Adaptive sampling for linear state estimation

When a sensor has continuous measurements but sends limited messages over a data network to a supervisor which estimates the state, the available packet rate fixes the achievable quality of state estimation. When such rate limits turn stringent, the sensor’s messaging policy should be designed anew. What are the good causal messaging policies ? What should message packets contain ? What is the lowest possible distortion in a causal estimate at the supervisor ? Is Delta sampling better than periodic sampling ? We answer these questions under an idealized model of the network and the assumption of perfect measurements at the sensor. For a scalar, linear diffusion process, we study the problem of choosing the causal sampling times that will give the lowest aggregate squared error distortion. We stick to finite-horizons and impose a hard upper bound on the number of allowed samples. We cast the design as a problem of choosing an optimal sequence of stopping times. We reduce this to a nested sequence of problems each asking for a single optimal stopping time. Under an unproven but natural assumption about the least-square estimate at the supervisor, each of these single stopping problems are of standard form. The optimal stopping times are random times when the estimation error exceeds designed envelopes. For the case where the state is a Brownian motion, we give analytically: the shape of the optimal sampling envelopes, the shape of the envelopes under optimal Delta sampling, and their performances. Surprisingly, we find that Delta sampling performs badly. Hence, when the rate constraint is a hard limit on the number of samples over a finite horizon, we should should not use Delta sampling.

💡 Research Summary

The paper tackles a fundamental problem in networked sensing: a sensor continuously measures a physical quantity, but the communication link to a remote supervisor can only carry a limited number of packets over a given time interval. Under this hard constraint the authors ask: which causal transmission policies are optimal, what should the packets contain, what is the minimum achievable distortion, and does the widely used Δ‑sampling outperform periodic sampling?

The authors model the plant as a scalar linear diffusion process, focusing on the special case of a Brownian motion (zero drift, constant diffusion coefficient). The supervisor employs a minimum‑mean‑square‑error (MMSE) estimator that is updated only when a new sample arrives; between arrivals the estimate is held constant. The performance metric is the integrated mean‑square error (MSE) over a finite horizon (