A complete greedy algorithm for infinite-horizon sensor scheduling

doi:10.1016/j.automatica.2017.04.018

Automatica

Volume 81, July 2017, Pages 335-341

https://doi.org/10.1016/j.automatica.2017.04.018 Get rights and content

Abstract

In this paper we study the problem of scheduling sensors to estimate the state of a linear dynamical system. The estimator is a Kalman filter and our objective is to optimize the a posteriori error covariance over an infinite time horizon. We focus on the case where a fixed number of sensors are selected at each time step, and we characterize the exact conditions for the existence of a schedule with uniformly bounded estimation error covariance. Using this result, we construct a scheduling algorithm that guarantees that the error covariance will be bounded if the existence conditions are satisfied. We call such an algorithm complete. Finally, we provide simulations to compare the performance of the algorithm against other known techniques.

Introduction

One technique for monitoring an environmental process is to deploy a sensor network. Each sensor can be equipped with the ability to make a range of measurements. Sensor networks have been used in various applications including determining a robot’s state (Hovland & McCarragher, 1997), tracking the position of a target (Isler & Bajcsy, 2005), selecting the frequency in radar and sonar applications, or monitoring tasks such as chemical processes (Kookos & Perkins, 1999), seismic activity or toxin levels at a factory. Sensor scheduling techniques can also be applied to problems such as adaptive compressed sensing (Liu, Chong, & Scharf, 2012).

The collection of data can be done by operating every sensor continuously; however, the network may be required to have a long life span and so this strategy may not be viable due to energy and communication constraints. To overcome these restrictions, sensors can alternate between awake and asleep modes. Unless the network provides enough redundancy, this method could result in an incomplete picture of the phenomenon of interest. Therefore, a sensing schedule has to be constructed in an intelligent way in order to obtain as much information as possible. This is, in essence, the sensor scheduling problem.

The sensor scheduling problem has received considerable attention in recent years. In the context of linear Gaussian systems, a Kalman filter is the optimal estimator in that it produces an estimate with the least mean square error. Thus, the Kalman filter is commonly used as the basis for the sensor scheduling problem. An exception is Ilkturk (2015), where the condition number of the sequence observability matrix is used as a metric to find a sensor schedule. In this paper we will use a metric on the error covariance of the Kalman filter as our objective function. With this setting, the infinite horizon sensor scheduling problem is studied in Zhang, Vitus, Hu, Abate, and Tomlin (2010). Under some mild conditions, it is shown that the optimal infinite horizon schedule is independent of the initial covariance. Also, it is shown that given an optimal schedule, its cost can be estimated arbitrarily closely by a periodic schedule, with a finite period. However, if the optimal schedule is not known, the analysis does not provide a constructive method for efficiently computing an approximate periodic schedule.

Numerous approaches have been proposed to tackle the sensor scheduling problem. The results in Zhang et al. (2010) serve as a reason to find optimal periodic schedules for infinite horizon scheduling problem. The authors in Shi and Chen (2013b) find a periodic schedule using a branch-and-bound approach. In Shi and Chen (2013a) the authors find an optimal periodic schedule by approximating the objective function of the sensor scheduling problem. A locally optimal solution to periodic scheduling was proposed in Liu, Fardad, Varshney, and Masazade (2014) with constraints on the number of times each sensor can be used in a period. Their objective function incorporates both the estimation error and the number of sensors used per time step. A drawback to these approaches is that the optimal period is unknown, and thus the desired period must be given as an input.

Optimal and semi-optimal algorithms for the finite horizon problem that use tree pruning techniques are provided in Vitus, Zhang, Abate, Hu, and Tomlin (2012). In Gupta, Chung, Hassibi, and Murray (2004), three different approaches (sliding window, greedy thresholding and random selection) are empirically compared. The authors further develop the random selection method in Gupta, Chung, Hassibi, and Murray (2006), where a strategy for stochastically selecting measurements based on an intelligently constructed probability distribution is described and bounded. In Maheswararajah, Halgamuge, and Premaratne (2009), a few different approaches are studied, including a best step look ahead algorithm, an approach based on the Viterbi algorithm and another by casting the problem as a duality problem. The algorithms are described and empirically compared in terms of performance and computation time.

A convex relaxation based approach is discussed in Weimer, Sinopoli, and Krogh (2008) and applied to the monitoring of ${CO}_{2}$ using a wireless sensor network. Another convex relaxation approach is given in Joshi and Boyd (2009) along with solution dependent bounds. In Shamaiah, Banerjee, and Vikalo (2010), this approach is, however, empirically shown to be worse than a greedy algorithm. In Jawaid and Smith (2015), authors studied some properties of greedy sensor scheduling algorithms and their relation to submodular set functions.

A general framework for the sensor scheduling problem is presented in Mo, Ambrosino, and Sinopoli (2011). A number of problems can be addressed in this framework such as minimizing the final covariance over a time horizon, the average covariance, the variance of a single state, or even the cost of a finite horizon LQG regulator. A number of network constraints can also be included. The problem is framed as a relaxed quadratic program, and a greedy approach is described although the error bound is not necessarily uniformly bounded for unstable systems. In Maity and Baras (2015), a continuous time sensor scheduling problem is considered for an objective capturing both estimation error and sensor switching costs.

In this paper we consider infinite-horizon sensor scheduling. Based on the discussion above, existing approaches for this problem are (1) to fix a period and compute a periodic schedule; (2) to repeatedly apply a finite-horizon algorithm; or (3) to greedily select sensors at each time step. For each of these methods, there are no guarantees that the resulting schedule will produce a uniformly bounded sequence of covariance matrices. In fact, we do not know of any results that characterize the exact conditions under which an infinite horizon sensor schedule exists that results in a uniformly bounded sequence of covariance matrices.

Contributions: We give necessary and sufficient conditions for the existence of an infinite horizon sensor schedule with a bounded error covariance (Section 4). We then provide a complete algorithm for sensor scheduling (Section 5): That is, our algorithm outputs a uniformly bounded sensor schedule if one exists. The algorithm has the same runtime as the simple greedy algorithm and we show in simulations (Section 6) that our proposed algorithm outperforms the greedy algorithm, and can be used to efficiently compute schedules for high-dimensional linear systems with a large number of sensors.

A preliminary version of this paper was presented in Jawaid and Smith (2014). Relative to this early version, we now provide a more efficient algorithm along with details on its implementation. We also extend both the algorithm and analysis to the general problem of $k$ sensors per time step, and provide complete proofs of the correctness of the proposed greedy algorithm. Finally, we present more extensive simulation results on high-dimensional linear systems, including a system obtained by discretizing the heat equation.

Section snippets

Preliminaries

Consider the discrete-time linear stochastic system $\begin{matrix} x_{t + 1} & = & A x_{t} + w_{t}, x_{t} \in R^{n}, \\ y_{t} & = & C_{t} x_{t} + v_{t}, y_{t} \in R^{k} \end{matrix}$ where $A \in R^{n \times n}$ and $C \in R^{m \times n}$ . The matrix $C_{t}$ is a subset of $k$ rows of $C$ . This is the standard sensor selection model, as in Mo et al. (2011) and Vitus et al. (2012). The process noise $w_{t}$ and measurement noise $v_{t}$ are zero mean Gaussian noise vectors with covariance matrices $W, V \in R^{n \times n}$ , respectively, with $W ⪰ 0$ and $V ≻ 0$ . We assume that the noises are independent over time.

For the case $C_{t} = C$ (LTI system), the system is said

Problem statement

Consider the dynamical system (1). Each row of $C$ corresponds to a single sensor in the sensor network. For the sensor scheduling problem, we pick a set of $k$ sensors at every time step to make a measurement (i.e., $k$ rows of $C$ ). A Kalman filter uses noisy measurements to estimate the state of the system. The a posteriori and a priori covariances using the Kalman filter are given by $Σ_{t | t} = Σ_{t | t - 1} - Σ_{t | t - 1} C_{t}^{⊤} {(C_{t} Σ_{t | t - 1} C_{t}^{⊤} + V_{t})}^{- 1} C_{t} Σ_{t | t - 1}$ $Σ_{t + 1 | t} = A Σ_{t | t} A^{⊤} + W$ where $V_{t}$ comprises of the $k$ rows and columns of $V$

Existence of a bounded sensor schedule

The question we now ask is, given an LTI system, does there exist a sequence of measurements that results in uniformly bounded error covariance? Using Lemma 3, it is equivalent to asking if there exists a uniformly detectable sequence of measurements. In this section we address this question.

Proposition 4

There exists a sequence of measurements resulting in bounded Kalman filter error covariance if and only if $(A, C)$ is detectable.

Proof

Sufficient Condition: To prove this result, we first constructively show in

A complete sensor selection algorithm

We define a complete scheduling algorithm as follows.

Definition 7 Complete Scheduling Algorithm

A sensor scheduling algorithm is complete if for every detectable LTI system $(A, C)$ , the resulting sequence of error covariance matrices is uniformly bounded for all time.

A complete sensing schedule can be naively constructed using the periodic sequence

σ_{C}

. It will in general result in very large values of the covariance metric

F (σ_{C})

. The greedy schedule presented in Shamaiah et al. (2010) chooses the sensors at each step that minimize the

Simulations

We perform several simulation experiments to investigate the performance of Algorithm 1. In these simulations, we run the algorithms for $T$ time steps. We let the covariance metric $F$ be the average trace of the a posteriori error covariance, so at each time step $t$ , $F (σ) = \frac{1}{t} \sum_{i = 1}^{t} trace (Σ_{i | i}) .$

For this section, we refer to the greedy algorithm as G and the detectableGreedy algorithm as DG. We use a sliding window approximation for comparison with these algorithms. The slidingWindow (SW) algorithm is

Conclusions and future directions

We gave conditions for the existence of a bounded sensor schedule and then presented an algorithm that outputs a bounded sensor schedule if one exists. The algorithm attains the same asymptotic runtime as the greedy algorithm, but we show empirically that it obtains better performance.

We are interested in quantifying the quality of the schedule given by the algorithm relative to the optimal. Another problem to consider is determining the minimum number of sensors required per time step that

Ahmad Bilal Asghar received his B.Sc. in Electrical Engineering from Lahore University of Management Sciences, Pakistan in 2012, and M.A.Sc. in Electrical Engineering with a specialization in Systems and Control from University of Waterloo, Canada in 2015. He is currently a Ph.D. student at the University of Waterloo under the supervision of Professor Stephen L. Smith. His general research interests are in the area of control and optimization, with a focus on path planning for robotic

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Syed Talha Jawaid received his B.A.Sc. in electrical engineering from the University of Waterloo, Waterloo, ON, in 2011. He proceeded to complete his M.A.Sc. in electrical engineering with a specialization in systems and control, specifically in the area of informative path planning, under the supervision of Professor Stephen L. Smith. He is currently working as a software engineer in Austin, TX.

Stephen L. Smith received the B.Sc. degree from Queen’s University, Canada in 2003, the M.A.Sc. degree from the University of Toronto, Canada in 2005, and the Ph.D. degree from the University of California, Santa Barbara in 2009. From 2009 to 2011 he was a postdoctoral researcher with the Computer Science & Artificial Intelligence Lab at the Massachusetts Institute of Technology. He is currently an Associate Professor in the Department of Electrical and Computer Engineering at the University of Waterloo, Canada. He is the recipient of several awards including the 2016 Early Researcher Award from the Ontario Ministry of Research and Innovation, the NSERC Discovery Accelerator Supplement Award, and an Outstanding Performance Award from the University of Waterloo. His main research interests lie in control and optimization for autonomous systems, with a particular emphasis on robotic motion planning and coordination.

^☆: This research is partially supported by the Natural Sciences and Engineering Research Council of Canada (NSERC). The material in this paper was partially presented at the 2014 American Control Conference, June 4–6, 2014, Portland, OR, USA. This paper was recommended for publication in revised form by Associate Editor Hyeong Soo Chang under the direction of Editor Ian R. Petersen.

¹: This work was performed while S.T. Jawaid was at the University of Waterloo.

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Brief paperA complete greedy algorithm for infinite-horizon sensor scheduling☆

Abstract

Introduction

Section snippets

Preliminaries

Problem statement

Existence of a bounded sensor schedule

A complete sensor selection algorithm

Simulations

Conclusions and future directions

Automatica

Automatica

Automatica

Automatica

Systems & Control Letters

Automatica

Detectability and stabilizability of time-varying discrete-time linear systems

SIAM Journal on Control and Optimization

Observability Methods in Sensor Scheduling

Brief paper
A complete greedy algorithm for infinite-horizon sensor scheduling☆