  Feedback control is an essential component of many modern technologies and
provides a key capability for emergent quantum technologies. We extend existing
approaches of direct feedback control in which the controller applies a
function directly proportional to the output signal (P feedback), to strategies
in which feedback determined by an integrated output signal (I feedback), and
to strategies in which feedback consists of a combination of P and I terms. The
latter quantum PI feedback constitutes the analog of the widely used
proportional-integral feedback of classical control. All of these strategies
are experimentally feasible and require no complex state estimation. We apply
the resulting formalism to two canonical quantum feedback control problems,
namely, generation of an entangled state of two remote qubits, and
stabilization of a harmonic oscillator under thermal noise under conditions of
arbitrary measurement efficiency. These two problems allow analysis of the
relative benefits of P, I, and PI feedback control. We find that for the
two-qubit remote entanglement generation the best strategy can be a combined PI
strategy when the measurement efficiency is less than one. In contrast, for
harmonic state stabilization we find that P feedback shows the best performance
when actuation of both position and momentum feedback is possible, while when
only actuation of position is available, I feedback consistently shows the best
performance, although feedback delay is shown to improve the performance of a P
strategy here.
