# Identification of High-Order Nonlinear Coupled Systems Using a Data-Driven Approach
## 6 Conclusions  
This study presents a modification of the classical SINDY algorithm, facilitating the identification of mathematical models for high-order nonlinear systems with coupled dynamics. Despite its negligible computational cost compared to the benefits it offers, a key requirement is the prior knowledge of the nonlinear model's structure to generate the  
Figure 5: Quadcopter’s four degrees of freedom with the linear stabilizer. Figure 6: Input signals produced by the linear stabilizer. template \(\Xi_{0}\) for correctly placing the nonlinear functions within the respective differential equations. The efficacy of the proposed approach was confirmed through the application of a linear controller, designed based on the identified model, to the real system. Looking ahead, applying this approach to real-time quadcopter data, as opposed to the data provided by MATLAB, would be advantageous. However, this endeavor exceeds the current scope of the study. Conceptualization, [PERSON] and [PERSON]-C.; formal analysis, [PERSON], [PERSON], [PERSON] and [PERSON]; investigation, [PERSON]-A., [PERSON] and [PERSON]-C.; methodology, [PERSON]-A., [PERSON]-H. and [PERSON]-C.; project administration, J.A.M.-C.; writing--original draft, J.A.M.-C.; writing--review and editing, [PERSON], [PERSON], [PERSON], [PERSON] and [PERSON]-C. All authors have read and agreed to the published version of the manuscript. This research received no external funding. Not applicable. Data Availability StatementData are contained within the article. The authors gratefully acknowledge the partial support provided for this research by the Consejo Nacional de Humanidades Ciencias y Tecnologias (CONAHCYT) through the SNI (Sistema Nacional de Investigadores) scholarship. Additionally, the Instituto Politecnico Nacional contributed to this work through research projects 20230023 and 20240025, as well as by providing scholarships under the programs EDI (Estimulo al Desempen de los Investigadores), COFAA (Comision de Operacion y Fomento de Actividades Academicas), and BEIFI (Beca de Estimulo Institucional de Formacion de Investigadores). The authors declare no conflicts of interest. Definitions  
The following is a verbal description of Algorithm A1:  
1. Given a series of snapshots \(x\in\mathbb{R}^{n}\), \(u\in\mathbb{R}^{p}\) and the corresponding time derivatives \(\dot{x}\) of a dynamical system \(\dot{x}=f(x,u)\), arrange them into matrices \(X,\dot{X}\in\mathbb{R}^{m\times n}\), and \(U\in\mathbb{R}^{m\times p}\) where \(m\) is the number of samples, \(n\) is the dimension of the state vector, and \(p\) is the dimension of the input vector. ```
0: Data matrix \(X\), \(U\), and derivative matrix \(\dot{X}\)
0: Sparse model \(\dot{x}=\Xi\Theta(X,U)\)
1: Construct a library of candidate functions \(\Theta(X,U)\)
2: Solve the sparse regression problem \(\dot{X}=\Theta(X,U)\Xi\) using a sparsity-promoting technique (e.g., LASSO)
3: Identify the active terms and coefficients in \(\Xi\) that form the model
2. Construct a library of nonlinear candidate functions \(\Theta(X,U)\) of size \(m\times q\), where \(q\) is the number of candidate functions. These functions can be constant, polynomial, trigonometric, or more exotic functions of the \(x\) and \(u\). 3. Solve the sparse regression problem \(\dot{X}=\Theta(X,U)\Xi\), where \(\Xi\) is a matrix of coefficients of size \(q\times n\), by minimizing the objective function \[\min_{\Xi}\|\dot{X}-\Theta(X,U)\Xi\|_{F}^{2}+\lambda\|\Xi\|_{1},\] (1) where \(\|\cdot\|_{F}\) is the Frobenius norm, \(\|\cdot\|_{1}\) is the \(l_{1}\) norm, and \(\lambda\) is a regularization parameter that controls the sparsity of \(\Xi\). 4. Identify the sparse set of active terms in \(\Theta(X,U)\) by selecting the rows of \(\Xi\) that have nonzero entries. These terms form the governing equations of the dynamical system: \[\dot{x}=f(x,u)=\sum_{i=1}^{q}\xi_{i}\theta_{i}(x,u),\] (2) where \(\xi_{i}\) is the i-th row of \(\Xi\) and \(\theta_{i}(x,u)\) is the i-th candidate function.