# Evasion-Faced Fast Adaptive Neural Attitude Control for Generic Hypersonic Vehicles with Structural and Parametric Uncertainties

Authors: Tian Yan; Yuanli Cai; Caisheng Wei
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

In general, the evasion task requires the hypersonic vehicles (HSVs) to quickly complete the attitude maneuver in a short time. Moreover, the rapid variation of the flight modes often induces the structural and parametric uncertainties as well as the highly dynamic disturbances of the HSVs. The peculiar and complex characteristics of the evasion process make it difficult to design the evasion-faced flight control systems. In this work, we investigate the fast adaptive control design problem for the generic HSVs under the evasion task. By introducing several especial nonlinear functional vectors and properly designing the adaptive laws, the high dynamic disturbances and uncertainties can be suppressed. To deal with the completed unknown parts of the structural uncertainties and aerodynamic uncertainties caused by evasion maneuver, two radial basis function neural networks (RBFNNs) are introduced as real-time approximators. Furthermore, to improve the response speed of the flight control system, a super-twisting (STW) algorithm-based predictor is used as a feed-forward term of the controller. Consequently, a novel evasion-faced fast adaptive feed-forward control structure has been established for the HSVs. It has been proven that all the signals of the closed-loop system are bounded with satisfactory tracking velocity. Finally, the simulation experiment has been set up to show the effectiveness and advantages of the proposed control method.


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## Body

## 1. Introduction

Hypersonic vehicle has been an active issue for several years due to its splendid maneuverability, strong penetration ability, and fast reaction ability. Compared with traditional ballistic missiles, hypersonic vehicles possess significant advantages such as satisfactory combat effectiveness [1–3]. The main functionality of the hypersonic vehicle control system is to achieve attitude control during flight and complete the tasks transmitted from the guidance subsystem. Because of high flight speed, the accurate information of the dynamic characteristics and atmospheric parameters is difficult to be obtained, which brings difficulties to the design of the flight controller. Moreover, the strong coupling and the complex uncertainties of the HSVs drive the control design difficulties to a higher degree. In summary, the design of the flight controller plays an important role in the development of hypersonic aircraft [4, 5].In recent years, many methods have been applied to the design of hypersonic vehicle controllers, including dynamic inverse method [6], sliding mode control method [7], active disturbance rejection control method [8], back-stepping control, [9] and adaptive control method [4]. Senll et al. proposed a dynamic inverse control method for the nonlinear and strong coupling characteristics of the aircraft during super-maneuvering at a high angle of attack [10]. In [11], a dynamic inverse control method has been proposed for the HSVs, for the purpose of suppressing the intrinsic nonlinear characteristics existing in the HSV system. Li and his colleagues [12] used the dynamic inverse method to achieve complete linearization of the longitudinal model of the HSVs and achieve good control of the hypersonic vehicle’s altitude and velocity maneuvers. First order and second order sliding mode control methods are also used in the control design of HSVs [13–15], dealing with the matched uncertainties and improving the robustness of the control systems. To achieve the antiwindup tracking control, the technique of terminal sliding mode control has been applied to a class of air-breathing HSVs in [16]. As further development, for the purpose of avoiding the undesired chattering phenomenon, a high-order sliding mode control structure has been designed for the HSVs in [17]. By proposing a sliding mode differentiator, a novel robust back-stepping control scheme has been constructed for the longitudinal dynamics of HSV, eliminating the problem of explosion of terms [18]. In [19–21], the active disturbance rejection control method is used to estimate and compensate the strong uncertainty, channel coupling, and external disturbance of hypersonic vehicles. To solve the actuator nonlinearities, as well as the mismatch between the mathematical model of hypersonic vehicle and the actual dynamic model in complex environment, the adaptive control method is employed in [22, 23]. Existing robust adaptive control methods include [1, 24–26]. The model reference adaptive controller for the HSVs can be found in [27]. In literatures [13, 28–33], several fuzzy adaptive controllers and three adaptive sliding mode controllers of the HSVs have been designed, respectively. Furthermore, adaptive back-stepping control structure is proposed in [34, 35]. Fault-tolerant adaptive control [36–38] has also been mentioned. In [1], a robust adaptive control law which possesses strong self-adaptability and robustness has been proposed for the HSVs, guaranteeing the stability of the attitude control system. With the aid of the parameter projection technique, a nonlinear robust adaptive control structure has been established for the HSV [25]. In order to adapt to the model change of long-span flight, the model reference adaptive control is used in [27] to make the controlled object track the ideal reference model, so as to obtain the ideal control effect and improve the performance of the control system. In literature [30], fuzzy adaptive method is used to approach the coupling uncertainty of attitude angle to achieve accurate tracking. In [13], a new attitude control algorithm for hypersonic vehicle is proposed by applying fuzzy design to adaptive sliding mode control algorithm. Moreover, in the control of HSVs, the disturbance observers are often used to observe and approximate the disturbances existing in the system, or to compensate the mismatched uncertainties in different channels [11]. It should be pointed out that, in the research on control methods of hypersonic vehicle, various control methods are often used together to solve the control problems of HSVs [6]. In [37], the high-order sliding mode control theory has been combined with the adaptive back-stepping control technique, enables the HSV to track speed and height commands stably and effectively, and reduces chattering phenomenon at the same time. In [34], an asymmetric barrier Lyapunov function has been utilized in the adaptive back-stepping controller design such that attack angle constraints of the HSVs can be guaranteed during the control process. In [39], a simplified adaptive back-stepping controller using neural networks has been proposed for the longitudinal dynamics of an AHV. Considering the modeling uncertainties and actuator saturation or backlash, an adaptive fault-tolerant controller has been designed in [36]. The state constrained controller of the HSVs can be found in [9, 40].However, few control schemes have been investigated for the HSVs under evasion task. In the evasion scenario, the design of the flight control system becomes more challenging because of the complex flight environment, the rapidly changing aerodynamic parameters, and the structural uncertainties. During evasion flight, the drastic change of aerodynamic parameters was a serious challenge to the aircraft control. Because of the peculiar and complex characteristics of the evasion task, a lot of flight controllers are not capable of achieving desired control performance anymore. The control methods mentioned above can hardly satisfy fast tracking requirement of the evasion task. In [41], for a class of uncertain nonlinear systems, a novel sliding mode controller on the basis of LMI has been proposed to satisfy the presence of the sliding mode around the linear surface in the finite time. Furthermore, this method has been extended for the uncertain nonlinear systems with disturbances [42, 43], time-delays, and input nonlinearities, while guaranteeing the robustness and transient performance. In [44], a multivariable super-twisting differentiator has been proposed, which can provide the prediction ability and drastically improve the response speed of the control system. In [45], by using the super-twisting control algorithm, a novel integral sliding mode control strategy has been proposed for the magnetic systems. Compared with the traditional sliding mode control, the symbol function of discontinuous high-frequency switching is replaced by the STW control law such that the output control signal is continuous and the chattering can be essentially eliminated. In [46], a super-twisting observer is designed to estimate the actual value of the brake cylinder, so as to weaken the time-varying disturbance and improve the suppression ability of the system to external disturbance. In [47], an adaptive sliding mode super-twisting differentiator scheme has been proposed, possessing significant estimation and prediction capability. Therefore, it can be known that the STW algorithm can be utilized to construct the feed-forward loop of the controller for the evasion-faced HSVs to enhance the ability of prediction and the response speed. Unfortunately, to the best of the author’s knowledge, few control schemes have been proposed for the HSVs under the evasion task, and the evasion-faced fast adaptive flight controller using STW predictor has never been investigated.Therefore, this paper focuses on the evasion-faced fast adaptive attitude control design problem of the HSVs. Several nonlinear function vectors have been utilized and the high dynamic disturbances and uncertainties can be handled. The RBFNNs are employed as real-time approximators to deal with the completed unknown uncertainties induced by evasion large maneuver. Moreover, an STW algorithm-based predictor is introduced as a feed-forward term to improve the response speed. Compared with the conventional control methods, the proposed control methods possess the following features:(i)
As far as the authors know, it is the first evasion-faced fast adaptive flight controller for the HSVs, suffering from structural and parametric uncertainties and high dynamic disturbances.(ii)
By using the STW algorithm-based predictor, the response speed of the flight controller can be enhanced and the requirement of the evasion task can be satisfied.(iii)
With the aid of the nonlinear adaptive method and the RBFNNs, the highly dynamic disturbances and unknown uncertainties caused by evasion large maneuver can be effectively suppressed.


## 2. Problem Formulation and Preliminaries

### 2.1. The Attitude Dynamics of HSV

Based on the results of [2], the attitude dynamics of the HSVs can be described by(1)η˙=A11ω+A12CF+g¯+d1t,ω˙=fω+BCM+d2t,where η=α,β,γvT denote the vector of the attitude angles, composed of the angle of attack, the sideslip angle, and the bank angle. ω=ωx,ωy,ωzT is the vector of the attitude angular rates. CF,CM∈ℝ^3 represent the coefficients of aerodynamic force and aerodynamic moments, respectively. d1t,d2t∈ℝ^3 represent the uncertainties in the attitude angle dynamics and attitude angular velocity dynamics. A11,A12,B∈ℝ^3×3 are the system matrices, defined as follows:(2)A11=−tanβcosα−tanβsinα1sinα−cosα0secβcosαsecβsinα0,A12=qS⋅0−1mVcosβ0001mV0tanβ+tanθsinγvmVtanθcosγvmV,B=qSL⋅Diag1Ix,1Iy,1Iz,where Ix,Iy,Iz represent the moments of inertia of roll, yaw, and pitch channel; the physical sense of V,g,m,θ,q,S, and L can be founded in [2]. g¯∈ℝ^3 is the nonlinear vector caused by the gravitational force; fω∈ℝ^3 is the nonlinear terms related to the inertia parameters. g¯ and fω can be formulated by(3)g¯=gcosθcosγvVcosβ,gcosθsinγvV,−gcosθtanβcosγvVT,fω=Iy−IzIxωyωz,Iz−IxIyωxωz,Ix−IyIzωxωyT.The coefficients of aerodynamic forceCF can be computed as(4)CF=CF,0+CF,1η.The aerodynamic moment coefficientsCM can be computed as(5)CM=CM,0+CM,1η+CM,2ω+CM,3δ,where δ=δz,δy,δx are the reflections of the elevator, rudder, and aileron, representing the vector of control inputs. CF,0,CM,0∈ℝ^3 are the baseline aerodynamic coefficients. CF,1,CM,1,CM,2,CM,3∈ℝ3×3 are the aerodynamic partial derivatives.Accordingly, the aerodynamic uncertainties can be modeled is(6)CF,i=C¯F,i+ΔCF,i,i=0,1,CM,i=C¯M,i+ΔCM,i,i=0,1,2,3,where C¯F,i and C¯M,i are known baseline values; ΔCF,i,i=0,1 and ΔCM,i,i=0,1,2,3 denote the aerodynamic perturbations.In practical, there often exist the modeling errors and structural uncertainties. LetΔA11,ΔA12, Δfω, and ΔB denote the unknown structural uncertainties. Hence, by taking the structural uncertainties and aerodynamic uncertainties into consideration, the following attitude control model can be obtained:(7)η˙=A11ω+ΔA11ω+A12C¯F,0+C¯F,1η+g¯+fηV,θ,η+d1t,ω˙=fω+Δfω+BC¯M,0+C¯M,1η+C¯M,2ω+fωV,θ,η+ℬδ+Δℬδ+d2t,where ℬ=BC¯M,3,fηV,θ,η,fωV,θ,η,ω are the lumped aerodynamic uncertainties, those can be formulated by(8)fηV,θ,η=ΔA12C¯F,0+C¯F,1η+A12+ΔA12ΔCF,0+ΔCF,1η,fωV,θ,η,ω=ΔBC¯M,0+C¯M,1η+C¯M,2ω+B+ΔBΔCM,0+ΔCM,1η+CM,2ω,where Δℬ are the composite uncertainties caused by modeling errors and aerodynamic perturbations, which can be described by(9)Δℬ=BΔCM,3+ΔBC¯M,3+ΔCM,3.The control objective can be given as follows.Problem 1.
Given the HSV system (7) under the evasion task, it is required to propose a dynamic control law δt to force the angle of attack α, the sideslip angle β, and the bank angle γV to track the evasion guidance command signals.


### 2.2. Assumptions and Preliminaries

Assumption 1.
It is supposed that the matricesA11 and ℬ are invertible.Assumption 2.
There exist positive constantsΔ¯F,Δ¯M such that the aerodynamic uncertainties satisfy ∀t≥0,d1t≤Δ¯F,d2t≤Δ¯M.Assumption 3.
In the vicinity of the equilibrium, the uncertain part ofA11 and ℬ remains unchanged and cannot change the positive or negative characteristics of A11 and ℬ, respectively. In other words, it is assumed that dΔA11/dt≈0,dΔℬ/dt≈0,ΛA=ΛAT,Λℬ=ΛℬT,λminΛA>0 and λminΛℬ>0. Notice that it is defined that ΛA=A11+ΔA11/A11 and Λℬ=ℬ+Δℬ/ℬ.Lemma 1 (see [3]).
Given anyε>0 and z∈ℝ, it can be known that(10)0≤z−zTzzTz+ε2<ε.Lemma 2 (see[4]).
Consider any time-varying variableσ,ς∈ℝp satisfying that(11)σ˙=−k11oσσ1/2−k12oσ+ς,ς˙=−k21oσσ−k21o+ϕt,where ϕt is a bounded vector, then there exist positive constants k11o,k12o,k21o,k22o such that σ and ς can be forced to zero in finite time and remain zero for all subsequent time.Based on the results provided by [5, 6], it can be known that the RBFNNs can approximate the unknown nonlinearities existing in the dynamical systems. As is well known, for any z∈Ω and constant ϵ>0, given any continuous function hz:Ω⟶ℝ, we can find a RBFNN ΘTΦz, satisfying supz∈Ωhz−ΘTΦz<ϵ. In the RBFNN, Θ∈ℝl denotes the weight vector, l represents the amount of nodes, Φz=ϕ1z,…,ϕlzT, and ϕiz=exp−z−vi2/bi2,i=1,…,l, bi denotes the width of the sigmoid function, and vi=vi,1,…,vt,nT is the center parameter. Define Θ∗ as the optimal weight vector; denote ε∗ as the optimal approximation error. Therefore, we know that(12)hz=Θ∗TΦz+ε∗.In the following text, the RBFNNs will be utilized to approximate and compensate the unknown nonlinearities existing in the hypersonic dynamics.


## 2.1. The Attitude Dynamics of HSV

Based on the results of [2], the attitude dynamics of the HSVs can be described by(1)η˙=A11ω+A12CF+g¯+d1t,ω˙=fω+BCM+d2t,where η=α,β,γvT denote the vector of the attitude angles, composed of the angle of attack, the sideslip angle, and the bank angle. ω=ωx,ωy,ωzT is the vector of the attitude angular rates. CF,CM∈ℝ^3 represent the coefficients of aerodynamic force and aerodynamic moments, respectively. d1t,d2t∈ℝ^3 represent the uncertainties in the attitude angle dynamics and attitude angular velocity dynamics. A11,A12,B∈ℝ^3×3 are the system matrices, defined as follows:(2)A11=−tanβcosα−tanβsinα1sinα−cosα0secβcosαsecβsinα0,A12=qS⋅0−1mVcosβ0001mV0tanβ+tanθsinγvmVtanθcosγvmV,B=qSL⋅Diag1Ix,1Iy,1Iz,where Ix,Iy,Iz represent the moments of inertia of roll, yaw, and pitch channel; the physical sense of V,g,m,θ,q,S, and L can be founded in [2]. g¯∈ℝ^3 is the nonlinear vector caused by the gravitational force; fω∈ℝ^3 is the nonlinear terms related to the inertia parameters. g¯ and fω can be formulated by(3)g¯=gcosθcosγvVcosβ,gcosθsinγvV,−gcosθtanβcosγvVT,fω=Iy−IzIxωyωz,Iz−IxIyωxωz,Ix−IyIzωxωyT.The coefficients of aerodynamic forceCF can be computed as(4)CF=CF,0+CF,1η.The aerodynamic moment coefficientsCM can be computed as(5)CM=CM,0+CM,1η+CM,2ω+CM,3δ,where δ=δz,δy,δx are the reflections of the elevator, rudder, and aileron, representing the vector of control inputs. CF,0,CM,0∈ℝ^3 are the baseline aerodynamic coefficients. CF,1,CM,1,CM,2,CM,3∈ℝ3×3 are the aerodynamic partial derivatives.Accordingly, the aerodynamic uncertainties can be modeled is(6)CF,i=C¯F,i+ΔCF,i,i=0,1,CM,i=C¯M,i+ΔCM,i,i=0,1,2,3,where C¯F,i and C¯M,i are known baseline values; ΔCF,i,i=0,1 and ΔCM,i,i=0,1,2,3 denote the aerodynamic perturbations.In practical, there often exist the modeling errors and structural uncertainties. LetΔA11,ΔA12, Δfω, and ΔB denote the unknown structural uncertainties. Hence, by taking the structural uncertainties and aerodynamic uncertainties into consideration, the following attitude control model can be obtained:(7)η˙=A11ω+ΔA11ω+A12C¯F,0+C¯F,1η+g¯+fηV,θ,η+d1t,ω˙=fω+Δfω+BC¯M,0+C¯M,1η+C¯M,2ω+fωV,θ,η+ℬδ+Δℬδ+d2t,where ℬ=BC¯M,3,fηV,θ,η,fωV,θ,η,ω are the lumped aerodynamic uncertainties, those can be formulated by(8)fηV,θ,η=ΔA12C¯F,0+C¯F,1η+A12+ΔA12ΔCF,0+ΔCF,1η,fωV,θ,η,ω=ΔBC¯M,0+C¯M,1η+C¯M,2ω+B+ΔBΔCM,0+ΔCM,1η+CM,2ω,where Δℬ are the composite uncertainties caused by modeling errors and aerodynamic perturbations, which can be described by(9)Δℬ=BΔCM,3+ΔBC¯M,3+ΔCM,3.The control objective can be given as follows.Problem 1.
Given the HSV system (7) under the evasion task, it is required to propose a dynamic control law δt to force the angle of attack α, the sideslip angle β, and the bank angle γV to track the evasion guidance command signals.


## 2.2. Assumptions and Preliminaries

Assumption 1.
It is supposed that the matricesA11 and ℬ are invertible.Assumption 2.
There exist positive constantsΔ¯F,Δ¯M such that the aerodynamic uncertainties satisfy ∀t≥0,d1t≤Δ¯F,d2t≤Δ¯M.Assumption 3.
In the vicinity of the equilibrium, the uncertain part ofA11 and ℬ remains unchanged and cannot change the positive or negative characteristics of A11 and ℬ, respectively. In other words, it is assumed that dΔA11/dt≈0,dΔℬ/dt≈0,ΛA=ΛAT,Λℬ=ΛℬT,λminΛA>0 and λminΛℬ>0. Notice that it is defined that ΛA=A11+ΔA11/A11 and Λℬ=ℬ+Δℬ/ℬ.Lemma 1 (see [3]).
Given anyε>0 and z∈ℝ, it can be known that(10)0≤z−zTzzTz+ε2<ε.Lemma 2 (see[4]).
Consider any time-varying variableσ,ς∈ℝp satisfying that(11)σ˙=−k11oσσ1/2−k12oσ+ς,ς˙=−k21oσσ−k21o+ϕt,where ϕt is a bounded vector, then there exist positive constants k11o,k12o,k21o,k22o such that σ and ς can be forced to zero in finite time and remain zero for all subsequent time.Based on the results provided by [5, 6], it can be known that the RBFNNs can approximate the unknown nonlinearities existing in the dynamical systems. As is well known, for any z∈Ω and constant ϵ>0, given any continuous function hz:Ω⟶ℝ, we can find a RBFNN ΘTΦz, satisfying supz∈Ωhz−ΘTΦz<ϵ. In the RBFNN, Θ∈ℝl denotes the weight vector, l represents the amount of nodes, Φz=ϕ1z,…,ϕlzT, and ϕiz=exp−z−vi2/bi2,i=1,…,l, bi denotes the width of the sigmoid function, and vi=vi,1,…,vt,nT is the center parameter. Define Θ∗ as the optimal weight vector; denote ε∗ as the optimal approximation error. Therefore, we know that(12)hz=Θ∗TΦz+ε∗.In the following text, the RBFNNs will be utilized to approximate and compensate the unknown nonlinearities existing in the hypersonic dynamics.


## 3. Main Results

### 3.1. Inner Loop Fast Adaptive NN Controller Design

Letαd,βd,γv,d represent the desired trajectories of α,β,γv. Define ηd=αd. First of all, we can design the following auxiliary system:(13)ησ.=−k11oησ−ηdησ−ηd1/2−k12oησ−ηd+ς,ς˙=−k21oησ−ηdησ−ηd−k22oησ−ηd,where eσd=ησ−ηd,eς=ς−η˙d; it can be easily obtained that(14)e˙σd=−k11oeσdeσd1/2−k12oeσd+eς,e˙ς=−k21oeσdeσd−k22oeσd−η¨d.Then, according to Lemma2, it can be known that by properly adjusting k11o,k12o,k21o,k22o, eς can be forced to zero in finite time and remain zero for all subsequent time. In other words, ς converges to η˙d in finite time and remains here, and besides, there exists a constant ϵς such that ∀t≥0,eς≤ες.Accordingly, we introduce the following auxiliary signal:(15)ηp=Tpς+ηd,where Tp>0 is a design constant. In the control law, ηp acts as the command signal, instead of ηd.Let system outputη track the predicted desired signal ηp. Define the tracking error as eη=η−ηp. Define eη,d=η−ηd,eη,p=ηp−ηd. It is obvious that eη,d=eη,p+eη. By using (15), it can be known that η˙p=Tpς˙+ς−eς.In view of the attitude dynamic model (7), the dynamic equation of inner loop tracking error can be written as(16)e˙η=A11ΛAω+A12C¯F,0+C¯F,1η+g¯+fηV,θ,η+d1t−Tpς˙−ς+eς.In this subsection, an adaptive controller is designed to forceeη to converge to a small vicinity of zero. To approximate the unknown aerodynamic uncertainties fηV,θ,η, we introduce a RBFNN ΘηTΦηV,θ,η. Hence, it can be obtained that fηV,θ,η=ΘηTΦηV,θ,η+εΦ,η, where εΦ,η is the bounded approximation error. Define ξη=supt≥0εΦ,η+ες+d1t. Let Θ^η,ξ^η be the estimations of Θη,ξη.Considering the inner loop tracking error dynamics (16), the indirect virtual control signal is designed as(17)ωc=A11−1−k1eη−A12C¯F,0+C¯F,1η−ξ^ηφηeη−Θ^ηTΦηV,θ,η−g¯+Tpς˙+ς,where k1>0 is the control gain of the inner loop, φηeη=eη/eηTeη+εξ,η2∈ℝ^3, and εξ,η>0 is a design constant.It is easy to know that only usingωc cannot achieve the inner loop stability, and the induced input uncertainties have to be handled. Hence, we define the final virtual control signal of the inner loop as ωv, which is designed as(18)ωv=ωc+ωa,where ωa is an adaptive compensation term of the input uncertainties, designed as(19)ωa=−W^Aωc,where W^A is the estimate of WA, WA=ΛA−I/ΛA. By substituting (17) and (18) into (16), we can get that(20)e˙η=A11ΛAeω+A11ΛA−Iωc+A11ΛAωa−k1eη−Θ˜ηTΦηV,θ,η+εΦ,η+d1+eς−ξ^ηφηeη,where eω=ω−ωv. The adaptive laws of Θ^η,W^A, and ξ^ηT will be obtained in the following text.Consider the following Lyapunov function:(21)Vη=12eη,pTeη,p+12eηTeη+12ΓΘηTrΘ˜ηTΘ˜η+12ΓWATrW˜ATΛAW˜A+12Γξηξ˜η2,where Θ˜η=Θ^η−Θη,W˜A=W^A−WA,ξ˜η=ξ^η−ξη.ΓΘηΓWA,Γξη>0 are the adaptive gains, and TrA denotes the trace of matrices A. Along (20), we can take the differential of Vη as(22)V˙η=−eη,pTeη,pTp+eη,pTεp+eηTA11ΛAeω+eηTA11ΛA−Iωc+eηTA11ΛAωa−k1eηTeη−eηTΘ˜ηTΦηV,θ,η+eηTεΦ,η+d1+eς−ξ^ηeηTφηeη+1ΓΘηTrΘ˜ηTΘ^˙η+1ΓWAtrW˜ATΛAW^˙A+1Γξηξ˜ηξ^˙η,where εp=Tpς˙+ς. By using Lemma 1, we can get that(23)eηTεΦ,η+d1+eς≤eηξη≤ξηeηTφηeη+ξηεξ,η.With the aid of (19) and (23), we can rewrite (22) as(24)V˙η≤−eη,pTeη,pTp+eη,pTεp+eηTA11ΛAeω−eηTA11ΛAW˜aωc−k1eηTeη−eηTΘ˜ηTΦηV,θ,η−ξ˜ηeηTφηeη+ξηεξ,η+1ΓΘηTrΘ˜ηTΘ^˙η+1ΓWATrW˜ATΛAW^˙A+1Γξηξ˜ηξ^˙η.By using the Young’s inequality, it can be known that for any constantcp>0, the following inequality holds:(25)eη,pTεp≤cpeη,pTeη,p+14cpε¯pTε¯p,where ε¯p=supt≥0εp. Hence, we can rewrite (24) as(26)V˙η≤−1Tp−cpeη,pTeη,p+eηTA11ΛAeω−eηTA11ΛAW˜aωc−k1eηTeη−eηTΘ˜ηTΦηV,θ,η−ξ˜ηeηTφηeη+ξηεξ,η+14cpε¯pTε¯p+1ΓΘηTrΘ˜ηTΘ^˙η+1ΓWATrW˜ATΛAW^˙A+1Γξηξ˜ηξ^˙η.Design the update laws forΘ^η,W^A, and ξ^η as(27)Θ^˙η=ΓΘηΦηV,θ,ηeηT−σΘηΘ^η,W^˙A=ΓWAωceηTA11−σWAW^A,ξ^˙η=ΓξηeηTφηeη−σξηξ^η,where σΘη,σWA,σξη>0. Therefore, by substituting the adaptive laws (27) into (26), we can get that(28)V˙η≤−1Tp−cpeη,pTeη,p+eηTA11ΛAeω−k1eηTeη+ξηεξ,η+14cpε¯pTε¯p−σΘηTrΘ˜ηTΘ^η−σWATrW˜ATΛAW^A−σξηξ˜ηξ^η.Since(29)−TrΘ˜ηTΘ^η≤−12TrΘ˜ηTΘ˜η+12TrΘηTΘη−TrW˜ATΛAW^A≤−12TrW˜ATΛAW˜A+12TrWATΛAWA−ξ˜ηξ^η≤−12ξ˜η2+12ξη2,we can finally get that(30)V˙η≤−1Tp−cpeη,pTeη,p−k1eηTeη+eηTA11ΛAeω−σΘη2TrΘ˜ηTΘ˜η−σWA2TrW˜ATΛAW˜A−σξη2ξ˜η2+εη,where(31)εη=σΘη2TrΘηTΘη+σWA2TrWATΛAWA+σξη2ξη2+ξηεξ,η+14cpε¯pTε¯p.Defineϑ=supt≥0A11ΛA2. Then, it is easy to know that for any positive constant c, the following inequality holds:(32)eηTA11ΛAeω≤14ceηTeη+cϑeωTeω.Hence, it follows from (30) that(33)V˙η≤−1Tp−cpeη,pTeη,p−k1−14ceηTeη+cϑeωTeω−σΘη2TrΘ˜ηTΘ˜η−σWA2TrW˜ATΛAW˜A−σξη2ξ˜η2+εη.


### 3.2. Outer Loop Adaptive NN Controller Design

With the aid of second equation of (7), it can be obtained that(34)e˙ω=fω+Δfω+BC¯M,0+C¯M,1η+C¯M,2ω+fωV,θ,η+ℬδ+Δℬδ+d2t−ω˙v.In this section, we design a robust adaptive control structure to ensure closed-loop stability of (34). Identical to the former subsection, we introduce a RBFNN such that Δfω+fωV,θ,η=ΘωTΦωV,θ,η+εΦ,ω, where ϵΦ,ω is the bounded approximation error. Let ξω=supt≥0εΦ,ω+d2t. Let Θ^ω,ξ^ω,ϑ^ be the estimations of Θω,ξω,ϑ. Several adaptive laws will be designed to update the parameters Θ^ω,ξ^ω,ϑ^.Hence, the indirect control signal can be given by(35)δc=ℬ−1−k2eω−cϑ^eω−fω−BC¯M,0+C¯M,1η+C¯M,2ω−Θ^ωTΦωV,θ,η,ω−ξ^ωφωeω+ω˙v,where k2>0,φωeω=eω/eωTeω+εξ,ω2,εξ,ω>0.It is easy to know that only usingδc cannot achieve the inner loop stability, and the induced input uncertainties have to be handled. Hence, we design the final control signal as(36)δ=δc+δa,where δa is an adaptive compensation term of the input uncertainties, designed as(37)δa=−W^ℬδc,where W^ℬ is the estimate of Wℬ, Wℬ=Λℬ−I/Λℬ. By substituting (35) and (36) into (34), we can get that(38)e˙ω=−k2eω−cϑ^eω−Θ^ωTΦωV,θ,η,ω+ϵΦ,ω+ℬΛℬ−Iδc+ℬΛℬδa+d2t−ξ^ωTφωeω.Select the following Lyapunov function,(39)Vω=12eωTeω+12ΓΘωTrΘ˜ωTΘ˜ω+12ΓWℬTrW˜ℬTΛℬW˜ℬ+12Γξωξ˜ω2+12Γϑθ˜2,where Θ˜ω=Θ^ω−Θω,W˜ℬ=W^ℬ−Wℬ,ξ˜ω=ξ^ω−ξω.ΓΘωΓWℬ,Γξω>0. Along (38), the derivative of Vω can be taken as(40)V˙ω=−k2eωTeω−cϑ^eωTeω−eωTΘ˜ωTΦωV,θ,η,ω+eωTℬΛℬ−Iδc+eωTℬΛℬδa+eωTεΦ,ω+d2t−ξ^ωeωTφωeω+1ΓΘωTrΘ˜ωTΘ^˙ω+1ΓWℬTrW˜ℬTΛℬW^˙ℬ+1Γξωξ˜ωξ^˙ω+1Γϑϑ˜ϑ^˙.According to Lemma1, the following inequality can be obtained:(41)eωTεΦ,ω+d2t≤eωξω≤ξωeωTφωeω+ξωεξ,ω.Then, from (41), we know that(42)V˙ω≤−k2eωTeω−cϑ˜eωTeω−cϑeωTeω−eωTΘ˜ωTΦωV,θ,η,ω−eωTℬΛℬW˜ℬδc−ξ˜ωeωTφωeω+ξωεξ,ω+1ΓΘωTrΘ˜ωTΘ^˙ω+1ΓWℬTrW˜ℬTΛℬW^˙ℬ+1Γξωξ˜ωξ^˙ω+1Γϑϑ˜ϑ^˙.In view of (42), the update laws for Θ^ω,W^ℬ,ϑ^, and ξ^ω are designed as follows:(43)Θ^˙ω=ΓΘωΦωV,θ,η,ωeωT−σΘωΘ^ω,W^˙ℬ=ΓWℬδceωTℬ−σWℬW^ℬ,ξ^˙ω=ΓξωeωTφωeω−σξωξ^ω,ϑ^˙=ΓϑceωTeω−σϑϑ^,where σΘω,σWℬ,σξω,σϑ>0 are the design constants. Then, from (42) and (43), it can be proved that(44)V˙ω≤−k2eωTeω−cϑeωTeω+ξωεξ,ω−σΘωTrΘ˜ωTΘ^ω−σWℬTrW˜ℬTΛℬW^ℬ−σξωξ˜ωξ^ω−σϑϑ˜ϑ^.By using the following inequalities,(45)−TrΘ˜ωTΘ^ω≤−12TrΘ˜ωTΘ˜ω+12TrΘωTΘω−TrW˜ℬTΛℬW^ℬ≤−12TrW˜ℬTΛℬW˜ℬ+12TrWℬTΛℬWℬ−ξ˜ωξ^ω≤−12ξ˜ω2+12ξω2,−ϑ˜ϑ^≤−12ϑ˜2+12ϑ2,we can finally get that(46)V˙ω≤−k2eωTeω−cϑeωTeω−σΘω2TrΘ˜ωTΘ˜ω−σWℬ2TrW˜ℬTΛℬW˜ℬ−σξω2ξ˜ω2−σϑ2ϑ˜2+εω,where(47)εω=ξωεξ,ω+σΘω2TrΘωTΘω+σWℬ2TrWℬTΛℬWℬ+σξω2ξω2+σϑ2ϑ2.The structure of the proposed evasion-faced fast attitude control method is given by Figure1.Figure 1
The structure of the proposed evasion-faced fast attitude control method.


### 3.3. Stability Analysis

Theorem 1.
Consider the hypersonic attitude dynamics (1). If the virtual control laws are designed as (17)–(19), the actual control laws are designed as (35)–(37), the adaptive laws are selected as (27) and (43), then the closed-loop system consisting of the adaptive controller, the auxiliary system (13), and the parameter update laws (27) and (43) is stable; all the signals and the equilibrium of closed-loop system are bounded, and the angle of attack α, the sideslip angle β, and the bank angle γv will converge to the desired values asymptotically.Proof.
Consider the following Lyapunov function:(48)V=Vη+Vω,where Vη and Vω have been given in (21) and (39), respectively. Then, according to the analysis provided in Sections 2.1 and 2.2, we can arrive at that(49)V˙≤−1Tp−cpeη,pTeη,p−k1−14ceηTeη−k2eωTeω−σΘη2TrΘ˜ηTΘ˜η−σWA2TrW˜ATΛAW˜A−σξη2ξ˜η2+εη−σΘω2TrΘ˜ωTΘ˜ω−σWℬ2TrW˜ℬTΛℬW˜ℬ−σξω2ξ˜ω2−σϑ2ϑ˜2+εω.
Therefore, it can be easily obtained that(50)V˙≤−λV+ε,where(51)λ=min21Tp−cp,2k1−12c,2k2,ΓΘησΘη,σWAΓWA,σξηΓξη,σΘωΓΘω,σWℬΓWℬ,σξωΓξω,σϑΓϑ,ε=εη+εω.
Hence, it is easy to know that∀t≥0,0≤Vt≤maxε/2λ,V0. As a consequence, the boundness of all the signals in the closed-loop system can be ensured. Then, by considering the definition of eη,p and eη, it can be proved that η can converge to a small neighborhood of ηd asymptotically. In other words, the angle of attack α, the sideslip angle β, and the bank angle γv can track the desired values asymptotically. The proof is completed.


## 3.1. Inner Loop Fast Adaptive NN Controller Design

Letαd,βd,γv,d represent the desired trajectories of α,β,γv. Define ηd=αd. First of all, we can design the following auxiliary system:(13)ησ.=−k11oησ−ηdησ−ηd1/2−k12oησ−ηd+ς,ς˙=−k21oησ−ηdησ−ηd−k22oησ−ηd,where eσd=ησ−ηd,eς=ς−η˙d; it can be easily obtained that(14)e˙σd=−k11oeσdeσd1/2−k12oeσd+eς,e˙ς=−k21oeσdeσd−k22oeσd−η¨d.Then, according to Lemma2, it can be known that by properly adjusting k11o,k12o,k21o,k22o, eς can be forced to zero in finite time and remain zero for all subsequent time. In other words, ς converges to η˙d in finite time and remains here, and besides, there exists a constant ϵς such that ∀t≥0,eς≤ες.Accordingly, we introduce the following auxiliary signal:(15)ηp=Tpς+ηd,where Tp>0 is a design constant. In the control law, ηp acts as the command signal, instead of ηd.Let system outputη track the predicted desired signal ηp. Define the tracking error as eη=η−ηp. Define eη,d=η−ηd,eη,p=ηp−ηd. It is obvious that eη,d=eη,p+eη. By using (15), it can be known that η˙p=Tpς˙+ς−eς.In view of the attitude dynamic model (7), the dynamic equation of inner loop tracking error can be written as(16)e˙η=A11ΛAω+A12C¯F,0+C¯F,1η+g¯+fηV,θ,η+d1t−Tpς˙−ς+eς.In this subsection, an adaptive controller is designed to forceeη to converge to a small vicinity of zero. To approximate the unknown aerodynamic uncertainties fηV,θ,η, we introduce a RBFNN ΘηTΦηV,θ,η. Hence, it can be obtained that fηV,θ,η=ΘηTΦηV,θ,η+εΦ,η, where εΦ,η is the bounded approximation error. Define ξη=supt≥0εΦ,η+ες+d1t. Let Θ^η,ξ^η be the estimations of Θη,ξη.Considering the inner loop tracking error dynamics (16), the indirect virtual control signal is designed as(17)ωc=A11−1−k1eη−A12C¯F,0+C¯F,1η−ξ^ηφηeη−Θ^ηTΦηV,θ,η−g¯+Tpς˙+ς,where k1>0 is the control gain of the inner loop, φηeη=eη/eηTeη+εξ,η2∈ℝ^3, and εξ,η>0 is a design constant.It is easy to know that only usingωc cannot achieve the inner loop stability, and the induced input uncertainties have to be handled. Hence, we define the final virtual control signal of the inner loop as ωv, which is designed as(18)ωv=ωc+ωa,where ωa is an adaptive compensation term of the input uncertainties, designed as(19)ωa=−W^Aωc,where W^A is the estimate of WA, WA=ΛA−I/ΛA. By substituting (17) and (18) into (16), we can get that(20)e˙η=A11ΛAeω+A11ΛA−Iωc+A11ΛAωa−k1eη−Θ˜ηTΦηV,θ,η+εΦ,η+d1+eς−ξ^ηφηeη,where eω=ω−ωv. The adaptive laws of Θ^η,W^A, and ξ^ηT will be obtained in the following text.Consider the following Lyapunov function:(21)Vη=12eη,pTeη,p+12eηTeη+12ΓΘηTrΘ˜ηTΘ˜η+12ΓWATrW˜ATΛAW˜A+12Γξηξ˜η2,where Θ˜η=Θ^η−Θη,W˜A=W^A−WA,ξ˜η=ξ^η−ξη.ΓΘηΓWA,Γξη>0 are the adaptive gains, and TrA denotes the trace of matrices A. Along (20), we can take the differential of Vη as(22)V˙η=−eη,pTeη,pTp+eη,pTεp+eηTA11ΛAeω+eηTA11ΛA−Iωc+eηTA11ΛAωa−k1eηTeη−eηTΘ˜ηTΦηV,θ,η+eηTεΦ,η+d1+eς−ξ^ηeηTφηeη+1ΓΘηTrΘ˜ηTΘ^˙η+1ΓWAtrW˜ATΛAW^˙A+1Γξηξ˜ηξ^˙η,where εp=Tpς˙+ς. By using Lemma 1, we can get that(23)eηTεΦ,η+d1+eς≤eηξη≤ξηeηTφηeη+ξηεξ,η.With the aid of (19) and (23), we can rewrite (22) as(24)V˙η≤−eη,pTeη,pTp+eη,pTεp+eηTA11ΛAeω−eηTA11ΛAW˜aωc−k1eηTeη−eηTΘ˜ηTΦηV,θ,η−ξ˜ηeηTφηeη+ξηεξ,η+1ΓΘηTrΘ˜ηTΘ^˙η+1ΓWATrW˜ATΛAW^˙A+1Γξηξ˜ηξ^˙η.By using the Young’s inequality, it can be known that for any constantcp>0, the following inequality holds:(25)eη,pTεp≤cpeη,pTeη,p+14cpε¯pTε¯p,where ε¯p=supt≥0εp. Hence, we can rewrite (24) as(26)V˙η≤−1Tp−cpeη,pTeη,p+eηTA11ΛAeω−eηTA11ΛAW˜aωc−k1eηTeη−eηTΘ˜ηTΦηV,θ,η−ξ˜ηeηTφηeη+ξηεξ,η+14cpε¯pTε¯p+1ΓΘηTrΘ˜ηTΘ^˙η+1ΓWATrW˜ATΛAW^˙A+1Γξηξ˜ηξ^˙η.Design the update laws forΘ^η,W^A, and ξ^η as(27)Θ^˙η=ΓΘηΦηV,θ,ηeηT−σΘηΘ^η,W^˙A=ΓWAωceηTA11−σWAW^A,ξ^˙η=ΓξηeηTφηeη−σξηξ^η,where σΘη,σWA,σξη>0. Therefore, by substituting the adaptive laws (27) into (26), we can get that(28)V˙η≤−1Tp−cpeη,pTeη,p+eηTA11ΛAeω−k1eηTeη+ξηεξ,η+14cpε¯pTε¯p−σΘηTrΘ˜ηTΘ^η−σWATrW˜ATΛAW^A−σξηξ˜ηξ^η.Since(29)−TrΘ˜ηTΘ^η≤−12TrΘ˜ηTΘ˜η+12TrΘηTΘη−TrW˜ATΛAW^A≤−12TrW˜ATΛAW˜A+12TrWATΛAWA−ξ˜ηξ^η≤−12ξ˜η2+12ξη2,we can finally get that(30)V˙η≤−1Tp−cpeη,pTeη,p−k1eηTeη+eηTA11ΛAeω−σΘη2TrΘ˜ηTΘ˜η−σWA2TrW˜ATΛAW˜A−σξη2ξ˜η2+εη,where(31)εη=σΘη2TrΘηTΘη+σWA2TrWATΛAWA+σξη2ξη2+ξηεξ,η+14cpε¯pTε¯p.Defineϑ=supt≥0A11ΛA2. Then, it is easy to know that for any positive constant c, the following inequality holds:(32)eηTA11ΛAeω≤14ceηTeη+cϑeωTeω.Hence, it follows from (30) that(33)V˙η≤−1Tp−cpeη,pTeη,p−k1−14ceηTeη+cϑeωTeω−σΘη2TrΘ˜ηTΘ˜η−σWA2TrW˜ATΛAW˜A−σξη2ξ˜η2+εη.


## 3.2. Outer Loop Adaptive NN Controller Design

With the aid of second equation of (7), it can be obtained that(34)e˙ω=fω+Δfω+BC¯M,0+C¯M,1η+C¯M,2ω+fωV,θ,η+ℬδ+Δℬδ+d2t−ω˙v.In this section, we design a robust adaptive control structure to ensure closed-loop stability of (34). Identical to the former subsection, we introduce a RBFNN such that Δfω+fωV,θ,η=ΘωTΦωV,θ,η+εΦ,ω, where ϵΦ,ω is the bounded approximation error. Let ξω=supt≥0εΦ,ω+d2t. Let Θ^ω,ξ^ω,ϑ^ be the estimations of Θω,ξω,ϑ. Several adaptive laws will be designed to update the parameters Θ^ω,ξ^ω,ϑ^.Hence, the indirect control signal can be given by(35)δc=ℬ−1−k2eω−cϑ^eω−fω−BC¯M,0+C¯M,1η+C¯M,2ω−Θ^ωTΦωV,θ,η,ω−ξ^ωφωeω+ω˙v,where k2>0,φωeω=eω/eωTeω+εξ,ω2,εξ,ω>0.It is easy to know that only usingδc cannot achieve the inner loop stability, and the induced input uncertainties have to be handled. Hence, we design the final control signal as(36)δ=δc+δa,where δa is an adaptive compensation term of the input uncertainties, designed as(37)δa=−W^ℬδc,where W^ℬ is the estimate of Wℬ, Wℬ=Λℬ−I/Λℬ. By substituting (35) and (36) into (34), we can get that(38)e˙ω=−k2eω−cϑ^eω−Θ^ωTΦωV,θ,η,ω+ϵΦ,ω+ℬΛℬ−Iδc+ℬΛℬδa+d2t−ξ^ωTφωeω.Select the following Lyapunov function,(39)Vω=12eωTeω+12ΓΘωTrΘ˜ωTΘ˜ω+12ΓWℬTrW˜ℬTΛℬW˜ℬ+12Γξωξ˜ω2+12Γϑθ˜2,where Θ˜ω=Θ^ω−Θω,W˜ℬ=W^ℬ−Wℬ,ξ˜ω=ξ^ω−ξω.ΓΘωΓWℬ,Γξω>0. Along (38), the derivative of Vω can be taken as(40)V˙ω=−k2eωTeω−cϑ^eωTeω−eωTΘ˜ωTΦωV,θ,η,ω+eωTℬΛℬ−Iδc+eωTℬΛℬδa+eωTεΦ,ω+d2t−ξ^ωeωTφωeω+1ΓΘωTrΘ˜ωTΘ^˙ω+1ΓWℬTrW˜ℬTΛℬW^˙ℬ+1Γξωξ˜ωξ^˙ω+1Γϑϑ˜ϑ^˙.According to Lemma1, the following inequality can be obtained:(41)eωTεΦ,ω+d2t≤eωξω≤ξωeωTφωeω+ξωεξ,ω.Then, from (41), we know that(42)V˙ω≤−k2eωTeω−cϑ˜eωTeω−cϑeωTeω−eωTΘ˜ωTΦωV,θ,η,ω−eωTℬΛℬW˜ℬδc−ξ˜ωeωTφωeω+ξωεξ,ω+1ΓΘωTrΘ˜ωTΘ^˙ω+1ΓWℬTrW˜ℬTΛℬW^˙ℬ+1Γξωξ˜ωξ^˙ω+1Γϑϑ˜ϑ^˙.In view of (42), the update laws for Θ^ω,W^ℬ,ϑ^, and ξ^ω are designed as follows:(43)Θ^˙ω=ΓΘωΦωV,θ,η,ωeωT−σΘωΘ^ω,W^˙ℬ=ΓWℬδceωTℬ−σWℬW^ℬ,ξ^˙ω=ΓξωeωTφωeω−σξωξ^ω,ϑ^˙=ΓϑceωTeω−σϑϑ^,where σΘω,σWℬ,σξω,σϑ>0 are the design constants. Then, from (42) and (43), it can be proved that(44)V˙ω≤−k2eωTeω−cϑeωTeω+ξωεξ,ω−σΘωTrΘ˜ωTΘ^ω−σWℬTrW˜ℬTΛℬW^ℬ−σξωξ˜ωξ^ω−σϑϑ˜ϑ^.By using the following inequalities,(45)−TrΘ˜ωTΘ^ω≤−12TrΘ˜ωTΘ˜ω+12TrΘωTΘω−TrW˜ℬTΛℬW^ℬ≤−12TrW˜ℬTΛℬW˜ℬ+12TrWℬTΛℬWℬ−ξ˜ωξ^ω≤−12ξ˜ω2+12ξω2,−ϑ˜ϑ^≤−12ϑ˜2+12ϑ2,we can finally get that(46)V˙ω≤−k2eωTeω−cϑeωTeω−σΘω2TrΘ˜ωTΘ˜ω−σWℬ2TrW˜ℬTΛℬW˜ℬ−σξω2ξ˜ω2−σϑ2ϑ˜2+εω,where(47)εω=ξωεξ,ω+σΘω2TrΘωTΘω+σWℬ2TrWℬTΛℬWℬ+σξω2ξω2+σϑ2ϑ2.The structure of the proposed evasion-faced fast attitude control method is given by Figure1.Figure 1
The structure of the proposed evasion-faced fast attitude control method.


## 3.3. Stability Analysis

Theorem 1.
Consider the hypersonic attitude dynamics (1). If the virtual control laws are designed as (17)–(19), the actual control laws are designed as (35)–(37), the adaptive laws are selected as (27) and (43), then the closed-loop system consisting of the adaptive controller, the auxiliary system (13), and the parameter update laws (27) and (43) is stable; all the signals and the equilibrium of closed-loop system are bounded, and the angle of attack α, the sideslip angle β, and the bank angle γv will converge to the desired values asymptotically.Proof.
Consider the following Lyapunov function:(48)V=Vη+Vω,where Vη and Vω have been given in (21) and (39), respectively. Then, according to the analysis provided in Sections 2.1 and 2.2, we can arrive at that(49)V˙≤−1Tp−cpeη,pTeη,p−k1−14ceηTeη−k2eωTeω−σΘη2TrΘ˜ηTΘ˜η−σWA2TrW˜ATΛAW˜A−σξη2ξ˜η2+εη−σΘω2TrΘ˜ωTΘ˜ω−σWℬ2TrW˜ℬTΛℬW˜ℬ−σξω2ξ˜ω2−σϑ2ϑ˜2+εω.
Therefore, it can be easily obtained that(50)V˙≤−λV+ε,where(51)λ=min21Tp−cp,2k1−12c,2k2,ΓΘησΘη,σWAΓWA,σξηΓξη,σΘωΓΘω,σWℬΓWℬ,σξωΓξω,σϑΓϑ,ε=εη+εω.
Hence, it is easy to know that∀t≥0,0≤Vt≤maxε/2λ,V0. As a consequence, the boundness of all the signals in the closed-loop system can be ensured. Then, by considering the definition of eη,p and eη, it can be proved that η can converge to a small neighborhood of ηd asymptotically. In other words, the angle of attack α, the sideslip angle β, and the bank angle γv can track the desired values asymptotically. The proof is completed.


## 4. Simulation Study

In the section, two simulation experiments are set up to show the effectiveness and advantages of the proposed control method. The first simulation considered the HSVs suffering from different uncertainties, and the effectiveness and robustness of the proposed adaptive neural network control method can be revealed. The second simulation involved the proposed fast adaptive NN control (FANNC) method, the adaptive NN control (ANNC) methods without super-twist predictor, and the conventional model reference adaptive control method which are also included for comparison.In the simulation, the parameters of the hypersonic vehicle are selected as Table1. The initial attitude dynamics of the hypersonic vehicle are α=β=γv=0,ωx=ωy=ωz=0. Considering the evasion requirement, we select the desired signal as(52)αdt=5sin2πt10+5sinπt10,βdt=0,γv,dt=5sin2πt10+3.6cosπt10.The uncertainties of the three cases considered in the simulations are given in Tables 2 and 3.Table 1
The parameters of the hypersonic vehicle.
mSLIxx4900kg7.565m24.55m2345kg⋅m2IyyIzzv24706kg⋅m224309kg⋅m21860m/sTable 2
The uncertainties of the three cases.
ΔA11ΔA12Case 10.010000.010000.010.010000.010000.01Case 20.030000.040000.0120.020000.0050000.01Case 30.050000.040000.030.060000.0450000.015Table 3
The uncertainties of the three cases.
ΔfΔBCase 10.10.10.10.010000.010000.01Case 20.050.20.250.0150000.010000.03Case 30.350.240.250.020000.040000.03The initial values of the adaptive control parameters matrix areΘ^η0,Θ^ω0=09×3, W^A0,W^B0=03×3, ησ0,ς0=03×1, and ξ^η0,ξ^ω0,ϑ^0=0. The fixed value parameters of the hypersonic vehicle control system are chosen as Tp=0.01, c=1, k11=0.5, k12=3, k21=5, k22=10, k1=12, k2=18, ΓΘη=5, ΓWA=5, ΓΘω=5, ΓWB=5, Γξη=5, Γξω=5, Γϑ=5, σΘη=3, σWA=3, σΘω=3, σWB=3, σξη=3, σξω=3, and σϑ=3. The initial auxiliary system states are defined as follows: ησ0=0,0,0T and ς0=0,0,0T.The simulation results are given in Figures2–6. The tracking results in cases 1–3 are respectively given in Figures 2–4, from which we can see that the attitude angle of the three channels all track the expected signal quickly and steadily in different cases. The weights and the adaptive parameters are given in Figures 5 and 6. It is obvious that the adaptive parameters are all bounded. The robustness properties of the proposed algorithm can be revealed therefore.Figure 2
The tracking performance ofαdt under three cases.Figure 3
The tracking performance ofβdt under three cases.Figure 4
The tracking performance ofγv,dt under three cases.Figure 5
The norm of the NN weights matrix under three cases.Figure 6
The adaptive parameters under three cases.
(a)(b)To illustrate the superiority of the proposed adaptive control scheme, we further apply the adaptive NN control methods without using the super-twist predictor and the conventional model reference adaptive control method to the attitude control of the HSVs. Figures7–9 respectively show the tracking results under three control schemes; Table 4 provides the comparative control performance, from which we can conclude that the tracking error is unacceptable without proper adaptive compensation, and the phase lag may be induced if the STW predictor is deprecated. The reason for this difference is that the proposed method introduces adaptive neural networks and the STW predictor, which can tolerate the uncertainty in the flight process and improve the tracking velocity. The superiority of the proposed control scheme can be verified therefore.Figure 7
The tracking performance ofαdt of the three methods.Figure 8
The tracking performance ofβdt of the three methods.Figure 9
The tracking performance ofγv,dt of the three methods.Table 4
The performance comparison of the three control methods.
FANNCMRACANNCPitch channelMaximal time delay 0.01Maximal time delay 0.5Maximal time delay 0.35Maximal tracking error 0.05Maximal tracking error 0.6Maximal tracking error 0.4Yaw channelMaximal time delay 0Maximal time delay 0Maximal time delay 0Maximal tracking error 0.005Maximal tracking error 0.09Maximal tracking error 0.36Roll channelMaximal time delay 0.02Maximal time delay 0.4Maximal time delay 0.45Maximal tracking error 0.05Maximal tracking error 0.95Maximal tracking error 0.8


## 5. Conclusions

In this paper, a novel evasion-faced fast adaptive feed-forward control structure has been synthesized. By using a super-twisting (STW) algorithm-based predictor, the requirement of the response speed of the flight control system can be satisfied. By employing two RBFNNs as real-time approximators, the unknown uncertainties existing in the fight control system can be handled. Moreover, the adaptive laws have been properly designed such that the difficulties caused by the highly dynamic disturbances can be circumvented. As a consequence, the peculiar and complex attitude control problem for the evasion tasks can be addressed. Simulation results show that by using the proposed fast adaptive feed-forward control structure, the tracking error can be forced to converge into a desired domain and the phase lag can be reduced. Our future work includes the evasion-faced integrated guidance and control design for the HSVs and the parametric optimization for the evasion-faced flight control system of the HSVs.



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*Source: 2514073-2021-01-27.xml*