Fast Sensor Scheduling with Communication Costs for Sensor Networks

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1 2010 American Control Conference Marriott Waterfront, Baltimore, MD, USA June 30-July 02, 2010 WeA08.5 Fast Sensor Scheduling with Communication Costs for Sensor Networks Shogo Arai, Yasushi Iwatani, and Koichi Hashimoto Abstract This paper is devoted to sensor scheduling for a class of sensor networks whose sensors are spatially distributed and measurements are influenced by state dependent noise. Sensor scheduling is reuired to achieve power saving since each sensor operates with a battery power source. The sensor scheduling problem is formulated as model predictive control which minimizes a uadratic cost function with communication costs, since communications among sensors take much power. A fast and optimal sensor scheduling algorithm is proposed for a class of sensor networks. From a theoretical standpoint, we present that computation time of the proposed algorithm increases exponentially with the number of sensor types, while that of standard algorithms is exponential in the number of the sensors. The proposed algorithm is faster than standard one, since the number of sensor types is always less than or eual to the number of sensors. I. INTRODUCTION A sensor network is a collection of spatially distributed sensors that communicate with each other. Applications of sensor networks include habitat monitoring, animal tracking, forest-fire detection, precision farming, and disaster relief applications [16], [18]. Sensor networks have been also implemented in control systems such as robot control systems [4], [12] and target tracking systems [10], [22]. Sensors in a sensor network are connected wirelessly, and each sensor operates with a battery power source. It is therefore desired for each sensor to prolong the battery life, or euivalently, to achieve power saving [11]. It is important to restrict communications of sensors for power saving since communications take much power. One of approaches to solve the problem is to select available sensors dynamically [2], [14]. This process is called sensor scheduling. One of the major problems on sensor scheduling is to reduce computation time, since the number of possible sensor seuences increases exponentially with the number of the sensors. In particular, a predictive control method [5], a branch and bound method [7], a stochastic scheduling algorithm [9], and a sub-optimal method based on relaxed dynamic programming [1] have been proposed for sensor scheduling. In addition, a sensor scheduling strategy for continuous-time systems has been provided in [13]. These approaches assume that sensor characteristics are different from each other, that is, each sensor observes a different state or the covariance matrices for the sensor model are different from each other. S. Arai and K. Hashimoto are with the Department of System Information Sciences, Tohoku University, Aramaki Aza Aoba , Sendai, Japan ( arai@ic.is.tohoku.ac.jp; koichi@ic.is.tohoku.ac.jp). Yasushi Iwatani is with the Graduate School of Science and Technology, Hirosaki University, Bunkyo 3, Hirosaki, Aomori, Japan ( iwatani@cc.hirosaki-u.ac.jp). The existing approaches cannot be applied to sensor scheduling for sensor networks for the following two reasons. First, a sensor network usually consists of a few types of sensors [18]. In other words, many sensors in a sensor network have the same characteristics, while the existing approaches assume that sensors have different characteristics as mentioned before. Thus the existing approaches cannot provide a reasonable solution for the sensor scheduling problems for sensor networks. Second, sensors in a sensor network are spatially distributed. Each measurement noise may depend on the position of a target relative to the position of the sensor. In particular, measurements taken by cameras or radar sensors are influenced by state dependent noise [6], [15], [17]. The existing works have not provided any optimal sensor scheduling algorithm for systems with state dependent noise. To solve the problems, we have proposed a fast and optimal sensor scheduling algorithm for sensor networks whose measurements are influenced by state dependent noise and which have various kind of sensor nodes [3]. The sensor scheduling problem has been formulated as a model predictive control problem with single sensor measurement per time. The proposed scheduling algorithm minimizes a given uadratic cost function at each time, the concept of sensor types has been introduced to classify sensor characteristics. From a theoretical standpoint, we have shown that its computation time increases exponentially with the number of sensor types, while the computation time of a primitive algorithm is exponential in the number of sensors. Thus computational cost of the proposed algorithm is expected to be much less than a primitive one since a networked control system usually consists of a few sensors types. However, the proposed algorithm does not consider any communication costs. Thus, it cannot provide appropriate sensor scheduling when communication costs are considered. This paper is devoted to sensor scheduling with communication costs and observational ranges. An optimal scheduling algorithm is proposed for a class of sensor networks. Computation time of the proposed algorithm increases exponentially with the number of the sensor types, while a primitive one is represented by an exponential function whose base is the number of sensors and whose exponent is twice the length of the prediction horizon. Thus it is guaranteed that computation time of our algorithm is less or eual to that of a primitive one. This paper uses the following notation. The Kronecker delta is denoted by δ ml. The expectation operator is represented by E[ ]. For a suare matrix A, tr[a] stands for the trace of A /10/$ AACC 295

2 Controller Controlled Object Sensor i Sensor Network Fig. 2. Definitions of θ i and r i. Fig. 1. A networked control system whose sensors communicate with each other when N = 7. Camera II. SYSTEM DESCRIPTION This paper considers a class of systems with N sensors as illustrated in Fig. 1. The sensors are labeled from 1 to N. Each sensor can communicate with some other sensors. To simplify notations, it is assumed that one sensor is available for observation at each time. However our algorithm presented in this paper is easily extended for the systems in which multiple sensors are available at each time, when all possible combinations of sensors are labeled. The observation data is transmitted over the sensor network, and one of the sensors sends the data to the controller. Details of the network are described in the next section. A. Sensor Model Let us first describe the sensor model considered in this paper. The sensor model is of the form: i y i (k) = C i x p (k) + d il (x(k))v il (k), (1) l=1 y i (k) R p i is the measurement taken by sensor i, x p (k) R np is the state of the controlled object, d il is a vector function that is differentiable, x is defined by x(k) = [ x p (k) x c (k) ], (2) and x c is the state of the controller. The noise v i (k) = [v i1 (k),...,v ii (k)] is white, Gaussian and zero mean with a covariance matrix E[v i (k)v i (τ)] = V i δ kτ. (3) The vector function d il (x) is a function of x not of only x p. This helps to develop a camera model presented in Example 2. Note that state-dependent sensor scheduling is reuired due to the function d il (x) even when C i = C l and V i = V l hold for all pairs of (i,l). The sensor model (1) is more general than models considered in other papers. For example, [7] considers a class of systems described by y i (k) = C i x(k) + D i v i (k) (4) C i and D i are constant matrices. Our models (1) and (9) presented in Section II-B include the above class. In addition, our approach is available for systems with both sensor models (1) and (4). The following examples show several sensors whose mathematical models can be written by (1). Fig. 3. Camera Camera system. Example 1: Consider radar sensors that measure the position of a target in the (x,y) plane. Two standard models have been developed for radar sensors. One is given by y i = [ x y] + [ ][ ] cos θi sinθ i 1 0 v sin θ i cos θ i 0 r i (5) i [17], and the other is given by [ ] [ ][ ] x cos θi sin θ y i = + a(r y i ) i 1 0 v sin θ i cos θ i 0 b i (6) [6], y i is the measurement, θ i the angle between the x axis and the vector joining sensor i to the target, r i the distance from sensor i to the target, a(r i ) a function such that a 2 (r i ) is a uadratic function of r i, b a constant, and v i a white, Gaussian and zero mean noise (see Fig. 2). It is clear that (5) and (6) are described by (1). Example 2 ([3]): Let a target and two cameras be set at (x,y), (p lx, p ly ) and (p mx, p my ), respectively, as illustrated in Fig. 3. The optical axes of cameras l and m are parallel to the y axis and the x axis, respectively. The combination of cameras l and m is labeled by i. Then the camera model is represented by y i = [ x y] + 1 f [ ] yc p ly 0 v 0 x c p i, (7) mx f is the focal length of the cameras and (x c,y c ) is an estimate of (x,y). Euation (7) is represented by (1), since d il is a function of x. B. Sensor Type The concept of sensor types is a key to developing a fast sensor scheduling algorithm. A function σ is defined as a surjection from {1,2,...,N} := N onto {1,2,...,M} =: M for some M that satisfies C i = C l and V i = V l σ(i) = σ(l). (8) Each value of σ represents a sensor type. In other words, a set of sensors with the same measurement matrix and the same covariance matrix constitutes a type of sensors. The number of sensor types is always less than or eual to the number of 296

3 sensors, that is, M N. A sensor network usually consists of a few sensor types [18]. Thus M is far less than N for a class of sensor networks. The concept of sensor types is effective for our scheduling algorithm proposed in Section IV-B. In order to clarify the sensor types, we rewrite C i and V i as C σ(i) and V σ(i), respectively. Hence, we have y i (k) = C σ(i) x p (k) + σ(i) l=1 d il (x(k))v il (k), (9) E[v i (k)v i (τ)] = V σ(i) δ kτ. (10) The sensor type σ(i) is sometimes simply written by σ. C. Closed Loop System The controlled object is represented as a discrete-time linear time-invariant system x p (k + 1) = A p x p (k) + B p u(k) + w(k), (11) x p (k) R n p is the state vector, u(k) R r the control input, and w(k) the process noise. The noise w(k) is white, Gaussian and zero mean with a covariance matrix W. The time index k is sometimes omitted to simplify notation. The initial state x p (0) is a random variable whose expectation value and covariance matrix are known constants. The controller is given by x c (k + 1) = A c x c (k) + B 1c y i(k) (k) + B 1c u(k), (12) u(k) = C c x c (k) + D c y i(k) (k). (13) The closed loop system described by (9) (13) is of the form: σ(i) [ ] w(k) x(k + 1) = A σ x(k) + B σ d il (x)v il (k) +, 0 l=1 (14) [ ] A A σ = p + B p D cσk C σ B p C cσ, B 1cσ C σ + B 2cσ D cσ C σ A cσ + B 2cσ C cσ (15) [ ] B B σ = p D cσ. (16) B 1cσ + B 2cσ D cσ We also define n := n p + n c and x 0 := E[x(0)]. III. SENSOR SCHEDULING PROBLEM The network considered in this paper works as follows. Sensor i(k) is used for observation at time k. The observation data is transmitted over the network, and sensor j(k) sends it to the controller. Sensors available for observation are restricted. The set of available sensors for observation at time k is denoted by I c (k). Sensors that can communicate with the controller are also restricted, and the set of such sensors at time k is denoted by J c (k). It is assumed that an optimal path from sensor i to sensor j is known for all pairs of i and j. The communication cost between sensors i and j is denoted by γ ij ( 0). The communication cost is given by γ ij =, when sensors i and j are disconnected. The sensor scheduling problem is formulated as follows. Problem 1: Find optimal pairs of ι (k) = (i (k),j (k)) such that {ι (0),ι (1),,ι (T)} = arg min ι(0) K(0),,ι(T) K(T) J c, (17) K(k) = I c (k) J c (k), (18) [ T { J c = E x p (k)q p x p (k) + u (k)ru(k) + (x(k) m j(k) ) (k)γ(x(k) m j(k) ) } ] T + x p (T + 1)Πx p (T + 1) + γ i(k)j(k), (19) T is a positive integer, Q p R np np, R R r r and Π R n p n p are positive definite symmetric matrices and Γ R n n is a semi-positive definite symmetric matrix and m j R n is a constant vector. The cost J c includes a communication cost between the controller and sensor j represented by the third term in the right hand side of (19). If there exist no communication costs, then Γ = 0 and γ ij = 0 for all i and j. In this case, we obtain a finite-time uadratic cost function [ T { J = E x p (k)q p x p (k) + u (k)ru(k) } ] + x p (T + 1)Πx p (T + 1). (20) In our previous paper [3], a fast sensor scheduling algorithm that minimizes J has been derived on the assumption that i(k) = j(k) and I c (k) = J c (k) = N for all k. This paper assumes that model predictive control is implemented. Problem 1 is solved at each time. Thus a fast algorithm for solving Problem 1 is desired. One primitive method to solve Problem 1 is as follows: Calculate values of the cost function for all possible seuences of pairs of sensors from time 0 to T and compare these values. This is called the exhaustive search in this paper. The exhaustive search reuires N 2(T+1) sensor seuences to determine the optimal solution, when I c (k) = N and J c (k) = N for all k. The exhaustive search is not suitable for model predictive control from the point of view of computation time. When model predictive control is applied, E[x(k)] is reuired to solve Problem 1 at time k. Therefore we have to estimate E[x p (k)]. We use (12) as an observer for estimation in a numerical example presented in this paper. Then x c converges to E[x p ] when σ is fixed and all the eigenvalues of A σ are in the open unit disk, since IV. E[x(k + 1)] = A σ E[x(k)]. (21) FAST SENSOR SCHEDULING A. Fast Sensor Selection in a Given Seuence of Sensor Types The key idea to obtain a fast sensor scheduling algorithm is to separate the original sensor scheduling problem into 297

4 two scheduling problems: Scheduling of sensor types and scheduling of sensors in a given seuence of sensor types. The later is formulated as follows. Problem 2: Suppose that a sensor type σ k is selected for observation and a sensor type ρ k is chosen for communication with the controller at time k {0,1,,T }. Find {ι c(0),ι c(1),,ι c(t)} = arg min ι c (0) Kt(0),,ι(T) Kt(T) J c (22) K t (k) = {I σk I c (k)} {I ρk J c (k)}, (23) I σk = {i σ(i) = σ k }, (24) I ρk = {i σ(i) = ρ k }. (25) This section proposes a fast sensor selection method that gives ι c(k) in (22). It is assumed throughout this paper that one of the following two holds to derive theoretical results. 1) There exist constant matrices S σl R p σ n and s il R pσ such that d il (x) = S σl x + s il, i {1,2,,N}, l {1,2,, σ }. (26) 2) There exist constant matrices S il R p σ n and s il R p σ such that d il (x)d im(x) = s il s im + S il (x x 0 )s im + s il (x x 0 ) S im, i {1,2,,N}, l,m {1,2,, σ }. (27) Roughly speaking, (28) implies that the magnitude of the sensor noise is proportional to the distance between the controlled object and sensor i, and the coefficient matrices S σl do not depend directly on sensor i. The sensor model (7) satisfies (28), although (28) does not hold for (5) or (6). We can derive all the results that will be provided in the paper when we replace (26) with the following: κ ilm d il (x)d im(x) = (S σlj x + s ilj )(S σmj x + s imj ), j=1 i {1,2,,N}, l,m {1,2,, σ }. (28) Note that (26) is not euivalent to (28). For example, the sensor model (9) with σ = 1 and d i1 = (x pix ) 2 + (y p iy ) 2 satisfies (28) but not (26). The second assumption (27) is valid for all systems in a neighborhood of x = x 0. In fact, we obtain (27) when we set s il =d il ( x 0 ), S il = d il (29) x x= x 0 and use a linear approximation of d il (x)d im (x) around x 0. Hence the proposed sensor scheduling algorithm with the linear approximation of the sensor model is always available if d il is differentiable. The following theorem gives a solution for Problem 2. Theorem 1: Suppose that sensor types σ k and ρ k are selected at time k {0,1,,T } for observation and communication with the controller, respectively. 1) If (26) holds, then the optimal pair of sensors ι c(k) satisfies ι c(k) = arg min ι(k) K t(k) { tr[p(k)ψ i (k)] + tr[φ i (k)] +g j (k) + γ ij }, (30) g j (k) = m j Γm j m j ΓA(k) x 0 (m j ΓA(k) x 0 ), (31) [ ] Π 0 P(T + 1) =, (32) 0 0 P(k) = Q σk + A σ k P(k + 1)A σk Ψ i (k) = σk + σk σk σk V σk lms σ k mb σ k P(k + 1)B σk S σk l, V σk lmb σk (S σk la(k) x 0 s im (33) + s il (S σk ma(k) x 0 ) + s il s im)b σ k + Ω (34) Φ i (k) = D cσ k RD cσk σk σk V σk lm{s σk la(k) x 0 s im + s il (S σk ma(k) x 0 ) + s il s im}, (35) [ ] [ ] Qp 0 C Q σk = + σk Dcσ k 0 0 Ccσ R [ ] D cσk C σk C cσk k Ω = σk + σk V σk lms σ k md cσ k RD cσk S σk l + Γ, (36) [ ] W 0, (37) 0 0 A(k) = A σk 1...A σ1 A σk0, (38) with A(0) = I and V σk lm is the (l,m) element of V σk. 2) Suppose that (27) is satisfied. Then (30) is also true when P, Ψ i, Φ i and Q σk are replaced with Ψ i (k) = P(k) = Q σk + A σ k P(k + 1)A σk, (39) σk σk V σk lmb σk {S il (A(k) I) x 0 s im + s il (S im (A(k) I) x 0 ) + s il s im}b σ k + Ω, (40) 298

5 Φ i (k) = D cσ k RD cσk σk σk V σk lm {S il (A(k) I) x 0 s im + s il (S im (A(k) I) x 0 ) + s il s im}, (41) [ ] [ ] Qp 0 C Q σk = + σk Dcσ k 0 0 Ccσ R [ ] D cσk C σk C cσk k + Γ. (42) Proof: The results can be derived in the same manner as Lemma 1 in [3]. Theorem 1 gives a fast method to solve Problem 2. Note that P(k) and A(k) can be computed before sensor scheduling, since they do not depend directly on sensor selection or the state x 0. Also, Ψ i (k), Φ i (k), g j and γ ij depend only on ι(k). They are independent of ι( k) for k k. Hence no combinatorial explosions occur when Theorem 1 is used to solve Problem 2. B. Fast Sensor Scheduling Applying Theorem 1 to the set of all possible seuences of sensor types yields a solution of Problem 1. Details of the algorithm are follows. 1) Define the set of all possible seuences of pairs of sensor types by O := {σ k, ρ k σ k I c (k),ρ k J c (k)}, (43) and set m = 0. 2) Repeat the following operations a), b) and c) until O =. a) m = m + 1. b) For O m O, derive the optimal sensor scheduling by using Theorem 1 and compute J m = min J c. (44) ι(k) K(k) c) O = O/O m. 3) Compute m = arg min m J m. 4) Obtain {ι c(0),, ι c(t)} for O m, and it is the optimal sensor seuence. Let us now evaluate the computational cost of the proposed algorithm when I c (k) = J c (k) = N for all k. Let σ mk and ρ mk denote the selected sensor types at time k in O m for observation and communication with the controller, respectively. In addition, the number of sensors that belong to σ is represented by µ(σ). Then T µ(σ mk)µ(ρ mk ) values of the cost function are computed to obtain J m for each m. Hence the number of seuences of sensor types computed in Step 2 is given by M 2(T+1) m=1 T µ(σ mk )µ(ρ mk ) = T M 2(T+1) m=1 µ(σ mk )µ(ρ mk ) = (T + 1)N 2 M 2T. (45) Furthermore, Step 3 compares M 2(T+1) values to obtain m. Thus we obtain the following theorem. Theorem 2: Assume I c (k) = J c (k) = N for all k. Then the computational cost of the proposed algorithm is given by O(TN 2 M 2T ). Recall that the exhaustive search reuires N 2(T+1) sensor seuences to determine the optimal sensor seuence. Thus the proposed algorithm is effective for fast sensor scheduling for sensor networks such that the number of sensor types, M, is far less than the number of sensors, N. Example 3: Consider a vehicle that travels on the twodimensional plane. Let (x, y) be the position of the vehicle. The goal here is to control the vehicle using a sensor network which consists of forty nine radar sensors. No controller is mounted on the vehicle. One of the sensors has a controller, and the input signal is sent over the network 1. The state euation of the vehicle in continuous time is given by x p = x p ū + w, (46) x := [x y ẋ ẏ] R 4 and ū R 2. The covariance matrix of w is set to 0.001I 4. The state euation (46) is discretized with a sampling period 0.1. An observer (12) and a controller (13) are designed such that the poles of (14) are set to 0.91 ± 0.055j, 0.92 ± 0.030j, 0.86 ± 0.091j and 0.86 ± 0.091j, j denotes the imaginary unit. Radar sensors 1,2,,49 with model (5) are set at (0,0), (0,0.25),,(0,1.5), (0.25,0),,(1.5,1.5), as illustrated in Fig. 4. The covariance matrices are given by [ ] V i =, (47) for all i {1,2,,49}. In this example, the number of sensors is forty nine, and the number of sensor types is one. It is assumed that each sensor can communicate with four neighborhood sensors, and the communication cost of the network is set to γ ij = 0.4 ( p ix p jx + p iy p jy ), (48) (p ix,p iy ) denotes the position of sensor i. The observation range of the sensors is defined by [ ] [ ] x pix 0.4, (49) y p iy for all i, and I c (k) = {i (49) holds}. The controller is set at (0.75,0), and j(k) = 22, or euivalently, J c (k) = {22} for all k. Other parameters are set to Q p = I 4, R = I 2, Π = I 4, Γ = 0, T = 4. 1 In a networked control system, the controller and the controlled object are sometimes connected through a network [21]. 299

6 Controller Time Fig. 4. Left: Sensor location and a sample path of the vehicle. The numbers represent the labels of the sensors. Right: Seuences of the selected sensors. ; scheduling with communication costs. ; scheduling without communication costs. The radar sensor model is linearized by (29), so that the proposed algorithm can be applied. The left of Fig. 4 shows a sample path of the vehicle for x p (0) = [ ], x c (0) = [ ], and E[x p (0)] = x c (0). The right of Fig. 4 illustrates the suboptimal sensor seuences with the communication costs and without the communication costs, that is, γ ij = 0 and Γ = 0. The suboptimal sensor seuences are given by , with the communication costs, and , without the communication costs. Each selected sensor in scheduling with the communication costs is eual to or closer to the controller than one without the communication costs to reduce the communication costs. The proposed method needs to compare at most 12 values of the cost function to obtain ι (0) at each time, and it takes [s] to get i (0) on average. On the other hand, the exhaustive search method takes more than one hour to get i (0) Monte Carlo method is implemented to compute the values of the cost function. The proposed method is useful for large-scale sensor networks. The programs ran in MATLAB 7.1 on a PentiumD 3.2 GHz PC with 2 GB of RAM. V. CONCLUSION In this paper, a sensor scheduling problem with communication costs for a class of sensor networks whose measurements are influenced by state dependent noise was addressed. The sensor scheduling problem was formulated as a model predictive control problem, and we proposed a fast and optimal sensor scheduling algorithm for a class of sensor networks. The computation time of the proposed algorithm increases exponentially with the number of sensor types, while the computation time of a primitive algorithm is represented by an exponential function whose base is the number of sensors and whose exponent is twice the length of the prediction horizon. A numerical example demonstrated that the proposed algorithm is effective for large-scale sensor networks. REFERENCES [1] P. Alriksson, and A. Rantzer: Sub-Optimal Sensor Scheduling with Error Bounds; Proc. of 16th IFAC World Congress (2005) [2] M. S. Andersland: On the Optimality of Open-Loop LQG Measurement Scheduling; IEEE Trans. on Automatic Control, Vol. 40, No. 10, pp (1995) [3] S. Arai, Y. Iwatani, and K. Hashimoto: Fast Sensor Scheduling for Large-Scale Networked Sensor Systems; Proc. of American Control Conf. (2009) [4] M. A. Batalin, G. S. Sukhatme, and M. Hattig: Mobile Robot Navigation Using a Sensor Network; Proc. of IEEE Conf. on Robotics and Automation (2004) [5] T. H. Chung, V. Gupta, B. Hassibi, J. W. Burdick, and R. M. Murray: Scheduling for Distributed Sensor Networks with Single Sensor Measurement per Time Step; Proc. of IEEE Conf. on Robotics and Automation (2004) [6] T. H. Chung, V. Gupta, B. Hassibi, J. W. Burdick, and R. M. Murray: On a Decentralized Active Sensing Strategy Using Mobile Sensor Platforms in a Network; Proc. of IEEE Conf. on Decision and Control (2004) [7] Z. G. Feng, K. L. Teo, and Y. Zaho: Branch and Bound Method for Sensor Scheduling in Discrete Time; Journal of Industrial and Management Optimization, Vol. 1, No. 4, pp (2005) [8] L. E. Ghaoui: State-Feedback Control of Systems with Multiplicative Noise via Linear Matrix Ineualities; Systems and Control Letters, Vol. 24, pp (1995) [9] V. Gupta, T. Chung, B. Hassibi, and R. M. Murray: On a Stochastic Sensor Selection Algorithm with Applications in Sensor Scheduling and Sensor Coverage; Automatica, Vol. 42, No. 2, pp (2006) [10] Y. He, and K. P. Chong: Sensor Scheduling for Target Tracking in Sensor Networks; Proc. of IEEE Conf. on Decision and Control (2006) [11] S. Kagami, and M. Ishikawa: A Sensor Selection Method Considering Communication Delays; Proc. of IEEE Conf. on Robotics and Automation (2004) [12] K. Kotay, R. Peterson, and D. Rus: Experiments with Robots and Sensor Networks for Mapping and Navigation; Proc. of Intl. Conf. on Field and Service Robotics (2005) [13] H. W. J. Lee, K. L. Teo, and A. E. B. Lim: Sensor Scheduling in Continuous Time; Automatica, Vol. 37, pp (2001) [14] L. Meier, J. Peschon, and R. M. Dressler: Optimal Control of Measurement Subsystems; IEEE Trans. on Automatic Control, Vol. 12, No. 5, pp (1967) [15] Y. Ma, S. Soatto, J. Košecká, and S. S. Sastry: An Invitation to 3-D Vision; Springer (2004) [16] H. Morikawa: Ubiuitous Sensor Networks; Proc. of US-Japan Workshop on Sensors, Smart Structures and Mechatronic Systems (2005) [17] K. V. Ramachandra: Kalman Filtering Techniues for Radar Tracking; Marcel Dekker (2000) [18] I. Stojmenović: Sensor Networks; Wiley InterScience (2005) [19] F. Wang and V. Balakrishnan: Robust Kalman Filters for Linear Time-Varying Systems with Stochastic Parametric Uncertainties; IEEE Trans. on Signal Processing., Vol. 50, No. 4, pp (2002) [20] J. L. Willems and J. C. Willems: Robust Stabilization of Uncertain Systems; SIAM Journal of Control Optimization, Vol. 21, pp (1983) [21] W. S. Wong and R. W. Brockett: Systems with Finite Communication Bandwidth Constraints; IEEE Trans. Automatic Control, Vol. AC-42, No. 9, pp (1997) [22] F. Zaho, J. Shin, and J. Reich: Information-Driven Dynamic Sensor Collaboration for Tracking Applications; IEEE Signal Processing Mag., Vol. 19, No. 2, pp (2002) 300