that approximately commute with elements of $A$. Our
main results show that these groups are related to Kasparov's $KK$-groups by a
Milnor exact sequence, in such a way that R\o{}rdam's $KL$-group is identified
with an inverse limit of our controlled $KK$-groups.
  In the case that the $C^*$-algebras involved satisfy the UCT, our Milnor
exact sequence agrees with the Milnor sequence associated to a $KK$-filtration
in the sense of Schochet, although our results are independent of the UCT.
Applications to the UCT will be pursued in subsequent work.
