Sensors 009, 9, 3078-3089; o:10.3390/s90403078 OPEN ACCESS sensors ISSN 144-80 www.mp.com/journal/sensors Artcle Networke Estmaton for Event-Base Samplng Sstems wth Packet Dropouts Vnh Hao Nguen an Young Soo Suh * Department of Electrcal Engneerng, Unverst of Ulsan, Namgu, Ulsan 680-749, Korea; E-Mal: vnhhao@hcmut.eu.vn (V.H.N.) * Author to whom corresponence shoul be aresse. E-Mal: suh@eee.org; Tel. 8-5-59-196; Fax: 8-5-59-1686 Receve: 3 March 009; n revse form: 1 Aprl 009 / Accepte: 4 Aprl 009 / Publshe: 4 Aprl 009 Abstract: Ths paper s concerne wth a networke estmaton problem n whch sensor ata are transmtte over the network. In the event-base samplng scheme known as levelcrossng or sen-on-elta (SOD), sensor ata are transmtte to the estmator noe f the fference between the current sensor value an the last transmtte one s greater than a gven threshol. Event-base samplng has been shown to be more effcent than the tmetrggere one n some stuatons, especall n network banwth mprovement. However, t cannot etect packet ropout stuatons because ata transmsson an recepton o not use a perocal tme-stamp mechansm as foun n tme-trggere samplng sstems. Motvate b ths ssue, we propose a mofe event-base samplng scheme calle mofe SOD n whch sensor ata are sent when ether the change of sensor output excees a gven threshol or the tme elapses more than a gven nterval. Through smulaton results, we show that the propose mofe SOD samplng sgnfcantl mproves estmaton performance when packet ropouts happen. Kewors: Networke estmaton; event-base samplng; sen-on-elta; packet ropout. 1. Introucton Recent works have scusse event-rven alternatves to tratonal tme-trggere samplng schemes. It has been shown to be more effcent than tme-trggere one n some stuatons, especall n network banwth mprovement. In [1-7], event-base samplng scheme was apple b ajustng
Sensors 009, 9 3079 the threshol value at each sensor noe, ata transmsson rate s reuce so that the network can be use for other traffc. However, analss an smulaton n the the works on event-rven samplng scheme were performe uner eal communcaton network contons: no elas or packet ropouts are assume, but n realstc applcatons, network nuce elas an packet losses o happen. The ssues of network elas an packet ropouts n tme-trggere sstems have been aresse an solve b researchers n [8-14]. In [8] the stablt of the Kalman flter n relaton to the ata arrval rate s nvestgate. It s shown that there exsts a crtcal ata arrval rate for an unstable sstem so that the mean flterng error covarance wll be boune for an ntal conton. In a ver recent stu [13], the optmal H flterng problems assocate respectvel wth possble ela of one samplng pero, uncertan observatons an multple packet ropouts are stue uner a unfe framework. The H -norm of sstems wth stochastc parameters s efne an compute va a Lapunov equaton an a stea-state flter s esgne va an LMI approach. In [14], the authors aopt a moel smlar to that of [13] for multple packet ropouts to nvestgate fnte-horzon optmal lnear flterng, precton an smoothng problems. In conventonal event-base samplng sstems, also calle sen-on-elta (SOD) samplng [5-7], the ssues of network ela an packet loss are ffcult to solve because ata transmsson an recepton o not use a perocal tme-stamp mechansm as n the tme-trggere samplng sstems. Motvate b those ssues, n ths paper, we ntrouce a mofe SOD samplng scheme n whch the event-rven samplng s combne wth a tme-trggere samplng scheme to etect packet ropouts. Then, a networke estmator base on a Kalman flter s formulate to estmate states of the sstem perocall even when the sensor noes o not transmt ata. The propose SOD samplng scheme has propertes nherte from the conventonal SOD samplng: so the benefts from event-rven samplng are stll hol. Through theoretcal analss an smulaton results, we show that the propose SOD samplng scheme gves better estmaton performance than the conventonal SOD one when packet loss happens.. Mofe SOD Samplng Scheme Conser a networke control sstem escrbe b the lnear contnuous-tme moel: x& ( t) = Ax( t) + Bu( t) + w( t) ( t) = Cx( t) + v( t) (1) n p where x( t) Î R s the state of the plant, u s the etermnstc nput sgnal, ( t) Î R s the measurement output whch s sent to the estmator noe b the sensor noes. w( t ) s the process nose wth covarance Q, an v( t ) s the measurement nose wth covarance R. We assume that w( t ) an v( t ) are uncorrelate, zero mean whte Gaussan ranom processes. The mofe SOD samplng scheme llustrate n Fgure 1b s state as follows: Let last, ( 1 p ) be the last transmtte value of the -th sensor output at nstant t last,. A new sensor value wll be sent to the estmator noe f one of two followng contons s satsfe: ( t) - > (a) last,,
Sensors 009, 9 3080 t - t > (b) last, t, where,, t, are the gven magntue, tme threshol values respectvel at the -th sensor noe. Fgure 1. Prncple of SOD an mofe SOD samplng schemes. (a) SOD samplng ( t ) ( t) last, ( t) (b) Mofe SOD samplng ( t ) ( t) last, ( t) t t t t t Usng the mofe SOD samplng scheme above we wll obtan some benefts. Frstl, the estmator can etect sgnal oscllatons or stea-state error f the fference of output value remans wthn the threshol range urng a long tme. Seconl, the estmator can etect multple packet ropouts f t oes not receve sensor ata wthn the nterval (0, t, ). Thrl, theoretcal analss for SOD samplng s stll apple for the mofe SOD samplng. However, ths scheme has one savantage that sensor ata transmsson rate wll be ncrease ue to conton (b). If t, s small, the estmator etects packet ropouts fast but ata transmsson rate s ncrease. If t, s large, transmsson rate s small but the estmator etects packet ropouts slowl. Therefore, an optmal t, value s necessar to compromse these constrants..1. Multple packet ropouts etecton The estmator noe etects packet ropouts of -th sensor ata b checkng the nstant -th sensor ata arrve. If there s no -th sensor ata arrvng, the estmator noe for the tme t - tlast, > t,, then the estmator noe knows that one-packet ropout happene at the -th sensor noe. Smlarl, f there s no -th sensor ata arrvng for t - tlast, > t,, then two-consecutve-packet ropout happene. We state the general case for multple packet ropouts as follows: If the estmator noe oes not receve -th sensor ata for tme ( t - tlast, ) > t, ( = 1,, 3,...) then the estmator knows that there have been at least consecutve packet ropouts at the -th sensor noe snce the tme recevng last,. Note that the estmator just etects at least consecutve packet ropouts, not precse consecutve packet ropouts because there exsts a ela nterval n etectng packet ropouts. As llustrate n Fgure, although packet loss happens wthn the tme range ( t, t + ), the last, t, last, last, t, estmator onl etects t at a tme ( tlast, + t, ). Thus, f there s more than one packet ropout wthn the tme range ( tlast,, tlast, + t, ), the estmator also etects onl one packet ropout at tme ( t + ). Ths s an nevtable flaw of the mofe SOD samplng scheme. We can constrant ths
Sensors 009, 9 3081 flaw b reucng the t, value, but sensor ata transmsson rate wll be ncrease. Therefore, an optmal t, value s necessar to compromse between the two constrants. Fgure. Multple packet ropout etecton. ( t) t t t t t t t t 3. State Estmaton wth Mofe SOD Transmsson Metho The networke estmaton problem applng mofe SOD transmsson metho can be escrbe as follows: 1. Measurement output ( 1 p ) are sample at the pero T but ther ata are onl sent to the estmator noe when (a) or (b) s satsfe.. For smplct n the problem formulaton, transmsson ela from the sensor noes to the estmator noe s gnore. 3. The estmator noe estmates states of the plant regularl at the pero T regarless of whether or not sensor ata arrve. If there s no -th sensor ata receve for ( t - t ) >, the last, t, estmator noe consers that the measurement value of the -th sensor output ( t ) s stll equal to last, but the measurement nose ncreases from v( t ) to vn, ( t) = v ( t) + D ( t, tlast, ). Note that f = 0 then there s no packet ropout, the estmator acts lke a conventonal SOD flter [5]. To formulate a state estmaton problem, the bounr of D ( t, tlast, ) nees to be etermne as ¹ 0 (packet ropouts happen). In the next secton, we wll compute the covarance of vn, ( t ) when ¹ 0 an then a mofe Kalman flter s apple for state estmaton. 3.1. Measurement nose ncrease ue to multple packet ropouts We know from (a) that ( t) - last,, as long as the estmator noe oes not receve a new -th sensor ata value. If one packet ropout happens, the -th sensor output value has change more than,. The estmator shoul know that: ( t) - + last,,,
Sensors 009, 9 308 For general cases, as shown n Fgure 3, f there are consecutve packet ropouts then: D ( t, t ) = ( t) - ( + 1). (3) last, last,, Note that (3) s also apple to the case of no packet ropout [5] b lettng = 0. Assumng that D ( t, tlast, ) has a unform strbuton wth (3), varance of D ( t, tlast, ) wll be: E é ( t, tlast, ) ù ë D û = 0 ( ) E é ( t, t, ) ù ë D last û = ( + 1), / 3 Var éd ( t, tlast, ) ù = E éd ( t, tlast, ) ù - E éd( t, tlast, ) ù ë û ë û ë û (, ) = ( + 1) / 3 Therefore, f there s no -th sensor ata receve for t > t last,, varance of measurement nose s ncrease from R(, ) to R(, ) + (( + 1) ) / 3., Fgure 3. Measurement nose ncrease ue to multple packet ropouts. (4) ( t) consecutve packet ropouts, ( t) - < ( + 1) last,,, last, tlast, t t 3.. State estmaton A mofe Kalman flter for state estmaton x ˆk at step k, where there s a change n the measurement upate part of the screte Kalman flter algorthm [15], s gven as n the Fgure 4. We use the scretze sstem moel sample at pero T : where AT Ar A = e, B = ò e Br, Q s the process nose covarance of the scretze sstem: T 0 T Ar A r Q = ò e Qe r, an last s the vector of p last receve sensor values: last = é last,1 last,... ù ê last, p ë ú. û 0
Sensors 009, 9 3083 Fgure 4. Structure of the mofe Kalman flter. Set xˆ, P last - - 0 0 = Cxˆ - 0 R (, ) = R(, ) k last, = ( kt) Solve : ( t - t ) > last, t, R (, ) = R(, ) + (( + 1) ) / 3 k, z k = last ( ) - - -1 k = k k + k K P C CP C R - - xˆ = xˆ + K ( z -Cxˆ ) k k k k k P = ( I - K C ) P - k k k xˆ, xˆ,... 0 1 xˆ = A xˆ + B u - k + 1 k k P = A P A + Q - k + 1 k In the mofe Kalman flter n Fgure 4, the states of the plant are estmate regularl at ever pero T, regarless of whether or not sensor ata arrve. If -th sensor ata arrve then D ( t, tlast, ) = 0, the mofe Kalman flter acts lke the conventonal Kalman flter. Otherwse, f - th sensor ata o not arrve ue to packet loss, t uses last, as the measurement value an R(, ) = R(, ) + (( 1) ) +, / 3 as measurement nose covarance for state estmaton. As state n [8], f the sstem (1) s unstable an a packet loss rate s hgh, the propose flter coul verge. For example, f all packets are lost, wll ncrease an thus R wll become nfnte. Thus P n Fgure 4 coul become nfnte. 4. Optmal δ t, Computng Problem As mentone n Secton 3, δ t, s a trae-off parameter between sensor ata transmsson rate an the response of packet ropouts etecton. The response of packet ropout etecton guarantees estmaton performance. Because SOD samplng s more effcent than the tme-trggere one n network banwth mprovement, we shoul choose δ t, such that sensor ata transmsson rate s reuce to promote ablt of SOD samplng. In the next secton, we wll nvestgate the relaton of δ t,
Sensors 009, 9 3084 wth transmsson rate an the effect of δ t, on estmaton performance. Then an optmzaton problem s formulate to fn the optmal δ t, value accorng to the gven estmaton performance. 4.1. Sensor ata transmsson rate b conton (b) The total sensor ata transmsson rate cause b conton (b) n a tme unt: where p s the number of sensor output p 1 f ( t, ) @ å (5) = 1 t, 4.. Estmaton error covarance ue to packet ropouts Let x ( 0 x < 1) be the packet loss rate at the -th sensor noe, x = 0 correspons to no packet loss. Let D T be the average transmttng tme per packet of the -th sensor noe n the conventonal SOD metho. Note that D T s epenent on the gven, value, but nepenent on δ t, value. D T s compute b runnng the smulaton moel n analss. In practce, t can be compute b lettng = an montorng the number of packets n a tme unt. t, The average number of packet ropouts n the conventonal SOD samplng per a tme unt: wll be: x @ (6) DT In the propose SOD samplng, the average number of packet ropouts wthn the tme nterval t, t, x (7) = D T We know from Secton 4.1 that the larger number of consecutve packet ropouts s, the larger measurement nose covarance s. Measurement nose covarance s largest f packets are consecutvel lost. Followng the ea n (4), f there s covarance shoul be ncrease as follows: 4.3. Optmal t, computaton ( ) R = R + ( + 1) / 3 (, ) (, ), æ t, x ö (, ) ç, çè DT ø = R + + 1 / 3 packet loss, the measurement nose (8) In ths secton, δ t, value s compute. Usng (8), we assume that the measurement nose covarance s gven b: æ æ t,1x1 ö æ t, pxp ö ö R = R + Dag 1,1 / 3,..., 1, p / 3 çç ç + T + 1 T (9) çè è D ø èç D p ø ø
Sensors 009, 9 3085 The estmaton performance n ths case can be compute from the followng screte algebrac Rccat equaton: - ( ) 1 P = A PA + Q - A PC CPC + R CPA (10) Note that (10) oes not prove the actual estmaton error covarance of the flter. The man purpose of (10) s to evaluate how δ t, affects the estmaton performance. We can see that f δ t, s large, the estmaton error covarance P ncreases. The soluton of (10) s enote b P ( ). In the followng optmzaton algorthm to fn δ, we tr t, to reuce the sensor transmsson rate cause b conton (b) subject to the gven estmaton performance constrant: δ t Optmzaton Problem mn f ( ) t, t, subject to DagP ( ) mp t, 0 where P 0 s the upper boun error covarance wth gven value, an no packet ropout (soluton of (10) as = Dag(0,..., 0) ). P 0 s also the estmaton performance of the conventonal SOD. m s the rato to the estmaton performance of conventonal SOD flter n case of no packet ropout. If m s large, the t, optmzaton problem (11) s one wth weaker estmaton performance constrants. (11) 5. Smulaton To verf the propose flter, we conser an example of the secon-orer sstem wth step nput where the output s sample b the SOD an mofe SOD samplng: é 0 1 ù é 0 ù x& ( t) = x( t) + u( t) + w( t) -1/ a -b / a M / a ëê ûú ëê ûú ( t) = é1 0 ù ê x( t) + v( t) ë úû Q = 0.01, R = 0.01, T = 10ms where the sstem parameters for performance evaluaton are gven b M = 30, a = 5, b = 1 (unerampe sstem). The smulaton process s mplemente for 50 secons. Choose m = 5 for the optmzaton problem (11). The soluton t,1, t, of (11) along wth, an x are shown n Fgures 5 an 6, respectvel. We see that t, s proportonal to, an reversel proportonal to x. It means that when, s large, the -th sensor ata transmsson rate s small, thus t, s also small to keep the overall transmsson rate small. But f packet ropouts ncrease ( x s large), t, value s lowere. As the result, the overall sensor ata transmsson rate s ncrease to guarantee estmaton performance.
Sensors 009, 9 3086 Fgure 5. t,1 of (11) along wth,1 an x 1. Fgure 6. t, of (11) along wth, an x. Table 1. Estmaton error along wth packet loss rate n two flters. Packet loss rate x = x 1 n (SOD) t, n (mofe SOD) e (SOD) e (mofe SOD) 0.05(5%) 0.1(10%) 0.15(15%) 0.(0%) t,1 = 4.1 t, = 4.69 n 1 = 101 n = 36 e 1 = 0.0383 e = 0.0167 e 1 = 0.0075 e = 0.0096 n 1 = 95 n = 31 t,1 =.08 t,1 = 1.73 t, =.31 t, = 1.91 n 1 = 109 n 1 = 11 n = 44 n = 47 e 1 = 0.0384 e 1 = 0.0386 e = 0.0168 e = 0.0169 e 1 = 0.0064 e 1 = 0.0039 e = 0.0089 e = 0.008 t,1 = 1.5 t, = 1.66 n 1 = 115 n = 50 e 1 = 0.0391 e = 0.017 e 1 = 0.000 e = 0.0069 Table 1 shows the estmaton error n two flters (SOD flter an mofe SOD flter) as = =, m = 5 an x1, x are varng 5%, 10%, 15%, 0%. Estmaton error s evaluate,1, 0.5 b:
Sensors 009, 9 3087 N 1 e ( ) = å xk, - xˆ k, (1) N k = 1 where x s the reference state, x ˆ s the estmate state, an N = 5,000. In Table 1, we see that when applng the mofe SOD flter, the estmaton error s sgnfcantl mprove. For nstance, n the case x1 = x = 0.05, the total number of sensor ata transmssons n the mofe SOD (# 137) s just slghtl greater than that n conventonal SOD (# 16) but the estmaton error s reuce so much ((e 1 = 0.0075, e = 0.0096) compare to (e 1 = 0.0383, e = 0.0167)). Fgure 7. Estmaton error n two flters as x1 = x = 0.05. Fgure 8. Instants the sensor noe transmts ata ue to conton (b). Fgure 7 ntutvel shows the estmaton error n two flters as x1 = x = 0.05,,1 =, = 0.5, t,1 = 4.1, t, = 4.69. The bounr of e 1 n the mofe SOD flter (SODa) s much smaller than that n the conventonal SOD flter. Fgure 8 shows the nstants the sensor noe transmts ata to the
Sensors 009, 9 3088 estmator noe ue to conton (b). We see that the number of sensor ata transmssons cause b conton (b) s ver small n comparson wth the total number of sensor ata transmssons [(n 1 = 7, n = 7) compare to (n 1 = 101, n = 36)]. When the mofe SOD samplng s apple, the total number of sensor ata transmssons s slghtl ncrease, but the estmaton error s sgnfcantl reuce. Therefore, the mofe SOD samplng sgnfcantl mproves estmaton performance wth onl a lttle ncrease n the ata transmsson rate. Notce that f we just conser the transmsson conton (Equaton a), estmaton error of the propose metho s worse for sstems that the output vares slowl. However, an ssue of conventonal event-base samplng s that t can not etect sgnal oscllatons or stea-state error f the fference of output value remans wthn the threshol range (because the output vares slowl). Ths fact causes estmaton error to be ncrease. Whereas, the propose metho uses the transmsson conton (Equaton b) not onl to etect packet ropouts but to reuce the error n case the output changes slowl. As llustrate n Fgures 7 an 8, where the estmaton error of the propose metho (top-rght graph of Fgure 7) an of the conventonal metho (top-left graph of Fgure 7) are shown accorng to the output 1 (top-left graph of Fgure 8). We see when 1 vares slowl (tme nterval from 0s to 50s), the propose metho gves much smaller estmaton error than the conventonal one. In case the output changes fast, t s obvous that gnorng packet ropout wll ntrouce extremel ncorrect result because we stll use the wrong ol measurement nose value even when we o not know how much the output value changes. 5. Conclusons In ths paper, the state estmaton problem wth mofe SOD transmsson metho over networks, n whch an event-base samplng s combne wth a tme-trggere samplng to etect packet loss stuatons, has been consere. We have shown that when usng the propose mofe SOD flter, estmaton performance s sgnfcantl mprove wth a small ncrease n sensor ata transmsson. If multple packet ropouts happen, the estmator noe wll etect an compensate for them wth an amount of atve measurement nose to mprove estmaton performance. Ths metho s ver useful for networks where ata transmsson s unrelable ue to nose. Acknowlegements Ths work was supporte b the Korea Research Founaton Grant D00059 ( I00048 ). The secon author woul lke to thank Mnstr of Knowlege Econom an Ulsan Metropoltan Ct whch supporte ths research through the Network-base Automaton Research Center (NARC) at the Unverst of Ulsan
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