1. Introduction

2. Mathematical Tools

3. Identification of a High-Order Nonlinear Systems with Coupled Dynamics Using the Classical SINDY Algorithm

Now, the MATLAB code for the classical SINDY algorithm, with $Theta=\Theta (X,U)$, $dXdt=\dot{X}$, $lambda=1\times {10}^{-3}$, and $n=12$, is:

Listing 1: Matlab function for the classical SINDY

4. Identification of a high-order nonlinear systems with coupled dynamics through the modified SINDY algorithm

4.1. GenesisXi algorithm

At this point, it is obvious that, in addition to knowing the functions that should appear in the model to be identified, it is important to know in which differential equation each of them should appear. This transforms the task of identifying the nonlinear model into a coefficient identification problem, which proves highly advantageous for a wide range of real-world nonlinear problems.

Therefore, the modification to the classical SINDY algorithm involves incorporating this additional information in a manner that the obtained result approaches the known models, specifically in this case, the models of the quadrotors.

To achieve this objective, it is necessary to formulate an initial matrix ${\Xi }_0$ wherein the positions of all candidate functions contained in $\Theta (X,U)$ within the differential equations of the mathematical model are defined. To automate this process, the Algorithm 2 for obtaining the aforementioned matrix ${\Xi }_0$ is proposed.

Algorithm 2 GenesisXi algorithm

Require: Data vector ${\Theta }_L$ and n Ensure: Initial matrix ${\Xi }_0$ 1: ${\Xi }_0\leftarrow 0_{[q,n]}$ 2: for$k=1$toqdo 3:     ${\Xi }_0(k,{\Theta }_L\left(k\right))\leftarrow 1$ 4: end for

The following is the description of the Algorithm 2:

1.

Given the vector ${\Theta }_L\in {\mathbf{R}}^q$, whose elements contain the number of the differential equation where the corresponding candidate function should be located, and n, which is the dimension of the nonlinear system to be estimated, then the matrix ${\Xi }_0\in {\mathbf{R}}^{q\times n}$ is set to zero. For this case, ${\Theta }_L=\left[9123654789101011111212\right]$.

2.

The vector ${\Theta }_L$ is traversed from its first element to the last using the index k, simultaneously placing a "1" at the position $(k,{\Theta }_L\left(k\right))$ in the matrix ${\Xi }_0$.

The MATLAB code corresponding to Algorithm 2, with $Theta\_L={\Theta }_L$, is given in Listing 2.

Listing 2: MATLAB function for the GenesisXi algorithm

Thus, after applying Algorithm 2, and in accordance with (7)-(19), the resulting matrix ${\Xi }_0$ is:

$${\Xi }_0=\left[\begin{array}{cccccccccccc}\hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 1& \hfill 0& \hfill 0& \hfill 0\\ \hfill 1& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0\\ \hfill 0& \hfill 1& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0\\ \hfill 0& \hfill 0& \hfill 1& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0\\ \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 1& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0\\ \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 1& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0\\ \hfill 0& \hfill 0& \hfill 0& \hfill 1& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0\\ \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 1& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0\\ \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 1& \hfill 0& \hfill 0& \hfill 0& \hfill 0\\ \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 1& \hfill 0& \hfill 0& \hfill 0\\ \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 1& \hfill 0& \hfill 0\\ \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 1& \hfill 0& \hfill 0\\ \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 1& \hfill 0\\ \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 1& \hfill 0\\ \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 1\\ \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 1\end{array}\right].$$

(28)

Notice that the columns of ${\Xi }_0$ represent the differential equations in the nonlinear model, while the rows of ${\Xi }_0$ represent the candidate functions included in $\Theta (X,U)$. Thus, the matrix (28) indicates that $f_1$ appears in the ninth differential equation, $f_2$ appears only in the second differential equation, and so forth.

4.2. Modified SINDY Algorithm

Now, the location of the candidate functions in the matrix ${\Xi }_0$ is passed as a parameter to the modified SINDY function, as illustrated in the MATLAB code presented in Listing 3.

Listing 3: MATLAB function with modifications to classical SINDY

Roughly speaking, the modifications consist of removing the functions that wrongly appear in the differential equations after the first guess of the matrix $\Xi$ in line 4. To achieve this, the MATLAB operator "$.*$" is applied to multiply each element of the matrix $\Xi$ obtained in line 4 by the corresponding element of the matrix ${\Xi }_0$. In other words, ${\Xi }_0$ is used as a template for the desired matrix $\Xi$. The remainder of the program follows exactly the structure of the classical SINDY function presented in Listing 1.

The matrix $\Xi$ obtained by applying the MATLAB function provided in Listing 3 to the data described in the previous section is:

$$\begin{array}{ccc}\hfill \Xi & =& \left[\begin{array}{cccccc}\hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0\\ \hfill 1& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0\\ \hfill 0& \hfill 1& \hfill 0& \hfill 0& \hfill 0& \hfill 0\\ \hfill 0& \hfill 0& \hfill 1& \hfill 0& \hfill 0& \hfill 0\\ \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 1\\ \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 1& \hfill 0\\ \hfill 0& \hfill 0& \hfill 0& \hfill 1& \hfill 0& \hfill 0\\ \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0\\ \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0\\ \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0\\ \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0\\ \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0\\ \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0\\ \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0\\ \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0\\ \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0\end{array}\right.\hfill \\ \hfill & & \left.\begin{array}{cccccc}\hfill 0& \hfill 0& \hfill 4.725& \hfill 0& \hfill 0& \hfill 0\\ \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0\\ \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0\\ \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0\\ \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0\\ \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0\\ \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0\\ \hfill 14.32& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0\\ \hfill 0& \hfill -14.29& \hfill 0& \hfill 0& \hfill 0& \hfill 0\\ \hfill 0& \hfill 0& \hfill 7.302& \hfill 0& \hfill 0& \hfill 0\\ \hfill 0& \hfill 0& \hfill 0& \hfill 8169.0& \hfill 0& \hfill 0\\ \hfill 0& \hfill 0& \hfill 0& \hfill -2.442& \hfill 0& \hfill 0\\ \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 5939& \hfill 0\\ \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill -7.961& \hfill 0\\ \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 4087\\ \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0.09223\end{array}\right].\hfill \end{array}$$

(29)

This matrix corresponds to the sparse identification of the high-order nonlinear system with coupled dynamics, specifically, the drone. The identified nonlinear system is given by:

$$\begin{array}{ccc}\hfill {\dot{x}}_1& =& x_7\hfill \end{array}$$

(30)

$$\begin{array}{ccc}\hfill {\dot{x}}_2& =& x_8\hfill \end{array}$$

(31)

$$\begin{array}{ccc}\hfill {\dot{x}}_3& =& x_9\hfill \end{array}$$

(32)

$$\begin{array}{ccc}\hfill {\dot{x}}_4& =& x_{12}\hfill \end{array}$$

(33)

$$\begin{array}{ccc}\hfill {\dot{x}}_5& =& x_{11}\hfill \end{array}$$

(34)

$$\begin{array}{ccc}\hfill {\dot{x}}_6& =& x_{10}\hfill \end{array}$$

(35)

$$\begin{array}{ccc}\hfill {\dot{x}}_7& =& 14.32{\beta }_1(sin\left(x_4\right)sin\left(x_6\right)+cos\left(x_4\right)cos\left(x_6\right)sin\left(x_5\right))\hfill \end{array}$$

(36)

$$\begin{array}{ccc}\hfill {\dot{x}}_8& =& -14.29{\beta }_1(cos\left(x_4\right)sin\left(x_6\right)-cos\left(x_6\right)sin\left(x_4\right)sin\left(x_5\right))\hfill \end{array}$$

(37)

$$\begin{array}{ccc}\hfill {\dot{x}}_9& =& 7.302{\beta }_{1}cos\left(x_5\right)cos\left(x_6\right)+4.725\hfill \end{array}$$

(38)

$$\begin{array}{ccc}\hfill {\dot{x}}_{10}& =& 8169{\beta }_2-2.442x_{11}{x}_{12}\hfill \end{array}$$

(39)

$$\begin{array}{ccc}\hfill {\dot{x}}_{11}& =& 5939{\beta }_3-7.961x_{10}{x}_{12}\hfill \end{array}$$

(40)

$$\begin{array}{ccc}\hfill {\dot{x}}_{12}& =& 4087.0{\beta }_4+0.09223x_{10}{x}_{11}.\hfill \end{array}$$

(41)

Notice that the gravity value in equation (38) is calculated as 4.725, which is 2.076 times smaller than 9.81. This implies that the entire equation is scaled by $2.{076}^{-1}$. However, as per equations (14)-(16), when the identified model is analyzed at the equilibrium, i.e., when all linear and angular displacements, velocities, and accelerations are zero, and the mass m is computed using equations (36), (37), and (38) multiplied by 2.076, it results in $m=0.0698$, $m=0.0699$, and $m=0.0659$, respectively, which are very close to the nominal value found in literature of $m=0.068$ [30,31].

Now, based on these equations, a linear stabilizer is designed. To achieve this, equations (30)-(41) are linearized at the equilibrium $x_{eq}={\left[000000000000\right]}^T$, resulting in:

$$\dot{x}=Ax+Bu,$$

(42)

with

$$A=\left[\begin{array}{cccccccccccc}\hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 1& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0\\ \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 1& \hfill 0& \hfill 0& \hfill 0& \hfill 0\\ \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 1& \hfill 0& \hfill 0& \hfill 0\\ \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 1\\ \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 1& \hfill 0\\ \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 1.0& \hfill 0& \hfill 0\\ \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill -9.27& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0\\ \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 9.247& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0\\ \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0\\ \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0\\ \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0\\ \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0\end{array}\right],$$

(43)

and

$$B={\left[\begin{array}{cccccccccccc}\hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 7.302& \hfill 0& \hfill 0& \hfill 0\\ \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 8169& \hfill 0& \hfill 0\\ \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 5939& \hfill 0\\ \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 4087\end{array}\right]}^T.$$

(44)

The stabilizer $u=-Kx$ is obtained using the Linear Quadratic Regulator (LQR) approach, which combines the control energy and the state deviations in the cost function of the form:

$$J={\int }_0^{\infty }\left(x^{T}Qx+u^{T}Ru\right)dt,$$

(45)

where Q and R are positive definite matrices determining the compromise between minimizing the state deviations and the control energy [32]. In this case, $Q=I_{[12\times 12}]$, and $R=1\times {10}^6·I_{[4\times 4}]$. Thus, with matrices A, B, Q, and R as above, the gain K is computed using the MATLAB function $lqr$, resulting in:

$$\begin{array}{ccc}\hfill K& =& \left[\begin{array}{cccccc}\hfill 0& \hfill 0& \hfill 0.001& \hfill 0& \hfill 0& \hfill 0\\ \hfill 0& \hfill 0.001& \hfill 0& \hfill 0& \hfill 0& \hfill 0.006795\\ \hfill -0.001& \hfill 0& \hfill 0& \hfill 0& \hfill 0.007291& \hfill 0\\ \hfill 0& \hfill 0& \hfill 0& \hfill 0.001& \hfill 0& \hfill 0\end{array}\right.\hfill \\ \hfill & & \left.\begin{array}{cccccc}\hfill 0& \hfill 0& \hfill 0.01658& \hfill 0& \hfill 0& \hfill 0\\ \hfill 0& \hfill 0.001571& \hfill 0& \hfill 0.001632& \hfill 0& \hfill 0\\ \hfill -0.001604& \hfill 0& \hfill 0& \hfill 0& \hfill 0.001859& \hfill 0\\ \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0.00122\end{array}\right]\hfill \end{array}$$

(46)

Then, the stabilizer $u=-Kx$ is applied to the nonlinear model within the Simulink Support Package for Parrot Minidrones to assess the effectiveness of the identified model. In this example, the quadcopter reaches an altitude of 1m using the linear stabilizer. The numerical results are presented in Figure 5 and Figure 6. Figure 5 illustrates the drone’s response to the linear stabilizer, while Figure 6 displays the corresponding input signals.

Please note that the goal of this work is not to design sophisticated controllers aiming for complex behaviors in the drone. Instead, the primary contribution of this work is to introduce an alternative method for obtaining the mathematical model of high-order nonlinear system with coupled dynamics based on real-time measured data.

5. Discussion

1

In the specific case study, the outcomes achieved through the classical SINDY algorithm do not align with physics-based identification. Moreover, they do not qualify as sparse identification, as all candidate functions are present in the matrix $\Xi$.

2

Based on the knowledge of the structure of the complex system to be identified, a more precise algorithm has been introduced without a significant increase in computational cost.

3

Although the identified model appears to have significant differences from the related model found in the literature, it enables the design of a controller capable of stabilizing the original high-order nonlinear system with coupled dynamics. This, in turn, demonstrates its notable approximation degree.

4

The primary drawback of the proposed approach is that the identification problem transforms into a coefficient identification task. Nevertheless, in many real-world applications, this outcome may still be appealing because, in numerous cases, it is impossible to precisely determine the coefficients of a specific system, even when the structure of its mathematical model is available.

6. Conclusions

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