  It is known that under fixed-rate information constraints, adaptive
quantizers can be used to stabilize an open-loop-unstable linear system on
$\mathbb{R}^n$ driven by unbounded noise. These adaptive schemes can be
designed so that they have near-optimal rate, and the resulting system will be
stable in the sense of having an invariant probability measure, or ergodicity,
as well as boundedness of the state second moment. Although structural results
and information theoretic bounds of encoders have been studied, the performance
of such adaptive fixed-rate quantizers beyond stabilization has not been
addressed. In this paper, we propose a two-part adaptive (fixed-rate) coding
scheme that achieves state second moment convergence to the classical optimum
(i.e., for the fully observed setting) under mild moment conditions on the
noise process. The first part, as in prior work, leads to ergodicity (via
positive Harris recurrence) and the second part ensures that the state second
moment converges to the classical optimum at high rates. These results are
established using an intricate analysis which uses random-time state-dependent
Lyapunov stochastic drift criteria as a core tool.
