1 Quadratic Optimal Control of Wirele Senor Network Deployment Ratko R. Selmic 1, Jinko Kanno, Jack Buchart 1, and Nichola Richardon 3 1 Department of Electrical Engineering, Department of Mathematic 3 Department of Computer Science Louiiana Tech Univerity Ruton, LA 717, USA Tel: 318-57-4641 Fax: 318-57-49 Email: relmic@latech.edu, jkanno@coe.latech.edu, jgb0595@gmail.com, nmr00@latech.edu Abtract Thi paper preent a method for an optimal wirele enor network deployment with a quadratic cot function. It i aumed that the network i ymmetric with an equal radio tranmiion range acro the network. The optimization problem i formulated uing a convex quadratic cot function that conider ditance between mobile node and both tatic node and the target(). The network contraint related to the network/graph connectivity condition are preented. It i aumed that the bae tation can directly communicate with node that are to be deployed and with the mobile node. Simulation reult are preented uing a newly developed imulation platform for enor network uboptimal deployment. Key word wirele enor network, optimal control, deployment, coverage. I. INTRODUCTION Wirele enor network (WSN) have integrated computing, toring, networking, ening, and actuating capabilitie [7], [1], [13], [14], [17], [18], [], [9], [33]. Thee network conit of a number of enor node (tatic and mobile) with multiple enor per node that communicate with each other and the bae tation through wirele radio link (ee Figure 1). The bae tation, or the gateway, i ued for data proceing, torage, and control of the enor network. Senor node are uually battery powered; hence the whole enor network i limited by fundamental tradeoff between ampling rate and battery lifetime []. Thee limited reource in term of cot, power, and bandwidth require the ue of optimality principle in enor network deployment. An effective and optimal coverage of the enor field reduce cot, improve enor network functionality, increae reilience to failure, and enable novel application in the enor network. We dicu here optimal wirele enor network coverage control applied to both tatic and mobile enor node. A novel method for optimal enor network coverage control i preented that utilize a linear quadratic cot function. The optimal enor node deployment problem ha been formulated under network connectivity contraint. Senor network and related coverage control problem have recently gained ignificant interet [3], [5], [6], [9], [10], [11], [16]. Several optimal control problem related to enor network were formulated in [5]. Thi urvey paper alo provide an excellent overview of exiting reult in thi area. An optimal coverage under contraint of imprecie detection and terrain propertie where the number of enor node i minimized wa preented in [10]. Reference [16] dicue the problem ariing in maintaining coordination and communication between the group of robot and olution to thee problem. The paper preented three model of coordination (i.e., deployment) in order to maximize the coverage area within the cloe range of the robot, deployment to maximize the probability of detecting a ource, and deployment to maximize the viibility of the network. A tutorial paper [9] provide an overview of control method in multivehicle cooperative control uing graph theory. UAV node Mobile node Gateway Static node Fig. 1. Ad-hoc wirele enor network with tatic and mobile node.
Optimal coverage control for mobile enor network wa preented in [9]. The paper ue a Voronoi partitioning and Lloyd decent algorithm but doe not conider network connectivity contraint. Weighted directed graph that relate averaging protocol in conenu problem and coverage control law over Voronoi graph were given in [15]. Two location function that characterize coverage performance were provided in [10]. The paper tudie their gradient propertie via nonmooth analyi. In all the above reult the feaibility et are aumed to be convex and the network (i.e., optimization) i not contrained. General textbook in the wirele enor network area include recent publication [4], [19], [4], [34]; however none deal with the pecific problem of network deployment optimization. A linear programming problem for enor deployment wa formulated in [3]. Moreover, the paper preent an excellent review of exiting deployment algorithm. The conidered optimization problem i a convex one and enor are contrained into a grid-type of network topology. Reult in [31] howed a fundamental relationhip between coverage and connectivity by proving that if the communication radiu r c of the node i at leat twice their ening radiu r, then complete ening coverage (1- coverage) of a convex region implie that the related network graph i connected. Additionally, it i hown that if r c r, then K -coverage of a convex region implie K -connectivity of the enor node. Reference [0] minimize the number of enor to cover an area of interet. However, they aume that the ening and communication range are identical, while our approach can eaily be extended to a general cae of different communication and ening range. A virtual force algorithm for enor network deployment that maximize the enor field coverage wa propoed in [35]. The approach require a grid-baed enor node placement and alo require that all node are mobile. Thi paper provide an excellent energy model for the network; however, i doe not conider the network connectivity a a contraint in the optimization proce. A determinitic and tatitical enor network coverage algorithm uing Voronoi diagram for enor field partitioning wa tudied in [3]. A graph theory abtraction ha been applied to tudy the coverage problem and to provide an algorithm complexity. II. BACKGROUND WSN are topologically imilar to graph. Thi imilarity can be ued to map different deployment cenario and condition into graph theory problem. In uch a mapping, every enor node, including the bae tation, correpond to a vertex in a graph that varie in time, i.e. G ( t) = ( N ( t), E ( t) )., (1) where N(t) i a et of vertice and E(t) i a et of edge. Node or vertice in a graph are connected with a graph edge if there i a direct communication link between correponding node. With direct communication link we mean that the node communicate directly, without multi-hopping. Note alo that the graph G(t) i, in general, a function of time ince node and link can be time dependant. Thi i epecially true in a enor network with mobile node a the et of edge will certainly change a the mobile node move throughout the network. Illutrative mapping i hown in Figure where a WSN for chemical agent monitoring application, developed with Crobow mote i repreented a a graph. Bae Station SN SN 1 Radio Link SN 0 SN 3 SN4 SN 5 Fig.. Wirele enor network map into a graph. A. k-vertex Connectivity and k-edge Connectivity A graph G i aid to be connected if there i a path between any two vertice of G. A graph G i called k-connected if the minimum number of vertice of G whoe removal will diconnect G i at leat k. A k-connected network graph will remain connected with the failure of (k-1) node. A a reult of Menger Theorem, a graph G i k-connected if and only if there are k vertex-dijoint path between any two vertice in G. The graph on the left (Figure 3) i 1-connected ince the removal of vertex v would diconnect the ubgraph from the bae tation. The graph on the right i -connected ince any ingle vertex can be removed and the graph will remain connected. Note that there are two vertex dijoint path between any vertex v i (i=1,,, 5) and the bae tation, namely one path i B-v 1 -v - -v i and another i B-v 5 -v 4 - -v i. B v B 1 0 v 1 v Fig. 3. The graph on the left i 1-connected, and the graph on the right i - connected. III. SENSOR NETWORK MODEL AND PROBLEM FORMULATION Wirele enor network (WSN) conit of enor that communicate between themelve and the bae tation uing multi-hop routing protocol. Information hop between the v 5 v 4 3 4 5 v 3
3 node toward the bae tation uing wirele on-board radio that have a limited tranmiion radiu, often adjutable by microcontroller algorithm. In order to map uch a network to an equivalent graph model, everal aumption are needed. Aumption 1. The WSN conit of enor node all having an equal wirele radio tranmiion range r. Wirele radio and enor node have omnidirectional antenna. While thi aumption i not alway fulfilled in practical application, one can alway conider the minimum available tranmiion range a the leat-common tranmiion range for enor node [1]. Aumption. The WSN conit of ymmetric link between the enor node, meaning that if node i can tranmit information to a node j, then node j can alo end information to node i. Aumption 1 and allow the ue of undirected graph in the enor network modeling and implify analyi by uing uniform radio tranmiion range r. A. Senor Network Connectivity and Localization Condition A wirele enor network i connected if the correponding graph G(t) i at leat 1-connected, which implie that data from any node in the network can multi-hop it data to the bae tation. A WSN i 1-connected (but not -connected) if there i a ingle node whoe failure in the enor network will diconnect the network. It i clear that uch connectivity doe not provide ytem robutne and fault redundancy in cae of node failure. Bredin et al. [3] conidered vertex k- connectivity algorithm to be ued for robut and fault tolerant network. In uch cae up to k-1 node may fail and the network will till be connected. We will conider the enor network deployment under the k-connectivity condition. Localization of enor node i an important problem in the wirele enor network area. Senor node can be equipped with GPS receiver in which cae the location i known to the degree of accuracy of the receiver. If GPS i not available, a variety of method exit to locate the enor node in the wirele enor network environment. A two-phae proce for localization i a common choice, which include gathering range information between node in the firt phae and etimating poition in the econd phae. Range information can be obtained by exploiting ignal trength, a RADAR [1] and APS [5], time difference of arrival of two different ignal, a Cricket [8], AHLoS [30], and angle of arrival, a given in [6]. However, thi may be a challenging problem ince a ranging error may propagate through the whole network. Condition needed to localize the unknown enor node alo have their equivalent counterpart in graph theory. Conider a enor network with tatic and mobile enor node where poition of tatic enor node are known. Either a field technician i aware of the poition of the deployed tatic wirele enor node or they have GPS receiver to report their geo-location. In the cae that the mobile node do not have GPS receiver, it i required to etimate the poition of a mobile enor node that i lowly moving in the enor network field. In the cae of -dimenional movement, a mobile node hould be in the communication neighborhood of at leat 3 node that know their poition in order to uniquely determine it poition. Such an etimate i baed on the received radio ignal trength. Equivalently, each mobile vertex mut have 3 incident edge (i.e. they mut be of degree at leat 3). Mobile Node Localization. Given a graph defined a in (1) that include tatic and mobile node, we define M(t) to be a ubet of N(t) where M(t) i the et of mobile node. The localization condition mean that for each mobile node in M(t), the degree mut be at leat 3. Connectivity condition in enor network do not uually follow crip logic, i.e., when inter-node ditance are le than r, the node are connected, otherwie they are diconnected. One natural quetion arie what happen with connectivity when the ditance between node i cloe to r? In order to mathematically deal with thee kind of ituation we define trong and weak connectivity. A panning ubgraph of a graph G i a ubgraph that contain all vertice of G. Definition (Strong Connectivity). The graph G(t) i trongly connected if there i a panning ubgraph H(t) uch that H(t) i k-connected and all edge of H(t) are le than or equal to λr, where 0 < λ 1 and r i a enor node tranmiion radiu. Definition (Weak Connectivity). The graph G(t) i weakly connected if it i connected and it i not trongly connected. For example, let λ=0.75 for ome graph G. If there i a panning k-connected ubgraph of G with every edge le than or equal to 0.75r, then G i trongly connected. If no uch ubgraph exit then G i weakly connected. B. Optimal Senor Network Deployment Problem Many practical application uing wirele enor network have a common optimization problem that preent a tradeoff between oppoite goal, e.g., uniformly ditribute enor node over the monitoring area while focuing on a pecific area or phenomenon (Figure 4). A fundamental quetion that motivate thi reearch can be expreed a follow: how to optimally poition a large number of enor node over the area of interet, while providing extenive coverage of the focu area of interet (e.g., a contamination cloud in cae of chemical agent monitoring). While it i deired to uniformly monitor an area of interet, it i alo required to focu additional attention on an intereting phenomenon that ha been detected. We introduce a novel mathematical problem formulation of a uboptimal enor network coverage control problem, namely a linear quadratic (LQ) type of optimal coverage control [], [1]. The LQ cot function have been deigned to balance the tradeoff between uniform enor network coverage and focued coverage of pecific area. Uing graph theory terminology, the problem can be ummarized a follow: find the optimal enor network deployment that minimize a given index of performance uch that the correponding enor network graph G(t) i vertex k- connected and atifie the mobile node localization condition.
4 Note that if k i at leat 3, the mobile node localization condition i met by default, and we only need to check for k- connectivity in thi cae. The problem formulation i given below. impoe two contraint on Problem 1, namely the k- connectivity requirement and the mobile node localization condition requirement. Again, if k> the mobile node localization condition will, by default, be atified. The next ection provide aumption needed to pecify network connectivity contraint. It alo provide a olution for the above preented optimization problem. Fig. 4. Focued coverage v. uniform coverage of the monitoring area of interet. Problem 1 (Optimal Senor Node Placement Without Network Contraint). Let the enor radio tranmiion range be given by r. Aume there i a focu point (X F (t), Y F (t)) where everal mobile enor node hould converge. Given the compact coverage area of interet S, a et of wirele enor node N(t), and a ubet of mobile enor node M(t) N(t), find an optimal vertex location of the mobile node M(t) uch that the following cot function i minimized min J 1 = R dit [( xi, y i ), ( X F ( t), YF ( t)) ] + i M () Q ( x, y ), ( x, y ) [ i i j j ] i M, j N where R and Q are control deign parameter. The parameter R reward cloene to the focu point and the parameter Q reward uniform ditribution of vertice acro the et S. For example, Q >> R mean that the uer i much more concerned with the uniform vertex ditribution than to cover the pecific area of interet, Figure 4 (right). On the other hand, R >> Q indicate that the uer want extenive coverage of the focu point rather than uniform vertex ditribution, Figure 4 (left). Note that the cot function in () wa choen becaue the problem we have propoed i a multi-objective optimization problem. One common way to olve thi type of problem i to combine the weighted um of each individual objective function into an aggregate objective function (AOF). Specifically, the two objective are to minimize the ditance between the mobile node and the focu point and to maximize the ditance between the each mobile node and all other node (including other mobile node). Rather than maximizing thi ditance, however, we have choen to minimize the invere of thi ditance o that we could eaily combine the two term into an AOF. Problem (Optimal Senor Node Placement With Network Contraint). Given the compact coverage area of interet S, a focu point (X F (t), Y F (t)), the graph G(t) = (N(t),E(t)), and the mobile ubgraph G m (t) = (M(t),E m (t)), find an optimal vertex location of the mobile ubgraph G m (t) uch that () i minizmized, the graph G(t) tay k-connected, and the mobile node localization condition i atified. Thu we IV. OPTIMAL WIRELESS SENSOR NETWORK DEPLOYMENT The preented indice of performance are quadratic convex function. In optimization theory there are two ubclae of uch problem, depending on the contraint condition: a problem can be convex or non-convex, depending on the contraint and the correponding contraint et or feaibility et. A feaibility et i a et of allowable olution, i.e., allowable enor node location in cae of the problem decribed in Section 3. Aumption 3. The bae tation i one hop away from all mobile enor node in the network. Note that thi aumption mean that the bae tation can directly ee all mobile enor node, therefore retricting the problem to a convex feaibility et, ee Figure 5. A imilar aumption wa ued in the literature [3] where it i aumed that all node communicate with the bae tation directly. Fig. 5. A uboptimal deployment of a mobile node (red) that preerve the network connectivity i a convex optimization in cae the bae tation can directly communicate with mobile node (one hop away). Otherwie, it i a non-convex optimization problem. A. Senor Network Placement Algorithm There are many algorithm that can be ued for contrained (convex) optimization problem, ee [7] for intance. We ummarize two method that can be eaily implemented. Sequential Grid Search. Thi method provide a ueful olution for problem with a relatively mall number of moving UAV node, for intance le than 10, ee [8]. For a preelected grid at the area of interet, the cot function i evaluated in the immediate neighborhood of the preent tate in m-dimenional pace and deciion i made about where the B
5 next tate i, where m i the number of mobile UAV node, i.e. the order of the et M. The algorithm i a follow. 1. Place a grid of preelected ize x for each variable (mobile enor node) in the pace of interet S.. Select the tarting point for all mobile enor node in S. 3. Evaluate the cot function J in 9 m urrounding point including the exiting point, where m i the number of mobile node. 4. A a next tate elect the point with the mallet cot function J. 5. Check if the next tate i in S and if it atifie the enor network contraint. If ye, accept that point a a new tate for mobile node and repeat. One can ee that the equential grid earch method will work fine for a few mobile node, but if a larger number of mobile node needed to be optimally poitioned, the method i inefficient in term of computational requirement. More efficient and eaier to implement i a gradient decent method that only require a gradient computation at every tep. Gradient Decent Algorithm for One Mobile Senor Node Deployment. If there i only one mobile enor node, the previou problem formulation can be expreed in equivalent form: find an optimal vertex location (x 1, y 1 ) uch that the following cot function i minimized. min J 1 = R dit Q [( x, y ), ( X, Y )] 1 1 F [( x, y ), ( x, y )] 1 j j N The gradient decent algorithm i then given a follow. 1. Let deignate the -th iteration (where =0, 1,, 3,...) and calculate ( x x ) = R( x X ) Q 1 F x1 j N [( x x ) + ( y y ) ] ( y y ) = R( y Y ) + Q 1 F y1 j N [( x x ) + ( y y ) ]. Let the tart point be given by (x 10, y 10 ) and let the tolerance be given by ε. 3. Chooe tep ize λ to minimize J, that i J ( x λ, y λ ) x y = min J ( x λ, y λ ) λ x y 4. The calculate the next point (x 1(+1), y 1(+1) ) a x = x λ 1( + 1) x y + = y λ 1( 1) 1 y 5. Terminate the calculation if F + (3) J x, y ) J ( x, y ε. ( ) 1( + 1) 1( + 1) Then, we take (x 1(+1), y 1(+1) ) to be the optimal poition of the enor node we are eeking; otherwie return to Step. Simulation reult preented in the next ection how an implementation of the gradient decent algorithm. Due to it implicity, thi algorithm i alo uitable for fat, real-real time execution. V. SIMULATION RESULTS Self-Adjuting Connected Senor Network Simulation (SACSeNS) i a imulation program that can tet different cot function and their effectivene to ditribute tationary and mobile enor node in a given area, ee Figure 6. The program alo provide graph tatitic uch a connectivity and coverage percentage. Currently the imulation make aumption that all node have the ame communication range, the mobile node move at the ame rate, and the mobile node can move in any direction. Uer can pecify a communication range; however, it i the ame for all node in the network. Stationary and mobile node can be deployed imultaneouly. The tationary node are moved to a pecific location by the uer and will remain there during the imulation. The mobile node automatically move into a poition to minimize the cot function baed on the poition of the tationary node, other mobile node, and the focu point. The movement of mobile node follow the imple gradient decent algorithm from the previou ection uch that the cot function i minimized. From thee poition, an additional contraint of connectivity will be placed on the ytem that will require the node to maintain connectivity to the network. If a node movement will violate the network connectivity contraint, the program will kip the elected node poition and check the other poition for the bet movement of the node. The program ue a panning tree algorithm to determine if the network graph i connected or diconnected. The focu/tracking point i omething that we may want additional enor node to monitor and track. Baed on parameter given in the cot function, mobile node can move to increae enor coverage and track a target. There i a performance index graph over time that i updated periodically and how the relative change baed on the new enor node poition. Thi chart how how well the node ditribute themelve baed on the cot function.
6 Mobile enor node (blue) Target to be tracked Network graph tatitic Static enor node (green) Cot function in time Fig. 6. Self-Adjuting Connected Senor Network Simulation (SACSeNS) program for imulation of wirele enor network. Figure 7 how two oppoite cae dicued previouly, i.e., uniform v. focued coverage. There i one target/focu point in the network. The enor network i connected, however, weakly connected ince communication link between a few enor node are weak. Preented reult are an initial development of SACSeNS, while future tep will include: conider cae where mobile node do not have ame peed, do not have ame communication range, power, etc. implement variou cot function and optimization algorithm develop a web-baed uer interface uch that the oftware can be ued over the Internet. Currently, the firt verion of SACSeNS i built in C++. VI. CONCLUSIONS A novel linear quadratic-type of a cot function wa preented for a wirele enor network deployment problem under network connectivity contraint. The convex optimization problem wa olved uing a gradient decent algorithm that wa implemented a a part of new enor placement imulation oftware for wirele enor network that i being developed at Louiiana Tech. The imulation tool wa preented a well a imulation reult for focued and uniform network coverage. Future reearch will conider the more general non-convex optimization problem, different cot function, and novel feature in the imulation oftware. Fig. 7. SACSeNS naphot of a uniform coverage and deployment of mobile (blue) enor node (left) and a focued coverage/tracking of mobile node (right). The red link emphaize weak connectivity of the network graph. ACKNOWLEDGMENT Thi project i upported in part by the ARFL Senor Directorate through contract FA8650-05-D-191 and by Louiiana Board of Regent through PKSFI grant LEQSF(007-1)-ENH-PKSFI-PRS-03. The author alo acknowledge help by graduate tudent Grain Zhang. REFERENCES [1] P. Bahl and V. N. Padmanabhan, RADAR: An In-Building RF-Baed Uer Location and Tracking Sytem, IEEE InfoCom 000, March 000.W.-K. Chen, Linear Network and Sytem (Book tyle). Belmont, CA: Wadworth, 1993, pp. 13 135. [] D. P. Berteka, Nonlinear Programming, Athena Scientific, 1995. [3] J. L. Bredin, E. D. Demaine, M. T. Hajiaghayi, and D. Ru, Deploying enor network with guaranteed capacity and fault tolerance, Proc. the 6th ACM International Sympoium on Mobile ad hoc Networking and Computing, Urbana-Champaign, IL, USA, May 005. [4] E. H. Callaway Jr., Wirele Senor Network: Architecture and Protocol, CRC Pre LLC, Boca Raton, Florida, 004. [5] M. Cardei and Jie Wu, Coverage in wirele enor network, in M. Ilya and I. Magboub, ed., Handbook of Senor Network, CRC Pre, 004. [6] C. G. Caandra and W. Li, Senor network and cooperative control, European Journal of Control, vol. 11, pp. 436-463, 005.
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