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

Authors: Shiyi Huang; Lulu Rong; Xiaofei Chang; Zheng Wang; Zhaohui Yuan; Caisheng Wei
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

In this paper, a BLSTM-based adaptive finite-time control structure has been constructed for a class of aerospace unmanned systems (AUSs). Firstly, a novel neural network structure possessing both the time memory characteristics and high learning speed, broad long short-term memory (BLSTM) network, has been constructed. Secondly, several nonlinear functions are utilized to transform the tracking errors into a novel state vector to guarantee the output constraints of the AUSs. Thirdly, the fractional-order control law and the corresponding adaptive laws are designed, and as a result, the adaptive finite-time control scheme has been formed. Moreover, to handle the uncertainties and the faulty elevator outputs, an inequality of the multivariable systems is utilized. Consequently, by fusing the output of 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. The simulation results show that the proposed BLSTM-based adaptive control algorithm can achieve favorable control results for the AUSs with multiple uncertainties.


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## Body

## 1. 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 [1–5]. 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 fixed-wing 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 multi-output (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 [18–20]. 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 [23–25], model reference adaptive control (MRAC) [26], adaptive dynamic inversion control [27], adaptive backstepping control [28–30], 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 [34–36]. 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 finite-time 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 [37–40]. 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 fixed-time 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 [47–50]. The 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 [56–58]. 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. Therefore, 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.Therefore, in this work, we propose a BLSTM-based adaptive finite-time control structure for the AUSs under output constraints. The broad long short-term memory network, which possesses both the time memory characteristics and high learning speed, has been constructed. The 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)
The 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 finite-time control law, the convergence time can be ensured and the control accuracy can be improved.


## 2. Problem Formulation and Preliminaries

### 2.1. The System Model of Aerospace Unmanned System

According to reference [63], the kinematic and dynamic equations of the aerospace unmanned system can be expressed by(1)ϑ˙=ωysinγ+ωzcosγ.ψ˙=ωycosγ−ωzsinγcosϑ,γ˙=ωx−tanϑωycosγ−ωzsinγ,ω˙=J−1M−ω×J⋅ω,where ϑ, ψ, and γ 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.(2)I=Ixx0Ixz0Iyy0Izx0Izz,M=qSLCzCyCx,Cz=Cz0+Czδa1δa1+Cxδa2δa2+Czδb1δb1+Czδb2δb2+Czαα,Cy=Cyδyδy+Cyββ,Cx=Cxδa1δa1+Cxδa2δa2+Cxδb1δb1+Cxδb2δb2+Cxββ.The definitions ofq, ρ, V, S, L, Cz, Cy, Cx, Cz0, δa1, δa2 , δb1, δb2, and δy can be found in [64]. α, β are the angle of attack and sideslip angle, respectively, with the following dynamic equations:(3)α˙=ωz+ρVSCzα2mα,β˙=−ωy+ρVSCyβ2mβ.With the external disturbances, the attitude dynamics of the AUS system can be modeled as(4)θ˙t=Aθωt+dθt,ω˙t=Bωδt+Cω+dωt,where θt=ϑ,ψ,γT and ωt=ωz,ωy,ωxT; δte=δb1,δb2,δy,δa1,δa2T; yout is the output of the AUS system; and dθt and dωt∈ℝ3×1 are the matched and mismatched disturbances.(5)Aθ=cosγsinγ0−sinγcosϑcosγcosϑ0tanϑsinγ−tanϑcosγ1,Bω=qSLI−1CM,Cω=I−1qSLCz0+CzααCyββCxββ+ωzωyωx×Iωzωyωx,where(6)CM=Czδb1Czδb20Czδa1Czδa200Cyδy00Cxδb1Cxδb20Cxδa1Cxδa2.Considering these factors, there often exist structural uncertainties and modeling errors, which can be rewritten as(7)θ˙t=Aθ+ΔAθωt+fθ+dθt,ω˙t=Bω+ΔBωδt+Cω+fθ,ω+dωt,where fθ, fθ,ω represent the unknown lumped aerodynamic uncertainties. Let ΔAθ represent the structural uncertainty of Aθ and let ΔBω represent the composite uncertainties.(8)ΔBω=ζ⋅ΔCM+Δζ⋅CM,where Δζ is the uncertainty of ζ.Therefore, the control objective of this paper is to force the attitude angleθ to follow the desired signals in spite of the structural uncertainties and the aerodynamic uncertainties.


### 2.2. Preliminaries

Assumption 1.
It is supposed thatAθ and Bω are invertible or generalized invertible.Assumption 2.
It is supposed thatdθt, dωt are both bounded, e.g., there exist unknown positive constants D¯θ and D¯ω such that dθt≤D¯θ, dωt≤D¯ω.The following lemmas are required.Lemma 1 (see [65]).
Given anya1, a2 and positive constants z1, z2, z3, the following inequality can be satisfied:(9)a1z1a2z2≤z1z1+z2z3a1z1+z2+z2z1+z2z3−z1/z2a2z1+z2.Lemma 2 (see [66]).
Given any constantε>0 and any variable x∈ℝ, the following inequality can be satisfied:(10)0<x−xTxxTx+ε<ε.Lemma 3 (see [42]).
Consider systemX˙=fX,U. Given any two differential positive definite functions VX and c>0, 0<l<1, ε>0 such that(11)V˙X≤−cVlX+ε,t≥0,the nonlinear system X˙=fX,U is semi-globally practical finite-time stable (SGPFS).


### 2.3. Broad Long Short-Term Memory (BLSTM) Network

The structure of BLSTM network is presented in Figure1.Figure 1
The structure of BLSTM.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 areht−1,xt, and σW1fiht−1+xt+b1fi, i=1,2,…,n, is utilized to generate the ith mapped features, Zti, where Wfi is the random weights with the proper dimensions. Denote Zi≡Zt1,…,Zti, which is the concatenation of all the first i groups of mapping features. Similarly, the jth group of enhancement nodes, σZt1Zt2,…,ZtnW2fj+b2fj, j=1,2,…,N, is denoted as Htj, and the concatenation of all the first j groups of enhancement nodes is denoted as Hj=Ht1,…,Htj. Hence, the broad model of forget gate ft can be represented as the equation of the form(12)ft=Zt1,…,Ztn|σZt1W2f1+b2f1,…,σZtnW2fn+b2nW1=Zt1,…,Ztn|Ht1,…,HtmW1=Ztn|HtmW1,where W1=Ztn|Htm+ft represents the connecting weights for the broad structure. Similarly, we can get the broad model of memory state c˜t as c˜t=Qtn|RtmW2, where W1=Qtn|Rtm+c˜t. The mathematical expressions of input gate, update gate, and output gate are(13)it=σWiht−1,xt+bi,ct=ft∘ct−1+it∘c˜t,ot=σWoht−1,xt+bo,ht=ot∘tanhct,where Wf, Wi, Wc, Wo, bf, bi, bc, and bo are the weight matrices and ∘ is the Hadamard product.


## 2.1. The System Model of Aerospace Unmanned System

According to reference [63], the kinematic and dynamic equations of the aerospace unmanned system can be expressed by(1)ϑ˙=ωysinγ+ωzcosγ.ψ˙=ωycosγ−ωzsinγcosϑ,γ˙=ωx−tanϑωycosγ−ωzsinγ,ω˙=J−1M−ω×J⋅ω,where ϑ, ψ, and γ 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.(2)I=Ixx0Ixz0Iyy0Izx0Izz,M=qSLCzCyCx,Cz=Cz0+Czδa1δa1+Cxδa2δa2+Czδb1δb1+Czδb2δb2+Czαα,Cy=Cyδyδy+Cyββ,Cx=Cxδa1δa1+Cxδa2δa2+Cxδb1δb1+Cxδb2δb2+Cxββ.The definitions ofq, ρ, V, S, L, Cz, Cy, Cx, Cz0, δa1, δa2 , δb1, δb2, and δy can be found in [64]. α, β are the angle of attack and sideslip angle, respectively, with the following dynamic equations:(3)α˙=ωz+ρVSCzα2mα,β˙=−ωy+ρVSCyβ2mβ.With the external disturbances, the attitude dynamics of the AUS system can be modeled as(4)θ˙t=Aθωt+dθt,ω˙t=Bωδt+Cω+dωt,where θt=ϑ,ψ,γT and ωt=ωz,ωy,ωxT; δte=δb1,δb2,δy,δa1,δa2T; yout is the output of the AUS system; and dθt and dωt∈ℝ3×1 are the matched and mismatched disturbances.(5)Aθ=cosγsinγ0−sinγcosϑcosγcosϑ0tanϑsinγ−tanϑcosγ1,Bω=qSLI−1CM,Cω=I−1qSLCz0+CzααCyββCxββ+ωzωyωx×Iωzωyωx,where(6)CM=Czδb1Czδb20Czδa1Czδa200Cyδy00Cxδb1Cxδb20Cxδa1Cxδa2.Considering these factors, there often exist structural uncertainties and modeling errors, which can be rewritten as(7)θ˙t=Aθ+ΔAθωt+fθ+dθt,ω˙t=Bω+ΔBωδt+Cω+fθ,ω+dωt,where fθ, fθ,ω represent the unknown lumped aerodynamic uncertainties. Let ΔAθ represent the structural uncertainty of Aθ and let ΔBω represent the composite uncertainties.(8)ΔBω=ζ⋅ΔCM+Δζ⋅CM,where Δζ is the uncertainty of ζ.Therefore, the control objective of this paper is to force the attitude angleθ to follow the desired signals in spite of the structural uncertainties and the aerodynamic uncertainties.


## 2.2. Preliminaries

Assumption 1.
It is supposed thatAθ and Bω are invertible or generalized invertible.Assumption 2.
It is supposed thatdθt, dωt are both bounded, e.g., there exist unknown positive constants D¯θ and D¯ω such that dθt≤D¯θ, dωt≤D¯ω.The following lemmas are required.Lemma 1 (see [65]).
Given anya1, a2 and positive constants z1, z2, z3, the following inequality can be satisfied:(9)a1z1a2z2≤z1z1+z2z3a1z1+z2+z2z1+z2z3−z1/z2a2z1+z2.Lemma 2 (see [66]).
Given any constantε>0 and any variable x∈ℝ, the following inequality can be satisfied:(10)0<x−xTxxTx+ε<ε.Lemma 3 (see [42]).
Consider systemX˙=fX,U. Given any two differential positive definite functions VX and c>0, 0<l<1, ε>0 such that(11)V˙X≤−cVlX+ε,t≥0,the nonlinear system X˙=fX,U is semi-globally practical finite-time stable (SGPFS).


## 2.3. Broad Long Short-Term Memory (BLSTM) Network

The structure of BLSTM network is presented in Figure1.Figure 1
The structure of BLSTM.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 areht−1,xt, and σW1fiht−1+xt+b1fi, i=1,2,…,n, is utilized to generate the ith mapped features, Zti, where Wfi is the random weights with the proper dimensions. Denote Zi≡Zt1,…,Zti, which is the concatenation of all the first i groups of mapping features. Similarly, the jth group of enhancement nodes, σZt1Zt2,…,ZtnW2fj+b2fj, j=1,2,…,N, is denoted as Htj, and the concatenation of all the first j groups of enhancement nodes is denoted as Hj=Ht1,…,Htj. Hence, the broad model of forget gate ft can be represented as the equation of the form(12)ft=Zt1,…,Ztn|σZt1W2f1+b2f1,…,σZtnW2fn+b2nW1=Zt1,…,Ztn|Ht1,…,HtmW1=Ztn|HtmW1,where W1=Ztn|Htm+ft represents the connecting weights for the broad structure. Similarly, we can get the broad model of memory state c˜t as c˜t=Qtn|RtmW2, where W1=Qtn|Rtm+c˜t. The mathematical expressions of input gate, update gate, and output gate are(13)it=σWiht−1,xt+bi,ct=ft∘ct−1+it∘c˜t,ot=σWoht−1,xt+bo,ht=ot∘tanhct,where Wf, Wi, Wc, Wo, bf, bi, bc, and bo are the weight matrices and ∘ is the Hadamard product.


## 3. 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. The 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. The structure of the proposed BLSTM-based intelligent adaptive control method is depicted in Figure2.Figure 2
The structure of the proposed control scheme.


### 3.1. BLSTM-Based Inner Loop Finite Control Law

Definee1,ϑ=ϑ−ϑd, e1,ψ=ψ−ψd, e1,γ=γ−γd. Therefore, the inner loop error dynamic equation can be given by(14)e˙1=AθΩθω+fθ+dθt−θ˙d,where Ωθ=Aθ+ΔAθ/Aθ.The following nonlinear functions are utilized to guarantee the output constraints:(15)sϑ=logℓa,1+e1,ϑℓa,2+e1,ϑ,sψ=logℓb,1+e1,ψℓb,2+e1,ψ,sγ=logℓc,1+e1,γℓc,2+e1,γ.Definesθ=sϑ,sψ,sγT. Accordingly, we can get that(16)s˙θ=TθAθΩθω+fθ+dθt−θ˙d,where(17)Tθ=Diagesϑ+e−sϑ+2ℓa,1+ℓa,2esψ+e−sψ+2ℓb,1+ℓb,2esγ+e−sγ+2ℓc,1+ℓc,2.Considering the uncertaintiesfθ, a BLSTM network Ot,θtanhCt,θ is applied to the inner controller. Clearly, fθ=Ot,θ∘tanhCt,θ+εθ, where εθ is the bounded approximation error. Define the BLSTM node number in the inner loop as N1. We can design the self-adjust strategy for the nodes number as(18)N1=N1+n1ad,∫t1−1t1sθτdτ>χ1up,N1=N1,χ1low≤∫t1−1t1sθτdτ≤χ1up,N1=N1−n1ad,∫t1−1t1sθτdτ<χ1low,where n1ad represents the adjustment and it is a designed positive integer; χ1low, χ1up are designed positive constants, denoting the adjustment threshold; t1 represents the time series to complete the judgment and adjustment; and O^t,θ is the estimated value of Ot,θ.Defineξθ=supt≥0εθ+dθt. Considering (16), the indirect virtual control signal is designed as(19)ωvc=Aθ−1−k1sθ2l−1Tθ−O^t,θtanhCt,θ−ξ^θφθ+θ˙d,φθ=ΛTθΛθΛTθ+εξ,Λ2,∀εξ,Λ>0,where ωvc is the virtual control signal of ω, k1∈ℝ3×3 is the control gain, ξ^θ is the estimated value of ξθ, and Λθ=sθTsθsθTTθ.Defineλ^A as the estimated value of λA. Therefore, we design the inner control law as(20)ωv=−λ^Aωvcϒθλ^Aωvcϒθλ^Aωvc2+εA2,∀εA>0,where ϒθ=Λθ,Aθ∈ℝ1×3. Substituting (19) into (16) generates(21)s˙θ=−k1sθ2l−1+TθAθΩθe2+AθΩθωv−Aθωvc−O˜t,θtanhCt,θ−ξ^θφθ+εθ+dθt,where e2=ω−ωv.The following Lyapunov function is selected:(22)V1=12sθTsθ+12ΓOt,θO˜t,θTO˜t,θ+12Γξθξ˜θ2+12ΓλAλAλ˜A2,where O˜t,θ=O^t,θ−Ot,θ, ξ˜θ=ξ^θ−ξθ, λ˜A=λ^A−λA, and ΓOt,θ, Γξθ, ΓλA>0 is the adaptive gain. Take the differential of V1 as follows:(23)V˙1=sθTs˙θ+1ΓOt,θO˜t,θTO^˙t,θ+1Γξθξ˜θξ^˙θ+1ΓλAλAλ˜Aλ^˙A.Substituting (21) into (23) yields(24)V˙1=−k1sθTsθ2l−1+ΛθAθΩθe2+AθΩθωv−Aθωvc−O˜t,θtanhCt,θ−ξ^θφθ+εθ+dθt+1ΓOt,θO˜t,θTO^˙t,θ+1Γξθξ˜θξ^˙θ+1ΓλAλAλ˜Aλ^˙A.By using Lemma1, it can be checked that(25)Λθεθ+dθt≤Λθξθ≤ξθΛθΛTθΛθΛTθ+εξ,Λ2+ξθεξ,Λ,where εξ,Σ>0. Substituting (25) into (24), we can obtain that(26)V˙1≤−k1sθTsθ2l−1+ΛθAθΩθe2+AθΩθωv−Aθωvc−O˜t,θtanhCt,θ−ξ˜θΛθΛTθΛθΛTθ+εξ,Λ2+ξθεξ,Λ+1ΓOt,θO˜t,θTO^˙t,θ+1Γξθξ˜θξ^˙θ+1ΓλAλAλ˜Aλ^˙A.Then, from (20) and Lemma 1, we know that(27)ΛθAθΩθωv=ϒθΩθωv=−Ωθϒθλ^Aωvc2ϒθλ^Aωvc2+εA2≤−ϒθλ^Aωvc2λAϒθλ^Aωvc2+εA2≤−1λAϒθλ^Aωvc−εA≤1λAϒθλ^Aωvc+εA.Therefore,(28)ΛθAθΩθωv−ΛθAθωvc≤λ˜AλAϒθωvc+εAλA.Obviously,(29)ΛθAΩθe2=ϒθΩθe2≤λmax2Ωθ+12ϒθϒTθe2Te2.Combining (26), (28), and (29), it can be proven that(30)V˙1≤−k1sθTsθ2l−1−ΛθO˜t,θtanhCt,θ−ξ˜θΛθφθ+λmax2Ωθ+12ϒθϒTθe2Te2+λ˜AλAϒθωvc+1ΓOt,θO˜t,θTO^˙t,θ+1Γξθξ˜θξ^˙θ+1ΓλAλAλ˜Aλ^˙A+εAλA+ξθεξ,Λ.Considering (30), we design the adaptive laws for O^t,θ, ξ^θ, λ^A as(31)O^˙t,θ=ΓOt,θΛTθ∘tanhCt,θ−σOt,θO^t,θ,ξ^˙θ=ΓξθΛθφθ−σξθξ^θ,λ^˙A=ΓλA−ϒθωvc−σλAλ^A,where σOt,θ, σξθ, σλA>0. Therefore, by using the update law (31), we know that(32)V˙1≤−k1sθTsθ2l−1+λmax2Ωθ+12ϒθϒTθe2Te2−σOt,θO˜t,θTO^t,θ−σξθξ˜θξ^θ−σλAλ˜Aλ^AλA+εAλA+ξθεξ,Λ.Since(33)−σOt,θO˜t,θTO^t,θ≤−12σOt,θO˜t,θF2+12σOt,θOt,θF2−σξθξ˜θξ^θ≤−12σξθξ˜θ2+12σξθξθ2−1λAσλAλ˜Aλ^A≤−12λAσλAλ˜A2+12λAσλAλA2,substituting (33) into (32), we can rewrite (32) as(34)V˙1≤−k1sθ2l+12ϒθϒTθe2Te2−12σOt,θO˜t,θF2−12σξθξ˜θ2−12λAσλAλ˜A2+λmax2Ωθ+12σOt,θOt,θF2+12σξθξθ2+12λAσλAλA2+εAλA+ξθεξ,Λ.According to Lemma3, the following inequality holds:(35)12σOt,θO˜t,θF2l≤ι1−l+12σOt,θO˜t,θF2,12σξθξ˜θ2l≤ι1−l+12σξθξ˜θ2,12λAσλAλ˜A2l≤ι1−l+12λAσλAλ˜A2.Therefore, we can get that(36)V˙1≤−λmink1sθ2l+12ϒθϒTθe2Te2−12σOt,θO˜t,θF2l−12σξθξ˜θ2l−12λAσλAλ˜A2l+εϒ,1,where(37)εϒ,1=12σOt,θOt,θF2+12σξθξθ2+12λAσλAλA2+λmax2Ωθ+3ι1−l+εAλA+ξθεξ,Λ.


### 3.2. BLSTM-Based Input Uncertainty Suppressed Outer Loop Finite-Time Controller Design

According to (7), we can get that(38)e˙2=Bω+ΔBωδ+Cω+fθ,ω+dωt−ω˙v.A BLSTM networkOt,ωtanhCt,ω is utilized to handle fθ,ω. Define the BLSTM node number in the outer loop as N2. We can also design the self-adjust strategy for the nodes number as(39)N2=N2+n2ad,∫t2−1t2e2τdτ>χ2up,N2=N2,χ2low≤∫t2−1t2e2τdτ≤χ2up,N2=N2−n2ad,∫t2−1t2e2τdτ<χ2low,where εω represents the adjustment and it is a designed positive integer; χ2low, χ2up are designed positive constants, denoting the adjustment threshold; and t2 represents the time series to complete the judgment and adjustment.According to (38), we design(40)δvc=Bω−1−k2e22l−1−12e2ϒθϒTθ−Cω−O^t,ωtanhCt,ω−ξ^ωφe2+ω˙v,φe2=e2e2Te2+εξ,e22,∀εξ,e2>0,where δvc is the virtual control signal of δ, k2 is the control gain, Bω−1=BωT/BωBωT, and ξ^ω represents the estimations of ξω (ξω=supt≥0εω+dωt).To deal with the perturbations ofBω, we define λB=1/infλminΩω. Accordingly, we design(41)δv=−λ^Bδvce2TBωλ^Bδvce2TBωλ^Bδvc2+εB2,∀εB>0,where λ^B is the estimation of λB. Substituting (40) and (41) into (38) yields(42)e˙2=−12e2ϒθϒTθ−k2e22l−1+BωΩωδ−Bωδvc−O˜t,ωtanhCt,ω−ξ^ωφe2+dωt+εω.Select the following Lyapunov function:(43)V2=12e2Te2+12ΓOt,ωO˜t,ωTO˜t,ω+12Γξωξ˜ω2+12ΓλBλBλ˜B2,where O˜t,ω=O^t,ω−Ot,ω, ξ˜ω=ξ^ω−ξω, λ˜B=λ^B−λB, ΓOt,ω, Γξω, ΓλB>0. We can take the differential of V2 as(44)V˙2=e2Te˙2+1ΓOt,ωO˜t,ωTO^˙t,ω+1Γξωξ˜ωξ^˙ω+1ΓλBλBλ˜Bλ^˙B=e2TBωΩωδ−Bωδvc−12e2ϒθϒTθ−k2e22l−1−O˜t,ωtanhCt,ω−ξ^ωφe2+εω+dωt+1ΓOt,ωO˜t,ωTO^˙t,ω+1Γξωξ˜ωξ^˙ω+1ΓλBλBλ˜Bλ^˙B.By using Lemma1, we know that(45)e2Tεω+dωt≤ξωe2≤ξωe2Tφe2+ξωεξ,e2.Then, from (41) and Lemma 1, we can obtain that(46)e2TBωΩωδv=−Ωωe2TBωλ^Bδvce2TBωλ^Bδvce2TBωλ^Bδvc2+εB2≤−1λBe2TBωλ^Bδvc−εB≤λ^BλBe2TBωδvc+εBλB.Therefore,(47)e2TBωΩωδ−e2TBωδvc≤λ˜BλBe2TBωδvc+εBλB.By substituting (45) and (47) into (44), it can be obtained that(48)V˙2≤−k2e22l−12e2Te2ϒθϒTθ+λ˜BλBe2TBωδvc+εBλB−e2TO˜t,ωtanhCt,ω−ξ˜ωe2Tφe2+ξωεξ,e2+1ΓOt,ωO˜t,ωTO^˙t,ω+1Γξωξ˜ωξ^˙ω+1ΓλBλBλ˜Bλ^˙B.In view of (41), the adaptive laws for O^t,ω, ξ^ω, λ^B are designed as(49)O^˙t,ω=ΓOt,ωe2∘tanhCt,ω−σOt,ωO^t,ω,ξ^˙ω=Γξωe2Tφe2−σξωξ^ω,λ^˙B=−ΓλBe2TBωδvc−σλBλ^B,where σOt,ω, σξω, σλB>0. Combining (48) with (49) generates(50)V˙2≤−k2e22l−12e2Te2ϒθϒTθ−σOt,ωO˜t,ωTO^t,ω−σξωξ˜ωξ^ω−σλBλ˜Bλ^BλB+ξωεξ,e2+εBλB.Therefore,(51)V˙2≤−12e2Te2ϒθϒTθ−λmink2e22l−12σOt,ωO˜t,ωF2+12σOt,ωOt,ωF2−12σξωξ˜ω2+12σξωξω2−12λBσλBλ˜B2+12λBσλBλB2+ξωεξ,e2+εBλB.Next, by applying Lemma3, we can obtain(52)12σOt,ωO˜t,ωF2l≤1−lι+12σOt,ωO˜t,ωF2,12σξωξ˜ω2l≤1−lι+12σξωξ˜ω2,12λBσλBλ˜B2l≤1−lι+12λBσλBλ˜B2.Hence, we can get that(53)V˙2≤−12e2Te2ϒθϒTθ−λmink2e22l−12σOt,ωO˜t,ωF2l−12σξωξ˜ω2l−12λBσλBλ˜B2l+εϒ,2,where(54)εϒ,2=12σOt,ωOt,ωF2+12σξωξω2+12λBσλBλB2+ξωεξ,e2+εBλB+31−lι.


### 3.3. Stability Analysis of the Proposed BLSTM-Based Finite-Time Control Law

The 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 sθ 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), we define V=V1+V2. Thus,(55)V˙=V˙1+V˙2≤−λmink1sθTsθ2l−λmink2e22l−12σOt,θO˜t,θF2l−12σξθξ˜θ2l−12λAσλAλ˜A2l−12σOt,ωO˜t,ωF2l−12σξωξ˜ω2l−12λBσλBλ˜B2l+εϒ1+εϒ2.
Define(56)σ=min4λmink1,2λmink2,σOt,θΓOt,θl,σξθΓξθl,σλAΓλAl,σOt,ωΓOt,ωl,σξωΓξωl,σλBΓλBl,εϒ=εϒ1+εϒ2=12σOt,θOt,θF2+12σξθξθ2+12λAσλAλA2+λmax2Ωθ+εAλA+ξθεξ,Λ+12σOt,ωOt,ωF2+12σξωξω2+12λBσλBλB2+ξωεξ,e2+εBλB+61−lι.
We can define that(57)V˙≤−σVl+εϒ.
By solving (55), we can get that(58)Vt≤V1−l0−ασ1−lt1/1−l,where 0<α<1. Meanwhile, let(59)T=1ασ1−lV1−l0−εϒ1−ασ1/1−l,where V0 represents the initial value of Vt. According to Lemma 2, for t≥T, Vl≤εϒ/1−ασ, that is, all the signals of the closed-loop system are SGPFS. Furthermore, according to the definition of V, it can be inferred that for ∀t≥T,(60)sθTsθ≤2εϒ1−ασ1/2lholds. In other words, after the finite time T, sθ remains in a small neighborhood of the origin. Thus, the tracking errors can converge to a desired compact set in a finite time and never violate the predefined constraint. The proof is completed.


## 3.1. BLSTM-Based Inner Loop Finite Control Law

Definee1,ϑ=ϑ−ϑd, e1,ψ=ψ−ψd, e1,γ=γ−γd. Therefore, the inner loop error dynamic equation can be given by(14)e˙1=AθΩθω+fθ+dθt−θ˙d,where Ωθ=Aθ+ΔAθ/Aθ.The following nonlinear functions are utilized to guarantee the output constraints:(15)sϑ=logℓa,1+e1,ϑℓa,2+e1,ϑ,sψ=logℓb,1+e1,ψℓb,2+e1,ψ,sγ=logℓc,1+e1,γℓc,2+e1,γ.Definesθ=sϑ,sψ,sγT. Accordingly, we can get that(16)s˙θ=TθAθΩθω+fθ+dθt−θ˙d,where(17)Tθ=Diagesϑ+e−sϑ+2ℓa,1+ℓa,2esψ+e−sψ+2ℓb,1+ℓb,2esγ+e−sγ+2ℓc,1+ℓc,2.Considering the uncertaintiesfθ, a BLSTM network Ot,θtanhCt,θ is applied to the inner controller. Clearly, fθ=Ot,θ∘tanhCt,θ+εθ, where εθ is the bounded approximation error. Define the BLSTM node number in the inner loop as N1. We can design the self-adjust strategy for the nodes number as(18)N1=N1+n1ad,∫t1−1t1sθτdτ>χ1up,N1=N1,χ1low≤∫t1−1t1sθτdτ≤χ1up,N1=N1−n1ad,∫t1−1t1sθτdτ<χ1low,where n1ad represents the adjustment and it is a designed positive integer; χ1low, χ1up are designed positive constants, denoting the adjustment threshold; t1 represents the time series to complete the judgment and adjustment; and O^t,θ is the estimated value of Ot,θ.Defineξθ=supt≥0εθ+dθt. Considering (16), the indirect virtual control signal is designed as(19)ωvc=Aθ−1−k1sθ2l−1Tθ−O^t,θtanhCt,θ−ξ^θφθ+θ˙d,φθ=ΛTθΛθΛTθ+εξ,Λ2,∀εξ,Λ>0,where ωvc is the virtual control signal of ω, k1∈ℝ3×3 is the control gain, ξ^θ is the estimated value of ξθ, and Λθ=sθTsθsθTTθ.Defineλ^A as the estimated value of λA. Therefore, we design the inner control law as(20)ωv=−λ^Aωvcϒθλ^Aωvcϒθλ^Aωvc2+εA2,∀εA>0,where ϒθ=Λθ,Aθ∈ℝ1×3. Substituting (19) into (16) generates(21)s˙θ=−k1sθ2l−1+TθAθΩθe2+AθΩθωv−Aθωvc−O˜t,θtanhCt,θ−ξ^θφθ+εθ+dθt,where e2=ω−ωv.The following Lyapunov function is selected:(22)V1=12sθTsθ+12ΓOt,θO˜t,θTO˜t,θ+12Γξθξ˜θ2+12ΓλAλAλ˜A2,where O˜t,θ=O^t,θ−Ot,θ, ξ˜θ=ξ^θ−ξθ, λ˜A=λ^A−λA, and ΓOt,θ, Γξθ, ΓλA>0 is the adaptive gain. Take the differential of V1 as follows:(23)V˙1=sθTs˙θ+1ΓOt,θO˜t,θTO^˙t,θ+1Γξθξ˜θξ^˙θ+1ΓλAλAλ˜Aλ^˙A.Substituting (21) into (23) yields(24)V˙1=−k1sθTsθ2l−1+ΛθAθΩθe2+AθΩθωv−Aθωvc−O˜t,θtanhCt,θ−ξ^θφθ+εθ+dθt+1ΓOt,θO˜t,θTO^˙t,θ+1Γξθξ˜θξ^˙θ+1ΓλAλAλ˜Aλ^˙A.By using Lemma1, it can be checked that(25)Λθεθ+dθt≤Λθξθ≤ξθΛθΛTθΛθΛTθ+εξ,Λ2+ξθεξ,Λ,where εξ,Σ>0. Substituting (25) into (24), we can obtain that(26)V˙1≤−k1sθTsθ2l−1+ΛθAθΩθe2+AθΩθωv−Aθωvc−O˜t,θtanhCt,θ−ξ˜θΛθΛTθΛθΛTθ+εξ,Λ2+ξθεξ,Λ+1ΓOt,θO˜t,θTO^˙t,θ+1Γξθξ˜θξ^˙θ+1ΓλAλAλ˜Aλ^˙A.Then, from (20) and Lemma 1, we know that(27)ΛθAθΩθωv=ϒθΩθωv=−Ωθϒθλ^Aωvc2ϒθλ^Aωvc2+εA2≤−ϒθλ^Aωvc2λAϒθλ^Aωvc2+εA2≤−1λAϒθλ^Aωvc−εA≤1λAϒθλ^Aωvc+εA.Therefore,(28)ΛθAθΩθωv−ΛθAθωvc≤λ˜AλAϒθωvc+εAλA.Obviously,(29)ΛθAΩθe2=ϒθΩθe2≤λmax2Ωθ+12ϒθϒTθe2Te2.Combining (26), (28), and (29), it can be proven that(30)V˙1≤−k1sθTsθ2l−1−ΛθO˜t,θtanhCt,θ−ξ˜θΛθφθ+λmax2Ωθ+12ϒθϒTθe2Te2+λ˜AλAϒθωvc+1ΓOt,θO˜t,θTO^˙t,θ+1Γξθξ˜θξ^˙θ+1ΓλAλAλ˜Aλ^˙A+εAλA+ξθεξ,Λ.Considering (30), we design the adaptive laws for O^t,θ, ξ^θ, λ^A as(31)O^˙t,θ=ΓOt,θΛTθ∘tanhCt,θ−σOt,θO^t,θ,ξ^˙θ=ΓξθΛθφθ−σξθξ^θ,λ^˙A=ΓλA−ϒθωvc−σλAλ^A,where σOt,θ, σξθ, σλA>0. Therefore, by using the update law (31), we know that(32)V˙1≤−k1sθTsθ2l−1+λmax2Ωθ+12ϒθϒTθe2Te2−σOt,θO˜t,θTO^t,θ−σξθξ˜θξ^θ−σλAλ˜Aλ^AλA+εAλA+ξθεξ,Λ.Since(33)−σOt,θO˜t,θTO^t,θ≤−12σOt,θO˜t,θF2+12σOt,θOt,θF2−σξθξ˜θξ^θ≤−12σξθξ˜θ2+12σξθξθ2−1λAσλAλ˜Aλ^A≤−12λAσλAλ˜A2+12λAσλAλA2,substituting (33) into (32), we can rewrite (32) as(34)V˙1≤−k1sθ2l+12ϒθϒTθe2Te2−12σOt,θO˜t,θF2−12σξθξ˜θ2−12λAσλAλ˜A2+λmax2Ωθ+12σOt,θOt,θF2+12σξθξθ2+12λAσλAλA2+εAλA+ξθεξ,Λ.According to Lemma3, the following inequality holds:(35)12σOt,θO˜t,θF2l≤ι1−l+12σOt,θO˜t,θF2,12σξθξ˜θ2l≤ι1−l+12σξθξ˜θ2,12λAσλAλ˜A2l≤ι1−l+12λAσλAλ˜A2.Therefore, we can get that(36)V˙1≤−λmink1sθ2l+12ϒθϒTθe2Te2−12σOt,θO˜t,θF2l−12σξθξ˜θ2l−12λAσλAλ˜A2l+εϒ,1,where(37)εϒ,1=12σOt,θOt,θF2+12σξθξθ2+12λAσλAλA2+λmax2Ωθ+3ι1−l+εAλA+ξθεξ,Λ.


## 3.2. BLSTM-Based Input Uncertainty Suppressed Outer Loop Finite-Time Controller Design

According to (7), we can get that(38)e˙2=Bω+ΔBωδ+Cω+fθ,ω+dωt−ω˙v.A BLSTM networkOt,ωtanhCt,ω is utilized to handle fθ,ω. Define the BLSTM node number in the outer loop as N2. We can also design the self-adjust strategy for the nodes number as(39)N2=N2+n2ad,∫t2−1t2e2τdτ>χ2up,N2=N2,χ2low≤∫t2−1t2e2τdτ≤χ2up,N2=N2−n2ad,∫t2−1t2e2τdτ<χ2low,where εω represents the adjustment and it is a designed positive integer; χ2low, χ2up are designed positive constants, denoting the adjustment threshold; and t2 represents the time series to complete the judgment and adjustment.According to (38), we design(40)δvc=Bω−1−k2e22l−1−12e2ϒθϒTθ−Cω−O^t,ωtanhCt,ω−ξ^ωφe2+ω˙v,φe2=e2e2Te2+εξ,e22,∀εξ,e2>0,where δvc is the virtual control signal of δ, k2 is the control gain, Bω−1=BωT/BωBωT, and ξ^ω represents the estimations of ξω (ξω=supt≥0εω+dωt).To deal with the perturbations ofBω, we define λB=1/infλminΩω. Accordingly, we design(41)δv=−λ^Bδvce2TBωλ^Bδvce2TBωλ^Bδvc2+εB2,∀εB>0,where λ^B is the estimation of λB. Substituting (40) and (41) into (38) yields(42)e˙2=−12e2ϒθϒTθ−k2e22l−1+BωΩωδ−Bωδvc−O˜t,ωtanhCt,ω−ξ^ωφe2+dωt+εω.Select the following Lyapunov function:(43)V2=12e2Te2+12ΓOt,ωO˜t,ωTO˜t,ω+12Γξωξ˜ω2+12ΓλBλBλ˜B2,where O˜t,ω=O^t,ω−Ot,ω, ξ˜ω=ξ^ω−ξω, λ˜B=λ^B−λB, ΓOt,ω, Γξω, ΓλB>0. We can take the differential of V2 as(44)V˙2=e2Te˙2+1ΓOt,ωO˜t,ωTO^˙t,ω+1Γξωξ˜ωξ^˙ω+1ΓλBλBλ˜Bλ^˙B=e2TBωΩωδ−Bωδvc−12e2ϒθϒTθ−k2e22l−1−O˜t,ωtanhCt,ω−ξ^ωφe2+εω+dωt+1ΓOt,ωO˜t,ωTO^˙t,ω+1Γξωξ˜ωξ^˙ω+1ΓλBλBλ˜Bλ^˙B.By using Lemma1, we know that(45)e2Tεω+dωt≤ξωe2≤ξωe2Tφe2+ξωεξ,e2.Then, from (41) and Lemma 1, we can obtain that(46)e2TBωΩωδv=−Ωωe2TBωλ^Bδvce2TBωλ^Bδvce2TBωλ^Bδvc2+εB2≤−1λBe2TBωλ^Bδvc−εB≤λ^BλBe2TBωδvc+εBλB.Therefore,(47)e2TBωΩωδ−e2TBωδvc≤λ˜BλBe2TBωδvc+εBλB.By substituting (45) and (47) into (44), it can be obtained that(48)V˙2≤−k2e22l−12e2Te2ϒθϒTθ+λ˜BλBe2TBωδvc+εBλB−e2TO˜t,ωtanhCt,ω−ξ˜ωe2Tφe2+ξωεξ,e2+1ΓOt,ωO˜t,ωTO^˙t,ω+1Γξωξ˜ωξ^˙ω+1ΓλBλBλ˜Bλ^˙B.In view of (41), the adaptive laws for O^t,ω, ξ^ω, λ^B are designed as(49)O^˙t,ω=ΓOt,ωe2∘tanhCt,ω−σOt,ωO^t,ω,ξ^˙ω=Γξωe2Tφe2−σξωξ^ω,λ^˙B=−ΓλBe2TBωδvc−σλBλ^B,where σOt,ω, σξω, σλB>0. Combining (48) with (49) generates(50)V˙2≤−k2e22l−12e2Te2ϒθϒTθ−σOt,ωO˜t,ωTO^t,ω−σξωξ˜ωξ^ω−σλBλ˜Bλ^BλB+ξωεξ,e2+εBλB.Therefore,(51)V˙2≤−12e2Te2ϒθϒTθ−λmink2e22l−12σOt,ωO˜t,ωF2+12σOt,ωOt,ωF2−12σξωξ˜ω2+12σξωξω2−12λBσλBλ˜B2+12λBσλBλB2+ξωεξ,e2+εBλB.Next, by applying Lemma3, we can obtain(52)12σOt,ωO˜t,ωF2l≤1−lι+12σOt,ωO˜t,ωF2,12σξωξ˜ω2l≤1−lι+12σξωξ˜ω2,12λBσλBλ˜B2l≤1−lι+12λBσλBλ˜B2.Hence, we can get that(53)V˙2≤−12e2Te2ϒθϒTθ−λmink2e22l−12σOt,ωO˜t,ωF2l−12σξωξ˜ω2l−12λBσλBλ˜B2l+εϒ,2,where(54)εϒ,2=12σOt,ωOt,ωF2+12σξωξω2+12λBσλBλB2+ξωεξ,e2+εBλB+31−lι.


## 3.3. Stability Analysis of the Proposed BLSTM-Based Finite-Time Control Law

The 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 sθ 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), we define V=V1+V2. Thus,(55)V˙=V˙1+V˙2≤−λmink1sθTsθ2l−λmink2e22l−12σOt,θO˜t,θF2l−12σξθξ˜θ2l−12λAσλAλ˜A2l−12σOt,ωO˜t,ωF2l−12σξωξ˜ω2l−12λBσλBλ˜B2l+εϒ1+εϒ2.
Define(56)σ=min4λmink1,2λmink2,σOt,θΓOt,θl,σξθΓξθl,σλAΓλAl,σOt,ωΓOt,ωl,σξωΓξωl,σλBΓλBl,εϒ=εϒ1+εϒ2=12σOt,θOt,θF2+12σξθξθ2+12λAσλAλA2+λmax2Ωθ+εAλA+ξθεξ,Λ+12σOt,ωOt,ωF2+12σξωξω2+12λBσλBλB2+ξωεξ,e2+εBλB+61−lι.
We can define that(57)V˙≤−σVl+εϒ.
By solving (55), we can get that(58)Vt≤V1−l0−ασ1−lt1/1−l,where 0<α<1. Meanwhile, let(59)T=1ασ1−lV1−l0−εϒ1−ασ1/1−l,where V0 represents the initial value of Vt. According to Lemma 2, for t≥T, Vl≤εϒ/1−ασ, that is, all the signals of the closed-loop system are SGPFS. Furthermore, according to the definition of V, it can be inferred that for ∀t≥T,(60)sθTsθ≤2εϒ1−ασ1/2lholds. In other words, after the finite time T, sθ remains in a small neighborhood of the origin. Thus, the tracking errors can converge to a desired compact set in a finite time and never violate the predefined constraint. The proof is completed.


## 4. 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 ish=1000m, and the initial attitude angles and angular velocities are set as zero. The desired signals are selected as ϑd=20°, ψd=15°, and γd=4°, and the simulation time is T=10s. The parameters of AUS are provided in Table 1.Table 1
The parameters of the AUS.
V=10m/sm=28kgb=3.1mc=0.58mS=1.8m2ρ=1.29kg/m3Ixx=2.56kg⋅m2Izx=0.5kg⋅m2Izz=11.3kg⋅m2Iyy=10.9kg⋅m2Ixz=0.5kg⋅m2The uncertainties are chosen asξθ=0.006,0.002,0.003T and ξω=0.007,0.001,0.004T. Simultaneously, the uncertainties of the four cases considered in the simulations are given in Table 2. In this paper, the parameters of BLSTM network are given in Table 3.Table 2
The uncertainties of the four cases.
fθ0.002+0.3×sin6.28t/100.001+0.2×sin6.28t/100.002+0.01×sin6.28t/10fθ,ω0.007+0.6×sin6.28t/100.001+0.2×sin6.28t/100.004+0.3×sin6.28t/10Table 3
The parameters of the system based on BLSTM networks.
The parameters based on BLSTM networks in the inner loopWii=0.1,Woi=1,bii=2,boi=2,Ninner0=7The parameters based on BLSTM networks in the outer loopWio=0.1,Woo=1,bio=5,boo=12,Nouter0=10The simulation results are given in Figures3–11. 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. That 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. Therefore, 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. The 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. Therefore, it can be concluded that the proposed BLSTM-based adaptive control algorithm can achieve favorable control results for the AUSs with multiple uncertainties.Figure 3
The performance ofϑ under four cases.Figure 4
The performance ofψ under four cases.Figure 5
The performance ofγ under four cases.Figure 6
Trajectory of system stateϑ in the case of fault-tolerant controller δv and controller δvc.Figure 7
Trajectory of system stateψ in the case of fault-tolerant controller δv and controller δvc.Figure 8
Trajectory of system stateγ in the case of fault-tolerant controller δv and controller δvc.Figure 9
The trajectories of adaptive parameters in inner loop with the proposed method.Figure 10
The trajectories of adaptive parameters in outer loop with the proposed method.Figure 11
The trajectory of BLSTM node number.


## 5. Conclusions

In this paper, we propose a BLSTM-based adaptive finite-time control structure for a class of AUSs. The novel BLSTM which possesses both the memory function and rapid learning speed has been constructed. The fractional-order 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. The 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 BLSTM-based adaptive finite-time control method proposed in this paper will be used to deal with the AUSs with stochastic disturbances.



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