M Labbadi
email: moussa.labbadi@uphf.fr

A J Muñoz-Vázquez

M Djemai

Y Boukal

M Zerrougui

M Cherkaoui

Fractional-Order Nonsingular Terminal Sliding Mode Controller for a Quadrotor with disturbances

Keywords: Quadrotor UAV, Backstepping, Terminal sliding mode control

This paper proposes a finite-time controller for an unmanned aerial vehicle in the presence of disturbances/uncertainties using fractional-order terminal sliding mode. First, contrary to existing fractional-order backstepping sliding mode controllers, this paper introduces a new control approach for quadrotor position control, which is based on fractional-order fast terminal backstepping sliding mode control. Using an appropriate sliding surface, the position tracking error converges to zero in finite time while providing good robustness properties in different complex path scenarios under unknown disturbances. Then, a novel fractional-order fast terminal sliding mode control scheme is developed for quadrotor attitude control, which provides good properties in terms of robustness against unknown disturbances, and in terms of convergence time, etc.

Finally, simulation results are presented to discuss the advantages of the hybrid control approach proposed in this work for quadrotors under unknown disturbances compared to other existing controllers.

Finite-time stabilization, Fractional-order control.

Nomenclature

Acronyms

QUAS

Quadrotor unmanned aerial system TSMC Terminal sliding mode control

Introduction

Advanced control techniques are implemented for quadrotor unmanned aerial systems (QUASs) to ensure a good path tracking such as hybrid controllers [START_REF] Cabecinhas | Robust Landing and Sliding Maneuver Hybrid Controller for a Quadrotor Vehicle[END_REF],

fast nonsingular terminal sliding mode controllers [START_REF] Silva | Fast Nonsingular Terminal Sliding Mode Flight Control for Multirotor Aerial Vehicles[END_REF][START_REF] Labbadi | Robust adaptive nonsingular fast terminal sliding-mode tracking control for an uncertain quadrotor UAV subjected to disturbances[END_REF], backstepping terminal sliding mode controllers (TSMC) [START_REF] Labbadi | Robust adaptive backstepping fast terminal sliding mode controller for uncertain quadrotor UAV[END_REF], aperiodic sampling based output feedback trajectory tracking controllers [START_REF] Guerrero-Sánchez | Nonlinear control strategies for a UAV carrying a load with swing attenuation[END_REF], finite-time reliable attitude tracking control [START_REF] Harshavarthini | Finite-time reliable attitude tracking control design for nonlinear quadrotor model with actuator faults[END_REF], adaptive finite-time control [START_REF] Jiang | Composite adaptive finite-time control for quadrotors via prescribed performance[END_REF], finite-time control [START_REF] Jiang | Finite-time control for small-scale unmanned helicopter with disturbances[END_REF], and state-dependent differential Riccati equation controllers [START_REF] Shao | Improved prescribed performance antidisturbance control for quadrotors[END_REF], etc.

To provide good performance with respect to unknown wind gust disturbances and time-varying payloads, a robust adaptive prescribed performance controller has been derived in [START_REF] Hua | Adaptive prescribed performance control of QUAVs with unknown time-varying payload and wind gust disturbance[END_REF]. The controller system was decoupled into position/attitude subsystems. A sliding mode control (SMC) with an adaptive law has been developed for translational subsystem while an adaptive backstepping technique has been designed to track the desired attitude. In [START_REF] Jiang | Composite adaptive finite-time control for quadrotors via prescribed performance[END_REF], a finite-time control method was proposed for the quadrotor system under perturbations. It has been designed using a learning approach which consists in an adaptive neural control with a disturbance observer and a quaternion-based backstepping technique. The authors in [START_REF] Yoo | Hybrid Reinforcement Learning Control for a Micro Quadrotor Flight[END_REF] have proposed a combination of deterministic controllers and reinforcement learning for the safe flight of a micro quadrotor. In [START_REF] Shao | Event-triggered robust control for quadrotors with preassigned time performance constraints[END_REF], an event-triggered control approach has been presented to reduce communication resources. The study in [START_REF] Qi | MUDEbased Control of Quadrotor for Accurate Attitude Tracking[END_REF] has addressed the problems of actuator disturbance, time delay, and uncertainties/disturbances in attitude control for a quadrotor system. In this paper, a disturbance estimator-based attitude control law has been presented. A robust adaptive type-2 fuzzy neural controller optimized for quadrotor systems has been presented in [START_REF] Shirzadeh | Trajectory tracking of a quadrotor using a robust adaptive type-2 fuzzy neural controller optimized by cuckoo algorithm[END_REF].

A superior and innovative sliding surface should be carefully built with this goal in mind. It has been shown that fractional-order sliding mode is well-suite since fast response time and tiny overshoot can be accomplished at the same time, thanks to the memory effect in fractional calculus [START_REF] Sun | Practical tracking control of linear motor via fractional-order sliding mode[END_REF]. Many works on the control of engineering systems have been developed using fractional-order (FO) integral/derivative operators. To obtain good performances and a more flexible structure of proportioanl integral derivative (PID) controller, a fractional-order error manifold has been proposed in [START_REF] Muõz-Vázquez | Fractional P D -I λ D µ error manifolds for robust tracking control of robotic manipulators[END_REF] for robot manipulators. In [START_REF] Wang | Practical Tracking Control of Robot Manipulators with Continuous Fractional-Order Nonsingular Terminal Sliding Mode[END_REF], the authors proposed to combine the concept of nonsingular terminal SMC and continuous FO dynamics for robot manipulators. The stabilization problem for Caputo-type FO systems under perturbation was addressed in [START_REF] Guo | Adaptive sliding mode control for a class of Caputo type fractional-order interval systems with perturbation[END_REF] using a finite-time control based on sliding mode. A continuous fractional sliding mode control was proposed in [START_REF] Muõz-Vázquez | Continuous fractional sliding mode-like control for exact rejection of non-differentiable Hölder disturbances[END_REF] to deal with non-differentiable Hölder disturbances.

Moreover, various versions of fractional-order SMC have also been developed for QUAS in order to enhance performances and robustness against dis-turbances [START_REF] Oliva-Palomo | A Fractional Nonlinear PI-structure Control for Robust Attitude Tracking of Quadrotors[END_REF]. In [START_REF] Shi | Design of Fractional-Order Backstepping Sliding Mode Control for Quadrotor UAV[END_REF], the authors have combined the concept of backstepping technique and SMC with fractional order dynamics to provide good robustness properties of quadrotor systems for complex path under disturbances. In [START_REF] Hua | Fractional-order sliding mode control of uncertain QUAVs with time-varying state constraints[END_REF],

a robust FOSM control technique has been proposed for attitude control and state constrained control approach has been designed for position control under Figures/uncertainties.

Motivated by the previous papers, and inspired by Refs. [START_REF] Shi | Design of Fractional-Order Backstepping Sliding Mode Control for Quadrotor UAV[END_REF][START_REF] Guo | Global sliding mode with fractional operators and application to control robot manipulators[END_REF], the current paper presents a FO hybrid finite-time control (FOHFTC) scheme with fractional order dynamics to deal with the trajectory tracking problem of quadrotor system. The disturbances and the variation of the drag coefficients of rotational and translational subsystems are considered in this study. The proposed FO-HFTC is based on two loops. The first one is the position loop. It is based on a fractional order backstepping fast terminal SMC in order to obtain finite time stability of the position error. The second one is the attitude loop. It is based on a flexible FO fast terminal SMC. Besides, compared with the previous results, it can be seen that the proposed FOHFTC provides better performance such as fast response time, reduction of chattering phenomenon, rejection of Figures/uncertainties, etc. Compared to [START_REF] Shi | Design of Fractional-Order Backstepping Sliding Mode Control for Quadrotor UAV[END_REF], where a classical fractional-order SMC was used, the current work presents a novel fractional-order terminal SMC for the outer-loop to track the desired position. Compared to [START_REF] Guo | Global sliding mode with fractional operators and application to control robot manipulators[END_REF], where a FOSMC was applied, the proposed approach is based on FO fast terminal SMC which increases the tracking performances. Furthermore, the FOHFTC proposed in this paper guarantees good disturbance rejection The contributions of the paper are highlighted as follows:

• Design of two new fractional order finite-time control strategies;

• Accurate tracking of a quadrotor with uncertainties and disturbances is obtained;

• The proposed control scheme ensures finite-time convergence of both the attitude and position velocity errors.

The paper is structured as follows; Section II presents the problem definition and formulation. Section III introduces the design of the proposed controller.

Some simulation results are presented in Section IV. Finally, Section V provides the conclusion. The notation sig a (y) represents |y| a sign(y).

Problem definition and formulation

F F F F X E O B O E F Z B d Y E Z E X B Y B

Attitude dynamics

I Ω = -Ω T × IΩ + M + M a + M c + D Ω (t) Position dynamics m P = R T u F + T d + T g + D P (t) Aerodynamic friction torque M a = diag -K 1 φ2 (t) -K 2 θ2 (t) -K 3 ψ2 (t) T Gyroscopic effects M c = - i=4 i=1 Ω T J r 0 0 (-1) i+1 ω i T Aerodynamic friction force T d = diag K 4 ẋ(t) -K 5 ẏ(t) -K 6 ż(t)
T Gravity force T g = 0 0 -g T One can defined the link between the angular velocities and control laws as:

        u F M xx M yy M zz         =         b b b b 0 b 0 -b -b 0 b 0 -c c -c c                 ω 2 1 ω 2 2 ω 2 3 ω 2 4         (1)
The position subsystem of the QUAS is underacted with one input and three outputs. Three virtual control signals are designed to tackle this issue:

v =      vx vy vz      =      (sin θ(t) cos φ(t) cos ψ(t) + sin φ(t) sin ψ(t)) u F m (sin θ(t) cos φ(t) sin ψ(t) -sin φ(t) cos ψ(t)) u F m (-g + 1 m (cos θ(t) cos φ(t) u F m ))      (2)
Thus, the left thrust, and desired tilting angles can be defined using these virtual controls as:

φ r (t) = arctan cos θ r (t) sin ψ r (t)vx -cos ψ r (t)vy vz + g (3a) θ r (t) = arctan cos ψ r (t)vx + sin ψ r (t)vy vz + g (3b) uF = m v 2 x + v 2 y + (vz + g) 2 (3c)
The acontrol objective is to design the torque M and virtual signal controls to guarantee tracking of the desired trajectories:

x r (t) y r (t) z r (t) φ r (t) θ r (t) ψ r (t)

T .

Figure 2: The proposed control scheme for QUAV.

Controller Design

The control structure of the QUAS in this paper is divided into an outerloop (position loop), and an inner-loop (attitude loop), which includes the roll, pitch, and yaw angles. In this section, two nonlinear controllers are designed for the QUAS system. The first controller is a FO backstepping fast terminal sliding mode control, while the second one is a FO fast terminal SMC for the inner-loop.

Position Controller

In the following, the virtual control signals v x , v y will be designed. Let us first consider the altitude dynamics which can be considered as:

   Ż1 (t) = Z 2 (t) Ż2 (t) = v z -K6 m Ż1 (t) -g + D z (t) (4)
where Z 1 (t) = z(t) and Z 2 (t) = ż(t) represent the state variable of altitude subsystem.

Define the tracking error of the altitude as:

e z (t) = Z 1 (t) -z r (t) (5)
The derivative of e z (t) is

ėz (t) = Ż1 (t) -żr (t) (6)
Define the Lyapunov function and its derivative as:

V z1 (t) = 1 2 e 2 z (t), Vz1 (t) = ėz (t)e z (t) (7)
The derivative of V z1 (t) can be rewritten as:

Vz1 (t) = ėz (t)e z (t) = e z (t)(Z 2 (t) -żr (t)) (8)
Let us set the following virtual control input Z r 2 as e zz (t) = Ż1 -Z r 2 . Differentiating V z1 (t), one gets,

Vz1 (t) = e z (t)(Z 2 (t) -żr (t)) (9)
= e z (t)(e zz (t

) + Z r 2 (t) -żr (t)) (10)
The virtual input can be defined as: Z r 2 (t) = żr (t)-c 1 e z (t) where c 1 is a positive constant. So, using this virtual input Z r 2 , one obtains,

Vz1 (t) = e z (t)e zz (t) + e z (t)(Z r 1 -c 1 e z (t) -Z r 1 ) (11)
= e z (t)e zz (t)

-c 1 e 2 z (t) (12)
To consider the term e z (t)e zz (t), a nonlinear sliding manifold is defined as:

S z (t) = e z (t) + k z e zz (t) + sig(e z (t)) µz [START_REF] Shao | Event-triggered robust control for quadrotors with preassigned time performance constraints[END_REF] with k z > 0.

The equivalent law can be designed as:

v ze = -(c 1 + k z ) ėz (t) + g + K 6 m Ż1 (t) + Zr (t) (14)
To achieve robustness against D z (t), a fractional-order switching law is designed as:

v zs = -h z1 S z (t) -h z2 D qz sign(S z (t)) ( 15
)
where h z1 , h z2 > 0 and q z ∈]0, 1[. The term D qz represents FO derivative, which q z is the fractional operator.

Remark 1. Discontinuity at the origin prevents the differentiation of sign(S z (t)),

It is replaced by tanh(kS z (t)) function with k > 0.

The ultimate altitude control is given by: a positive but unknown constant. We assumed that the perturbations affecting the quadcopter system are bounded.

v z = v zz + v zs (16
Lemma 1. [START_REF] Yin | Fractional-order sliding mode based extremum seeking control of a class of nonlinear systems[END_REF] Consider the l derivative of the ϕ(t)

D q ϕ(t) = 1 Γ(1 -q) d dt t 0 ϕ(τ ) (t -τ ) q dτ, 0
q < 1, the following inequalities are obtained

D q sign(S i (t)) =    > 0, if S i (t) > 0, t > 0 < 0, if S i (t) < 0, t > 0 ( 17
)
where Γ(.) is the Gamma function.

Remark 2. Consider the following two cases: : (i) S(t) > 0 ∀t > 0, we have 0 < D q sgn(kS i (t)) < t -q Γ(1-q) ; (ii) S(t) < 0 ∀t > 0, one has t -q Γ(1-q) < D q sgn(kS i (t)) < 0. Then, D q sgn(kS i (t)) is bounded in both cases. Because the above situations have a higher absolute value than the others, D q sgn(kS i (t)) ∀S i (t) is bounded, i.e. there exists ς > 0 such that |D q sgn(kS i (t))| < ς. On the other hand, due to the onboard power's restriction, the variation in the quadrotor flight states' attitude and trajectory subsystems cannot vary at a supernormal pace. There is also an upper limit to the amount of disturbances since turbulence and gusts, which make up the environment, are energy restricted and cannot last for a long period.

Theorem 1. The origin of position system presented in Table 1, using Assumption 1 under the FO backstepping fast terminal sliding mode control designed in [START_REF] Sun | Practical tracking control of linear motor via fractional-order sliding mode[END_REF] and the nonlinear sliding variable S z (t), is finite-time stable.

Proof. Let us consider the Lyapunov function candidate

V z2 (t) = V z1 (t) + 1 2 S 2 z (t) (18)
Differentiating V z2 (t) yields

Vz2 (t) = -c 1 e 2 z (t) + e z (t)e zz (t) + S z (t)[(k z + c 1 ) ėz (t) -g + v z -Zr + D z (t)] (19)
According to Assumption 1 and replacing [START_REF] Sun | Practical tracking control of linear motor via fractional-order sliding mode[END_REF]

One can define the following matrix in order to analyze the stability :

Q φ =   c 1 + h z1 k 2 z h z1 k z -0.5 h z1 k z h z1   , E 1 =   e z (t) e zz (t)   (21)
If the Q φ is positive matrix, then the derivative of the Lyapunov can be rewritten,

Vz2 (t) ≤ -E T 1 Q φ E 1 ≤ 0 (22)
In order to explicate the expression of the reaching time, the switching controller can be defined in the integer order form as:

v zs = -h z1 S z (t) - h z2 sign(S z (t)). Thus, we could obtain S z (t) Ṡz (t) = -h z1 S 2 z (t) -h z2 |S z (t)|.
There are two scenarios based on the initial condition of S z (0).

• Case 1: S z (0) > 0, one has Ṡz (t)) = -h z1 S z (t) -h z2 . The analytical solution is S z (t) S z (0) + hz2 hz1 e -hz1t -hz2 hz1 , then the reaching-time is t rz = 1 hz2 ln hz2Sz(0)+hz1 hz1 > 0; • Case 2: S z (0) ≤ 0, one has Ṡz (t)) = -h z1 S z (t) + h z2 . The analytical solution is S z (t) -S z (0) + hz2 hz1 e -hz1t + hz2 hz1 , then the reaching-time is t rz = 1 hz2 ln -hz2Sz(0)+hz1 hz1 > 0
From the two cases, the reaching time can be defined as

t rz = 1 hz2 ln hz2|Sz(0)|+hz1 hz1 > 0. Simultaneously, the reaching time of the switching FO controller v zs = -h z1 S z (t) -h z2 D qz sign(S z (t)
) can be defined by two cases.

• Case 3: S z (0) > 0, one has Ṡz (t) = -h z1 S z (t) -h z2 D qz sign(S z (t)).
Therefore, one obtain S z (t)e hz2t = e hz2t ( Ṡz (t) + h z2 S z (t)). Therefore, S z (t)e hz2t = S z (0) -hz1

Γ(1-qz) 0 t τ -qz e hz2τ dτ . Also, we can handle

S z (t reach )e hz2t = S z (0) -hz1 Γ(1-qz) t reach 0 τ -qz e hz2τ dτ = Sz(0)Γ(1-qz) hz1 = 0, since S z (t reach ) = 0. Hence, the settlement of S z (0) = hz1 Γ(1-qz) 0 t τ -qz e hz2τ dτ is t reach . Let, Υ(t) = 0 t τ -qz e hz2τ dτ = Sz(0)Γ(1-qz) hz1 > 0. Clearly, Υ(t), t ∈ [0, inf) is an increasing function. According to intermediate value theorem, there exists t * such that Λ(t * ) = Sz(0)Γ(1-qz) hz1
. Applying numerical approximation, we can gain t reach = t * .

• Case 4 :

S z (0) < 0, we can have Ṡz (t) = -h z1 S z (t) -h z2 D qz sign(S z (t)). One obtain S z (t)e hz2t = e hz2t ( Ṡz (t) + h z2 S z (t)) = hz1 Γ(1-qz) t -qz e hz2t . We can handle S z (t reach )e hz2t = S z (0) + hz1 Γ(1-qz) t reach 0 τ -qz e hz2τ dτ = Sz(0)Γ(1-qz) hz1 = 0. Let, Υ(t) = 0 t τ -qz e hz2τ dτ = Sz(0)Γ(1-qz) hz1 > 0. Similar to case 3, there exists t * > 0 such that Λ(t * ) = -Sz(0)Γ(1-qz) hz1
. Applying numerical approximation, we can gain t reach = t * . Therefore, t teach of FO switching controller can be obtained from Υ(t) = 0 t τ -qz e hz2τ dτ =

Sz(0)Γ(1-qz) hz1
.

Similar to v z , one can defined the following two controllers. For the y(t)

channel:

vy = vye + vys = -hy1Sy(t) -hy2D qy sign(Sy(t)) -(c2 + ky) ėy(t) + K5 m Ẏ1(t) + Ÿr (t) (23)
and

vx = vxe + vxs = -hx1Sx(t) -hx2D qx sign(Sx(t)) -(c3 + kx) ėx(t) + K4 m Ẋ1(t) + Ẍ r (t) ( 24
)
where

h x1 , h x2 > 0, q x , q y ∈]0, 1[ 2 c 1 , c 2 > 0 and h y1 , h y2 > 0. X 1 (t) and X 2 (t) = Ẋ1 (t) represent the state variable of x(t)-subsystem. The notations Y 1 (t) and Y 2 (t) = Ẏ1 (t) represent the state variable of y(t)-subsystem.
Theorem 2. The origin of position system presented in Table 1, using Assumprion 1 under the FO backstepping fast terminal sliding mode controls designed in (16), ( 23), [START_REF] Guo | Global sliding mode with fractional operators and application to control robot manipulators[END_REF] and the nonlinear sliding variable the nonlinear sliding variables S x (t), S y (t), and S z (t), is finite-time stable.

Proof. The Lyapunov function for the position subsystem can be used as follows:

VT = Vz2(t) + Vx2(t) + Vy2(t) = Vz1(t) + 1 2 S 2 z (t) + Vx1(t) + 1 2 S 2 x (t) + Vy1(t) + 1 2 S 2 y (t) (25)
The derivative of V T is,

VT = -c1e 2 z (t) + ez(t)ezz(t) + Sz(t)[(kz + c1) ėz(t) -g + vz -Zr + Dz(t)] -c3e 2 x (t) + ex(t)exx(t) + Sx(t)[(kx + c3) ėx(t) + vx -Ẍ r + Dx(t)] -c5e 2 y (t) + ey(t)eyy(t) + Sx(t)[(ky + c5) ėx(t) + vy -Ÿr + Dx(t)]
From ( 16), ( 22), ( 23), [START_REF] Guo | Global sliding mode with fractional operators and application to control robot manipulators[END_REF], we can obtain,

VT ≤ -E T 1 Q φ E 1 -E T 2 Q θ E 2 -E T 3 Q ψ E 3 ≤ 0 ( 26
)
where

e xx (t) = Ẋ1 -X r 2 , e yy (t) = Ẏ1 -Y r 2 , X r 2 (t) = ẋr (t) -c 3 e x (t
) and Y r 2 (t) = ẏr (t) -c 5 e y (t). According to the aforesaid analysis, the outer-loop stability is ensured.

Attitude Controller

In this section, a FO fast terminal sliding mode controller will be designed for attitude subsystem. First, let us consider the roll subsystem as :

Φ1 (t) = Φ 2 (t) Φ2 (t) = I 2 -I 3 I 1 θ(t) ψ(t) - J r I 1 θ(t) - K 1 I 1 φ2 (t) + 1 I 1 M xx + D φ (t) (27)
where Φ 1 (t) and Φ 2 (t) represent the state variable of roll subsystem. Let the roll error be as:

e φ (t) = Φ 1 (t) -φ r (t) (28)
A FOTSM manifold is presented as,

S φ (t) = β φ I α φ e φ (t) + γ φ D 1-α φ [K φ1 e φ (t) + K φ2 sig(e µ φ φ (t))] + D 1-α φ ėφ (t) (29)
where α φ ∈]0, 1[, β φ and γ φ , K φ1 , and K φ2 are positive coefficients.

Tacking fractional derivative of the sliding variable yields:

D α φ S φ (t) = β φ e φ (t) + γ φ D 1 [K φ1 e φ (t) + K φ2 sig(e µ φ φ (t))] + ëφ (t) (30)
Hence, one has

D α φ S φ (t) = β φ e φ (t) + γ φ D 1 [K φ1 e φ (t) + K φ2 sig(e µ φ φ (t))] + I2 -I3 I1 θ(t) ψ(t) - Jr I1 θ(t) - K1 I1 φ2 (t) + 1 I1 Mxx + D φ (t) (31)
From ( 30), one gets the following equivalent control law

M xxe = -I 1 [β φ e φ (t) + γ φ D 1 [K φ1 e φ (t) + K φ2 sig(e µ φ φ (t))] + I 2 -I 3 I 1 θ(t) ψ(t) - J r I 1 θ(t) - K 1 I 1 φ2 (t)] (32)
Assumption 2. The disturbance on the roll dynamics is satisfied

D φ ≤ I 1-α φ δ D φ or D 1-α φ D φ ≤ δ D φ where δ D φ is a positive function.
There is an upper constraint on the time derivative of disturbances because during quadrotor flight, the environment disturbances, such as turbulence and gusts, are energy constrained and cannot last for a long period. A FO derivative of disturbances is taken into consideration in this paper.

The following Lemma is introduced in order to get the approximated form of the fractional calculus answer.

Lemma 2. In Ref. [START_REF] Mitrinovic | Analytic inequalities[END_REF], suppose that 1 , 2 > 1, and

1 1 + 1 2 = 1, if |κ 1 ( * )| 1 , |κ 2 ( * )| 2 ∈ L 1 (D), then κ 1 ( * )κ 2 ( * ) ∈ L 1 (D) and D |κ 1 (x)κ 2 (x)|dx ≤ D |κ 1 (x)| 1 dx 1 1 D |κ 2 (x)| 2 dx 1 2 (33)
where L 1 (D) stands for the Banach space of all Lebesgue measurable functions κ

1 : D → R with D |κ 1 (x)|dx < ∞.
Since fractional calculus has a memory, it is necessary to know the previous values of all the variables throughout an infinite time span in order to calculate the fractional derivative of the variable x(t). Consequently, the ensuing presumption is presented. Remark 3. Assumption 2 states that the perturbation is bounded by a known function to be designed using Lemma 2. It is based on the fact that most input physical systems are limited by motor converters. Hence, the control input can change over time but cannot be infinite. As mentioned in [START_REF] Guo | Global sliding mode with fractional operators and application to control robot manipulators[END_REF], although being conservative, it may nonetheless cover a wide range of practical applications. We should also mention that Assumption 2 is based on the fact that the generalized version of the second-order nonlinear dynamical system has been studied like as the subsystem of the quadrotor. It should be noted that this Assumption 2 is strict as it can still be applied to a variety of dynamical systems in the literature, including robotic systems [START_REF] Guo | Global sliding mode with fractional operators and application to control robot manipulators[END_REF] and quadrotor systems [START_REF] Pouzesh | Event-triggered fractional-order sliding mode control technique for stabilization of disturbed quadrotor unmanned aerial vehicles[END_REF][START_REF] Labbadi | Fractional-order global sliding mode controller for an uncertain quadrotor UAVs subjected to external disturbances[END_REF]. Remark 4. Assumption 2 studies the bounded disturbances using a positive function to present the upper bounded of the disturbance and uncertainty on the QUAV dynamics. Similar assumptions can be found in [START_REF] Guo | Global sliding mode with fractional operators and application to control robot manipulators[END_REF][START_REF] Pouzesh | Event-triggered fractional-order sliding mode control technique for stabilization of disturbed quadrotor unmanned aerial vehicles[END_REF][START_REF] Labbadi | Fractional-order global sliding mode controller for an uncertain quadrotor UAVs subjected to external disturbances[END_REF].

To improve the tracking performance against disturbances, we define the switching law as follows:

Mxxs = -I1[I 1-α φ δD φ + ε φ sig q φ (S φ (t)) + σ φ I 1-α φ S φ (t)] (34)
where q φ ∈]0, 1[, ε φ and σ φ are positive parameters. Thus, the proposed control law for the roll subsystem under model uncertainties/disturbances is

M xx = M xxe + M xxs (35)
Theorem 3.

The origin of roll attitude system presented in Table 1, with Assumption 2, under the designed controller in [START_REF] Malti | Object-oriented crone toolbox for fractional differential signal processing[END_REF] and using the the sliding variable S φ (t), is finite-time stable with t r ≤=

1 σ φ (1-q φ ) ln(1 + σ φ ε φ S φ (0) 1-q φ 2
).

Proof. Define the LF candidate for the φ(t)-subsytem as

V φ (t) = 1 2 S 2 φ (t) (36)
The time-derivative of V φ (t) is,

Vφ (t) = S φ (t) Ṡφ (t) = S φ (t)D 1-α φ (D α φ S φ (t)) (37)
Substituting ( 31) and ( 35) into (37), one gets

Vφ (t) = S φ (t)D 1-α φ (D φ (t) -I 1-α φ δ D φ -I 1-α φ ε φ sig q φ S φ (t) -σ φ I 1-α φ S φ (t)) (38) = S φ (t)(D 1-α φ D φ (t) -δ D φ -ε φ sig q φ S φ (t) -σ φ S φ (t))
Using Assumption 2, one get

Vφ (t) ≤ -ε φ |S φ (t)| 1+q φ -σ φ S 2 φ (t) (39)
Hence, one obtains

Vφ (t) ≤ -ε φ |S φ | 1+q φ (t) (40)
The above inequality (40) can be rewritten as

Vφ (t) ≤ -ε φ S φ (t) 1+q φ 2 (41)
Using Eq. ( 39), one has

Vφ (t) = -ε φ |S φ (t)| 1+q φ -σ φ S φ (t) 2 = -ε φ (2V φ (t)) 1+q φ 2 -σ φ (2V φ (t)) (42)
As a result, a simple calculation yields,

dt = - dV φ (t) ε φ (2V φ (t)) 1+q φ 2 + σ φ (2V φ (t)) = - 1 2 (2V φ (t)) -1 2 d(2V φ (t)) ε φ (2V φ (t)) q φ 2 + σ φ (2V φ (t)) 1 2 = - d(2V φ (t)) 1 2 ε φ (2V φ (t)) q φ 2 + σ φ (2V φ (t)) 1 2 = d S φ (t) 2 ε φ S φ (t) q φ 2 + σ φ S φ (t) 2 = S φ (t) -q φ 2 d S φ (t) 2 ε φ + σ φ S φ (t) 1-q φ 2 = 1 σ φ (1 -q φ ) d(σ φ S φ (t) 1-q φ 2 ε φ + σ φ S φ (t) 1-q φ 2 (43)
Taking integral of both sides of Eq. (43) from 0 to t r and S φ (t r ) = 0, one gets

tr 0 dt = 1 σ φ (1 -q φ ) tr 0 d(σ φ S φ (t) 1-q φ 2 ε φ + σ φ S φ (t) 1-q φ 2 t r = 1 σ φ (1 -q φ ) ln(1 + σ φ ε φ S φ (0) 1-q φ 2 ) ( 44
)
This completes the proof.

Similarly to the roll controller, the pitch and yaw control laws can be given as:

M yy = -I 2 [β θ e θ (t) + γ θ D 1 [K θ1 e θ (t) + K θ2 sig(e µ θ θ (t))] + I 3 -I 1 I 2 φ(t) ψ(t) + J r I 2 φ(t) - K 2 I 2 θ2 (t)] -I 2 [I 1-α θ δ D θ + ε θ sig q θ (S θ (t)) + σ θ I 1-α θ S θ (t)] (45)
and

M zz = -I 3 [β ψ e ψ (t) + γ ψ D 1 [K ψ1 e ψ (t) + K ψ2 sig(e µ ψ ψ (t))] + I 1 -I 2 I 3 φ(t) θ(t) - K 3 I 3 ψ2 (t)] -I 3 [I 1-α ψ δ D ψ + ε ψ sig q ψ (S ψ (t)) + σ θ I 1-α ψ S ψ (t)] ( 46
)
where Proof. Consider the candidate Lyapunov function for the attitude as follows: The time-derivative of V R is given by:

α θ , α ψ ∈]0, 1[ 2 . β φ , γ φ , K φ1 , K φ2 > 0. q θ , q ψ ∈]0, 1[ 2 ,
V R = 1 2 S 2 φ (t) + 1 2 S 2 θ (t) + 1 2 S 2 ψ (t) (47)
VR =S φ (t) Ṡφ (t) + S θ (t) Ṡθ (t) + S ψ (t) Ṡψ (t) = S φ (t)D 1-α φ (D α φ S φ (t)) + S θ (t)D 1-α θ (D α θ S θ (t)) + S ψ (t)D 1-α ψ (D α ψ S ψ (t))
According to Assumption 2 and from ( 31), ( 35), ( 39), ( 45), (46), we can obtain,

VR ≤ -ε φ |S φ (t)| 1+q φ -σ φ S 2 φ (t) -ε θ |S θ (t)| 1+q θ -σ θ S 2 θ (t) -ε ψ |S φ (t)| 1+q ψ -σ ψ S 2 ψ (t) ≤ 0 (48)
The reaching condition of attitude loop stability is ensured. Then, using the results of Theorem 3, of the origin of the attitude loop is finite-time stable.

Corollary 1. Considered overall system presented in Table 1 in the presence of disturbances using the ultimate control laws, presented in (16), ( 23), ( 24), ( 35), (45), and (46), the sliding mode variables be reached in the finite-time, then the tracking errors for both attitude and position converge to zero in a finite-time.

Proof. The Lyapunov function for the quadrotor system is chosen as follows:

V S = V R + V T (49)
The time-derivative of V S is given by:

VS = VR + VT (50)
Based on the results of proof of the theorem 2 and 4 in order to proof Corollary 1. Then from [START_REF] Mitrinovic | Analytic inequalities[END_REF] and (48), we have,

VS ≤ -ε φ |S φ (t)| 1+q φ -σ φ S 2 φ (t) -ε θ |S θ (t)| 1+q θ -σ θ S 2 θ (t) -ε ψ |S φ (t)| 1+q ψ -σ ψ S 2 ψ (t) -E T 1 Q φ E1 -E T 2 Q θ E2 -E T 3 Q ψ E3 ≤ 0 (51)
The Lyapunov technique is used to demonstrate the global stability of rotational and translational tracking errors.

M i = sat(M iL , Mi , τ iL ) =          M iL Mi < M iL Mi M iL < M iL < M iU M iU Mi > M iU (52)
where M iL and M iU are the four control inputs' lower and upper limit bounds, respectively. In this paper, the saturation function (52) applied for the four inputs is used to reduce oscillation and amplitudes of the inputs of the system Ref. [START_REF] Li | Adaptive and robust control of quadrotor aircrafts with input saturation[END_REF].

Remark 7. In this paper two nonlinear surfaces are proposed in order to obtain the finite-time convergence of the quadrotor states. Remark 9. The proposed control method's implementation challenges for quadrotor position and attitude will be examined from four simulations. First, the complexity of the problem is due to the fact that for position and attitude, tracking performance such as convergence time and steady-state performance are considered, while the time-varying disturbances, parameter uncertainties (such as air drag coefficients), and external disturbances exist. Second, the novelty and difficulty of the proposed method can be broken down to the following two points.

• For position control, an flexible FO backstepping TSMC is constructed and employed different from [START_REF] Shi | Design of Fractional-Order Backstepping Sliding Mode Control for Quadrotor UAV[END_REF], which enhanced the tracking performance against random external disturbances and random drag coefficients.

• For attitude control, a finite-time controller based on fractional-order fast terminal sliding mode control is developed, which enhanced the robustness and accuracy the rotational subsystem.

The disadvantages of the proposed scheme can be defined as: requires complex adaption laws, complexity in coefficient calculation, and selection of nonlinear surfaces is cumbersome in order to avoid the singularity problem.

Simulation results and discussions

In this part, the protocol of simulations will be discussed. Simulator implements dynamics presented in Table 1. This dynamic included the forces and momentums. The variation of the drag coefficients and external disturbances are taken account to make simulate realistic quadrotor in order validate the proposed FO controllers. The mathematical models of the sensors, actuators, and the plant refer to [START_REF] Labbadi | Robust adaptive nonsingular fast terminal sliding-mode tracking control for an uncertain quadrotor UAV subjected to disturbances[END_REF][START_REF] Raffo | Automatica An integral predictive / nonlinear H ∞ control structure for a[END_REF] and the selected physic parameters of the QUAS are have been considered in all scenarios. To obtain signification tracking performance, the FOHFTC parameters are tuned. In order to highlight the present results, a comparison study is presented using two controllers. In the following subsections, we present the results these scenarios with discussion.

Control parameters selection

In this part, the parameters selection of the suggested control method will be discussed in order to implement this controller.

(A) For outer loop (position controller)

-Selection of c j for j = 1, 2, 3: this parameter increases the stability of tracking errors. The latter depends on the initial conditions of the state of the system.

-Selection of k j , µ j : these parameters affect directly the sliding mode dynamics as presented in [START_REF] Shao | Event-triggered robust control for quadrotors with preassigned time performance constraints[END_REF]. A faster convergence to zero of the tracking errors can be obtained by choosing a big value of those parameters but increase the input signals.

-Selection of c i : it is fractional operator which used to improve the robustness.

-Selection of h j1 , h j2 : these parameters are positive gains designed in the reaching law [START_REF] Shirzadeh | Trajectory tracking of a quadrotor using a robust adaptive type-2 fuzzy neural controller optimized by cuckoo algorithm[END_REF] to achieve a high level of robustness. -Selection of β i , α i , γ i , k i , µ i for i = φ, θ, ψ for i = φ, θ, ψ: these parameters affect directly the sliding mode dynamics as presented in [START_REF] Li | Adaptive and robust control of quadrotor aircrafts with input saturation[END_REF]. A faster convergence to zero of the tracking errors can be obtained by choosing a big value of those parameters but increase the input signals.

-Selection of σ i , ε i , q i : these coefficients are designed in the reaching control law [START_REF] Oustaloup | La Derivation Non Entiere: Synthese et Applications[END_REF] to obtain a high level of robustness against parameters uncertainties and disturbances.

(C) Selection of parameters in the simulations. The following remark gives the method used to obtain the parameter controllers.

Remark 10. The controller settings are fine-tuned to meet the quadrotors performance requirements. Furthermore, the Simulink software's optimization toolbox is utilized to determine the optimum values for these parameters. (see Ref. ([31]).

The control parameters are given in Table 2.

According to Lemma 2, we can write the fractional estimation of "disturbance term" as

δ Di = ( π Γ(αi) )(Γ(α i )(1 -α 2 i )Γ(α 2 i )) 1 1+α i ( t 1+α i 1+αi ) α i 1+α i .

Approximation of the FO derivatives

The FO approximation that was used in the simulation will be introduced in this part of the paper. Complications in practical implementation of fractionalorder derivative and integrator terms are one of the challenges that have been encountered in applications and simulations of these types of operators. This explains why only integer-order derivatives and integrals are used to approximate fractional-order derivative and integrator terms [START_REF] Oustaloup | La Commande CRONE[END_REF][START_REF] Vinagre | Some Approximations of Fractional Order Operators used in Control Theory and Applications[END_REF]. In the literature, the Crone approximation is the most used method to simulate the FO operators developed in the proposed controllers (see Refs. [START_REF] Oustaloup | La Derivation Non Entiere: Synthese et Applications[END_REF][START_REF] Malti | Object-oriented crone toolbox for fractional differential signal processing[END_REF] and the references therein). A recursive distribution of poles and zeros within a frequency range

[ω l , ω h ] is used in this approximation. The higher the order N, the more exact the approximation of fractional dynamics in the frequency range [ω l , ω h ] becomes. Original mathematical concepts based on this approximation were included in the CRONE toolbox, which has been developed by the CRONE team since the 1990s and is a MATLAB toolbox dedicated to the fractional calculus and its applications in automatic control and signal processing.

Remark 11. In this paper, the Oustaloup modified filter is used to approximate the FO terms. The frequency range is 0.01 to 100 rad/s with 10 orders.

To show clearly the superiority of the suggested hybrid control method and to verify its advantages, four scenarios in terms of disturbances are considered under different paths following.

Simulations with slowly time-varying disturbances

In this subsection, we conduct a simulation in the case of random uncertainties on the drag coefficients, including both translational and rotational subsystems. The effect is shown in Fig. 4, which approximates to a real flight. The chosen disturbances in this case are

D x (t) =    0 m/s 2 t ∈ [0, 10] 1 2 m/s 2 t ∈ [0, 155] D y (t) =          0.4 m/s 2 t ∈ [0, 20] -0.2 m/s 2 t ∈ [20, 50] 0.2 m/s 2 t ∈ [50, 155] D z (t) =          0 m/s 2 t ∈ [0, 10] -2 10 -1 m/s 2 t ∈ [10, 40] 3 10 -1 m/s 2 t ∈ [40, 155] D φ (t) = 1 2 cos 1 2 t + tanh(0.3t) rad.s -2 D θ (t) = 1 2 cos(t) + cos(0.3t) rad.s -2 D ψ (t) = 1 2 sin(0.7t) + 2 rad.s -2 (53)
Figure 5 shows the results of position tracking, and Figure 6 depicts the attitude tracking. Based on these results, the QUAS tracks the desired path with accuracy under disturbances and the change of drag coefficients after a short period. Figure 7 shows the responses of the control inputs are chattering-free, and its amplitudes have appropriate values. The path following in 2D and 3D spaces is depicted respectively in the Figures 10 and9. As can be seen, the suggested controller is capable of controlling the QUAS in the presence of perturbation/uncertainty.

The disturbances and uncertainties applied on both rotational and translational subsystems decrease tracking performance in path following of the QUAS using classical control methods. As result, the FO control method proposed here has better tracking performances and the ability to cope with these perturbations compared with [START_REF] Shi | Design of Fractional-Order Backstepping Sliding Mode Control for Quadrotor UAV[END_REF] and [START_REF] Jia | Integral backstepping sliding mode control for quadrotor helicopter under exter-nal uncertain disturbances[END_REF] as shown in Fig. 5.

Simulations with random Gaussian disturbances

In this subsection, another scenario is proposed tacking account the random Gaussian disturbances for both attitude/position as shown in Fig. 11. The reference trajectory is chosen as :

x r (t) = 1 2 cos π 20 m, y r (t) = 1 2 sin π 20 m (54) z r (t) = 2 -2 cos π 2 m, ψ r = 1 2 rad (55)
The results in this case using the FOHFTC proposed in this work, control methods proposed in [START_REF] Shi | Design of Fractional-Order Backstepping Sliding Mode Control for Quadrotor UAV[END_REF], and [START_REF] Jia | Integral backstepping sliding mode control for quadrotor helicopter under exter-nal uncertain disturbances[END_REF] are plotted in Figs. 12 to 16. As can be seen, these errors converge to the origin with a short finite time and [START_REF] Shi | Design of Fractional-Order Backstepping Sliding Mode Control for Quadrotor UAV[END_REF] and [START_REF] Jia | Integral backstepping sliding mode control for quadrotor helicopter under exter-nal uncertain disturbances[END_REF] control methods.

The 2D trajectory-tracking performance is depicted in Fig. 16 and 3D pathfollowing is shown in Fig. 17. As can be seen, the trajectories in 2D and 3D spaces track the desired trajectories with high precision. In addition, the effect of the random Gaussian disturbances applied on the translational and rotational loops can be precisely compensated using the designed FOHFTC. More importantly, the proposed control method achieves a better tracking of the desired trajectory without chattering phenomenon and enhances the tracking performance against random Gaussian disturbances.

4.3.

Comparison with fractional-order backstepping sliding mode control scheme proposed in [START_REF] Shi | Design of Fractional-Order Backstepping Sliding Mode Control for Quadrotor UAV[END_REF] In summary, the tracking performance obtained in scenario 1, Figs. 5,6, and 8 show good of the control scheme proposed in this work. Nonetheless, for the control objective of rejecting the disturbances during the flight and following the desired trajectory, the proposed nonlinear control law was found to perform better than the FO control law developed in [START_REF] Shi | Design of Fractional-Order Backstepping Sliding Mode Control for Quadrotor UAV[END_REF]. Figures 9 and 10 demonstrate this performance in term of tracking performance.

The convergence obtained by the proposed control method is better than the results obtained from the control scheme suggested in [START_REF] Shi | Design of Fractional-Order Backstepping Sliding Mode Control for Quadrotor UAV[END_REF]. For example, the settling time produced by the proposed control method is small than other FOBSMC as presented in Figs. 5 and9. It can be seen from Fig. refAttitude1, the orientation dynamics are quickly regulated using the proposed control approach.

Concerning the scenario 2, the results obtained by the proposed controller is better than the FOBSMC [START_REF] Shi | Design of Fractional-Order Backstepping Sliding Mode Control for Quadrotor UAV[END_REF]. From the Figs. 12, 13, and 15 we can observe more advantages in the proposed control scheme, in the sense that it has better transient response.

Comparison with another nonlinear controller

The proposed controller was compared with the integral backstepping sliding mode control proposed in [START_REF] Jia | Integral backstepping sliding mode control for quadrotor helicopter under exter-nal uncertain disturbances[END_REF]. Both controllers were used on a quadrotor subjected to random disturbances. The performance responses of both controllers are shown in Fig. 5. Comparing the proposed controller to [START_REF] Jia | Integral backstepping sliding mode control for quadrotor helicopter under exter-nal uncertain disturbances[END_REF] in Fig. 5, it is clear that the proposed controller performs marginally better in terms of position response. The proposed control method performs better in terms of rejection disturbances. However, the tracking results of FOBSMC and IBSMC approaches are depicted in 12 to 17. As can be observed the used control methods as comparison are not effective disturbance compensation, and fail in accomplishing trajectory-tracking, especially in the first-period flight.

Therefore, using the proposed controller the tracking performance in the QUAS system is enhanced in the presence of disturbances compared to existing control techniques for the QUAS system.

Quantitative analysis of the controllers

The integral of the square value of the error (ISE)

t f
ti e 2 dt is used for quantitative comparison. The ISE is a numerical representation of tracking-error performance. The ISE performance of three controllers for scenarios 1 and 2 is shown respectively in Table 3 and4 3 and4. The control approach presented in this study has higher tracking control performance, which has been proven. It outperforms FOBSMC and IBSMC methods in terms of tracking accuracy, convergence rate, and robustness against time-varying/random disturbances.

On the other hand, for all control signals in three control strategies, the integral absolute derivative control signal (IADU) index [START_REF] Raffo | Automatica An integral predictive / nonlinear H ∞ control structure for a[END_REF] has been computed. This performance index is ideal for determining the smoothness of control signals. The HFTFSMC method creates smoother input control signals than the other strategy, as shown in Table 5, highlighting the suggested HFTFSMC structure's quality and viability. Table 5 shows the outcomes of the simulation. The proposed control method for all control signals improves smoothness, as can be demonstrated. Table 5 shows that, when compared to FOBSMC and IBSMC methods, the proposed control can guarantees the least amount of transmission data controller-to-motor ends, significantly reducing consumption energy then reducing unnecessary costs. Another key characteristic of the proposed hybrid control is the ability to achieve convergence in a finite time. In fact, this feature has a significant benefit over other algorithms since it forces the system states to zero at a finite time, the value of which can be pre-assigned by the user based on the tun requirements.

Simulations of the proposed controller and the controller designed [36] using the same conditions

To further evaluate the performance of the controller presented in [START_REF] Jia | Integral backstepping sliding mode control for quadrotor helicopter under exter-nal uncertain disturbances[END_REF] and the proposed control strategy, another simulation is presented in this part. In this simulation, we use the same test set including the initial conditions and the parameters of the quadrotor. Also, the same disturbances proposed in the paper [START_REF] Jia | Integral backstepping sliding mode control for quadrotor helicopter under exter-nal uncertain disturbances[END_REF] are used in the simulation. The quadrotor parameters are given as [START_REF] Jia | Integral backstepping sliding mode control for quadrotor helicopter under exter-nal uncertain disturbances[END_REF] :

m = 0.25kg, I = diag[2.
The results of this case are shown in Figs. 18 and19, it is demonstrated that the suggested finite-time controller has a greater tracking accuracy than the conventional including IBSMC while starting the maneuvering flight, i.e. the take-off and landing phase. The provided control approach has a higher tracking efficacy than others, as can be shown in the results. A four simulation collection has been carried out of the proposed controller and controller presented in [START_REF] Shi | Design of Fractional-Order Backstepping Sliding Mode Control for Quadrotor UAV[END_REF] with the same conditions like quadrotor parameters, condition initials, and disturbances. The quadrotor parameters refer to [START_REF] Shi | Design of Fractional-Order Backstepping Sliding Mode Control for Quadrotor UAV[END_REF] are : m = 0.53kg, I = diag[6.28, 6.28, 6.28]10 -3 kg.m 2 , g = 9.81m/s 2 , b = 3.1310 -5 N s 2 , c = 7.510 -7 N ms 2 and d = 0.232m. The random disturbances used in this simulation are plotted in Fig. 21, which have the same amplitude and white noise as given in [START_REF] Shi | Design of Fractional-Order Backstepping Sliding Mode Control for Quadrotor UAV[END_REF]. disturbances by a square trajectory. The simulation begins with the quadrotor hovering using the same conditions for both the proposed controller and the technique proposed in [START_REF] Shi | Design of Fractional-Order Backstepping Sliding Mode Control for Quadrotor UAV[END_REF]. The origin of the three-dimensional space above the ground is where the quadrotor is placed. As result, the proposed controller provided fast responses against random disturbances compared to the controller proposed in [START_REF] Shi | Design of Fractional-Order Backstepping Sliding Mode Control for Quadrotor UAV[END_REF].

Conclusions

In this paper, a trajectory-tracking control for the QUAS under external

Figure 1 :

 1 Figure 1: Geometry of a QUAS.

Figure 3 :

 3 Figure 3: The flowchart diagram of the proposed FOFTSMC controller.

Remark 5 .Remark 6 .

 56 The proposed control method is based on backstepping with FTSMC, wich show finite-time convergence of the QUAS state variables and robustness against parametric uncertainties and disturbances; FO control have the capability to choose the fractional operators; FOFTSMC provides the finite-time stability and compensates the effects of disturbances. The proposed attitude control is an improved version of FOSMC presented by the authors of [24]. In addition, the proposed control laws presented in (35), (45), and (46) have a flexible choice of FO parameters of the derivatives and integrators (α and q). The control law signals v x , v y , v z , M xx , M yy , and M zz are modified by the saturation function below

Remark 8 .

 8 The advantages of the proposed fractional-order finite-time sliding mode controllers: finite-time convergence of the system states, finite-time convergence in the reaching phase, reduction of chattering phenomenon, offers an extra FO differential element into the existing one insensitive to bounded external disturbances and parametric uncertainties.

[ 28 ]

 28 , m = 2kg, I = diag[16, 16, 32]10 -3 kg.m 2 , KP = diag[0.1, 0.1, 0.1]10 -1 N sm -1 , KΨ = diag[12, 12, 12]10 -3 N srad -1 , g = 9.8m/s 2 , b = 2.984210 -3 N s 2 , and c = 3.232010 -2 N ms 2 . The initial conditions [0.5, 0.1, 0]10 -1 m and [1, 1, 1]10 -2 rad

Figure 4 :

 4 Figure 4: The imitated drag coefficients.

Figure 5 :

 5 Figure 5: Position tracking performance comparison with[START_REF] Shi | Design of Fractional-Order Backstepping Sliding Mode Control for Quadrotor UAV[END_REF] and[START_REF] Jia | Integral backstepping sliding mode control for quadrotor helicopter under exter-nal uncertain disturbances[END_REF] control methods.

Figure 6 :Figure 7 :

 67 Figure 6: Attitude tracking performance comparison with[START_REF] Shi | Design of Fractional-Order Backstepping Sliding Mode Control for Quadrotor UAV[END_REF] and[START_REF] Jia | Integral backstepping sliding mode control for quadrotor helicopter under exter-nal uncertain disturbances[END_REF] control methods.

Figure 8

 8 Figure 8 depicts the time-history of the tracking errors. The simulation demonstrates the uniform tracking performances in terms of steady states, rejection of disturbances, fast responses and speed. Figure 7 shows the responses

Figure 8 :

 8 Figure 8: Tracking performance errors comparison with[START_REF] Shi | Design of Fractional-Order Backstepping Sliding Mode Control for Quadrotor UAV[END_REF] and[START_REF] Jia | Integral backstepping sliding mode control for quadrotor helicopter under exter-nal uncertain disturbances[END_REF] control methods.

Figure 9 :Figure 10 :

 910 Figure 9: 3D trajectory tracking performance comparison with[START_REF] Shi | Design of Fractional-Order Backstepping Sliding Mode Control for Quadrotor UAV[END_REF] and[START_REF] Jia | Integral backstepping sliding mode control for quadrotor helicopter under exter-nal uncertain disturbances[END_REF] control methods.

Figure 11 :

 11 Figure 11: Gaussian random disturbances.

Figure 12 :Figure 13 :

 1213 Figure 12: Position tracking performance comparison with[START_REF] Shi | Design of Fractional-Order Backstepping Sliding Mode Control for Quadrotor UAV[END_REF] and[START_REF] Jia | Integral backstepping sliding mode control for quadrotor helicopter under exter-nal uncertain disturbances[END_REF] control methods.

FFigure 14 :

 14 Figure 14: Control inputs.

Figure 14 Figure 15 :

 1415 Figure 14 depicts the applied control inputs. As can be observed from these results, the inputs are smooth and converge to their values.

Figure 16 :Figure 17 :

 1617 Figure16: 3D trajectory tracking performance comparison with[START_REF] Shi | Design of Fractional-Order Backstepping Sliding Mode Control for Quadrotor UAV[END_REF] and[START_REF] Jia | Integral backstepping sliding mode control for quadrotor helicopter under exter-nal uncertain disturbances[END_REF] control methods.

  . In comparison to FOBSMC and IBSMC, the proposed control scheme shows that the ISE indices are less important. The suggested control gives lower ISE error values in terms of both position and orientation of the quadrotor, as shown by the quantitative analysis in Tables

Figure 18 :

 18 Figure 18: Position trajectory tracking performance comparison with [36] control method.

Figure 19 :

 19 Figure 19: 3D trajectory tracking performance comparison with [36] control method.

Figure 20 :

 20 Figure 20: Energy consumption of two controllers.

4. 7 .

 7 Simulations of the proposed controller and the controller designed[START_REF] Shi | Design of Fractional-Order Backstepping Sliding Mode Control for Quadrotor UAV[END_REF] using the same conditions

Figure 21 :

 21 Figure 21: Gaussian random disturbances.

Figures 22 and 23

 23 Figures 22 and 23 show the tracking results for rectangular trajectory tracking under Gaussian random disturbances. The simulation results show that even in the presence of wind fields, the system can track the reference trajectory. The system can fast converge and stay stable, as seen in Figures 22 and 23. The simulation's aim is to move the quadrotor quickly and with random

Figure 22 :Figure 23 :

 2223 Figure 22: Position trajectory tracking performance comparison with [22] control method.

  disturbances and uncertainties is presented based on the hybrid fractional-order finite-time control. The control law of the position subsystem is yielded from FO backstepping fast terminal SMC algorithm designed in this study. To succeed in path following, the orientation control induces a flexible controller FOFTSMC deployed in this research that reject the unknown perturbation/uncertainty, inducing the finite-time stability. The proposed sliding manifold for the attitude subsystem reduces the effect of the chattering during the control phase. The concept of the suggested controllers is based on the Lyapunov stability theory which verifies and ensures the stability of the system under external disturbances. The advantages of the proposed HFOFTC are highlighted by a comparison study using simulation results. The proposed FOHFTC law has a higher performance in suppressing disturbances and gaining a lower stabilization time than the other control laws, as shown by the performance indices in the tables 3, 4, and 5.

Table 1 :

 1 Quadrotor Modelling.

  )

	Assumption 1. The disturbances D Ω (t) and D P (t) represent the dis-
	turbances and uncertainties parameters of quadrotor, such as aero-
	dynamic coefficients, mass, moments of inertia and the external fac-
	tors. There exists γ > 0 such that D i (t) is satisfied with |D i | ≤
	γ|D q tanh(kS i (t))|. According to Lemma 1, one get |D

i | ≤ γ with γ is

Table 2 :

 2 Control system parameters.

	Position parameters loop	Values
	c j , h 1 , h 2	4, 1, 6
	µ j , q j , k	0.9, 0.7, 10
	Attitude parameters loop	Values
	σ, α, β,	0.6, 0.98, 0.1224
	q i , ε, γ i	0.7, 35.6089, 39.3958
	µ i , K i1 , K i2	0.9, 5, 12
	(B) For inner loop (attitude controller)

Table 3 :

 3 Indices de performance ISE of the scenario 1

	Variable Proposed method FOBSMC IBSMC
	x(t)	0.0474	0.7673	0.2437
	y(t)	0.0355	0.2687	0.1474
	z(t)	0.02	0.1718	0.0491
	ψ(t)	0.0028	0.0031	34.85

Table 4 :

 4 Indices de performance ISE of the scenario 2

	Variable Proposed method FOBSMC IBSMC
	x(t)	0.0534	0.1068	0.0783
	y(t)	0.0004	0.00066	0.0007
	z(t)	7.5 e-5	0.0003	0.0006
	ψ(t)	0.0028	0.0029	0.0041
	In comparison to the results of the other approaches, the ISE values for the
	tracking errors are lower. All of these findings show that the proposed control
	method achieves better tracking performance, including high precision tracking,
	quick response, smooth control commands, and high robustness.

Table 5 :

 5 IADU index performance analysis of the scenario 2

	Control signals Proposed method FOBSMC IBSMC
	Total torques	0.0257	0.0772	0.0438
	Total thrust	19.9	20.9	20.16

  35, 2.35, 52.6]10 -3 kg.m 2 , g = 9.81m/s 2 , b = 6.1310 -5 N s 2 , c = 2.510 -6 N ms 2 and d = 0.2m. The perturbations are selected to refer[START_REF] Jia | Integral backstepping sliding mode control for quadrotor helicopter under exter-nal uncertain disturbances[END_REF] :

	D x,y,z (t) =	    	0.25 + 0.01 sin(t) m/s 2 t ∈ [50, 60] 0 m/s 2
		   	0.01 + 0.005 sin(t) m/s 2 t ∈ [90, 120]
	D φ,θ,ψ (t) =	    	0.25 + 0.01 sin(t) rad.s -2 t ∈ [50, 60] 0 rad.s -2
		   	0.01 + 0.005 sin(t) rad.s -2 t ∈ [90, 120]

Acknowledgement

The first author would like to thank Editor, Associate Editor, and the anonymous reviewers and Dr. Michael Defoort for their much time and constructive comments to review and improve this paper.