Is it correct to omit the subject of a subordinate clause if it coincides with the subject of the main clause, for example,
I felt drowsy when (I) woke up?
Can you give me a reference to the rules?
This is one of the big differences between tensed and untensed subordinate clauses. There are two untensed types -- infinitive clauses and gerund clauses -- and both can often appear without subjects, but only by one of the various rules that achieve that effect.
The subjects, even if they aren't present, are still understood, and they aren't always the same as the subject of the matrix predicate, either. Everything depends on the predicate of the superordinate clause and the syntactic rule being applied.
However, this is only true of untensed clauses. Tensed subordinate clauses like when I woke up or which didn't appeal to me or that I realized who he was can't delete their subjects under any rule. Tensed clauses must have subjects; this is why rules like Extraposition leave a dummy subject behind (it in the case of Extraposition) to keep the tensed verb company.
Some examples of the kinds of rules and the untensed clauses they apply to, with solved exercises, can be found here.
You can say:
I woke up feeling drowsy
but omitting the subject directly doesn't seem correct English.