An event-triggered approach to state estimation with multiple point- and set-valued measurements

doi:10.1016/j.automatica.2014.04.004

Automatica

Volume 50, Issue 6, June 2014, Pages 1641-1648

https://doi.org/10.1016/j.automatica.2014.04.004 Get rights and content

Abstract

In this work, we consider state estimation based on the information from multiple sensors that provide their measurement updates according to separate event-triggering conditions. An optimal sensor fusion problem based on the hybrid measurement information (namely, point- and set-valued measurements) is formulated and explored. We show that under a commonly-accepted Gaussian assumption, the optimal estimator depends on the conditional mean and covariance of the measurement innovations, which applies to general event-triggering schemes. For the case that each channel of the sensors has its own event-triggering condition, closed-form representations are derived for the optimal estimate and the corresponding error covariance matrix, and it is proved that the exploration of the set-valued information provided by the event-triggering sets guarantees the improvement of estimation performance. The effectiveness of the proposed event-based estimator is demonstrated by extensive Monte Carlo simulation experiments for different categories of systems and comparative simulation with the classical Kalman filter.

Introduction

Event-based estimation strategy provides the possibility to maintain estimation performance under limited communication resources (Åström & Bernhardsson, 2002) and has attracted considerable attention in the control community for the past few years. For scalar linear systems, Imer and Basar (2005) and Rabi, Moustakides, and Baras (2006) studied the optimal event-based finite-horizon sensor transmission scheduling problems in continuous and discrete time, respectively. Li, Lemmon, and Wang (2010) extended the results to vector linear systems by relaxing the zero mean initial conditions and considering measurement noises. In Li and Lemmon (2011), the tradeoff between performance and average sampling period was analyzed, where a sub-optimal event-triggering scheme with a guaranteed least average sampling period was proposed. Rabi, Moustakides, and Baras (2012) considered the adaptive sampling for state estimation of linear continuous-time systems. In Wu, Jia, Johansson, and Shi (2013), the Minimum Mean Squared Error (MMSE) estimator was derived, and the tradeoff between communication rate and performance was analyzed. Shi, Chen, and Shi (2014) studied the likelihood estimation problem for a level-based event-triggering scheme, and the evaluation of upper and lower bounds on communication rates was discussed. Sijs and Lazar (2012) formulated a general description of event sampling, and a state estimator with a hybrid update was proposed to reduce the computational complexity.

The above results consider the scenario that only one event detector is used to process the measured state information from the sensor. There also exist many applications (e.g., in the context of wireless sensor networks) where multiple sensors with multiple event detectors are equipped to measure the state of the process. These invariably lead to sensor scheduling/fusion issues, which have been extensively studied for the case of periodic sampled systems (Alriksson and Rantzer, 2005, Mo et al., 2011, Shi and Chen, 2013). However, the effect of multiple event detectors on the MMSE estimates still remains unexplored, which is the basic motivation of our research. In this work, we consider the scenario that the process is measured by a network of sensors and that each sensor chooses to provide its latest measurement update according to its own event-triggering condition. In this case, the hybrid information is provided by the whole group of sensors as well as the event-triggering sets. For the sensors whose event-triggering conditions are satisfied, the exact values of the sensor outputs are known, providing “point-valued measurement information” to the estimator; for sensors that the event-triggering conditions are not satisfied, some information contained in the event-triggering sets is known to the estimator as well, to which we refer as “set-valued measurement information” in this paper. The basic goal is to find the MMSE estimate given the hybrid measurement information. As will be addressed later, the main issues arise from the computational aspect, due to the non-Gaussianity of the a posteriori distributions. Therefore we focus on the derivation of an approximate (due to the Gaussian assumption) MMSE estimate that possesses a simple structure but still inherits the important properties of the exact optimal estimate. In Sijs and Lazar (2012), a sum of Gaussians approach was utilized to solve the MMSE problem under a uniform distribution assumption; for the single-channel case, an alternative approach was proposed by Nguyen and Suh (2007), where an adaptive scheduling algorithm was developed to adjust the virtual moments of the measurement noises to achieve the improved estimation performance. The difference is that the aforementioned results would add an additional covariance matrix to the measurement noise covariance, while the present approach introduces a scalar weight when updating the estimation error covariance matrix (see Theorem 7). The main contributions are summarized as follows:

(1) An approximate MMSE estimate induced by the hybrid measurement information provided by a sequence of sensors has been derived. We show that the estimate is determined by the conditional mean and covariance of the innovations. The results are valid for general event-triggering schemes and reduce to the results obtained in Wu et al. (2013) if only one sensor and the level-based event-triggering conditions are considered.

(2) Insights on the optimal estimate when each sensor has only one channel are provided. In this case, closed-form recursive state estimate update equations are obtained. Utilizing the recent results on the partial order of uncertainty and information (Chen, 2011), we show that the exploration of the set-valued information guarantees the improved estimation performance in terms of smaller estimation error covariance. The results are equally applicable to multiple-channel sensors with uncorrelated/correlated measurement noises but separate event-triggering conditions on each channel.

(3) Extensive Monte Carlo experiments are performed to test the effectiveness of the proposed estimator. Compared with the Kalman filter that only exploits the received point-valued measurements, the proposed estimator provides almost-guaranteed improved performance, which is not sensitive to the sensor sequence used.

The rest of the paper is organized as follows: Section 2 presents the system description and problem setup. Section 3 presents the main results. Experimental verification using Monte Carlo simulation is provided in Section 4, followed by the concluding remarks in Section 5.

Section snippets

System description and problem setup

Consider a linear time-invariant process that evolves in discrete time driven by white noise: $x_{k + 1} = A x_{k} + w_{k},$ where $x_{k} \in R^{n}$ is the state, and $w_{k} \in R^{n}$ is the process noise, which is zero-mean Gaussian with covariance $Q \geq 0$ . The initial value $x_{0}$ of the state is Gaussian with $E (x_{0}) = μ_{0}$ , and covariance $P_{0}$ . The state information is measured by a number of battery-powered sensors, which communicate with the state estimator through a wireless channel, and the output equations are $y_{k}^{i} = C^{i} x_{k} + v_{k}^{i},$ where $v_{k}^{i} \in R^{m}$ is

Optimal fusion of sequential event-triggered measurement information

In this section, Problem 2 is studied in detail. Define $z_{k}^{i} = y_{k}^{i} - C^{i} {\hat{x}}_{k}^{0}$ . Since ${\hat{x}}_{k}^{0}$ is known at time $k$ by the estimator, this relationship maps the set $Ξ_{k}^{i}$ to a unique set $Ω_{k}^{i} ≔ {z_{k}^{i} : z_{k}^{i} = y_{k}^{i} - C^{i} {\hat{x}}_{k}^{0}, y_{k}^{i} \in Ξ_{k}^{i}}$ . Define $L_{k}^{i + 1} ≔ P_{k}^{i} {(C^{i + 1})}^{⊤} {[C^{i + 1} P_{k}^{i} {(C^{i + 1})}^{⊤} + R^{i + 1}]}^{- 1}$ , and $e_{k}^{i} ≔ x_{k} - {\hat{x}}_{k}^{i}$ . We have the following result:

Theorem 3

(1)
The optimal prediction ${\hat{x}}_{k}^{0}$ of the state $x_{k}$ and the corresponding covariance $P_{k}^{0}$ are given by ${\hat{x}}_{k}^{0} = A {\hat{x}}_{k - 1}^{M}, P_{k}^{0} = h (P_{k - 1}^{M}) .$
(2)
For $i \in N_{0 : M - 1}$ , the fusion of information from the $(i + 1)$ th sensor leads to the

Experimental verification of the proposed results based on Monte Carlo simulations

In this section, we test the efficiency of the proposed results by Monte Carlo simulation. Specifically, we consider the practical “send on delta” communication strategy (Miskowicz, 2006), namely, at time $k$ , sensor $i$ decides whether to send new measurement updates to the remote estimator according to the following condition: $γ_{k}^{i} = {\begin{matrix} 1 & if | y_{k}^{i} - y_{τ_{k}^{i}}^{i} | \geq δ^{i}, \\ 0 & otherwise, \end{matrix}$ where $τ_{k}^{i}$ denotes the last instance when the measurement of sensor $i$ is transmitted. To study the applicability of the results, we

Conclusion

In this work, the problem of optimal fusion of hybrid measurement information for event-based estimation is studied. For a fixed sensor sequence, we show that the optimal MMSE estimate depends on the conditional mean and variance of the innovations. When each sensor has only one channel, a closed-form representation for the MMSE estimate is developed, and it is proved that exploring the set-valued information always improves estimation performance. The results are equally applicable to

Acknowledgments

The authors would like to thank the Associate Editor and the anonymous reviewers for their suggestions which have improved the quality of the work.

Dawei Shi was born in Zibo, Shandong Province, China. He received B.Eng. Degree in Electrical Engineering and Automation from Beijing Institute of Technology in 2008. From 2008 to 2010, he was a Ph.D. Student in School of Automation, Beijing Institute of Technology. Since 2010, he has been a Ph.D. Student in the Department of Electrical and Computer Engineering, University of Alberta. He was the recipient of the Best Student Paper Award at the 2009 IEEE International Conference on Automation

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Tongwen Chen is presently a Professor of Electrical and Computer Engineering at the University of Alberta, Edmonton, Canada. He received the B.Eng. degree in Automation and Instrumentation from Tsinghua University (Beijing) in 1984, and the M.A.Sc. and Ph.D. degrees in Electrical Engineering from the University of Toronto in 1988 and 1991, respectively.

His research interests include computer- and network-based control systems, process safety and alarm systems, and their applications to the process and power industries. He has served as an Associate Editor for several international journals, including IEEE Transactions on Automatic Control, Automatica, and Systems and Control Letters. He is a Fellow of IFAC and IEEE.

Ling Shi received his B.S. degree in Electrical and Electronic Engineering from the Hong Kong University of Science and Technology in 2002 and Ph.D. degree in Control and Dynamical Systems from California Institute of Technology in 2008. He is currently an Assistant Professor at the Department of Electronic and Computer Engineering at the Hong Kong University of Science and Technology. His research interests include networked control systems, wireless sensor networks and distributed control.

^☆: This work was supported by Natural Sciences and Engineering Research Council of Canada; in addition, the work by L. Shi was supported by an HK RGC GRF grant 618612. The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Martin Enqvist under the direction of Editor Torsten Söderström.

¹: Tel.: +1 780 886 5780; fax: +1 780 492 1811.

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Brief paperAn event-triggered approach to state estimation with multiple point- and set-valued measurements☆

Abstract

Introduction

Section snippets

System description and problem setup

Optimal fusion of sequential event-triggered measurement information

Experimental verification of the proposed results based on Monte Carlo simulations

Conclusion

Acknowledgments

Automatica

Automatica

Automatica

Evaluation of convergence rate in the central limit theorem for the Kalman filter

IEEE Transactions on Automatic Control

Optimal filtering

Cubature Kalman filters

IEEE Transactions on Automatic Control

A partial order on uncertainty and information

Journal of Theoretical Probability

Gaussian filters for nonlinear filtering problems

IEEE Transactions on Automatic Control

Brief paper
An event-triggered approach to state estimation with multiple point- and set-valued measurements☆