COVERAGE CONTROL ON MULTI- AGENT SYSTEM

Transcription

1 Artkel Reguler COVERAGE CONTROL ON MULTI- AGENT SYSTEM Reka Inovan 1, Adha Imam Cahyad 2, Sgt Basuk Wbowo 3 Abstract In ths work, we study the problem of maxmzng the coverage of a Moble Sensor Network under an unknown nterest functon through dstrbuted means. To maxmze the coverage, we adopt the dstrbuted verson of the Vorono locatonal optmzaton method. The stablty results of the method s dscussed. Numercal experment s conducted to study the effect of network topology and nterest functon. To estmate the nterest functon, we used the dstrbuted Recursve Least Square method based on Alternatng Mnmzaton Algorthm Framework. The correctness and effectveness of the method s proved through numercal smulaton. Then, we am to combne those two method. We also ntroduced some modfcaton to the RLS algorthm to allevate the problem of the lack of exctaton. The combned algorthm s then smulated to show ts effectveness. The result of ths work shows that the modfed method perform slghtly worse compared wth the case of centralzed locatonal optmzaton wth known nterest feld due to the lack of exctaton. Keywords Locatonal Optmzaton, Vorono based optmzaton, Dstrbuted Estmaton, Recursve Least Square, Alternatng Mnmzaton Algorthm, Consensus buldng. I. INTRODUCTION Moble wreless sensor networks s a new paradgm whch am to combne the beneft of cheap and multple sensng gven by a sensor network and augment t wth moblty capablty to further ncrease t effectveness. Asde from the problem whch ths approach nherts from Wreless Sensor Network (WSN) approach, ths approach also ntroduce a new knd of possblty and ts unque problem from the mult-agent nature of the system. A comprehensve survey of the wreless sensor network can be found on [9], whle the survey of problem n mult-agent system can be seen n [6]. In ths work we am to solve one of the problems, namely the problem of contnuously optmzng the locaton of the network to provde the optmal coverage of the sensed functon. On classcal wreless sensor WSN, the problem of coverage control s also known as the node deployment problem. Ths name s gven because the actual motvaton of the problem s to optmze a deployment of WSN node n a known envronment, a survey of ths 1 Mahasswa, Jurusan Teknk Elektro dan Teknolog Informas, Fakultas Teknk Unverstas Gadjah Mada 2,3 Dosen, Jurusan Teknk Elektro dan Teknolog Informas, Fakultas Teknk Unverstas Gadjah Mada lne of research s n [1]. For classcal WSN, ths namng s approprate because after the node s deployed, the node do not posses capablty to reconfgure ts poston. By augmentng the node n WSN wth moblty capablty, ths problem acqure a new dmenson as t s now possble for each node to reconfgure ts poston. The possble applcatons for ths approach are numerous. For example, let us consder a network of underwater vehcles surveyng the ocean bed. The other example s a rover whch s desgned for sensng the sgn of volcanc erupton. In solvng ths problem, we am to create a strategy whch s not depend on the avalablty of a central node whch coordnates the behavor of the network. There are several establshed researches whch also am to solve the same problem, for example by [2], [8], [3], and [7]. But these researches suffer from the assumpton that each agent has the actual characterstc of the envronment. Ths assumpton s hard to defend realstcally as t s generally not possble for the agent to characterze the envronment before deployment. The model utlzed n ths work s only a smplfcaton of the actual WSN model. For example, n ths work the communcaton characterstc of the network s largely gnored. Ths delberate smplfcaton s amed to mantan the focus of ths work on the development of dstrbuted algorthm to acheve the optmal coverage. However n the actual mplementaton, t s necessary to factor the effect of network on the algorthm performance. II. PROBLEM FORMULATION In ths problem we consder the problem of optmzng coverage for moble WSN nodes wth nodes numbered from 1 to N. The network s assumed to be a dscrete system wth, -th node s descrbed by ts two dmensonal poston x R 2. In ths work we only consder the case of node dynamc as smple accumulator. Therefore the poston dynamc for each node can be formulated as x[n+ 1] = x[n]+ γu[n] (1) Where γ s constant and u[n] R 2 represent the velocty command to the network. The node operates on a two dmensonal plane defned as the operatng regon, V R 2, wth a sensng regon V V. An nterest functon ϕ(x) : V R s defned to quantfes the relatve mportance of each pont n the operatng regon. 117 Volume 1 Nomor 2, Jul 2014

2 Jurnal Peneltan Teknk Elektro dan Teknolog Informas We also defne a sensng functon f(x,y) : V V R, whch defnes the unrelablty of the sensng n the pont x f measured from sensor poston y. Therefore, we can defne a global unrelablty as N =0 φ (y) f(x, y) dy. (2) V In ths work, we assume that the sensng functon s L2 norm, f(x,y) = x-y 2. We also defne the sensng regon for each node as V = {y y V, f(x,y) < f (xj,y),j [1..N ] } (3) It can be proven that the operatng regon decomposton generated by ths rule s general for a certan class of sensng functon. In ths work, we also assume that the nterest feld s ntally unknown. However, each node s equpped wth the ablty to measure these nterest functons. We denote the result of the measurement as z[n] = ϕ(x[n]) (4) To estmate the nterest feld, we assume that the nterest feld ϕ(y) can be parameterzed by θ, ϕ(y;θ). Therefore, we can defne an estmatng algorthm n the form of θ[n+ 1] = g(z[n],θ[n]) (5) and the control law, n the form of u[n] = h(θ[n], x[n]). (6) Our objectve n ths work s to fnd the algorthm g and h such that the objectve functon n eq. (2) s mnmal under the constrant of the descrbed scenaro. A. Locatonal Optmzaton III. ALGORITHM DESIGN The frst step n desgnng the algorthm s to desgn the algorthm h. Our approach n desgnng ths algorthm s by usng the classcal Lloyd-Max algorthm, frst proposed n [4]. In ths algorthm, the optmzaton process s dvded nto two part. The frst part nvolves n updatng the sensor poston to optmze the global relablty under the assumpton of fxed sensng regon. The second part s to update the sensng regon to optmze the global unrelablty for the current sensor poston. To solve the frst part, t can be proven that the sensor poston s optmal f t s located n the centrod of ts sensng regon. We defne the centrod as c = φ(y;θ )y dy φ(y;θ)dy (7) For the second part, we re-compute the vorono regon for each node under the new poston. The sensng regon s defned to be equal to the newly computed vorono regon. The proof of the optmalty of the algorthm s omtted. The problem n decentralzng the algorthm s because each node has to posses the locaton for all other nodes n ths system. However, a closer observaton wll reveal that the algorthm actually need only the mmedate neghbor to compute the vorono regon. Therefore, we can convert ths algorthm nto a dstrbuted algorthm by only sendng the poston of the mmedate neghbor n a broadcast manner. The control sgnal s generated as follow u[n] = c[n]- x[n ] (8) It can be proven that ths control sgnal wll drve the system nto the most optmal confguraton for the case of smple accumulator. Although n ths work we only study the case of smple accumulator, t can be proven that the followng control sgnal wll also be drve the system nto optmalty f the dynamcs can be reduced nto the frst order system. B. Feld Estmaton The feld estmaton method used n ths work s an extenson of the regular RLS algorthm by usng the Alternatng Mnmzaton Algorthm to decentralze ts operaton. Ths method s frst proposed n [5]. Ths method s also known as AMA-RLS algorthm. Ths algorthm ams to optmze the followng augmented objectve functon J(θ [n] = n k=0 (z k φ(x [k]; θ [n])) + k N v k [n] θ θ k 2 (9) The algorthm proposed by [5] to optmze ths objectve functon can be dvded nto 3 steps : 1. Update the Lagrangan Multpler v k [n + 1] = v k [n ] + τ(θ k [n] θ [n]) (10) 2. Update the estmate θ [n + 1] = P [n]r [n] + k (N) P [n](v [n] v k [n]) (11) 3. Update the covarance matrx P [n + 1] = P [n] + P [n]ψ [n]ψ T [n]p [n] ψ T [n]p [n]ψ [n] (12) Wth P defned as the nverse covarance matrx, R as the autovarance matrx, ψ as the pror argument. In ths algorthm, only the value of current estmate θ and the value of Lagrange multpler vk that has to be send between nodes. C. Improvng the stablty Although the feld estmaton algorthm s shown to be stable under the normal condton, the estmate wll grow wthout bound on the condton of lack of exctaton. Ths defcency s sgnfcant for the correct operaton of the algorthm, as n the latter phase of locatonal optmzaton, the movement of the nodes s Volume 1 Nomor 2, Jul

3 Artkel Reguler farly lmted. Ths ntroduce a problem of lack of exctaton n the algorthm. To allevate ths problem, we ntroduce the followng modfcaton to the estmaton update step of the algorthm. θ [n + 1] = P [n]r [n] + P [n] (v P [n ] k (N) k [n] v k [n]). (13) Ths modfcaton s justfed by the fact that under the condton of lack of exctaton, the covarance matrx tend to become a sngular matrx, thus ncreasng the value of ts nverse to grow wthout bound. As the norm s equal to the largest egenvalue of the matrx, we can assume that the contrbuton of the Lagrange multpler s mnmzed. To compensate for the lack of consensus enforcng term n the estmaton update, we ntroduce a new term, such that θ [n + 1] = P [n]r [n] + ( P [n] + τ) (v[n] P [n] k (N) vk[n ]). (14) Wth τ s a constant. IV. NUMERICAL EXPERIMENT DESIGN We developed a numercal smulaton testbed n MATLAB. Ths desgn for numercal smulaton testbed s descrbed by fg. 1. 3: θ = P R + τp v o 4: P = P + P ψ T ψ P ψ T P ψ 5: Calculate u = φ(y;θ ) y dy φ(y;θ )dy x 6: Calculate new Vorono decomposton V usng neghbor poston The system smulate a dscrete system, such that for each tme-step the transmt secton s called and supply a message to the communcaton network. Ths message s receved by the receve secton and stored n the local memory of each node. A control sgnal s then generated by the control secton whch then used by the smulator to update the state of all nodes n the system. V. RESULTS There are three experment that we conducted to test the effectveness of the algorthm. The frst experment s to compare the result of Lloyd-Max under the constrant of lmted communcaton range. In ths experment, the nterest functon s known beforehand n the form of ϕ(y) = 1. We compare the result of Lloyd-Max algorthm under ths constrant aganst the condton of full nformaton. The result of ths comparson can be seen n fg. 2. The trajectory of ths algorthm s shown n fg. 3. Fg. 2. Unrelablty Comparson between dfferent topology Fg. 1. Scheme of Smulaton Testbed In ths desgn, the mplemented algorthm s broken down nto three parts. The pseudo-code for each secton s gven n the alg. 2, 1, 3. Algorthm 1. Receve Secton for node from node j 1: v j = v j + c 2 (s s j ) 2: Store the value of v j and x j Algorthm 2. Transmt Secton for node 1: To each neghborng node j, send v j 2: To all neghborng node, send (x, θ ) Algorthm 3. Control Secton for node 1: For all par of v j and v j, v o = v o + (v j v j ) 2: F = F + τx ψ Fg. 3. Trajectory of Lloyd-Max Algorthm The results of ths experment ndcates that there are no notceable dfference between the dstrbuted verson of Lloyd-Max algorthm and the centralzed verson of the algorthm. The trajectory of the system also shows the ntended behavour as system are 119 Volume 1 Nomor 2, Jul 2014

4 Jurnal Peneltan Teknk Elektro dan Teknolog Informas supposed to decompose the operatng regon unformly under the constant nterest functon In the second experment, we want to observe the performance of the dstrbuted RLS algorthm. In ths experment, the nterest feld s unknown beforehand. The underlyng nterest feld s gven by the followng equaton n the regular x - y coordnate. f (x, y) = x 2y + 3xy (15) The pror used by the system s as follows f(x, y) = p 1 + p 2 x + p 3 y + p 4 xy (16) Each node s gven a random control functon. The result of the parameter convergence s gven n fg. 4. fg. 8 and the regular AMA-RLS fg. 7. We can see that the modfefd algorthm manage to attenuate the oscllaton on the parameter value and ensure the stablty of the parameter. Fg. 5. Unrelablty Curve for the AMA-RLS Fg. 4. Parameter Value The estmate s convergng to the correct value for each nodes. Ths ndcates the ablty of the algorthm to estmate the underlyng feld n a dstrbuted manner. In the thrd experment, we combne the result of the prevous experment to enable the system to smultaneously optmze ts locaton and estmatng ts underlyng feld. In ths experment, we use the underlyng feld n the form of f(x, y) = 1 + x + 0.6y 0.1x 2 0.1y 2 (17) wth the pror f(x, y) = p 1 + p 2 x + p 3 y + p 4 x 2 + p 5 y 2 (18) The unrelablty graph of the system s gven by fg. 5. We can see from the graph that the result of the dstrbuted verson of RLS s slghtly worse compared to the case of known feld. The result also show some sort of oscllaton n the unrelablty value. We argue that ths oscllaton s due to the condton of the lack of exctaton that experenced by the system due to the lack of movement n the latter phase of locatonal optmzaton. We compare the result of ths experment and the smulaton f the modfcaton that we propose n ths work s ncluded. The comparson of unrelablty value s shown n fg. 6. It can be seen that ntroducton of normalzaton manage to supress the oscllaton, however t drves the unrelablty value to be equal to regular RLS, because the lack of consensus. The unrelablty value s mproved after we ntroduce the consensus enforcng term. he dfference s more pronounced f we compare the evoluton of weght between the modfed algorthm Fg. 6. Unrelablty Curve for Modfed AMA-RLST VI. CONCLUSION In ths work we have dscussed the algorthm for locatonal optmzaton n an unknown nterest feld. We consder a scenaro where a moble sensor network s optmzng the qualty of ts coverage. In the conventonal applcaton, an nterest feld s assgned to determne the ntended densty of the network. However, n ths scenaro, the network s unknown beforehand. Therefore the network has to estmate ts nterest feld, whle smultaneously form a specfc spatal confguraton to optmze ts coverage. For locatonal optmzaton, we use a modfed verson of the Lloyd-Max algorthm whch only requre the exchange of poston nformaton between the neghborng nodes. It permt the algorthm to be mplemented n a dstrbuted verson. The result of the experment has ndcated that the performance of ths modfed algorthm s almost smlar to the centralzed verson n whch the nodes know all the value. We estmate the nterest feld by usng a dstrbuted verson of RLS whch based on AMA approach. The smulaton s conducted to verfy the correctness of the algorthm. It has shown that the algorthm s able to estmate the nterest feld parameter whle at the same tme enforcng the consensus n the network. We combne those two algorthm to enable the network to perform locatonal optmzaton under an unknown nterest feld. It has shown that the man problem n ths step s the lack of exctaton durng the latter phase of locatonal optmzaton. We ntroduce a Volume 1 Nomor 2, Jul

5 Artkel Reguler normalzaton and consensus enforcng scheme to mprove the performance of the algorthm. It has been shown through smulaton that ths modfcaton mproves the performance of the algorthm. Fg. 7. Parameter Evoluton for the orgnal AMA-RLS Fg. 8. Parameter Evoluton for the modfed AMA-RLS REFERENCES [1] Jan Beutel, Kay Rmer, Matthas Rngwald, and Matthas Woehrle. Deployment technques for sensor networks. In Ganlug Ferrar, edtor, Sensor Networks, pages Sprnger Berln Hedelberg, Berln, Hedelberg, [2] Islam I. Hussen and Dusan M. Stpanovc. Effectve coverage control for moble sensor networks wth guaranteed collson avodance. IEEE Transactons on Control Systems Technology, 15(4): , July [3] We L and C.G. Cassandras. Dstrbuted cooperatve coverage control of sensor networks. pages IEEE. [4] S. Lloyd. Least squares quantzaton n PCM. IEEE Transactons on Informaton Theory, 28(2): , March [5] G. Mateos and G. B. Gannaks. Dstrbuted recursve leastsquares: Stablty and performance analyss. IEEE Transactons on Sgnal Processng, 60(7): , July [6] We Ren, R.W. Beard, and E.M. Atkns. A survey of consensus problems n mult-agent coordnaton. pages IEEE. [7] Mac Schwager, Jean-Jacques Slotne, and Danela Rus. Decentralzed, adaptve control for coverage wth networked robots. pages IEEE, Aprl [8] R. Xong, Y. Q. Ba, W. Sun, and X. Lu. Moble sensor network node deployment va central vorono tessellaton. pages IEEE, July [9] Jennfer Yck, Bswanath Mukherjee, and Dpak Ghosal. Wreless sensor network survey. Computer Networks, 52(12): , Volume 1 Nomor 2, Jul 2014