A Robust Control Scheme for a PVTOL System Subject to Wind Disturbances

Tecnológico Nacional de México, Instituto Tecnológico de Iztapalapa, Departamento de Ciencias Básicas, Ciudad de México 09208, Mexico CIC, Instituto Politécnico Nacional, Unidad Profesional Adolfo López Mateos, Ciudad de México 07700, Mexico SEPI, UPIITA, Instituto Politécnico Nacional, La Laguna Ticoman, Ciudad de México 07340, Mexico Instituto Politécnico Nacional, UPIICSA, SEPI, Mexico City 08400, Mexico Tecnológico Nacional de México, Instituto Tecnológico de Iztapalapa, Departamento de Sistemas y Computación, Ciudad de México 09208, Mexico SEPI, ESCOM, Instituto Politécnico Nacional, Unidad Profesional Adolfo López Mateos, Ciudad de México 07700, Mexico


Introduction
e 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 [1], which has been used as a test bed for automatic control applications. In fact, the PVTOL aerial vehicle is a simplified model that embodies the behavior of several actual vertical take-off and landing aircraft, which ultimately makes it a suitable benchmark to test new and existing controllers. erefore, there is a vast literature on the subject. However, a current and open challenge is the design of robust controllers for the PVTOL system under wind disturbance in order to pursue outdoor applications. us, this paper presents a robust control scheme for a PVTOL system subjected to disturbance due to wind. e works considered most relevant and closely related to the control problem treated in this study are mentioned as follows. In [1], an input-output linearization to achieve trajectory tracking control for the nonminimum phase nonlinear system is presented. In [2], the authors developed a nonlinear output-feedback controller for the trajectory tracking of a reference model by using a global exponential observer, coordinate transformations, the Lyapunov's method, and an extension of backstepping. In [3], a global stabilization control derived from nonlinear combinations of linear saturation functions was presented, whereas in [4], a nonlinear controller for taking-off, hovering, tracking of a straight line, and landing of the PVTOL system was introduced. e corresponding experimental implementation of the controller was reported in [5], where a camera was used to estimate the position and orientation of the PVTOL vehicle. Also, in [6], a global configuration stabilization for the VTOL aircraft with a strong input coupling using a smooth static state feedback was reported. An alternative feedback-based stabilization law to the one introduced in [6] was presented in [7]. is law simplifies the term that connects the state vector with the dynamics error. In [8], the authors designed a stabilization control by transforming the system into an integrator chain plus a nonlinear disturbance, after which the saturation technique was used so that neither backstepping/forwarding approaches nor small gain analysis is required. Furthermore, a nonlinear controller for a PVTOL vehicle, based on prediction and partial feedback linearization, was designed in [9]. A robust and linear state-feedback gainscheduled control to achieve hovering of a PVTOL system with uncertainties in the mass, the momentum of inertia, and the parasitic coupling parameter was introduced in [10], while a nested set stabilization approach to locally solve path following for the PVTOL system was introduced in [11]. e system center of mass was constrained to lie on the path, and the roll angle should be specified at any given point on the path. In [12], the authors developed a bounded backstepping method to achieve input-to-state stability, with respect to the actuator errors, and to force all trajectories of the system to track a reference trajectory for all initial configurations. Also, Zavala-Río et al. [13] introduced a finite-time observer-based output-feedback control for the global stabilization of both taking-off and landing of the PVTOL system. In [14][15][16], a solution for the regulation of a simplified version of the PVTOL system was reported, which consisted of two control actions that act simultaneously. Aguilar-Ibáñez et al. [14] used, as first action, a feedback linearization along with a saturation function to asymptotically stabilize the vertical position. For the second action, backstepping was exploited to stabilize both the horizontal and the angle positions. Similarly, Aguilar-Ibañez et al. [15] utilized as first action a feedback linearization in combination with a nonlinear controller to stabilize the vertical variable. e other action stabilizes the horizontal and angular variables to the desired rest position through an energy-control method, whereas Aguilar-Ibañez [16] employed again a feedback linearization with a saturation function to stabilize the vertical variable, while a PD-controller and a sliding-mode controller were used to stabilize both the horizontal and angular variables. Moreover, Yu-Chan et al. [17] dealt with the stabilization of the PVTOL system with unknown model parameters by applying a slidingmode technique to design a state feedback control law. Recently, in [18], a controller for the stabilization of the PVTOL vehicle was designed on the basis of the immersion and invariance control technique. e controller gives priority to the control of the aircraft's altitude before controlling the lateral displacement. More recently, in [19], a cascade active disturbance rejection controller was introduced to counteract the adverse effects caused by an actuator failure in the PVTOL aircraft, while Escobar et al. [20] were focused on finding conditions to determine local asymptotic stability using a feedback linearization control for the PVTOL platform, so that reaching any singularity due to the transformation of the system is prevented. Aguilar-Ibanez et al. [21] introduced an output-feedback regulation control law for a PVTOL aircraft, based on a version of the matching control energy method. Such a control was improved to compensate bounded, smooth, and matching perturbations with a suitable finite time-varying identificator. Finally, Aguilar-Ibanez et al. [22] proposed a robust controller to solve the trajectory-tracking control problem of PVTOL aircraft under crosswind by applying an input-output feedback linearization to the PVTOL model under no crosswind conditions. us, the resulting linearized system under the crosswind effects is controlled using an active disturbance rejection control approach to counteract the effects of these perturbations.
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. erefore, few controllers are robust under unknown model parameters, actuator failure, and crosswind. us, 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. e 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. en, 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. is is carried out by satisfying stability conditions obtained from application of the second method of Lyapunov. erefore, 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. e remaining of the paper is organized as follows. In Section 2, the PVTOL system and its dynamics are introduced. In Section 3 the design of the backstepping controller to perform trajectory tracking for the height position of the system is presented. In Section 4, the controller based on nested saturation functions for stabilization of the horizontal position and the roll angle is designed. e extended state observer is introduced in Section 5. In Section 6, the outcome of numerical simulations that show the behavior of the proposed control scheme is reported. Finally, Section 7 is devoted to the concluding remarks.

PVTOL System
Here, the PVTOL system and its dynamic model considering a disturbance due to wind are presented.
e 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 1, where θ 1 is the roll angle, x 1 is the horizontal position in the x axis, y 1 is the vertical position in the y axis, f 1 and f 2 are the forces produced by the motors, L is the distance between the center of the rigid structure to the center of the motors, m is the mass of the system, and g is the gravitational acceleration. e representation in state variables of the PVTOL dynamic model, when L, m, and g are normalized, has been previously reported in [1] and used in [14,23,24], which is given by where 1 are the state variables, u 1 � f 1 + f 2 and u 2 � f 1 − f 2 are the control inputs, ϵ is the coefficient giving the coupling between the rolling moment and the lateral acceleration, and A L is the rolling moment due to the air, defined by Gomes and Ramos [25] as with ρ being the air density, C l being the nondimensional coefficient of the rolling moment in the standard convention for airships, U being the air speed, and V a being the airship model volume. at is, A L is considered as a disturbance due to wind.

Height Position Control
To obtain the height position control, we apply the following global coordinate change [26] to model (1): Also, we introduce A L � A L − A L , where A L is an estimation of the disturbance due to wind carried out by an extended state observer, which is described later in Section 5.
us, model (1) is transformed into the following system: where u 1 � u 1 − ϵθ 2 2 is a new control input. 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: with y 1 d being the desired height position. en, the method of Lyapunov is used, considering the following candidate function: which is positive definite and whose time derivative results in To ensure the stabilization of e y , the auxiliary control, y 2 , is proposed as with α 1 > 0, so that (7) results in the following negative semidefinite expression: x O y y 1 Figure 1: Diagram of the PVTOL system. Complexity en, it is proceeded with the following change of variables: So, the augmented Lyapunov function is given by whose time derivative is determined by To facilitate the proposal of u 1 that ensures stabilization of e 2y , let us make (12) equal to zero for a moment and solve for u 1 . Since € y 1 d � 0 is considered because it is the desired acceleration of the height position, u 1 is found as follows: Note that in order to avoid indeterminate (13), − (π/2) < θ 1 < + (π/2) is required.

Control of the Horizontal Position and Roll Angle
In this section, a nested saturation function-based controller for the stabilization of the horizontal position, x 1 , and roll angle, θ 1 , is developed [27]. is technique has been used for the stabilization of nonlinear systems that can be approximately expressed as integrator chain [28][29][30]. To solve the PVTOL system stability problem, first a linear transformation to propose the stabilizing controller is used. en, it is showed that the proposed controller guarantees the boundedness of all states and, after a finite time, the closedloop system is asymptotically stable. Before developing the control strategy, the definition of a saturation function is introduced.
Definition 1 (see [31]). A linear saturation function σ b (s): R ⟶ R is defined as with b > 0 being the upper bound of the function.

System as an Integrator
Chain. After applying the controller u 1 , the transformed system (4) can be reduced to the subsystem (x 1 , θ 1 ), that is: To express system (17) as an integrator chain, with a nonlinear perturbation, and to propose a controller for the stabilization of the subsystem (x 1 , θ 1 ), it is proceeded similarly as in [32], so that the following global nonlinear transformation is defined: ] � − u 2 sec 2 θ 1 + 2θ 2 2 tanθ 1 sec 2 θ 1 .

(18)
Hence, the transformed system as an integrator chain is given by whose matrix representation can be expressed as where ξ � (x 1 , x 2 , w 1 , w 2 ) is the new state vector,

Nested Saturation Function-Based Controller.
In order to obtain the stabilizing controller for system (20) and inspired by [27], the linear transformation q � Sξ is used, in which S must satisfy e matrix S that achieves the aforementioned equalities is given by [31] S � us, q results in which transforms system (20) into for which, the following nested saturation function-based stabilizing controller is proposed: where σ α (·), σ β (·), and σ c (·) are linear saturation functions as defined in (16) and α, β, and c are the upper bounds of each nested saturation function. Finally, departing from (18) and using (26), u 2 can be constructed as follows: 4.3. Boundedness of All States. Now, it is proved that the proposed closed-loop system, (25) with (26), ensures that all the states are bounded and that the bound of each of them directly depends on the design parameters of the controller.
Step 1: to show that the state q 4 is bounded, the following positive definite function is defined: whose time derivative is expressed as It is clear that _ V 4 < 0 is accomplished when |q 4 | ≥ α + (A L sec 2 θ). erefore, there exists a finite time T 1 > 0, such that Step 2: now, the behavior of q 3 is analyzed. For this, a positive definite function is introduced as follows: Differentiating it with respect to time and after substituting (26) into _ q 3 , the following is obtained: To ensure _ V 3 < 0 is achieved, the following conditions must be satisfied: en, there exists a finite time T 2 > T 1 after which Complexity q 3 (t) < β + A L sec 2 θ 1 , ∀t > T 2 .
us, when conditions in (33) are satisfied, q 3 is bounded and the stabilization controller (26) takes the following structure: Step 3: substituting (35) into the second differential equation of (25), the following is obtained: en, the following definite positive function is defined: whose first time derivative is obtained using (36), as follows: With the purpose of performing _ V 2 < 0, it is required that β and c satisfy the below conditions: Hence, there exists a finite time T 3 > T 2 , after which Consequently, q 2 is bounded and the control ] turns out to be Step 4: substituting (41) into the first equation of (25), the following is obtained: To demonstrate that q 1 is bounded, a definite positive function is defined as follows: Differentiating V 1 along the trajectories of (42), the following is obtained: where c must be selected so that c > − 3ϵA L secθ 1 + A L sec 2 θ 1 and |q 1 | > − 3ϵA L secθ 1 + A L sec 2 θ 1 to achieve _ V 1 < 0. erefore, there exists a finite time T 4 > T 3 , such that Consequently, q 1 is also bounded. Finally, the values of the parameters α, β, and c can be determined as follows: Since secθ 1 > secθ 1 − ϵ > secθ 1 − 3ϵ, r > |A L sec 2 θ 1 | > 0 can be introduced, which is directly related to the magnitude of the system disturbance, to select the upper bounds of the saturation functions as

Convergence of All States to Zero.
Here, we prove that the closed-loop system, provided by (25) and (26) satisfying (47), is asymptotically stable. Note that after t > T 4 , controller (26) is no longer saturated. at is, and the closed-loop system turns out to be To demonstrate convergence to zero of all the states, the following Lyapunov function is used: and differentiating it along the trajectories of (49), the following is obtained: is positive definite with λ min � (1/2). erefore, (51) is strictly negative, when A L ⟶ 0. en, the vector of states q exponentially converges to zero after some time t > T 4 .

Extended State Observer
In this section, the extended state observer needed to estimate the disturbance due to wind, A L , is introduced [33].
Consider only the disturbed coordinate: e following extended state observer is designed: where θ 2 is the estimate of the roll velocity, A L is the estimate of the disturbance due to wind, λ 1 and λ 2 are the gains of the observer, which must satisfy the following condition: λ 1 ≪ λ 2 , and lastly u 2 is redefined and proposed as where u 2 � − (1/sec 2 θ 1 )] + 2θ 2 2 tanθ 1 represents a fictitious controller, acting on the coordinate θ 2 .

Simulation Results
In this section, the outcomes of some numerical tests are presented in order to validate that the proposed control scheme successfully achieves that (x 1 , x 2 , y 1 , y 2 , θ 1 , θ 2 ) ⟶ (0, 0, y 1 d , 0, 0, 0). at is, the control scheme carries out height position regulation, through performing trajectory tracking task, and stabilization of the horizontal position and roll angle for the PVTOL system in the presence of random disturbance due to wind. e simulations were performed with the normalized model (1) in MATLAB-Simulink, using Euler's numerical method with fixed step and a sample time of 1 ms. In that direction, the coefficient giving the coupling between the rolling moment and the lateral acceleration was chosen as in [1], i.e., ϵ � 0.001. Also, the desired trajectory, y 1 d , was proposed as the following Bézier polynomial: where v 1 � 0 m and v 2 � 2 m are the constant values, P d (t) is defined by Regarding the disturbance due to wind given in (2), the air density of the Mexico City was used for ρ and the aerodynamic coefficient C l was characterized as the following linear approximation: with m 1 being the slope and θ 1 being the roll angle of the PVTOL system in degrees. It is important to mention that this linear approximation was determined from the results of C l , obtained with respect to the variation of the roll angle of an aircraft similar to the PVTOL system, by using a wind Complexity 7 chamber [34]. Furthermore, the air speed U was generated as follows: where A U is a random amplitude, t is time, P is a random period, ϕ is a random phase, and a is a random offset. erefore, (2) is random and can be rewritten as e whole parameters to construct A L are shown in Table 1.
On the other hand, the tuning parameters of u 1 were set at e parameter r � 1 was chosen for u 2 so that e gains implemented for the extended state observer were selected as (64) e initial conditions of the PVTOL system were set as indicated in Table 2. e corresponding simulation results are shown in Figure 2. With the intention of comparing the performance of the proposed control scheme with a classical controller, Figure 2 also presents simulation results when using a PID structure for u 1 and u 2 to carry out regulation of the system. For the simulation of the normalized system in closed loop with the PID controllers, (61) with parameters in Table 1 was preserved, the initial conditions in Table 2 were set, the gains of the observer were maintained, and the gains of the controllers were tuned as for u 1 and for u 2 . To distinguish the results, the ones associated with the PID controllers are denoted with the subscript PID.
In Figure 2, it can be observed that the proposed control scheme allows achieving successfully the height position regulation, through the trajectory tracking task, and stabilization of the horizontal position and roll angle when the system is subjected to random disturbance due to wind. at is, (x 1 , x 2 , y 1 , y 2 , θ 1 , θ 2 ) ⟶ (0, 0, y 1 d , 0, 0, 0) is accomplished. Note that the execution of the trajectory tracking task in the vertical position provides maneuverability when taking-off the PVTOL system. However, with the PID controllers, the regulation of state x 1 cannot be achieved, but only for the positions y 1 and θ 1 .
at is, stabilization of the whole system is not achieved. Although PID structure can achieve the height position regulation when the system is far from the desired height, it requires excessive values for u 1 and u 2 and does not allow the taking-off of the system in a controlled way.
us, advantages of the proposal presented herein are maneuverability and whole stabilization, when the system is under the undesired effect caused by wind.

Conclusions
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. us, 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. e stability analysis of the closedloop 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. us, experimental implementation of the control scheme proposed herein is considered as a future work.

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

Conflicts of Interest
e authors declare that there are no conflicts of interest regarding the publication of this article.  Complexity