# Identification of High-Order Nonlinear Coupled Systems Using a Data-Driven Approach
## 2 Mathematical Tools
### Simulink Support Package for Parrot Minidrones  
The Simulink Support Package for Parrot Minidrones version 20.2.3 is a toolbox that enables users to design and deploy flight control algorithms for Parrot minidrones using Simulink. Users can wirelessly connect to the minidrones via Bluetooth and access their onboard sensors and camera. Users can also use additional tools such as the Aerospace Blockset and Simulink Coder to enhance their simulations and code generation. The toolbox also includes an example that shows how to model and simulate the 6-DOF equations of motion for the minidrones [29; 30]. In Figure 2, the physical quadcopter simulated by the Simulink Support Package for Parrot Minidrones is depicted. Figure 1: Body diagram of the quadcopter. Figure 2: Rolling Spider by Parrot Minidrones. This package includes a Controller Project template designed as a framework for crafting a personalized controller or adapting it to specific needs. The template integrates a robust plant model simulation for both the Parrot Rolling Spider and Parrot Mambo drones. This simulation streamlines the evaluation of model outcomes before deployment, empowering users to scrutinize and refine their controller designs prior to applying them to the physical hardware. The template supports the modeling of equations of motion with six degrees of freedom (6-DOF), facilitating the simulation of aircraft behavior under diverse flight and environmental conditions. Figure 3 depicts the VRML environment employed by the Simulink Package for Parrot Minidrones. It is worth mentioning that the Simulink Support Package for Parrot Minidrones has undergone extensive validation, confirming its high accuracy in approximating the real-time behavior of the Parrot Minidrones. As a result, the identification of the quadcopter's mathematical model in this study relies heavily on the results generated by the Simulink package. Identification of a High-Order Nonlinear Systems with Coupled Dynamics Using the Classical SINDY Algorithm  
In this section, the Algorithm A1 is used to obtain the mathematical model of a quadcopter from the data generated by the Simulink Support Package for Parrot Minidrones. For the sake of simplicity, throughout the remainder of this study, it is assumed that the Rolling Spider MiniDrone depicted in Figure 2 can be effectively modeled using Equations (5)-(17). In other words, the drone's controls are considered to be \(\beta_{1}\), \(\beta_{2}\), \(\beta_{3}\), and \(\beta_{4}\), instead of \(u_{1}\), \(u_{2}\), \(u_{3}\), and \(u_{4}\). Adopting this perspective streamlines the required library of functions for SINDY, specifically the matrix \(\Theta(X,U)\). Despite this simplification, the resulting nonlinear system maintains sufficient complexity to reveal the limitations of the conventional SINDY approach. This underscores the advantages of the proposed modification, particularly in the identification of high-order nonlinear systems with coupled dynamics. The Simulink Support Package for Parrot Minidrones is employed to simulate a 10-s flight sequence of the drone, with the sampling time \(T=0.005\) s. Throughout this duration, the drone undergoes takeoff and ascends to an altitude of 1.5 m. Meanwhile, the controls \(\beta_{1}\), \(\beta_{2}\), \(\beta_{3}\), and \(\beta_{4}\) are subtly perturbed to excite all four independent degrees of freedom of the drone. These inputs are shown in Figure 4.  
Figure 3: Capture of the VRML environment provided by the Simulink Support Package for Parrot Minidrones. During this simulated flight, the vectors \(x_{1},\ldots,x_{12},\dot{x}_{1},\ldots,\dot{x}_{12},\beta_{1},\ldots,\beta_{4}\in \mathbb{R}^{m}\) are formed by reading the linear and angular displacements, linear and angular velocities, linear and angular accelerations, and controls. Clearly, \(m=2001\).