Robust unknown input observer for state and fault estimation in discrete-time Takagi-Sugeno systems

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. 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