Predefined-time anti-saturation fault-tolerant attitude control for tailless aircraft with guaranteed output constraints

This work investigates the anti-saturation attitude tracking control for the tailless aircraft with guaranteed output constraints, in the presence of uncertain inertia parameters, bounded external disturbance, and actuator faults/failures. A predefined-time adaptive backstepping attitude control scheme has been proposed, the main features of this scheme lie in (a) designing a predefined-time filter to deal with the ‘explosion of complexity’ and singularity problem; (b) introducing a nonlinear state-dependent function to handle the asymmetric time-varying output constraints; (c) compensating for the impact of the actuator faults/failures and input saturation by a nonlinear function and bounded estimation simultaneously. Moreover, the proposed control scheme can ensure all signals in the closed-loop system converge to a residual set around the origin within a predefined time, and this time constant can be set freely by the designer, independently of initial conditions. Finally, numerical simulations have been conducted to verify the performance of the proposed predefined-time fault-tolerant control scheme.


Introduction
Unconventional tailless flying wing layout aircraft has a lightweight body with a large payload capability, besides flexible maneuverability and good stealth. This concept has been the focus of the control and aerial research [1][2][3][4], besides being the future line of development of the unmanned combat aerial vehicle, large transport aircraft, and bombers. When compared with the conventional layout aircraft, tailless aircraft has many merits, e.g., lower structural weight, higher liftto-drag ratio, smaller radar cross section, less oil consumption, and bigger load-carrying capacity. However, the cost of employing a tailless layout are the difficulties in the control and stability, particularly in the attitude control, of the aircraft. Furthermore, the tailless layout leads to the lower effectiveness of the control surfaces; hence, numerous redundant control surfaces are required to coherently generate the moment [1], which may increase the risk of actuator failure. Therefore, the design of a fault-tolerant controller for tailless aircraft is an important and challenging task.
Faults from actuators, sensors, or other components will ensue system instability or even catastrophic consequences. Therefore, the fault-tolerant flight control is particularly important for flight safety. Recently, the fault-tolerant control has attracted the global attention of researchers [5][6][7][8][9][10]. To enhance the reliability and availability of the attitude control for tailless aircraft suffering from actuator faults, a multiple time-scale reconfigurable sliding mode control has been proposed in [11]; nevertheless, only a linear model was considered. In [12], an adaptive optimal fault-tolerant control (FTC) has been proposed, based on an incremental fault detection and diagnosis (FDD) scheme, which incur a complex structuration and large computational burden. In [13], a robust adaptive nonlinear FTC was proposed via norm estimation, but only the partial loss of effectiveness (PLOE) was considered. However, the above works all used ideal actuator model. A common obstacle for the aforementioned methodologies is that the practical actuator deflection has magnitude constraints. Thus, it is necessary to consider the actuator failures and input constraints together.
In addition, it is necessary to consider the attitude constraints problem in a practical flight, to prevent the loss of control of the aircraft. Barrier Lyapunov function (BLF) has become an effective scheme for the output constrained problem [14], such as log-type BLF [15], tan-type BLF [16], and integral-type BLF [17]; on this basis, some adaptive control schemes have been developed for nonlinear systems [14][15][16][17][18]. Furthermore, to deal with asymmetric time-varying constraints, the asymmetric time-varying BLF was first proposed in [19]. It should be noted that no matter which BLF is chosen, there exists a feasibility condition on virtual controllers, which complicates the selection of the design parameter [20,21]. A nonlinear state-dependent function was proposed for full state constraints in [21], which converts a constrained system into a non-constrained system, and the feasibility conditions are removed.
However, all the methods mentioned above rarely address the issue of the convergence time of the closedloop system, which implies that the convergence time of those systems may be relatively long. To achieve the desired maneuverability of the aircraft, the convergence time should be finite. Consequently, several finite-time control schemes have been developed for the actual systems, such as hypersonic vehicles [22], spacecraft [23], and quadrotor UAV [24]. Nevertheless, the convergence time of finite-time control schemes is dependent on the initial conditions. Furthermore, a fixed-time method was presented in [25], and it can achieve a finite convergence time independent of the initial conditions. Therefore, the fixed-time control schemes have been extensively applied for the uncertain nonlinear systems [26][27][28]. However, it is often difficult to find an explicit relation between the upper bound of the convergence time and the design parameters. In certain cases, the bound of the convergence time cannot be featured as less than a fixed constant, even by tuning the design parameters. To adjust the bound of the convergence time more conveniently, the concept of the predefined-time control has been formulated [29][30][31]. It has the advantage of fixed-time stability, besides the free setting of the convergence time. Recently, some latest researches have been done on predefined time stability control. In [32], a predefined-time backstepping exact tracking control scheme has been proposed for a class of mechanical systems. In [33], a Lyapunov-like characterization of predefined-time stability has been studied Motivated by the above discussions, the problem naturally arises: how to design a predefined-time FTC for tailless aircraft with input and output constraints. To our best knowledge, in the existing literature there have not been results which provide solution for this problem.
This article investigates the predefined-time adaptive fault-tolerant control problem for a tailless aircraft with the input and output constraints when it experiences uncertain inertia parameters and bounded external disturbance. The main contributions of this work are listed below.
1. A novel predefined-time adaptive backstepping attitude control scheme is presented for tailless aircraft. Moreover, with the newly designed predefined-time filter, the 'explosion of complexity' and singularity problem are avoided. 2. By utilizing a state transformation function, the asymmetric time-varying output constraints problem is solved. Compared with the asymmetric timevarying BLF, this method removes the feasibility conditions on the virtual controls and facilitates the selection of control parameters. 3. A nonlinear function is introduced to represent the saturation degree of the actuators, which can be unified with the actuator failure model, and the impact of the actuator failures and input saturation is compensated simultaneously.

Preliminaries
The notations in this article are given in the following, · denotes the Euclidean 2-norm of vectors or matrixes, R m represents the real m-vector and R m×n represents m × n real matric. λ min (B) denote the minimum eigenvalues of the matrix B. For a given Consider a nonlinear systeṁ where S is the domain of attraction, and f : S → R n is a continuous nonlinear function, with f (0) = 0. The initial condition is denoted by Definition 1 [34] (1) The equilibrium of (1) is finitetime stable if it is asymptotically stable, and T (x 0 ) < +∞ such that for all t T (x 0 ), x(t; x 0 ) = 0. Here, T (x 0 ) is defined as the settling-time function of (1).
Definition 2 [29] Define a predefined constant T c > 0, and the equilibrium of (1) is predefined-time stable if it is fixed-time stable, and settling time function T (x 0 ) < T c , where T c is called predefined time.
Lemma 1 [35,36] Suppose there is a radially unbounded Lyapunov function V : S → R + for (1) such thaṫ where 0 < υ < 1 is a system parameter, and T c > 0 is a predefined-time constant. Then, the equilibrium of system (1) is predefined-time stable with predefined time T c .
for all the solutions x(t; x 0 ) of (1), then the trajectory of systemẋ = f (x) is practically predefined-time stable, and the residual set of the solution of system (1) is given by where 0 < ϑ < 1, and the predefined time can be given by Integrating from 0 to the settling time T (x 0 ) both sides, it has Note that sup x 0 ∈S arctan b a V υ/2 (x 0 ) = π 2 , hence the supremum of the settling-time function follows When is considered, and there exist a 0 < ϑ < 1 such that the inequality (3) can be rewritten aṡ oṙ Then, the system trajectory will fall into the residual set in predefined time T pc . Similarly, the system trajectory of (8) will fall into the residual set Combining (9) and (10), the residual set can be given as ⎫ ⎬ ⎭ Remark 1 According to Definitions 1 and 2, in contrast to the finite-time stable, the convergence time of the fixed-time stable is bounded and independent of the initializations. However, it is difficult to find a relationship directly between T max (x 0 ) and the system parameters. It means that the settling-time function cannot be arbitrarily set through the system tunable parameters. Compared with the fixed-time stable, the predefinedtime stable has the advantages of the fixed-time stable; whereas, the upper bound of the convergence time can be directly selected through an appropriate selection of the system parameters.
Remark 2 : From Lemma 1, when compared with the predefined-time stable, the practically predefined-time stable has bounded stability but not asymptotic stability in predefined time. However, the form (3) with the term is a more general condition for the practical systems, and the predefined-time stable is a special case of the practically predefined-time stable.
Lemma 2 [37] For any given positive constants a 1 , a 2 , and a 3 , it holds that Lemma 3 [38] For any state variable x i ∈ R, i = 1, · · · , n, and positive constant υ > 0, satisfy Lemma 4 [39] For any y x, x, y ∈ R, and positive constant υ > 0, it has Lemma 5 [28] For any x ∈ R, and positive constant Lemma 6 [40] For any x ∈ R, and positive constant

Kinematics and dynamics of tailless aircraft
Considering a tailless flying wing unmanned aerial vehicle similar to the B-2 layout. It has eight control surfaces, including two split drag rudders and six elevons, and the configuration of tailless aircraft is given in Fig. 1. A 6-DOF model of the tailless aircraft can be written as [3], [41,42]: where V is the airspeed, χ the velocity heading angle, γ the flight path angle, and m the mass of the aircraft. Further, F X , F Y , and F Z represent the total aerodynamic and propulsion forces along the body axes X , Y , and Z , respectively, which can be measured by an accelerometer along the body axes. g = 0, 0, g z T , represents the gravitational acceleration in the Earth coordinates, μ is velocity bank angle, α is the angle of attack, β is the sideslip angle, and we define ω = [ p, q, r ] T , represents the angular velocity. J is the rotary inertia, expressed as C h/b denotes the direction cosine matrix from the body coordinates to the heading coordinates, and C h/e denotes the direction cosine matrix from the Earth coordinates to the heading coordinates, given as following where sn and cs are the abbreviation of sine and cosine, respectively. Here where q = 1 2 ρV 2 denotes the dynamic pressure, S is the wing area, and b,c represent the wingspan and the mean aerodynamic chord, respectively, C l0 , C m0 , C n0 are the dimensionless moment coefficients, {δ li , i = 1, 2, 3, 4}, {δ ri , i = 1, 2, 3, 4} denote the control surfaces on the left side and right side, respectively, and B ∈ R 3×8 represents the control effectiveness matrix.

Actuator faults and saturation
The actuator failures model in this article is given as [6,7,40]: where δ ci represents the input of the ith actuator subject to saturation, and ρ i ∈ [0, 1) represents the effectiveness of the ith actuator.δ i represents an unknown bounded failure constant. Three cases are considered in this article, viz., 2. Partial Loss of Effectiveness (PLOE), where ρ i ∈ (0, 1). 3. Total Loss of Effectiveness (TLOE), where ρ i = 0 andδ i = 0, the ith actuator stuck at an unknown bounded value.
As for the actuator saturation nonlinearities, we consider the following the control inputs whereδ ci is to be designed in the following paper, and f i ∈ (0, 1] is introduced to represent the saturation nonlinearities of the actuators From a practical point of view,δ ci is bounded, hence, f i (δ ci , δ i,max ) ∈ (0, 1] and the dynamics of the actuator with failures and saturation nonlinearities can be expressed as where = diag(ρ 1 f 1 (.), · · · , ρ 8 f 8 (.)), andδ = δ 1 , · · · ,δ 8 T .

Control-oriented model of attitude system
Tailless aircraft is a typical over-actuated system, and the control commands could be achieved through multiple combinations of actuators to maximize its maneuverability. Pseudo-inverse, which is the most general linear control allocation algorithm, is employed in this article to solve the following optimization problem.
The solution to (19) can be expressed as where Q is a nonsingular weighting positive definite matrix. Define = [μ, α, β] T that represents the attitude angle. Considering the influence of the unknown external disturbances d 1 ∈ R 3×1 and the change of inertia parameter J ∈ R 3×3 . Therefore, the control-oriented model of the attitude system can be written as: where Substituting (22) into (25), we havė represents the lumped uncertainty.

Assumption 1
The external disturbance d 1 is bounded, then d ω is bounded, and satisfy | d ωi | D i , i = 1, 2, 3, where D i is an unknown positive constant.

Assumption 3
The time-varying constraints k 1i (t), i = 1, 2, 3 and k 2i (t), i = 1, 2, 3 are smooth and positive functions, and its first derivatives are continuous and bounded. The attitude reference signal d = [μ d , α d , β d ] T and its first derivatives are continuous and bounded. Furthermore, k 1i (t), k 2i (t) and d sat-

Remark 3
The external disturbance d 1 is assumed to be bounded, and the uncertain part of the inertia J is a small change, which is primarily caused by the fuel consumption and variations of payloads. Thus, it can imply that d ω is bounded, and Assumption 1 is reasonable and a common conclusion in the existing papers [43][44][45][46][47][48][49]. Assumption 2 is utilized to ensure the controllability of the tailless aircraft during the occurrence of actuator failures and saturation, it implies up to n − 3 TLOE simultaneously, and the remaining actuators are capable of generating the desired torque to achieve the control objectives. Since B is the control efficiency matrix, hence, B B T is positive definite, we can surmise that there exists a constant λ 0 such that

Nonlinear state-dependent function
To deal with the asymmetric time-varying output constraints, a nonlinear state-dependent function is introduced as where k 1 (t) and k 2 (t) are smooth and positive timevarying constraints function. From (27), we can see that that the new variable ς(t) is dependent on the state x, and it implies that if the initial value x(0) satisfying . Therefore, as long as ς(t) is bounded, the constraints of the state will be guaranteed.
By utilizing the nonlinear state-dependent function, the constrained system is transformed into a nonconstrained system, we only need to guarantee the boundedness of the new variable ς(t), and the timevarying constraints −k 1 (t) < x(t) < k 2 (t) will be satisfied. When compared with the BLF, which converts the state constraints into the constraints of boundary error, this scheme removes the feasibility conditions on the virtual controllers.
Taking the derivative of (27), it haṡ where Applying the state transformation in (27) to , one haṡ where ς = ς μ , ς α , ς β Control Objective: The control objective of the attitude tracking in this paper is to design a controller for tailless aircraft system such that attitude tracking error and all the signals of the closed-loop system in (29) are bounded and practically predefinedtime stable, when suffering from uncertain inertia parameters, bounded external disturbance, and actuator faults/failures. Meanwhile, the actuator satisfies {δ i min δ li , δ ri δ i max , i = 1, 2, 3, 4} and attitude angular satisfies {−k 1i x i k 2i , i = μ, α, β}.

Controller design
Here, we present an adaptive nonsingular predefinedtime fault-tolerant control scheme for the system (29), and the actual control lawδ c will be designed on the framework of backstepping.
To solve the problems of 'explosion of complexity' and singularity, a novel predefined-time filter (PTF) is proposed as follows: where φ di is the output of the PTF for the virtual control law φ i , 0 < υ < 1, and T f > 0 is the predefined time of PTF. Then, the error of the filter can be defined as by using −z f iφi υ , and select the appropriate parameters, then (32) can be rewritten aṡ υ , and select the appropriate parameters, then (34) can be rewritten aṡ Comparing (33) and (35), we find that the two cases have the same form, according to Corollary 1, the filter error z f i is practically predefined-time stable, and it converges to the residual set f : where 0 < ϑ < 1, and the predefined time can be given by

Remark 4
The novel PTF proposed in this article has two functions. First, PTF is used to avoid the 'explosion of complexity' problem on the framework of the backstepping approach, compared with the traditional firstorder filter, PTF is practically predefined-time stable and the convergence time T f can be set directly. Furthermore, PTF is utilized to overcome the singularity problem when taking the time derivatives of the virtual control inputs, and this will be explained in detail in the backstepping recursive design procedure.

Remark 5
As we can see in (40), the control singular problem may arise from the first term k 11 sig 1−υ (ς ) while taking the time derivative of the virtual control inputs α 1 based on traditional backstepping design. Instead, the proposed PTF could solve this singularity problem for the predefined-time backstepping control design. Define the angular velocity tracking errorω = ω − ω c ,ω = ω 1 ,ω 2 ,ω 3 T , and then the error of the filter can be written as: Select a Lyapunov function as: with respect to (39) and (42), the derivative of V 1 yieldṡ Substituting (40) and (41) into (43), and according to the Assumption 4 and Lemma 3, one haṡ Considering that 1 2 ς 2 ≤ 1 2 ( ς 2−υ + ς 2+υ ) and 1+ h 1 g 2 where Step 2. DefineD = D −D, whereD represents the estimate of D. According to the Assumption 2 and Remark 3, define Considering the following Lyapunov function: whereξ = ξ −ξ denotes the estimation error withξ , γ > 0 and χ > 0 are design parameters.

Stability analysis
Theorem 2 Consider the tailless aircraft system (29) with Assumption 1, 2, 3, and 4. If a predefined-time filter is selected as (30), then the virtual control law is designed as (40), and the intermediate control law is constructed as (48). Further, the actual control law is designed as (50), and the adaptive laws are chosen as (51) and (52). By selecting appropriate parameters, we have (i) All the signals of the closed system in (29) are bounded and practically predefined-time stable, the predefined time can be given by (ii) Attitude tracking error˜ i is bounded and practically predefined-time stable, the attitude angle satisfies the output constraints.
Proof See the Appendix.

Remark 6
To guarantee that Theorem 2 holds and the attitude control system has satisfactory tracking performance, the selection principle of controller parameters and its influence on the attitude control system performance are given here. Firstly, determine the predefined time T c based on the task requirement and physical limits, then select the predefined time T f such that T f T c . Secondly, the control gains k i1 and k i2 can be chosen as large values to improve the convergence rate, while too large values of k i1 and k i2 may cause input saturation. Thirdly, the exponential term υ is a bridge between a discontinuous feedback term (υ = 0 ) and a linear feedback term (υ = 0) and is related to the robustness of the attitude control system and the degree of the chattering of the control inputs. When υ → 1, the attitude control system becomes more robust to uncertainties, and when υ → 0, the control inputs become more smooth. Finally, one should select a large value for the parameters γ and χ and a small value for η i of the adaptive laws to improve the convergence precision.
To the effectiveness and the superiorities of the proposed control scheme, we make a comparison between the proposed predefined-time control (PTC) scheme and the finite-time control (FTC) scheme presented in [22] under the same conditions and simulation sample time. The controller parameters are given as k 11 Table 1. Figure 2 shows that the reference attitude can be followed successfully and the trajectory of attitude satisfies the output constraints. Further, the attitude tracking errors converge to the residual set around the origin within a predefined time T c , regardless of the external disturbance, uncertainty of the inertia matrix, and actuator failures. When compared with FTC scheme, the PTC scheme can reach the steady-state faster with significantly less energy. Meanwhile, the PTC scheme has smaller tracking error at the steady-state stage. Figure 3 shows the angular velocity and its tracking errors in two different control schemes, and the angular velocity tracking errors converge to the residual set within a predefined time T c , and besides the angular velocity being proximate to zero value.   The response of the lumped uncertainty estimation D and the adaptive parameterξ in PTC scheme are given in Fig. 4, it shows that the lumped uncertainty and adaptive parameterξ are bounded. The actuator deflection is plotted in Fig. 5, and they are bounded by the saturation nonlinearities. The simulation results show that the theoretical results can be obtained by utilizing the control law proposed in this article. Furthermore, compared with FTC scheme, the proposed scheme is more superior in tracking accuracy, convergence time, and energy consumption.
To show the impact of the initial conditions, changing the initial states of the tailless aircraft to 1 = [−1 deg, 6 deg, 1 deg] T , ω 1 = [2.5 deg /s, 2.5 deg /s, 2.5 deg /s] T , the controller parameters and other simulation conditions remain constant. Attitude tracking performance of the two methods is given in Fig. 6. The PTC/FTC scheme under initial conditions ( 0 , ω 0 ) and ( 1 , ω 1 ) are denoted as 'PTC0/FTC0' and 'PTC1/FTC1,' respectively. The actuator energy index and tracking performance indexes of the two methods are shown in Table. 2.
From Fig. 6, we see that the PTC scheme is able to track the reference attitude within the predefined time, whereas the convergence time of the FTC scheme is longer than the previous one. It verifies the convergence time of the PTC scheme is independent of the initial value. According to Table. 2, when the initial error increases, the energy consumption of the PTC scheme is much less than the FTC scheme, despite the increase in the energy consumption of both the schemes, and it implies that both the attitude and angular velocity tracking performance of PTC scheme is superior to FTC scheme. Thus, we can conclude that the proposed scheme has higher precision and less energy consumption, and has the ability to converge within the predefined time.

Conclusion
This article proposes a predefined-time adaptive attitude tracking control scheme for a tailless aircraft with the input and output constraints experiencing uncer-(a) Actuator deflection δ l1 ,δ r1 ,δ l2 ,δ r2 (b) Actuator deflection δ l3 ,δ r3 ,δ l4 ,δ r4 .  tain inertia parameters, bounded external disturbance, and actuator failure. By introducing a nonlinear statedependent function, the asymmetric time-varying output constraints have been solved. Further, by designing a nonlinear function and a bounded estimation, the influence of the faults and saturation nonlinearities is compensated simultaneously. Thereafter, with the help of the novel predefined-time filter, the 'explosion of complexity' and singularity problem are avoided. Finally, the proposed PTC scheme is compared with the FTC scheme. The simulation results show that the proposed scheme has higher precision and less energy consumption; besides, all signals of the system can converge into the residual set within a predefined time independent of the initial value. On the one hand, our future works will be extended to the infinite number of time-varying actuator failures; on the other hand, deep learning-based hierarchical active fault-tolerant control will be adopted to estimate and compensate the actuator faults to enhance the fault tolerance of the system.

Author contributions
All authors contributed to the study conception and design. Material preparation, data collection, and analysis were performed by ZY, YL, ML, JC, and BP. The first draft of the manuscript was written by ZY, and all authors commented on previous versions of the manuscript. All authors read and approved the final manuscript.
Funding This work was supported in part by Natural Science Foundation of China under Grant 62003252 and 62103440.

Data availability
The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.

Conflict of interest
The authors have no relevant financial or non-financial interests to disclose.
Ethics approval This research does not include any human participants and/or animals.

Consent for publication
All authors agree to the publication of this research.

Appendix A: A Proof of Theorem 2
Proof Define a Lyapunov function as (A. 1)