This is an Accep ed Manusc ip o an a icle published by Taylo &
F ancis in “In e na ional Jou nal o Sys ems Science” on 06 h Ap il
2016, a ailable online:
h p://wwww. and online.com/10.1080/00207721.2016.1165898

Robus unknown inpu obse e o s a e and aul es ima ion in disc e e- ime
Takagi-Sugeno sys ems
Damiano Ro ondoa,∗, Ma cin Wi czakb, Vicenç Puiga,c, Fa iha Nejja ia, Ma cin Paze ab
aAu oma ic Con ol Depa men , Uni e si a Poli ècnica de Ca alunya (UPC), Rambla de San Neb idi 11,
08222 Te assa, Spain.
bIns i u e o Con ol and Compu a ion Enginee ing, Uni e si y o Zielona Go a, ul. Podgó na 50, 65-246 Zielona
Gó a, Poland.
cIns i u de Robò ica i In o mà ica Indus ial (IRI), UPC-CSIC, Ca e de Llo ens i A igas 4-6, 08028
Ba celona, Spain.
(Recei ed 00 Mon h 20XX; inal e sion ecei ed 00 Mon h 20XX)
In his pape , a obus unknown inpu obse e (UIO) o he join s a e and aul es ima ion in disc e e- ime
Takagi-Sugeno (TS) sys ems is p esen ed. The p oposed obus UIO, by applying he H∞ amewo k, leads o a
less es ic i e design p ocedu e wi h espec o ecen esul s ound in he li e a u e. The esul ing design p oce-
du e aims a achie ing a p esc ibed a enua ion le el wi h espec o he exogenous dis u bances, while ob aining
a he same ime he con e gence o he obse e wi h a desi ed bound on he decay a e. An ex ension o he case
o unmeasu able p emise a iables is also p o ided. Since he design condi ions educe o a se o linea ma ix
inequali ies (LMIs), ha can be sol ed e icien ly using he a ailable so wa e, an e iden ad an age o he p o-
posed app oach is i s simplici y. The inal pa o he pape p esen s an academic example and a eal applica ion
o a mul i- ank sys em, which exhibi clea ly he pe o mance and e ec i eness o he p oposed s a egy.
Keywo ds: S a e es ima ion, aul diagnosis, unknown inpu obse e s (UIO), Takagi-Sugeno (TS) uzzy
sys ems.
1. In oduc ion
Faul de ec ion and isola ion (FDI) sys ems ha e been a e y ac i e a ea o esea ch in he las decades
and, consequen ly, many schemes o FDI ha e been de eloped (see Zhang and Jiang (2008); Hwang
e al. (2010); Samy e al. (2011)). The FDI app oaches, such as neu al-ne wo k-based me hods (Pa an
e al. 2008) and iden i ica ion-based me hods (Simani e al. 2003), a e gene ally classi ied in o model-
based/da a-based and quan i a i e/quali a i e echniques (Zhang and Jiang 2008). A quan i a i e model-
based FDI scheme u ilizes a ma hema ical model, o en known as analy ical edundancy, o ca y ou
FDI in eal- ime.
Among he p oposed solu ions o aul diagnosis sys ems, he obse e -based ones ha e gained a
lo o in e es . These aul es ima ion me hods a emp o econs uc he aul a he han o de ec i s
p esence, and p o ide a di ec es ima e o i s magni ude and se e i y, which is impo an in many ap-
plica ions, especially when an ac i e aul - ole an con ol (FTC) s a egy is implemen ed (Mahmoud
e al. 2003;Nou a e al. 2009;Wi czak 2014). Among hese echniques, he e a e Kalman il e -based
schemes (Kelle and Da ouach 1999), minimum- a iance es ima o s (Gillijns and Moo 2007), adap i e
es ima o s (Zhang e al. 2010), sliding mode obse e s (Xu e al. 2012;B ahim e al. 2015) and adap i e
obse e s (Ro ondo e al. 2014).
∗Co esponding au ho . e-mail: [email p o ec ed]
1
Takagi-Sugeno (TS) sys ems, as in oduced by Takagi and Sugeno (1985), p o ide an e ec i e way
o ep esen ing nonlinea sys ems wi h he aid o uzzy se s, uzzy ules and a se o local linea models
which a e smoo hly connec ed by uzzy membe ship unc ions (Feng 2006). TS uzzy models a e uni e -
sal app oxima o s since hey can app oxima e any smoo h nonlinea unc ion o any deg ee o accu acy
(Johansen e al. 2000), such ha hey can ep esen complex nonlinea sys ems. Di e en obse e de-
sign echniques ha e been de eloped in he li e a u e o TS sys ems (Ichalal e al. 2010,2009;Chadli
e al. 2009;Boua ou e al. 2010;Moodi and Fa okhi 2013;K. Zhang and Shi 2009).
The s a e obse a ion o dynamic sys ems wi h unknown inpu s o dis u bances has become o
pa amoun impo ance bo h om he heo e ical and he p ac ical poin s o iew. Fo his eason, s a ing
om he seminal wo k by Wang e al. (1975), he esea ch in e es has been a ac ed by he p oblem o
designing unknown inpu obse e s (UIOs), and a lo o e o has been pu in o de eloping his ech-
nique in he las decades (see Wi czak (2007), Wi czak (2014), and he e e ences he ein). The capaci y
o es ima ing he s a e in he p esence o unknown inpu s has pa icula ele ance in he design o FDI
schemes, as sugges ed in ecen wo ks (Chen and Sai 2007,2010;Jia e al. 2011;Fonod e al. 2014).
In pa icula , he design o UIOs o TS sys ems has been an in e es ing opic o esea ch in ecen yea s
(Chadli 2010;Chadli and Ka imi 2013).
In his pape , a obus UIO o he join s a e and aul es ima ion in disc e e- ime TS sys ems is p o-
posed. The esul ing design p ocedu e aims a achie ing a p esc ibed a enua ion le el wi h espec o
he exogenous dis u bances, while ob aining a he same ime he con e gence o he obse e wi h a
desi ed bound on he decay a e. The p oblem is add essed in bo h he cases o measu able and un-
measu able p emise a iables. One ad an age o he p oposed app oach is i s simplici y in educing he
design condi ions o a se o linea ma ix inequali ies (LMIs), ha can be sol ed e icien ly using he
a ailable so wa e. An academic example and a eal applica ion o a mul i- ank sys em show clea ly he
e ec i eness o he p oposed s a egy.
This pape is s uc u ed as ollows. Sec ion 2 o mula es he p oblem, and e isi s he ecen esul
de eloped in Chadli and Ka imi (2013), in o de o show he limi a ions ha a e o e come by he p o-
posed app oach. Sec ion 3p esen s he main esul s o he pape , i.e. he de ini ion o he UIO, he design
p ocedu e wi h and wi hou con e gence a e speci ica ions, and he aul es ima ion. In Sec ion 4, wo
illus a i e examples a e used o show he e ec i eness o he echnique. Finally, he main conclusions
a e d awn in Sec ion 5.
2. P elimina ies and p oblem o mula ion
Conside he ollowing TS uzzy model:
xk+1 =A(sk)xk+B(sk)uk+B(sk) k+W1(sk)wk(1)
=
M
X
i=1
hi(sk)Aixk+Biuk+Bi k+Wi
1wk
yk=C(sk)xk+W2(sk)wk(2)
=
M
X
i=1
hi(sk)Cixk+Wi
2wk
wi h:
hi(sk)≥0∀i= 1, . . . , M
M
X
i=1
hi(sk) = 1 (3)
2
whe e xk∈Rns ands o he s a e, yk∈Rmis he ou pu , uk∈R deno es he nominal con ol inpu ,
k∈R is he ac ua o aul , and wk∈l2is a an exogenous dis u bance ec o sa is ying:
l2={w∈Rn|kwkl2<+∞} (4)
kwkl2= ∞
X
k=0 kwkk2!1
2
(5)
The ac i a ion unc ions hi(·)depend on he ec o o p emise a iables sk=[s1
k, s2
k, . . . , sp
k]T, which
is assumed o depend on measu able a iables, e.g. sys em ou pu s and known inpu s (Takagi and Sugeno
1985) (howe e , his assump ion will be la e elaxed by conside ing he case o unmeasu able p emise
a iables).
No ice ha (2) desc ibes sys ems wi h a ime- a ying ou pu equa ion and hus is a mo e gene al ep-
esen a ion han he one wi h cons an ma ices Cand W, which cons i u es a special case o (2). Fo R3-1
ins ance, cases o which his gene aliza ion could be o in e es comp ehend sys ems wi h nonlin-
ea senso s (e.g. Co on and Wilamowski (2010)) o s a e-space models iden i ied using black-box
iden i ica ion (Vize e al. 2013).
I is desi ed o achie e he ollowing goals:
• o ob ain an es ima ion o he s a es using an UIO, aking in o accoun he ac ua o aul kas an
unknown inpu ;
• o es ima e he ac ua o aul kusing he s a e es ima ion p o ided by he UIO.
He ea e , a sho e iew o he ecen esul (Chadli and Ka imi 2013) is pe o med, in o de o
compa e i wi h he app oach p esen ed in he emaining o he pape . The TS uzzy models conside ed
in Chadli and Ka imi (2013) a e ep esen ed by:
xk+1 =
M
X
i=1
hi(sk)[Aixk+Biuk+Bi k+Wi
1wk](6)
yk=Cxk+F k+W2wk(7)
while he associa ed UIO is:
zk+1 =
M
X
i=1
hi(sk)[Nizk+Giuk+Liyk](8)
ˆxk=zk−Eyk(9)
Le us de ine he s a e es ima ion e o ek=xk−ˆxk, which aking in o accoun (6)-(9) gi es:
ek+1 =
M
X
i=1
hi(sk)Niek+ (TAi−KiC−Ni)xk+ (TBi−Gi)uk
+(TBi−KiF) k+ (TWi
1−KiW2)wk+EF k+1 +EW2wk+1(10)
wi h:
T=I−EC, Ki=NiE+Li(11)
3
which unde :
Ni=TAi−KiC(12)
TBi−Gi= 0 (13)
TBi−KiF= 0 (14)
E[F W2]=0 (15)
TWi
1−KiW2= 0 (16)
boils down o:
ek+1 =
M
X
i=1
hi(sk)Niek(17)
Subsequen ly, Chadli and Ka imi (2013) show ha he design p ocedu e, which gua an ees ha ek
con e ges asymp o ically o ze o, can be educed o sol ing a ela i ely simple se o LMIs.
The app oach p oposed by Chadli and Ka imi (2013) has an incon es able appeal, also due o he ac
ha i conside s an unknown inpu in he ou pu equa ion, which may ep esen a senso aul . Howe e ,
i has he ollowing limi a ions:
• he ma ices Cand W2in he ou pu equa ion (7) a e cons an , whe eas he ma ices in (2) a e
ime- a ying combina ions o Ciand Wi
2;
• he ex e nal dis u bance wkis elimina ed om (10), which equi es ha (15)-(16) hold. This can be
ealized unde pe ec knowledge abou W1and W2, which is a he un ealis ic o ha e in p ac ice;
•a single ma ix Thas o sa is y (14) o i= 1, . . . , M;
•no solu ion o es ima ing kis p o ided in Chadli and Ka imi (2013).
In he ollowing sec ion, a no el app oach ha o e comes hese limi a ions will be p oposed.
3. Main esul s
3.1 Full ank condi ion
Following Gillijns and Moo (2007) and Wi czak (2007,2014), le us assume ha o (1)–(2) he ank
condi ion:
ank (C(sk+1)B(sk)) = ank (B(sk)) = ∀sk(18)
is sa is ied. As demons a ed in he subsequen pa o he pape , unde he abo e ank condi ionR3-2
i is possible o de i e an exac algeb aic o mulae, which uniquely desc ibes he aul . This gua -
an ees uniqueness and iden i iabili y o he aul . I his condi ion we e no sa is ied, one could use
an adap i e app oach (see, e.g., (Wi czak e al. 2015) and he e e ences he ein) o decompose he
o iginal e m B(sk)in o:
B(sk) = B1(sk)B2(sk)(19)
wi h B1(sk)ha ing he desi ed ank p ope y.
No ice ha he ank condi ion (18) is equi alen o:
ank 

M
X
j=1
hj(sk+1)
M
X
i=1
hi(sk)CjBi
= ank M
X
i=1
hi(sk+1)Bi!= (20)
4

Then, he p oblem boils down o checking he ull ank p ope y o all con ex combina ions o Bi, i =
1, . . . , M as well as CjBi,i= 1, . . . , M,j= 1, . . . , M.
Le us conside he p oblem o checking he ull ank p ope y o all con ex combina ions o Bi,i=
1, . . . , M. No ice ha he ask o checking he ull ank p ope y o CjBi,i= 1, . . . , M,j= 1, . . . , M
can be done in he same way.
Fi s o all, le us ecall ha a ma ix Ξ∈Rn×nis called a P-ma ix i all i s p incipal mino s a e
posi i e (Elsne e al. 2002). On he o he hand, a ma ix Ξ∈Rn×nis a block P-ma ix wi h espec o a
pa i ion N(λ)o N={1, . . . , n}in o λ∈[1, n]pai wise disjoin non oid subse s Nio ca dinali y ni,
i= 1, . . . , λ, i o any T∈ Tλ
n(see (Wi czak e al. 2015) o a de ailed explana ion):
de (TΞ+(I−T)) 6= 0 (21)
whe e Tλ
nis he se o all diagonal ma ices T∈Rn×nsuch ha T[Ni] = iI, i∈[0,1],i= 1, . . . , λ,
whe e T[Ni]is he p incipal subma ix o Twi h ow and column indices in Ni(Elsne e al. 2002). A
P-ma ix is also block P-ma ix wi h espec o any pa i ion (Elsne e al. 2002). P-ma ices and block
P-ma ices play an impo an p ope y in s udying he nonsingula i y, Schu and Hu wi z s abili y o
con ex combina ions o ma ices (Johnson and Tsa some os 1995;Elsne and Szulc 1998,2002).
Le us assume ha he ma ices Bi,i= 1, . . . , M, a e ull ank (i no , i can be al eady concluded
ha he ull ank p ope y does no hold), and le us de ine:
Qp,p =BpT Bp, p = 1, . . . , M (22)
Qp,a =BpT Ba+BaT Bp−BaT Ba−BpT Bp o p < a (23)
Rp
a,b =










Qp,p i (a, b) = (1,1)
Qb−1,p i a= 1 ∧b= 2, . . . , p
Ii a=b∧1< b < k
−Ii b= 1 ∧a=p+ 1
0o he wise
(24)
Theo em 1.(Kolodziejczak and Szulc 1999) The ollowing a e equi alen :
(a) All con ex combina ions o B1, . . . , BMha e ull ank.
(b) BMhas ull ow ank and he (M−1)Mn-by-(M−1)Mn ma ix:
V=





R1R−1
MV1,2V1,3. . . V1,4
−IMn IMn 0Mn . . . 0Mn
0Mn −IMn IMn . . . 0Mn
. . . . . . . . . . . . . . .
0Mn . . . 0Mn −IMn IMn






(25)
whe e V1,2= (R2−R1)R−1
M,V1,3= (R3−R2)R−1
Mand V1,4= (RM−1−RM−2)R−1
Mis a block P-
ma ix (Kolodziejczak and Szulc 1999) wi h espec o he pa i ion {F1, . . . , FM−1}o {1, . . . , (M−
1)Mn}, wi h Fi={(M−1)Mn + 1, . . . , iMn},i= 1, . . . , M −1.
P oo . See Kolodziejczak and Szulc (1999). 
Ha ing a ool o checking condi ion (20), i is possible o de i e he obse e design p ocedu e, which
is he main esul o his pape .
3.2 Unknown inpu obse e
By combining (1) and (2), he ollowing is ob ained:
C(sk+1)B(sk) k=yk+1−C(sk+1)A(sk)xk−C(sk+1)B(sk)uk−C(sk+1)W1(sk)wk−W2(sk+1)wk+1
(26)
5
No ice ha (26) is an iden i y, since o gi en ma ices A(sk),B(sk),C(sk+1), and ec o s k,xk,uk,
wk,wk+1, he alue o he ec o yk+1 canno be a bi a y, bu is de e mined by (1)-(2). I ollows ha i
kwas conside ed an unknown a iable, he linea sys em o equa ions esul ing om (26) would admi
a solu ion (i.e. he ac ual alue o k), ha could be ob ained as:
k=H(sk, sk+1)yk+1 −H(sk, sk+1)C(sk+1)A(sk)xk−uk(27)
−H(sk, sk+1)C(sk+1)W1(sk)wk−H(sk, sk+1)W2(sk+1)wk+1
whe e H(sk, sk+1)deno es he Moo e-Pen ose pseudoin e se o C(sk+1)B(sk).
Due o he ank condi ion (18), (27) is he unique solu ion o he linea sys em ob ained om (26).
Mo eo e , H(sk, sk+1)can be calcula ed easily as:
H(sk, sk+1) = (C(sk+1)B(sk))†=h(C(sk+1)B(sk))TC(sk+1)B(sk)i−1(C(sk+1)B(sk))T(28)
whe e †deno es he Moo e-Pen ose pseudoin e se.
In oducing (27) in o (1) leads o:
xk+1 =¯
A(sk, sk+1)xk+¯
H(sk, sk+1)yk+1 +¯
W1(sk, sk+1)wk+¯
W2(sk, sk+1)wk+1 (29)
o , al e na i ely:
xk=¯
A(sk−1, sk)xk−1+¯
H(sk−1, sk)yk+¯
W1(sk−1, sk)wk−1+¯
W2(sk−1, sk)wk(30)
wi h:
¯
A(sk−1, sk)=(I−B(sk−1)H(sk−1, sk)C(sk)) A(sk−1)(31)
¯
H(sk−1, sk) = B(sk−1)H(sk−1, sk)(32)
¯
W1(sk−1, sk)=(I−B(sk−1)H(sk−1, sk)C(sk)) W1(sk−1)(33)
¯
W2(sk−1, sk) = B(sk−1)H(sk−1, sk)W2(sk)(34)
Then, using some echnique, e.g. he well-known sec o nonlinea i y app oach (Tanaka and Wang
2001;Ro ondo e al. 2015), i is possible o ob ain a TS model o (29) and (2), as ollows1:
xk=¯
A(ςk)xk−1+¯
H(ςk)yk+¯
W1(ςk)wk−1+¯
W2(ςk)wk(35)
=
¯
M
X
i=1
ρi(ςk)¯
Aixk−1+¯
Hiyk+¯
Wi
1wk−1+¯
Wi
2wk
yk=ˇ
C(ςk)xk+ˇ
W2(ςk)wk=
¯
M
X
i=1
ρi(ςk)ˇ
Cixk+ˇ
Wi
2wk(36)
1No ice ha he p emise a iables deno ed by ςka e no he same as sk. Also, he ma ices ˇ
Ci,ˇ
Wi
2,i= 1,..., ¯
Ma e di e en om he
ma ices Ci,Wi
2,i= 1, . . . , M.
6
wi h:
ρi(ςk)≥0∀i= 1,..., ¯
M
¯
M
X
i=1
ρi(ςk) = 1 (37)
Then, he ollowing UIO is p oposed o he sys em (35)-(36):
ˆxk=¯
A(ςk)ˆxk−1+¯
H(ςk)yk+K(ςk)(yk−1−ˆyk−1)(38)
=
¯
M
X
i=1
ρi(ςk)¯
Aiˆxk−1+¯
Hiyk+Ki(yk−1−ˆyk−1)
ˆyk=ˇ
C(ςk)ˆxk=
¯
M
X
i=1
ρi(ςk)ˇ
Ciˆxk(39)
Example: Le us conside a TS sys em as in (1)-(2), wi h:
A(sk) = 
−0.4−0.1sk0.2 + 0.2sk0.3−0.1sk
0.3−0.2sk−0.6 + 0.2sk0.3−0.3sk
0.4 + 0.4sk0.2 + 0.5sk0.6−sk
B=

1
2
1
C(sk) = 1 0 −1
1 + sk2 1 
W1(sk) = 

1 + sk0000
0 1000
0 0100
W2(sk) = 0001+sk0
0 0 0 0 1
and sk∈[0,1]. I is s aigh o wa d o check ha :
C(sk+1)B=0
6 + sk+1 
has ank 1, i.e. he same ank o B,∀sk+1 ∈[0,1], such ha (18) holds and i is possible o calcula e
H(sk, sk+1)using (28), ha leads o:
H(sk, sk+1) = 01
6+sk+1 
Hence, he ma ices ¯
A(sk−1, sk),¯
H(sk−1, sk),¯
W1(sk−1, sk)and ¯
W2(sk−1, sk), calcula ed as (31)-
(34), ake he ollowing o m:
¯
A(sk−1, sk) =


−0.5sk−1−3 0.1sk−1+ 2 1.1sk−1+ 0.3
−sk−1+ 1.1sk+ 0.6−0.2sk−1sk−sk−1−sk−2−0.1sk−1sk+ 1.6sk−1−0.3sk−1.2
0.5sk−1sk+ 2.5sk−1+ 0.8sk+ 1.8 0.3sk−1sk+ 1.9sk−1+ 2 −0.9sk−1sk−4.3sk−1+ 0.3sk+ 2.1


6 + sk(40)
¯
H(sk−1, sk) =


0 1
0 2
0 1

6 + sk
(41)
7
¯
W1(sk−1, sk) =


5+5sk−1−2−1 0 0
−2sk−1sk2 + sk−2 0 0
−sk−1sk−sk−1−sk−1−2 5 + sk0 0

6 + sk
(42)
¯
W2(sk−1, sk) =


00001
00002
00001

6 + sk
(43)
By de ining he new p emise a iables ς1=sk−1,ς2=skand ς3=1
6+sk, and conside ing ha
ς1∈[0,1],ς2∈[0,1] and ς3∈[1/7,1/6], (35)-(36) becomes a se o ¯
M= 8 subsys ems, wi h he
ollowing ma ices
¯
A1=
−0.4286 0.2857 0.0429
0.0857 −0.2857 −0.1714
0.2571 0.2857 0.3000 
¯
A2=
−0.5000 0.3333 0.0500
0.1000 −0.3333 −0.2000
0.3000 0.3333 0.3500 

¯
A3=
−0.4286 0.2857 0.0429
0.2429 −0.4286 −0.2143
0.3714 0.2857 0.3429 
¯
A4=
−0.5000 0.3333 0.0500
0.2833 −0.5000 −0.2500
0.4333 0.3333 0.4000 

¯
A5=
−0.5000 0.3000 0.2000
−0.0571 −0.4286 0.0571
0.6143 0.5571 −0.3143
¯
A6=
−0.5833 0.3500 0.2333
−0.0667 −0.5000 0.0667
0.7167 0.6500 −0.3667

¯
A7=
−0.5000 0.3000 0.2000
0.1000 −0.6000 0
0.8000 0.6000 −0.4000
¯
A8=
−0.5833 0.3500 0.2333
0.1167 −0.7000 0
0.9333 0.7000 −0.4667

¯
H1=¯
H3=¯
H5=¯
H7=

0 0.1429
0 0.2857
0 0.1429
¯
H2=¯
H4=¯
H6=¯
H8=

0 0.1667
0 0.3333
0 0.1667

¯
W1
1=

0.7143 −0.2857 −0.1429 0 0
−0.2857 0.2857 −0.2857 0 0
−0.1429 −0.2857 0.7143 0 0
¯
W2
1=

0.8333 −0.3333 −0.1667 0 0
−0.3333 0.3333 −0.3333 0 0
−0.1667 −0.3333 0.8333 0 0

¯
W3
1=

0.7143 −0.2857 −0.1429 0 0
−0.5714 0.4286 −0.2857 0 0
−0.2857 −0.2857 0.8571 0 0
¯
W4
1=

0.8333 −0.3333 −0.1667 0 0
−0.6667 0.5000 −0.3333 0 0
−0.3333 −0.3333 1.0000 0 0

8
a e θ > 0 o he s a e es ima ion e o (84), he H∞obse e design p oblem o he sys em (35)–(36)
wi h ˇ
Ci=ˇ
Cand ˇ
Wi
2=ˇ
W2,i= 1, . . . , ¯
M, and he obse e (82)-(83) is sol able i he e exis ma ices
Pi0,Ni(i= 1,..., ¯
M) and Usuch ha he ollowing inequali y is sa is ied o all i, l = 1, . . . , ¯
M:
Υl
i,j =





I−τPi+ε1ρI +ε2σI 0 0 ϕε2−ε1
2I Ai
1UT
0−µ2I0 0 ˜
Wi
1UT
0 0 −µ2I0¯
Wi
2UT
ϕε2−ε1
2I0 0 −ε2I UT
UAi
1U˜
Wi
1U¯
Wi
2U Pl−U−UT






(89)
wi h τ=e−2θ>0,µ=ω√2and UAi
1,U˜
Wi
1de ined as:
UAi
1=U¯
Ai−UKiˇ
C=U¯
Ai−Niˇ
C(90)
U˜
Wi
1=U¯
Wi
1−UKiˇ
W2=U¯
Wi
1−Niˇ
W2.(91)
P oo . Le us conside he ollowing Lyapuno unc ion:
Vk=
¯
M
X
i=1
ρi(ˆςk)eT
kPiek, Pi0(92)
and le us assume ha (70) holds (see p oo o Co olla y 3.2). Then, by de ining k−1=
[eT
k−1, wT
k−1, wT
k,∆(·)T], i can be shown ha he condi ion (70) is equi alen o:
¯
M
X
i=1
ρi(ˆςk−1)
¯
M
X
l=1
ρl(ˆςk) T
k−1Φl
i k−1<0(93)
whe e:
Φl
i=



(Ai
1)TPlAi
1+I−τP i(Ai
1)TPl˜
Wi
1(Ai
1)TPl¯
Wi
2(Ai
1)TPl
(˜
Wi
1)TPlAi
1(˜
Wi
1)TPl˜
Wi
1−µ2I(˜
Wi
1)TPl¯
Wi
2(˜
Wi
1)TPl
(¯
Wi
2)TPlAi
1(¯
Wi
2)TPl˜
Wi
1(¯
Wi
2)TPl¯
Wi
2−µ2I(¯
Wi
2)TPl
PlAi
1Pl˜
Wi
1Pl¯
Wi
2Pl




(94)
F om (87), we ge ρeT
kek−eT
k∆(·)≥0. The e o e, o any ε1>0:
ε1ek−1
∆(·)TρI −I
2
−I
2−Iek−1
∆(·)≥0(95)
Simila ly, om (88), we ha e o any ε2>0:
ε2ek−1
∆(·)TσI ϕI
2
ϕI
2−Iek−1
∆(·)≥0(96)
By combining (93)-(94) wi h (95)-(96), he ollowing is ob ained:
Φl
i=



(Ai
1)TPlAi
1+I−τP i+ε1ρI +ε2σI (Ai
1)TPl˜
Wi
1(Ai
1)TPl¯
Wi
2(Ai
1)TPl+ϕε2−ε1
2I
(˜
Wi
1)TPlAi
1(˜
Wi
1)TPl˜
Wi
1−µ2I(˜
Wi
1)TPl¯
Wi
2(˜
Wi
1)TPl
(¯
Wi
2)TPlAi
1(¯
Wi
2)TPl˜
Wi
1(¯
Wi
2)TPl¯
Wi
2−µ2I(¯
Wi
2)TPl
PlAi
1+ϕε2−ε1
2I Pl˜
Wi
1Pl¯
Wi
2Pl−ε2I



≺0
(97)
15

Applying Lemma 3.1 o (97) leads o (89), which comple es he p oo . 
4. Illus a i e examples
4.1 Academic example
Le us conside he TS sys em p o ided in he example a he end o Sec ion 3.2, and le us no ice ha
he app oach p oposed by Chadli and Ka imi (2013) canno be applied o his example, since (15) leads
o E= 0 which, combined wi h (11) and (14), gi es:
TBi−KiF=Bi= 0 (98)
which is alse. On he o he hand, applying he design p ocedu e desc ibed in Sec ion 3.3, he ollowing
UIO ma ices a e ob ained wi h µ= 4.6615:
K1=
−0.4473 0.1226
0.1880 −0.1324
0.0788 0.1570 
K2=
−0.5237 0.1383
0.2313 −0.1431
0.0790 0.1685 
K3=
−0.4872 0.1039
0.4589 −0.1651
0.0063 0.1620 

K4=
−0.5657 0.0992
0.5328 −0.1693
0.0164 0.1855 
K5=
−0.5416 0.1205
0.1190 −0.1616
0.3205 0.2266 
K6=
−0.6522 0.1273
0.1654 −0.1708
0.3576 0.2639 

K7=
−0.5802 0.0860
0.4185 −0.1661
0.2673 0.2233 
K8=
−0.6246 −0.0264
0.5515 −0.0403
0.3749 0.2220 

In o de o show he pe o mance o he p oposed app oach, a simula ion has been pe o med wi h
x0= (5,−5,2)T,uk= sin(k/20),wi∈[−0.1,0.1],i= 1, . . . , 5, whe e each wiis gene a ed as a
andom sequence, s∈[0,1] as shown in Figu e 1(co esponding o he ac i a ion unc ions ρi,i=
1, . . . , 8, depic ed in Figu e 2), and kas ollows:
k=


0k≤20
0.5 20 < k ≤40
0.5 + 0.5 sin(2πk/40) else
As expec ed, he esul s shown in Figu es 3and 4demons a e he con e gence o he UIO in he
es ima ion o bo h he s a e xkand he aul k, hus p o ing he e ec i eness o he de eloped me hod.
4.2 Mul i- ank sys em
Le us conside a mul i- ank sys em po ayed in Fig. 5. I consis s o h ee sepa a e anks placedR2-1 R3-4
one abo e he o he and equipped wi h d ain al es and le el senso s based on hyd aulic p essu e
measu emen (Wi czak 2014). Each o hem has a di e en c oss-sec ion in o de o e lec sys em
nonlinea i ies. The lowe bo om ank is a wa e ese oi o he sys em. A a iable speed wa-
e pump is used o ill he uppe ank and he wa e ou lows he anks due o he g a i y. The
16
0 20 40 60 80 100
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Sample k
s(k)
Figu e 1. P emise a iable s(k).
0 50 100
0
0.05
0.1
0.15
0.2
ρ1(ςk)
0 50 100
0
0.5
1
ρ2(ςk)
0 50 100
0
0.05
0.1
0.15
0.2
ρ3(ςk)
0 50 100
0
0.05
0.1
0.15
0.2
ρ4(ςk)
0 50 100
0
0.05
0.1
0.15
0.2
ρ5(ςk)
Sample k 0 50 100
0
0.05
0.1
0.15
0.2
ρ6(ςk)
Sample k 0 50 100
0
0.05
0.1
0.15
0.2
ρ7(ςk)
Sample k 0 50 100
0
0.05
0.1
0.15
0.2
ρ8(ςk)
Sample k
Figu e 2. Ac i a ion unc ions ρ(k).
nonlinea disc e e- ime model o he mul i- ank sys em is gi en by (Wi czak 2014):





h1(k+ 1) = h1(k)−c1h1(k)α+bu(k)
h2(k+ 1) = h2(k)+(c3+c4h2(k))−1(c2h1(k)α−c5h2(k)α)
h3(k+ 1) = h3(k) + c7−(c8−h3(k))2−0.5(c6h2(k)α−c9h3(k)α)
(99)
whe e he eal da a-based pa ame e s we e iden i ied as ollows: b= 1.14,α= 0.5,c1= 1.15·10−4,
c2= 1.01·10−6,c3= 3.5·10−3,c4= 3.48·10−2,c5= 1.20 ·10−6,c6= 3.42·10−5,c7= 1.33·10−1,
c8= 3.5·10−1and c9= 2.80 ·10−5.
By conside ing ac ua o aul s kand an exogenous dis u bance ec o wken e ing in o he
sys em h ough he ma ix:
W1=

0.001 0 0
0 0 0
0 0 0

17
0 20 40 60 80 100
−5
0
5
x1(k)
0 20 40 60 80 100
−5
0
5
x2(k)
0 20 40 60 80 100
−5
0
5
10
Sample k
x3(k)
eal
es ima ion
Figu e 3. S a e xkand i s es ima ion ˆxkusing he p oposed UIO.
0 20 40 60 80 100
−0.2
0
0.2
0.4
0.6
0.8
1
1.2
Sample k
k
eal
es ima ion
Figu e 4. Faul kand i s es ima ion ˆ
kusing he p oposed UIO.
one can ew i e (99)as (1), wi h xk= [h1(k), h2(k), h3(k)]T,uk=u(k), and:
A(sk) = 


s(1)
k0 0
s(2)
ks(3)
k0
0s(4)
ks(5)
k


B=

b
0
0

18
Figu e 5. Mul i-Tank sys em.
whe e:
s(1)
k=1 −c1h1(k)α−1
s(2)
k=c2(c3+c4h2(k))−1h1(k)α−1
s(3)
k=1 −c5(c3+c4h2(k))−1h2(k)α−1
s(4)
k=c6c7−(c8−h3(k))2−0.5h2(k)α−1
s(5)
k=1 −c9c7−(c8−h3(k))2−0.5h3(k)α−1
I is assumed ha noisy measu emen s o h1(k)and h2(k)a e a ailable, i.e., he ou pu equa ion
(2)is cha ac e ized by he ma ices:
C=100
010W2=0 0.003 0
000.003
I is easy o check ha (18)is e i ied, since:
CB =b
0
has ank 1, which allows calcula ing Husing (28):
H=1
b0
Hence, i ollows om (31)-(34) ha :
¯
A(sk−1) = 


000
s(2)
k−1s(3)
k−10
0s(4)
k−1s(5)
k−1



¯
H=

1 0
0 0
0 0

19
¯
W1=

0 0 0
0 0 0
0 0 0
¯
W2=

0 0.003 0
0 0 0
0 0 0

By de ining he new p emise a iables ς1=s(2)
k−1,ς2=s(3)
k−1,ς3=s(4)
k−1and ς4=s(5)
k−1and
conside ing ha hi∈[hmin, hmax],i= 1,2,3,(35)-(36)becomes a se o ¯
M= 16 subsys ems,
ob ained conside ing all he possible combina ions o ex eme alues o he in e als [ςmin
i, ςmax
i],
i= 1,2,3,4, calcula ed as ollows:
ςmin
1=c2(c3+c4+hmax)−1hα−1
max ςmax
1=c2(c3+c4+hmin)−1hα−1
min
ςmin
2= 1 −c5(c3+c4hmin)−1hα−1
min ςmax
2= 1 −c5(c3+c4hmax)−1hα−1
max
ςmin
3=c6c7−(c8−hmax)2−0.5hα−1
max ςmax
3=c6c7−(c8−hmin)2−0.5hα−1
min
ςmin
4= 1 −c9c7−(c8−hmin)2−0.5hα−1
min ςmax
4= 1 −c9c7−(c8−hmax)2−0.5hα−1
max
Due o he noise and he una ailabili y o measu emen s o h3(k), he case o unmeasu able
p emise a iables should be conside ed, as desc ibed in Sec ion 3.6. In his case, (86) eads as
ollows:
∆(·) = 



0
s(2)
k−ˆs(2)
kh1(k) + ˆs(3)
k−s(3)
kh2(k)
s(4)
k−ˆs(4)
kh2(k) + ˆs(5)
k−s(5)
kh3(k)




The p ope ies o ∆(·)being one-sided Lipschi z and quad a ic inne -bounded, i.e. (87)-(88),
ha e been e i ied wi h ρ= 0.0012,σ= 10−11 and ϕ= 0.2396.
Applying he design p ocedu e desc ibed in Sec ion 3.6, a easible solu ion has been ob ained
wi h τ= 5,µ= 1.35 ·10−3,ε1= 6.3829 and ε2= 26.89.
All o he expe imen s ha e been pe o med wi h he eal sys em using he ollowing pa ame e s:
x0= [0.001,0.001,0.001]T,ˆx0= [0.05,0.05,0.01]T,uk= 9 ·10−5, and kde ined as:
k=−0.33 ·uk10001 < k ≤15000
0o he wise
Fig. 6shows he ac i a ion unc ions ρ(k)o he desc ibed sys em. Figs. 7–9p esen he s a es and hei
es ima es. Conside ing ha he wa e le el in he hi d ank is unmeasu able and ha i a ies in he
in e al [0,0.35 (m)] he esul s a e sa is ac o y, as i can be clea ly obse ed in Fig. 10 depic ing he
s a e es ima ion e o . Indeed, he es ima ion e o does no exceed 1cm which, aking in o accoun he
pe manen wa e low as well as he senso measu emen imp ecision, should be pe cei ed as a good
esul . Fig. 11 shows he ac ua o loss o e ec i eness aul kand i s obus es ima e ˆ
k( ed dashed
line). The ob ained esul s ha e been compa ed wi h he ones ob ained using a linea UIO, as p oposed
by Wi czak (2014). In his case, he ma ices desc ibing he linea model ha e been aken om he
documen a ion (INTECO 2013). The aul es ima ion ob ained using he linea UIO is shown in Fig. 11
(black dashed line). F om hese esul s, i is clea ha he p oposed UIO pe o ms signi ican ly be e
han he linea one. Indeed, Fig. 12 clea ly shows ha he aul es ima ion e o associa ed wi h he
p oposed app oach is ela i ely small.
20

0 0.5 1 1.5 2 2.5
x 104
0
0.5
1
ρ1(ςk)
0 0.5 1 1.5 2 2.5
x 104
0
0.5
1
ρ2(ςk)
0 0.5 1 1.5 2 2.5
x 104
0
0.5
1
ρ3(ςk)
0 0.5 1 1.5 2 2.5
x 104
0
0.5
1
ρ4(ςk)
0 0.5 1 1.5 2 2.5
x 104
0
0.5
1
ρ5(ςk)
0 0.5 1 1.5 2 2.5
x 104
0
0.5
1
ρ6(ςk)
0 0.5 1 1.5 2 2.5
x 104
0
0.5
1
ρ7(ςk)
0 0.5 1 1.5 2 2.5
x 104
0
0.5
1
ρ8(ςk)
0 0.5 1 1.5 2 2.5
x 104
0
0.5
1
ρ9(ςk)
0 0.5 1 1.5 2 2.5
x 104
0
0.5
1
ρ10(ςk)
0 0.5 1 1.5 2 2.5
x 104
0
0.5
1
ρ11(ςk)
0 0.5 1 1.5 2 2.5
x 104
0
0.5
1
ρ12(ςk)
0 0.5 1 1.5 2 2.5
x 104
0
0.5
1
ρ13(ςk)
Sample k 0 0.5 1 1.5 2 2.5
x 104
0
0.5
1
ρ14(ςk)
Sample k 0 0.5 1 1.5 2 2.5
x 104
0
0.5
1
ρ15(ςk)
Sample k 0 0.5 1 1.5 2 2.5
x 104
0
0.5
1
ρ16(ςk)
Sample k
Figu e 6. Ac i a ion unc ions ρ(k).
0 0.5 1 1.5 2 2.5
x 104
0
0.05
0.1
0.15
0.2
0.25
0.3
Sample k
x1(k)
eal
es ima ion
0 50 100
0
0.005
0.01
Figu e 7. S a e x1,k and i s es ima e using he p oposed UIO - i s ank.
5. Conclusions
In his pape , an UIO o TS sys ems has been p oposed, wi h he goal o join ly es ima ing he s a e
and he unknown ac ua o aul s. Di e en ly om ecen esul s appea ed in he li e a u e, he p oposed
app oach can deal wi h TS sys ems whose ou pu equa ion ma ices a e no cons an . Mo eo e , ins ead
o elimina ing he ex e nal dis u bances, hei in luence on he UIO pe o mance is minimized, wi h he
ad an age o no equi ing he pe ec knowledge o he dis u bance dis ibu ion ma ices. The whole
design p ocedu e, ha aims a achie ing con e gence o he es ima ion e o ei he wi h any a e, o wi h
a gua an eed decay a e, boils down o sol e a se o LMIs, a p oblem ha can be e icien ly sol ed using
he sol e s a ailable nowadays. The case o ac i a ion unc ions which depend on unmeasu able p emise
a iables has been also conside ed. The e ec i eness o he p oposed me hod has been demons a ed
using an academical example and a eal sys em applica ion. In pa icula , a mul i- ank sys em was em-
ployed o pe o m a compa ison o he p oposed TS UIO wi h a linea one. The ob ained esul s clea ly
show ha he p oposed app oach pe o ms signi ican ly be e han he linea one.
21
0 0.5 1 1.5 2 2.5
x 104
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
Sample k
x2(k)
eal
es ima ion
0 50 100
0
0.005
0.01
Figu e 8. S a e x2,k and i s es ima e using he p oposed UIO - second ank.
0 0.5 1 1.5 2 2.5
x 104
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
Sample k
x3(k)
eal
es ima ion
0 50 100
0
0.005
0.01
Figu e 9. S a e x3,k and i s es ima e using he p oposed UIO - hi d ank.
Acknowledgmen s
This wo k has been unded by he Na ional Science Cen e in Poland unde he g an
2013/11/B/ST7/01110, by he Spanish Go e nmen (MINECO) h ough he p ojec CICYT ECOCIS
( e . DPI2013-48243-C2-1-R), by MINECO and FEDER h ough he p ojec CICYT HARCRICS ( e .
DPI2014-58104-R), by AGAUR h ough he con ac s FI-DGR 2014 ( e . 2014FI_B1 00172) and FI-
DGR 2015 ( e . 2015FI_B2 00171), and by he DGR o Gene ali a de Ca alunya (SAC g oup Re .
2014/SGR/374).
Re e ences
Abbaszadeh, M., and Ma quez, H.J. (2009), “LMI op imiza ion app oach o obus H∞obse e design and s a ic
ou pu eedback s abiliza ion o disc e e- ime nonlinea unce ain sys ems,” In e na ional Jou nal o Robus
and Nonlinea Con ol, 19(3), 313–340.
Boua ou , M., Chadli, M., El Hajjaji, A., and Chaabane, M. (2010), “Es ima ion o s a e, ac ua o and senso aul s
o TS models,” in P oceedings o he 49 h IEEE Con e ence on Decision and Con ol (CDC), pp. 1613–1618.
B ahim, A., Dhah i, S., Hmida, F., and Sellami, A. (2015), “An H∞sliding mode obse e o Takagi–Sugeno
22
0 0.5 1 1.5 2 2.5
x 104
−0.01
−0.008
−0.006
−0.004
−0.002
0
0.002
0.004
0.006
0.008
0.01
Sample k
S a e x3 es ima ion e o
Figu e 10. S a e es ima ion e o using he p oposed UIO - hi d ank.
0 0.5 1 1.5 2 2.5
x 104
−1.5
−1
−0.5
0
0.5
1
1.5 x 10−4
Sample k
F(k)
eal
es ima ion
linea es ima ion
Figu e 11. Faul kand i s es ima ion ˆ
kusing he p oposed and linea UIO.
nonlinea sys ems wi h simul aneous ac ua o and senso aul s,” In e na ional Jou nal o Applied Ma hema ics
and Compu e Science, 25(3), 547–559.
Chadli, M. (2010), “An LMI app oach o design obse e o unknown inpu s Takagi-Sugeno uzzy models,” Asian
Jou nal o Con ol, 12(4), 524–530.
Chadli, M., Akhenak, A., Rago , J., and Maquin, D. (2009), “S a e and unknown inpu es ima ion o disc e e ime
mul iple model,” Jou nal o he F anklin Ins i u e, 346(6), 593–610.
Chadli, M., and Ka imi, H.R. (2013), “Robus obse e design o unknown inpu s Takagi-Sugeno models,” IEEE
T ansac ions on Fuzzy Sys ems, 21(1), 158–164.
Chen, W., and Sai , M. (2007), “Design o a TS based uzzy nonlinea unknown inpu obse e wi h aul diagnosis
applica ion,” in P oceedings o he 24 h Ame ican Con ol Con e ence, pp. 2545–2550.
Chen, W., and Sai , M. (2010), “Fuzzy nonlinea unknown inpu obse e design wi h aul diagnosis applica ions,”
Jou nal o Vib a ion and Con ol, 16(3), 377–401.
Co on, N.J., and Wilamowski, B.M. (2010), “Compensa ion o senso s nonlinea i y wi h neu al ne wo ks,” in
P oceedings o he 24 h IEEE In e na ional Con e ence on Ad anced In o ma ion Ne wo king and Applica ions,
pp. 1210–1217.
de Oli ei a, M.C., Be nussou, J., and Ge omel, J.C. (1999), “A new disc e e- ime obus s abili y condi ion,” Sys-
ems and Con ol Le e s, 37(4), 261–265.
Duca d, G., Faul - ole an ligh con ol and guidance sys ems: p ac ical me hods o small unmanned ae ial ehi-
23
0 0.5 1 1.5 2 2.5
x 104
−4
−3
−2
−1
0
1
2
3
4x 10−5
Sample k
Faul es ima ion e o
Figu e 12. Faul es ima ion e o using he p oposed UIO.
cles, Be lin: Sp inge -Ve lag (2009).
Elsne , L., Mono , V., and Szulc, T. (2002), “On some p ope ies o con ex ma ix se s cha ac e ized by P-ma ices
and block P-ma ices,” Linea and Mul ilinea Algeb a, 50, 199–218.
Elsne , L., and Szulc, T. (1998), “Con ex combina ions o ma ices - nonsingula i y and Schu s abili y cha ac e i-
za ions,” Linea and Mul ilinea Algeb a, 44, 301–312.
Elsne , L., and Szulc, T. (2002), “Con ex se s o Schu s able and s able ma ices,” Linea and Mul ilinea Algeb a,
48, 1–19.
Feng, G. (2006), “A su ey on analysis and design o model-based uzzy con ol sys ems,” IEEE T ansac ions on
Fuzzy Sys ems, 14(5), 676–697.
Fonod, R., Hen y, D., Cha bonnel, C., and Bo nschlegl, E. (2014), “A class o nonlinea unknown inpu obse e
o aul diagnosis: applica ion o aul ole an con ol o an au onomous spacec a ,” in P oceedings o he 10 h
UKACC In e na ional Con e ence on Con ol, pp. 19–24.
Gillijns, S., and Moo , B.D. (2007), “Unbiased minimum- a iance inpu and s a e es ima ion o linea disc e e-
ime sys ems wi h di ec eed h ough,” Au oma ica, 43(5), 934–937.
Gue a, T.M., K uszewski, A., and Laube , J. (2009), “Disc e e Takagi-Sugeno models o con ol: whe e a e we?,”
Annual Re iews in Con ol, 33(1), 37–47.
Hwang, I., Kim, S., Kim, Y., and Seah, C.E. (2010), “A su ey o aul de ec ion, isola ion, and econ igu a ion
me hods,” IEEE T ansac ions on Con ol Sys ems Technology, 18(3), 636–653.
Ichalal, D., Ma x, B., Rago , J., and Maquin, D. (2009), “An app oach o he s a e es ima ion o Takagi-Sugeno
models and applica ion o senso aul diagnosis,” in P oceedings o he 48 h IEEE Con e ence on Decision and
Con ol (CDC), pp. 7789–7794.
Ichalal, D., Ma x, B., Rago , J., and Maquin, D. (2010), “S a e es ima ion o Takagi-Sugeno sys ems wi h unmea-
su able p emise a iables,” IET Con ol Theo y and Applica ions, 4(5), 897–908.
INTECO„ Mul i ank Sys em - Use ’s manual, www.in eco.com.pl (2013).
Jia, Q.X., Zhang, Y.C., Guan, Y., and Wu, L.N. (2011), “Robus nonlinea unknown inpu obse e -based aul
diagnosis o sa elli e a i ude con ol sys em,” in P oceedings o he 10 h UKACC In e na ional Con e ence on
Con ol, pp. 345–350.
Johansen, T.A., Sho en, R., and Mu ay-Smi h, R. (2000), “On he in e p e a ion and iden i ica ion o dynamic
Takagi-Sugeno models,” IEEE T ansac ions on Fuzzy Sys ems, 8(3), 297–313.
Johnson, C.R., and Tsa some os, M.J. (1995), “Con ex se s o nonsingula and P-ma ices,” Linea and Mul ilinea
Algeb a, 38, 233–239.
K. Zhang, B.J., and Shi, P. (2009), “A new app oach o obse e -based aul - ole an con olle design o Takagi-
Sugeno uzzy sys ems wi h s a e delay,” Ci cui s, Sys ems and Signal P ocessing, 28(5), 679–697.
Kelle , J.Y., and Da ouach, M. (1999), “Two-s age Kalman es ima o wi h unknown exogenous inpu s,” Au oma -
ica, 35(2), 339–342.
Kolodziejczak, B., and Szulc, T. (1999), “Con ex combina ions o ma ices - Full ank cha ac e iza ion,” Linea
Algeb a and i s Applica ions, 287(1-3), 215–222.
24