BLSTM-Based Adaptive Finite-Time Output-Constrained Control for a Class of AUSs with Dynamic Disturbances and Actuator Faults

Software Institute, East China Jiaotong University, Nanchang, China School of Astronautics, Northwestern Polytechnical University, Xi’an 710072, China National Key Laboratory of Aerospace Flight Dynamics, Northwestern Polytechnical University, Xi’an, China Research Center for Unmanned System Strategy Development, Northwestern Polytechnical University, Xi’an, China Unmanned System Research Institute, Northwestern Polytechnical University, Xi’an, China School of Aeronautics and Astronautics, Central South University, Changsha 410083, China


Introduction
As is well known, the adaptive control methodology can adjust control parameters automatically according to the variation of the environment and the system uncertainties and can improve the adaptability with respect to the uncertain factors mentioned above [1][2][3][4][5]. In recent years, a lot of research has been done on adaptive control in China. In [6], a novel adaptive switching dynamic surface control (DSC) strategy is first presented for fractional-order nonlinear systems in the nonstrict feedback form with unknown dead zones and arbitrary switching. In [6], a novel adaptive prescribed performance tracking control scheme has been proposed for stochastic nonlinear systems with unmodeled dynamics in finite-time prescribed performance. In [7], a novel event-triggered-based fuzzy adaptive finite-time control method has been proposed for stochastic nonlinear nonstrict feedback systems with unmodeled dynamics. In [8], a novel fault-tolerant IGC structure has been constructed for a class of skid-to-turn (STT) missiles subjected to rapidly changing actuator failures and coupled multisource uncertainties. In [9], a novel adaptive fault-tolerant attitude control approach has been designed based on the long short-term memory (LSTM) network for the fixedwing UAV subject to the high dynamic disturbances and actuator faults. In [10], a novel evasion-faced fast adaptive feed-forward control structure has been established for the hypersonic vehicles. Adaptive robust control is also a typical representative of adaptive control methodology, which can effectively realize the stable control of the system. In [11], a vehicle mass estimation algorithm based on the least squares method and sliding mode observer is proposed for variable mass unmanned aircraft systems. In [12], the fuzzy neural network (FNN) control method using impedance learning has been proposed for coordinated multiple constrained robots carrying a common object. In [13], the adaptive control-based design strategies have been proposed for a pressurized water (PWR) nuclear power plant (NPP) system. In [14], a reinforcement learning (RL) structure control strategy based on actor-critic has been developed for a flexible two-link manipulator (FTLM) system. In [15], robust fault-tolerant control for a class of second-order nonlinear systems using an adaptive third-order sliding mode control was proposed. In [16], an adaptive control scheme under the hierarchical framework has been proposed for the trajectory tracking problem of flapping-wing microaerial vehicles (FWMAVs). In [17], the adaptive robust tracking control scheme based on backstepping technique has been proposed for a class of multi-input and multioutput (MIMO) nonaffine systems with uncertain structure and parameters, unknown control direction, and unknown external disturbance. Furthermore, the adaptive control is usually used together with some other control laws, inheriting the original advantages of these control laws, as well as taking advantages of the automatic regulation of the adaptive control [18][19][20]. In practice, since the attitude control performance of the aerospace unmanned systems is always affected by aerodynamic uncertainties and external disturbances, the adaptive control method has been introduced into the control design process of AUS. For instance, the adaptive sliding mode control [21,22], adaptive fuzzy control [23][24][25], model reference adaptive control (MRAC) [26], adaptive dynamic inversion control [27], adaptive backstepping control [28][29][30], adaptive super-twisting approach [31], and adaptive anti-disturbance control [32,33] have all been applied to the attitude control design of the AUSs. Nevertheless, it is difficult to utilize the aforementioned control technique to guarantee rapidity of the closed-loop control system, especially when the external circumstance is fickle.
In order to tackle this problem, the concept of finite-time control (FTC) was put forward in the middle of the last century [34][35][36]. Given a limit condition of the system initial value, if the system states never exceed a certain threshold within a finite-time interval, the system is described as finitetime stable. In the past decade, many studies have been built on infinite time asymptotic convergence and can be utilized to guarantee the system convergence under the fickle external circumstance [37][38][39][40]. Due to the outstanding advantages under time-constraint situations, a series of papers have explored the potential of finite-time control laws. In [37], the finite-time Lyapunov stability theorem has been established. In [38], through output feedback based on a finite-time separation principle, the global finite-time stability has been realized for the double integrator system. In [39], a finite-time adaptive controller has been designed for a class of strict feedback nonlinear systems. Moreover, by the combination of a robust controller [40] and an adaptive controller [41], the finite-time tracking task was completed and the difficulties caused by state constraints and dead zone were conquered. In addition, FTC has also been employed in space missions. In [42], the finite-time stability of spacecraft large-angle maneuver under disturbances was achieved, avoiding the chattering phenomenon by virtue of a robust adaptive controller. In [43], a finite-time control strategy based on a finite-time auxiliary system has been designed for a 12-rotor UAV. Compared with the finite-time control law, the fixed-time control law has the outstanding advantage in the specific situation, and it can effectively realize the stable control at a given time. In [44], the event-triggered fixedtime adaptive fuzzy control method has been proposed for state-constrained stochastic nonlinear systems. In [45], a novel adaptive fixed-time control strategy has been proposed for a class of nonlinear bilateral teleoperation systems. In [46], a light fixed-time controller has been designed for the issues of chattering effects and multiple parameters.
Moreover, intelligent methods, algorithms, and theories have achieved remarkable progress recently, providing a useful way to improve the response speed and the tracking performance of the control systems [47][48][49][50]. e broad learning system (BLS), which utilizes the self-adjusting mechanism of node number to achieve better learning performance, has been firstly proposed in [51]. In [52], based on the broad learning method, a framework that can learn and fuse two modal characteristics has been constructed, possessing better stability and rapidity. Combining the Takagi-Sugeno (TS) fuzzy modeling method with the BLS, the fuzzy broad learning system (BLS) has been established in [53], which shows advantages over neuro-fuzzy models. By combining a convolutional neural network with broad learning system, a framework for license plate recognition has been reported in [54]. In [55], the least p-norm-based BLS (LP-BLS) using the fixed-point iteration strategy has been proposed. Furthermore, BLS has also been used to solve practical engineering problems [56][57][58]. In [56], a novel quaternion broad learning system has been proposed. In [57], the BLS has been introduced into hyper-spectral image analysis area, providing new ideas and technical reserves for a variety of hyper-spectral image analysis problems. Based on BLS, Zhongdong et al. [58] proposed a method of landscape capacity allocation and obtained a capacity allocation result that met the total investment cost and minimized network active power loss. Moreover, in [59], the long short-term memory (LSTM) network, which possesses the storage functions and can effectively approximate the time accumulative nonlinearities in the control systems, has been established. In [60,61], the effectiveness and robustness of the LSTM have been demonstrated. In [62], a LSTM-based intelligent control structure is constructed, showing the better approximating accuracy of the LSTM compared to the common neural networks.
However, up to know, the broad learning methods have rarely been introduced into the control design of the AUSs. According to [51,52], the broad learning methods could be made use of to enhance the control performance and reduce the control complexity. erefore, it is of remarkable importance to develop the BLS-based adaptive control methods for the AUSs. Furthermore, in the actual flight process, the nonlinearities and uncertainties of the AUSs usually possess time-varying features and accumulative effect, and as a result, the LSTM should be introduced together with the BLS simultaneously. erefore, in this work, we propose a BLSTM-based adaptive finite-time control structure for the AUSs under output constraints. e broad long short-term memory network, which possesses both the time memory characteristics and high learning speed, has been constructed. e fractional-order control law and the tracking error transformation are utilized to guarantee the finite-time convergence and output constraints. Compared with the existing results, this work possesses the following features: (i) It is the first BLSTM-based adaptive finite-time control structure for the AUSs. (ii) e proposed broad long short-term memory network can be applied to a plenty of practical engineering systems to improve the approximation, classification, or recognition performance. (iii) By introducing the BLSTM into the adaptive finitetime control law, the convergence time can be ensured and the control accuracy can be improved.

e System Model of Aerospace Unmanned
System. According to reference [63], the kinematic and dynamic equations of the aerospace unmanned system can be expressed by where ϑ, ψ, and c denote the pitch angle, yaw angle, and roll angle, respectively; ω is the attitude angular velocity; I is the inertia matrix of the AUS; and M is the total moment of force.
e definitions of q, ρ, V, S, L, C z , C y , C x , C z0 , δ a1 , δ a2 , δ b1 , δ b2 , and δ y can be found in [64]. α, β are the angle of attack and sideslip angle, respectively, with the following dynamic equations: With the external disturbances, the attitude dynamics of the AUS system can be modeled as where y out is the output of the AUS system; and d θ (t) and d ω (t) ∈ R 3×1 are the matched and mismatched disturbances. where Considering these factors, there often exist structural uncertainties and modeling errors, which can be rewritten as where f(θ), f(θ, ω) represent the unknown lumped aerodynamic uncertainties. Let ΔA θ represent the structural uncertainty of A θ and let ΔB ω represent the composite uncertainties.
where Δζ is the uncertainty of ζ. erefore, the control objective of this paper is to force the attitude angle θ to follow the desired signals in spite of Mathematical Problems in Engineering the structural uncertainties and the aerodynamic uncertainties.

Preliminaries
Assumption 1. It is supposed that A θ and B ω are invertible or generalized invertible.
e following lemmas are required.
Lemma 1 (see [65]). Given any a 1 , a 2 and positive constants z 1 , z 2 , z 3 , the following inequality can be satisfied: Lemma 2 (see [66]). Given any constant ε > 0 and any variable x ∈ R, the following inequality can be satisfied: Lemma 3 (see [42]). Consider system _ X � f(X, U). Given any two differential positive definite functions V(X) and the nonlinear system _ X � f(X, U) is semi-globally practical finite-time stable (SGPFS).

Broad Long Short-Term Memory (BLSTM) Network.
e structure of BLSTM network is presented in Figure 1. Note that the broad learning thought has been introduced to build the structure of BLSTM, and the concept of "gate" is inherited. In BLSTM, the input data are

Main Results
In this section, the controller design will be divided into inner loop and outer loop of AUS subject to finite-time convergence, respectively. In the inner loop, several nonlinear functions are used to constrain the state error to predefined boundness. In view of the external disturbances, the BLSTM network is introduced to approximate the unknown nonlinear functions. e adaptive laws are designed to compensate the structural uncertainties. Meanwhile, in the outer loop, an innovative adaptive finite-time control law is designed to deal with the input uncertainties. e structure of the proposed BLSTM-based intelligent adaptive control method is depicted in Figure 2.
where Ω θ � (A θ + ΔA θ )/A θ . e following nonlinear functions are utilized to guarantee the output constraints: Define s θ � [s ϑ , s ψ , s c ] T . Accordingly, we can get that where Considering the uncertainties f(θ), a BLSTM network O t,θ tanh(C t,θ ) is applied to the inner controller. Clearly, where ε θ is the bounded approximation error. Define the BLSTM node number in the inner loop as N 1 . We can design the self-adjust strategy for the nodes number as where n 1 ad represents the adjustment and it is a designed positive integer; χ 1 low , χ 1 up are designed positive constants, denoting the adjustment threshold; t 1 represents the time series to complete the judgment and adjustment; and O t,θ is the estimated value of O t,θ . Define ξ θ � sup t≥0 ‖ε θ + d θ (t)‖. Considering (16), the indirect virtual control signal is designed as where ω vc is the virtual control signal of ω, k 1 ∈ R 3×3 is the control gain, ξ θ is the estimated value of ξ θ , and Λ(θ) � (s T θ s θ )s T θ T(θ). Define λ A as the estimated value of λ A . erefore, we design the inner control law as   Mathematical Problems in Engineering where Υ(θ) � Λ(θ), A θ ∈ R 1×3 . Substituting (19) into (16) generates where e 2 � ω − ω v . e following Lyapunov function is selected: is the adaptive gain. Take the differential of V 1 as follows: Substituting (21) into (23) yields By using Lemma 1, it can be checked that where ε ξ,Σ > 0. Substituting (25) into (24), we can obtain that en, from (20) and Lemma 1, we know that erefore, Obviously, Combining (26), (28), and (29), it can be proven that Considering (30), we design the adaptive laws for O t,θ , ξ θ , λ A as where σ O t,θ , σ ξ θ , σ λ A > 0. erefore, by using the update law (31), we know that Since 6 Mathematical Problems in Engineering substituting (33) into (32), we can rewrite (32) as According to Lemma 3, the following inequality holds: erefore, we can get that where

BLSTM-Based Input Uncertainty Suppressed Outer Loop
Finite-Time Controller Design. According to (7), we can get that A BLSTM network O t,ω tanh(C t,ω ) is utilized to handle f(θ, ω). Define the BLSTM node number in the outer loop as N 2 . We can also design the self-adjust strategy for the nodes number as where ε ω represents the adjustment and it is a designed positive integer; χ 2 low , χ 2 up are designed positive constants, denoting the adjustment threshold; and t 2 represents the time series to complete the judgment and adjustment. According to (38), we design where δ vc is the virtual control signal of δ, k 2 is the control gain, To deal with the perturbations of B ω , we define λ B � 1/inf(λ min (Ω ω )). Accordingly, we design where λ B is the estimation of λ B . Substituting (40) and (41) into (38) yields Select the following Lyapunov function: We can take the differential of V 2 as

Mathematical Problems in Engineering
By using Lemma 1, we know that en, from (41) and Lemma 1, we can obtain that erefore, By substituting (45) and (47) into (44), it can be obtained that In view of (41), the adaptive laws for O t,ω , ξ ω , λ B are designed as where σ O t,ω , σ ξ ω , σ λ B > 0. Combining (48) with (49) generates erefore, Next, by applying Lemma 3, we can obtain Hence, we can get that where

Stability Analysis of the Proposed BLSTM-Based Finite-Time Control
Law. e main result on global adaptive tracking is stated in this section.

Theorem 1.
Consider the closed-loop system consisting of the AUS (7), the indirect virtual controllers (19) and (40), the final virtual controllers (20) and (41), and the adaptive update laws (31) and (49) and suppose that Assumptions 1 and 2 are satisfied. Moreover, it is guaranteed that the nonlinear mapping error s θ is bounded and can converge in finite time, and thus the tracking error can converge to a small compact set in finite time in the presence of the uncertainties and the faulty elevator outputs.
Proof. According to (36) and (53) us, Define 8 Mathematical Problems in Engineering We can define that By solving (55), we can get that where 0 < α < 1. Meanwhile, let where V(0) represents the initial value of V(t). According to Lemma 2, for t ≥ T, V l ≤ ε Υ /(1 − α) σ, that is, all the signals of the closed-loop system are SGPFS.

Mathematical Problems in Engineering
Furthermore, according to the definition of V, it can be inferred that for ∀t ≥ T, holds. In other words, after the finite time T, s θ remains in a small neighborhood of the origin. us, the tracking errors can converge to a desired compact set in a finite time and never violate the predefined constraint. e proof is completed.

Simulation Study
In this section, four cases of numerical examples, including the proposed control method, one without BLSTM, one without output constraint, and one without finite time, are carried out in the simulation.
In the simulation, the AUS flight height is h � 1000 m, and the initial attitude angles and angular velocities are set as zero. e desired signals are selected as ϑ d � 20°, ψ d � 15°, and c d � 4°, and the simulation time is T � 10 s. e parameters of AUS are provided in Table 1. Ref. [9] Ref. [10] Ref. [11] proposed method  Ref. [10] Ref. [11] proposed method   Table 2. In this paper, the parameters of BLSTM network are given in Table 3.
e simulation results are given in Figures 3-11. Meanwhile, the control performances under four cases are shown in Figures 3-5, from which we can see that the attitude angles of the four methods are quiet different. In Case 4, we apply the proposed method according to the analysis in Section 4, and it can track the desired signal closely in a finite time. However, for comparison, in Case 1, it shows that the proposed method can induce more overshoot without output constraint. Moreover, in Case 2, we apply the proposed control method with LSTM instead of BLSTM, which may induce tracking errors. at is to say, the BLSTM network can guarantee more accurate tracking accuracy. In Case 3, it is assumed that without finite time, the proposed method will converge in a longer time. erefore, we know that the proposed BLSTM-based control law can achieve satisfactory control performance. From Figures 6-8, It can be seen that the proposed method with fault compensation has more advantages in control effect than the method without fault compensation and can effectively reduce the oscillation and realize stable control. e node number of BLSTM network is shown in Figure 11. Also, it can be seen that the nodes are variable with the tracking error, which will converge to a fixed value when the tracking error is stable.  erefore, it can be concluded that the proposed BLSTMbased adaptive control algorithm can achieve favorable control results for the AUSs with multiple uncertainties.

Conclusions
In this paper, we propose a BLSTM-based adaptive finitetime control structure for a class of AUSs. e novel BLSTM which possesses both the memory function and rapid learning speed has been constructed. e fractionalorder control law and the corresponding adaptive laws are designed. Combining the BLSTM, the transformation of the tracking errors, and the adaptive finite-time control law, a novel BLSTM-based intelligent adaptive finite-time control structure has been established for the AUSs under output constraints. e simulation results demonstrate the effectiveness of the proposed BLSTM-based intelligent adaptive finite-time control method and reveal that the introduction of the BLSTM can improve the control performance for the AUSs. In future work, the BLSTMbased adaptive finite-time control method proposed in this paper will be used to deal with the AUSs with stochastic disturbances.

Data Availability
e data used to support the findings of this study are included within the article.

Conflicts of Interest
e authors declare that they have no conflicts of interest.