. A movng horzon scheme for dsrbued sae esmaon arcello Farna, Gancarlo Ferrar-Trecae and Rccardo Scaoln The research of.f. and R.S. has receved fundng from European Communy hrough FP7/7-3 under gran agreemen n. 3854 ( Herarchcal and Dsrbued odel Predcve Conrol of Large Scale Sysems, HD-PC projec).. Farna and R. Scaoln are wh he Dparmeno d Eleronca e Informazone, Polecnco d lano, lan, Ialy farna,rccardo.scaoln @ele.polm. G. Ferrar-Trecae s wh he Dparmeno d Informaca e Ssemsca, Unversa degl Sud d Pava, Ialy gancarlo.ferrar @unpv.
CONTENTS I Inroducon 5 II Problem formulaon and soluon 6 II-A Local, regonal and collecve models........................ 6 II-B The dsrbued esmaon algorhm......................... 7 II-C Compuaon of he marces Π N and esmaon procedure.......... 8 III Convergence properes of DHE IV Example V Conclusons 3 References 3 Appendx 4 A Proof of Proposon................................. 4 B Furher noaon and defnons: collecve esmaon................ 5 C Frs convergence resul................................ 6 D The convergence of he proposed algorhm..................... 9 E Proof of Theorem..................................
I. INTRODUCTION A sensor nework consss of a se of elecronc devces, wh sensng and compuaonal capables, whch coordnae her acvy hrough a communcaon nework. They can be employed n wde range of applcaons, such as monorng, exploraon, survellance or o rack arges over specfc regons. The dffuson of sensor neworks s parly due o he recen developmens n wreless communcaons and o he avalably of low cos devces. On he oher hand, many heorecal and echnologcal challenges have sll o be be ackled n order o fully explo her poenales. Among he open problems, he use of sensor neworks for dsrbued sae esmaon s of paramoun mporance. The problem can be descrbed as follows. Assume ha any sensor of he nework measures some varables, compues a local esmae of he overall sae of he sysem under nvesgaon, and ransms o s neghbors boh he measured values and he compued sae esmaon. Then, he man challenge s o provde a mehodology whch guaranees ha all he sensor asympocally reach a common relable esmae of he sae varables,.e. he local esmaes reach a consensus. Ths goal mus be acheved even f he measuremens performed by any sensor are no suffcen o guaranee observably of he process sae (namely, he local observably), provded ha all he sensors, f pu ogeher, guaranee such propery (namely, he collecve observably). The ransmsson of measuremens and of esmaes among he sensors mus lead o he wofold advanage of enhancng he propery of observably of he sensors and of reducng he uncerany of all he nodes sae observaon. Consensus algorhms for dsrbued sae esmaon based on Kalman flers have recenly been descrbed n [3], [], [9], [7], [4], [8], [6]. In parcular, n [9], [7], [4], consensus on he measuremens s used o reduce her uncerany and Kalman flers are appled by each agen. In [8], hree algorhms for dsrbued flerng are proposed. The frs algorhm s smlar o he one descrbed n [7], save for he fac ha paral measure of he sae vecor s allowed o he sensors. The second approach reles on communcang he sae esmaes among neghborng agens (consensus on he sae esmaes). The hrd algorhm, named erave Kalman consensus fler, s based on he dscree-me verson of a connuous-me Kalman fler plus a consensus sep on he sae esmaes, whch s proved o be sable. Despe s neresng properes, for he dscree-me verson of he algorhm, sably s no proved and opmaly of he esmaes s no addressed. Recenly, convergence n mean of he local sae esmaes obaned wh he algorhm presened n [7] s proved n [6], provded ha he observed process s sable. In [] consensus on he esmaes s used ogeher wh Kalman flers. The weghs of he sensors esmaes n he consensus sep and he Kalman gan are opmzed (n wo subsequen seps) o mnmze he esmaon error covarance: however, n so dong opmaly s no guaraneed. A wo-sep procedure s also used n [3], where he consdered observed sgnal s a random walk. In he proposed algorhm flerng and consensus are performed subsequenly, and he esmaon error s mnmzed wh respec o boh he observer gan and he consensus weghs. Ths guaranees opmaly of he soluon. Recenly, an neresng soluon o he problem of dsrbued esmaon of a parameer vecor, wh nosy lnear measuremens, has been proposed n []. The algorhm accouns for dynamcally changng nerconnecons among sensors, unrelable communcaon lnks, and fauls. Asympoc convergence of he esmae o he rue parameer s proved, under suable hypohess of dynamcal graph connecvy. In hs paper we propose a dsrbued algorhm based on he concep of ovng Horzon Esmaon (HE), [], []. Ths approach has many advanages; frs of all, he observer s opmal n a sense, snce a suable mnmzaon problem mus be solved on-lne a each me nsan. Furhermore, we prove ha, under weak observably condons, convergence of he sae esmae s guaraneed n a deermnsc framework. Fnally, consrans on he nose are aken no accoun, as s common n recedng horzon approaches n conrol and esmaon, [5]. The paper s srucured as follows. In Secon II we nroduce he dynamcal sysem we wan o observe and he srucure of he sensor nework. We defne a number of observably properes of hs nework and we descrbe he dsrbued sae esmaon algorhm. In Secon III we nvesgae he convergence properes of he algorhm. In Secon IV we presen a smulaon example, whle Secon V repors some concludng remarks and hns for fuure developmens. All he proofs are repored n he Appendx. 5
II. PROBLE FORULATION AND SOLUTION The observed process s descrbed by he lnear dscree-me dynamcs: x Ax w () n ( ) represens a dsurbance wh varance n s a gaussan random varable wh mean µ and covarance marx Π. easuremens on he sae vecor are performed by sensors, accordng o he sensng model (n general dfferen from sensor o sensor): where x n s he sae vecor and w Q n n. We assume ha he se s convex. The nal condon x y C x v () where he erm v p s a whe nose wh varance R p p and he marx C s non null. The communcaon nework among sensors s descrbed by he dreced graph G V E, where he nodes n V represen he sensors and he edge j n he se E V V models ha sensor j can ransm nformaon o sensor. We assume E, V. oreover, we denoe by V he se of neghbors o node,.e., V j : j E. We assocae o he graph he sochasc marx K wh enres k j! f j E k j oherwse (3) j" k j In he followng, we assume he marx K s gven,.e. wll be no consdered as a desgn parameer for sae esmaon. A. Local, regonal and collecve models We assume ha, a a generc me nsan, a gven sensor can collec he measuremen produced by self and by s neghborng sensors. In oher words, we assume sensors communcae only once whn a samplng nerval. We can now dsngush hree ypes of quanes: local, regonal, and collecve. Wh reference o sensor, a quany s referred o as local when s relaed o he node solely, whle s called regonal f s relaed o he nodes n V. Fnally, we say ha a quany s collecve, f s relaed o he whole nework. For he sake of clary, we use dfferen noaons for local, regonal and collecve varables. Namely, gven a varable z, z represens s local verson, z s s regonal counerpar, and z he collecve verson. For nsance, we refer o y n () as local measuremen. On he oher hand, f V # j,..., j v, he regonal measuremen of node s: where ȳ %$ y j T y j v T& T, C (' C j T ȳ C x v (4) C j v T) T, and v *$ T. The dmenson of vecors ȳ and v, and he number of rows of marx C s p v v j T k" p j k v j v T&. Furhermore, we denoe by. R p + p, he covarance marx relaed o he regonal nose v on sensor,.e., R dag R j, R j v Accordng o he adoped ermnology, hree dfferen observably noons can be nroduced n hs framework, namely local, regonal and collecve observably. Defnon : The sysem s locally observable by sensor f he par A C s observable. The sysem s regonally observable by sensor f he par A C s observable. The sysem s collecvely observable f he par A C-. s observable, where C- / C T C T T. Noce ha, for a gven sensor, local observably mples regonal observably, and regonal observably of any sensor mples collecve observably, whle all oppose mplcaons are no rue. Gven a sngle sensor model ()-(), he -h sensor regonal observably marx O n s: O n 43 C T C A T 6 C A n T5 T (5)
Le P NO be he orhogonal projecon marx on ker O n, ha s he regonally unobservable subspace. Smlarly, le P O be he orhogonal projecon on he regonal observably subspace ker O n,6. Nex, we recall how P O and P NO can be compued. Le r rank O n and denoe wh ξ r,, ξ n an orhonormal bass of ker O n. Le also ξ,, ξ r be an orhonormal bass of ker O n 6 and defne he non-sngular marx: Defnng he marces S O and S NO as: T 7/ ξ ξ n S O 8$ I r & 9 n r : r S NO r 9 n r <; : I n r = we have P O T S O S O T T and P NO T S NO S NO T T. Furhermore, defnng T dag > T? T @, S O dag > S O? S O @, and S NO dag > SNO, SNO@, he collecve projecon marces are P O TS O S T T O and P NO TS NO S T T NO. Noe ha S NO s empy when he sysem s regonally observable by sensor. In hs case we assume ha P NO n n. B. The dsrbued esmaon algorhm Our am s o desgn, for a generc sensor V, an algorhm for compung a relable esmae of he sysem s sae based on regonal measuremens ȳ and furher peces of nformaon provded by sensors j V. The proposed soluon reles on he use of HE, see [], [], [5], n vew of s capably o handle nose consrans. ore specfcally, we propose a Dsrbued HE (DHE) scheme where each sensor solves a HE problem. For a gven esmaon horzon N A, each node V a me solves he consraned mnmzaon problem HE- defned as under he consrans Θ- mn J ˆx B NC D ŵ ke B G N ˆx N ŵ ˆ v Γ N (6) kf B N where k G N, and he local cos funcon J s ˆx k A ˆx k ŵ k (7a) ȳ k C ˆx k ˆ v k (7b) ŵ k (7c) J H I NJ J ˆx K NJ ŵ J ˆ v J ΓK NLN O kp K N Q ˆ v kq RR S K kp K N Q ŵ kq QR T Γ S K H N ˆx K N ; ˆ x K NU K LWV In (8) and hereafer, he noaon X zx S sands for z T Sz, where S s a posve-semdefne marx. We denoe by ˆx N and Y ŵ k Z k" he opmzers o (6) and wh N ˆx k, k G N he local sae sequence semmng from ˆx N and Y ŵ k Z k". Furhermore, ˆ x N N denoes he weghed average of sae esmaes produced by sensors j V,.e. (8) ˆ x N k j ˆx j" j N (9) 7
e J In (8), he funcon Γ N ˆx N ; ˆ x N s he so called nal penaly, defned as follows: Γ N ˆx N ; ˆ x N Γ NOC N Γ OC N ˆx N ; ˆ x N ˆx N ; ˆ x N (a) where: Γ NO[ K N H ˆx K N; ˆ x K NU K LN Γ O[ K N H ˆx K N; ˆ x K NU K LN Q P H NO ˆx K N I ˆ x K NU K L Q \ Π R N] R ^ R Q ˆx K N I ˆ x K NU K Q \ P NO^ \ T Π R N] R ^ R P NO Q P H O ˆx K N I ˆ x K NU K L Q \ Π R N] R ^ R S Θ_ K Q ˆx K N I ˆ x K NU K Q \ P O^ \ T Π R N] R ^ R P O S Θ_ K (b) (c) where he consan erm Θ- s defned n (6) and known a me. For hs reason, could be negleced when solvng he opmzaon problem. However, snce plays a major role n esablshng he man convergence properes of he proposed DHE, s here mananed for clary of presenaon. Noe ha ˆx j compued by sensor j a me G and herefore, n vew of N s he esmae of x?` N he defnon of k j n (3), Γ depends only upon regonal quanes. Snce also he cos (8) and he consrans (7) depend?` only upon regonal varables, he overall esmaon scheme s decenralzed. Fnally, noce ha Γ embodes a consensus erm, n he sense ha penalzes devaons of ˆx N from he weghed average of he sae esmaes produced by he neghbors o sensor. Consensus, besdes ncreasng accuracy of he local esmaes, s fundamenal o guaranee convergence of he sae esmaes o he sae of he sysem even f regonal observably does no hold. In fac, allows sensor o reconsruc componens of he sysem sae ha canno be esmaed by he -h regonal model. The posve defne symmerc marx Π N appearng n () plays he role of a covarance marx and s a desgn parameer whose choce wll be dscussed n deals n he followng. C. Compuaon of he marces Π N and esmaon procedure Le us defne: Π dag a Π? Π b () We requre he marx Π N o sasfy he followng Lnear arx Inequaly (LI): K T a P T Π O N P O P T Π NO N P NO b K c Π N () n n, he symbol d where K K d I n denoes he Kronecker produc, I n s he n n deny marx and Π N dag a Π N, Π N b. The marx Π N, V, s gven by one eraon of he dfference Rcca equaon assocaed o he Kalman fler for he sysem x N Ax N w N z N ON x N V N 8
S S S S where marx O N s defned n (5) and C N gfh h jlk k C...... C A N C A N m 3 C p N n 9 N : (3) R N dag > R? R p @ N p N (4) Q N n dag Q, Q 9 N : 9 n N : (5) Cov/ w Q (6) Cov/ V R N CNQ N CN T s se as The uncerany of he esmae ˆx N Π_ K NK U K HnH P OL T H ΠK NK U K L K P O H S P NOL T H ΠK NK U K L and yelds he Rcca equaon updae Π K NU K o R Π_ K NK U K ;QJ R N S AΠ_ K NK U K AT S q o C Π_ K NK U K H C T L Q I AΠ_ K P NOL H C T L H R L K K C CNQ NK H CNL K NK U K H C T L q R p K C Π_ K NK U K AT (7) (8) Tp (9) In he followng we skech he seps ha have o be carred ou, n pracce, n order o apply he proposed algorhm: r A all nodes sore he marx Π and he esmae ˆx µ of x, where µ s gven. r f c c N, he esmaon horzon N s reduced o Ñ and node V performs he followng seps: compue Π Ñ Π from Π accordng o (), for all V, solve he problem HE-, wh nal penaly Γ K Ñ Q P H O ˆx I ˆ x U K Q \ Π ] R ^ R S Q P H NO ˆx I ˆ x U K L Q \ Π ] R ^ R J r f! N, a each me nsan: compue Π N from Π N accordng o (), (8) and (9), for all V, for all nodes V, solve he problem HE-, wh nal penaly Γ K N Q P H O ˆx K N I ˆ x K NU K L Q \ Π R N] R ^ R S Q P H NO ˆx K N I ˆ x K NU K L Q \ Π R N] R ^ R J The LI () deserves a few commens. Frs, condon () s requred o guaranee convergence of he DHE scheme. Second, he choce of Π N verfyng () s no unque. Inuvely, marces Π N model he uncerany one has abou he erm ˆx N G ˆ x N and herefore one would make he lef hand sde of () as close as possble o zero. A way for achevng hs s o solve he LI problem subj. o () () mn a race Π N b where Π N has he srucure gven n (). Noce ha () can be solved by each sensor snce, smlarly o he formula for updang covaraces n Kalman flerng, he compuaon of Π N does no depend upon he colleced measuremens. 9
s However, problem () has a cenralzed flavor snce each sensor needs o know he marces PO and of all sensors, and he marx K encodng he graph opology. These lmaons are severe snce, P NO for nsance, he LI () has sze n whch mples ha he compuaonal burden requred a each sensor for solvng () scales wh he number of sensors, hence hamperng he applcaon of DHE o large neworks. The nex proposon provdes a way o crcumven hs problem. Proposon : The marces Π N whch sasfy, V K u H POL T H ΠK NU K L K P O S H P NOL T H Π K NU K L k jp j Π K j NU K () also sasfy he LI (). Therefore, he mnmzaon problem () can be replaced by he followng decenralzed one, performed by each sensor V : K P NO mn a race Π N b, subjec o (). Noce ha, n he soluon provded by Proposon, each node compues Π N of he nformaon provded by s neghbors, ha amouns o he marces Π j N, j V. III. CONVERGENCE PROPERTIES OF DHE solely on he bass When he nework s composed by a sngle sensor one has K and DHE reduces o he HE scheme, for whch convergence and sably have been esablshed n []. The man purpose of hs Secon s o exend he convergence resuls of [] o he proposed DHE scheme. Defnon : Le Σ be sysem () wh w and denoe by x he sae reached by Σ a me sarng from nal condon x. Assume ha he rajecory all. DHE s convergen f X ˆx G x wx x s feasble,.e., x v for x Gzy for all V. Noe ha, as n [], convergence s defned assumng ha he model generang he daa s noseless, bu he possble presence of nose s aken no accoun n he sae esmaon algorhm. Now, f we defne he collecve vecors ˆx / ˆx T ˆx T T n and x { n d x he followng nermedae resul can be saed Lemma : If marces Π N are compued as n Secon II-C, N A n G and N A, hen he dynamcs of he sae esmaon error provded by he DHE scheme s gven by where A dag A A ˆx K NU I H I NJ x LN ΦH ˆx K NK U K I H I N I J x LnL S n n, and α s an asympocally vanshng erm,.e. X α X α () Φ P NO KAP NO (3) x Gzy. The nex heorem provdes condons for convergence of DHE. Theorem : Under he assumpons of Lemma, DHE s convergen f and only f he marx Φ s Schur. The fundamenal assumpon ha marx Φ s Schur s movaed by Lemma and does no requre ha sysem () s asympocally sable. oreover, Theorem does no hnge on observably properes. For nsance, embraces he case where a componen of he sae of sysem () s no observed by any sensor bu he esmaon error decays o zero because he sae componen has he same propery. Neverheless, undersandng how observably condons affec he error dynamcs s a opc of grea neres. As a movaon, consder he problem of desgnng a lnear sae esmaor when he nework
} s composed by a sngle sensor. I s well known ha f sysem () s deecable he error dynamcs wll nher he unobservable egenvalues of A. However, f sysem () s observable one can desgn esmaors such ha he egenvalues of he error dynamcs do no depend upon he egenvalues of A. In our framework, hs rses he problem of sudyng condons for guaraneeng ha he marx Φ does no nher some egenvalues of A. ore formally, le λa and v A be he egenvalues and he egenvecors of A, respecvely, wh n. Noe ha, n vew of s defnon, he egenvalues of A are λa (? n), each one wh mulplcy. oreover, denoe by e j ( j ) he canoncal bass, so ha he egenspace relaed o λa s span(e d v A e d v A ). We consder he followng vecors of propery. Propery : For all n and for all x span(e d v A e d v A ), λ A and x are no an egenvalue/egenvecor par for Φ. Before gvng he man resuls, we nroduce he defnon of solaed subgraph. If he graph G s no srongly conneced (.e., s reducble), one can paron G no l nonempy rreducble subgraphs G j - N j - E j-, j? l (see e.g. [4]). If p N - G- s solaed. Remark ha, f G s srongly conneced, s also solaed. Theorem : Consder a paron of G no he rreducble subgraphs G -, solaed srongly conneced subgraphs G- holds j~ N and q V p mples ha q N - ker O n j we say ha, l. If for all he hen Propery s verfed. In he case of srongly conneced graphs we have he followng resul. Corollary : If G s srongly conneced and he sysem s collecvely observable, hen Propery s verfed. An example can provde beer undersandng of he meanng of solaed subgraphs, see Fg.. G G G3 3 Fg.. The graph s decomposed no hree conneced subgraphs: G, G and G 3. Noce ha he node (of G ) s a neghbor of node 3 of G 3. From hs, follows ha graph G 3 s no solaed. Analogously, he graph G s no solaed, n ha he node (of G ) s a neghbor of node. Accordng o he defnon, he subgraph G s solaed. Assumpon (??) saes ha collecve observably (see Secon II-A) s requred o he nodes of he subgraph G solely. As a parcular case, assume ha all sensors enjoy regonal observably and are arranged n a srongly conneced graph G. Ths yelds P NO Φ n n and convergence of DHE follows from Theorem. oreover, snce he sysem s also collecvely observable, Corollary guaranees ha Propery holds.
We consder he fourh order sysem x fh IV. EXAPLE j k 996 949 G 949 389 m x w (4) G 98 where x / x C x C x 3C x 4C T. Noce ha he egenvalues of he sysem s marx A are 964, 457, 99 4795 and, snce 99 4795!, he sysem s unsable. Le e R 4, be whe nose wh covarance Q e dag 38 38. In he followng we consder wo cases A. w e, Q Q e and 4 (unconsraned npu nose) B. w e, Q Q e and 4 (consraned npu nose) In boh cases, we se µ 7/ T, Π I n and N n he DHE algorhm. The sae of (4) s measured by 4 sensors wh sensng model y / x v f y / x v f 3 4 where Var v R,? 4. Sensors are conneced accordng o he graph n Fg., where he marx K s also gven. I s apparen ha he nformaon avalable, a each nsan, o node consss 4 3 Fg.. Communcaon nework and assocaed marx K of he measuremens of x C and x 3 C (ransmed by sensor 4). Analogously, he nformaon avalable o node 3 consss of x C (ransmed by sensor ) and x 3 C. I s easy o check ha he sysem s regonally observable by sensors and 3. On he oher hand, a each me nsan sensor can only use wo dfferen measuremens of x C (produced by sensors and ). Smlarly, sensor 4 can only use wo dfferen measures of x 3 C (produced by sensors 3 and 4). Therefore, he sysem s no regonally observable by sensors and 4. In fac, P NO dag, P NO 4 dag. The egenvalues of he marx Φ defned n (3) are, 463, 58 and 495 397. Snce Φ s Schur, convergence of DHE s guaraneed by Theorem. oreover, snce he graph s srongly conneced and collecve observably holds, Corollary guaranees ha also Propery holds. In Fg. 3 he esmaon errors produced by all sensors n he scenaro A are shown. I s worh nocng ha he esmaes produced by sensors and 4, relave o saes x 3 C, x 4C and x C, x C, respecvely, dsplay bg errors for ƒ 6. In fac, hese saes canno be observed by hese sensors usng regonal measuremens. Noneheless, he esmaon errors of all sensors asympocally end o he same values, hanks o he consensus acon emboded n he proposed algorhm. Fg. 4 depcs he evoluon of he egenvalues of marces Π N over me. Noe ha hese marces are he same n he cases A and B. Indeed, he updae procedure descrbed n Secon II-C does no depend on he esmaes and can be run off-lne. The esmaon errors for case B are depced n Fg. 5. Analogously o case A, convergence of DHE can be noced.
4 6 8 4 3 4 e () (=,...,4) e () (=,...,4) e () (=,...,4) 3 e () (=,...,4) 4 4 6 8 4 6 8 Fg. 3. Case A: componens of he esmaon error e C e C e 3 C e T 4 " x ˆx of he dfferen sensors. " (sold lne), " (doed lne), " 3 (dashed lne), " 4 (dash-doed lne). Sensor 5 Sensor 5 Sensor 3 5 Sensor 4 4 6 8 4 6 8 Fg. 4. Evoluon of he four egenvalues (for each sensor) of he marces Π ˆ NB, " CŠ Š Š ŠC 4. V. CONCLUSIONS any generalzaons of he DHE scheme descrbed n he paper can be consdered n order o enhance s poenales. A frs one s he developmen of DHE schemes enjoyng convergence even n presence of sae consrans. In addon we wll sudy how o: () explo he degrees of freedom n he choce of he marx K n order o mprove he speed of convergence of he sae esmaes provded by sensors; () exend he DHE scheme o me-varyng graphs; () use properly mulple ransmssons beween sensors whn a samplng me. REFERENCES [] P. Alrksson and A. Ranzer. Dsrbued Kalman flerng usng weghed averagng. In Proc. of he 7h Inernaonal Symposum on ahemacal Theory of Neworks and Sysems, Kyoo, Japan, 6. [] G.C. Calafore and F. Abrae. Dsrbued lnear esmaon over sensor neworks. Inernaonal Journal of Conrol, acceped. [3] R. Carl, A. Chuso, L. Schenao, and S. Zamper. Dsrbued Kalman flerng based on consensus sraeges. IEEE journal on Seleced Areas In Communcaons, pages 6 633, 7. [4] L. Farna and S. Rnald. Posve Lnear Sysems. J. Wley and Sons,. [5] G.C. Goodwn,.. Seron, and J.A. De Doná. Consraned Conrol and Esmaon. Sprnger, New Jersey, 5. [6]. Kamgarpour and C. Tomln. Convergence properes of a decenralzed Kalman fler. Proc. Conference on Decson and Conrol, pages 35 3, 8. [7] R. Olfa-Saber. Dsrbued Kalman fler wh embedded consensus flers. Proc. Conference on Decson and Conrol - European Conrol Conference, 5. 3
; 8 6 4 4 3 4 e () (=,...,4) e () (=,...,4) e 3 () (=,...,4) e 4 () (=,...,4) 4 6 8 4 6 8 Fg. 5. Case B: componens of he esmaon error e C e C e 3 C e T 4 " x ˆx of he dfferen sensors.. " (sold lne), " (doed lne), sensor " 3 (dashed lne), " 4 (dash-doed lne). [8] R. Olfa-Saber. Dsrbued Kalman flerng for sensor neworks. Proc. Conference on Decson and Conrol, 7. [9] R. Olfa-Saber and J. Shamma. Consensus flers for sensor neworks and dsrbued sensor fuson. Proc. Conference on Decson and Conrol - European Conrol Conference, pages 6698 673, 5. [] C. V. Rao and J.B. Rawlngs. Nonlnear movng horzon sae esmaon. n F.Allgöwer and A. Zheng, edors, Nonlnear odel Predcve Conrol, Progress n sysems and Conrol Theory, Brkhauser, pages 45 7,. [] C. V. Rao, J.B. Rawlngs, and J.H Lee. Sably of consraned lnear movng horzon esmaon. Proc. Amercan Conrol Conference, pages 3387 339, 999. [] C. V. Rao, J.B. Rawlngs, and J.H. Lee. Consraned lnear sae esmaon - a movng horzon approach. Auomaca, 37:69 68,. [3] C. V. Rao, J.B. Rawlngs, and D.Q ayne. Consraned sae esmaon for nonlnear dscree-me sysems: Sably and movng horzon approxmaons. IEEE Trans. Auomac Conrol, 48():46 58, 3. [4] D.P. Spanos, R. Olfa-Saber, and R.. urray. Approxmae dsrbued Kalman flerng n sensor neworks wh quanfable performance. Proc. Conference on Decson and Conrol - European Conrol Conference, 5. A. Proof of Proposon Proof: Frs defne, for smplcy, and recall ha Π- N ha, usng Schur complemen, s equvalen o APPENDIX Π- N P T O Π N P O P T NO Π N P NO (5) has a block dagonal form, see () and (5). The LI () s equvalen o: Π N G K T Π- N K A (6) Π N K T K Π- N = A (7) In vew of hs, beng marces Π- N and Π N posve defne, (7) s equvalen o: Π- N G K Π N KT A (8) For all vecors x / x T x T T n, from (8) holds ha x T Π- N x A xt K Π N KT x (9) 4
X Noce ha he j-h block of K T x corresponds o " k j x. So, he rgh hand sde of equaon (9) can be wren as: Usng he rangle nequaly we oban: X j" " k j x X Π B j N B X K T xx Π B N B j" c j" " k j x X Π " j B N B k j X x X Π B j N B X x X " jf k Π j B j N B In vew of hs, o sasfy nequaly (9) s suffcen o requre ha, for all, : Π- N ' PO T Π N PO Ths concludes he proof. P NO T Π N P NO ) A k j j" Π j N B. Furher noaon and defnons: collecve esmaon We denoe he overall cos funcon J as he sum (for all he sensors) of he local cos funcons J (8): J?` We defne he followng collecve vecors: ˆx fh ˆx ṭ. ˆx jlk m ˆ v fh J G N ˆx N ŵ ˆ v Γ N (3) " ˆ v ṭ. ˆ v jlk m F p ŵ fh ŵ. ŵ jlk m and he marces?` R dag > R? R @ F p F p, Q dag Q, Q cos funcon J can now be rewren as: wh where J G N ˆx N ŵ ˆ v Γ N X ˆ v k X RB k" N n n n. The overall X ŵ k X QB Œ Γ N ˆx N k" ; ˆx N (3) N Γ N ˆx N ; ˆx N Γ NO N ˆx N ; ˆx N Γ O N ˆx N ; ˆx N N Γ NO Γ O N ˆx N ; ˆx N ˆx N ; ˆx N X ˆx N G Kˆx N X P B T NO Π B N B P NO X ˆx N G Kˆx N X P B T O Π B N B Θ- P O n n where Θ- " Θ-. Recallng A dag A A and defnng C dag C C F p n, also consrans (7) can be wren n he followng collecve form: (3a) (3b) ˆx k A ˆx k ŵ k (33a) ȳ k ŵ k C ˆx k ˆ v k 5 (33b) (33c)
wh k G N. I s mporan o noe ha solvng he followng problem: mn ˆx B NC D ŵ ke B J G N ˆx N ŵ ˆ v Γ N (34) kf B N s equvalen o solve he HE G problems (6), n he sense ha ˆx N ŵ k k" s a soluon N o (6) f and only f ˆx N ŵ k k" N s a soluon o (34), where ŵ k / ŵ k T ŵ k T T. Remark : when he sae s generaed by sysem Σ (see Defnon ), he sae sequence ˆx x { n d x and he nose sequences w and ˆ v, A verfy he consrans (33), A N. We defne he rans cos n a generc pon x Ž n, compued a nsan (wh G N c c ) n he followng way: Ξ x mn ˆx B NC D ŵ ke B Y J G N ˆx N ŵ ˆ v Γ N subj. o (33) and ˆx x Z (35) kf B N?` The mnmum value of he oal cos funcon J s: Θ- mn ˆx B NC D ŵ ke B kf B N J G N ˆx N ŵ ˆ v Γ N subjec o (33) C. Frs convergence resul The proof of Theorem uses classcal resuls for HE, [], [], [], [3] and addonal resuls we provde n hs Secon. Frs we nroduce wo assumpons on nal penales: (A) Γ O N ˆx N ; ˆx N A Θ-,, (A) Γ N ; ˆx N c Ξ N Assumpons (A) and (A) are smlar o he assumpons (C) and (C) n [3], ha are fundamenal for provng convergence of HE when appled o nonlnear consraned sysems. However, hree dfferences arse: frs, n (A) jus he observable componen of he nal penaly s used. Also, n (A), he rans cos nsead of he arrval cos appears, and he bound s requred only when Γ N and Ξ N are evaluaed n he sae sequence provded by Σ (see Defnon ). Theorem 3: If N A n G (wh N A ), and Assumpons (A) and (A) hold, hen he dynamcs of he sae esmaon error provded by he DHE scheme s gven by (). Snce he proof of Theorem 3 requres a number of prelmnary resuls, s organzed n a number of seps. Lemma : Assume ha he nal penales verfy Assumpon (A). Then: Θ- G Θ- A X ˆ v k k" X RB N X ŵ k k" X QB Γ NO N ˆx N ; ˆx N Œ (36) N Proof: [Proof of Lemma ] Under Assumpon (A), he nequaly (36) can be proved by drec calculaon, usng he fac ha:?`?`?` Γ O N Γ NO N A Γ NO N Θ- Lemma 3: If Assumpon (A) s sasfed, hen: Then: Θ- c Γ x ;x for all (37) where x 7/ x T x T T n and x { d µ. Proof: [Proof of Lemma 3] By opmaly, we have 6
G oreover, from Remark, holds Ξ N Θ- c Ξ N G N x G N x c J G N Γ N Noe ha, from (3), one has J G N Γ N Γ N and, n vew of Assumpon (A), Θ- c Ξ N c Γ N We can furher erae hs procedure and prove, by nducon, equaon (37). Lemma 4: Assume ha N A n G, wh N A, and ha: max k" NC C X ˆ v k X X ŵ k X Γ NO N ; ˆx N G N G x ; ˆx N. ˆx N ; ˆx N x Gzy (38) Then he dynamcs of he sae esmaon error provded by he DHE scheme s gven by (). Proof: [Proof of Lemma 4] In he noseless case, for any sensor and for any, he oupu sgnal s ȳ k C x. Smlarly o Lemma 4.3 n []: X ˆ v k X k" N X ȳ k G k" N C ˆx k X A The frs erm a he rgh hand sde of (39) s: X ȳ k G C A k 9 N : ˆx N X k" N X C A k 9 N : ˆx N G C ˆx k X (39) k" N X ȳ k G C A k 9 N : ˆx N X X C A k 9 N : G ˆx N wx k" N k" N X O N G ˆx N wx (4) where O N s he exended observably marx (5) (of N rows). From he recursve sysem (7), one has: ˆx k A k 9 N : ˆx N k 9 N : j" A j ŵ k j The second erm a he rgh hand sde of (39) can be bounded as: k" N X C A k 9 N : ˆx N G C ˆx k X c k" N X C k 9 N j" : A j ŵ k j X c X C X k" N k 9 N j" : X AX j X ŵ k j X (4) By replacng eqs. (4) and (4) no (39), one obans: X O N From (38), one has: ˆx N G wx c X P X ˆ v k X X C X k" N ˆx N G wx k" N k 9 N : j" X AX j X ŵ k j X (4) x Gzy (43) 7
?` Furhermore, noce ha mnmzng J mples ha he unobservable nal penaly Γ NO N zero,.e., Noe ha, for k G N G : s se o Γ NO N ˆx N ; ˆx N (44) ˆx k Aˆx k ŵ k (45) and ha ŵ k y as y, from (38). One has: From now on, we nroduce, for smplcy of noaon, erms α j varables,.e., X α j X x Gzy ˆx k G Aˆx k y as y (46) o ndcae asympocally vanshng, for all j V. Formulae (43), (44) and (45) are equvalen o: P Oˆx N P O α P NOˆx N P NO Kˆx N ˆx N A ˆx N α Recall ha, by defnon, P O P NO I. Therefore: (47a) (47b) (47c) ˆx N P O ˆx N P NO ˆx N (48) In (48), we replace erms P O ˆx N and P NOˆx N accordng o (47a) and (47b), respecvely. ˆx N P O P NO Kˆx N α 3 (49) Snce P O P NO I, we wre P O G P NO, and we oban ha: ˆx N G P NO Kˆx N G α 3 (5) Frs recall ha, snce K s sochasc and x Σ { d, K G N x. Then noce ha A G N G x and recall (47c). One obans: ˆx N G P NO KA ˆx N G G N G x α 4 P NO KA P NO P O ˆx N G G N G x α 4 (5) See (47a). I mples ha he erm P O ˆx N G G N G x s an asympocally vanshng erm. We oban equaon () from (5). Fnally he Theorem 3 can be proved. Proof: [Proof of Theorem 3] From Assumpon (A), (36) holds n vew of Lemma. Furhermore, (37) s proved n Lemma 3 n vew of Assumpon (A). I follows ha X ˆ v k k" X RB N X ŵ k k" X QB Γ NO N ˆx N ; ˆx N Œ N x Gzy and hence (38) s verfed: Ths, n urn, mples (from Lemma 4) ha he dynamcs of sae esmaon error provded by he DHE scheme s gven by () 8
D. The convergence of he proposed algorhm In Theorem 3 we have shown ha, f we verfy Assumpons (A) and (A), hen asympoc convergence of DHE can be nferred by he egenvalues of Φ. Here, we show ha Assumpons (A) and (A) are me as long as () on Π N s fulflled. Specfcally, we prove he followng lemma: Lemma 5: If () s sasfed, hen he Assumpons (A) and (A) are guaraneed. Proof: [Proof of Lemma 5] The man seps of he proof of Lemma 5 are he followng: r Le he unconsraned rans cos be defned as Ξ u N x N mn ˆx B NC D ŵ ke B J G N ˆx N ŵ ˆ v Γ N subjec o (33a), (33b) kf B N and ˆx N x N (5) ha, dfferenly from Ξ N n (35), does no accoun for npu and sae consrans. Noce ha: Ξ u N x N where he sngle sensor unconsraned rans coss are Ξ C u N r We prove ha: x N Ξ C u N Ξ C u " N x N mn ˆx B NC D ŵ ke B J G N ˆx N ŵ ˆ v Γ N subjec o (7a), (7b) kf B N and ˆx N x N (53) x N where ˆx C u N r We use Lemma 4 and Lemma n [] o prove ha: X x N G ˆx C u N X 9 Π B Nˆ : B and Π N are gven by a suable Kalman fler updae (smlarly o [5]). Ξ u N x N c Ξ N Accordng o hs skech, we frs compue explcly Ξ C u wre: N x N V N C Ȳ N C G O N ˆx N G C NW N C G N x. Recall (7). We can where marces CN and O N are defned accordng o (3) and (5), respecvely, V N C ˆ v T T, Ȳ N C ȳ T T, and / ˆ v N T? / ȳ N T, W N C 7/ ŵ N T? ŵ T T. We can rewre he -h sensor s cos funcon as:?` J X Ȳ N C G O N ˆx N G CNW N C X 9 R N: B X W N C X 9 Q NB : B X ŵ N X QB X ˆx N G k j ˆx j" j N X 9 P T 9 O: Π B N B : B P 9 O P NO: T 9 Π B N B : B P NO X ȳ N G C ˆx N X 9 R : B (54) where marces?` R N and Q N L?` are defned n (4) and (5), respecvely. We denoe wh he mnmum of J wh respec o vecor W N C,.e., L mn D ŵ ke B?` J kf B Nˆ 9
s e S We wre: J H š L Wœ K N [ K ž Ÿ T I H CNL T H R NL The vecor W N C C op W N C C op > CN T R N CN K o Ȳœ K N [ ž I O N ˆx K N I CNWœ K N [ K ž p whch solves (55) s: By replacng (56) no (54) one obans: Q N @ H Q NK L K Wœ K N [ K ž (55) C N T R N a Ȳ N C G O N ˆx N b L X Ȳ N C G O N ˆx N X 9 R N CN Q NB 9 CN: T : B X ŵ N X QB X ˆx N G j" k j ˆx j N X 9 P O: T 9 Π B N B : B P 9 O P NO: T 9 Π B N B : B P X ȳ NO N G C ˆx N X 9 R : B The soluon of he opmzaon problem (53) can be compued as a soluon of a Kalman fler, wh respec o he modfed dynamcal sysem: ˆx N A ˆx N w N Ȳ N C ON ˆx N V N C (57) where w has covarance equal o Q, he covarance of V N C s R N CN Q N CN T, and Π- N P O T Π N P O P NO T Π N P NO C T R C (8) s he uncerany of he nal condon guess: ˆx_ K NU K Π_ K NU K H P OL T H ΠK NU K L K P O S H P NOL T H Π K NU K L In hs way we can wre he rans cos as follows (see [5]): Ξ C u N x K P NO k jp j ˆx H K j NU K S C T L H R L K ȳ K N Ÿ (56) X x G ˆx N X 9 Π B Nˆ : B (58) where he covarance marx Π N s compued by eraon of he Rcca dfference equaon, wh nal condon Π- N, as n (9). Remark ha he unobservable subspace of sysem (57) concdes wh ha of sysem ()-() for all V. Ths follows from he fac ha he oupu marx of (57) s he -h sensor exended observably marx O N. Noce ha: Γ N ; ˆx N X G Kˆx N X P B T O Π B N B P O P B T NO Π B N B P NO X K a X The equaly can be obaned, snce K x x. From (58) Ξ u N G ˆx N b X P T O Π B B N B P O P T NO Π B B N B P NO G ˆx N X K T P B T O Π x { n d x where Π N dag Π N Π N Γ N s equvalen o sasfy he LI (). B N B P O P B T NO Π B N B P NO K x and he marx K s sochasc. Therefore X x G ˆx N X 9 Π B Nˆ : B (59). Hence, o sasfy: ; ˆx N c Ξu N (6)
j m & Le us fnally consder he case of consraned esmaon. In he paper [] Lemma 4 and Lemma prove ha, snce ses v and are convex: Ξ C u N x c Ξ N x, for all V (6) From (6), he sably condon for he consraned case s sasfed f (6) holds, ha s: Γ N ; ˆx N c Ξ N whch corresponds o Assumpon (A). Fnally, Assumpon (A) s sasfed by defnon of Γ O N. Fnally, he proof of Lemma and Theorem can be carred ou. Proof: [Proof of Lemma ] From Lemma 5, snce he LI () s sasfed, Assumpons (A) and (A) are verfed. Hence, all he assumpons of Theorem 3 are gven. Ths, n urns, proves Lemma. Proof: [Proof of Theorem ] From Lemma, he dynamcs of he esmaon error s proved o be gven by equaon (). So, asympoc convergence s guaraneed f and only f Φ s Schur. E. Proof of Theorem Proof: [Proof of Theorem ] If he graph G s no srongly conneced we paron he graph no k rreducble subgraphs G -, G -,..., G - k of cardnaly m,..., m k, and k " m. By exchangng poson o he sensors no marx K (by cluserng he sensors accordng o he paron no communcang subgraphs), K can be brough n a block lower rangular form (wh k square dagonal blocks K,...K kk, of dmensons m,..., m k, respecvely), whou loss of generaly. Noce ha he block K s sochasc f and only f K j for j ƒ. In hs case, he nodes of he subgraph G- have no neghbors belongng o oher subgraphs and G- s solaed. Recall ha, f G s srongly conneced, s an solaed graph self. So, f a subgraph G- s solaed, he block K s sochasc and has a sngle Frobenus egenvalue equal o. On he oher hand, f a graph G- s no solaed, K s rreducble bu no sochasc (specfcally, he sum of he enres of a leas a row s smaller han ) and s Frobenus egevalue has absolue value smaller han. Noce ha he egenvalues of K are he egenvalues of K,...K kk. So, he number of egenvalues of K equal o equals he number of solaed graphs n he nework. Now, recall ha marx Φ P NO KAP NO, ha P NO TS NO S T T NO and ha T AT A KO dag A KO, A KO. arx A KO ou by sensor, ha s: We oban ha: A KO $ s he observably Kalman decomposon of A carred & A O A A NO Φ P NO Kà where à Tà KO T, à KO dag à KO? à KO, and: à KO $ A NO Now we prove ha x span(e d v A e d v A ) s no an egenvecor of Φ assocaed o egenvalue λ A. In general, gven a vecor α Ž, wh α, we oban ha he egenvecor x of A can be wren as x α d v A. We oban ha: jlk Ãx dag Ã, à f α v A. α v A fh α à v A. α à v A Ths follows from he hrd Gerschgorn heorem, dealng wh rreducble marces. Specfcally, an egenvalue of an rreducble marx (n our case K ), whch s on he boundary of a Gerschgorn crcle, s suaed on he boundary of all he Gershgorn crcles. Snce here s a leas a row of K such ha he sum of s enres s smaller han, canno be an egenvalue of K, and hence all he egenvalues of K fnd hemselves nsde he un crcle, from he frs Gerschgorn heorem. m
} e By consrucon à j v A λ A v A f v A belongs o he unobservable subspace of sensor j. Oherwse à j v A. We wre, n general à j v A f j f j λ A v A, where: f v A belongs o he unobservable subspace of sensor j oherwse. we defne vecor f 7/ f, f T. So, we can wre: Recall ha K K d I n. We oban ha: Recall ha A d B Fnally, we oban: C d D Φx λ A P NO Ãx λ A Φx λa P NO BD. So: AC d dag f α d v A K d I n 3 Kdag f α d v A λ A dag f α d v A 5 dag f Kdag f α d v A Φ α d v A λa dag f Kdag f α d v A So, α d v A s an egenvecor of Φ, wh egenvalue λ A f and only f dag f Kdag f α α. There exss α sasfyng he prevous equaon f and only f dag f Kdag f has a leas one egenvecor equal o. Ths occurs f and only f f j for all j belongng o an solaed subgraph. Ths means ha all he sensors of an solaed subgraph have a leas a common unobservable egenvecor. So, ) s no an egenvecor of Φ f: x span(e d v A e d v A j~ G Fnally, he proof of Corollary s gven. ker O n j / for all solaed srongly conneced subgraphs G- Proof: [Proof of Corollary ] The sysem s sad o be collecvely observable f he par A C- s observable, where C- #/ C T C T T. Ths condon holds f and only f he observably marx K of ō A C- s such ha he followng s verfed: Noce ha, by commung he rows of K, we oban ō 3 equvalen o: } ker K ō / (6) O n T O n T5 T. So, condon (6) s ~ V ker O n / whch s equvalen o v of Theorem. Ths concludes he proof.