  In this paper, we consider a discrete time linear quadratic Gaussian (LQG)
control problem in which state information of the plant is encoded in a
variable-length binary codeword at every time step, and a control input is
determined based on the codewords generated in the past. We derive a lower
bound of the rate achievable by the class of prefix-free codes attaining the
required LQG control performance. This lower bound coincides with the infimum
of a certain directed information expression, and is computable by semidefinite
programming (SDP). Based on a technique by Silva et al., we also provide an
upper bound of the best achievable rate by constructing a controller equipped
with a uniform quantizer with subtractive dither and Shannon-Fano coding. The
gap between the obtained lower and upper bounds is less than $0.754r+1$ bits
per time step regardless of the required LQG control performance, where $r$ is
the rank of a signal-to-noise ratio matrix obtained by SDP, which is no greater
than the dimension of the state.
