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arxiv: 2311.13056 · v2 · pith:TKT5VS4Fnew · submitted 2023-11-21 · 📡 eess.SY · cs.SY

Simultaneous Online System Identification and Control using Composite Adaptive Lyapunov-Based Deep Neural Networks

Omkar Sudhir Patil , Emily J. Griffis , Wanjiku A. Makumi , Warren E. Dixon This is my paper

Pith reviewed 2026-05-25 09:07 UTC · model grok-4.3

classification 📡 eess.SY cs.SY
keywords adaptive controldeep neural networkssystem identificationLyapunov stabilitynonlinear systemstrajectory trackingonline adaptation
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The pith

Lyapunov-based adaptation of all DNN layers enables simultaneous online system identification and trajectory tracking for nonlinear systems.

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

The paper develops update laws for every layer of a deep neural network that adapt online using both tracking error and state-derivative estimates. A single Lyapunov function is constructed to prove that tracking error, derivative estimation error, and all weight errors remain uniformly ultimately bounded. When a persistence of excitation condition holds, the same analysis shows exponential convergence of tracking and weight errors to a neighborhood whose size depends on the gains and the excitation level. The resulting identified model can then be run forward to compensate for missing measurements.

Core claim

This paper provides the first result on simultaneous online system identification and trajectory tracking control of nonlinear systems using adaptive updates for all layers of the DNN. A combined Lyapunov-based stability analysis is provided, which guarantees that the tracking error, state-derivative estimation error, and DNN weight estimation errors are uniformly ultimately bounded. Under the persistence of excitation (PE) condition, the tracking and weight estimation errors are shown to exponentially converge to a neighborhood of the origin, where the rate of convergence and the size of this neighborhood depends on the gains and a factor quantifying PE, thus achieving system identification

What carries the argument

Composite adaptive update laws for DNN weights that use both tracking error and state-derivative estimation error inside a single Lyapunov function.

Load-bearing premise

The persistence of excitation condition must hold for the exponential convergence result.

What would settle it

If tracking and weight errors remain only ultimately bounded and do not converge exponentially when the persistence of excitation condition is satisfied, the stronger claim is falsified.

Figures

Figures reproduced from arXiv: 2311.13056 by Emily J. Griffis, Omkar Sudhir Patil, Wanjiku A. Makumi, Warren E. Dixon.

Figure 1
Figure 1. Figure 1: Comparative plots of the tracking and function approximation error  [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Comparative plots of the RMS function approximation error norm  [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
read the original abstract

Although deep neural network (DNN)-based controllers are popularly used to control uncertain nonlinear dynamic systems, most results use DNNs that are pretrained offline and the corresponding controller is implemented post-training. Recent advancements in adaptive control have developed controllers with Lyapunov-based update laws (i.e., control and update laws derived from a Lyapunov-based stability analysis) for updating the DNN weights online to ensure the system states track a desired trajectory. However, the update laws are based on the tracking error, and offer guarantees on only the tracking error convergence, without providing any guarantees on system identification. This paper provides the first result on simultaneous online system identification and trajectory tracking control of nonlinear systems using adaptive updates for all layers of the DNN. A combined Lyapunov-based stability analysis is provided, which guarantees that the tracking error, state-derivative estimation error, and DNN weight estimation errors are uniformly ultimately bounded. Under the persistence of excitation (PE) condition, the tracking and weight estimation errors are shown to exponentially converge to a neighborhood of the origin, where the rate of convergence and the size of this neighborhood depends on the gains and a factor quantifying PE, thus achieving system identification and enhanced trajectory tracking performance. As an outcome of the system identification, the DNN model can be propagated forward to predict and compensate for the uncertainty in dynamics under intermittent loss of state feedback. Comparative simulation results are provided on a two-link manipulator system and an unmanned underwater vehicle system with intermittent loss of state feedback, where the developed method yields significant performance improvement compared to baseline methods.

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 / 0 minor

Summary. The manuscript proposes a composite adaptive Lyapunov-based approach for simultaneous online system identification and trajectory tracking control of uncertain nonlinear systems using deep neural networks (DNNs) with adaptive updates for all layers. A combined Lyapunov analysis is used to prove uniform ultimate boundedness of the tracking error, state-derivative estimation error, and weight estimation errors. Under a persistence of excitation (PE) condition, exponential convergence of the tracking and weight errors to a neighborhood of zero is shown, with the neighborhood size and convergence rate depending on gains and a PE factor. The approach is demonstrated in simulations on a two-link manipulator and an unmanned underwater vehicle with intermittent state feedback loss, showing improved performance over baselines.

Significance. If the stability claims hold, this would be a notable contribution as it provides the first such result for full DNN layer adaptation enabling both control and identification, with practical implications for systems experiencing feedback interruptions. The simulation results lend support to the performance claims.

major comments (2)
  1. [Abstract] The stronger claims of exponential convergence and system identification hinge on the persistence of excitation (PE) condition for the composite regressor. The paper states this assumption explicitly but offers no indication that the control or composite update laws are constructed to satisfy or monitor PE for the DNN in closed loop; without it, the result reduces to UUB only, which is a load-bearing concern for the identification guarantee.
  2. [Lyapunov Analysis] The abstract describes a combined Lyapunov-based stability analysis producing the UUB and exponential convergence claims, but the full derivation is needed to verify the composite update law construction and the precise handling of the state-derivative estimation error term to ensure the bounds are correctly derived without unaccounted residuals.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive feedback on our manuscript. We address each major comment below.

read point-by-point responses

  1. Referee: [Abstract] The stronger claims of exponential convergence and system identification hinge on the persistence of excitation (PE) condition for the composite regressor. The paper states this assumption explicitly but offers no indication that the control or composite update laws are constructed to satisfy or monitor PE for the DNN in closed loop; without it, the result reduces to UUB only, which is a load-bearing concern for the identification guarantee.

    Authors: We agree that the exponential convergence and associated identification guarantees are conditional on the PE assumption for the composite regressor, which is explicitly stated in the manuscript. As is standard in adaptive control, the laws are not constructed to enforce or monitor PE in closed loop, since guaranteeing PE for general nonlinear systems without additional probing signals (which could degrade tracking) remains an open problem. The primary contribution is the composite adaptation framework that yields UUB unconditionally and exponential convergence under PE. We will revise the abstract, introduction, and conclusions to more explicitly distinguish the UUB result from the conditional exponential result and add a remark on the PE assumption. revision: yes

  2. Referee: [Lyapunov Analysis] The abstract describes a combined Lyapunov-based stability analysis producing the UUB and exponential convergence claims, but the full derivation is needed to verify the composite update law construction and the precise handling of the state-derivative estimation error term to ensure the bounds are correctly derived without unaccounted residuals.

    Authors: The combined Lyapunov analysis, including derivation of the composite update laws for all DNN layers and explicit bounding of the state-derivative estimation error (along with all cross terms and residuals) to obtain the UUB result, is provided in Sections III-B and III-C. The exponential result under PE follows directly from the same Lyapunov function with an additional PE-based term. To address the request for verification, we will add a supplementary appendix containing the complete step-by-step expansion of the Lyapunov derivative with all intermediate bounds shown. revision: yes

Circularity Check

0 steps flagged

No circularity: Lyapunov analysis and PE assumption are independent of paper's own fitted quantities or self-definitions

full rationale

The paper's central claims rest on a combined Lyapunov-based stability analysis that guarantees UUB for tracking, derivative estimation, and weight errors, with exponential convergence under the external PE condition. The PE requirement is stated explicitly as an assumption for the stronger result and is not derived from or reduced to the paper's equations, update laws, or any fitted parameters. No self-definitional steps, fitted inputs renamed as predictions, or load-bearing self-citations appear in the derivation chain. The analysis follows standard adaptive control techniques and remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on two standard domain assumptions in neural adaptive control plus the PE condition; no free parameters or new invented entities are introduced in the abstract.

axioms (2)
  • domain assumption The DNN is capable of approximating the system uncertainty with a bounded reconstruction error.
    Invoked to close the Lyapunov analysis for boundedness of weight estimation errors.
  • domain assumption Persistence of excitation (PE) condition holds on the regressor signals.
    Required for the exponential convergence claim stated in the abstract.

pith-pipeline@v0.9.0 · 5820 in / 1397 out tokens · 39134 ms · 2026-05-25T09:07:02.792985+00:00 · methodology

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Reference graph

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