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Nonadaptive Output Regulation of Second-Order Nonlinear Uncertain Systems

T0 review · 2 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The paper proves that a nonadaptive internal-model control law drives the tracking error to zero for a class of second-order nonlinear uncertain systems with unknown exosystems, with global asymptotic stability established via a strict…

desk verdict Solid extension of generic internal models to second-order systems, but Theorem 1 needs an explicit lower-bound assumption on b(v,w) before it is true. read the letter →

arxiv 2505.21838 v1 pith:LE4QJD5L submitted 2025-05-28 eess.SY cs.AIcs.SYmath.OCnlin.CD

Maobin Lu , Martin Guay , Telema Harry , Shimin Wang , Jordan Cooper This is my paper
classification eess.SYcs.AIcs.SYmath.OCnlin.CD MSC 93C1093D0593D3093C15
keywords outputregulationnonlinearuncertainsystemsinternalmodelprinciplenonadaptivecontrolrobustLyapunovstabilitysecond-orderDuffingsystem
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper claims that robust output regulation for a class of second-order nonlinear uncertain systems with an unknown exosystem can be achieved without any adaptive parameter estimation. Instead of identifying the exosystem's parameters online, the authors construct generic internal models—fixed linear filters with a smooth nonlinear readout—that reproduce the steady-state state and input for all admissible uncertainties. A coordinate transformation turns the tracking problem into the stabilization of an augmented error system, and a control law with a strict Lyapunov function is shown to drive the tracking error to zero while keeping all signals bounded. If correct, this removes the bursting phenomenon and the need for Barbalat-lemma-based convergence arguments in this class of problems.

What carries the argument

The central object is the generic internal model (12), a fixed linear filter $\dot{\eta}_i=M_i\eta_i+N_i u_i$ with a smooth readout map $\chi_i(\eta_i)$ that reproduces the steady-state input and state as functions of the filter state. It is constructed using the generalized Sylvester equation (11) and a Hankel-matrix-based estimate $\check a_i$; the nonsingularity of $\Xi_i$ (Assumption 3) makes $\chi_i$ globally well-defined through a smooth pseudo-inverse. This internal model converts the output regulation problem into the stabilization of the augmented error system (16), whose Lyapunov analysis is carried out with a strict Lyapunov function.

What would settle it

Run the Duffing example with the reference signal extended by a non-polynomial component, such as a slowly decaying exponential $v_3(t)=e^{-0.1t}$; the finite-dimensional polynomial internal model cannot reproduce the steady-state input, so the tracking error should fail to converge to zero. A second check: allow the input gain $b(v,w)$ to vanish for some $v,w$ in the admissible set; then the gain $k_0$ cannot be chosen to dominate $\beta^*/(2b_*)$, and the strict Lyapunov inequality (22) no longer holds.

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Extended reading notes

Core claim

The paper's central claim is Theorem 1: for the second-order system (1) with exosystem (2), under Assumptions 1–3 (simple eigenvalues on the imaginary axis; steady-state input and state polynomial in the exogenous signal; nonzero Fourier coefficients), the nonadaptive dynamic controller (23) solves the robust output regulation problem. The controller consists of the two internal-model filters (13) and the static nonlinear feedback $u=-k_0 k(\zeta)\zeta+\chi_2(\eta_2)$, where $\zeta=x_2-\chi_1(\eta_1)+\rho(e)e$. The proof route is: Lemma 2 establishes an input-to-state Lyapunov function for the $(\eta_1,x_1,\eta_2)$-subsystem using the changing supply rate technique; Lemma 3 combines this with a high-gain-like term in $\zeta$ to get a strict Lyapunov function for the whole augmented system; Theorem 1 then follows directly from Lemma 3. The authors omit the theorem's proof because they view it as a direct corollary of Lemma 3. A Duffing-system simulation shows the tracking error vanishing.

Load-bearing premise

The load-bearing premise is Assumption 2: the steady-state input and state must be polynomials in the exogenous signal, so the reference or disturbance must be generated by finitely many harmonics; if a non-polynomial component appears, the finite-dimensional generic internal model may not exist.

Editorial extensions

If this is right

  • The tracking error tends to zero for every initial condition, with no parameter-update law running online.
  • The closed-loop system is globally asymptotically stable in the Lyapunov sense, so the validity does not rely on Barbalat's lemma or persistence-of-excitation conditions.
  • The same controller structure applies to any second-order system whose steady-state state and input are polynomial functions of the exogenous signal, including Duffing-type oscillators with unknown frequencies, phases, and amplitudes.
  • Because no parameter estimates are adjusted, the bursting phenomenon typical of adaptive designs is avoided.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Assumption 2 limits the exosystem to signals whose steady-state response is a finite sum of sinusoids (polynomial in $v$). An immediate testable extension would be to allow non-polynomial components such as a damped exponential; the finite-dimensional generic internal model would then be insufficient, and the paper's analysis would not apply.
  • The proof of Theorem 1 is omitted; the chain from Lemma 3 to Theorem 1 is asserted to be direct. A complete proof would need to verify that the stabilization of the transformed error system (16) indeed implies $e(t)\to 0$ for the original output, especially since $x_2$ is not measured and only $\chi_1(\eta_1)$ is available.
  • The control gain $k_0$ in (21e) depends on a positive lower bound $b_*$ of the input gain $b(v,w)$, but no such bound is listed among the assumptions. If $b(v,w)$ can cross zero, the Lyapunov argument in Lemma 3 fails, so adding an explicit 'b is bounded away from zero with known sign' assumption would be needed for the statement as written.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 4 minor

Summary. The paper studies robust output regulation for a class of second-order nonlinear uncertain systems with an unknown exosystem. It constructs generic internal models for the steady-state state and input, introduces a coordinate transformation that converts the problem into a nonadaptive stabilization problem for an augmented system, and designs a control law together with a strict Lyapunov function. The main result, Theorem 1, claims global asymptotic stability of the closed-loop equilibrium and convergence of the tracking error under Assumptions 1–3. A simulation on a Duffing system illustrates the approach. The proof of Theorem 1 is omitted, but is said to follow directly from Lemma 3.

Significance. If the result holds, the paper offers a useful nonadaptive alternative to adaptive internal-model designs for a relevant class of second-order systems, with the claimed benefit of avoiding bursting and providing a strict Lyapunov function that gives robustness to unmodeled disturbances. The construction of generic internal models for both the steady-state state and the input, following prior work in [27,28], is a meaningful extension. The central Lyapunov argument is plausible and the Duffing example provides a concrete demonstration. However, as detailed below, the main theorem currently rests on an unstated assumption on the input gain, and the proof of a key lemma has presentation/correctness gaps that need to be repaired.

major comments (2)
  1. [Section III.C, Lemma 3 and Theorem 1]
  2. [Section III.B, Lemma 2 proof]
minor comments (4)
  1. [Section III.C, Theorem 1]
  2. [Section III.C, Lemma 3 statement]
  3. [Section II, Eq. (6)]
  4. [Section III.A, Eq. (8b)]

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the claimed output regulation theorem is supported by a new Lyapunov stabilization argument, and the self-citations supply internal-model building blocks rather than the central result.

full rationale

The derivation chain is not circular in the sense of the checklist. The internal model construction in Section III-A imports generic internal-model existence results from the authors' prior work [27,28], e.g., 'Under Assumptions 1, 2 and 3, by Lemma 3 in [27], we have eta_i^*(v(t),sigma,w) = theta_i'. These are self-citations, since Wang and Guay are coauthors, but they are used as lemmas or building blocks and are not the claimed output-regulation theorem. The central claim, Theorem 1, rests on the nonadaptive stabilization result in Lemma 3, whose proof constructs a strict Lyapunov function with no fitted parameters and no equation that is equivalent to its input by construction. Theorem 1 is stated to follow directly from Lemma 3 together with the standard reduction in Lemma 1; the omitted proof is a presentation choice rather than a circular reduction. The gain k0 in (21e) is chosen using a lower bound b* of b(v,w), but no assumption is stated that guarantees b(v,w) >= b* > 0 with known sign; this is a correctness gap in the theorem's hypotheses, not a circularity, because the missing condition is not an input being repackaged as an output. No fitted-input-called-prediction, renaming-known-result, ansatz-smuggled-in-via-citation, or uniqueness-imported-from-authors pattern is present. The self-citations therefore do not make the central claim circular; score 2 reflects only minor, non-load-bearing self-citation.

Assumptions & free parameters 3 free parameters · 7 assumptions · 0 invented entities

The control design uses no fitted parameters; the free parameters are systematic design choices such as Hurwitz filter coefficients, gain functions, and simulation constants. The axioms are standard output regulation assumptions and cited mathematical facts, plus one silent assumption about the sign and lower bound of the input gain. No new physical entities are introduced.

free parameters (3)
  • m1 and m2 coefficients = m1 = (10,18,15,6), m2 = (1,5,13,22,26,22,13,5)
    Coefficients of Hurwitz polynomials chosen by the designer to define the internal model filters in the simulation. No data fitting is involved.
  • rho(s), k(s), k0 = rho(s)=10+4s^4, k(s)=s^2+1, k0=1
    Smooth functions and gain selected in the simulation to satisfy the theoretical inequalities. They are design choices, not fitted to data.
  • epsilon = 0.1
    Parameter in the smooth mapping O(Theta) in Remark 2, chosen for the simulation.
assumptions (7)
  • domain assumption Assumption 1: all eigenvalues of S(sigma) are simple with zero real parts
    Ensures the exosystem state v(t) is bounded and the steady-state generator has a finite-dimensional structure.
  • domain assumption Assumption 2: u(v,w,sigma) and x2(v,w,sigma) are polynomials in v(t)
    This is the key structural condition that enables the finite-dimensional generic internal model. If the steady-state signals are not polynomial, the internal model may not exist.
  • domain assumption Assumption 3: coefficients Ci,j in the polynomial representation are nonzero
    A genericity assumption needed to guarantee nonsingularity properties in the internal model construction, inherited from reference [27].
  • ad hoc to paper The input gain b(v,w) is bounded away from zero with known sign b_* > 0
    This is used implicitly when the gain k0 is set with b_* in (21e), and when deriving the regulator equation (4c) which divides by b. No explicit assumption is stated in the paper.
  • standard math The matrix Xi_i(a_i) is nonsingular under Assumptions 1-3
    Taken from Lemma 3 of reference [27] without proof in this paper. The paper relies on this to define the internal model.
  • standard math Any C^1 function vanishing at the origin admits quadratic bounds with smooth positive coefficients
    Invoked via Lemma 11.1 of reference [33] to obtain the inequalities used in the Lyapunov proofs of Lemmas 2 and 3.
  • standard math Changing supply rate technique for ISS systems
    Used to construct the Lyapunov function U_eta_i with desired decay rates in Lemma 2, following reference [34].

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Cite this review

Pith. "Pith review of Nonadaptive Output Regulation of Second-Order Nonlinear Uncertain Systems." pith.science (2026). https://pith.science/paper/LE4QJD5L

@misc{pith2026250521838,
  author       = {Pith},
  title        = {Pith review of: Nonadaptive Output Regulation of Second-Order Nonlinear Uncertain Systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LE4QJD5L}},
  note         = {Machine review of arXiv:2505.21838}
}
read the original abstract

This paper investigates the robust output regulation problem of second-order nonlinear uncertain systems with an unknown exosystem. Instead of the adaptive control approach, this paper resorts to a robust control methodology to solve the problem and thus avoid the bursting phenomenon. In particular, this paper constructs generic internal models for the steady-state state and input variables of the system. By introducing a coordinate transformation, this paper converts the robust output regulation problem into a nonadaptive stabilization problem of an augmented system composed of the second-order nonlinear uncertain system and the generic internal models. Then, we design the stabilization control law and construct a strict Lyapunov function that guarantees the robustness with respect to unmodeled disturbances. The analysis shows that the output zeroing manifold of the augmented system can be made attractive by the proposed nonadaptive control law, which solves the robust output regulation problem. Finally, we demonstrate the effectiveness of the proposed nonadaptive internal model approach by its application to the control of the Duffing system.

Figures

Figures reproduced from arXiv: 2505.21838 by the authors.

Figure 1
Figure 1. Trajectory of (t, x1, x2) of the Duffing system 0 10 20 30 40 50 time (s) -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 0.5 [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. Tracking error of the Duffing system 0 5 10 15 20 25 30 35 40 45 50 time (s) -1 0 1 2 0 5 10 15 20 25 30 35 40 45 50 time (s) -2 -1 0 1 2 [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Parameter estimation error of non-adaptive method [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗

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Works this paper leans on

35 extracted references · 33 canonical work pages

  1. [1]

    Output regulation of nonlin ear systems,

    A. Isidori and C. I. Byrnes, “Output regulation of nonlin ear systems,” IEEE Transactions on Automatic Control , vol. 35, no. 2, pp. 131–140, 1990

    work page 1990

  2. [2]

    Huang, Nonlinear output regulation: theory and applications

    J. Huang, Nonlinear output regulation: theory and applications . SIAM, 2004

    work page 2004

  3. [3]

    Isidori, L

    A. Isidori, L. Marconi, and A. Serrani, Robust autonomous guidance: an internal model approach . Springer Science & Business Media, 2003. Fig. 1. Trajectory of (t, x1, x2) of the Duffing system 0 10 20 30 40 50 time (s) -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 0.5 Fig. 2. Tracking error of the Duffing system 0 5 10 15 20 25 30 35 40 45 50 time (s) -1 0 1 2 0 5 10 15 ...

    work page 2003

  4. [4]

    Approximate nonline ar regulation via identification-based adaptive internal models,

    M. Bin, P . Bernard, and L. Marconi, “Approximate nonline ar regulation via identification-based adaptive internal models,” IEEE Transactions on Automatic Control, vol. 66, no. 8, pp. 3534–3549, 2020

    work page 2020

  5. [5]

    Adaptive internal models in neuroscience,

    M. E. Broucke et al. , “Adaptive internal models in neuroscience,” F oundations and Trends® in Systems and Control, vol. 9, no. 4, pp. 365– 550, 2022

    work page 2022

  6. [6]

    Adaptive network traffic c ontrol with an integrated model-based and data-driven approach an d a decen- tralised solution method,

    Z. Su, A. H. Chow, and R. Zhong, “Adaptive network traffic c ontrol with an integrated model-based and data-driven approach an d a decen- tralised solution method,” Transportation Research Part C: Emerging Technologies, vol. 128, p. 103154, 2021

    work page 2021

  7. [7]

    The internal model princi ple of control theory,

    B. A. Francis and W. M. Wonham, “The internal model princi ple of control theory,” Automatica, vol. 12, no. 5, pp. 457–465, 1976

    1976

  8. [8]

    The linear multivariable regulator prob lem,

    B. A. Francis, “The linear multivariable regulator prob lem,” SIAM Journal on Control and Optimization , vol. 15, no. 3, pp. 486–505, 1977

    work page 1977

Show all 35 references
  1. [9]

    The robust control of a servomechanism prob lem for linear time-invariant multivariable systems,

    E. Davison, “The robust control of a servomechanism prob lem for linear time-invariant multivariable systems,” IEEE Transactions on Automatic Control, vol. 21, no. 1, pp. 25–34, 1976

    1976

  2. [10]

    On a nonlinear multivariable se rvomechanism problem,

    J. Huang and W. J. Rugh, “On a nonlinear multivariable se rvomechanism problem,” Automatica, vol. 26, no. 6, pp. 963–972, 1990

    1990

  3. [11]

    On a robust nonlinear servomech anism prob- lem,

    J. Huang and C.-F. Lin, “On a robust nonlinear servomech anism prob- lem,” IEEE Transactions on Automatic Control, vol. 39, no. 7, pp. 1510– 1513, 1994

    1994

  4. [12]

    Asymptotic tracking and disturbance reject ion in uncertain nonlinear systems,

    J. Huang, “Asymptotic tracking and disturbance reject ion in uncertain nonlinear systems,” IEEE Transactions on Automatic Control , vol. 40, no. 6, pp. 1118–1122, 1995

    1995

  5. [13]

    A general framework for tackling t he output regulation problem,

    J. Huang and Z. Chen, “A general framework for tackling t he output regulation problem,” IEEE Transactions on Automatic Control , vol. 49, no. 12, pp. 2203–2218, 2004

    2004

  6. [14]

    Limit sets, zero dynamics, and internal models in the problem of nonlinear output regulation,

    C. I. Byrnes and A. Isidori, “Limit sets, zero dynamics, and internal models in the problem of nonlinear output regulation,” IEEE Transac- tions on Automatic Control , vol. 48, no. 10, pp. 1712–1723, 2003

    2003

  7. [15]

    Nonlinear internal models for output regulation,

    C. I. Byrnes and A. Isidori, “Nonlinear internal models for output regulation,” IEEE Transactions on Automatic Control , vol. 49, no. 12, pp. 2244–2247, 2004

    2004

  8. [16]

    Remarks on the robust output regulation prob lem for nonlin- ear systems,

    J. Huang, “Remarks on the robust output regulation prob lem for nonlin- ear systems,” IEEE Transactions on Automatic Control , vol. 46, no. 12, pp. 2028–2031, 2001

    2001

  9. [17]

    St ructurally stable output regulation of nonlinear systems,

    C. I. Byrnes, F. D. Priscoli, A. Isidori, and W. Kang, “St ructurally stable output regulation of nonlinear systems,” Automatica, vol. 33, no. 3, pp. 369–385, 1997

    1997

  10. [18]

    Adaptive non-linear tracking with co mplete compen- sation of unknown disturbances,

    V . O. Nikiforov, “Adaptive non-linear tracking with co mplete compen- sation of unknown disturbances,” European journal of control , vol. 4, no. 2, pp. 132–139, 1998

    1998

  11. [19]

    Output regulation for linear sy stems via adaptive internal model,

    R. Marino and P . Tomei, “Output regulation for linear sy stems via adaptive internal model,” IEEE Transactions on Automatic Control , vol. 48, no. 12, pp. 2199–2202, 2003

    2003

  12. [20]

    Adaptive sinusoidal distu rbance cancella- tion for unknown LTI systems despite input delay,

    H. I. Basturk and M. Krstic, “Adaptive sinusoidal distu rbance cancella- tion for unknown LTI systems despite input delay,” Automatica, vol. 58, pp. 131–138, 2015

    2015

  13. [21]

    Semi-global no nlinear output regulation with adaptive internal model,

    A. Serrani, A. Isidori, and L. Marconi, “Semi-global no nlinear output regulation with adaptive internal model,” IEEE Transactions on Auto- matic Control, vol. 46, no. 8, pp. 1178–1194, 2001

    2001

  14. [22]

    Parameter convergence an d minimal internal model with an adaptive output regulation problem,

    L. Liu, Z. Chen, and J. Huang, “Parameter convergence an d minimal internal model with an adaptive output regulation problem, ” Automatica, vol. 45, no. 5, pp. 1306–1311, 2009

    2009

  15. [23]

    Nonlinear output regulation wit h adaptive conditional servocompensator,

    R. Li and H. K. Khalil, “Nonlinear output regulation wit h adaptive conditional servocompensator,” Automatica, vol. 48, no. 10, pp. 2550– 2559, 2012

    2012

  16. [24]

    Adaptive tracking control of uncertain euler–lagrange systems subject to external disturbances,

    M. Lu, L. Liu, and G. Feng, “Adaptive tracking control of uncertain euler–lagrange systems subject to external disturbances, ” Automatica, vol. 104, pp. 207–219, 2019

    2019

  17. [25]

    Uniform practical nonlinear o utput regula- tion,

    L. Marconi and L. Praly, “Uniform practical nonlinear o utput regula- tion,” IEEE Transactions on Automatic Control, vol. 53, no. 5, pp. 1184– 1202, 2008

    2008

  18. [26]

    Robust design of n onlinear internal models without adaptation,

    A. Isidori, L. Marconi, and L. Praly, “Robust design of n onlinear internal models without adaptation,” Automatica, vol. 48, no. 10, pp. 2409–2419, 2012

    2012

  19. [27]

    A nonparamet ric learning framework for nonlinear robust output regulation,

    S. Wang, M. Guay, Z. Chen, and R. D. Braatz, “A nonparamet ric learning framework for nonlinear robust output regulation,” IEEE Transactions on Automatic Control , DOI: 10.1109/TAC.2024.3470065, 2024

    2024

  20. [28]

    Nonparametric stead y-state learning for robust output regulation of nonlinear output f eedback systems,

    S. Wang, M. Guay, and R. D. Braatz, “Nonparametric stead y-state learning for robust output regulation of nonlinear output f eedback systems,” arXiv preprint arXiv:2402.16170 , 2024

    2024 arXiv

  21. [29]

    State and pa rameter esti- mation: A nonlinear Luenberger observer approach,

    C. Afri, V . Andrieu, L. Bako, and P . Dufour, “State and pa rameter esti- mation: A nonlinear Luenberger observer approach,” IEEE Transactions on Automatic Control , vol. 62, no. 2, pp. 973–980, 2016

    2016

  22. [30]

    A new solution to the generalize d Sylvester matrix equation A V -EVF= BW,

    B. Zhou and G.-R. Duan, “A new solution to the generalize d Sylvester matrix equation A V -EVF= BW,” Systems & Control Letters , vol. 55, no. 3, pp. 193–198, 2006

    2006

  23. [31]

    Output stabiliza tion via nonlinear luenberger observers,

    L. Marconi, L. Praly, and A. Isidori, “Output stabiliza tion via nonlinear luenberger observers,” SIAM Journal on Control and Optimization , vol. 45, no. 6, pp. 2277–2298, 2007

    2007

  24. [32]

    Nonlinear observers fo r autonomous Lipschitz continuous systems,

    G. Kreisselmeier and R. Engel, “Nonlinear observers fo r autonomous Lipschitz continuous systems,” IEEE Transactions on Automatic Con- trol, vol. 48, no. 3, pp. 451–464, 2003

    2003

  25. [33]

    Chen and J

    Z. Chen and J. Huang, Stabilization and regulation of nonlinear systems . Springer, 2015

    2015

  26. [34]

    Changing supply functions in inp ut/state stable systems,

    E. Sontag and A. Teel, “Changing supply functions in inp ut/state stable systems,” IEEE Transactions on Automatic Control , vol. 40, no. 8, pp. 1476–1478, 1995

    1995

  27. [35]

    Cooperative adaptive output regula tion for a class of nonlinear uncertain multi-agent systems with unkn own leader,

    Y . Su and J. Huang, “Cooperative adaptive output regula tion for a class of nonlinear uncertain multi-agent systems with unkn own leader,” Systems & Control Letters , vol. 62, no. 6, pp. 461–467, 2013

    2013

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