  In this letter, we consider a Linear Quadratic Gaussian (LQG) control system
where feedback occurs over a noiseless binary channel and derive lower bounds
on the minimum communication cost (quantified via the channel bitrate) required
to attain a given control performance. We assume that at every time step an
encoder can convey a packet containing a variable number of bits over the
channel to a decoder at the controller. Our system model provides for the
possibility that the encoder and decoder have shared randomness, as is the case
in systems using dithered quantizers. We define two extremal prefix-free
requirements that may be imposed on the message packets; such constraints are
useful in that they allow the decoder, and potentially other agents to uniquely
identify the end of a transmission in an online fashion. We then derive a lower
bound on the rate of prefix-free coding in terms of directed information; in
particular we show that a previously known bound still holds in the case with
shared randomness. We generalize the bound for when prefix constraints are
relaxed, and conclude with a rate-distortion formulation.
