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Nonadaptive Learning in Robust Nonlinear Output Regulation
Shimin Wang, Martin Guay, Richard D. Braatz
Intelligence
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Summary
This paper proposes a robust nonadaptive control framework for general nonlinear systems with arbitrarily high relative degree in an output-feedback setting. The method combines an input-driven filter, a generic internal model, and a recursive backstepping law to achieve global asymptotic regulation by stabilizing an augmented error system. Unlike adaptive schemes, it avoids linearly parameterized regressors and specific Lyapunov derivative constraints, relying instead on verifiable inequalities for gain selection. The approach is validated on a controlled Duffing system.
Entities (8)
Relation Signals (8)
Nonadaptive Learning → comprises → Input-Driven Filter
confidence 95% · We develop a nonadaptive design that combines an input-driven filter and a generic internal model
Nonadaptive Learning → comprises → Generic Internal Model
confidence 95% · We develop a nonadaptive design that combines an input-driven filter and a generic internal model
Nonadaptive Learning → comprises → Recursive Backstepping Law
confidence 95% · We develop a nonadaptive design that combines an input-driven filter and a generic internal model with a recursive backstepping law
Nonadaptive Learning → solves → Robust Nonlinear Output Regulation
confidence 95% · This paper considers robust nonadaptive regulation for general nonlinear systems
Nonadaptive Learning → recastsproblemas → Robust Input-to-State Stabilization
confidence 93% · thereby recasting the regulation problem as the robust input-to-state stabilization of an augmented error system.
Nonadaptive Learning → demonstratedon → Duffing System
confidence 92% · The effectiveness of the theoretical results is demonstrated using a benchmark controlled Duffing system.
Nonadaptive Learning → contrastswith → Adaptive Schemes
confidence 90% · Unlike adaptive schemes, the proposed method does not rely on linearly parameterized regressors
Generic Internal Model → →
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Abstract
Abstract:This paper considers robust nonadaptive regulation for general nonlinear systems in an output-feedback setting with arbitrarily high relative degree. We develop a nonadaptive design that combines an input-driven filter and a generic internal model with a recursive backstepping law, thereby recasting the regulation problem as the robust input-to-state stabilization of an augmented error system. Unlike adaptive schemes, the proposed method does not rely on linearly parameterized regressors and does not require the construction of Lyapunov functions having merely nonpositive derivatives. Under standard assumptions on the exosystem, including purely imaginary and simple eigenvalues, together with a minimum-phase input-to-state stability condition on the internal dynamics, we establish global asymptotic regulation and derive explicit, verifiable inequalities for selecting the design gains. The resulting nonadaptive framework guarantees convergence of the estimation and tracking errors even when the controlled-system dynamics are complex or only partially known. The effectiveness of the theoretical results is demonstrated using a benchmark controlled Duffing system.
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- Source: https://arxiv.org/abs/2608.17262v1
- Canonical: https://arxiv.org/abs/2608.17262v1
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NONADAPTIVE LEARNING IN ROBUST NONLINEAR OUTPUT REGULATION ∗ SHIMIN WANG † , MARTIN GUAY ‡ ,ANDRICHARD D. BRAATZ § Abstract.This paper considers robust nonadaptive regulation for general nonlinear systems in an output-feedback setting with arbitrarily high relativedegree. We develop a nonadaptive design that combines an input-driven filter and a generic internal model with a recursive backstepping law, thereby recasting the regulation problem as the robust input-to-state stabilization of an augmented error system. Unlike adaptive schemes, the proposed methoddoes not rely on linearly parameterized regressors and does not require the construction of Lyapunov functions having merely nonpositive derivatives. Under standard assumptions on the exosystem,including purely imaginary and simple eigenvalues, together with a minimum-phase input-to-state stability condition on the internal dynam- ics, we establish global asymptotic regulation and derive explicit, verifiable inequalities for selecting the design gains. The resulting nonadaptive framework guarantees convergence of the estimation and tracking errors even when the controlled-system dynamics are complex or only partially known. The effectiveness of the theoretical results is demonstrated using a benchmark controlled Duffing system. Key words.nonlinear output regulation, nonadaptive learning, robust control, output feedback, internal model, data-driven control AMS subject classifications.93C10, 93B52, 93D15, 93C40 1. Introduction.Output regulation is an essential issue in the design of control systems [15, 32, 28]. It aims to have a system track a class of desired signals while rejecting the external disturbance [24, 15, 11]. In the linear case, regulation reduces to a pole-assignment problem because the steady-state trackingerror depends lin- early on the exogenous signals [10]. For nonlinear plants, however, the steady-state error is a nonlinear function of the exogenous signals [14, 17], and linear feedforward or linear internal-model designs fail in the presence of nonlinearitiesor parametric uncertainties. To address general nonlinear output regulation problems, variousinternal-model structures have been developed over the past decades. Canonical linear internal mod- els have been effectively used with adaptive methods to handle uncertain linear exosys- tems [27, 29, 11, 26] and more recently for disturbance rejectionin Euler–Lagrange systems [35]. For nonlinear exosystems generating non-sinusoidalsignals without uncertainty, nonlinear internal models were introduced in [2] and [4].A significant milestone is the generic internal model proposed in [24], which removes structural assumptions on the steady-state input and accommodates both minimum-phase and non-minimum-phase systems. A broader overview can be found in [13]. Adaptive internal-model-based methods provide a mechanism to address para- metric uncertainties but suffer from structural limitations. Specifically, they require Lyapunov functions with non-positive derivatives and rely on explicitregressors de- termined by the system and exosystem structure [9, 21]. Such designs offer weaker robustness, as highlighted by counterexamples demonstrating that boundedness may fail even under small external inputs [3], and they apply primarily to systems with ∗ This research was supported by the U.S. Food and Drug Administration under the FDA BAA- 22-00123 program, Award Number 75F40122C00200. † School of Data Science, Lingnan University, Hong Kong, and Massachusetts Institute of Tech- nology, Cambridge, MA 02139, USA. Corresponding author. ‡ Department of Chemical Engineering, Queen’s University, Kingston, ON K7L 3N6, Canada (martin.guay@queensu.ca). § Massachusetts Institute of Technology, Cambridge, MA 02139, USA (braatz@mit.edu). 1 arXiv:2608.17262v1 [eess.SY] 18 Aug 2026 2S. WANG, M. GUAY, AND R. D. BRAATZ parametric uncertainty in a suitable regression form. These challenges restrict the applicability of adaptive approaches in general nonlinear output regulation. Nonadaptive methods alleviate these difficulties by avoiding explicit parameter adaptation [16, 25], but the generic internal-model approach still depends on an ex- plicit nonlinear continuous mapping that characterizes the steady-state input. This mapping is assumed to exist [19] but is generally unknown, and no general ana- lytic expression is available.Existing solutions often rely on numerical least-squares techniques to approximate this function [24, 25], which introduces additional tuning requirements and may limit robustness or generality. Recent work [34] addresses this issue for nonlinear output regulation problems of relative degree one by constructing the steady-state mapping nonparametrically, without requiring parametric regressors or restrictive exosystem assumptions. Motivated by these developments, the present article addressesthe nonlinear ro- bust output regulation problem for general nonlinear output-feedback systems with arbitrary relative degree by employing nonadaptive learning methods. This setting is considerably more challenging than the relative-degree-one case treated in [34], as the controller no longer has access to derivatives of the regulated output and the inter- nal model and stabilization structures must be redesigned accordingly. The proposed approach differs from existing techniques based on adaptive control [33, 21, 23, 6], which typically require explicit regressors, parameter adaptation laws, and Lyapunov functions with non-positive derivatives, whose applicability is often restricted to para- metric uncertainty structures. Output regulation for uncertainnonlinear systems with higher relative degree remains an active research topic [33, 5], with recent work focus- ing on low-complexity, approximation-free output-feedback designs capable of achiev- ing prescribed transient and steady-state performance [5]. In contrast, the nonadap- tive learning framework developed in this article addresses the global robust output regulation problem through the construction of a linear generic internal model based on the existence of a continuous nonlinear steady-state mapping.This construction transforms the nonlinear robust output regulation problem into a robust nonadap- tive stabilization problem for an augmented system with Input-to-State Stable (ISS) dynamics. The rest of this paper is organized as follows. Section 2 introduces some standard assumptions and lemmas. Section 3 is devoted to the presentation of the main results. This is followed by simulation examples in Section 4 and brief conclusions inSection 5. Notation:k·kis the Euclidean norm.Id:R→Ris an identity function. For X i ∈R n i ×m withi= 1,...,N, let col(X 1 ,...,X N ) = [X ⊤ 1 ,...,X ⊤ N ] ⊤ and diag(X 1 ,...,X N ) = X 1 . . . X N . A functionα:R ≥0 →R ≥0 is of classKif it is continuous, positive definite, and strictly increasing. The notationK ∞ identifies the subclasses of unboundedKfunctions. For functionsf 1 (·) andf 2 (·) with compatible dimensions, their compositionf 1 (f 2 (·)) is denoted byf 1 ◦f 2 (·). For a matrixX, Adj[X] denotes its adjugate matrix. 2. Problem Formulation and Assumptions.We consider a class of nonlinear control systems of the form: ̇z=f(z,y,v,w),(2.1a) ̇x=A c x+g(z,y,v,w) +B c bu,(2.1b) NONADAPTIVE LEARNING IN ROBUST NONLINEAR OUTPUT REGULATION3 y=C c x,(2.1c) e=y−h(v,w),(2.1d) where (z,x)∈R n z ×R r is the vector of state variables withr≥1 with az-subsystem of fully nonlinear dynamics and ax-subsystem of partially structured linear dynamics with an additional nonlinear termg(z,y,v,w) = col(g 1 (z,y,v,w),...,g r (z,y,v,w)), y∈Ris the output of the system,h(v,w)is explicitly thereference outputgenerated by the exosystem (2.2),e∈Ris the tracking error of the system,u∈Ris the input,w∈W⊂R n w is an uncertain parameter vector withWbeing an arbitrarily prescribed subset ofR n w containing the origin,bis a positive constant, the functions h(·),f(·),g i (·) are globally defined and sufficiently smooth and satisfyf(0,0,0,w) = 0, andg i (0,0,0,w) = 0 for allw∈W, andv(t)∈R n v is an exogenous signal representing the reference input and disturbance, which is generated by the exosystem ̇v=S(σ)v,(2.2) whereσ∈S⊂R n σ represents the uncertainties in the exosystem withS(σ) being a constant matrix. The matricesA c ∈R r×r ,C c ∈R 1×r , andB c ∈R r have the form A c = 0I r−1 00 , C ⊤ c = col(1,0 r−1 ), B c = col(0 r−1 ,1), whereI r−1 and0 r−1 = col(0,...,0) are the identity matrix and zero vector ofr−1 dimension, respectively. The nonlinear robust output regulation problem in this article is formulated below Problem2.1.Given the nonlinear system(2.1)–(2.2)and any compact subsets S∈R n σ ,W∈R n w , andV∈R n v withWandVcontaining the origin, design a control law such that for all initial conditionsv(0)∈V,σ∈Sandw∈W, and any initial statescol(z(0),x(0))∈R n z +r , the solution of the closed-loop system exists and is bounded for allt≥0, andlim t→∞ e(t) = 0. Before proceeding with the main results, we state the assumptions. Assumption2.2. For allσ, all the eigenvalues ofS(σ) are simple with zero real parts. Assumption2.3. There exists a globally defined smooth functionz(v,w,σ) : R n v ×R n w ×R n σ 7−→R n z such that ∂z(v,w,σ) ∂v S(σ)v=f(z(v,w,σ),h(v,w),v,w)(2.3) for all (v,w,σ)∈V×W×Swithz(0,w,σ) = 0. Remark2.4. The termz(v,w,σ) denotes the steady state of thez-subsystem (2.1a), which is the solution to the associated regulator equation (2.3). Assumption 2.2 is a standard assumption appearing in [16, 12, 29, 34] which limits the exogenous signalvgenerated in (2.2) to be arbitrarily large constant signals and multi-tone sinusoidal signals with arbitrarily unknown initial phases, amplitudes and arbitrarily known frequencies. Assumption2.5 (Minimum-phase condition). The translated inverse system ̇ ̄z=f( ̄z+z(μ),e+h(v,w),v,w)−f(z(μ),h(v,w),v,w)(2.4) 4S. WANG, M. GUAY, AND R. D. BRAATZ is input-to-state stable with state ̄z=z−z(μ),μ= col(v,w,σ) and inputein the sense of [31]. In particular, there exists a continuous functionV ̄z ( ̄z) satisfying α ̄z (k ̄zk)≤V ̄z ( ̄z)≤ α ̄z (k ̄zk) for some classK ∞ functionsα ̄z (·) and α ̄z (·) such that, for anyv∈V, along the trajectories of the ̄zsubsystem, ̇ V ̄z ≤−α ̄z (k ̄zk) +γ(e), whereα ̄z (·) is some known classK ∞ function satisfying lim sup ς→0 + α −1 ̄z (ς 2 )/ς <+∞, andγ(·) is some known smooth positive definite function. Remark2.6. Assumption 2.5 guarantees that the ̄z-system (2.4) is input-to-state stable with state ̄zand inpute. In addition, by the changing supply function technique [30], there exists a continuous function ̄ V ̄z ( ̄z) satisfyingα ̄z (k ̄zk)≤ ̄ V ̄z ( ̄z)≤ ̄α ̄z (k ̄zk), for some classK ∞ functionsα ̄z (·) and ̄α ̄z (·) such that for any smooth function ∆ ̄z ( ̄z)>0 and anyμ∈V×W×S, the time derivative of ̄ V ̄z ( ̄z) along the trajectory (2.4) satisfies the inequality ̇ ̄ V ̄z ( ̄z)≤−∆ ̄z ( ̄z)k ̄zk 2 +δ ̄z γ ̄z (e)e 2 , whereδ ̄z andγ ̄z (·) are some positive constant and positive smooth function, respec- tively. Under Assumption 2.3, there exists globally defined smooth functions x(v,w,σ),z(v,w,σ),u(v,w,σ)withx 1 (v,w,σ) =h(v,w) satisfying ∂x(μ) ∂v S(σ)v=A c x(μ) +g(z(μ),x 1 (μ),μ) +B c bu(μ), u(μ) =b −1 × ∂x r (μ) ∂v S(σ)v−g r (z,x 1 (μ),μ) , withx(μ) = col(x 1 (μ),...,x r (μ)). For convenience, letx≡x(μ),z≡z(μ) and u≡u(μ). The control system (2.1) has relative degreer≥2. Motivated by [18], we define the input-driven filter ̇ ˆx=Aˆx+B c u,(2.5) where ˆx∈R r is an estimate ofxin system (2.1), andA=A c −λC c withλ= col(λ 1 ,...,λ r ) such thatAis Hurwitz. We perform the coordinate transformation ̃x i =b −1 x i −ˆx i ,i= 1,...,r, to obtain ̇z=f(z,y,v,w),(2.6a) ̇ ̃x=A ̃x+b −1 ×(λy+g(z,y,v,w)),(2.6b) ̇y=bˆx 2 +b ̃x 2 +g 1 (z,y,v,w),(2.6c) ̇ ˆx i = ˆx i+1 −λ i ˆx 1 , i= 2,...,r−1,(2.6d) ̇ ˆx r =u−λ r ˆx 1 ,(2.6e) whereg(z,y,v,w) = col(g 1 (z,y,v,w),...,g r (z,y,v,w)) and ̃x= col( ̃x 1 ,..., ̃x r ). Assumption2.7. The functionu(v,σ,w) is a polynomial invwith coefficients depending onwandσ. NONADAPTIVE LEARNING IN ROBUST NONLINEAR OUTPUT REGULATION5 Lemma2.8.Suppose Assumption 2.3 holds. If the functionu(v,σ,w)is a poly- nomial invwith coefficients depending onwandσ, then there exist a dimen- sionn τ , a vector of monomialsτ u (v)∈R n τ , and matricesΦ u (σ)∈R n τ ×n τ and Γ u (σ,w)∈R 1×n τ such that: ∂τ u (v) ∂v S(σ)v= Φ u (σ)τ u (v),(2.7) u(μ) = Γ u (μ)τ u (v), where all eigenvalues ofΦ u (σ)have zero real part. Furthermore, if the Hurwitz matrix Ashares no common eigenvalues withΦ u (σ), there exists a unique matrixP u ∈R r×n τ such thatˆx(μ) : =P u τ u (v)satisfies the regulator equation: (2.8) ̇ ˆx(μ) =Aˆx(μ) +B c u(μ). Proof.Sinceu(v,σ,w) is a polynomial inv, there exists an integern u such that ucan be expanded as: u(v,σ,w) = n u X l=1 U l (σ,w)v [l] = U 1 (σ,w)·U n u (σ,w) | z Γ u (σ,w) col(v [1] ,...,v [n u ] ) where Γ u (σ,w) is a suitable coefficient matrix andv [l] represents the vector of mono- mials of degreel: v [l] = col(v l 1 ,v l−1 1 v 2 ,...,v l−1 1 v n u ,...,v l n u ), l= 1,...,n u . Letτ u (v) = col(v [1] ,...,v [n u ] ). Following Chapter 4 of [11], the time derivative of the monomial vector satisfies linear dynamics driven by a matrix Φ u (σ) with eigenvalues on the imaginary axis, proving (2.7). SinceAis Hurwitz and Φ u (σ) has eigenvalues with zero real parts, they share no common eigenvalues. Thus, the generalized Sylvester equation: P u Φ u (σ) =AP u +B c Γ u (μ) admits a unique solutionP u . Definingˆx(μ) =P u τ u (v) and taking its time derivative along the trajectories of the exosystem yields: ̇ ˆx(μ) =P u ∂τ u (v) ∂v S(σ)v =P u Φ u (σ)τ u (v) = (AP u +B c Γ u )τ u (v) =Aˆx(μ) +B c u(μ). This completes the proof. LetE(μ) =b −1 x(μ)−ˆx(μ); then the regulator equation solution associated with the composite systems (2.2) and (2.6) is z(μ),E(μ),y(μ),ˆx(μ),u(μ). 6S. WANG, M. GUAY, AND R. D. BRAATZ Remark2.9. The second element of the vectorˆx(μ) = col(ˆx 1 (μ),...,ˆx r (μ)) with μ= col(v,σ,w) is denoted by ˆ x 2 (μ). Letˆxbe the steady state associated with (2.5) and,ˆx(μ), the solution to the regulator equation (2.8). From [21], under Assumptions 2.2 and 2.7, for the functionˆx 2 (v,σ,w), there exist integersn >0 such thatˆx 2 (v,σ,w) can be expressed as ˆx 2 (v(t),σ,w) = X n j=1 C j (v(0),w,σ)e ıˆω j t (2.9) for some functionsC j (v(0),w,σ)∈C, whereıis the imaginary unit and ˆω j are distinct real numbers for 1≤j≤n. The minimal zeroing polynomial ofˆx 2 (v(t),σ,w) is Π n j=1 (s+ıˆω j ). Assumption2.10. The initial conditionv(0)∈Vand any parameter vectorsw∈ Wandσ∈Ssatisfy the coefficients satisfyC j (v(0),w,σ)6= 0 for all 1≤j≤n. Remark2.11 (On Assumption 2.10). Assumption 2.10 imposes a nondegeneracy condition on the coefficientsC j (v(0),w,σ) appearing in the steady-state generator. Its purpose is to ensure that the nonlinear mapping used in the construction of the generic internal model is well defined and does not encounter singularities for any admissible exosystem initial condition or parameter values. This prevents the gen- eralized Sylvester-type equation from losing rank and guaranteesthe existence and uniqueness of the steady-state input required by the internal model. Similar nonde- generacy assumptions appear in classical nonlinear output regulation theory, e.g., in the solvability conditions of the nonlinear regulator equations and steady-state map- pings in [15, 11] and [24], as well as in our previous nonparametric framework for relative-degree-one systems [34]. Assumption 2.10 should therefore be understood as a mild structural condition that avoids singularities rather than a stability or ISS requirement. 2.1. Generic internal model design.Under Assumptions 2.2 and 2.7, there exists a positive integernsuch thatˆx 2 (μ) satisfies, for allμ∈V×W×S, d n ˆx 2 (μ) dt n +a 1 (σ)ˆx 2 (μ) +·+a n (σ) d n−1 ˆx 2 (μ) dt n−1 = 0,(2.10) wherea 1 (σ),...,a n (σ) belong toR. Under Assumptions 2.2 and 2.7, equation (2.10) yields the polynomial ς n +a 1 (σ) +a 2 (σ)ς+·+a n (σ)ς n−1 whose roots are distinct with zero real parts for allσ∈S. Leta(σ) = col(a 1 (σ),...,a n (σ)),ξ(μ) = col ˆx 2 (μ), dˆx 2 (μ) dt ,..., d n−1 ˆx 2 (μ) dt n−1 , andξ≡ξ(μ), and define Φ(a(σ)) = 0 (n−1)×1 I n−1 −a 1 (σ)−a 2 (σ),...,−a n (σ) , Γ = 1 0·0 1×n . Then,ξ(μ), Φ(a(σ)) and Γ satisfy ̇ ξ(μ) = Φ(a(σ))ξ(μ),(2.11a) ˆx 2 (μ) = Γξ(μ).(2.11b) NONADAPTIVE LEARNING IN ROBUST NONLINEAR OUTPUT REGULATION7 System (2.11) is called a steady-state generator with output ˆx 2 as it can be used to produce the steady-state signalˆx 2 . Define the matrix pair (M,N) by M= 0 (2n−1)×1 I 2n−1 −m 1 −m 2 ,...,−m 2n ,(2.12a) N= 0·0 1 ⊤ 1×2n ,(2.12b) wherem 1 ,m 2 ,...,m 2n are chosen such thatMis Hurwitz, together with all the eigenvalues of Φ(a) being distinct with zero real parts, which results in the nonsingular matrix-valued function Ξ(a)≡Φ(a) 2n + X 2n j=1 m j Φ(a) j−1 ∈R n×n . Then, using Ξ(a)Φ(a) = Φ(a)Ξ(a) and col(Γ,ΓΦ(a),...,ΓΦ(a) n−1 ) =I n gives that Φ(a)Ξ(a) −1 = Ξ(a) −1 Φ(a) and col(Q 1 (a),Q 2 (a),...,Q n (a)) = col(ΓΞ(a) −1 ,ΓΞ(a) −1 Φ(a),...,ΓΞ(a) −1 Φ(a) n−1 ) = col(ΓΞ(a) −1 ,ΓΦ(a)Ξ(a) −1 ,...,ΓΦ(a) n−1 Ξ(a) −1 ) = col(Γ,ΓΦ(a),...,ΓΦ(a) n−1 ) | z I n Ξ(a) −1 (2.13) withQ j (a) = ΓΞ(a) −1 Φ(a) j−1 ∈R 1×n ,j= 1,...,n. Define the Hankel real matrix [1]: Θ(θ)≡ θ 1 θ 2 ·θ n θ 2 θ 3 ·θ n+1 . . . . . . . . . . . . θ n θ n+1 ·θ 2n−1 ∈R n×n , whereθ= col(θ 1 ,θ 2 ,...,θ 2n ) =Qξwith Q≡col(Q 1 ,...,Q 2n )∈R 2n×n ,(2.14) and Q j (a) = ΓΞ(a) −1 Φ(a) j−1 ∈R 1×n ,1≤j≤2n. Under Assumptions 2.2, 2.3 and 2.7, the matrices Φ(a(σ)),Q,M,N, and Γ satisfy the matrix equation (see [34]): MQ=QΦ(a(σ))−NΓ,(2.15) which is called theGeneralized Sylvester Matrix Equation. The explicit solutions can be found in [37]. As shown in [19, 25, 24] and [34], there exists a continuous nonlinearmapping χ(·) such that η ⋆ (v(t),σ,w)) = Z t −∞ e M(t−τ) Nˆx 2 (v(τ),σ,w)dτ,(2.16) 8S. WANG, M. GUAY, AND R. D. BRAATZ ˆx 2 (v(t),σ,w)) =χ(η ⋆ (v(t),σ,w)),η ⋆ ∈R n 0 , that satisfies the differential equations dη ⋆ (v(t),σ,w) dt =Mη ⋆ (v(t),σ,w) +Nˆx 2 (v(t),σ,w), ˆx 2 (v(t),σ,w)) =χ(η ⋆ (v(t),σ,w)),(2.17) namely, the steady-state generator ofˆx 2 with sufficiently large dimensionn 0 and some continuous mappingχ(·). Under Assumptions 2.2, 2.7, and 2.10, insertion of Generalized Sylvester Matrix Equation(2.15) into (2.16) and rearranging gives that η ⋆ =θ(see Lemma 3 in [34]). Then, system (2.17) leads the internal model ̇η=Mη+Nˆx 2 ,(2.18) which is the internal model associated with the signal ˆx 2 . 2.2. Error dynamics.Perform coordinate and input transformations on the composite systems (2.2), (2.6), and (2.18) to give ̄z=z−z, ̄x= ̃x−E, ̄η=η−η ⋆ −Nb −1 e,e=y−x 1 , which yields an error system in the form: ̇ ̄z= ̄ f( ̄z,e,μ),(2.19a) ̇ ̄x=A ̄x+b −1 ̄g( ̄z,e,μ) +λe ,(2.19b) ̇ ̄η=M ̄η−N ̄x 2 −b −1 e+b −1 ̄g 1 ( ̄z,e,μ) ,(2.19c) ̇e=b(ˆx 2 −ˆx 2 ) +b ̄x 2 + ̄g 1 ( ̄z,e,μ),(2.19d) ̇ ˆx i = ˆx i+1 −λ i ˆx 1 , i= 2,...,r−1,(2.19e) ̇ ˆx r =u−λ r ˆx 1 ,(2.19f) whereμ= col(σ,v,w), ̄ f( ̄z,e,μ) =f( ̄z+z,e+x 1 ,μ)−f(z,x 1 ,μ), ̄g( ̄z,e,μ) =g( ̄z+z,e+x 1 ,μ)−g(z,x 1 ,μ). It can be verified that, for allμ∈V×W×S, ̄ f(0,0,μ) = 0 and ̄g(0,0,μ) = 0. Problem 2.1 can be solved if a control law can be found to stabilize the system (2.19). Let ̄x c = col( ̄x, ̄η) and ̄ G c ( ̄z,e,μ) =b −1 col Ne−N ̄g 1 ( ̄z,e,μ), ̄g( ̄z,e,μ) +λe ; the system (2.19) can be rewritten into the form ̇ ̄z= ̄ f( ̄z,e,μ),(2.20a) ̇ ̄x c = M−NC c A c 0A | z M c ̄x c + ̄ G c ( ̄z,e,μ),(2.20b) ̇e=b(ˆx 2 −χ(η ∗ )) +b ̄x 2 + ̄g 1 ( ̄z,e,μ),(2.20c) ̇ ˆx i = ˆx i+1 −λ i ˆx 1 , i= 2,...,r−1,(2.20d) ̇ ˆx r =u−λ r ˆx 1 .(2.20e) NONADAPTIVE LEARNING IN ROBUST NONLINEAR OUTPUT REGULATION9 It can be verified that, for allμ∈V×W×S, ̄ G c (0,0,μ) =0and the matrixM c is Hurwitz. Hence, the ( ̄z, ̄x c )-subsystem in system (2.20) is in a similar form as the system (8) of [34]. As a result, the ( ̄z, ̄x c )-subsystem in system (2.20), under Assumptions 2.2, 2.3, and 2.5, admits the following properties (see Properties 1 and 2 in [34]): Property2.12.There exists a smooth input-to-state Lyapunov functionV 0 ≡ V 0 ( ̄z, ̄x c )satisfying α 0 (k ̄ Zk)≤V 0 ( ̄ Z)≤ ̄α 0 (k ̄ Zk), ̇ V 0 ≤−k ̄ Zk 2 + ̄γ ∗ ̄γ(e),(2.21) for some positive constant ̄γ ∗ and comparison functionsα 0 (·)∈K ∞ , ̄α 0 (·)∈K ∞ , and ̄γ(·)∈K ∞ with ̄ Z= col( ̄z, ̄x c ). Property2.13.There are positive smooth functionsγ g0 (·)andγ g1 (·)such that b 2 ̄x 2 2 +k ̄g 1 ( ̄z,e,μ)k 2 ≤γ g0 ( ̄ Z)k ̄ Zk 2 +e 2 γ g1 (e). Remark2.14. Since ̄ G c ( ̄z,e,μ) in (2.20b) is smooth and satisfies ̄ G c (0,0,μ) =0, for allμ∈V×W×S, by Lemma 7.8 in [11], kP c ̄ G c ̄z,e,v k 2 ≤π 1 ( ̄z)k ̄zk 2 +φ 1 (e)e 2 for some known smooth functionsπ 1 (·)≥1 andφ 1 (·)≥1, whereP c is a positive definite matrix such thatP c M c +M c P ⊤ c =−2I. By Remark 2.6, for any smooth function ∆ ̄z ( ̄z)>0, there exits a continuous function ̄ V ̄z ( ̄z) satisfyingα ̄z (k ̄zk)≤ ̄ V ̄z ( ̄z)≤ ̄α ̄z (k ̄zk) for some classK ∞ functionsα ̄z (·) and ̄α ̄z (·) such that, for any μ∈V×W×S, the time derivative of ̄ V ̄z ( ̄z) along the trajectory (2.4) satisfies ̇ ̄ V ̄z ( ̄z)≤−∆ ̄z ( ̄z)k ̄zk 2 +δ ̄z γ ̄z (e)e 2 , whereδ ̄z andγ ̄z (·) are some positive constant and positive function. LetV 0 ( ̄ Z) = ̄ V ̄z ( ̄z) + ̄x ⊤ c P c ̄x c , which satisfiesα 0 (k ̄ Zk)≤V 0 ( ̄ Z)≤ ̄α 0 (k ̄ Zk) for some classK ∞ functionsα 0 (·)∈ K ∞ and ̄α 0 (·)∈ K ∞ . By choosing ∆ ̄z ( ̄z)> π 1 ( ̄z) + 1, the time derivative ofV 0 ( ̄ Z) along the ̄ Z-subsystem of (2.20) satisfies ̇ V 0 ≤−∆ ̄z ( ̄z)k ̄zk 2 +δ ̄z γ ̄z (e)e 2 −kx c k 2 +kP c ̄ G c ̄z,e,v k 2 ≤−(∆ ̄z ( ̄z)−π 1 ( ̄z) | z >1 )k ̄zk 2 −kx c k 2 + (δ ̄z γ ̄z (e) +φ 1 (e))e 2 |z ̄γ ∗ ̄γ(e) . Sinceb 2 ̄x 2 2 +k ̄g 1 ( ̄z,e,μ)k 2 is smooth and vanishes at0when col( ̄x 2 , ̄z,e) = col (0,0,0), for allμ∈V×W×S, by using Lemma 7.8 in [11], Property 2.13 can be verified. 3. Main results. 3.1. Non-adaptive method in robust output regulation.The proposed non-adaptive framework for the solution of the robust output regulation problem is shown in Fig. 1. The analysis presented below is based on the backstepping recursive method introduced in [20]. This technique was further generalized toneural network control for strict-feedback nonlinear systems in [36]. This approach is suitable for the design of control systems that can handle the complexities and nonlinearities of the error system (2.20). Backstepping is a widely used approach for analyzing lower trian- gular systems and remains an active area of research in the control community. Very 10S. WANG, M. GUAY, AND R. D. BRAATZ u=α r (ǫ 1 ,ǫ 2 ,...,ǫ r ,k ∗ ,η,ˆx 1 ) Controller Internal Model Input-driven Filter Recursive Equations System Disturbances u ̇ ˆx=Aˆx+B c u ̇η=Mη+Nˆx 2 α 1 (ǫ 1 ,k ∗ ,η) α 2 (ǫ 1 ,ǫ 2 ,k ∗ ,η,ˆx 1 ) . . . α r (ǫ 1 ,...,ǫ r ,k ∗ ,η,ˆx 1 ) h(v,w) y − η ˆx 2 ˆx ǫ 1 =e α r Figure 1.Non-adaptive method in robust output regulation. recently, [7, 8] originally established the high-order fully actuated system approaches, and based on which the second- and higher-order methods of backstepping (recursive method) are effectively proposed for both uncertain second-order and higher-order strict-feedback nonlinear systems. By iterating the control design process, the recur- sive method ensures the convergence, robustness, and stabilityof the error system (2.20). The analysis adopts the expressions: ǫ 1 =e, ǫ i+1 = ˆx i+1 −α i (ǫ 1 ,...,ǫ i ,k ∗ ,η,ˆx 1 ), α 1 (ǫ 1 ,k ∗ ,η) =−k ∗ ρ(ǫ 1 )ǫ 1 +χ(η), α 2 (ǫ 1 ,ǫ 2 ,k ∗ ,η,ˆx 1 ) =−bǫ 1 −ǫ 2 +λ 2 ˆx 1 + ∂α 1 ∂η ̇η +b ∂α 1 ∂ǫ 1 (ǫ 2 −k ∗ ρ(ǫ 1 )ǫ 1 ) − 1 2 ǫ 2 ∂α 1 ∂ǫ 1 2 + ∂α 1 ∂k ∗ ̇ k ∗ , α i (ǫ 1 ,...,ǫ i ,k ∗ ,η,ˆx 1 ) =−ǫ i−1 −ǫ i +λ i ˆx 1 + ∂α i−1 ∂η ̇η + ∂α i−1 ∂ˆx 1 ̇ ˆx 1 + i−1 X j=2 ∂α i−1 ∂ǫ j ̇ǫ j +b ∂α i−1 ∂ǫ 1 (ǫ 2 −k ∗ ρ(ǫ 1 )ǫ 1 ) − 1 2 ǫ i ∂α i−1 ∂ǫ 1 2 + ∂α i−1 ∂k ∗ ̇ k ∗ , i= 3,...,r,(3.1) where ̇ k ∗ will be zero whenk ∗ is a constant, and ˆx 1 ,...,ˆx r andηare generated in (2.5) and (2.18), respectively. Theorem3.1.For the system(2.20)under Assumptions 2.2–2.10, there is a sufficiently large positive smooth functionρ(·)and a positive real numberk ∗ such that the controller u=α r (ǫ 1 ,ǫ 2 ,...,ǫ r ,k ∗ ,η,ˆx 1 )(3.2) solves Problem 2.1. In addition, there exists a continuous positive definite function NONADAPTIVE LEARNING IN ROBUST NONLINEAR OUTPUT REGULATION11 U r ( ̄ Z,ǫ 1 ,...,ǫ r )such that, for allμ∈S×V×W, ̇ U r ( ̄ Z,ǫ 1 ,...,ǫ r )≤− ̄ Z 2 − X r j=1 ǫ j .(3.3) Proof.From Property 2.12, the changing supply rate technique [30] can be applied to show that, given any smooth function ∆ Z ( ̄ Z)>0, there exists a continuous function V 1 ( ̄ Z) satisfying α 1 ̄ Z 2 ≤V 1 ̄ Z ≤ α 1 ̄ Z 2 for some classK ∞ functionsα 1 (·) and α 1 (·), such that, for allμ∈Σ, along the trajectories of theZsubsystem, ̇ V 1 ≤−∆ Z ( ̄ Z) ̄ Z 2 + ˆγ ∗ ˆγ(ǫ 1 )ǫ 2 1 , where ˆγ ∗ is known positive constant and ˆγ(·)≥1 is a known smooth positive definite function. Define the Lyapunov functionU 1 ( ̄ Z,ǫ 1 ) =V 1 ̄ Z +ǫ 2 1 . Then, the time derivative ofU 1 ≡U 1 ( ̄ Z,ǫ 1 ) along the trajectory ofǫ 1 -subsystem with ˆx 2 =ǫ 2 +α 1 andη= ̄η+η ⋆ +Nb −1 ǫ 1 leads to ̇ U 1 ( ̄ Z,ǫ 1 ) = ̇ V 1 ̄ Z + 2ǫ 1 ̇ǫ 1 ≤−∆ Z ( ̄ Z) ̄ Z 2 + ˆγ ∗ ˆγ(ǫ 1 )ǫ 2 1 + 2ǫ 1 ̄g 1 ( ̄z,ǫ 1 ,μ) + 2bǫ 1 (ǫ 2 +α 1 (ǫ 1 ,k ∗ ,η) | z ˆx 2 −χ(η ∗ )) + 2bǫ 1 ̄x 2 ≤−∆ Z ( ̄ Z) ̄ Z 2 + ˆγ ∗ ˆγ(ǫ 1 )ǫ 2 1 + 2ǫ 1 ̄g 1 ( ̄z,ǫ 1 ,μ) + 2bǫ 1 (ǫ 2 +α 1 (ǫ 1 ,k ∗ ,η) | z −k ∗ ρ(ǫ 1 )ǫ 1 +χ(η) −χ(η ∗ )) + 2bǫ 1 ̄x 2 ≤−∆ Z ( ̄ Z) ̄ Z 2 − 2bk ∗ ρ(ǫ 1 )−ˆγ ∗ ˆγ(ǫ 1 ) ǫ 2 1 + 2bǫ 1 (ǫ 2 − ̄χ( ̄η,ǫ 1 ,μ)) + 2bǫ 1 ̄x 2 + 2ǫ 1 ̄g 1 ( ̄z,ǫ 1 ,μ) ≤−∆ Z ( ̄ Z) ̄ Z 2 − 2bk ∗ ρ(ǫ 1 )−ˆγ ∗ ˆγ(ǫ 1 ) ǫ 2 1 + 2bǫ 1 ǫ 2 −2bǫ 1 ̄χ( ̄η,ǫ 1 ,μ) + 2bǫ 1 ̄x 2 + 2ǫ 1 ̄g 1 ( ̄z,ǫ 1 ,μ) ≤−∆ Z ( ̄ Z) ̄ Z 2 − 2bk ∗ ρ(ǫ 1 )−3−ˆγ ∗ ˆγ(ǫ 1 ) ǫ 2 1 + 2bǫ 1 ǫ 2 + ∆ 1 (ǫ 1 , ̄ Z,μ)(3.4) where ∆ 1 (ǫ 1 , ̄ Z,μ) =b 2 ̄x 2 2 + ̄g 1 ( ̄z,ǫ 1 ,μ) 2 +b 2 ̄χ( ̄η,ǫ 1 ,μ) 2 , ̄χ( ̄η,ǫ 1 ,μ)≡χ( ̄η+η ∗ +Nb −1 ǫ 1 )−χ(η ∗ ). Now letU 2 ( ̄ Z,ǫ 1 ,ǫ 2 ) =U 1 ( ̄ Z,ǫ 1 )+ǫ 2 2 . The time derivative ofU 2 ≡U 2 ( ̄ Z,ǫ 1 ,ǫ 2 ) along the trajectory ofǫ 2 -subsystem with ˆx 3 =ǫ 3 +α 2 is given by ̇ U 2 ≤ ̇ U 1 + 2ǫ 2 ̇ǫ 2 ≤−∆ Z ( ̄ Z) ̄ Z 2 − bk ∗ ρ(ǫ 1 )−3−ˆγ ∗ ˆγ(ǫ 1 ) ǫ 2 1 12S. WANG, M. GUAY, AND R. D. BRAATZ + 2bǫ 1 ǫ 2 + ∆ 1 (ǫ 1 , ̄ Z,η) + 2ǫ 2 (ǫ 3 +α 2 −λ 2 ˆx 1 − ̇α 1 ) ≤−∆ Z ( ̄ Z) ̄ Z 2 − bk ∗ ρ(ǫ 1 )−3−ˆγ ∗ ˆγ(ǫ 1 ) ǫ 2 1 + 2bǫ 1 ǫ 2 + ∆ 1 (ǫ 1 , ̄ Z,η) + 2ǫ 2 ǫ 3 +α 2 −λ 2 ˆx 1 − ∂α 1 ∂ǫ 1 ̇ǫ 1 − ∂α 1 ∂η ̇η− ∂α 1 ∂k ∗ ̇ k ∗ ≤−∆ Z ( ̄ Z) ̄ Z 2 − bk ∗ ρ(ǫ 1 )−3−ˆγ ∗ ˆγ(ǫ 1 ) ǫ 2 1 + 2bǫ 1 ǫ 2 + ∆ 1 (ǫ 1 , ̄ Z,η) + 2ǫ 2 ǫ 3 + 2ǫ 2 α 2 −λ 2 ˆx 1 − ∂α 1 ∂η ̇η−b ∂α 1 ∂ǫ 1 (ǫ 2 +α 1 | z ˆx 2 −χ(η)) − ∂α 1 ∂ǫ 1 b ̄χ( ̄η,ǫ 1 ,μ) +b ̄x 2 + ̄g 1 ( ̄z,ǫ 1 ,μ) − ∂α 1 ∂k ∗ ̇ k ∗ ≤−∆ Z ( ̄ Z) ̄ Z 2 − bk ∗ ρ(ǫ 1 )−3−ˆγ ∗ ˆγ(ǫ 1 ) ǫ 2 1 + ∆ 1 (ǫ 1 , ̄ Z,η) + 2ǫ 2 ǫ 3 + 2ǫ 2 α 2 −ǫ 2 +ǫ 2 +bǫ 1 −λ 2 ˆx 1 − ∂α 1 ∂η ̇η | −b ∂α 1 ∂ǫ 1 (ǫ 2 −k ∗ ρ(ǫ 1 )ǫ 1 +χ(η) | z α 1 −χ(η))− ∂α 1 ∂k ∗ ̇ k ∗ z −α 2 + 1 2 ǫ 2 ∂α 1 ∂ǫ 1 2 +ǫ 2 2 + b 2 ̄χ( ̄η,ǫ 1 ,μ) 2 +b 2 ̄x 2 2 + ̄g 1 ( ̄z,ǫ 1 ,μ) 2 | z ∆ 1 (ǫ 1 , ̄ Z,η) ≤−∆ Z ( ̄ Z) ̄ Z 2 + 2∆ 1 (ǫ 1 , ̄ Z,μ) + 2ǫ 2 ǫ 3 − 2bk ∗ ρ(ǫ 1 )−3−ˆγ ∗ ˆγ(ǫ 1 ) ǫ 2 1 −ǫ 2 2 . Now letU i ( ̄ Z,ǫ 1 ,...,ǫ i ) =U i−1 ( ̄ Z,ǫ 1 ,...,ǫ i−1 ) +ǫ 2 i . The time derivative of U i ≡U i ( ̄ Z,ǫ 1 ,...,e i ) along the trajectory ofǫ i -subsystem with ˆx i+1 =ǫ i+1 +α i is given by ̇ U i ≤−∆ Z ( ̄ Z) ̄ Z 2 +i∆ 1 (ǫ 1 , ̄ Z,μ) + 2ǫ i ǫ i+1 − 2bk ∗ ρ(ǫ 1 )−3−ˆγ ∗ ˆγ(ǫ 1 ) ǫ 2 1 − X i j=2 ǫ 2 j . Finally, ati=randǫ r+1 = 0 results in ̇ U r ≤−∆ Z ( ̄ Z) ̄ Z 2 +r∆ 1 (ǫ 1 , ̄ Z,μ) − 2bk ∗ ρ(ǫ 1 )−3−ˆγ ∗ ˆγ(ǫ 1 ) ǫ 2 1 − X r j=2 ǫ 2 j .(3.5) From [19],χ(·) is a continuously differentiable function defined in (2.16). Moreover, it can be verified that the function ̄χ( ̄η,ǫ 1 ,μ) is continuous and vanishes at col( ̄z,ǫ 1 , ̄η) = col(0,0,0) for allμ∈V×W×S. As a result, the function ∆ 1 (ǫ 1 , ̄ Z,μ) =b 2 ̄x 2 2 + ̄g 1 ( ̄z,ǫ 1 ,μ) 2 +b 2 ̄χ( ̄η,ǫ 1 ,μ) 2 is continuous differentiable and vanishes at col( ̄ Z,ǫ 1 ,μ) = NONADAPTIVE LEARNING IN ROBUST NONLINEAR OUTPUT REGULATION13 col(0,0,0) for allμ∈V×W×S. Following Lemma 11.1 of [4], there exist positive smooth functionsγ 1 (·) andγ 2 (·) such that k∆ 1 (ǫ 1 , ̄ Z,μ)k 2 ≤γ 1 ( ̄ Z)k ̄ Zk 2 +ǫ 2 1 γ 2 (ǫ 1 ) forμ∈V×W×S. We can then choose the functions ∆ Z ( ̄ Z) andρ(ǫ 1 ) and the constantk ∗ as ∆ Z ( ̄ Z)≥γ 1 ( ̄ Z) + 1, ρ(ǫ 1 )≥maxγ 2 (ǫ 1 ),ˆγ(ǫ 1 ),1,(3.6) k ∗ ≥(3 + ˆγ ∗ )/(2b), such that (3.3) is satisfied. That is, for allμ∈V×W×S, the equilibrium of the closed-loop system at the origin is globally asymptotically stable. This completes the proof. From Theorem 3.1, we can also use the adaptive method to estimate thek ∗ . Corollary3.2.For the system(2.20)under Assumptions 2.2–2.10, there is a sufficiently large enough positive smooth functionρ(·)such that the controller, u=α r (ǫ 1 ,ǫ 2 ,...,ǫ r , ˆ k,η),(3.7a) ̇ ˆ k=ρ(ǫ 1 )ǫ 2 1 ,(3.7b) solves Problem 2.1 with the functionsα 1 (ǫ 1 , ˆ k,η),α 2 (ǫ 1 ,ǫ 2 , ˆ k,η,ˆx 1 )and α i (ǫ 1 ,...,ǫ i , ˆ k,η,ˆx 1 )defined in(3.1), fori= 3,...,r. Remark3.3. The proof of Corollary 3.2 can easily proceed with the Lyapunov function V r ( ̄ Z,ǫ 1 ,...,ǫ r , ˆ k−k ∗ ) =U r ( ̄ Z,ǫ 1 ,...,ǫ r ) +b( ˆ k−k ∗ ) 2 . Therefore, the proof is omitted for the sake of brevity. By differentiatingV r (t) and Theorem 3.1, we can show that ̇ V r (t)≤0. ThusV r (t) is nonincreasing and bounded below, and therefore converges ast→ ∞. In the meantime, the updated law in (3.7b) does not generate an unbounded high-gain. Since ̇ V r (t) is uniformly continuous under the smooth closed-loop dynamics, Barbalat’s Lemma yields ̇ V r (t)→0, implying that the driving term of ̇ ˆ kvanishes asymptotically. Consequently, ˆ k(t) is uniformly bounded and converges to a finite limit. Notably, the control law (3.7) differs from [21] and [33] by utilizing a non-adaptive design framework. Remark3.4. Following Lemma 3 in [34], the existence of the nonlinear mapping χin (2.17) hinges on solving the time-varying linear equation Θ(η) ˇa(η) + col(η n+1 ,...,η 2n ) = 0. Since Θ(η(t)) may be singular at isolated time instants, we introduce a globally defined smooth approximate inverseO(Θ) :R n×n →R n×n and set ˇa(η),−O(Θ(η)) col(η n+1 ,...,η 2n ), with (3.8)O(Θ) = det(Θ) det(Θ) 2 + Ψ 1 + det(Θ) 2 −ε 2 adj(Θ), 14S. WANG, M. GUAY, AND R. D. BRAATZ Figure 2.State trajectory of the Duffing system (*: initial point) . whereε >0 is a design constant, adj(·) denotes the adjugate matrix, and Ψ(ς) = κ(ς) κ(ς) +κ(1−ς) , κ(s) = ( e −1/s , s >0, 0,s≤0. Note thatO(Θ) = Θ −1 when Θ is nonsingular and det(Θ) 2 ≫ε 2 , whereasO(Θ)→0 smoothly as det(Θ)→0, which ensures global smoothness. Finally, define χ(η) = Γ Ξ ˇa(η) col(η 1 ,...,η n ). 4. Application to Duffing’s system.Consider the nonlinear system modelled by a controlled Duffing system [22]: ̇x 1 =x 2 ,(4.1a) ̇x 2 =−c 1 x 1 −c 2 x 3 1 −c 3 x 2 +u+d(t),(4.1b) where col(x 1 ,x 2 )∈R 2 is the state;c 1 = 1.5,c 2 =−2, andc 3 = 0.5 are the coeffi- cients; and the external disturbance isd(t) =Acos(σt+ψ) with unknown amplitude, frequency, and phase, which can be generated by an uncertain exosystem in the form (2.2) with S(σ) = 0σ −σ0 , v= v 1 v 2 , e=y−h(v,w),(4.2) NONADAPTIVE LEARNING IN ROBUST NONLINEAR OUTPUT REGULATION15 020406080100120140160 time (s) -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 Figure 3.Time profile of the tracking errore=y−h(v, w)for the Duffing system with h(v, w) =v 1 . 020406080100120140160 Time(Second) -8 -6 -4 -2 0 2 4 6 Figure 4.Time profile of the control input for the Duffing system. whereh(v,w) =v 1 ,σ∈S=σ∈R:σ∈[0.1,1]andV=v∈R 2 :kvk ≤2.1 withσbeing the unknown constant 0.5. Under the stated assumptions, [21] showed that there exists a solution of ˆx 2 (v,σ,w), polynomial inv, satisfying d 4 ˆx 2 dt 4 +a 1 ˆx 2 +a 2 dˆx 2 dt +a 3 d 2 ˆx 2 dt 2 +a 4 d 3 ˆx 2 dt 3 = 0, 16S. WANG, M. GUAY, AND R. D. BRAATZ 020406080100120140160 time (s) -2 -1 0 1 2 3 4 0.5625 2.5 0 Figure 5.Estimated parameters of the steady-state dynamics for the Duffing system. 020406080100120140160 time (s) -3 -2 -1 0 1 2 Figure 6.Parameter estimation error of the steady-state dynamics for the Duffing system. with unknown true value vector a≡col(9σ 4 ,0,10σ 2 ,0) | z col(a 1 ,a 2 ,a 3 ,a 4 ) in (2.11). For the control law (3.7a), we can chooseρ(e) = 2 + 2e 2 based on (3.6) to make the the inequality (3.5) to be negative definite,λ 1 = 4 andλ 2 = 4 are chosen to make the matrixA=A c −λC c in (2.5) to be Hurwitz,m 1 = 1,m 2 = 5.1503,m 3 = 13.301,m 4 = 22.2016,m 5 = 25.7518,m 6 = 21.6013,m 7 = 12.8005, andm 8 = 5.2001 are chosen to makeMin (2.18) defined in (2.12) to be Hurwitz. The simulation starts NONADAPTIVE LEARNING IN ROBUST NONLINEAR OUTPUT REGULATION17 with the initial conditionsx(0) = col(1,1),v(0) = col(1,2), ˆx(0) =0 2 ,η(0) =0 8 , and ˆ k(0) = 0. The control law stabilizes the system, with the tracking error converging to nearly zero within 50 seconds (Figs. 2–3), and the control signal converging to a periodic signal (Fig. 4). The estimated parameters of the steady-state dynamics for the closed-loop Duffing system converge to zero within the same timeperiod, which further implies the the estimated parameters converge to the true values (Fig. 5–6). 5. Conclusion.This article proposes a nonadaptive nonlinear robust output regulation approach for general nonlinear output feedback systems with error out- put. The proposed nonadaptive framework transforms the robust output regulation problem into a robust non-adaptive stabilization method that is effective for systems with Input-to-State Stable dynamics. 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