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In this study, a control scheme that allows performing height position regulation and stabilization for an unmanned planar vertical take-off and landing aerial vehicle, in the presence of disturbance due to wind, is presented. To this end, the backstepping procedure together with nested saturation function method is used. Firstly, a convenient change of coordinates in the aerial vehicle model is carried out to dissociate the rotational dynamics from the translational one. Secondly, the backstepping procedure is applied to obtain the height position controller, allowing the reduction of the system and expressing it as an integrator chain with nonlinear disturbance. Therefore, the nested saturation function method is used to obtain a stabilizing controller for the horizontal position and roll angle. The corresponding stability analysis is conducted via the Lyapunov second method. In addition, to estimate the disturbance due to wind, an extended state observer is used. The effectiveness of the proposed control scheme is assessed through numerical simulations, from which convincing results have been obtained.

The Planar Vertical Take-Off and Landing (PVTOL) unmanned aerial vehicle is a representation of the Harrier Yab-8b aircraft when considering a minimum of inputs and outputs to obtain a vertical short take-off and landing behavior [

The works considered most relevant and closely related to the control problem treated in this study are mentioned as follows. In [

Having reviewed the literature, it was found that almost all the works mentioned above were developed to test the PVTOL system indoor, mainly to avoid the undesirable effect produced by the wind (instability and, even, the collapse of the PVTOL system), which is not easy to counteract. Therefore, few controllers are robust under unknown model parameters, actuator failure, and crosswind. Thus, with the intention of contributing to overcome wind undesirable effects, a robust control scheme that combines a backstepping approach and a nested saturation function-based controller is proposed herein to perform taking-off maneuvers in the presence of disturbance due to wind. The backstepping is used to carry out the trajectory tracking task over the vertical position of the PVTOL system and, consequently, to control the height position. Then, from a set of convenient linear transformations, the system is represented as an integrator chain with a nonlinear perturbation, for which a nested saturation function-based controller is developed to stabilize the horizontal position and roll angle. This is carried out by satisfying stability conditions obtained from application of the second method of Lyapunov. Therefore, boundedness of each state and asymptotic convergence to the origin are ensured. Lastly, to estimate the disturbance due to the wind, an extended state observer is used.

The remaining of the paper is organized as follows. In Section

Here, the PVTOL system and its dynamic model considering a disturbance due to wind are presented.

The PVTOL system emulates the vertical take-off and landing of an aerial vehicle, whereas it is automatically stabilized. Hence, in practice, this system has a rigid structure and two motors collocated at the ends of the structure, as can be seen in Figure

Diagram of the PVTOL system.

The representation in state variables of the PVTOL dynamic model, when

To obtain the height position control, we apply the following global coordinate change [

Also, we introduce

From this point, the backstepping procedure can be applied to force the system to track a desired trajectory and, consequently, to reach a desired height position. To this end, an error is defined as follows:

To ensure the stabilization of

Then, it is proceeded with the following change of variables:

So, the augmented Lyapunov function is given by

To facilitate the proposal of

Note that in order to avoid indeterminate (

Taking into account the previous result and proposing

In this section, a nested saturation function-based controller for the stabilization of the horizontal position,

Before developing the control strategy, the definition of a saturation function is introduced.

A linear saturation function

After applying the controller

To express system (

Hence, the transformed system as an integrator chain is given by

In order to obtain the stabilizing controller for system (

The matrix

Thus,

Finally, departing from (

Now, it is proved that the proposed closed-loop system, (

Step 1: to show that the state

whose time derivative is expressed as

It is clear that

Step 2: now, the behavior of

Differentiating it with respect to time and after substituting (

To ensure

Then, there exists a finite time

Thus, when conditions in (

Step 3: substituting (

Then, the following definite positive function is defined:

whose first time derivative is obtained using (

With the purpose of performing

Hence, there exists a finite time

Consequently,

Step 4: substituting (

To demonstrate that

Differentiating

Consequently,

Since

Here, we prove that the closed-loop system, provided by (

Note that after

To demonstrate convergence to zero of all the states, the following Lyapunov function is used:

In this section, the extended state observer needed to estimate the disturbance due to wind,

Consider only the disturbed coordinate:

The following extended state observer is designed:

In this section, the outcomes of some numerical tests are presented in order to validate that the proposed control scheme successfully achieves that

The simulations were performed with the normalized model (

Regarding the disturbance due to wind given in (

The whole parameters to construct

Parameters of

Parameter | Value |
---|---|

On the other hand, the tuning parameters of

The parameter

The gains implemented for the extended state observer were selected as

The initial conditions of the PVTOL system were set as indicated in Table

Initial conditions.

State | Value |
---|---|

The corresponding simulation results are shown in Figure

Simulation results.

In Figure

In this study, a nested saturation function-based controller, in combination with a backstepping controller, for stabilizing the PVTOL system under a disturbance due to wind was used. With this approach, the control design complexity of a higher-order system is reduced to design a control for a lower-order nonlinear subsystem of the original system. Thus, the proposed control approach allows designing a controller based on nested saturation functions, which contemplates perturbations, guaranteeing the convergence of the roll angle to zero within a finite time and, consequently, the convergence to zero of the horizontal state. The stability analysis of the closed-loop system was based on the second method of Lyapunov, using a simple candidate function. It is important to remark that the controller, based on backstepping and nested saturation functions, allows performing take-off maneuvers in the presence of exogenous disturbances, which are found when aircraft carries out actual maneuvers. Furthermore, an extended state observer is used to estimate the disturbance due to wind. Numerical simulations were carried out to test the effectiveness of the proposed controller, having obtained convincing results. Finally, the proposed scheme was compared with a classical controller, finding that the controller based on backstepping and nested saturation functions presented herein has better performance.

It is worth mentioning that an experimental platform that allows configuring the PVTOL system has been designed, whose construction is in process. Thus, experimental implementation of the control scheme proposed herein is considered as a future work.

The data used to support the conclusion of this study are available from the corresponding author upon request.

The authors declare that there are no conflicts of interest regarding the publication of this article.

This study was supported by the Secretaría de Investigación y Posgrado of the Instituto Politécnico Nacional under the research grants 20201829, 20200162, 20201623, and 20200671. C. A. Merlo-Zapata acknowledges financial support from Instituto Tecnológico de Iztapalapa. M. Antonio-Cruz and C. Márquez-Sánchez thank the support from the Sistema Nacional de Investigadores-CONACYT.