Adaptive Semiglobal Nonlinear Output Regulation:An Extended-State Observer Approach

Adapti v e Semiglobal Nonlinear Output Re gulation: An Extended-State Observer Approach Lei W ang, Christopher M. Kellett Abstract — This paper proposes a new extended-state observ er - based framewo rk f or adapti ve nonlinear regulator design of a class of nonlinear systems, in the general nonequi- librium theory . By augmenting an extended-state observe r with an internal model, one is able to obtain an estimate of the term containing un certain parameters, whi ch then makes i t p ossible to design an adapti ve internal mo del in the prese nce of a general nonlinearly parameterized i mmersion condition. I . I N T RO D U C T I O N The o u tput regulation pro blem aims at controllin g a dis- turbed system so as to achieve bou nded n ess o f the resulting trajectories and asymp totic con vergence of the ou tp ut to- wards a pre scr ibed tr ajectory . Se veral f rameworks have been established fo r this problem . Du e to its a b ility to cope with uncertainties, the interna l model- based method h as been re- garded as one of the most pro mising approac h es, particularly since the milestone contributions for linear systems in [5] and nonlinear sy stems in [2]. The main idea o f this metho d is to approp riately in corpo rate the controller with the structure of an exosystem tha t g e n erates th e d isturbance an d the trackin g trajectory . In the de sign of an intern al m odel-b a sed regulator , a ke y step is to design an appro priate intern a l m o del to generate the steady state input such that the inter nal model property is fulfilled. Several systematic design method s ha ve been developed such as in [10], [1 3], [14], [ 16], [17]. Am ong them, in terms of a construc tive design , significant atten tion has been attr a cted by th e “immersion condition”, which re- quires the solution o f the regulator eq uations to satisfy so m e specific d ifferential equa tio ns (i.e., th e im mersed dy n amics). It is noted that, if th ere exist parameter u ncertainties in the exosystem, the cor respond ing im mersed dynamics would be uncertain in general, which makes the design of internal model challengin g. T o cope with parameter un certainties, in [3] the internal mo d el is aug mented with an identifier, which is approp riately design ed via th e adap tiv e desig n methodo logy [19]. Mo tiv ate d by this adaptive framework, se veral r elev ant results ha ve been reported th a t differ in the kind of exosystems (linear [23], [22] and nonlinea r [21 ] exosystems), in the kind of a vailable inform ation (state and output feedb ack), and in the kin d of co ntrolled systems (linear [20] an d nonlinear [8] systems). On the other han d , the above m e ntioned “immersio n con ditions” are fo rmulated on an extra assumption that the regulator eq uations are L. W ang and C. Ke llett are w ith Faculty of Engineering and Built En viron- ment, Un iv ersity of Newcastl e, Australia. E-mail: (w anglei 0201@163.com; chris.ke llett @ne wcastle.edu.au). solvable. This f u ndame n tally limits the class of co ntrolled systems that can be handled. In [6], [7], th is extra assum ption is removed by taking advantage of the noneq uilibrium theor y of nonlin e a r ou tp ut regulation. In [9], the co rrespon ding ex- tension to adaptive n onlinear output regulation is addressed. Despite the afor emention ed efforts, r e search on adap tiv e internal mod el d esign is still at quite an early stag e . In fact, the immersion cond itio ns in th e existing d esign me thods ar e quite restrictive, at least in th e fo llowing two aspects. Firstly , the immersed dynamic s is usua lly requ ired to be linear , hence limiting the exosy stem to be linear generally . I t is noted that the on ly exception is [9], wh ere the im mersed dy n amics is assumed to be in the outp ut-feed b ack form. Besides, as in [19], the design of all adaptation laws, to the best k n owledge of the auth ors, is based o n the idea o f “cancellatio n”, th at is to cancel the term containin g the unknown par a m eters when co mputing the d eriv ati ve o f the L yapun ov fun ction, which usually req u ires a linearly p arameterize d imm ersion condition . This in tu rn fundamen tally limits th e class o f exogenou s an d controlled systems. In orde r to d e a l with a broad class of exogen ous and controlled systems, this paper studies the adaptiv e non linear output regulation problem with a general immersion condi- tion, in the gen eral noneq uilibrium theory of nonlinear outpu t regulation developed in [6], [9]. Inspired b y [1 5], [12], [18], a new exten d ed-state observer-based design pa r adigm is de- veloped to construct an adaptive nonlinear intern al m odel. By taking advantage of the extra state provid ed by th e extended- state ob server , one is able to obtain an estimate of the term containing the uncertain par ameter to be estimated , which then can be utilised to achiev e asymptotic id entification. It is no ted that th e propo sed meth o d allows a n o nlinearly parameteriz e d immersion condition . Mo re specifically , th e uncertain parameters in the immersed dynamic s can appear in a “m onoto nic-like structure”, with linear parameter ization as a particular case. The paper is organized as fo llows. Section I I gives th e problem for mulation an d some standin g assump tions. In Section III, th e main results are addressed by presenting the de sign o f the adap tive internal m odel and the stability analysis of the resulting closed-loo p system. An illu stra tive example is presen te d in Section IV to sh ow the validity of the prosed me thod. A br ie f con c lusion is made in Section V. Notatio ns: For any positive integer d , ( A d , B d , C d ) is used to den ote the matrix triplet in the prime fo rm. Nam ely , A d denotes a shift matrix of dim ension d × d whose all superdiag onal entries are one and other entries a re a ll zero, B d denotes a d × 1 vector whose entr ies are all zero except the last one which is equal to 1 , and C d is a 1 × d vector whose entries are all zer o except the first one which is eq ual to 1. A fu nction f : R + := [0 , ∞ ) → R + is of class K , if it is con tinuous, positive definite, and strictly increasing. A cla ss K function is of class K ∞ if it is unbo unded . A continuo us fu nction δ : R + × R + → R + is of class KL if, for each fixed t ≥ 0 , th e f unction δ ( · , t ) is of class K and, for each fixed s > 0 , δ ( s, · ) is strictly decr easing and lim t →∞ δ ( s, t ) = 0 . I I . P R E L I M I N A R I E S A. Pr oblem Statement Consider the system ˙ z = f 0 ( ρ, w , z ) + f 1 ( ρ, w , z , x ) x ˙ x = q ( ρ, w , z , x ) + b ( ρ, w , z , x ) u y e = x (1) with state z ∈ R n and x ∈ R , contro l inpu t u ∈ R , regu la te d output y e ∈ R , and in which ρ ∈ R p and w ∈ R s denote the exogenou s inp ut, genera te d b y the exosystem ˙ ρ = 0 ˙ w = s ( ρ, w ) , (2) with the initial conditions ρ and w 0 taking values from compact sets P ⊂ R p and W ⊂ R s , respec tiv ely . As customary in the field of ou tp ut regulation, it is assum ed that P × W is inv ariant f or (2), and there e xists a constant b 0 > 0 such that b ( ρ, w , z , x ) ≥ b 0 (3) holds for all ( ρ, w, z , x ) ∈ P × W × R n × R . Addition- ally , function s f 0 ( · ) , f 1 ( · ) , q ( · ) , b ( · ) , s ( · ) are assume d to b e sufficiently smooth. In th is framework, the ou tput regulation prob le m of in- terest can b e summarize d as below . Giv en any compact sets C z ⊂ R n , C x ⊂ R , all trajec tories of system (1)-(2), controlled b y an output feedback regu lato r of the form ˙ x c = ϕ c ( x c , y e ) , x c ∈ R n c u = γ c ( x c , y e ) , (4) with all initial co nditions ranging over P × W × C z × C x × C x c with C x c being any g iv en compact set in R n c , are bounded and lim t →∞ y e ( t ) = 0 . W ith this in m ind, it is ob served that by v iewing x as the output, system (1) cascaded with (2) has a well-d efined relativ e d egre e on e, an d the c o rrespon ding zero dyn amics, driven by the contro l input u = − q ( ρ, w , z , 0 ) b ( ρ, w , z , 0) , is given by ˙ ρ = 0 ˙ w = s ( ρ, w ) ˙ z = f 0 ( ρ, w , z ) . which, with z := ( ρ, w , z ) , can be com pactly rewritten as ˙ z = f ( z ) (5) According ly , we set Z := P × W × C z with C z ⊂ R n being any given co mpact set. Remark 1: This paper is main ly intere sted in n onlinear systems having nor mal form. Altho ugh system (1) has r e la- ti ve degree one, its extension to higher relative degree can be trivially ach iev ed as in [1] by redefining a regulated o u tput so as to red u ce the relati ve d egree to o ne. B. Standing Assumption s In orde r to de a l w ith a m ore gen eral class of n onlinear systems, following [9] we make some a ssum ptions on the zero dy namics (5). Assumption 1 : There exist a n onemp ty , c o mpact set A z ⊂ R n , and a class K L fun ction δ 1 ( · , · ) such that for all z 0 ∈ P × W × R n , dist ( z ( t, z 0 ) , Z c ) ≤ δ 1 ( dist ( z 0 , Z c ) , t ) for all t ≥ 0 where Z c := P × W × A z , and z ( t, z 0 ) denotes the solutio n of system (5) passing through z 0 at time t = 0 . Assumption 2 : There e xist constants M ≥ 1 , a > 0 , and δ 2 > 0 such that f o r all z 0 ∈ P × W × R n , dist ( z 0 , Z c ) ≤ δ 2 ⇒ dist ( z ( t, z 0 ) , Z c ) ≤ M e − a t dist ( z 0 , Z c ) . Remark 2: Assumptio n 1 indicates that Z c is an in vari- ant and asympto tically stable compact set un der (5). More specifically , in the sense of [6], Z c is the ω -limit set of P × W × R n under (5). It can also be seen that th ere exists a compact set Z such th at the solution of ( 5) satisfies z ( t, z 0 ) ∈ Z f or all t ≥ 0 , so long as z 0 ∈ Z . Assump tion 2 imp lies tha t Z c is locally exponentially stable for (5), which plays a significant role in the sub sequent analysis of asymptotic stability . Remark 3: Assumptio n 1 can be regarded as the minimum- phase assumptio n in general nonequ ilibrium the- ory . Compar ed with the conventional m inimum- phase as- sumption su c h as in [3], [ 14], th e main ben efit is that the extra assump tion o n the solvability of the regulator equ ations is rem oved, which broadens the class of systems that can b e addressed. T o this end, a g e n eral nonlinea rly parameterized immer- sion condition will be pr o posed, wh ic h leads to a constructive design o f the internal model. Assumption 3 : There exist p ositiv e integers d and q , a C 0 map θ : P → R q , ρ 7→ θ ( ρ ) , a C d map τ : Z → R d , z 7→ τ ( z ) , and a C 2 map φ : R p × R d → R such th at the f ollowing identities ∂ τ ∂ z f ( z ) = A d τ ( z ) + B d φ ( θ ( ρ ) , τ ( z )) q 0 ( z ) = C d τ ( z ) (6) with q 0 ( z ) = − q ( ρ, w , z , 0 ) b ( ρ, w , z , 0) , h old for all z ∈ Z c and ρ ∈ P . Remark 4: In Assumption 3, the imm ersed dynamics (6) is allowed to b e depend ent on the uncertain paramete r ρ , which motiv ates u s to incorpo r ate the in ternal model with an ide ntifier . Since ρ appear s only in the fun ction θ ( · ) , for convenience we regard θ as an un certain parameter to be estimated in the sequel, tho ugh this may result in overparameterization . In the literatur e, several imme rsion condition s fo r adap tiv e output regulation have been propo sed. It is worth n o ting that compared to the existing ones, Assumptio n 3 is mu ch weaker , at least in the f ollowing two aspects. In previous work, the im mersion map τ is requ ir ed to satisfy either a linear eq uation (e .g. [ 3]), or a nonlinear equatio n but in the “output-f eedback for m” (e.g. [9]). Fundamentally , all these for ms in [3], [9] can be transfo rmed to the form (6). Moreover , in all th e previous related liter ature, the imm e rsed dynamics (6) is required to be linearly param e terized, while this paper permits a non lin ear parameterizatio n, with linea r parameteriz a tion as a particular case. In this paper, we aim to hand le a more gen eral immersion pr op erty having a n onlinear ly par a meterized fun ction φ ( θ , τ ) in the uncertain param eter θ . W e require the following proper ties on φ ( · , · ) . Assumption 4 : There exists a smooth function β ( · ) : R d → R p having the pr operties: (i) The re exist ǫ 0 ,i > 0 , i = 1 , . . . , q such that for any r ∈ τ ( Z c ) 1 , and any s 1 , s 2 ∈ B q 0 := { θ ∈ R q : | θ i | ≤ a 0 ,i + ǫ 0 ,i } with a 0 ,i = max ρ ∈P | θ i ( ρ ) | , the inequality ( s 1 − θ ) ⊤ β ( r ) ∂ φ ( s 2 , r ) ∂ s 2 ( s 1 − θ ) ≤ 0 (7) holds, with θ i denoting th e i -th entry of vector θ ; (ii) For any z 0 ∈ Z c and s 1 , s 2 ∈ B q 0 , the persistent excitation (PE) con dition φ ( s 1 , τ ( z ( t, z 0 ))) − φ ( s 2 , τ ( z ( t, z 0 ))) = 0 = ⇒ s 1 = s 2 (8) is fulfilled, wher e z ( t, z 0 ) denotes the trajectory o f (5) passing thr o ugh z 0 at t = 0 . Remark 5: Assumptio n 4.(i) means that there exists a smooth functio n β ( r ) such that f o r all r ∈ τ ( Z c ) , the function β ( r ) φ ( s, r ) is mono tonically decr easing in s ∈ B q 0 . In this respect, we say th at the functio n φ ( s, r ) satis- fying Assumption 4.(i) is in the monoton ic-like structur e . If as in [3] , [9], the function φ is linearly p arameter- ized, tha t is φ ( s, r ) has th e form of s ⊤ ψ ( r ) fo r some function ψ ( · ) , th en Assumption 4.(i) can always be ful- filled by ch oosing β ( r ) = ψ ( r ) . Indee d, the class of function s φ ( r , s ) satisfyin g such a monoton icity condi- tion includ es not only all linearly param e terized f unction s, but also some nonlinea r ly param eterized function s, such as arctan ( s ⊤ ψ ( r )) or ψ 0 ( r ) P p i =1 θ i ψ i ( r ) + ψ p +1 ( r ) , wher e th e correspo n ding functio n β ( r ) can be ch osen as ψ ( r ) or − ( ψ 0 ( r ) ψ 1 ( r ) · · · ψ 0 ( r ) ψ p ( r ) ) ⊤ , re spectiv ely . 1 For simplicity , we use τ ( Z c ) to denote the set of τ ( z ) for all z ∈ Z c . It is o bserved that the maps φ ( s, r ) an d β ( r ) are continu- ously d ifferentiable a nd Assum ption 3 and 4 are r e spectiv ely made over the co m pact sets s ∈ B q 0 and ( s, r ) ∈ B q 0 × τ ( Z c ) . In view of this, the r e is n o loss o f gen e rality to suppose that f unction s φ ( · , · ) and β i ( · ) a re globally Lipschitz and bound ed, i.e., there exist a 1 > 0 and a 2 ,i > 0 , i = 1 , . . . , q such that inequalities | φ ( s, r ) | ≤ a 1 , | β i ( r ) | ≤ a 2 ,i (9) with β i denoting the i -th en tr y of vector β , h old for all s ∈ R q , r ∈ R d . I I I . A D A P T I V E R E G U L A T O R D E S I G N A. Adaptive Interna l Model W ith Assum p tion 3, if θ were k nown, then we cou ld design an intern al mod el of th e for m ˙ η = A d η + B d φ ( θ, η ) + v η (10) in which η ∈ R d , and v η ∈ R d denotes the inp u t of the internal m odel, and the contr o l input can be chosen as u = v u + C d η (11) where v u is the residual input. Howe ver , since θ is u n known, the internal model (1 0) is no t implementa b le. T o overcome this obstacle, an extra identifier ca n be used to provide an estimate of θ , deno ted by ˆ θ ∈ R q . I t is worth noting that, due to the presen c e o f the nonlinear para meterization , we canno t take advantage of the u sual “canc ellation” idea ( e .g. [3], [9]). Inspired b y various impo rtant results o n the design of extended-state ob servers (e.g. [15], [12], [18]), we pro pose a n ew adaptive in ternal m odel, having the form ˙ η = A d η + B d φ ( ˆ θ , η ) − satv (( A d + λI ) ˆ ξ ) − B d sat d +1 ( ˆ σ ) ˙ ˆ θ = β ( η ) sat d +1 ( ˆ σ ) − dzv ( ˆ θ ) ˙ ˆ ξ = A d ˆ ξ + B d ˆ σ − satv (( A d + λI ) ˆ ξ ) − B d sat d +1 ( ˆ σ ) − Λ ℓ G ( v u + ˆ ξ 1 ) ˙ ˆ σ = − ℓ d +1 g d +1 ( v u + ˆ ξ 1 ) (12) where ˆ ξ := col ( ˆ ξ 1 , . . . , ˆ ξ d ) , λ > 0 , Λ ℓ = diag ( ℓ, . . . , ℓ d ) , G = col ( g 1 , . . . , g d ) , f u nctions sat i ( · ) for i = 1 , . . . , d + 1 have th e form sat i ( s ) =      s , | s | ≤ l i s − sign ( s ) ( | s | − l i ) 2 2 , l i < | s | < l i + 1 l i + 1 2 , | s | ≥ l i + 1 , with saturation lev el l i , satv ( · ) : R d → R d denotes a vector- valued saturation fun ction, defined b y satv ( s 1 , . . . , s d ) = col ( sat 1 ( s 1 ) , . . . , sat d ( s d )) , and d zv ( · ) denotes a vector- valued dead-z o ne f unction , each elem ent of wh ich is a function o f the f orm dz i ( s ) =          0 , | s | ≤ a 0 ,i c i ( | s | − a 0 ,i ) 2 2 ǫ 0 ,i sign ( s ) , a 0 ,i < | s | < a 0 ,i + ǫ 0 ,i c i s − c i  a 0 ,i + ǫ 0 ,i 2  sign ( s ) , | s | ≥ a 0 ,i + ǫ 0 ,i . As it can be seen f rom Fig. 1 , fun c tions sat i and dz i are constructed to be smooth. All design parameters g i , l i , and c i will b e defined later in Proposition 1, (31), and (2 0), respectively . -10 -5 0 5 10 -6 -4 -2 0 2 4 6 -10 -5 0 5 10 -4 -2 0 2 4 Fig. 1. Left: plot of function sat i with l i = 3 ; and right: plot of function dz i with c i = 1 . 2 , a 0 ,i = 4 , ǫ 0 ,i = 2 . By cascading system (1) with the adap tiv e intern al mod el (12) and the contro l in put ( 11), we obtain a cascaded system of the form ˙ ρ = 0 ˙ w = s ( ρ, w ) ˙ z = f 0 ( ρ, w , z ) + f 1 ( ρ, w , z , x ) x ˙ η = A d η + B d φ ( ˆ θ , η ) − satv (( A d + λI ) ˆ ξ ) − B d sat d +1 ( ˆ σ ) ˙ ˆ θ = β ( η ) sat d +1 ( ˆ σ ) − dzv ( ˆ θ ) ˙ ˆ ξ = A d ˆ ξ + B d ˆ σ − satv (( A d + λI ) ˆ ξ ) − B d sat d +1 ( ˆ σ ) − Λ ℓ G ( v u + ˆ ξ 1 ) ˙ ˆ σ = − ℓ d +1 g d +1 ( v u + ˆ ξ 1 ) ˙ x = q ( ρ, w , z , x ) + b ( ρ, w, z , x )( C d η + v u ) (13) It is ob ser ved th at system (13), viewing v u as control inpu t and x as o utput, has a well- d efined relativ e degree one, and the c o rrespon ding extended zero dyna mics, forced by v u = − C d η − q ( ρ, w , z , 0 ) b ( ρ, w , z , 0) , (14) can be giv en by ˙ ρ = 0 ˙ w = s ( ρ, w ) ˙ z = f 0 ( ρ, w , z ) ˙ η = A d η + B d φ ( ˆ θ , η ) − satv (( A d + λI ) ˆ ξ ) − B d sat l d +1 ( ˆ σ ) ˙ ˆ θ = β ( η ) sat d +1 ( ˆ σ ) − dzv ( ˆ θ ) ˙ ˆ ξ = A d ˆ ξ + B d ˆ σ − satv (( A d + λI ) ˆ ξ ) − B d sat d +1 ( ˆ σ ) − Λ ℓ G  − C d η − q ( ρ, w , z , 0 ) b ( ρ, w , z , 0) + ˆ ξ 1  ˙ ˆ σ = − ℓ d +1 g d +1  − C d η − q ( ρ, w , z , 0 ) b ( ρ, w , z , 0) + ˆ ξ 1  (15) By simp le calculatio n s, it is observed that under Assump- tions 1 and 3, the adaptive controller (11)-(12) fulfills the internal mo del p r op erty , relative to the set Z c . Therefor e, in light of pr evious analysis, accor ding to [6], the desired adaptive ou tput regulation problem can b e solved by the adaptive con troller (11)-(1 2) with the residual control v u having the form v u = − κx , if the extended zero dyn a mics (15) can be shown to to possess an asym p totically (locally exponentially) stable co mpact attra ctor . Remark 6: As will be shown in next subsection, (12) contains an extended state o bserver, i.e., the ( ˆ ξ , ˆ σ ) d y namics, in which ˆ σ den o tes the extra estimate. Using this extra estimate, we are a b le to take advantage of the nonlin ear parameteriz a tion structure given in Assump tion 4 , which thus enables the ide n tifier ˆ θ -dy namics to achieve an asympto tic estimate o f the u ncertain par ameters θ . B. Stability Ana lysis of E xtended Zer o Dyn amics (15) In the previous sub section, with the design o f ( 12) for system (13), we obtain an extended zero dy namics (15), whose stability analysis will be presented in the sequel. As befo re, we write z = ( ρ, w, z ) . Consider the chang e of coordinates ˜ η = η − τ ( z ) . This, recalling (5), tr a nsforms (15) to the fo r m ˙ z = f ( z ) ˙ ˜ η = A d ˜ η + B d [ φ ( ˆ θ , ˜ η + τ ) − φ ( θ, τ )] − satv (( A d + λI ) ˆ ξ ) − B d sat d +1 ( ˆ σ ) + ς ( z ) ˙ ˆ θ = β ( ˜ η + τ ) sat d +1 ( ˆ σ ) − dzv ( ˆ θ ) ˙ ˆ ξ = A d ˆ ξ + B d ˆ σ − satv (( A d + λI ) ˆ ξ ) − B d sat d +1 ( ˆ σ ) +Λ ℓ G ( ˜ η 1 − ˆ ξ 1 ) ˙ ˆ σ = ℓ d +1 g d +1 ( ˜ η 1 − ˆ ξ 1 ) (16) where ς ( z ) = A d τ ( z ) + B d φ ( θ, τ ( z ) ) − ∂ τ ( z ) ∂ w s ( w ) − ∂ τ ( z ) ∂ z f 0 ( z ) is a term which v anishes in Z c by Assump tion 3. Let ς i ( z ) denote the i -th element of the vector ς ( z ) , and then set ξ := col ( ξ 1 , . . . , ξ d ) with ξ 1 = ˜ η 1 ξ 2 = ˜ η 2 + ς 1 ( z ) ξ i = ˜ η i + P i − 2 j =1 L i − j − 1 f ς j +1 ( z ) + ς i − 1 ( z ) , 3 ≤ i ≤ d with L denoting th e Lie deriv ati ve, which also suggests that ˜ η = ξ − ¯ ς ( z ) for a n appropriately d e fined fu nction ¯ ς ( z ) , satisfying ¯ ς ( z ) = 0 for all z ∈ Z c . In view of th e pre vious analy sis, (16) can be rewritten as ˙ z = f ( z ) ˙ ξ = A d ξ + B d [ φ ( ˆ θ , ξ + τ ( z ) − ¯ ς ( z )) − φ ( θ , τ ( z ))] − satv (( A d + λI ) ˆ ξ ) − B d sat d +1 ( ˆ σ ) + B d ν ( z ) ˙ ˆ θ = β ( ξ + τ ( z ) − ¯ ς ( z )) sat d +1 ( ˆ σ ) − dzv ( ˆ θ ) ˙ ˆ ξ = A d ˆ ξ + B d ˆ σ − satv (( A d + λI ) ˆ ξ ) − B d sat d +1 ( ˆ σ ) +Λ ℓ G ( ξ 1 − ˆ ξ 1 ) ˙ ˆ σ = ℓ d +1 g d +1 ( ξ 1 − ˆ ξ 1 ) (17) where ν ( z ) = d − 1 X i =1 L d − i f ς i ( z )+ ς d ( z ) . It is n oted th a t ν ( z ) = 0 for all z ∈ Z c and there exists a co nstant a 3 > 0 such that for all z ∈ Z , | ν ( z ) | ≤ a 3 . (18) It then can be seen that the ( ˆ ξ , ˆ σ ) d ynamics in (1 7) can be viewed as an extended -state o bserver of the ξ dyn amics, with observer states ˆ ξ an d ˆ σ respectively being used to e stima te the variables ξ , an d the “perturba tio n” term σ := φ ( ˆ θ , ξ + τ ( z ) − ¯ ς ( z )) − φ ( θ, τ ( z ) ) + ν ( z ) . (19) This observation thus motivates us to analy se th e asymptotic stability of the exten d ed zero d ynamics (1 7) by using the nonlinear separatio n prin ciple [1], but in the ge neral non e q ui- libirum th eory . Fix all coefficients of the de a d -zone fu nction dzv ( · ) as c i > 4 a 1 a 2 ,i + 2 a 2 ,i a 3 ǫ 0 ,i , i = 1 , . . . , d , (20) with con stan ts a 1 , a 2 ,i , a 3 , and ǫ 0 ,i being given by (9), (18), and Assump tion 4.(i). W ith th e above choic e of c i ’ s in mind , to apply the nonlinear separatio n pr inciple to analyze the asymptotic stability o f system (17), it is natural to first co nsider th e auxiliary system ˙ z = f ( z ) ˙ ξ = − λξ ˙ ˆ θ = β ( ξ + τ ( z ) − ¯ ς ( z ))[ φ ( ˆ θ , ξ + τ ( z ) − ¯ ς ( z )) − φ ( θ , τ ( z ))] + β ( ξ + τ ( z ) − ¯ ς ( z )) ν ( z ) − dzv ( ˆ θ ) (21) whose stability properties can be character ized as below . Lemma 1: Suppo se Assumptio ns 1, 3, and 4 hold . Then the set A a := Z c × { 0 } × { θ } is asymptotically stable under the flow (21), for every initial condition ( z 0 , ξ 0 , ˆ θ 0 ) rangin g over the set M := Z × R d × R p . Pr oo f: The proof is gi ven in App endix A. Lemma 2: Suppo se Assumptio ns 2, 3, and 4 hold . Then the set A a under the flow (21) is loc a lly expo nentially stable. Pr oo f: The proof is gi ven in App endix B. By setting ˜ θ = ˆ θ − θ and letting z ( t ) denote th e solution of system ˙ z = f ( z ) with initial co ndition ranging over Z , system ( 21) can be rewritten as a nonauto nomou s system ˙ ξ = − λξ ˙ ˜ θ = β ( ξ + τ ( z ( t )) − ¯ ς ( z ( t ))) · · h φ ( ˜ θ + θ , ξ + τ ( z ( t )) − ¯ ς ( z ( t ))) − φ ( θ , τ ( z ( t ))) i + β ( ξ + τ ( z ( t )) − ¯ ς ( z ( t ))) ν ( z ( t )) − dzv ( ˜ θ + θ ) . (22) W ith L emma 1 , and r ecalling A ssum ption 1 and the fact that z ( t ) are captured by the comp act set Z , we ca n conclud e the following resu lt, whose pr o of can be obtained by simply adap ting the proof o f [3, Theo rem 3.1] to the present fr amework and is thus o mitted. Cor ollary 1: Suppose Assumptions 1, 3, and 4 ho ld . The zero equilibrium o f nonauton omou s system (22) is uniform ly asymptotically stable, for all z 0 ∈ Z . By letting x a = col ( ξ , ˜ θ ) , system (22) can be compactly rewritten as ˙ x a = f a ( z ( t ) , x a ) (23) where f a ( z ( t ) , x a ) is continuou sly differentiable. It is worth noting that b y co nstructing f unctions β ( · ) and φ ( · , · ) to be globally boun ded and Lipschitz, and since z ( t ) ∈ Z for all t ≥ 0 , ther e exists a  0 > 0 such that     ∂ f a ( z ( t ) , x a ) ∂ x a     ≤  0 . According to [ 11, Theo rem 4 . 16], this p roperty , together with Lemma 1, indic a te s that there exist a smooth, po siti ve definite function W a ( t, x a ) , an d class K ∞ function s α 1 , α 2 , α 3 , an d α 4 such that α 1 ( | x a | ) ≤ W a ( t, x a ) ≤ α 2 ( | x a | ) ∂ W a ∂ t + ∂ W a ∂ x a ˙ x a ≤ − α 3 ( | x a | )     ∂ W a ∂ x a     ≤ α 4 ( | x a | ) . (24) W ith this in m in d, we tur n to system ( 17) and d efine the rescaled estimate errors as ˜ ξ = ℓ d +1 Λ − 1 ℓ ( ξ − ˆ ξ ) , ˜ σ = σ − ˆ σ . (25) T aking time de r iv a tiv es of these errors along (1 7) y ields ˙ ˜ ξ = ℓ ( A d − G C d ) ˜ ξ + ℓ B d ˜ σ (26) and ˙ ˜ σ = − ℓg d +1 ˜ ξ 1 + ˙ φ ( ˆ θ , ξ + τ ( z ) − ¯ ς ( z )) − ˙ φ ( θ , τ ( z )) = − ℓ g d +1 ˜ ξ 1 + ∆ e (27) where the term ∆ e is define d by ∆ e = ∂ φ ( ˆ θ , ξ + τ ) ∂ ˆ θ ˙ ˆ θ + ∂ φ ( ˆ θ , ξ + τ ) ∂ ξ ˙ ξ + " ∂ φ ( ˆ θ, ξ + τ − ¯ ς ) ∂ τ − ∂ φ ( θ , τ ) ∂ τ # ˙ τ ( z ) − ∂ φ ( ˆ θ, ξ + τ − ¯ ς ) ∂ ¯ ς ˙ ¯ ς ( z ) . (28) It is worth no ting that ∆ e = 0 for a ll ( z , x a ) ∈ A a and e = 0 , and due to the p resence of saturatio n function s, | ∆ e | is bou nded for all bound ed ( z , x a ) , un iformly in ( ˜ ξ , ˜ σ ) . Putting these equ ations together and letting e = col ( ˜ ξ , ˜ σ ) , we can compactly obtain ˙ e = ℓF e e + B d +1 ∆ e (29) where F e is define d by F e =  − G I d − g d +1 0  This allows us to rewrite (17) as ˙ z = f ( z ) ˙ x a = f a ( z , x a ) + Ξ( z ( t ) , x a , e ) ˙ e = ℓF e e + B d +1 ∆ e . (30) Thus, g iv en any compact set C x ∈ R p + d , c h oose c such that A c ⊃ C x with A c = { x a : α 1 ( | x a | ) ≤ c } , and let Ω c +1 = { x a : α 1 ( | x a | ) ≤ max x a ∈A c α 2 ( | x a | ) + 1 } . It is clear th a t A c ⊂ Ω c +1 . Then, choose the satur ation lev els as l i = m ax x a ∈ Ω c +1 | λξ i + ξ i +1 | + 1 , 1 ≤ i ≤ d − 1 l d = m ax x a ∈ Ω c +1 | λξ d | + 1 l d +1 = max ( z , x a ) ∈ Z × Ω c +1    φ ( ˆ θ , ϕ η ( ξ + τ ( z ) − ¯ ς ( z ( t )))) − φ ( θ, τ ( z )) | + 1 . (31) W ith th e ab ove choice of l i ’ s, it can b e o bserved th at f or all ( z , x a ) ∈ Z × Ω c +1 , Ξ( z , x a , e ) is bounded uniform ly in e , and Ξ( z , x a , 0) = 0 . Therefo re, from the standard argum ents of nonlin ear sep- aration prin c iples [ 1], semiglob al asymptotic stability of th e closed-loo p system (17) can be easily co ncluded as b elow . Pr op osition 1: Suppo se Assumptio n s 1, 2 and 4 hold. Giv en any compa ct sets C x ∈ R q + d and C e ∈ R d +1 , and choosing g i ’ s such that matrix F e is Hurwitz, ther e exists ℓ ∗ > 1 such that f o r all ℓ ≥ ℓ ∗ the set { ( z , ˆ θ, ξ , ˆ ξ , ˆ σ ) : z ∈ Z c , ξ = 0 , ˆ θ = θ , ˆ ξ = 0 , ˆ σ = 0 } under the flow (17) is loca lly expone ntially stab le, and asymptotically stable f or all initial co nditions in Z × C x × C e . C. Adaptive Outpu t Regulation W e n ow turn to the extende d system (13). As mentioned before, this system, viewed as a system with input v u and output y e = x , has relative degree o ne. By taking the change of variables ˇ ξ := ˆ ξ + Λ ℓ G Z x 0 1 b ( ρ, w , z , s ) ds ˇ σ := ˆ σ + ℓ d +1 g d +1 Z x 0 1 b ( ρ, w , z , s ) ds system ( 13) can be rewritten in “no rmal form” as ˙ ρ = 0 ˙ w = s ( ρ, w ) ˙ z = f 0 ( ρ, w , z ) + f 1 ( ρ, w , z , x ) x ˙ η = A d η + B d φ ( ˆ θ , η ) − satv (( A d + λI ) ˇ ξ ) − B d sat d +1 ( ˆ σ ) + µ 1 ( ρ, w , z , ˆ θ, η , ˇ ξ , ˇ σ , x ) x ˙ ˆ θ = β ( η ) sat d +1 ( ˇ σ ) + µ 2 ( ρ, w , z , ˆ θ, η , ˇ ξ , ˇ σ , x ) x ˙ ˇ ξ = A d ˇ ξ + B d ˇ σ − satv (( A d + λI ) ˇ ξ ) − B d sat d +1 ( ˇ σ ) +Λ ℓ G  C d η + q ( ρ, w , z , 0 ) b ( ρ, w , z , 0) − ˆ ξ 1  + µ 3 ( ρ, w , z , ˆ θ, η , ˇ ξ , ˇ σ , x ) x ˙ ˇ σ = ℓ d +1 g d +1  C d η + q ( ρ, w , z , 0 ) b ( ρ, w , z , 0) − ˇ ξ 1  + µ 4 ( ρ, w , z , ˆ θ , η , ˇ ξ , ˇ σ , x ) x ˙ x = q ( ρ, w , z , x ) − b ( ρ, w, z , x ) q ( ρ, w , z , 0 ) b ( ρ, w , z , 0) + b ( ρ, w , z , x )  C d η + q ( ρ, w , z , 0 ) b ( ρ, w , z , 0)  + b ( ρ, w, z , x ) v u (32) in wh ich µ i ( · ) , i = 1 , . . . , 4 are contin uous function s. Bearing in min d the resu lts in Prop osition 1 and recalling [9, Pro position 4], we can choose v u for system (32) as the form v u = − κx , (33) and the following co nclusion can be easily made. Pr op osition 2: Consider system (1) with exosystem (2) and con troller (4) having the for m (12) and (33). Supp ose Assumptions 1-4 ho ld. Given any compact sets C z ⊂ R n , C x ⊂ R and C x c ⊂ R 2 d + q +1 , and cho osing g i ’ s such that matrix F e is Hurwitz, there exist ℓ ∗ > 1 and a positi ve function κ ∗ ( · ) such that for all ℓ > ℓ ∗ and κ ≥ κ ∗ ( ℓ ) , the resulting tr a jectories of the closed- lo op system ar e bound ed and x ( t ) → 0 as t → ∞ , with the domain of attraction th at contains C z × C x × C x c . I V . A N I L L U S T R A T I V E E X A M P L E Consider the outpu t regulation problem for the nonlinear system ˙ ζ 1 = ρζ 1 − ( ζ 1 + w 1 ) 3 + w 2 + ζ 2 ˙ ζ 2 = ζ 3 ˙ ζ 3 = − w 1 + ζ 1 ζ 2 + u y e = ζ 1 (34) in which ( ζ 2 , ζ 3 ) are measurable states, an d the exogenou s variables w 1 , w 2 are ge n erated by an uncertain nonlinear oscillator ˙ w 1 = w 2 ˙ w 2 = − w 1 + (1 − w 2 1 ) w 2 1 + ρw 1 (35) where ρ is a con stant un known parameter satisfying ρ ∈ [ − 0 . 2 , 0 . 2] . T he trajectories of (3 5) at each ρ ∈ {− 0 . 2 , 0 , 0 . 2 } are giv en in Fig.2. It can be seen that for any ρ ∈ [ − 0 . 2 , 0 . 2] ther e exists a limit cycle, that is an inv ariant set W fo r ( 35), and particu la r ly W ⊂ { ( w 1 , w 2 ) : | w i | ≤ 3 , i = 1 , 2 } . -5 0 5 -4 -2 0 2 4 =0.2 -5 0 5 -4 -2 0 2 4 =0 -5 0 5 -4 -2 0 2 4 =-0.2 Fig. 2. Phase portrai t of (35) at ρ = 0 . 2 , ρ = 0 and ρ = − 0 . 2 . Note that, when w 1 = w 2 ≡ 0 , system (3 4), regarde d as a system with in put u and o utput y e , has relative degree 2 and a zero dynamics as ˙ ζ 1 = ρζ 1 − ζ 3 1 , whose zero equilibrium point is unstable wh en ρ > 0 and stable wh en ρ ≤ 0 . Th us, the con ventional method s [14], [3] based o n equilibrium theory can n ot be applied. Follo wing th e design para d igm propo sed in this pap er , we first set z 1 = ζ 1 , z 2 = ζ 2 and x = ζ 2 + ζ 3 , which redu ces the relativ e degree of system (34) to one, leading to the f orm ˙ z 1 = ρz 1 − ( z 1 + w 1 ) 3 + w 2 + z 2 ˙ z 2 = − z 2 + x ˙ x = − w 1 − z 2 + z 1 z 2 + x + u . (36) The zero dynamics o f system (35)-(36) with respect to input u a n d output x , forced by the control input u = w 1 + z 2 − z 1 z 2 , can be described as ˙ ρ = 0 ˙ w 1 = w 2 ˙ w 2 = − w 1 + (1 − w 2 1 ) w 2 1 + ρw 1 ˙ z 1 = ρz 1 − ( z 1 + w 1 ) 3 + w 2 + z 2 ˙ z 2 = − z 2 . Then, by some simple calculations, it can be seen that As- sumptions 1 and 2 are fulfilled for some ω -limit set on which z 2 = 0 . In v iew of this, we pro c e ed to verify Assumptions 3 and 4. Observe that in the present setting, Assumption 3 is fulfilled with the m a p τ := ( τ 1 , τ 2 ) = ( w 1 , w 2 ) satisfying the e q uations ˙ τ 1 = τ 2 , ˙ τ 2 = φ ( θ , τ ) where θ = ρ and function φ ( θ, τ ) = − ϕ s ( τ 1 ) + (1 − ϕ 2 s ( τ 1 )) ϕ s ( τ 2 ) 1 + ϕ s ( θ ) ϕ s ( τ 1 ) with ϕ s ( τ i ) = τ i , for | τ i | ≤ 3 ϕ s ( θ ) = θ , for | θ | ≤ 0 . 2 . Moreover , b y choo sin g β ( τ ) = (1 − ϕ 2 s ( τ 1 )) ϕ s ( τ 1 ) ϕ s ( τ 2 ) , it can be easily foun d that the function β ( τ ) φ ( θ , τ ) is strictly decreasing in | θ | ≤ 0 . 25 , for all τ ∈ W . In this w ay , Assumption 4 is also fulfilled. Therefo re, th e a d aptive internal model- based regulator (12) and (11) can be emp loyed to handle the n o nlinear o utput regulation pro blem at hand. Figure 3 shows simulatio n results for ρ = 0 . 2 , and the d esign parameter s ℓ = 10 and κ = 30 . It demonstra te s that the regulated output y e conv erges to zero asymptotically and th e parameter estimate ˆ θ co n verges to the real value. 0 5 10 15 Time(s) -0.4 -0.2 0 0.2 0.4 0.6 0.8 Regulated Output Parameter Estimate Fig. 3. Tra jectori es of regul ated output y e ( t ) and parameter estimate ˆ θ ( t ) . V . C O N C L U S I O N This paper studies th e adaptive ou tput regulation prob lem for a class of no nlinear systems using the gen e r al n onequ ilib- rium theor y developed in [6]. By incorpor ating an extended - state observer into the adaptive internal model, a new ap- proach is p roposed to d eal with ad aptive no nlinear regulatio n, which a llows for more genera l nonlinear ly parameterized immersion con ditions. A P P E N D I X A. Pr oof of Lemma 1 The proof mainly follows the n onequ ilibrium theory de- veloped in [6]. First, we will show that the tr a jectories o f system (21) are bounde d, i.e. there is no fin ite-time escape. By Assumption 1 and the choice of λ > 0 , it can be easily seen that z ( t ) and ξ ( t ) are b o unde d . T o show ˆ θ ( t ) is also bound ed, we let ˆ θ i denote the i -th element of vector ˆ θ an d choose V ˆ θ ,i = 1 2 | ˆ θ i | 2 , i = 1 , . . . , p . T aking the time d eriv ati ve of V ˆ θ ,i along th e b ottom equ a tion of (2 1) yields that ˙ V ˆ θ ,i = ˆ θ i β i [ φ ( ˆ θ , ξ + τ ( z ) − ¯ ς ( z )) − φ ( θ , τ ( z ))] + ˆ θ i β i ν ( z ) − ˆ θ i dz i ( ˆ θ i ) ≤ − ˆ θ i dz i ( ˆ θ i ) + (2 a 1 a 2 ,i + a 2 ,i a 3 ) | ˆ θ i | where (9) and (18) are used to get the inequality . If | ˆ θ i | ≥ a 0 ,i + ǫ 0 ,i , th e n ˙ V ˆ θ,i ≤ − c i ˆ θ i [ ˆ θ i − ( a 0 ,i + ǫ 0 ,i 2 )] + (2 a 1 a 2 ,i + a 2 ,i a 3 ) | ˆ θ i | ≤ − ǫ 0 ,i 2 ( c i − 4 a 1 a 2 ,i +2 a 2 ,i a 3 ǫ 0 ,i ) | ˆ θ i | . From (20), we can con c lu de th at ˙ V ˆ θ ,i < 0 fo r all | ˆ θ i | ≥ a 0 ,i + ǫ 0 ,i with i = 1 , . . . , d . This then indicates that in the presence of dead-zo ne functions dzv ( ˆ θ ) , th e trajectory ˆ θ ( t ) of (21) is g lo bally unifor mly bou nded, and will enter an d stay in sid e the closed cube B q 0 . W ith the bo u ndedn ess o f trajectories of system (21), it th us can be dedu ced that there exists an ω -limit set, de n oted by ω ( M ) , of M = Z × R d × R q under the flow o f (21), which is nonemp ty , compact and inv ar iant, and uniform ly attracts all trajecto ries of (21) with initial conditions in M . Now we proceed to in vestigate the stru cture of this ω - limit set ω ( M ) . Du e to the special triangula r structu re of (21), and by Assum ption 1 and th e fact that the ξ -sub sy stem is glob ally exponentially stab le at the origin, it immediately follows that on the points of ω ( M ) , necessarily z ∈ Z c and ξ = 0 . As a consequ ence, o n the ω -limit set ω ( M ) , ¯ ς ( z ) = 0 and ν ( z ) = 0 . In v iew o f the previous analy sis, to specify the structure of ω ( M ) , we still n eed to determine the value of ˆ θ . O n the other hand , wh en proving the boun d ness of ˆ θ ( t ) , we have shown that ˆ θ ( t ) will en ter and stay inside the closed cube B q 0 . Thus, by recalling that Z c is in variant under (5 ), the v alue of ˆ θ o n ω ( M ) is determined by the pro p erties o f the sy stem ˙ z = f ( z ) ˙ ˆ θ = β ( τ ( z ))[ φ ( ˆ θ , τ ( z )) − φ ( θ , τ ( z ))] − dzv ( ˆ θ ) (37) where the initial con dition z 0 ∈ Z c and ˆ θ 0 ∈ B q 0 . It is no ted that ˆ θ ( t ) ∈ B q 0 for all t ≥ 0 under (37). Then, ch o ose V ˜ θ = 1 2 | ˜ θ | 2 with ˜ θ = ˆ θ − θ , whose time deriv ativ e along (37) ca n be given by ˙ V ˜ θ = ( ˆ θ − θ ) ⊤ β ( τ ( z ))[ φ ( ˆ θ , τ ( z )) − φ ( θ , τ ( z ))] − ˜ θ ⊤ dzv ( ˆ θ ) . Bearing in mind the definition of dzv ( · ) , observe that ( ˆ θ − θ ( ρ )) ⊤ dzv ( ˆ θ ) ≥ 0 f o r all ˆ θ ∈ R p and ρ ∈ P . (38) This, togeth e r with the first part o f Assumption 4, implies that u nder the flo w (37), ˙ V ˜ θ ≤ 0 , (39) where the equality hold s if and only if ( ˆ θ − θ ) ⊤ β ( τ ( z ))[ φ ( ˆ θ , τ ( z )) − φ ( θ , τ ( z ))] = 0 ( ˆ θ − θ ) ⊤ dzv ( ˆ θ ) = 0 . Thus, ˆ θ ( t ) con verges to som e con stant value ˆ θ ∞ as t go es to infinity . By LaSalle’ s inv ariance theore m, this ˆ θ ∞ necessarily is such that ( ˆ θ ∞ − θ ) ⊤ β ( τ ( z ))[ φ ( ˆ θ ∞ , τ ( z )) − φ ( θ , τ ( z ))] = 0 ( ˆ θ ∞ − θ ) ⊤ dzv ( ˆ θ ∞ ) = 0 β ( τ ( z ))[ φ ( ˆ θ ∞ , τ ( z )) − φ ( θ , τ ( z ))] − dzv ( ˆ θ ∞ ) = 0 . (40) It is n oted that the second of (40) indic a tes that dzv ( ˆ θ ∞ ) = 0 . This f urther red uces (40) to β ( τ ( z ))[ φ ( ˆ θ ∞ , τ ( z )) − φ ( θ , τ ( z ))] = 0 . By Assump tion 4 . (ii), we hav e ˆ θ ∞ = θ . This com p letes th e proof .  B. Pr oof of Lemma 2 Due to the special cascad ed-structu re of system (21) and since fun ctions β and φ are construc ted to be globally Lipschitz a n d bou nded, with the choice of λ > 0 and Assumption 2, it is clear th at the proof is co mpleted if for any z 0 ∈ Z c , th e origin of the linear time- varying system ˙ ˜ θ = β ( τ ( z ( t, z 0 ))) ∂ φ ( θ , τ ( z ( t, z 0 ))) ∂ θ ˜ θ (41) with ˜ θ = ˆ θ − θ , is shown to be unifo rmly exponentially stable. Since z ( t, z 0 ) is the solu tion of the auto n omou s system (5) passing throu gh z 0 at t = 0 , (4 1) can be rewritten as a cascaded a utono mous system , having the form ˙ z = f ( z ) ˙ ˜ θ = β ( τ ( z ) ) ∂ φ ( θ , τ ( z )) ∂ θ ˜ θ . (42) W e then calculate the derivati ve of V ˜ θ as ˙ V ˜ θ = ˜ θ ⊤ β ( τ ( z )) ∂ φ ( θ , τ ( z )) ∂ θ ˜ θ ≤ 0 where the inequality is obtain e d by using Assumption 4.(i). Then, similar to the proof of Lemma 1, by LaSalle’ s in- variance theore m and Assumptio n 4.( ii) , we ca n conclud e that system (42) is u niform ly asymptotically stab le at the set Z c × { 0 } , for any initial cond ition ( z 0 , ˜ θ 0 ) ∈ Z c × R q . I n other words, for any ε > 0 and ( z 0 , ˜ θ 0 ) ∈ Z c × R q , th ere exists T ε > 0 such that | ˜ θ ( t ) | = d ist  ( z ( t ) , ˜ θ ( t )) , Z c × { 0 }  ≤ ε for all t ≥ T ε . (43) Therefo re, the zer o equilibriu m of the lin ear time-varying system (41) is unifo rmly asymptotically stable, which also indicates the desired expon ential stability .  R E F E R E N C E S [1] A. Isidori. Nonlinear Contr ol Systems II . New Y ork: Springer ,1999. [2] A. Isidori, C.I. Byrnes, “Output regula tion of nonlinear systems, ” IE EE T rans. Automatic Contr ol , vol.5, pp.131-140, 1990. [3] A. Serrani, A. Isidori, and L. Marconi, “Semiglobal nonlinear output regul ation with adapti ve internal model, ” IEEE Tr ans. Autom. Contr ol , vol. 46, no. 8, pp. 1178-1194, 2001. [4] A. T eel, and L. Praly . “T ools for semigl obal stab ilizat ion by partial state and output feedback, ” SIAM J. Contr . Optimiz. , vol. 33, pp. 1443- 1488, 1995. [5] B. A. Francis and W . M. W onham, “The inte rnal model principl e of control theory , ” Automatica , vol. 12, pp. 457-465, 1976. [6] C. Byrnes, A. Isidori, “Limit sets, zero dynamics, and internal models in the problem of nonlinea r output reg ulation , ” IEEE T rans. Autom. Contr ol , vol. 48, no. 10, pp. 1712-1723, 2003. [7] C. Byrne s, A. Isidori, “Nonlinea r int ernal models for output re gula- tion, ” IEEE T rans. Autom. Contr ol , vol. 49, no. 12, pp. 1712-1 723, 2004. [8] D. Xu, X. W ang, Z. Chen, “Output regulat ion of nonlinear output feedbac k syste ms with exponen tial parameter con ve rgence , ” Systems & Contr ol Letters , vol.88, pp. 81-90, 2016. [9] F . D. Priscoli, L . Marconi and A. Isidori, “ A ne w approach to adapti ve nonline ar regulati on, ” SIAM J . Contr ol Optimization , vol.45, no.3, pp.829-855, 2006. [10] H.K. Khalil, “Robust servomechan ism output feedback control lers for feedbac k linea rizable systems, Automatic a , vol. 30, pp. 1587-1589, 1994. [11] H. Khalil . Non linear Systems , 3rd ed. Upper Saddle Ri ver , NJ: Prentic e-Hall, 1996. [12] J. Han, “ A class of exte nded state observers for uncert ain systems, ” Contr ol Decis. , vol.10, no.1, pp. 85-88, 1995 (in Chinese). [13] J. Huang, Z . Chen, “ A genera l framewo rk for tackl ing the output regul ation problem, ” IEEE T rans. Autom. Contr ol , 49(12): 2203-2218, 2004. [14] J. Huang, Nonline ar Output Re gulati on: Theory and Application s , Philade lphia, USA, SIAM, 2004. [15] L.B. Freidovich, H.K. Khalil, “Performance recov ery of feedba ck- linea rizatio n based designs, ” IEEE T rans. Automat. Con tr ol , vo l.53, no.10, pp.2324-2334, 2008. [16] L. Marconi, L. Praly , A. Isidori. “Output stabilizat ion via nonlinear Luenber ger observers, ” SIAM J ournal on Contr ol and Optimizati on , vol.45, no.6, pp.2277-2298, 2007. [17] L. Marconi, and L. Praly , “Uniform practic al nonlinear output re g- ulati on, ” IEEE T rans. Autom. Con tr . , v ol. 53, no. 5, pp.1184-12 02, 2008. [18] L. W ang, A. Isidori, H. Su, “Output feedback stabiliza tion of nonlinear MIMO systems havi ng uncertai n high-freque ncy gain matrix, ” Systems & Contr ol Letters , vol. 83, pp. 1-8, 2015. [19] M. Krstic, I. Kanella kopoul os, P . Kokot ovic , Nonlinear and Adaptive Contr ol Design , John Will ey , New Y ork, 1995. [20] R. Marin o, P . T omei. “Output regul ation for line ar systems via adapti ve internal model, ” IEE E T rans. Automat. Contr ol , vol.48, no.12, pp.2199-2202, 2003. [21] R. Marino, P . T omei. “Global adapti ve regul ation of uncerta in nonlin- ear systems in output feedbac k form, ” IEE E T rans. Automat. Contr ol , vol.58, no.11, pp.2904-2909, 2013. [22] X. W ang, Z. Chen and D. Xu, “ A frame work for global robust output regula tion of nonlinear lower triangular systems with uncertain exosyst ems, ” IEEE Tr ans. Automat. Contr ol , vol.63, no.3, pp.894-901, 2016. [23] Z. Ding, “Global stabili zation and disturbanc e suppression of a class of nonline ar systems with uncertain interna l model, ” Automatic a , vol.39, no.3, pp.471-479, 2003.