基于增广拉格朗日方法的数据驱动系统神经反馈线性化学习

# Learning Neural Feedback Linearization for Data-driven Systems via Augmented Lagrangian
Source: [https://arxiv.org/html/2609.25163](https://arxiv.org/html/2609.25163)
Andreas Schwung††thanks:Lakshmi Priya P\. K\. and Andreas Schwung are with the Department of Automation Technology and Learning Systems, South Westphalia University of Applied Sciences,Lübecker Ring 2, Soest, 59494, Germany \(pondicherrykumar\.lakshmipriya@fh\-swf\.de; schwung\.andreas@fh\-swf\.de\)\.

###### Abstract

The paper proposes a novel data\-driven framework for designing and training a feedback linearizing controller by explicitly incorporating relative degree based conditions into the learning process\. This enables the conventional feedback controller components to be replaced by neural Lie derivatives, thereby facilitating a fully data\-driven feedback linearization framework\. Furthermore, practical closed\-loop stability is established by deriving sufficient conditions under which bounded identification errors lead to bounded tracking errors\. The derived theoretical results are validated through their application to an armature controlled DC motor\.

###### Index Terms:

Closed\-loop stability, data\-driven control systems, feedback\-linearization, neural Lie derivatives

## IIntroduction

Feedback linearization is a well established nonlinear control technique that converts a nonlinear dynamical system into an equivalent linear representation through a suitable change of coordinates and a nonlinear control law\. This allows for systematic controller design, stability analysis, and performance evaluation using established linear systems theory\. Consequently, feedback linearization has been widely applied to nonlinear systems characterized by complex dynamical behavior\[[3](https://arxiv.org/html/2609.25163#bib.bib3)\]\. However, its practical implementation relies on accurate knowledge of the system model, which may not be readily available for complex intricate systems\.

Data\-driven approaches address this challenge by using measurement data directly for controller synthesis and analysis\[[15](https://arxiv.org/html/2609.25163#bib.bib15)\]\. While this reduces reliance on explicit physical equations, it introduces additional challenges in experiment design, model structure selection, parameter estimation, and interpretation\[[25](https://arxiv.org/html/2609.25163#bib.bib25),[24](https://arxiv.org/html/2609.25163#bib.bib24)\]\. Early pioneering research in data\-driven control \(DDC\) can be found in\[[36](https://arxiv.org/html/2609.25163#bib.bib36),[9](https://arxiv.org/html/2609.25163#bib.bib9),[2](https://arxiv.org/html/2609.25163#bib.bib2),[28](https://arxiv.org/html/2609.25163#bib.bib28)\]\.

Due to recent major advancements in machine learning, data\-driven modeling have gained considerable attention and widespread adoption\[[19](https://arxiv.org/html/2609.25163#bib.bib19),[31](https://arxiv.org/html/2609.25163#bib.bib31)\]\. Neural networks provide an effective framework for data\-driven modeling as flexible function approximators capable of capturing complex nonlinear system dynamics directly from measured data\[[5](https://arxiv.org/html/2609.25163#bib.bib5)\]\. Their differentiable structure also enables to jointly learn the system representation and construct controller directly from data\[[14](https://arxiv.org/html/2609.25163#bib.bib14),[26](https://arxiv.org/html/2609.25163#bib.bib26),[30](https://arxiv.org/html/2609.25163#bib.bib30),[35](https://arxiv.org/html/2609.25163#bib.bib35)\]\. The increasing availability of measured process data has also motivated data\-driven feedback linearization approaches\[[21](https://arxiv.org/html/2609.25163#bib.bib21)\]\. These methods retain advantages of feedback linearization while extending its applicability to systems with unknown or difficult to derive governing dynamics\.

Since analytical tools in nonlinear control theory are grounded in differential\-geometric concepts, the corresponding feedback linearization requires the learned neural network model to preserve the structural properties necessary for enforcing relative degree condition\. However, an approach which enforces the relative degree condition during training, and links learning errors to closed\-loop tracking performance is missing in the literature\. Existing data\-driven feedback linearization methods either learn predictive models without explicitly incorporating the geometric structure required for feedback linearization or recover linearization transformations from predefined dictionaries using sparse regression\[[4](https://arxiv.org/html/2609.25163#bib.bib4),[7](https://arxiv.org/html/2609.25163#bib.bib7),[11](https://arxiv.org/html/2609.25163#bib.bib11)\]\. Furthermore, stability of the resulting closed\-loop controller is generally not addressed, particularly in the presence of bounded identification errors\.

We overcome these limitations by introducing a data\-driven feedback linearization framework using multilayer perceptron \(MLP\) neural network\. The differential geometric structure are incorporated directly into the learning process through an Augmented\-Lagrangian formulation\. The feedback\-linearizing controller is directly defined by the networks, promoting consistency between model and structural requirements of the controller design\. A practical stability result is established showing that bounded identification errors lead to ultimately bounded tracking errors, providing a closed\-loop stability characterization for the proposed learning\-based control\.

#### I\-1Related work

The existing literature can be broadly divided into direct and indirect control design\. Indirect approaches first identify a system model and subsequently use it for controller design\[[4](https://arxiv.org/html/2609.25163#bib.bib4)\]\. Sparse identification and Koopman\-based approaches establish structured data\-driven representations of nonlinear dynamics, providing a systematic foundation for subsequent control synthesis\[[11](https://arxiv.org/html/2609.25163#bib.bib11)\]\. Koopman generator least\-squares formulations recover linearizing state and control transformations from prescribed function dictionaries\[[8](https://arxiv.org/html/2609.25163#bib.bib8)\]\. The integration of learning techniques with feedback linearization has also been investigated using Gaussian processes\[[30](https://arxiv.org/html/2609.25163#bib.bib30)\]\. Robustness of sparse identification with feedback linearization and an additional observer for finite\-time compensation of modeling errors and disturbances is studied in\[[23](https://arxiv.org/html/2609.25163#bib.bib23)\]\. The authors of\[[35](https://arxiv.org/html/2609.25163#bib.bib35)\]use neural networks to approximate the unknown system online which are used to construct the feedback\-linearizing controller which ensures uniform ultimate boundedness of all closed\-loop signals via Lyapunov analysis\.

In direct data\-driven control, the controller is obtained directly from measured data without first identifying an explicit model of the plant\[[22](https://arxiv.org/html/2609.25163#bib.bib22),[20](https://arxiv.org/html/2609.25163#bib.bib20)\]\. This approach reduces the intermediate modeling step and can be useful when the main objective is controller synthesis rather than model reconstruction\. It supports the idea that system trajectories can be represented directly from data under persistency of excitation\[[34](https://arxiv.org/html/2609.25163#bib.bib34)\]\. Neural networks were initially used as nonlinear identifiers and adaptive control models, with subsequent extensions to architectures tailored towards controller synthesis\[[29](https://arxiv.org/html/2609.25163#bib.bib29),[14](https://arxiv.org/html/2609.25163#bib.bib14)\]\. Representative approaches include quasi\-ARX models and control\-affine neural NARX models\[[10](https://arxiv.org/html/2609.25163#bib.bib10),[32](https://arxiv.org/html/2609.25163#bib.bib32)\]\. In\[[27](https://arxiv.org/html/2609.25163#bib.bib27)\], the authors employed a neural network to derive state and control transformations, while\[[33](https://arxiv.org/html/2609.25163#bib.bib33)\]introduced an approach based on reinforcement learning\. Sampled\-data stabilization has been established for single\-input single\-output \(SISO\) systems without explicit model identification\[[7](https://arxiv.org/html/2609.25163#bib.bib7)\], whereas behavioral formulations provide data\-based trajectory representations for multiple\-input multiple\-output \(MIMO\) feedback\-linearizable systems under basis\-function approximation and noisy output measurements\[[1](https://arxiv.org/html/2609.25163#bib.bib1)\]\. A data\-driven dynamic output\-feedback controller was developed to robustly stabilize linear MIMO systems using only noisy input–output measurements\[[18](https://arxiv.org/html/2609.25163#bib.bib18)\]\. Data\-driven output\-feedback stabilization under measurement noise and neglected nonlinearities has been studied through auxiliary input–output representations and data\-dependent linear matrix inequalities, although not within a feedback\-linearization framework\[[6](https://arxiv.org/html/2609.25163#bib.bib6)\]\. A related result studies data\-driven feedback linearization using stacked regression with functional dictionaries\[[17](https://arxiv.org/html/2609.25163#bib.bib17)\]\.

#### I\-2Contributions

Current literature on data\-driven control approaches either learn system representations without explicitly preserving the differential geometric structure required for feedback linearization or derive linearizing transformations while closed\-loop behavior in the presence of identification errors remains insufficiently characterized\. To address these limitations, we make the following contributions:

- •We propose a data\-driven feedback\-linearization architecture for unknown nonlinear systems by using neural networks to approximate the system dynamics\.
- •We formulate a constrained optimization framework to enforce the required relative degree condition during training which is reformulated as an unconstrained optimization problem using augmented Lagrangian multiplier technique\.
- •A practical closed\-loop stability result is established showing that bounded identification error in the learned vector fields leads to ultimately bounded tracking error and the learning feedback\-linearizing control architecture remains stable\.
- •The proposed learning\-based feedback\-linearization framework and the resulting closed\-loop stability is validated on an armature\-controlled DC motor\.

The article is organized as follows: Section 2 reviews the fundamental concepts and introduces the neural identification framework\. Section 3 presents the main theoretical results, i\.e\. the data\-driven feedback linearizing control using an augmented Lagrangian framework\. Section 4 establishes practical closed\-loop stability\. Section 5 applies the approach to an armature\-controlled DC motor\. Section 6 concludes the paper\.

## IIBasic concepts and Preliminaries

In this section, we recall the foundational concepts underlying in identification and feedback control of model free systems using neural networks\. In particular, we discuss how a MLP architecture can be constructed to model a nonlinear control system solely from the process data\.

### II\-AData Collection and Model Construction

Consider the following nonlinear control affine system

x˙​\(𝓉\)\\displaystyle\\dot\{\\mathrm\{x\}\}\(\\mathpzc\{t\}\)=𝖿⁡\(x⁡\(𝓉\)\)\+𝗀⁡\(x⁡\(𝓉\)\)​𝗎​\(𝓉\),\\displaystyle=\\mathsf\{f\}\(\\mathrm\{x\}\(\\mathpzc\{t\}\)\)\+\\mathsf\{g\}\(\\mathrm\{x\}\(\\mathpzc\{t\}\)\)\\mathsf\{u\}\(\\mathpzc\{t\}\),𝗒⁡\(𝓉\)\\displaystyle\\mathsf\{y\}\(\\mathpzc\{t\}\)=𝖼⁡\(x⁡\(𝓉\)\)\.\\displaystyle=\\mathsf\{c\}\(\\mathrm\{x\}\(\\mathpzc\{t\}\)\)\.\(1\)The statex⁡\(𝓉\)∈ℛ𝓃,\\mathrm\{x\}\(\\mathpzc\{t\}\)\\in\\mathscr\{R\}^\{n\},𝗎⁡\(𝓉\)∈ℛ𝓃𝓊\\mathsf\{u\}\(\\mathpzc\{t\}\)\\in\\mathscr\{R\}^\{n\_\{u\}\}is the control input and𝗒⁡\(𝓉\)∈ℛ𝓃𝓎\\mathsf\{y\}\(\\mathpzc\{t\}\)\\in\\mathscr\{R\}^\{n\_\{y\}\}is the output of the system\. The system is considered in the SISO setting, withny=nu=1n\_\{y\}=n\_\{u\}=1, and is assumed to have a well defined, known relative degreer=nr=n\. We note, that the methodology can be generalized to MIMO systems while extension to the case with relative degree\(r=δ<n\)\(r=\\delta<n\)for unknown systems has been discussed in\[[17](https://arxiv.org/html/2609.25163#bib.bib17)\]using a sparse regression approach, which can be extended to the setting presented in this paper\. A finite number ofmmdata samples ofx⁡\(𝓉\),\\mathrm\{x\}\(\\mathpzc\{t\}\),𝗎⁡\(𝓉\)\\mathsf\{u\}\(\\mathpzc\{t\}\)and𝗒⁡\(𝓉\)\\mathsf\{y\}\(\\mathpzc\{t\}\)are collected and organized in a standard form\[[4](https://arxiv.org/html/2609.25163#bib.bib4)\]\. The collected process data are represented in the formX=\[x⁡\(𝓉1\),…,x⁡\(𝓉𝓂\)\]T∈ℛm×n,X˙=\[x˙​\(𝓉1\),…,x˙​\(𝓉𝓂\)\]T∈ℛm×n,𝖴=\[𝗎⁡\(𝓉1\),…,𝗎⁡\(𝓉𝓂\)\]T∈ℛm×1,𝖸=\[𝗒⁡\(𝓉1\),…,𝗒⁡\(𝓉𝓂\)\]T∈ℛm×1\\mathrm\{X\}=\\left\[\\mathrm\{x\}\(\\mathpzc\{t\}\_\{1\}\),\\ldots,\\mathrm\{x\}\(\\mathpzc\{t\}\_\{m\}\)\\right\]^\{T\}\\in\\mathscr\{R\}^\{m\\times n\},\\dot\{\\mathrm\{X\}\}=\\left\[\\dot\{\\mathrm\{x\}\}\(\\mathpzc\{t\}\_\{1\}\),\\ldots,\\dot\{\\mathrm\{x\}\}\(\\mathpzc\{t\}\_\{m\}\)\\right\]^\{T\}\\in\\mathscr\{R\}^\{m\\times n\},\\mathsf\{U\}=\\left\[\\mathsf\{u\}\(\\mathpzc\{t\}\_\{1\}\),\\ldots,\\mathsf\{u\}\(\\mathpzc\{t\}\_\{m\}\)\\right\]^\{T\}\\in\\mathscr\{R\}^\{m\\times 1\},\\mathsf\{Y\}=\\left\[\\mathsf\{y\}\(\\mathpzc\{t\}\_\{1\}\),\\ldots,\\mathsf\{y\}\(\\mathpzc\{t\}\_\{m\}\)\\right\]^\{T\}\\in\\mathscr\{R\}^\{m\\times 1\}\. If direct measurements of the derivatives are not readily available, they may be approximated numerically from the observed state data\[[16](https://arxiv.org/html/2609.25163#bib.bib16)\]\. The quality of the collected data plays a critical role in the predictive performance of the learned model\. The data set is assumed to be persistently exciting and to provide sufficient coverage of the considered operating region for reliable identification of the system dynamics\. Having constructed the dataset, the subsequent step is to develop the neural network architecture for the considered system\.

### II\-BNeural network architecture

We present the MLP parameterized approximations𝖿ϕ\\mathsf\{f\}\_\{\\upphi\},𝗀ϕ\\mathsf\{g\}\_\{\\upphi\}and𝖼ϕ\\mathsf\{c\}\_\{\\upphi\}using sampled data\. By learning directly from data, the network parameters capture the underlying structure of the system which enables the learned representation to approximate the observed state trajectories of the true system\. Let𝖿ϕ:Ω→ℛn,𝗀ϕ:Ω→ℛn,\\mathsf\{f\}\_\{\\upphi\}:\\Omega\\rightarrow\\mathscr\{R\}^\{n\},\\,\\mathsf\{g\}\_\{\\upphi\}:\\Omega\\rightarrow\\mathscr\{R\}^\{n\},\\,and𝖼ϕ:Ω→ℛny,Ω⊂ℛn\\mathsf\{c\}\_\{\\upphi\}:\\Omega\\rightarrow\\mathscr\{R\}^\{n\_\{y\}\},\\,\\Omega\\subset\\mathscr\{R\}^\{n\}is an open subset, then

x˙^​\(𝓉\)\\displaystyle\\hat\{\\dot\{\\mathrm\{x\}\}\}\(\\mathpzc\{t\}\)=𝖿ϕ​\(x⁡\(𝓉\)\)\+𝗀ϕ​\(x⁡\(𝓉\)\)​𝗎​\(𝓉\),\\displaystyle=\\mathsf\{f\}\_\{\\upphi\}\(\\mathrm\{x\}\(\\mathpzc\{t\}\)\)\+\\mathsf\{g\}\_\{\\upphi\}\(\\mathrm\{x\}\(\\mathpzc\{t\}\)\)\\mathsf\{u\}\(\\mathpzc\{t\}\),𝗒^​\(𝓉\)\\displaystyle\\hat\{\\mathsf\{y\}\}\(\\mathpzc\{t\}\)=𝖼ϕ​\(x​\(𝓉\)\)\.\\displaystyle=\\mathsf\{c\}\_\{\\upphi\}\(\\mathrm\{x\}\(\\mathpzc\{t\}\)\)\.\(2\)Fork=1,2,⋯,m,k=1,2,\\cdots,m,\\,x˙^k=𝖿ϕ​\(xk\)\+𝗀ϕ​\(xk\)​𝗎k∈ℛn\\hat\{\\dot\{\\mathrm\{x\}\}\}\_\{k\}=\\mathsf\{f\}\_\{\\upphi\}\(\\mathrm\{x\}\_\{k\}\)\+\\mathsf\{g\}\_\{\\upphi\}\(\\mathrm\{x\}\_\{k\}\)\\mathsf\{u\}\_\{k\}\\,\\in\\mathscr\{R\}^\{n\}defines the prediction dynamics per sample\. Letni​nn\_\{in\},nhn\_\{h\}, andnon\_\{o\}denote the number of neurons in the input, hidden, and output layers, respectively\. For anll\-layer MLP let𝒲\(1\),𝒱\(1\)∈ℛnh×ni​n,𝒲\(i\),𝒱\(i\)∈ℛnh×nh,i=2,…,l−1,𝒲\(l\)∈ℛno×nh,𝒱\(l\)∈ℛ1×nh,b\(i\),a\(i\)∈ℛnh,i=1,…,l−1,b\(l\)∈ℛno,a\(l\)∈ℛ\\mathscr\{W\}^\{\(1\)\},\\mathscr\{V\}^\{\(1\)\}\\in\\mathscr\{R\}^\{n\_\{h\}\\times n\_\{in\}\},\\mathscr\{W\}^\{\(i\)\},\\mathscr\{V\}^\{\(i\)\}\\in\\mathscr\{R\}^\{n\_\{h\}\\times n\_\{h\}\},\\;i=2,\\ldots,l\-1,\\mathscr\{W\}^\{\(l\)\}\\in\\mathscr\{R\}^\{n\_\{o\}\\times n\_\{h\}\},\\mathscr\{V\}^\{\(l\)\}\\in\\mathscr\{R\}^\{1\\times n\_\{h\}\},b^\{\(i\)\},a^\{\(i\)\}\\in\\mathscr\{R\}^\{n\_\{h\}\},\\;i=1,\\ldots,l\-1,b^\{\(l\)\}\\in\\mathscr\{R\}^\{n\_\{o\}\},a^\{\(l\)\}\\in\\mathscr\{R\}\. Then forx∈ℛn\\mathrm\{x\}\\in\\mathscr\{R\}^\{n\}, the drift, input\-vector\-filed and the output network are represented as

𝖿ϕ​\(x\)\\displaystyle\\mathsf\{f\}\_\{\\upphi\}\(\\mathrm\{x\}\)=𝒲𝖿\(l\)σ\(𝒲𝖿\(l−1\)σ\(⋯σ\(𝒲𝖿\(1\)x\+b𝖿\(1\)\)⋯\)\+b𝖿\(l−1\)\)\+b𝖿\(l\),\\displaystyle\\\!=\\\!\\mathscr\{W\}\_\{\\mathsf\{f\}\}^\{\(l\)\}\\\!\\sigma\\\!\\left\(\\mathscr\{W\}\_\{\\mathsf\{f\}\}^\{\(l\-1\)\}\\\!\\sigma\\\!\\left\(\\\!\\cdots\\\!\\sigma\\\!\\left\(\\\!\\mathscr\{W\}\_\{\\mathsf\{f\}\}^\{\(1\)\}\\\!\\mathrm\{x\}\\\!\+\\\!b\_\{\\mathsf\{f\}\}^\{\(1\)\}\\\!\\right\)\\\!\\cdots\\\!\\right\)\\\!\+\\\!b\_\{\\mathsf\{f\}\}^\{\(l\-1\)\}\\\!\\right\)\\\!\+\\\!b\_\{\\mathsf\{f\}\}^\{\(l\)\}\\\!,𝗀ϕ​\(x\)\\displaystyle\\mathsf\{g\}\_\{\\upphi\}\(\\mathrm\{x\}\)=𝒲𝗀\(l\)σ\(𝒲𝗀\(l−1\)σ\(⋯σ\(𝒲𝗀\(1\)x\+b𝗀\(1\)\)⋯\)\+b𝗀\(l−1\)\)\+b𝗀\(l\),\\displaystyle\\\!=\\\!\\mathscr\{W\}\_\{\\mathsf\{g\}\}^\{\(l\)\}\\\!\\sigma\\\!\\left\(\\mathscr\{W\}\_\{\\mathsf\{g\}\}^\{\(l\-1\)\}\\\!\\sigma\\\!\\left\(\\\!\\cdots\\\!\\sigma\\\!\\left\(\\\!\\mathscr\{W\}\_\{\\mathsf\{g\}\}^\{\(1\)\}\\\!\\mathrm\{x\}\\\!\+\\\!b\_\{\\mathsf\{g\}\}^\{\(1\)\}\\\!\\right\)\\\!\\cdots\\\!\\right\)\\\!\+\\\!b\_\{\\mathsf\{g\}\}^\{\(l\-1\)\}\\\!\\right\)\\\!\+\\\!b\_\{\\mathsf\{g\}\}^\{\(l\)\}\\\!,𝖼ϕ​\(x\)\\displaystyle\\mathsf\{c\}\_\{\\upphi\}\(\\mathrm\{x\}\)=𝒱\(l\)σ\(𝒱\(l−1\)σ\(⋯σ\(𝒱\(1\)x\+a\(1\)\)⋯\)\+a\(l−1\)\)\+a\(l\)\.\\displaystyle\\\!=\\\!\\mathscr\{V\}^\{\(l\)\}\\\!\\sigma\\\!\\left\(\\mathscr\{V\}^\{\(l\-1\)\}\\\!\\sigma\\\!\\left\(\\\!\\cdots\\\!\\sigma\\\!\\left\(\\mathscr\{V\}^\{\(1\)\}\\mathrm\{x\}\\\!\+\\\!a^\{\(1\)\}\\right\)\\\!\\cdots\\\!\\right\)\\\!\+\\\!a^\{\(l\-1\)\}\\right\)\\\!\+\\\!a^\{\(l\)\}\\\!\.
Note that forward equations of𝖼ϕ​\(x\)\\mathsf\{c\}\_\{\\upphi\}\(\\mathrm\{x\}\)through layers can be equivalently represented by

z𝖼\(1\)​\(x\)\\displaystyle z\_\{\\mathsf\{c\}\}^\{\(1\)\}\(\\mathrm\{x\}\)=𝒱\(1\)​x\+a\(1\)∈ℛnh,h𝖼\(1\)​\(x\)=σ⁡\(z𝖼\(1\)​\(x\)\)∈ℛnh,\\displaystyle\\\!=\\\!\\mathscr\{V\}^\{\(1\)\}\\mathrm\{x\}\\\!\+\\\!a^\{\(1\)\}\\\!\\in\\\!\\mathscr\{R\}^\{n\_\{h\}\},\\,h\_\{\\mathsf\{c\}\}^\{\(1\)\}\(\\mathrm\{x\}\)\\\!=\\\!\\sigma\\\!\\left\(\\\!z\_\{\\mathsf\{c\}\}^\{\(1\)\}\\\!\(\\mathrm\{x\}\)\\\!\\right\)\\\!\\\!\\in\\\!\\mathscr\{R\}^\{n\_\{h\}\},z𝖼\(i\)​\(x\)\\displaystyle z\_\{\\mathsf\{c\}\}^\{\(i\)\}\(\\mathrm\{x\}\)=𝒱\(i\)​h𝖼\(i−1\)​\(x\)\+a\(i\)∈ℛnh,h𝖼\(i\)​\(x\)=σ⁡\(z𝖼\(i\)​\(x\)\)∈ℛnh,\\displaystyle\\\!=\\\!\\mathscr\{V\}^\{\(i\)\}h\_\{\\mathsf\{c\}\}^\{\(i\-1\)\}\\\!\(\\mathrm\{x\}\)\\\!\+\\\!a^\{\(i\)\}\\\!\\\!\\in\\\!\\mathscr\{R\}^\{n\_\{h\}\},\\,h\_\{\\mathsf\{c\}\}^\{\(i\)\}\\\!\(\\mathrm\{x\}\)\\\!=\\\!\\sigma\\\!\\left\(\\\!z\_\{\\mathsf\{c\}\}^\{\(i\)\}\(\\mathrm\{x\}\)\\\!\\right\)\\\!\\\!\\in\\\!\\mathscr\{R\}^\{n\_\{h\}\},\\,i\\displaystyle i=2,…,l−1,𝖼ϕ​\(x\)=𝒱\(l\)​h𝖼\(l−1\)​\(x\)\+a\(l\)∈ℛ\.\\displaystyle\\\!=\\\!2,\\ldots,l\\\!\-\\\!1,\\quad\\mathsf\{c\}\_\{\\upphi\}\(\\mathrm\{x\}\)\\\!=\\\!\\mathscr\{V\}^\{\(l\)\}h\_\{\\mathsf\{c\}\}^\{\(l\-1\)\}\\\!\(\\mathrm\{x\}\)\\\!\+\\\!a^\{\(l\)\}\\\!\\\!\\in\\\!\\mathscr\{R\}\.
Having defined the neural network representation of the unknown system we define the loss function of \([2](https://arxiv.org/html/2609.25163#S2.Ex2)\) as the empirical mean squared error between the target values and the network predictions:

ℒ⁡\(ϕ\)\\displaystyle\\mathscr\{L\}\(\\upphi\)=‖X˙−𝖿ϕ​\(X\)−𝗀ϕ​\(X\)⊙𝖴‖22\+λ​‖𝖸−𝖼ϕ​\(X\)‖22,\\displaystyle\\\!=\\\!\\left\\\|\\dot\{\\mathrm\{X\}\}\\\!\-\\\!\\mathsf\{f\}\_\{\\upphi\}\(\\mathrm\{X\}\)\\\!\-\\\!\\mathsf\{g\}\_\{\\upphi\}\(\\mathrm\{X\}\)\\odot\\mathsf\{U\}\\right\\\|\_\{2\}^\{2\}\\\!\+\\\!\\lambda\\left\\\|\\mathsf\{Y\}\\\!\-\\\!\\mathsf\{c\}\_\{\\upphi\}\(\\mathrm\{X\}\)\\right\\\|\_\{2\}^\{2\},\(3\)where⊙\\odotdenotes the Hadamard product\. The parameterλ\>0,\\lambda\>0,is a weighting factor, determines the relative importance of the output prediction error compared to the state prediction error in the combined loss function\. For data\-driven feedback linearization the optimization \([3](https://arxiv.org/html/2609.25163#S2.E3)\) is not sufficient to ensure that the learned model satisfies the structural conditions required for feedback linearization\. Therefore, additional constraints have to be introduced to enforce the relative\-degree conditions\.

### II\-CFeedback linearization

This section introduces the preliminary concepts of feedback linearization\[[13](https://arxiv.org/html/2609.25163#bib.bib13)\]\. Feedback linearization transforms the original nonlinear system using a smooth and invertible coordinate transformation \(diffeomorphism\)𝓆:ℛ𝓃→ℛ𝓃\\mathpzc\{q\}:\\mathscr\{R\}^\{n\}\\rightarrow\\mathscr\{R\}^\{n\}by means of Lie derivativesℒ𝖺​𝖻​\(x\)=∂𝖻⁡\(x\)∂x​𝖺​\(x\)\\mathcal\{L\}\_\{\\mathsf\{a\}\}\\mathsf\{b\}\(\\mathrm\{x\}\)=\\frac\{\\partial\\mathsf\{b\}\(\\mathrm\{x\}\)\}\{\\partial\\mathrm\{x\}\}\\,\\mathsf\{a\}\(\\mathrm\{x\}\), yielding a new coordinate frame with linear closed\-loop dynamics\. The integerr,r,1≤r≤n1\\leq r\\leq n, for whichℒ𝗀ℒ𝖿p−1𝖼\(x\)=0,p=1,…,r−1,\\mathcal\{L\}\_\{\\mathsf\{g\}\}\\mathcal\{L\}\_\{\\mathsf\{f\}\}^\{p\-1\}\\,\\mathsf\{c\}\(\\mathrm\{x\}\)=0,\\,p=1,\\ldots,r\-1,andℒ𝗀​ℒ𝖿r−1​𝖼​\(x\)≠0,\\mathcal\{L\}\_\{\\mathsf\{g\}\}\\mathcal\{L\}\_\{\\mathsf\{f\}\}^\{r\-1\}\\,\\mathsf\{c\}\(\\mathrm\{x\}\)\\neq 0,hold is defined as the relative degree of the system\. Within the transformed framework, the feedback control is defined as

𝗎\\displaystyle\\mathsf\{u\}=−ℒ𝖿n​𝖼​\(x\)\+kn−1​ℒ𝖿n−1​𝖼​\(x\)\+⋯\+k0​𝖼​\(x\)−V​𝓌ℒ𝗀​ℒ𝖿n−1​𝖼​\(x\)\\displaystyle\\\!=\\\!\-\\dfrac\{\\mathcal\{L\}\_\{\\mathsf\{f\}\}^\{n\}\\mathsf\{c\}\(\\mathrm\{x\}\)\\\!\+\\\!k\_\{n\-1\}\\mathcal\{L\}\_\{\\mathsf\{f\}\}^\{n\-1\}\\mathsf\{c\}\(\\mathrm\{x\}\)\\\!\+\\\!\\cdots\\\!\+\\\!k\_\{0\}\\mathsf\{c\}\(\\mathrm\{x\}\)\\\!\-\\\!V\\mathpzc\{w\}\}\{\\mathcal\{L\}\_\{\\mathsf\{g\}\}\\mathcal\{L\}\_\{\\mathsf\{f\}\}^\{\\,n\-1\}\\mathsf\{c\}\(\\mathrm\{x\}\)\}\(4\)gives the closed\-loop input\-output dynamics, where the auxiliary input is introduced so that

V​𝓌=𝗒\(𝓃\)\+𝓀𝓃−1​𝗒\(𝓃−1\)\+⋯\+𝓀1​𝗒˙\+𝓀0​𝗒,V\\mathpzc\{w\}=\\mathsf\{y\}^\{\(n\)\}\+k\_\{n\-1\}\\mathsf\{y\}^\{\(n\-1\)\}\+\\cdots\+k\_\{1\}\\dot\{\\mathsf\{y\}\}\+k\_\{0\}\\mathsf\{y\},𝓌\\mathpzc\{w\}is the new external input to the linearized system andVVis a prefilter gain\. In the proposed framework, conventional control \([4](https://arxiv.org/html/2609.25163#S2.E4)\) is replaced with neural Lie derivative components, enabling the design and training of data\-driven controller\.

## IIIData\-driven feedback linearization for Composite neural network

We now formulates the data\-driven feedback linearization which is subdivided into a learning stage and a control stage, see Fig\.[1](https://arxiv.org/html/2609.25163#S3.F1)\. In the learning stage, process data collected from the nonlinear plant is used to train MLP representations of the unknown system, minimize the discrepancy between learned model and measured plant data\. In parallel, structural conditions to enforce the relative degree are imposed as constraints on the learning problem\. To handle the resulting nonconvex constraints, we use an augmented Lagrangian formulation, in which the constraints are incorporated into the learning objective through Lagrange multipliers\. In the control stage, the learned quantities are used to construct data\-driven feedback\-linearizing controller\. Proportional–integral action is incorporated to improve the closed\-loop tracking performance\.

Fig\. 1:Schematic of the proposed data\-driven feedback\-linearization framework, comprising a learning stage and a control stage\.### III\-AFeedback\-linearization framework

To enable smooth computation of the neural Lie derivatives required for data\-driven feedback\-linearization, the following assumption is imposed on the activation function:

###### Assumption 1

The activation functions satisfyσ∈𝒞∞\\sigma\\in\\mathpzc\{C\}^\{\\infty\}\.

In this work, the sigmoid activation function is chosen to satisfy this regularity condition\. We now state the main result:

###### Theorem 1

Forx∈ℛn\\mathrm\{x\}\\in\\mathscr\{R\}^\{n\}, consider the data\-driven neural network representation of the nonlinear control system\([2](https://arxiv.org/html/2609.25163#S2.Ex2)\)\(\\ref\{6\}\)with full relative degreer=nr=n\. Then, the corresponding learning\-based neural feedback controller is given by:

𝗎⁡\(x\)=−θn\(x\)\+kn−1θn−1\(x\)\+⋯k0θ0\(x\)−Vwθn−1′\(x\)𝗀ϕ\(x\)\.\\displaystyle\\mathsf\{u\}\(\\mathrm\{x\}\)\\\!=\\\!\-\\dfrac\{\\theta\_\{n\}\(\\mathrm\{x\}\)\+k\_\{n\-1\}\\theta\_\{n\-1\}\(\\mathrm\{x\}\)\+\\cdots k\_\{0\}\\theta\_\{0\}\(\\mathrm\{x\}\)\-V\\mathrm\{w\}\}\{\\theta\_\{n\-1\}^\{\{\}^\{\\prime\}\}\(\\mathrm\{x\}\)\\mathsf\{g\}\_\{\\upphi\}\(\\mathrm\{x\}\)\}\.\(5\)The functionθp​\(x\),p=0,1,⋯,n\\theta\_\{p\}\(\\mathrm\{x\}\),p=0,1,\\cdots,n, is recursively defined by

θ0\(x\)=𝖼ϕ\(x\)=𝒱\(l\)h𝖼\(l−1\)\+a\(l\),θp\(x\)=θp−1′\(x\)𝖿ϕ\(x\),\\theta\_\{0\}\(\\mathrm\{x\}\)=\\mathsf\{c\}\_\{\\upphi\}\(\\mathrm\{x\}\)=\\mathscr\{V\}^\{\(l\)\}h\_\{\\mathsf\{c\}\}^\{\(l\-1\)\}\+a^\{\(l\)\},\\hskip 2\.84544pt\\theta\_\{p\}\(\\mathrm\{x\}\)=\\theta\_\{p\-1\}^\{\{\}^\{\\prime\}\}\(\\mathrm\{x\}\)\\mathsf\{f\}\_\{\\upphi\}\(\\mathrm\{x\}\),whereθp′\(x\)=∂θp​\(x\)∂x\\theta\_\{p\}^\{\{\}^\{\\prime\}\}\(\\mathrm\{x\}\)=\\dfrac\{\\partial\\theta\_\{p\}\(\\mathrm\{x\}\)\}\{\\partial\\mathrm\{x\}\}denotes the Jacobian and the controller components are determined by the following constrained optimization problem:

minϕ⁡‖X˙−𝖿ϕ​\(X\)−𝗀ϕ​\(X\)⊙𝖴‖22\+λ​‖𝖸−𝖼ϕ​\(X\)‖22,\\displaystyle\\min\_\{\\upphi\}\\left\\\|\\dot\{\\mathrm\{X\}\}\-\\mathsf\{f\}\_\{\\upphi\}\(\\mathrm\{X\}\)\-\\mathsf\{g\}\_\{\\upphi\}\(\\mathrm\{X\}\)\\odot\\mathsf\{U\}\\right\\\|\_\{2\}^\{2\}\+\\lambda\\left\\\|\\mathsf\{Y\}\-\\mathsf\{c\}\_\{\\upphi\}\(\\mathrm\{X\}\)\\right\\\|\_\{2\}^\{2\},subject to\(6\)θp′\(xk\)𝗀ϕ\(xk\)=0,∀p∈\{0,…,n−2\},∀k∈\{1,…,m\}\.\\displaystyle\\theta\_\{p\}^\{\{\}^\{\\prime\}\}\(\\mathrm\{x\}\_\{k\}\)\\,\\mathsf\{g\_\{\\upphi\}\}\(\\mathrm\{x\}\_\{k\}\)\\\!=\\\!0,\\,\\forall\\,p\\\!\\in\\\!\\\{0,\\ldots,n\\\!\-\\\!2\\\},\\forall\\,k\\\!\\in\\\!\\\{1,\\ldots,m\\\}\.

###### Proof:

To establish that system \([2](https://arxiv.org/html/2609.25163#S2.Ex2)\) is feedback linearizable, it is sufficient to verify that the learned neural representation satisfies the required relative\-degree conditions\. The proof proceeds by first computing the gradient of the output neural map, which is then used to derive the corresponding neural Lie derivatives characterizing the relative\-degree conditions\. These are subsequently incorporated as constraints into the learning procedure\. Once these conditions are enforced, the resulting learned quantities are used to construct the data\-driven feedback\-linearizing control law\.

Since the governing dynamics of the system are expressed as a composition of nested nonlinear transformation, the statex\\mathrm\{x\}is embedded within the layers of the MLP\. Consequently, the gradient must be calculated by propagating into the network’s internal chain of weights and activations\. For the output function

𝖼ϕ​\(x\)=𝒱\(l\)​h𝖼\(l−1\)​\(x\)\+a\(l\),\\mathsf\{c\}\_\{\\upphi\}\(\\mathrm\{x\}\)=\\mathscr\{V\}^\{\(l\)\}h\_\{\\mathsf\{c\}\}^\{\(l\-1\)\}\(\\mathrm\{x\}\)\+a^\{\(l\)\},the gradient computed using the chain rule is

∂𝖼ϕ​\(x\)∂x\\displaystyle\\frac\{\\partial\\mathsf\{c\}\_\{\\upphi\}\(\\mathrm\{x\}\)\}\{\\partial\\mathrm\{x\}\}=∂𝖼ϕ​\(x\)∂h𝖼\(l−1\)​\(x\)∂h𝖼\(l−1\)​\(x\)∂z𝖼\(l−1\)​\(x\)∂z𝖼\(l−1\)​\(x\)∂h𝖼\(l−2\)​\(x\)⋯\\displaystyle=\\frac\{\\partial\\mathsf\{c\}\_\{\\upphi\}\(\\mathrm\{x\}\)\}\{\\partial h\_\{\\mathsf\{c\}\}^\{\(l\-1\)\}\(\\mathrm\{x\}\)\}\\frac\{\\partial h\_\{\\mathsf\{c\}\}^\{\(l\-1\)\}\(\\mathrm\{x\}\)\}\{\\partial z\_\{\\mathsf\{c\}\}^\{\(l\-1\)\}\(\\mathrm\{x\}\)\}\\frac\{\\partial z\_\{\\mathsf\{c\}\}^\{\(l\-1\)\}\(\\mathrm\{x\}\)\}\{\\partial h\_\{\\mathsf\{c\}\}^\{\(l\-2\)\}\(\\mathrm\{x\}\)\}\\cdots×∂z𝖼\(2\)​\(x\)∂h𝖼\(1\)​\(x\)​∂h𝖼\(1\)​\(x\)∂z𝖼\(1\)​\(x\)​∂z𝖼\(1\)​\(x\)∂x\\displaystyle\\quad\\times\\frac\{\\partial z\_\{\\mathsf\{c\}\}^\{\(2\)\}\(\\mathrm\{x\}\)\}\{\\partial h\_\{\\mathsf\{c\}\}^\{\(1\)\}\(\\mathrm\{x\}\)\}\\frac\{\\partial h\_\{\\mathsf\{c\}\}^\{\(1\)\}\(\\mathrm\{x\}\)\}\{\\partial z\_\{\\mathsf\{c\}\}^\{\(1\)\}\(\\mathrm\{x\}\)\}\\frac\{\\partial z\_\{\\mathsf\{c\}\}^\{\(1\)\}\(\\mathrm\{x\}\)\}\{\\partial\\mathrm\{x\}\}\(7\)=𝒱\(l\)σ′\(z𝖼\(l−1\)\(x\)\)𝒱\(l−1\)⋯𝒱\(2\)σ′\(z𝖼\(1\)\(x\)\)𝒱\(1\)\.\\displaystyle=\\mathscr\{V\}^\{\(l\)\}\\sigma^\{\\prime\}\\\!\(z\_\{\\mathsf\{c\}\}^\{\(l\-1\)\}\(\\mathrm\{x\}\)\)\\mathscr\{V\}^\{\(l\-1\)\}\\cdots\\mathscr\{V\}^\{\(2\)\}\\sigma^\{\\prime\}\\\!\(z\_\{\\mathsf\{c\}\}^\{\(1\)\}\(\\mathrm\{x\}\)\)\\mathscr\{V\}^\{\(1\)\}\.Forz𝖼\(i\)​\(x\)∈ℛnhz\_\{\\mathsf\{c\}\}^\{\(i\)\}\(\\mathrm\{x\}\)\\in\\mathscr\{R\}^\{n\_\{h\}\}, the sigmoid derivative ofjj\-th neuron in theii\-th layer is defined as

s𝖼,j\(i\)\(x\)=σ′\(z𝖼,j\(i\)\(x\)\)=σ\(z𝖼,j\(i\)\(x\)\)\[−σ\(z𝖼,j\(i\)\(x\)\)\],j=1,…,nh\.\\displaystyle s\_\{\\mathsf\{c\},j\}^\{\(i\)\}\(\\mathrm\{x\}\)\\\!=\\\!\\sigma^\{\{\}^\{\\prime\}\}\\\!\(z\_\{\\mathsf\{c\},j\}^\{\(i\)\}\(\\mathrm\{x\}\)\)\\\!=\\\!\\sigma\\\!\(z\_\{\\mathsf\{c\},j\}^\{\(i\)\}\(\\mathrm\{x\}\)\)\[1\\\!\-\\\!\\sigma\\\!\(z\_\{\\mathsf\{c\},j\}^\{\(i\)\}\(\\mathrm\{x\}\)\)\],j\\\!=\\\!1,\\ldots,n\_\{h\}\.Then we have fori=1,⋯,l−1,i=1,\\cdots,l\-1,

σ′​\(z𝖼\(i\)​\(x\)\)=\[s𝖼,1\(i\)​\(x\),s𝖼,2\(i\)​\(x\),…,s𝖼,nh\(i\)​\(x\)\]T,\\displaystyle\\sigma^\{\\prime\}\\\!\(z\_\{\\mathsf\{c\}\}^\{\(i\)\}\(\\mathrm\{x\}\)\)\\\!=\\\!\[s\_\{\\mathsf\{c\},1\}^\{\(i\)\}\(\\mathrm\{x\}\),s\_\{\\mathsf\{c\},2\}^\{\(i\)\}\(\\mathrm\{x\}\),\\ldots,s\_\{\\mathsf\{c\},n\_\{h\}\}^\{\(i\)\}\(\\mathrm\{x\}\)\]^\{T\},S𝖼\(i\)​\(x\)=∂h𝖼\(i\)​\(x\)∂z𝖼\(i\)​\(x\)=\[s𝖼,1\(i\)​\(x\)⋯00⋯0⋱0⋯s𝖼,nh\(i\)​\(x\)\]∈ℛnh×ℛnh\.\\displaystyle S\_\{\\mathsf\{c\}\}^\{\(i\)\}\(\\mathrm\{x\}\)\\\!=\\\!\\frac\{\\partial h\_\{\\mathsf\{c\}\}^\{\(i\)\}\(\\mathrm\{x\}\)\}\{\\partial z\_\{\\mathsf\{c\}\}^\{\(i\)\}\(\\mathrm\{x\}\)\}\\\!=\\\!\\\!\\begin\{bmatrix\}s\_\{\\mathsf\{c\},1\}^\{\(i\)\}\(\\mathrm\{x\}\)\\\!\\\!\\\!\\\!&\\\!\\cdots\\\!&\\\!0\\\\ 0&\\cdots&0\\\\ \\vdots&\\ddots&\\vdots\\\\ 0&\\cdots&\\\!\\\!\\\!\\\!s\_\{\\mathsf\{c\},n\_\{h\}\}^\{\(i\)\}\(\\mathrm\{x\}\)\\end\{bmatrix\}\\\!\\\!\\in\\\!\\mathscr\{R\}^\{n\_\{h\}\}\\\!\\\!\\times\\\!\\mathscr\{R\}^\{n\_\{h\}\}\\\!\.\(8\)Substituting \([8](https://arxiv.org/html/2609.25163#S3.E8)\) in \([7](https://arxiv.org/html/2609.25163#S3.Ex14)\) the preceding chain\-rule expression can be rewritten in compact form

∂𝖼ϕ​\(x\)∂x=𝒱\(l\)S𝖼\(l−1\)\(x\)𝒱\(l−1\)⋯𝒱\(2\)S𝖼\(1\)\(x\)𝒱\(1\)\.\\frac\{\\partial\\mathsf\{c\}\_\{\\upphi\}\(\\mathrm\{x\}\)\}\{\\partial\\mathrm\{x\}\}=\\mathscr\{V\}^\{\(l\)\}S\_\{\\mathsf\{c\}\}^\{\(l\-1\)\}\(\\mathrm\{x\}\)\\mathscr\{V\}^\{\(l\-1\)\}\\cdots\\mathscr\{V\}^\{\(2\)\}S\_\{\\mathsf\{c\}\}^\{\(1\)\}\(\\mathrm\{x\}\)\\mathscr\{V\}^\{\(1\)\}\.Therefore, the derivative of the output network is given by

\(∇𝖼ϕ\(x\)\)T=\(𝒱\(l\)S𝖼\(l−1\)\(x\)𝒱\(l−1\)⋯𝒱\(2\)S𝖼\(1\)\(x\)𝒱\(1\)\)e^,\\displaystyle\(\\nabla\\mathsf\{c\}\_\{\\upphi\}\(\\mathrm\{x\}\)\)^\{T\}=\(\\mathscr\{V\}^\{\(l\)\}S\_\{\\mathsf\{c\}\}^\{\(l\-1\)\}\(\\mathrm\{x\}\)\\mathscr\{V\}^\{\(l\-1\)\}\\cdots\\mathscr\{V\}^\{\(2\)\}S\_\{\\mathsf\{c\}\}^\{\(1\)\}\(\\mathrm\{x\}\)\\mathscr\{V\}^\{\(1\)\}\)\\hat\{\\mathrm\{e\}\},wheree^∈ℛ1×n\\hat\{\\mathrm\{e\}\}\\\!\\in\\\!\\mathscr\{R\}^\{1\\times n\}denotes the canonical unit vector ande^\\hat\{\\mathrm\{e\}\}embeds the output scalar derivative to the corresponding component of thenndimensional state gradient\. We compute the iterated Lie derivatives

ℒ𝖿ϕ1​𝖼ϕ​\(x\)\\displaystyle\\mathcal\{L\}^\{1\}\_\{\\mathsf\{f\}\_\{\\upphi\}\}\\mathsf\{c\}\_\{\\upphi\}\(\\mathrm\{x\}\)=\(𝒱\(l\)S𝖼\(l−1\)\(x\)⋯S𝖼\(1\)\(x\)𝒱\(1\)\)\\displaystyle=\(\\mathscr\{V\}^\{\(l\)\}S\_\{\\mathsf\{c\}\}^\{\(l\-1\)\}\(\\mathrm\{x\}\)\\cdots S\_\{\\mathsf\{c\}\}^\{\(1\)\}\(\\mathrm\{x\}\)\\mathscr\{V\}^\{\(1\)\}\)×e^​\(𝒲𝖿\(l\)​h𝖿\(l−1\)​\(x\)\+b𝖿\(l\)\)\\displaystyle\\quad\\times\\hat\{\\mathrm\{e\}\}\(\\mathscr\{W\}\_\{\\mathsf\{f\}\}^\{\(l\)\}h\_\{\\mathsf\{f\}\}^\{\(l\-1\)\}\(\\mathrm\{x\}\)\+b\_\{\\mathsf\{f\}\}^\{\(l\)\}\)=θ0′\(x\)𝖿ϕ\(x\)=θ1\(x\),\\displaystyle=\\theta\_\{0\}^\{\{\}^\{\\prime\}\}\(\\mathrm\{x\}\)\\mathsf\{f\}\_\{\\upphi\}\(\\mathrm\{x\}\)=\\theta\_\{1\}\(\\mathrm\{x\}\),ℒ𝖿ϕn−1​𝖼ϕ​\(x\)\\displaystyle\\mathcal\{L\}^\{n\-1\}\_\{\\mathsf\{f\}\_\{\\upphi\}\}\\mathsf\{c\}\_\{\\upphi\}\(\\mathrm\{x\}\)=θn−1​\(x\)\.\\displaystyle=\\theta\_\{n\-1\}\(\\mathrm\{x\}\)\.Next we proceed to examine the conditionsℒ𝗀ϕℒ𝖿ϕp𝖼ϕ\(x\)=0,p=0,…,n−2,\\mathcal\{L\}\_\{\\mathsf\{g\}\_\{\\upphi\}\}\\mathcal\{L\}\_\{\\mathsf\{f\}\_\{\\upphi\}\}^\{p\}\\,\\mathsf\{c\}\_\{\\upphi\}\(x\)=0,\\,p=0,\\ldots,n\-2,required for the system \([2](https://arxiv.org/html/2609.25163#S2.Ex2)\) to possess relative degreer=n\.r=n\.Therefore we enforce the following equality constraints represented by the neural Lie derivatives

ℒ𝗀ϕ​ℒ𝖿ϕ0​𝖼ϕ​\(x\)\\displaystyle\\mathcal\{L\}\_\{\\mathsf\{g\}\_\{\\upphi\}\}\\mathcal\{L\}^\{0\}\_\{\\mathsf\{f\}\_\{\\upphi\}\}\\mathsf\{c\}\_\{\\upphi\}\(\\mathrm\{x\}\)=θ0′\(x\)𝗀ϕ\(x\)=0,\\displaystyle=\\theta^\{\{\}^\{\\prime\}\}\_\{0\}\(\\mathrm\{x\}\)\\,\\mathsf\{g\}\_\{\\upphi\}\(\\mathrm\{x\}\)=0,ℒ𝗀ϕ​ℒ𝖿ϕn−2​𝖼ϕ​\(x\)\\displaystyle\\mathcal\{L\}\_\{\\mathsf\{g\}\_\{\\upphi\}\}\\mathcal\{L\}^\{n\-2\}\_\{\\mathsf\{f\}\_\{\\upphi\}\}\\mathsf\{c\}\_\{\\upphi\}\(\\mathrm\{x\}\)=θn−2′\(x\)𝗀ϕ\(x\)=0,\\displaystyle=\\theta\_\{n\-2\}^\{\{\}^\{\\prime\}\}\(\\mathrm\{x\}\)\\,\\mathsf\{g\}\_\{\\upphi\}\(\\mathrm\{x\}\)=0,which can be expressed in the following generalized form:

θp′\(x\)𝗀ϕ\(x\)=0,∀p∈\{0,1,2,⋯,n−2\}\.\\displaystyle\\theta\_\{p\}^\{\{\}^\{\\prime\}\}\(\\mathrm\{x\}\)\\,\\mathsf\{g\}\_\{\\upphi\}\(\\mathrm\{x\}\)=0,\\forall\\,p\\in\\\{0,1,2,\\cdots,n\-2\\\}\.Hence, we obtain optimization problem \([6](https://arxiv.org/html/2609.25163#S3.Ex11)\)\. Solving the constrained optimization problem and replacing the components of the traditional controller \([4](https://arxiv.org/html/2609.25163#S2.E4)\) with the neural Lie derivatives we obtain the following feedback linearized controller

𝗎⁡\(x\)=−θn\(x\)\+kn−1θn−1\(x\)\+⋯k0θ0\(x\)−Vwθn−1′\(x\)𝗀ϕ\(x\),\\displaystyle\\mathsf\{u\}\(\\mathrm\{x\}\)=\-\\dfrac\{\\theta\_\{n\}\(\\mathrm\{x\}\)\+k\_\{n\-1\}\\theta\_\{n\-1\}\(\\mathrm\{x\}\)\+\\cdots k\_\{0\}\\theta\_\{0\}\(\\mathrm\{x\}\)\-V\\mathrm\{w\}\}\{\\theta\_\{n\-1\}^\{\{\}^\{\\prime\}\}\(\\mathrm\{x\}\)\\mathsf\{g\}\_\{\\upphi\}\(\\mathrm\{x\}\)\},where the identityθn−1′\(x\)𝗀ϕ\(x\)≠0,\\theta\_\{n\-1\}^\{\{\}^\{\\prime\}\}\(\\mathrm\{x\}\)\\mathsf\{g\}\_\{\\upphi\}\(\\mathrm\{x\}\)\\neq 0,holds forx∈ℛn\.\\mathrm\{x\}\\in\\mathscr\{R\}^\{n\}\.∎

### III\-BConstraint Handling via the Augmented\-Lagrangian Method

The proposed optimization \([6](https://arxiv.org/html/2609.25163#S3.Ex11)\) problem is multidimensional and highly nonconvex owing to the large number of neural\-network parameters and the imposed equality constraints\. Standard gradient\-based training updates the network parameters primarily to reduce the prediction loss and does not preserve feasibility with respect to the constraints\. Contrarily, direct solution of the constrained optimization problem are computationally demanding due to the number of parameters\. To address this, we propose the augmented\-Lagrangian method which embeds the constraint residuals into the training objective through Lagrange\-multiplier and quadratic\-penalty terms\. The gradients of the resulting augmented objective can be readily evaluated using standard auto\-differentiation functions and subsequently used for gradient\-based optimization\. This enables the network parameters to be updated toward minimization prediction\-loss while satisfying the imposed constraints\. Let the constrained problem \([6](https://arxiv.org/html/2609.25163#S3.Ex11)\) be written as

minϕℒ\(ϕ\)s\.t\.ψp,k\(ϕ\)=0,p=0,…,n−2,k=1,…,m,\\displaystyle\\min\_\{\\upphi\}\\ \\mathscr\{L\}\(\\upphi\)\\ \\text\{s\.t\.\}\\ \\psi\_\{p,k\}\(\\upphi\)\\\!=\\\!0,\\ p\\\!=\\\!0,\\\!\\ldots\\\!,n\\\!\-\\\!2,\\ k\\\!=\\\!1,\\\!\\ldots\\\!,m,\(9\)whereψp,k​\(ϕ\)=θp′​\(xk\)​𝗀ϕ​\(xk\)\\psi\_\{p,k\}\(\\upphi\)=\\theta\_\{p\}^\{\\prime\}\(\\mathrm\{x\}\_\{k\}\)\\,\\mathsf\{g\}\_\{\\upphi\}\(\\mathrm\{x\}\_\{k\}\)are equality constraints indexed by both samplekkand structural indexppwhose collection determines the feasible set\. We solve \([9](https://arxiv.org/html/2609.25163#S3.E9)\) via the augmented\-Lagrangian method

ℒρ​\(ϕ,ν\)=ℒ⁡\(ϕ\)\+∑k=1m∑p=0n−2νp,k​ψp,k​\(ϕ\)\+ρ2​∑k=1m∑p=0n−2\(ψp,k​\(ϕ\)\)2\\mathscr\{L\}\_\{\\rho\}\(\\upphi,\\nu\)\\\!=\\\!\\mathscr\{L\}\(\\upphi\)\\\!\+\\\!\\sum\_\{k=1\}^\{m\}\\sum\_\{p=0\}^\{n\-2\}\\nu\_\{p,k\}\\,\\psi\_\{p,k\}\(\\upphi\)\\\!\+\\\!\\frac\{\\rho\}\{2\}\\sum\_\{k=1\}^\{m\}\\sum\_\{p=0\}^\{n\-2\}\\big\(\\psi\_\{p,k\}\(\\upphi\)\\big\)^\{2\}\(10\)whereνp,k∈ℛ\\nu\_\{p,k\}\\in\\mathscr\{R\}denotes the Lagrange multiplier associated with the equality constraintψp,k​\(ϕ\)=0\\psi\_\{p,k\}\(\\upphi\)=0, andρ\>0\\rho\>0is the penalty parameter\. The second term on the right hand\-side of \([10](https://arxiv.org/html/2609.25163#S3.E10)\) corresponds to the classical Lagrangian contribution, whereas the quadratic term penalizes violations of the equality constraints\. The original constrained problem is thereby replaced by the unconstrained optimization problem

minϕ⁡ℒρ​\(ϕ,ν\)\.\\displaystyle\\min\_\{\\upphi\}\\mathscr\{L\}\_\{\\rho\}\(\\upphi,\\nu\)\.\(11\)The solution of \([11](https://arxiv.org/html/2609.25163#S3.E11)\) is obtained by an iterative scheme, see Algorithm[1](https://arxiv.org/html/2609.25163#alg1)\. Letiidenote the outer iteration index and for fixed multipliersνi\\nu^\{i\}and penalty parameterρi\\rho\_\{i\}, the neural network parameters are updated by solving the subproblem

ϕi\+1≈arg⁡minϕ​ℒρi​\(ϕ,νi\),\\upphi^\{i\+1\}\\approx\\underset\{\\upphi\}\{\\arg\\min\}\\;\\mathscr\{L\}\_\{\\rho\_\{i\}\}\(\\upphi,\\nu^\{i\}\),\(12\)using a gradient\-based optimizer\. The multipliers and penalty parameters are subsequently updated according to

νp,ki\+1\\displaystyle\\nu\_\{p,k\}^\{\\,i\+1\}\\\!=νp,ki\+ρiψp,k\(ϕi\+1\),p=0,…,n−2,k=1,…,m,\\displaystyle=\\\!\\nu\_\{p,k\}^\{\\,i\}\\\!\+\\\!\\rho\_\{i\}\\,\\psi\_\{p,k\}\(\\upphi^\{i\+1\}\),\\,p\\\!=\\\!0,\\\!\\dots\\\!,n\\\!\-\\\!2,\\,k\\\!=\\\!1,\\\!\\dots\\\!,m,ρi\+1\\displaystyle\\rho\_\{i\+1\}=min⁡\{β​ρi,ρmax\},β\>1,\\displaystyle\\\!=\\\!\\min\\left\\\{\\beta\\rho\_\{i\},\\rho\_\{\\max\}\\right\\\},\\,\\beta\\\!\>\\\!1,whereρmax\\rho\_\{\\max\}limits excessive growth of the penalty parameter\. Thus, each outer iteration consists of minimizing the augmented objective with respect to the MLP parameters, followed by updates of the multipliers and the penalty parameter\. Repeating this procedure progressively reduces the identification error while enforcing the constraints\.

Algorithm 1Augmented\-Lagrangian Training\.Input:

x0,T,Δ​t,𝗎⁡\(t\),ρ0,β,ρmax,Nout,Nin\\mathrm\{x\}\_\{0\},T,\\Delta t,\\mathsf\{u\}\(t\),\\rho\_\{0\},\\beta,\\rho\_\{\\max\},N\_\{\\mathrm\{out\}\},N\_\{\\mathrm\{in\}\}
Output:Learned parameters

ϕ\\upphi
Simulate

x˙=f⁡\(x\)\+g⁡\(x\)​𝗎​\(t\)\\dot\{\\mathrm\{x\}\}=f\(\\mathrm\{x\}\)\+g\(\\mathrm\{x\}\)\\mathsf\{u\}\(t\),

𝗒=c⁡\(x\)\\mathsf\{y\}=c\(\\mathrm\{x\}\)
Construct

D=\{\(xk,𝗎k,x˙k,𝗒k\)\}k=1mD=\\\{\(\\mathrm\{x\}\_\{k\},\\mathsf\{u\}\_\{k\},\\dot\{\\mathrm\{x\}\}\_\{k\},\\mathsf\{y\}\_\{k\}\)\\\}\_\{k=1\}^\{m\}
Initialize

ϕ\\upphi,

νp,k←0\\nu\_\{p,k\}\\leftarrow 0, and

ρ←ρ0\\rho\\leftarrow\\rho\_\{0\}
for

i=1,…,Nouti=1,\\ldots,N\_\{\\mathrm\{out\}\}do

for

j=1,…,Ninj=1,\\ldots,N\_\{\\mathrm\{in\}\}do

Update

ϕ\\upphiby minimizing

ℒρ​\(ϕ,ν\)=ℒ⁡\(ϕ\)\+∑k=1m∑p=0n−2νp,k​ψp,k​\(ϕ\)\+ρ2​∑k=1m∑p=0n−2\(ψp,k​\(ϕ\)\)2\\\!\\\!\\\!\\\!\\\!\\\!\\\!\\\!\\\!\\\!\\\!\\\!\\\!\\mathscr\{L\}\_\{\\rho\}\(\\upphi,\\nu\)\\\!=\\\!\\mathscr\{L\}\(\\upphi\)\\\!\+\\\!\\sum\_\{k=1\}^\{m\}\\\!\\sum\_\{p=0\}^\{n\-2\}\\\!\\nu\_\{p,k\}\\psi\_\{p,k\}\(\\upphi\)\\\!\+\\\!\\frac\{\\rho\}\{2\}\\sum\_\{k=1\}^\{m\}\\\!\\sum\_\{p=0\}^\{n\-2\}\\\!\(\\psi\_\{p,k\}\(\\upphi\)\)^\{2\}
endfor

Evaluate

ψp,k​\(ϕ\)=θp′​\(xk\)​𝗀ϕ​\(xk\)\\psi\_\{p,k\}\(\\upphi\)=\\theta\_\{p\}^\{\\prime\}\(\\mathrm\{x\}\_\{k\}\)\\mathsf\{g\}\_\{\\upphi\}\(\\mathrm\{x\}\_\{k\}\)
Update

νp,k←νp,k\+ρ​ψp,k​\(ϕ\)\\nu\_\{p,k\}\\leftarrow\\nu\_\{p,k\}\+\\rho\\psi\_\{p,k\}\(\\upphi\)
Update

ρ←min⁡\{β​ρ,ρmax\}\\rho\\leftarrow\\min\\\{\\beta\\rho,\\rho\_\{\\max\}\\\}
endfor

Verify

θn−1′​\(xk\)​𝗀ϕ​\(xk\)≠0\\theta\_\{n\-1\}^\{\\prime\}\(\\mathrm\{x\}\_\{k\}\)\\mathsf\{g\}\_\{\\upphi\}\(\\mathrm\{x\}\_\{k\}\)\\neq 0,

k=1,…,mk=1,\\ldots,m
return

ϕ\\upphi

## IVPractical Closed\-loop Stability Analysis

This section establishes a practical closed\-loop stability result for the proposed data\-driven feedback\-linearizing controller in the presence of neural approximation errors\. Using bounded identification errors together with the universal function approximation property \(UFAP\) of neural networks\[[12](https://arxiv.org/html/2609.25163#bib.bib12)\], we first derive finite bounds on the errors of the learned feedback\-linearization quantities\. These bounds are then propagated into the closed\-loop dynamics to characterize the resulting perturbations and establish practical stability\.

###### Theorem 2

Consider a data\-driven nonlinear system of the form \([2](https://arxiv.org/html/2609.25163#S2.Ex2)\), in a compact operation domain𝒟⊆ℛn\.\\mathcal\{D\}\\subseteq\\mathscr\{R\}^\{n\}\.Assume the system has relative degreer=n,r=n,and the corresponding controller is given in \([5](https://arxiv.org/html/2609.25163#S3.E5)\)\. If the MLPs are sufficiently expressive, i\.e\. fulfill the UFAP, and the controller gains are selected such that the resulting characteristic polynomial is Hurwitz, then the closed\-loop tracking error is uniformly ultimately bounded\. Consequently, the learning\-based feedback\-linearized closed\-loop system is practically stable\.

###### Proof:

Consider the nonlinear SISO system \([1](https://arxiv.org/html/2609.25163#S2.Ex1)\) with relative degreer=nr\\\!=\\\!n\. Differentiating the outputnntimes yields

𝗒\(n\)​\(𝓉\)=α⁡\(x\)\+β⁡\(x\)​𝗎​\(𝓉\),\\mathsf\{y\}^\{\(n\)\}\(\\mathpzc\{t\}\)=\\alpha\(\\mathrm\{x\}\)\+\\beta\(\\mathrm\{x\}\)\\,\\mathsf\{u\}\(\\mathpzc\{t\}\),\(13\)withα⁡\(x\)=ℒ𝖿n​𝖼​\(x\)\\alpha\(\\mathrm\{x\}\)=\\mathcal\{L\}\_\{\\mathsf\{f\}\}^\{\\,n\}\\mathsf\{c\}\(\\mathrm\{x\}\),β⁡\(x\)=ℒ𝗀​ℒ𝖿n−1​𝖼​\(x\)\\beta\(\\mathrm\{x\}\)=\\mathcal\{L\}\_\{\\mathsf\{g\}\}\\mathcal\{L\}\_\{\\mathsf\{f\}\}^\{\\,n\-1\}\\mathsf\{c\}\(\\mathrm\{x\}\)\. If the unknown nonlinear mapping is approximated by neural network \([2](https://arxiv.org/html/2609.25163#S2.Ex2)\), then

αϕ​\(x\)\\displaystyle\\alpha\_\{\\upphi\}\(\\mathrm\{x\}\)=ℒ𝖿ϕn​𝖼ϕ​\(x\)=θn​\(x\),\\displaystyle=\\mathcal\{L\}\_\{\\mathsf\{f\}\_\{\\upphi\}\}^\{\\,n\}\\mathsf\{c\}\_\{\\upphi\}\(\\mathrm\{x\}\)=\\theta\_\{n\}\(\\mathrm\{x\}\),βϕ​\(x\)\\displaystyle\\beta\_\{\\upphi\}\(\\mathrm\{x\}\)=ℒ𝗀ϕ​ℒ𝖿ϕn−1​𝖼ϕ​\(x\)=θn−1′​\(x\)​𝗀ϕ​\(x\),\\displaystyle=\\mathcal\{L\}\_\{\\mathsf\{g\}\_\{\\upphi\}\}\\mathcal\{L\}\_\{\\mathsf\{f\}\_\{\\upphi\}\}^\{\\,n\-1\}\\mathsf\{c\}\_\{\\upphi\}\(\\mathrm\{x\}\)=\\theta\_\{n\-1\}^\{\\prime\}\(\\mathrm\{x\}\)\\mathsf\{g\}\_\{\\upphi\}\(\\mathrm\{x\}\),whereθ0​\(x\),⋯,θn−1​\(x\),θn​\(x\)\\theta\_\{0\}\(\\mathrm\{x\}\),\\cdots,\\theta\_\{n\-1\}\(\\mathrm\{x\}\),\\theta\_\{n\}\(\\mathrm\{x\}\)are defined as in Theorem[1](https://arxiv.org/html/2609.25163#Thmtheorem1)\. The data\-driven neural feedback\-linearizing controller \([5](https://arxiv.org/html/2609.25163#S3.E5)\) can be written as

𝗎⁡\(𝓉\)=𝒱​w−αϕ​\(x\)−𝓀𝓃−1​θ𝓃−1​\(x\)−⋯−𝓀0​θ0​\(x\)βϕ​\(x\),\\mathsf\{u\}\(\\mathpzc\{t\}\)=\\frac\{V\\mathrm\{w\}\-\\alpha\_\{\\upphi\}\(\\mathrm\{x\}\)\-k\_\{n\-1\}\\theta\_\{n\-1\}\(\\mathrm\{x\}\)\-\\cdots\-k\_\{0\}\\theta\_\{0\}\(\\mathrm\{x\}\)\}\{\\beta\_\{\\upphi\}\(\\mathrm\{x\}\)\},\(14\)where the coefficientsk0,…,kn−1k\_\{0\},\\ldots,k\_\{n\-1\}are selected such that the resulting equationsn\+kn−1​sn−1\+⋯\+k1​s\+k0s^\{n\}\+k\_\{n\-1\}s^\{n\-1\}\+\\cdots\+k\_\{1\}s\+k\_\{0\}is Hurwitz\. Substituting \([14](https://arxiv.org/html/2609.25163#S4.E14)\) in \([13](https://arxiv.org/html/2609.25163#S4.E13)\), we obtain

𝗒\(n\)\\displaystyle\\mathsf\{y\}^\{\(n\)\}=α⁡\(x\)\+β⁡\(x\)βϕ​\(x\)​\[V​w−αϕ​\(x\)−∑p=0n−1kp​θp​\(x\)\]\.\\displaystyle=\\alpha\(\\mathrm\{x\}\)\+\\frac\{\\beta\(\\mathrm\{x\}\)\}\{\\beta\_\{\\upphi\}\(\\mathrm\{x\}\)\}\\big\[V\\mathrm\{w\}\-\\alpha\_\{\\upphi\}\(\\mathrm\{x\}\)\-\\sum\_\{p=0\}^\{n\-1\}k\_\{p\}\\theta\_\{p\}\(\\mathrm\{x\}\)\\big\]\.\(15\)Adding∑p=0n−1kp​𝗒\(p\)\\sum\_\{p=0\}^\{n\-1\}k\_\{p\}\\mathsf\{y\}^\{\(p\)\}to both sides of \([15](https://arxiv.org/html/2609.25163#S4.E15)\), we obtain

𝗒\(n\)\+∑p=0n−1kp​𝗒\(p\)\\displaystyle\\mathsf\{y\}^\{\(n\)\}\+\\sum\_\{p=0\}^\{n\-1\}k\_\{p\}\\mathsf\{y\}^\{\(p\)\}=α⁡\(x\)\+β⁡\(x\)βϕ​\(x\)​\[V​w−αϕ​\(x\)−∑p=0n−1kp​θp\]\\displaystyle=\\alpha\(\\mathrm\{x\}\)\+\\frac\{\\beta\(\\mathrm\{x\}\)\}\{\\beta\_\{\\upphi\}\(\\mathrm\{x\}\)\}\\big\[V\\mathrm\{w\}\-\\alpha\_\{\\upphi\}\(\\mathrm\{x\}\)\-\\sum\_\{p=0\}^\{n\-1\}k\_\{p\}\\theta\_\{p\}\\big\]\+∑p=0n−1kp𝗒\(p\)=Vw\+d\(𝓉,x\),\\displaystyle\\quad\+\\sum\_\{p=0\}^\{n\-1\}k\_\{p\}\\mathsf\{y\}^\{\(p\)\}=V\\mathrm\{w\}\+d\(\\mathpzc\{t\},\\mathrm\{x\}\),\(16\)where

d⁡\(𝓉,x\)=\\displaystyle d\(\\mathpzc\{t\},\\mathrm\{x\}\)=α⁡\(x\)−αϕ​\(x\)\+∑p=0n−1kp​\[𝗒\(p\)−θp​\(x\)\]\\displaystyle\\alpha\(\\mathrm\{x\}\)\-\\alpha\_\{\\upphi\}\(\\mathrm\{x\}\)\+\\sum\_\{p=0\}^\{n\-1\}k\_\{p\}\\big\[\\mathsf\{y\}^\{\(p\)\}\-\\theta\_\{p\}\(\\mathrm\{x\}\)\\big\]\(17\)\+β​\(x\)−βϕ​\(x\)βϕ​\(x\)​\[V​w−αϕ​\(x\)−∑p=0n−1kp​θp​\(x\)\]\.\\displaystyle\+\\frac\{\\beta\(\\mathrm\{x\}\)\\\!\-\\\!\\beta\_\{\\upphi\}\(\\mathrm\{x\}\)\}\{\\beta\_\{\\upphi\}\(\\mathrm\{x\}\)\}\\big\[V\\mathrm\{w\}\\\!\-\\\!\\alpha\_\{\\upphi\}\(\\mathrm\{x\}\)\\\!\-\\\!\\sum\_\{p=0\}^\{n\-1\}k\_\{p\}\\theta\_\{p\}\(\\mathrm\{x\}\)\\big\]\.Therefore,

𝗒\(n\)\+kn−1​𝗒\(n−1\)\+⋯\+k1​𝗒˙\+k0​𝗒=V​w\+d⁡\(𝓉,x\)\.\\mathsf\{y\}^\{\(n\)\}\+k\_\{n\-1\}\\mathsf\{y\}^\{\(n\-1\)\}\+\\cdots\+k\_\{1\}\\dot\{\\mathsf\{y\}\}\+k\_\{0\}\\mathsf\{y\}=V\\mathrm\{w\}\+d\(\\mathpzc\{t\},\\mathrm\{x\}\)\.\(18\)The termd⁡\(x,𝓉\)d\(\\mathrm\{x\},\\mathpzc\{t\}\)denotes the residual perturbation induced by the mismatch between the true Lie derivatives and the identified Lie derivatives after applying the data\-driven feedback\-linearizing control law\. As the true system mappings and their neural approximations are continuous on the compact operating domain𝒟\\mathcal\{D\}, it follows from the UFAP of sufficiently expressive MLP, that the corresponding modeling errors are bounded on𝒟\\mathcal\{D\}\. Thus, there exist finite positive constantsεf,εg,εc\\varepsilon\_\{f\},\\varepsilon\_\{g\},\\varepsilon\_\{c\}such that

‖𝖿−𝖿ϕ‖≤εf,‖𝗀−𝗀ϕ‖≤εg,‖𝖼−𝖼ϕ‖≤εc,∀x∈𝒟\.\\\|\\mathsf\{f\}\-\\mathsf\{f\}\_\{\\upphi\}\\\|\\leq\\varepsilon\_\{f\},\\quad\\\|\\mathsf\{g\}\-\\mathsf\{g\}\_\{\\upphi\}\\\|\\leq\\varepsilon\_\{g\},\\quad\\\|\\mathsf\{c\}\-\\mathsf\{c\}\_\{\\upphi\}\\\|\\leq\\varepsilon\_\{c\},\\quad\\forall\\,\\mathrm\{x\}\\in\\mathcal\{D\}\.Since neural networks employ smooth sigmoid activation functions and the learned Lie derivatives are also continuously differentiable on𝒟\\mathcal\{D\}, every term on the right\-hand side of \([17](https://arxiv.org/html/2609.25163#S4.E17)\) is continuously bounded on the compact operating region and there exists positive constantsεα,εθ,εβ\\varepsilon\_\{\\alpha\},\\varepsilon\_\{\\theta\},\\varepsilon\_\{\\beta\}such that

\|α⁡\(x\)−αϕ​\(x\)\|\\displaystyle\\left\|\\alpha\(\\mathrm\{x\}\)\-\\alpha\_\{\\upphi\}\(\\mathrm\{x\}\)\\right\|≤εα,\\displaystyle\\leq\\varepsilon\_\{\\alpha\},max0≤p≤n−1⁡\|𝗒\(p\)​\(𝓉\)−θ𝓅​\(x\)\|\\displaystyle\\max\_\{0\\leq p\\leq n\-1\}\\left\|\\mathsf\{y\}^\{\(p\)\}\(\\mathpzc\{t\}\)\-\\theta\_\{p\}\(\\mathrm\{x\}\)\\right\|≤εθ\\displaystyle\\leq\\varepsilon\_\{\\theta\}\|β⁡\(x\)−βϕ​\(x\)\|\\displaystyle\\left\|\\beta\(\\mathrm\{x\}\)\-\\beta\_\{\\upphi\}\(\\mathrm\{x\}\)\\right\|≤εβ\.\\displaystyle\\leq\\varepsilon\_\{\\beta\}\.
By assumptionr=nr=n, there exists a constantδ\>0\\delta\>0such thatinfx∈𝒟\|βϕ​\(x\)\|≥δ\>0\.\\inf\\limits\_\{\\mathrm\{x\}\\in\\mathcal\{D\}\}\\left\|\\beta\_\{\\upphi\}\(\\mathrm\{x\}\)\\right\|\\geq\\delta\>0\.DefineNϕ​\(𝓉,x\)=𝒱​w−αϕ​\(x\)−∑𝓅=0𝓃−1𝓀𝓅​θ𝓅​\(x\),N\_\{\\upphi\}\(\\mathpzc\{t\},\\mathrm\{x\}\)=\\mathscr\{V\}\\,\\mathrm\{w\}\-\\alpha\_\{\\upphi\}\(\\mathrm\{x\}\)\-\\sum\_\{p=0\}^\{n\-1\}k\_\{p\}\\theta\_\{p\}\(\\mathrm\{x\}\),on a compact domain\|Nϕ​\(𝓉,x\)\|≤εN\.\\left\|N\_\{\\upphi\}\(\\mathpzc\{t\},\\mathrm\{x\}\)\\right\|\\leq\\varepsilon\_\{N\}\.Applying the triangle inequality and using the above bounds, we obtain

\|d⁡\(𝓉,x\)\|≤εα\+∑p=0n−1\|kp\|​εθ\+εβδ​εN=d¯\.\\left\|d\(\\mathpzc\{t\},\\mathrm\{x\}\)\\right\|\\leq\\varepsilon\_\{\\alpha\}\+\\sum\_\{p=0\}^\{n\-1\}\|k\_\{p\}\|\\varepsilon\_\{\\theta\}\+\\frac\{\\varepsilon\_\{\\beta\}\}\{\\delta\}\\varepsilon\_\{N\}=\\bar\{d\}\.Hence, we get,\|d⁡\(𝓉,x\)\|≤d¯\\left\|d\(\\mathpzc\{t\},\\mathrm\{x\}\)\\right\|\\leq\\bar\{d\}, i\.e\. bounded neural modeling errors imply bounded tracking errors\. Consequently, the learning based feedback\-linearized closed\-loop system is practically stable\. If the neural representation is exact, thend¯=0\\bar\{d\}=0\. ∎

## VArmature controlled DC motor

In this study, a nonlinear armature\-controlled DC motor is considered to validate the derived results, as illustrated in Fig\.[2](https://arxiv.org/html/2609.25163#S5.F2)\. Its main components are the armature, stator, commutator, and brushes, which collectively convert the applied DC electrical energy into rotational mechanical motion\.

![Refer to caption](https://arxiv.org/html/2609.25163v1/DC_motor.png)Fig\. 2:Schematic representation of an armature\-controlled DC motor\.The armature voltage is treated as the control input, the two state variables are armature currentx1\\mathrm\{x\}\_\{1\}and the angular speed of the motor shaftx2\\mathrm\{x\}\_\{2\}\. The focus is on speed regulation because the motor speed can be directly controlled by varying the armature voltage, which changes the armature current and the generated torque\. Considering nonlinear friction, the DC motor dynamics are expressed as

x˙1\\displaystyle\\dot\{\\mathrm\{x\}\}\_\{1\}=−RL​x1−KbL​x2\+1L​𝗎,\\displaystyle=\-\\frac\{R\}\{L\}\\mathrm\{x\}\_\{1\}\-\\frac\{K\_\{b\}\}\{L\}\\mathrm\{x\}\_\{2\}\+\\frac\{1\}\{L\}\\mathsf\{u\},x˙2\\displaystyle\\dot\{\\mathrm\{x\}\}\_\{2\}=KTJ​x1−BJ​x2−αJ​x23,\\displaystyle=\\frac\{K\_\{T\}\}\{J\}\\mathrm\{x\}\_\{1\}\-\\frac\{B\}\{J\}\\mathrm\{x\}\_\{2\}\-\\frac\{\\alpha\}\{J\}\\mathrm\{x\}\_\{2\}^\{3\},𝗒\\displaystyle\\mathsf\{y\}=x2,\\displaystyle=\\mathrm\{x\}\_\{2\},\(19\)whereJJ,BB,RR, andLLdenote the rotor moment of inertia, damping coefficient, armature resistance, and armature inductance\. Further,KbK\_\{b\},KTK\_\{T\}, and𝗎\\mathsf\{u\}represent the back\-emf constant, motor torque constant and applied armature voltage, respectively\. The parameters are given asR=2R=2,L=0\.5L=0\.5,Kb=0\.1K\_\{b\}=0\.1,KT=0\.1K\_\{T\}=0\.1,J=0\.02J=0\.02,B=0\.1B=0\.1, andα=0\.01\\alpha=0\.01\.

#### V\-1Data Generation

The training dataset is generated by simulating continuous system trajectories initiated from randomized initial conditions\. The system is persistently excited by pseudo\-random binary sequence control input𝗎k∈\{−1,\+1\}\\mathsf\{u\}\_\{k\}\\in\\\{\-1,\+1\\\}while the state trajectories evolve within the operating region\[−1,2\]×\[−1,2\]\[\-1,2\]\\times\[\-1,2\]\. Each trajectory is simulated over a time horizon ofT=5\.0T=5\.0s with a sampling interval ofΔ​t=0\.01\\Delta t=0\.01s, where the state derivatives are evaluated from the nonlinear plant model and the states are propagated using the explicit forward\-Euler integration scheme\. Consequently, each training sample consists of the tuple\{xk,𝗎k,x˙k,𝗒k\},\\left\\\{\\mathrm\{x\}\_\{k\},\\mathsf\{u\}\_\{k\},\\dot\{\\mathrm\{x\}\}\_\{k\},\\mathsf\{y\}\_\{k\}\\right\\\},resulting in an aggregate training dataset ofm=2,000m=2,000samples\.

To emulate realistic process and measurement uncertainties, zero\-mean Gaussian noise with a standard deviation ofσ=0\.2\\sigma=0\.2was introduced during the data generation stage\. Specifically, the noise was superimposed on both the numerically propagated state trajectories and the corresponding state derivative observations, resulting in substantially corrupted training data\. The augmented\-Lagrangian training was performed forNout=25N\_\{\\mathrm\{out\}\}=25outer iterations, withNin=40N\_\{\\mathrm\{in\}\}=40inner epochs per outer iteration, usingρ0=1\.0\\rho\_\{0\}=1\.0,β=1\.5\\beta=1\.5,ρmax=50\\rho\_\{\\max\}=50, zero\-initialized Lagrange multipliers, and an output\-loss weighting factor ofλ=0\.5\\lambda=0\.5\. The neural model was trained using Adam optimizer with a learning rate3×10−33\\times 10^\{\-3\}\.

#### V\-2Performance Evaluation of neural approximation

Figs\.[3](https://arxiv.org/html/2609.25163#S5.F3)and[4](https://arxiv.org/html/2609.25163#S5.F4)compare the noisy state trajectories and their corresponding state derivatives with the responses predicted by the learned neural model for four representative initial conditions\. The close agreement between the noisy measurements and the predicted trajectories demonstrates that the proposed framework accurately captures both state evolution and underlying system dynamics despite the presence of significant measurement noise\. Further, Fig\.[5](https://arxiv.org/html/2609.25163#S5.F5), shows the progressive reduction of the augmented Lagrangian training loss, with the constraint violation remaining consistently small throughout training\.

Fig\. 3:Validation simulation evaluating true continuous state trajectories against the trained model predictions across four distinct initial state configurations\. The near overlap between the true and predicted curves also suggests good generalization and a high\-quality fit in both transient and steady\-state regions\.Fig\. 4:Comparison profiles between the true analytical state derivatives and the neural network vector field predictions \(x˙1,x˙2\\dot\{\\mathrm\{x\}\}\_\{1\},\\dot\{\\mathrm\{x\}\}\_\{2\}\)\. The learned state\-derivative profiles closely follow the true derivatives across all tested initial conditions, showing that the neural model accurately captures the local dynamics of the system\. The small deviations visible only in a few early transient portions indicate minor approximation errors, while the overall overlap confirms strong derivative\-level consistency\.Fig\. 5:Training Convergence of data loss and constraint residual: Random initialization initially produces a small constraint residual\. As the network parameters are adjusted to improve data fitting, the residual temporarily increase, after which the augmented\-Lagrangian terms progressively enforce the constraint and drive the residual toward zero\.
#### V\-3Data\-driven Feedback control

To evaluate the closed\-loop performance of the learned neural model representing the DC motor, the desired closed loop poles are selected at−3,−5\-3,\-5, resulting in the gaink0=15,k\_\{0\}=15,andk1=8k\_\{1\}=8\. The resulting data\-driven feedback control is given as

𝗎⁡\(x\)=−θ2​\(x\)\+r¨​\(𝓉\)−8​\[θ1​\(x\)−𝓇˙​\(𝓉\)\]−15​\[θ0​\(x\)−𝓇⁡\(𝓉\)\]θ1′\(x\)𝗀ϕ\(x\)\.\\mathsf\{u\}\(\\mathrm\{x\}\)\\\!=\\\!\\frac\{\\\!\-\\theta\_\{2\}\(\\mathrm\{x\}\)\\\!\+\\\!\\ddot\{r\}\(\\mathpzc\{t\}\)\\\!\-\\\!8\\left\[\\theta\_\{1\}\(\\mathrm\{x\}\)\\\!\-\\\!\\dot\{r\}\(\\mathpzc\{t\}\)\\right\]\\\!\-\\\!15\\left\[\\theta\_\{0\}\(\\mathrm\{x\}\)\\\!\-\\\!r\(\\mathpzc\{t\}\)\\right\]\}\{\\theta\_\{1\}^\{\{\}^\{\\prime\}\}\(\\mathrm\{x\}\)\\mathsf\{g\}\_\{\\upphi\}\(\\mathrm\{x\}\)\}\.The state trajectories are randomly initialized away from the equilibrium to evaluate the transient regulation performance, while the corresponding control input is monitored to verify that the imposed control bounds are respected\. The closed\-loop tracking performance is evaluated with the reference trajectoryr⁡\(𝓉\)=1−ℯ−𝓉r\(\\mathpzc\{t\}\)=1\-e^\{\-\\mathpzc\{t\}\}, whose derivatives arer˙​\(𝓉\)=ℯ−𝓉\\dot\{r\}\(\\mathpzc\{t\}\)=e^\{\-\\mathpzc\{t\}\}andr¨​\(𝓉\)=−ℯ−𝓉\\ddot\{r\}\(\\mathpzc\{t\}\)=\-e^\{\-\\mathpzc\{t\}\}\. The exponential reference provides a smooth transition to the desired equilibrium with continuous first and second derivatives, making it suitable for the feedback\-linearizing controller while avoiding the excessive control effort associated with an ideal step input\. To compensate for accumulated tracking errors and suppress steady\-state offsets, an integral error compensation is incorporated into the feedback\-linearizing controller\. The simulation results presented in the Figs\.[6](https://arxiv.org/html/2609.25163#S5.F6)and[7](https://arxiv.org/html/2609.25163#S5.F7)demonstrate the closed\-loop performance of the proposed data\-driven neural feedback\-linearizing controller under noisy operating conditions\. As shown in Fig\.[6](https://arxiv.org/html/2609.25163#S5.F6)the state trajectories converge toward the desired equilibrium while the corresponding control input remains bounded\. Fig\.[7](https://arxiv.org/html/2609.25163#S5.F7)illustrates that the output follows the prescribed reference trajectory with a bounded tracking error and a bounded control effort\.

Fig\. 6:Closed\-loop states under noisy operating conditions\. The state trajectories converge toward the desired equilibrium while the corresponding control input remains bounded\.Fig\. 7:Closed\-loop reference tracking performance under noisy operating conditions\. The system output follows the reference trajectory with bounded tracking error while maintaining a bounded control input\.
#### V\-4Numerical Verification of Practical Stability

For the neural model identified from noisy data, Fig\.[8](https://arxiv.org/html/2609.25163#S5.F8)illustrates the residual modeling errord⁡\(𝓉,x¯\)d\(\\mathpzc\{t\},\\bar\{\\mathrm\{x\}\}\)along the simulated closed\-loop trajectories, evaluated according to Theorem[2](https://arxiv.org/html/2609.25163#Thmtheorem2)\. The maximum residual magnitude is approximatelyd¯≈1\.04\\bar\{d\}\\approx 1\.04, such that\|d⁡\(𝓉,x¯\)\|≤d¯,\\left\|d\(\\mathpzc\{t\},\\bar\{\\mathrm\{x\}\}\)\\right\|\\leq\\bar\{d\},throughout the closed\-loop operation\. The larger residual bound obtained for the noisy model is attributed to the process and measurement noise introduced during training\. The results verify the assumptions of the practical stability theorem and confirm that the learning\-based feedback\-linearized closed\-loop system remains practically stable\.

Fig\. 8:Residual modeling error under noisy operating conditions\. The modeling error decreases rapidly during the initial transient and gradually converges to a bounded steady\-state value, demonstrating consistent estimation performance despite the presence of noise\.

## VIConclusion

This work proposed a data\-driven framework for nonlinear system identification and feedback linearization using multilayer perceptron neural networks\. The Lie\-derivative structure required for feedback linearization was incorporated directly into the learning process through an Augmented\-Lagrangian formulation, thereby promoting consistency between the identified model and the structural requirements of the subsequent controller design\. The numerical results indicate that the learned model can reproduce the relevant nonlinear dynamics while enabling both stabilization and reference tracking under noisy operating conditions\. In addition, the established practical stability result shows that bounded identification errors lead to ultimately bounded tracking errors, providing a direct connection between model accuracy and closed\-loop performance\. The application to an armature\-controlled DC motor further demonstrates the applicability of the proposed approach in the presence of process and measurement uncertainties\. A limitation of the present framework is the assumption that the system possesses a known and uniform relative degree over the considered operating region\. Accordingly, the required Lie\-derivative constraints are prescribed a priori during training\. Future work will therefore focus on jointly identifying the relative\-degree structure and enforcing the corresponding feedback\-linearization conditions, including the nonvanishing decoupling condition\. Further developments will also investigate improved robustness against modeling uncertainty and measurement noise, reduced data requirements, and validation on experimental nonlinear systems\.

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