IEEE ICC 2012 - Signal Processing for Communications Symposium
Robust Power Allocation via Semidefinite Programming for Wireless Localization William Wei-Liang Li, Student Member, IEEE, Yuan Shen, Student Member, IEEE, Ying Jun (Angela) Zhang, Senior Member, IEEE, and Moe Z. Win, Fellow, IEEE
Abstract—In wireless localization systems, mobile nodes (agents) typically obtain their positions through ranging with respect to fixed infrastructure (anchors). Transmission power allocation not only affects network lifetime, throughput and interference, but also determines the localization accuracy. In this paper, we develop a robust anchor power allocation strategy to combat imperfect network topology parameters. We formulate the problem to minimize the squared position error bound (SPEB), which characterizes the fundamental limit of localization accuracy, and show that such formulation can be efficiently solved via semidefinite programming (SDP). The simulation results show that the proposed robust scheme significantly outperforms both non-robust scheme and uniform power allocation. Index Terms—Localization, power allocation, semidefinite programming, robust optimization.
I. I NTRODUCTION Positional information is of critical importance for future wireless networks, which will support an increasing number of location-based applications and services [1], [2], such as cellular positioning, rescue operation, blue-force tracking, etc. Typically, wireless localization is referred to as a process to determine the positions of mobile nodes (agents) based on the measurements with respect to fixed infrastructure (anchors), as illustrated in Fig. 1. With the rapid development of advanced wireless techniques, wireless localization has attracted increasing research interest in the past decades [3]–[7]. Localization accuracy is a critical performance measure in wireless location-aware networks. In recent work [4], [5], the fundamental limits of wideband localization has been derived in terms of the squared position error bound (SPEB). It shows that anchor power allocation affects localization accuracy, in addition to network lifetime, throughput, and interference. However, few work has addressed the problem of power resource allocation for localization. The authors in [8] investigated anchor power allocation for wideband localization systems, and derived the optimal solution only for single-agent network. In [9], it exploited the geometrical interpretation of This research was supported, in part, by the General Research Fund (Project number 419509) established under the University Grant Council of Hong Kong, the National Science Foundation under Grant ECCS-0901034, the Office of Naval Research under Grant N00014-11-1-0397, and MIT Institute for Soldier Nanotechnologies. W. W.-L. Li and Y. J. (A.) Zhang are with Department of Information Engineering, The Chinese University of Hong Kong, Shatin, New Territories, Hong Kong. Y. Shen and M. Z. Win are with the Laboratory for Information and Decision Systems (LIDS), Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139 USA (e-mail:
[email protected],
[email protected],
[email protected],
[email protected]).
978-1-4577-2053-6/12/$31.00 ©2012 IEEE
C
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B D Fig. 1: Location-aware networks: the anchors (A, B, C, and D) transmit signals to the two agents for localization.
localization information to minimize the maximum directional position error bound. In [10], it adopted the constraint relaxation and domain decomposition methods to obtain suboptimal solutions for MIMO radar localization. In general, how to optimally allocate the transmission power in location-aware networks still remains as an open problem. On the other hand, anchor power allocation needs to adapt to instantaneous network conditions, such as network topology and channel qualities, to optimize localization performance. Previous work often considers the deterministic optimization, i.e., assuming the agents’s positions and channel conditions perfectly known [8]–[10]. However, these network parameters are obtained through estimation and hence subject to errors. The power allocation schemes based on imperfect network parameters could often lead to sub-optimal or even infeasible solutions in realistic systems [11]. Therefore, it is essential to design a robust scheme to combat the uncertainties in network parameters. In this paper, we propose an optimization framework for power allocation in location-aware networks to tackle imperfect network topology parameters. Specifically, we treat the fundamental limit of localization accuracy, i.e., SPEB, as our objective, and formulate a convex optimization problem for anchor power allocation by minimizing the SPEB. The formulation can be equivalently converted into a semidefinite program (SDP). Moreover, we propose a robust method to combat imperfect network topology parameters, in which we minimize the worst-case SPEB under the uncertainties in network topology parameters. The proposed robust formulation retains the same structure of SDP. Notations: We use lowercase and uppercase bold symbols to
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denote vectors and matrices, respectively; det(A) and tr(A) denote the determinant and trace of matrix A, respectively; the superscript (·)T and · denote the transpose and Euclidean norm of its argument, respectively. We use calligraphic symbols, e.g., N , to denote sets. The notation E{·} and Pr{·} are the expectation and probability operators, respectively. II. S YSTEM M ODEL
coefficient denoting the amplitude loss exponent, and Pkj is the power of the transmitted waveform.2 Since the SPEB characterizes the limit of localization accuracy, and is a function of anchor transmission power, how to allocate power among anchors will be critical to the system aiming at improving its localization accuracy. Moreover, such a bound is achievable under high SNR regime, thus, we will use SPEB as a performance metric for location-aware network.
In this section, we describe the system model, and introduce a performance metric of location-aware networks.
III. O PTIMAL P OWER A LLOCATION VIA SDP
A. System Model Consider a location-aware network consisting of Na agents and Nb anchors. The anchors are the fixed infrastructure with known positions, and the agents are the mobile nodes. The position of each agent is determined based on the point-topoint measurements from Nb anchors. Denote the set of agents by Na = {1, 2, . . . , Na }, and the set of anchors by Nb = {Na + 1, Na + 2, . . . , Na + Nb }. The 2-D position of node k is denoted by pk = [xk yk ]T . The distance and angle between agent k and anchor j are given by dkj = pk − pj and φkj = arctan ((yk − yj )/(xk − xj )), respectively. We assume the measurements between anchors and agents do not interfere each other, which can be achieved by medium access control. Here, we consider a synchronized network that uses one-way transmission for inter-node distance estimation,1 and our work also applies in the round-trip transmission systems which do not require synchronization between anchors and agents. B. Position Error Bound The SPEB introduced in [4] is a performance metric that characterizes the localization accuracy and bound the position error, i.e., P(pk ) tr J−1 (1) p k − pk 2 e (pk ) ≤ E ˆ where Je (pk ) is the equivalent Fisher information matrix ˆ k is an unbiased (EFIM) for agent k’s position [4], and p estimate of the position pk . The EFIM in (1) can be written as a 2 × 2 matrix [4], Je (pk ) = λkj · Jr (φkj ) (2) j∈Nb
where Jr (φkj ) = qkj (φkj )qkj (φkj )T is a 2 × 2 matrix with qkj (φkj ) = [cos φkj sin φkj ]T , and λkj is defined as ranging information intensity of agent k with respect to anchor j, given by ξkj λkj = Pkj · 2β dkj in which ξkj is a positive coefficient determined by the properties of the channel and transmit signal, β is a positive 1 There are two common ways for inter-node distance estimation: one-way transmission (only anchor transmits) or round-trip transmission (both anchor and agent transmit). In one-way transmission system, anchors and agents need to be synchronized for estimating the inter-node distance.
In this section, we formulate the problem of anchor power allocation using SPEB as the objective function, and show that the formulation is a SDP. We consider finding the optimal power allocation which minimizes the total SPEB while the system is subject to a budget of power consumption. The problem can be formulated as Nb −1 ξkj P : min (3) tr P J (φ ) kj r kj 2β {Pkj } j=1 dkj k∈Na s.t. Pkj ≤ P tot (4) k∈Na k∈Nb
Pkj ≥ 0,
∀k ∈ Na , ∀j ∈ Nb
(5)
where the constraint (4) gives the upper bound of the total transmission power of all the anchors. Note that the problem P is convex in Pkj , since tr X−1 is convex in X 0, and the function inside tr{·} in (3) is linear in Pkj . Furthermore, we show that the problem P can be equivalently converted into a SDP, which can be solved efficiently [16], [17]. Specifically, we replace the EFIM’s in (3) with matrices Mk , and add another constraint as Mk J−1 e (pk ). Since Je (pk ) is a positive semidefinite matrix, due to the property of Schur complement, the above inequality is equivalent to
M I 0. I Je (pk ) Then, we obtain a SDP formulation P SDP equivalent to P, i.e., P SDP: min tr {Mk } {Pkj }, Mk
s.t.
k∈Na
Mk I k∈Nb
I ξkj P J (φ ) 0, ∀k ∈ Na d2β kj r kj kj
(4) – (5). Remark 1: We can impose additional linear constraints on anchor transmission power depending on the requirements of localization systems, e.g., Pkj ≤ P¯kj where P¯kj is the upper bound of the transmission power from anchor j to agent k, 2 Although derived based on the received waveforms for wideband systems in [4], the structure of SPEB is also observed in other TOA- or RSS-based localization systems, e.g., [12]–[15].
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or k∈Na Pkj ≤ Pjtot where Pjtot is the upper bound of the total transmission power on anchor j, etc. Due to the linearity of these constraints, the convexity of the problem is retained, and the optimal solution can be efficiently obtained via SDP. Remark 2: To obtain the optimal solution of P or P SDP , it requires the network topology parameters, i.e., the distance dkj and the angle φkj . However, dkj ’s and φkj ’s are usually not perfectly known in realistic systems, and only estimated values are available. When estimation errors exist, the formulation P (P SDP ) may fail to provide reliable services, since the actual SPEB is not necessarily optimized. Therefore, it is more important to design a power allocation scheme which is robust to the uncertainties in network topology parameters. IV. ROBUST P OWER A LLOCATION UNDER I MPERFECT N ETWORK T OPOLOGY PARAMETERS In this section, we consider the location-aware network with imperfect network topology parameters, and propose a robust optimization method to minimize the worst-case SPEB. Let dˆkj and φˆkj denote distance and angle estimates, respectively. We consider the actual distance and angle lie in linear sets, i.e.,3 d dkj ∈ Skj dkj dˆkj − εdkj ≤ dkj ≤ dˆkj + εdkj φ φkj ∈ Skj φkj φˆkj − εφkj ≤ φkj ≤ φˆkj + εφkj εdkj
εφkj
and are both small positive numbers denoting where the maximum estimation errors on the distance and angle estimates, respectively.4 To deal with the estimation errors, we adopt robust optimization techniques to consider the worst-case performance. Instead of using the estimated values, we consider minimizing the maximum SPEB over the possible set of actual network topology parameters, i.e., PR0 : min max tr J−1 e (pk ) {Pkj }
s.t.
φ d ,φ {dkj ∈Skj kj ∈Skj } k∈N a
(4) – (5).
Note that the maximization over dkj and φkj can be separated, and the maximization over dkj simply follows that d ˆ dkj arg max tr J−1 (6) e (pk ) = dkj + ε
We next consider a relaxation for the robust optimization with respect to {φkj } and introduce a new matrix Q(φˆkj , δkj ) = Jr (φˆkj ) − δkj · I to replace Jr (φkj ) in the SPEB. We will show that the worstcase SPEB over φkj can be bounded above by the new function for sufficiently large δkj in the following proposition. 2β ·Pkj Q(φˆkj , δkj ) 0 and Proposition 1: If j∈Nb ξkj /dkj φ δkj ≥ sin(εkj ), the maximum SPEB over the actual angle φkj is always bounded above as follows −1 ξkj P J (φ ) max tr 2β kj r kj φ {φkj ∈Skj } j∈Nb dkj −1 ξkj ˆkj , δkj ) . (7) ≤ tr P Q( φ kj 2β j∈Nb dkj Moreover, the tightest upper bound in (7) is attained by −1 ξkj ˆkj , δkj ) . P Q( φ sin(εφkj ) = arg min tr kj 2β δkj j∈Nb dkj Proof: See the proof in [18]. In the following, we take the minimizer δkj = sin(εφkj ) and denote the matrix Q(φˆkj ) = Jr (φˆkj ) − sin(εφkj ) · I by omitting the variable δkj in the matrix Q(φˆkj , δkj ) for simplicity. Then, we replace the matrix Jr (φkj ) with Q(φˆkj ) in the previous formulation, and propose a robust counterpart of P as follows: Nb −1 ξkj ˆ PR : min tr P Q(φkj ) 2β kj {Pkj } j=1 dkj k∈Na ξkj P Q(φˆkj ) 0, ∀k ∈ Na (8) s.t. 2β kj k∈Nb dkj (4) – (5). Again due to the property of Schur complement, the problem PR is equivalent to a SDP formulation, given by PRSDP: min tr {Mk } {Pkj }, Mk
Mk s.t. I k∈Nb
d } {dkj ∈Skj
since tr J−1 e (pk ) is a monotonically non-decreasing function of dkj . On the other hand, however, the maximization over φkj is not trivial, since the SPEB in (1) is not convex in φkj . Hence, it is difficult to obtain a close-form solution of {φkj } since it depends on {Pkj }. consider the distance dkj to be always positive, i.e., dˆkj − dkj > 0. estimation error in ξkj can be equivalently accounted for in the distance. Moreover, if uncertainties exist in anchor positions, it can be equivalently converted into the estimation errors in channel qualities [5]. 3 We
4 The
k∈Na
I ξkj P Q(φˆkj ) 0, ∀k ∈ Na d 2β kj kj
(9) (4) – (5). Note that from Proposition 1, the new formulation PR is a valid relaxation for PR0 when the condition (8) holds. ˆkj ) is not positive definite due to det Q(φˆkj ) = Since Q( φ sin(εφ ) sin(εφ ) − 1 ≤ 0, such a condition does not necessarily hold for all power allocation {Pkj }. However, we will show that it holds for the optimal power allocation of PR0 with high probability (w.h.p.) when the number of anchors is large or the estimation error is small.
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Fig. 2: Compare the SPEB per agent in multiple-agent network (Nb = 10) resulted by the optimal power allocation and uniform allocation.
Fig. 3: The actual SPEB with respect to the number of anchors, resulted by robust, non-robust schemes and uniform allocation with imperfect network topology parameters (εd = εφ = 0.2).
Proposition 2: Consider a random network where all the nodes are uniformly located in a R × R square region and the minimum distance between two nodes is r0 . The coefficient ξkj has a support on [ξmin ξmax ] where 0 < ξmin ≤ ξmax ≤ 1. ∗ Let {Pkj } be the optimal solution of PR0 , and δ = sin(εφ ) φ where ε = max{εφkj }, then
V. S IMULATION R ESULTS
(a) when Nb → ∞ and δ ≤ δmax , where δmax is the first positive root5 of equation 4δ 4 −4δ 2 −2ξmax /ξmin δ+1 = 0, it follows Pr
ξ kj ∗ ˆkj ) 0 = 1−O exp(−α·Nb ) , P Q( φ 2β kj d k∈Nb
kj
∀k ∈ Na where α is a fixed positive number; (b) when εφ → 0, it follows Pr
ξ kj ∗ ˆkj ) 0 = 1−O (εφ )Nb /2 , P Q( φ kj 2β d k∈Nb
kj
∀k ∈ Na . Proof: See the proof in [18]. Remark 3: Proposition 2 implies that the condition (8) holds w.h.p. at the rate indicated by the O notation, where O(f (n)) means that the function value is on the order of f (n) [19]. ∗ Remark 4: Note that Proposition 2 holds for {Pkj }, which implies that the optimal solution of the original robust formulation PR0 is included in the feasible set of the proposed formulation PR (or PRSDP ). 5 We give some numerical examples ξmax = 1; δmax = 0.096 when ξξmax ξmin min
as follows: δmax = 0.318 when = 5.
In this section, we investigate the localization performance by the proposed anchor power allocation schemes. We consider the networks with anchors and agents located in a squared region, i.e., [−10, 10] × [−10, 10]. The total anchor power for localization is normalized to P tot = 1. The proposed SDP formulations of anchor power allocation are solved by the standard optimization solver CVX [20]. First, we investigate the performance of the optimal power allocation with perfect network topology parameters. Specifically, we consider the network consisting of ten anchors, which are evenly placed on a circle with the distance of 8 toward the center of the squared region. Given the number of agents, we run Monte Carlo simulation to generate 103 deployments of agents that are uniformly distributed in the squared region, and then compute the average SPEBs obtained by the power allocation scheme. In Fig. 2, we compare the SPEBs resulted by the optimal anchor power allocation formulated in P SDP , and the uniform allocation which equally assigns P tot over all the anchors, as the number of agents changes. It shows that the optimal power allocation remarkably outperforms the uniform allocation. In addition, we observed that the SPEB per agent increases linearly with the number of agents. As indicated by the slope, the speed of SPEB increase of optimized allocation is about 60% slower than that of uniform allocation. Next, we investigate the performance of the power allocation with imperfect network topology parameters. We compared the robust allocation formulated in PRSDP , nonrobust allocation formulated in P SDP , and uniform allocation. We evaluate the actual SPEB of each scheme by Monte Carlo simulation. Specifically, we first solve the optimal power allocation decision of each scheme based on (dˆkj , φˆkj ). For each deployment of (dˆkj , φˆkj ), we uniformly generate 105 pairs of errors (edkj , eφkj )’s which are bounded by εd and εφ ,
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system with imperfect network topology parameters, and proposed robust power allocation scheme to combat the uncertainties in network topology parameters. The results demonstrated that the power allocation with robust optimization remarkably outperforms both the non-robust power allocation and uniform allocation.
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Fig. 4: The actual SPEB with respect to error size on network topology parameters (εd and εφ are set to be equal) resulted by robust, non-robust schemes and uniform allocation. respectively. There are totally 103 pairs of (dˆkj , φˆkj ) randomly generated in simulation. Then, we compute the actual SPEB under each realization of (dkj , φkj ) and determine its average value for comparison. In Fig. 3, we investigate the actual SPEB as the number of anchors changes. We consider the single-agent network where anchors are uniformly distributed, and set εd = εφ = 0.2. A decreasing tendency in SPEB is observed as the number of anchors increases. This is reasonable since the agent has more freedom to choose “good” anchors when there are more anchors. The result shows that the robust SPEB-based scheme (PRSDP ) outperforms the non-robust scheme (P SDP ) by 34%, and outperforms uniform allocation by 60%. In Fig. 4, we investigate the actual SPEB with respect to the error size on network topology parameters, i.e., εd and εφ .6 We consider a single-agent network consisting of 10 anchors with fixed positions on a circle (similar to the simulation settings in Fig. 2). As we observed, the actual SPEB of both nonrobust scheme and uniform allocation quickly increases with the error size, while the actual SPEB of the robust scheme increases slowly. In other word, the improvement in SPEB by robust optimization becomes more remarkable than that by non-robust optimization as the network topology parameters error goes larger. Both Fig. 3 and 4 have demonstrated the advantage of the proposed robust power allocation scheme in the practical localization systems with imperfect network topology parameters. VI. C ONCLUSION In this paper, we investigated the power allocation among anchors for localization based on the performance metric SPEB. We first showed the optimal power allocation can be efficiently obtained via SDP. Then, we considered the practical 6 The
[1] A. Sayed, A. Tarighat, and N. Khajehnouri, “Network-based wireless location: challenges faced in developing techniques for accurate wireless location information,” IEEE Signal Process. Mag., vol. 22, no. 4, pp. 24–40, 2005. [2] M. Z. Win, A. Conti, S. Mazuelas, Y. Shen, W. M. Gifford, D. Dardari, and M. Chiani, “Network localization and navigation via cooperation,” IEEE Commun. Mag., vol. 49, no. 5, pp. 56–62, May 2011. [3] D. Dardari, A. Conti, C. Buratti, and R. Verdone, “Mathematical evaluation of environmental monitoring estimation error through energyefficient wireless sensor networks,” IEEE Trans. Mobile Comput., vol. 6, no. 7, pp. 790–802, Jul. 2007. [4] Y. Shen and M. Z. Win, “Fundamental limits of wideband localization – Part I: A general framework,” IEEE Trans. Inf. Theory, vol. 56, no. 10, pp. 4956–4980, Oct. 2010. [5] Y. Shen, H. Wymeersch, and M. Z. Win, “Fundamental limits of wideband localization – Part II: Cooperative networks,” IEEE Trans. Inf. Theory, vol. 56, no. 10, pp. 4981–5000, Oct. 2010. [6] A. Conti, M. Guerra, D. Dardari, N. Decarli, and M. Z. Win, “Network experimentation for cooperative localization,” IEEE J. Sel. Areas Commun., vol. 30, no. 2, pp. 467–475, Feb. 2012. [7] S. Gezici, Z. Tian, G. B. Giannakis, H. Kobayashi, A. F. Molisch, H. V. Poor, and Z. Sahinoglu, “Localization via ultra-wideband radios: a look at positioning aspects for future sensor networks,” IEEE Signal Process. Mag., vol. 22, no. 4, pp. 70–84, Jul. 2005. [8] Y. Shen and M. Z. Win, “Energy efficient location-aware networks,” in Proc. IEEE Int. Conf. on Commun., Beijing, China, May 2008, pp. 2995–3001. [9] W. W.-L. Li, Y. Shen, Y. J. Zhang, and M. Z. Win, “Efficient anchor power allocation for location-aware networks,” in Proc. IEEE Int. Conf. on Commun., Kyoto, Japan, Jun. 2011, pp. 1–6. [10] H. Godrich, A. Petropulu, and H. V. Poor, “Power allocation strategies for target localization in distributed multiple-radar architectures,” IEEE Trans. Signal Process., vol. 59, no. 7, pp. 3226–3240, Jun. 2011. [11] T. Q. S. Quek, M. Z. Win, and M. Chiani, “Robust power allocation of wireless relay channels,” IEEE Trans. Commun., vol. 58, no. 7, pp. 1931–1938, Jul. 2010. [12] D. B. Jourdan, D. Dardari, and M. Z. Win, “Position error bound for UWB localization in dense cluttered environments,” IEEE Trans. Aerosp. Electron. Syst., vol. 44, no. 2, pp. 613–628, Apr. 2008. [13] H. Godrich, A. M. Haimovich, and R. S. Blum, “Target localization accuracy gain in mimo radar-based systems,” IEEE Trans. Inf. Theory, vol. 56, no. 6, pp. 2783–2803, Jun. 2010. [14] Y. Qi, H. Kobayashi, and H. Suda, “Analysis of wireless geolocation in a non-line-of-sight environment,” IEEE Trans. Wireless Commun., vol. 5, no. 3, pp. 672–681, 2006. [15] S. Mazuelas, R. Lorenzo, A. Bahillo, P. Fernandez, J. Prieto, and E. Abril, “Topology assessment provided by weighted barycentric parameters in harsh environment wireless location systems,” IEEE Trans. Signal Process., vol. 58, no. 7, pp. 3842–3857, Jul. 2010. [16] Z.-Q. Luo and W. Yu, “An introduction to convex optimization for communications and signal processing,” IEEE J. Sel. Areas Commun., vol. 24, no. 8, pp. 1426–1438, Aug. 2006. [17] Z.-Q. Luo, W.-K. Ma, A. M.-C. So, Y. Ye, and S. Zhang, “Semidefinite relaxation of quadratic optimization problems,” IEEE Signal Process. Mag., vol. 27, no. 3, pp. 20–34, May 2010. [18] W. W.-L. Li, Y. Shen, Y. J. Zhang, and M. Z. Win, “Robust power allocation for energy-efficient location-aware networks,” IEEE/ACM Trans. Netw., 2011, submitted for publication. [19] T. H. Cormen, C. E. Leiserson, R. L. Rivest, and C. Stein, Introduction to Algorithms. Cambridge, MA and Boston, MA: MIT Press and McGraw-Hill, 2001. [20] M. Grant and S. Boyd, “CVX: Matlab software for disciplined convex programming, version 1.21,” http://cvxr.com/cvx, Aug. 2010.
error size depends on the prior positions of the agents.
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