L 1 Adaptive Controller for Nonlinear Systems in the Presence of

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1 28 American Conrol Conference Wesin Seale Hoel, Seale, Washingon, USA June -3, 28 FrA4.3 L Adapive Conroller for Nonlinear Sysems in he Presence of Unmodelled Dynamics: Par II Chengyu Cao and Naira Hovakimyan Absrac This paper presens a novel adapive conrol mehodology for a class of uncerain nonlinear sysems in he presence of unmodelled dynamics. The adapive conroller ensures uniformly bounded ransien response for sysem s boh inpu and oupu signals simulaneously. The performance bounds can be sysemaically improved by increasing he adapaion gain. I. INTRODUCTION Reference [] considers a class of uncerain sysems in he presence of ime and sae dependen unknown nonlineariies and develops an adapive conrol mehodology ha ensures uniformly bounded ransien response for sysem s inpu/oupu signals simulaneously. This paper eends he resuls of [] o a class of uncerain nonlinear sysems in he presence of unmodeled dynamics. The L norm bounds for he error signals beween he closed-loop adapive sysem and he closed-loop reference sysem can be sysemaically reduced by increasing he adapaion gain. The paper is organied as follows. Secion II gives he problem formulaion. In Secion III, he novel L adapive conrol archiecure is presened. Sabiliy and uniform performance bounds are presened in Secion IV. In Secion V, simulaion resuls are presened, while Secion VI concludes he paper. II. PROBLEM FORMULATION Consider he following sysem dynamics: ẋ = A m + bωu + f,, = g o,, ẋ = g,,, y = c, =, where R n is he sysem sae vecor measurable, u R is he conrol signal, y R is he regulaed oupu, b,c R n are known consan vecors, A m is a known n n Hurwi mari, ω R is he unknown conrol effeciveness, and are he oupu and he sae vecor of he unmodelled dynamics, while f, g o, g are unknown nonlinear funcions. Assumpion : [Known sign for conrol effeciveness] There eis ω u > ω l > such ha ω l ω ω u. Assumpion 2: [Sabiliy of inernal dynamics] The - dynamics are bounded inpu bounded oupu BIBO sable, i.e. here eis L > and L 2 > such ha L L L + L 2. 2 Research is suppored by AFOSR under Conrac No. FA The auhors are wih Aerospace & Ocean Engineering, Virginia Polyechnic Insiue & Sae Universiy, Blacksburg, VA , {chengyu, nhovakim@v.edu} Furher, le X [ ]. Assumpion 3: [Semiglobal Lipschi condiion] For any δ >, here eis posiive K δ and B such ha fx, fx 2, K δ X X 2, 3 f, B 4 for all X i δ, i =,2, uniformly in. Assumpion 4: [Semiglobal uniform boundedness of parial derivaives] For any δ >, here eis d f δ >, and d f δ > such ha for any δ, he parial derivaives of fx, are piece-wise coninuous and bounded fx, d f δ, fx, d f δ. 5 X The conrol objecive is o design a full-sae feedback adapive conroller o ensure ha y racks a given bounded reference signal r boh in ransien and seady sae, while all oher error signals remain bounded. III. L ADAPTIVE CONTROLLER The design of L adapive conroller involves a sricly proper ransfer funcion Ds and a gain k R +, which leads o a sricly proper sable Cs = ωkds + ωkds wih DC gain C =. The simples choice of Ds is 6 Ds = s, 7 which yields a firs order sricly proper Cs in he following form: Cs = ωk s + ωk. 8 Le Hs = si A m b, 9 and r be he signal wih is Laplace ransformaion si A m. Since A m is Hurwi and is finie, r L is finie. Furher, for every δ > le where L δ δ δ K δ, δ ma{δ + γ, L δ + γ + L 2 }, and γ is an arbirary posiive consan /8/$ AACC. 499 Auhoried licensed use limied o: UNIVERSITY OF CONNECTICUT. Downloaded on February 2, 29 a 9:44 from IEEE Xplore. Resricions apply.

2 For he proofs of sabiliy and performance bounds, he choice of Ds and k furher needs o ensure ha here eiss ρ r such ha: Gs L < ρ r k g CsHs L r L where r L /ρ r L ρr + B, 2 Gs = Hs Cs, and k g = c A m b. 3 We consider he following sae predicor or passive idenifier for generaion of he adapive laws: ˆ = A mˆ + b ˆωu + ˆθ L + ˆσ ŷ = c ˆ, ˆ =. 4 The adapive esimaes ˆω, ˆθ, ˆσ are defined as: ˆθ = ΓProjˆθ, L Pb, ˆθ = ˆθ ˆσ = ΓProjˆσ, Pb, ˆσ = ˆσ 5 ˆω = ΓProjˆω, Pbu, ˆω = ˆω, where = ˆ, Γ R + is he adapaion gain, P is he soluion of he algebraic equaion A mp +PA m = Q, Q >, and he projecion operaor ensures ha he adapive esimaes ˆω, ˆθ, ˆσ remain inside he compac ses [ω l, ω u ], [ θ b, θ b ], [ σ b, σ b ], respecively, wih θ b, σ b defined as follows: θ b = L ρ, σ b = B + ǫ, 6 where ǫ is arbirary posiive consan, and ρ = ρ r + β, 7 wih arbirary β > γ. Remark : In he following analysis, we demonsrae ha ρ r and ρ characerie he domain of aracion of he closed loop reference sysem ye o be defined and he sysem in respecively. Since γ and β can be se arbirarily small, ρ can approimae ρ r arbirarily closely. The conrol signal is generaed hrough gain feedback of he following sysem: χs = Ds rs, us = kχs, 8 where k R + was inroduced in 6, while rs is he Laplace ransformaion of he signal r = ˆωu + ˆθ L + ˆσ k g r. 9 The complee L adapive conroller consiss of 4, 5 and 8, subjec o he L -gain upper bound in 2. As compared o he corresponding L adapive conrol archiecure for sysems wihou unmodeled dynamics in [], he only difference is ha here we use L in 4 and 5 insead of. Consequenly, he subsequen analysis is also principally differen from he one in []. IV. ANALYSIS OF L ADAPTIVE CONTROLLER A. Reference Sysem We now consider he following closed-loop reference sysem wih is conrol signal and sysem response being defined as follows: ẋ ref = A m ref + b f ref,, +ωu ref, ref =, 2 u ref s = Cs/ωk g rs r ref s, 2 y ref = c ref, 22 where r ref s is he Laplace ransformaion of he signal r ref = f ref,,, and k g is inroduced in 3. The ne Lemma esablishes he sabiliy of he closed-loop sysem in Lemma : For he closed-loop reference sysem in 2-22, subjec o he L -gain upper bound in 2, if < ρ r and hen L L ref L + γ + L 2, 23 where ρ r is inroduced in 2 and ref L < ρ r, 24 u ref L < ρ ur, 25 ρ ur = Cs/ω L ρ r L ρr + B + k g r L. Proof. I follows from 2-22 ha ref s = Gs r ref s+hscsk g rs+si A m. 26 Eample 5.2 in [2] page 99 implies ha refτ L Gs L r refτ L + k g CsHs L r τ L + r L. 27 If 24 is no rue, since ref = < ρ r and ref is coninuous, here eiss τ [, ] such ha I follows from 23 and 28 ha and hence refτ L ρ r, 28 ref τ = ρ r. 29 τ L L ρ r + γ + L 2, X τ L ρ r ma{ρ r + γ,l ρ r + γ + L 2 }. 3 Furher, i follows from Assumpion 3 ha r refτ L K ρr ρ r + B, and he redefiniion in leads o he following upper bound r refτ L L ρr ρ r + B. 3 Since r τ L r L, i follows from 27 ha refτ L Gs L L ρr ρ r + r L + k g CsHs L r L + Gs L B Auhoried licensed use limied o: UNIVERSITY OF CONNECTICUT. Downloaded on February 2, 29 a 9:44 from IEEE Xplore. Resricions apply.

3 The condiion in 2 can be solved for ρ r o obain he following upper bound leading o Gs L L ρr ρ r + k g CsHs L r L + r L + Gs L B < ρ r, 33 refτ L < ρ r, 34 which conradics 29 and proves he upper bound in 24. This furher implies ha he upper bound in 3 holds for any τ, i.e. r ref L < ρ r L ρr + B. 35 Eample 5.2 in [2] page 99 leads o he following bound u ref L < Cs/ω L ρ r L ρr + B + k g r L, 36 which proves 25. B. Equivalen Linear Time-Varying Sysem In his secion we demonsrae ha he sysem wih unmodelled dynamics in can be ransformed ino a linear sysem wih unknown ime-varying parameers. To sreamline he subsequen analysis, we need o inroduce several noaions. Le γ be he desired performance bound for L, and β be an arbirary posiive consan verifying he following equaliy Cs L Gs L K ρr γ + β = γ, 37 where ρ r was defined in 3, and γ was inroduced in. I follows from 2 ha Gs L L ρr <. Since K ρr < L ρr, we have Gs L K ρr >, which implies ha he condiion in 37 can be always saisfied by selecing γ and β sufficienly small. Furher, le ρ u = ρ ur + γ 2, 38 Cs γ 2 = K ρr γ Cs ω + γ L ω c o H o s c o 39 L I follows from Lemma 4 in [3] ha here eiss c o R n such ha c o Hs = N ns N d s, 4 where he order of N d s is one more han he order of N n s, and boh N n s and N d s are sable polynomials. Lemma 2: For he sysem in, if L ρ, u L ρ u, 4 here eis differeniable θτ and στ wih bounded θτ and στ over τ [, ] such ha θτ < θ b, 42 στ < σ b, 43 fτ,τ,τ = θτ τ L + στ. 44 The proof is similar o he proof of Lemma 2 in [] and is herefore omied. If 4 holds, Lemma 2 implies ha he sysem in can be rewrien over τ [, ] as: ẋτ = A m τ + bθτ τ L + ωuτ + στ, yτ = c τ, =, 45 where θτ, στ are unknown ime-varying signals subjec o for all τ [, ], and heir derivaives are also uniformly bounded for all τ [, ]: θτ d θ ρ,ρ u <, στ d σ ρ,ρ u <. 46 Le θτ ˆθτ θτ, ωτ ˆωτ ω, στ ˆστ στ. 47 I follows from 4 and 45 ha for all τ [,] τ = A m τ + b ωτuτ + θτ τ L + στ 48 and =. C. Tracking error signal Lemma 3: For he sysem in and he L adapive conroller in 4, 5 and 8, if hen where L ρ, u ρ u, 49 L θ m ρ,ρ u λ min PΓ, 5 θ m ρ,ρ u 4θ 2 b + 4σ 2 b + ω u ω l 2 +4 λ map λ min Q θ bd θ ρ,ρ u + σ b d σ ρ,ρ u. 5 The proof is similar o Lemma 3 in []. D. Transien and Seady-Sae Performance The block diagram of he closed-loop sysem wih L adapive conroller and he reference sysem is illusraed in Figure. We noe ha he reference sysem is no implemenable since i uses he unknown signal and he unknown funcion f. This closed-loop sysem is only used for analysis purposes and does no affec he implemenaion of L adapive conroller. In he following Theorem, we prove he sabiliy and he ransien performance of he closed-loop sysem wih L adapive conroller. Theorem : Consider he closed-loop sysem wih L adapive conroller defined via 4, 5, 8, subjec o 2, and he reference sysem in If ρ r, 52 and he adapive gain verifies he lower bound Γ > θ mρ,ρ u λ min Pγ 2, 53 4 Auhoried licensed use limied o: UNIVERSITY OF CONNECTICUT. Downloaded on February 2, 29 a 9:44 from IEEE Xplore. Resricions apply.

4 Reference Sysem Fig.. sysem. we have: f ref,, u Cs + + ωu ref Sysem = g,, & g,, o = f,, + ω u L Adapive Conroller k g si A m si A m b b ref Closed-loop sysem wih L adapive conroller and reference ref L ρ r, 54 u ref L ρ ur, 55 r r L γ, 56 ref L < γ, 57 u u ref L < γ 2, 58 where γ 2 is defined in 39. Proof. The proof will be done by conradicion. Assume ha are no rue. Then, since ref = γ, u u ref =, and τ, ref τ, uτ, u ref τ are coninuous, here eiss such ha while ref = γ, or u u ref = γ 2, 59 ref L γ, u u ref L γ 2. 6 On he oher hand, i follows from Assumpion 2 ha L L ref L + γ + L 2. 6 Consequenly, Lemma implies ha ref L ρ r, u ref L ρ ur. 62 Taking ino consideraion he definiions from 7 and 38, i follows from 6 ha L ρ r + γ ρ, u L ρ ur + γ 2 = ρ u. 63 Choosing he adapive gain according o 53, Lemma 3 implies ha L γ. 64 Le rτ = ωτuτ + θτ τ L + στ, r τ = θτ τ L + στ. I follows from 8 ha χs = Dsωus+r s k g rs+ rs, where rs and r s are he Laplace ransformaions of signals rτ and r τ. Consequenly χs = Ds + kωds r s k g rs + rs, us = kds + kωds r s k g rs + rs. 65 Using he definiion of Cs from 6, we can wrie ωus = Csr s k g rs + rs, 66 and he sysem in consequenly akes he form: s = Hs Csr s + Csk g rs Cs rs + si A m. 67 Le eτ = τ ref τ. Then from 26 we have es = Hs Csr 2 s Cs rs, e =, 68 where r 2 s is he Laplace ransformaion of he signal r 2 τ = θτ τ L refτ L. 69 Since 44 implies ha fτ,τ,τ = θτ τ L + στ, f ref τ,τ,τ = θτ refτ L + στ, i follows from 69 ha r 2 τ = fτ,τ,τ f ref τ,τ,τ. 7 Eample 5.2 in [2] page 99 gives he following upper bound: e L Gs L r 2 L + r 3 L, 7 where r 3 τ is he signal wih is Laplace ransformaion being r 3 s = CsHs rs. From he error dynamics in 48 we have s = Hs rs, which leads o r 3 s = Cs s, and hence r 3 L Cs L L. Subsiuing 62 ino 6, we obain and hence L L ρ r + γ + L 2, X L ma{ρ r + γ,l ρ r + γ + L 2 }. 72 Similarly, we have X ref L X L ma{ρ r +γ,l ρ r +γ +L 2 }, 73 where X ref = [ ref ]. I follows from 72, 73 ha Xτ ρ r, X ref τ ρ r, over τ [, ], and hence Assumpion 3 implies ha r 2 τ K ρr eτ over τ [, ], which implies ha r 2 L K ρr e L. 74 From 7 we have e L Gs L K ρr e L + Cs L L. The upper bound in 64 and he L - gain upper bound in 2 lead o he following upper bound 42 Auhoried licensed use limied o: UNIVERSITY OF CONNECTICUT. Downloaded on February 2, 29 a 9:44 from IEEE Xplore. Resricions apply.

5 e L γ Cs L / Gs L K ρr, which along wih 37 leads o e L γ β < γ. 75 We noice ha from 2 and 66 ha one can derive us u ref s = Cs ω r 2s r 4 s, where r 4 s = Cs ω rs. Therefore, i follows from Eample 5.2 in [2] page 99 ha u u ref L Cs/ω L r 2 L + r 4 L, and hence u u ref L Cs/ω L K ρr e L + r 4 L. 76 Cs We have r 4 s = ω c o Hsc o Hs rs = Cs ω c o Hsc o s, where c o is inroduced in 4. Using he polynomials from 4, we can wrie ha Cs Cs N ω = d s c o Hs ω N ns. Since Cs is sable and sricly proper, he complee sysem Cs c o Hs is proper and sable, which implies ha is L gain eiss and is finie. Hence, we have r 4 L Cs ω c o Hsc o L. L The upper bound in 63 leads o r 4 L Cs ω c o Hs c o γ. 77 L I follows from 75, 76 and 77 and he definiion of γ 2 in 39 ha u u ref L K ρr Cs/ω L γ β + Cs ω c o Hs c o γ < γ L We noe ha he upper bounds in 75 and 78 conradic he equaliy in 59, which proves Since are uniform bounds for all, he upper bound in 56 follows from 64 direcly, while he upper bounds in follow from 62 correspondingly. I follows from 53 ha we can achieve arbirarily small γ by increasing he adapive gain. V. SIMULATIONS Consider he dynamics of a single-link robo arm roaing on a verical plane: I q + Fq, q,, = u, 79 where q and q are measured angular posiion and velociy, respecively, u is he inpu orque, I is he unknown momen of ineria, represens he oupu of unmodelled dynamics, Fq, q,, is an unknown nonlineariy ha lumps he forces and orques due o graviy, fricion, disurbance, oher eernal sources and unmodelled dynamics. The conrol objecive is o design u o achieve racking of bounded reference inpu r by q, where r L. Le = [ 2 ] = [q q]. The sysem in 79 can be presened in he canonical form: ẋ = A m + bωu + f,, y = c, [ ] where b = [ ], c = [ ], A m =, ω =.4 /I is he unknown conrol effeciveness, and he unknown funcion f is given by: f, = [.4] F 2,,,. Le he unknown conrol effeciveness, he unknown Fig. 2. Gs L L ρ wih respec o ωk. nonlineariy be given by ω = /I = and F, 2,, = , while is he oupu of he unmodelled dynamics, given by: s = s s 2 + 3s + 2 σs, σ = sin.2 + 2, so ha he compac ses can be conservaively chosen according o he following upper and lower bounds ω l =.5, ω u = 2, θ b = 2,σ b = Time a solid, ˆ dashed, and rdoed ime b Time-hisory of u Fig. 3. Performance of L adapive conroller for σ = sin In he implemenaion of[ he L adapive conroller, ] we se Q = 2I and hence P = Firs, we need o verify he condiion in 2. Leing s Ds = /s, we have Gs = s+ωkhs, Hs = s [ s 2 +.4s+ s 2 +.4s+ ]. We choose conservaive L ρ = 2. In Fig. 2, we plo Gs L L ρ as a funcion of ωk and compare i o. We noice ha for ωk > 3, we have 43 Auhoried licensed use limied o: UNIVERSITY OF CONNECTICUT. Downloaded on February 2, 29 a 9:44 from IEEE Xplore. Resricions apply.

6 Time a solid, ˆ dashed, and rdoed unmodelled dynamics. The conrol signal and he sysem response approimae he same signals of a closed-loop reference sysem, which can be designed o achieve desired specificaions. REFERENCES [] C. Cao and N. Hovakimyan. L adapive conroller for a class of sysems wih unknown nonlineariies: Par I. Submied o American Conrol Conference, 28. [2] H. K. Khalil. Nonlinear Sysems. Prenice Hall, Englewood Cliffs, NJ, 22. [3] C. Cao and N. Hovakimyan. Guaraneed ransien performance wih L adapive conroller for sysems wih unknown ime-varying parameers: Par I. American Conrol Conference, pages , ime b Time-hisory of u Fig Performance of L adapive conroller for σ = sin Time a solid, ˆ dashed, and rdoed ime b Time-hisory of u Fig. 5. Performance of L adapive conroller for σ = sin sin5. Gs L L ρ <. Since ω >.5, we se k = 6. We se he adapive gain Γ c =. The simulaion resuls of he L adapive conroller are shown in Figures 3a-3b for he reference inpu r = cos.5. Ne, we consider he same sysem in he presence of σ = sin The simulaion resuls, wihou any reuning of he conroller, are shown in 4a-4b. Finally, we change σ = sin sin5. The simulaion resuls are shown in 5a-5b. We noe ha he L adapive conroller guaranees smooh and uniform ransien performance in he presence of differen unknown nonlineariies and does no require any reuning. We also noice ha and ˆ are almos he same in Figs. 3a, 4a and 5a. VI. CONCLUSION A novel L adapive conrol archiecure is presened ha has guaraneed ransien response in addiion o sable racking for uncerain nonlinear sysems in he presence of 44 Auhoried licensed use limied o: UNIVERSITY OF CONNECTICUT. Downloaded on February 2, 29 a 9:44 from IEEE Xplore. Resricions apply.