Optimal scheduling of mobile sensor networks for detection and

Optimal scheduling of mobile sensor networks for detection and localization of stationary contamination sources Maciej Patan

Dariusz Uci´nski

Institute of Control and Computation Engineering, University of Zielona G´ora, 65-246 Zielona G´ora, Poland Email: [email protected]

Institute of Control and Computation Engineering, University of Zielona G´ora, 65-246 Zielona G´ora, Poland Email: [email protected]

Abstract— The problem of optimal detection and localization of contamination sources in a distributed parameter system is formulated as that of maximizing the power of a parametric hypothesis test which checks whether or not system parameters have nominal values. We consider a setting where mobile nodes with sensing capabilities form a network aimed at collecting the most valuable measurements for parameter estimation of a distributed system. The framework is based on the use of the Ds -optimality criterion defined on the Fisher information matrix associated with the estimated parameters as a measure of the information content in the measurements. The approach consists in converting the problem to a canonical optimal control one in Mayer form, in which the control forces of the sensors are optimized. Numerical solutions can then be obtained using widespread numerical packages dedicated to optimal control tasks. A simulation example is presented which clearly demonstrates the presented ideas.

Keywords: Distributed parameter systems, parameter estimation, sensor location, source detection and localization. I. I NTRODUCTION The importance of measurement system design for estimation of unknown coefficients in distributed parameter systems (DPSs), i.e., systems described by partial differential equations (PDEs), has been recognized for a long time, but relatively few attempts have been made at solving this problem, cf. surveys in [1]–[6]. It has generated special interest in areas such as the design of air quality monitoring systems, groundwater-resources management, recovery of valuable minerals and hydrocarbon, model calibration in meteorology and oceanography, chemical engineering, hazardous environments and smart materials, cf. [2], [7]–[15]. In the context of these numerous engineering examples, the problem of identifying contaminating sources has been receiving significant research attention due to its applications in the fields of security, environmental and industrial monitoring, pollution control, etc., see, e.g., [16]–[20]. Examples include the detection of a potential biochemical attack, e.g., from a crop-duster spreading toxins in aerosol, sensing explosives mounted on a vehicle, detecting leakage of dangerous biochemical materials from a tank carried by a train, or detecting a contaminant source dropped into a water reservoir. In a

typical scenario, after some type of biological or chemical contamination has occured, there is a developing cloud of dangerous or toxic material. Its evolution is governed by partial differential equations which include diffusion and transport phenomena, as well as forcing functions such as the prevailing weather conditions, boundary conditions related to the possibly complicated surrounding geography. Now, assume that a number of sensors is deployed in the spatial area in question to form a sensor network and they can measure the concentration of the contaminant. In modern military systems, sensors can be located on various platforms and these platforms can be highly dynamic in motion. Then the goal is to use these observations to detect and localize contamination sources, determine the space-time concentration distribution of the chemical dispersion, and predict its cloud envelope evolution. The inherent ill-posedness of the resulting inverse problem makes its solution extremely difficult. It is even more complicated if we realize that from the online implementation viewpoint a recursive procedure would be preferred for estimating time-dependent parameters characterizing the sources so that the estimate tracks the measurement data and so that the new measurements can be effectively incorporated. Unfortunately, no universal solutions to the above problems have been proposed so far. Endowing nodes in a sensor network with mobility drastically expands the spectrum of the network’s capabilities. Although the number of sensor placement techniques developed to manage the problems of a practical scale is very limited (cf. [1], [2], [5], [21]), some effective approaches has been proposed to cover a number of different experimental settings related to parameter estimation, including stationary [4], [7], [22], [23], scanning [24]–[26] or moving observations [8], [10], [27]–[29]. It is worth pointing out that the optimal design of moving sensor trajectories is increasingly attracting attention in the context of sensor networks which play a role of importance in the research community, cf. [30]–[35]. Technological advances in communication systems and the growing ease in making small, low power and inexpensive mobile systems now make it feasible to deploy a group of

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networked vehicles in a number of environments, cf. [36]. A cooperated and scalable network of vehicles, each of them equipped with a single sensor, has the potential to substantially improve the performance of the observation systems. The purpose of the investigations reported here was to establish a practical approach to properly formulate and solve the problem of guiding mobile nodes of a sensor network so as to accurately estimate DPS parameters and maximize the reliability of detection and localization of contamination sources within a given spatial environment. In spite of the rapid development of fault detection and localization methods for dynamic systems [37]–[39], there are no effective methods tailored to spatiotemporal systems. Some successful attempts at exploiting the Ds -optimality criterion were reported in [22], [24], [40] and some adaptive threshold techniques in [7], [8], [10]. The approach presented here is based on the formulation of the sensor motion planning task in terms of an optimalcontrol problem for which solutions are then obtained with the use of the M ATLAB toolbox RIOTS 95, a high-performance tool for solving optimal control problems, cf. [41]. The performance of the delineated approach is illustrated via numerical simulations regarding a two-dimensional advection-diffusionreaction process. II. S ENSING STRATEGY FOR SOURCE DETECTION

z j (t) =y(xj (t), t; θ) + εj (t), t ∈ T = [0, tf ], j = 1, . . . , N,

Let Ω ⊂ Rd be a bounded domain with sufficiently smooth boundary Γ and T = (0, tf ] a bounded time interval, where tf < ∞ denotes a fixed observation horizon. Consider a ¯ ⊂ Rd and DPS whose scalar state at a spatial point x ∈ Ω time instant t ∈ T¯ is denoted by y(x, t). Mathematically, its evolution can be described by the PDE
∂y = F x, t, y, θ in Ω × T , (1) ∂t where F is a well-posed, possibly nonlinear, differential operator which involves first- and second-order spatial derivatives. By definition, F may include terms accounting for various mass transport phenomena such as convective diffusion, wind effects, gravitation and/or terms describing the chemical activity of the underlying substances as well as those representing forcing inputs, e.g., contamination sources [42]. The PDE (1) is supplemented by the appropriate boundary and initial conditions on Γ × T, in Ω × {t = 0},

(2) (3)

such that the existence of a sufficiently smooth and unique solution is guaranteed. Respectively, B is an operator acting on the boundary Γ and y0 = y0 (x) a given function. It is assumed that the forms of F and B are given explicitly up to an m-dimensional vector of unknown constant parameters θ which must be estimated using observations of the system (these coefficients can parameterize, e.g., convective diffusivity, locations and intensities of sources, etc.).

(4)

with εj ( · ) being the measurement noise uncorrelated in space and time [43]–[45] such that E[εj (t)] = 0,

var[εj (t)] = σ 2 .

(5)

Although white noise is a physically impossible process, it constitutes a reasonable approximation to a disturbance whose adjacent samples are uncorrelated at all time instants for which the time increment exceeds some value which is small compared with the time constants of the DPS. The white-noise assumption is consistent with most of the literature on the subject. B. Source detection reliability measure The estimate θb of the unknown parameter vector θ is determined as a global minimizer of the least-squares criterion N Z X  j 2 J (ϑ) = z (t) − y(x` , t; ϑ) dt (6) `=1

A. System description

B(x, t, y, θ) = 0 y = y0

The concentration y is observed by N moving sensors, which take measurements in continuous time over the interval T along the observation curves x1 (t), . . . , xN (t). Then, the observation equation can be formally denoted as

T

where y( · , · ; ϑ) denotes the solution to (1)–(3) for a given value of the parameter vector ϑ. An elementary idea of source detection is to compare the resulting parameter estimates with the corresponding known nominal values, treating possible differences as residuals which contain information about potential contamination sources. Based on some thresholding techniques, the appropriate decision making system could be constructed to detect abnormal situations in system functioning [22], [24]. Basically, only a subset of system parameters describes the properties of contamination sources. This accounts for partitioning the parameter vector into two subsets. With no loss of generality, we may write    T θ = θ1 . . . θs θs+1 . . . θm = αT β T , (7) where α is a vector of s parameters which are essential for a proper source detection and localization, being directly or indirectly related to the emission intensities and location coordinates of contamination sources. Vector β contains some unknown parameters which are important part of the model but are useless for source detection and/or localization. Based on the observations, it is possible to test the simple null hypothesis H0 : α = α 0 ,

(8)

where α0 is the nominal value for the vector α corresponding to the normal system performance. For a fixed significance level (i.e. fixed probability of rejecting H0 when it is true), the power of the likelihood ratio test for the alternative hypothesis of the form HA : α 6= α∗ (i.e.

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1 − the probability of accepting H0 when HA is true) can be made large by maximizing the Ds -optimality criterion (see [22] for details) −1 T Ψs [M ] = log det[Mαα − Mαβ Mββ Mαβ ],

III. P ROPOSED FORMULATION BASED ON OPTIMAL CONTROL APPROACH

A. Equations of sensor motion Introducing notation

(9)


s(t) = x1 (t), . . . , xN (t) ,

m×m

where M ∈ R stands for the so-called Fisher Information Matrix (FIM) which is decomposed as   Mαα Mαβ , (10) M = T Mαβ Mββ s×(m−s)

such that Mαα ∈ R , Mαβ ∈ R , Mββ ∈ R(m−s)×(m−s) . The FIM is widely used in optimum experimental design theory for lumped systems [46]–[50]. In our settings, the FIM is given by [43] M=

N Z X j=1

0

 g(x, t) =

∂y(x, t; ϑ) ∂ϑ

T (12) ϑ=θ 0

stands for the so-called sensitivity vector, θ0 being a prior estimate to the unknown parameter vector θ [2], [15], [27]. Such a formulation is generally accepted in optimum experimental design for DPSs, since the inverse of the FIM constitutes, up to a constant multiplier, the Cram´er-Rao lower bound on the covariance matrix of any unbiased estimator of θ [49]. When the time horizon is large, the nonlinearity of the model with respect to its parameters is mild and the measurement errors are independently distributed and have small magnitudes, it is legitimate to assume that our estimator is efficient (minimumvariance) in the sense that the parameter covariance matrix achieves the lower bound [28]. Observe that for the partition   Dαα Dαβ , M −1 =  T (13) Dαβ Dββ

(17)

U = {u : T → Rr | u( · ) is measurable, umin ≤ u(t) ≤ umax a.e. on T }, (18)

(11)

where

s(0) = s0 ,

where f : Rn × Rr → Rn is a known function (required to be continuously differentiable), s0 ∈ ΩN denotes the initial configuration of sensors and u is a control function belonging to the set of admissible controls

tf

g(xj (t), t)g T (xj (t), t) dt,

(16)

setting n = dim(s(t)), and assuming without loss of generality that all sensors are conveyed by identical vehicles, it is possible to write the general equation of sensors motion in the following form: s(t) ˙ = f (s(t), u(t)) a.e. on T,

s×s

∀t ∈ T,

for some fixed vectors umin and umax representing technical constraints present in a majority of practical applications. In particular, there exist various choices for (17) in the literature, including different forms of first and second order linear equations. However, in the considered case, where the sensors dynamics is not of paramount importance and our attention is primarily focused on the trajectories themselves, the following simple variant may be exploited: s(t) ˙ = u(t),

s(0) = s0 ,

(19)

i.e., direct control of sensor velocities [2]. In real situations, some additional restrictions on the motions are inevitable. Firstly, all sensors should stay within the admissible region Ωad where measurements are allowed. We assume that it is a compact set defined as follows Ωad = {x ∈ Ω ∪ ∂Ω | bi (x) ≤ 0, i = 1, . . . , I}

(20)

where bi ’s are given continuously differentiable functions. Accordingly, the conditions bi (xj (t)) ≤ 0,

∀t ∈ T

(21)

must be fulfilled, where 1 ≤ i ≤ I and 1 ≤ j ≤ N . where Dαα ∈ Rs×s , Dαβ ∈ Rs×(m−s) , Dββ R(m−s)×(m−s) , we have [51, Fact 2.8.7, p.44] −1 T Dαα = Mαα − Mαβ Mββ Mαβ


−1

∈

(14)

and further [51, Fact 2.15.8, p.73] det(Dαα ) =

det(Mββ ) . det(M )

(15)

Consequently, maximization of the Ds -optimality criterion amounts to minimization of det(Dαα ), which is proportional to the determinant of the covariance matrix for α.

B. Optimal control formulation In general, to increase the degree of optimality in the considered approach, the initial locations of sensors s0 can also be treated as a decision variables to be determined in addition to the control u. Consequently, this leads to the following formulation of the optimal sensor scheduling problem for source detection and localization: Problem 1: Determine a pair (s?0 , u? ) ∈ ΩN × U, which maximizes J(s0 , u) = Ψs [M (s)], (22) subject to the constraints (17) and (21).

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It is convenient to reformulate Problem 1 into an optimal control problem with inequality-constrained trajectories and a nonlinear endpoint cost [52]. This paves the way to the possibility of employing existing software packages designed for numerically solving dynamic optimization and optimal control problems. Indeed, defining the quantity N Z t X Π(t) = g(xj (τ ), τ )g T (xj (τ ), τ ) dτ, (23) j=1

0

we get M = Π(tf ).

(24)

Thus the task of solving Problem 1 is equivalent to the following problem: Problem 2: Find a pair (s?0 , u? ) ∈ ΩN × U, which minimizes J (s0 , u) = −Ψs [Π(tf )], (25) subject to the constraints (17), (21) and N

X d Π(t) = g(xj , t)g T (xj , t), Π(0) = 0. (26) dt j=1 Thus, we have an optimal-control problem in Mayer form. The problem formulated in this way can be solved using existing packages for numerically solving dynamic optimization problems, such as R IOTS 95 [41], [53] or M ISER [54]. In our simulation example presented in the next section the first of them, i.e., R IOTS 95, which is designed as a M ATLAB toolbox was used.

v(x, t) within this interval varies in space and time according to the following model (cf. Fig. 1) v(x, t) = (y − x − t, tx + y − 1).

(29)

As for prior estimates of the unknown parameters µ and κ the values 10.0 and 0.06 were used, respectively. From among the system parameters θ = (x01 , x02 , µ, κ), the detection and localization of the pollution source is based on the knowledge of the first three, namely the coordinates of the source and its emission intensity. In this simulation study, our goal was to determine Ds -optimal motion trajectories for mobile sensors in order to maximize the reliability of simultaneous detection and localization of the contamination source. In order to verify the proposed approach, a M ATLAB program was written using a PC equipped with Pentium M740 processor (1.73GHz, 1 GB RAM) running Windows XP and M ATLAB 7 (R14). First, the system of PDEs was solved using efficient solvers of the COMSOL environment based on the finite element method. Calculations were performed for a spatial mesh composed of 3032 triangles and 1565 nodes and evenly partitioned time interval (40 subintervals). The sensitivity coefficients were then linearly interpolated and stored. Finally, for determining the optimal trajectories, the package RIOTS 95 [41], [53] based on the M ATLAB environment and dedicated to solving the optimal control problems hwas applied. From among its three main optimization procedures the routine riots was used, which is based on the SQP algorithm. As regards the sensors dynamics, the simple model (19) was applied and the following bounds for u were used:

IV. S IMULATION EXAMPLE As a suitable example illustrating the delineated approach, consider a pollutant transport-chemistry process over a spatial domain being normalized to a unit circle. An active source of a pollutant is located at point x0 = (0.0, 0.5) and produces changes in the pollutant concentration y(x, t). The entire process over the observation interval T can be described by the advection-diffusion-reaction equation:
∂y(x, t) +∇ · v(x, t)y(x, t) ∂t
=∇ · κ∇u(x, t) + f (x),

(27) (x, t) ∈ Ω × T

subject to the boundary and initial conditions:   ∂y(x, t) = 0, (x, t) ∈ Γ × T, ∂n y(x, 0) = 0, x ∈ Ω, 2

(28)

where the term f (x) = µe−100kx−x0 k represents a model of an active source of the pollutant with the emission intensity coefficient µ, κ is a turbulent diffusion coefficient and ∂y/∂n stands for the partial derivative of y with respect to the outward normal to the boundary Γ. Due to the numerical reasons the real-time observation interval has been appropriately rescaled and is assumed to be T = [0, 2]. Further, the velocity field

−0.5 ≤ ui (t) ≤ 0.5,

∀t ∈ T.

(30)

Since usually it is not possible to freely deploy sensors within the considered domain, the sensors are assumed to approach the contaminated area starting from the boundary and their initial positions are arbitrarily fixed and in this example are not optimized. To simplify the numerical task, the constraints on the potential collisions of sensors were neglected. However, any real sensor platform possesses some amount of autonomy allowing for avoiding such critical situations. In order to avoid convergence to local optima, the simulations were restarted several times from different initial starting points. Each simulation took about 15 minutes of computation time. The results are shown in Fig. 2, where the sensor paths are shown for different numbers of vehicles. The behaviour of sensors is not that intuitive, since the concentration changes are quite complex due to the combination of different mass transport processes. However, it is clear that if sensors have enough time and power to reach the location of the contamination source, they finish in its close vicinity where they can collect most informative observations (this concerns sensors starting from the top left-hand part of boundary). Otherwise they attempt to reach the areas with high pollutant concentration as quickly as possible (sensors starting from the bottom part of boundary).

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However, there still remain some open questions which need close attention. The first is the extension of the presented approach to the case of moving contamination sources. Another delicate issue is the dependence of the optimal solution on the system parameters and therefore some robust approaches to parameter uncertainty of the model are required. This leads directly to the notion of the so-called robust design which would attempt to make the optimal solutions independent of the parameters to be identified.

The problem of sensor scheduling was considered for a monitoring network with mobile nodes providing proper diagnostic information about the presence and location of a contamination source in given spatial domain. It was demonstrated that the relevant optimization task can be fit to the framework of an optimal control problem and readily solved using existing efficient numerical packages for dynamic optimization.

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Temporal changes in the wind velocity field and pollutant concentration.

V. C ONCLUSION

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Optimal trajectories of sensors (starting positions marked with open circles and final with triangles) for: three (a), four (b) and five (c) sensors.

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Optimal controls for the case of N = 3 sensors.

ACKNOWLEDGMENT Effort sponsored by the Air Force Office of Scientific Research, Air Force Material Command, USAF, under grant number FA8655–08–1–3014. The U.S. Government is authorized to reproduce and distribute reprints for Government purpose notwithstanding any copyright notation thereon. R EFERENCES [1] C. S. Kubrusly and H. Malebranche, “Sensors and controllers location in distributed systems — A survey,” Automatica, vol. 21, no. 2, pp. 117–128, 1985. [2] D. Uci´nski, Optimal Measurement Methods for Distributed-Parameter System Identification. Boca Raton, FL: CRC Press, 2005.

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