Resul s on dis ibu ed s a e es ima ion o
LTI sys ems acing communica ion ailu es
A. Rod ´ıguez del Nozal ∗L. O ihuela ∗∗ P. Mill´an ∗∗
A. Seu e ∗∗∗ L. Zacca ian ∗∗∗,∗∗∗∗
∗Depa men o Elec ical Enginee ing, Uni e sidad de Se illa, Se ille,
Spain (e-mail: [email p o ec ed])
∗∗ Depa amen o de Ingenie ´ıa, Uni e sidad Loyola Andaluc´ıa, Se illa,
Spain (e-mail: {do ihuela, pmillan}@uloyola.es)
∗∗∗ LAAS-CNRS, Uni e si ´e de Toulouse, CNRS, Toulouse, F ance
(e-mail: {aseu e ,zacca ian}@laas. )
∗∗∗∗ Depa men o Indus ial Enginee ing, Uni e si y o T en o, I aly
Abs ac : We add ess dis ibu ed es ima ion o he s a e o a linea plan by a se o agen s.
The p oblem is cas in a se ing whe e he communica ion capabili ies o an agen migh be
deac i a ed om ime o ime, due o ailu es in he communica ion de ices o malicious a acks.
An obse e a chi ec u e is p oposed o achie e ou es ima ion goal, based on a mul i-hop
subspace decomposi ion, which allows each agen o iden i y i s obse able and unobse able
subspaces and asymp o ically es ima e he plan s a e by using i s own measu emen and he
in o ma ion exchanged wi h he neighbo ing agen s. Uni o m exponen ial con e gence o ze o
o he es ima ion e o s is p o en in he p esence o communica ion ailu es, unde a pe sis ence
o exci a ion assump ion. Finally, he obse e pe o mance is e alua ed in simula ion, showing
he me i s o he p oposed me hod and sugges ing di ec ions o u u e de elopmen s.
Keywo ds: Dis ibu ed es ima ion, Link ailu es, Mul i-agen sys ems, Linea ime-in a ian
sys ems, Swi ching opologies.
1. INTRODUCTION
T adi ional con ol sys ems consis o a cen alized con-
olle ha collec s all he measu emen s om a plan
and ca ies ou he necessa y compu a ions o assign
he con ol inpu . To accomplish his goal, i is o c u-
cial impo ance o place he con olle node close o he
plan , namely bo h nea he senso s and he ac ua o s.
An al e na i e is o in oduce poin - o-poin communi-
ca ion ne wo ks exclusi ely de o ed o he con ol loop.
In con as o his cen alize amewo k, he concep o
Ne wo ked Con ol Sys ems (NCS) a ose a he end o he
las cen u y. NCSs a e spa ially dis ibu ed sys ems o
which he communica ion among he senso s, ac ua o s
and con olle s is suppo ed by a sha ed communica ion
ne wo k. The use o a mul ipu pose sha ed ne wo k o
connec spa ially dis ibu ed elemen s esul s in lexible
a chi ec u es and gene ally educes bo h ins alla ion and
main enance cos s. Despi e o he appa en ad an ages o
NCS, new challenges a ise o he scien i ic communi y,
some o hem being summa ized in Zhang e al. (2015);
Ge e al. (2017).
One o he p oblems ha ecei ed a g ea deal o a en ion
in he las yea s is dis ibu ed es ima ion, as a i s
s ep owa ds achie ing dis ibu ed con ol, o simply o
moni o ing pu poses. The dis ibu ed es ima ion goal is
o es ima e he s a e o a plan by a ne wo k o agen s
ha need o sha e (pa ial) in o ma ion o accomplish a
collec i e es ima ion goal. The eade is e e ed o Rego
e al. (2019) o a ecen su ey o esul s in his a ea.
The abo e-men ioned su ey pape s emphasize ha dis-
ibu ed es ima o s mus deal, among se e al issues ela ed
o communica ions, wi h he occu ence o ailu es in he
communica ion links/de ices. These ailu es may suddenly
isola e one agen o limi i s abili y o ecei e and send
in o ma ion. The e o e, p oposing a dis ibu ed es ima o
capable o coping wi h link and communica ion ailu es,
is he main pu pose o his pape . The p oblem i sel
i s pe ec ly one o he challenges o he cybe -physical
sys ems communi y, which is he ope a ion con inuum, see
Engell e al. (2015).
The li e a u e on dis ibu ed es ima ion is huge and many
app oaches can be ound. This e ision will s a by
p esen ing he main ac i e esea ch lines in he las yea s
om a b oad poin o iew. La e , he scope will be
na owed o he end o aming he pape con ibu ions
agains he la es ad ances in his a ea.
When conside ing pe u bed sys ems, he e a e mainly
h ee amilies o dis ibu ed obse e s, namely, dis ibu ed
Kalman, H∞and se -membe ship il e s, each one alid
and op imized o speci ic models o dis u bances and
noises. Dis ibu ed Kalman il e s (DKF) ( i s p esen ed
in Ol a i-Sabe (2007)) p o ide an op imal s a e es ima-
ion when he sys em model and he measu emen s o
he agen s a e a ec ed by Gaussian noises. Howe e , an
accu a e model o hese noise dis ibu ions is needed and
some imes his is di icul o ob ain. The H∞ il e ing
heo y is used o de elop dis ibu ed es ima o s p o iding
s a e es ima es wi h gua an eed pe o mance. This s a -
egy ha e been success ully applied in Ug ino skii (2011)
and Shen e al. (2010). Usually, dis ibu ed H∞ il e s ely
on cos ly LMI design me hods. Finally, se -membe ship
app oaches aim a inding a compac se whe e he plan
s a e is ce ainly con ined, see O ihuela e al. (2018);
Wang e al. (2017). They a e conse a i e app oaches ha
a e adequa e when he exogenous signals sa is y known
bounds.
Rega ding unpe u bed and noiseless sys ems, di e en
modi ica ions o dis ibu ed Luenbe ge obse e s ha e
been p oposed. In pu suing o a decen alized design o he
obse e s wi h minimum in o ma ion, he au ho s o Pa k
and Ma ins (2017) use s a e augmen a ions. Wi h simila
objec i es, bu using subspace decomposi ions (obse able
and unobse able modes), ecen esul s ha e been p e-
sen ed in del Nozal e al. (2019); Kim e al. (2016); Mi a
and Sunda am (2018). The mos in e es ing ea u e o
hese app oaches is ha hey can p o ide necessa y and
su icien design condi ions, based on de ec abili y p op-
e ies accoun ing o he p esence o he communica ion
ne wo k, and o exploi his ac in he p oposed obse e
s uc u e.
The li e a u e o dis ibu ed obse e s dealing wi h link
ailu es and communica ion losses is less dense. The com-
munica ion ailu es a e modeled wi h di e en me hods.
Fo ins ance, in Ug ino skii (2013), Ma ko p ocesses a e
employed o model andom communica ion opologies.
Howe e , he local mode in o ma ion o he en i e ne wo k
opology is non-Ma ko ian, which complica es he p ob-
lem sol abili y. To o e come his di icul y, Ug ino skii
(2013) emploies a wo-s ep design p ocedu e. The co e-
sponding solu ion equi es sol ing linea ma ix inequali-
ies subjec o ank cons ain s, which a e gene ally di -
icul . A di e en app oach o model he communica ion
ailu es can be ound in Liu e al. (2017) whe e Be noulli
a iables a e used. Liu e al. (2017) in oduces a weigh ed
ma ix in he consensus s eps in o de o implemen dis-
ibu ed il e ing. In addi ion, boundedness p ope ies a e
ho oughly in es iga ed, using s a is ic in o ma ion o he
andom link ailu es. Rega ding he s a egy used o deal
wi h he dis ibu ed es ima ion p oblem wo main ap-
p oaches can be ound in he li e a u e. On he one hand,
he use o H∞s a egies as discussed in Yan e al. (2017)
and Yu e al. (2013). In Yan e al. (2017), neu al ne -
wo ks a e used o es ima e he sys em s a e using lea ning
me hods o he co esponding ma ices. Ins ead, Yu e al.
(2013) designs a il e on each node in he senso ne wo k
ensu ing ha he dynamics o he il e ing e o is mean-
squa e s able and he p esc ibed a e age H∞pe o mance
cons ain is me . On he o he hand, he beha iou o
he Kalman il e dealing wi h communica ion p oblems
has been also s udied. In Ba ilo i e al. (2018) a ailu e
de ec ion de ice is in oduced in e e y agen o de ec link
ailu es in he ne wo k a he ecei ing side. In addi ion,
by using he maximum a pos e io i p obabili y decision
ule, he au ho s p opose a me hod o iden i y online
he gene ally co ela ed mul iple- alued s ochas ic ou pu
delay which gua an ees (wi h some app oxima ion) he
minimum p obabili y o e o , gi en he a ailable obse a-
ions. Finally, he algo i hm p esen ed in Alonso-Rom´an
and Be e ull-Lozano (2016) p o ides unbiased es ima ions
when he s eady-s a e alue o he a e age consensus p o-
cess becomes a andom a iable.
Wi hin his se ing, we ocus he e on dis ibu ed es ima-
ion in he p esence o communica ion losses, and p o ide:
•A dis ibu ed obse e s uc u e based on a mul i-hop
decomposi ion, which decomposes he s a e space in he
obse able subspace o each agen and he inno a ion in-
oduced by he neighbo s a each hop.
•A su icien condi ion on he dis ibu ed obse e gains
ensu ing uni o m global exponen ial s abili y o he e o
dynamics o e all possible communica ion losses sa is ying
a sui able pe sis ence o exci a ion assump ion.
•A se o assump ions (some necessa y and some su i-
cien ) unde which i is always possible o design he gains
o he dis ibu ed obse e s in o de o mee he abo e
men ioned su icien condi ions.
This pape is o ganized as ollows. Sec ion 2 s a es he
main p oblem and p esen s he necessa y assump ions.
Sec ion 4 p esen s he p oposed obse a ion s uc u e and
he main esul s o he pape conce ning s abili y and
easibili y. Sec ion 5 shows he obse e pe o mance in
simula ions. Finally, conclusions a e d awn in Sec ion 6.
No a ion. A g aph is a pai G= (V,E) comp ising a se
V={1,2, . . . , p}o e ices o agen s, and a se E ⊂ V ×V
o edges o links. A di ec ed g aph is a g aph in which
edges ha e o ien a ions, so ha i (j, i)∈ E, hen agen
iob ains in o ma ion om agen j. A di ec ed pa h om
node i1 o node ikis a sequence o edges such as (i1, i2),
(i2, i3), . . ., (ik−1, ik) in a di ec ed g aph. The neighbo hood
o i,Ni,{j: (j, i)∈ E}, is de ined as he se o nodes
wi h edges incoming o node i. Gi en ρ∈Z+, he ρ-hop
eachable se o i,Ni,ρ, is de ined as he se o nodes wi h
a di ec pa h o iin ol ing ρedges. No e ha he 1-hop
eachable se o ico esponds o he neighbo hood o iand
he 0-hop eachable se o ima ches wi h i.
Ope a o s col(·,·), ow(·,·) s acks subsequen ma ices in o
a column/ ow ec o , e.g. o Aand Bo app op ia e
dimensions, col(A, B)=[A>B>]>and ow(A, B) =
[A B]. |x|is he Euclidean no m o ec o x.||A|| s ands
o he induced ma ix no m o ma ix A.
2. PROBLEM STATEMENT
Conside a se o agen s V={1,2, . . . , p}in ending o
dis ibu edly es ima e he s a e o he disc e e- ime LTI
sys em
x(k+ 1) = Ax(k),(1)
yi(k) = Cix(k),∀i∈ V,(2)
whe e xis he s a e ec o , Ais he sys em ma ix,
yi∈Rmiis he ou pu locally measu ed by agen iand
Ci∈Rmi×nis i s ou pu ma ix.
Since he agen s a e no able o econs uc he whole
s a e xbased only on he local measu emen yi(i.e.
de ec abili y o (Ci, A) is no assumed), a communica ion
ne wo k among hem is equi ed.
Thus, le G= (V,E) ep esen s he di ec ed g aph mod-
elling he communica ion ne wo k whe e no communica-
ion ailu es a e allowed. Fo his di ec ed g aph, E ⊂ V×V
ep esen s e e y communica ion channel be ween pai s o
agen s.
2.1 Collec i e de ec abili y assump ion
We in oduce he e some key concep s, use ul o he
de elopmen s o he es o he pape .
De ini ion 1. Fo he communica ion g aph G, he ρ-hop
ou pu ma ix o agen i,Ci,ρ, is de ined as:
Ci,ρ := Ci,ρ−1
col(Cj,ρ−1)j∈Ni,∀ρ≥1,(3)
whe e Ci,0:= Ci.
In ui i ely speaking, he ρ-hop ou pu ma ix Ci,ρ o agen
i, ecu si ely de ined in (3), comp ises i s ou pu ma ix
Ciand he ou pu ma ices o all he agen s jwi h a
di ec pa h o iin ol ing ρo less edges. Based on his
concep , we can o mula e ou i s assump ion. A simila
assump ion was in oduced in del Nozal e al. (2019).
De ini ion 2. Sys em (1)-(2) is collec i ely de ec able i ,
o any i∈ V, he e exis s a ini e numbe o hops `i∈Z>0
such ha pai (Ci,`i, A) is de ec able.
Assump ion 1. Sys em (1)-(2) is collec i ely de ec able.
As shown in del Nozal e al. (2019), Assump ion 1 is
necessa y o he exis ence o a con e ging dis ibu ed
s a e es ima o . Acco ding o De ini ion 2, sys em (1)-(2)
is collec i ely de ec able i , o each agen , he comple e
in o ma ion p o ided by he ne wo k ( ha is, he ρ-hop
ou pu ma ix wi h ρsu icien ly la ge) is su icien o build
a con e ging s a e obse e . Recall ha a s ongly con-
nec ed communica ion ne wo k is no equi ed in con as
wi h o he app oaches as Wang e al. (2019)
2.2 Communica ion model and pe sis ence assump ion
In his pape , we conside ha he opology o he ne wo k
Gcan a y wi h ime due o ailu es in he communica-
ion de ices, jamming a acks (see Jin (2010)) o packe
d opou s.
To his end, a logic a iable k7→ δi(k)∈ {0,1}is
associa ed o each node i o ep esen communica ion
ailu es o agen ia ime k. Using δi, he se o links
poin ing o agen i, (i, j) : ∀j∈ Ni, a e ac i e a ime k
i and only i δi(k) = 1. O he wise, when δi(k) = 0, hese
links a e inac i e due o a communica ion ailu e. As a
esul , E(k) and G(k) = (V,E(k)) ep esen , espec i ely,
he se o ac i e links a ime kand cha ac e ize he ime-
a ying g aph a ime k.
Whene e a loss occu s, agen icanno ecei e in o ma ion
om i s neighbo hood. Consequen ly, i would be eason-
able o ope a e based only on he sys em model and on
he local measu emen yi. Howe e , i his si ua ion is
ex ended in ime, he plan s a e canno be de ec ed in
he di ec ions ha a e no obse able om ha ou pu . To
ule ou his scena io, we assume he ollowing pe sis ence
o exci a ion p ope y.
De ini ion 3. G aph k7→ G(k) = (V,E(k)) enjoys a
uni o m local pe sis ence p ope y i o each i∈ V
he e exis a ini e ime ho izon τi∈Z>0and an in ege
nτi∈Z>0such ha
τi−1
X
h=0
δi(k+h)≥nτi,∀k∈Z≥0,(4)
namely, o each ime in e al {k, . . . , k +τi−1},k∈Z>0,
he e exis a leas nτidis inc alues o s∈ {k, . . . , k +
τi−1}sa is ying δi(s) = 1.
Assump ion 2. The g aph k7→ G(k) = (V,E(k)) enjoys a
uni o m local pe sis ence p ope y.
Assump ion 2 equi es ha each agen iexpe iences no
mo e han τi−nτicommunica ion losses in each ime
window spanning τi ime ins an s. This implies ha he
agen ecei es enough in o ma ion om he neighbo ing
agen s, which co esponds o some kind o pe sis ence o
exci a ion. While Assump ion 2 is no necessa y, in gen-
e al, we emphasize ha i does no hold ue only in cases
whe e he communica ion ins an s (when δi= 1) become
inc easingly a e as ime lows. Such a scena io is qui e
undesi able i one wan s o achie e uni o m con e gence
p ope ies like hose in ou p oblem s a emen below.
2.3 P oblem s a emen
Based on ou s anding Assump ions 1 and 2, we a e eady
o s a e ou p oblem s a emen .
P oblem 1. Conside sys em (1)-(2) and an in e connec-
ion g aph G(k)=(V,E(k)). Unde Assump ions 1 and 2,
design a dis ibu ed obse e p o iding, a each node i,
an es ima e ˆxio he s a e xo (1)-(2), such ha hese
es ima es con e ge uni o mly and exponen ially o x. In
pa icula , o each δi,i= 1, . . . , p, sa is ying Assump ion
2, he e mus exis scala s M > 0 and λ∈(0,1) such ha ,
o any x(0) and any ˆxi(0), i∈ V,
|x(k)−ˆxi(k)|2≤Mλk
p
X
j=1
|x(0) −ˆxj(0)|2,∀i∈ V.(5)
The dis ibu ed obse e ha we design o sol e he p ob-
lem abo e gene alizes he linea ime-in a ian solu ion
in del Nozal e al. (2019). The no el y ha we p opose
he e is ha we ocus on linea dynamics subjec o he
“ex e nal” ime- a ying logical inpu s δi(k). Due o hese
ex a inpu s, he linea cascaded a gumen s o del Nozal
e al. (2019) canno be adop ed, bu we may eso o
nonlinea Inpu o S a e S abili y (ISS) esul s o ime-
a ying sys ems.
3. MULTI-HOP SUBSPACE DECOMPOSITION
Be o e p esen ing he obse e dynamics, we ecall he ba-
sic concep s behind he mul i-hop subspace decomposi ion
p esen ed in del Nozal e al. (2019).
The e always exis s a coo dina e ans o ma ion ma ix
¯
Vi,ρ Vi,ρ∈Rn×nassocia ed o pai (Ci,ρ, A), such ha
he change o a iable [ ¯
Vi,ρ Vi,ρ]>x∈Rn ans o ms he
o iginal s a e-space ep esen a ion in o he obse abili y
s ai case o m Hespanha (2009). No e ha ¯
Vi,ρ ∈Rn×n¯o
i,ρ
is composed by n¯o
i,ρ column ec o s in Rn ha o m an
o hogonal basis o he unobse able subspace o pai
(Ci,ρ, A). Co espondingly, Vi,ρ ∈Rn×no
i,ρ is an o hogonal
basis o i s o hogonal complemen .
De ini ion 4. The ρ-hop unobse able subspace om agen
i, deno ed ¯
Oi,ρ, is composed o all sys em modes ha
canno be obse ed om he ou pu locally measu ed by
agen iand hose measu ed by all he agen s belonging
o he s-hop eachable nodes om i,∀s∈ {0, . . . , ρ}.
Equi alen ly, he ρ-hop unobse able subspace om agen
iis he unobse able subspace ela ed o pai (Ci,ρ, A)
using he abo e coo dina e ans o ma ion:
¯
Oi,ρ := Im( ¯
Vi,ρ).
The o hogonal complemen o ¯
Oi,ρ, wi h some abuse o
no a ion, is deno ed ρ-hop obse able subspace om agen
i,Oi,ρ := Im(Vi,ρ). We deno e no
i,ρ = dim(Oi,ρ).
Acco ding o De ini ion 4, i is clea ha :
Oi,ρ−1⊆ Oi,ρ,∀i∈ V, ρ ≥0.(6)
whe e we conside Oi,−1=∅. Then, he ec o s o he
“inno a ion” basis ha gene a es Oi,ρ ∩(Oi,ρ−1)⊥can be
s acked in o a ma ix Wi,ρ ∈Rn×ni,ρ , whe e ni,ρ =no
i,ρ −
no
i,ρ−1, in such a way ha :
Im(Wi,ρ) := Oi,ρ ∩(Oi,ρ−1)⊥, ρ ≥0,(7)
Le us de ine `i∈Z>0, o be selec ed la e , as an a bi a y
numbe o hops. F om hese de ini ions i is clea ha o
all ρ∈ {0, . . . , `i}and all i∈ V, i holds ha
Im(Vi,ρ) = Im ([Wi,ρ Vi,ρ−1]) ,(8)
Im( ¯
Vi,ρ−1) = Im Wi,ρ ¯
Vi,ρ,(9)
wi h ¯
Vi,−1:= In.
I is wo h poin ing ou ha Im(Wi,ρ) co esponds o he
inno a ion in oduced by he ρ-hop eachable se Ni,ρ o
agen i, ha is, he obse able modes o agen ia hop ρ
ha a e no obse able a hop ρ−1. Acco dingly
he ans o ma ion ma ix Ti∈Rn×n, de ined as Ti=
[¯
Vi,`iVi,`i], can be pa i ioned as ollows, using he inno-
a ions a each hop:
Ti:= | {z }
¯
Vi,ρ
¯
Vi,`iWi,`i· · · Wi,ρ+1
| {z }
Vi,ρ
Wi,ρ · · · Wi,0,(10)
o all ρ∈ {0, . . . , `i}, whe e i is easy o iden i y he
obse able and unobse able subspaces o he sys em by
agen ia hop ρ. No e also ha Tiis o hogonal by
cons uc ion, namely T−1
i=T>
i.
The ollowing lemma, p o en in (del Nozal e al., 2019,
Lemma 3), in oduces some impo an p ope ies ha a e
cen al o he de i a ions o his pape .
Lemma 1. (del Nozal e al., 2019, Lemma 3) Fo each
agen i∈ V, and any `i>Z>0, he nex p ope ies hold,
∀ρ,ρ0∈ {1, . . . , `i}such ha ρ6=ρ0:
(i) W>
i,ρWi,ρ0= 0,
(ii) Im(Wj,ρ−1)⊆Im(Vi,ρ),∀j∈ Ni,
(iii) Im(Wi,ρ)⊆L
j∈Ni
Im(Wj,ρ−1).
4. OBSERVER DESIGN FOR STABILITY DEALING
WITH COMMUNICATION FAILURES
This sec ion con ains he main esul s o he pape . Fi s
we p esen he obse e s uc u e and hen we de i e he
ensuing e o dynamics. Finally, we p o ide design ules o
he obse e gains sol ing P oblem 1 and we show ha ,
unde he p esc ibed assump ions, he design o hese gains
is easible.
4.1 Obse e s uc u e and e o dynamics
Fo any agen i, we p opose he ollowing obse e s uc-
u e:
ˆxi(k+ 1) = Aˆxi(k) + Wi,0Li(yi(k)−ˆyi(k))
+δi(k)
`i
X
ρ=0 X
j∈Ni
Wi,ρNi,j,ρW>
j,ρ−1(ˆxj(k)−ˆxi(k)),(11)
whe e Liand Ni,j,ρ a e, espec i ely, a local gain and
consensus gains o be selec ed la e in such a way ha
P oblem 1 is sol ed. The alue o `iis chosen so ha
collec i e de ec abili y is ul illed as pe Assump ion 1.
This s uc u e was p esen ed in del Nozal e al. (2019)
o δi(k) = 1,∀k. Fo a mo e de ailed explana ion o he
p oposed obse e s uc u e, he eade is e e ed o ha
pape .
Fo each agen i∈ V, le us de ine he co esponding
es ima ion e o ei(k) := x(k)−ˆxi(k). Simila ly, i is
possible o de ine he ans o med es ima ion e o as
εi:= col(εi,`i+1, . . . , εi,0) := T>
iei,(12)
using he mul i-hop subspace decomposi ion (10) in-
oduced in Sec ion 2. Mo e speci ically, he es ima-
ion e o o agen i∈ V, a hop ρ, is de ined as:
εi,ρ(k) := W>
i,ρei(k),∀ρ= 0, . . . , `i+ 1,whe e we
deno e Wi,`i+1 =¯
Vi,`ico esponding o he collec i ely
unobse able bu de ec able sys em modes.
The ollowing p oposi ion cla i ies he dynamics o hese
es ima ion e o s. I s p oo is a s aigh o wa d ex ension
o he esul s in del Nozal e al. (2019) and is he e o e
omi ed.
P oposi ion 1. Conside he ne wo k o agen s desc ibed
by he g aph G(k), whe e e e y agen iimplemen s he
obse e s uc u e (11) o es ima e he s a e o sys em
(1). Then he dynamics o he e o s in (12) co esponds
o
εi,0(k+ 1) = (W>
i,0AWi,0−LiCiWi,0)εi,0(k),(13)
εi,ρ(k+ 1) =
ρ
X
=0
Di,(ρ, )(δi)εi, (k) (14)
+δi(k)X
j∈Ni
Ni,j,ρεj,ρ−1(k), ρ ∈ {1, . . . , `i},
wi h
Di,(ρ, )(δi) = W>
i,ρAWi, −δiX
j∈Ni
Ni,j,ρW>
j,ρ−1Wi, .
The dynamics in P oposi ion 1 o ρ∈ {1, . . . , `i}can be
compac ly w i en as (we emo e he dependence on k o
simplici y):
ε+
i,0= (W>
i,0AWi,0−LiCiWi,0)εi,0,(15a)
ε+
i,ρ =Di,(ρ,ρ)(δi)εi,ρ +Bi,ρ(δi)ui,ρ,i ρ6= 0 (15b)
whe e
Bi,ρ(δi) = h ow Di,(ρ, )(δi) ∈{0,...,ρ−1}|
| ow (δiNi,j,ρ)j∈Nii,
ui,ρ(k) = col(εi, ) ∈{0,...,ρ−1}
col(εj,ρ−1)j∈Ni,
which shows an in e es ing cascaded s uc u e exploi ed in
ou main esul s o he nex sec ion.
4.2 Main esul and uning o he obse e gains
By exploi ing he cascaded dynamics (15), his sec ion
p esen s a design equi emen ha will be p o en o be
su icien o gua an ee he exponen ial es ima ion p ope -
ies o P oblem 1.
No e ha , since he e olu ion o he ans o med es ima-
ion e o a hop ρ= 0 does no depend on he agen s
connec i i y, he local gain Lican be easily uned o
ensu e uni o m exponen ial con e gence o ze o o he
solu ions o (15a). Ins ead, he connec i i y p ope ies in
Assump ion 2 a e undamen al o he e ec i eness o he
design o he consensus gains Ni,j,ρ, o which he cascade
s uc u e e ealed wi h he mul i-hop decomposi ion be-
comes c ucial.
P ope y 1. Fo each agen i, he local gain Liand con-
sensus gains Ni,j,ρ o hops ρ∈ {1, . . . , `i}a e designed in
such a way ha he ma ix
(W>
i,0AWi,0−LiCiWi,0) (16)
is Schu , and he ollowing inequali ies a e me :
Di,(ρ,ρ)(δi)>Pi,ρDi,(ρ,ρ)(δi)< µi,ρ(δi)Pi,ρ, δi∈ {0,1}
(17)
µi,ρ := µi,ρ(0)τi−nτiµi,ρ(1)nτi<1,(18)
whe e µi,ρ(δi), δi= 0,1 a e wo a scala pa ame e s
depending on he swi ching signal δi, sa is ying µi,ρ(1) ≤
µi,ρ(0) and Pi,ρ is a posi i e de ini e ma ix wi h app o-
p ia e dimensions.
Now, we a e in posi ion o in oduce he main esul o
he pape in Theo em 1, which es ablishes ha obse e
(11) sol es P oblem 1 whene e P ope y 1 is sa is ied. I s
p oo is pos poned o Sec ion 4.3 o a oid b eaking he
low o he exposi ion.
Theo em 1. Conside plan (1) obse ed by a se o agen s
ha can measu e hei local ou pu s (2), each o hem
implemen ing he obse e s uc u e (11). Unde Assump-
ions 1 and 2, i he obse e gains a e designed acco ding
o P ope y 1, hen P oblem 1 is sol ed, namely he es i-
ma ion e o s sa is y (5).
The nex heo em comple es he s a emen o Theo em 1.
Theo em 2. I is always possible, unde Assump ions 1
and 2, o design ma ices Li, Ni,j,ρ,∀i, ρ and j∈ Ni, ha
sa is y P ope y 1.
P oo . Acco ding o (del Nozal e al., 2019, Theo em
14), in he absence o communica ion ailu es, namely
δi(k) = 1 o all iand o all k, unde Assump ion 1 i
is possible o design gain ma ices Li, and Ni,j,ρ o ix
he con e gence a e o he es ima o a bi a ily as (a
de ailed design me hod is p esen ed he e). In o he wo ds
he alue µi,ρ(1) can be selec ed a bi a ily close o ze o
by app op ia e choices o he gains. Due o he ac ha
he alue o µi,ρ(0) is de e mined by he open-loop sys em
dynamics, and he e o e independen o he obse e gains,
i is hen possible o choose µi,ρ(1) su icien ly small o
ensu e µi,ρ(0)τi−nτiµi,ρ(1)nτi<1 o any gi en pai τi, nτi
om Assump ion 2. 2
4.3 P oo o Theo em 1
Be o e p o ing he heo em, we in oduce some p elimi-
na y esul s. Ou p oo is based on Inpu o S a e S abili y
(ISS) p ope ies o sys ems o he o m
ξ(k+ 1) = (ξ(k), u(k), k),(19)
which well ep esen dynamics (15b). In pa icula , we
make use o he ollowing p ope y.
De ini ion 5. Sys em (19) is uni o mly globally exponen-
ially ini e-gain ISS wi h espec o ui he e exis scala s
M > 0, λ∈(0,1) and γ > 0, such ha o any ini ial ime
k0, any ini ial condi ion ξ(k0) and any uni o mly bounded
inpu k7→ u(k), he co esponding solu ion k7→ ξ(k)
sa is ies 1
|ξ(k)|2≤Mλk−k0|ξ(k0)|2+γkuk2
∞,∀k≥0,
whe e kuk∞:= supk≥k0|u(k)|deno es he l∞no m o he
inpu u.
To p o e he abo e ISS p ope y o each one o he
subsys ems in (15b), o each i∈ V and each ρ∈
{1, . . . , `i}we will use he ollowing quad a ic Lyapuno
unc ion
Vi,ρ(εi,ρ(k)) = εi,ρ(k)>Pi,ρεi,ρ(k),(20)
whe e Pi,ρ is a symme ic posi i e de ini e ma ix. Wi h
some abuse o no a ion, we will use Vi,ρ(k) in place o
Vi,ρ(εi,ρ(k)) in he nex de i a ions.
Lemma 2. I P ope y 1 holds, hen o each i∈ V and
each ρ∈ {1, . . . , `i} he e exis s a scala γi,ρ ∈R+such
ha he Lyapuno unc ion (20) sa is ies
Vi,ρ(k+ 1) < µi,ρ(δi)Vi,ρ(k) + γi,ρ |ui,ρ(k)|2(21)
along he solu ions o (15b).
P oo . Le us w i e he de i a i e o he Lyapuno unc-
ion (20) along he e o dynamics in (15). To his end, and
o keep he no a ion mo e compac , le us deno e Vi,ρ(k)
me ely by Vi,ρ and Vi,ρ(k+ 1) by V+
i,ρ (and simila ly o δi
and εi,ρ and ui,ρ). We hen ha e
V+
i,ρ = (ε+
i,ρ)>Pi,ρε+
i,ρ
=ε>
i,ρDi,(ρ,ρ)(δi)>Pi,ρDi,(ρ,ρ)(δi)εi,ρ
+ 2ε>
i,ρDi,(ρ,ρ)(δi)>Pi,ρBi,ρ(δi)ui,ρ
+u>
i,ρBi,ρ(δi)>Pi,ρBi,ρ(δi)ui,ρ
≤ε>
i,ρDi,(ρ,ρ)(δi)>Pi,ρDi,(ρ,ρ)(δi)εi,ρ
+ 2 Di,(ρ,ρ)(δi)>Pi,ρBi,ρ(δi)· |εi,ρ|·|ui,ρ|
+Bi,ρ(δi)>Pi,ρBi,ρ(δi)· |ui,ρ|2.
1No e ha he s anda d de ini ion o ISS does no include squa e
powe s in signals no ms. Ne e heless, i his exp ession is ul illed i
is i ial o go back o he s anda d de ini ion.

By comple ing squa es and using he Cauchy–Schwa z
inequali y, he ollowing bound holds o any selec ion o
ηi,ρ ∈R+:
2|εi,ρ| |ui,ρ| ≤ ηi,ρ |εi,ρ|2+1
ηi,ρ
|ui,ρ|2(22)
Mo eo e , om P ope y 1, he e exis s a posi i e scala
νi,ρ ∈R+such ha
Di,(ρ,ρ)(δi)>Pi,ρDi,(ρ,ρ)(δi)< µi,ρ(δi)Pi,ρ −νi,ρI.
The e o e, selec ing ηi,ρ in (22) small enough o sa -
is y Di,(ρ,ρ)(δi)>Pi,ρBi,ρ(δi)ηi,ρ < νi,ρ, we may com-
bine he p e ious bounds o p o e (21) wi h he se-
lec ion γi,ρ = max
δi∈{0,1}η−1
i,ρ Di,(ρ,ρ)(δi)>Pi,ρBi,ρ(δi)+
Bi,ρ(δi)>Pi,ρBi,ρ(δi).2
Based on he p e ious lemma, we can now p o e an ISS
p ope y, in he sense o De ini ion 5, o dynamics (15b)
o each i∈ V and each ρ∈ {1, . . . , `i}. This is es ablished
nex .
Lemma 3. I P ope y 1 and Assump ion 2 hold, hen o
each i∈ V and each ρ∈ {1, . . . , `i} he e o sys em wi h
dynamics (15b) is uni o mly globally exponen ially ini e-
gain ISS wi h espec o ui,ρ.
P oo . The p oo is based on demons a ing ha he sys-
em e o dynamics mee s De ini ion 5 and consequen ly,
i is exponen ially ini e-gain ISS wi h espec o ui,ρ.
Fi s , exploi ing µi,ρ(1) ≤µi,ρ(0) in P ope y 1 and using
Assump ion 2, we may ecu si ely apply exp ession (21)
o τisuccessi e ime ins an s o ob ain
Vi,ρ(k+τi)< µi,ρVi,ρ(k) + γi,ρ||ui,ρ||2
∞,(23)
whe e ||ui,ρ||∞= supk≥0|ui,ρ(k)| he bound on ui,ρ(k),∀k,
µi,ρ <1 is de ined in (18), and
γi,ρ = τi−2
X
=0 τi−1
Y
s= +1
µi,ρ(δi(k+s))!+ 1!γi,ρ,
By ecu si ely applying equa ion (23) e alua ed a imes
o k=hτi, we ob ain (we ocus on he case k0= 0 because
he ex ension o he case k06= 0 is s aigh o wa d)
Vi,ρ((h+ 1)τi)< µi,ρVi,ρ(hτi) + γi,ρ||ui,ρ||2
∞
.
.
. (24)
< µh+1
i,ρ Vi,ρ(0) + γi,ρ||ui,ρ||2
∞,
whe e we used he ac ha µi,ρ <1 implies ha he
ollowing geome ic se ies con e ges:
γi,ρ := γi,ρ
1
1−µi,ρ
=γi,ρ
+∞
X
s=0
µs
i,ρ ≥
h
X
s=0
µs
i,ργi,ρ.
Conside now he in e sample beha io o Vi,ρ and no e
ha linea i y o he dynamics (15b) implies ha he e
exis s a la ge enough scala σsuch ha
Vi,ρ(hτi+s)≤σVi,ρ(hτi) + σ||ui,ρ||2
∞,∀s∈ {0, . . . , τi−1}.
(25)
Combining bounds (24) and (25), we ob ain he ollowing
bound o some sui able λV∈(0,1), MV>0 and γV>0:
Vi,ρ(k)≤MVλk
VVi,ρ(0) + γVkuk2
∞,∀k≥0.(26)
Finally, om s anda d p ope ies o posi i e de ini e ma-
ices, we ge
λmin(Pi,ρ)|εi,ρ(k)|2≤Vi,ρ(k)≤λmax(Pi,ρ)|εi,ρ(k)|2,
which can be used wice in (26) o p o e he desi ed ISS
bound
|εi,ρ(k)|2<λM(Pi,ρ)
λm(Pi,ρ)MVλk
V|εi,ρ(0)|2+γV
λm(Pi,ρ)||ui,ρ||2
∞,
as o be p o en. 2
F om Lemma 3, he p oo o Theo em 1 can be p esen ed.
P oo o Theo em 1. Since ma ix (16) in P ope y 1
is Schu by assump ion, he dynamics o he es ima ion
e o a hop ρ= 0 is exponen ially s able o all agen s,
namely he e exis M0>0 and λ0∈(0,1) such ha
|εi,0(k)| ≤ M0λk
0|εi,0(0)| o all k > 0 and all i∈ V.
Using his bound, and due o he cascaded-like exp ession
o ui,ρ in (15b), we may conca ena e he ISS bounds
es ablished in Lemma 3 o ob ain ha he e exi s λε∈
(0,1) and Mε>0 such ha ec o ε= col(ε1, . . . , εp)
sa is ies he ISS bound
|ε(k)|2≤Mελk
ε|ε(0)|2.(27)
Since εis equi alen ( h ough linea ans o ma ion) o
e= col(e1, . . . , ep) (whe e we ecall ha ei=x−ˆxi), hen
he p e ious bound implies bound (5) in P oblem 1, hus
comple ing he p oo . 2
Rema k 1. We emphasize ha he p oo echnique o his
sec ion, based on he ime- a ying dynamics (19), en-
su es ha o each pe sis en ly exci ing selec ion o δi,
i= 1, . . . , p, as pe De ini ion 3, he e exis Mεand λεsa -
is ying (27) (equi alen ly (5) in P oblem 1). Howe e , we
don’ gi e he e a gua an ee ha hose scala s be uni o m
o e he in ini ely many pe sis en ly exci ing selec ions
o δi. Ne e heless, we conjec u e ha a di e en p oo
echnique may be used o p o e a uni o m exponen ial
bound, alid o all such selec ions. P o ing his uni o m
exponen ial con e gence p ope y is le as u u e wo k.
5. SIMULATION RESULTS
This sec ion p esen s some simula ion esul s ha demon-
s a e he e ec i eness o he p oposed es ima ion algo-
i hm. To his end, le us conside he ollowing LTI
au onomous sys em:



x1
x2
x3
x4



+
=


0.95 0 0 0
0 0.8606 −1.3368 0
0 0.1485 0.9315 0
0 0 0 1.015





x1
x2
x3
x4


.
The sys em is obse ed by a se o h ee agen s ha
communica e acco ding o he ollowing diag am 1 ↔2↔
3. Then, he in e connec ion g aph is composed by V=
{1,2,3}and he se o edges is E={(1,2),(2,1),(2,3),
(3,2)}, gene a ing g aph G= (V,E). In all o he examples
discussed below, we conside he ollowing ou pu ma ices
o he h ee agen s:
C1=


1 0
0 0
0 0
0 1



>
, C2=


0
1
0
0



>
, C3=


1 0
0 0
0 0
0 1



>
.
Example 1. Le us assume a scena io whe e he agen s
expe ience communica ion ailu es. We assume ha a
e e y ime in e al {k, . . . , k+τi−1}, whe e τi= 100,∀i∈
V, he e exis s a leas nτi= 20 imes when e e y agen s
i= 1,2,3, can communica e wi h hei neighbo hood.
Acco ding o (11), when he agen icanno communica e,
he obse e dynamics only uses he local measu emen s
yi. Thus, he e o dynamics o he locally unobse able
subspaces will e ol e in open loop. In he case o uns able
dynamics, he es ima ion e o will acco dingly g ow.
0
0.5
1
1.5
2
2.5
3
10 20 30 40 50 60 70 80 90 100
1
0
Fig. 1. Es ima ion e o o agen 2 in Example 1.
Figu e 1 shows he e olu ion o he es ima ion e o o
agen 2, es ima ing s a es x1and x4. Recall ha hese
s a es a e measu ed by agen s 1 and 3 and, acco ding o
he communica ion opology, hese agen s a e one hop
away om agen 2. When agen 2 can communica e,
he es ima ion e o dec eases signi ican ly. This is a
consequence o he as con e gence a e ixed in he
obse e design. Howe e , when agen 2 is no able o
communica e wi h i s neighbo s he es ima ion e o g ows
acco ding o he uns able open-loop dynamics.
Example 2. This second example shows he uns able e-
sponse o he s a e es ima ion e o when P ope y 1 is
no me . The consensus ma ices designed in he p e ious
example o τi= 100, nτi= 20, do no sa is y P ope y 1
in he wo sened scena io wi h τi= 100, nτi= 2, o all
i∈ V, namely wi h inc eased communica ion losses.
By pe o ming a pa allel simula ion o he one o he
p e ious example, we now obse e a di e ging e o e-
sponse. In pa icula , Figu e 2 shows he e olu ion o he
es ima ion e o o agen 2 es ima ing s a e x4. No e ha
he es ima ion e o dec eases when he communica ions
a e ac i e. Howe e , his is no enough o s abilize he
es ima ion e o .
Example 3. Le us p esen in his example an ex ension o
he wo k p oposed in his pape in which, ins ead o o al
communica ion losses, we conside ha each link in he
ne wo k can indi idually ail. In his case one may selec
he same obse e s uc u e (11) by se ing δi(k) = 0 when
jus one link incoming o agen iis ailing. Howe e his
app oach may be qui e conse a i e.
He e we p opose an al e na i e obse e s uc u e whe e
he ailu e- ela ed logical a iables δa e associa ed o he
links ( ha is, we call hem δi,j (k)) a he han o he
nodes. In pa icula , we may modi y (11) as ollows
1
2
3
4
5
10 20 30 40 50 60 70 80 90 100
1
0
Fig. 2. Es ima ion e o o agen 2 es ima ing s a e x4in
Example 2.
0
1
2
0
0.5
1
10 20 30 40 50 60 70 80 90 100
0
0.5
1
(4)
Fig. 3. Es ima ion e o o Agen 2 es ima ing s a e x4in
Example 2.
ˆxi(k+ 1) = Aˆxi(k) + Wi,0Li(yi(k)−ˆyi(k))
+
`i
X
ρ=0 X
j∈Ni
δi,j(k)Wi,ρNi,j,ρW>
j,ρ−1(ˆxj(k)−ˆxi(k)),
whe e δi,j(k) is a bina y a iable indica ing whe he link
(i, j) is ac i e (δi,j (k) = 1) o no (δi,j (k) = 0) a ime k.
Due o he opology conside ed in ou example, only
agen 2 can bene i om he p oposed modi ica ion. Hence,
Figu e 3 shows he e olu ion o he es ima ion e o
o Agen 2 es ima ing s a e x4when using his e ised
s uc u e. Wi h he same gains Ni,j,ρ used in he p e ious
examples, he new s uc u e success ully es ima es he
ou h s a e.
This las example encou ages he au ho s o wo k owa ds
a gene aliza ion o ou app oach o he case whe e he
agen s ha e pa ial communica ion o link ailu es, bu
s ill can make use o he in o ma ion ecei ed om he
es hei neighbo s.
6. CONCLUSIONS
In his pape , he dis ibu ed s a e es ima ion p oblem o
an au onomous LTI sys em by a lossy ne wo k o agen s
has been add essed. By using an obse e s uc u e based
on a mul i-hop subspace decomposi ion, each agen in-
ol ed in he ne wo k can iden i y i s obse able subspace
and he inno a ions in oduced (whene e a communica-
ion loss does no occu ) by i s neighbo s a each hop.
Unde some easonable assump ions on he ne wo k con-
nec i i y, we ha e shown ha i is always possible o ind
obse e gains gua an eeing uni o m exponen ial con e -
gence o ze o o all he es ima ion e o s. Ou a chi ec u e
has been es ed by means o simula ions and he main
esul s o he pape ha e been illus a ed by a ne wo k
o h ee agen s. Fu u e wo k includes p o ing a uni o m
e sion o ou exponen ial bound.
ACKNOWLEDGEMENTS
Resea ch pa ially suppo ed by g an TEC2016-80242-
P ounded by AEI/FEDER h ough he Labo a o io de
Simulaci´on Ha dwa e-in- he-loop, and by ANR ia g an
HANDY, numbe ANR-18-CE40-0010.
REFERENCES
Alonso-Rom´an, D. and Be e ull-Lozano, B. (2016). Adap-
i e consensus-based dis ibu ed Kalman il e o WSNs
wi h andom link ailu es. In 2016 In e na ional Con-
e ence on Dis ibu ed Compu ing in Senso Sys ems
(DCOSS), 187–192. IEEE.
Ba ilo i, S., Cacace, F., d’Angelo, M., and Ge mani,
A. (2018). Dis ibu ed Kalman il e ing o e senso
ne wo ks wi h unknown andom link ailu es. IEEE
con ol sys ems le e s, 2(4), 587–592.
del Nozal, A.R., Mill´an, P., O ihuela, L., Seu e , A., and
Zacca ian, L. (2019). Dis ibu ed es ima ion based on
mul i-hop subspace decomposi ion. Au oma ica, 99,
213–220.
Engell, S., Paulen, R., Renie s, M., Sonn ag, C., and
Thompson, H. (2015). Co e esea ch and inno a ion
a eas in cybe -physical sys ems o sys ems. In In e na-
ional Wo kshop on Design, Modeling, and E alua ion
o Cybe Physical Sys ems, 40–55. Sp inge .
Ge, X., Yang, F., and Han, Q. (2017). Dis ibu ed ne -
wo ked con ol sys ems: A b ie o e iew. In o ma ion
Sciences, 380, 117–131.
Hespanha, J.P. (2009). Linea sys ems heo y. P ince on
uni e si y p ess.
Jin, H. (2010). Handbook o Resea ch on De elopmen s and
T ends in Wi eless Senso Ne wo ks: F om P inciple o
P ac ice. IGI Global.
Kim, T., Shim, H., and Cho, D.D. (2016). Dis ibu ed
Luenbe ge obse e design. In P oceedings o he IEEE
Con e ence on Decision and Con ol (CDC), 6928–6933.
Las Vegas.
Liu, Q., Wang, Z., He, X., and Zhou, D. (2017). On
Kalman-consensus il e ing wi h andom link ailu es
o e senso ne wo ks. IEEE T ansac ions on Au oma ic
Con ol, 63(8), 2701–2708.
Mi a, A. and Sunda am, S. (2018). Dis ibu ed obse e s
o LTI sys ems. IEEE T ansac ions on Au oma ic
Con ol, 63(11), 3689–3704.
Ol a i-Sabe , R. (2007). Dis ibu ed Kalman il e ing o
senso ne wo ks. In P oceedings o he IEEE Con e ence
on Decision and Con ol (CDC), 5492–5498. LA.
O ihuela, L., Mill´an, P., Roshany-Yamchi, S., and Ga c´ıa,
R.A. (2018). Nego ia ed dis ibu ed es ima ion wi h
gua an eed pe o mance o bandwid h-limi ed si ua-
ions. Au oma ica, 87, 94–102.
Pa k, S. and Ma ins, N.C. (2017). Design o dis ibu ed
LTI obse e s o s a e omniscience. IEEE T ansac ions
on Au oma ic Con ol, 62(2), 561–576.
Rego, F., Pascoal, A., Aguia , A., and Jones, C. (2019).
Dis ibu ed s a e es ima ion o disc e e- ime linea
ime in a ian sys ems: A su ey. Annual Re iews in
Con ol.
Shen, B., Wang, Z., and Hung, Y.S. (2010). Dis ibu ed
H∞-consensus il e ing in senso ne wo ks wi h mul iple
missing measu emen s: The ini e-ho izon case. Au o-
ma ica, 46(10), 1682–1688.
Ug ino skii, V. (2011). Dis ibu ed obus il e ing wi h
H∞consensus o es ima es. Au oma ica, 47(1), 1–13.
Ug ino skii, V. (2013). Dis ibu ed obus es ima ion
o e andomly swi ching ne wo ks using H∞consensus.
Au oma ica, 49(1), 160–168.
Wang, L., Liu, J., Mo se, A.S., and Ande son, B.D.O.
(2019). A dis ibu ed obse e o a disc e e- ime linea
sys em. In 2019 Con e ence on Decision and Con ol,
Nice, F ance.
Wang, Y., Alamo, T., Puig, V., and Cemb ano, G. (2017).
Dis ibu ed zono opic se -membe ship s a e es ima ion
based on op imiza ion me hods wi h pa ial p ojec ion.
IFAC-Pape sOnLine, 50(1), 4039–4044.
Yan, H., Yang, Q., Zhang, H., Yang, F., and Zhan, X.
(2017). Dis ibu ed H∞s a e es ima ion o a class o
il e ing ne wo ks wi h ime- a ying swi ching opolo-
gies and packe losses. IEEE T ansac ions on Sys ems,
Man, and Cybe ne ics: Sys ems, 48(12), 2047–2057.
Yu, H., Zhuang, Y., and Wang, W. (2013). Dis ibu ed
H∞ il e ing in senso ne wo ks wi h andomly occu ed
missing measu emen s and communica ion link ailu es.
In o ma ion Sciences, 222, 424–438.
Zhang, X., Han, Q., and Yu, X. (2015). Su ey on
ecen ad ances in ne wo ked con ol sys ems. IEEE
T ansac ions on Indus ial In o ma ics, 12(5), 1740–
1752.