Adaptive Sampling for Linear State Estimation

Abstract

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. If the state process is a scalar, or a vector of low dimension, then we can ignore sample quantization. If in addition, we can ignore jitter in the transmission delays over the network, then our search for efficient messaging policies simplifies. Firstly, each message packet should contain the value of the state at that time. Thus a bound on the number of data packets becomes a bound on the number of state samples. Secondly, the remaining choice in messaging is entirely about the times when samples are taken. 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 N 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. arXiv:0904.4358v2 [math.OC]

An Iterative Algorithm for Optimal Event-Triggered Estimation*

“…Event-triggered sampling is also referred to as adaptive sampling in Rabi et al [2012], Lebesgue sampling inÅström and Bernhardsson [2002] and dead-band control in Otanez et al [2002], Hirche et al [2005]. The event-trigger is a preprocessing unit situated at the sensor which decides upon its available information, whether to update the estimator with current information.…”

“…Examples for such networked control systems are given by sensor networks, multi-robot systems and distributed power generation networks. The work inÅström and Bernhardsson [2002] and Rabi et al [2012] showed that event-triggered sampling outperforms periodic sampling with respect to the state estimation error of a first-order linear system in the presence of two different communication constraints. In Rabi et al [2012], the communication constraint is induced by limiting the number of transmissions during a finite interval, whereas the work inÅström and Bernhardsson [2002] limits the average transmission rate.…”

“…The work inÅström and Bernhardsson [2002] and Rabi et al [2012] showed that event-triggered sampling outperforms periodic sampling with respect to the state estimation error of a first-order linear system in the presence of two different communication constraints. In Rabi et al [2012], the communication constraint is induced by limiting the number of transmissions during a finite interval, whereas the work inÅström and Bernhardsson [2002] limits the average transmission rate. Differing to these approaches, we extend the standard minimum mean square estimator problem by an additional communication penalty to reflect the communication constraint in the optimization problem.…”

“…A similar problem is also studied in Xu and Hespanha [2004] and Lipsa and Martins [2011]. Opposed to the aforementioned work which either fixes the estimator, such asÅström and Bernhardsson [2002], Rabi et al [2012], Xu and Hespanha [2004] or computes the estimator from the choice of the event-trigger, such as Lipsa and Martins [2011], we aim at the joint optimal design of the estimator and the event-trigger. Built on previous work that identifies structural properties of the optimal estimator Molin and Hirche [2010a,b], we formulate a twoplayer team problem with a nested information pattern, where the players are given by the event-trigger and the estimator.…”

Decentralized event-triggered medium access control for networked control systems

“…However, in practice, often only noisy measurements are accessible. The event-trigger can be obtained by formulating the design as an optimal stochastic control problem [7], [10], [12]. These works show that eventtriggered sampling outperforms time-triggered schemes also in the multiple-loop NCSs.…”