Optimal Fusion Filtering in Multisensor Stochastic Systems with Missing Measurements and Correlated Noises

Hindawi Publishing Co po a ion Ma hema ical P oblems in Enginee ing Volume 2013, A icle ID 418678, 14 pages h p://dx.doi.o g/10.1155/2013/418678 Resea ch A icle Op imal Fusion Fil e ing in Mul isenso S ochas ic Sys ems wi h Missing Measu emen s and Co ela ed Noises R. Caballe o-Águila,1I. Ga cía-Ga ido,2and J. Lina es-Pé ez2 1Depa amen o de Es ad´ ıs ica, Uni e sidad de Ja´ en, Pa aje Las Lagunillas, 23071 Ja´ en, Spain 2Depa amen o de Es ad´ ıs ica, Uni e sidad de G anada, A enida Fuen enue a, 18071 G anada, Spain Co espondence should be add essed o R. Caballe o-´ Aguila; [email protected] Recei ed 30 Janua y 2013; Accep ed 28 Ap il 2013 Academic Edi o : Weihai Zhang Copy igh © 2013 R. Caballe o-´ Aguila e al. This is an open access a icle dis ibu ed unde he C ea i e Commons A ibu ion License, which pe mi s un es ic ed use, dis ibu ion, and ep oduc ion in any medium, p o ided he o iginal wo k is p ope ly ci ed. The op imal leas -squa es linea es ima ion p oblem is add essed o a class o disc e e- ime mul isenso linea s ochas ic sys ems wi h missing measu emen s and au oco ela ed and c oss-co ela ed noises. The s ochas ic unce ain ies in he measu emen s coming om each senso (missing measu emen s) a e desc ibed by scala andom a iables wi h a bi a y disc e e p obabili y dis ibu ion o e he in e al [0,1]; hence, a each single senso he in o ma ion migh be pa ially missed and he di e en senso s may ha e di e en missing p obabili ies. The noise co ela ion assump ions conside ed a e (i) he p ocess noise and all he senso noises a e one-s ep au oco ela ed; (ii) di e en senso noises a e one-s ep c oss-co ela ed; and (iii) he p ocess noise and each senso noise a e wo-s ep c oss-co ela ed. Unde hese assump ions and by an inno a ion app oach, ecu si e algo i hms o he op imal linea il e a e de i ed by using he wo basic es ima ion usion s uc u es; mo e speci ically, bo h cen alized and dis ibu ed usion es ima ion algo i hms a e p oposed. The accu acy o hese es ima o s is measu ed by hei e o co a iance ma ices, which allow us o compa e hei pe o mance in a nume ical simula ion example ha illus a es he easibili y o he p oposed il e ing algo i hms and shows a compa ison wi h o he exis ing il e s. 1. In oduc ion Fo a long ime, he leas -squa es (LS) es ima ion p oblem in linea s ochas ic sys ems om measu emen s pe u bed by addi i e noises has ecei ed conside able a en ion in he scien i ic communi y due o i s wide applicabili y in many p ac ical si ua ions (e.g., ideo and lase acking sys ems, sa elli e na iga ion, ada and me eo ological applica ions, e c. [1]). As i is well known, one o he majo con ibu ions made o sol e his p oblem is he Kalman il e , which p o- ides a ecu si e algo i hm o he op imal LS es ima o when headdi i ewhi enoisesand heini ials a ea eGaussian and mu ually independen (o , equi alen ly, unco ela ed due o he Gaussiani y assump ion) and, he e o e, he op imal LS es ima o is he op imal LS linea es ima o . F om he publica ion o he Kalman il e [2] in 1960, nume ous esul s and se e al solu ion me hods ha e been epo ed in he li e a u e o add ess he s a e es ima ion p oblem om noisy obse a ions, which depend on models ep esen ing possible ela ionships be ween he unknown s a e and he obse able a iables and also on he noise p ocesses assump ions. Speci ically, du ing he pas decades, he e has been an inc easing in e es in he il e ing p oblem in mul isenso sys ems, whe e senso ne wo ks a e used o ob ain he whole a ailable in o ma ion on he sys em s a e and i s es ima ion mus be ca ied ou om he obse a ions p o ided by all he senso s. A basic ma e o his class o sys ems is how o use he measu emen da a om he di e en senso s o add ess he es ima ion p oblem. Commonly, wo me hods a eused op ocess hemeasu edda acoming ommul iple senso s:cen alizedanddis ibu ed usionme hods.In he cen alized usionme hodall hemeasu edda a omsenso s a e communica ed o he usion cen e o being p ocessed; ne e heless, as is widely known, cen alized es ima o s ha e many compu a ional disad an ages, which mo i a e he esea ch in o o he usion me hods. In he dis ibu ed usion me hod, each senso es ima es he s a e based on i s own single measu emen da a, and hen i sends such 2Ma hema ical P oblems in Enginee ing es ima e o he usion cen e o usion acco ding o a ce ain in o ma ion usion c i e ion. Al hough he use o senso ne wo ks o e s se e al ad an ages, he un eliable ne wo k cha ac e is ics usually cause p oblems du ing da a ans- mission om senso s o he usion cen e , such as missing measu emen s, andom communica ion packe losses and/o delays. Taking in o accoun hese ne wo k unce ain ies, he models ep esen ing he ela ionships be ween he s a e and measu emen s do no allow o apply he Kalman il e , and modi ica ions o con en ional es ima ion algo i hms ha e been p oposed (see e.g., [3–9] and e e ences he ein). As in he Kalman il e , independen whi e noises a e con- side ed in all he men ioned pape s; howe e , his assump ion may no be ealis ic and can be a limi a ion in many eal- wo ld p oblems in which noise co ela ion may be p esen . This p oblem a ises, o example, when a a ge is aking an elec onic coun e measu e, o example, noise jamming [10], o i he p ocess noise and he senso measu emen noises a e dependen on he sys em s a e, hen he e may be c oss-co ela ion be ween di e en senso noises and c oss- co ela ion be ween p ocess noise and senso noises. Also, i all he senso s a e obse ed in he same noisy en i onmen , he measu emen noises o di e en senso s a e usually co ela ed. Fo hese easons, he es ima ion p oblem in sys ems wi h co ela ed noises has ecei ed signi ican esea ch in e es in ecen yea s. Fo example, he op imal Kalman il e ing usion p oblem in sys ems wi h c oss-co ela ed senso noises is add essed in [10], while [11,12]s udy hesamep oblemin sys ems wi h c oss-co ela ed p ocess noises and measu e- men noises; in hese pape s co ela ed noises a he same sampling ime a e conside ed. In gene al, he assump ion o co ela ion and c oss-co ela ion o he noise p ocess and measu emen noises in di e en sampling imes makes di - icul he iden i ica ion o op imal es ima o s; his limi a ion has encou aged a wide esea ch in o subop imal Kalman- ype es ima ion p oblems. In [13], a Kalman- ype ecu si e il e is p esen ed o sys ems wi h ini e-s ep co ela ed p ocess noises, and he il e ing p oblem wi h mul is ep co ela ed p ocess and measu emen noises is in es iga ed in [14]. The op imal obus non agile Kalman- ype ecu si e il e ing p oblem is s udied in [15] o aclasso unce ain sys ems wi h ini e-s ep au oco ela ed measu emen noises and mul iple packe d opou s. The p oblem o dis ibu ed weigh ed obus Kalman il e usion is s udied in [16] o a class o unce ain sys ems wi h au oco ela ed and c oss- co ela ed noises. In [17], a s ochas ic singula sys em wi h co ela ed noises a he same sampling ime is ans o med in o an equi alen nonsingula sys em wi h co ela ed noises a he same and neighbo ing sampling imes. Also, in [18], an augmen ed pa ame e ized sys em wi h co ela ed noises a he same and neighbo ing sampling imes is used o desc ibe he senso delay, packe d opou , and unce ain obse a ion phenomenons. On he o he hand, as no ed abo e, he use o communi- ca ion ne wo ks o ansmi ing measu ed da a mo i a es he need o conside ing s ochas ic unce ain ies. Missing mea- su emen s ha e been widely ea ed due o i s applicabili y o model a la ge class o eal-wo ld p oblems, such as ading phenomena in p opaga ion channels, a ge acking o , in gene al, si ua ions whe e he e exis in e mi en ailu es in he obse a ion mechanism, acciden al loss o some mea- su emen s, o inaccessibili y o he da a du ing ce ain imes. The s a e es ima ion p oblem om missing measu emen ansmi ed by mul iple senso s has been s udied based on he assump ion ha all he senso s a e iden ical (see, e.g., [19–22]);howe e , hisassump ioncanbeun easonable since some eal sys ems usually in ol e mul iple senso s wi h di e en cha ac e is ics. Recen ly, he il e ing p oblem using missing measu emen s whose s a is ical p ope ies a e assumed no o be he same in all he senso s has been add essed by se e al au ho s unde di e en app oaches andhypo heseson hep ocessesin ol ed(see,e.g.,[23– 27]). In all he abo e pape s, Be noulli andom a iables a e used o model he missing measu emen s phenomenon, and hence, i is assumed ha he measu emen signal is ei he comple ely los (i he co esponding Be noulli a iable akes he alue ze o) o success ully ans e ed (when he Be noulli a iable is equal o one). Recen ly, his missing measu emen model has been gene alized consid- e ing any disc e e dis ibu ion on he in e al [0,1],which allows o co e some p ac ical applica ions whe e only pa ial in o ma ion is missing (see [28,29] and e e ences he ein). Mo i a ed by he abo e conside a ions, ou a en ion is ocused on in es iga ing he op imal LS linea cen alized and dis ibu ed usion es ima ion p oblems in mul isenso sys ems wi h missing measu emen s and au oco ela ed and c oss-co ela ed noises. In each senso , he missing measu e- men phenomenon is go e ned by a scala andom a iable wi h a bi a y disc e e p obabili y dis ibu ion o e he in e - al [0,1], and he di e en senso s may ha e di e en missing p obabili ies. Assume ha he p ocess noise and all he senso noises a e one-s ep au oco ela ed; di e en senso noises a e one-s ep c oss-co ela ed; and he p ocess noise and each senso noise a e wo-s ep c oss-co ela ed. This pape makes a wo old subs an ial no el con ibu ion: (1) unlike mos p e ious esul s wi h co ela ed noises, in which subop imal Kalman- ype es ima o s a e p oposed, in his pape op imal LS linea es ima o s a e ob ained by using an inno a ion app oach, which p o ides a simple de i a ion o he es ima- ion algo i hms due o he ac ha he inno a ions cons i u e a whi e p ocess; and (2) ou missing measu emen model conside s a each senso he possibili y o obse a ions con- aining only pa ial in o ma ion abou he s a e, o e en only noise. The pape is o ganized as ollows. In Sec ion 2 he sys em model wi h au oco ela ed and c oss-co ela ed noises and missing measu emen s coming om mul iple senso s is desc ibed.Also, hesui ablep ope ieson hes a eand noise p ocesses a e speci ied and a b ie desc ip ion o he inno a ion app oach o he op imal LS linea es ima ion p oblem is included. In Sec ion 3 a ecu si ealgo i hm o he cen alized op imal linea il e is p esen ed o he conside ed model ( he de i a ion has been de e ed o Appendix 6). Nex , in Sec ion 4, he local LS linea il e s and hei co esponding e o co a iance ma ices be ween any wo local es ima es a e p o ided, and hen Ma hema ical P oblems in Enginee ing 3 he dis ibu ed op imal weigh ed usion es ima o s and hei e o co a iance ma ices a e ob ained by apply- ing he op imal in o ma ion usion c i e ion weigh ed by ma ices in he linea minimum a iance sense. Finally, in Sec ion 5, a nume ical simula ion example is p esen ed o show he e ec i eness o he es ima ion algo i hms p oposed in hecu en pape ,andsomeconclusionsa ed awnin Sec ion 6. No a ion. The no a ion used h oughou he pape is s an- da d. Fo any ma ix 𝐴, heno a ionsymbols𝐴𝑇and 𝐴−1 ep esen i s anspose and in e se, espec i ely; R𝑛deno es he 𝑛-dimensional Euclidean space and R𝑚×𝑛 is he se o all eal ma ices o dimension 𝑚×𝑛.Thesho hand Diag(𝑎1,...,𝑎𝑟)deno es a diagonal ma ix whose diagonal en ies a e 𝑎1,...,𝑎𝑟. I he dimensions o ma ices a e no explici ly s a ed, hey a e assumed o be compa ible o algeb aic ope a ions. 𝛿𝑘−𝑠 is he K onecke del a unc ion, which is equal o one, i 𝑘=𝑠,andze oo he wise. Mo eo e , o a bi a y andom ec o s 𝛼and 𝛽,wewill deno e Co [𝛼,𝛽] = 𝐸[(𝛼−𝐸[𝛼])(𝛽−𝐸[𝛽])𝑇]and Co [𝛼] = Co [𝛼,𝛼],whe e𝐸[⋅] s ands o he ma hema ical expec a- ion ope a o . Finally,  𝛼deno es he es ima o o 𝛼and  𝛼= 𝛼− 𝛼 he es ima ion e o . 2. P oblem Fo mula ion Ou aim is o ob ain ecu si e algo i hms o he op imal LS linea il e ing p oblem in a class o disc e e- ime s ochas ic sys ems wi h missing measu emen s coming om mul iple senso s, by using cen alized and dis ibu ed usion me hods. In his sec ion, i s ly he sys em model and he assump ions abou he s a e and noise p ocesses a e p esen ed and, sec- ondly, he op imal LS linea es ima ion p oblem is o mula ed using an inno a ion app oach. 2.1. S ochas ic Sys em Model. Conside a disc e e- ime linea s ochas ic sys em wi h au oco ela ed and c oss-co ela ed noises and missing measu emen s coming om 𝑟senso s. The phenomenon o missing measu emen s occu s andomly and, o each senso , a di e en sequence o scala andom a iables wi h disc e e dis ibu ion o e he in e al [0,1]is used o model his phenomenon. Speci ically, he ollowing sys em is conside ed: 𝑥𝑘=𝐹 𝑘−1𝑥𝑘−1 +𝑤𝑘−1,𝑘≥1, (1) whe e 𝑥𝑘∈R𝑛is he s a e, {𝑤𝑘;𝑘≥0}is he p ocess noise, and 𝐹𝑘, o 𝑘≥0,a eknownma iceswi hcompa ible dimensions. Conside 𝑟senso s which, a any ime 𝑘,p o idescala measu emen s o he sys em s a e, pe u bed by addi i e and mul iplica i e noises acco ding o he ollowing model: 𝑦𝑖 𝑘=𝜃𝑖 𝑘𝐻𝑖 𝑘𝑥𝑘+V𝑖 𝑘, 𝑘≥1, 𝑖=1,2,...,𝑟, (2) whe e {𝑦𝑖 𝑘;𝑘≥1}a e he measu ed da a; {V𝑖 𝑘;𝑘≥1}a e measu emen noises; {𝜃𝑖 𝑘;𝑘≥1}a e scala andom a iables sequences; 𝐻𝑖 𝑘, o 𝑘≥1, a e known ime- a ying ma ices wi h compa ible dimensions; supe sc ip 𝑖deno es he 𝑖 h senso , and 𝑟is he numbe o senso s. Nex , he s a is ical p ope ies assumed abou he ini ial s a e and noise p ocesses in ol ed in (1)and(2) a e speci ied. (i) The ini ial s a e 𝑥0is a andom ec o wi h 𝐸[𝑥0]=𝑥0 and Co [𝑥0]=𝑃 0. (ii) The p ocess noise, {𝑤𝑘;𝑘≥0}, and he measu emen noises, {V𝑖 𝑘;𝑘 ≥ 1},𝑖 = 1,2,...,𝑟, a e ze o-mean sequences wi h co a iances and c oss-co a iances: Co [𝑤𝑘,𝑤𝑠]=𝑄𝑘,𝑘𝛿𝑘−𝑠 +𝑄𝑘,𝑠𝛿𝑘−𝑠+1 +𝑄𝑘,𝑠𝛿𝑘−𝑠−1, Co [V𝑖 𝑘,V𝑗 𝑠]=𝑅𝑖𝑗 𝑘,𝑘𝛿𝑘−𝑠 +𝑅𝑖𝑗 𝑘,𝑠𝛿𝑘−𝑠+1 +𝑅𝑖𝑗 𝑘,𝑠𝛿𝑘−𝑠−1, Co [𝑤𝑘,V𝑖 𝑠]=𝑆𝑖 𝑘,𝑘𝛿𝑘−𝑠 +𝑆𝑖 𝑘,𝑠𝛿𝑘−𝑠+1 +𝑆𝑖 𝑘,𝑠𝛿𝑘−𝑠+2. (3) (iii) The mul iplica i e noises {𝜃𝑖 𝑘;𝑘≥1},𝑖=1,2,...,𝑟, a e whi e sequences o scala a iables wi h disc e e dis ibu ion o e he in e al [0,1],wi h𝐸[𝜃𝑖 𝑘]=𝜃𝑖 𝑘 and Va [𝜃𝑖 𝑘]=𝑉𝜃𝑖 𝑘. (i ) The ini ial s a e 𝑥0and he mul iplica i e noises {𝜃𝑖 𝑘;𝑘≥1}, o 𝑖 = 1,2,...,𝑟,a emu ually independen , and hey a e independen o he addi i e noises {𝑤𝑘;𝑘≥0}and {V𝑖 𝑘;𝑘≥1}, o 𝑖=1,2,...,𝑟. Rema k 1. F om assump ion (ii) he ollowing co ela ion p ope ies o he addi i e noises a e easily deduced. (1) The noise ec o s 𝑤𝑘and 𝑤𝑠a eco ela eda consec- u i e sampling imes, |𝑘 − 𝑠| = 1, and independen o he wise; he co a iance ma ices o 𝑤𝑘wi h 𝑤𝑘−1, and 𝑤𝑘+1 a e 𝑄𝑘,𝑘−1,and𝑄𝑘,𝑘+1, espec i ely. (2) Fo 𝑖,𝑗=1,2,...,𝑟, he measu emen noises V𝑖 𝑘and V𝑗 𝑠a e c oss-co ela ed a he same sampling ime and a consecu i e sampling imes, |𝑘−𝑠| = 0,1,and independen o he wise; he c oss-co a iances o V𝑖 𝑘 wi h V𝑗 𝑘,V𝑗 𝑘−1 and V𝑗 𝑘+1 a e 𝑅𝑖𝑗 𝑘,𝑘,𝑅𝑖𝑗 𝑘,𝑘−1 and 𝑅𝑖𝑗 𝑘,𝑘+1, espec i ely. (3) Fo 𝑖 = 1,2,...,𝑟, he measu emen noises V𝑖 𝑘 a e co ela ed wi h he noise ec o s 𝑤𝑠, o 𝑠= 𝑘, 𝑘−1, 𝑘−2, and independen o he wise; he c oss- co a iance ma ices o V𝑖 𝑘wi h 𝑤𝑘,𝑤𝑘−1 and 𝑤𝑘−2 a e 𝑆𝑖 𝑘,𝑘,𝑆𝑖 𝑘−1,𝑘 and 𝑆𝑖 𝑘−2,𝑘, espec i ely. The co ela ion condi ions o he p ocess noise and he measu emen noises conside ed in his pape a e he same as hose in [16]. Sys ems wi h only ini e-s ep co ela ed p ocess noises o mul is ep co ela ed p ocess and measu emen noises a e conside ed in [13–15], among o he s. The cu en s udy can be ex ended o mo e gene al sys ems in ol ing ini e-s ep au oco ela ed and c oss-co ela ed noises wi h no di icul y, excep o a g ea e complexi y in he ma hema ical de i a ions. 4Ma hema ical P oblems in Enginee ing Rema k 2. F om he s a e equa ion (1) and assump ions (ii) and (i ), i is easy o deduce ha 𝐷𝑘=𝐸[𝑥𝑘𝑥𝑇 𝑘]is ecu si ely calcula ed by 𝐷𝑘=𝐹 𝑘−1𝐷𝑘−1𝐹𝑇 𝑘−1 +𝑄𝑘−1,𝑘−1 +𝐹 𝑘−1𝑄𝑘−2,𝑘−1 +𝑄𝑘−1,𝑘−2𝐹𝑇 𝑘−1, 𝑘≥2, 𝐷1=𝐹 0𝐷0𝐹𝑇 0+𝑄0,0,𝐷 0=𝑃 0+𝑥0𝑥𝑇 0. (4) Also, i is easy o see ha he s a e 𝑥𝑘is co ela ed wi h he measu emen noises V𝑖 𝑘, o 𝑖 = 1,2,...,𝑟,and he expec a ions 𝐸𝑖 𝑘=𝐸[𝑥𝑘V𝑖 𝑘]sa is y 𝐸𝑖 𝑘=𝐹 𝑘−1𝑆𝑖 𝑘−2,𝑘 +𝑆𝑖 𝑘−1,𝑘, 𝑘≥2; 𝐸𝑖 1=𝑆𝑖 0,1.(5) Rema k 3. Acco ding o assump ion (iii), he scala andom a iables 𝜃𝑖 𝑘 ake alues o e he in e al [0,1] and hey can sa is y any a bi a y disc e e p obabili y dis ibu ion o e such in e al, o ins ance, a Be noulli dis ibu ion. Usually, Be noulli andom a iables ha e been used o model he phenomenon o missing measu emen s (see, e.g., [25]and e e ences he ein), wi h 𝜃𝑖 𝑘=1meaning ha he s a e 𝑥𝑘is p esen in he measu emen 𝑦𝑖 𝑘coming om he 𝑖 h senso a ime 𝑘, while 𝜃𝑖 𝑘=0means ha he s a e is missing in he measu ed da a a ime 𝑘o , equi alen ly, ha such obse a ion only con ains addi i e noise V𝑖 𝑘.Howe e , in p ac ice, he in o ma ion ansmi ed a a sampling ime can usually be nei he comple ely missing no comple ely success ul, bu only pa o he in o ma ion can go h ough; in such si ua ions, only pa ial in o ma ion is missing and he p opo ion o missed da a a one momen is a ac ion o he han 0 o 1 (see, e.g., [28,29] and e e ences he ein). 2.2. S acked Measu emen Equa ion. As no ed abo e, ou aimis osol e heop imalLSlinea es ima ionp oblem o he s a e 𝑥𝑘basedon hemeasu emen s{𝑦𝑖 1,𝑦𝑖 2,...,𝑦𝑖 𝑘}, o 𝑖 = 1,2,...,𝑟, by using cen alized and dis ibu ed usion me hods o p ocess he measu ed senso da a. The cen alized usion me hod conside s ha all he measu emen da a coming om 𝑟senso s a e ansmi ed o a usion cen e o being p ocessed; o his pu pose and o simpli y he no a ion, he measu emen equa ion (2)is ew i enina s acked o m as ollows: 𝑦𝑘=Θ𝑘𝐻𝑘𝑥𝑘+V𝑘, 𝑘≥1, (6) whe e 𝑦𝑘=(𝑦 1 𝑘,...,𝑦𝑟 𝑘)𝑇,V𝑘=(V1 𝑘,...,V𝑟 𝑘)𝑇,𝐻𝑘=(𝐻 1𝑇 𝑘,..., 𝐻𝑟𝑇 𝑘)𝑇,andΘ𝑘=Diag(𝜃1 𝑘,...,𝜃𝑟 𝑘). The ollowing p ope ies o he noises in (6)a eeasily in e ed om he model assump ions (ii)–(i ) p e iously s a ed. (i) The addi i e noise {V𝑘;𝑘≥1}is a ze o-mean p ocess sa is ying: Co [V𝑘,V𝑠]=𝑅𝑘,𝑘𝛿𝑘−𝑠 +𝑅𝑘,𝑠𝛿𝑘−𝑠+1 +𝑅𝑘,𝑠𝛿𝑘−𝑠−1, Co [𝑤𝑘,V𝑠]=𝑆𝑘,𝑘𝛿𝑘−𝑠 +𝑆𝑘,𝑠𝛿𝑘−𝑠+1 +𝑆𝑘,𝑠𝛿𝑘−𝑠+2,(7) whe e 𝑅𝑘,𝑠 =(𝑅𝑖𝑗 𝑘,𝑠)𝑖,𝑗=1,2,...,𝑟 and 𝑆𝑘,𝑠 =(𝑆1 𝑘,𝑠,...,𝑆𝑟 𝑘,𝑠). (ii) The s a e ec o 𝑥𝑘and he measu emen noise ec o V𝑘a e co ela ed wi h 𝐸𝑘=𝐸[𝑥𝑘V𝑇 𝑘]sa is ying 𝐸𝑘=𝐹 𝑘−1𝑆𝑘−2,𝑘 +𝑆𝑘−1,𝑘,𝑘≥2,𝐸 1=𝑆0,1.(8) (iii) The andom ma ices {Θ𝑘;𝑘≥1}sa is y 𝐸[Θ𝑘]= Θ𝑘=Diag(𝜃1 𝑘,...,𝜃𝑟 𝑘)and 𝐸[(Θ𝑘− Θ𝑘)2]= Diag(𝑉𝜃1 𝑘,...,𝑉𝜃𝑟 𝑘); also, deno ing 𝜃𝑘=(𝜃 1 𝑘,...,𝜃𝑟 𝑘)𝑇, i is clea ha Co [𝜃𝑘]=Diag(𝑉𝜃1 𝑘,...,𝑉𝜃𝑟 𝑘). Mo eo e , o any andom ma ix 𝐺independen o {Θ𝑘;𝑘≥1}, i is easily deduced ha 𝐸[(Θ𝑘−Θ𝑘)𝐺(Θ𝑘−Θ𝑘)]= Co [𝜃𝑘]∘𝐸[𝐺],(9) whe e ∘deno es he Hadama d p oduc [23]. (i ) The ini ial s a e 𝑥0and {Θ𝑘;𝑘≥1}a e independen , and hey a e independen o {𝑤𝑘;𝑘≥0}and {V𝑘;𝑘≥ 1}. 2.3. Inno a ion App oach o he Op imal LS Linea Es i- ma ion P oblem. To add ess he op imal LS linea es ima- ion p oblem o he s a e 𝑥𝑘based on he measu emen s {𝑦𝑖 1,𝑦𝑖 2,...,𝑦𝑖 𝑘},𝑖=1,2,...,𝑟, he cen alized and dis ibu ed usion me hods will be used. In bo h cases, ecu si e algo- i hms o he LS linea es ima o s will be es ablished using an inno a ion app oach and he o hogonal p ojec ion Lemma (OPL); mo e speci ically we ha e he ollowing. Cen alized Fusion Es ima ion P oblem. Ou aim is o ob ain he op imal LS linea il e ,  𝑥𝑘/𝑘,o hes a e𝑥𝑘based on he measu emen s {𝑦1,𝑦2,...,𝑦𝑘},gi enin(6), by ecu si e algo i hms. Asknown, heLSlinea il e  𝑥𝑘/𝑘 is he o hogonal p ojec ion o he s a e 𝑥𝑘o e he linea space spanned by {𝑦1,𝑦2,...,𝑦𝑘}. These obse a ions a e gene ally nono hog- onal ec o s, bu he G am-Schmid o hogonaliza ion p o- cedu e allows us o subs i u e hem by a se o o hogonal ec o s, called inno a ions, de ined as he di e ence be ween each obse a ion and i s one-s age p edic o . Due o he o hogonali y p ope y o he inno a ions and since he inno a ion p ocess is uniquely de e mined by he obse a- ions, he LS linea il e ,  𝑥𝑘/𝑘, can be calcula ed as linea combina ion o he inno a ions; namely,  𝑥𝑘/𝑘 =𝑘 ∑ 𝑠=1 X𝑘,𝑠Π−1 𝑠,𝑠𝜇𝑠, 𝑘≥1, (10) whe e 𝜇𝑠=𝑦 𝑠− 𝑦𝑠/𝑠−1 a e he inno a ion ec o s, wi h  𝑦𝑠/𝑠−1 he one-s age obse a ion p edic o , Π𝑠,𝑠 = 𝐸[𝜇𝑠𝜇𝑇 𝑠],and X𝑘,𝑠 =𝐸[𝑥𝑘𝜇𝑇 𝑠]. Dis ibu ed Fusion Es ima ion P oblem. To add ess he dis- ibu ed usion es ima ion p oblem, i s ly, ecu si e algo- i hms o ob ain local LS linea il e s,  𝑥𝑖 𝑘/𝑘, o 𝑖=1,2,...,𝑟, and he e o c oss-co a iance ma ices be ween any wo local es ima es, a e de i ed. Secondly, he dis ibu ed usion il e ,  𝑥𝐷 𝑘/𝑘, is es ablished by applying he op imal in o ma ion Ma hema ical P oblems in Enginee ing 5 usion c i e ion weigh ed by ma ices in he linea minimum a iance sense [30]. Analogously o (10), deno ing 𝜇𝑖 𝑠=𝑦 𝑖 𝑠− 𝑦𝑖 𝑠/𝑠−1,Π𝑖𝑖 𝑠,𝑠 = 𝐸[𝜇𝑖 𝑠𝜇𝑖 𝑠],andX𝑖 𝑘,𝑠 = 𝐸[𝑥𝑘𝜇𝑖 𝑠], helocal il e  𝑥𝑖 𝑘/𝑘 is exp essed as  𝑥𝑖 𝑘/𝑘 =𝑘 ∑ 𝑠=1 X𝑖 𝑘,𝑠(Π𝑖𝑖 𝑠,𝑠)−1𝜇𝑖 𝑠, 𝑘≥1. (11) 3. Op imal LS Linea Cen alized Fusion Es ima ion In his sec ion a ecu si e algo i hm o he cen alized op imal (unde he LS c i e ion) linea il e ,  𝑥𝑘/𝑘 is de i ed. Such algo i hm is deduced using (10) and he OPL, and i is p esen ed in Theo em 5. Fi s ly, in o de o simpli y he p oo o Theo em 5, he ollowing lemma is es ablished. Lemma 4. Unde assump ions (i)–(i ), he ollowing esul s hold: W𝑘,𝑘 =𝐸[𝑤 𝑘𝜇𝑇 𝑘]=𝑄𝑘,𝑘−1𝐻𝑇 𝑘Θ𝑘+𝑆𝑘,𝑘,𝑘≥1, (12) V𝑘,𝑘−1 =𝐸[V𝑘𝜇𝑇 𝑘−1]=𝑆𝑇 𝑘−2,𝑘𝐻𝑇 𝑘−1Θ𝑘−1 +𝑅𝑘,𝑘−1, 𝑘≥2. (13) P oo . Since 𝑤𝑘is independen o 𝑦1,...,𝑦𝑘−1,𝐸[𝑤𝑘 𝑦𝑇 𝑘/𝑘−1]= 0and hence W𝑘,𝑘 = 𝐸[𝑤𝑘𝑦𝑇 𝑘].Now,using(1)and(6), W𝑘,𝑘 canbecalcula edas ollows: W𝑘,𝑘 =𝐸[𝑤 𝑘(Θ𝑘𝐻𝑘𝑥𝑘+V𝑘)𝑇] =𝐸[𝑤 𝑘𝑥𝑇 𝑘]𝐻𝑇 𝑘Θ𝑘+𝑆𝑘,𝑘 =𝐸[𝑤 𝑘(𝐹𝑘−1𝑥𝑘−1 +𝑤𝑘−1)𝑇]𝐻𝑇 𝑘Θ𝑘+𝑆𝑘,𝑘 =𝑄 𝑘,𝑘−1𝐻𝑇 𝑘Θ𝑘+𝑆𝑘,𝑘. (14) Taking in o accoun ha V𝑘is independen o 𝑦1,...,𝑦𝑘−2, he calcula ion o V𝑘,𝑘−1 is simila o ha o W𝑘,𝑘, and hence he p oo is omi ed. Theo em 5. Fo he sys em model (1)and measu emen model (6), unde assump ions (i)–(i ), he op imal LS linea il e  𝑥𝑘/𝑘 is ob ained as  𝑥𝑘/𝑘 = 𝑥𝑘/𝑘−1 +X𝑘,𝑘Π−1 𝑘,𝑘𝜇𝑘, 𝑘≥1,  𝑥0/0 =𝑥0,(15) whe e hes a ep edic o , 𝑥𝑘/𝑘−1,sa is ies  𝑥𝑘/𝑘−1 =𝐹 𝑘−1  𝑥𝑘−1/𝑘−1 +W𝑘−1,𝑘−1Π−1 𝑘−1,𝑘−1𝜇𝑘−1, 𝑘≥2,  𝑥1/0 =𝐹 0 𝑥0/0.(16) The inno a ion, 𝜇𝑘,isgi enby 𝜇𝑘=𝑦 𝑘−Θ𝑘𝐻𝑘 𝑥𝑘/𝑘−1 −V𝑘,𝑘−1Π−1 𝑘−1,𝑘−1𝜇𝑘−1, 𝑘≥2, 𝜇1=𝑦 1−Θ1𝐻1 𝑥1/0.(17) The ma ix X𝑘,𝑘 =𝐸[𝑥𝑘𝜇𝑇 𝑘]is calcula ed by X𝑘,𝑘 =𝑃 𝑘/𝑘−1𝐻𝑇 𝑘Θ𝑘+𝐸𝑘−X𝑘,𝑘−1Π−1 𝑘−1,𝑘−1V𝑇 𝑘,𝑘−1, 𝑘≥2, X1,1 =𝑃 1/0𝐻𝑇 1Θ1+𝐸1,(18) whe e X𝑘,𝑘−1 =𝐸[𝑥𝑘𝜇𝑇 𝑘−1]sa is ies X𝑘,𝑘−1 =𝐹 𝑘−1X𝑘−1,𝑘−1 +W𝑘−1,𝑘−1, 𝑘≥2. (19) The p edic ion e o co a iance ma ix, 𝑃𝑘/𝑘−1,isob ainedby 𝑃𝑘/𝑘−1 =𝐹 𝑘−1𝑃𝑘−1/𝑘−1𝐹𝑇 𝑘−1 +𝑄𝑘−1,𝑘−1 +𝐹 𝑘−1J𝑘−1 +J𝑇 𝑘−1𝐹𝑇 𝑘−1 −W𝑘−1,𝑘−1Π−1 𝑘−1,𝑘−1W𝑇 𝑘−1,𝑘−1, 𝑘≥2, 𝑃1/0 =𝐹 0𝑃0/0𝐹𝑇 0+𝑄0,0,(20) whe e J𝑘=𝐸[ 𝑥𝑘/𝑘𝑤𝑇 𝑘]is calcula ed by J𝑘=𝑄𝑘−1,𝑘 −X𝑘,𝑘Π−1 𝑘,𝑘W𝑇 𝑘,𝑘,𝑘≥1. (21) The il e ing e o co a iance ma ix, 𝑃𝑘/𝑘,isgi enby 𝑃𝑘/𝑘 =𝑃 𝑘/𝑘−1 −X𝑘,𝑘Π−1 𝑘,𝑘X𝑇 𝑘,𝑘, 𝑘≥1, 𝑃 0/0 =𝑃 0.(22) The inno a ion co a iance ma ix, Π𝑘,𝑘,sa is ies Π𝑘,𝑘 =Co (𝜃𝑘)∘(𝐻𝑘𝐷𝑘𝐻𝑇 𝑘)+𝑅𝑘,𝑘 +Θ𝑘𝐻𝑘X𝑘,𝑘 +X𝑇 𝑘,𝑘𝐻𝑇 𝑘Θ𝑘−Θ𝑘𝐻𝑘𝑃𝑘/𝑘−1𝐻𝑇 𝑘Θ𝑘 −V𝑘,𝑘−1Π−1 𝑘−1,𝑘−1V𝑇 𝑘,𝑘−1, 𝑘≥2, Π1,1 =Co (𝜃1)∘(𝐻1𝐷1𝐻𝑇 1)+𝑅1,1 +Θ1𝐻1X1,1 +X𝑇 1,1𝐻𝑇 1Θ1−Θ1𝐻1𝑃1/0𝐻𝑇 1Θ1. (23) The ma ices 𝐷𝑘,𝐸𝑘,W𝑘,𝑘,andV𝑘,𝑘−1 a e gi en in (4),(8), (12),and(13), espec i ely. P oo . See Appendix 6. Rema k 6. In con en ional es ima ion p oblems in sys ems wi h missing measu emen s and unco ela ed addi i e whi e noises, he one-s age s a e and obse a ion p edic o s a e calcula ed as  𝑥𝑘/𝑘−1 =𝐹 𝑘−1  𝑥𝑘−1/𝑘−1 and  𝑦𝑘/𝑘−1 =Θ𝑘𝐻𝑘 𝑥𝑘/𝑘−1, espec i ely.Howe e , hisisno ue o hep oblema hand since, due o he co ela ion assump ion (ii), he noise es i- ma o s  𝑤𝑘−1/𝑘−1 and  V𝑘/𝑘−1 mus be aken in o accoun o he de i a ion o he p edic o s. Besides he ac o conside ing missing measu emen s, his is he main di e ence be ween he op imal es ima o s p oposed in he cu en pape and he subop imal Kalman- ype ones p oposed in [16], whe e he noise es ima o s a e conside ed o be equal o ze o. 6Ma hema ical P oblems in Enginee ing 4. Dis ibu ed Fusion Es ima ion One o he main disad an ages o he cen alized usion es ima o s de i ed in Sec ion 3 is ha hey may ha e a high compu a ional cos due o augmen a ion. Mo eo e , as is widely known, he cen alized app oach has se e al o he d awbacks, such as aul de ec ion, isola ion, poo eliabili y,andso o h.Too e come hesedisad an ages, ou aim in his sec ion is o add ess he op imal dis ibu ed usion es ima ion p oblem, in which each single senso p o ides i s local LS linea es ima o and hei es ima ion e o co a iance ma ices, and hen hese local es ima o s along wi h he co a iances and c oss-co a iance ma ices o he es ima ion e o s be ween any wo senso s a e sen o he usion cen e o usion based on he ma ices-weigh ed usion es ima ion c i e ion in he linea minimum a iance sense [30]. 4.1. Local LS Linea Fil e ing Algo i hms. Fo each single sen- so subsys em o sys ems (1)and(2), he ollowing heo em p o ides ecu si e o mulas o he local LS linea il e s,  𝑥𝑖 𝑘/𝑘, and hei co esponding e o co a iance ma ices, 𝑃𝑖𝑖 𝑘/𝑘. Theo em 7. Fo he 𝑖 h senso subsys em o sys ems (1)and (2)unde assump ions (i)–(i ), he local LS linea il e ,  𝑥𝑖 𝑘/𝑘,is calcula ed by  𝑥𝑖 𝑘/𝑘 = 𝑥𝑖 𝑘/𝑘−1 +X𝑖 𝑘,𝑘(Π𝑖𝑖 𝑘,𝑘)−1𝜇𝑖 𝑘, 𝑘≥1,  𝑥𝑖 0/0 =𝑥0, (24) whe e he local LS linea p edic o ,  𝑥𝑖 𝑘/𝑘−1,sa is ies  𝑥𝑖 𝑘/𝑘−1 =𝐹 𝑘−1  𝑥𝑖 𝑘−1/𝑘−1 +W𝑖 𝑘−1,𝑘−1(Π𝑖𝑖 𝑘−1,𝑘−1)−1𝜇𝑖 𝑘−1, 𝑘≥2,  𝑥𝑖 1/0 =𝐹 0 𝑥𝑖 0/0,(25) wi h W𝑖 𝑘,𝑘 =𝜃𝑖 𝑘𝑄𝑘,𝑘−1𝐻𝑖𝑇 𝑘+𝑆𝑖 𝑘,𝑘,𝑘≥1. The inno a ion, 𝜇𝑖 𝑘,isgi enby 𝜇𝑖 𝑘=𝑦𝑖 𝑘−𝜃𝑖 𝑘𝐻𝑖 𝑘 𝑥𝑖 𝑘/𝑘−1 −V𝑖𝑖 𝑘,𝑘−1(Π𝑖𝑖 𝑘−1,𝑘−1)−1𝜇𝑖 𝑘−1,𝑘≥2, 𝜇𝑖 1=𝑦𝑖 1−𝜃𝑖 1𝐻𝑖 1 𝑥𝑖 1/0,(26) wi h V𝑖𝑖 𝑘,𝑘−1 =𝜃𝑖 𝑘−1𝑆𝑖𝑇 𝑘−2,𝑘𝐻𝑖𝑇 𝑘−1 +𝑅𝑖𝑖 𝑘,𝑘−1,𝑘≥2. The ec o X𝑖 𝑘,𝑘 =𝐸[𝑥𝑘𝜇𝑖 𝑘]is calcula ed om he ollowing exp ession X𝑖 𝑘,𝑘 =𝜃𝑖 𝑘𝑃𝑖𝑖 𝑘/𝑘−1𝐻𝑖𝑇 𝑘+𝐸𝑖 𝑘−X𝑖 𝑘,𝑘−1(Π𝑖𝑖 𝑘−1,𝑘−1)−1V𝑖𝑖 𝑘,𝑘−1, 𝑘≥2, X𝑖 1,1 =𝜃𝑖 1𝑃𝑖𝑖 1/0𝐻𝑖𝑇 1+𝐸𝑖 1,(27) whe e X𝑖 𝑘,𝑘−1 =𝐹 𝑘−1X𝑖 𝑘−1,𝑘−1 +W𝑖 𝑘−1,𝑘−1,𝑘≥2. The local p edic ion e o co a iance ma ix, 𝑃𝑖𝑖 𝑘/𝑘−1,is ob ained by 𝑃𝑖𝑖 𝑘/𝑘−1 =𝐹 𝑘−1𝑃𝑖𝑖 𝑘−1/𝑘−1𝐹𝑇 𝑘−1 +𝑄𝑘−1,𝑘−1 +𝐹 𝑘−1J𝑖 𝑘−1 +J𝑖𝑇 𝑘−1𝐹𝑇 𝑘−1 −W𝑖 𝑘−1,𝑘−1(Π𝑖𝑖 𝑘−1,𝑘−1)−1W𝑖𝑇 𝑘−1,𝑘−1, 𝑘≥2, 𝑃𝑖𝑖 1/0 =𝐹 0𝑃𝑖𝑖 0/0𝐹𝑇 0+𝑄0,0,(28) whe e J𝑖 𝑘=𝑄 𝑘−1,𝑘 −X𝑖 𝑘,𝑘(Π𝑖𝑖 𝑘,𝑘)−1W𝑖𝑇 𝑘,𝑘,𝑘≥1,and𝑃𝑖𝑖 𝑘/𝑘, he il e inge o co a iancema ix,isgi enby 𝑃𝑖𝑖 𝑘/𝑘 =𝑃𝑖𝑖 𝑘/𝑘−1 −X𝑖 𝑘,𝑘(Π𝑖𝑖 𝑘,𝑘)−1X𝑖𝑇 𝑘,𝑘, 𝑘≥1, 𝑃𝑖𝑖 0/0 =𝑃 0. (29) The inno a ion a iance, Π𝑖𝑖 𝑘,𝑘,sa is ies Π𝑖𝑖 𝑘,𝑘 =𝑉 𝜃𝑖 𝑘𝐻𝑖 𝑘𝐷𝑘𝐻𝑖𝑇 𝑘+𝑅𝑖𝑖 𝑘,𝑘 +𝜃𝑖 𝑘𝐻𝑖 𝑘X𝑖 𝑘,𝑘 +𝜃𝑖 𝑘X𝑖𝑇 𝑘,𝑘𝐻𝑖𝑇 𝑘 −(𝜃𝑖 𝑘)2𝐻𝑖 𝑘𝑃𝑖𝑖 𝑘/𝑘−1𝐻𝑖𝑇 𝑘−(V𝑖𝑖 𝑘,𝑘−1)2(Π𝑖𝑖 𝑘−1,𝑘−1)−1, 𝑘≥2, Π𝑖𝑖 1,1 =𝑉 𝜃𝑖 1𝐻𝑖 1𝐷1𝐻𝑖𝑇 1+𝑅𝑖𝑖 1,1 +𝜃𝑖 1𝐻𝑖 1X𝑖 1,1 +𝜃𝑖 1X𝑖𝑇 1,1𝐻𝑖𝑇 1−(𝜃𝑖 1)2𝐻𝑖 1𝑃𝑖𝑖 1/0𝐻𝑖𝑇 1. (30) The ma ix 𝐷𝑘and he ec o 𝐸𝑖 𝑘a e gi en in (4)and (5), espec i ely. P oo . Thep oo ,basedon heinno a ionapp oachand he OPL, is omi ed o being analogous o ha o Theo em 5. Ne e heless, i should be indica ed ha , in his p oo , he Hadama d p oduc is no used since, ins ead o he diagonal s ochas ic ma ix Θ𝑘, hescala a iable𝜃𝑖 𝑘is now in ol ed in he de i a ion o he es ima o s. Rema k 8. As indica ed in Rema k 6 o he cen alized es ima o s, i mus be no ed ha , due o he co ela ion assump ion (ii) o he addi i e noises {𝑤𝑘}and {V𝑖 𝑘}, he es ima o s  𝑤𝑖 𝑘−1/𝑘−1 =W𝑖 𝑘−1,𝑘−1(Π𝑖𝑖 𝑘−1,𝑘−1)−1𝜇𝑖 𝑘−1 and  V𝑖 𝑘/𝑘−1 = V𝑖𝑖 𝑘,𝑘−1(Π𝑖𝑖 𝑘−1,𝑘−1)−1𝜇𝑖 𝑘−1 a e no equal o ze o, and hence he op imal local s a e p edic o ,  𝑥𝑖 𝑘/𝑘−1 =𝐹 𝑘−1  𝑥𝑖 𝑘−1/𝑘−1+ 𝑤𝑖 𝑘−1/𝑘−1, and he obse a ion p edic o ,  𝑦𝑖 𝑘/𝑘−1 = 𝜃𝑖 𝑘𝐻𝑖 𝑘 𝑥𝑖 𝑘/𝑘−1 + V𝑖 𝑘/𝑘−1, a e qui e di e en om con en ional il e ing algo i hms wi h unco ela ed whi e noises. This issue, along wi h he conside a ion o missing measu emen s a each single senso , cons i u es he main di e ence be ween he cu en op imal local es ima o s and he subop imal local es ima o s p o- posed in [16]. 4.2. C oss-Co a iance Ma ices o Local Es ima ion E o s. To apply he op imal usion c i e ion weigh ed by ma ices in Ma hema ical P oblems in Enginee ing 7 he linea minimum a iance sense, he il e ing, 𝑃𝑖𝑗 𝑘/𝑘,and p edic ion, 𝑃𝑖𝑗 𝑘/𝑘−1, e o c oss-co a iance ma ices be ween local es ima o s o any wo subsys ems mus be calcula ed. Fo simplici y, besides he no a ion o Theo em 7, o 𝑖 =𝑗, 𝑖,𝑗=1,2...,𝑟, we in oduce he ollowing no a ion: 𝐿𝑖𝑗 𝑘=𝐸[ 𝑥𝑖 𝑘/𝑘−1𝜇𝑗 𝑘], Π𝑖𝑗 𝑘,𝑠 =𝐸[𝜇𝑖 𝑘𝜇𝑗 𝑠], V𝑖𝑗 𝑘,𝑘−1 =𝐸[V𝑖 𝑘𝜇𝑗 𝑘−1]. (31) Also,ino de osimpli y hecalcula iono hee o c oss- co a iance ma ices, he ollowing lemmas a e gi en. Lemma 9. Unde assump ions (i)–(i ), he ollowing esul s hold. (a) The expec a ion 𝐸[ 𝑥𝑖 𝑘/𝑘−1𝜇𝑗 𝑘−1]sa is ies 𝐸[ 𝑥𝑖 𝑘/𝑘−1𝜇𝑗 𝑘−1]=𝐹 𝑘−1𝐿𝑖𝑗 𝑘−1 +X𝑖 𝑘,𝑘−1(Π𝑖𝑖 𝑘−1,𝑘−1)−1Π𝑖𝑗 𝑘−1,𝑘−1, 𝑘≥2. (32) (b) The expec a ion 𝐸[ 𝑥𝑖 𝑘/𝑘−1V𝑗 𝑘]sa is ies 𝐸[ 𝑥𝑖 𝑘/𝑘−1V𝑗 𝑘]=X𝑖 𝑘,𝑘−1(Π𝑖𝑖 𝑘−1,𝑘−1)−1V𝑗𝑖 𝑘,𝑘−1, 𝑘≥2, (33) whe e V𝑗𝑖 𝑘,𝑘−1 =𝜃𝑖 𝑘−1𝐻𝑖 𝑘−1𝑆𝑗 𝑘−2,𝑘 +𝑅𝑖𝑗 𝑘−1,𝑘. (c) The expec a ion 𝐸[V𝑖 𝑘𝜇𝑗 𝑘]sa is ies 𝐸[V𝑖 𝑘𝜇𝑗 𝑘]=𝜃𝑗 𝑘𝐸𝑖𝑇 𝑘𝐻𝑗𝑇 𝑘+𝑅𝑖𝑗 𝑘,𝑘 −V𝑖𝑗 𝑘,𝑘−1(Π𝑗𝑗 𝑘−1,𝑘−1)−1(𝜃𝑗 𝑘𝐻𝑗 𝑘X𝑗 𝑘,𝑘−1 +V𝑗𝑗 𝑘,𝑘−1)𝑇, 𝑘≥2. (34) P oo . (a) F om (25) o  𝑥𝑖 𝑘/𝑘−1 and (24) o  𝑥𝑖 𝑘−1/𝑘−1,weha e 𝐸[ 𝑥𝑖 𝑘/𝑘−1𝜇𝑗 𝑘−1] =𝐹 𝑘−1𝐸[ 𝑥𝑖 𝑘−1/𝑘−1𝜇𝑗 𝑘−1]+W𝑖 𝑘−1,𝑘−1(Π𝑖𝑖 𝑘−1,𝑘−1)−1Π𝑖𝑗 𝑘−1,𝑘−1 =𝐹 𝑘−1𝐸[ 𝑥𝑖 𝑘−1/𝑘−2𝜇𝑗 𝑘−1]+𝐹 𝑘−1X𝑖 𝑘−1,𝑘−1 ×(Π𝑖𝑖 𝑘−1,𝑘−1)−1Π𝑖𝑗 𝑘−1,𝑘−1 +W𝑖 𝑘−1,𝑘−1(Π𝑖𝑖 𝑘−1,𝑘−1)−1Π𝑖𝑗 𝑘−1,𝑘−1 =𝐹 𝑘−1𝐿𝑖𝑗 𝑘−1 +(𝐹 𝑘−1X𝑖 𝑘−1,𝑘−1 +W𝑖 𝑘−1,𝑘−1) ×(Π𝑖𝑖 𝑘−1,𝑘−1)−1Π𝑖𝑗 𝑘−1,𝑘−1,(35) and since X𝑖 𝑘,𝑘−1 =𝐹 𝑘−1X𝑖 𝑘−1,𝑘−1 +W𝑖 𝑘−1,𝑘−1, exp ession (32) is p o ed. (b) Analogously, aking in o accoun ha 𝐸[ 𝑥𝑖 𝑘−1/𝑘−2V𝑗 𝑘]= 0,weha e 𝐸[ 𝑥𝑖 𝑘/𝑘−1V𝑗 𝑘]=𝐹 𝑘−1𝐸[ 𝑥𝑖 𝑘−1/𝑘−1V𝑗 𝑘] +W𝑖 𝑘−1,𝑘−1(Π𝑖𝑖 𝑘−1,𝑘−1)−1𝐸[𝜇𝑖 𝑘−1V𝑗 𝑘] =(𝐹𝑘−1X𝑖 𝑘−1,𝑘−1+W𝑖 𝑘−1,𝑘−1)(Π𝑖𝑖 𝑘−1,𝑘−1)−1V𝑗𝑖 𝑘,𝑘−1, (36) and exp ession (33) is immedia ely ob ained. Finally, he de i a ion o exp ession V𝑗𝑖 𝑘,𝑘−1 = 𝜃𝑖 𝑘−1𝐻𝑖 𝑘−1𝑆𝑗 𝑘−2,𝑘 +𝑅 𝑖𝑗 𝑘−1,𝑘 is simila o ha o (13) and hence i is omi ed. (c) Taking in o accoun exp ession (26) o 𝜇𝑗 𝑘,wi h(2) o 𝑦𝑗 𝑘,weha e 𝐸[V𝑖 𝑘𝜇𝑗 𝑘]=𝜃𝑗 𝑘𝐸𝑖𝑇 𝑘𝐻𝑗𝑇 𝑘+𝑅𝑖𝑗 𝑘,𝑘 −𝜃𝑗 𝑘𝐸[V𝑖 𝑘 𝑥𝑗𝑇 𝑘/𝑘−1]𝐻𝑗𝑇 𝑘 −V𝑖𝑗 𝑘,𝑘−1(Π𝑗𝑗 𝑘−1,𝑘−1)−1V𝑗𝑗 𝑘,𝑘−1, 𝑘≥2, (37) and using (33) o 𝐸[V𝑖 𝑘 𝑥𝑗𝑇 𝑘/𝑘−1], exp ession (34)isob ained. Lemma 10. Unde assump ions (i)–(i ), o 𝑖 =𝑗,𝑖,𝑗 = 1,2...,𝑟, heexpec a ions𝐿𝑖𝑗 𝑘=𝐸[  𝑥𝑖 𝑘/𝑘−1𝜇𝑗 𝑘]a e ecu si ely ob ained by 𝐿𝑖𝑗 𝑘= 𝜃𝑗 𝑘(𝑃𝑗𝑗 𝑘/𝑘−1 −𝑃𝑖𝑗 𝑘/𝑘−1)𝐻𝑗𝑇 𝑘−𝐹 𝑘−1𝐿𝑖𝑗 𝑘−1(Π𝑗𝑗 𝑘−1,𝑘−1)−1V𝑗𝑗 𝑘,𝑘−1 +X𝑖 𝑘,𝑘−1(Π𝑖𝑖 𝑘−1,𝑘−1)−1 ×(V𝑗𝑖 𝑘,𝑘−1 −V𝑗𝑗 𝑘,𝑘−1(Π𝑗𝑗 𝑘−1,𝑘−1)−1Π𝑖𝑗 𝑘−1,𝑘−1), 𝑘≥2, (38) wi h ini ial condi ion 𝐿𝑖𝑗 1=0. P oo . Taking in o accoun exp ession (26) o 𝜇𝑗 𝑘,wi h(2) o 𝑦𝑗 𝑘,weha e 𝐿𝑖𝑗 𝑘= 𝜃𝑗 𝑘𝐸[ 𝑥𝑖 𝑘/𝑘−1𝑥𝑇 𝑘]𝐻𝑗𝑇 𝑘+𝐸[ 𝑥𝑖 𝑘/𝑘−1V𝑗 𝑘] −𝜃𝑗 𝑘𝐸[ 𝑥𝑖 𝑘/𝑘−1  𝑥𝑗𝑇 𝑘/𝑘−1]𝐻𝑗𝑇 𝑘 −𝐸[ 𝑥𝑖 𝑘/𝑘−1𝜇𝑗 𝑘−1](Π𝑗𝑗 𝑘−1,𝑘−1)−1V𝑗𝑗 𝑘,𝑘−1, 𝑘≥2. (39) F om he OPL, 𝐸[ 𝑥𝑖 𝑘/𝑘−1𝑥𝑇 𝑘]=𝐸[  𝑥𝑖 𝑘/𝑘−1  𝑥𝑖𝑇 𝑘/𝑘−1]; hen, aking in o accoun (32) o 𝐸[ 𝑥𝑖 𝑘/𝑘−1𝜇𝑗 𝑘−1],and(33) o 𝐸[ 𝑥𝑖 𝑘/𝑘−1V𝑗 𝑘], i is enough o p o e ha 𝐸[ 𝑥𝑖 𝑘/𝑘−1  𝑥𝑖𝑇 𝑘/𝑘−1]−𝐸[ 𝑥𝑖 𝑘/𝑘−1  𝑥𝑗𝑇 𝑘/𝑘−1]=𝑃𝑗𝑗 𝑘/𝑘−1 −𝑃𝑖𝑗 𝑘/𝑘−1,(40) 8Ma hema ical P oblems in Enginee ing which is easily deduced since 𝐸[ 𝑥𝑖 𝑘/𝑘−1  𝑥𝑗𝑇 𝑘/𝑘−1] =𝑃𝑖𝑗 𝑘/𝑘−1 −𝐷𝑘+𝐸[ 𝑥𝑖 𝑘/𝑘−1  𝑥𝑖𝑇 𝑘/𝑘−1]+𝐸[ 𝑥𝑗 𝑘/𝑘−1  𝑥𝑗𝑇 𝑘/𝑘−1], 𝐸[ 𝑥𝑗 𝑘/𝑘−1  𝑥𝑗𝑇 𝑘/𝑘−1]=𝐷𝑘−𝑃𝑗𝑗 𝑘/𝑘−1.(41) Lemma 11. Unde assump ions (i)–(i ), o 𝑖 =𝑗,𝑖,𝑗 = 1,2...,𝑟, he inno a ion c oss-co a iance Π𝑖𝑗 𝑘,𝑘 = 𝐸[𝜇𝑖 𝑘𝜇𝑗 𝑘] sa is ies Π𝑖𝑗 𝑘,𝑘 = 𝜃𝑖 𝑘𝐻𝑖 𝑘(X𝑗 𝑘,𝑘 −𝐿𝑖𝑗 𝑘)+𝜃𝑗 𝑘𝐸𝑖𝑇 𝑘𝐻𝑗𝑇 𝑘+𝑅𝑖𝑗 𝑘,𝑘 −V𝑖𝑖 𝑘,𝑘−1(Π𝑖𝑖 𝑘−1,𝑘−1)−1Π𝑖𝑗 𝑘−1,𝑘 −V𝑖𝑗 𝑘,𝑘−1(Π𝑗𝑗 𝑘−1,𝑘−1)−1(𝜃𝑗 𝑘𝐻𝑗 𝑘X𝑗 𝑘,𝑘−1 +V𝑗𝑗 𝑘,𝑘−1)𝑇, 𝑘≥2, Π𝑖𝑗 1,1 =𝜃𝑖 1𝐻𝑖 1X𝑗 1,1 +𝜃𝑗 1𝐸𝑖 1𝐻𝑗𝑇 1+𝑅𝑖𝑗 1,1,(42) whe e Π𝑖𝑗 𝑘−1,𝑘 =𝐸[𝜇𝑖 𝑘−1𝜇𝑗 𝑘]is gi en by Π𝑖𝑗 𝑘−1,𝑘 = 𝜃𝑗 𝑘(X𝑖 𝑘,𝑘−1 −𝐹 𝑘−1𝐿𝑗𝑖 𝑘−1 −X𝑗 𝑘,𝑘−1(Π𝑗𝑗 𝑘−1,𝑘−1)−1Π𝑗𝑖 𝑘−1,𝑘−1)𝑇𝐻𝑗𝑇 𝑘 +V𝑗𝑖 𝑘,𝑘−1 −Π𝑖𝑗 𝑘−1,𝑘−1(Π𝑗𝑗 𝑘−1,𝑘−1)−1V𝑗𝑗 𝑘,𝑘−1,𝑘≥2. (43) P oo . Taking in o accoun exp ession (26) o 𝜇𝑖 𝑘,wi h(2) o 𝑦𝑖 𝑘,weha e Π𝑖𝑗 𝑘,𝑘 = 𝜃𝑖 𝑘𝐻𝑖 𝑘𝐸[𝑥𝑘𝜇𝑗 𝑘]+𝐸[V𝑖 𝑘𝜇𝑗 𝑘]−𝜃𝑖 𝑘𝐻𝑖 𝑘𝐸[ 𝑥𝑖 𝑘/𝑘−1𝜇𝑗 𝑘] −V𝑖𝑖 𝑘,𝑘−1(Π𝑖𝑖 𝑘−1,𝑘−1)−1𝐸[𝜇𝑖 𝑘−1𝜇𝑗 𝑘] =𝜃𝑖 𝑘𝐻𝑖 𝑘(X𝑗 𝑘,𝑘 −𝐿𝑖𝑗 𝑘)+𝐸[V𝑖 𝑘𝜇𝑗 𝑘] −V𝑖𝑖 𝑘,𝑘−1(Π𝑖𝑖 𝑘−1,𝑘−1)−1Π𝑖𝑗 𝑘−1,𝑘, 𝑘≥2, (44) and, om (34) o 𝐸[V𝑖 𝑘𝜇𝑗 𝑘], exp ession o Π𝑖𝑗 𝑘,𝑘 is clea . Analogously, aking in o accoun exp ession (26) o 𝜇𝑗 𝑘, wi h (2) o 𝑦𝑗 𝑘,weha e Π𝑖𝑗 𝑘−1,𝑘 = 𝜃𝑗 𝑘𝐸[𝜇𝑖 𝑘−1𝑥𝑇 𝑘]𝐻𝑗𝑇 𝑘+𝐸[𝜇𝑖 𝑘−1V𝑗 𝑘] −𝜃𝑗 𝑘𝐸[𝜇𝑖 𝑘−1  𝑥𝑗𝑇 𝑘/𝑘−1]𝐻𝑗𝑇 𝑘 −𝐸[𝜇𝑖 𝑘−1𝜇𝑗 𝑘−1](Π𝑗𝑗 𝑘−1,𝑘−1)−1V𝑗𝑗 𝑘,𝑘−1, =𝜃𝑗 𝑘(X𝑖 𝑘,𝑘−1 −𝐸[𝜇𝑖 𝑘−1  𝑥𝑗𝑇 𝑘/𝑘−1])𝐻𝑗𝑇 𝑘+V𝑗𝑖 𝑘,𝑘−1 −Π𝑖𝑗 𝑘−1,𝑘−1(Π𝑗𝑗 𝑘−1,𝑘−1)−1V𝑗𝑗 𝑘,𝑘−1,𝑘≥2, (45) and, om (32) o 𝐸[𝜇𝑖 𝑘−1  𝑥𝑗𝑇 𝑘/𝑘−1], exp ession o Π𝑖𝑗 𝑘−1,𝑘 is immedia ely de i ed. In he ollowing heo em, ecu si e o mulas o calcula e he il e ing and p edic ion e o c oss-co a iance ma ices, 𝑃𝑖𝑗 𝑘/𝑘 and 𝑃𝑖𝑗 𝑘/𝑘−1, espec i ely, a e de i ed. Theo em 12. Unde assump ions (i)–(i ), he c oss-co a iance ma ices, 𝑃𝑖𝑗 𝑘/𝑘, o he il e ing e o s be ween he 𝑖 h and he 𝑗 h senso subsys ems a e ecu si ely compu ed by 𝑃𝑖𝑗 𝑘/𝑘 =𝑃 𝑖𝑗 𝑘/𝑘−1 +X𝑖 𝑘,𝑘(Π𝑖𝑖 𝑘,𝑘)−1Π𝑖𝑗 𝑘,𝑘(Π𝑗𝑗 𝑘,𝑘)−1X𝑗𝑇 𝑘,𝑘 −(X𝑗 𝑘,𝑘 −𝐿𝑖𝑗 𝑘)(Π𝑗𝑗 𝑘,𝑘)−1X𝑗𝑇 𝑘,𝑘 −X𝑖 𝑘,𝑘(Π𝑖𝑖 𝑘,𝑘)−1(X𝑖 𝑘,𝑘 −𝐿𝑗𝑖 𝑘)𝑇, 𝑘≥1, 𝑃𝑖𝑗 0/0 =𝑃 0, (46) whe e 𝑃𝑖𝑗 𝑘/𝑘−1, he c oss-co a iance ma ix o he p edic ion e o be ween he 𝑖 h and he 𝑗 h senso subsys ems, sa is ies 𝑃𝑖𝑗 𝑘/𝑘−1 =𝐹 𝑘−1𝑃𝑖𝑗 𝑘−1/𝑘−1𝐹𝑇 𝑘−1 +𝑄𝑘−1,𝑘−1 +𝐹 𝑘−1J𝑖 𝑘−1 +J𝑗𝑇 𝑘−1𝐹𝑇 𝑘−1 +W𝑖 𝑘−1,𝑘−1(Π𝑖𝑖 𝑘−1,𝑘−1)−1Π𝑖𝑗 𝑘−1,𝑘−1 ×(Π𝑗𝑗 𝑘−1,𝑘−1)−1W𝑗𝑇 𝑘−1,𝑘−1 −G𝑖𝑗 𝑘−1(Π𝑗𝑗 𝑘−1,𝑘−1)−1W𝑗𝑇 𝑘−1,𝑘−1 −W𝑖 𝑘−1,𝑘−1(Π𝑖𝑖 𝑘−1,𝑘−1)−1G𝑗𝑖𝑇 𝑘−1, 𝑘≥2, 𝑃𝑖𝑗 1/0 =𝐹 0𝑃𝑖𝑗 0/0𝐹𝑇 0+𝑄0,0,(47) whe e G𝑖𝑗 𝑘=W𝑗 𝑘,𝑘 +𝐹𝑘(X𝑗 𝑘,𝑘 −𝐿𝑖𝑗 𝑘−X𝑖 𝑘,𝑘(Π𝑖𝑖 𝑘,𝑘)−1Π𝑖𝑗 𝑘,𝑘),𝑘≥1. The ec o s 𝐿𝑖𝑗 𝑘and he inno a ion c oss-co a iances Π𝑖𝑗 𝑘,𝑘 a e gi eninLemmas10 and 11, espec i ely. P oo . By using (24) o  𝑥𝑖 𝑘/𝑘 and  𝑥𝑗 𝑘/𝑘,weha e 𝑃𝑖𝑗 𝑘/𝑘 =𝑃 𝑖𝑗 𝑘/𝑘−1 +X𝑖 𝑘,𝑘(Π𝑖𝑖 𝑘,𝑘)−1Π𝑖𝑗 𝑘,𝑘(Π𝑗𝑗 𝑘,𝑘)−1X𝑗𝑇 𝑘,𝑘 −𝐸[(𝑥𝑘− 𝑥𝑖 𝑘/𝑘−1)𝜇𝑗𝑇 𝑘](Π𝑗𝑗 𝑘,𝑘)−1X𝑗𝑇 𝑘,𝑘 −X𝑖 𝑘,𝑘(Π𝑖𝑖 𝑘,𝑘)−1𝐸[𝜇𝑖 𝑘(𝑥𝑘− 𝑥𝑗 𝑘/𝑘−1)𝑇]. (48) Taking in o accoun ha 𝐸[𝑥𝑘𝜇𝑗 𝑘]=X𝑗 𝑘,𝑘 and 𝐸[ 𝑥𝑖 𝑘/𝑘−1𝜇𝑗 𝑘]= 𝐿𝑖𝑗 𝑘, he ecu si e exp ession o he c oss-co a iance ma ices o he local il e ing e o s is immedia ely deduced. Ma hema ical P oblems in Enginee ing 9 Following an analogous easoning, using now (25)and aking in o accoun ha 𝐸[(𝑥𝑘− 𝑥𝑖 𝑘/𝑘)𝑤𝑇 𝑘]=J𝑖 𝑘and 𝐸[𝜇𝑖 𝑘𝑤𝑇 𝑘]=W𝑖𝑇 𝑘,𝑘,i iseasy osee ha 𝑃𝑖𝑗 𝑘/𝑘−1 =𝐹 𝑘−1𝑃𝑖𝑗 𝑘−1/𝑘−1𝐹𝑇 𝑘−1 +𝑄𝑘−1,𝑘−1 +𝐹 𝑘−1J𝑖 𝑘−1 +J𝑗𝑇 𝑘−1𝐹𝑇 𝑘−1 +W𝑖 𝑘−1,𝑘−1(Π𝑖𝑖 𝑘−1,𝑘−1)−1Π𝑖𝑗 𝑘−1,𝑘−1 ×(Π𝑗𝑗 𝑘−1,𝑘−1)−1W𝑗𝑇 𝑘−1,𝑘−1 −(W𝑗 𝑘−1,𝑘−1 +𝐹 𝑘−1𝐸[ 𝑥𝑖 𝑘−1/𝑘−1𝜇𝑗 𝑘−1]) ×(Π𝑗𝑗 𝑘−1,𝑘−1)−1W𝑗𝑇 𝑘−1,𝑘−1 −W𝑖 𝑘−1,𝑘−1(Π𝑖𝑖 𝑘−1,𝑘−1)−1 ×(W𝑖 𝑘−1,𝑘−1 +𝐹 𝑘−1𝐸[ 𝑥𝑗 𝑘/𝑘𝜇𝑖 𝑘−1])𝑇. (49) Finally, using again (24) o  𝑥𝑖 𝑘−1/𝑘−1, and since 𝐸[ 𝑥𝑖 𝑘−1/𝑘−2𝜇𝑗 𝑘−1]=𝐿𝑖𝑗 𝑘−1,weha e 𝐸[ 𝑥𝑖 𝑘−1/𝑘−1𝜇𝑗 𝑘−1] =X𝑗 𝑘−1,𝑘−1 −𝐿𝑖𝑗 𝑘−1 −X𝑖 𝑘−1,𝑘−1(Π𝑖𝑖 𝑘−1,𝑘−1)−1Π𝑖𝑗 𝑘−1,𝑘−1 (50) and heexp ession o hec oss-co a iancema iceso he localp edic ione o siseasilyob ained. 4.3. Dis ibu ed Fusion Fil e ing Es ima o s. Once he local LS linea il e ing es ima o s  𝑥𝑖 𝑘/𝑘 and hei e o co a iance ma ices 𝑃𝑖𝑖 𝑘/𝑘,gi eninTheo em 7, along wi h he e o c oss- co a iance ma ices, 𝑃𝑖𝑗 𝑘/𝑘,gi eninTheo em 12,a ea ailable, he dis ibu ed op imal weigh ed usion es ima o s and hei e o co a iance ma ices a e ob ained by applying he op imal in o ma ion usion c i e ion weigh ed by ma ices in he linea minimum a iance sense [30]. Theo em 13. Fo he sys em model (1)and measu emen model (2), unde assump ions (i)–(i ), he dis ibu ed op imal usion il e ,  𝑥𝐷 𝑘/𝑘,isgi enby  𝑥𝐷 𝑘/𝑘 =𝐴1 𝑘 𝑥1 𝑘/𝑘 +⋅⋅⋅+𝐴𝑟 𝑘 𝑥𝑟 𝑘/𝑘, 𝑘≥0, (51) whe e he local es ima o s  𝑥𝑖 𝑘/𝑘,𝑘 ≥ 0(𝑖 = 1,2...,𝑟)a e calcula ed by he ecu si e algo i hm es ablished in Theo em 7. The op imal ma ix weigh s 𝐴𝑖𝑘(𝑖 = 1,2,...,𝑟) a e compu ed by 𝐴𝑘=Σ−1 𝑘/𝑘𝑒(𝑒𝑇Σ−1 𝑘/𝑘𝑒)−1,(52) whe e he ma ices 𝐴𝑘=[𝐴1 𝑘,...,𝐴𝑟 𝑘]𝑇and 𝑒=[𝐼,...,𝐼]𝑇a e bo h 𝑛𝑟×𝑛ma ices, and Σ𝑘/𝑘 =𝐸[( 𝑥1 𝑘/𝑘,..., 𝑥𝑟 𝑘/𝑘)( 𝑥1 𝑘/𝑘,..., 𝑥𝑟 𝑘/𝑘)𝑇] =(𝑃𝑖𝑗 𝑘/𝑘)𝑖,𝑗=1,2,...,𝑟 (53) is an 𝑛𝑟×𝑛𝑟 posi i e de ini e symme ic block ma ix, whose 𝑛×𝑛ma ix en ies 𝑃𝑖𝑗 𝑘/𝑘 a e gi en in Theo ems 7and 12. The e o co a iance ma ices o he dis ibu ed weigh ed usion il e ing es ima o s a e compu ed by 𝑃𝐷 𝑘/𝑘 =(𝑒𝑇Σ−1 𝑘/𝑘𝑒)−1,𝑘≥0, (54) and he ollowing inequali y holds: 𝑃𝐷 𝑘/𝑘 ≤𝑃𝑖𝑖 𝑘/𝑘,𝑖=1,2,...,𝑟. P oo . The p oo is omi ed because i ollows di ec ly om he op imal in o ma ion c i e ion weigh ed by ma ices in he linea minimum a iance sense [30]. Rema k 14. The p oposed dis ibu ed op imal LS linea usion il e equi es he compu a ion o an 𝑛𝑟 × 𝑛𝑟 in e se ma ix, wi h 𝑛 he dimension o he sys em s a e and 𝑟 he numbe o senso s. Consequen ly, he p oposed dis- ibu ed usion me hod has a compu a ional complexi y o 𝑂[(𝑛𝑟)3], equal o ha o he dis ibu ed Kalman- ype il e in [16] and less han ha o he dis ibu ed usion il e sbasedon hes a eaugmen a ionapp oach.Hence,ou dis ibu ed usionme hodissupe io o he il e p oposed in [16](sincei has hesamecompu a ionbu denbu be e accu acy) and also o he dis ibu ed usion il e s based on s a e augmen a ion (since i has less compu a ional complexi y). 5. Nume ical Simula ion Example In his sec ion, a nume ical simula ion example is p esen ed o illus a e he e ec i eness o he cen alized and dis ibu ed il e ing algo i hms p oposed in his pape . Conside a scala i s -o de au o eg essi e model wi h missing mea- su emen s coming om wo senso s wi h au oco ela ed and c oss-co ela ed noises. Acco ding o he p oposed obse a ion model, wo di e en independen sequences o andom a iables wi h a ce ain p obabili y dis ibu ion o e he in e al [0,1] a eused omodel hemissingphe- nomenon. Speci ically, he ollowing model is conside ed as ollows: 𝑥𝑘=0.95𝑥𝑘−1 +𝑤𝑘−1, 𝑘≥1 𝑦𝑖 𝑘=𝜃𝑖 𝑘𝑥𝑘+V𝑖 𝑘,𝑘≥1,𝑖=1,2, (55) whe e he ini ial s a e 𝑥0is a ze o-mean Gaussian a iable wi h a iance 𝑃0=1. The noise p ocesses {𝑤𝑘;𝑘≥0}and {V𝑖 𝑘;𝑘≥1},𝑖=1,2, a e de ined by 𝑤𝑘=0.6(𝜂𝑘+1 +𝜂𝑘+2), V𝑖 𝑘=𝑐 𝑖(𝜂𝑘+𝜂𝑘+1), 𝑖=1,2, (56) whe e he sequence o a iables {𝜂𝑘;𝑘≥1}is a ze o-mean Gaussian whi e p ocess wi h a iance 0.5. Clea ly, acco ding