Decentralized Adaptive Fault-Tolerant Cooperative Control for Multiple UAVs with Input Saturation and State Constraints

This paper proposes a fault-tolerant cooperative control (FTCC) scheme for multiple UAVs in a distributed communication network against input saturation, full-state constraints, actuator faults, and unknown dynamics. Firstly, by considering physical limitations, an auxiliary control signal is designed to simplify the analysis process. Secondly, to avoid the di ﬃ culty in the backstepping design caused by full-state constraints, virtual control signals are constructed to transform constrained variables, while the dynamic surface control is adopted to avoid the phenomenon of “ di ﬀ erential explosion. ” Thirdly, a disturbance observer (DO) is designed to estimate the unknown uncertainty caused by parameter uncertainty and actuator fault. Moreover, a recurrent wavelet fuzzy neural network (RWFNN) is used to compensate for the estimation errors of DO. Finally, it is proved that all states are uniformly ultimately bounded (UUB) by Lyapunov and invariant set theory. The e ﬀ ectiveness of the proposed scheme is further demonstrated by the simulation results.


Introduction
In recent years, the development of unmanned aerial vehicle (UAV) technology has led to wide applications. However, single UAV provides limited capabilities, which may not be applicable to some highly complex tasks. Inspired by multiagent technology, researchers begin to investigate the application technology of multiple UAVs (multi-UAVs). Compared with a single UAV, multi-UAVs have more benefits in terms of forest fire monitoring, area detection, and disaster assistance. Unlike a single UAV, the cooperative control of multi-UAVs need the information from neighboring UAVs, which significantly increasing the control design challenge.
As the basis of multi-UAVs control, the cooperative control design is an important task. In the past few years, numerous research results of cooperative control have been reported. In [1], a cooperative control strategy for motion control of multiple unmanned vehicles was proposed, which can keep the formation during the motion. In [2], a novel distributed intermittent control framework for containment control of multiagent system was proposed, which can reduce the communication burden via a directed graph. In [3], the obstacle avoidance problem of multi-UAVs in multiple obstacle environment was studied. In [4], a robust adaptive control strategy for cooperative control of UAVs under the decentralized communication network was proposed against uncertainty. In [5], the authors investigated the cooperative transport control problem using multirotor UAVs. In [6], a system analysis method was proposed for the tracking control problem of multi-UAVs. The distributed framework was used to describe the dynamic model of UAV, and the information of nodes and networks were considered in the distributed control design. [7] studied the output feedback formation control of multi-UAVs without velocity and angular velocity sensors, which were obtained via the state observer. However, the above researches only focused on the distributed control of the first-order or second-order systems, and there exist few research on the cooperative control of fixed-wing UAVs with high-order nonlinear characteristics.
In addition, the number of components in the multi-UAV system is more than that of a single UAV. Therefore, designed to transform the restricted input. To handle the full-state constraint problem, virtual control signals are defined to replace the constraints, which can simplify the back-stepping design. For the unknown nonlinear dynamics caused by actuator faults and other unknown uncertainties, disturbance observer (DO) is designed for providing the estimation, while a recurrent wavelet fuzzy neural network (RWFNN) is adopted to further compensate the estimation error. In the RWFNN, the online adaptive learning strategy of parameters is designed based on the Lyapunov theory. Compared with other existing works, the main contributions of this paper are as follows: (1) In [21][22][23], actuator faults, input saturation, output constraints, and external disturbances were considered, while the state constraints were not taken into account. To obtain satisfactory control performance against such factors, the FTCC scheme is designed in this paper by simultaneously considering the state constraints, actuator faults, and external disturbances.
(2) Compared with [24][25][26][27], which assessed uncertainty dynamics by designing a DO without compensation of the DO estimation error, this work further adopts an RWFNN to offset the estimation error, in which the parameters are updated by the proposed online learning strategy.
The organization of this paper is arranged as follows. Section 2 describes the preliminaries and problem statement. The design process of the FTCC scheme and the stability analysis are given in Section 3. Section 4 shows the simulation results and analysis. Finally, the conclusion is drawn in Section 5.

Preliminaries and Problem Statement
2.1. System Dynamics. In this paper, the cooperative control of N UAVs is investigated. The set of UAVs is denoted as Ω = f1, 2, ⋯, Ng, and the position dynamics of the ith UAV is described as where i ∈ Ω, x i , y i , and z i are the positions. V i , γ i , and χ i are velocity, flight path angle, and heading angle, respectively. The aerodynamic force equations are given by where i ∈ Ω, m i and g are the mass of the ith UAV and 2 International Journal of Aerospace Engineering gravitational coefficient, respectively; T i , D i , L i , Y i are the thrust, drag, lift, and lateral forces, respectively, and μ i , α i , and β i are the bank angle, angle of attack, and sideslip angle, respectively. The attitude kinematic model is expressed as where i ∈ Ω. p i , q i , and r i are the angular rates. The angular rate model is given as where i ∈ Ω. L i , M i , and N i are roll, pitch, and yaw moments, respectively. The forces T i , D i , L i , and Y i and the aerodynamic moments L i , M i , and N i are expressed as where Q i = ρV 2 i /2 is the dynamic pressure and s i , b i , and c i represent the wing area, wing span, and mean aerodynamic chord, respectively. T i max and δ T i are the maximum thrust and instantaneous thrust throttle setting, respectively. C iL , C iD , C iY , C iL , C iM , and C iN are given by where δ ia , δ ie , and δ ir are aileron, elevator, and rudder deflections, respectively.
and C iN r are aerodynamic derivatives. The definition of the inertial terms c ij ðj = 1, 2, ⋯, 9Þ in (4) can be found in [28].

Control-Oriented Model.
By defining X i1 = ½μ i , α i , β i T , X i2 = ½p i , q i , r i T , and U i = ½δ 1a , δ 1e , δ 1r T and substituting (5), (6), and (7) into (1), (2), (3), and (4), then it follows that where F i1 , G i1 , F i2 , and G i2 are given by It can be seen that F i2 and G i2 have many aerodynamic parameters. However, it is difficult to obtain accurate aerodynamic parameters of UAVs in practical engineering applications. To facilitate the design of the controller, F i2 and G i2 can be decomposed into known items F i20 , G i20 and unknown items ΔF i2 , ΔG i2 , respectively.
Then, the attitude model can be described as where ΔF i2 , and ΔG i2 are unknown nonlinear functions caused by the uncertain parameters, while F i1 , G i1 , F i20 , and G i20 are known functions.
Remark 1. Due to the physical constraints, the sideslip angle β i ≠ ±π/2, and det ðG i1 Þ = −sec β i , so G i1 is invertible. In addition, G i2 is related to aerodynamic parameters so that it is invertible in the flight envelope.
2.3. Actuator Fault and Input Saturation. In this paper, the actuator fault is considered, which includes gain and bias failures. Therefore, the fault model can be expressed as [29] where i ∈ Ω, U i0 = ½u i01 , u i02 , u i03 T represents the designed control signal, and represents the gain fault matrix, and U if ∈ ℝ 3 represents bias fault vector.
In the practical application, the output of the actuator is limited. In order to avoid the incredible phenomenon caused by actuator saturation, the designed control signal U i0 needs to satisfy the following constraint: where τ = f1, 2, 3g, u i0τ max is a positive constant and u i0τ min is a negative constant, which is the maximum and minimum allowable values for the actuator, respectively.
To solve input saturation problem, an auxiliary signal v i = ½v i1 , v i2 , v i3 T is used to get control signal U i0 ðv i Þ, and U i0 ðv i Þ = ½u i01 ðv i1 Þ, u i02 ðv i2 Þ, u i03 ðv i3 Þ T , which is expressed as By substituting (13) and (15) into (12), then the attitude model can be expressed as where is an unknown nonlinear function. Due to D i being related to auxiliary control signal v i , designing the observer of unknown function D i for generating the control signal v i will cause the problem of "algebraic ring," which is solved by introducing the following first-order filter: where Λ is a diagnonal matrix with positive eigenvalue and ξ is an auxiliary control signal.
Remark 2. As shown in (13), the fault that occurs in the actuator will diminish its ability to provide control input. For example, the range of motion of the rudder surface can reach −25~25 deg in the normal state, while it may deteriorate into −20~20 deg after the fault occurs. It seems in the fault conditions the actuator is more likely to occur saturation, i.e., cannot reach to the expected control value. In this paper, the upper and lower boundaries of the control input are fixed to the values in the normal state of the actuator, and a hyperbolic function is used to prevent actuator saturation as shown in (15). Meanwhile, using virtual control signal v i and ξ to generate the expected control signal U i0 and using RWFNN to evaluate uncertainty item D i which contains the actuator bias section, even though the fault could occur, the system can still keep stable.

State Constraints.
The states X i1 = ½μ i , α i , β i T and X i2 = ½p, q, r T generally have constraints in the practical application. In this paper, such a problem has been concerned. Due to the fact that states X i1 and X i2 have limits, inspired by works [30,31], a transformation is used to convert the restricted states X i1 and X i2 to unrestricted states Z i1 and Z i2 : International Journal of Aerospace Engineering where τ = 1, 2, 3. X i1τ , X i2τ , Z i1τ and Z i2τ represent the τth element of X i1 , X i2 , Z i1 , and Z i2 , respectively. X i1τ and X i2τ represent maximum allowable range of the τth element of X i1 and X i2 , respectively, while the X i1τ and X i2τ represent minimum allowable range of the τth element of X i1 and X i2 , respectively. Remark 3. Since (19) and (20) are bijective, X i1 and X i2 will always stay in their own limits if Z i1 and Z i2 are bounded on ∀t ≥ 0.
2.5. Basic Graph Theory. In this paper, an undirected graph G = fV , C, Ag is used to describe the formation flight of N UAVs. The set of UAVs is described by V = fv 1 , v 2 ,⋯, v N g, C ⊆ V × V represents the communication links between UAVs, and A ∈ R N×N is the adjacency matrix. If the link between the ith UAV and jth UAV exists, a ij = a ji = 1, which are the elements of A. A path from the ith UAV to the kth UAV can be a sequence If there exists a path between any two UAVs, then G is a connected graph. A set is defined 2.6. Control Objective. In this paper, the control objective is to design an FTCC scheme for N UAVs, such that the attitude tracking error of each UAVs can be finally uniformly bounded, while the attitude X i1 and X i2 of all UAVs are always in limits, even when a portion of UAVs is subjected to actuator saturation and actuator faults.

Fault-Tolerant Cooperative Controller Design and Stability Analysis
In this section, the process of designing the FTCC scheme for N UAVs will be described. A main method adopted in the design is transforming the individual tracking error of each UAV to the synchronization tracking error.

Fault-Tolerant Cooperative Controller
Design. Define the independent tracking error of ith UAV asZ i1 = Z i1 − Z i1d , then the cooperative tracking error of ith UAV is defined as where E i1 = ½E i11 , E i12 , E i13 T , λ 1 and λ 2 are positive parameters, which are used to regulate the cooperative tracking performance.
Using the Kronecker product " ⊗ ", and define E 1 = , then the cooperative tracking error of all UAVs can be expressed as By recalling Lemma 5, it yields kZ 1 k = k½ðλ 1 + λ 2 LÞ −1 ⊗ I 3 E 1 k ≤ 1/ðσ min ðλ 1 + λ 2 LÞÞkE 1 k, where σ min ð·Þ represents the minimum singular value of matrix "·." Therefore, Using (22), the synchronization error of each UAV E i1 can be expressed as Differentiating (24) yields where Substituting (11) into (25) yields Based on the back-stepping control architecture, (27) can be expressed as where E i2 = Z i2 − Z i2d and Z i2d is a virtual control signal. By using a low-pass filter, one has 5 International Journal of Aerospace Engineering where k ϵ 1 is a positive constant and Z i2d is an auxiliary signal, designed as where K i1 is a positive diagonal matrix. Defining the filtering error as By substituting (30) into (28), one can obtain where h i1 is a positive constant and k·k represents 2-norm of vector.
To estimate the unknown function D i of each UAV, the following DO is designed for the ith UAV: where k 1 and k 2 are positive parameters,D i is the estimate Taking the derivative of (35) and using (33) give From (36), it can be known that the estimation errorD i will not converge to zero since _ D ≠ 0. In order to estimate the unknown function D i more accurate, a five-layer RWFNN is used to estimate the errorD i of the DO with The RWFNN structure is illustrated in Figure 1, including five layers (input layer, membership layer, rule layer, composite layer, and output layer) [33]. The components of the RWFNN are introduced as follows: Layer 1-Input Layer: Input layer is the first layer, where r i = ½r i1 , r i2 , ⋯, r iv 1 T is the input features of RWFNN. The output of layer 1 is expressed as where j ∈ f1, 2, ⋯, v 1 g, v 1 is the dimension of input features, and y ð1Þ ij represents the output of jth neuron of Layer 1. Layer 2-Membership Layer: Layer 2 has v 1 rows and v 2 columns, and its output can be described as where j ∈ f1, 2, ⋯, v 1 g, k ∈ f1, 2, ⋯, v 2 g. v 2 is a positive constant, depending on the number of neurons. y ijk denotes the neuron of layer 2 in row j, column k.
Layer 3-Rule Layer: Layer 3 has v 2 neurons, and the output of layer 3 is described as where k ∈ f1, 2, ⋯, v 2 g, and y ð3Þ ik denotes the kth neuron of Layer 3.
Layer 4-Composite Layer: Layer 4 also has v 2 neurons, and the input of Layer 4 consists the output of the wavelet layer, recurrent layer, and Layer 2, where the output of wavelet layer is described as where y i2 (4) y i2 (5) y iv 2 (3) y iv 3 (4) y iv 3 (5) y i21 (2) y i31 (2) y i12 y i32 y i22 II II Figure 1: The structure of the RWFNN for each UAV. 6 International Journal of Aerospace Engineering the jth neuron of Layer 1 to kth neuron of wavelet layer. ψ ik represents the output of kth neuron of wavelet. w F ijk , a ijk , and b ijk represent the connecting weight, translation, and dilation variables, respectively.
The output of Layer 4 is expressed as where y ik is the output of kth neuron of Layer 4. w r ik and y ð4Þ ik ðt − 1Þ represent the recurrent weight and the output at the previous time step of the kth neuron, respectively.
Layer 5-Output Layer: Layer 5 has v 3 neurons, which determines the dimension of the final output. Each neuron's output of this layer is given by where l ∈ f1, 2, ⋯, v 3 g, w ikl is the connecting weight, and y ð5Þ il denotes the lth neuron of Layer 5.
Using (42), one can express y ð5Þ i in the following vector form: , which are expressed as In this paper, v 3 is set as 3 due to the fact that the estimated variable Δ i is three-dimensional. Therefore, there exist optimal values ω where ε i1 is the approximation error.
To design the adaptive law of weights for estimating the unknown item, it is needed to obtain the gradient of y ð5Þ i of its variables firstly.
Differentiating both sides of y ð5Þ i , one can obtain To yield dy ð4Þ i , by the same way, differentiating both sides of y ð4Þ ik , then For term dy ð3Þ ik , one can obtain it by combining (37), (38), and (39), that is where "⋄" represents dot division between matrices, and "ð * Þ n " represent the aligned " * " itself does n times dot product. And the c ik and σ ik have been defined in (44). For the term dψ ik and dðω r ik y ð4Þ ik ðt − 1ÞÞ, there exists Then, by combining (50), (48), and (49), dy ik can be expressed as Using (50) and (51), one can further write dy ð4Þ i as where , , , Moreover, using the property of Kronecker product, the term dω ð5Þ i · y ð4Þ i in (58) can be transformed to the following column vector form: where " ⊗ " represents Kronecker product, and "vecð * Þ" represents the operation that converts the aligned " * " to a column vector, that means vec ω To simplify the representation, using ω ! ð5Þ i to represent vecðω ð5ÞT i Þ. Employing (58), (52), and (54) yeilds the following total differential equation: where On the other hand, the total derivative of dy ð5Þ i can be expressed as the following total differential form: Hence, one can derive that Taking into account (58) and (59) and using the Taylor expansion, y ð5Þ i can be expressed as International Journal of Aerospace Engineering Using b Δ i to estimate the unknown item Δ i , and it is expressed as where Γ H i = diag ðsign ðE T i2 g i2 ðX i2 ÞÞÞ andĤ i is an estimated value and will be introduced later.
Taking the derivative of E i2 and using (17) and (20), one can obtain where g i2 ðxÞ and x = ½x 1 , By using the back-stepping method and defining v id as a virtual control signal, then _ E i2 can be expressed as where E i3 = v i − v id . In order to reduce computational burden of taking time derivative for virtual control signal v id , a filter is used to obtain v id , which is given by where k ϵ 2 is a positive constant, and v id is an auxiliary signal, designed as where K i2 is a positive diagonal matrix. Define the filter error as ϵ i2 = v id − v id , then one can obtain from (66) that Substituting (65) with (67) and (62), with considering j ε i1 − ε i2 j ≤ H i in which j * j represents the absolute value of the matrix " * ", while definingĤ i as the estimation of H i andH i = H −Ĥ i as the estimation error, one can obtain Taking the time derivative of E i3 and using (18), one can obtain Design the auxiliary control signal ξ as where K i3 is a positive diagonal matrix, and h i1 and h i2 are positive constants. By combining (70) and (71), one has Finally, the adaptive laws of RWFNN for the ith UAV are developed as where γ i1~γi5 are discontinuous switching constants to prevent the weights to infinity and γ i6 is a positive constant, 9 International Journal of Aerospace Engineering where the switching constants are designed as where ω ð5Þ , c, σ, ω F i , and ω r i are positive constants, which represent the boundness of weight ω ! ð5Þ To this end, the proposed FTCC scheme is shown in Figure 2 to better illustrate the design principle and functional components in the control system. Remark 6. Many papers choose multiplication on input and recurrence data as an operation on the neuron of the composite layer in RWFNN. However, it is sometimes problematic. For example, when inputs from layer 3 are minuscule, the outputs of the composite neuron will also become exceedingly small under multiplication. Under the limitation of computational precision, the outputs are equal to zero. Since the outputs will loop to the next multiplication, the outputs will always be zero, which causes neuron inactivation. Therefore, this paper uses the addition operation as an alternative, and the back-propagation gradient is deduced in detail using vectorized expressions, i.e., (46)-(59).

Stability Analysis
Theorem 7. Consider the N UAVs (1)-(4) under the distributed communication network against the actuator faults (13), states constraints, and input saturation (15), if the control laws are chosen as (30), (67), and (71), the disturbance observers are developed as (33), and (34), and the adaptive laws are constructed as (73), (74a); then, all the states in the system are ultimately uniformly bounded and strictly confined within the limits.

International Journal of Aerospace Engineering
Under the condition of (74a), the term γ i1 e ω ! ð5ÞT i ω ! i has the following property: The reason is as follows: When On the other hand, when ω ð5Þ < kω ! ð5Þ Hence, by combining (78) and (79), one can obtain (77). By the same way, one can conclude that Furthermore, the term γ i6H T iĤi in (76) satisfies that Substituting (76) with (77), (80), and (81) then yeilds where δ is By choosing the parameters K i1 and K i2 as respectively, where K i10 and K i20 are positive diagonal

12
International Journal of Aerospace Engineering    Figure 9: Control input signals δ ia , δ ie , and δ ir of each UAVs (i = 1, 2, 3, 4). loop dynamics E i2 and ϵ i1 are converge rapidly, then, it can be further represented as _ E i1 ≈ −K i10 E i1 , so that the settling time of the dynamic E i1 approximately equal to 3/λ min ð K i10 Þ. There exists a tradeoffa larger value K i10 helps to reduce the settling time while it needs larger change of the state X i2 , which may cause saturation, so K i10 is taken as diag f5, 5, 5g. Similarly, parameters K i20 and K i3 decide the convergence rate of the inner-loop control errors E i2 and E i3 , respectively, and their values normally take several times as the K i10 to get faster response. Moreover, for parameters η i1~ηi6 , their inverses serve as learning rates of neural network, which usually take as a value among 1~1000 for fixed rate learning algorithms. In addition, parameters k ϵ i1 and k ϵ i2 are adjustment factors of dynamic surface filter, which are usually taken as a large value. Finally, for μ ib , it can adjust the astringency of the error E i3 , which is the smaller the better, but the too small value will tend to cause virtual control signal "explosion".
The response of bank angle μ i , angle of attack α i , and sideslip angle β i of each UAV are shown in Figures 4-6, respectively. It can be seen that all UAVs can track their ref-erences μ id , α id , and β id . Although the fault occurs to UAV 1, UAV 2, UAV 3, and UAV 4 at 8 s, 10 s, 12 s, and 14 s, respectively, all UAVs can quickly track individual references again.
The response of state X i2 is shown in Figure 7. It can be seen that the states p i , q i , and r i of all UAVs never exceed its upper or lower bound. Meanwhile, if the state constraints are not considered in the FTCC scheme, the states q 1 and q 3 will exceed the lower bound, which is shown in Figure 8. In addition, the control input signals δ ia , δ ie , and δ ir are presented in Figure 9. Since the actuator constraint scheme (15) is adopted, the input signals never exceed their upper and lower bounds.

Conclusion and Future Work
This paper has explored an FTCC scheme for multi-UAVs under the distributed communication network, in which the issues including input saturation, state constraints, actuator faults, and unknown disturbances have all been taken into account.
It can be noted that the proposed FTCC scheme only considers fixed and undirected communication network. In addition, communication delay and communication interferences are not considered, and finite-time convergence technology has not been considered in the FTCC scheme, so the control performance cannot be achieved in finite time. Moreover, compared to the Euler attitude angles, the airflow attitude angles are necessary and easy to combine with the UAV's outer loop for position control, hence in this paper, it is directly used in the attitude control. However, using the airflow attitude angles for feedback control is less reliable than the former. Furthermore, sensor fault may occur at the same time, which perhaps outweigh the risk of actuator fault, so it deserves more attention and investigation. Finally, state measurements have been directly used in the control law without considering noise filtering, so that the performance may be degraded when sensor measurements have severe noises. Taking into account the noise filtering algorithms and sensor faults simultaneously will significantly increases the difficulty of proving the closed-loop system stability, which makes the issue challenging.
Therefore, in future work, the essence of communication delays, finite-time convergence technology, the reliability of using airflow attitude angle, sensor fault, and noise filtering will be taken into account on the basis of existing research. Besides, based on the simulation results from MATLAB/ Simulink, the hardware-in-the-loop verification scheme will be adopted to further verify the proposed control scheme towards more practical applications.

Data Availability
No data were used to support this study.

Conflicts of Interest
The authors declare that they have no conflicts of interest. 16 International Journal of Aerospace Engineering