Adaptive Semiglobal Nonlinear Output Regulation:An Extended-State Observer Approach
aa r X i v : . [ c s . S Y ] J a n Adaptive Semiglobal Nonlinear Output Regulation:An Extended-State Observer Approach
Lei Wang, Christopher M. Kellett
Abstract — This paper proposes a new extended-stateobserver-based framework for adaptive nonlinear regulatordesign of a class of nonlinear systems, in the general nonequi-librium theory. By augmenting an extended-state observer withan internal model, one is able to obtain an estimate of the termcontaining uncertain parameters, which then makes it possibleto design an adaptive internal model in the presence of a general nonlinearly parameterized immersion condition.
I. I
NTRODUCTION
The output regulation problem aims at controlling a dis-turbed system so as to achieve boundedness of the resultingtrajectories and asymptotic convergence of the output to-wards a prescribed trajectory. Several frameworks have beenestablished for this problem. Due to its ability to cope withuncertainties, the internal model-based method has been re-garded as one of the most promising approaches, particularlysince the milestone contributions for linear systems in [5] andnonlinear systems in [2]. The main idea of this method is toappropriately incorporate the controller with the structure ofan exosystem that generates the disturbance and the trackingtrajectory.In the design of an internal model-based regulator, a keystep is to design an appropriate internal model to generatethe steady state input such that the internal model propertyis fulfilled. Several systematic design methods have beendeveloped such as in [10], [13], [14], [16], [17]. Amongthem, in terms of a constructive design, significant attentionhas been attracted by the “immersion condition”, which re-quires the solution of the regulator equations to satisfy somespecific differential equations (i.e., the immersed dynamics).It is noted that, if there exist parameter uncertainties in theexosystem, the corresponding immersed dynamics would beuncertain in general, which makes the design of internalmodel challenging. To cope with parameter uncertainties,in [3] the internal model is augmented with an identifier,which is appropriately designed via the adaptive designmethodology [19]. Motivated by this adaptive framework,several relevant results have been reported that differ in thekind of exosystems (linear [23], [22] and nonlinear [21]exosystems), in the kind of available information (state andoutput feedback), and in the kind of controlled systems(linear [20] and nonlinear [8] systems). On the other hand,the above mentioned “immersion conditions” are formulatedon an extra assumption that the regulator equations are
L. Wang and C. Kellett are with Faculty of Engineering and Built Environ-ment, University of Newcastle, Australia. E-mail: (wanglei [email protected];[email protected]). solvable. This fundamentally limits the class of controlledsystems that can be handled. In [6], [7], this extra assumptionis removed by taking advantage of the nonequilibrium theoryof nonlinear output regulation. In [9], the corresponding ex-tension to adaptive nonlinear output regulation is addressed.Despite the aforementioned efforts, research on adaptiveinternal model design is still at quite an early stage. In fact,the immersion conditions in the existing design methods arequite restrictive, at least in the following two aspects. Firstly,the immersed dynamics is usually required to be linear, hencelimiting the exosystem to be linear generally. It is noted thatthe only exception is [9], where the immersed dynamics isassumed to be in the output-feedback form. Besides, as in[19], the design of all adaptation laws, to the best knowledgeof the authors, is based on the idea of “cancellation”, thatis to cancel the term containing the unknown parameterswhen computing the derivative of the Lyapunov function,which usually requires a linearly parameterized immersioncondition. This in turn fundamentally limits the class ofexogenous and controlled systems.In order to deal with a broad class of exogenous andcontrolled systems, this paper studies the adaptive nonlinearoutput regulation problem with a general immersion condi-tion, in the general nonequilibrium theory of nonlinear outputregulation developed in [6], [9]. Inspired by [15], [12], [18],a new extended-state observer-based design paradigm is de-veloped to construct an adaptive nonlinear internal model. Bytaking advantage of the extra state provided by the extended-state observer, one is able to obtain an estimate of the termcontaining the uncertain parameter to be estimated, whichthen can be utilised to achieve asymptotic identification.It is noted that the proposed method allows a nonlinearlyparameterized immersion condition. More specifically, theuncertain parameters in the immersed dynamics can appearin a “monotonic-like structure”, with linear parameterizationas a particular case.The paper is organized as follows. Section II gives theproblem formulation and some standing assumptions. InSection III, the main results are addressed by presentingthe design of the adaptive internal model and the stabilityanalysis of the resulting closed-loop system. An illustrativeexample is presented in Section IV to show the validity ofthe prosed method. A brief conclusion is made in Section V.
Notations:
For any positive integer d , ( A d , B d , C d ) isused to denote the matrix triplet in the prime form. Namely, A d denotes a shift matrix of dimension d × d whose allsuperdiagonal entries are one and other entries are all zero, d denotes a d × vector whose entries are all zero exceptthe last one which is equal to 1, and C d is a × d vectorwhose entries are all zero except the first one which is equalto 1. A function f : R + := [0 , ∞ ) → R + is of class K ,if it is continuous, positive definite, and strictly increasing.A class K function is of class K ∞ if it is unbounded. Acontinuous function δ : R + × R + → R + is of class KL if, for each fixed t ≥ , the function δ ( · , t ) is of class K and, for each fixed s > , δ ( s, · ) is strictly decreasing and lim t →∞ δ ( s, t ) = 0 .II. P RELIMINARIES
A. Problem Statement
Consider the system ˙ z = f ( ρ, w, z ) + f ( ρ, w, z, x ) x ˙ x = q ( ρ, w, z, x ) + b ( ρ, w, z, x ) uy e = x (1)with state z ∈ R n and x ∈ R , control input u ∈ R , regulatedoutput y e ∈ R , and in which ρ ∈ R p and w ∈ R s denote theexogenous input, generated by the exosystem ˙ ρ = 0˙ w = s ( ρ, w ) , (2)with the initial conditions ρ and w taking values fromcompact sets P ⊂ R p and W ⊂ R s , respectively. Ascustomary in the field of output regulation, it is assumedthat P × W is invariant for (2), and there exists a constant b > such that b ( ρ, w, z, x ) ≥ b (3)holds for all ( ρ, w, z, x ) ∈ P × W × R n × R . Addition-ally, functions f ( · ) , f ( · ) , q ( · ) , b ( · ) , s ( · ) are assumed to besufficiently smooth.In this framework, the output regulation problem of in-terest can be summarized as below. Given any compactsets C z ⊂ R n , C x ⊂ R , all trajectories of system (1)-(2),controlled by an output feedback regulator 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 conditions ranging over P ×W ×C z ×C x ×C x c with C x c being any given compact set in R n c , are boundedand lim t →∞ y e ( t ) = 0 .With this in mind, it is observed that by viewing x asthe output, system (1) cascaded with (2) has a well-definedrelative degree one, and the corresponding zero dynamics,driven by the control input u = − q ( ρ, w, z, b ( ρ, w, z, , is given by ˙ ρ = 0˙ w = s ( ρ, w )˙ z = f ( ρ, w, z ) . which, with z := ( ρ, w, z ) , can be compactly rewritten as ˙ z = f ( z ) (5) Accordingly, we set Z := P × W × C z with C z ⊂ R n beingany given compact set. Remark 1:
This paper is mainly interested in nonlinearsystems having normal form. Although system (1) has rela-tive degree one, its extension to higher relative degree can betrivially achieved as in [1] by redefining a regulated outputso as to reduce the relative degree to one.
B. Standing Assumptions
In order to deal with a more general class of nonlinearsystems, following [9] we make some assumptions on thezero dynamics (5).
Assumption 1:
There exist a nonempty, compact set A z ⊂ R n , and a class KL function δ ( · , · ) such that for all z ∈P × W × R n ,dist ( z ( t, z ) , Z c ) ≤ δ ( dist ( z , Z c ) , t ) for all t ≥ where Z c := P × W × A z , and z ( t, z ) denotes the solutionof system (5) passing through z at time t = 0 . Assumption 2:
There exist constants M ≥ , a > , and δ > such that for all z ∈ P × W × R n ,dist ( z , Z c ) ≤ δ ⇒ dist ( z ( t, z ) , Z c ) ≤ M e − a t dist ( z , Z c ) . Remark 2:
Assumption 1 indicates that Z c is an invari-ant and asymptotically stable compact set under (5). Morespecifically, 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 thereexists a compact set Z such that the solution of (5) satisfies z ( t, z ) ∈ Z for all t ≥ , so long as z ∈ Z . Assumption2 implies that Z c is locally exponentially stable for (5),which plays a significant role in the subsequent analysis ofasymptotic stability. Remark 3:
Assumption 1 can be regarded as theminimum-phase assumption in general nonequilibrium the-ory. Compared with the conventional minimum-phase as-sumption such as in [3], [14], the main benefit is that theextra assumption on the solvability of the regulator equationsis removed, which broadens the class of systems that can beaddressed.To this end, a general nonlinearly parameterized immer-sion condition will be proposed, which leads to a constructivedesign of the internal model.
Assumption 3:
There exist positive integers d and q , a C map θ : P → R q ,ρ θ ( ρ ) , a C d map τ : Z → R d , z τ ( z ) , and a C map φ : R p × R d → R such that the followingidentities ∂τ∂ z f ( z ) = A d τ ( z ) + B d φ ( θ ( ρ ) , τ ( z )) q ( z ) = C d τ ( z ) (6)with q ( z ) = − q ( ρ, w, z, b ( ρ, w, z, , hold for all z ∈ Z c and ρ ∈ P . emark 4: In Assumption 3, the immersed dynamics (6)is allowed to be dependent on the uncertain parameter ρ ,which motivates us to incorporate the internal model withan identifier. Since ρ appears only in the function θ ( · ) ,for convenience we regard θ as an uncertain parameterto be estimated in the sequel, though this may result inoverparameterization.In the literature, several immersion conditions for adaptiveoutput regulation have been proposed. It is worth notingthat compared to the existing ones, Assumption 3 is muchweaker, at least in the following two aspects. In previouswork, the immersion map τ is required to satisfy either alinear equation (e.g. [3]), or a nonlinear equation but inthe “output-feedback form” (e.g. [9]). Fundamentally, allthese forms in [3], [9] can be transformed to the form (6).Moreover, in all the previous related literature, the immerseddynamics (6) is required to be linearly parameterized, whilethis paper permits a nonlinear parameterization, with linearparameterization as a particular case.In this paper, we aim to handle a more general immersionproperty having a nonlinearly parameterized function φ ( θ, τ ) in the uncertain parameter θ . We require the followingproperties on φ ( · , · ) . Assumption 4:
There exists a smooth function β ( · ) : R d → R p having the properties:(i) There exist ǫ ,i > , i = 1 , . . . , q such that for any r ∈ τ ( Z c ) , and any s , s ∈ B q := { θ ∈ R q : | θ i | ≤ a ,i + ǫ ,i } with a ,i = max ρ ∈P | θ i ( ρ ) | , the inequality ( s − θ ) ⊤ β ( r ) ∂φ ( s , r ) ∂s ( s − θ ) ≤ (7)holds, with θ i denoting the i -th entry of vector θ ;(ii) For any z ∈ Z c and s , s ∈ B q , the persistentexcitation (PE) condition φ ( s , τ ( z ( t, z ))) − φ ( s , τ ( z ( t, z ))) = 0= ⇒ s = s (8)is fulfilled, where z ( t, z ) denotes the trajectory of (5)passing through z at t = 0 . Remark 5:
Assumption 4.(i) means that there exists asmooth function β ( r ) such that for all r ∈ τ ( Z c ) , thefunction β ( r ) φ ( s, r ) is monotonically decreasing in s ∈B q . In this respect, we say that the function φ ( s, r ) satis-fying Assumption 4.(i) is in the monotonic-like structure .If as in [3], [9], the function φ is linearly parameter-ized, that is φ ( s, r ) has the form of s ⊤ ψ ( r ) for somefunction ψ ( · ) , then Assumption 4.(i) can always be ful-filled by choosing β ( r ) = ψ ( r ) . Indeed, the class offunctions φ ( r, s ) satisfying such a monotonicity condi-tion includes not only all linearly parameterized functions,but also some nonlinearly parameterized functions, suchas arctan ( s ⊤ ψ ( r )) or ψ ( r ) P pi =1 θ i ψ i ( r ) + ψ p +1 ( r ) , where thecorresponding function β ( r ) can be chosen as ψ ( r ) or − ( ψ ( r ) ψ ( r ) · · · ψ ( r ) ψ p ( r ) ) ⊤ , respectively. For simplicity, we use τ ( Z c ) to denote the set of τ ( z ) for all z ∈ Z c . It is observed that the maps φ ( s, r ) and β ( r ) are continu-ously differentiable and Assumption 3 and 4 are respectivelymade over the compact sets s ∈ B q and ( s, r ) ∈ B q × τ ( Z c ) .In view of this, there is no loss of generality to supposethat functions φ ( · , · ) and β i ( · ) are globally Lipschitz andbounded, i.e., there exist a > and a ,i > , i = 1 , . . . , q such that inequalities | φ ( s, r ) | ≤ a , | β i ( r ) | ≤ a ,i (9)with β i denoting the i -th entry of vector β , hold for all s ∈ R q , r ∈ R d .III. A DAPTIVE R EGULATOR D ESIGN
A. Adaptive Internal Model
With Assumption 3, if θ were known, then we could designan internal model of the form ˙ η = A d η + B d φ ( θ, η ) + v η (10)in which η ∈ R d , and v η ∈ R d denotes the input of theinternal model, and the control input can be chosen as u = v u + C d η (11)where v u is the residual input.However, since θ is unknown, the internal model (10)is not implementable. To overcome this obstacle, an extraidentifier can be used to provide an estimate of θ , denotedby ˆ θ ∈ R q . It is worth noting that, due to the presence ofthe nonlinear parameterization, we cannot take advantage ofthe usual “cancellation” idea (e.g. [3], [9]).Inspired by various important results on the design ofextended-state observers (e.g. [15], [12], [18]), we proposea new adaptive internal model, 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 )˙ˆ σ = − ℓ d +1 g d +1 ( v u + ˆ ξ ) (12)where ˆ ξ := col ( ˆ ξ , . . . , ˆ ξ d ) , λ > , Λ ℓ = diag ( ℓ, . . . , ℓ d ) , G = col ( g , . . . , g d ) , functions sat i ( · ) for i = 1 , . . . , d + 1 have the formsat i ( s ) = s , | s | ≤ l i s − sign ( s ) ( | s | − l i ) , l i < | s | < l i + 1 l i + , | s | ≥ l i + 1 , with saturation level l i , satv ( · ) : R d → R d denotes a vector-valued saturation function, defined by satv ( s , . . . , s d ) = col ( sat ( s ) , . . . , sat d ( s d )) , and dzv ( · ) denotes a vector-valued dead-zone function, each element of which is afunction of the formdz i ( s ) = , | s | ≤ a ,i c i ( | s | − a ,i ) ǫ ,i sign ( s ) , a ,i < | s | < a ,i + ǫ ,i c i s − c i (cid:16) a ,i + ǫ ,i (cid:17) sign ( s ) , | s | ≥ a ,i + ǫ ,i . s it can be seen from Fig. 1, functions sat i and dz i areconstructed to be smooth. All design parameters g i , l i , and c i will be defined later in Proposition 1, (31), and (20),respectively. -10 -5 0 5 10-6-4-20246-10 -5 0 5 10-4-2024 Fig. 1. Left: plot of function sat i with l i = 3 ; and right: plot of functiondz i with c i = 1 . , a ,i = 4 , ǫ ,i = 2 . By cascading system (1) with the adaptive internal model(12) and the control input (11), we obtain a cascaded systemof the form ˙ ρ = 0˙ w = s ( ρ, w )˙ z = f ( ρ, w, z ) + f ( 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 )˙ˆ σ = − ℓ d +1 g d +1 ( v u + ˆ ξ )˙ x = q ( ρ, w, z, x ) + b ( ρ, w, z, x )( C d η + v u ) (13)It is observed that system (13), viewing v u as control inputand x as output, has a well-defined relative degree one, andthe corresponding extended zero dynamics, forced by v u = − C d η − q ( ρ, w, z, b ( ρ, w, z, , (14)can be given by ˙ ρ = 0˙ w = s ( ρ, w )˙ z = f ( 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 (cid:18) − C d η − q ( ρ, w, z, b ( ρ, w, z,
0) + ˆ ξ (cid:19) ˙ˆ σ = − ℓ d +1 g d +1 (cid:18) − C d η − q ( ρ, w, z, b ( ρ, w, z,
0) + ˆ ξ (cid:19) (15)By simple calculations, it is observed that under Assump-tions 1 and 3, the adaptive controller (11)-(12) fulfills the internal model property , relative to the set Z c . Therefore,in light of previous analysis, according to [6], the desiredadaptive output regulation problem can be solved by theadaptive controller (11)-(12) with the residual control v u having the form v u = − κx , if the extended zero dynamics (15) can be shown to to possess an asymptotically (locallyexponentially) stable compact attractor. Remark 6:
As will be shown in next subsection, (12)contains an extended state observer, i.e., the ( ˆ ξ, ˆ σ ) dynamics,in which ˆ σ denotes the extra estimate. Using this extraestimate, we are able to take advantage of the nonlinearparameterization structure given in Assumption 4, which thusenables the identifier ˆ θ -dynamics to achieve an asymptoticestimate of the uncertain parameters θ . B. Stability Analysis of Extended Zero Dynamics (15)
In the previous subsection, with the design of (12) forsystem (13), we obtain an extended zero dynamics (15),whose stability analysis will be presented in the sequel.As before, we write z = ( ρ, w, z ) . Consider the changeof coordinates ˜ η = η − τ ( z ) . This, recalling (5), transforms(15) to the form ˙ 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 − ˆ ξ )˙ˆ σ = ℓ d +1 g d +1 (˜ η − ˆ ξ ) (16)where ς ( z ) = A d τ ( z ) + B d φ ( θ, τ ( z )) − ∂τ ( z ) ∂w s ( w ) − ∂τ ( z ) ∂z f ( z ) is a term which vanishes in Z c by Assumption 3.Let ς i ( z ) denote the i -th element of the vector ς ( z ) , andthen set ξ := col ( ξ , . . . , ξ d ) with ξ = ˜ η ξ = ˜ η + ς ( z ) ξ i = ˜ η i + P i − j =1 L i − j − f ς j +1 ( z ) + ς i − ( z ) , ≤ i ≤ d with L denoting the Lie derivative, which also suggests that ˜ η = ξ − ¯ ς ( z ) for an appropriately defined function ¯ ς ( z ) ,satisfying ¯ ς ( z ) = 0 for all z ∈ Z c .In view of the previous analysis, (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 − ˆ ξ )˙ˆ σ = ℓ d +1 g d +1 ( ξ − ˆ ξ ) (17)where ν ( z ) = d − X i =1 L d − i f ς i ( z )+ ς d ( z ) . It is noted that ν ( z ) = 0 for all z ∈ Z c and there exists a constant a > such thatfor all z ∈ Z , | ν ( z ) | ≤ a . (18)t then can be seen that the ( ˆ ξ, ˆ σ ) dynamics in (17) can beviewed as an extended-state observer of the ξ dynamics, withobserver states ˆ ξ and ˆ σ respectively being used to estimatethe variables ξ , and the “perturbation” term σ := φ (ˆ θ, ξ + τ ( z ) − ¯ ς ( z )) − φ ( θ, τ ( z )) + ν ( z ) . (19)This observation thus motivates us to analyse the asymptoticstability of the extended zero dynamics (17) by using thenonlinear separation principle [1], but in the general nonequi-libirum theory.Fix all coefficients of the dead-zone function dzv ( · ) as c i > a a ,i + 2 a ,i a ǫ ,i , i = 1 , . . . , d , (20)with constants a , a ,i , a , and ǫ ,i being given by (9), (18),and Assumption 4.(i).With the above choice of c i ’s in mind, to apply thenonlinear separation principle to analyze the asymptoticstability of system (17), it is natural to first consider the 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 characterized as below. Lemma 1:
Suppose Assumptions 1, 3, and 4 hold. Thenthe set A a := Z c × { } × { θ } is asymptotically stable underthe flow (21), for every initial condition ( z , ξ , ˆ θ ) rangingover the set M := Z × R d × R p . Proof:
The proof is given in Appendix A.
Lemma 2:
Suppose Assumptions 2, 3, and 4 hold. Thenthe set A a under the flow (21) is locally exponentially stable. Proof:
The proof is given in Appendix B.By setting ˜ θ = ˆ θ − θ and letting z ( t ) denote the solutionof system ˙ z = f ( z ) with initial condition ranging over Z ,system (21) can be rewritten as a nonautonomous 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)With Lemma 1, and recalling Assumption 1 and thefact that z ( t ) are captured by the compact set Z , we canconclude the following result, whose proof can be obtainedby simply adapting the proof of [3, Theorem 3.1] to thepresent framework and is thus omitted. Corollary 1:
Suppose Assumptions 1, 3, and 4 hold. Thezero equilibrium of nonautonomous system (22) is uniformlyasymptotically stable, for all z ∈ Z .By letting x a = col ( ξ, ˜ θ ) , system (22) can be compactlyrewritten as ˙ x a = f a ( z ( t ) , x a ) (23)where f a ( z ( t ) , x a ) is continuously differentiable. It is worthnoting that by constructing functions β ( · ) and φ ( · , · ) to be globally bounded and Lipschitz, and since z ( t ) ∈ Z for all t ≥ , there exists a ̟ > such that (cid:13)(cid:13)(cid:13)(cid:13) ∂ f a ( z ( t ) , x a ) ∂ x a (cid:13)(cid:13)(cid:13)(cid:13) ≤ ̟ . According to [11, Theorem 4.16], this property, together withLemma 1, indicates that there exist a smooth, positive definitefunction W a ( t, x a ) , and class K ∞ functions α , α , α , and α such that α ( | x a | ) ≤ W a ( t, x a ) ≤ α ( | x a | ) ∂W a ∂t + ∂W a ∂ x a ˙ x a ≤ − α ( | x a | ) (cid:12)(cid:12)(cid:12)(cid:12) ∂W a ∂ x a (cid:12)(cid:12)(cid:12)(cid:12) ≤ α ( | x a | ) . (24)With this in mind, we turn to system (17) and define therescaled estimate errors as ˜ ξ = ℓ d +1 Λ − ℓ ( ξ − ˆ ξ ) , ˜ σ = σ − ˆ σ . (25)Taking time derivatives of these errors along (17) yields ˙˜ ξ = ℓ ( A d − G C d ) ˜ ξ + ℓ B d ˜ σ (26)and ˙˜ σ = − ℓg d +1 ˜ ξ + ˙ φ (ˆ θ, ξ + τ ( z ) − ¯ ς ( z )) − ˙ φ ( θ, τ ( z ))= − ℓg d +1 ˜ ξ + ∆ e (27)where the term ∆ e is defined 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 noting that ∆ e = 0 for all ( z , x a ) ∈ A a and e = 0 , and due to the presence of saturation functions, | ∆ e | is bounded for all bounded ( z , x a ) , uniformly in ( ˜ ξ, ˜ σ ) .Putting these equations together and letting e = col ( ˜ ξ, ˜ σ ) ,we can compactly obtain ˙ e = ℓF e e + B d +1 ∆ e (29)where F e is defined by F e = (cid:18) − G I d − g d +1 (cid:19) 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, given any compact set C x ∈ R p + d , choose c suchthat A c ⊃ C x with A c = { x a : α ( | x a | ) ≤ c } , and let Ω c +1 = { x a : α ( | x a | ) ≤ max x a ∈A c α ( | x a | ) + 1 } . t is clear that A c ⊂ Ω c +1 . Then, choose the saturation levelsas l i = max x a ∈ Ω c +1 | λξ i + ξ i +1 | + 1 , ≤ i ≤ d − l d = max x a ∈ Ω c +1 | λξ d | + 1 l d +1 = max ( z , x a ) ∈ Z × Ω c +1 (cid:12)(cid:12)(cid:12) φ (ˆ θ, ϕ η ( ξ + τ ( z ) − ¯ ς ( z ( t )))) − φ ( θ, τ ( z )) | + 1 . (31)With the above choice of l i ’s, it can be observed that for all ( z , x a ) ∈ Z × Ω c +1 , Ξ( z , x a , e ) is bounded uniformly in e ,and Ξ( z , x a ,
0) = 0 .Therefore, from the standard arguments of nonlinear sep-aration principles [1], semiglobal asymptotic stability of theclosed-loop system (17) can be easily concluded as below.
Proposition 1:
Suppose Assumptions 1, 2 and 4 hold.Given any compact sets C x ∈ R q + d and C e ∈ R d +1 , andchoosing g i ’s such that matrix F e is Hurwitz, there exists ℓ ∗ > such that for 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 locally exponentially stable, andasymptotically stable for all initial conditions in Z ×C x ×C e . C. Adaptive Output Regulation
We now turn to the extended system (13). As mentionedbefore, this system, viewed as a system with input v u andoutput y e = x , has relative degree one. By taking the changeof variables ˇ ξ := ˆ ξ + Λ ℓ G Z x b ( ρ, w, z, s ) ds ˇ σ := ˆ σ + ℓ d +1 g d +1 Z x b ( ρ, w, z, s ) ds system (13) can be rewritten in “normal form” as ˙ ρ = 0˙ w = s ( ρ, w )˙ z = f ( ρ, w, z ) + f ( 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 ( ρ, w, z, ˆ θ, η, ˇ ξ, ˇ σ, x ) x ˙ˆ θ = β ( η ) sat d +1 (ˇ σ ) + µ ( 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 (cid:18) C d η + q ( ρ, w, z, b ( ρ, w, z, − ˆ ξ (cid:19) + µ ( ρ, w, z, ˆ θ, η, ˇ ξ, ˇ σ, x ) x ˙ˇ σ = ℓ d +1 g d +1 (cid:18) C d η + q ( ρ, w, z, b ( ρ, w, z, − ˇ ξ (cid:19) + µ ( 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, b ( ρ, w, z, b ( ρ, w, z, x ) (cid:18) C d η + q ( ρ, w, z, b ( ρ, w, z, (cid:19) + b ( ρ, w, z, x ) v u (32)in which µ i ( · ) , i = 1 , . . . , are continuous functions.Bearing in mind the results in Proposition 1 and recalling[9, Proposition 4], we can choose v u for system (32) as theform v u = − κx , (33) and the following conclusion can be easily made. Proposition 2:
Consider system (1) with exosystem (2)and controller (4) having the form (12) and (33). SupposeAssumptions 1-4 hold. Given any compact sets C z ⊂ R n , C x ⊂ R and C x c ⊂ R d + q +1 , and choosing g i ’s such thatmatrix F e is Hurwitz, there exist ℓ ∗ > and a positivefunction κ ∗ ( · ) such that for all ℓ > ℓ ∗ and κ ≥ κ ∗ ( ℓ ) , theresulting trajectories of the closed-loop system are boundedand x ( t ) → as t → ∞ , with the domain of attraction thatcontains C z × C x × C x c .IV. A N ILLUSTRATIVE E XAMPLE
Consider the output regulation problem for the nonlinearsystem ˙ ζ = ρζ − ( ζ + w ) + w + ζ ˙ ζ = ζ ˙ ζ = − w + ζ ζ + uy e = ζ (34)in which ( ζ , ζ ) are measurable states, and the exogenousvariables w , w are generated by an uncertain nonlinearoscillator ˙ w = w ˙ w = − w + (1 − w ) w ρw (35)where ρ is a constant unknown parameter satisfying ρ ∈ [ − . , . . The trajectories of (35) at each ρ ∈{− . , , . } are given in Fig.2. It can be seen that for any ρ ∈ [ − . , . there exists a limit cycle, that is an invariantset W for (35), and particularly W ⊂ { ( w , w ) : | w i | ≤ , i = 1 , } . -5 0 5-4-2024 =0.2 -5 0 5-4-2024 =0 -5 0 5-4-2024 =-0.2 Fig. 2. Phase portrait of (35) at ρ = 0 . , ρ = 0 and ρ = − . . Note that, when w = w ≡ , system (34), regarded as asystem with input u and output y e , has relative degree anda zero dynamics as ˙ ζ = ρζ − ζ , whose zero equilibriumpoint is unstable when ρ > and stable when ρ ≤ . Thus,the conventional methods [14], [3] based on equilibriumtheory cannot be applied.Following the design paradigm proposed in this paper, wefirst set z = ζ , z = ζ and x = ζ + ζ , which reduces therelative degree of system (34) to one, leading to the form ˙ z = ρz − ( z + w ) + w + z ˙ z = − z + x ˙ x = − w − z + z z + x + u . (36)he zero dynamics of system (35)-(36) with respect to input u and output x , forced by the control input u = w + z − z z , can be described as ˙ ρ = 0˙ w = w ˙ w = − w + (1 − w ) w ρw ˙ z = ρz − ( z + w ) + w + z ˙ z = − z . 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 = 0 . In view of this, we proceed to verify Assumptions3 and 4. Observe that in the present setting, Assumption 3is fulfilled with the map τ := ( τ , τ ) = ( w , w ) satisfyingthe equations ˙ τ = τ , ˙ τ = φ ( θ, τ ) where θ = ρ and function φ ( θ, τ ) = − ϕ s ( τ ) + (1 − ϕ s ( τ )) ϕ s ( τ )1 + ϕ s ( θ ) ϕ s ( τ ) with ϕ s ( τ i ) = τ i , for | τ i | ≤ ϕ s ( θ ) = θ , for | θ | ≤ . . Moreover, by choosing β ( τ ) = (1 − ϕ s ( τ )) ϕ s ( τ ) ϕ s ( τ ) , itcan be easily found that the function β ( τ ) φ ( θ, τ ) is strictlydecreasing in | θ | ≤ . , for all τ ∈ W . In this way,Assumption 4 is also fulfilled.Therefore, the adaptive internal model-based regulator (12)and (11) can be employed to handle the nonlinear outputregulation problem at hand. Figure 3 shows simulation resultsfor ρ = 0 . , and the design parameters ℓ = 10 and κ = 30 . Itdemonstrates that the regulated output y e converges to zeroasymptotically and the parameter estimate ˆ θ converges to thereal value. Time(s) -0.4-0.200.20.40.60.8
Regulated OutputParameter Estimate
Fig. 3. Trajectories of regulated output y e ( t ) and parameter estimate ˆ θ ( t ) . V. C
ONCLUSION
This paper studies the adaptive output regulation problemfor a class of nonlinear systems using the general nonequilib-rium theory developed in [6]. By incorporating an extended-state observer into the adaptive internal model, a new ap-proach is proposed to deal with adaptive nonlinear regulation,which allows for more general nonlinearly parameterizedimmersion conditions. A
PPENDIX
A. Proof of Lemma 1
The proof mainly follows the nonequilibrium theory de-veloped in [6]. First, we will show that the trajectories ofsystem (21) are bounded, i.e. there is no finite-time escape.By Assumption 1 and the choice of λ > , it can be easilyseen that z ( t ) and ξ ( t ) are bounded. To show ˆ θ ( t ) is alsobounded, we let ˆ θ i denote the i -th element of vector ˆ θ andchoose V ˆ θ,i = | ˆ θ i | , i = 1 , . . . , p . Taking the time derivativeof V ˆ θ,i along the bottom equation of (21) 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 a ,i + a ,i a ) | ˆ θ i | where (9) and (18) are used to get the inequality.If | ˆ θ i | ≥ a ,i + ǫ ,i , then ˙ V ˆ θ,i ≤ − c i ˆ θ i [ˆ θ i − ( a ,i + ǫ ,i )] + (2 a a ,i + a ,i a ) | ˆ θ i |≤ − ǫ ,i ( c i − a a ,i +2 a ,i a ǫ ,i ) | ˆ θ i | . From (20), we can conclude that ˙ V ˆ θ,i < for all | ˆ θ i | ≥ a ,i + ǫ ,i with i = 1 , . . . , d . This then indicates that in thepresence of dead-zone functions dzv (ˆ θ ) , the trajectory ˆ θ ( t ) of (21) is globally uniformly bounded, and will enter andstay inside the closed cube B q .With the boundedness of trajectories of system (21), it thuscan be deduced that there exists an ω -limit set, denoted by ω ( M ) , of M = Z × R d × R q under the flow of (21), whichis nonempty, compact and invariant, and uniformly attractsall trajectories of (21) with initial conditions in M .Now we proceed to investigate the structure of this ω -limit set ω ( M ) . Due to the special triangular structure of(21), and by Assumption 1 and the fact that the ξ -subsystemis globally exponentially stable at the origin, it immediatelyfollows that on the points of ω ( M ) , necessarily z ∈ Z c and ξ = 0 . As a consequence, on the ω -limit set ω ( M ) , ¯ ς ( z ) = 0 and ν ( z ) = 0 . In view of the previous analysis, to specify thestructure of ω ( M ) , we still need to determine the value of ˆ θ . On the other hand, when proving the boundness of ˆ θ ( t ) ,we have shown that ˆ θ ( t ) will enter and stay inside the closedcube B q . Thus, by recalling that Z c is invariant under (5),the value of ˆ θ on ω ( M ) is determined by the properties ofthe system ˙ z = f ( z )˙ˆ θ = β ( τ ( z ))[ φ (ˆ θ, τ ( z )) − φ ( θ, τ ( z ))] − dzv (ˆ θ ) (37)where the initial condition z ∈ Z c and ˆ θ ∈ B q . It is notedthat ˆ θ ( t ) ∈ B q for all t ≥ under (37).Then, choose V ˜ θ = | ˜ θ | with ˜ θ = ˆ θ − θ , whose timederivative along (37) can 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 ) ≥ for all ˆ θ ∈ R p and ρ ∈ P . (38)his, together with the first part of Assumption 4, impliesthat under the flow (37), ˙ V ˜ θ ≤ , (39)where the equality holds 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 ) converges to some constant value ˆ θ ∞ as t goes toinfinity. By LaSalle’s invariance theorem, this ˆ θ ∞ necessarilyis 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 noted that the second of (40) indicates that dzv (ˆ θ ∞ ) = 0 .This further reduces (40) to β ( τ ( z ))[ φ (ˆ θ ∞ , τ ( z )) − φ ( θ, τ ( z ))] = 0 . By Assumption 4.(ii), we have ˆ θ ∞ = θ . This completes theproof. (cid:4) B. Proof of Lemma 2
Due to the special cascaded-structure of system (21) andsince functions β and φ are constructed to be globallyLipschitz and bounded, with the choice of λ > andAssumption 2, it is clear that the proof is completed if forany z ∈ Z c , the origin of the linear time-varying system ˙˜ θ = β ( τ ( z ( t, z ))) ∂φ ( θ, τ ( z ( t, z ))) ∂θ ˜ θ (41)with ˜ θ = ˆ θ − θ , is shown to be uniformly exponentiallystable.Since z ( t, z ) is the solution of the autonomous system(5) passing through z at t = 0 , (41) can be rewritten as acascaded autonomous system , having the form ˙ z = f ( z )˙˜ θ = β ( τ ( z )) ∂φ ( θ, τ ( z )) ∂θ ˜ θ . (42)We then calculate the derivative of V ˜ θ as ˙ V ˜ θ = ˜ θ ⊤ β ( τ ( z )) ∂φ ( θ, τ ( z )) ∂θ ˜ θ ≤ where the inequality is obtained by using Assumption 4.(i).Then, similar to the proof of Lemma 1, by LaSalle’s in-variance theorem and Assumption 4.(ii), we can concludethat system (42) is uniformly asymptotically stable at the set Z c × { } , for any initial condition ( z , ˜ θ ) ∈ Z c × R q . Inother words, for any ε > and ( z , ˜ θ ) ∈ Z c × R q , thereexists T ε > such that | ˜ θ ( t ) | = dist (cid:16) ( z ( t ) , ˜ θ ( t )) , Z c × { } (cid:17) ≤ ε for all t ≥ T ε . (43)Therefore, the zero equilibrium of the linear time-varyingsystem (41) is uniformly asymptotically stable, which alsoindicates the desired exponential stability. (cid:4) R EFERENCES[1] A. Isidori.
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