  We consider a discrete-time linear quadratic Gaussian networked control
setting where the (full information) observer and controller are separated by a
fixed-rate noiseless channel. The minimal rate required to stabilize such a
system has been well studied. However, for a given fixed rate, how to quantize
the states so as to optimize performance is an open question of great
theoretical and practical significance. We concentrate on minimizing the
control cost for first-order scalar systems. To that end, we use the Lloyd-Max
algorithm and leverage properties of logarithmically-concave functions and
sequential Bayesian filtering to construct the optimal quantizer that greedily
minimizes the cost at every time instant. By connecting the globally optimal
scheme to the problem of scalar successive refinement, we argue that its gain
over the proposed greedy algorithm is negligible. This is significant since the
globally optimal scheme is often computationally intractable. All the results
are proven for the more general case of disturbances with
logarithmically-concave distributions and rate-limited time-varying noiseless
channels. We further extend the framework to event-triggered control by
allowing to convey information via an additional "silent symbol", i.e., by
avoiding transmitting bits; by constraining the minimal probability of silence
we attain a tradeoff between the transmission rate and the control cost for
rates below one bit per sample.
