Decentralized Adaptive Regulation for Nonlinear Systems with iiss Inverse Dynamics

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1 Vol. 3, No. ACTA AUTOMATICA SINICA February, 8 Decentralzed Adaptve Regulaton for Nonlnear Systems wth ISS Inverse Dynamcs YAN Xue-Hua, XIE Xue-Jun, LIU Ha-Kuan Abstract Ths paper consders the decentralzed adaptve regulaton va output-feedback for nonlnear systems wth ntegral nputto-state stable ISS nverse dynamcs, nonlnear uncertantes, and unknown control drecton. It s shown that all the sgnals n the closed-loop system obtaned are bounded, and the asymptotc regulaton s acheved. A numercal example shows the effectveness of the desgn. Key words Large-scale systems, decentralzed adaptve control, ISS, backsteppng, output-feedback The class of nput-to-state stable ISS systems has been extensvely nvestgated and has been playng an mportant role n the recent lterature of nonlnear control theory. For nstance, that cascades of ISS systems or ISS s wdely used n stablzaton, and the ISS small-gan theorem also becomes a popular tool to establsh the stablty of feedback nterconnecton of ISS systems. However, t s sometmes the case that feedback desgn does not render ISS behavor or that only a weaker than ISS property s verfed n recursve desgn. Such a weaker, but stll very meanngful, property was gven the name of ntegral ISS ISS n []. Sontag showed that ISS s, n general, strctly weaker than ISS, and he provded a very conservatve Lyapunov-type suffcent condton []. Several foundatonal results were provded n [], showng that the ISS property s the most natural one to be expected for well-behaved nonlnear systems, and admttng elegant Lyapunov-theoretc characterzaton. Stablty crtera smlar to the ISS small-gan theorem have been developed for nterconnecton nvolvng ISS systems [3 7]. Pepe and Jang further extended the ISS and the ISS theores to nonlnear tme-delay systems [8]. Recently, Jang et al. presented a unfyng framework for the problem of robust global regulaton va output feedback for nonlnear systems wth ISS nverse dynamcs [9]. Motvated by [9], ths paper extends the framework to practcally mportant classes of large-scale systems. Our man contrbutons are composed of two parts. We accomplsh varable separaton from the nput of ISS nverse dynamcs. Moreover, a desgn functon ψ s chosen to satsfy ψ. All these brng about convenence to deal wth the nteracton terms effectvely. By combnng Nussbaum-type gan approach, backsteppng desgn technque, and a subtle analyss approach [], we propose for the frst tme a decentralzed adaptve control scheme for a class of large-scale systems n the presence of uncertan nonlnear functons, unmeasured ISS nverse dynamcs and unknown drecton control coeffcents. It s shown that all the sgnals n the closed-loop system obtaned are bounded, and asymptotc regulaton s acheved. A numercal example demonstrates the effectveness of the desgn. Receved August, 6; n revsed form January 5, 7 Supported by Program for New Century Excellent Talents n Unversty of Chna NCET-5-67, Natonal Natural Scence Foundaton of Chna 677, Program for Fundamental Research of Natural Scences n Unverstes of Jangsu Provnce 7KJB5, Program for Summt of Sx Types of Talents of Jangsu Provnce 7- A-. School of Electrcal Engneerng & Automaton, Xuzhou Normal Unversty, Xuzhou 6, P. R. Chna. Insttute of Automaton, Qufu Normal Unversty, Qufu 7365, P. R. Chna DOI:.37/SP.J Ths paper s organzed as follows. Secton begns wth some mathematcal prelmnares. Secton presents the correspondng output feedback control desgn procedure. An example s gven n Secton 3. Fnally, the paper s concluded n Secton. Mathematcal prelmnares The followng notatons wll be used throughout ths paper. R + denotes the set of all nonnegatve real numbers. R n denotes the real n-dmensonal space. For a gven vector or matrx X, X T denotes ts transpose; X denotes the Eucldean norm of vector X; X denotes the nduced matrx norm of matrx X. A functon γ : R + R + s of class K f γ = and γ s contnuous and strctly ncreasng; t s of class K f addtonally t s unbounded; a functon βs, t : R + R + R + s of class KL f t s of class K for each fxed t, and decreases to zero as t for each fxed s. Consder a system wth the form of ẋx = ft, x, u, x R n, u R m, t R + where f : R n R m R n s locally Lpschtz. Defnton. System s ISS wth respect to u f there exst functons α K, β KL, and γ K such that for each ntal condton x R n and each measurable, locally essental bounded functon u : R + R m, the soluton xt exsts for each t > and satsfes α xt β x, t + γ uτ dτ In vew of [], ISS property can be equvalently characterzed usng the Lyapunov functon. Proposton. System s ISS f and only f there exsts a postve defnte and proper functon V, called ISS- Lyapunov functon, such that α x V t, x α x 3 V t + V ft, x, u α x + γ u x where α s a postve defnte contnuous functon, α, α K, and γ K. Motvated by [9], we gve the followng techncal result, whch wll be used later. For smplcty, we use σ s = Oσ s to mean that σ s cσ s for some constant c > and all s n a small neghborhood of the orgn. Proposton. Consder an ISS system wth an ISS-Lyapunov functon V t, x satsfyng 3 and, and take any smooth functon φ wth the followng property φ s = Oαs 5

2 68 ACTA AUTOMATICA SINICA Vol. 3 Moreover, when α s bounded, the followng addtonal condton holds φ s lm s αs < 6 Then, there always exst a postve-defnte functon σ and class-k functons ϕ =,, m such that φ xτ dτ σ x + m = ϕ u τ dτ 7 Moreover, f the ISS-gan γ n satsfes γs = Os, so does ϕ. Proof. For the proof of ths proposton, the reader can refer to Proposton of [9]. Output-feedback control desgn Consder a class of large-scale nonlnear systems composed of N nterconnected subsystems wth relatve degree descrbed by η = q t, η, y + ẋ j = x,j+ + g jt, η, y + ẋ = β u + g t, η, y + f k t, y k h j k t, y k h k t, y k y = x, N, j 8 where η R n and x = x,, x R are the states, u R and y R are nput and output of the th subsystem, respectvely; f k R n and h k R denote the nteractons from the kth subsystem to the th subsystem. It s assumed that y s measurable and the uncertan functons q, g j, f k, and h j k j are locally Lpschtz. Here, β s an unknown nonzero constant wth ndefnte sgn. In ths secton, the followng hypotheses are made on system 8. H. The η -subsystem of 8 s ISS wth respect to y = y,, y N T n the sense that there exsts an ISS- Lyapunov functon V such that α η V t, η α η V + V t, η t η q t, η, y + f k t, y k α η + γ y 9 where α s a postve defnte contnuous functon, α, α K, and γ K. H. For each j, there exst two unknown postve constants, p j and p j, and two known postve semdefnte, smooth functons φ j and φ j such that g jt, η, y p jφ j y + p jφ j η H3. f k t, y k and h k t, y k satsfy f k t, y k r k y k, h k t, y k r k y k where r k and r k are unknown constants denotng the strengths of nteractons. Remark. The lnear-growth condton n hypothess H3 s made only for smplfyng the presentaton and hghlghtng the man contrbuton n ths paper. In the sprt of [ ], t can actually be relaxed by a nonlnear-growth condton. The control objectve s to desgn a decentralzed adaptve controller for each subsystem so that all the sgnals of the closed-loop system are bounded over [,, and all the states, nputs and outputs can be regulated to zero.. Adaptve backsteppng controller desgn Frst, the followng flters are ntroduced to rebuld the unmeasured partal-states x,, x, ˆξ j = ˆξ,j+ L j ˆξ, j ˆξ = u L ˆξ where L = L,, L T s chosen such that A = I L, s asymptotcally stable. For each,, j, by denotng wth p = max follows that ξ j = β x j 3 e j = ξj ˆξ j p, p j, p j β β e = A e + G t, η p, y + p β β, p j, t h k t, y k 5 wth G t, η, y = colg t, η, y /β + L y /β,, g t, η, y /β + L y /β. Because A s asymptotcally stable, there exsts a P = P T > such that P A + A T P = I 6 Along the solutons of 5, dfferentatng the quadratc functon V e = e T P e yelds V e e + P φ j y + L j y + j= P φ j η + N P j= N r ky kψ k 7 To summarze, the complete system can be expressed as η = q t, η, y + ėe = A e + p f k t, y k G t, η, y + p β ẏ = β ˆξ + β p e + g t, η, y + ˆξ j = ˆξ,j+ L j ˆξ, j h k t, y k h kt, y k ˆξ = u L ˆξ 8

3 No. YAN Xue-Hua et al.: Decentralzed Adaptve Regulaton for Nonlnear Systems wth ISS Inverse Dynamcs 69 The next task s to desgn a controller for 8. Step. Begn wth the y -subsystem of 8 and consder ˆξ as the vrtual control nput. One can choose for the th subsystem the followng vrtual control law α = c N k ψ y y, k = Γ ψ y y 9 where N s a smooth Nussbaum-type functon ntroduced to counteract the lack of a pror knowledge of the hgh-frequency-gan sgn sgnβ, c, Γ > are desgn constants, and desgn functon ψ. A Nussbaum-type functon N possesses the propertes lm k sup k Nsds =, k lm k nf k Nsds =. In ths paper, we choose k N : s s coss. Settng z = ˆξ α k, y, one obtans ż = υ y β ˆξ + β p e + g + h k where υ = ˆξ 3 L ˆξ k Γ ψ y y. Denote V = y + λ ˆp p, where λ > s a desgn parameter, β, β, p + β p ε + ˆp s the estmate of p = max N, < ε < s a small desgn parameter. Step. Consder the augmented system composed of the y -subsystem and n whch υ or equvalently ˆξ 3 s vewed as the vrtual control nput. The dervatve of the Lyapunov functon V = V + z along the solutons of 8 satsfes V = c β N k ψ y y + β y z + β p e y + g y + [ h ky + z υ β α + z + g + y β p e + h k ] + λ ˆp p ˆp By H and H3, wth Young s nequalty, one obtans β y z β z α + z y + c N ψ β y y ψ + z+ ψ β + z y z β p e + g y ε e + p + βp z + g ε y p 3 h ky z h k y k= r k y kψ k + N y z By settng Φ t, e, η, y = ε e + β p e y + g t, η, y y + β y ψ y + g, 5 p α = c z + L ˆξ + Γ ψ y y k + c N ψ y z ψ ˆp + y z 6 z = ˆξ 3 α, 7 τ = λ + z 8 y where c > s a desgn parameter. becomes V c β N k ψ y y + Φ + z z c z + ˆp p ˆp τ + ry k kψ k 9 λ k= and z satsfes ż = υ 3 ˆp β ˆξ + β p e + g + ˆp y 3 h k where υ 3 = ˆξ L 3 ˆξ k Γ ψ y y s= ˆξ ˆξ,s+ s L s ˆξ. Step l = 3,,. It s easy to obtan the followng concluson by nducton that the tme dervatve of the augmented functon V l = V,l + z,l satsfes V l c β N k ψ y y + Φ,l + z,l z l c l + z l s= csz s + λ ˆp p l s= λzs s ˆp ˆp τ,l + N k= rk,l ykψ k 3 for any l = 3,,, where Φ,l = Φ,l + β y ψ y + ε e + g p 3 α l = c,l z,l + L l ˆξ +,l k Γ ψ y y + l s=,l ˆξ,s+ ˆξ L s ˆξ z,l + s,l τ,l ˆp c N y,l ψ z,l z,l + l s z s λ ˆp y s=,l ˆp z,l, s l 33 y

4 7 ACTA AUTOMATICA SINICA Vol. 3 z s = ˆξ,s+ α s k, y, ˆξ,, ˆξ s, ˆp 3 Such a functon ψ always exsts because of the condtons of Theorem. Then, wth 7, 37,, and, t,l follows from that τ,l = τ,l + λ z,l 35 y V c c β N k ψ y y c + z wth c,l >. Therefore, at step, one obtans the smooth dynamc c szs ε e + [ β + output feedback law k = Γ ψ y y, ˆp = τ, u = α k, y, ˆξ,, ˆξ, ˆp 36 such that the tme dervatve of the functon V = y + s= z s + λ ˆp p satsfes V c β N k ψ y y + Φ, c + z c szs + r, k ykψ k 37 s=. Man result k= Now, we state the man theorem n ths paper. Theorem. Assume that the hypotheses H and H hold wth the followng propertes φ js = Oα s, j 38 and that n the case, where α s bounded, lm sup φ js <, j 39 s α s If γ s = Os n H, then the solutons of 8 and 36 are well-defned and bounded over [, for approprately chosen smooth functon ψ. Furthermore, lm x t + η t + u t t = Proof. Consder the functon as follows V c = V e + V Wth the help of H, by 5 and 3, one has Φ, ε e + β y ψ y + φ η + θ y + φ y y + φ y β p where θ max ε + p H and Proposton, φ j η s ds σ j η + k=, p, p p. By ϕ jk y k s ds 3 holds, where σ j j are postve defnte, and ϕ jk j, k N are of class K and quadratc near the orgn. Take a smooth functon ψ so that ψ y y max y + y φ y + φ y, ϕ mj y s= θ + P ] y ψ y + φ η + P φ j η + r k ykψ k y k 5 j= k= Integratng both sdes of 5, and usng 9, 3, and, one has V ct c Γ k t β N sds + m= r m Γ m k mt + d 6 where r m β + θ = + 8 P + + r, m = + P + r m, m k d = V c γ m Γ m k m c Γ β N sds + m= σ η + P σ j η j= For the proof of the frst statement on the boundedness property and the second statement on the convergence property, the reader can refer to Theorem of [9] and Secton III-B of []. 3 A smulaton example Accordng to the desgn procedure gven n Secton, ths secton consders the control desgn for the followng nterconnected system consstng of two relatve degree-two subsystems and llustrates the dynamcs behavors of all the closed-loop sgnals. η = c η + q y η + f y ẋ = x + g η, y + h y ẋ = β u y = x η = c η + q y η + f y ẋ = x + g η, y + h y ẋ = β u y = x 7 Let uncertan functons and nterconnectons be as follows. q y =.y, f y = y sny g η, y = η + η cosy, h y = y q y =.y, f y = y sny η g η, y = cosy, h y = + η φ j y + L j y, γ m y, j m N For η -subsystem, one employs V = ln + η =,, s =, and s =. It follows from 3 and that ψ y = =,. The desgn parameters are

5 No. YAN Xue-Hua et al.: Decentralzed Adaptve Regulaton for Nonlnear Systems wth ISS Inverse Dynamcs 7 chosen as c = c = β = β =, c = c =.5, L = L = L = L =, c = c =.5, λ = λ =.5, Γ =.5, Γ =.3, and the ntal condtons are ˆξ = ˆξ = ˆξ = ˆξ =, η = η =., y =.5, x =., y =.8, x =., k = k =, ˆp =, ˆp =.9. From Fg., one can see that for lnear growth nterconnectons, the desgned decentralzed adaptve controllers are robust to the nonlnear unmodeled dynamcs and can acheve good regulaton performance. Fg. Concluson a b Responses of adaptve output feedback system In ths paper, a decentralzed adaptve output regulaton problem s addressed for a class of large-scale nonlnear systems wth ISS nverse dynamcs, nonlnear uncertantes, and unknown control drecton. The man contrbutons are to deal wth the nterconnectons tactfully by usng varable separaton technque and choosng proper desgn functons, and to propose a constructve decentralzed adaptve control scheme by combnng Nussbaum-type gan approach, backsteppng desgn technque and the subtle analyss approach []. It s shown that all the sgnals n the closed-loop system obtaned are bounded and the asymptotc regulaton s acheved. References Sontag E D. Comments on ntegral varants of ISS. Systems and Control Letters, 998, 3-: 93 Angel D, Sontag E D, Wang Y. A characterzaton of ntegral nput-to-state stablty. IEEE Transactons on Automatc Control,, 56: Ito H. Stablty crtera for nterconnected ISS systems and ISS systems usng scalng of supply rates. In: Proceedngs of the Amercan Control Conference. Boston, USA: ACC, Ito H. Establshng ISS property of nterconnected systems va parametrzaton of supply rates. In: Proceedngs of the th IEEE Conference on Decson and Control, and the European Control Conference. Sevlle, Span: IEEE, Ito H. A parametrzaton of supply rates for small-gan condtons for global stablty of nterconnected ISS and ISS systems. In: Proceedngs of Amercan Control Conference. Portland, USA: ACC, Ito H. State-dependent scalng problems and stablty of nterconnected ISS and ISS systems. IEEE Transactons on Automatc Control, 6, 5: Ito H. Explct solutons to state-dependent scalng problems for nterconnected ISS and ISS nonlnear systems. In: Proceedngs of Amercan Control Conference. Portland, USA: ACC, Pepe P, Jang Z P. A Lyapunov-Krasovsk methodology for ISS and ISS of tme-delay systems. Systems and Control Letters, 6, 55: 6 9 Jang Z P, Mareels I, Hll D J, Huang J. A unfyng framework for global regulaton va nonlnear output feedback: from ISS to ISS. IEEE Transactons on Automatc Control,, 9: Ye X D. Decentralzed adaptve regulaton wth unknown hgh-frequency-gansgns. IEEE Transactons on Automatc Control, 999, : 7 76 Sontag E, Teel A. Changng supply functons n nput/state stable system. IEEE Transactons on Automatc Control, 995, 8: Wu Z J, Xe X J, Zhang S Y. Robust decentralzed adaptve stablzaton for a class of nterconnected systems wth unmodelled dynamcs. Internatonal Journal of Systems Scence,, 357: Jan S, Khorram F. Decentralze adaptve output feedback desgn for large-scale nonlnear systems. IEEE Transactons on Automatc Control, 997, 5: Ye X D, Jang J P. Adaptve nonlnear desgn wthout a pror knowledge of control drectons. IEEE Transactons on Automatc Control, 998, 3: 67 6 YAN Xue-Hua Master student at Qufu Normal Unversty. Her research nterest covers nonlnear system control. E-mal: huaxue@63.com XIE Xue-Jun Receved hs Ph. D. degree from Insttute of Systems Scence, Chnese Academy of Scences n 999. Hs research nterest covers stochastc nonlnear control systems and adaptve control. Correspondng author of ths paper. E-mal: xxj@mal.qfnu.edu.cn LIU Ha-Kuan Professor at Xuzhou Normal Unversty. Hs research nterest covers robust control, power source technology, and computer ntegrated system wth applcaton. E-mal: luhakuan96@63.com