  State estimation for discrete-time linear systems with quantized measurements
is addressed. By exploiting the set-theoretic nature of the information
provided by the quantizer, the problem is cast in the set membership estimation
setting. Motivated by the possibility of suitably tuning the quantizer
thresholds in sensor networks, the optimal design of adaptive quantizers is
formulated in terms of the minimization of the radius of information associated
to the state estimation problem. The optimal solution is derived for
first-order systems and the result is exploited to design adaptive quantizers
for generic systems, minimizing the size of the feasible output signal set.
Then, the minimum number of sensor thresholds for which the adaptive quantizers
guarantee asymptotic boundedness of the state estimation uncertainty is
established. Threshold adaptation mechanisms based on several types of outer
approximations of the feasible state set are also proposed. The effectiveness
of the designed adaptive quantizers is demonstrated on numerical tests
involving a specific case study and randomly generated systems, highlighting
the trade off between the resulting estimation uncertainty and the
computational burden required by recursive set approximations.
