Distributed Moving Horizon State Estimation of Nonlinear Systems. Jing Zhang

Transcription

1 Dstrbuted Movng Horzon State Estmaton of Nonnear Systems by Jng Zhang A thess submtted n parta fufment of the requrements for the degree of Master of Scence n Chemca Engneerng Department of Chemca and Materas Engneerng Unversty of Aberta c Jng Zhang, 214

2 Abstract Large-scae compex chemca processes ncreasngy appear n the modern process ndustry due to ther economc effcency. Such a arge-scae compex chemca process usuay conssts of severa unt operatons (subsystems), whch are connected together through matera and energy fows. Because of the ncreased process scae and the sgnfcant nteractons between dfferent subsystems, t poses great chaenges n the desgn of automatc contro systems for such arge-scae compex chemca processes whch are desred to fuf the fundamenta safety, envronmenta sustanabty and proftabty requrements. In recent years, predctve process contro has emerged as an attractve contro approach to hande the scae and nteractons of arge-scae compex chemca processes. It has been demonstrated that predctve process contro can acheve mproved cosed-oop performance compared wth decentrazed contro whe preservng the fexbty of the decentrazed framework. However, amost a of the exstng dstrbuted predctve process contro desgns are deveoped under the assumpton that the state measurements of subsystems or the entre system are avaabe. Ths assumpton does not hod n many appcatons. Ths thess presents a robust dstrbuted movng horzon state estmaton (DMHE) scheme that s approprate for output feedback dstrbuted predctve contro of nonnear systems as we as approaches for reducng the communcaton demand of the proposed DMHE scheme and a strategy for handng deays n the communcaton between subsystem estmators. Frst, the proposed robust DMHE scheme s presented for a cass of nonnear systems that are composed of severa subsystems. It s assumed that the subsystems nteract wth each other va ther states ony. Subsequenty, two trggered communcaton agorthms are ntroduced for the proposed DMHE scheme to reduce the number of nformaton transmssons between subsystems. Foowng ths, an approach s proposed to hande the potenta tme-varyng deays n the communcaton between the subsystem estmators. The appcabty and effectveness of the proposed approaches are ustrated va ther appcatons to dfferent chemca process exampes.

3 Acknowedgements It gves me great peasure n acknowedgng the patence, support and hep of Professor Jnfeng Lu, as we as hs frendshp wth me durng my graduate studes. Ths thess woud have remaned a dream had t not been for the gudance of Prof. Lu and hs mentorshp was sgnfcant n assstng to bud up my ong-tme career goas and experences. As an undergraduate student graduated from the Department of Poymer Scence and Engneerng, I know tte about mathematcs and programmng. I was not sure f I coud make t to transfer from expermenta scence to computatona scence and I was ready to get my master s degree n three years. To hep me better nvoved n ths specaty, Prof. Lu even gave me ectures on the bascs of math, contro theores and programmng. I woud ke to gve my apprecaton to Prof. Lu agan for hs trust, cutvaton and comprehensveness. I woud ke to thanks Professor Bao Huang for hs hep n system dentfcaton n my project and the access to Prof. Huang s genera group meetng. I am ndebted to my many coeagues n CPC groups who supported me: Tanbo Lu, Shunng L, Kangkang Zhang, Ruben Gonzaez, Yaoje Lu, Mng Ma, Lu Lu, Zhankun X, Muang Chen, Chen L, Su Lu, Yuja Zhao, Ruomu Tan and Ouyang Wu. I aso wsh to thank my frends who makes my graduate fe so wonderfu: Ran L, Qan Fu, Q Zhang, X Huang and Steven Myronuk. I owe my deepest grattude to my parents and sbngs. Wthout ther support and understandng of my graduate studes, I woud never accompshed such a huge mprovement n my career. I hope They d ke to be proud of me.

4 Contents 1 Introducton Motvaton Background Thess outne and contrbutons Dstrbuted Movng Horzon State Estmaton for Nonnear Systems wth Bounded Uncertantes Introducton Notaton System descrpton Nonnear observers The DMHE scheme Dstrbuted estmaton agorthm Loca MHE desgn Stabty anayss Appcaton to a reactor-separator process Process descrpton and modeng Loca MHE desgn Smuaton resuts Concusons Two Trggered Communcaton Agorthms for Dstrbuted Movng Horzon State Estmaton Introducton Modeng of measurements The DMHE scheme wth trggered communcatons DMHE wth the frst communcaton trgger v

5 3.4.1 Impementaton agorthm The frst trggerng condton Loca MHE formuaton Stabty anayss DMHE wth the second communcaton trgger Impementaton agorthm The second trggerng condton Stabty anayss Appcaton to the reactor-separator process Smuaton settngs Smuaton resuts Concusons Dstrbuted Movng Horzon State Estmaton Subject to Communcaton Deays Introducton Modeng of measurements and communcatons The DMHE scheme subject to communcaton deays Dstrbuted state estmaton agorthm State predcton Reference state estmate cacuaton Subsystem MHE desgn Stabty anayss Appcaton to the reactor-separator process Smuaton settngs Smuaton resuts Concusons Concusons 76 Bbography 78 v

6 Lst of Tabes 2.1 Process varabes for the reactors Process parameters for the reactors Mean evauaton tmes of the predctor, observer (4.2) and oca MHE for each subsystem v

7 Lst of Fgures 2.1 The proposed DMHE desgn Reactor-separator process wth a recyce stream Trajectores of the actua system state (sod nes), the estmates gven by the proposed DMHE (dashed nes) and the nonnear observers of Eq. (2.8e) mpemented foowng Agorthm 2 (dash-dotted nes) Trajectores of the estmaton error norm of the proposed DMHE (dashed ne) and of the nonnear observers of Eq. (2.8e) mpemented foowng Agorthm 2 (dash-dotted ne) Trajectores of the actua system state (sod nes), the estmates gven by the proposed DMHE (dashed nes) and a decentrazed MHE n whch the subsystem MHEs do not communcate and the nteractons between subsystems are compensated for usng ther steady-state vaues (dash-dotted nes) Trajectores of the estmaton error norm of the proposed DMHE (dashed ne) and of a decentrazed MHE n whch the subsystem MHEs do not communcate and the nteractons between subsystems are compensated for usng ther steady-state vaues (dash-dotted nes) Trajectores of the actua system state (sod nes), the estmates gven by the proposed DMHE (dashed nes) and the proposed DMHE wth the correcton gans n (2.8e) beng zero vectors (dash-dotted nes) Trajectores of the estmaton error norm of the proposed DMHE (dashed ne) and of the proposed DMHE wth the correcton gans n (2.8e) beng zero vectors (dash-dotted ne) Trajectores of the actua system state (sod nes) and the estmates gven by the proposed DMHE (dashed nes) subject to mode parameter uncertantes Scheme of the proposed DMHE desgn wth trggered communcaton Damped snusoda nputs to the three subsystems v

8 3.3 State trajectores of the actua system state(sod nes) and the state estmates gven by the proposed DMHE mpemented foowng Agorthm 14 based on trggerng condton (3.1) wth ɛ =1.,=1, 2, 3 (dashed nes) nstants when subsystem, =1, 2, 3, sent out ts nformaton by the proposed DMHE mpemented foowng Agorthm 14 based on trggerng condton (3.1) State trajectores of the actua system state(sod nes) and the state estmates gven by the proposed DMHE mpemented foowng Agorthm 18 based on trggerng condton (3.3) wth ɛ 2, =1.,=1, 2, 3 (dashed nes) nstants when subsystem, =1, 2, 3, sent out ts nformaton by the proposed DMHE mpemented foowng Agorthm 18 based on trggerng condton (3.3) Trajectores of the estmaton error norm of the proposed DMHE mpemented foowng Agorthm 14 (dashed nes) based on trggerng condton (3.1) and Agorthm 18 based on trggerng condton (3.3) (sod nes) Average number of communcatons and performance ndex of the proposed DMHE mpemented foowng Agorthm 14 based on trggerng condton (3.1) wth ɛ, =1, 2, 3, varyng from to 3 (sod nes) and the dashed nes denote the number of communcatons and performance of the proposed DMHE wth the subsystems exchangng nformaton every sampng tme Average number of communcatons and performance ndex of the proposed DMHE mpemented foowng Agorthm 18 based on trggerng condton (3.3) wth ɛ 2,, =1, 2, 3, varyng from to 3 (sod nes) and the dashed nes denote the number of communcatons and performance of the proposed DMHE wth the subsystems exchangng nformaton every sampng tme Scheme of the proposed DMHE desgn consderng communcaton deays Trajectores of the actua process states (sod nes) and the estmates gven by the proposed DMHE (dashed nes) when the communcaton deays between subsystems aways equa to the maxmum possbe deay D wth D = Trajectores of the actua process states (sod nes) and the estmates gven by the DMHE (dashed nes) n Chapter 2 when the communcaton deays between subsystems aways equa to the maxmum possbe deay D wth D = v

9 4.4 Trajectores of the actua process states (sod nes) and the estmates gven by nonnear observer (4.2) (dashed nes) when the communcaton deays between subsystems aways equa to the maxmum possbe deay D wth D = Trajectores of the norm of the estmaton errors of the proposed DMHE (sod ne) and the DMHE n Chapter 2 (dashed ne) and nonnear observer (4.2) (dash-dotted ne) when the communcaton deays between subsystems aways equa to the maxmum possbe deay D wth D = Communcaton deay sequences Trajectores of the norm of the estmaton errors of the proposed DMHE (sod ne) and the DMHE n Chapter 2 (dashed ne) and nonnear observer (4.2) (dash-dotted ne) Trajectores of the norm of the estmaton error of the proposed DMHE (sod nes) and the approxmated bound that utmatey bounds the estmaton error (dashed nes) when (a) D =1,(b)D =2,(c)D =3,(d)D =5,(e) D = 7; and (f) the bounds n one fgure x

10 Chapter 1 Introducton 1.1 Motvaton Due to the ncreasng goba competton, arge-scae compex chemca processes s common appearances n the modern process ndustry due to ther economc effcency. In recent years, dstrbuted predctve contro has emerged as an attractve contro approach to hande the scae and nteractons of arge-scae compex chemca processes. It has been demonstrated that dstrbuted predctve contro can acheve mproved (sometmes the centrazed) cosedoop performance whe preservng the fexbty of the decentrazed framework. However, amost a of the exstng dstrbuted predctve contro desgns were deveoped under the assumpton that the state measurements of subsystems or the entre system are avaabe. It s n genera dffcut to measure a the state varabes n a process system. In order to mantan the structura fexbty of dstrbuted predctve contro, dstrbuted or decentrazed state estmaton systems shoud be used nstead of centrazed observers. There are many exstng resuts on decentrazed determnstc observer desgns for dfferent casses of systems (e.g., [1, 2, 3, 4]) and dstrbuted Kaman fterng based on consensus agorthms wth appcatons to sensor networks (e.g, [5, 6, 7, 8]). These resuts are prmary deveoped n the context of near systems. Recenty, n [9], a dstrbuted state estmaton approach for near systems was deveoped n the framework of movng horzon estmaton (MHE) whch was extended to nonnear systems n [1]. However, these desgns are not approprate for output-feedback contro. Motvated by the above consderatons, n ths thess, we present a robust dstrbuted MHE (DMHE) desgn for a cass of nonnear systems wth bounded output measurement nose and process dsturbances. The proposed DMHE desgn has the potenta to be used n output-feedback contro. 1

11 1.2 Background Large-scae compex chemca processes ncreasngy appear n the modern process ndustry due to ther economc effcency. Such a arge-scae compex chemca process usuay conssts of severa unt operatons (subsystems), whch are connected together through matera and energy fows. Because of the ncreased process scae and the sgnfcant nteractons between dfferent subsystems, t poses great chaenges n the desgn of automatc contro systems for such arge-scae compex chemca processes whch are desred to fuf the fundamenta safety, envronmenta sustanabty and proftabty requrements. Tradtonay, contro and state estmaton of arge-scae systems has been studed prmary wthn the centrazed and the decentrazed frameworks. Whe the centrazed framework s shown to provde the best performance, t s not favorabe from the computatona and faut toerance vew ponts. In a decentrazed framework, the nteractons between subsystems n genera are ether not taken nto account or accounted for n conservatve fashons such as worst case scenaros (e.g., [11, 12] and references theren). Decentrazed framework n genera has a reduce compexty n the controer and observer desgn and mpementaton. However, t may ead to deterorated performance or even ost of cosed-oop stabty. In recent years, dstrbuted mode predctve contro (DMPC) has emerged as an attractve contro approach to hande the scae and nteractons of arge-scae compex chemca processes; pease see [13, 14, 15] for revews of resuts on DMPC. The exstng DMPC agorthms can be broady cassfed nto non-cooperatve and cooperatve DMPC agorthms based on the cost functon used n the oca controer optmzaton probem [13]. In a noncooperatve DMPC agorthm, each oca controer optmzes a oca cost functon whe n a cooperatve DMPC agorthm, a oca controer optmzes a goba cost functon. Noncooperatve DMPC agorthms ncude [16, 17, 18, 19, 2, 21]. Cooperatve DMPC was frst proposed n [22] and was deveoped n [13, 23, 24]. Lyapunov-based cooperatve DMPC agorthms for nonnear systems were aso deveoped n [25, 26] n recent years. It has been demonstrated that DMPC has the potenta to acheve the performance of the centrazed contro whe preservng the fexbty of decentrazed frameworks [23, 15]. In addton to DMPC, other mportant work wthn process contro ncudes the deveopment of a quasdecentrazed contro framework for mut-unt pants that acheves the desred cosed-oop objectves wth mnma cross communcaton between the pant unts under state feedback contro [27]. However, amost a of the above resuts are derved under the assumptons that the system states are avaabe a the tmes or that a centrazed state observer s 2

12 avaabe. These assumptons, however, ether fa n many appcatons or are nconsstent wth the dstrbuted framework whch s not favorabe from a faut toerance pont of vew. Therefore, t s desrabe to deveop state estmaton schemes n the dstrbuted framework. In the terature, a majorty of the exstng resuts on state observer desgns are derved n the centrazed framework. For near systems, Kaman fters and Luenberger observers are standard soutons. In the context of nonnear systems, observer desgns ncudng hgh-gan observers for dfferent specfc casses of nonnear systems are avaabe (e.g., [28, 29, 3, 31, 32, 33, 34, 35, 36, 37]). In a recent work [38], observers for systems wth deayed measurements were aso deveoped. It s worth notng that the capabty of hghgan observers to be used n output feedback contro desgns has made hgh-gan observers very popuar n output feedback contro of nonnear systems (e.g., [39, 4, 41, 42, 43, 44]). In another ne of work, MHE has become popuar because of ts abty to hande expcty nonnear systems and constrants on decson varabes (e.g., [45, 46, 47, 48]). In MHE, the state estmate s determned by sovng onne an optmzaton probem that mnmzes the sum of squared errors. In order to have a fnte dmensona optmzaton probem, the horzon (estmaton wndow sze nto the past) of MHE s n genera chosen to be fnte. At a sampng tme, when a new measurement s avaabe, the odest measurement n the estmaton wndow s dscarded, and the fnte horzon optmzaton probem s soved agan to get the new estmate of the state [49, 45]. In a recent work [5], a robust MHE scheme was deveoped whch effectvey ntegrates determnstc (hgh-gan) observers nto the MHE framework. The resutng robust MHE scheme gves bounded estmaton error and has a tunabe convergence rate. In order to mantan the structura fexbty of DMPC, dstrbuted or decentrazed state estmaton systems shoud be used nstead of centrazed observers. There are many exstng resuts on decentrazed determnstc observer desgns for dfferent casses of systems (e.g., [1, 2, 3, 4]) and dstrbuted Kaman fterng based on consensus agorthms wth appcatons to sensor networks (e.g, [5, 6, 7, 8]). These resuts are prmary deveoped n the context of near systems. Recenty, n [9], a dstrbuted state estmaton approach for near systems was deveoped n the framework of movng horzon estmaton (MHE) whch was extended to nonnear systems n [1]. The dstrbuted MHE (DMHE) schemes deveoped n [9, 1] were aso based on consensus agorthms whch requre the use of the entre system mode n each ndvdua MHEs. Aong ths ne of work, n [51, 52], DMHE schemes based on subsystem modes were deveoped for both near and nonnear systems. Snce the above DMHE schemes were deveoped based on the cassca centrazed MHE 3

13 [45, 46, 47], they mantan the advantages of MHE ncudng the abty to hande nonneartes, constrants and optmaty consderatons expcty. However, as n the centrazed MHE, the convergence of the above DMHE schemes to the actua system state requres a reabe approxmaton of the arrva cost. Even though dfferent approaches ncudng the extended Kaman fterng [53], the unscented Kaman fterng [54] and partce fters [55, 56] have been deveoped to approxmate the arrva cost, t s n genera a dffcut task to determne the arrva cost for constraned nonnear systems. Aso, when there are bounded measurement nose and process dsturbances, t s n genera dffcut to ensure the boundedness of the estmaton error [47]. Moreover, the convergence rates of the estmates gven by the above DMHE schemes to the actua system states are not tunabe and s not favorabe from an output feedback contro pont of vew. 1.3 Thess outne and contrbutons Ths thess s organzed as foows: In Chapter 2, a robust DMHE desgn for a cass of nonnear systems wth bounded output measurement nose and process dsturbances s presented. In ths DMHE, each subsystem MHE communcates wth subsystems that t nteracts wth every sampng tme. In the desgn of each subsystem MHE, an auxary determnstc nonnear observer s taken advantage of to cacuate a confdence regon that contans the actua system state every sampng tme. The subsystem MHE s ony aowed to optmze ts state estmate wthn the confdence regon. Ths strategy was demonstrated to guarantee the convergence and utmate boundedness propertes of the estmaton error. In Chapter 3, two agorthms are proposed to reduce the number of communcatons between subsystem based on the DMHE framework deveoped n Chapter 2. In partcuar, event-trggered approach s adopted to reduce the number of communcaton between subsystems. Specfcay, n the frst proposed agorthm, a subsystem sends out ts current nformaton when a trggerng condton based on the dfference between the current state estmate and a prevousy transmtted state estmate s satsfed; n the second proposed agorthm, the transmsson of nformaton from a subsystem to other subsystems s trggered by the dfference between the current measurement of the output and ts dervatves and a prevousy transmtted measurement of the output and ts dervatves. Because of the trggered communcaton, a subsystem may not have the atest nformaton of the other subsystems. The appcaton to a chemca process ustrates the effectveness of the proposed approaches n reducng the number of communcatons between the subsystems whe 4

14 mantanng the estmaton performance. In Chapter 4, the DMHE deveoped n Chapter 2 s extended to hande tme-varyng communcaton deays. In partcuar, an open-oop state predctor s desgned for each subsystem to provde predctons of unavaabe subsystem states. In the desgn of each predctor, the centrazed system mode s used. Based on the state predctons, an auxary nonnear observer s used to generate a reference subsystem state estmate for each subsystem every sampng tme. Based on the reference subsystem state estmate as we as the oca output measurement, a confdence regon s constructed for the actua state of a subsystem. A subsystem MHE s ony aowed to optmze ts state estmate wthn the correspondng confdence regon at a sampng tme. The proposed DMHE s proved to gve decreasng and utmatey bounded estmaton errors under the assumpton that there s an upper bound on the tme-varyng deay. The theoretca resuts are ustrated va the appcaton to a reactor-separator chemca process. Chapter 5 summarzes the man resuts of ths thess and dscusses future research drectons. 5

15 Chapter 2 Dstrbuted Movng Horzon State Estmaton for Nonnear Systems wth Bounded Uncertantes 2.1 Introducton In ths chapter, we propose a DMHE scheme for a cass of nonnear systems wth bounded output measurement nose and process dsturbances. Specfcay, we consder a cass of nonnear systems that are composed of severa subsystems and the subsystems nteract wth each other va ther subsystem states. Frst, a dstrbuted estmaton agorthm s desgned whch specfes the nformaton exchange protoco between the subsystems and the mpementaton strategy of the DMHE. Subsequenty, a oca MHE scheme s desgn for the each subsystem. In the desgn of each subsystem MHE, an auxary nonnear determnstc observer that can asymptotcay track the correspondng nomna subsystem state when the subsystem nteractons are absent s taken advantage of. For each subsystem, the nonnear determnstc observer together wth an error correcton term s used to cacuate confdence regons for the subsystem states every sampng tme. Wthn the confdence regons, the subsystem MHE s aowed to optmze ts estmates. The proposed DMHE scheme s proved to gve bounded estmaton errors. It s aso possbe to tune the convergence rate of the state estmate gven by the DMHE to the actua system state. The appcabty and effectveness of the proposed DMHE are ustrated va the appcaton to a reactor-separator process exampe. Ths chapter s a revsed verson of J. Zhang and J. Lu, Dstrbuted movng horzon state estmaton for nonnear systems wth bounded uncertantes. Journa of Process Contro, 23: ,

16 2.2 Notaton Throughout ths thess, we operator denotes Eucdean norm of a scaar or a vector whe 2 Q ndcates the square of the weghted Eucdean norm of a vector, defned as x 2 Q = xt Qx where Q s a postve defnte square matrx. A functon f(x) s sad to be ocay Lpschtz wth respect to ts argument x f there exsts a postve constant L x f such that f(x ) f(x ) L x f x x for a x and x n a gven regon of x and L x f s the assocated Lpschtz constant. A contnuous functon α :[,a) [, ) s sad to beong to cass K f t s strcty ncreasng and satsfes α() =. A functon β(r, s) s sad to be a cass KL functon f for each fxed s, β(r, s) beongs to cass K wth respect to r, and for each fxed r, t s deceasng wth respect to s, andβ(r, s) ass. The symbo dag(v) denotes a dagona matrx whose dagona eements are the eements of vector v. The symbo \ denotes set subtracton such that A \ B := {x A,x / B}. The superscrpt (s) denotes the s-th order tme dervatve of a functon. The matrx (or vector) A + denotes the pseudonverse of a matrx (or vector) A. The set I = {1,...,m}. 2.3 System descrpton Throughout ths thess, we consder a cass of nonnear systems composed of m nterconnected subsystems where the -th subsystem can be descrbed by the foowng state-space mode: ẋ (t) = f (x (t),w (t)) + f (X (t)) y (t) = h (x )+v (t) (2.1) where I, x (t) R nx denotes the vector of state varabes of subsystem, w (t) R nw denotes dsturbances assocated wth subsystem, and the vector functon f characterzes the dependence of the dynamcs of x on tsef and the assocated dsturbances. The vector functon f characterzes the nteractons between subsystem and other subsystems. The state vector X (t) R n X denotes the vector of states that nvoved n characterzng the nteractons. The vector y R ny s the measured output of subsystem and v R nv s the measurement nose vector. Ths system w aso be used n Chapter 3 and Chapter 4. In the foowng dscusson, we use I I, I, to denote the set of subsystem ndces whose correspondng subsystem states are nvoved n X. For exampe, f X 1 contans states of subsystem 1, subsystem 3 and subsystem 4, then I 1 = {1, 3, 4}. Itsassumedthatthe sets I, I, areknown. 7

17 It s assumed that the subsystem states x, I, satsfy the constrant: x X (2.2) where X, I, are convex compact sets and the system dsturbances and measurement nose are bounded such as w W and v V, I, where W := {w R nw : w θ w } (2.3) V := {v R nv : v θ v } wth θ w and θ v, I, known postve rea numbers. The entre nonnear system state vector and measured output vector are denoted as x and y whch are compostons of the states and outputs of the m subsystems, respectvey. That s, x =[x T 1 xt xt m] T R nx and y =[y T 1 yt y T m] T R ny. The entre system can be descrbed as foows: ẋ(t) = f(x(t),w(t)) + f(x(t)) y(t) = h(x(t)) + v(t) (2.4) where f, f, w, h, and v are approprate compostons of f, f, w, h, and v, I, respectvey. The outputs of the m subsystems, y, I, are assumed to be samped synchronousy and perodcay at tme nstants {t k } such that t k = t +kδwtht = the nta tme, Δ a fxed sampng tme nterva and k postve ntegers. For each subsystem, a state estmator (observer) w be desgned n the framework of MHE to estmate ts state. It s assumed that the estmator assocated wth subsystem has drect access to the measurements of y and can communcate wth the other subsystems when necessary to exchange ther subsystem output measurements and state estmates. Remark 1 In order to ustrate the system mode representaton, consder the foowng system wth three one-dmensona subsystems: ẋ 1 = x 1 + x 1 x 3 = f 1 (x 1 )+ f 1 (X 1 ) ẋ 2 =.5x 2 + x 2 x 1 + x 2 3 = f 2 (x 2 )+ f 2 (X 2 ) ẋ 3 = x 3 + x 1 x 2 +.1x 3 2 = f 3 (x 3 )+ f 3 (X 3 ) In ths exampe, X 1 =[x 1,x 3 ] T wth I 1 = {1, 3}, X 2 =[x 1,x 2,x 3 ] T wth I 2 = {1, 2, 3} and X 3 =[x 1, x 2 ] T wth I 3 = {1, 2}. Note that n order to smpfy the dscusson but wthout oss of generaty, nputs of the system are not consdered n (2.4). 2.4 Nonnear observers Throughout ths thess, an auxary oca nonnear determnstc observer for each subsystem w be taken advantage of to cacuate a confdence regon for the actua system state 8

18 every sampng tme. In the context of nonnear systems, there are extensve studes on nonnear determnstc observers focusng on the desgn of centrazed observers [57, 58, 31, 3, 29, 35, 59, 6, 37] wth many successfu appcatons to dfferent areas ncudng the contro and montorng of nonnear chemca processes [42, 43, 32, 33, 34, 61, 44]. One mportant cass of nonnear observers s the so-caed hgh-gan observers [29, 3, 31, 39] whch aow for effectve separaton prncpes n output feedback contro desgns. However, tte attenton has been pad to the desgn of nonnear decentrazed or dstrbuted determnstc observers. Takng ths fact nto account, we assume that there exsts a nonnear determnstc observer for subsystem, I, of the foowng form: ż (t) =F (z (t),h (x (t))) (2.5) such that f f (X (t)), w (t) for a t, thenz asymptotcay approaches x for a the states x X. Ths assumpton mpes that f f (X (t)), w (t) for a t, there exsts a KL functon β such that: z (t) x (t) β ( z () x (),t) (2.6) where z () and x () are the nta states of the observer and the subsystem. The above observabty assumpton aso mpes that [62]: rank(o (x )) = n x (2.7) wth O (x )= dφ (x ) for a x X. Note that the convergence property of the nonnear dx observer (2.5) s obtaned based on contnuous nose-free output measurements. We aso assume that F, I, are ocay Lpschtz functons. It s further assumed that the entre system of Eq. (2.4) s ocay observabe whch essentay mpes that the nteractons between the subsystems do not damage the coectve observabty of the subsystems. Note that n the above assumpton of the nonnear observers, the nteractons between the subsystems are assumed to be absent (.e., f (X (t)) for a t). Note aso that the convergence propertes of the nonnear observers are obtaned based on contnuous nosefree output feedback. In the proposed DMHE dscussed n the next secton, we w dscuss n detas on how to take advantage of observer (2.5) and to compensate for the nteractons between the subsystems usng nformaton exchanged between subsystems. 2.5 The DMHE scheme A schematc of the proposed DMHE desgn whch ncudes m oca MHEs for the nonnear system of Eq. (2.4) s shown n Fg In the proposed DMHE desgn, each subsystem s 9

19 Fgure 2.1: The proposed DMHE desgn. assocated wth an MHE whch s evauated every sampng tme. We aso assume that a oca MHE has mmedate access to the output measurements of ts assocated subsystem and can communcate wth the other subsystems to exchange ther subsystem output measurements and state estmates whch are used to compensate for the nteractons between subsystems to mprove ther state estmates. The MHE assocated wth subsystem ( I) w be referred to as MHE. In the remander of ths chapter, we w frst ntroduce the proposed dstrbuted estmaton agorthm; subsequenty, we present the desgn of the oca MHEs and fnay derve suffcent condtons under whch the proposed DMHE gves bounded estmaton error Dstrbuted estmaton agorthm The proposed DMHE uses the foowng dstrbuted state estmaton agorthm to get an estmate of the entre system. Agorthm 2 Dstrbuted state estmaton agorthm 1. At t =, a the MHEs are ntazed wth the nta subsystem state guesses ˆx (), I, and the actua subsystem output measurements y (), I. 2. At t k >, carry out the foowng steps: 2.1. Each MHE receves the output measurement of the subsystem that t s assocated wth; that s, MHE receves y (t k ) Each MHE requests and receves the output measurements and subsystem state estmates of the prevous tme nstant from subsystems that drecty affect ts 1

20 dynamcs; that s, MHE requests and receves y (t k 1 ) and ˆx (t k 1 ) (whch denotes the state estmate of subsystem at t k 1 ) for a I Based on both the oca measurement and nformaton from other subsystems, each MHE cacuates the estmate of ts subsystem s state; that s, MHE cacuates ˆx (t k ). The estmate of the entre system state s ˆx(t k )=[ˆx 1 (t k ) T...ˆx m (t k ) T ] T Go to Step 2.1 at the next sampng tme t k+1. From Agorthm 2, t can be seen that t s a non-teratve agorthm. At a sampng tme, the MHEs are evauated ony once n parae. Ths mpementaton compensates for the nteractons between subsystems based on the state estmates and output measurements at the prevous tme nstant. In addton, Agorthm 2 does not requre an a-to-a communcaton between the MHEs. From Step 2.2, t can be seen that an MHE ony communcates wth subsystems that t nteracts drecty. For exampe, f the subsystems are connected n seres, an MHE ony has to receve nformaton from ts drecty connected upstream subsystem and to send nformaton to ts drecty connected downstream subsystem. Remark 3 Note that an teratve mpementaton agorthm may be used for the proposed DMHE desgn based on the current output measurements and the state estmates obtaned at the prevous teraton of the current sampng tme. In ths case, t s possbe to acheve the convergence property of the state estmates by redesgnng the oca MHEs accordngy. An teratve mpementaton agorthm may ead to mproved dstrbuted estmaton performance f the teratons converge to the goba optmum whch, however, s not ensured for genera nonnear systems due to the non-convexty of the optmzaton probems. Moreover, when an teratve mpementaton agorthm s used, t may sgnfcanty ncrease the computatona compexty of the proposed DMHE desgn Loca MHE desgn In the desgn of a oca MHE, the subsystem mode of Eq. (2.1), the correspondng nonnear determnstc observer of Eq. (2.5) together wth the nformaton receved from other subsystems are used. A confdence regon that contans the actua subsystem state w be cacuated every sampng tme takng nto account the boundedness of the measurement nose and process dsturbances. The oca MHE s ony aowed to optmze the subsystem state estmate wthn the confdence regon. 11

21 Specfcay, the proposed desgn of MHE at tme nstant t k s formuated as foows: k 1 k mn w (t q ) 2 + v Q 1 (t q ) 2 + V R 1 (t k N ) (2.8a) x (t k N ),..., x (t k ) q=k N q=k N s.t. x (t) =f ( x (t),w (t q )) + f ( ˆX (t q )), t [t q,t q+1 ],q= k N,...,k 1 (2.8b) v (t q )=y (t q ) h ( x (t q )), q= k N,...,k (2.8c) w (t q ) W,v (t q ) V, x (t q ) X,q= k N,...,k (2.8d) ż (t) =F (z (t),y (t k 1 )) + f ( ˆX (t k 1 )) + I K, (ˆx )(y (t k 1 ) h (ˆx (t k 1 ))) z (t k 1 )=ˆx (t k 1 ) x (t k ) z (t k ) κ y (t k ) h (z (t k )) (2.8e) (2.8f) (2.8g) where N s the estmaton horzon, Q and R are the covarance matrces of w and v respectvey, V (t k N ) denotes the arrva cost whch summarzes past nformaton up to t k N, x s the predcted x n the above optmzaton probem, ˆx s the optma estmate of x at prevous tme nstants, K, for I are gan matrces whch are functons of ˆx, and κ s a postve constant. The roes of K, and κ w be made cear n the foowng dscusson. Once probem (2.8) s soved, an optma trajectory of the system states, x (t k N),..., x (t k), s obtaned. The ast eement x (t k) s used as the optma estmate of the state of subsystem at t k and s denoted as ˆx (t k ). That s, ˆx (t k )= x (t k ). (2.9) Note that n the optmzaton probem (2.8), w and v are assumed to be pece-wse constant varabes wth sampng tme Δ to ensure that (2.8) s a fnte dmensona optmzaton probem. In optmzaton probem (2.8), constrant (2.8a) s the cost functon that needs to be mnmzed. The arrva cost V (t k N ) summarzes the past nformaton that s not covered n the estmaton horzon. Constrant (2.8b) s the mode of subsystem. Because ony state estmates at the sampng tmes are avaabe, f ( ˆX (t q )) s used to approxmate f (X (t)) from t q to t q+1. Ths aso mpes that each MHE shoud be abe to store the prevousy receved nformaton from other MHEs. Constrant (2.8d) are bounds on the dsturbances, nose and subsystem state. Constrants (2.8e)-(2.8g) are used to cacuate a confdence regon that contans the actua subsystem state based on the determnstc nonnear observer, a correcton term 12

22 and the nformaton receved from other subsystems. The estmate of the current subsystem state s ony aowed to be optmzed wthn the confdence regon. Specfcay, constrant (2.8e) s an augmented nonnear observer takng nto account expcty the nteractons between subsystem and the other subsystems. The frst term on the rght-handsde of (2.8e) comes from observer (2.5); the second term on the rght-hand-sde of (2.8e) s based on the nteracton mode and the prevous state ˆX (t k 1 ) whch s the atest avaabe nformaton avaabe to subsystem ; and the ast term s used to correct the errors n the nteracton mode. The gan K, depends on ˆx and s cacuated at each sampng tme as foows: K, = f ( ) + h (2.1) x x ˆx (t k 1 ) for I and 1,...,m. The above cacuaton of K, mpes that the estmaton error caused by the nteracton s compensated by ts nearzed approxmaton. Ths dea w be made expct n Secton Constrant (2.8g) expcty defnes the confdence regon based on the parameter κ and the estmate gven by the nonnear observer (2.8e) as we as the actua output measurement. The parameter κ depends on the propertes of the system. Condtons that the vaue of κ ( I) needs to satsfy w be derved n Secton It w aso be proven n Secton that the proposed approach eads to bounded estmaton errors. Remark 4 In the proposed DMHE desgn, the nteractons between subsystems (.e., f (X ), I) are compensated for based on the nteracton modes as we as subsystem state estmates (.e., ˆX ). The error caused by the dfference between ˆX and X s further compensated for usng the correcton term I K, (y (t k 1 h (ˆx (t k 1 ))) wth K,, I, determned foowng (2.1). The correcton term wth K, essentay compensates for the near part of the error dynamcs caused by the dfference between ˆX and X and negects the hgher order dynamcs whch w be made cear n the proof of Proposton 5 n Secton In many appcatons, a frst order correcton term ke the one used n the present work s suffcent to acheve desred estmaton performance. Pease see Secton 2.6 for the appcaton of the proposed approach to a reactor-separator chemca process. If n an appcaton t s necessary to compensate for the hgher order error dynamcs n system nteractons, the proposed approach can be extended n a straghtforward manner. 13

23 2.5.3 Stabty anayss In ths secton, we study the robustness and stabty propertes of the proposed DMHE. Specfcay, we frst nvestgate the boundedness of the estmaton error gven by the nonnear observer (2.8e) wth K, determned foowng (2.1) takng nto account measurement nose and process dsturbances. Foowng ths, we state the stabty and utmate boundedness of the estmaton error of the proposed DMHE. Suffcent condtons w be provded. Proposton 5 Consder the nonnear observer of Eq. (2.8e) for subsystem, I, wth nta condton z (t k )=ˆx (t k ) and output measurement y (t k ). If K, for I and I are determned as n (2.1) and K, are bounded, then the devaton of the observer state z n one sampng tme Δ (.e., at t k+1 ) from the actua subsystem state x s bounded for a x X, I, asfoows: e z, (t k+1 ) β ( e z, (t k ), Δ) + γ (Δ) + L, Δ e z, (t k ) 2 (2.11) I where e z, = z x, I, andγ (τ) =L y F L h M τ 2 /2+L y F θ v τ +L w f θ w τ + I M K, θ v τ + I L x M f τ 2 /2 and L, = H f + M K, H h wth L y F, L h, L w f,andl x beng the Lpschtz f constants of F wth respect to y, h wth respect to x, f wth respect to w,and f wth respect to x, respectvey, and M, M K,, I and I, beng constants that bound ẋ n X and K, n X, respectvey, and H f, Hh beng postve constants that assocated wth the Tayor expansons of f and h. Proof: We consder the oca MHE of Eq. (2.8) ( I) and defne e z, = z x where z denotes the trajectory of observer (2.8e) and x s the state trajectory of the actua subsystem of Eq. (2.1). In ths proof, we consder the tme nterva from t = t k to t = t k+1 and the nta condton z (t k )=ˆx (t k ). The dervatve of e z, s evauated as foows: ė z, (t) = F (z (t),y (t k )) f (x (t),w (t)) + f ( ˆX (t k )) f (X (t)) + I K, (ˆx )(y (t k ) h (ˆx (t k ))). (2.12) From the Lpschtz propertes of F, f and h, the fact that y (t k )=h (x (t k )) + v (t k ), and v (t k ) θ v, w θ w, the foowng nequaty can be obtaned from (2.12): ė z, (t) F (z (t),h (x (t))) f (x (t), ) + L y F L h x (t) x (t k ) + L y F θ v + L w f θ w + f ( ˆX (t k )) f (X (t)) + I K, (ˆx )(y (t k ) h (ˆx (t k ))) (2.13) where L y F, L h and L w f are the Lpschtz constants assocated wth F, h and f, respectvey. 14

24 Usng Tayor seres expanson, the foowng nequates can be obtaned: f (X (t k )) = f ( ˆX (t k )) + I f x (x (t k ) ˆx (t k )) + H.O.T f, h (x (t k )) = h (ˆx (t k )) + h (x (t k ) ˆx (t k )) + H.O.T h x (2.14) where H.O.T f and H.O.T h are hgh order terms assocated wth the expansons of f and h. These hgh order terms satsfy the foowng constrants: H.O.T f H f X (t k ) ˆX (t k ) 2, H.O.T h H h x (t k ) ˆx (t k ) 2 (2.15) for a x X ( I) wthh f, I, andhh, =1,...,m, are postve constants. Let us defne A (t k )= f ( ˆX (t k )) f (X (t k )) + I K, (ˆx )(h (x (t k )) h (ˆx (t k ))). From (2.14), the foowng equaton can be wrtten: A (t k )= ( f ) h (x (t k ) ˆx (t k )) + K, (x (t k ) ˆx (t k )) + K, H.O.T h ˆx ˆx H.O.T f. I I (2.16) If K, s determned foowng (2.1), from (2.15) and (2.16), t can be obtaned that: A (t k ) H f X (t k ) ˆX (t k ) 2 + I K, H h x (t k ) ˆx (t k ) 2. (2.17) From (2.13), usng the trange nequaty and takng nto account (2.17), X (t k ) ˆX (t k ) 2 = I x (t k ) ˆx (t k ) 2 as we as y (t k )=h (x (t k ))+v (t k ), and from the Lpschtz property of f wth respect to x ( I ), the foowng nequaty can be obtaned: ė z, (t) F (z (t),h (x (t))) f (x (t), ) + L y F L h x (t) x (t k ) + L y F θ v + L w + (H f ) + K, H h e z, (t k ) 2 + K, θ v + L x x f (t) x (t k ) I I I f θ w (2.18) where e z, (t k )=ˆx (t k ) x (t k ). Takngntoaccountthatz (t k )=ˆx (t k ) and condton (2.6) and ntegratng (2.18) from t = t k to t = t k+1, the foowng nequaty can be obtaned: e z, (t k+1 ) β ( e z, (t k ), Δ) + L y F L h M Δ 2 /2+L y F θ v Δ+L w f θ w Δ + (H f ) + M K, H h e z, (t k ) 2 Δ+ M K, θ v Δ+ L x M f Δ 2 /2 I I I (2.19) where M, I, are constants that bounds ẋ n X (.e., ẋ M )andm K,, I,are constants that bounds K, n X (.e., K, M K, ). If γ (τ) andl, aredefnedasn Proposton 5, (2.19) can be wrtten n the form of Eq. (2.11). Ths proves Proposton 5. 15

25 From the formuaton of the oca MHE of Eq. (2.8), t can be seen that observer (2.8e) s operated n open-oop snce ony nformaton at t k 1 s used. The resut of Proposton 5 shows that wthn one sampng tme, the estmaton error gven by the open-oop observer (2.8e) s bounded and the upper bound depends on the Lpschtz propertes of the system, the property of observer (2.5), the sampng tme, the uncertantes nvoved n the system, and the nteractons between the subsystems. Theorem 6 beow takes advantage of the boundeness property of observer (2.8e) and provdes suffcent condtons on the convergence and utmate boundedness of the estmaton error of the proposed DMHE. Theorem 6 Consder system (2.4) wth the outputs of ts subsystems y samped at tme nstants {t k }. If the proposed DMHE mpemented foowng Agorthm 2 wth subsystem MHE desgned as n (2.8) based on determnstc nonnear observers satsfyng (2.6) and K, determned foowng (2.1) that are bounded, and f there exst concave functons g ( ), I, suchthat: g ( e ) β ( e, Δ) (2.2) for a e d and f there exst vector constants d s,, d such that d s, d,and postve constants a 1, b >, andɛ >, such that: d s, a g (d s, )+γ (Δ) + L, Δd 2 b θ v ɛ (2.21) I for a I, andfκ for a I, are pcked as foows: κ mn{(a 1)/L h,b }, (2.22) then the estmaton error e = ˆx x ( I) s a decreasng sequence f e () d for a I and s utmatey bounded as foows: m t sup e (t) d,mn (2.23) for I wth d,mn =max{ e (t +Δ) : e (t) d s, } for a e () d and x X. Ths aso mpes that the entre system state estmaton error s utmatey bounded. Proof: We prove that the evouton of the estmaton error of each subsystem state e = ˆx x, I, under the proposed DMPC wth the oca MHE of Eq. (2.8) s a decreasng sequence and s utmatey bounded n a sma regon. The decrease and utmate boundedness of subsystem estmaton errors mpy the decrease and utmate boundedness 16

26 of the entre system state estmaton error. Specfcay, we frst focus on MHE, I, for the tme nterva from t k to t k+1 andthenextendtothegeneracase. From constrant (2.8g) for MHE, t can be wrtten that: ˆx (t k+1 ) z (t k+1 ) κ y (t k+1 ) h (z (t k+1 )). (2.24) From the Lpschtz property of h, the fact that y = h (x )+v and v θ v, t s obtaned that: ˆx (t k+1 ) z (t k+1 ) κ L h x (t k+1 ) z (t k+1 ) + κ θ v (2.25) where L h s the Lpschtz constant of h as defned n Proposton 5. Usng the trange nequaty ˆx x ˆx z + z x, t s obtaned from (2.25) that: ˆx (t k+1 ) x (t k+1 ) (1 + κ L h ) x (t k+1 ) z (t k+1 ) + κ θ v. (2.26) From Proposton 5 and (2.26), and notcng that e (t k )=e z, (t k ), the foowng nequaty can be obtaned: e (t k+1 ) (1 + κ L h ) β ( e (t k ), Δ) + γ (Δ) + L, Δ e (t k ) 2 + κ θ v. (2.27) I If condton (2.2) s satsfed, from (2.27), t can be obtaned that: e (t k+1 ) (1 + κ L h ) g ( e (t k ) )+γ (Δ) + L, Δ e (t k ) 2 + κ θ v. (2.28) I If there exsts d s, satsfy (2.21) and κ s pcked foowng (2.22), then (2.21) hods for a d s, e d takng nto account that g ( ) s a concave functon; that s: e (1 + κ L h ) g ( e )+γ (Δ) + L, Δ e 2 κ θ v ɛ (2.29) I for a d s, e d and e d ( I ). From (2.28) and (2.29), t can be obtaned that: e (t k+1 ) e (t k ) ɛ (2.3) for a d s, e d.if e d s, for a the tme from to t k, usng (2.3) recursvey, t can be obtaned that: e (t k ) e () kɛ (2.31) for a d s, e (t k ) d. Ths mpes that e decreases every sampng tme and w become smaer than d s, n fnte steps. Once e s, <d s,, t w reman to satsfy e (t) 17

27 d,mn whch s ensured by the defnton of d,mn ; that s, m sup e (t) d,mn. Note that t the above proof hods for a I. The utmate boundedness of subsystem state estmaton error mpes the utmate boundedness of the entre system state estmaton error. Ths can be seen from the nequaty e e whch mpes m that: =1 Ths proves Theorem 6. m sup e m d,mn. (2.32) t =1 Remark 7 The purpose of the ntroducton of g as n condton (2.2) s to ensure that when condton (2.21) s satsfed for e = d s,,tsasosatsfedford s, e d whch mpes that f e d s,, e w be decreasng. For many of the exstng nonnear observers (e.g., [31, 3, 33, 37]) that provde exponentay convergence rates such that e (t) λ e () exp( α t) wth λ and α postve numbers, condton (2.2) can be easy satsfed. Remark 8 Referrng to condton (2.21) n Theorem 6 (or (2.29) n the proof), the term g ( e (t k ) ) denotes an upper bound on the error vaue after one sampng tme (.e., e (t k+1 ) ) f the nta error vaue s e (t k ) for the nomna subsystem wthout nteractons under contnuous output y feedback; the term γ (Δ) represents the effects of samped-and-hod mpementaton of nonnear observer (2.5), measurement nose and process dsturbances; the term I L, Δd 2 bounds the effect of subsystem nteractons; and the term κ θ v represents the uncertanty ntroduced nto condton (2.8g) due to measurement nose. Condton (2.21) essentay requres that the assumed nonnear observer (2.5) for the nomna system wthout nteractons converges to the actua nomna subsystem state fast enough such that ts contrbuton to the decease of the estmaton error domnates the effects caused by other factors that contrbute to the ncrease of the estmaton error. Remark 9 Note that Theorem 6 provdes a set of suffcent condtons that essentay decoupe the error dynamcs of each subsystem. Condton (2.21) nvoves the nta estmaton error d, I, of the nteractng subsystems of subsystem. Ths set of suffcent condtons requres that the nta estmaton errors of the subsystems shoud be suffcenty sma. In other words, the convergence rates of the nonnear observers of Eq. (2.5) shoud be hgh enough to reject the effects of the nta estmaton errors. 18

28 Remark 1 From Theorem 6 as we as Remark 8, t can be seen that the convergence rates of the nonnear observers of Eq. (2.5) for the nomna subsystems wthout nteractons pay an mportant roe n the convergence rate of the estmates of the proposed DMHE to the actua system states. Ths mpes that t s possbe to tune the convergence rate of the proposed DMHE by tunng the convergence rates of the nonnear observers of Eq. (2.5). By examnng condton (2.21), t can be found that when we tune the nonnear observers to ncrease ther convergence rates (.e., g (e ) decreases), the negatve effects caused by the samped-and-hod mpementaton (.e., γ (Δ)) ncrease at the same tme whch mpes the ncrease of the vaue of d,mn. Ths s because wth the ncrease of the convergence rates of the nonnear observers, ther Lpschtz constants (.e., L y F )ncrease. In order to overcome the above ssue, two approaches may be used. Frst, a nonnear observer of Eq. (2.5) wth swtched convergence rates s used n the desgn of the MHE of Eq. (2.8) for each subsystem. Specfcay, a hgh convergence rate s adopted when the estmaton error s arge and a ow convergence rate s used when the estmaton error s sma. By appyng ths approach, a hgh convergence rate of each oca MHE can be acheved whe keepng the vaue of d,mn sma [63]. A second approach that may be used to further mprove the performance of the frst approach s to use/requre contnuous output measurements for the tme perod when the hgh convergence rate s used n the evauaton of the nonnear observer of Eq. (2.5). Ths approach can sgnfcanty reduce the negatve effects caused by the samped-and-hod mpementaton of the nonnear observer n the case of hgh convergence rate. Remark 11 The proposed DMHE scheme ntegrates determnstc nonnear observer desgn technques and MHE. It can ncrease the robustness and reabty of the observer over ether determnstc observers or cassca MHE as w be demonstrated n Secton 2.6 (see aso [5, 64]). Ths s due to constrant (2.8g) n each subsystem MHE desgn whch ensures that the MHE nherts the robustness of the determnstc nonnear observer. The proposed approach, however, requres more efforts n the nta desgn stage and n the tunng of the parameters. Remark 12 Note that n ths work, we consder a type of bounded mode msmatch (.e., process dsturbances). Other types of mode msmatches, such as uncertantes n mode parameters or mode structure, can be consdered n a smar fashon as ong as the mode msmatches are bounded and the auxary nonnear observers are desgned foowng the assumptons n Secton 2.4. Note aso that even we do not expcty consder mode ms- 19