An adaptive sliding mode fault‐tolerant control of a quadrotor unmanned aerial vehicle with actuator faults and model uncertainties

An adaptive sliding mode fault‐tolerant control strategy is proposed for a quadrotor unmanned aerial vehicle in this article to accommodate actuator faults and model uncertainties. First, a new reaching law is proposed, with which a sliding mode control (SMC) law is constructed. The proposed reaching law is made up of a sliding variable and the distance between it and a designated boundary layer, and it can effectively suppress the unexpected control chattering while preserving the necessary system tracking performance. Then, an adaptive SMC scheme is proposed to further solve the fault and uncertainty compensation problem. The proposed adaptation law helps to prevent overestimation of the adaptive control parameters, as well as avoiding control chattering. Finally, a number of comparative simulation tests are carried out to validate the effectiveness and superiority of the proposed control strategy. The demonstrated quantitative comparison results confirm its advantages.

An overview of the existing FTC strategies for quadrotor UAVs is presented in Reference 19.In general, the commonly used FTC strategies for quadrotor UAVs can be divided into two categories: passive and active FTC approaches.In Reference 20, a sliding mode-based passive FTC scheme is designed for a quadrotor UAV to tolerate actuator faults and model uncertainties.In Reference 21, both passive and active FTC methods are designed for a commercial quadrotor under actuator faults based on sliding mode technique, and their advantages and disadvantages are investigated and compared through analytical results.Essentially, passive FTC approaches use the inherent robustness of the designed control algorithm to mitigate prescribed faults and they may sacrifice tracking control performance under fault-free conditions.In contrast to passive FTC approaches, active FTC methods can reconfigure the controller in real time to compensate for faults based on the fault information provided by the fault diagnosis unit.In Reference 22, by combining a recurrent neural network-based fault estimation technique with a sliding mode feedback controller, an active FTC mechanism is created for actuator fault accommodation.In Reference 23, to compensate for the loss of effectiveness actuator defect caused by battery depletion, an active FTC approach based on linear parameter varying control technique is provided.Although active FTC system can effectively compensate actuator faults, its performance heavily depends on the level of accuracy of the estimated fault magnitude.The fault estimation error and delay may degrade the FTC performance.To resolve this issue, many researchers focus on investigating adaptive FTC strategies in recent years.In Reference 24, to compensate the actuator faults for a quadrotor UAV without the necessity for a fault estimation mechanism, a nonlinear adaptive FTC approach is developed for altitude and attitude tracking.In Reference 25, the position and attitude tracking control of a quadrotor UAV is built using an indirect neural network-based adaptive FTC method that can compensate actuator faults and external disturbances.In Reference 26, an adaptive radial basis function neural network fuzzy sliding mode control (SMC) scheme is built to accommodate actuator faults for a coaxial octorotor UAV.In Reference 27, an adaptive fuzzy backstepping FTC scheme is presented for a modified quadrotor UAV to achieve accommodation of actuator faults.Despite the fact that the aforementioned control strategies are capable of achieving desired tracking performance of quadrotor UAVs in the face of actuator faults, the issues of model uncertainties are not adequately investigated.
In addition, fault-tolerance capability is not the only factor to consider when it comes to ensuring the safety control of quadrotor UAVs.When UAVs are deployed for practical applications, many uncertain factors often occur which may affect their safety and reliability.One of the challenges for safety control of quadrotor UAVs is to deal with model uncertainties.To enhance the quadrotor UAVs' capability for tolerating certain level of model uncertainties, a number of control strategies have been developed.In Reference 28, to manage the attitude and position of a quadrotor UAV while attenuating parameter changes and uncertainties, a model-free-based terminal SMC technique is designed.In Reference 29, an adaptive fuzzy terminal SMC strategy is presented to overcome the quadrotor's model uncertainties and provide a robust path tracking performance.In Reference 30, a complementary control strategy based on approximate dynamic programming is developed to improve the quadrotor UAV tracking performance under model uncertainties.It is worth noting that the robust tracking control performance of the above mentioned control strategies can only be achieved under fault-free conditions.To further ensure robustness of the developed control strategies, the uncertain factors including actuator faults and model uncertainties need to be jointly considered.
In this sense, the developed control system for quadrotor UAVs is anticipated to tolerate the adverse impact produced by actuator faults and model uncertainties for the sake of safety control of quadrotor UAVs.In Reference 31, to mitigate actuator faults and model uncertainties for a quadrotor UAV, a dual adaptive FTC strategy is developed.In Reference 32, an active FTC scheme is presented for a quadrotor UAV, in which the model uncertainty is compensated with adaptive neural network mechanism, and actuator fault information is provided by a fault estimation unit and compensated in an active way.In Reference 33, a sliding mode observer is designed to estimate the uncertainties induced by actuator faults and unknown parameters and integrated with the feedback SMC to compensate the adverse effect of actuator faults.SMC is well known for its robustness against the so-called matched uncertainties.It has been successfully applied in a variety of fields. 34,35However, the robustness of conventional SMC can only be ensured by selecting a large discontinuous control gain, which may lead to unanticipated control chattering. 36To solve this problem, a number of adaptive SMC approaches for accommodating actuator faults and model uncertainties have been presented.In Reference 37, an adaptive SMC with projection method is designed that ensures constrained estimations of uncertain inertial parameters for a quadrotor.In Reference 38, for a class of nonlinear systems with actuator faults a multivariable SMC scheme with the aid of an adaptive single-hidden-layer feedforward network is designed.As reported in Reference 39, a fractional order SMC with adaptive fuzzy approximation technique is designed for the attitude system of a spacecraft subject to inertia uncertainty and multiple type actuator faults.In Reference 40, to compensate actuator faults and uncertainties for high-speed trains, an adaptive fault-tolerant SMC scheme is presented.In Reference 41, a continuous adaptive integral SMC strategy is built for a space manipulator to achieve trajectory tracking control against actuator uncertainties.
Although substantial research studies have been done towards safety control of quadrotor UAVs, there still exist several problems that need to be solved in order to enhance the safety and reliability of quadrotor UAVs: (1) most of the previous works using SMC to deal with actuator faults and model uncertainties usually employ a constant discontinuous control gain, which may lead to a conservative tracking performance in order to avoid control chattering; (2) when utilizing adaptive SMC technique to mitigate actuator faults and model uncertainties, the discontinuous control component of SMC is usually overused resulting in unanticipated control chattering; (3) because of the inherent tracking errors, if sliding variable is used for constructing the adaptive schemes to mitigate actuator faults and uncertainties, it may not be able to guarantee the convergence of adaptive control parameters.
In an effort to overcome the aforementioned challenges in achieving safety control of quadrotor UAVs, this article proposes a unique adaptive sliding mode FTC strategy with a new reaching law to jointly compensate both actuator faults and model uncertainties.The major contributions of this article can be summarized as follows: (1) A new reaching law is proposed in this article, which can adapt to the variations of the distance between the sliding variable and the designated boundary layer.It features that when the sliding variable is far away from the desired sliding surface, a large discontinuous control gain is generated for a faster reaching time, and when the sliding variable approaches to the desired sliding surface, the discontinuous control gain will gradually decrease to zero to suppress control chattering.(2) In the existing adaptive sliding mode FTC designs 37,42 for quadrotor UAVs, the adaptation law is commonly constructed with sliding variable.Because of the inevitable tracking errors, it is possible that the adaptive control parameters will not be able to converge.To address this issue, inspired by Xu et al., 43 this article proposes an adaptation law that can prevent overestimation of adaptive control parameters and suppress control chattering.(3) To achieve actuator fault tolerance, instead of merely adaptively changing the discontinuous control component of SMC in contrast to the works in References 38 and 44, with the developed control strategy, the continuous and discontinuous control components are both constructed with adaptive parameters.In this case, the utilization of the discontinuous control component will be decreased, which can help to prevent control chattering.
The rest of this article is organized as follows.In Section 2, the modeling and system formulation of the researched quadrotor UAV are presented.In Section 3, the detailed design procedures of the proposed adaptive FTC strategy are explained.Then, the effectiveness of the proposed control strategy is validated and compared in Section 4 through a series of simulation tests.Finally, Section 5 summarizes the general conclusions of this article.

PROBLEM FORMULATION
This section describes the mathematical modeling of the quadrotor UAV under consideration, as well as the corresponding system formulation.Figure 1 depicts the researched quadrotor UAV, that is produced by Quanser.It is assembled with four 10-inch propellers driven by brushless motors in a plus configuration.The motor #1 and #2 spin clockwise (CW), while the motor #3 and #4 rotate counter-clockwise (CCW).The motors are connected to electric speed controllers that receive commands from the flight controller in the form of pulse-width modulation (PWM) outputs.The generated thrust of each individual propeller is varied to control the quadrotor UAV.

Modeling of quadrotor UAV
Two reference frames, the body-fixed reference frame and the earth-fixed reference frame, are first defined for the purpose to determine the equations of motion of the quadrotor UAV.The center of gravity of the quadrotor UAV is chosen as the origin of the body-fixed reference frame, whereas the takeoff point is chosen as the origin of the earth-fixed reference frame.The axes of the body-fixed reference frame and the earth-fixed reference frame are represented by (o b , x b , y b , z b ) and (o e , x e , y e , z e ), respectively.Concentrating on the fault-tolerant attitude control of the quadrotor UAV, the dynamic equations of motion of the quadrotor UAV are derived as follows using the Newton-Euler formulation: where I is a diagonal matrix denoting the quadrotor UAV's moment of inertia that is defined as T , and  B = [p, q, r] T represent the vector of resultant torques and the vector of angular velocities, respectively, with respect to the body-fixed reference frame.
The resultant torques operating on the quadrotor UAV are composed of the propeller-generated torques ( T ) and the rotational motion induced torques ( f ), that can be formulated as follows: where   ,   ,   are the propeller-generated torques along x b -, y b -, and z b -axis, respectively., ,  are the roll, pitch, and yaw angle of the quadrotor UAV, and k di (i = 1, 2, 3) is the aerodynamic drag coefficient caused by rotational motion.
The relationship between the propeller-generated torques and the PWM inputs of the electric motors can be formulated as per the illustrated configuration of the quadrotor UAV in Figure 1 as follows: where u i (i = 1, 2, 3, 4) is the PWM input of the ith motor whose range is [0, 0.05].A command of 0 corresponds to zero throttle, which will cause the motors to stop.A command of 0.05 corresponds to full throttle.K u is the thrust gain related to the propeller-generated force, K y is the torque gain related to the reaction torque generated by the propellers, L is the distance between the individual motor and center of gravity of the quadrotor UAV.
In this sense, substituting Equations ( 2) and (3) into Equation ( 1), one can obtain: Furthermore, the following equations can be used to define the relationship between the angular velocities and the Euler-angle rates: where R E B is a transformation matrix from the body-fixed reference frame to the earth-fixed reference frame.
Assume that the changes of the roll and pitch angles are small to facilitate the controller design.Under this assumption, the transformation matrix R E B will be close to an identity matrix, and the angular velocities (p, q, r) can be simply replaced by the Euler-angle rates ( φ, θ, ψ).Note that, this assumption is only made for controller design.In this situation, Equation (4) can be rewritten as follows:

System formulation
Consider a nonlinear quadrotor UAV system with actuator faults and model uncertainties as: where ) is a diagonal matrix that represents the level of remaining control effectiveness of the actuators, where l ci (t If the ith actuator encounters certain level of fault, then 0 ≤ l ci (t) < 1, otherwise, the ith actuator works well.In the subsequent sections, the notation (t) is removed for convenience of expression.For example, u(t) is expressed as u.
In order to facilitate the fault-tolerant attitude control design for the quadrotor UAV, define the state vector of the system as x = [, φ, , θ, , ψ] T .Thus, the quadrotor UAV system described in Equation ( 6) can be expressed as a general integral-chain nonlinear system as follows using this state vector: where i = 1, 2, 3 represents the roll, pitch, and yaw subsystem, respectively, T .Δh 1i and Δh 2i represent the model uncertainties of the system induced by the uncertain moments of inertia of the quadrotor UAV, which are bounded as Remark 1.The diagonal matrix L c is mainly used to model the investigated actuator faults.In the context of adaptive FTC framework, the main priority is to compensate the fault and may not need to know the actual faulty level of the actuators.Therefore, the unknown matrix L c is not used in the developed control law and the following equation  i = B ui u is employed to calculate the control law.

ADAPTIVE SLIDING MODE FTC STRATEGY
In this section, an adaptive sliding mode FTC strategy is designed to simultaneously compensate actuator faults and model uncertainties for the investigated quadrotor UAV. Figure 2 depicts an overview of the proposed control strategy.To achieve the desired attitude control performance, two issues need to be addressed.The first issue is to design a reaching law-based SMC scheme that ensures the desired system tracking performance while minimizing control chattering under Overview of the proposed adaptive sliding mode FTC scheme.
nominal conditions.The second task is to investigate the adaptive control schemes in order to determine the adaptive control parameters without overestimation and then to combine them with the developed control law for accommodation of actuator faults and model uncertainties.

Design of reaching law-based SMC
In general, the SMC design typically consists of two steps.The initial step involves establishing a sliding surface on which the system performance can be attained as expected.The second step is to construct an appropriate control law that will push the sliding variable towards the designed sliding surface, ensuring that the system will satisfy the sliding mode reaching requirement.
The desired time-varying system trajectory is denoted as x d 1i , and its derivative is assumed to be bounded.The corresponding tracking error states can be described as x e 1i = x 1i − x d 1i and x e 2i = x 2i − ̇xd 1i .With the defined tracking error states, the integral sliding surface for the considered attitude control system of the quadrotor UAV is designed as: 45 where s i0 is the linear combination of the tracking error states which is formulated as s i0 = c i x e 1i + x e 2i , and z i includes the integral term that is determined by Equation (10).
where k c1i and k c2i denote the adjustable design parameters, and t 0 represents the initial time instant.Then, combining the above equations, the following equation can be obtained as: Following the construction of the sliding surface, the next step is to design a suitable control law for satisfying the reaching requirement of the sliding mode.The conventional SMC law is usually made up of two control components, namely the continuous control component and the discontinuous robust control component.However, a large discontinuous control gain will lead to the occurrence of control chattering even though it can increase the robustness of the SMC.
In order to address this problem, this article proposes a novel reaching law as follows: where k c3i and  i are positive design parameters, 0 <  < 1, Δs i = s i − Φ i sat(s i ), and with Φ i representing the defined boundary layer thickness.
Remark 2. As can be observed from the afore-defined variable Δs i , it represents the algebraic distance between the current sliding variable and the designated boundary layer.Δs i is equal to zero when the sliding variable is inside the designated boundary layer.
Remark 3.With the proposed reaching law, it can be observed from Equation ( 12) that if Δs i increases, the function will converge to the value of k c3i ∕ i , which is larger than k c3i .As can be seen, a large discontinuous control gain is used for a faster reaching time if the sliding variable is far away from the specified sliding surface where the tracking performance is unsatisfactory.

Remark 4. With the decrease of the variable |Δs
When the sliding variable is driven onto the desired sliding surface s i = 0 by the subsequent designed control law, the discontinuous control gain will decrease to zero.In this sense, with the sliding variable approaching the desired sliding surface, the discontinuous control gain will gradually decrease to zero to suppress control chattering.
Based on the defined sliding surface in Equation ( 11), its time derivative can be calculated as: By recalling the definition of x e 1i and x e 2i , substituting Equations ( 8) and (12) into Equation ( 14) leads to: Therefore, by solving Equation ( 15), the corresponding control law can be designed as follows: where ]. Furthermore, model uncertainties are common in quadrotor UAVs.They might compromise the tracking control performance of quadrotor UAVs or possibly cause instability of the system.With this in mind, an adaptive control law is developed to compensate model uncertainties as: The corresponding adaptation laws for estimating the uncertain parameters are then designed in the following manner: where  1i and  2i are positive parameters that control the adaptation rate.
Theorem 1.Consider an integral-chain nonlinear system in Equation ( 8) with model uncertainties, given the defined sliding surface in Equation (11) and the proposed reaching law in Equation ( 12), an adaptive feedback control law is designed in Equation ( 17) that is updated by Equation (18).Therefore, the desired sliding motion will be obtained and system tracking performance can be ensured with the discontinuous control gain chosen as k Proof.Consider the Lyapunov function as: Then, considering the condition that Δs i ≠ 0, that is, the sliding variable is outside the boundary layer, and recalling the control law in Equation ( 17) and the adaptation law in Equation ( 18), the time derivative of V 1 can be calculated as: As a result, in the presence of certain level of model uncertainties with bounded Δ 1 and Δ 2 , the system's stability can be guaranteed by using the developed control law and adaptation law with reasonable adaptation rate.▪

Adaptive fault-tolerant SMC
Considering the occurrence of actuator faults in the quadrotor UAV, the diagonal matrix L c , which measures the fault severity of the actuators, is not an identity matrix.One way to achieve FTC purpose of compensating the negative effect of actuator faults is to use a fault diagnosis unit to estimate the actual value of L c and incorporate it with the developed control law.In practice, however, there may be a significant time delay in obtaining the essential fault information.Moreover, there may be fault estimation inaccuracies, which would have an impact on the relevant control performance.In this section, an adaptive FTC technique is proposed for the studied quadrotor UAV to tolerate actuator faults without the need for fault information.Under this condition, Equation ( 8) can be rewritten as: where I c is an identity matrix with the same dimension as L c ,  i =  d i +  e i denotes the actual virtual control signal generated by the individual actuators under the actuator faulty condition,  d i = B ui I c u represents the desired virtual control signal derived from the developed control law without considering the actuator faults,  e i = −B ui (I c − L c )u represents the virtual control signal error due to the occurrence of actuator faults.
As can be observed from Equation (21), to mitigate the negative effects of virtual control signal error induced by actuator faults while maintaining the desired system tracking control performance, one intuitive way is to adaptively adjust the parameter g i .In this case, let g i  e i = gi  d i , and Equation ( 21) can be reconstructed as follows: where ĝi represents the estimation of g i , and gi = ĝi − g i denotes the estimation error.
Under this condition, instead of using g i for developing the feedback control law, the estimated parameter ĝi should be employed to compensate the actuator faults and maintain the original tracking control performance of the closed-loop system.Moreover, recalling the developed control law in Equation ( 17), the parameters ĥ1i and ĥ2i need to be utilized in conjunction with ĝi to develop the control law for compensation of actuator faults and model uncertainties.Therefore, by denoting Γi = ĝ−1 i ĥ1i , Υi = ĝ−1 i ĥ2i , and Ψi = ĝ−1 i , the feedback control law can be developed as: The corresponding adaptation laws for uncertain parameter estimation are designed as: where  1i ,  2i , and  3i are positive parameters for tuning the adaptation rate.
Remark 5.As can be seen from the developed feedback control law in Equation ( 23), rather than adaptively adjusting the discontinuous control gain to compensate actuator faults, Ψi and k * c3i are treaded together as the discontinuous control gain and Ψi is adaptively estimated.Since Ψi is employed in both continuous and discontinuous control components, both of them will contribute to compensate the actuator faults after fault occurrence.In this sense, the utilization of the discontinuous control component will be decreased, which can contribute to preventing control chattering.(8) with actuator faults and model uncertainties.Given the proposed reaching law in Equation (12), by utilizing the developed feedback control law in Equation ( 23) and the adaptation law in Equation ( 24), the desired system tracking control performance can be achieved in spite of actuator faults and model uncertainties.

Theorem 2. Consider an integral-chain nonlinear quadrotor UAV system in Equation
Proof.A Lyapunov function is selected as: where Γi = Γi − Γ i , Υi = Υi − Υ i , and Ψi = Ψi − Ψ i .For simplicity of analysis, V 2 is divided into four components as: where First, recalling Equation ( 23), the time derivative of V 21 is calculated as: Then, employing the adaptation law in Equation ( 24), one can obtain that: Combining Equations ( 27) and ( 28) and considering the condition that Δs i ≠ 0, it gives: For the above equation, we have the following two conditions.If s i > Φ i , then sign(s i ) = 1, sat(s i ) = 1, and sign(s i ) In this case, we can get that sign(s i ) In this case, we can obtain that sign(s i )Δs i = −Δs i = |Δs i |.Based on above analysis, we can get that no matter Δs i is positive or negative, we can always guarantee that Δs i sign(s i ) = |Δs i |.Therefore, Equation ( 29) can be rewritten as: As a result, under the influence of certain level of actuator faults and model uncertainties with bounded Δ 1 and Δ 2 , system stability can be maintained from any initial conditions by using the proposed adaptive control strategy with reasonable adaptation rate.▪ Remark 6.As per the definition of the designed variable Δs i , by employing this variable for constructing the adaptation law in Equation (24), the adaptation will be ceased as long as the sliding variable is inside the defined boundary, which can help to avoid overestimation of the discontinuous control gain.As a result, the proposed adaptive scheme can avoid control chattering as compared to the existing adaptive SMC methods in the literature, in which it is difficult to guarantee the adaptation convergence due to the use of sliding variable for adaptation law formulation and the inevitable tracking errors.

SIMULATION RESULTS AND DISCUSSIONS
A series of simulation tests are undertaken to quantitatively examine the effectiveness of the proposed adaptive fault-tolerant control method (PAFTC) for mitigating actuator faults and model uncertainties of the quadrotor UAV.Table 1 lists the physical specifications of the quadrotor UAV under investigation.For the sake of comparison and demonstrating the advantages of PAFTC, the performance of the SMC constructed with the proposed reaching law (SMCRL) and a nominal adaptive SMC whose adaptation law is formulated with sliding variable (NASMC) 42 are investigated as well.The control parameters of PAFTC are set as follows: where [t 0 , t 1 ] spans the entire time frame of the simulation test.
In Scenario 1, a 20% loss of control effectiveness fault is injected into actuator #1 at 10 s and actuator #4 at 20 s, the uncertain parameters for drag forces are set to k di = 0.2 (i = 1, 2, 3), and the model uncertainties for inertial moments are set to ΔI = 0.5I and injected into the attitude system at 15 s.The detailed tracking performance of roll, pitch, and yaw motions are demonstrated in Figure 3.After faults occurrence, the PAFTC can instantaneously change both the discontinuous control gain as shown in Figure 3D and the adaptive control parameters as illustrated in Figure 4 to mitigate the negative impact caused by actuator faults and retain the original system tracking performance.In addition, both the compared SMCRL and NASMC can retain the appropriate system tracking performance as well.However, as shown in Figure 5, the compared NASMC induces control chattering because of the employment of a sliding variable in the adaptation law formulation.Figure 3 shows that the PAFTC developed in this article outperforms the compared SMCRL and NASMC with less tracking errors and avoidance of control chattering.Moreover, both the PAFTC and SMCRL use the same reaching law for SMC development, however, the PAFTC has less tracking errors than the SMCRL.This is because in the PAFTC, the developed adaptation law is incorporated in both continuous and discontinuous control components of SMC, which can contribute to compensating actuator faults without merely relying on the robustness of the discontinuous control component of SMC.
In Scenario 2, a larger actuator fault with 35% loss of control effectiveness is injected into actuator #1 at 10 s and actuator #4 at 20 s, respectively, to further test the effectiveness and capability of the proposed control method.Moreover, when compared to Scenario 1, the uncertain parameters for drag forces are adjusted with a bigger amplitude and time-varying property.The uncertain parameters are set as k di = 10 sin(0.2t)(i = 1, 2, 3), and the uncertain inertial moments are the same as those in Scenario 1.The corresponding tracking performance of roll, pitch, and yaw motions are demonstrated in Figure 6.Compared to Scenario 1, due to the increased severity of actuator faults and model uncertainties, the compared SMCRL is more affected and exhibits large tracking errors.Nevertheless, under this faulty and uncertain condition, although the compared NASMC can track the desired trajectory, it exhibits oscillatory tracking errors.As can be observed from Figure 8, the compared NASMC uses more control effort and induces severe control chattering.In contrast, the PAFTC is still able to retain the necessary tracking performance with modest tracking deviations.Figures 7 and 6D demonstrate the change of adaptive control parameters and discontinuous control gain, which are bigger than those in Scenario 1 because of the increased severity of actuator faults and model uncertainties.
Table 2 summarizes the performance indices related to the roll, pitch, and yaw tracking.Taking roll motion control in Scenario 1 as an example, compared to SMCRL and NASMC, the PAFTC has improved the RMSE of roll angle from 0.1466 • and 0.1297 • to 0.0611 • with the enhancement rate of 58.32% and 52.89%, respectively.As per the performance indices in Table 2, the PAFTC can ensure the desired tracking performance of the quadrotor UAV even under the unfavorable conditions involving actuator faults, inertial moment variations, and time-varying drag forces.In addition, the quantitative comparison results confirm the effectiveness and superiority of the PAFTC.

CONCLUSIONS
An adaptive sliding mode FTC strategy for a quadrotor UAV is proposed in this article to compensate both actuator faults and model uncertainties.First, a new reaching law is proposed to achieve a fast reaching time while avoiding control chattering.Based on the proposed reaching law, a baseline SMC law is developed, which can effectively suppress the unexpected control chattering.Then, to further address the problem of compensating actuator faults and model uncertainties, an adaptive FTC scheme is proposed to mitigate the adverse impact of both actuator faults and model uncertainties.With the aid of the proposed adaptation law, overestimation of the uncertain adaptive control parameters can be prevented.Finally, the effectiveness and advantages of the proposed control strategy are demonstrated in a series of comparative simulation tests, in which the quadrotor UAV is exposed to different level of actuator faults, inertial moment variations, and time-varying drag forces.

TA B L E 1 2 LΦ 3 = 0. 1 ,
Physical specifications of the researched quadrotor UAV.Symbols Interpretations Values I = diag([I xx , I yy , I zz ]) Moment of inertia I = diag([0.03,0.03, 0.04]) kg ⋅ m Distance between motor and center of gravity 0 13 =  23 =  33 = 1, Γ3 (0) = Υ3 (0) = 0, Ψ3 (0) = 0.04.The non-adaptive parameters of SMCRL and NASMC are chosen the same as those of PAFTC.The following two scenarios are used to demonstrate the effectiveness of the proposed control strategy.Furthermore, for the purpose of quantitatively evaluating the tracking performance of the demonstrated control approaches, an index indicating the root mean squared error (RMSE) of the attitude tracking is specified as:

3 F I G U R E 4 5 F I G U R E 6
Tracking performance of quadrotor UAV in Scenario 1. (A) Roll motion responses; (B) pitch motion responses; (C) yaw motion responses; (D) change of discontinuous control gain.Change of adaptive control parameter and sliding variable of PAFTC in Scenario 1. (A) Roll motion related adaptive control parameter; (B) pitch motion related adaptive control parameter; (C) yaw motion related adaptive control parameter; (D) change of sliding variable.Control inputs of quadrotor UAV in Scenario 1. (A) control inputs with SMCRL; (B) control inputs with NASMC; (C) control inputs with PAFTC.Tracking performance of quadrotor UAV in Scenario 2. (A) Roll motion responses; (B) pitch motion responses; (C) yaw motion responses; (D) change of discontinuous control gain.

7 F I G U R E 8
Change of adaptive control parameter and sliding variable of PAFTC in Scenario 2. (A) Roll motion related adaptive control parameter; (B) pitch motion related adaptive control parameter; (C) yaw motion related adaptive control parameter; (D) change of sliding variable.Control inputs of quadrotor UAV in Scenario 2. (A) Control inputs with SMCRL; (B) control inputs with NASMC; (C) control inputs with PAFTC.
Comparison of performance indices.
TA B L E 2