Networked Estimation with an Area-Triggered Transmission Method

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1 Sensors 2008, 8, sensors ISSN by MDPI Full Paper Neworked Esmaon wh an Area-Trggered Transmsson Mehod Vnh Hao Nguyen and Young Soo Suh * Deparmen of Elecrcal Engneerng, Unversy of Ulsan, Namgu, Ulsan , Korea E-mal: vnhhao@hcmu.edu.vn. E-mal: suh@eee.org * Auhor o whom correspondence should be addressed. Emal: suh@eee.org Receved:18 January 2008 / Acceped: 7 February 2008 / Publshed: 15 February 2008 Absrac: Ths paper s concerned wh he neworked esmaon problem n whch sensor daa are ransmed over he nework. In he even-drven samplng scheme known as level-crossng or send-on-dela, sensor daa are ransmed o he esmaor node f he dfference beween he curren sensor value and he las ransmed one s greaer han a gven hreshold. The even-drven samplng generally requres less ransmsson han he me-drven one. However, he ransmsson rae of he send-on-dela mehod becomes large when he sensor nose s large snce sensor daa varaon becomes large due o he sensor nose. Movaed by hs ssue, we propose anoher even-drven samplng mehod called area-rggered n whch sensor daa are sen only when he negral of dfferences beween he curren sensor value and he las ransmed one s greaer han a gven hreshold. Through heorecal analyss and smulaon resuls, we show ha n he ceran cases he proposed mehod no only reduces daa ransmsson rae bu also mproves esmaon performance n comparson wh he convenonal even-drven mehod. Keywords: Neworked esmaon, even-drven, level-crossng, send-on-dela. 1. Inroducon The radonal way o ransm sensor daa n neworked conrol sysems s o sample sensor sgnals equdsan n me. Ths approach s wdely used snce analyss and desgn are smple and nhered from well-esablshed conrol sysem heores. If he nework speed s hgh and he raffc s sparse, he perodc samplng approach has many mers because nework-nduced delay s small and can be

2 Sensors 2008, gnored. Bu when he nework bandwdh s lmed due o execung asks of several nodes, me delay becomes large and randomly varyng. To avod hese problems, he sensor daa ransmsson rae should be reduced. Recen works have dscussed even-drven alernaves o radonal me-rggered samplng scheme. I has been shown o be more effcen han me-rggered one n some suaons, especally n nework bandwdh mprovemen. For nsance n [1], he auhors provded a comparson of merggered mpulse conrol and level-rggered one where he level-rggered scheme gave lower average error for he same average rae of mpulse. In [2, 3], an adjusable deadband was defned on each node o reduce nework raffc. The node does no broadcas a new message f s sgnal s whn he deadband. A level-crossng samplng scheme n assocaon wh an 1-b codng and decodng sraegy o reduce daa sze was nroduced n [4]. Ths codng samplng scheme becomes hghly effcen under daa-rae consrans snce he nodes only ransm one b, 0 or 1, per sample. In [5], an opmal levelrggered samplng desgn problem was proposed n order o mnmze he dsoron of a fler over a fne horzon. A smlar samplng scheme named send-on-dela (SOD) ransmsson was explored n [6-8]. By adjusng he hreshold value a each sensor node, daa ransmsson rae s reduced n order for he nework bandwdh o ncrease and can be used for oher raffc. Alhough he even-drven schemes menoned above have several names, he basc prncple of samplng s he same. Tha s, n framework of neworked conrol sysem, a sensor node ransms daa only f he dfference Δ beween he curren sensor value and he las ransmed one s greaer han a gven hreshold. Hereafer, for ease n represenaon we refer o hem as SOD samplng. One major ssue of SOD samplng mehod as poned ou n [9] s ha does no deec sgnal oscllaons or seady-sae error f he dfference Δ remans whn he hreshold range durng a long me. Furhermore, n hghly nosy sysems, he sensor value could exceed he hreshold no because he rue value exceeds he hreshold bu because sensor nose exceeds he hreshold. In ha case, here could be many unnecessary sensor daa ransmssons. Movaed by he perspecve resuls of he even-based negral samplng n [9] and he neworked esmaon problem usng SOD ransmsson mehod n [7], we propose a novel samplng scheme called area-rggered or send-on-area (SOA) n whch sensor daa are sen only when he negral of Δ s greaer han a gven hreshold. The proposed SOA samplng scheme s slghly dfferen from [9] o mprove bandwdh nework n case of large nose. Then, a neworked esmaor based on Kalman fler s formulaed o esmae saes of he sysem perodcally even when he sensor nodes do no ransm daa. Through heorecal analyss and smulaon resuls, we show ha n he ceran cases he proposed mehod gves beer esmaon performance han he SOD mehod n [7]. 2. Area-rggered samplng scheme Consder a neworked conrol sysem descrbed by he lnear connuous-me model: x () = Ax () + Bu () + w () y () = Cx () + v () (1) where x R n s he sae of he plan, u s he deermnsc npu sgnal, y R p s he measuremen oupu whch s sen o he esmaor node by he sensor nodes. w () s he process nose wh

3 Sensors 2008, covarance Q, and v () s he measuremen nose wh covarance R. We assume ha w () and v() are uncorrelaed, zero mean whe Gaussan random processes. The area-rggered samplng scheme llusraed n Fg.1b s saed as follows: ( 1 p The area formed by he curren oupu y() and y las, along me: Le y las, ) be he las ransmed value of he -h sensor oupu a nsan las, Π (, ) ( y() y ) d. las, las, las,. A new sensor value wll be sen o he esmaor node f he followng condon s sasfed: Π (, ) > α (3) where α las, s he gven hreshold value a he -h sensor node. Noe ha n he SOD samplng scheme [7] shown n Fg.1a, a new sensor value wll be sen f Δ (, las, ) > δ, where δ s he gven hreshold value. (2) y( ) y las, y la s y( ), + _ Δ δ Comparaor Δ > δ T s + _ Δ Π α Comparaor Π > α T s a. SOD samplng scheme b. SOA samplng scheme Fgure 1. SOD and SOA samplng schemes 2.1. Effec of nose on sensor daa ransmsson rae In hs secon, we assume u() = 0 o nvesgae effec of nose on he sysem. The dfference beween he curren sensor value and he las ransmed one s defned by: Δ (, ) y() y las, las, = C [ Φ(, ) I ] x( ) + C Φ (, r) w( r) dr + v () v ( ) las, las, las, las, where Φ (, las, ) = exp( A ( las, )). From (2) we have: Inserng (4) no (5), we oban (6): Π (, las, ) = Δ(, las, ) dr. (5) las, (4)

4 Sensors 2008, Π (, ) = Δ (, ) dr las, las, las, = C [ Φ(, ) I ] x( ) dr las, + C Φ (, rwrdrdu ) ( ) + [ v() v( )] dr las, las, las, las, = C [ Φ(, ) I ] x( ) dr las, + C Φ(,) u r du w() r dr + v() dr v( las, ) dr. las, r las, las, Assumng x ( las, ) = 0, we can derve he varance of Π(, las, ) as follows E{ ΠΠ } = C θ(, r) w( r) w ( r) θ (, r) drc + 2 R(, ) dr las, las, las, u las, las, las, las, = C θ(, r) Qθ (, r) drc + 2 R(, ) ( ) where R (, ) s he (, ) -h elemen of R and θ( r, ) s defned by Noe ha he varance of las, θ ( r, ) Φ( urdu, ). E{ ΔΔ } = C r) C Φ(, QΦ (, r) dr + 2 R(, ). (8) From (7) and (8) we see ha boh E { ΠΠ } and E { ΔΔ } consss of wo erms. The frs erm depends on he process nose w () and he second one depends on he measuremen nose v (). Therefore, we beleve E ΠΠ } (n he case of SOA) plays he same role as E ΔΔ } (n he case of { r las, Δ (, ) n he SOD samplng scheme [7] s gven by las, SOD) n nvesgaon of daa ransmsson rae. Ths assumpon wll be verfed n he nex secon 2.3. In he analyss n [7], we know ha daa ransmsson rae s proporonal o E{ ΔΔ }. In he hghly nosy sysems, large R value makes E { ΔΔ } { ransmsson rae when applyng he SOD samplng scheme. Bu wh he SOA samplng scheme, we can consran he effec of R on E{ ΠΠ } n (7) by lowerng he facor ( las, ). Ths suaon can be acheved by choosng he α hreshold value such ha ( las, ) 1s. In oher words, f he me nerval of wo consecuve sensor daa packes s less han one second, hen he SOA mehod can be appled. Ths s no a so src condon n neworked conrol sysems. Π 2.2. compuaon and SOA samplng n dscree me a he sensor nodes (6) (7) n (8) large. I leads o ncrease daa Le T be samplng me of he sgnals y( ), yk, ( k = 1,2, 3,...) be he k -h sampled value y( kt) from me he -h sensor node ransms. If he sensor oupu has no nose, les on he y las, y k,

5 Sensors 2008, smooh curve Cx (). Bu under he effec of measuremen nose, y k, wll be n he vcny of Cx () as llusraed n Fg.2. Supposng ha condon (3) s sasfed a = 3,, Π(, las, ) shown approxmaely by he slashed area s calculaed: y( ) y 2, y y 3, las, Cx () y las, y 1, y 1, las, T T T T 1, 2, 3, las, 1, Fgure 2. Sensor oupu wh nose n dscree me. 1, 2, 3, Π = ( y ( ) y ) d + ( y ( ) y ) d + ( y ( ) y ) d las, las, las, las, 1, 2, ( y y ) T /2 + ( y + y 2 y ) T /2 + ( y + y 2y 1, las, 1, 2, las, 2, 3, las, ) T /2 The prncple of compung he slashed area n (9) s o dvde no several small pars. Each par s a rapezod wh he hegh T and wo sdes ( y y ), ( y y excep for he frs par whch s a rangle. However, usng (9) s nconvenen f condon (3) s no sasfed for a long me nerval because he sensor node has o spend much memory on sorng y k,. An algorhm o calculae and ransm sensor daa n whch we only use 3 memory uns o sore ( Π, y, y ) a every Π perod T s proposed as follows: 2.3. Effec of nose on sgnal dsoron k, las, k+ 1, las, ) Π = 0; k = 0; y = y ; 0, las, whle k = k + 1 Π = Π + ( y + y 2y ) T /2 y k, 0, las, 0, k, end whle Π < α = y y = y ; Transm y las, k, las, N las, 0, 2 D ( yr, ( kt) ylas, ( kt) ) (11) where yr, ( kt) = Cx( kt ) s he rue oupu value of he -h sensor whou measuremen nose and ( kt) s he -h sensor value receved a he esmaor node. y las, We defne a performance ndex called squared error dsoron D for each sensor node: k = 1 (9) (10)

6 Sensors 2008, a. Case 1: δ = 0.005, α = b. Case 2: δ = 0.01, α = c. Case 3: δ = 0.05, α = Fgure 3. Effec of R on daa ransmsson rae and dsoron for y() = v ().

7 Sensors 2008, a. Case 1: δ = 0.005, α = b. Case 2: δ = 0.01, α = c. Case 3: δ = 0.05, α = Fgure 4. Effec of R on daa ransmsson rae and dsoron for y() = 5(1 e ) + v ().

8 Sensors 2008, I s very dffcul o derve an explc expresson of sgnal dsoron n wo mehods SOD and SOA by heorecal analyss because depends no only on he gven hreshold bu also he sysem model and nose. Thus, n hs secon we evaluae he effec of measuremen nose on dsoron by consderng an example nsead. 0.1 Consder wo sensor oupus y () = v () and y () = 5(1 e ) + v (), where R s varyng. We wll see how daa ransmsson rae and dsoron vary as R vares n boh mehods SOD and SOA. The evaluaon s mplemened wh T = 0.01s n 50 seconds. The hreshold values δ and α are gven n 3 cases: Case 1. δ = : small hreshold Case 2. δ = 0.01 : medum hreshold Case 3. δ = 0.05 : large hreshold In he above example, α s chosen accordng o δ such ha number of daa ransmssons n wo mehods s dencal as v ()= 0 (whou nose). From he resuls n he Fg. 3 and Fg. 4, he effec of R on daa ransmsson rae and dsoron s summarzed as follows: - When he hreshold value s small as n case 1, dsoron of SOA s equvalen o ha of SOD bu daa ransmsson rae of SOA s smaller han ha of SOD. In hs case, usng SOA wll reduce daa ransmsson rae. - When he hreshold value ncreases as n case 2, dsoron of SOA s slghly smaller han ha of SOD and daa ransmsson rae of SOA s much smaller han ha of SOD. In oher words, we ge benef of daa ransmsson rae reducon and a lle dsoron reducon from SOA mehod. - When he hreshold value s large as n case 3, SOA reduces no only daa ransmsson rae bu also dsoron. The larger R value s, he more reducon s. Ths resul s remarkable! I means ha n he SOA mehod, sgnal dsoron s no degraded along wh he reducon of daa ransmsson rae even when sysem nose s large. Therefore, we hope ha esmaon performance wll be sgnfcanly mproved when applyng a fler. 3. Sae esmaon wh SOA Transmsson mehod The neworked esmaon problem applyng SOA ransmsson mehod s depced as follows: 1. Measuremen oupus y ( 1 p ) are sampled a he perod T bu her daa are only ransmed o he esmaor node when (3) s sasfed. 2. For smplcy n he problem formulaon, ransmsson delay from he sensor nodes o he esmaor node s gnored. 3. The esmaor node esmaes saes of he plan regularly a he perod T regardless of wheher or no sensor daa arrve. If here s no -h sensor daa receved for > las,, he esmaor node consders ha he measuremen value of he -h sensor oupu y() s sll equal o y las, bu he measuremen nose ncreases from v() o vn, () = v() + Δ (, l as, ). To formulae a sae esmaon problem, he bound of Δ(, las, ) needs o be deermned. In he nex secon, we wll compue he bound of Δ (, ) and hen a modfed Kalman fler s appled for sae esmaon. las,

9 Sensors 2008, Bound of Δ (, ): A me = 1, las, (3) and (9) we know ha: herefore A me or = 2,, f he esmaor node does no receve -h sensor daa as llusraed n Fg.2, from Π (, ) = ( y y ) T/2 < α, (12) 1, las, 1, las, 2 α / T < ( y1, ylas, ) < 2 α / T. (13), f he esmaor node does no receve -h sensor daa ye hen: ( y1, ylas, ) T /2 + ( y1, + y2, 2 ylas, ) T/2 < α, (14) 2 α / T 2 ( y y ) < ( y y ) < 2 α / T 2( y y ). (15) 1, las, 2, las, 1, las, Inserng (13) no (15) we oban: 6 α / T < ( y2, ylas, ) < 6 α / T. (16) Compung smlarly for nsans = k, ( k = 3,4,...), we always have he followng nequaly: 6 α / T < ( yk, ylas, ) < 6 α / T. (17) In oher words, he bound of Δ (, ) s gven by Noe ha Δ (, ) = 0 las, las, Δ (, las, ) < 6 α / T. (18) a me he esmaor node receves -h sensor daa. Then, he process s repeaed as (12)-(17) as long as he esmaor node does no receve daa from he -h sensor node ye Sae esmaon Assumng ha Δ (, ) has a unform dsrbuon wh (18), varance of Δ (, ) wll be las, ( 6 α / ) 2 T /3. Thus, f here s no -h sensor daa receved for >,, varance of measuremen nose s ncreased from R (, ) o R (,) + ( 6 α / ) 2 T /3. A modfed Kalman fler for sae esmaon x a sep k, where here s a change n he measuremen updae par of he dscree Kalman fler algorhm [10], s gven as n he Fg.5. We use he dscrezed sysem model sampled a perod T : Q d las T AT Ar, d r Ad = e B = e Bd 0 s he process nose covarance of he dscrezed sysem: Qd = T Ar A e Qe r dr 0 and y s he vecor of p las receved sensor values:,, y = y y... y las las,1 las,2 las, p. In he modfed Kalman fler n Fg.5, he saes of he plan are esmaed regularly a every perod T regardless of wheher or no sensor daa arrve. If -h sensor daa arrve, Δ (, las, ) = 0, he modfed Kalman fler acs lke he convenonal Kalman fler. Oherwse, uses as he measuremen value and esmaon. ˆk las las, y las, R (, ) = R (, ) + ( 6 α / T) 2 / 3 as measuremen nose covarance for sae

10 Sensors 2008, Inalzaon Se xˆ, P y las 0 0 = Cxˆ 0 Yes -h sensor daa arrve? No R (, ) = R(, ) k y las, = y ( kt) R (, ) = R(, ) + ( 6 α / T) 2 /3 k z k = y las Measuremen updae ( ) K = P C CP C + R k k k k xˆ = xˆ + K ( z Cxˆ ) k k k k k P = ( I K C) P k k k 1 xˆ, xˆ, Projec ahead xˆ = Axˆ + B u P + = APA + Q k + 1 d k d k k 1 d k d d Fgure 5. Srucure of he modfed Kalman fler. 4. Smulaon To verfy he proposed fler, we consder an example of he second-order sysem wh sep npu where he oupu s sampled by he SOD and SOA mehods: x () = x () u () w () 1/ a b/ a + M / a + y () = 1 0 x () + v () Q = 0.01, R = 0.01, T = 10ms The sysem parameers are gven n he followng wo cases for performance evaluaon:

11 Sensors 2008, Case 1. (underdampled sysem) M = 30, a = 5, b = 1. Case 2. (undamped sysem) M = 30, a = 5, b = 0. The smulaon process s mplemened for 50 seconds. In each case, we use wo mehods (SOD and SOA) for performance comparson. Esmaon performance s evaluaed by he average dsoron: N N D = ( ˆ ) 2 ek, xk, xk, N = N (19) k= 1 k= 1 where e ( = 1,2) s he esmaon error, x s he reference sae, and N = The smulaon resuls wh dfferen hreshold values for he wo cases are shown n Table 1 and Table 2, where α value s chosen accordng o δ value such ha n (number of sensor daa ransmssons) s dencal n he wo mehods. In boh cases, we see ha when δ s small (.e. δ = 0.1, 0.3 ), esmaon performance of SOD and SOA s almos he same. Rgorously speakng, SOD s slghly beer han SOA. However, when δ s ncreasng, SOA mehod shows o be ouperform sgnfcanly. For example as δ = 0.9, SOA ouperforms SOD by he D reducon of 5 mes (.e vs n case 1, and vs n case 2). As llusraed n Fg. 6 and Fg. 7, he esmaon error of SOA s much smaller han ha of SOD. Table 1. Esmaon performance of 2 mehods wh dfferen hreshold values n case 1 δ α n D 1 by SOD 6.08e D 1 by SOA 6.40e D 2 by SOD D 2 by SOA Table 2. Esmaon performance of 2 mehods wh dfferen hreshold values n case 2 δ α n D 1 by SOD 3.49e e D 1 by SOA 3.65e e D 2 by SOD D 2 by SOA

12 Sensors 2008, Fgure 6. Esmaon error as δ = 0.9, α = n case 1 Fgure 7. Esmaon error as δ = 0.9, α = n case 2. Through smulaon resuls, we see ha he SOA mehod provdes beer esmaon performance han he SOD mehod. The key reason s ha n he SOD mehod, nose-conanng sensor daa no only make ransmsson rae ncrease bu also degrade esmaon performance. Whereas, hanks o he negral block whch acs as a nose fler n he SOA samplng scheme, he sensor node does no send hghly nosy daa, bu ransms daa only when he oupu ndeed changes value. For ha reason, he ransmed sensor daa n he SOA mehod s more relable. Ths helps esmaon performance beer. 5. Concluson In hs paper, he sae esmaon problem wh SOA ransmsson mehod over nework has been consdered. We have shown ha when ncreasng he hreshold value o mprove bandwdh nework, he SOA mehod gves much beer esmaon performance han he SOD mehod. Ths s very useful n he realsc applcaons where sensor daa ransmsson rae needs o be lowered due o jonng of many sensor nodes or for power savng n wreless neworks. The SOA mehod has been also proven o be more effcen han he SOD mehod n he hghly nosy sysems.

13 Sensors 2008, Acknowledgemens Ths work was suppored by he Korea Scence and Engneerng Foundaon (KOSEF) gran funded by he Korea governmen (MOST) (No. R ). The auhors also would lke o hank Mnsry of Commerce, Indusry and Energy and Ulsan Meropolan Cy whch parly suppored hs research hrough he Nework-based Auomaon Research Cener (NARC) a Unversy of Ulsan. References and Noes 1. Asrom, K.J.; Bernhardsson, B.M. Comparson of Remann and Lebesgue samplng for frs order sochasc sysems. In Proceedngs 41s Conference on Decson and Conrol; Las Vegas, U.S.A., 2002; pp Oanez, P.G.; Moyne, J.R.; Tlbury, D.M. Usng deadbands o reduce communcaons n neworked conrol sysems. In Proceedngs of he Amercan Conrol Conference; Anchorage, U.S.A., 2002; pp Sandra, H.; Peer, H.; Eckehard, S.; Marn, B. Nework Traffc Reducon n Hapc Telepresence Sysems by Deadband Conrol. In Proceedngs IFAC World Congress; Inernaonal Federaon of Auomac Conrol: Prague, Czech Republc, Kofman, E.; Braslavsky, J.H. Level Crossng Samplng n Feedback Sablzaon under Daa-Rae Consrans. In Proceedngs 45h Conference on Decson and Conrol; San Dego, U.S.A., 2006; pp Rab, M.; Mousakdes, G.V.; Baras, J.S. Mulple Samplng for Esmaon on a Fne Horzon. In Proceedngs 45h Conference on Decson and Conrol; San Dego, U.S.A., 2006; pp Mkowcz, M. Send-On-Dela Concep: An Even-Based Daa Reporng Sraegy. Sensors 2006, 65, Suh, Y.S.; Nguyen, V.H.; Ro, Y.S. Modfed Kalman fler for neworked monorng sysems employng a send-on-dela mehod. Auomaca 2007, 43(2), Suh, Y.S. Send-On-Dela sensor daa ransmsson wh a lnear predcor. Sensors 2007, 7, Mkowcz, M. Asympoc Effecveness of he Even-Based Samplng Accordng o he Inegral Creron. Sensors 2007, 7, Brown, R.G.; and Hwang, P.Y.C. Inroducon o Random Sgnals and Appled Kalman Flerng; John Wley & Sons: New York, Asrom, K. J. Inroducon o Sochasc Conrol Theory; Academc Press: New York, by MDPI (hp:// Reproducon s permed for noncommercal purposes.