Observer-Based Adaptive Fuzzy Finite-Time Attitude Control for Quadrotor UAVs

This article presents an observer-based adaptive fuzzy finite-time attitude control strategy for quadrotor unmanned aerial vehicles (UAVs). To estimate the information of angular velocity with the finite-time property, an adaptive neural network observer is first developed. Subsequently, an adaptive fuzzy-logic-system (FLS)-based nonsingular fast terminal sliding mode controller is proposed to compensate for the lumped disturbance and adjust the control gain online. To cope with the input saturation, an auxiliary system without the boundedness of the saturation difference is constructed. The theoretical analysis proves that all the system signals are bounded and the tracking errors can converge to small neighborhoods in finite time. Finally, comparative simulations and experiments are performed to manifest the feasibility and superiority of the proposed control strategy, in terms of strong robustness, singularity avoidance, free-chattering, fault tolerance, and saturation attenuation.


I. INTRODUCTION
In recent years, quadrotor unmanned aerial vehicles (UAVs) have been widely applied in various fields [1], [2], [3], [4], [5] due to some distinct advantages such as hovering capability, simple structure, and vertical take-off and landing.However, there are many difficulties such as the complex atmosphere, high nonlinearity, underactuation, and various disturbances [6], [7], [8], [9].The above difficulties greatly increase the challenge of the controller design and reduce the dynamic performance.As we know, the accuracy of attitude control directly affects the accuracy of position control, so how to realize the high performance of attitude control is the top priority.
Nowadays, many control strategies have been developed [10], [11], [12], [13], [14], [15], [17], [18], [19], [20], [21], [22].Due to its fast response and strong robustness, the sliding mode control (SMC) scheme was designed in [10] to compensate for various uncertainties, but large control gain causes severe chattering.To improve the performance of SMC, the study [2] presented an adaptive SMC-based observer, but the disturbance's derivative is required to be bounded.Attributed to the strong approximation ability, radial basis function neural networks (RBFNNs) show remarkable performance for nonlinear systems [11].An adaptive NN-based dynamic surface control (DSC) policy was proposed to solve the "exponential explosion" problem [12].Tripathi et al. [13] designed a disturbance observer (DO)-based SMC and conducted the experiments to verify its validity.Fang's [14] team presented a new NN-based hybrid mode-switching controller for the flapping wing aerial vehicle.As an alternative intelligence method, fuzzy logic system (FLS) can work reasonably well in the face of nonlinear systems [15], [16], [17], [18], [19].Besides, FLS provides an effective human-interpretable solution that is very easy to understand, especially from the perspective of laypersons.Li et al. [19] proposed a T-S fuzzy-based event-triggering attitude control for the spacecraft systems to reduce the communication burden.Ma et al. [20] presented a hybrid flight controller for unmanned helicopters, where the FLS was constructed to cope with the parametric uncertainties.The fuzzy SMCs were proposed to realize the chattering attenuation and improve the disturbance rejection capability [17], [21], [22].Although the accurate system dynamics is unnecessary, many logic rules are required for the fuzzy controller design, which would lead to structural complexity and computational pressure.As a result, there remains an ongoing challenge to investigate an adaptive fuzzy approximation control with a simpler fuzzy structure and fewer logic rules.
Recently, the finite-time control has attracted considerable attention [23], [24], [25].Its primary advantage is that the system convergence can be guaranteed in finite time instead of infinite time.In [26], a linear-quadratic regulation attitude controller was presented, but it may be impossible to realize the desired control effect for the quadrotor UAV in the presence of high nonlinearity.Moreover, it is less robust and has a high sensitivity to unanticipated disturbances.To solve these difficulties, a terminal SMC (TSMC), which adopts a nonlinear surface to replace a linear surface, was presented to realize the finite-time convergence [27], [28].In contrast to the linear SMC (LSMC), the TSMC has a slower convergence speed and an inevitable singularity.Hence, fast TSMC (FTSMC) [29] and nonsingular TSMC (NTSMC) [30] were proposed for the quadrotor attitude system.However, individual FTSMC [29] or NTSMC [30] schemes cannot handle the above two disadvantages in the TSMC.To solve the above two disadvantages, some nonsingular FTSMCs (NFTSMCs) were presented in [9], [31], [32], [33], and [34].In [9], an adaptive NFTSMC was developed to obtain the desired performance.Xu et al. [33] studied a fault-tolerant control (FTC) method based on NFTSMC to achieve strong robustness and fast response.Nevertheless, the chattering problem is difficult to avoid.To this end, several solutions such as the boundary layer technique [29], [31], continuous controller [10], [27], and observer/approximator [25], [35], have been developed.Although these approaches are constructive, they would lose the fast finite-time nature, and there is just theoretical assurance for the control precision.These important requirements correspond to our goal.
Most attitude controllers require the acquirability of angular velocity.However, it is difficult to measure accurately due to sensor failure and measurement noise, and it costs expensive expenses to install extra velocity sensors.Hence far, many works have been deeply studied to obviate this problem [23], [37], [38], [39], [40], [41], [42].The motion capture system [37] was used to numerically estimate the velocity signals.However, the obtained velocity signals contain the noises and errors and, therefore, cannot be used to make rigorous stability analysis.By applying the immersion and invariance technique, an exponentially convergent velocity observer was designed for mechanical systems, but it ignores the unknown parameters and disturbances [38].Even though the velocity estimation is a critical issue in the quadrotor UAV, the key is to consider the existence of unknown parameters and external disturbances in the design of the state observer.The finite-time observer was presented via a high-order sliding mode mechanism [23] to detect the angular velocity, but it requires the boundedness of the disturbance and its first-order derivative.To solve this challenge, in [39] and [40], the authors designed a state observer-based FLS/NN, but the finite-time convergence cannot be guaranteed.In [41], a distributed finitetime homogeneous controller was proposed to achieve the aim of velocity-free.However, the finite-time convergence analysis is unclear when the system is subject to various disturbances and actuator faults.To address this challenge, a model-free velocity observer was developed in [42] to realize the velocity-free attitude control, but its estimation speed is relatively slow.Notably, the precise reconstruction of angular velocity is important for the system stability, and this is also one of our aims.
In reality, the actuator faults and input saturation should be considered, otherwise it could cause mechanical failure and unpredictable consequences.Current results on how to solve the input saturation mainly include the following: 1) The small-gain approach is used to reduce the input amplitude [43]; 2) The auxiliary system or observer technique is constructed to compensate for the saturation difference [44], [45], [46]; 3) The continuous function is adopted to approximate the discontinuous signal [47].
However, there are some problems that need to be improved, such as free singularity, fast saturation elimination, and compensation-error ability.For another problem, many FTCs have been developed to guarantee the safety of the system.In [48], an active FTC-based observer was developed to tackle the actuator faults while the fault-tolerant ability depends on the accuracy of the fault estimation.As a key component of the active FTC, the fault detection and isolation (FDI) could increase the complexity of the FTC.To solve the above problems, some passive FTCs were proposed in [22], [49], [50], and [51].By combining the benefits of FLS and DO, the parametric uncertainties and fault components were resolved [22], [49].Harshavarthini et al. [50] proposed a finite-time FTC via the Lyapunov-Krasovskii function.Xiao et al. [51] designed a projectionbased adaptive algorithm to overcome the actuator faults while ensuring the boundedness of the adaptive parameters.Compared with the active FTC, the passive FTC can address the actuator faults without any FDI process and is robust in solving a group of considered faults.
Although diverse results have been obtained, solving all the aforesaid factors simultaneously brings major challenges.This study develops an observer-based adaptive fuzzy finite-time attitude controller for quadrotor UAVs with unavailable angular velocity, external disturbances, parametric uncertainties, actuator faults, and input saturation, which can realize free singularity, satisfactory robustness, chattering avoidance, fault tolerance, and saturation elimination.First, an adaptive finite-time NN observer (AFTNNO) is proposed to estimate the accurate information of angular velocity.Then, an adaptive FLS-based NFTSMC is designed to improve the system robustness.Besides, an auxiliary system is constructed to solve the input saturation.By comparison, the main contributions of this study are summarized as follows.
1) In contrast to the previous state observers in [20], [23], [38], and [41], the designed AFTNNO can estimate the information of angular velocity without the accurate knowledge of the system dynamics.Meanwhile, the AFTNNO not only keeps the basic property of the controller with the full-state measurable compared with the state observer in [23] but also realizes the finite-time convergence rather than the exponential convergence in [38], [39], and [40] and provides a faster convergence speed than the state observer in [42].These infer the AFTNNO can achieve better estimation performance.2) In comparison with the conventional fuzzy SMCs in [3], [18], and [22] and fuzzy logic controllers in [15], [16], [17], [19], [20], and [21], the designed adaptive FLS-based NFTSMC strategy not only has a simpler fuzzy structure, fewer logic rules and free singularity but also updates fuzzy gain automatically, achieves the finite-time stability, and improves the convergence speed, which helps overcome the undesired chattering and enhance the steady-state performance.When the tracking errors are close to the sliding mode surface, the fuzzy switching control part is removed to decrease the unnecessary energy loss.Furthermore, compared with the current works in [10], [13], [17], [22], [26], and [35], where the lumped disturbance D and its derivative Ḋ are both bounded [17], [22], [26], Ḋ changes slowly (i.e., Ḋ = 0) [13], or the bound of D needs to be known [10], [35], this work just requires D to be bounded.These could release the application limitation and possess higher adaptability.3) Compared with the previous approaches in [44], [45], and [47] to tackle the input saturation, the designed auxiliary system not only guarantees the finitetime stability without the boundedness of the saturation region but also overcomes the singularity issue and the saturation-compensation error, which is beneficial to improve the saturation rejection capability.Extensive comparative simulations and realtime experiments are executed to demonstrate the effectiveness and advantages of the developed control strategy.
The rest of this article is organized as follows.Section II describes some preliminaries.Section III gives the controller development and the stability analysis.Section IV performs comparative simulations and experiments.Finally, Section V concludes this article.
Notations: I p×p , 0 p×p , and tr(•) are the p × p identity matrix, the p × p zero matrix, and the matrix's trace, respectively.λ min (•) and λ max (•) stand for the minimum and maximum singular values of a matrix.a transformation of a vector z = [z 1 ; z 2 ; z 3 ] to a skewsymmetric matrix, which can be written by

II. PRELIMINARIES AND PROBLEM FORMULATION A. System Description
The physical structure of the quadrotor UAV is vividly shown in Fig. 1, where the coordinate frames A and B stand for the Earth-fixed and body-fixed frames, respectively.The relation of attitude angle = [φ; θ; ϕ] and angular velocity where S i sin(i), C i cos(i), and T i tan(i), i = φ, θ, ϕ.
Notice that the Euler angles φ and θ are limited to (− π 2 , π 2 ), which is physically meaningful to ensure the quadrotor UAV never be overturned and to prevent the singular issue in the Euler angle propagation equations [1], [13], [20].
The quadrotor attitude dynamics can be modeled via Euler-Lagrangian methodology, as [4], [9] where u = [u 1 ; u 2 ; u 3 ] and J = diag{J x , J y , J z } are the control input and inertial matrix, respectively; denotes the disturbance factor.
ASSUMPTION 1 (see [21]) With the consideration of the structural flexibility and load changes, the inertial matrix J can be described as J = J 0 + J , where J 0 = [J 0,x ; J 0,y ; J 0,z ] and J = [J ,x ; J ,y ; J ,z ] denote the ideal part and uncertain part of J, respectively, which is reasonable to assume J ≤ J with J > 0 denoting an unknown scalar.
ASSUMPTION 2 [21] The external disturbance d is unknown but bounded by an unknown constant d > 0, i.e., d ≤ d.

B. Analysis of Actuator Faults and Input Saturation
The control input with actuator faults and input saturation can be generally expressed by [22], [44] where ] is the designed control input.
The additive fault u f = [u f ,1 ; u f ,2 ; u f ,3 ] represents the uncontrollable portion of the control input that is unmeasurable and time-varying.In a real quadrotor UAV system, u f means the external input, such as wind disturbances, the dampings from various frictions, or the force bias induced by the electric regulator errors of the motors.E = diag{e 1 , e 2 , e 3 } is the actuation effectiveness matrix.This study considers the following types of actuator faults.
1) Type 1: If e i = 1 and u f ,i = 0, it means that the ith actuator is healthy.2) Type 2: If e i = 1 and u f ,i = 0, it means that the ith actuator is an additive fault.3) Type 3: If e i ∈ (0, 1) and u f ,i = 0, it means that the ith actuator is partial effectiveness.4) Type 4: If e i ∈ (0, 1) and u f ,i = 0, it means that the ith actuator is partial effectiveness and additive failure.
ASSUMPTION 3 (see [32], [44]) The parameters u f ,i and e i satisfy the conditions such that |u f ,i | < ∞ and 0 < e i ≤ 1.
The actual input sat(u o,i (t )) can be characterized by where u i < 0 and ūi > 0 are, respectively, the known lower and upper bounds on u o,i (t ), and g r,i (•) and g l,i (•) are unknown nonlinearities.Thus, the attitude model of the quadrotor UAV can be expressed by where R t is the inverse matrix of R s (i.e., R t = R −1 s ), It is worth emphasizing that since the Euler angle θ is constrained to − π 2 < θ < π 2 , R t is nonsingular.On the basis of this, each elements of N 1 and N 2 are, respectively, written by and Thus, it can be concluded that N −1  1 and N −1 2 are nonsingular.ASSUMPTION 4 The lumped disturbance D satisfies D ≤ D, where D > 0 is an unknown scalar.REMARK 1 In this article, the problems of the external disturbances, uncertain parameters, actuator faults, input saturation, and unmeasurable angular velocity are considered simultaneously in the attitude dynamics.Because this could lead to the complexity of the controller development and model establishment, most prior studies in [7], [8], [9], [10], and [12] only considered part of the aforesaid issues.Moreover, in contrast to the previous study in [6], the pitch angle and roll angle are not assumed to vary near zero and be relatively small.Therefore, the considered situations are more realistic.
REMARK 2 For Assumptions 1-4, it is necessary to make further discussions: 1) The uncertain inertia matrix is usually caused by the deployment of sensors, the structural flexibility, and the change in payloads, Assumption 1 is, thus, general [21].2) Since the external disturbances such as wind gusts, aerodynamic friction, and gyroscopic effect are constantly changing and have the limited energy influence, the external disturbances acting on the quadrotor UAV can, thus, be regarded as unknown time-varying yet bounded commands.3) In fact, insufficient battery power would lead to the degradation of the actuator effectiveness, and due to the limited energy and the avoidance of infinite control gain, Assumption 3 is standard to describe the actuator faults [32], [44].4) Since the lumped disturbance D contains the external disturbance, uncertain inertia, and actuator faults, it is reasonable to assume that D is bounded.Besides, in contrast with [10], [13], [17], [22], [26], [35], where the lumped disturbance D and its derivative Ḋ are both bounded [17], [22], [26], Ḋ changes slowly (i.e., Ḋ = 0) [13], or the bound of D is known [10], [35], this study only requires that d is bounded by an unknown scalar.Thus, Assumption 4 removes the application restriction.

C. RBFNN Approximation
Any unknown nonlinear function F (Z) : R w → R can be approximated by the following RBFNN: where is the approximation error and holds δ(Z) ≤ η with η being a positive scalar, and h(Z) ∈ R p can be written by [12] where C i ∈ R w and κ i > 0 represent the center of the receptive field and the width of h i (Z), respectively.In addition, W * ∈ R p expresses the ideal weight vector calculated by [14] where Ŵ * ∈ R p is the estimation of W * , W is the upper bound of W * , p > 1 is the node number of RBFNN.Note that the ideal network weight W * is unknown and used only for analysis objectives, and it needs to be estimated in the design procedure.

D. FLS Design
The fuzzy inference engine uses a set of IF-THEN rules to perform a mapping from the input m where m i and g i are described by the following: 1) m i [antecedent proposition]: P (positive), N (negative), Z (zero); 2) g i [consequent proposition]: PI (positive influence), NI (negative influence), ZI (zero influence).
In this study, the fuzzy linguistic rule bases are given by the following.1) Rule 1: If m i belongs to P, then g i belongs to PI. 2) Rule 2: If m i belongs to Z, then g i belongs to ZI.
3) Rule 3: If m i belongs to N, then g i belongs to NI.The membership functions of m i and g i are shown in Fig. 2, and the choice of m i,r and m i,l is based on the performance demands.From the perspective of simple calculation and intuitive credibility, the singleton fuzzification with triangular membership function and center of gravity defuzzification scheme is employed.As a result, one obtains where v i, j ∈ [0, 1] is the firing strength of Rule j.The fuzzy gains r i,1 , r i,2 and r i,3 need to be chosen suitably, and r i,1 = r a,i , r i,2 = 0, and r i,3 = −r a,i are the centers of PI, ZI, and NI, respectively; the relation 1 is true since it meets the special situation of triangular membership function.Next, this study will analyze only four possible situations, which are as follows.

Thus, one can get a conclusion (v
Furthermore, the final output of the designed FLS can be written as g REMARK 3 To realize low-computation fuzzy approximation, the number of fuzzy rules is unexpected to be large.In this study, the total number of fuzzy rules is less than in previous studies on the attitude control of the quadrotor UAV [3], [15], [17].Particularly, this study performs comparative simulations and experiments to verify that even though the number of fuzzy rules is reduced, good control performance can still be achieved. LEMMA 1 (SEE [36]) For a scalar β > 0 and any two matrices P and Q with appropriate dimensions, it follows that LEMMA 2 (SEE [4]) For a scalar h ∈ (0, 1] and any variable x i , the following inequality holds: LEMMA 3 (SEE [32]) For the nonlinear system ẋ = f (x), f (0) = 0, x ∈ R n , suppose there exists a Lyapunov function V (x) and some scalars 0 < η < ∞, π 1 > 0, π 2 > 0, and Then, the system is fast practically finite-time stable and the function V (x) converges to the following bounded region, as where ∈ (0, 1), and the convergence time T is bounded by where t 0 is the initial time.

III. CONTROLLER DESIGN AND STABILITY ANALYSIS
First, an AFTNNO is proposed to estimate the angular velocity.Then, this study designs an observer-based adaptive fuzzy finite-time attitude controller to deal with various disturbances.Finally, the stability analysis is given.For the convenience of the reader, the overall block diagram for the attitude control system is vividly shown in Fig. 3.

A. AFTNNO Design
First, by letting a new state variable as w = ˙ , one gets where Then, an AFTNNO is designed as where ˆ = [ φ; θ ; φ], ŵ = [ φ; θ ; φ] and Ŵ * stand for the estimations of , w and W * , respectively; k 1 > 0 and k 2 > 0 are design parameters; Here should satisfy 4  5 < l < 1 and 2l − 1 = l 1 l 2 , where l 1 and l 2 are positive odd integers.In this study, Ŵ * can be adjusted by where ϒ ∈ R 3×3 is a positive-definite matrix and k 4 > 0. By denoting w = w − ŵ, W * = W * − Ŵ * and h = h( , w) − ĥ( ˆ , ŵ), it follows that For simplicity, we denote a new estimation error as ξ Then, the time derivative of ξ along ( 16)-( 19) is given by ξ where THEOREM 1 For the attitude system ( 16) and the presented AFTNNO ( 17) and ( 18), the estimation errors ˜ , w and W * converge to the bounded regions in finite time, if the positive-definite matrix C holds the linear matrix inequalities, as where Taking the derivative of ( 22) yields With consideration of Lemma 5, Assumptions 1-4, and μ max with μ max being an unknown positive scalar.Hence, (23) can be rewritten as where 21) and E = B T − CA 3 , (24) can be transformed to be max .In view of Lemma 2, ( 25) is rewritten as as long as k 3 > (2νλ 2 max (CA 3 ))/(k 3 λ min (K 2 )) and k 4 > ( h2 /k 3 ), the inequality (26) will be satisfied.
Afterward, we can transform (26) into the following: where 0 < 1 < 1.Based on ( 27) and (28), it follows that when the function ; and when the function V 1 satis- Thus, based on Lemma 3, it can be obtained that the function V 1 shall converge into the following region, as −1) .(29) By letting 1 = 1 /( 1 (1 − 1 )) and 2 = 1 /((1 − 1 ) 2 ) l 2l−1 , it follows that ˜ , w and W * will drive into the following regions, as and the setting time t is bounded by where t 0 is the initial time.

B. Controller Development
Since the signals and ˙ can be substituted by their estimations ˆ and ˙ , the estimated errors can be written by where d and ˙ d are the desired attitude signal and its firstorder derivative.Then, an NFTSM surface is introduced as where s n = [s n,1 ; s n,2 ; s n,3 ] expresses the NFTSM surface, where and p 0 > 0 and q 0 > 0 denote odd numbers with 0 < p 0 /q 0 < 1.When the state is close to the equilibrium, e ensures fast transient convergence.When the state is far from the equilibrium, the term p a sign( e ) p c would accelerate the error e to zero.From Fig. 4 , it can be observed that FTSM provides a faster convergence speed than that of LSM and TSM surfaces, and the convergence speed of the FTSM surface can be accelerated by selecting a larger value p d .Furthermore, in contrast to the FTSM surface, the NFTSM surface can solve the singularity problem due to the fractional order m a m b − 1 > 0. The derivative of (33) along ( 5) is first formulated as The equivalent controller u o,eq can be obtained by solving (35) without consideration of D, as where 5) are nonsingular and based on the definition of R t , it is not hard to know that P −1 1 and P −1 2 are nonsingular.Since the lumped disturbance is inevitable in practice, this study puts forward the following switching controller, as where u o,sw,1 is a fast switching control part, k a > 0 and k b > 0, k c > 0, and k d > 0 denote odd integers with k c < k d ; u o,sw,2 is a fuzzy logic inference mechanism part.
To overcome the problem of the input saturation, this article presents an auxiliary system with the following form: with a smooth and nonsingular function (χ) being where n a > 1 and n b > 0; δ a > 0 and δ b > 0 are arbitrarily small design constants; u = u o − sat(u).Therefore, the saturation compensation controller u o,sa is constructed by By recalling the previous development, the composite attitude control law is given by Since the upper bound of the lumped uncertainty is hard to obtain accurately, a larger r a needs to be chosen.Unfortunately, this causes more energy consumption and chattering.To overcome this challenge, this article develops the following adaptive mechanism to update the parameter r a : where ra = [r a,1 ; ra,2 ; ra,3 ], ra,i (0) ≥ 0, μ 1 > 0, μ 2 > 0, 0 < μ 3 < 1, and σ > 0 can adjust the estimation rate of ra .From ( 41) and ( 42), an observer-based adaptive fuzzy finite-time attitude controller can be deduced to the following: THEOREM 2 For the attitude system (5), the proposed attitude control law in ( 38), ( 39), (42), and ( 43) can guarantee that all the system signals can bounded and tracking errors converge to sufficiently small bounded regions in finite time.
PROOF Select a composite Lyapunov function as where ra,i = ra,i − ra,i denotes the estimation error, and ra,i is the upper bound of ra,i .Without loss of generality, ra,i is supposed to be ra,i = D c 1 −c 3 + r 0,i , and in which r 0,i is a very small positive constant and belongs to an element of r o ∈ R 3 .
The stability analysis are given by the following three cases.
On one hand, it follows from (53) that when 3 .Based on Lemma 3, one can know that all the system signals will drive into the following: where 3 = ( 2 /(1 − 2 )L * ), and the convergence time T 1 is bounded by where T 0 denotes the initial time.
On the other hand, it follows from (54) that when Then, all the system signals converge to the following regions: . The upper bound of the convergence time T 2 is calculated by To sum up, we can know that all the closed-loop signals will converge to the following regions: and the convergence time T reach is bounded by Next, the error convergence for e is studied by transforming (33) into the following form: The system states will be maintained in the FNTSM surface if the following conditions are held: From (62), the tracking error e converges into the following: where = √ 2min{ 3 , 4 }.The total convergence time T total is T total = T reach + T sliding , where T reach has been given earlier, and T sliding is can be computed according to [31,Remark 4], that is, and −β a e (0) m c −1 < 0 and Lemma 4, it follows that the function (•) is convergent.Thus, Theorem 2 is proved.REMARK 5 In (41), u o,eq is the equivalent control signal to drive the system states into the sliding mode surface under the ideal conditions, u o,sw,1 is designed to realize the fast finite-time convergence and alleviate the undesired chattering, u o,sw,2 is the fuzzy control signal to stem from the time-varying external disturbances, uncertain parameters and actuator faults, and u o,sa is the saturation compensation control law to restrain the negative effects of the input saturation.
REMARK 6 For the auxiliary system in (38) and (39), there are two advantages to be pointed out: (1) Different from the work in [44], the singularity problem can be effectively avoided in this study when the state ξ closes to zero.(2) Unlike [45], this study designs an auxiliary system without the boundedness information of u. (3) Compared with the work in [46], the output of the auxiliary system is smooth.These help improve the antisaturation ability and release the application limitation.REMARK 7 In contrast to the well-known conclusions on adaptive fuzzy control approaches, the main differences of this study are described as follows: 1) The system states in [17] and [18] are supposed to be known, and our results are derived without requiring the system states to be known.2) The control performance in [17], [21], [22], and [46] can be achieved when the convergence time is infinite while our work can achieve the control objective when the convergence time is finite.
3) The control framework of adaptive fuzzy control has a simpler fuzzy structure and fewer fuzzy gains to handle the lumped disturbance compared to the existing works on the attitude control of the quadrotor UAV [3], [15], [17], which helps to reduce the computational burden of fuzzy approximation.
REMARK 8 To achieve better attitude control performance, the main design parameters should be carefully selected according to the following criteria: 1) Choice of parameters m i and l j , (i = a, b, c; j = 1, 2): Under the conditions 1 < m a m b < 2 and m c > m a m b , larger values m a m b and m c can improve the convergence speed.From (30) and 4  5 < l < 1, a smaller value l is useful for providing high-accuracy state estimation, but in turn, it tends to cause high-frequency oscillations.Moreover, positive odd integers l 1 and l 2 are chosen based on 2l − 1 = l 1 l 2 .2) Choice of parameters β i and n i , (i = a, b): In order to realize shorter convergence time and smaller convergence regions, large values β a and β b are usually selected.A larger value n a > 1 can quickly overcome the effect of the saturation error, while the parameter n b > 0 plays a key role in achieving the finite-time convergence property.3) Choice of parameters σ and μ i , (i = 1, 2, 3): The parameter σ is introduced to adjust the update rate of ra , which is usually chosen to be small enough to accelerate the update speed, but it may cause the overestimation problem.Besides, an appropriate value μ i can guarantee the finite-time convergence and avert the drift of the parameter ra .4) Choice of parameters k i , (i = a, b, c, d ): Since this study designs an adaptive fuzzy control part, the values of the control gains k a and k b can be selected to be smaller than the work [10], which helps to reduce the input amplitude.The term sign(s n ) kc k d with k c < k d can enhance the system robustness, where a larger value k c k d helps reduce the chattering phenomenon but decreases the system robustness.Thus, it should make a balance in the choice of the value k c k d .
In particular, there is no standard procedure for selecting these design parameters.They are currently chosen by trial and error until satisfactory control results are achieved.

IV. SIMULATIONS AND EXPERIMENTS
In addition to the aforementioned theoretical discussion and analysis, extensive simulations and experiments are performed in this section to verify the validity of the proposed controller, by comparing the proposed controller without an auxiliary system, proportion-differentiation (PD), finite-time DSC [18], DO-NTSMC [35], adaptive fuzzy finite-time control [22], FTC-NFTSM [33], and adaptive NFTSMC [34].The comparison of the proposed controller, the proposed controller without an auxiliary system, PD, finite-time DSC [18], DO-NTSMC [35], and adaptive fuzzy finite-time control [16] is to verify that the proposed controller can achieve saturation elimination, fault tolerance, and strong robustness while the comparison of the proposed controller without an auxiliary system, FTC-NFTSM [33], and adaptive NFTSMC [34] is to verify that the proposed switching control part is capable of realizing free chattering.With the help of the auxiliary system, only the proposed controller can solve the input saturation.

A. Simulation Analysis
Simulation: The reference attitude command d (t ) = [φ d (t ); θ d (t ); ϕ d (t )] is predefined by and χ(0) = 0.The fuzzy rules are set as follows: if s n,i > 1.2, then v i,1 = 1, v i,2 = 0 and v i,3 = 0; if 0 < s n,i ≤ 1.2, then v i,1 = 0.5, v i,2 = 0.5 and v i,3 = 0; if −1.2 < s n,i ≤ 0, then v i,1 = 0, v i,2 = 0.5 and v i,3 = 0.5; else if s n,i ≤ −1.2, then v i,1 = 0, v i,2 = 0 and v i,3 = 1, where i = 1, 2, 3.The input saturation sat(u o,i ) is described as The control results of comparative simulations are visually shown in Figs.6-9.As depicted in Fig. 6, we can see that the proposed controllers with/without an auxiliary system can resume to closely track the desired attitude signals after a short transient period even if larger disturbances suddenly happen, and have smaller oscillations than other controllers.It should be noticed that although other comparative controllers also have the ability to stabilize the attitude system and achieve acceptable attitude control performance when the disturbances do not increase, the output attitude signals are difficult to accurately track the desired signals when the disturbances become suddenly large at the time t = 10 [s].These results reflect that the proposed controllers with/without an auxiliary system can improve the ability of disturbance suppression.To be precise, it can be found from Fig. 7 that the tracking errors of the proposed controllers with/without an auxiliary system are smaller than those of other remaining controllers, which fully illustrates that the proposed controllers with/without an auxiliary system are obviously less affected by system uncertainties, timevarying disturbances, and actuator faults.From Fig. 8, the time responses of control inputs infer that the proposed controller can effectively overcome the input saturation and is protected from the chattering influence while the FTC-NTSMC and adaptive NFTSMC methods suffer from serious chattering issues and the DO-NFTSMC method has a slightly chattering phenomena.This reason is that the switching control part u o,sw in (37) containing the fast-type control u o,sw,1 and the compensation control u o,sw,2 can reduce control gain and accelerate the convergence rate.As shown in Fig. 9(a), it can be observed that the designed AFTNNO can realize the precise estimation of the angular velocity even in the presence of the lumped disturbance and input saturation.weight, respectively.In addition, Fig. 9(d) displays the evolution of adaptive fuzzy gain, and we can see that there is no parameter drift problem.

B. Experiment Analysis
To better demonstrate the superiority of the proposed controller, comparative experiments are performed in this section.The hardware configuration of the quadrotor UAV is shown in Fig. 10, where the personal computer is used to detect the quadrotor's states, a digital signal processor is utilized as the onboard control computer to execute the control algorithm, the attitude information can be estimated by an inertial measurement unit, the desired attitude signals 0) = 0, and χ(0) = 0.The fuzzy rules are set as follows: if s n,i > 1.5, then v i,1 = 1, v i,2 = 0 and v i,3 = 0; if 0 < s n,i ≤ 1.5, then v i,1 = 0.5, v i,2 = 0.5 and v i,3 = 0; if −1.5 < s n,i ≤ 0, then v i,1 = 0, v i,2 = 0.5 and v i,3 = 0.5; else if s n,i ≤ −1.5, then v i,1 = 0, v i,2 = 0 and v i,3 = 1, where i = 1, 2, 3.
The obtained results of Experiment 1 are depicted in Figs.11-  tracking errors with time, respectively.One can see from Figs. 11 and 12 that the proposed controllers with/without an auxiliary system not only overcome the effect of the disturbances in a short time but also avert large oscillations during the control process.In particular, the curves of the proposed controllers with/without an auxiliary system are almost the same.These visually exhibit satisfactory robustness of the proposed controller even in the presence of input saturation.From the three figures on the left-hand side of Fig. 13, with the help of the designed auxiliary system, the control inputs of the proposed controller are constrained within the specified range.To avoid the interference of the auxiliary system, we compare with the proposed controller, FTC-NTSMC, and adaptive NFTSMC to infer that the chattering issue can be solved, as depicted in the three figures on the right-hand side of Fig. 13.From Fig. 14(a), the designed AFTNNO can estimate the information of angular velocities accurately.Fig. 14(b)-(c) shows the curves of the sliding mode surfaces and the norm of the RBFNN weight, respectively.The time evolution of the adaptive fuzzy gains is shown in Fig. 9(d), which indicates that the problem of the parameter drifting can be avoided.The control performance of Experiment 1 is consistent with the theoretical analysis and simulation results.
Experiment 2: Based on Experiment 1, Experiment 2 considers stronger disturbances and persistent actuator faults.Apart from this, other conditions such as the desired attitude signal and control parameters are the same as Experiment 1.The effectiveness matrix E and additive actuator fault u f are set as E = diag{0.85+ 0.15 sin(0.5t); 0.9 + 0.1 cos(t ); 0.75 + 0.25 cos(1.2t)} and u f = [3 − cos(2t ); 2 + sin(0.5t); 2 cos(t )], respectively.In the experimental tests, the actuator faults are injected by software via the thrust signals before being sent to the quadrotor UAV. 15 and 16, respectively.From Figs. 15 and 16, one can find that although the quadrotor UAV is subjected to bigger disturbances, persistent actuator faults, and input saturation, the proposed controller still makes the real attitude signals rapidly return to the desired attitude signals and maintains satisfactory tracking effects.These imply that the proposed controller is almost unaffected by the above negative factors and has stronger system robustness than other controllers.As shown in Fig. 17, it can be seen that the control inputs of the proposed controller are still within the prescribed range even when the saturation issue occurs in comparison to the proposed controller without an auxiliary system, and  the proposed controller can avoid the chattering issue in comparison to the FTC-NTSMC and adaptive NFTSMC.The control results in terms of fault tolerance, saturation elimination, and free chattering are attributed to the adaptive fuzzy mechanism and the auxiliary system.Fig. 18(a) shows that the AFTNNO can precisely estimate the information of angular velocity online.As shown in Fig. 18(b), the dynamics of the sliding mode surface s n can be quickly stabilized around zero.Besides, the time responses of the adaptive RBFNN weight Ŵ * and the adaptive fuzzy gain ra are clearly shown in Fig. 18(c) and (d), respectively, which means that the designed adaptive parameters would ultimately converge.

C. Numerical Analysis
To quantitatively evaluate the performance of different controllers, four performance indices [49] are employed in this study and are concretely described as where N is the total time of the simulation operation.Since the tracking errors and control inputs may be negative, these indices are mostly described by using either the absolute value or the square value.Especially, the smaller values of μ SE , μ AE , μ TWAE , and μ TEC illustrate less persistent oscillations, quicker convergence rate, higher steady-state precision, and less energy consumption, respectively.
The values of the above performance indices are recorded in Tables I and II to better explain the superiority of the proposed controller.From Table I, it can be seen that all the index values of the proposed controllers with/without an auxiliary system are smaller than those of the other six controllers, which strongly implies that the proposed control framework can provide a higher precision.By comparing the proposed controller and adaptive NFTSMC, it can be illustrated that the adaptive fuzzy control part can improve the robustness of the system to various disturbances.In contrast to the proposed controller without an auxiliary system, the proposed controller successfully addresses the problem of input saturation with the help of the designed auxiliary system.Actually, the input saturation limits the growth of the control input, which inevitably leads to slower tracking speed.However, the control precision is slightly reduced, but not significantly.The reason for this is that the designed auxiliary system could limit the control input beyond the specified requirements.In addition,  II that when external disturbances become larger and the actuator faults occur, the proposed controllers with/without an auxiliary system are much less affected than the other controllers when the control parameters are not reset, which means that the proposed control framework has higher adaptability and stronger robustness.Through experimental implementation and numerical analysis, it fully demonstrates the feasibility and superiority of the proposed controller in terms of disturbance suppression, free-chattering, fault tolerance, and saturation elimination.

V. CONCLUSION
This study designs an observer-based adaptive fuzzy finite-time attitude control strategy for quadrotor UAVs under unavailable angular velocity, external disturbances, uncertain dynamics, actuator faults, and input saturation.First, an AFTNNO is constructed to estimate the information of angular velocity online.Subsequently, an adaptive FLS-based NFTSMC scheme is proposed to automatically adjust the control gain.Furthermore, an auxiliary system with free singularity is designed to overcome the input saturation.Although a series of comparative simulations and real-time experiments authenticate the advantages and effectiveness of the proposed control strategy, the following limitations are worth investigating in the future.
1) Although Remark 8 gives detailed guidance on how to choose the control parameters, it is necessary to study the reliable tuning mechanism so as to obtain their optimal values.
2) Although the designed FLS provides a simpler fuzzy structure and fewer logic rules, the design complexity of the whole control system has not significantly reduced, which is one of the future studies.
3) Since the problems of sensor failures and state constraints are not considered, our research direction will also focus on how to ensure the safer flight in the ever-changing environments.
For any vector y ∈ R m and a scalar b > 1, sign(y) b is defined as sign(y) b = [|y 1 | b sign(y 1 ); . . .; |y m | b sign(y m )], which can be proved that d dt (sign(y) b ) = b|y| b−1 ẏ.The subscript × is

Fig. 2 .
Fig. 2. Illustration of Membership functions.(a) Input fuzzy sets for m.(b) Output fuzzy sets for r i .

Fig. 3 .
Fig. 3. Block diagram of the presented control framework.
K 1 and K 2 are arbitrary positive-definite matrices, k 1 and k 2 are positive constants, k 3 > 2νλ 2 max (CA 3 ) k 3 λ min (K 2 ) with ν 1 > 0 being an arbitrary parameter, and k 4 > h2 k 3 .PROOF Construct the Lyapunov function as and m a and m b denote positive odd integers with 1 < m a m b < 2 and m c > m a m b .REMARK 4 The mathematical expressions of LSM, TSM, and FTSM surfaces are described by LSM surface: s l = β c e + ˙ e (34a) TSM surface: s t = β d e + ˙ surface: s f = ˙ e + p a sign( e ) p c + p b sign( e ) p d (34c)

Fig. Simulation:
Fig. Simulation: Time response of attitude signals.
Fig. 9(b) and (c) shows the time responses of sliding mode surfaces and the norm of adaptive RBFNN

Fig. 9 .
Fig. 9. Simulation: Time response of various system states under the proposed controller.(a) Estimation of the attitude angular velocity based on the designed AFTNNO.(b) NFTSM surface.(c) Adaptive RBFNN weight.(d) Adaptive fuzzy gains.

Fig.
Fig. Experiment Time response of attitude signals.
14. Figs.11 and 12 display the tracking performance of attitude angles and the change of the attitude

Fig. 18 .
Fig. 18.Experiment 2: Time response of various states under the proposed controller.(a) Estimation of the attitude angular velocity based on the designed AFTNNO.(b) NFTSM surface.(c) Adaptive RBFNN weight.(d) Adaptive fuzzy gains.

TABLE I
Comparison of Attitude Control Performance TABLE II Change of Performance Indices in Experiment 2 to Experiment 1 one can see from Table