Cooperative Parameter Identification of Advection-diffusion Processes Using a Mobile Sensor Network

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1 207 Ameran Control Conferene Sheraton Seattle Hotel May 2 26, 207, Seattle, USA Cooperatve Parameter Identfaton of Adveton-dffuson Proesses Usng a Moble Sensor Networ Je You, Yufe Zhang, Mnghen L, Kun Su, Fumn Zhang, and Wenen Wu Abstrat Onlne parameter dentfaton of advetondffuson proesses s performed usng a moble sensor networ. A onstraned ooperatve Kalman flter s developed to provde estmates of the feld values and gradents along the trajetores of the moble sensor networ so that the temporal varatons of the feld values an be estmated. Utlzng the state estmates from the onstraned ooperatve Kalman flter, a reursve least square (RLS) algorthm s desgned to estmate the unnown parameters of the adveton-dffuson proess. We provde bas analyss of the RLS n the paper. In addton to valdatng the proposed algorthm n smulated adveton-dffuson felds, we buld a ontrollable CO 2 adveton-dffuson feld n a lab and desgn a sensor grd that ollets the feld onentraton over tme to allow the valdaton of the proposed algorthm n the CO 2 feld. Expermental results demonstrate robustness of the algorthm under realst unertantes and dsturbanes. I. INTRODUCTION Many omplated spato-temporal proesses have been observed n dverse felds nludng physal, hemal, and bologal systems []. These spato-temporal proesses are often vewed as dstrbuted parameter systems (DPSs), whh are mathematally desrbed by partal dfferental equatons (PDEs) n model-based shemes. In many pratal problems, the parameters of PDEs suh as the dffuson oeffent may be unnown or naurate. Therefore, to better understand the proesses, there s a need to use parameter dentfaton methods to refne, update, or estmate these unnown parameters [2], [3]. On the other hand, the proedure of parameter dentfaton wll also provde mportant nsghts nto the analyss, desgn, and ontrol of DPSs under study. Many dentfed models have been used n applatons []. Varous aspets of parameter dentfaton of DPSs have been nvestgated n [5] [7] and referenes theren. The dentfaton of PDEs from dsrete samples an be done at least n two ways: ndret method [7] and dret method [2], [8]. A typal ndret method s based on a wea formulaton usng a Galern-le fnte element proedure [7]. The most attratve feature of the ndret method s ts flexblty to deal wth PDEs wth arbtrary ntal ondtons and omplex geometr boundares. In a dret method, the spato-temporal varables are usually dsretzed wth respet to both tme and spae. Dervatves of the funtons at eah dsretzaton node have to be approxmated usng some standard fnte The researh wor s supported by NSF grant CNS-66. Je You, Yufe Zhang, Mnghen L, Kun Su, and Wenen Wu are wth the Department of Eletral, Computer, and Systems Engneerng, Rensselaer Polytehn Insttute, Troy, NY , USA. Fumn Zhang s wth the Shool of Eletral and Computer Engneerng, Georga Insttute of Tehnology, Atlanta, GA 3038, USA youj@rp.edu, zhangy7@rp.edu,lm@rp.edu,su2@rp.edu, fumn@gateh.edu,wuw8@rp.edu dfferene approxmatons suh as fnte dfferene and fnte volume method [2], [8]. The dret method an be readly used n all nds of PDEs and mantan a straghtforward ln to the physal propertes of the orgnal DPS system. Many of these studes requre large numbers of stat sensors to ollet data n the whole doman. Due to the lmted number of atuators and sensors n pratal sensng, n a very large and omplex feld, preferable to employ moble sensor networs (MSNs), whh onsst of groups of robot agents wth omputatonal, ommunaton, sensng, and loomotve apabltes [9] [], to perform parameter dentfaton. Although there exsome ontrbutons on the ssue of parameter dentfaton of PDEs usng moble sensor networs [3], [2], [3], most of these studes are based on an offlne sheme and requre hgh omputatonal loads wth few exeptons that nvestgate the onlne parameter dentfaton [2], [], [5]. There are a number of dffultes nherent n the onlne parameter dentfaton of PDEs. Frst, a hallengng nverse problem, whh requres the dentfaton of system parameters from olleted fnte-dmensonal measurements. Seond, onlne parameter dentfaton usng a moble sensor networ requres a ombnaton of ooperatve ontrol and ooperatve sensng. In our prevous wor [2], we desgned a ooperatve flterng sheme for onlne parameter estmaton of dffuson proesses usng four sensng agents arranged n a symmetr formaton. The sheme onssts of two parts: a ooperatve Kalman flter and a reursve leasquare (RLS) estmator. We proved the onvergene of the ooperatve Kalman flter and valdated the algorthm n smulatons. In ths paper, we nvestgate onlne parameter dentfaton for 2D advetondffuson proesses. By usng the fnte volume method, we extend the ooperatve flterng sheme [2] to the ase wth N agents n an arbtrary formaton to allow flexblty n pratal senaros. Utlzng the state estmates from the ooperatve flterng sheme, a RLS algorthm s desgned to estmate the unnown model parameters of the advetondffuson proess. We provde neessary bas analyss of the proposed method. Addtonally, we buld a ontrollable CO 2 adveton-dffuson feld n a lab and desgn a sensor grd that ollets the feld onentraton over tme to allow the valdaton of the proposed algorthm n the CO 2 feld. Expermental results show satsfatory performane. The problem s formulated n Seton II. Seton III presents the fnte volume approxmaton model and Seton IV shows the ooperatve Kalman flterng. Seton V llustrates the RLS and bas analyss of the proposed method. Experments results are presented n Seton VI and /$ AACC 3230

2 onlusons follow n Seton VII. III. THE FINITE VOLUME APPROXIMATION II. PROBLEM FORMULATION In ths seton, we formulate the problem of onlne parameter estmaton of adveton-dffuson proesses usng moble sensor networs. A. The model We assume thaystem dynams s desrbed by the followng two-dmensonal (2D) adveton-dffuson proess defned on a doman W =[0,L x ] [0,L y ] 2 R 2 : z(r,t) = q 2 z(r,t)+v T z(r,t), r 2 W, () where z(r,t) s the onentraton funton, q > 0 s a onstant dffuson oeffent, represents the gradent operator, 2 represents the Laplaan operator, and v s a onstant vetor representng the flow veloty, whh s supposed to be nown through measurements. The ntal and boundary ondtons for Equaton () are assumed as z(r,0) =z 0 (r), and z(r,t)=z b (r,t), r 2 W, where z 0 (r) and z b (r,t) are the arbtrary ntal ondton and Drhlet boundary ondton, respetvely. Many natural proesses an be desrbed by the adveton-dffuson equaton (). In many senaros, q s unnown or naurate, whh requres dentfaton. B. Sensor dynams Consder a formaton of N oordnated sensng agents movng n the feld, eah of whh arres a sensor that taes pont measurements of the feld z(r, t). We onsder the sensng agents wth sngle-ntegrator dynams gven by ṙ (t)=u (t), =,2,...,N, where r (t) and u (t) R 2 are the poston and the veloty of the th agent, respetvely. In most applatons, the sensor measurements are taen dsretely over tme. Let the moment when new measurements are avalable be t, where s an nteger ndex. Denote the poston of the th agent at the moment t be r and the feld value at r be z(r,). The measurement of the th agent an be modeled as p(r,)=z(r,)+n, (2) where n s assumed to be..d. Gaussan nose. We have the followng assumpton for the sensng agents. Assumpton II. Eah agent an measure ts poston r and onentraton value z(r,), and share these nformaton wth other agents. The problem s formulated as: ) Under Assumpton II., develop a ooperatve flterng sheme that estmate the states z(r,t), z(r,t), 2 z(r,t), and z(r,t) based on the olleted measurements n Equaton (2) usng a moble sensor networ movng n the adveton-dffuson feld. 2) Utlzng the estmated state, develop an onlne parameter dentfaton algorthm that estmates the unnown onstant dffuson oeffent q of the adveton-dffuson equaton (). Under Assumpton II., the proposed parameter dentfaton algorthm s based on the dsrete measurements taen by moble agents over tme. In the followng, we wll frst buld a fnte volume approxmaton model of Equaton (). Suppose the urrent tme step s. Let r =[r,x,r,y] T be the enter of the formaton at the moment t,.e., r = N N = r. We dsretze the adveton-dffuson PDE () at the formaton enter r as, z(r, + ) z(r,) v T z(r t,)=q 2 z(r,), (3) s where s the samplng perod. If we an get the estmates of z(r, + ), z(r,), z(r,), and 2 z(r,), then q an be estmated usng RLS based on the sem-dsrete model (3). To obtan these state estmates, n Seton IV, we wll develop a onstraned ooperatve Kalman flter to estmate z(r, + ), z(r,), and z(r,) along the movng trajetory of a moble sensor networ. On the other hand, the Laplaan term 2 z(r,) also requres to be estmated smultaneously. One smple and straghtforward way s to use fnte dfferene method to approxmate 2 z(r,)= = z(r,) z(r,) Dr, where Dr s the spatal nterval [2]. Unfortunately, ths method only wors for the ase when four agents are arranged n a symmetr formaton, whh lmts ts apablty for usage n atual applatons. In the followng seton, we wll employ a fnte volume method to allow the estmaton of 2 z(r,) wth N agents n an arbtrary formaton. We summarze the bas proedures followng the fnte volume method [8]. We frst denote the ells of agents as C,C 2,...C N, and the orrespondng ell-enters as r,r 2,... r N. We further denote the ells of the formaton enter r as C. 323 The volume of the formaton enter ell C s denoted as, whh s a fnte volume. Let the surfae area of be S = S ˆn, where ˆn s the outward unt vetor. To llustrate the dea and for notaton onvenene, we onsder the ase when N = n the followng dervatons. But our sheme an be straghtforwardly extended to the ase when N, whh wll be spefed n Remar III.. As llustrated n Fg., we arrange four agents n an arbtrary formaton. In ths ase, the surfae area S s the quadrlateral ABCD and s the volume of ABCD. The orrespondng outward unt vetor ˆn for the edge AB s ˆn AB = r r, whh means AB?r r. In a smlar way, we have BC?r2 r, CD?r3 r, and DA?r r. Fg.. Fnte-volume onstruton for a moble sensor networ n 2D.

3 By ntegratng Equaton () over a fnte volume, we an have the followng expresson: Z Z I z(r,t) vt z(r,t) d + F ˆndS = 0, () where F = q z(r,t) s obtaned by applyng the Green s theorem [6]. The ntegraton () over a ell area ABCD shown n Fg. results n the sem-dsrete equaton as follows: z(r,t) v T z(r,t)= (F AB S AB + F BC S BC f aes + F CD S CD + F DA S DA ), (5) where F AB S AB s the ontnuous flux on the edge AB, whh s expressed as dffusve terms as follows, Z F AB S AB = q z(r,t) ˆn AB dl, (6) AB where ˆn AB s the unt outer normal on the edge AB. The flux terms wth respet to the other edges F BC S BC, F CD S CD, and F DA S DA have the smlar defntons. Next, we wll derve z(r,t), r 2 AB n Equaton () at tme step. Wth r beng lose to r, z(r,) an be loally approxmated by usng the Taylor expanson as, z(r,) z(r,) (r r ) T z(r,) (7) + Z 0 (H r H r )x dx, r 2 AB, where H r =(r r) T H x r +( x )r, (r r) wth H x r +( x )r, beng the Hessan matrx at the pont x r +( x )r, r 2 AB. The other notatons H r2, H r3, H r, and H r have the smlar defnton. Hene, by reorganzng Equaton (7), we an obtan the gradent term z(r,) for the edge r 2 AB. Substtutng the expresson of z(r,) nto R AB q z(r,) ˆn AB dl gves, Z q z(r,) ˆn AB dl = q A B r r z(r,) z(r,) Z Z q r r (H r H AB 0 r )x dx dl. (8) AB q R R Denote E AB = r r AB 0 (H r H r )x dx dl. Sne E AB s the ntegraton of the dfferenes of two Hessan matres, t s obvous that E AB = O(h 2 ), whh s the hgher order term of the grd sze h = sup dam(c )/2. Here, dam(c ) s the =N dameter of ell C. By substtutng (8) nto (6), we an obtan F AB S AB = q A B r r z(r,) z(r,) + E AB. (9) In a smlar way, we an obtan the normal flux on B C the other sdes, F BC S BC = q z(r2,) z(r,) + E BC, F CD S CD = q C D r3 r r 2 r z(r 3,) S z(r,) + E CD, and D A F DA S DA = q z(r r r,) z(r,) +E DA. Then we an rewrte Equaton (5) as follows, z(r, + ) z(r,) v T z(r t,)=q [a AB z(r s W,)+a BC z(r2,)+a CD z(r3,)+a DA z(r,)+a enter z(r,)] + e(r,), (0) where the a oeffents are as follows: a AB = A B r r,a BC = B C r2 r, a CD = C D r 3 r,a DA = D A r r, () a enter = A B r r B C r2 r C D r3 r D A r r, and e(r,)= W (E AB + E BC + E CD + E DA ) s the approxmaton error from omttng hgher order terms usng fntevolume method. Ths then allows us to assume that the modelng error e(r,) s an ndependent nose sequene wth zero mean and fnte varane. For notaton smplfaton, let us denote = a s z(r,)=a AB z(r,)+a BC z(r2,)+a CD z(r3,)+a DA z(r,). Then a fnte volume approxmaton of the advetondffuson PDE () an be wrtten as follows: z(r, + ) z(r,) v T z(r t,)=q [ s = a s z(r,) + a enter z(r,)] + e(r,). (2) We an observe that Equaton (2) s also a dsretzed verson of Equaton () wth a replaement of 2 z(r,) wth [ = a s z(r,)+a enter z(r,)] n Equaton (3). Ihould be noted that must obey the nequaltes apple q v 2 and apple q h2 for the dsretzaton method to onverge [8]. Remar III. Even though we only onsder four agents n the above dervaton, the fnte volume approxmaton of the adveton-dffuson model () an be readly extended to the ase when N followng the mplementaton of the standard fnte-volume method outlned n [8]. IV. COOPERATIVE FILTERING FOR PARAMETER IDENTIFICATION In ths seton, we show how to desgn a ooperatve flterng sheme to sequentally estmate the states z(r, + ), z(r,), z(r,) and = z(r,) over tme. A. Informaton dynams for the ooperatve Kalman flter We frst ntrodue the motvaton of desgnng a ooperatve flter by pontng out the dfferene between z(r, + ) and z(r,) n Equaton (2). By desgnng a ooperatve flter smlar to the one developed n [7], z(r,) may be dretly estmated by ombnng the measurements taen by the sensng agents at tme step. However, at tme step +, the formaton enter of the group s at poston r +. Therefore, the ooperatve flter desgn n [7] an only provde the estmate of z(r +,+), not z(r,+). In order 3232

4 to estmate the temporal varatons of the feld value along the trajetory, we frst need to derve a ooperatve flter to estmate both z(r,) and z(r, + ). To onstrut a ooperatve Kalman flter to obtan the estmates of z(r,) and z(r, + ), we frst analyze the dynams of the adveton-dffuson feld value along the trajetory of the formaton enter r aordng to ż(r,t)= z(r,t) dr r dt + z(r,t) = z(r,t) ṙ + z(r,t), (3) where z(r,t) s the gradent of z(r,t). To dsretze Equaton (3), the fnte dfferenes of eah term of (3) at tme t = t and at poston r = r gve: ż(r,t) t=t,r =r z(r,t) ṙ t=t,r =r z(r,) z(r, ), () (r r ) T z(r, ). Substtutng Equaton () and the fnte volume equaton (2) nto Equaton (3) gves the nformaton dynams of z(r,) as! z(r,)= + a enter ˆq z(r, ) ˆq = a s (5) z(r, )+(r r + v ) T z(r, )+w(r,), where ˆq s the estmate of q, whh an be obtaned from the RLS method that wll be ntrodued n Seton V. w(r,) s the error term, whh aounts for postonng errors, estmaton errors for the Hessan matrx, and errors aused by hgher-order terms omtted from the fnte volume sheme. Smlarly, we also obtan the dynams of z(r, + ) by dsretzng Equaton (3) at t = t and r = r.! z(r, + )= + a enter ˆq z(r,) ˆq = a s z(r,)+(r r + v ) T z(r,)+w(r,). (6) Furthermore, we are also nterested n estmatng z(r,t) sne the gradent estmate s not only neessary for the RLS method, but also used n the moton ontrol that wll be ntrodued n Seton IV-D. We derve the total tme dervatve of z(r,t) as z(r,t)=h(r,t) ṙ + z(r,t), (7) where H(r,t) s the Hessan matrx, and z(r,t) s the hgher order term, whh an be onsdered as nose. By dsetzng Equaton (7) at t = t,r = r and t = t,r = r, respetvely, we an get that z(r,) and z(r, + ) evolve aordng to the followng equatons: z(r,)= z(r, )+H(r, )(r r ), z(r, + )= z(r,)+h(r,)(r r ). (8) Defne the nformaton state as X( + ) = [z(r,), z(r,),z(r, + ), z(r, + )] T. By ombng (5), (6), and (8), the nformaton state evolves aordng to the followng equaton: X( + )=A ˆq ()X()+U()+w(), (9) where w() =[w(r, ),0,w(r,),0] T represents the model error terms n Equaton (5) and (6). We denote the ovarane matrx of w() as E[w()w() T ]=W. The matres A ˆq () and U() are defned by 2 A ˆq ()= + a enter ˆq (r r + v ) T 0 I a enter ˆq (r r + v ) T 5. (20) 0 I ˆq = a s z(r, ) H(r U()= 6, )(r r ) ˆq = a s z(r,) 7 5, (2) H(r,)(r r ) where H(r,) s the Hessan matrx. We observe that U() s determned by the values of z(r, ), z(r,), and the Hessan matrx, whh wll be spefed n Seton IV-C. A measurement equaton s also requred for the ooperatve Kalman flter. By applyng formaton ontrol, r and r an be ontrolled to be lose to r. Therefore, the onentraton an be loally approxmated by a Taylor seres up to seond order as z(r, ) z(r, )+(r r ) T z(r, ) + 2 (r r ) T H(r, )(r r ), z(r,) z(r,)+(r r ) T z(r,) (22) + 2 (r r ) T H(r,)(r r ). Let P() = [p(r, ) p(rn, ) p(r,) p(r N,)]T. By ombng Equaton (2) and Equaton (22), the measurement equaton an be modelled n a vetor form as, P() =C() X() +D()Ĥ() +D()e() +n(), (23) where Ĥ()=[Ĥ(r, ) Ĥ(r,)] T s a olumn vetor obtaned by rearrangng elements of the estmate of Hessan terms, e() represents the error n the estmaton of the Hessan matres, n() s the Gaussan measurement n n a vetor form, E[e()e() T ]=Q, E[n()n() T ]=R. D() s a matrx wth ts frst N rows defned by [ 2 ((r r ) N (r r )) T 0] and last N rows defned by [0 2 ((r r ) N (r r )) T ], where =,2,,N and N s the Kroneer produt. C() s a matrx wth ts frst N rows defned by [ (r r ) T 00] and last N rows defned by [0 0(r r ) T ] for =,2,,N. 3233

5 we use subsrpt ( ) to ndate predtons and (+) to ndate updated estmates. If we assume the number of sensor N and the formaton s not o-lnear, we have P() =C() ˆX () +D()Ĥ(). The Hessan estmate an be solved by usng the least mean square method, Ĥ() = D() T D() D() T P() C() ˆX (). Fg. 2. Blo dagram of the relatonshp between z(r, +) and z(r,). B. The PDE state onstrant We observe that the matres A ˆq (), C(), and D() are blo dagonal matres. That means the nformaton dynams (9) was obtaned here as a dret ombnaton of the semdsrete ODE (6) for the state z(r, + ) and ODE (5) for the state z(r,). As a matter of fat, the state z(r, + ) and z(r,) are the future and present state estmates at a gven poston r = r, whh are dsretzed terms of z(r,t) n Equaton (2). Hene, there s a PDE onstrant between the state z(r, + ) and z(r,) at eah step, whh s shown n Fg. 2. By rewrtng Equaton (2), we an obtan the PDE onstrant: z(r, + )+( a enter ˆq = ˆq = )z(r,)+v T z(r,) a s z(r,) (2) The state equalty onstrant an be rewrtten as follow: G() X()=d(), (25) where G() =[( a enter ˆq ) 0 v T ] and d() = ˆq = a s z(r, ). We observe that the proposed ooperatve Kalman flter s based on the tme-varyng nformaton dynams (9) wth the state equalty onstrant (25). Ths type of flter has been prevously nvestgated n [8]. By followng anonal proedures n [8], the equatons for the ooperatve Kalman flter wth state equalty onstrants an be obtaned. Detals of the ooperatve Kalman flter an be found n [2]. C. Cooperatve estmaton of the Hessan Estmates of z(r,), z(r, ), and the Hessan Ĥ() n the matrx U() (2) are needed to enable the ooperatve Kalman flter. ) Estmates of z(r,) and z(r, ): Sne the sensor measurements p(r,) and p(r, ) are avalable n the measurement vetor P(), one straghtforward and smple way s to replae z(r,) and z(r, ) wth the sensor measurements p(r,) and p(r, ), whh s adopted n ths paper. 2) Cooperatve estmaton of the Hessan: By tme step, we have obtaned an estmate of ˆX + ( ) from the ooperatve Kalman flter. Usng the omputed estmates ˆX + ( ) and U( ), before the arrval of measurements at tme step, we an obtan a predton for X() as ˆX () =A ˆq ( ) ˆX + ( )+U( ). Here, D. Formaton and moton ontrol Control laws for the velotes of the agents are requred so that the moble sensor networ an move along a ertan trajetory whle mantanng a desred formaton. Thus, the fnte volume, as well as the oeffents a AB, a BC, a CD, a DA, and a enter an be onsdered as onstants. We vew the entre formaton as a deformable body. Thus, there are two parts of ontrol: moton ontrol and formaton ontrol. Wth the gradent estmates provded by the ooperatve Kalman flter, the moton ontrol for the agents an be easly realzed by settng the velotes of the agents to be algned wth the estmated gradent dreton. Thus, the moble sensor networ an aheve smultaneously parameter estmaton and gradent lmbng. Furthermore, there exsts several results about the formaton ontrol for moble agents [7] [9]. We omt the detaled desgn of ontrol here due to spae lmtaton. Interested readers an refer to [7] [9]. V. RECURSIVE LEAST SQUARE ESTIMATION A. The RLS method In ths seton, we use the RLS method to teratvely update the estmate of q based on the dsretzed model (2). We do ths usng the nformaton state ˆX( + ) = [ẑ(r,), ẑ(r,),ẑ(r, + ), ẑ(r, + )] T obtaned from the ooperatve Kalman flter to alulate the temporal varatons of the feld value ẑ(r,+) ẑ(r,). By ombng the left terms of Equaton (2), we denote the term Ŷ (r,) as Ŷ (r,) = ẑ(r,+) ẑ(r,) v T ẑ(r,), where the hat notaton ndates that Ŷ (r,) s the estmate from the ooperatve Kalman flter. Note that the tme ndex s the same as the ndex n the ooperatve Kalman flter. Sne the onvergene of the ooperatve Kalman flter has already been proved n [2], we an have Ŷ (r,)=y (r,)+v(r,), (26) where Ŷ (r,) s the estmate of Y (r,), all elements of whh ome from the ooperatve Kalman flter and V(r,) s a Gaussan nose wth zero mean and bounded ovarane. Wth the ombnaton of Equaton (2) and (26), the fnte volume approxmaton model an be represented as Ŷ (r,)=q [ = a s z(r,)+a enter z(r,)] + V(r,)+e(r,)=Pẑq + h(), (27) where h() = V(r,) + e(r,) + (P z Pẑ)q, P z = [ = a s z(r,) + a enter z(r,)], and Pẑ = W [ = a s p(r,) +a enter ẑ(r,)]. The RLS parameter dentfaton s based on mnmzng the mean 323

6 squared error rteron J = E[h()2 ], where E[.] denotes the expetaton value. Therefore, based on the ooperatve flterng sheme, the dffuson oeffent an be dretly estmated wthout the need of numerally solvng the dffuson equaton. Gven an ntal estmate for the dffuson oeffent, a smple applaton of the RLS method an teratvely update the estmate of q. Followng the anonal proedure of RLS estmaton outlned n [20], we derve the followng equatons to update the estmate q. q = q + g() Y (r, ) Pz q ; (28) ; g() = h( )Pz T Pz h( )Pz T + Re h() = (I g() Pz ) h( ), where g() s the estmator gan matrx, h() s the estmaton error ovarane matrx, and Re s the nose ovarane. In the above framewor, we an observe that the proposed reursve ooperatve flterng sheme s based on two subsystems: ooperatve Kalman flterng subsystem (Equatons (9) and (23)) and RLS subsystem n (28). In the ooperatve Kalman flterng subsystem, assume that the parameter q s nown, we run the ooperatve Kalman flter to estmate the states based on the olleted measurements. In the RLS subsystem, assume that the estmated states an tra the true values, we employ the RLS method to teratvely update the estmate of q. Ihould be noted that the onvergene of the proposed losed loop reursve sheme heavly depends on the property that the onvergene of the Kalman flter s ndependent of the estmated parameter q, whh s used n the Kalman flter n Equaton (9). In other words, the estmated states from the ooperatve Kalman flterng an suessfully tra the true values even though the estmated parameter q s based. Ths part of onvergene proof has been publshed n our prevous wor [2]. VI. E XPERIMENT In ths seton, we ntrodue the desgn of a ontrollable CO2 dffuson feld n our lab. By valdatng the proposed algorthm n ths real feld, we demonstrate that the algorthm s robust under realst unertantes and dsturbanes. A. Generatng and vsualzng a dffuson Feld A referene CO2 dffuson feld s produed n our lab n an area of m2. When experment begns, 5 CFH ( f t 3 /h) amount of CO2 gas s released from an outlet 0.9 meters above the area for 8 mnutes. Then the release stops and the gas dffuses freely for 0 mnutes untl the gas onentraton n the room dereases ba to normal values. Sne CO2 s a transparent and nvsble gas, a sensor grd s assembled to measure the onentraton of the gas over the area. We llustrate the struture of the sensor grd n Fg. 3, whh onssts of 2 CO2 sensors, 8 ARM-mbed mroontrollers, and an H shaped steel frame. In ths experment, the sensors are evenly dstrbuted as an asters shape as ndated n Fg. 3. The mroontrollers are used to ollet and store the data from the sensors and send them to a entral omputer. The H shaped steel frame s bult to support the sensor grd. We hoose K-30 CO2 sensors to apture the gas onentraton. The range of the sensor measuremen [0, 0000] ppm. The measurng frequeny of the sensors s set to 0.5Hz, whh an guarantee the suessfully trang of the dynams of CO2 gas. The dffuson proess obtaned from the real feld s shown n Fg.. CO2 begns dffusng atep = 0 and ends at = 625. The omputatonal tme step s seond. B. The bas analyss of the RLS method We further provde the bas analyss of the RLS method and have the followng proposton. Proposton V. Consder the RLS updated laws n (28). Under the assumpton that the modellng error e(r, ) s an ndependent nose wth zero mean and fnte varane, the RLS algorthm produes a based estmaton of q n the presene of Gaussan nose V (r, ). (a) Fg. 3. (b) The llustraton of the sensor grd. Proof: The RLS algorthm n (28) gves rse to the estmate of q as, q = E[Pz T Pz ] E[Pz T Y (r, + )]. (29) Sne V (r, ) and e(r, ) are..d. Gaussan noses, then E[Pz T h(r, + )] = E[Pz T e(r, + )] + E[Pz T e(r, )] (30) + E[Pz T (Pz Pz )]q = E[Pz T (Pz Pz )]q, whh s generally not zero and yelds a based estmaton of q, even f we assume the modellng error e(r, + ) s a Gaussan nose, as follows q = q + E[Pz T Pz ] E[Pz T (Pz Pz )]q. (3) Fg The dffuson feld olleted and vsualzed by MATLAB.

7 B. Expermental results We perform two dfferent experments for dffuson oeffent dentfaton wth four sensng agents deployed n the feld. We hoose two dfferentartng ponts for the agents: northeast (NE) startng pont and southeast (SE) startng pont. The robots are ontrolled to move along the gradent dreton of the feld estmated from the ooperatve Kalman flter whle eepng a onstant formaton. In Fg. 5, the ontours represent the level urves of the dffuson feld, the olored stars represent the four sensng agents, the red lne and blue lne represent the trajetores of the enter of the moble sensor networ starng from NE and SE, respetvely. Sne the feld s pre-olleted n MATLAB, t an be repeatably used for both experments. As we an observe from the fgure, the robots trae the gradent of the dffuson feld n both experments to fnd the dffuson soure of the CO 2 gas, whh s the pont wth the hghest CO 2 onentraton. Both of the two groups arrve at the soure around step = 550. Whle the moble sensor networ s searhng for the soure, t also aheves real-tme dentfaton of the dffuson oeffent by mplementng the ooperatve Kalman flter and the RLS. The estmaton results of the dffuson oeffent are shown n Fg. 6. As we an observe from Fg. 6 that, the estmates of the parameter onverge to stablzed values n both experments. The two values dffer by an amount of Ths dfferene may be aused by the nfluene of the veloty omponents of the CO 2 flow. Nevertheless, seen that the proposed algorthm s robust under realst unertantes and dsturbanes. Fg. 5. The trajetores of the robots n the two experments. Fg. 6. The estmated dffuson oeffent. VII. CONCLUSION We propose a novel flterng sheme for performng onlne parameter estmaton for adveton-dffuson proesses utlzng a moble sensor networ. By usng the fnte volume approxmaton, the proposed sheme an deal wth the ase when N agents are arranged n an arbtrary formaton. Theoretal justfatons are provded for the based analyss of RLS. Experment results based on a real CO 2 feld show satsfatory performane. Future wor nludes extendng the proposed algorthm to other types of PDEs. REFERENCES [] R. Ghez, Dffuson Phenomena. Kluwer Aadem/ Plenum Publshers, 2nd edton, 200. [2] J. You, F. Zhang, and W. Wu, Cooperatve flterng for parameter dentfaton of dffuson proesses, n Pro. of IEEE Conferene on Deson and Control, 206, pp [3] D. Uńs and M. Patan, Sensor networ desgn for the estmaton of spatally dstrbuted proesses, Int. J. Appl. Math. Comput. S., vol. 20, no. 3, pp. 59 8, 200. [] L. Ross, B. Krshnamahar, and C. C. J. Kuo, Dstrbuted parameter estmaton for montorng dffuson phenomena usng physal models, n Sensor and Ad Ho Communatons and Networs, IEEE SECON, Frst Annual IEEE Communatons Soety Conferene, 200, pp [5] D. Uńs, Optmal measurment methods for dstrbuted parameter system dentfaton. Boa Raton, FL: CRC Press, 200. [6] L. Z. Guo, S. A. Bllngs, and H. L. We, Estmaton of spatal dervatves and dentfaton of ontnuous spato-temporal dynamal systems, Internal Journal of Control, vol. 79, no. 9, pp. 8 35, [7] H. L and C. Q, Modelng of dstrbuted parameter systems for applaton - a syntheszed revew from tme-spae separaton, Journal of Proess Control, vol. 20, pp , 200. [8] J. Dronou, Fnte volume shemes for dffuson equatons: ntroduton to and revew of modern methods, Mathematal Models and Methods n Appled Senes, vol. 2, no. 8, pp , 20. [9] B. Groholsy, J. Keller, V. Kumar, and G. Pappas, Cooperatve ar and ground survellane, IEEE Robots & Automaton Magazne, vol. 3, no. 3, pp. 6 25, [0] Z. Tang and U. Ozguner, Moton plannng for multtargeurvellane wth moble sensor agents, IEEE Transatons on Robots, vol. 2, no. 5, pp , [] A. I. Mours and S. I. Roumelots, Performane analyss of multrobot ooperatve loalzaton, IEEE Transatons on Robots, vol. 22, no., pp , [2] S. Martnez and F. Bullo, Optmal sensor plaement and moton oordnaton for target trang, Automata, vol. 2, no., pp , [3] M. A. Demetrou and I. I. Hussen, Estmaton of spatally dstrbuted proess usng moble spatally dstrbuted sensor networ, SIAM Journal on Control and Optmzaton, vol. 8, no., pp , [] V. N. Chrstopoulos and S. Roumelots, Adaptve sensng for nstantaneous gas release parameter estmaton, n Proeedngs of the 2005 IEEE Internatonal Conferene on Robots and Automaton, 2005, pp [5] J. You and W. Wu, Onlne passve dentfer for spatally dstrbuted systems usng moble sensor networs, IEEE Transatons on Control Systems Tehnology, 207, aepted. [6] M. A. Demetrou, N. A. Gatsons, and J. R. Court, Coupled ontrolomputatonal fluds approah for the estmaton of the onentraton form a movng gaseous soure n a 2-D doman wth a lyapunovguded sensng aeral vehle, IEEE Transatons on Control Systems Tehnology, vol. 22, no. 3, pp , 20. [7] F. Zhang and N. E. Leonard, Cooperatve ontrol and flterng for ooperatve exploraton, IEEE Transatons on Automat Control, vol. 55, no. 3, pp , 200. [8] D. Smon and T. L. Cha, Kalman flterng wth state equalty onstrants, IEEE Transatons on Aerospae and Eletron Systems, vol. 38, no., pp , [9] W. Ren and R. W. Beard, Dstrbuted onsensus n mult-vehle ooperatve ontrol, ommunatons and ontrol engneerng seres. London: Sprnger-Verlag, [20] L. Ljung, System Identfaton, 2nd ed. Prente-Hall, Englewood Clffs,