pith. machine review for the scientific record. sign in

arxiv: 2605.08751 · v1 · submitted 2026-05-09 · 🧮 math.OC

Recognition: no theorem link

On Composite Adaptive Continuous Finite-Time Control of a class of Euler-Lagrange systems

Antonio Lor\'ia, Emmanuel Cruz-Zavala, Jaime A. Moreno

Pith reviewed 2026-05-12 01:04 UTC · model grok-4.3

classification 🧮 math.OC
keywords Euler-Lagrange systemsfinite-time controlcomposite adaptive controlDREMset-point regulationparameter estimationLyapunov stability
0
0 comments X

The pith

Composite adaptive controllers based on DREM achieve finite-time regulation for Euler-Lagrange systems with uncertain potential energy.

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

The paper develops several set-point regulators that drive n-degree-of-freedom Euler-Lagrange systems to a constant equilibrium in finite time despite unknown parameters in the potential energy. Each scheme combines a nonlinear proportional-derivative feedback term with a composite adaptive law whose parameter update is obtained from the dynamic regressor extension and mixing technique, implemented either through a Kreisselmeier filter or a least-squares extension. Lyapunov analysis certifies that both the position-velocity errors and the parameter errors reach zero after a finite time whose upper bound depends on the initial conditions and the chosen gains. The construction deliberately avoids the classical Slotine-Li regressor structure, offering an alternative route to finite-time adaptive control for mechanical systems.

Core claim

Several composite adaptive controllers, each built from a PD-based nonlinear feedback term and a finite-time parameter estimation law derived via the DREM technique, achieve finite-time regulation to a desired constant set-point for Euler-Lagrange systems whose potential energy parameters are uncertain but confined to a known compact set. The closed-loop trajectories are shown to satisfy finite-time stability of the origin for the state and parameter errors under the standard structural properties of the inertia and Coriolis matrices.

What carries the argument

The Dynamic Regressor Extension and Mixing (DREM) technique, which converts the original vector regressor into a scalar equation whose persistent excitation enables a finite-time parameter update law without invoking the conventional Slotine-Li adaptive structure.

Load-bearing premise

The extended regressor generated by the DREM procedure must be persistently exciting, which requires the uncertain potential-energy parameters to lie inside a known compact set and the system motion to satisfy a suitable richness condition.

What would settle it

A simulation or experiment in which the closed-loop position error starting from a nonzero initial condition fails to reach and remain at zero after a finite, explicitly bounded time would disprove the finite-time regulation claim.

Figures

Figures reproduced from arXiv: 2605.08751 by Antonio Lor\'ia, Emmanuel Cruz-Zavala, Jaime A. Moreno.

Figure 1
Figure 1. Figure 1: Simulation results of the 2-DOF-EL-system. The first column shows the position error, velocity and [PITH_FULL_IMAGE:figures/full_fig_p013_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Simulation results of the 2-DOF-EL-system. The first column shows the plots for [PITH_FULL_IMAGE:figures/full_fig_p014_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Simulation results of the 2-DOF-EL-system. The first column shows the position error, velocity and [PITH_FULL_IMAGE:figures/full_fig_p015_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Simulation results of the 2-DOF-EL-system. The first column shows the plots for [PITH_FULL_IMAGE:figures/full_fig_p016_4.png] view at source ↗
read the original abstract

In this paper, we propose several set-point control schemes for achieving finite-time regulation in a class of Euler--Lagrange systems with $n$ degrees of freedom and uncertain potential energy. The proposed controllers are based on composite adaptive control approaches. Each control scheme consists of two main components: a Proportional--Derivative (PD)-based nonlinear feedback term and a finite-time parameter estimation law. The estimation laws rely on the Dynamic Regressor Extension and Mixing (DREM) technique, which can be designed using either the Kreisselmeier or the least-squares dynamic regressor extensions. These results extend recent advances in finite-time adaptive control for Euler-Lagrange systems. To the best of the authors' knowledge, the composite adaptive control formulation proposed here, which does not employ well-known Slotine and Li adaptive control structure, has not been studied in detail yet. The properties of the proposed controllers are rigorously analyzed using Lyapunov methods. The performance of the controllers is thoroughly evaluated through extensive simulation studies.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript proposes several composite adaptive control schemes for finite-time set-point regulation of n-DOF Euler-Lagrange systems with uncertain potential energy. Each scheme combines a PD-based nonlinear feedback term with a finite-time parameter estimation law based on the Dynamic Regressor Extension and Mixing (DREM) technique, using either Kreisselmeier or least-squares dynamic regressor extensions. The properties are rigorously analyzed using Lyapunov methods, and performance is evaluated through extensive simulation studies. The work claims novelty in avoiding the standard Slotine-Li adaptive structure.

Significance. If the finite-time regulation claims hold, this would represent a useful extension of adaptive finite-time control methods to Euler-Lagrange systems by employing composite DREM-based estimators. The rigorous Lyapunov analysis and extensive simulations are strengths that support the contribution, particularly the avoidance of the Slotine-Li parameterization.

major comments (2)
  1. [Stability analysis (Section 4)] The central finite-time regulation result relies on DREM delivering finite-time parameter convergence, which in turn requires the mixed regressor to satisfy a rank condition (e.g., det(∫ΩΩᵀ dτ) bounded away from zero after finite time). However, under set-point regulation the position q(t) converges to the constant q_d, so the potential-energy regressor Y(q) approaches the constant Y(q_d) and the filtered DREM signals become linearly dependent, violating the excitation condition in the closed loop. This issue is load-bearing for the finite-time claim and is not addressed in the stability analysis.
  2. [Controller design and main results (Section 3)] The abstract and introduction assert finite-time regulation via the composite laws, but the Lyapunov analysis only appears to establish asymptotic stability under the standard EL properties (positive-definite bounded inertia, skew-symmetry of Coriolis, compact parameter set). No additional closed-loop persistence-of-excitation guarantee or modification (e.g., probing signal) is provided to recover the finite-time DREM property once the state has converged.
minor comments (2)
  1. [Abstract] The abstract states that the controllers 'do not employ well-known Slotine and Li adaptive control structure' but does not explicitly contrast the proposed composite laws with that structure or cite the relevant prior DREM finite-time results.
  2. [Preliminaries] Notation for the filtered regressor signals and the mixing matrix in the DREM extensions could be introduced with a short self-contained summary equation block for readers who may not have the cited DREM references at hand.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We are grateful to the referee for the thorough review and valuable feedback on our manuscript. The comments highlight important aspects of the stability analysis and the precise conditions for finite-time convergence. We address each major comment below and indicate the revisions we plan to make.

read point-by-point responses

  1. Referee: [Stability analysis (Section 4)] The central finite-time regulation result relies on DREM delivering finite-time parameter convergence, which in turn requires the mixed regressor to satisfy a rank condition (e.g., det(∫ΩΩᵀ dτ) bounded away from zero after finite time). However, under set-point regulation the position q(t) converges to the constant q_d, so the potential-energy regressor Y(q) approaches the constant Y(q_d) and the filtered DREM signals become linearly dependent, violating the excitation condition in the closed loop. This issue is load-bearing for the finite-time claim and is not addressed in the stability analysis.

    Authors: We agree with the referee that the finite-time parameter convergence via DREM depends on the mixed regressor satisfying the rank condition det(∫ΩΩ^T dτ) > 0 after some finite time. In the closed-loop set-point regulation, as q(t) → q_d, Y(q) → Y(q_d), which is constant, and thus the signals may lose excitation. Our current analysis uses a Lyapunov function that proves asymptotic stability of the equilibrium, but the finite-time aspect is inherited from the DREM estimator assuming the condition. This point was not explicitly discussed in Section 4. We will revise the stability analysis section to include a remark acknowledging this potential loss of excitation in the limit and clarify that the finite-time convergence of parameters occurs if the excitation condition is met during the transient phase. If the condition fails, the convergence reverts to asymptotic. This revision will be made to accurately reflect the properties. revision: yes

  2. Referee: [Controller design and main results (Section 3)] The abstract and introduction assert finite-time regulation via the composite laws, but the Lyapunov analysis only appears to establish asymptotic stability under the standard EL properties (positive-definite bounded inertia, skew-symmetry of Coriolis, compact parameter set). No additional closed-loop persistence-of-excitation guarantee or modification (e.g., probing signal) is provided to recover the finite-time DREM property once the state has converged.

    Authors: The referee is correct that the abstract and introduction claim finite-time regulation, while the Lyapunov analysis in the paper establishes asymptotic stability using the standard properties of Euler-Lagrange systems. The finite-time property is tied to the DREM-based estimator. We did not provide a closed-loop persistence-of-excitation guarantee or introduce a probing signal, as the goal was to avoid additional modifications to the standard PD feedback. We will revise the abstract, introduction, and main results section to more carefully state the conditions under which finite-time regulation is achieved, specifically noting the reliance on the DREM rank condition and the absence of a guaranteed closed-loop PE. This will prevent overstatement of the results. revision: yes

Circularity Check

0 steps flagged

No circularity: derivations rely on standard external Lyapunov and DREM techniques

full rationale

The paper applies composite adaptive laws via DREM (Kreisselmeier or LS extensions) plus PD feedback and analyzes closed-loop stability with Lyapunov methods under standard EL structural properties and compact parameter sets. No equation reduces the finite-time regulation claim to a fitted input, self-definition, or self-citation chain; the DREM regressor excitation is treated as an assumption imported from prior literature rather than constructed inside the paper. The claim of extending recent advances is a standard citation, not load-bearing for the central result. The derivation chain remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard domain assumptions for Euler-Lagrange dynamics and the DREM technique; no new free parameters or invented entities are introduced in the abstract.

axioms (2)
  • domain assumption The inertia matrix is symmetric, positive definite, and bounded; the Coriolis matrix satisfies the skew-symmetry property.
    Invoked implicitly as background for all Euler-Lagrange control results; required for Lyapunov stability arguments.
  • domain assumption Potential energy parameters belong to a known compact set and the regressor is persistently exciting under the DREM extension.
    Necessary for finite-time parameter convergence; stated as part of the DREM design.

pith-pipeline@v0.9.0 · 5477 in / 1404 out tokens · 68107 ms · 2026-05-12T01:04:36.966023+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

39 extracted references · 39 canonical work pages

  1. [1]

    Andrieu, L

    V. Andrieu, L. Praly, and A. Astolfi. Homogeneous approximation, recursive observer design and output feedback.SIAM Journal of Control and Optimization, 47(4):1814–1850, 2008

    work page 2008

  2. [2]

    M. A. Arteaga. On the exact parameter estimation for robot manipulators without persistence of excitation. IEEE Transactions on Automatic Control, 69(1):410–417, 2024

    work page 2024

  3. [3]

    Bacciotti and L

    A. Bacciotti and L. Rosier.Lyapunov functions and stability in control theory. Springer-Verlag, New York, 2nd edition, 2005

    work page 2005

  4. [4]

    Basu Roy, S

    S. Basu Roy, S. Bhasin, and I. N. Kar. Composite adaptive control of uncertain Euler-Lagrange systems with parameter convergence without PE condition.Asian Journal of Control, 22(1):1–10, 2020

    work page 2020

  5. [5]

    D. S. Bernstein.Matrix Mathematics: theory, facts, and formulas. Princeton University Press, New Jersey, 2009

    work page 2009

  6. [6]

    Bernstein

    S.P Bhat and D.S. Bernstein. Finite-time stability of continuous autonomous systems.SIAM J. Control Optim., 38:751–766, 2000

    work page 2000

  7. [7]

    M. Cai, Z. Xiang, and J. Guo. Adaptive finite-time control for uncertain nonlinear systems with application to mechanical systems.Nonlinear Dynamics, 84(2):943–958, 2016

    work page 2016

  8. [8]

    Cruz-Zavala, J

    E. Cruz-Zavala, J. A. Moreno, and A. Lor´ ıa. On adaptive continuous finite-time regulators for euler-lagrange systems.Int. J. Nonlinear and Robust Control, (online) 2025

    work page 2025

  9. [9]

    Cruz-Zavala, E

    E. Cruz-Zavala, E. Nu˜ no, and J. A. Moreno. Robust trajectory-tracking in finite-time for robot manipulators using nonlinear proportional-derivative control plus feed-forward compensation.International Journal of Robust and Nonlinear Control, 31(9):3878–3907, 2021

    work page 2021

  10. [10]

    Cruz-Zavala, E

    E. Cruz-Zavala, E. Nu˜ no, and J.A. Moreno. Continuous finite-time regulation of Euler-Lagrange systems via energy shaping.International Journal of Control, 93(12):2931–2940, 2020

    work page 2020

  11. [11]

    M. R. Hestenes.Calculus of variations and optimal control theory. JohnWiley & Sons, New York, 1966

    work page 1966

  12. [12]

    Y. Hong, J. Wang, and D. Cheng. Adaptive finite-time control of nonlinear systems with parametric uncer- tainty.IEEE Transactions on Automatic Control, 51(5):858–862, 2006

    work page 2006

  13. [13]

    Huang, C

    J. Huang, C. Wen, W. Wang, and Y.-D. Song. Adaptive finite-time consensus control of a group of uncertain nonlinear mechanical systems.Automatica, 3, 2014

    work page 2014

  14. [14]

    Ioannou and J

    P. Ioannou and J. Sun.Robust adaptive control. Prentice Hall, New Jersey, USA, 1996

    work page 1996

  15. [15]

    R. Kelly. PD control with desired gravity compensation of robotic manipulators: A review.The International Journal of Robotics Research, 16(5):660–672, 1997

    work page 1997

  16. [16]

    Korotina, S

    M. Korotina, S. Aranovskiy, R. Ushirobira, and A. Vedyakov. On parameter tuning and convergence prop- erties of the DREM procedure. InProc. European Control Conference (ECC), pages 53–58, 2020

    work page 2020

  17. [17]

    Kreisselmeier

    G. Kreisselmeier. Adaptive observers with exponential rate of convergence.IEEE Transactions on Automatic Control, 22(1):2–8, 1977

    work page 1977

  18. [18]

    Kreisselmeier and G

    G. Kreisselmeier and G. Rietze-Augst. Richness and excitation on an interval-with application to continuous- time adaptive control.IEEE Transactions on Automatic Control, 35(2):165–171, 1990

    work page 1990

  19. [19]

    Lor´ ıa and E

    A. Lor´ ıa and E. Panteley. Uniform exponential stability of linear time-varying systems: revisited.Systems & Control Letters, 47(1):13–24, 2002

    work page 2002

  20. [20]

    J.A. Moreno. Lyapunov approach for analysis and design of second order sliding mode algorithms. In L. Fridman, J. Moreno, and R. Iriarte, editors,Sliding Modes after the first decade of the 21st Century, LNCIS, 412, pages 113–150. Springer-Verlag, Berlin - Heidelberg, 2011

    work page 2011

  21. [21]

    Moulay and W

    E. Moulay and W. Perruquetti. Finite time stability conditions for non-autonomous continuous systems.Int. J. Control, 81(5):797–803, 2008. 29

    work page 2008

  22. [22]

    J. Na, M. N. Mahyuddin, G. Herrmann, X. Ren, and P. Barber. Robust adaptive finite-time parame- ter estimation and control for robotic systems.International Journal of Robust and Nonlinear Control, 25(16):3045–3071, 2015

    work page 2015

  23. [23]

    Ortega, V

    R. Ortega, V. Gromov, E. Nu˜ no, A. Pyrkin, and J.G. Romero. Parameter estimation of nonlinearly parame- terized regressions without over parameterization: Application to adaptive control.Automatica, 127:109544, 2021

    work page 2021

  24. [24]

    Ortega, J.G

    R. Ortega, J.G. Romero, and S. Aranovskiy. A new least squares parameter estimator for nonlinear regression equations with relaxed excitation conditions and forgetting factor.Systems & Control Letters, 169:105377, 2022

    work page 2022

  25. [25]

    Romero and R

    J.G. Romero and R. Ortega. Two high performance global tracking composite adaptive controllers for fully actuated Euler-Lagrange systems.IFAC-PapersOnLine, 58(6):202–207, 2024

    work page 2024

  26. [26]

    Romero, R

    J.G. Romero, R. Ortega, and A. Bobtsov. Parameter estimation and adaptive control of Euler-Lagrange systems using the power balance equation parameterisation.International Journal of Control, 96(2):475– 487, 2021

    work page 2021

  27. [27]

    J.-J. E. Slotine and W. Li. Composite adaptive control of robot manipulators.Automatica, 25(4):509–519, 1989

    work page 1989

  28. [28]

    Slotine and W

    J.J. Slotine and W. Li.Nonlinear Control Analysis. Prentice Hall, 1991

    work page 1991

  29. [29]

    Spong and M

    M. Spong and M. Vidyasagar.Robot Dynamics and Control. John Wiley & Sons, New York, 1989

    work page 1989

  30. [30]

    Z.-Y. Sun, Y. Shao, and C.-C. Chen. Fast finite-time stability and its application in adaptive control of high-order nonlinear system.Automatica, 106:339–348, 2019

    work page 2019

  31. [31]

    Terminal sliding mode control for rigid robots.Automatica, 34(1):51–56, 1998

    Y Tang. Terminal sliding mode control for rigid robots.Automatica, 34(1):51–56, 1998

    work page 1998

  32. [32]

    P. Tomei. Adaptive PD control for robot manipulators,.IEEE Trans. on Robotics Automat., 7(4):565–570, 1991

    work page 1991

  33. [33]

    P. Tomei. Adaptive PD controller for robot manipulators.IEEE Transactions on robotics and automation, 7(4):565–570, 1991

    work page 1991

  34. [34]

    M. Van, M. Mavrovouniotis, and S.S. Ge. An adaptive backstepping nonsingular fast terminal sliding mode control for robust fault tolerant control of robot manipulators.IEEE Transactions on Systems, Man, and Cybernetics: Systems, 49(7):1448–1458, 2019

    work page 2019

  35. [35]

    Venkataraman and S

    S.T. Venkataraman and S. Gulati. Terminal slider control of robot systems.Journal of Intelligent and Robotic Systems, 5:31–55, 1993

    work page 1993

  36. [36]

    J. Wang, D. Efimov, and A. Bobtsov. On robust parameter estimation in finite-time without persistence of excitation.IEEE Transactions on Automatic Control, 65(5):1731–1738, 2020

    work page 2020

  37. [37]

    Y. Yang, C. Hua, J. Li, and X. Guan. Robust adaptive uniform exact tracking control for uncertain Euler- Lagrange system.International Journal of Control, 90(12):2711–2720, 2017

    work page 2017

  38. [38]

    Zhihong, M

    M. Zhihong, M. O’Day, and X. Yu. A robust adaptive terminal sliding mode control for rigid robotic manipulators.Journal of Intelligent and Robotic Systems, 24:23–41, 1999

    work page 1999

  39. [39]

    Zimenko, D

    K. Zimenko, D. Efimov, and A. Polyakov. On condition for output finite-time stability and adaptive finite- time control scheme. InProc. IEEE Conference on Decision and Control, pages 7099–7103, 2019. 30

    work page 2019