Resource-aware sensor activity scheduling for parameter estimation of distributed systems

Proceedings of the 18th World Congress The International Federation of Automatic Control Milano (Italy) August 28 - September 2, 2011

Resource-aware sensor activity scheduling for parameter estimation of distributed systems ⋆ Maciej Patan ∗ Dariusz Uci´ nski ∗ ∗

Institute of Control and Computation Engineering University of Zielona G´ ora, Poland e-mail: {M.Patan,D.Ucinski}@issi.uz.zgora.pl

Abstract: The problem under consideration is to determine an activation policy of discrete scanning sensors for identification of unknown parameters in distributed systems in the situation when some resource related constraints have to be simultaneously satisfied. An observation policy is proposed to provide a proper distribution of sensors over the considered spatial domain for the purpose of maximizing the parameter identification accuracy using a minimal number of sensors and reducing the accumulated activation time. The problem is cast as a multiobjective optimization one and solved using the goal attainment method which applies sequential quadratic programming and an efficient guided search algorithm based on the branchand-bound method. Keywords: Distributed parameter systems, optimal experiment design, parameter estimation, sensor network. 1. INTRODUCTION The rapidly increasing complexity of modern monitoring systems forces engineers to look for more systematic approaches to the problem of optimization of the measurement process based on various data sources. In particular, one of the crucial design issues in parameter estimation of systems governed by partial differential equations (PDEs), commonly termed ‘Distributed Parameter Systems’ (DPSs), is the problem of how to deploy and schedule the activity of the measurement sensors. The accurate parameter estimation is important as the estimates are then often used in optimal control, quality monitoring, fault diagnosis or to give an indication of the basic mechanism underlying the spatiotemporal process. From a technical point of view, this observation should be now easier due to the emergence of distributed sensor networks which have recently become an important re¨ search area regarding spatio-temporal phenomena (Ogren et al., 2004; Chong and Kumar, 2003; Cassandras and Li, 2005; Mart´ınez and Bullo, 2006). A sensor network may comprise thousands of inexpensive, miniature and lowpower sensor nodes that can be deployed throughout a physical space and connect through a multi-hop wireless network, providing dense sensing close to physical phenomena. In this context, sensor location problem for DPSs has already attained a lot of attention (for reviews, see papers Kubrusly and Malebranche (1985); Uci´ nski (2000); van de Wal and de Jager (2001) and comprehensive monographs Uci´ nski (2005); Patan (2004)), however most contributions usually rely on exhaustive search over a predefined set ⋆ The work was partially supported by the Polish Ministry of Science and Higher Education under grants N N514 230537 and N N519 297133.

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of candidates and the combinatorial nature of the design problem is taken into account very occasionally (van de Wal and de Jager, 2001). Therefore, the number of sensor placement techniques developed to manage the problems of practical scale is very limited (Uci´ nski, 2005; Kubrusly and Malebranche, 1985; Patan, 2004). However, some effective approaches have been proposed to cover various experimental settings, including stationary (Nehorai et al., 1995; Uci´ nski, 2000; Point et al., 1996), scanning (Patan and Uci´ nski, 2005; Uci´ nski and Patan, 2007; Uci´ nski and Demetriou, 2004; Patan, 2008) or moving observations (Jeremi´c and Nehorai, 2000; Rafajlowicz, 1986; Uci´ nski and Chen, 2005; Patan et al., 2008; Demetriou and Hussein, 2009; Song et al., 2009). On the other hand, sensor networks imply an additional layer of complexity. For example, if communication connectivity is to be maintained, we must ensure that each node remains within the range of some other nodes. We must also take into account that sensor activity consumes a considerable amount of energy, which amplifies the need for various forms of power control. As a result, the existing activity scheduling algorithms for wireless sensor networks abstract away from the mathematical models of the observed physical phenomena and the classical techniques invented for optimal sensor location in DPSs are not adapted to take account of various practical constraints associated with the operation of sensor networks. The aim of the research reported here is to develop a practical approach to sensor activation which, while being independent of a particular model of the dynamic DPS in question, would be versatile enough to cope with practical monitoring networks consisting of many stationary sensors. In particular, the efficient approach to sensor selection reported in (Uci´ nski and Patan, 2007; Patan and

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Uci´ nski, 2008; Uci´ nski, 2009) can be adapted and extended to the more general setting of multi-criteria scheduling for scanning sensor networks, where the observation system comprises multiple stationary sensors located at already specified locations and it is desired to activate only a subset of them during a given time interval while the other sensors remain dormant (Demetriou, 2000). We formulate the scheduling problem as a multiobjective optimization one, in which we are interested in simultaneously maximizing a scalar measure of identification accuracy based on the Fisher information matrix (FIM) associated with the parameters to be identified, minimizing the number of time slices each node is active and minimizing the number of sensors which guarantee the coverage of whole monitored area. The problem is solved using the goal attainment method which applies an efficient guided search algorithm based on the branch-and-bound method. To illustrate the use of our algorithm, we include some numerical experience on a sensor network design problem regarding a two-dimensional convection-diffusion process. 2. SYSTEM DESCRIPTION

3. SENSOR SCHEDULING PROBLEM 3.1 Estimation accuracy measure The most widely used formulation of the parameter estimation problem is as follows: Given the model (1)–(3) and ℓ the outcomes of the measurements zm ( · ), ℓ = 1, . . . , nk on b a global minimizer of time intervals Tk , estimate θ by θ, the output least-squares criterion nk Z K X X  ℓ 2 J (ϑ) = zm (t) − y(xℓk , t; ϑ) dt (5) k=1 ℓ=1

Consider a bounded spatial domain Ω ⊂ Rd with sufficiently smooth boundary Γ, a bounded time interval T = (0, tf ], and a distributed parameter system (DPS) ¯ ⊂ Rd and time whose scalar state at a spatial point x ∈ Ω instant t ∈ T¯ is denoted by y(x, t). Mathematically, the system state is governed by the partial differential equation (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 and may include terms accounting for forcing inputs specified a priori. The PDE (1) is accompanied by the appropriate boundary and initial conditions B(x, t, y, θ) = 0 on Γ × T, (2) y = y0 in Ω × {t = 0}, (3) respectively, B being an operator acting on the boundary Γ and y0 = y0 (x) a given function. Conditions (2) and (3) complement (1) such that the existence of a sufficiently smooth and unique solution is guaranteed. We assume that the forms of F and B are given explicitly up to an m-dimensional vector of unknown constant parameters θ which are supposed to be estimated based on observations of the system behaviour. Let us partition the time horizon T into subintervals Tk = (tk−1 , tk ], k = 1, . . . , K, where 0 = t0 < t1 < · · · < tK = tf . The state y can be observed (possibly indirectly) by N pointwise sensors, but among them only nk ≥ n are activated on Tk . Here n constitutes a given minimum allowable number of active sensors. Forming such an arbitrary partition on the time interval T , the considered ‘scanning’ observation strategy can be formally represented as ℓ zm (t) =y(xℓk , t; θ) + ε(xℓk , t), t ∈ Tk , ℓ = 1, . . . , nk , k = 1, . . . , K,

subinterval Tk , X signifies the part of the spatial domain Ω where the measurements can be made and ε(xℓk , t) denotes the measurement noise, which is customarily assumed to be zero-mean, Gaussian, spatial uncorrelated and white. 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.

(4)

ℓ where zm (t) is the scalar output and xℓk ∈ X stands for the location of the ℓ-th active sensor on the time

T

where y( · , · ; ϑ) denotes the solution to (1)–(3) for a given value of the parameter vector ϑ. b of the above leastInevitably, the covariance matrix cov(θ) squares estimator depends on the active sensor locations xℓk . This fact suggests that we may attempt to select them so as to yield best estimates of the system parameters. To form a basis for a comparison of different locations, a quantitative measure of the ‘goodness’ of particular sensor configurations is required. Such a measure is customarily based on the concept of the Fisher Information Matrix (FIM) which is widely used in optimum experimental design theory for lumped systems (Walter and Pronzato, 1997; Atkinson et al., 2007). In our setting, owing to the character of noise in (4), the FIM is given by Z nk K X X 1 g(xℓk , t)g T (xℓk , t) dt, (6) M= tf Tk k=1 ℓ=1

where



T ∂y(x, t; ϑ) ∂y(x, t; ϑ) (7) , ..., ∂ϑ1 ∂ϑm ϑ=θ 0 stands for the so-called sensitivity vector, θ0 being the nominal value of the parameter vector θ (Uci´ nski, 2005; Rafajlowicz, 1983). Up to a constant scalar multiplier, the inverse of the FIM constitutes a good approximation of b provided that the time horizon is large, the nonlincov(θ) earity of the model with respect to its parameters is mild, and the measurement errors are independently distributed and have small magnitudes (Walter and Pronzato, 1997). g(x, t) =

As for a specific form of the criterion which is supposed to quantify the identification accuracy, various options exist (Walter and Pronzato, 1997; Atkinson et al., 2007), but the most popular criterion to be minimized, called the Doptimality criterion, is the negative of the log-determinant of the FIM: Ψ1 (M ) = − log det(M ). (8) The resulting D-optimum sensor configuration leads to the minimum volume of the uncertainty ellipsoid for the estimates (Walter and Pronzato, 1997).

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any of the sensors should not be activated much more frequently than others. Since the total activation time for PK i the i-th node is τi = k=1 vk ∆tk , where ∆tk = tk − tk−1 , in order to provide a balanced sensor schedule, the criterion Ψ3 = max τi (11) i=1,...,N

has to be minimized. Rewriting the FIM in (6) as K N X X

vki Mki ,

(12)

g(xi , t)g T (xi , t) dt

(13)

M (v1 , . . . , vK ) =

i=1 k=1

Fig. 1. Example of monitoring regions and sensor coverage

where Mki

3.2 Minimum cover The optimal sensor scheduling problem considered in what follows consists in seeking, for each time subinterval Tk , the best subset of nk locations from among the N given potential ones, so as to minimize the criterion (8) associated with the expected accuracy of the parameters to be estimated. Thus the problem is reduced to a combinatorial one. However, as one of crucial issues for sensor networks is to assure that the communication connectivity will be maintained during the monitoring process, therefore we must ensure that each node remains within the range of at least some other nodes. Moreover, often it is required to achieve this with possible the lowest cost of operation and equipment. In order to formulate this mathematically, for each possible location xi (i = 1, . . . , N ) introduce a set of variables vki s, each of them taking the value 1 or 0 depending on whether or not a sensor residing at xi is activated during Tk , respectively. Given a sensor set S = {s1 , . . . , sN } and a partitioning Ω1 , . . . , ΩL of the domain Ω, such that Ω = ∪L ℓ=1 Ωℓ and Ωℓ s are disjoint sets, we can introduce the area coverage matrix CL×N , where each element ciℓ is 1 if the Ωℓ lies within the range of sensor si and 0 otherwise. The idea is illustrated in Fig. 1 where, e.g., the area Ω2 is covered with sensors 1 and 2 only, so that the second row of C will be (1, 1, 0, 0, 0, 0). Attaching to each sensor si its associated cost bi of operation, a minimum cover can be understood as sensor configuration minimizing the maximum cost over the given observational subintervals N X vki bi (9) Ψ2 (v) = max k=1,...,K

subject to N X ciℓ vki ≥ 1,

i=1

ℓ = 1, . . . , L, k = 1, . . . , K.

(10)

i=1

3.3 Balanced operation scheduling Another critical resource constraint which cannot be neglected results from the fact that the available battery power for each node is limited. Therefore, the activation of sensor nodes along the time slots Tk has to be properly balanced. This means that, if possible, sensor nodes should equally share the effort of providing measurements, i.e.,

1 = tf

Z

Tk

and vk = (vk1 , . . . , vkN ), our design problem may be cast as a multiobjective optimization one: Problem 1.
Find a sequence v = (v1 , . . . , vK ) to minimize Ψ1 M (v) , Ψ2 (v) and Ψ3 (v) subject to N X

ciℓ vki ≥ 1,

ℓ = 1, . . . , L, k = 1, . . . , K,

(14)

i = 1, . . . , N, k = 1, . . . , K.

(15)

i=1

vki = 0 or 1,

This constitutes a 0–1 integer programming problem whose solution is highly nontrivial due to the nonlinearity of all criteria. Moreover, the minimax character of Ψ2 and Ψ3 makes a direct application of nonlinear integer programming techniques even more difficult. In the following a general sequential computational scheme is proposed to determine a solution based on the goal attainment method combined with the branch-and-bound search. 4. GOAL ATTAINMENT FORMULATION AND SOLUTION VIA BRANCH AND BOUND The majority of methods developed for solving a multiobjective optimization problems is based on the concept of the Pareto optimal solution which is understood as a solution which cannot be improved in the sense of any objective function without simultaneous decrease of at least one other objective. The main idea of such a group of techniques is to generate a set of Pareto optimal solutions and use some additional rule to select from this set one particular solution to become the ultimate one (Rao, 2009). Within this class of methods the goal attainment method is one of the most popular due to its flexibility in applications and the intuitive formulation leading to a single aggregate objective function (Rao, 2009). For each objective Ψj a goal ψj is set on its value and a nonnegative weight αj expressing the importance of the j-th objective function relative to the other criteria in meeting the goal ψj . The goals can be set a priori or can be determined as optimal solutions to the problem when only one particular objective function is selected. Applying these ideas to the sensor network design formulated in Problem 1 we can restate the task in terms of single objective optimization:

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Problem 2. (P). Find a sequence v to minimize P(v, bmax , τmax , γ) := γ subject to
Ψ1 M (v) − γα1 ≤ ψ1 , bmax − γα2 ≤ ψ2 , τmax − γα3 ≤ ψ3 ,

(16) (17) (18) (19)

N X

vki bi ≤ bmax ,

K X

vki ∆tk ≤ τmax ,

i = 1, . . . , N,

(21)

ciℓ vki ≥ 1,

ℓ = 1, . . . , L, k = 1, . . . , K,

(22)

k = 1, . . . , K,

(20)

j=1

k=1 N X

i=1 vki =

0 or 1, i = 1, . . . , N, k = 1, . . . , K α1 + α2 + α3 = 1.

The key assumption in the BB method is that for every nonterminal node V (I0 , I1 ), i.e., the node for which I0 ∪ I1 6= I, there is an algorithm that determines a lower bound P(I0 , I1 ) to the minimum criterion value over V (I0 , I1 ), and a feasible solution v¯ ∈ V for which P(¯ v ) can serve as an upper bound to the minimum criterion value over V . The value P(I0 , I1 ) can be determined through solving the following relaxed problem: Problem 3. (R(I0 , I1 )). Find a sequence v to minimize (16) subject to (17)–(22) and 0 ≤ vki ≤ 1, ∀(i, k) ∈ I \ (I0 ∪ I1 ), (26) vki = 1, ∀(i, k) ∈ I1 .

The result of solving Problem R(I0 , I1 ) can be used to construct a branching rule for the binary BB tree. We adopt here the approach in which the node/subset V (I0 , I1 ) is expanded (i.e., partitioned) by first picking out all fractional values from among the values of the relaxed variables, and then rounding to 0 and 1 a value which is the most distant from both 0 and 1. Combination of this branching rule with a search strategy to incrementally explore all the nodes of the BB tree constitutes a sequential procedure which starts from the root corresponding to V (∅, ∅) = V and the fully relaxed problem.

(23) (24)

In Uci´ nski and Patan (2007) a general sequential computational scheme was proposed to solve a similar but much simpler problem based on the branch-and-bound (BB) method which is a standard technique for finding optimal solutions of various optimization tasks and stands as one of classical approaches to combinatorial problems (Floudas, 2001; Bertsekas, 1999; Eckstein, 2006). Its proper application may lead to algorithms of high average performance.  Let I denote the set of index pairs (i, k)| i = 1, . . . , N and k = 1, . . . , K identifying sensor locations and time subintervals. Consider a relaxation of Problem P, which starts by partition of the feasible set V consisting of all the sequences (v1 , . . . , vK ) satisfying constraints (17)-(22) into subsets of the form V (I0 , I1 ) = v ∈ V | vki = 0, ∀(i, k) ∈ I0 , (25) vki = 1, ∀(i, k) ∈ I1 , where I0 and I1 are disjoint subsets of I. Consequently, V (I0 , I1 ) is the subset of V such that a sensor is activated at the locations and time subintervals specified with indices in I1 , no sensor is activated at the time and locations with indices in I0 , and a sensor may be activated or stay dormant at the remaining time moments and locations. Each subset V (I0 , I1 ) can be directly identified with a node in the associated BB tree.

vki = 0, ∀(i, k) ∈ I0 ,

As for the upper bound v¯, we can specify it as the best feasible solution (i.e., an element of V ) found so far. If no solution has been found yet, we can either set the lower bound to −∞, or use any initial guess about the optimal solution.

(27)

In Problem R(I0 , I1 ) all 0–1 constraints on the variables vki are relaxed by allowing them to take any value in the interval [0, 1], except that the variables vki , (i, k) ∈ I0 ∪ I1 are fixed at either 0 or 1.

5. SIMULATION EXAMPLE As an illustration of the presented approach, consider the problem of sensor configuration for parameter estimation in the process of air pollutant transport over a given urban area Ω, being a square with a side of length 1 km. In this domain, two active sources of pollution are present, which yields the pollutant spatial concentration y = y(x, t). The evolution of y over the observation interval T = (0, 1000] (in seconds) is described by the following advection-diffusion-reaction equation:
∂y(x, t) + ∇ · v(x, t)y(x, t) + βy(x, t) ∂t
= ∇ · κ∇y(x, t) + f1 (x) + f2 (x), x ∈ Ω, (28) subject to the boundary and initial conditions: ∂y(x, t) = 0, on ∂Ω × T, (29) ∂n y(x, 0) = y0 , in Ω, (30)
ℓ 2 where terms fℓ (x) = µℓ exp − 100kx − ξ k , ℓ = 1, 2 represent the pollutant sources with emission intensities µℓ located at the points ξ ℓ = (ξ1ℓ , ξ2ℓ ), ℓ = 1, 2, and ∂y/∂n stands for the partial derivative of y with respect to the outward normal to the boundary ∂Ω. The average spatiotemporal changes in the wind velocity field over Ω were approximated according to the model (scaled in [km/h])   v(x, t) = 7.2· (x1 +x2 −t·10−3 ), (2x1 −1)t·10−3 +x2 −1 . Furthermore, κ = 50 m2 /s denotes a turbulent diffusion coefficient and β = 0.01 s−1 stands for the absorption rate modelling a slow decay of the pollutant. Figure 2 illustrates the resulting complex process dynamics.

Our goal is to identify the intensities of the sources, i.e., the vector of θ = (µ1 , µ2 ) using a homogenous sensor network with scanning nodes with the range of 300 m. Simultaneously, we are interested in minimizing the number of sensors used per time slot (i.e. cost per sensor is assumed as bi = 1) and the number of time slots each sensor is activated. The weights for particular objectives were set to α1 = 0.5, α2 = 0.3 and α3 = 0.2. The observation horizon was split into 5 evenly partitioned subintervals Tk = (200(k − 1), 200k], k = 1, . . . , 5. In order to verify the proposed approach, a Matlab program

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18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011

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Fig. 2. Temporal changes in the wind velocity field and pollutant concentration. decentralized strategy based on both decomposition of the problem and parallelization of the computations on a grid of computers, cf. Zi¸eba and Uci´ nski (2008). REFERENCES Atkinson, A.C., Donev, A.N., and Tobias, R. (2007). Optimum Experimental Design, with SAS. Oxford University (a)

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The observation grid was assumed to be located on the internal points of a spatial triangulation mesh (there was 52 such points, which are indicated with dots in Fig. 3(a)). A separate triangulation mesh composed of 68 triangles was used to generate the partitioning of the domain. The size of mesh should be a compromise between reasonable small number of constraints and good decomposition of the domain giving sensors possibility to cover many different subdomains. The potential sensor locations and area partitioning is shown in Fig. 3(a). As an effective solver for the relaxed problem within the BB search the fgoalattain function from the MATLAB Optimization Toolbox was applied. The algorithm started from a randomly generated initial solution and converged within 98 recursive calls to the procedure solving the relaxed optimization problem. Optimal sensor configurations at the consecutive time slots are shown in Fig. 3. From the point of view of the estimation accuracy the sensors should tend to form patterns reflecting the areas of greatest changes in the pollutant concentration, but due to the limited activation time and the limited number of nodes used, the sensors switch to other areas and it is not trivial to interpret the changes in the observation strategy. Evidently, the cover of the monitored are is guaranteed with no more than 12 sensors used (the goal for this objective was 9) and no more than 3 time slots allocated for each sensor (the goal was 1). The value of the determinant of the information matrix dropped to 52% of that achieved when the D-optimality criterion was the only performance index.

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6. CONCLUSION The sensor scheduling problem in view of accurate parameter estimation for distributed-parameter systems subject to constraints on the total node activity has been addressed. The presented results show that some methods of optimum experimental design and discrete optimization can be extended to the considered setting leading to effective numerical schemes. The ongoing work is related to an extension of the proposed approach in the direction of a

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was written using a PC equipped with Intel Core i7 processor (1.6GHz, 4 GB RAM) running Windows 7 and Matlab 2009b. First, the system of PDEs was solved using efficient solvers of the COMSOL environment based on the finite element method COMSOL AB (2007). The nominal values of the system parameters were assumed to be θ = (10 kg/s, 15 kg/s) and the sources are assumed to be located at (0.5 km, 0.75 km) and (0.75 km, 0.5 km).

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18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011

Press, Oxford. Bertsekas, D.P. (1999). Nonlinear Programming. Optimization and Computation Series. Athena Scientific, Belmont, MA, 2nd edition. Cassandras, C.G. and Li, W. (2005). Sensor networks and cooperative control. European Journal of Control, 11(4– 5), 436–463. Chong, C.Y. and Kumar, S.P. (2003). Sensor networks: Evolution, opportunities, and challenges. Proceedings of the IEEE, 91(8), 1247–1256. COMSOL AB (2007). COMSOL Multiphysics Modelling Guide, ver. 3.4. Demetriou, M.A. (2000). Activation policy of smart controllers for flexible structures with multiple actuator/sensor pairs. In A. El Jai and M. Fliess (eds.), Proc. 14th Int. Symp. Mathematical Theory of Networks and Systems, Perpignan, France, 19–23 June 2000. Published on CD-ROM. Demetriou, M.A. and Hussein, I.I. (2009). Estimation of spatially distributed processes using mobile spatially distributed sensor network. SIAM Journal on Control and Optimization, 48(1), 266–291. Eckstein, J. (2006). Massively parallel mixed-integer programming: Algorithms and applications. In M.A. Heroux, P. Raghavan, and H.D. Simon (eds.), Parallel Processing for Scientific Computing, 323–340. SIAM Books, Philadelphia, USA. Floudas, C.A. (2001). Mixed integer nonlinear programming, MINLP. In C.A. Floudas and P.M. Pardalos (eds.), Encyclopedia of Optimization, volume 3, 401–414. Kluwer Academic Publishers, Dordrecht, The Netherlands. Jeremi´c, A. and Nehorai, A. (2000). Landmine detection and localization using chemical sensor array processing. IEEE Transactions on Signal Processing, 48(5), 1295– 1305. Kubrusly, C.S. and Malebranche, H. (1985). Sensors and controllers location in distributed systems — A survey. Automatica, 21(2), 117–128. Mart´ınez, S. and Bullo, F. (2006). Optimal sensor placement and motion coordination for target tracking. Automatica, 42, 661–668. Nehorai, A., Porat, B., and Paldi, E. (1995). Detection and localization of vapor-emitting sources. IEEE Transactions on Signal Processing, 43(1), 243–253. ¨ Ogren, P., Fiorelli, E., and Leonard, N.E. (2004). Cooperative control of mobile sensor networks: Adaptive gradient climbing in a distributed environment. IEEE Transactions on Automatic Control, 49(8), 1292–1302. Patan, M. (2004). Optimal Observation Strategies for Parameter Estimation of Distributed Systems. Zielona G´ ora: University Press. May be viewed at http://www.zbc.zgora.pl. Patan, M. (2008). A parallel sensor scheduling technique for fault detection in distributed parameter systems. Lecture Notes in Computer Science: Euro-Par 2008: Parallel Processing, 5168, 833–843. Patan, M., Chen, Y., and Tricaud, C. (2008). Resourceconstrained sensor routing for parameter estimation of distributed systems. In Proc. 17th IFAC World Congress, Seoul, South Korea, 4–8 July 2008. Published on DVD-ROM.

Patan, M. and Uci´ nski, D. (2005). Optimal activation strategy of discrete scanning sensors for fault detection in distributed-parameter systems. In Proc. 16th IFAC World Congress, Prague, Czech Republic, 4–8 July 2005. Published on CD-ROM. Patan, M. and Uci´ nski, D. (2008). Configuring a sensor network for fault detection in distributed parameter systems. International Journal of Applied Mathematics and Computer Science, 18(4), 513–524. Point, N., Vande Wouwer, A., and Remy, M. (1996). Practical issues in distributed parameter estimation: Gradient computation and optimal experiment design. Control Engineering Practice, 4(11), 1553–1562. Rafajlowicz, E. (1983). Optimal experiment design for identification of linear distributed-parameter systems: Frequency domain approach. IEEE Transactions on Automatic Control, 28(7), 806–808. Rafajlowicz, E. (1986). Optimum choice of moving sensor trajectories for distributed parameter system identification. International Journal of Control, 43(5), 1441– 1451. Rao, S.S. (2009). Engineering Optimization. Theory and Practice. John Wiley & Sons, New Jersey. Song, Z., Chen, Y., Sastry, C., and Tas, N. (2009). Optimal Observation for Cyber-physical Systems: A Fisherinformation-matrix-based approach. Springer-Verlag, Berlin Heidelberg. Uci´ nski, D. (2000). Optimal selection of measurement locations for parameter estimation in distributed processes. International Journal of Applied Mathematics and Computer Science, 10(2), 357–379. Uci´ nski, D. (2005). Optimal Measurement Methods for Distributed-Parameter System Identification. CRC Press, Boca Raton, FL. Uci´ nski, D. (2009). D-optimum sensor activity scheduling for distributed parameter systems. In Proc. 15th IFAC Symposium on System Identification, SYSID 2009, July 6-8, Saint-Malo , France. Published on CD-ROM. Uci´ nski, D. and Chen, Y. (2005). Time-optimal path planning of moving sensors for parameter estimation of distributed systems. In Proc. 44th IEEE Conference on Decision and Control, and the European Control Conference 2005, Seville, Spain. Published on CDROM. Uci´ nski, D. and Demetriou, M.A. (2004). An approach to the optimal scanning measurement problem using optimum experimental design. In Proc. American Control Conference, Boston, MA. Published on CD-ROM. Uci´ nski, D. and Patan, M. (2007). D-optimal design of a monitoring network for parameter estimation of distributed systems. Journal of Global Optimization, 39, 291–322. van de Wal, M. and de Jager, B. (2001). A review of methods for input/output selection. Automatica, 37, 487–510. ´ and Pronzato, L. (1997). Identification of ParaWalter, E. metric Models from Experimental Data. Communications and Control Engineering. Springer-Verlag, Berlin. Zi¸eba, T. and Uci´ nski, D. (2008). Mobile sensor routing for parameter estimation of distributed systems using the parallel tunneling method. International Journal of Applied Mathematics and Computer Science, 18(3), 307–318.

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