# Controlling the PVTOL Aircraft System with an Inverted Pendular Load by means of Nested Saturation Functions and GPI Controller

Authors: Cesar Alejandro Villaseñor Rios; Octavio Gutiérrez-Frías; Carlos Aguilar-Ibanez; Miguel S. Suarez-Castanon
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

This paper presents a control scheme that allows height position regulation and stabilization for an unmanned planar vertical takeoff and landing aircraft system with an inverted pendular load. The proposed controller consists of nested saturations and a generalized proportional integral (GPI). The GPI controls the aircraft height and the roll attitude; the latter is used as the fictitious input control. Next, the system is reduced through linear transformations, expressing it as an integrator chain with a nonlinear perturbation. Finally, the nested saturation function-based controller stabilizes the aircraft’s horizontal position and the pendulum’s angle. Obtaining the control approach was a challenging task due to the underactuated nature of the aircraft, particularly ensuring the pendulum’s upright position. The stability analysis was based on the second method of Lyapunov using a simple candidate function. The numerical simulation confirmed the control strategy’s effectiveness and performance. Additionally, the numerical simulation included a comparison against a PD controller, where its corresponding performance indexes were estimated, revealing that our controller had a better response in the presence of unknown disturbances.


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## Body

## 1. Introduction

The control of underactuated mechanical systems is a widely studied field and continuously increases knowledge, mainly because controlling this kind of system is challenging since it has fewer controllers than degrees of freedom. The inverted pendulum system is a classic example of an underactuated system. It consists of a freely spinning load around an axis and attached to a base that freely moves forward and backward—multiple authors have proposed control laws to stabilize this kind of system in its inverted position. Block and Spong [1] proposed the partial feedback collocated and noncollocated control law for the stabilization around the origin of the acrobot and the pendubot. Fantoni and Lozano [2] proposed a nested saturation control for the wheeled inverted pendulum that enables stabilization around the origin. Ibañez and Frias [3] proposed a nested saturation control for the nonlinear perturbed wheeled inverted pendulum, expressed it as a chain of integrators. The proposed control demonstrates asymptotical stability around the origin through the Lyapunov method when the pendulum angle is in the upper half-plane. The Furuta pendulum is another challenging system that has attracted the attention of several researchers, who have presented many interesting approaches to control this kind of pendulum. For instance, in [4], the authors proposed a Lyapunov-based control method for the stabilization around the origin, Furuta pendulum stabilization around the origin, while in [5], the authors introduced an active disturbance rejection-based control and its stability analysis. Another exciting example of underactuated systems is the unmanned aerial vehicle (UAV) such that this kind of system has been of high interest in the present century because of wide applications for different fields such as farming, photography, exploration, and military [6–8]. A typical example of a UAV is the planar vertical takeoff and landing (PVTOL) aircraft, a simplified model of the actual vertical takeoff and landing aircraft [9], which encompasses almost all the dynamics found in real UAVs. The PVTOL has been used as a suitable benchmark to test new and existing controllers because it behaves like the well-known quadrotor in a two-dimensional plane. There exist many works that tackle the PVTOL stabilization problem. For instance, Aguilar-Ibanez et al. [10] proposed a control scheme using controlled Lagrangian for PVTOL stabilization. Fantoni et al. [11] introduced a control scheme using the PVTOL attitude as fictitious control and nested saturation technique for the stabilization at the origin. In [12], Lozano et al. showed that once controlling the PVTOL aircraft height, the system resembles an inverted pendulum, and using a change of coordinates, it can be seen as an integrator chain with nonlinear disturbance. Also, in this study, the authors present a nested saturation-based control and demonstrated asymptotic stabilization at the origin. In [13], based on the image-based visual servoing method and the backstepping technique, the authors presented a novel control strategy to force a vertical takeoff and landing (VTOL) aircraft. Works [14, 15] can be valuable sources for the readers interested in this problem.Combining the inverted pendulum and UAV systems adds a degree of freedom, obtaining a system noneasy to stabilize. Hehn et al. [16] proposed this problem in 2011 and named it the flying inverted pendulum, consisting of an inverted pendulum attached to a quadcopter. The control goals of this study are stabilization at the origin and tracking a circular trajectory for the quadcopter, using linear quadratic regulator (LQR) control in both cases. The fact that the pendulum weights less than 5% of the UAV weight allows us to separate its corresponding dynamics. Some works found in the specialized literature deal with the control of the flying inverted pendulum using different control ideas [17–20], and some others consider the control of UAVs carrying loads (commonly known as the flying crane). This problem is closely related and relevant to the central control problem in this study. For instance, Nicotra et al. [21] showed that the linearized model of a quadcopter with a suspended load could be stabilized at the origin using nested saturation functions. Pizetta et al. in [22] proposed a total feedback control with an auxiliary controller to accomplish tracking trajectory; almost all the references therein were developed to test the PVTOL system indoors, mainly to avoid counteracting the undesirable effect of the wind, which is not an easy task. However, techniques for nonlinear systems can be applied to obtain robust controllers, as the controller developed in this study. Of these techniques, perhaps the most used are backstepping control [23, 24], fuzzy control [25–27], active disturbance rejection control approach [28], and others [29]. To the authors’ knowledge, a general solution for stabilizing this type of system has not yet been reported in the literature.In this context, we propose a control scheme for a PVTOL aircraft system with an inverted pendular load (PVTOL-ASIPL). This scheme mainly consists of a generalized proportional integral (GPI) controller and a nested saturation-based control. The GPI controller accomplishes height and roll attitude control, using the roll attitude angle as fictitious input control. Then, proposing a set of convenient linear transformations and reducing the system can be expressed as a chain of integrators with a nonlinear perturbation. Finally, we design a nested saturation function-based controller to stabilize the horizontal position and the pendulum angle. Applying the second method of Lyapunov assures boundedness of the whole state and asymptotic convergence to the origin.The main contributions of this study are as follows:(i)
An algorithm control that uses a fictitious control, and we propose a combination of a GPI controller and a controller-based saturation for the takeoff and landing maneuvers(ii)
A set of convenient transformations, in which a high-order system can be expressed as a various low-order system(iii)
A control strategy for the PVTOL aircraft system with an inverted pendular load controls the height, roll attitude, horizontal position, and roll angle simultaneously, even in the presence of exogenous disturbanceThe organization of the rest of this study is as follows. Section2 introduces the model of PVTOL-ASIPL externally perturbed, obtained from the Euler–Lagrange formalism. Section 3 develops the control scheme design, while Section 4 presents the results of the numerical simulations. Finally, Section 5 is devoted to the concluding remarks and future work.


## 2. Dynamic Model

This section presents the dynamic model of the PVTOL aircraft system with an inverted pendular load (see Figure1). The dynamic equations were obtained by the Euler–Lagrange formalism as follows:(1)ℒ=Ek−Ep,where Ek and Ep are, respectively, the system kinetic and potential energies. Besides, the inverted pendular load base is in the PVTOL aircraft center of mass, Pv=xv,yv, α is defined as the angle between the PVTOL aircraft and the horizontal axis, and θ is the angle between the inverted pendular load and the vertical axis. Finally, the inverted pendular load center of mass PP=xp,yp is defined as(2)xp=xv−lpsinθ,yp=yv+lpcosθ.Figure 1
The PVTOL aircraft system with an inverted pendular load.Therefore, Lagrangian of the system can be expressed as(3)ℒ=mvP˙vP˙v⊺2+mpP˙pP˙p⊺2+α2iv2+θ2ip2−mvgyv−mpgyp,where mv is the PVTOL aircraft mass, mp is the pendular load mass, g is the gravity force, iv is the PVTOL aircraft inertia, and ip is the inverted pendular load inertia.The Euler–Lagrange equations of motion for the PVTOL-ASIPL system are in the form of(4)ddt∂ℒ∂q˙−∂ℒ∂q=τ+D,where q=xv,yv,α,θ is the generalized coordinate vector, τ is the input control vector, and D is the external disturbance vector; without loss of generality, iv=1 and ip=0.Developing Euler–Lagrange equation (4) leads to(5a)lpmpsinθθ˙2+mv+mpx¨v−lpmpcosθθ¨=−u1sinα+Dx,(5b)−lpmpcosθθ˙2+mv+mpg+y¨v−lpmpsinθθ¨=u1cosα+Dy,(5c)ivα˙=u2+Dα,(5d)−lpmpcosθx¨v−lpmpsinθg+y¨v+ip+lp2mpθ¨=0.Then, the equations in (5a) represent the PVTOL-ASIPL, and they can be expressed in a compact form as(6)Mq¨+Cq,q˙q˙+G=U+D,with(7)M=mv+mp00−lpmpcosθ0mv+mp0−lpmpsinθ00i0−lpmpcosθ−lpmpsinθ0ip+lp2mp,C=000lpmpsinθθ˙000−lpmpcosθθ˙00000000,G=0mv+mp0−lpmpsinθ,U=−u1sinαu1cosαu20⊺,D=DxDyDαDθ⊺,where M is a positive semidefinite matrix, C is the Coriolis matrix, G is the gravity matrix, U is the input control matrix, and D is the external disturbance matrix. The terms Dx, Dy, and Dα are assumed to be unmodelled owing to the external disturbances and are defined as follows [30, 31]:(8a)Dx=axx˙v,(8b)Dy=ayy˙v,(8c)Dα=aαα˙,(8d)Dθ=0.Finally, becauseM is not a singular matrix, it is possible to represent the dynamical model of the PVTOL aircraft system with an inverted pendular load as(9a)x¨v=ex1+ex22mvmv+mp,(9b)y¨v=ey1+ey22mvmv+mp,(9c)α¨=u2+aαα˙,(9d)θ¨=−u1sinα−θ+axcosθx˙v+aysinθy˙vlpm,where(10)ex1=−u12mv+mpsinα+mpsinα−2θ,ex2=ax2mv+mp+mpcos2θx˙v+aympsin2θy˙v−2lpmvmpsinθθ˙2,ey1=−2gmvmv+mp+2mv+mpu1cosα−mpu1cosα−2θ,ey2=axmpsin2θx˙v+ay2mv+mp−mpcos2θy˙v+2lpmvmpcosθθ˙2.


### 2.1. Problem Statement

The control goal consists in proposing an algorithm to accomplish stabilization at the origin of the PVTOL aircraft system with an inverted pendular load. The maneuvers are trajectory tracking tasks involving a step-by-step procedure, consisting of (1) the stabilization of the coordinateα; (2) the stabilization of the coordinate yv; and (3) further stabilization of coordinates xv and θ, even in the presence of disturbances due to the aerodynamic effects.We have expressed the system in its minimal representation form, which will allow us to decouple it and simplify it. Instead of working with a complex system, we work with a few simple systems that embody the dynamics of the original one. We are now in a position to design and propose the control scheme in the following section.


## 2.1. Problem Statement

The control goal consists in proposing an algorithm to accomplish stabilization at the origin of the PVTOL aircraft system with an inverted pendular load. The maneuvers are trajectory tracking tasks involving a step-by-step procedure, consisting of (1) the stabilization of the coordinateα; (2) the stabilization of the coordinate yv; and (3) further stabilization of coordinates xv and θ, even in the presence of disturbances due to the aerodynamic effects.We have expressed the system in its minimal representation form, which will allow us to decouple it and simplify it. Instead of working with a complex system, we work with a few simple systems that embody the dynamics of the original one. We are now in a position to design and propose the control scheme in the following section.


## 3. Control Scheme

This section establishes the framework to solve the main control problem. To this end, please consider that the input controlu2 acts over α plus disturbance Dα. So, it is possible to design a control law for α using a fictitious controller for the PVTOL aircraft with an inverted pendular load [9, 11]. Then, through linear transformations, a GPI law is used for the PVTOL takeoff and landing maneuvers. Once the GPI law stabilized the PVTOL aircraft height, a change of coordinates allows expressing the system as a chain of integrators nonlinearly perturbed, allowing to propose the nested saturation function-based stabilizing controller. Figure 2 presents the schematic diagram of the closed-loop system.Figure 2
Control scheme of the PVTOL aircraft system with an inverted pendular load.


### 3.1. Controlling the PVTOL Aircraft Attitudeα

It is clear that the third equation (9c) consists of a double-chain integrator with nonlinear disturbance Dα and control input u2. Then, a control law for tracking trajectory is searched for α.


#### 3.1.1. Control Statement

The dynamical equation for the roll attitude angleα can be expressed as(11)α¨=u2+ξt,where ξt is a lumped generalized disturbance input.Also, according to Lozano Hernández et al. and Fliess et al. [31, 32], to overcome the lack of available measurements of α˙, an integral reconstructor α˙^=∫0tuτdτ can be proposed, and using the local approximation of the disturbance input, it is possible to propose the control input u2 as(12)u2=α¨d−kα4e˙α−kα3eα−kα2∫τ1=0teαdτ1−kα1∫τ1=0t∫τ2=0τ1eαdτ2dτ1−kα0∫τ1=0t∫τ2=0τ1∫τ3=0τ2eαdτ3dτ2dτ1,where αd is a smooth reference signal for the state α, which is twice differentiable. Also, e˙α=α˙^−α˙d and eα=α−αd, and the relation between the actual value of α˙ and α˙^ is expressed by(13)α˙=α˙^+∫0tξτdτ+α˙0.Substituting (12) into (11) and expressing the resulting dynamics in terms of the tracking error, the following dynamics are obtained:(14)eα5+kα4eα4+kα3eα3+kα2e¨α+kα1e˙α+kα0eα=0,where kαi, with i=0,…,4, are selected such that characteristic polynomial s5+kα4s4+kα3s3+kα2s2+kα1s+kα0=0 is Hurwitz, reducing the undesirable effects of the nonlinear disturbances [33, 34]. So, the dynamic error is exponentially asymptotically stable at the origin. This fact allows using αd as a fictitious control for subsystems (9a), (9b), and (9d). This proposal was previously used in other works dealing with the stabilization of PVTOL aircrafts [9, 12].


### 3.2. Simplified PVTOL Aircraft System with an Inverted Pendular Load

Afterα⟶αd and applying the controller u2, the following system of equations represents the PVTOL aircraft system with an inverted pendular load:(15a)x¨v=ex1+ex22mvmv+mp,(15b)y¨v=ey1+ey22mvmv+mp,(15c)θ¨=−u1sinαd−θ+axcosθx˙v+aysinθy˙vlpm,with(16)ex1=−u12mv+mpsinαd+mpsinαd−2θ,ex2=ax2mv+mp+mpcos2θx˙v+aympsin2θy˙v−2lpmvmpsinθθ˙2,ey1=−2gmvmv+mp+2mv+mpu1cosαd−mpu1cosαd−2θ,ey2=axmpsin2θx˙v+ay2mv+mp−mpcos2θy˙v+2lpmvmpcosθθ˙2,where αd and u1 are the input controls.Thus, we propose the following control laws [12]:(17a)u1=r12+r22,(17b)αd=arctanr2r1,where r1>0andr2>0 are auxiliary control inputs.To obtain the dynamic model, the following change of coordinates is applied to model (15a)–(15c):(18a)xv=x^v2mvmv+mp,(18b)yv=y^v2mvmv+mp,(18c)θ=θ^lpmv.Therefore, models (15a)–(15c) transform into the following system:(19)x^¨v=−2mv+mpr2−axx˙v+mpcos2θ−r2+axx^˙v+mpsin2θr1+ayy^˙v−2lpmvmpsinθθ˙2,y^¨v=−2gmvmv+mp+2mv+mpr1−mpr1cos2θ+r2sin2θ,(20)+axmpsin2θx^˙v+ay2mv+mp−mpcos2θy^˙v+2lpmvmpcosθθ˙2θ^¨=cosθ−r2+axx^˙v+sinθr1+ayy^˙v.Hence, system (19) takes the compact form(21)x^¨vy^¨v=M2r1r2+C2,with(22)M2=mpsin2θ−2mv+mp+mpcos2θ2mv+mp−mpcos2θ−mpsin2θ,C2=−2lpmvmpsinθθ˙2−2gmvmv+mp+2lpmvmpcosθθ˙2−2gmvmv+mp.Then, given system (19) and ignoring the effect of the disturbances, the following partial feedback control can be proposed [1, 9]:(23)r1r2=M2−1−C2+v1v2,where v1 and v2 are new auxiliary control inputs. Thus, the system defined by equation (19), in a closed loop with control model (23), leads us to obtain the following system:(24a)x^¨v=v1+ax2mv+mp+mpcos2θx^˙v+aympsin2θy^˙v,(24b)y^¨v=v2+axmpsin2θx^˙v+ay2mv+mp−mpcos2θy^˙v,(24c)θ¨=cosθv1+2axmv+mpx^˙+sinθ2gmvmv+mp+v2+2aymv+mpy^˙2mv+mp.Notice that the above system is the reduced model for the PVTOL aircraft system with an inverted pendular load, withv1 and v2 as the control inputs.


### 3.3. Stabilization of Heightyv

A GPI controller with a saturation function is applied to obtain the height position control, allowing the tracking control to accomplish the takeoff and landing of the PVTOL aircraft system with an inverted pendular load.


#### 3.3.1. Control Statement

Consider the vertical displacementyv described by (25), and let v2 be the control input defined as (26):(25)v2=σay¨d−ky4e˙y−ky3ey−ky2∫τ1=0teydτ1−ky1∫τ1=0t∫τ2=0τ1eydτ2dτ1−ky0∫τ1=0t∫τ2=0τ1∫τ3=0τ2eydτ3dτ2dτ1=σaugpi,where yd is a smooth reference signal twice derivable and ey=yd−y^v.Also,σLq is a saturation function defined as(26)σaq=ifq>a,a,ifq<−a,−a,if−a<q<a,q.Thus, the tracking trajectory errorey is exponentially asymptotically stable.Proof.
The proof is split into two parts. First, it is proven that a saturation function in finite time bounds the error dynamicsey. Then, we demonstrate exponentially asymptotic stability.


#### 3.3.2. Error Bounded

The tracking trajectory error is defined as(27)ey=yd−y^v,where yd is the desired height position.Let us define the following state variables as(28)e1=ey;e2=e˙y.Therefore, the dynamic error is transformed into the following system:(29a)e˙1=e2,(29b)e˙2=y¨d−v2−axmpsin2θx^˙v−ay2mv+mp−mpcos2θy^˙v,where v2 is the control input.Now, to use the second method of Lyapunov, consider the following candidate function:(30)Vy=e122+e222,which is positive definite, with time derivative(31)V˙y=e1e2+e2e˙2=e1e2+e2y¨d−σaugpi−axmpsin2θx^˙v−ay2mv+mp−mpcos2θy^˙v.Thus, the first and second time derivatives ofyd are known and bounded as Myd. Besides, in a close neighbourhood where e1 and e2 are such that e1≤δe1 and e2≤δe2, the saturation function fulfills sσas⩽0. So, parameters were designed as kyi in ugpi such that e2 is dominant, and e2σaugpi>0. Finally, the last terms of equation (31) satisfy the following:(32)axmpsin2θ≤δx,(33)ay2mv+mp−mpcos2θ≤δy.Therefore, computing the time derivative of the candidate Lyapunov function leads to(34)V˙y=δe1δe2+δe2Myd−e2σaugpi−δe2δxx^˙v−δe2δyy^˙v,and if x^˙v and y^˙v are small enough such that e2σaugpi>δe1δe2+δe2Myd−δe2δxx^˙v−δe2δyy^˙v, then V˙y<0 and y^v are bounded in finite time. Thus, after time T, when this condition is satisfied, control law (26) takes the following structure:(35)v2=y¨d−ky4e˙y−ky3ey−ky2∫τ1=0teydτ1−ky1∫τ1=0t∫τ2=0τ1eydτ2dτ1−ky0∫τ1=0t∫τ2=0τ1∫τ3=0τ2eydτ3dτ2dτ1.From the above, equation (24b) is expressed as(36)y^¨v=v2+ξ1t,where ξ1t is a lumped generalized disturbance input.Then, to overcome the lack of available measurements ofy^˙v, the following integral reconstructor is introduced [31]:(37)y^˙¯v=∫0tv2τdτ.Using the local approximation of the disturbance input, the relation between the actual value ofy^˙v and y^˙¯v is expressed by(38)y^˙v=y^˙¯v+∫0tξτdτ+y^˙v0.Now, differentiating equation (38) and substituting into (36), the control law v2 takes the form(39)v2=y¨d−ky4e¯˙y−ky3e¯y−ky2∫τ1=0te¯ydτ1−ky1∫τ1=0t∫τ2=0τ1e¯ydτ2dτ1−ky0∫τ1=0t∫τ2=0τ1∫τ3=0τ2e¯ydτ3dτ2dτ1,with e¯y=yv−yd and e¯˙y=y^˙¯v−y˙d.Substituting (39) into (36) is possible to express the dynamic error e¯y as follows:(40)e¯y5+ky4e¯y4+ky3e¯y3+ky2e¯¨y+ky1e¯˙y+ky0e¯y=0.The coefficientsky4,ky3,ky2,ky1,andky0 are selected such that the polynomial S5+ky4S4+ky3S3+ky2S2+ky1S+ky0 is Hurwitz, eliminating the nonlinear disturbances [33]. Then, tracking trajectory error e¯y is exponentially asymptotically stable to zero.


### 3.4. Control for (xv, θ)

This section proposes a nested saturation-based controller strategy to stabilize the PVTOL-ASIPL horizontal position and roll angle. Notice that the nested saturation-based controller strategy allows stabilizing nonlinear systems that can be approximately expressed as a chain of integrators [35–37]. Thus, our stability problem will be solved as follows. First, we introduce a linear transformation for the stabilizing controller. Then, we demonstrate that the proposed controller guarantees boundedness and converges to zero, in finite time, of the whole state.


### 3.5. Expressing the PVTOL-ASIPL as a Chain of Integrators

After the application of the controllerv2, system (25) can be reduced to the subsystem (x^v, θ) as follows:(41)x^¨v=v1+ax2mv+mp+mpcos2θx^˙v,θ¨=2gmvmv+mpsinθ+cosθv1+2axmv+mpx^˙v2mv+mp.Then, introduce the following coordinates’ change:(42)x^v=xv−2gmvmv+mp,θ^=θgmv,v1=2mv+mpgmvvsecθ−tanθ−axx˙v.System (42) takes the following form:(43)x¨v=vsecθ−tanθ−ϵxsinθ2x˙v,θ¨=v,where ϵxsinθ2x˙v is considered as a nonlinear disturbance and ϵx is an unknown constant that depends on variables mv,mp,g, and ax (Note that ϵx=mp/g∗mv∗mv+mpax.). Notice that subsystem (43) is similar to the cart-pole system plus a nonlinear disturbance [38].


#### 3.5.1. Control Statement

Based on work [3], we define the following state variables x1=xv, x2=x˙v, θ1=θ, and θ2=θ.. Then, we express the dynamic system as(44)x˙1=x2,x˙2=−vsecθ1+tanθ1+ϵxsinθ12x2,θ˙1=θ2,θ˙1=v.To express system (44) as a chain of integrators with a nonlinear perturbation and propose a controller for the stabilization of the subsystem (xv, θ), applying the decoupling theorem [3, 39], the following global nonlinear transformation is introduced:(45a)z1=1+tanθ1/21−tanθ1/2+x1,(45b)z2=θ2cosθ1+x2,(45c)ω1=tanθ1,(45d)ω2=θ2sec2θ1,(45e)vf=sec2θ1v+2θ22tanθ1sec2θ1.Hence, the transformed system into a chain of integrators is given by(46)z˙1=z2,z˙2=ω1+ω1ω121+ω12+ϵxω2ω121+ω123/2−ϵxz2ω121+ω12,ω˙1=ω2,ω˙2=vf.


### 3.6. Nested Saturation Function-Based Controller

Inspired by Teel [40], a linear transformation is proposed to obtain the stabilizing controller for system (46) as follows:(47)q1q2q3q4=1331012100110001z1z2ω1ω2.Applying transformation (47) to system (46) leads to(48)q˙1=vf+q2+q3+q4+3G1q3−q4q42+3G2q3−q4q4ϵx+3G3q3−q4q2−2q3+q4ϵx,q˙2=vf+q3+q4+G1q3−q4q42+G2q3−q4q4ϵx+G3q3−q4q2−2q3+q4ϵx,q˙3=vf+q4,q˙4=vf,for which the following nested saturation function-based stabilizing controller is proposed:(49)vf=−σc1q4+σc2q3+σc3q2+σc4q1,where σci is defined by equation (26), and Gi with i=1,2,3 is given by(50a)G1s=s1+s23/2,(50b)G2s=s21+s23/2,(50c)G3s=s21+s2.Remark 1.
Note thatmaxG1s=K1=2/33 when s=1/2 and minG1s=−2/33 when s=−1/2 and lims⟶infG1s=0. Besides, maxG2s=K2=2/33 when s=±2 and minG1s=0 when s=0 and lims⟶infG2s=0. Additionally, minG3s=0 when s=0 and lims⟶infG3s=K3=1.


### 3.7. Whole State Boundedness

Now, we prove that the proposed controller (49) ensures whole state boundedness. Moreover, the bound of each state directly depends on the designed parameters of the controller.Step 1.
A positive definite function is defined as(51)V4=q422.
The time derivative ofV4 is expressed by(52)V˙4=q4q˙4=q4vf=−q4σc1q4+σc2q3+σc3q2+σc4q1,where c1 and c2 are selected such that c1>2c2. It is clear that V˙4<0 when ∣q4∣≥c2; therefore, there exists a finite time T1>0 such that(53)q4≤c2,∀t>T1.
Thus, when this condition is satisfied, the control law (49) takes the following structure:(54)vf=−q4−σc2q3+σc3q2+σc4q1.Step 2.
Behaviour analysis ofq3: let us consider the following positive definite function:(55)V3=q322.
Differentiating it with respect to time and after substituting (54) into q˙3, the following expression is obtained:(56)V˙3=q3q˙3=q3vf+q4=−q3σc2q3+σc3q2+σc4q1.
To ensure thatV˙3<0 is achieved, the following conditions must be satisfied:(57)c2>2c3,q3≥c3.
Then, there exists a finite timeT2>T1 after which(58)q3≤c3,∀t>T2.
Therefore,q3 is bounded, and the control law takes the following form:(59)vf=−q4−q3−σc3q2+σc4q1.Step 3.
The following positive definite function is introduced:(60)V2=q222.
DifferentiatingV2 and after substituting (60) into the second equation of (48), the following is obtained:(61)V˙2=q2q˙2=−q2σc3q2+σc4q1−G1q3−q4q42−G2q3−q4q4ϵx−G3q3−q4q2−2q3+q4,where c3 and c2 must satisfy the relation c3>2c4+G1q3−q442+G2q3−q4q4ϵx+G3q3−q4q2−2q3+q4. It is easy to observe that if |q2|>c4+G2q3−q4q4ϵx+G3q3−q4q2−2q3+q4, V˙2<0. Hence, there exists a finite time T3>T2 after which(62)q2≤c4+G2q3−q4q4ϵx+G3q3−q4q2−2q3+q4;∀t>T3.
Therefore,q2 is bounded, and the control vf is revealed to be(63)vf=−q4−q3−q2−σc4q1.Step 4.
Substituting (63) into the first equation of (48), we obtain(64)q˙1=−σc4q1+3G1q3−q4q42+3G2q3−q4q4ϵx+3G3q3−q4q2−2q3+q4ϵx.
To demonstrate thatq1 is bounded, a positive definite function is defined as follows:(65)V1=q122.
DifferentiatingV1 along the trajectories of (66) leads to(66)V˙1=q1q˙1=−q1σc4q1−3G1q3−q4q42−3G2q3−q4q4ϵx−3G3q3−q4q2−2q3+q4,where c4 must be selected so that c4>3G1q3−q4q42+3G2q3−q4q4ϵx+3G3q3−q4q2−2q3+q4 and q1>3G1q3−q4q42+3G2q3−q4q4ϵx+3G3q3−q4q2−2q3+q4 to achieve V1<0. Therefore, there exists a finite time T4>T3 such that(67)q1<3G1q3−q4q42+3G2q3−q4q4ϵx+3G3q3−q4q2−2q3+q4,∀t>T4.
Consequently,q1 is also bounded. Finally, the values of parameters c1,c2,c3, and c4 can be determined by the following inequalities:(68)c1>2c2,c2>2c3,c3>2c4+G1q3−q4q42+G2q3−q4q4ϵx+G3q3−q4q2−2q3+q4,c4>3G1q3−q4q42+3G2q3−q4q4ϵx+3G3q3−q4q2−2q3+q4.
From the above conditions, the set of control parameters can be selected as(69)c1=28r,c2=14r,c3=7r,c4=3r,where r is directly related to the magnitude of the system disturbances.


### 3.8. Whole State Convergence to Zero

Here, we prove that the closed-loop system, provided by (48) and (49) and satisfying (70), is asymptotically stable.Notice that, aftert>T4, the control law (49) is no longer saturated; that is,(70)vf=−q4−q3−q2−q1,and the closed-loop system turns out to be(71)q˙1=−q1+3G1q3−q4q42+3G2q3−q4q4ϵx+3G3q3−q4q2−2q3+q4ϵx,q˙2=−q1−q2+G1q3−q4q42+G2q3−q4q4ϵx+G3q3−q4q2−2q3+q4ϵx,q˙3=−q1−q2−q3,q˙4=−q1−q2−q3−q4.To demonstrate whole state convergence to zero, the following Lyapunov function is used:(72)V=12q⊺q,where q=q1,q2,…,q3,q2⊺, and differentiating V along the trajectories of equation (72), it is obtained that(73)V˙=−q⊺Mq+3q1+q2G1q3−q4q42+G2q3−q4q4ϵx+G3q3−q4q2−2q3+q4ϵx,and(74)M=1121212121121212121121212121,with M being positive definite with λminM1/2.From Remark1, it is shown that the following relations are satisfied:(75)3q1+q2G1q3−q4q42<K123q1+q2q42<K123q1+q22+q44,3q1+q2G2q3−q4q4ϵx<K2ϵx23q1+q2q4<K2ϵx23q1+q22+q42,3q1+q2G3q3−q4q2ϵx<K3ϵx23q1+q2q2<K3ϵx23q1+q22+q22,3q1+q2G3q3−q4q4ϵx<K3ϵx23q1+q2q4<K3ϵx23q1+q22+q42.So,V˙ fulfills(76)V˙<−12q12−K13q1+q22−K2ϵx3q1+q22−2K3ϵx3q1+q22−12q421−K1q42−K3ϵx−12q221−K2ϵx.Hence, the previous inequality is strictly negative definite, and the following constraints are obtained:(77)q12−K13q1+q22−K2ϵx3q1+q22−2K3ϵx3q1+q22>0,1−K1q42−K3ϵx>0,1−K2ϵx>0.Therefore, if the restrictions of (77) are fulfilled, V˙ is strictly negative, and the state vector q exponentially converges to zero after t>T4.The following proposition summarizes the previous discussion, which is the main result of our study.Proposition 1.
Consider the PVTOL aircraft system with an inverted pendular load as described in (9) and in a closed loop with controllers (12), (17a) and (17b), (23), (25), and (49). Then, the closed-loop system is exponentially asymptotically stable provided that the control parameters c1, c2, c3, and c4 satisfy inequalities (69), and kαi and kyi are selected such that the characteristic polynomial is Hurwitz.
Finally, in Figure3, the steps’ sequence is shown, obtained from the control laws u1 and u2.
Having designed the control scheme for the PVTOL aircraft system with an inverted pendular load and carried out its convergence analysis, we test its effectiveness through numerical simulations in the following section. It is worth mentioning that it would be ideal for testing the scheme by conducting actual experiments—unfortunately, the construction of the needed prototype is still in progress.Figure 3
Control strategy sequence.


## 3.1. Controlling the PVTOL Aircraft Attitudeα

It is clear that the third equation (9c) consists of a double-chain integrator with nonlinear disturbance Dα and control input u2. Then, a control law for tracking trajectory is searched for α.


### 3.1.1. Control Statement

The dynamical equation for the roll attitude angleα can be expressed as(11)α¨=u2+ξt,where ξt is a lumped generalized disturbance input.Also, according to Lozano Hernández et al. and Fliess et al. [31, 32], to overcome the lack of available measurements of α˙, an integral reconstructor α˙^=∫0tuτdτ can be proposed, and using the local approximation of the disturbance input, it is possible to propose the control input u2 as(12)u2=α¨d−kα4e˙α−kα3eα−kα2∫τ1=0teαdτ1−kα1∫τ1=0t∫τ2=0τ1eαdτ2dτ1−kα0∫τ1=0t∫τ2=0τ1∫τ3=0τ2eαdτ3dτ2dτ1,where αd is a smooth reference signal for the state α, which is twice differentiable. Also, e˙α=α˙^−α˙d and eα=α−αd, and the relation between the actual value of α˙ and α˙^ is expressed by(13)α˙=α˙^+∫0tξτdτ+α˙0.Substituting (12) into (11) and expressing the resulting dynamics in terms of the tracking error, the following dynamics are obtained:(14)eα5+kα4eα4+kα3eα3+kα2e¨α+kα1e˙α+kα0eα=0,where kαi, with i=0,…,4, are selected such that characteristic polynomial s5+kα4s4+kα3s3+kα2s2+kα1s+kα0=0 is Hurwitz, reducing the undesirable effects of the nonlinear disturbances [33, 34]. So, the dynamic error is exponentially asymptotically stable at the origin. This fact allows using αd as a fictitious control for subsystems (9a), (9b), and (9d). This proposal was previously used in other works dealing with the stabilization of PVTOL aircrafts [9, 12].


## 3.1.1. Control Statement

The dynamical equation for the roll attitude angleα can be expressed as(11)α¨=u2+ξt,where ξt is a lumped generalized disturbance input.Also, according to Lozano Hernández et al. and Fliess et al. [31, 32], to overcome the lack of available measurements of α˙, an integral reconstructor α˙^=∫0tuτdτ can be proposed, and using the local approximation of the disturbance input, it is possible to propose the control input u2 as(12)u2=α¨d−kα4e˙α−kα3eα−kα2∫τ1=0teαdτ1−kα1∫τ1=0t∫τ2=0τ1eαdτ2dτ1−kα0∫τ1=0t∫τ2=0τ1∫τ3=0τ2eαdτ3dτ2dτ1,where αd is a smooth reference signal for the state α, which is twice differentiable. Also, e˙α=α˙^−α˙d and eα=α−αd, and the relation between the actual value of α˙ and α˙^ is expressed by(13)α˙=α˙^+∫0tξτdτ+α˙0.Substituting (12) into (11) and expressing the resulting dynamics in terms of the tracking error, the following dynamics are obtained:(14)eα5+kα4eα4+kα3eα3+kα2e¨α+kα1e˙α+kα0eα=0,where kαi, with i=0,…,4, are selected such that characteristic polynomial s5+kα4s4+kα3s3+kα2s2+kα1s+kα0=0 is Hurwitz, reducing the undesirable effects of the nonlinear disturbances [33, 34]. So, the dynamic error is exponentially asymptotically stable at the origin. This fact allows using αd as a fictitious control for subsystems (9a), (9b), and (9d). This proposal was previously used in other works dealing with the stabilization of PVTOL aircrafts [9, 12].


## 3.2. Simplified PVTOL Aircraft System with an Inverted Pendular Load

Afterα⟶αd and applying the controller u2, the following system of equations represents the PVTOL aircraft system with an inverted pendular load:(15a)x¨v=ex1+ex22mvmv+mp,(15b)y¨v=ey1+ey22mvmv+mp,(15c)θ¨=−u1sinαd−θ+axcosθx˙v+aysinθy˙vlpm,with(16)ex1=−u12mv+mpsinαd+mpsinαd−2θ,ex2=ax2mv+mp+mpcos2θx˙v+aympsin2θy˙v−2lpmvmpsinθθ˙2,ey1=−2gmvmv+mp+2mv+mpu1cosαd−mpu1cosαd−2θ,ey2=axmpsin2θx˙v+ay2mv+mp−mpcos2θy˙v+2lpmvmpcosθθ˙2,where αd and u1 are the input controls.Thus, we propose the following control laws [12]:(17a)u1=r12+r22,(17b)αd=arctanr2r1,where r1>0andr2>0 are auxiliary control inputs.To obtain the dynamic model, the following change of coordinates is applied to model (15a)–(15c):(18a)xv=x^v2mvmv+mp,(18b)yv=y^v2mvmv+mp,(18c)θ=θ^lpmv.Therefore, models (15a)–(15c) transform into the following system:(19)x^¨v=−2mv+mpr2−axx˙v+mpcos2θ−r2+axx^˙v+mpsin2θr1+ayy^˙v−2lpmvmpsinθθ˙2,y^¨v=−2gmvmv+mp+2mv+mpr1−mpr1cos2θ+r2sin2θ,(20)+axmpsin2θx^˙v+ay2mv+mp−mpcos2θy^˙v+2lpmvmpcosθθ˙2θ^¨=cosθ−r2+axx^˙v+sinθr1+ayy^˙v.Hence, system (19) takes the compact form(21)x^¨vy^¨v=M2r1r2+C2,with(22)M2=mpsin2θ−2mv+mp+mpcos2θ2mv+mp−mpcos2θ−mpsin2θ,C2=−2lpmvmpsinθθ˙2−2gmvmv+mp+2lpmvmpcosθθ˙2−2gmvmv+mp.Then, given system (19) and ignoring the effect of the disturbances, the following partial feedback control can be proposed [1, 9]:(23)r1r2=M2−1−C2+v1v2,where v1 and v2 are new auxiliary control inputs. Thus, the system defined by equation (19), in a closed loop with control model (23), leads us to obtain the following system:(24a)x^¨v=v1+ax2mv+mp+mpcos2θx^˙v+aympsin2θy^˙v,(24b)y^¨v=v2+axmpsin2θx^˙v+ay2mv+mp−mpcos2θy^˙v,(24c)θ¨=cosθv1+2axmv+mpx^˙+sinθ2gmvmv+mp+v2+2aymv+mpy^˙2mv+mp.Notice that the above system is the reduced model for the PVTOL aircraft system with an inverted pendular load, withv1 and v2 as the control inputs.


## 3.3. Stabilization of Heightyv

A GPI controller with a saturation function is applied to obtain the height position control, allowing the tracking control to accomplish the takeoff and landing of the PVTOL aircraft system with an inverted pendular load.


### 3.3.1. Control Statement

Consider the vertical displacementyv described by (25), and let v2 be the control input defined as (26):(25)v2=σay¨d−ky4e˙y−ky3ey−ky2∫τ1=0teydτ1−ky1∫τ1=0t∫τ2=0τ1eydτ2dτ1−ky0∫τ1=0t∫τ2=0τ1∫τ3=0τ2eydτ3dτ2dτ1=σaugpi,where yd is a smooth reference signal twice derivable and ey=yd−y^v.Also,σLq is a saturation function defined as(26)σaq=ifq>a,a,ifq<−a,−a,if−a<q<a,q.Thus, the tracking trajectory errorey is exponentially asymptotically stable.Proof.
The proof is split into two parts. First, it is proven that a saturation function in finite time bounds the error dynamicsey. Then, we demonstrate exponentially asymptotic stability.


### 3.3.2. Error Bounded

The tracking trajectory error is defined as(27)ey=yd−y^v,where yd is the desired height position.Let us define the following state variables as(28)e1=ey;e2=e˙y.Therefore, the dynamic error is transformed into the following system:(29a)e˙1=e2,(29b)e˙2=y¨d−v2−axmpsin2θx^˙v−ay2mv+mp−mpcos2θy^˙v,where v2 is the control input.Now, to use the second method of Lyapunov, consider the following candidate function:(30)Vy=e122+e222,which is positive definite, with time derivative(31)V˙y=e1e2+e2e˙2=e1e2+e2y¨d−σaugpi−axmpsin2θx^˙v−ay2mv+mp−mpcos2θy^˙v.Thus, the first and second time derivatives ofyd are known and bounded as Myd. Besides, in a close neighbourhood where e1 and e2 are such that e1≤δe1 and e2≤δe2, the saturation function fulfills sσas⩽0. So, parameters were designed as kyi in ugpi such that e2 is dominant, and e2σaugpi>0. Finally, the last terms of equation (31) satisfy the following:(32)axmpsin2θ≤δx,(33)ay2mv+mp−mpcos2θ≤δy.Therefore, computing the time derivative of the candidate Lyapunov function leads to(34)V˙y=δe1δe2+δe2Myd−e2σaugpi−δe2δxx^˙v−δe2δyy^˙v,and if x^˙v and y^˙v are small enough such that e2σaugpi>δe1δe2+δe2Myd−δe2δxx^˙v−δe2δyy^˙v, then V˙y<0 and y^v are bounded in finite time. Thus, after time T, when this condition is satisfied, control law (26) takes the following structure:(35)v2=y¨d−ky4e˙y−ky3ey−ky2∫τ1=0teydτ1−ky1∫τ1=0t∫τ2=0τ1eydτ2dτ1−ky0∫τ1=0t∫τ2=0τ1∫τ3=0τ2eydτ3dτ2dτ1.From the above, equation (24b) is expressed as(36)y^¨v=v2+ξ1t,where ξ1t is a lumped generalized disturbance input.Then, to overcome the lack of available measurements ofy^˙v, the following integral reconstructor is introduced [31]:(37)y^˙¯v=∫0tv2τdτ.Using the local approximation of the disturbance input, the relation between the actual value ofy^˙v and y^˙¯v is expressed by(38)y^˙v=y^˙¯v+∫0tξτdτ+y^˙v0.Now, differentiating equation (38) and substituting into (36), the control law v2 takes the form(39)v2=y¨d−ky4e¯˙y−ky3e¯y−ky2∫τ1=0te¯ydτ1−ky1∫τ1=0t∫τ2=0τ1e¯ydτ2dτ1−ky0∫τ1=0t∫τ2=0τ1∫τ3=0τ2e¯ydτ3dτ2dτ1,with e¯y=yv−yd and e¯˙y=y^˙¯v−y˙d.Substituting (39) into (36) is possible to express the dynamic error e¯y as follows:(40)e¯y5+ky4e¯y4+ky3e¯y3+ky2e¯¨y+ky1e¯˙y+ky0e¯y=0.The coefficientsky4,ky3,ky2,ky1,andky0 are selected such that the polynomial S5+ky4S4+ky3S3+ky2S2+ky1S+ky0 is Hurwitz, eliminating the nonlinear disturbances [33]. Then, tracking trajectory error e¯y is exponentially asymptotically stable to zero.


## 3.3.1. Control Statement

Consider the vertical displacementyv described by (25), and let v2 be the control input defined as (26):(25)v2=σay¨d−ky4e˙y−ky3ey−ky2∫τ1=0teydτ1−ky1∫τ1=0t∫τ2=0τ1eydτ2dτ1−ky0∫τ1=0t∫τ2=0τ1∫τ3=0τ2eydτ3dτ2dτ1=σaugpi,where yd is a smooth reference signal twice derivable and ey=yd−y^v.Also,σLq is a saturation function defined as(26)σaq=ifq>a,a,ifq<−a,−a,if−a<q<a,q.Thus, the tracking trajectory errorey is exponentially asymptotically stable.Proof.
The proof is split into two parts. First, it is proven that a saturation function in finite time bounds the error dynamicsey. Then, we demonstrate exponentially asymptotic stability.


## 3.3.2. Error Bounded

The tracking trajectory error is defined as(27)ey=yd−y^v,where yd is the desired height position.Let us define the following state variables as(28)e1=ey;e2=e˙y.Therefore, the dynamic error is transformed into the following system:(29a)e˙1=e2,(29b)e˙2=y¨d−v2−axmpsin2θx^˙v−ay2mv+mp−mpcos2θy^˙v,where v2 is the control input.Now, to use the second method of Lyapunov, consider the following candidate function:(30)Vy=e122+e222,which is positive definite, with time derivative(31)V˙y=e1e2+e2e˙2=e1e2+e2y¨d−σaugpi−axmpsin2θx^˙v−ay2mv+mp−mpcos2θy^˙v.Thus, the first and second time derivatives ofyd are known and bounded as Myd. Besides, in a close neighbourhood where e1 and e2 are such that e1≤δe1 and e2≤δe2, the saturation function fulfills sσas⩽0. So, parameters were designed as kyi in ugpi such that e2 is dominant, and e2σaugpi>0. Finally, the last terms of equation (31) satisfy the following:(32)axmpsin2θ≤δx,(33)ay2mv+mp−mpcos2θ≤δy.Therefore, computing the time derivative of the candidate Lyapunov function leads to(34)V˙y=δe1δe2+δe2Myd−e2σaugpi−δe2δxx^˙v−δe2δyy^˙v,and if x^˙v and y^˙v are small enough such that e2σaugpi>δe1δe2+δe2Myd−δe2δxx^˙v−δe2δyy^˙v, then V˙y<0 and y^v are bounded in finite time. Thus, after time T, when this condition is satisfied, control law (26) takes the following structure:(35)v2=y¨d−ky4e˙y−ky3ey−ky2∫τ1=0teydτ1−ky1∫τ1=0t∫τ2=0τ1eydτ2dτ1−ky0∫τ1=0t∫τ2=0τ1∫τ3=0τ2eydτ3dτ2dτ1.From the above, equation (24b) is expressed as(36)y^¨v=v2+ξ1t,where ξ1t is a lumped generalized disturbance input.Then, to overcome the lack of available measurements ofy^˙v, the following integral reconstructor is introduced [31]:(37)y^˙¯v=∫0tv2τdτ.Using the local approximation of the disturbance input, the relation between the actual value ofy^˙v and y^˙¯v is expressed by(38)y^˙v=y^˙¯v+∫0tξτdτ+y^˙v0.Now, differentiating equation (38) and substituting into (36), the control law v2 takes the form(39)v2=y¨d−ky4e¯˙y−ky3e¯y−ky2∫τ1=0te¯ydτ1−ky1∫τ1=0t∫τ2=0τ1e¯ydτ2dτ1−ky0∫τ1=0t∫τ2=0τ1∫τ3=0τ2e¯ydτ3dτ2dτ1,with e¯y=yv−yd and e¯˙y=y^˙¯v−y˙d.Substituting (39) into (36) is possible to express the dynamic error e¯y as follows:(40)e¯y5+ky4e¯y4+ky3e¯y3+ky2e¯¨y+ky1e¯˙y+ky0e¯y=0.The coefficientsky4,ky3,ky2,ky1,andky0 are selected such that the polynomial S5+ky4S4+ky3S3+ky2S2+ky1S+ky0 is Hurwitz, eliminating the nonlinear disturbances [33]. Then, tracking trajectory error e¯y is exponentially asymptotically stable to zero.


## 3.4. Control for (xv, θ)

This section proposes a nested saturation-based controller strategy to stabilize the PVTOL-ASIPL horizontal position and roll angle. Notice that the nested saturation-based controller strategy allows stabilizing nonlinear systems that can be approximately expressed as a chain of integrators [35–37]. Thus, our stability problem will be solved as follows. First, we introduce a linear transformation for the stabilizing controller. Then, we demonstrate that the proposed controller guarantees boundedness and converges to zero, in finite time, of the whole state.


## 3.5. Expressing the PVTOL-ASIPL as a Chain of Integrators

After the application of the controllerv2, system (25) can be reduced to the subsystem (x^v, θ) as follows:(41)x^¨v=v1+ax2mv+mp+mpcos2θx^˙v,θ¨=2gmvmv+mpsinθ+cosθv1+2axmv+mpx^˙v2mv+mp.Then, introduce the following coordinates’ change:(42)x^v=xv−2gmvmv+mp,θ^=θgmv,v1=2mv+mpgmvvsecθ−tanθ−axx˙v.System (42) takes the following form:(43)x¨v=vsecθ−tanθ−ϵxsinθ2x˙v,θ¨=v,where ϵxsinθ2x˙v is considered as a nonlinear disturbance and ϵx is an unknown constant that depends on variables mv,mp,g, and ax (Note that ϵx=mp/g∗mv∗mv+mpax.). Notice that subsystem (43) is similar to the cart-pole system plus a nonlinear disturbance [38].


### 3.5.1. Control Statement

Based on work [3], we define the following state variables x1=xv, x2=x˙v, θ1=θ, and θ2=θ.. Then, we express the dynamic system as(44)x˙1=x2,x˙2=−vsecθ1+tanθ1+ϵxsinθ12x2,θ˙1=θ2,θ˙1=v.To express system (44) as a chain of integrators with a nonlinear perturbation and propose a controller for the stabilization of the subsystem (xv, θ), applying the decoupling theorem [3, 39], the following global nonlinear transformation is introduced:(45a)z1=1+tanθ1/21−tanθ1/2+x1,(45b)z2=θ2cosθ1+x2,(45c)ω1=tanθ1,(45d)ω2=θ2sec2θ1,(45e)vf=sec2θ1v+2θ22tanθ1sec2θ1.Hence, the transformed system into a chain of integrators is given by(46)z˙1=z2,z˙2=ω1+ω1ω121+ω12+ϵxω2ω121+ω123/2−ϵxz2ω121+ω12,ω˙1=ω2,ω˙2=vf.


## 3.5.1. Control Statement

Based on work [3], we define the following state variables x1=xv, x2=x˙v, θ1=θ, and θ2=θ.. Then, we express the dynamic system as(44)x˙1=x2,x˙2=−vsecθ1+tanθ1+ϵxsinθ12x2,θ˙1=θ2,θ˙1=v.To express system (44) as a chain of integrators with a nonlinear perturbation and propose a controller for the stabilization of the subsystem (xv, θ), applying the decoupling theorem [3, 39], the following global nonlinear transformation is introduced:(45a)z1=1+tanθ1/21−tanθ1/2+x1,(45b)z2=θ2cosθ1+x2,(45c)ω1=tanθ1,(45d)ω2=θ2sec2θ1,(45e)vf=sec2θ1v+2θ22tanθ1sec2θ1.Hence, the transformed system into a chain of integrators is given by(46)z˙1=z2,z˙2=ω1+ω1ω121+ω12+ϵxω2ω121+ω123/2−ϵxz2ω121+ω12,ω˙1=ω2,ω˙2=vf.


## 3.6. Nested Saturation Function-Based Controller

Inspired by Teel [40], a linear transformation is proposed to obtain the stabilizing controller for system (46) as follows:(47)q1q2q3q4=1331012100110001z1z2ω1ω2.Applying transformation (47) to system (46) leads to(48)q˙1=vf+q2+q3+q4+3G1q3−q4q42+3G2q3−q4q4ϵx+3G3q3−q4q2−2q3+q4ϵx,q˙2=vf+q3+q4+G1q3−q4q42+G2q3−q4q4ϵx+G3q3−q4q2−2q3+q4ϵx,q˙3=vf+q4,q˙4=vf,for which the following nested saturation function-based stabilizing controller is proposed:(49)vf=−σc1q4+σc2q3+σc3q2+σc4q1,where σci is defined by equation (26), and Gi with i=1,2,3 is given by(50a)G1s=s1+s23/2,(50b)G2s=s21+s23/2,(50c)G3s=s21+s2.Remark 1.
Note thatmaxG1s=K1=2/33 when s=1/2 and minG1s=−2/33 when s=−1/2 and lims⟶infG1s=0. Besides, maxG2s=K2=2/33 when s=±2 and minG1s=0 when s=0 and lims⟶infG2s=0. Additionally, minG3s=0 when s=0 and lims⟶infG3s=K3=1.


## 3.7. Whole State Boundedness

Now, we prove that the proposed controller (49) ensures whole state boundedness. Moreover, the bound of each state directly depends on the designed parameters of the controller.Step 1.
A positive definite function is defined as(51)V4=q422.
The time derivative ofV4 is expressed by(52)V˙4=q4q˙4=q4vf=−q4σc1q4+σc2q3+σc3q2+σc4q1,where c1 and c2 are selected such that c1>2c2. It is clear that V˙4<0 when ∣q4∣≥c2; therefore, there exists a finite time T1>0 such that(53)q4≤c2,∀t>T1.
Thus, when this condition is satisfied, the control law (49) takes the following structure:(54)vf=−q4−σc2q3+σc3q2+σc4q1.Step 2.
Behaviour analysis ofq3: let us consider the following positive definite function:(55)V3=q322.
Differentiating it with respect to time and after substituting (54) into q˙3, the following expression is obtained:(56)V˙3=q3q˙3=q3vf+q4=−q3σc2q3+σc3q2+σc4q1.
To ensure thatV˙3<0 is achieved, the following conditions must be satisfied:(57)c2>2c3,q3≥c3.
Then, there exists a finite timeT2>T1 after which(58)q3≤c3,∀t>T2.
Therefore,q3 is bounded, and the control law takes the following form:(59)vf=−q4−q3−σc3q2+σc4q1.Step 3.
The following positive definite function is introduced:(60)V2=q222.
DifferentiatingV2 and after substituting (60) into the second equation of (48), the following is obtained:(61)V˙2=q2q˙2=−q2σc3q2+σc4q1−G1q3−q4q42−G2q3−q4q4ϵx−G3q3−q4q2−2q3+q4,where c3 and c2 must satisfy the relation c3>2c4+G1q3−q442+G2q3−q4q4ϵx+G3q3−q4q2−2q3+q4. It is easy to observe that if |q2|>c4+G2q3−q4q4ϵx+G3q3−q4q2−2q3+q4, V˙2<0. Hence, there exists a finite time T3>T2 after which(62)q2≤c4+G2q3−q4q4ϵx+G3q3−q4q2−2q3+q4;∀t>T3.
Therefore,q2 is bounded, and the control vf is revealed to be(63)vf=−q4−q3−q2−σc4q1.Step 4.
Substituting (63) into the first equation of (48), we obtain(64)q˙1=−σc4q1+3G1q3−q4q42+3G2q3−q4q4ϵx+3G3q3−q4q2−2q3+q4ϵx.
To demonstrate thatq1 is bounded, a positive definite function is defined as follows:(65)V1=q122.
DifferentiatingV1 along the trajectories of (66) leads to(66)V˙1=q1q˙1=−q1σc4q1−3G1q3−q4q42−3G2q3−q4q4ϵx−3G3q3−q4q2−2q3+q4,where c4 must be selected so that c4>3G1q3−q4q42+3G2q3−q4q4ϵx+3G3q3−q4q2−2q3+q4 and q1>3G1q3−q4q42+3G2q3−q4q4ϵx+3G3q3−q4q2−2q3+q4 to achieve V1<0. Therefore, there exists a finite time T4>T3 such that(67)q1<3G1q3−q4q42+3G2q3−q4q4ϵx+3G3q3−q4q2−2q3+q4,∀t>T4.
Consequently,q1 is also bounded. Finally, the values of parameters c1,c2,c3, and c4 can be determined by the following inequalities:(68)c1>2c2,c2>2c3,c3>2c4+G1q3−q4q42+G2q3−q4q4ϵx+G3q3−q4q2−2q3+q4,c4>3G1q3−q4q42+3G2q3−q4q4ϵx+3G3q3−q4q2−2q3+q4.
From the above conditions, the set of control parameters can be selected as(69)c1=28r,c2=14r,c3=7r,c4=3r,where r is directly related to the magnitude of the system disturbances.


## 3.8. Whole State Convergence to Zero

Here, we prove that the closed-loop system, provided by (48) and (49) and satisfying (70), is asymptotically stable.Notice that, aftert>T4, the control law (49) is no longer saturated; that is,(70)vf=−q4−q3−q2−q1,and the closed-loop system turns out to be(71)q˙1=−q1+3G1q3−q4q42+3G2q3−q4q4ϵx+3G3q3−q4q2−2q3+q4ϵx,q˙2=−q1−q2+G1q3−q4q42+G2q3−q4q4ϵx+G3q3−q4q2−2q3+q4ϵx,q˙3=−q1−q2−q3,q˙4=−q1−q2−q3−q4.To demonstrate whole state convergence to zero, the following Lyapunov function is used:(72)V=12q⊺q,where q=q1,q2,…,q3,q2⊺, and differentiating V along the trajectories of equation (72), it is obtained that(73)V˙=−q⊺Mq+3q1+q2G1q3−q4q42+G2q3−q4q4ϵx+G3q3−q4q2−2q3+q4ϵx,and(74)M=1121212121121212121121212121,with M being positive definite with λminM1/2.From Remark1, it is shown that the following relations are satisfied:(75)3q1+q2G1q3−q4q42<K123q1+q2q42<K123q1+q22+q44,3q1+q2G2q3−q4q4ϵx<K2ϵx23q1+q2q4<K2ϵx23q1+q22+q42,3q1+q2G3q3−q4q2ϵx<K3ϵx23q1+q2q2<K3ϵx23q1+q22+q22,3q1+q2G3q3−q4q4ϵx<K3ϵx23q1+q2q4<K3ϵx23q1+q22+q42.So,V˙ fulfills(76)V˙<−12q12−K13q1+q22−K2ϵx3q1+q22−2K3ϵx3q1+q22−12q421−K1q42−K3ϵx−12q221−K2ϵx.Hence, the previous inequality is strictly negative definite, and the following constraints are obtained:(77)q12−K13q1+q22−K2ϵx3q1+q22−2K3ϵx3q1+q22>0,1−K1q42−K3ϵx>0,1−K2ϵx>0.Therefore, if the restrictions of (77) are fulfilled, V˙ is strictly negative, and the state vector q exponentially converges to zero after t>T4.The following proposition summarizes the previous discussion, which is the main result of our study.Proposition 1.
Consider the PVTOL aircraft system with an inverted pendular load as described in (9) and in a closed loop with controllers (12), (17a) and (17b), (23), (25), and (49). Then, the closed-loop system is exponentially asymptotically stable provided that the control parameters c1, c2, c3, and c4 satisfy inequalities (69), and kαi and kyi are selected such that the characteristic polynomial is Hurwitz.
Finally, in Figure3, the steps’ sequence is shown, obtained from the control laws u1 and u2.
Having designed the control scheme for the PVTOL aircraft system with an inverted pendular load and carried out its convergence analysis, we test its effectiveness through numerical simulations in the following section. It is worth mentioning that it would be ideal for testing the scheme by conducting actual experiments—unfortunately, the construction of the needed prototype is still in progress.Figure 3
Control strategy sequence.


## 4. Numerical Simulations

To test the controllers’ performance, we carried out some numerical simulations using the MATLAB-Simulink program, and the results were obtained based on the numerical method of Runge–Kutta of the fourth order with a fixed step of 0.01 s. The physical parameters of the system aremp=0.2 kg, mv=0.8 kg, lp=0.2 m, and g=9.8m/s2. Also, the tuning parameters proposed for each controller are listed in Table 1. Notice that the controller parameter values u2 and v2 were selected such that the error dynamics is equal to the desired closed-loop polynomial s2+2ζωns+ωn2s+β. Besides, the parameters of vf were chosen such that ci<1/2ci+1 [40]. Finally, the initial conditions were set as follows: xv0=0.2 m,yv0=0 m,α=0 rad, θ=0.1 rad, x˙v=0.15 m/s,y˙v=0 m/s, α˙=0 rad/s, and θ.=0.05 rad/s.Table 1
Tuning parameters for controllers.
ControllerControl parameterGainu2L3kα0167kα1299kα2183kα362kα411a1.45ky0167ky1299v2ky2183ky362ky41c10.11vfc20.24c30.49c40.99In the first experiment, the control strategy simultaneously carries out height position by performing trajectory regulation, stabilizing the horizontal position and roll angle for the PVTOL and the pendulum angle. In this case, the desired trajectory,yd, was proposed as(78)yd=11+e20−2t1+e−60+2t.To test the effectiveness of the control strategy introduced in this study, we carried out a comparison test against the classical PD controller, with gains tuned askp=1 and kd=4. Figure 4 shows the outcome of this experiment, where we can observe that both strategies successfully achieve the height position regulation through the trajectory tracking task and the stabilization of the horizontal position, the roll angle, and the pendulum angle. Also, the plot in this figure shows that our controller converges to the desired values faster than the PD controller does. Figure 5 shows the behaviours of the angular velocities, and Figure 6 shows the tracking error between α and αd and the control inputs u1 and u2.Figure 4
Comparison between the position closed-loop responses of the proposed controller and PD controller. (a) Behaviour of statexv. (b) Behaviour of state yv. (c) Behaviour of state α. (d) Behaviour of state θ.
(a)(b)(c)(d)Figure 5
Angular velocities’ closed-loop response of the proposed controller. (a) Behaviour of statex˙v. (b) Behaviour of state y˙v. (c) Behaviour of state α˙. (d) Behaviour of state θ˙.
(a)(b)(c)(d)Figure 6
Tracking error of the stateα and behavior of controllers u1 and u2. (a) Error tracking dynamics for α. (b) Control action u1. (c) Control action u2.
(a)(b)(c)Using the same setup as before, we run a second experiment, but in this case, the system is affected by external disturbances with parameters fixed asax=1.4, ay=0.8, and aθ=2. Figure 7 shows the corresponding plots, where we can see that our controller is capable of accomplishing height tracking and position regulation and, simultaneously, stabilizing the horizontal position, roll angle, and pendulum angle. Also, it can be seen that the PD controller used in the first experiment regulates the system slower than our controller does and exhibits undesirable oscillations. Therefore, our controller has better performance, maneuverability, and whole stabilization, even when the system is affected by external disturbances. Figure 8 shows the system angular velocities, and Figure 9 shows the tracking error behaviour between α and αd and the control inputs u1 and u2. Please notice that the proposed control has adequate energy management, according to the obtained performance index ∫ui2 shown in Figure 10.Figure 7
Position closed-loop response of the proposed controller in the presence of external disturbances. (a) Behaviour of statex˙v. (b) Behaviour of state y˙v. (c) Behaviour of state α˙. (d) Behaviour of state θ˙.
(a)(b)(c)(d)Figure 8
Angular velocities’ closed-loop responses of the proposed controller in the presence of external disturbances. (a) Behaviour of statex˙v. (b) Behaviour of state y˙v. (c) Behaviour of state α˙. (d) Behaviour of state θ˙.
(a)(b)(c)(d)Figure 9
The behaviour of the error tracking of the stateα and controllers u1 and u2, respectively, owing to external disturbances. (a) Error tracking dynamics for α. (b) Control action u1. (c) Control action u2.
(a)(b)(c)Figure 10
The performance index of the controllersu1 and u2, respectively, owing to external disturbances. (a) Performance index of the controller u1. (b) Performance index of the controller u2.
(a)(b)


## 5. Conclusions

This work presents a nested saturation function-based controller, combined with a GPI controller, to stabilize the PVTOL aircraft system with an inverted pendular load. The model of this system was derived using Euler–Lagrange formalism. The main contribution consists of using a fictitious control, and then a GPI controller is proposed for the aircraft angleα. Several linear transformations and coordinate changes were introduced to express the original system to a minimal representation. To accomplish the takeoff and landing maneuvers, we propose a GPI controller to track the desired trajectory. After stabilizing the PVTOL height, the system was represented as a chain of integrators plus nonlinear disturbance, allowing us to use nested saturation functions to design a controller to stabilize the horizontal position and pendulum angle. The stability analysis was carried out using the second method of Lyapunov, using a simple candidate function. Designing the control scheme was not an easy task because the PVTOL system with an inverted pendular load is underactuated, and ensuring the pendulum’s upright position makes this problem even harder to solve. We ran numerical simulations to assess the performance of our control scheme, having obtained convincing results. These simulations included a comparison against the well-established PD control strategy. The performance index of both controllers was computed to compare them, and the outcome revealed that our strategy has a better performance than the PD controller. It is important to note that the performance indexes were estimated in the presence of nonlinear perturbations, which means that the proposed controller behaves well even in this undesirable yet unavoidable realistic scenario. It is important to note that the controller, based on a GPI controller and nested saturation functions, allows us to perform takeoff maneuvers in the presence of exogenous disturbances.In future work, we will explore a design to estimate the disturbance due to wind or robust techniques for parametric uncertainties of the system. In addition, it is worth mentioning that an experimental platform that allows configuring the PVTOL with an inverted pendular load system has been designed, whose construction is in process, Thus, experimental implementation of the control scheme proposed herein is considered.



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