Nonlinear H∞ Measurement Feedback Control of Euler-Lagrange Systems

NONLINEAR H∞MEASUREMENT FEEDBACK CONTROL OF EULER-LAGRANGE SYSTEMS Ca los Vi as Venegas ∗F ancisco R. Rubio ∗,1 ∗Dep . Ingenie ´ıa de Sis emas y Au om´a ica. Escuela Supe io de Ingenie os. Uni e sidad de Se illa. Camino de los Descub imien os s/n. 41092-Se illa. SPAIN e-mail: { i as, ubio}@ca uja.us.es Abs ac : This pape conside s he p oblem o designing explici measu emen eedback H∞con ol laws o a class o Eule -Lag ange sys ems. Fo hese sys ems he join posi ions a e assumed as ou pu s o he sys em, while eloci y measu es a e o be es ima ed om an obse e +con olle s uc u e. The main con ibu ion o his wo k lies in he explici o mula ion o he dynamic s uc u e o a joined obse e +con olle ha gua an ees local asymp o ic s abili y as well as a enua ion o dis u bances acco ding o an H∞ amewo k. In o de o illus a e his me hodology, expe imen al esul s a e shown on a 2 do gy os abilized pla o m. Copy igh c 2005 IFAC. Keywo ds: Non-linea H∞Con ol, Measu emen eedback, Hamil on-Jacobi-Isaacs equa ion, A ac ion basin. 1. INTRODUCTION The as majo i y o cu en con ol echniques o elec omechanical sys ems a e based on com- ple e eedback o he sys em a iables. Ac ually, e en simple con ol laws as PD con ol, equi e measu emen o all he s a e a iables, posi ions and eloci ies. None heless, mos p ac ical elec omechanical sys- ems equen ly omi eloci y senso due o sa - ings in cos s, olume o weigh ha can be ob- ained in his way. As a consequence, only mea- su es o he join displacemen s a e usually a ail- able. This ac ypically yields high-p ecision low- noise posi ion signals, and by con as , he e- loci y mus be ob ained om nume ical es ima- ion om hese posi ion signals. This esul s in a noisy eloci y signal ha mus be ca e ully il e ed 1The au ho s wish o hank CICYT o unding his wo k unde g an DPI2004-06419. o be used as eedback o he con olle . Mo e- o e , e en in he case whe e eloci y senso s a e p esen ( achome e s), hese o en p o ide low- quali y noisy signals because o i s manu ac u - ing echnology. Typically, discon inui ies in he magne ic ield o he achome e s a o a low equencies and o he high equency phenomena educe he quali y o he measu ed eloci y signal. In p ac ice, his ci cums ances may deg ade he dynamic pe o mance o he con olled sys em since noisy signals impose limi s on he maximum a ainable bandwid h o he con olled sys ems, hence educing he alues o he maximum con- olle gains ha can be used. Thus, his pape add esses he p oblem o de- signing a combined obse e +con olle s uc u e o a class o Eule -Lag ange sys ems, such ha he L2-gain o he mapping om he exogenous inpu noise o he penal y ou pu is minimized, o gua an eed o be less han o equal o a p esc ibed Copy igh (c) 2005 IFAC. All igh s ese ed 16 h T iennial Wo ld Cong ess, P ague, Czech Republic 391 alue, γ. As i is well known, his p oblem can ex- p essed acco ding o a nonlinea H∞ amewo k. The design o obse e s o elec omechanical sys- ems is e y complex, due o he nonlinea and coupled s uc u e o he associa ed dynamic mod- els. So a in he li e a u e, he e ha e been a numbe o app oaches o his p oblem. Some el- e an esul s can be ound in (K ene , A.J. and Isido i, A., 1983; Walco , B.A.; Co less, M.J. and Zak, S.H., 1987). Mos o hese me hods p o ide condi ions unde which he o iginal sys em can be ans o med, ia nonlinea change o coo dina es, in o special canonical o ms whe e he obse e can be designed. None heless, hese condi ions a e somehow es ic i e and a e no me by many physical sys ems, as is he case o elec omechan- ical sys ems. I is possible o ob ain less es ic i e condi- ions when local es ima ion o he s a e ec- o is conside ed (Baumann, W.T. and Rugh, W.J., 1986; Nicosia, S. and Tomei, P. and To - nambe, A., 1989). The main d awback o hese app oaches is ha he es ima o can be used only in he neighbo hood o he design ope a ing poin , and mo eo e , complex in e se ans o ma ions a e equi ed o ge he s a e ec o exp essed in physical a iables. These obse e s a e somehow uni e sal in he sense ha a e designed ega dless o he unde lying con ol s a egy implemen ed. This o en causes ha he es ima ed s a e, when used in conjunc ion wi h a con en ional s a e eed- back con olle , does no gua an ee s abili y o he o e all con olled sys em. This ac mo i a ed he de elopmen o com- bined con ol-obse e design s a egies, such ha he s abili y o he sys em is gua an eed. Re- ma kable esul on his espec a e, o example, (Canudas de Wi , C. and Fixo , N. and ˚ As ¨om, K.J., 1992), whe e a modi ied compu ed o que echnique wi h an embedded obse e s uc u e is p oposed, o (Tomei, P., 1989; Nicosia, S. and Tomei, P., 1990), whe e a con ol s uc u e o lexible join s obo is p oposed, aking in o ac- coun he dynamics o he obse e , such ha he joined con olle +obse e sys em gua an ees s abili y assuming he obse e gains sa is y ce - ain es ic ions. None heless, hese esul s do no ake pe o mance o he sys em in o conside a ion, and assume a pe ec knowledge o he sys em dynamics, so obus ness is no conside ed ei he . Mo e ecen ly, he so-called passi i y-based ap- p oach, (O ega and Spong, 1989), has gained much a en ion. This me hodology exploi s he sys em’s physical s uc u e o eshape i s na u al ene gy unc ion, such ha he con ol objec i e is achie ed. This con ol philosophy is adop ed in (Be guis, H. and Nijmeije , H., 1994), whe e a pas- si i y based app oach ha embeds he obse e dynamics in he con ol s uc u e is p oposed . The design o obse e s uc u es wi hin he non- linea H∞ amewo k was ini ia ed in (Isido i and As ol i, 1992; Van de Scha , 1991) wi h la e de- elopmen s in (Rei e al., 1999; Ki iakidis, 2002). The main d awback o his app oach lies in he di icul y o inding explici solu ions o he se o coupled PDE Hamil on-Jacobi-Isaacs equa ions (HJIE) inhe en o he p oblem o mula ion. This has mo i a ed ew applica ions o his me hodol- ogy o eal p oblems, despi e i s po en ial good p ope ies in e ms o dis u bance ejec ion o obus ness. In his pape he p oblem o designing explici measu emen eedback H∞con ol laws o a class o Eule -Lag ange sys ems is conside ed. Fo hese sys ems he join posi ions a e assumed acces- sible as ou pu s o he sys em, while eloci y measu es a e o be es ima ed om a combined obse e +con olle s uc u e. This wo k ex ends p e ious esul s (Isido i and As ol i, 1992) on he opic o he case o ime- a ying sys ems, wi h applica ions o e e ence acking p oblems o Eule -Lag ange sys ems. Fo hese sys ems, an explici o mula ion o he dynamic s uc u e o a combined obse e +con olle is gi en, while a - enua ion o dis u bances is gua an eed acco ding o he H∞ o malism. 2. GENERAL FORMULATION Conside a dynamical sys em in he o m ˙x= (x, ) + g1(x, )ω+g2(x, )u(1) z=h1(x, ) + k12(x, )u(2) y=h2(x, ) (3) whe e he equa ion (1) desc ibes he nonlinea plan dynamics in Rnwi h s a e ec o x( ). u( )∈Rmu ep esen s he con ol ac ion and ω∈Rmωis an exogenous dis u bance ac ing on he sys em. Addi ionally, equa ion (2) de ines a penalizing unc ion z∈Rmz, and y∈Rmpin (3), is con- side ed he accessible ou pu o he sys em. Ad- di ionally, x= 0 is assumed o be an equilib ium poin o he unpe u bed unac ua ed sys em (1), which implies (0, ) = 0, h1(0, ) = 0 y h2(0, ) = 0. Simila ly, he unc ions (x, ), g1(x, ), g2(x, ), h1(x, ), h2(x, ) y k12(x, ) a e assumed o be su icien ly smoo h. The dynamic con ol s uc u e conside ed in his pape akes he o m 392 ˙ ξ=η(ξ, y) (4) u=θ(ξ) whe e ξis he con olle s a e in a neighbo hood Ξ o he o igin o he sys em in R and η: Ξ × Rmp→Rm ,θ: Ξ →Rmua e smoo h unc ions. Addi ionally, η(0,0) = 0 and θ(0) = 0 is sa is ied o gua an ee ha he o igin is an equilib ium poin o he sys em as equi ed. In o de o simpli y he exp essions o he con- olle he ollowing hypo hesis a e also assumed •(H1) hT 1(x, )k12(x, ) = 0 •(H2) kT 12(x, )k12(x, ) = R=RT≥0 Wi h hese de ini ions, he con ol objec i e can be exp essed as:gi en a dynamical sys em in he o m (1)-(3), ob ain a dynamic ou pu eedback con ol law, o equi alen ly, he unc ions ηyθin (4), ha locally asymp o ically s abilize he o igin, sa is ying an L2-gain a enua ion less han γ o he mapping ω7→ z. Tha is, he con ol law umus sa is y he dissi- pa i i y inequali y J∞(u, γ) = 1 2Z∞ 0 kz(x, u, )k2d −γ2 2Z∞ 0 kω( )k2d ≤0 (5) 3. SOLUTION FOR THE GENERAL CASE In o de o o mula e a gene al solu ion o he p oposed p oblem, i s a s anda d esul (Van de Scha , 1991) on he ull s a e eedback solu ion is summa ized. 3.1 The s a e eedback case Theo em 1: Assume he e exis a posi i e de ini e unc ion V(x, ), de ined in a neighbo hood o x= 0, such ha sa is ies de HJIE ∂V ∂ +∂V ∂x −1 2u∗TRu∗+γ2 2ω∗Tω∗+1 2hT 1h1≤0 (6) whe e ω∗(x, ) = 1 γ2gT 1 ∂V T ∂x u∗(x, ) = −R−1gT 2 ∂V T ∂x (7) hen, he s a e eedback con ol law u(x, ) = u∗(x, ) locally asymp o ically s abilizes sys em (1), e i ying he L2-gain a enua ion (5) o he mapping ω7→ z. 3.2 Nonlinea H∞measu emen eedback In his sec ion, he p e ious esul on s a e eed- back H∞con ol is used, as well as some addi- ional esul s, o ex end p e ious esul s (Isido i and As ol i, 1992) on he opic o he case o ime- a ying sys ems. Fi s , le ’s in oduce some ai ly s anda d no a- ion in his con ex . Thus, ˜y=y−ˆyis he ou pu obse a ion e o , wi h y he measu ed ou pu acco ding o (3), and ˆy he obse e es ima ed ou pu . I he es ima ed sys em s a e is deno ed ˆx, he obse e e o dynamics can be exp essed in e ms o he a iable ξ=x−ˆx. Wi h hese de ini ions, he ollowing esul can be s a ed Theo em 2: Assume wo posi i e de ini e unc- ions, V(x, ) and W(x, ξ, ), de ined in a neigh- bo hood o x= 0 and (x, ξ) = (0,0) espec i ely. I he ollowing condi ions a e sa is ied •(i) V(x, ) sa is ies equa ion (6) •(ii) W(x, ξ, ) sa is ies ∂W ∂ +∂W ∂x e1+∂W ∂ξ e2+1 2hT ehe+γ2 2ΦTΦ≤0 (8) whe e e(x, ξ, ) =  e1(x, ξ, ) e2(x, ξ, )= (9) = (x, ) + g1(x, )ω∗(x, ) + g2(x, )υ∗(ξ, ) o(ξ, ) + go(ξ, ˆy, u, , Γ)  he(x, ξ, ) = υ∗(ξ, )−u∗(x, ) (10) Φ(x, ξ, ) = 1 γ2∂W(x, ξ, ) ∂x g1(x, )T (11) wi h ω∗(x, ) de ined as in (7), and υ∗(ξ, ), a ealizable app oxima ion o u∗(x, ) in (7), and Γ a cons an ma ix alue. •(iii) The subsys em ˙x= (x, ) ˙ ξ= o(ξ, ) + go(ξ, ˆy, 0, , Γ) is locally asymp o ically s able. Then, he con ol law ugi en by ˙ ξ= o(ξ, ) + go(ξ, ˆy, u, , Γ) u=υ∗(ξ, ) (12) locally asymp o ically s abilizes sys em (1) e i- ying he a enua ion ela ion in (5). P oo : Due o space limi a ions, he p oo mus be un o una ely omi ed he e. This esul can none heless be p o ed ollowing simila a gumen s o hose used in (Isido i and As ol i, 1992) o ime in a ian sys ems. Fo ime- a ying sys ems, as is he p esen case, he key a gumen o he p oo lies on an app op ia e applica ion o he well known Ba bala heo em. 393 4. PARTICULARIZATION FOR EULER-LAGRANGE SYSTEMS The esul in heo em 2 equi es ob aining he solu ions o wo coupled HJIE, o equi alen ly, inding unc ions V(x, ) and W(x, ξ, ) ha sa - is ies pa ial di e en ial inequali ies (6) and (8). This is a ha d p oblem in he gene al case, so in o de o p o ide wi h explici solu ions, he p ob- lem is pa icula ized o Eule -Lag ange sys ems as is desc ibed in he ollowing sec ions 4.1 Eule -Lag ange sys ems Le ’s conside in his sec ion Eule -Lag ange sys- ems ha can be exp essed as M(q)¨q+C(q, ˙q) ˙q+G(q) = τ+ω( ) (13) whe e, as is usual no a ion, M(q) is he posi- i e de ini e ine ia ma ix, C(q, ˙q) ep esen s he Co iolis-cen i ugal e ms, and G(q) is he po en- ial ene gy e m. The sys em is ac ua ed by gen- e alized o ce- o que ec o τ, unde he in luence o exogenous dis u bances ω( ). I he s a e ec o is aken o be ˜x∈Rnas ˜x=˙q−˙q q−q  and assuming ha q ( ) is a ime a ying e e ence o be ollowed, i can be easily in e p e ed as an s acked measu e o he acking posi ion and eloci y e o s. Using he ollowing ans o ma ion om (Johansson, 1990) z=T0˜x T0=ρI T12 0I(14) and applying he con ol ac ion change τ=M(q)¨q +C(q, ˙ ˆq) ˙q +G(q)−1 ρM(q)T12 ˙ ˆ ˜q −1 ρC(q, ˙ ˆq)T12 ˜q+1 ρu(15) whe e T1= (ρI T12), sys em (13) can be ex- p essed as ˙ ˜x= (˜x, ) + g1(˜x, )ω+g2(˜x, )(u+u ) (16) wi h u = (ρC(q, ˙q )−M(q)T12 −C(q, T12 ˜q)) ˙ ˜y(17) I is wo h o men ion ha his ans o ma ion yields an applicable con ol law since (15) only depends on accessible magni udes, while nonacce- sible componen s ( ˙q) a e lumped on he u e m in (17). The es o e ms in (16) can be easily p o en o ake he o m (˜x, ) = T−1 0 −M−1(q)( 1 2 ˙ M(q, ˙q) + N(q, ˙q))) 0 1 ρI−1 ρT12 !T0˜x (18) and g1(˜x, ) = g2(˜x, ) = T−1 0M−1(q) 0(19) I addi ionally, equa ions (2) and (3) o he gene al o mula ion a e pa icula ized as z(˜x, u) = 1 2˜xTQ˜x+1 2uTRu wi h Qand Rposi i e de ini e ma ices o app o- p ia e dimensions, and y=q−q =0I˜x he ollowing obse e s uc u e can be s a ed 4.2 Obse e s uc u e Wi h hese de ini ions, he gene ic obse e s uc- u e in (4) can ake he o m ˙ ˆx1 ˙ ˆx2=    −M−1C(q, ˆx1)ˆx1−M−1G+M−1+ +1 ρM−1C(q, ˆx1)T12 ˜y+ Γ2˜y ˆx1−1 ρT12 ˜y+γ1˜y     ˆq= ˆx2(20) whe e unc ional dependencies on M(q) and G(q) ha e been omi ed o he sake o compac ness, and Γ1=γ1Innand Γ2∈Rnnis a posi i e de ini e ma ix. Thus, i ˜ ξ=˙ ˜yT˜yTTis de ined, he obse e e o dynamics can be exp essed ˙ ˜ ξ=T−1 0−M−1(q)C(q, ˙q) 0 1 ρ−1 ρT12 T0˜ ξ+ (21) +   1 ρM−1(q)C(q, ˙ ˜y)T12 ˜y−M−1(q)Γ2˜y−γ1˙ ˜y− −M−1(q)C(q, ˙ ˆq)˙ ˜y 0    4.3 Obse e explici o mula ion Exp ession (21) gi es he dynamical s uc u e o sys em’s obse e p o ided app op ia e ma ices Γ2,T12 and scala s ρ≥0, γ1can be ound. The ollowing esul makes use o he gene ic s uc u e de eloped in heo ems 1 and 2, o p o ide analy - ical condi ions o his unknowns o be ound. Theo em 3 : Assume ma ices K1≥0, T12 and scala ρ≥0 can be ound such ha 394 •0K1 K10−TT 1(R−1−1 γ2I)T1+Q≤0 •A1=   ρ2¯ RI 1 2K1+ρ¯ RT12 1 2K1+ρ¯ RTT 12 ¯ RTT 12T12   ≤0 •Γ2>T2 12Mkck˙ ˜yk ρT12m I;γ1>kck Mm o k˙q ( )k ≤ k ∀ ≥0 wi h ¯ R=1 γ2I−R−1and kcsa is ying, as is inhe en o Eule -Lag ange sys ems, he p ope y kC(q, ˙q)k ≤ kck˙qk. Addi ionally, (·)Mand (·)m deno e espec i ely he maximum and minimum eigen alue o he co esponding ma ix. I hese condi ions a e sa is ied, he con ol law υ∗(ξ, ) = ρ˙ ˆ ˜y+T12 ˜ylocally asymp o ically s abi- lizes he con olle +obse e sys em sa is ying he equi ed L2-gain a enua ion ela ion associa ed o he H∞p oblem. Mo eo e , i is possible o gi e an es ima ion o he a ac ion basin o he combined con- olle +obse e sys em as S=nkχk< κmin{1 kc (ρ2(Mmγ1−kck )),ρT12mkpm T2 12Mkco wi h χ=˜xT˜ ξTTand =1 √2qLm LM P oo : Due o space limi a ions, only a b ie ske ch o he p oo is gi en. Theo em 3 is he esul o pa icula izing he unc ions V(x, ) and W(x, ξ, ) in heo em 2, o he ollowing exp es- sions V(˜x, ) = 1 2˜xTTT 0M(q) 0 0K1T0˜x(22) and W(˜x, ˜ ξ, ) = 1 2˜xTTT 0M(q) 0 0K1T0˜x+ +1 2 ˜ ξTTT 0M(q) 0 0K2T0˜ ξ(23) Thus, he i s condi ion o he heo em is ob- ained by di ec subs i u ion o (22) in (6). Taking now U(x, ξ, ) = V(x, ) + W(x, ξ, ) as joined Lyapuno unc ion o he con olle + obse e sys em, i is easy o show ha dU(x, ξ, ) d ≤ − 1 2kh1(x)k2−1 2υ∗T(ξ, )Rυ∗(ξ, )− −1 2γ2kω∗(x) + Φk2+∂V ∂˜x+∂W ∂˜xg1u +∂W ∂˜ ξgo(24) exp ession which has been con enien ly simpli ied by using he ela ions ob ained om subs i u ing (22) and (23) in (6) and (8) espec i ely. Exp ession (24) can be o ced o be nega i e i he igh mos e ms sa is y ∂V ∂˜x+∂W ∂˜xg1u +∂W ∂˜ ξgo≤0 (25) This exp ession can be expanded by using he de i- ni ions in (22) and (23) such ha i is ans o med in an exp ession o he o m ˜xT˜ ξTA1A3 0A2˜x ˜ ξ≤0 (26) whe e A1,A2, and A3 unc ional ma ices o ˜x, ˜ ξand . Applying a his poin he well known Schu complemen esul , i is possible o ob ain he second and hi d condi ions o he heo em om imposing A1≤0 and A2≤0 espec i ely. The addi ional esul on he a ac ion basin can be ob ained by compu ing an uppe bound on exp ession (26), and assuming bounded e e ence eloci y (k˙q ( )k ≤ k ∀ ≥0). 5. EXPERIMENTAL RESULTS To e i y he heo e ical analysis, a se ies o expe - imen s we e pe o med on a gy os abilized pla - o m. The pla o m has wo deg ees o eedom, such ha can o ien a e any a ached de ice ac- co ding o a ange o o ien a ion and ele a ion co- o dina es. The sys em has a couple o gy oscopic de ices o p o ide a i ude ele an in o ma ion o eedback con ol. Figu e 1 shows he acking pe o mance on he ele a ion axis o he pla o m, o a e e ence a- jec o y consis ing in a se ies o s eps linked by smoo h i h o de polynomial in e pola ions. The 012345678910 −0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 Re e ence Ele a ion axis posi ion Time (s) q ( ), qel( ) ( ad) T acking Fig. 1. T acking beha io o he con olled pla - o m eloci y es ima o was gi en ini ially a pe u bed es ima ion o eloci y. This causes he acking o be a he poo o he i s ew seconds. None he- less, i can be obse ed how he con ol s uc u e g adually co ec s he ini ial e o , d i ing he sys- em sa is ac o ily a e he second s ep. I is wo h o men ion ha he sligh esidual acking e o 395 ha can be obse ed in he g aphics is no a con- sequence o he con ol echnique employed, bu o he unmodeled ic ion phenomena. In e es ingly, 012345678910 −0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5 Veloci y by di ec de i a ion Time (s) ˙qel( ), ( ad/s) 012345678910 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5 Es ima ed eloci y Time (s) ˙ ˆqel( ) ( ad/s) Fig. 2. Es ima ed eloci y s. measu ed eloci y igu e 2 shows he beha io o bo h, he es ima ed eloci y ob ained om he obse e s uc u e, and he eloci y ob ained om i s o de de i a ion o he posi ion in o ma ion o he sys em, which esul s in mo e noisy signal. 6. CONCLUSIONS This pape has p esen ed an app oach o de- sign nonlinea measu emen eedback H∞con- ol laws. The pape gene alizes p e ious esul s on he opic o he case o ime- a ying sys- ems, such ha a combined con olle +obse e s uc u e can be designed ha gua an ees local asymp o ic s abili y o he o e all sys em while keeping bounded e ec s o dis u bances ac ing on he sys em. Mo e p ecisely, he con ol e i ies an L2-gain a enua ion o he mapping ω( )7→ zless han a gi en cons an alue, γ. Addi ionally, a solu ion o he pa icula case o he e e ence acking p oblem in Eule -Lag ange sys ems is p o ided. Fo his kind o sys ems he abo e men ioned esul s a e pa icula ized, such ha explici condi ions o he exis ence o he con olle a e gi en. Finally, expe imen al esul s o he p oposed ech- nique a e p esen ed wi h applica ion o a gy- os abilized pla o m, showing good esul s o he acking p oblem p oposed. REFERENCES Baumann, W.T. and Rugh, W.J. (1986). Feedback con ol o nonlinea sys ems by ex ended lin- ea iza ion. IEEE T ansac ions on Au oma ic Con ol 31, 40–46. Be guis, H. and Nijmeije , H. (1994). Robus con ol o obo s ia linea es ima ed s a e eedbacks. IEEE T ansac ions on Au oma ic Con ol 39(10), 2159 – 2162. Canudas de Wi , C. and Fixo , N. and ˚ As ¨om, K.J. (1992). T ajec o y acking in obo ma- nipula o s ia nonlinea es ima ed s a e eed- back. IEEE T ans. Au oma . Con ol 8, 138– 144. Isido i, A. and A. As ol i (1992). Dis u bance a enua ion in H∞-con ol ia measu emen eedback. IEEE T ans. Au oma . Con ol (37), 1283–1293. Johansson, R. (1990). Quad a ic op imiza ion o mo ion coo dina ion and con ol. IEEE T ans. Au oma . Con ol 35(11), 1197–1208. Ki iakidis, K. (2002). H∞op imal il e s o a class o nonlinea models. P oc. o he Ame - ican Con ol Con e ence 3, 2336–2339. K ene , A.J. and Isido i, A. (1983). Linea iza ion by ou pu injec ion and nolinea obse e s. Sys . Con . Le . 3, 47–52. Nicosia, S. and Tomei, P. (1990). Robo Con- ol by using only posi ion measu emen . IEEE T ansac ions on Au oma ic Con ol 35(9), 1058–1061. Nicosia, S. and Tomei, P. and To nambe, A. (1989). An app oxima e obse e o a class o nonlinea sys ems. Sys . Con . Le . 12, 43– 51. O ega, R. and M.W. Spong (1989). Adap i e mo ion con ol o igid obo s: A u o ial. Au oma ica 25, 877–888. Rei , K., F. Sonnemann and R. Unbehauen (1999). Nonlinea s a e obse a ion using H∞- il- e ing Ricca i design. IEEE T ansac ions on Au oma ic Con ol 44(1), 203–208. Tomei, P. (1989). Design o nonlinea obse e s o elas ic join obo s. IFAC symp. nonlinea con ol. Van de Scha , A.J. (1991). A s a e space ap- p oach o nonlinea H∞con ol. Sys . Con ol Le . Walco , B.A.; Co less, M.J. and Zak, S.H. (1987). Compa a i e s udy o nonlinea s a e obse - a ion echniques. In . J. Con ol 45, 2109– 2132. 396