  We show that in the regime of ground-state cooling, simple expressions can be
derived for the performance of resolved-sideband cooling --- an example of
coherent feedback control --- and optimal linear measurement-based feedback
cooling for a harmonic oscillator. These results are valid to leading order in
the small parameters that define this regime. They provide insight into the
origins of the limitations of coherent and measurement-based feedback for
linear systems, and the relationship between them. These limitations are not
fundamental bounds imposed by quantum mechanics, but are due to the fact that
both cooling methods are restricted to use only a linear interaction with the
resonator. We compare the performance of the two methods on an equal footing
--- that is, for the same interaction strength --- and confirm that coherent
feedback is able to make much better use of the linear interaction than
measurement-based feedback. We find that this performance gap is caused not by
the back-action noise of the measurement but by the projection noise. We also
obtain simple expressions for the maximal cooling that can be obtained by both
methods in this regime, optimized over the interaction strength.
