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arxiv: 2604.06338 · v2 · submitted 2026-04-07 · 🧮 math.OC · cs.SY· eess.SY

Adaptive Control with Sparse Identification of Nonlinear Dynamics

Trivikram Satharasi , Tochukwu E. Ogri , Muzaffar Qureshi , Kyle Volle , Rushikesh Kamalapurkar This is my paper

Pith reviewed 2026-05-10 18:39 UTC · model grok-4.3

classification 🧮 math.OC cs.SYeess.SY
keywords sparse identificationintegral concurrent learningadaptive controlnonlinear dynamicsLyapunov analysissliding modesparameter estimationL1 regularization
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The pith

A sparsity-promoting integral concurrent learning adaptation law identifies sparse parameters online in uncertain nonlinear systems while ensuring ultimately bounded closed-loop trajectories.

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

The paper introduces an online parameter update law called SP-ICL that combines integral concurrent learning with an L1 regularization term to promote sparsity in the identified dynamics. This integration is achieved through sliding modes, allowing the system to learn which parameters are zero without prior knowledge. Non-smooth Lyapunov analysis establishes that the closed-loop trajectories remain ultimately bounded under this law. The approach is relevant for control applications where the underlying dynamics are sparse, as it can simplify models and improve efficiency during real-time operation. Simulations demonstrate its ability to recover sparse structures while tracking trajectories.

Core claim

This paper develops a sparsity-promoting integral concurrent learning (SP-ICL) adaptation law for a linearly parametrized uncertain nonlinear control-affine system. The unknown parameters are learned using ICL with sparsity-promoting ℓ1 regularization. The use of ℓ1 regularization for sparsity promotion is integrated with ICL via sliding modes to create an online parameter update law. Using the SP-ICL update law, the trajectories of the closed-loop system are ultimately bounded as shown via non-smooth Lyapunov analysis. Simulations verify the effectiveness of the sparsity penalty in recovering sparse dynamics during trajectory tracking.

What carries the argument

The SP-ICL update law, which integrates the ℓ1 regularization penalty with integral concurrent learning via sliding modes to promote sparsity in the parameter estimates for adaptive control.

If this is right

  • The closed-loop system trajectories are ultimately bounded despite uncertainties in the nonlinear dynamics.
  • The method enables online recovery of sparse parameter structures in the system model.
  • Integration via sliding modes avoids excessive chattering while maintaining stability.
  • Effective performance is shown in trajectory tracking tasks for uncertain systems.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • This could lead to reduced model complexity in real-time control by automatically setting irrelevant parameters to zero.
  • The approach might extend to other forms of regularization or learning methods in adaptive control.
  • Applications in robotics or autonomous systems could benefit from faster convergence to minimal models.

Load-bearing premise

The unknown parameters can be learned online by integrating the L1 regularization penalty with integral concurrent learning via sliding modes without causing instability or excessive chattering in the system.

What would settle it

An experiment or simulation in which applying the SP-ICL law causes the closed-loop trajectories to become unbounded or introduces significant chattering, or fails to identify the true sparse parameters.

Figures

Figures reproduced from arXiv: 2604.06338 by Kyle Volle, Muzaffar Qureshi, Rushikesh Kamalapurkar, Tochukwu E. Ogri, Trivikram Satharasi.

Figure 1
Figure 1. Figure 1: Parameter estimation error norm ∥θ˜(t)∥ (log scale) for different values of the regularization parameter λ. history stack is constructed online using the data selection procedure described in Algorithm 1. The desired trajectory is defined as xd(t) =  sin(t) + 0.12 sin(3t) − 0.04 sin(5t) 0.95 sin(2t) + 0.08 sin(4t)  . For the experiments, the selected proportional error gain is K = 10I2, the relative conc… view at source ↗
Figure 3
Figure 3. Figure 3: Confusion matrix statistics for sparse term recovery. Classification performance is reported for nonzero (positive) and zero (negative) terms using [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
read the original abstract

This paper develops a sparsity-promoting integral concurrent learning (SP-ICL) adaptation law for a linearly parametrized uncertain nonlinear control-affine system. The unknown parameters are learned using ICL with sparsity-promoting $\ell_1$ regularization. The use of $\ell_1$ regularization for sparsity promotion is common in system identification and machine learning; however, unlike existing approaches, this paper develops an online parameter update law that integrates the regularization penalty with ICL via sliding modes. Using the SP-ICL update law, we show via non-smooth Lyapunov analysis that the trajectories of the closed-loop system are ultimately bounded. Simulations verify the effectiveness of the sparsity penalty in the SP-ICL update law on recovering sparse dynamics during trajectory tracking.

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 develops a sparsity-promoting integral concurrent learning (SP-ICL) adaptation law for linearly parametrized uncertain nonlinear control-affine systems. It combines standard ICL with an ℓ1 regularization penalty realized through sliding modes, claims to prove ultimate boundedness of closed-loop trajectories via non-smooth Lyapunov analysis, and reports simulation results showing improved recovery of sparse dynamics during trajectory tracking.

Significance. If the non-smooth stability argument is rigorous, the result would offer a practical online method for promoting sparsity in adaptive control without destabilizing the closed loop, which is relevant for applications requiring interpretable models. The simulation verification of sparsity effects is a positive feature, but the absence of explicit error bounds or detailed data in the high-level description limits evaluation of practical impact.

major comments (2)
  1. [non-smooth Lyapunov analysis (likely §4)] The central claim of ultimate boundedness rests on non-smooth Lyapunov analysis of the SP-ICL law. The manuscript must explicitly construct the Filippov regularization of the sliding-mode realization of the nondifferentiable ℓ1 term and verify that the generalized gradient of the Lyapunov function contains the convex hull of the set-valued vector field outside a compact set; otherwise the passage to a negative-definite derivative (and thus the existence of the ultimate bound) does not follow.
  2. [SP-ICL update law (likely §3)] The update law definition must be shown to avoid both instability and excessive chattering. Provide the explicit differential inclusion induced by the sliding-mode ℓ1 term and demonstrate that the switching does not persist indefinitely or grow with the sliding gain, as this directly affects whether the boundedness result remains uniform.
minor comments (2)
  1. [Abstract and §1] The abstract and introduction should clarify the precise class of systems (control-affine, linearly parametrized) and state any assumptions on the regressor matrix or persistence of excitation that are used in the boundedness proof.
  2. [Simulations] Simulation section should include quantitative metrics (e.g., parameter error norms, sparsity level achieved, comparison against plain ICL) and specify the numerical integration method used for the sliding-mode dynamics.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments on our manuscript. We address each major comment below, indicating the revisions we will incorporate to strengthen the rigor of the non-smooth analysis and the explicit characterization of the update law.

read point-by-point responses

  1. Referee: [non-smooth Lyapunov analysis (likely §4)] The central claim of ultimate boundedness rests on non-smooth Lyapunov analysis of the SP-ICL law. The manuscript must explicitly construct the Filippov regularization of the sliding-mode realization of the nondifferentiable ℓ1 term and verify that the generalized gradient of the Lyapunov function contains the convex hull of the set-valued vector field outside a compact set; otherwise the passage to a negative-definite derivative (and thus the existence of the ultimate bound) does not follow.

    Authors: We agree that the non-smooth analysis would benefit from greater explicitness. In the revised manuscript we will add the explicit Filippov regularization of the sliding-mode ℓ1 term, state the associated differential inclusion, and verify that the generalized gradient of the candidate Lyapunov function contains the convex hull of the set-valued vector field outside a compact set. This will be inserted as a new lemma in §4 immediately preceding the ultimate-boundedness theorem, thereby making the passage to the negative-definite derivative fully rigorous. revision: yes

  2. Referee: [SP-ICL update law (likely §3)] The update law definition must be shown to avoid both instability and excessive chattering. Provide the explicit differential inclusion induced by the sliding-mode ℓ1 term and demonstrate that the switching does not persist indefinitely or grow with the sliding gain, as this directly affects whether the boundedness result remains uniform.

    Authors: We will augment §3 with the explicit differential inclusion generated by the sliding-mode realization of the ℓ1 penalty. We will also prove that the measure of the switching set remains uniformly bounded (independent of the sliding gain) by exploiting the integral concurrent-learning term and the positive-definiteness of the regressor matrix along the trajectory. This establishes that chattering does not persist indefinitely and that the ultimate bound derived in §4 remains uniform with respect to the regularization parameter. revision: yes

Circularity Check

0 steps flagged

No significant circularity; boundedness follows from independent non-smooth Lyapunov analysis of the novel update law.

full rationale

The paper defines a new SP-ICL adaptation law by combining standard ICL with an l1 penalty realized through sliding modes, then applies non-smooth Lyapunov analysis to the resulting closed-loop dynamics to establish ultimate boundedness. This chain does not reduce any claimed result to its own inputs by construction, nor does it rely on load-bearing self-citations whose content is unverified or tautological. The stability argument uses standard Filippov regularization and generalized gradients on the introduced dynamics, without redefining boundedness in terms of fitted parameters or renaming prior empirical patterns. The derivation remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the assumption that the plant is linearly parametrized and control-affine, plus standard results from non-smooth analysis; no new entities are postulated and no free parameters are explicitly fitted in the abstract.

axioms (2)
  • domain assumption The uncertain nonlinear system is linearly parametrized and control-affine.
    Explicitly stated as the class of systems for which the SP-ICL law is developed.
  • standard math Non-smooth Lyapunov analysis can be used to prove ultimate boundedness for the closed-loop system with the proposed update law.
    Invoked to establish stability of the trajectories.

pith-pipeline@v0.9.0 · 5445 in / 1292 out tokens · 42780 ms · 2026-05-10T18:39:46.170908+00:00 · methodology

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