Adaptive Threshold Selection for Set Membership State Estimation with Quantized Measurements

State estimation for discrete-time linear systems with uniformly 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. Assuming the possibility of suitably tuning the quantizer range and resolution, 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 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, highlighting the tradeoff between the resulting estimation uncertainty and the computational burden required by recursive set approximations.