Adaptive Barrier Fast Terminal Sliding Mode Actuator Fault Tolerant Control Approach for Quadrotor UAVs

Abstract

This paper proposes an adaptive barrier fast terminal sliding mode control (ABFTSMC) approach for quadrotor unmanned aerial vehicles (UAV). Its main objectives are to mitigate the external disturbances, parametric uncertainties, and actuator faults. An adaptive barrier function is considered in the design to ensure the finite-time convergence of the output variables to a predefined locality of zero, independent of the disturbance bounds. A fast terminal sliding mode control (FTSMC) approach is designed to speed up the convergence rate in both reaching and sliding phases. The design considers hyperbolic tangent functions in the adaptive control law to drastically reduce the chattering effect, typically associated with the standard SMC. The performance of the proposed approach was assessed using a quadrotor UAV subject to external disturbances and sudden actuator faults. The obtained results show that the trajectory and the sliding surface converge to the origin in a finite time, without being affected by the high disturbance and actuator faults. In this method, due to the substitution of the discontinuous function by the hyperbolic tangent function, the chattering effect has also been highly reduced.

1. Introduction

Unmanned aerial vehicles (UAVs) are progressively being deployed in the various army and civilian applications, such as topography, fire monitoring, aerial imaging, surveillance, pipeline inspection, reconnaissance, the defeat of enemy air shield, etc. [1,2,3,4,5,6,7]. They owe this popularity to their autonomy, lower cost, simple configuration, ease of control, small size, and ability to navigate hazardous terrains and hard-to-reach locations, to list a few [8,9]. However, from a control point of view, their small size and underactuated design make them vulnerable to faults and exogenous disturbances such as wind gusts [10]. Faults are defined as deviations in the system’s structure or parameters from the nominal situation. Fault-tolerant controls (FTCs) are control systems that can maintain satisfactory operation under faulty conditions and prevent faults from developing into failures that might jeopardize the system’s safety [11]. Actuators are said to be faulty when they behave abnormally as a result of loss of effectiveness or actuator bias [12,13]. FTC techniques are commonly organized into active and passive strategies. Active FTCs require a fault detection and isolation (FDI) unit to detect and evaluate the faults explicitly, whereas passive FTCs depend exclusively on the use of robust control schemes to ensure system insensitivity to faults without such forthright detection [14]. PFTC systems are designed to be robust against the assumed faults. This method does not need to diagnose, isolate, or know the fault type. Compared to PFTCs, AFTCs require a lot of computing authority. Between fault detection, fault isolation, and controller reconfiguration, there is also a time delay, which is one of the weaknesses of this method [15]. These disadvantages are the main motivation for investigating a robust PFTC in this paper. Various control approaches have been investigated for the design of PFTCs, such as linear matrix inequality (LMI) schemes [16], H_∞ control [17], adaptive sliding mode control (ASMC) [18,19,20,21], sliding mode control (SMC) [8,9], fuzzy logic control [22,23], neural networks (NN) [24,25], model predictive control (MPC) [26,27], etc. Among the above-listed approaches, SMC has drawn much attention due to its robustness, ability to eliminate parametric uncertainties and external disturbances, and effectively mitigate faults [4,28,29,30,31,32,33,34,35,36,37,38]. SMC’s main ideas can be described as follows: First, a sliding surface is defined to fulfill the desired motion properties of the closed-loop system. Second, a control law is designed to force the states to reach the sliding surface and remain on it thereafter [39]. In order to eliminate the effects of parametric uncertainties, external disturbances, and input saturations in the quadrotor system, [40] proposed practical finite-time adaptive robust flight control and [41] used the nonsingular finite-time adaptive robust saturated command filtered control approach.

An ASMC approach was proposed in [42] to mitigate the effects of actuator faults in the framework of FTC. However, like the standard SMC, this approach suffers from limitations such as finite-time convergence and the chattering phenomenon. An integral terminal sliding mode control (ITSMC) was proposed in [43] to ensure the convergence of the states to the closest vicinity of zero in the finite time. An adaptive fuzzy state-observer approach was considered in [43] to approximate the limitless states to troubleshoot actuator faults, external disturbances, and actuators’ saturation bound. In [44], to eliminate the impacts of external disturbances and actuator faults in a quadrotor system, the fault-tolerant controller based on the internal type-2 fuzzy sliding mode control approach was used. In [45], an adaptive type-2 fuzzy backstepping fault tolerant controller was proposed to remove the effects of the parameter uncertainties, external disturbances, and actuator faults. In [46], an LMI-based adaptive barrier global sliding mode control approach was used to reduce the parametric uncertainties, external disturbances, and actuator faults. In [47], two proportional-derivative (PD) controllers and an adaptive fuzzy TSMC was proposed to stabilize a quadrotor. In order to determine the preferred attitude of the quadrotor, PD controllers were used, and TSMC was used to adjust the rotors’ rotation speed. In [48], a nonlinear ASMC with two separate controllers was utilized to eliminate the effects of wing damage. In the event of simultaneous actuator faults, the stability of the system is reduced. The second controller employs an adaptive control method to guarantee the stability of the closed-loop system and eliminate the drawbacks of the controller. In [49], it was shown that the model predictive control is a suitable fault-tolerant control approach to mitigate actuator faults in quadrotor UAV systems. Fault tolerance control has been achieved with the help of moving horizon estimation (MHE) and an unscented Kalman filter (UKF) and their integration with MPC. In [50], an SMC-based FTC approach was proposed for a quadrotor subject to the butterfly damage and actuator fault situation. In [51], in order to reject the impressions of external disturbances and actuator additive faults, a nonsingular fast terminal sliding mode fault-tolerant control scheme based on the disturbance observer (DO) for a quadrotor UAV was used. Both passive and active FTC designs have been considered to mitigate the actuator faults. Ref. [52] proposed a nonsingular TSMC to eliminate the parametric uncertainties, external disturbances, and unexpected actuator faults. The mentioned approach has achieved the finite-time convergence in both reaching and sliding phases. In [53], in order to remove the effects of the multiple actuator faults and multisource disturbances, a finite-time controller based on the composite barrier Lyapunov function was employed.

In [2,54], a nonlinear robust FTC method aiming to mitigate wind disturbances and actuator faults in the quadrotor system was investigated. A continuous FTSMC was suggested for accurate tracking and limiting the convergence time based on the estimated information. In [1], an adaptive sliding mode controller was mentioned to eliminate the actuator faults, parametric uncertainties, and time delays. In this approach, to improve the robustness of the control system, the integral term was added. With the help of this method, the adaptive law can estimate and eliminate the actuator fault without the need of fault detection and isolation. This method guarantees the asymptotic stability of the sliding dynamics. In [3], in order to reject the existence disturbance effects of the six-degrees-of-freedom (6DoF) quadrotor system, an adaptive barrier terminal sliding mode controller (ABTSMC) was used. In order to improve the methods of [1,2,3], the researchers could ensure the convergence of the output variable by adding a comparative barrier function. The main idea of choosing the barrier function is to use the cost functions that prevent undesirable areas. Loop insensitivity is guaranteed by SMC depending on the faults and disturbances in the system and finite-time convergence. However, the implementation of this controller requires the knowledge of upper bounds of perturbations.

This paper proposes an adaptive barrier fast terminal sliding mode control (ABFTSMC) approach for elimination of actuator faults, external disturbances, and parametric uncertainties in quadrotor unmanned aerial vehicles (UAV) in the finite time. The main contributions of this paper are organized as follows:

• A fast terminal sliding mode control (FTSMC) approach speeds up the convergence rate in both reaching and sliding phases, and fast finite-time robust performance is achieved.

• A design method that provides a barrier function without information about the upper bounds of perturbations and actuator faults makes the quadrotor track the desired trajectory.

• An SMC technique relies on the hyperbolic tangent function in order to achieve the chattering-free responses.

The remainder of this paper is organized as follows: Section 2 provides a brief description of the quadrotor’s dynamic model. Section 3 derives the proposed controller and discusses the stabilization analysis. The simulation results are presented in Section 4, and finally, conclusions are given in Section 5.

2. Quadrotor Model

The quadrotor UAV can be represented using the following second-order time-varying nonlinear system [52]:

(1)$\dot{X}=f\left(X,t\right)+\left(g\left(X,t\right)+\left(\zeta -g\left(X,t\right)\right)\kappa \right)u+D_x$

where $X\in {ℝ}^{12}$ denotes the vector of the states to be controlled, $f\left(X,t\right)\in {ℝ}^{12}$ is a nonlinear system function, and $\zeta =g\left(X,t\right)L\in {ℝ}^{12\times 4}$ signifies a non-singular post-fault dynamic. $L=diag\left(\left[l_1,\cdots ,l_4\right]\right)$ displays the loss of actuator effectiveness coefficients, with the scalars $l_j{}_{\left(j=1,\cdots ,4\right)}$ satisfying $0\le l_j\le 1$, where $l_j=1$, indicating a faulty $j\mathrm{th}$ actuator; $0<l_j<1$, indicating a certain level of loss of actuator effectiveness; and $l_j=0$, denoting the complete failure of the $j\mathrm{th}$ actuator. $\kappa \in R$ is designed as a step function to model the abrupt fault during fly. To be specific, $t<t_f$, $\kappa =0$ shows the fault-free case, and $t>t_f$, $\kappa =1$ symbolizes the post-fault condition. The terms $X,f\left(X,t\right)$ and $g\left(X,t\right)$ are assumed as follows:

(2)$$\begin{gathered} X={[x,y,z,\phi ,\theta ,\psi ,\dot{x,}\dot{y,}\dot{z},\dot{\phi },\dot{\theta },\dot{\psi }]}^T \\ f\left(X,t\right)=\left[\begin{array}{c}x\\ y\\ z\\ \phi \\ \theta \\ \psi \\ 0\\ 0\\ -g\\ \frac{I_y-I_z}{I_x}\dot{\theta }\dot{\psi }+\frac{J}{I_x}\dot{\theta }\Omega \\ \frac{I_z-I_x}{I_y}\dot{\phi }\dot{\psi }-\frac{J}{I_y}\dot{\phi }\Omega \\ \frac{I_x-I_y}{I_z}\dot{\phi }\dot{\theta }\end{array}\right] , g\left(X,t\right)=\left[\begin{array}{cccc}0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ {\Lambda }_x& 0& 0& 0\\ {\Lambda }_y& 0& 0& 0\\ {\Lambda }_z& 0& 0& 0\\ 0& \frac{1}{I_x}& 0& 0\\ 0& 0& \frac{1}{I_y}& 0\\ 0& 0& 0& \frac{1}{I_z}\end{array}\right] \end{gathered}$$

where

${\Lambda }_x=\frac{1}{m}\mathit{cos}(\phi )\mathit{sin}(\theta )\mathit{cos}(\psi )+\mathit{sin}(\phi )\mathit{sin}(\psi )$

${\Lambda }_y=\frac{1}{m}\mathit{cos}(\phi )\mathit{sin}(\theta )\mathit{sin}(\psi )-\mathit{sin}(\phi )\mathit{cos}(\psi )$

${\Lambda }_z=-\frac{1}{m}\mathit{cos}(\phi )\mathit{cos}(\theta )$

The angular velocities of the rotors are obtained by combining the control inputs and are written as follows:

(3)$\left\{\begin{array}{c}{\Omega }_1^2=\frac{1}{4b}{u}_z-\frac{1}{2b\pounds }{u}_{\phi }-\frac{1}{4p}{u}_{\psi }\\ {\Omega }_2^2=\frac{1}{4b}{u}_z-\frac{1}{2b\pounds }{u}_{\phi }+\frac{1}{4p}{u}_{\psi }\\ {\Omega }_3^2=\frac{1}{4b}{u}_z+\frac{1}{2b\pounds }{u}_{\theta }-\frac{1}{4p}{u}_{\psi }\\ {\Omega }_1^2=\frac{1}{4b}{u}_z+\frac{1}{2b\pounds }{u}_{\theta }+\frac{1}{4p}{u}_{\psi }\end{array}\right.$

The inputs of the system are also obtained by combining the angular velocity of the rotors and are expressed as follows:

(4)$u_i=\left[\begin{array}{c}b\left({\Omega }_1^2+{\Omega }_2^2+{\Omega }_3^2+{\Omega }_4^2\right)\\ b\pounds \left({\Omega }_2^2-{\Omega }_4^2\right)\\ b\pounds \left({\Omega }_1^2-{\Omega }_3^2\right)\\ p\left({\Omega }_2^2-{\Omega }_1^2+{\Omega }_4^2-{\Omega }_3^2\right)\end{array}\right]$

where $\left[\begin{array}{cccc}{u}_z& u_{\phi }& u_{\theta }& u_{\psi }\end{array}\right]$ denote the rotor inputs after applying the controller; $I_x$, $I_y$, and $I_z$ denote the moments of inertia; $b$ and $p$ are the thrust and drag coefficients, respectively, $\pounds$ denotes the distance between the quadrotor center and rotor center; $m$ denotes the quadrotor mass; and $J$ represents the rotor inertia. The terms ${\Omega }_{\mathrm{i}\left(\mathrm{i}=1,\dots ,4\right)}^2$ denote the square angular velocities. $x$, $y$, and $z$ are the quadrotor positions and $\phi$, $\theta$, and $\psi$ signify the quadrotor attitudes. $\dot{x}$, $\dot{y}$, and $\dot{z}$ represent the position derivation of the quadrotor, and $\dot{\phi }$, $\dot{\theta }$, and $\dot{\psi }$ represent the angular velocities. Ω is a disturbance term that is presented in the system depending on the rotor speeds. It represents the overall residual propeller speed by [55]:

$\Omega =-{\Omega }_1+{\Omega }_2-{\Omega }_3+{\Omega }_4$

Considering the parametric uncertainties and external disturbances, the vector $D_x$ is defined as follows:

(5)$D_x=\Delta f+\Delta gu+d$

where $\Delta f\in {ℝ}^{12}$ and $\Delta g\in {ℝ}^{12}$ denote parametric uncertainties, and $d\in {ℝ}^{12}$ represents external disturbances. The main control objectives are to force the state variables to track the desired trajectory in the presence of faults and disturbances.

3. Adaptive Barrier Fast Terminal Sliding Mode Control

Figure 1 shows the control technique of the quadrotor. The thrust input ($u_z$) controls the altitude of the quadrotor. The roll ($\phi$), pitch ($\theta$), and yaw ($\psi$) angles are controlled via the roll, pitch, and yaw controller, respectively. Their desired values are (${\phi }_d$), (${\theta }_d$), and (${\psi }_d$), respectively. The actual positions of $x$, $y$, and $z$, are acquired via a Global Positioning System (GPS) unit and transformed from the altitude signals. The actual $\phi$, $\theta$, and $\psi$. angles are measured from the inertial measurement corps. The rotation motion of the quadrotor is used to control the control inputs $u_{\phi }, u_{\theta }$, and $u_{\psi }$.

The following assumptions were considered to design the ABFTSMC technique:

Assumption 1.

The main control objective is to eliminate the actuator faults, external disturbances, and parametric uncertainties. For this purpose, to apply the effect of the actuator faults, we consider $\kappa =1$.

Assumption 2.

The parametric uncertainty and external disturbance term $D_x$ is bounded and is less than ${\eta }_k$ , i.e.,:

(6)$\left|\right|D_x\left|\right|<{\eta }_k$

Assumption 3.

For non-zero and positive values of ${\sigma }_1$and ${\sigma }_2$, the following relation is established:

(7)${\sigma }_1\le 1\le {\sigma }_2$

Lemma 1.

For every given scalar $x$and positive scalar $y$the following inequality holds [56]:

(8)$xtanh\left(yx\right)=\left|xtanh\left(yx\right)\right|=\left|x\right|\left|tanh\left(yx\right)\right|\ge 0$

Proof. 

From the mathematical definition of the tanh(.) function, we have

(9)$xtanh\left(yx\right)=x\frac{e^{yx}-e^{-yx}}{e^{yx}+e^{-yx}}$

Multiplying the above equation by $\frac{e^{yx}}{e^{y{x}^{\prime }}}$, one can obtain

(10)$xtanh\left(yx\right)=\left(\frac{1}{e^{2yx}+1}\right)x\left(e^{2yx}-1\right)$

According to the conditions $\left\{\begin{array}{c}{e}^{2yx}-1\ge 0 if x\ge 0\\ e^{2yx}-1<0 if x<0\end{array}\right.$, one can obtain

(11)$x\left(e^{2yx}-1\right)\ge 0$

Based on $\left(\frac{1}{e^{2yx}+1}\right)>0$, and from Equation (11), we have

(12)$xtanh\left(yx\right)=\left(\frac{1}{e^{2yx}+1}\right)x\left(e^{2yx}-1\right)\ge 0$

Therefore, from the fact that for every scalar z and v, if $zv\ge 0$ then $zv=\left|zv\right|=\left|z\right|\left|v\right|\ge 0$ holds, one can conclude that

(13)$xtanh\left(yx\right)=\left|xtanh\left(yx\right)\right|=\left|x\right|\left|tanh\left(yx\right)\right|\ge 0$

This completes the proof. □

Lemma 2.

Suppose the positive definite continuous function $v\left(t\right)$achieves the following inequality:

(14)$\dot{v}\left(t\right)\le -\alpha v\left(t\right)-\beta v^{\eta }\left(t\right) \forall t\ge t_0, v\left(t_0\right)\ge 0$

where $\eta$ is the fraction of two odd positive numbers with $0<\eta <1$, and the $\alpha$ and $\beta$ are positive parameter coefficients. Then the $t_r$ is a finite-time convergence obtained by Lyapunov, such as:

(15) $t_r=t_0+\frac{1}{\alpha \left(1-\eta \right)}ln\frac{\alpha v^{1-\eta }\left(t_0\right)+\beta }{\beta }$

The desired quadrotor trajectory is represented by $X_d=\left[x_d,y_d,z_d,{\phi }_d,{\theta }_d,{\psi }_d,{\dot{x}}_d,{\dot{y}}_d,{\dot{z}}_d,{\dot{\phi }}_d,{\dot{\theta }}_d,{\dot{\psi }}_d\right]$; moreover, the tracking error can be represented as:

(16)$e=X-X_d$

The sliding surface of the FTSMC can be described as follows [2,57,58]:

(17)$\delta =e+b_1{\left|e\right|}^{\alpha }sign\left(e\right)+b_2{\left|\dot{e}\right|}^{\beta }sign\left(\dot{e}\right)$

The time differentiation of Equation (17) is as follows:

(18)$\dot{\delta }=\dot{e}+\alpha b_1{\left|e\right|}^{\alpha -1}\dot{e}+\beta b_2{\left|\dot{e}\right|}^{\beta -1}\left(f\left(X,t\right)+g\left(X,t\right)Lu+D_x-{\dot{X}}_d\right)$

by placing $\dot{\delta }=0$, the equivalent input control rule is given by:

(19)$u_{eq}=-{(g\left(X,t\right)L)}^{†}\left(f\left(X,t\right)+D_x-{\ddot{X}}_d+\frac{1}{\beta b_2}{\left|e\right|}^{\beta -1}\left(1+\alpha b_1{\left|e\right|}^{\alpha -1}\right)\dot{e}\right)$

where the term $g\left(x,t\right)L$ is a non-zero (non-singular) expression, and ${(g\left(x,t\right)L)}^{†}$ is the pseudo-inverse of $g\left(x,t\right)L$, i.e.,: ${(g\left(x,t\right)L)}^{†}$=${\left[L^{\mathrm{T}}g{\left(x,t\right)}^{\mathrm{T}}g\left(x,t\right)L\right]}^{-1}{L}^{\mathrm{T}}g{\left(x,t\right)}^{\mathrm{T}}$.

Remark 1.

In order to find the coefficients$\alpha$,$\beta$,$b_1$, and $b_2$, it is necessary to consider the following conditions with respect to the time-derivative of the fast terminal sliding surface [59,60]:

• The multiplication of$\alpha$and$b_1$must be a positive value.

• $\beta$multiplied by$b_2$must be a positive value.

• The terms$\alpha$−1 and$\beta$−1 should also be positive values.

An adaptive controller based on the barrier function is presented. The external disturbances can be approximated more effectively by expending an adaptive barrier SMC, and the closed-loop system can evolve more steadily. In order to employ the barrier function approach, the switching control law can be formed as:

(20)$u_{sw}=-{(g\left(X,t\right)L)}^{†}\left(\widehat{Q}\left(t\right)+\mu \right)sign\left(\delta \right)$

with

(21)$\widehat{Q}\left(t\right)=\left\{\begin{array}{c}{Q}_a if 0<t<\overline{t}\hfill \\ Q_{psb} if t>\overline{t}\hfill \end{array}\right.$

where $\overline{t}$ is the time that the state trajectories consolidate to the environs of the fast terminal sliding mode consistency $\delta$, and the term $g\left(X,t\right)L$ is a non-zero (non-singular) expression.

The adaptive-tuning rule and the positive-semi-definite barrier function are provided by [61,62]:

(22)$\left\{\begin{array}{c}{\dot{Q}}_a=\rho \left|\right|\delta \left|\right|\hfill \\ Q_{psb}=\frac{\left|\right|\delta \left|\right|}{\epsilon -\left|\right|\delta \left|\right|}\hfill \end{array}\right.$

where $\epsilon , \rho >0$.

Using adaptation law (21), the control gain $\widehat{Q}$ is adjusted to advance until the state trajectory reaches the neighborhood $\epsilon$ of the fast terminal sliding surface $\delta$ at the time $\overline{t}$. Then, for the times after $\overline{t}$, the adaptive control gain $\widehat{Q}$ switches to the positive-semi-definite barrier function, diminishing the convergence region and preserving the system states there. The system’s stability is confirmed in two situations as follows:

(23)$\left\{\begin{array}{c}\left(a\right) 0<t<\overline{t}\\ \left(b\right) t>\overline{t}\end{array}\right.$

Remark 2.

$\overline{t}$is the time that the state trajectories take to reach the vicinity of the sliding surface $\delta$. The boundary of the sliding surface is equal to the value $\epsilon$. The value of $\overline{t}$is equal to the amount of time that the system spends to reach the boundary $\epsilon$. The time $\overline{t}$cannot be entered manually or by test because it causes a chattering problem. The value of$\epsilon$is defined in the system and a control algorithm is designed that can calculate the value of time $\overline{t}$.

The final sliding mode control input is:

(24)$u=u_{eq}+u_{sw}$

Theorem 1.

Consider the disturbed nonlinear system (1). By combining the adaptive control law (20) with the equivalent controller (19) and the discontinuous controller (21) assuming $\widehat{Q}=Q_a$, then in a finite-time, the tracking trajectories of system states converge to the vicinity of the fast terminal sliding surface.

Proof. 

Consider the following Lyapunov function:

(25)$v_1=\frac{1}{2}({\delta }^T\delta +\rho {\vartheta }^{-1}\left(Q_a-Q{)}^2\right).$

where $\vartheta >0$, and $Q$ is a positive unknown constant; ${\delta }^T$ is transpose vector of $\delta$. The time-differentiation of $v_1$ is

(26)${\dot{v}}_1={\delta }^T\dot{\delta }+\rho {\vartheta }^{-1}\left(Q_a-Q\right){\dot{Q}}_a$

where substituting Equation (18) into the above equation, we obtain:

(27)$$\begin{gathered} {\dot{v}}_1={\delta }^T(\dot{e}+\alpha b_1{\left|e\right|}^{\alpha -1}\dot{e}+\beta b_2{\left|\dot{e}\right|}^{\beta -1}(f\left(X,t\right) \\ +g\left(X,t\right)Lu+D_x-{\ddot{X}}_d))+\rho {\vartheta }^{-1}\left(Q_a-Q\right)||\delta || \end{gathered}$$

Substituting the adaptive control law (24) in the above equation, one has:

(28)$$\begin{gathered} {\dot{v}}_1=-{\delta }^T\left(\left(Q_a+\mu \right)sign\left(\delta \right)-D_x\right)+\rho {\vartheta }^{-1}\left(Q_a-Q\right)||\delta || \\ \le -\mu ||\delta ||+||\delta ||||D_x||-{\delta }^T{Q}_{a}sign\left(\delta \right)+\rho {\vartheta }^{-1}\left(Q_a-Q\right)||\delta || \\ \le ||\delta ||||D_x||-Q_a||\delta ||+\rho {\vartheta }^{-1}\left(Q_a-Q\right)||\delta ||-\mu ||\delta ||\le ||\delta ||||D_x|| \\ -Q_a||\delta ||+\rho {\vartheta }^{-1}\left(Q_a-Q\right)||\delta ||+Q||\delta ||-Q||\delta || \\ \le -\left(Q-||D_x||\right)||\delta ||-\left(1-\rho {\vartheta }^{-1}\left(Q_a-Q\right)\right)||\delta || \end{gathered}$$

where $||D_x||<{\eta }_k$. Because $Q-||D_x||>0$ and $\rho {\vartheta }^{-1}<1$, Equation (28) is written as

(29)$$\begin{gathered} {\dot{v}}_1\le -\sqrt{2}\left(Q-D_x\right)\frac{||\delta ||}{\sqrt{2}}-\sqrt{2}\vartheta \left(1-\rho {\vartheta }^{-1}\right)\frac{\left(Q_a-Q\right)}{\sqrt{2}\vartheta } \\ \le -\mathit{min}\left\{\sqrt{2}\left(Q-||D_x||\right),\sqrt{2}\vartheta \left(1-\rho {\vartheta }^{-1}\right)||\delta ||\right\}\left(\frac{||\delta ||}{\sqrt{2}}+\frac{\Vert Q_a-Q\Vert }{\sqrt{2}\vartheta }\right) \\ \le -\chi v_1^{\frac{1}{2}} \end{gathered}$$

□

Theorem 2.

For disturbed nonlinear system (1), using the adaptive control law (20) with the comparable controller (19) and the intermittent controller (21) assuming $\widehat{Q}=Q_{psb}$(Equation (24)), i.e.,:

(30)$$\begin{gathered} u=-{(g\left(X,t\right)L)}^{†}\left(f\left(X,t\right)+D_x-{\ddot{X}}_d+\frac{1}{\beta b_2}{\left|e\right|}^{1-\beta }\left(1+\alpha b_1{\left|e\right|}^{\alpha -1}\right)\dot{e}\right) \\ -{(g\left(X,t\right)L)}^{†}\left(Q_{psb}+\mu \right)sign\left(\delta \right) \end{gathered}$$

Then, the system’s states arrive at the convergence region $||\delta ||<\epsilon$ in finite-time.

Proof. 

Consider the Lyapunov function as:

(31)$v_2=\frac{1}{2}({\delta }^T\delta +\left(Q_{psb}-Q_{psb}\left(0\right){)}^2\right)$

Now, differentiating the Lyapunov Function (31), concerning time, we have:

(32)${\dot{v}}_2={\delta }^T\dot{\delta }+\left(Q_{psb}-Q_{psb}\left(0\right)\right){\dot{Q}}_{psb}$

where substituting $\dot{\delta }$, and $Q_{psb}\left(0\right)=0$ into the above equation, we obtain:

(33)$$\begin{gathered} {\dot{v}}_2={\delta }^T({(g\left(X,t\right)L)}^{-1}(\dot{e}+\alpha b_1{\left|e\right|}^{\alpha -1}\dot{e}+\beta b_2{\left|\dot{e}\right|}^{\beta -1}\times \\ \left(f\left(X,t\right)+g\left(x,t\right)Lu+D_x-{\ddot{X}}_d\right)+Q_{psb}{\dot{Q}}_{psb} \end{gathered}$$

Substituting the control law (30) into (33) gives:

(34)$$\begin{gathered} {\dot{v}}_2=-{\delta }^T\left(\left(Q_{psb}+\mu \right)sign\left(\delta \right)-D_x\right)+Q_{psb}{\dot{Q}}_{psb} \\ \le -\mu ||\delta ||+||\delta ||||D_x||-Q_{psb}||\delta ||+Q_{psb}{\dot{Q}}_{psb} \\ \le -\left(Q_{psb}-||D_x||\right)||\delta ||+Q_{psb}\left(\frac{\epsilon }{{(\epsilon -||\delta ||)}^2}\right)sign\left(\delta \right)\dot{\delta } \\ \le -\mu ||\delta ||-\left(Q_{psb}-||D_x||\right)||\delta ||-Q_{psb}\left(\frac{\epsilon }{{(\epsilon -||\delta ||)}^2}\right)sign\left(\delta \right)\times \\ \left(\left(Q_{psb}+\mu \right)sign\left(\delta \right)-D_x\right) \le -\left(Q_{psb}-||D_x||\right)||\delta ||- \\ \frac{\epsilon }{{(\epsilon -||\delta ||)}^2}\left(Q_{psb}-||D_x||\right)||\delta ||Q_{psb } \end{gathered}$$

where, since $Q_{psb}>||D_x||$ and $\frac{\epsilon }{{(\epsilon -||\delta ||)}^2}>0$, one finds:

(35)$$\begin{gathered} { \dot{v} }_2\le -\sqrt{2}\left(Q_{\mathit{psb}}-||D_x||\right)\frac{||\delta ||}{\sqrt{2}}-\frac{\sqrt{2}\epsilon }{{(\epsilon -||\delta ||)}^2}\left(Q_{\mathit{psb}}-||D_x||\right)\frac{{||\delta ||Q}_{\mathit{psb}}}{\sqrt{2}} \\ \le -\sqrt{2}\left(Q_{\mathit{psb}}-||D_x||\right)\mathit{min}\left\{1,\frac{\epsilon }{{(\epsilon -||D_x||)}^2}\right\}\left(\frac{||\delta ||}{\sqrt{2}}+\frac{{||\delta ||Q}_{\mathit{psb}}}{\sqrt{2}}\right) \end{gathered}$$

where $\Omega =\sqrt{2}\left(Q_{psb}-||D_x||\right)\mathit{min}\left\{1,\frac{\epsilon }{{(\epsilon -||D_x||)}^2}\right\}.$

The undesired response results from using the sign(.) function in the control law (20), which causes the chattering phenomenon in the system. To mitigate this problem, the discontinuous signum function is replaced by the continuous hyperbolic tangent function. If the steepness coefficients increase, the chattering phenomena has occurred drastically using the discontinuous sign(.) function. However, by estimating this function and replacing that method with the continuous tanh(.) function, the problem is significantly solved and the chattering phenomenon is reduced [63]. The chattering avoidance idea is to decrease the steepness of the function tanh(.). Therefore, via the continuous hyperbolic tangent function, the auxiliary control Function (20) can be defined as:

(36)$u_{sw}=-(g\left(X,t\right)L{)}^{†}\left(\widehat{Q}\left(t\right)+\mu \right)\mathit{tanh}(\frac{\delta }{\zeta })$

where $sign\left(\frac{\delta }{\zeta }\right)\approx \mathit{tanh}(\frac{\delta }{\zeta })$, and $\zeta$ is a boundary layer thickness ratio. □

Remark 3.

The hyperbolic functions are utilized in Equation (20), and the fast terminal sliding surface $\delta \left(t\right)$will not be equipollent to zero for all time. The adaptive parameters will increase to alleviate this drawback. An adaptive barrier rule is modified by [64]:

(37) $\widehat{Q}\left(t\right)=\left\{\begin{array}{c}0 if\left|\delta \right|<\zeta \hfill \\ \rho ||\delta || if\left|\delta \right|>\zeta 0<t<\overline{t}\hfill \\ \frac{||\delta ||}{\epsilon -||\delta ||} if\left|\delta \right|>\zeta t>\overline{t}\hfill \end{array}\right.$

According to Equation (21), the barrier function has two criteria, so the reason for replacing sign(.) with tanh(.) must be defined and proved separately for both $\widehat{Q}\left(t\right)$ criteria of the Lyapunov function.

Theorem 3.

Using the adaptive controller (24) with $\widehat{Q}\left(t\right)=Q_a\left(t\right)$, the state trajectories reach the neighborhood$\epsilon$of the sliding surface in the finite time. The hyperbolic tangent function is applied instead of sign(.) to reduce the chattering. This is also done by reducing the steepness of the hyperbolic tangent function.

Proof. 

Considering Lyapunov Function (25), the time-derivative of the Lyapunov function is found as

${\dot{v}}_1=-{\delta }^T\left(\left(Q_a+\mu \right)sign\left(\delta \right)-D_x\right)+\rho {\vartheta }^{-1}\left(Q_a-Q\right)||\delta ||$

Substituting the sign function with the hyperbolic tangent, the following equation is obtained:

(38)$$\begin{gathered} {\dot{v}}_1=-{\delta }^T\left(\left(Q_a+\mu \right)\mathit{tanh}(\frac{\delta }{\zeta }\right)-D_x)+\rho {\vartheta }^{-1}\left(Q_a-Q\right)||\delta || \\ \le ||\delta ||||D_x||-{\delta }^T(\left(Q_a+\mu \right)\mathit{tanh}(\frac{\delta }{\zeta }))+\rho {\vartheta }^{-1}\left(Q_a-Q\right)||\delta || \end{gathered}$$

Considering Lemma 1, the subsequent equation is obtained:

(39)$-\left(Q_a+\mu \right)\mathit{tanh}(\frac{\delta }{\zeta })\le 0$

Considering Assumption 3 and multiplying Equation (7) by $u^2$ yields:

(40)${\sigma }_1{u}^2\le u^2\le {\sigma }_2{u}^2$

From Equations (38) and (40), the following condition is obtained:

(41)${\sigma }_1\left[\frac{1}{{\sigma }_1^2}{(Q_a+\mu )}^2(\mathit{tanh}(\frac{\delta }{\zeta }){)}^2\right]\le u\left(t\right)\times u\left(t\right)=-\frac{1}{{\sigma }_1}\left(Q_a+\mu \right)\mathit{tanh}(\frac{\delta }{\zeta })u\left(t\right)$

By multiplying ${(\frac{\delta }{\zeta })}^2>0$ to both sides of the above equation, one has

(42)$\frac{\delta }{\zeta }u\left(t\right)\le -\left(Q_a+\mu \right)||\frac{\delta }{\zeta }||$

whereby combining Equations (38) and (42), we have:

(43)$$\begin{gathered} { \dot{v} }_1\le ||\delta ||||D_x||-{\delta }^T\left(Q_a+\mu \right)||\frac{\delta }{\zeta }||+\rho {\vartheta }^{-1}\left(Q_a-Q\right)||\delta ||+Q||\delta ||-Q||\delta || \\ \le -\mu ||\frac{\delta }{\zeta }||-\left(Q-||D_x||\right)||\delta ||-(\frac{1}{\zeta }-\rho {\vartheta }^{-1}\left(Q_a-Q\right))||\delta || \end{gathered}$$

with $||D_x||<{\eta }_k$. Because $Q-||D_x||>0$ and $\rho {\vartheta }^{-1}<1$, Equation (43) is written as

(44)$$\begin{gathered} { \dot{v} }_1\le -\sqrt{2}\left(Q-||D_x||\right)\frac{||\delta ||}{\sqrt{2}}\vartheta \left(\frac{1}{\zeta }-\rho {\vartheta }^{-1}\right)\frac{\left(Q_a-Q\right)}{\sqrt{2}\vartheta } \\ \le -\mathit{min}\left\{\sqrt{2}\left(Q-||D_x||\right),\sqrt{2}\vartheta \left(\frac{1}{\zeta }-\rho {\vartheta }^{-1}\right)||\delta ||\right\}\left(\frac{||\delta ||}{\sqrt{2}}+\frac{\Vert Q_a-Q\Vert }{\sqrt{2}\vartheta }\right) \\ \le -{\chi v}_1^{\frac{1}{2}} \end{gathered}$$

□

Theorem 4.

Using the adaptive controller (24) with $\widehat{Q}\left(t\right)=Q_{psb}$, the state trajectories reach the neighborhood$\epsilon$of the sliding surface in the finite time.

Proof. 

Considering the Lyapunov Function (31) and its time derivative and embedding the control law (30), the following equation is obtained:

${\dot{v}}_2=-{\delta }^T\left(\left(Q_{psb}+\mu \right)sign\left(\delta \right)-D_x\right)+Q_{psb}{\dot{Q}}_{psb}$

Substituting the sign function with tanh(.), the following equation is obtained:

(45)$$\begin{gathered} {\dot{v}}_2=-{\delta }^T(\left(Q_{psb}+\mu \right)tanh(\frac{\delta }{\zeta })-D_x)+Q_{psb}{\dot{Q}}_{psb} \\ \le -\mu ||\delta ||+||\delta ||||D_x||-Q_{psb}||\delta ||+Q_{psb}{\dot{Q}}_{psb} \\ \le -\left(Q_{psb}-||D_x||\right)||\delta ||+Q_{psb}(\frac{\epsilon }{{(\epsilon -||\delta ||)}^2})\mathit{tanh}(\frac{\delta }{\zeta })\dot{\delta } \\ \le -\mu ||\delta ||-\left(Q_{psb}-||D_x||\right)||\delta ||-Q_{psb}(\frac{\epsilon }{{(\epsilon -||\delta ||)}^2})\times \mathit{tanh}(\frac{\delta }{\zeta }) \\ \times (\left(Q_{psb}+\mu \right)\mathit{tanh}(\frac{\delta }{\zeta })-D_x) \end{gathered}$$

Considering Assumption 3 and multiplying $u^2$ to Equation (7), we have

(46)$$\begin{gathered} { \dot{v} }_2\le -\mu ||\delta ||-\left(Q_{\mathit{psb}}-||D_x||\right)||\delta ||-Q_{\mathit{psb}}(\frac{\epsilon }{{(\epsilon -||\delta ||)}^2})(\frac{\delta }{\zeta })(\left(Q_{\mathit{psb}}+\mu \right)(\frac{\delta }{\zeta })-D_x) \\ \le -\left(Q_{psb}-||D_x||\right)||\delta ||-\frac{\epsilon }{{\zeta }^2{(\epsilon -\delta )}^2}\left(Q_{psb}-||D_x||\right)||\delta ||Q_{psb } \end{gathered}$$

where, since $Q_{psb}>||D_x||$, and $\frac{\epsilon }{{(\epsilon -||\delta ||)}^2}>0$, one finds

(47)$$\begin{gathered} {\dot{v}}_2\le -\sqrt{2}\left(Q_{psb}-||D_x||\right)\frac{||\delta ||}{\sqrt{2}}-\frac{\sqrt{2}\epsilon }{{\zeta }^2{(\epsilon -||\delta ||)}^2}\left(Q_{psb}-||D_x||\right)\frac{||\delta ||Q_{psb}}{\sqrt{2}} \\ \le -\sqrt{2}\left(Q_{psb}-||D_x||\right)\mathit{min}\left\{1,\frac{\epsilon }{{\zeta }^2{(\epsilon -||D_x||)}^2}\right\}\left(\frac{||\delta ||}{\sqrt{2}}+\frac{||\delta ||Q_{psb}}{\sqrt{2}}\right)\le -\Omega v_2{}^{\frac{1}{2}} \end{gathered}$$

□

4. Simulation Results

Before starting the scenarios and showing the simulation results, the general schematic of the system and the location of the fault and disturbance terms as well as the way that the controller works in the quadrotor system should be described. It should be stated that the quadrotor consists of several parts:

• Input controller;

• Quadrotor actuators;

• General quadrotor system;

• System output.

In the part of the actuators, as shown in Figure 1, the actuator fault reduced the effectiveness of the rotors and the output from the actuators was entered into the quadrotor system as the input of the system, and as a result, the output of the system was obtained. It should be noted that the perturbation was applied to the output part of the system according to Equation (1). By comparing the system output and the desired path and getting its error, the error signal was entered as the input of the sliding surface to the controller. With the help of the proposed control method (ABFTSMC), the resulting error was reduced and the effects of actuator fault, output disturbance, and parameter uncertainties were eliminated.

The performance of the adaptive barrier fast terminal sliding mode control (ABFTSMC) is assessed in this section. For performance analysis, in addition to the perturbations that enter the system, two types of actuator faults were considered. For comparison purposes, the performance of the proposed approach is compared with three methods: The first method is an adaptive sliding mode controller in [1] and the second method is the fast terminal sliding mode controller in [2]. Additionally, we performed a comparative analysis with the ABTSMC in [3]. The parameters of the quadrotor in this study are illustrated in Table A1. The controller parameters are also given in Table 1. The results are illustrated by using the following scenarios:

Scenario 1: In the first scenario, the performance comparison was conducted with a 40% loss of control effectiveness by multiplicative sinusoidal fault in actuators 2 and 3. The faults were injected at the 20th second. One of the reasons for this type of fault is cracking or breakage of the rotor blades. In this scenario, 40% of the multiplicative sinus fault was applied to actuators 2 and 3. The desired trajectories and the amount of output perturbation are given in Table 2. The simulation results are illustrated in Figure 1, Figure 2 and Figure 3 via the comparison of methods [1,2,3] and the proposed ABFTSMC technique based on the tanh(.) function.

Figure 2 shows a comparison between the sliding surfaces of all three methods. In the case of variable z, methods [1,2,3] had high chattering. The proposed ABFTSMC method based on the tanh(.) function drastically reduced this value and damage to the system. This method worked similarly in other cases ($\phi ,\theta ,\psi$) and converged the sliding surface to zero with more minor error. Figure 3 and Figure 4 display the performance tracking of the quadrotor attitude and position.

Figure 3 shows the performance tracking of the quadrotor position $\left(x,y,z\right)$. As can be seen, the proposed controller, which is an adaptive barrier fast terminal sliding mode control and modified by a hyperbolic tangent, significantly reduced output chattering. The perturbation and fault of the actuator imposed so much instability on the system that the proposed method had better performance compared to methods [1,2,3] and followed the defined path with a slight error, where Figure 3 shows that the initial time reached the desired path in the finite time. The desired path tracked faster and with the least chattering and error. Figure 4 shows the $\left(x,y,z\right)$ performance tracking of the quadrotor attitude. As is known, the proposed controller had a more successful performance than the methods. The proposed method sped up the system response, converged to the desired trajectories, and tracked the trajectory with minimal error and chattering. Three examples of performance indicators, such as the integral of the absolute value of error (IAE), integral of the squared error (ISE), and integral of the time-weighted absolute error (ITAE), are examined in Table 3, which confirms the simulation results of Figure 3 and Figure 4 and the proper performance of the proposed method compared to the other methods. In all three comparative cases, the values of IAE, ISE, and ITAE in all six output modes in the proposed method were lower than with the other approaches.

Scenario 2: In this scenario, a loss of 50% control effectiveness by a multiplicative step fault in actuators 1 and 2 was imposed. These faults were injected at the 20th second. In this scenario, actuators 1 and 2 received a 50% step fault. The desired path is given in Table 2. In each scenario, the value of $\overline{t}$ in Equation (21) for each subsystem is written as follows:

${\overline{t}}_{{\left[u_z,u_{\phi },u_{\theta },u_{\psi }\right]}^T}=\left[\begin{array}{c}0.865\\ 2.1\\ 1.4\\ 1.38\end{array}\right]$

The simulation results are similar to the results of the first scenario, where the results of control methods [1,2,3] are compared with those of the proposed ABFTSMC method based on the hyperbolic tangent function, and are shown in Figure 5, Figure 6 and Figure 7, respectively. In all four sliding surfaces that correspond to the $\left(z,\phi ,\theta ,\psi \right)$ modes, it is clear that the sliding surface of the proposed method had the least chattering. In case z, papers [1,2,3] had high chattering, which caused serious damage, but the proposed method greatly reduced the chattering phenomenon.

The ABTSMC method [3] had higher chattering in the sliding surfaces compared to the other two methods. However, the proposed method reduced the chattering to approximately zero with the least error. According to Figure 6 and Figure 7, it is clear that all six system outputs experienced a lot of instability and chattering by applying the actuator fault after the 20th second. All three methods were able to solve this problem to some extent. However, the method mentioned in this paper, firstly, due to the barrier function, was able to limit the chattering due to output disturbances, parametric uncertainties, and actuator faults. Secondly, the main advantage of this method is the use of the continuous hyperbolic tangent function instead of the discontinuous sign function, which was able to reduce the chattering caused by the sign function and other disorders compared to other methods. Table 4 compares the indicator performance of the three methods applied to the quadrotor. This table shows that in all three indicators, IAE, ITAE, and ISE, the numbers obtained from the proposed method had the lowest value in all six output modes, indicating the superior performance of the proposed controller.

Remark 4.

It should be noted that scenarios 1 and 2 have the following differences:

• -. In the first scenario, 40% of the fault was entered into the system, but in scenario 2, the amount of this fault was 50% in the form of loss of rotor efficiency.

• -. In the first scenario, the type of sinusoidal fault was intermittent, which sometimes results from cracking of the rotor blades, but the type of fault in the second scenario was a fixed-step function.

• -. In the first scenario, fault was entered in actuators 2 and 3, but in the second scenario, it was applied to actuators 1 and 2.

5. Conclusions and Future Works

This paper proposed an adaptive barrier FTSMC based on the hyperbolic tangent function for quadrotor UAVs. The approach aims to eliminate the influences of external disturbances and mitigate the effects of the actuator faults and nonlinear uncertainties. This approach achieves three important objectives: (1) FTSMC provides the convergence performance in both reaching and sliding phases in the finite time, (2) an adaptive barrier function is used to ensure the convergence of the output variables independent of high gain of the disturbance and without overshoot, and (3) in order to reduce the severe chattering phenomenon, a hyperbolic tangent function is used instead of the sign function, which is effective in the simulation results. The simulation results and the performance index table show the effectiveness of this method in mitigating the actuator faults. A comparison analysis with the adaptive sliding mode control and fast terminal sliding mode control approaches proposed in [1,2,3] shows that the proposed method outperforms the other approaches. In future work, fuzzy-based model-predictive control with higher-order sliding surfaces will be employed for robust tracking control of quadrotor UAVs.

Author Contributions

Conceptualization, A.N., M.T.V. and S.M.; formal analysis, A.N., A.F., J.H.A. and M.T.V.; funding acquisition, A.F.; investigation, S.M., A.F., A.N. and J.H.A.; methodology, A.N. and S.M.; writing—original draft, M.T.V. and A.N.; writing—review and editing, and supervision, S.M., A.F., A.N. and M.T.V. All authors have read and agreed to the published version of the manuscript.

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

The data that support the findings of this study are available within the article.

Conflicts of Interest

The authors declare no conflict of interest.

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Figures and Tables

Figure 1. The control scheme of the quadrotor.

Figure 2. Sliding surface under 40% sinusoidal fault to actuators 2 and 3.

Figure 2. Sliding surface under 40% sinusoidal fault to actuators 2 and 3.

Figure 3. Quadrotor position under 40% sinusoidal fault to actuators 2 and 3.

Figure 4. Quadrotor attitude under 40% sinusoidal fault in actuators 2 and 3.

Figure 4. Quadrotor attitude under 40% sinusoidal fault in actuators 2 and 3.

Figure 5. Sliding surface under 50% step fault in actuators 1 and 2.

Figure 5. Sliding surface under 50% step fault in actuators 1 and 2.

Figure 6. Quadrotor position under 50% step fault to actuators 1 and 2.

Figure 6. Quadrotor position under 50% step fault to actuators 1 and 2.

Figure 7. Quadrotor attitude under 50% step fault to actuator 1 and 2.

Figure 7. Quadrotor attitude under 50% step fault to actuator 1 and 2.

Table 1

Desired path and disturbances.

Variable	Value	Variable	Value
$b_1$ = $b_2$for ($u_z$)	5	$b_1$ = $b_2$for ($u_{\phi }$,$u_{\theta },u_{\psi }$)	9
$\alpha$	2	$Q_{psb}$	0
$\beta$	3	$\mu$	5
$\rho$	5	$\zeta$	1

Table 2

Desired path and disturbances.

State	Desired Path	Disturbance
$x_d$	$\left|\mathit{cos}(0.2t)\right|$	$10\mathit{sin}(0.0015t)$
$y_d$	$\left|\mathit{sin}(0.4t)\right|$	$5\mathit{sin}(0.0015t)$
$z_d$	3	$10\mathit{sin}(0.0015t)$
${\phi }_d$	$\left|\mathit{cos}(0.2t)\right|$	$10\mathit{sin}(0.0015t)$
${\theta }_d$	$\left|\mathit{sin}(0.4t)\right|$	$10\mathit{sin}(0.015t)$
${\psi }_d$	$\left|\mathit{cos}(0.3t)\right|$	$10\mathit{sin}(0.0015t)$

Table 3

Performance indices under 40% sinusoidal fault actuators 2 and 3.

	States	$\mathit{x}$	$\mathit{y}$	$\mathit{z}$	$\mathit{\phi }$	$\mathit{\theta }$	$\mathit{\psi }$
	ASMC [1]	$90.17$	$915$	$1.188$.	$1.236$	$0.002245$	$0.5868$
ISE	FTSMC [2]	$11,040$	$390.15$	$3.254$	$0.5127$	$0.004933$	$0.5559$
ISE	ABTSMC [3]	$13.35$	$0.2575$	$1.211$	$1.414$	$0.001146$	$0.6035$
	Proposed method	$11.6$	$0.01381$	$1.173$	$0.3965$	$0.00081$	$0.4428$
	ASMC [1]	$27.35$	$3.129$	$1.162$	$1.268$	$0.07129$	$0.7905$
	FTSMC [2]	$828.4$	$62.1$	$1.496$	$1.133$	$0.05958$	$0.774$
IAE	ABTSMC [3]	$19.268$	$3.1$	$1.405$	$1.847$	$0.0506$	$0.776$
	Proposed method	$11.27$	$0.2956$	$1.055$	$0.89$	$0.03881$	$0.7498$
	ASMC [1]	$868$	$76.37$	$17.76$	$3.761$	$1.369$	$0.4188$
	FTSMC [2]	$5312$	$81.76$	$13.79$	$9.218$	$0.5146$	$0.3896$
ITAE	ABTSMC [3]	$479.3$	$73.26$	$25.51$	$5.113$	$0.9975$	$0.3771$
	Proposed method	$433.6$	$12.56$	$0.37818$	$1.096$	$0.2571$	$0.3672$

Table 4

Performance indices under 50% step fault in actuators 1 and 2.

	States	$\mathit{x}$	$\mathit{y}$	$\mathit{z}$	$\mathit{\phi }$	$\mathit{\theta }$	$\mathit{\psi }$
	ASMC [1]	$95.92$	$1.791$	$1.187$	$1.419$	$0.0008$	$0.6196$
ISE	FTSMC [2]	$11,045$	$3960$	$1.162$	$0.5541$	$0.04088$	$0.6026$
ISE	ABTSMC [3]	$15.52$	$0.2473$	$1.207$	$1.239$	$0.00229$	$0.6031$
	Proposed method	$11.51$	$0.01593$	$0.4741$	$0.4741$	$0.0007$	$0.442$
	ASMC [1]	$29.12$	$3.033$	$1.364$	$1.282$	$0.07129$	$0.8093$
	FTSMC [2]	$826.1$	$16.23$	$1.523$	$1.33$	$0.1405$	$1.078$
IAE	ABTSMC [3]	$19.26$	$2.932$	$1.162$	$1.926$	$0.08221$	$0.778$
	Proposed method	$11.48$	$0.3401$	$1.044$	$0.9746$	$0.04334$	$0.775$
	ASMC [1]	$951.6$	$8116$	$68.18$	$9.631$	$2.387$	$0.4136$
	FTSMC [2]	$5216$	$74.67$	$18.185$	$12.35$	$4.432$	$0.3764$
ITAE	ABTSMC [3]	$476$	$65.18$	$17.62$	$5.766$	$3.492$	$0.386$
	Proposed method	$442.4$	$14.49$	$13.46$	$3.969$	$0.7044$	$0.3448$

Appendix A

The parametric values of the model were taken from [54] and are given in Table A1.

Table A1

Quadrotor parameters [54].

Parameter	Value	Unit	Parameter	Value	Unit
$g$	9.81	$\frac{\mathrm{m}}{{\mathrm{s}}^2}$	$J_r$	$90\times {10}^{-6}$	$\mathrm{kg}·{10}^{-6}$
$m$	1	$\mathrm{kg}$	$l$	0.24	$\mathrm{m}$
$I_x$	$8.1\times {10}^{-3}$	$\mathrm{kg}·{\mathrm{m}}^2$	$b$	$54\times {10}^{-6}$	$\mathrm{N}·{\mathrm{m}}^2$
$I_y$	$8.1\times {10}^{-3}$	$\mathrm{kg}·{\mathrm{m}}^2$	$d$	$1.1\times {10}^{-6}$	$\mathrm{N}·\mathrm{m}·{\mathrm{s}}^2$
$I_z$	$14\times {10}^{-3}$	$\mathrm{kg}·{\mathrm{m}}^2$

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