Robust Intelligent Fault-Tolerant Based Finite-Time Attitude Control for Quadrotor UAV Without Angular Velocity Measurements

This study considers the problem of ﬁnite-time attitude control for quadrotor unmanned aerial vehicles (UAVs) subject to parametric uncertainties, external disturbances, input saturation, and actuator faults. Under the strong approximation of radial basis function neural networks (RBFNN), an adaptive ﬁnite-time NN observer is ﬁrst presented to obtain the accurate information of unavailable angular velocity. More importantly, an adaptive mechanism to adjust the output gain of the fuzzy logic system (FLS) is developed to avoid the selection of larger control gains, and can even work well without the prior information on the bound of the lumped disturbance. Based on the nonsingular fast terminal sliding mode manifold, a novel switching control law is designed by incorporating the adaptive FLS and fast continuous controller in order to remove the undesired chattering phenomenon and solve the negative eﬀects induced from the parametric uncertainty, external disturbance, and actuator fault. To deal with the input saturation, an auxiliary system is constructed. The rigorous theoretical analysis is given to prove that all the signals in the closed-loop system are uniformly bounded, and tracking errors converge into bounded neighborhoods near the origin in ﬁnite time. feasibility control


Introduction
During the past decade, quadrotor unmanned aerial vehicles (UAVs) have witnessed a boost and are favored solutions to be deployed in military and civil fields, for instance, aviation reconnaissance, package delivery, aerial photography, agricultural production, wildlife protection, and so on [1][2][3][4]. In the process of such tasks, the high-performance attitude control is a significant part [5,6]. Compared with conventional aircraft, the quadrotor possesses some specific properties such as simple structure, vertical takeoff/landing, hovering capability, and agile operation in the cluttered environment. But, there are confronted with many difficulties including the multiple-input-multiple-output (MIMO), high nonlinearity, under-actuation, and extremely complicated atmosphere, and the quadrotor system is sensitive to several uncertainties and/or disturbances [7]. Apart from increasing the challenge of the controller design, the foregoing factors pose multiple handicaps in the stable response and task success. To this end, it is imperative to develop a reliable flight framework in the control community.
Recently, several advanced control strategies are developed, such as adaptive control, sliding mode control (SMC), neural network (NN), fuzzy logic system (FLS), Weaknesses asymptotically convergence on the sliding surface, serious chattering phenomenon, and big tracking error non-monotone convergence on the sliding surface, complex design, and large overshoot complex computation due to control term with fractional power, slow convergence rate, and singularity problem and the references therein. For a review of literature, an immersion and invariance-based adaptive controller was proposed in [2,8] to estimate the parametric uncertainty, without requiring a linear parameterization assumption. Despite the provision of fast response, easy implementation, and preferable robustness, SMCs have to select sufficiently large sliding gains for compensating the lumped uncertainty, and/or need a prior knowledge for the upper bound of lumped uncertainty [9,10]. However, this is unrealistic for the complexity of system uncertainty and could cause more energy loss and system chattering. For solving these problems to enhance the SMC performance, an adaptive SMC-based observer was widely applied in the recent stage [11][12][13]. In the work of [14], a high-order SMC is comprehensively discussed and analyzed. In [15], a novel high-order SMC strategy was proposed to address a class of arbitrary order sliding mode systems with mismatched terms. Attributed to the strong nonlinear approximation ability, the quadrotor system shows evident inhibition effects for various uncertainties [3]. In [16], an improved NN with slight online computational load is presented to avoid the explosion of online learning parameters. Due to the possibility to convey the ambiguity of human knowledge, FLS as an alternative intelligent policy is in a position to cope with the unknown nonlinearity and can provide an excellent performance. As pointed out in [17], the explosion trouble existing in the traditional backstepping method is solved by designing the adaptive fuzzy dynamic surface control. The fuzzy SMC was proposed to tackle disturbances with the chattering avoidance [18,19]. In this way, one can gain the extra benefit that accuracy physical model and prior information of disturbances are unnecessary. Despite the advantages of the above-mentioned approaches, integrating the adaptive technique into a hybrid system including SMC, NN and FLS may destroy the stability of the whole system. In this setting, there still remains an open challenging to design a more efficient scheme without sacrificing robust performance.
Now along with the development of the control theory and the increasing demand for the high-performance tracking control, the finite-time control strategy has attracted extensive attention from researchers and engineers [6,8,13,20,21]. Its major merit is that the system state can converge to the equilibrium point after finite time rather than asymptotic convergence. To list a few, an adaptive multivariable finite-time control method was skillfully constructed in [8] to overcome the overestimation phenomenon and unknown disturbance. In [21,22], a finite-time disturbance observer was proposed to remove the negative effects from compound disturbances. As given in [23,24], terminal sliding mode control (TSMC), which adopts nonlinear surfaces to take the place of linear surfaces, was utilized to obtain a finite-time convergence, better transient response, and higher tracking precision. As shown in Table 1, the performance comparison of various SMC methods is given. Compared with the linear SMC, the conventional TSMC has two shortcomings that provide a slower convergence rate and exists a singularity. Therefore, the fast TSMC (FTSMC) [25][26][27] and nonsingular TSMC (NTSMC) [22,28,29] have been proposed successively. However, individual methods based on FTSMC or NTSMC can only overcome one of the above-mentioned shortcomings. Simultaneously solving two troubles, a new SMC called nonsingular FTSMC (NFTSMC) is developed and applied in different fields [21,[30][31][32][33]. However, no matter which manifold structure has a common problem in the face of the unmodeled high-frequency dynamics of output signals called the chattering phenomenon. In practical, it is particularly unexpected since it may lead to performance degradation and unforeseen instability.
To get rid of the chattering problem existing in SMCs, several solutions have been developed in succession, which can be generally divided into the following types: 1) The first solution is to employ a socalled boundary layer technique (BLT) that replaces the discontinuous term [19,30,31,34], but the asymptotic stability inside the boundary layer is unable to be guaranteed, and the inadequate choice of BLT may produce large steady-state errors. 2) The second approach is to develop a continuous controller [10,23,24,33], however, it could lead to deterioration accuracy and low robustness. 3) Because the degree of the unexpected chattering is proportional to the magnitude of switching gains, some observers/approximators and adaptive approaches are designed to avoid choosing big switching gains [7,11,19,20,29,[34][35][36]. The afore-mentioned approaches so far solve the unwanted chattering, however, the fast finite-time nature disappears. For this important demand, it corresponds to the pursuit of our study. Moreover, most attitude control strategies for the quadrotor UAV are developed according to an idea assumption that requires the acquirability of the attitude orientation and angular velocity. From the perspective of cost limitations, sensor malfunction, and implementation considerations, angular velocity measurement may be unavailable and inaccurate. Nevertheless, this is unrealistic for the practical engineering, and so it is highly desirable to realize a free angular velocity measurement. Hence far, this issue has been deeply studied in many literature [20,26,[38][39][40]. For instance, a learning observer-based adaptive algorithm was constructed to approximate the feedback velocity, which helps reduce the number of equipped sensors [26]. It has also been indicated that from the work [38], the extended state observer can accomplish the evident inhibition effects for various disturbances. In spite of that, the lumped uncertainty and its derivative require to be bounded together, and the finite-time property fails to realize. In [39], a saturated robust controller with only position measurement is proposed for a class of the motion system to overcome the problems of unknown friction, parametric uncertainty and actuator saturation. In addition to the aforesaid approaches, different finitetime observers are introduced in [20,40] as well, however the parameter uncertainty is not considered for the design of velocity observer. As we know, the rapid and precise reconstruction of angular velocity is very important for timely accommodation of fault and saturation; otherwise, it could cause system instability, even worse, catastrophic accidents.
As a matter of fact, due to environmental changes, unexpected problems such as actuator faults and input saturation may happen at any time, especially in a sud-den moment and a long-duration flight. These problems could lead to mechanical failure, large energy consumption, and unpredictable motion. Currently, research results corresponding to the input saturation mainly focus on three aspects: 1) The use of small-gain method is to reduce the input amplitude [21,39,41]. 2) The use of smooth function is to approximate the saturation function [42]. 3) The use of an auxiliary system is to compensate for the saturation region [43][44][45]. Despite these efforts, the issues concerning the strong robustness, fast anti-saturation speed, and avoidable approximation/compensation errors have to face. Moreover, the great majority of present conclusions may end up with performance degradation under actuator faults. In general, fault-tolerant control (FTC) methods can be mainly classified into two categories, that is, passive FTC (PFTC) [19,28,33,46,47] and active FTC (AFTC) [48]. Compared with the AFTC, PFTC has the ability to overcome the adverse influence of actuator faults quicker, and adapt to complex structures and real-time performance, because the feedback information from the fault diagnosis observer is needless. In this respect, this brief takes a lot of effort on PFTC for the quadrotor attitude control. The existence of two risk factors cause the attitude control problem to be more difficult and intricate. This is also another intention of our study.
Motivated by the remarkable importance and advantages, this study concentrates on designing a reliable attitude control scheme for the quadrotor UAV. Since the information of angular velocity can not be obtained directly, an adaptive finite-time NN observer (AFTNNO) is firstly constructed. By borrowing advantages from the NFTSM surface, a continuous attitude controller is developed to achieve the fast finite-time convergence, fault avoidance, and chattering attenuation. Since the upper bounds of time-varying disturbances, unknown uncertainties, and actuator faults are difficult to obtain precisely, a novel compensation term as a part of the switching controller is applied with the help of the adaptive algorithm and FLS. Additionally, an auxiliary system is developed to address the input saturation. By comparing previous conclusions comprehensively, the main contribution can be summarized as the following threefold: 1) The attitude control scheme designed in this study does not require the measurement of the angular velocity. Hence, the cost related to the velocity measurement sensors can be reduced for the controller development. Moreover, this study considers more complex situations, such as the external disturbance, parametric uncertainty, input saturation, and actuator fault, which makes the designed algorithm more available for practical implementation. Other similar studies in [1-5, 7, 8, 13, 17-19, 19-22, 38, 39, 48] either demand the measurement of the angular velocity or only consider one of these circumstances. 2) As an alternative strategy to eliminate the chattering phenomenon and compensate for the effect of the lumped disturbance, this study presents a novel adaptive fuzzy-based fast switching controller, and which not only guarantees the finite-time stability with fast convergence speed and high tracking precision, but also does not require a stringent assumption that the lumped disturbance and its derivative are both bounded by a constant [8,13,21,22,[27][28][29]44]. This signifies that the proposed control scheme provides a widespread potential for application. In addition, the auxiliary design system is designed to deal with the problem of input saturation under the premise of finite-time stability. 3) Compared with existing observer techniques [20,26,27,[38][39][40][49][50][51], the velocity observer is developed based on the adaptive NN technique, which can not only achieve the fast finite-time error convergence but also accurately estimate the information of angular velocity. It is worth mentioning that all the tracking signals in the closed-loop system are uniformly bounded and tracking errors can converge into small neighborhoods of the origin after a finite time. To further guarantee the practicability of the developed controller, several guidelines about the parameter adjusting are given in detail. This is very helpful for engineers to accomplish a high-precision attitude tracking of quadrotor UAV in real.
The rest of this paper is laid out in the following manner: Section 2 describes some helpful notations and preliminaries including the model of quadrotor UAV, actuator fault and saturation analysis, NN approximations, FLS, and the control objective. Next, Section 3 details the development of the angular velocity observer and attitude controller design, and gives the relevant stability analysis. Section 5 provides simulations to illustrate the effectiveness of control scheme, followed by conclusive statements and future works in Section 6.

Notations
Throughout this work, some notations are imposed. R n and R n×m express the n− dimensional Euclidean space and n × m real matrix, respectively. I n denotes a n × n identity matrix. | · | and · are the absolute value of a scalar and the Euclidean norm of a vector, respectively, while diag{•} and tr(•) are the diagonal matrix and the trace of a matrix, respectively.λ(•) and λ(•) indicate the maximum and minimum singular values of a matrix, respectively. For a given vector y = [y 1 , y 2 , y 3 ] T , a superscript × is a transformation of y to skew-symmetric matrix, i.e., [y] × = [0, −y 3 , y 2 ; y 3 , 0, −y 1 ; −y 2 , y 1 , 0].

Definitions and Lemmas
Definition 1: For any real number y, we define the hyperbolic tangent function (HTF), as follows: tanh(y) := e y − e −y e y + e −y .
Then, for any vector y = [y 1 , · · · , y n ] T , tanh(y) ∈ R n can be defined as tanh(y) = [tanh(y 1 ), · · · , tanh(y n )] T . Definition 2: For any given vector x = [x 1 , · · · , x n ] T and a positive number α, we define sign(x) α ∈ R n as In order to improve the readability of this article, we define the power of a vector x ∈ R n , as follows: x m :=[x m 1 , · · · , x m n ] T ∈ R n , x m :=[ẋ m 1 , · · · ,ẋ m n ] T ∈ R n , |x| m :=diag{|x 1 | m , · · · , |x n | m } ∈ R n×n , |ẋ| m :=diag{|ẋ 1 | m , · · · , |ẋ n | m } ∈ R n×n . Definition 3 [6]: Consider a nonlinear system aṡ wherein x and u are state and input vectors, respectively, and f : W → R n is continuous on an open neighbourhood W . The equilibrium x = 0 is practical finite-time stable for the given initial condition x(t 0 ) with t 0 denoting the initial time, there exist ǫ > 0 and Lemma 1 [52]: For any two matrices P and Q that have the appropriate dimensions, it follows that 2P T Q ≤ βP T P + β −1 Q T Q with β being a positive scalar. Lemma 2 [40]: For any real number x i ∈ R 1 , there exists a positive scalar h ∈ (0, 1] that holds the following inequality, that is, (

System Description
The schematic configuration of a quadrotor UAV with the structure of symmetrical and rigid is vividly shown in Fig. 1, where the coordinate frames A = {O a , x a , b a , z a } and B = {O b , x b , y b , z b } are the earth frame with respect to the ground and the boy frame attached to the quadrotor UAV, respectively. The attitude motion for the quadrotor UAV can be controlled by adjusting the rotation speeds of four rotors appropriately. The attitude angle vector is denoted as Θ = [φ, θ, ϕ] T in the frame A, while Ω = [p b , q b , r b ] T is the angular velocity vector in the frame B. The relationship between Θ and Ω can be described in the form of where the rotation matrix R s (Θ) and its inverse matrix R t (Θ) are respectively written as follows: where S (·) sin(·), C (·) cos(·), and T (·) tan(·).
Using the Euler-Lagrangian methodology, the dynamics equation for the quadrotor attitude system can be expressed as (see among others [2,7,18]) where u = [u 1 , u 2 , u 3 ] T and J = diag{J x , J y , J z } denote the control input and the inertial matrix, respectively; d is the lumped disturbance including the uncertain inertial matrix, external disturbance, aerodynamic friction and gyroscopic effect. Suppose that the rotor thrust is proportional to the square of the rotational speed of the rotor, that is, u i = κ a w 2 i , (i = 1, 2, 3, 4), where κ a > 0 and w i are the lift coefficient and the rotary speed of the i rotor, respectively. The relationship between the rotor speed w i and the control input u i can be expressed by using the following matrix manner as where κ b > 0 is the drag factor; l d is the distance between an equipped rotor and the center of the mass. Before proceeding, some mild assumptions are given in the subsequent development. Assumption 1 To prevent the high-frequency oscillations of motion states, the desired attitude command Θ d , and its time derivatives ( i.e.,Θ d andΘ d ) are assumed to be bounded. Assumption 2 To avoid the singularity at θ = ± π 2 and guarantee that the quadrotor never be overturned, angles φ and θ are both limited to |φ| < π 2 and |θ| < π 2 . Assumption 3 Due to the deployment of sensors, the structural flexibility, the variations in payloads, and other factors, the inertial matrix J is uncertain for the controller design. Thus, suppose that J is represented known nominal part of J , and J ∆ = [J ∆,x , J ∆,y , J ∆,z ] T regarded as the corresponding uncertain part of J is bounded such that J ∆ ≤J withJ being an unknown positive constant.

Actuator Fault and Saturation Analysis
In the presence of actuator fault and input saturation, the output torque of ith rotor can be expressed as In this paper, the above conditions are taken into account, except the condition (4). For the stuck fault, it often requires that the number of actuators is more than that of control outputs, and the control allocation scheme should be further investigated. As shown in Fig. 2, the actual control input sat(u o,i ) satisfies the non-symmetric input saturation, which is described as below: where u i < 0 andū i > 0 represent the known lower and upper bounds on u o,i , respectively. Fig. 3 Description of the architecture of the RBFNN used in this study, where the grey circle is the output of the RBFNN, the orange circle is the output of the hidden layer, and the blue circle is the input of the RBFNN.
According to previous modeling processes and Assumptions 1-3, the complete attitude equation of the quadrotor UAV is transformed into the following form Remark 1 Since the nature flight environment is constantly changing and has finite energy, the external disturbance d acting on the quadrotor UAV can be regarded as the unknown time-varying yet bounded command. Based on the definition of R t and the recall of Assumptions 2 and 3, it is hence reasonable to give Assumption 4. Compared with the existing conclusions given in [8-10, 13, 21, 22, 27-29, 44], where D andḊ are both bounded orḊ is the steady-state vanishing type, this work just needs that the upper bound of D exists. Thus, Assumption 4 is also very loose and more general.

RBFNN Approximation
Since RBFNN can accurately approximate any nonlinear function, it has been extensively employed to achieve high-performance control for complex nonlinear systems. As a consequence, the smooth nonlinear function F(Z) : R w → R can be approximated by where Z = [z 1 , . . . , z w ] T ∈ Ω Z ⊂ R w denotes the input vector of the RBFNN with w being the input number of RBFNN; δ(Z) signifies the function approximation error; h(Z) = [h 1 (Z), . . . , h p (Z)] ∈ R p is the output of the hidden layer, where p is the number of neural nodes and h i (Z) is the Gaussian function defined as where C = [c 1 , . . . , c w ] T ∈ R w and κ ∈ R 1 are the center and width of the Gaussian function, respectively.
for the input Z, which can be calculated by the following formulation Remark 2 With respect to the RBFNN, here are three aspects that need to be stated: (i) In practice, W * is unknown and needs to be estimated by designing the neural estimator [36] or adaptive algorithm [16], which is constrained by a positive constantW , that is, W * ≤W ; (ii) Because the function exp(·) is a monotonically increasing and − Z − C 2 ≤ 0, there is a positive constanth such that h(Z) ≤h. (iii) In the light of Stone-Weierstrass theorem, one can get an uniform result that δ(Z) holds δ(Z) ≤δ withδ being a positive constant. For a more visual display, the simplified structure of the RBFNN employed in this study is shown in Fig. 3.

FLS Design
In this work, we design a FLS to handle the lumped disturbance. The fuzzy inference engine utilizes a set of fuzzy IF-THEN rules to perform a mapping from The fuzzy linguistic rule base is described as follows: Taking account into the simple calculation and intuitive credibility, the singleton fuzzification with triangular membership functions and center of gravity defuzzification way are used accordingly. The membership functions of input and output fuzzy sets are presented in Fig. 4. The values of v a and v b are selected based on the system performance requirements. Thus, the output of FLS can be written as where 0 ≤ c j ≤ 1, (j = 1, 2, 3), denote the firing strengths of the ith rule; r a = [r a,1 , . . . , r a,n ] T ∈ R n represents a fuzzy vector to be selected suitably; r i,1 = −r a,i , r i,2 = 0 and r i,3 = r a,i indicate the center of membership functions PE, ZE, and NE, respectively; the relation c 1 + c 2 + c 3 = 1 is valid in conformity to the special condition of triangular membership functions. There exists only four possible cases that need to be discussed.
-Case 1: Only Rule 1 is satisfied (that is, v i > v a , c 1 = 1, and c 2 = c 3 = 0), it follows that f i = r ai . -Case 2: Both Rules 1 and 2 are satisfied (that is, Recalling the afore-mentioned Cases 1-4, it follows that z i (c 1 − c 3 ) = |z i (c 1 − c 3 )| ≥ 0. Taking a summary, one has Control Objective: This study concentrates on designing a robust intelligent fault-tolerant based finitetime attitude controller for the quadrotor UAV with only available attitude information, such that all the commands in the closed-loop system are ultimately uniformly bounded and tracking errors will converge into small bounded regions around the origin in finite time, despite the presence of the uncertain inertia matrix, unknown disturbances, actuator faults, and input saturation.

Controller Design Methodology and Stability Analysis
In this section, an adaptive finite-time NN observer (AFTNNO) is first developed to measure the angular velocity. Then, our study investigates a novel NFTSMC based adaptive FLS scheme, to eliminate the negative effects induced by the lumped uncertainty, actuator fault and the undesired chattering.

AFTNNO Design
To begin with definition of a new state variable, i.e., w =Θ, it follows that It should be noticed that since the subitem g(Θ, w) contains the uncertain factor J ∆ , it can not be directly utilized to design the control law.
To estimate the unavailable angular velocity in the presence of the uncertain inertial matrix and other uncertain factors, we design an AFTNNO in the form of whereΘ = [φ,θ,φ] T ,ŵ = [φ,θ,φ] T andŴ * express the estimations of Θ, w and W * , respectively. k 1 and k 2 are positive constants to be determined later.
(Θ)),Θ = Θ −Θ, and k 3 is a positive design constant. Here should satisfy the conditions such that 4 5 < l < 1 and 2l − 1 = l1 l2 , where l 1 and l 2 denote positive odd integers. In this study,Ŵ * can be adjusted online by using the following adaptive mechanism: where Υ ∈ R p×p is a positive definite matrix and k 4 > 0. By lettingw = w −ŵ,W * = W * −Ŵ * and h = h(Θ, w) − h(Θ,ŵ), the corresponding dynamics of estimation errors can be hence derived as follows: where Ξ = g(Θ, w) + p 2 (Θ)(u f + d). For the sake of simplicity, we denote a new estimation error vector as The time derivative of ξ is given bẏ In the following, the main theorem about AFTNNO is provided in details.

Remark 3
In many existing works [1-5, 7, 13, 17], the attitude angular velocity is assumed to be measurable. This could cause these approaches to no longer useful in practical application, and thus an AFTNNO is developed in this study to tackle this challenge. Although different kinds of observers have been developed in [49][50][51] recently, the application of these control frameworks is required to know the accurate inertia matrix. To the best of our knowledge, the fast and accurate reconstruction of angular velocity looks forward to providing significant support for the timely solution of fault and saturation; otherwise, it is prone to task failure, even worse, serious flight accidents.

Robust Intelligent Fault-Tolerant based Finite-Time Attitude Controller Design
For the purpose of the finite-time convergence, preferable robustness, and rapid response, a NFTSM surface s n = [s n,1 , s n,2 , s n,3 ] T containing the attitude-tracking error and angular velocity tracking error is introduced as [30,32] s n = Θ e + β a sign(Θ e ) mc + β b sign(Θ e ) ma m b (27) where β a and β b are positive design parameters; m a and m b denote positive odd integers with the relationships of 1 < ma m b < 2 and m c > ma m b . Particularly, since this study can estimate the unavailable angular velocity signal and attitude signal, Θ and its derivativeΘ are substituted by their estimation valuesΘ andΘ. From the AFTNNO in (15), the estimated tracking errors Θ e andΘ e in (27) is hence defined as Θ e =Θ − Θ d anḋ Θ e =ŵ −Θ d , respectively.

Remark 4
The fast finite-time convergence for the control strategy-based NFTSM manifold is interpreted as follows: When the system state approaches the neighborhood of the equilibrium point, the lower-order term of Θ e plays a leading role to ensure the faster reaching rate. During this phase, the convergence speed of the system state on the NFTSM manifold is analogous to that on the NTSM manifold described byΘ e = −(1/β b ) m b /ma sign(Θ e ) ma/m b . When the system state is far from the equilibrium state, the higher-order term of Θ e has a great effect on improving the convergence rate. During this phase, the convergence rate of the system state on the NFTSM manifold is quicker than that on the NTSM manifold described byΘ e = −(β a /β b ) m b /ma sign(Θ e ) ma/m b . By doing so, one can conclude that the system state can converge to the equilibrium state faster on the NFTSM manifold than on the NTSM manifold. In contrast to the FTSM manifold described by s F T SM =Θ e + p a sign(Θ e ) pc + p b sign(Θ e ) p d with p a > 0, p b > 0, p c ≥ 1 and 0 < p d < 1, the NFTSM manifold has a merit to solve the singular problem since m a /m b > 1 can prevent any negative power induced by the derivative of s n .
To design the attitude control law, the time derivative of (27) along (8) is first formulated aṡ Thereby, we obtain the equivalent controller u o,eq by computing the equationṡ n = 0 in the absence of the lumped disturbance and actuator fault.
where P 1 = M −1 1 M 2 and P 2 = M −1 1 R T t E. Since the lumped disturbance and actuator faults are inevitable in practical engineering, the switching controller for dealing with this issue are put forward in the following manner: where k a and k b are positive design parameters, k c and k d are positive odd integers with k c < k d , and the definitions of c 1 , c 3 , and r a are already given in (12).
To eliminate the negative effect from the input saturation, an auxiliary system is constructed aṡ (31) where ∆u = u o − sat(u), n a > 1 and n b > 0. Thus, the saturation compensation controller u o,sa is designed as Recalling the pervious design development, the composite attitude controller is given by Consider the following Lyapunov function as Substituting (33) into the time derivative of (34), and utilizing the Lemma 1 that s T n χ ≤ 1 2 (s T n s n + χ T χ) and χ T ∆u ≤ 1 2 (χ T χ + ∆u T ∆u), one can obtain thaṫ According to the detailed analysis about FLS and using s n as the input vector of FLS, it can be confirmed that s n (c 1 − c 3 ) is a positive vector. Thus, we have If each element of r a satisfies r ai ≥ |D c1−c3 |, the inequality (35) can be transformed based on Lemma 2, aṡ where n a > 1, 2l−1 (37), one can illustrate that the system is the practical finite-time stable with the aid of Lemma 3. Since the accuracy upper bound of lumped disturbance is hard to obtain in practice, a bigger r a needs to be chosen in order to ensure the system stability. However, it will cause even more energy loss. So as a solution to this dilemma, an adaptive algorithm is developed to adjust the parameter vector r a online, as follows: wherer a = [r a,1 ,r a,2 ,r a,3 ] T ,r a (0) ≥ 0, µ 1 > 0, µ 2 > 0, 0 < µ 3 < 1, and σ represents a positive constant determining the rate of the estimated boundr a . With the aforesaid preparations, the designed control law can be deduced to Remark 5 In (33), the subitem u o,eq is nominal feedback control, the subitem u o,sw is a fast-type reaching control to suppress the undesired chattering and guarantee strong robustness against the lumped disturbance, and the subitem u o,sa as a compensation control serves for dealing with the input saturation.

Remark 6
For the previous finite-time control strategies developed in [8,13,[20][21][22]25,27,28,31,42,47], where only part of the input saturation, actuator faults, parametric uncertainty, and external disturbance is taken into account. In this study, all these factors are considered into the controller development. Therefore, our study is of both theoretical significance and practical worth to pave the way for achieving the high-performance attitude tracking of the quadrotor UAV.
The overall block diagram for the attitude control system of the quadrotor UAV is delicately shown in Fig. 5. In the following, we are able to give the stability analysis of the closed-loop system in Theorem 2.
Theorem 2 Consider the attitude system transformation described in (8) and Assumptions 1-4. By applying the designed controller (39) with an adaptive mechanism (38), all the signals in the closed-loop system can be guaranteed to be bounded uniformly, and tracking errors will converge to bounded regions near the origin in finite time.
Proof: Consider the Lyapunov function as wherer a,i =r a,i −r a,i is the estimation error, and r a,i represents an upper bound ofr a,i . Without loss of generality,r a,i , (i = 1, 2, 3), are suppose to ber a,i = |D c1−c3 |+r 0,i , and in which the scalar r 0,i is a very small positive constant and belongs to the element of r o ∈ R 3 . Evaluating the derivative of (40) along (38) and (39), results iṅ By adding and subtracting a termr a (c 1 − c 3 ) to the right-hand side of (41), then (41) can be turned intȯ With the help of Lemmas 1 and 5, one can derive   where 0 < µ 4 < 1.

Adaptive Weight Vector
Combing (42) and (43), leads tȯ where L 5 = min{2λ min (L 1 ), 2(n a − 1), Ψ 1 , µ 1 }, L 6 = min 2 kc +k d , where 0 < ǫ 0 < 1 and T 0 is the initial time. Moreover, it readily follows that the tracking signals of the whole system shall drive into following bounded regions near the origin, as follows: where 0 < ǫ 2 < 1. Next, the finite-time error convergence for Θ e is further studied. First, we notice that (30) can be transformed into the following form With attention to Lemma 5 and Remark 5 given in [30], it follows that (46) can be maintained in the form of FNTSM manifold if the following conditions are held: From (47), it can be concluded that the tracking error Θ e will eventually converge into the bounded region, as

∆2
(1−ǫ2)L5 . Further, the total convergence time T total is given by where T reach represents the time that it spends to arrive at the convergence region of NFTSM manifold s n and it has been directly given earlier, and meanwhile, T sliding represents the time that it spends to arrive at the convergence region of tracking error Θ e and which can be calculated according to Remark 5 given in [30], (mc−1)ma , −β a Θ e (0) mc−1 . Due to the facts that (1+ 1 2 ) and −β a Θ e (0) mc−1 < 0, it follows that by means of Lemma 4, the function Λ(·) will keep convergent. Therefore, the proof of Theorem 2 is completed.

Control Parameters Selection
Since the attitude tracking performance is usually compromised with actuator faults, constrained inputs, and especially, with various uncertainties and disturbances, some control parameters should be thoroughly selected. In this study, the error convergence regions (49) can be adjusted to arbitrary small neighbourhoods by selecting tunable parameters suitably. As a result, the relevant selection criteria of controller parameters are clearly stated, as follows: other way round, this may cause more serious chattering. Thus, we should make an acceptable tradeoff between the convergence speed and the control chattering. Luckily, a fast-type reaching control law is designed in this study, which is conducive to the chattering suppression and the improvement of convergence speed. 2) Selections of parameters β i , n i , (i = a, b): The increase of parameters β a and β b can achieve a shorter setting time and the smaller error convergence regions. However, overlarge β a may lead to a larger control input in (39). Under the condition that the parameter n a must satisfy n a > 1, a large value n a can quickly eliminate the saturation error, while it is easy to cause overcompensation if the value n a is too big. For the parameter n b , it only plays a role in achieving a finite-time convergence. 3) Selections of parameters σ and µ i , (i = 1, 2, 3): The parameter σ determines the update rate for adaptive variabler a , which can be allowed to choose small enough in order to improve the update rate. In turn, this leads to overestimation and input saturation. In practice, the parameter σ should be adjusted based on a trial and error manner. Furthermore, the appropriate choice of parameter µ i can ensure the finite-time convergence property and avoid the drift of variabler a . 4) Selections of parameters k i , (i = a, b, c, d): In many references [32,33,37], the bigger gains k a and k b should be selected to overcome the big lumped disturbance. It should be noted that the overlarge gains are prone to input saturation and energy waste, while too small gains fail to attain a fast finite-time convergence. Fortunately, we design an adaptive FLS to compensate for the lumped disturbance, and hence the smaller values k a and k b can be selected. Moreover, because the term sign(s n ) kc k d with kc k d ∈ (0, 1) is designed to enhance the system robustness, and displays like a bridge between the linear control if kc k d → 1 and the discontinuous control if kc k d → 0 [24], a bigger value kc k d tends to reduce the signal chattering but at the price of lower robustness. Therefore, we should conduct a satisfactory balance between chattering elimination and strong robustness for the selection of kc k d .

Parameter Setting
The numerical simulations are performed in the MAT-LAB/Simulink software environment by employing a J0,x, J0,y, J0,z Inertial coefficients 0.01175, 0.01175, 0.02229 (N · m · s 2 /rad) κa Lift coefficient 2 κ b Drag factor 5 fixed-step Runge-Kutta solver and are compared with PD, DSC, and AFTC methods, where the sampling frequency is set as 100 Hz. The physical parameters of the quadrotor UAV used in this study are specifically given in Table 2.

Result Analysis
The control results of the comparative simulation are depicted from Fig. 7 to Fig. 13. As can be seen from Fig.  7, it is obvious that the PD approach has a larger oscillations than the other remaining controllers in terms of attitude angles, which illustrates that the PD controller is susceptible to uncertain parameters, external disturbances, and actuator faults. To be precise, the tracking errors under various controllers are plotted in Fig. 8. From Fig. 8, it is clear that the tracking errors of the AFTC and proposed methods are smaller than that of the PD and DSC methods. The control input commands are presented in Fig. 9, which reflects that the proposed controller was capable of achieving the chattering elimination and saturation suppression. Fig. 10 shows that the sliding surfaces are smooth and quickly reach near zero. The evolution of the adaptive gain in the proposed controller is provided in Fig. 11, and the problem of parameter drifting be avoided effectively. From Fig. 12, the designed AFTNNO can realize the  Notation: (i) The mathematical form of PD control is expressed in this study as u P D = P Θ e + DΘ e ; (ii) Based on the work of [5], the expression of DSC strategy is given as follows: Compared with the AFTC scheme, the proposed method with an additional auxiliary system has the ability to deal with the problem of input saturation. estimation for the attitude angular velocity even under these adverse effects. In addition, the adaptive weight parameter is presented in Fig. 13, in which the parameters can accomplish a rapid convergence performance.
To quantitatively analyze the tracking performance of various controllers, four performance indices are employed in this study and are concretely described in the following forms: 1) Average squared error (ASE): Since the ASE index penalizes a bigger error more than a smaller one, one can illustrate that the designed control algorithm provides a fast convergent speed if the ASE value is lower. 2) Average absolute error (AAE): In contrast to the ASE index, the AAE index possesses a slower convergent rate but with less persistent oscillation. 3) Average time-weighted absolute error (ATAE): i|φ e |(i) + i|θ e (i)| + i|ψ e (i)| . (52) The ATAE index pays close attention to steadystate error, but it thinks little of the initial errors. This problem can be overcome based on the simultaneous consideration of ASE and AAE indices. 4) Total energy consumption (TEC): The smaller TEC value reflects the developed control scheme can reduce the energy consumption.
where N denotes the total time of the simulation operation. Therefore, these performance indices are applied  to fairly evaluate the tracking performance of the different controllers. Moreover, it should be especially emphasized that these indices are expected to be as small as possible.  The statistical conclusions in terms of aforesaid performance indices are visually presented in Fig. 14 to better observe the superiority of the designed control scheme. By analyzing Fig. 13, we can obtain the follow-  ing results: (i) The values of all performance indices for PD control are the largest, which reflects that PD control provides a worse robustness; (ii) Although the performance of DSC strategy is better than that of PD con-   controller successfully overcomes the input saturation without the cost of robustness degradation, for this point, the proposed controller is almost the same in ISE, IAE, and ITAE indices, but has a great enhancement in the TEC index. Consequently, the above analyses illustrate that despite the existence of parametric uncertainty, external disturbance, actuator fault, and input saturation, the proposed controller can still achieve high-performance attitude tracking of quadrotor UAV in aspects of strong robustness, chattering elimination, saturation alleviation, and fault-tolerant.

Conclusions and Future Works
In this study, a robust intelligent fault-tolerant based finite-time attitude tracking control strategy for quadrotor UAVs subject to the parametric uncertainty, external disturbance, input saturation, and actuator faults, which is not only of research value but also academically challenging. Firstly, an AFTNNO is developed to measure the information of angular velocity after a finite time. To deal with the problem of parametric uncertainty, external disturbance, and actuator faults, a novel switching control law including the adaptive FLS and the continuous controller is constructed, which can avoid the choice of big gains and does not the prior information of the lumped disturbance bounds. In addition, to attenuate the effect of actuator constraint, an auxiliary system is further developed. Comprehensive comparisons show that the designed control framework shows remarkably superior performance. Recognizing these advantages, in further research, we are ready to perform the real-time experiments to better verify the practical feasibility of the proposed control strategy.