# Adaptive Neural Control of Hypersonic Vehicles with Actuator Constraints

Authors: Changxin Luo; Humin Lei; Dongyang Zhang; Xiaojun Zou
Journal: International Journal of Aerospace Engineering
 (2018)

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## Abstract

An adaptive neural control method is proposed in this paper for the flexible air-breathing hypersonic vehicle (AHV) with constraints on actuators. This scheme firstly converts the original control problem with input constraints into a new control problem without input constraints based on the control input saturation function. Secondly, on the basis of the implicit function theorem, the radial basis function neural network (RBFNN) is introduced to approximate the uncertain items of the model. And the minimal-learning-parameter (MLP) technique is adopted to design the adaptive law for the norm of network weight vector, which significantly reduces calculations. Meanwhile, the finite-time convergence differentiator (FD) is introduced, through which the model state variables and their derivatives are accurately estimated to ensure the control effect. Finally, it is theoretically proved that the closed-loop control system is stable. And the effectiveness of the designed controller is verified by simulation.


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## Body

## 1. Introduction

Air-breathing hypersonic vehicle (AHV) is a new type of aircraft flying in the near space at a speed of more than Mach 5. It has outstanding advantages in terms of high speed, high maneuverability, and large flight envelope that traditional aircrafts do not have, which possesses potential applications in both civilian and military fields [1–3]. But it also has more complex coupling, stronger nonlinearity, more severe elastic vibration, and tighter control input constraints than ordinary aircraft due to the flat and slender body design of the AHV and the integration of scramjet engine, which make the flight control of AHV a frontier issue in today’s control field [4, 5].In the design process of the control law for AHV, by introducing a neural network to approximate the model’s unknown dynamics or control laws that are difficult to be directly implemented, the nonlinear, strong coupling, and uncertainties of the model can be handled well to ensure the robust performance of the control law [6, 7]. For the rigid body model of AHV, the continuous and discrete adaptive neural controllers are designed, respectively, in [7, 8] by expressing original model as a strict feedback form and introducing RBFNN to approximate the model’s unknown function, which guarantee that the closed-loop signals are uniformly ultimately bounded but fail to consider the influence of flexible states. Different from references [7, 8], a nonsingular direct neural control is proposed in [9] based on backstepping method by firstly designing the ideal backstepping control law for each subsystem of AHV and then using the RBFNN to approximate the designed ideal control law online instead of the unknown function of the model. However, the design process and form of the control law in [9] are cumbersome and complex, which restrict its applications in engineering. Two novel neural backstepping control strategies are proposed in [10, 11], which are designed with improved backstepping methods, respectively. Simulations show that the proposed methods in [10, 11] have strong robustness and superior control effects, but both of them assume that the model is affine for control input.On the other hand, considering that AHV controls the height and attitude of longitudinal motion with elevators, as the flying height increases, the efficiency of the elevator will decrease significantly [12]. In addition, the AHV can be affected by unknown airflows such as gust and turbulence during the flight process. Therefore, it is easy to encounter the phenomenon of elevator saturation when flying at high altitude. Once the actuators reach saturation, it may cause failure of the control system [13]. Thence, it requires urgent development of antisaturation control studies for AHV. However, the related research in this respect is still relatively little. Only considering the throttle setting constraint of the engine, a neural control method based on time-scale decomposition is designed in [14]. In [13], the neural network is used to approximate the saturation characteristics of the control law, which effectively solves the problem of control input constraint. However, the neural network weight parameters strongly depend on the model and are difficult to select. The tracking error is corrected through the designed auxiliary error compensation system in [15]. Although the simulation result shows that it has certain feasibility in dealing with the problem of actuator constraints, it cannot theoretically guarantee that the tracking error is bounded. The method of [15] is further extended to AHV flight control where both control input and flight attitudes are constrained. However, in the actual project, AHV has no corresponding actuators to directly limit its flight attitude.There are two shortcomings in the above studies. First, forcing AHV’s nonaffine motion model into an affine model will inevitably result in the loss of certain key dynamic characteristics. The control law designed based on the simplified affine model will have the risk of partial or complete failure. The second is that the difficulty of the controller design is increased to some extent by introducing the auxiliary system or using the neural network to approach the actuator saturation characteristic when dealing with the actuator constraints of the AHV.Based on the above shortcomings, it is imperative to design a simple and effective nonaffine control law to solve the control problem of AHV when the actuators are constrained. This paper studies the control problem of AHV with actuator constraints and proposes an adaptive neural control method. By designing the hyperbolic tangent input saturation function, the original control problem with input constraints is transformed into a control problem without input constraints. Different from the above references, the AHV model is regarded as a pure feedback system with nonaffine control input that is closer to the actual situation of AHV. Based on the implicit function theorem, the RBFNN is introduced to accurately approximate the unknown function of the model. The MLP technique is adopted to adaptively adjust the norm of the weight vector, which greatly reduces the amount of adaptive calculations. And through employing FD to achieve effective estimation of the system state variables, the control accuracy is ensured. Simulation examples verify the effectiveness and superiority of the design method.


## 2. AHV Model and Preliminaries

### 2.1. Model Description

Parker, a scholar at the US Air Force Research Laboratory, based on the study of AHV models by Bolender and Doman [16] and by neglecting some slow dynamics and weak coupling of AHV, establishes the following AHV longitudinal motion model [17]:
(1)V̇=Tcosθ−γ−Dm−gsinγ,(2)ḣ=Vsinγ,(3)γ̇=L+Tsinθ−γmV−gVcosγ,(4)θ̇=Q,(5)Q̇=M+ψ~1η¨1+ψ~2η¨2Iyy,(6)k1η¨1=−2ζ1ω1η̇1−ω12η1+N1−ψ~1MIyy−ψ~1ψ~2η¨2Iyy,(7)k2η¨2=−2ζ2ω2η̇2−ω22η2+N2−ψ~2MIyy−ψ~2ψ~1η¨1Iyy,where
(8)k1=1+ψ~1Iyy,k2=1+ψ~2Iyy,ψ~1=∫−Lf0m^fξϕfξdξ,ψ~2=∫0Lam^aξϕaξdξ,where velocity V, altitude h, flight-path γ, pitch angle θ, and pitch rate Q are the five rigid body states; m and Iyy represent vehicle mass and moment of inertia, respectively; the four flexible modes η1, η̇1, η2, and η̇2 denote the first two bending modes of the fuselage; ζi and ωii=1,2 represent the damping ratio and natural frequency for flexible modes, respectively; Lf and La represent the length of forward beam and aft beam; m^f and m^a are the mass distribution of forward beam and aft beam, respectively; and ϕf⋅ and ϕa⋅ are structural mode shapes [16]. The force map of an AHV model is shown in Figure 1. The approximations of thrust T, drag D, lift L, pitching moment M, and the generalized forces Nii=1,2 are expressed as [17]
(9)T≈CTα3α3+CTα2α2+CTαα+CT0,D≈q¯SCDα2α2+CDαα+CDδe2δe2+CDδeδe+CD0,L≈q¯SCLαα+CLδeδe+CL0,M≈zTT+q¯Sc¯CM,αα2α2+CM,ααα+CM,α0+ceδe,N1≈N1α2α2+N1αα+N10,N2≈N2α2α2+N2αα+N2δeδe+N20,CTα3=β1h,q¯Φ+β2h,q¯,CTα2=β3h,q¯Φ+β4h,q¯,CTα=β5h,q¯Φ+β6h,q¯,CT0=β7h,q¯Φ+β8h,q¯,q¯=12ρ¯V2,ρ¯=ρ¯0exph0−hhs,where α=θ−γ is the angle of attack; fuel equivalence ratio Φ and elevator angular deflection δe are control inputs; c¯ and S are the aerodynamic chord and reference area, respectively; q¯ represents dynamic pressure; ρ¯ is the air density at height h; zT means thrust moment arm; ce is the elevator coefficient; h0 and ρ¯0 represent nominal altitude and air density at the altitude h0, respectively; 1/hs is the air density decay rate; CTαii=1,2,3 is the ith order coefficient of α in T; CD∗ii=1,2;∗=α,δe represents the ith order coefficient of ∗ in D; CL∗∗=α,δe means coefficient of ∗ in L; C∗0∗=T,D,L is the constant coefficient in ∗; CM,ααii=0,1,2 represents the ith order coefficient of α in M; Njαii=0,1,2;j=1,2 is the ith order contribution of α to Nj; and N2δe is the contribution of δe to N2; βih,q¯i=1,2,…,8 is the ith trust fit parameter. For more detailed definitions of the model geometric parameters and coefficients, the reader could refer to [17].Figure 1
Force map of an AHV model.


### 2.2. Input Constraint Problem and Model Conversion

According to [18] and combining (1), (2), (3), (4), (5), and (9), we know that velocity V is mainly controlled by equivalence ratio Φ since the thrust T is directly affected by Φ. On the other hand, altitude h is mainly controlled by elevator angular deflection δe since δe directly affects the pitch rate Q and then affects the pitch angle θ and flight-path γ, ultimately controlling the change of h. Therefore, we can firstly decompose the AHV model into a velocity subsystem (1) and altitude subsystem ((2), (3), (4), and (5)) and then design the control law separately [19].Taking the actual situation into account, the executable ranges ofΦ and δe have certain limits, which are described as follows:
(10)Φ∈Φmin,Φmax,δe∈δemin,δemax,where Φmin and Φmax≥0 are the respective upper and lower bound of Φ; δemin and δemaxδemin=−δemax stand for the upper and lower bound of δe, respectively.In the above situation of actuator constraints, for velocity subsystem and altitude subsystem, the control objective is to design the respective control lawΦ and δe under the constraints of (10) such that V and h track their reference commands Vref and href.Obviously,Φ and δe have saturation characteristics under the constraints of (10). In order to convert the original control problem with input constraints into a new control problem without input constraints, here we use hyperbolic tangent function to approximate Φ and δe
(11)ΦΦ∗=Φmax−Φmin2tanh2Φ∗Φmax−Φmin−1+Φmax+Φmin2,δeδe∗=δemaxtanhδe∗δemax,where Φ∗ and δe∗∈R are the actual input of actuator saturation loop. By (11) and the character of hyperbolic tangent function, we can know that no matter what values Φ∗ and δe∗ take, both Φ and δe satisfy the constraint of (10).At this point, the original control objective can be converted to design unconstrainedΦ∗ and δe∗ such that V and h track their reference commands Vref and href.Remark 1.
In order to maintain the normal operating mode of scramjet engine, a restriction is imposed on the value range ofΦ, which is generally taken as Φ∈0.05,1.5; taking the physical limit of the deflection angle of elevator into account, usually δe∈−20∘,20∘ [20].Remark 2.
Although the problem of actuator saturation is often considered when planning the trajectory, AHV can be affected by unknown airflows such as gusts and turbulences as well as sudden actuator failures during the flight process, which will cause the actuator to be saturated instantly such that the actual executable range of it will be even smaller than its theoretical value [13].


## 2.1. Model Description

Parker, a scholar at the US Air Force Research Laboratory, based on the study of AHV models by Bolender and Doman [16] and by neglecting some slow dynamics and weak coupling of AHV, establishes the following AHV longitudinal motion model [17]:
(1)V̇=Tcosθ−γ−Dm−gsinγ,(2)ḣ=Vsinγ,(3)γ̇=L+Tsinθ−γmV−gVcosγ,(4)θ̇=Q,(5)Q̇=M+ψ~1η¨1+ψ~2η¨2Iyy,(6)k1η¨1=−2ζ1ω1η̇1−ω12η1+N1−ψ~1MIyy−ψ~1ψ~2η¨2Iyy,(7)k2η¨2=−2ζ2ω2η̇2−ω22η2+N2−ψ~2MIyy−ψ~2ψ~1η¨1Iyy,where
(8)k1=1+ψ~1Iyy,k2=1+ψ~2Iyy,ψ~1=∫−Lf0m^fξϕfξdξ,ψ~2=∫0Lam^aξϕaξdξ,where velocity V, altitude h, flight-path γ, pitch angle θ, and pitch rate Q are the five rigid body states; m and Iyy represent vehicle mass and moment of inertia, respectively; the four flexible modes η1, η̇1, η2, and η̇2 denote the first two bending modes of the fuselage; ζi and ωii=1,2 represent the damping ratio and natural frequency for flexible modes, respectively; Lf and La represent the length of forward beam and aft beam; m^f and m^a are the mass distribution of forward beam and aft beam, respectively; and ϕf⋅ and ϕa⋅ are structural mode shapes [16]. The force map of an AHV model is shown in Figure 1. The approximations of thrust T, drag D, lift L, pitching moment M, and the generalized forces Nii=1,2 are expressed as [17]
(9)T≈CTα3α3+CTα2α2+CTαα+CT0,D≈q¯SCDα2α2+CDαα+CDδe2δe2+CDδeδe+CD0,L≈q¯SCLαα+CLδeδe+CL0,M≈zTT+q¯Sc¯CM,αα2α2+CM,ααα+CM,α0+ceδe,N1≈N1α2α2+N1αα+N10,N2≈N2α2α2+N2αα+N2δeδe+N20,CTα3=β1h,q¯Φ+β2h,q¯,CTα2=β3h,q¯Φ+β4h,q¯,CTα=β5h,q¯Φ+β6h,q¯,CT0=β7h,q¯Φ+β8h,q¯,q¯=12ρ¯V2,ρ¯=ρ¯0exph0−hhs,where α=θ−γ is the angle of attack; fuel equivalence ratio Φ and elevator angular deflection δe are control inputs; c¯ and S are the aerodynamic chord and reference area, respectively; q¯ represents dynamic pressure; ρ¯ is the air density at height h; zT means thrust moment arm; ce is the elevator coefficient; h0 and ρ¯0 represent nominal altitude and air density at the altitude h0, respectively; 1/hs is the air density decay rate; CTαii=1,2,3 is the ith order coefficient of α in T; CD∗ii=1,2;∗=α,δe represents the ith order coefficient of ∗ in D; CL∗∗=α,δe means coefficient of ∗ in L; C∗0∗=T,D,L is the constant coefficient in ∗; CM,ααii=0,1,2 represents the ith order coefficient of α in M; Njαii=0,1,2;j=1,2 is the ith order contribution of α to Nj; and N2δe is the contribution of δe to N2; βih,q¯i=1,2,…,8 is the ith trust fit parameter. For more detailed definitions of the model geometric parameters and coefficients, the reader could refer to [17].Figure 1
Force map of an AHV model.


## 2.2. Input Constraint Problem and Model Conversion

According to [18] and combining (1), (2), (3), (4), (5), and (9), we know that velocity V is mainly controlled by equivalence ratio Φ since the thrust T is directly affected by Φ. On the other hand, altitude h is mainly controlled by elevator angular deflection δe since δe directly affects the pitch rate Q and then affects the pitch angle θ and flight-path γ, ultimately controlling the change of h. Therefore, we can firstly decompose the AHV model into a velocity subsystem (1) and altitude subsystem ((2), (3), (4), and (5)) and then design the control law separately [19].Taking the actual situation into account, the executable ranges ofΦ and δe have certain limits, which are described as follows:
(10)Φ∈Φmin,Φmax,δe∈δemin,δemax,where Φmin and Φmax≥0 are the respective upper and lower bound of Φ; δemin and δemaxδemin=−δemax stand for the upper and lower bound of δe, respectively.In the above situation of actuator constraints, for velocity subsystem and altitude subsystem, the control objective is to design the respective control lawΦ and δe under the constraints of (10) such that V and h track their reference commands Vref and href.Obviously,Φ and δe have saturation characteristics under the constraints of (10). In order to convert the original control problem with input constraints into a new control problem without input constraints, here we use hyperbolic tangent function to approximate Φ and δe
(11)ΦΦ∗=Φmax−Φmin2tanh2Φ∗Φmax−Φmin−1+Φmax+Φmin2,δeδe∗=δemaxtanhδe∗δemax,where Φ∗ and δe∗∈R are the actual input of actuator saturation loop. By (11) and the character of hyperbolic tangent function, we can know that no matter what values Φ∗ and δe∗ take, both Φ and δe satisfy the constraint of (10).At this point, the original control objective can be converted to design unconstrainedΦ∗ and δe∗ such that V and h track their reference commands Vref and href.Remark 1.
In order to maintain the normal operating mode of scramjet engine, a restriction is imposed on the value range ofΦ, which is generally taken as Φ∈0.05,1.5; taking the physical limit of the deflection angle of elevator into account, usually δe∈−20∘,20∘ [20].Remark 2.
Although the problem of actuator saturation is often considered when planning the trajectory, AHV can be affected by unknown airflows such as gusts and turbulences as well as sudden actuator failures during the flight process, which will cause the actuator to be saturated instantly such that the actual executable range of it will be even smaller than its theoretical value [13].


## 3. Adaptive Neural Controller Design

### 3.1. RBFNN Approximation

RBFNN has the characteristics of simple structure, strong learning, and fault tolerance and has the ability of globally approximating nonlinear continuous functions [21]. It can be expressed as a mapping from input to output.
(12)y=WThX,where X=X1,X2,…,XnT∈Rn is the input vector (n represents the dimension of the input vector), W=w1,w2,…,wNT∈RN stands for the weight vector (N is the number of hidden layer nodes), hX=h1X,h2X,…,hNXT∈RN, hiX, denotes the activation function. Here, hiX is chosen as the following Gaussian function:
(13)hiX=exp−X−ci22bi2,i=1,2,…,N,where b=b1,b2,…,bNT∈RN, bi, is the width of the ith Gaussian function; c=c1,c2,…,cN∈Rn×N (ci represents the center of the ith Gaussian function), which can be expressed as follows:
(14)c=c11⋯c1N⋮⋱⋮cn1⋯cnN.For any unknown nonlinear continuous functionFX, using the RBFNN and by selecting enough nodes (selecting a sufficiently large N), there must be an ideal weight vector W∗=w1∗,w2∗,…,wN∗T∈RN that satisfies [21]
(15)FX=W∗ThX+μ,μ≤μM,where μ∈R is the approximation error and μM∈R+ stands for the upper bound of approximation error. When taking N large enough, μM can be arbitrarily small.Remark 3.
Sinceexp⋅ is a strictly monotonically increasing and positive function and −X−ci2/2bi2≤0, there is 0<hiX≤hi0=1. Therefore, there must be a bounded constant h¯∈R+ such that hX≤h¯.


### 3.2. Controller Design for Velocity Subsystem

Based on the research conclusion of [7], here velocity subsystem is further expressed as a more general nonaffine form of control input
(16)V̇=FVV,Φ.Combined with (11), the above formula can be rewritten as
(17)V̇=FVV,Φ∗,where FVV,Φ∗ is a completely unknown continuously differentiable function.In order to design the control lawΦ∗, the following theorem is given firstly.Theorem 1.
For anyV,Φ∗∈ΩV×R, the following inequality is established:
(18)∂FVV,Φ∗∂Φ∗>0,where ΩV is a controllable area.Proof 1.
From (11), (16), and (17), we can know
(19)∂FVV,Φ∗∂Φ∗=∂FVV,Φ∗∂Φ·∂Φ∂Φ∗=∂FVV,Φ/∂Φcosh22Φ∗/Φmax−Φmin−1.
According to [17], there is
(20)∂FVV,Φ∂Φ>0.
Therefore,(21)∂FVV,Φ∗∂Φ∗>0.Define velocity tracking error(22)V~=V−Vref.Taking time derivative along (22) and using (17) yield
(23)V~̇=V̇−V̇ref=FVV,Φ∗−V̇ref.Let(24)FV∗V,Φ∗=FVV,Φ∗−kVΦ∗.where kV∈R+ is the design parameter.Combine (23) and (24)
(25)V~̇=V̇−V̇ref=FV∗V,Φ∗+kVΦ∗−V̇ref.Design the control lawΦ∗ as
(26)Φ∗=kV−1Φ0∗−Φ1∗,where
(27)Φ0∗=−kV1V~−kV2∫0tV~dτ+V̇ref.kV1 and kV2∈R+ stand for design parameters and Φ1∗ is the neural control law to be designed to counteract the influence of uncertain term FV∗V,Φ∗.In order to facilitate the subsequent design process, the implicit function theorem is introduced here [22].Theorem 2.
Assume that the implicit functionΨ:Rl×Rr→Rl is continuously differentiable at each point ς,σ on the open set Y⊂Rl×Rr. ς0,σ0 is the point in Y; Ψς0,σ0=0 and the Jacobian matrix ∂Ψ/∂ςς0,σ0 is nonsingular. Then, for any σ∈G, a neighborhood U⊂Rl of ς0 and a neighborhood G⊂Rr of σ0 can make the equation Ψς,σ=0 which has a unique solution ς∈U, and the solution can be expressed as ς=g0σ, where g0⋅ is a continuously differentiable function on σ=σ0.Remark 4.
Theorem2 shows that once the implicit function Ψς,σ satisfies all conditions in the theorem, ς can be expressed as a continuously differentiable function of σ, that is, ς=g0σ. At this time, using RBFNN to approximate Ψς,σ, only σ, instead of ς, is used as the input signal of the neural network to obtain a satisfactory approximation effect. This is where the special meaning of the implicit function theorem lies.Let(28)ΓV,Φ0∗,Φ1∗≜FV∗V,Φ∗−Φ1∗=FV∗V,kV−1Φ0∗−Φ1∗−Φ1∗.To illustrate thatΓV,Φ0∗,Φ1∗ satisfies the implicit function theorem, the following theorem is given.Theorem 3.
Define(29)kV>12∂FVV,Φ∗∂Φ∗.
Then there are a controllable areaΩV⊂R and a unique Φ1∗ for any V,Φ1∗∈ΩV×R; Φ1∗ satisfies
(30)ΓV,Φ0∗,Φ1∗=0.Proof 2.
According to [23], a sufficient condition for the existence of Φ1∗ is that the following inequality holds:
(31)∂FV∗V,Φ∗∂Φ1∗<1.
Consider (18), (24), (26), and (29), there are
(32)∂FV∗V,Φ∗∂Φ1∗=∂FVV,Φ∗−kVΦ∗∂Φ1∗=∂FVV,Φ∗−kVΦ∗∂Φ∗∂Φ∗∂Φ1∗=∂FVV,Φ∗∂Φ∗−kV1kV=1kV∂FVV,Φ∗∂Φ∗−1<1.
Therefore,Φ1∗ exists.Further,(33)∂ΓV,Φ0∗,Φ1∗∂Φ1∗=∂FV∗V,Φ∗−Φ1∗∂Φ1∗=∂FVV,Φ∗−kVΦ∗∂Φ1∗−1=∂FVV,Φ∗−kVΦ∗∂Φ∗∂Φ∗∂Φ1∗−1=∂FVV,Φ∗∂Φ∗−kV−1kV−1=−1kV∂FVV,Φ∗∂Φ∗,where Φ∗=kV−1Φ0∗−Φ1∗.Combine (18) and (33)
(34)∂ΓV,Φ0∗,Φ1∗∂Φ1∗<0.According to Theorem3 and (34), ΓV,Φ0∗,Φ1∗ satisfies the implicit function theorem. Therefore, Φ1∗ can be regarded as a function of Φ0∗ and V, and further FV∗V,Φ∗ can be regarded as a function of Φ0∗ and V.DefineX1=V,Φ0∗T∈R2 as the input vector of RBFNN and introduce RBFNN to approximate FV∗V,Φ∗
(35)FV∗V,Φ∗=WV∗ThX1+ε,ε≤εM,where WV∗T=wV1∗,wV2∗,…,wVN1∗T∈RN1 is the weight vector, N1 is the number of nodes, and ε and εM are approximation errors and its upper bounds, respectively. And hX1=h1X1,h2X1,…,hN1X1T∈RN1, where hiX1 is the activation function; here, hiX1 is chosen as the Gaussian function.The following is based on the MLP technique to adaptively adjust the norm of the RBFNN weight vector.DefineωV=WV∗2 and design Φ1∗ as
(36)Φ1∗=12V~ω^VhTX1hX1,where ω^V is the estimation of ωV, and its adaptive law is designed as
(37)ω^̇V=μV2V~2hTX1hX1−2ω^V,where μV∈R+ is the design parameter.Substitute (27) and (36) into (26) and finally get the control law Φ∗
(38)Φ∗=kV−1−kV1V~−kV2∫0tV~dτ+V̇ref−12V~ω^VhTX1hX1.Remark 5.
Different from [24] in which the weight vector of the neural network is directly adjusted online, this paper regards WV∗ as a whole based on the MLP technique. Adaptive adjustment of ωV requires only one online learning parameter ω^V, and the computational complexity of the approximation algorithm is significantly reduced.


### 3.3. Controller Design for Altitude Subsystem

Define altitude tracking error(39)h~=h−href.Taking time derivative along (39) and using (2) yield
(40)h~̇=Vsinγ−ḣref.The reference trajectory ofγ is chosen as
(41)γd=arcsin−kγ1h~−kγ2∫0th~dτ+ḣrefV,where kγ1, and kγ2∈R+ are design parameters. If γ→γd, known by (40) and (41), the dynamic response of h~ is
(42)h~¨+kγ1h~̇+kγ2h~=0.Performing Laplace transformation on both sides of (42) can get its characteristic equation
(43)s2+kγ1s+kγ2=0.The two characteristic roots−kγ1−kγ12−4kγ2/2 and −kγ1+kγ12−4kγ2/2 of (43) are negative real numbers, so the system (42) is stable and h~ is exponentially convergent. Therefore, as long as γ→γd, h can track href stably. In this way, the control objective of the altitude subsystem turns to ensure that γ tracks γd.Definex1=γ, x2=θ, and x3=Q. Based on the research conclusion of [7], the remaining part of the altitude subsystem ((3), (4), and (5)) is further expressed as the following more general nonaffine form:
(44)ẋ1=f1x1,x2,ẋ2=x3,ẋ3=f3x,δe.Considering (11), the above formula can be rewritten as
(45)ẋ1=f1x1,x2,ẋ2=x3,ẋ3=f3x,δe∗,where x=x1,x2,x3T. f1x1,x2 and f3x,δe∗ are completely unknown continuously differentiable functions.Remark 6.
Due to the strong coupling between the rigid body states and the flexible modes in the AHV model ((1), (2), (3), (4), (5), (6), and (7)), here the dynamic characteristics of strong nonlinearity and strong coupling in the original model are regarded as completely unknown continuous differentiable functions ((16) and (44)) by referring to the method of [25]. The proposed method in this paper will use RBFNN to accurately approximate these continuous differentiable functions and then complete the design of the control law and ensure the stability of the closed-loop control system.In order to design the control lawδe∗, the following theorem is given firstly.Theorem 4.
For anyx,δe∗∈Ωx×R, the following inequalities are established:
(46)∂f1x1,x2∂x2>0,∂f3x,δe∗∂δe∗>0,where Ωx is a controllable area.Proof 3.
According to the results in [17], we can know
(47)∂f1x1,x2∂x2>0,∂f3x,δe∂δe>0.
Consider (11), (44) and (45), there is
(48)∂f3x,δe∗∂δe∗=∂f3x,δe∗∂δe·∂δe∂δe∗=∂f3x,δe/∂δecosh2δe∗/δemax.
Combining (47) and (48), we can see that (46) is true.In order to avoid the cumbersome and complicated design process of the traditional backstepping method in designing the control law, the following equivalent transformation is performed on the system (45).Step 1.
Letz1=x1=γ and z2=ż1=f1x1,x2. From (45), the time derivative of z2 is derived as
(49)ż2=∂f1x1,x2∂x1ẋ1+∂f1x1,x2∂x2ẋ2=∂f1x1,x2∂x1f1x1,x2+∂f1x1,x2∂x2x3≜fh1x.Step 2.
Letz3=ż2=fh1x. From (45), the time derivative of z3 is derived as
(50)ż3=∂fh1x∂x1ẋ1+∂fh1x∂x2ẋ2+∂fh1x∂x3ẋ3=∂fh1x∂x1f1x1,x2+∂fh1x∂x2x3+∂fh1x∂x3f3x,δe∗≜fh2x,δe∗.
After the above transformation, system (45) is as follows:
(51)ż1=z2,ż2=z3,ż3=fh2x,δe∗,where fh2x,δe∗ is a completely unknown continuously differentiable function.Remark 7.
From (46), (49) and (50), there is
(52)∂fh2x,δe∗∂δe∗=∂fh1x∂x3∂f3x,δe∗∂δe∗=∂f1x1,x2∂x2∂f3x,δe∗∂δe∗>0.
Define flight-path tracking errore and error function E
(53)e=γ−γd=z1−γd,E=ddt+λ3∫0tedτ=e¨+3λė+3λ2e+λ3∫0tedτ,where λ∈R+ stands for the design parameter. Since s+λ3 is a Hurwitz polynomial, when E is bounded, e must be bounded.According to (51) and (53), we obtain
(54)ė=ż1−γ̇d=z2−γ̇d,e¨=ż2−γ¨d=z3−γ¨d,e⃛=ż3−γ⃛d=fh2x,δe∗−γ⃛d.Considering thatz2 and z3 are unknown, we can know that z2=γ̇ and z3=γ¨ from the previous model transformation process. A new finite-time convergence differentiator (FD) will be used below to accurately estimate differential signal. By taking γ as the input signal of FD (take n=4), we can get the estimated values of z2 and z3, which are expressed as z^2 and z^3, respectively. Similarly, the estimations of γ̇d, γ¨d, and γ⃛d can be obtained via taking γd as the input signal of FD (take n=4), which are expressed as γ^̇d, γ^¨d, and γ^⃛d.Therefore, the estimations of the first three derivatives ofe can be expressed as
(55)e^̇=z^2−γ^̇d,e^¨=z^3−γ^¨d,e^⃛=fh2x,δe∗−γ^⃛d.From (53) and (55), we obtain the estimation of E
(56)E^=e^¨+3λe^̇+3λ2e+λ3∫0tedτ.Let(57)Fhx,δe∗=fh2x,δe∗−khδe∗,where kh∈R+ is the design parameter.Combine (55), (56), and (57)
(58)E^̇=khδe∗+Fhx,δe∗−γ^⃛d+3λe^¨+3λ2e^̇+λ3e.Design the control lawδe∗ as
(59)δe∗=kh−1δe0∗−δe1∗,where δe0∗=−kh1E^+γ^⃛d−3λe^¨−3λ2e^̇−λ3e, kh1∈R+ represents the design parameter, and δe1∗ is the neural control law to be designed to counteract the influence of the uncertain term Fhx,δe∗.Let(60)Hx,δe0∗,δe1∗≜Fhx,δe∗−δe1∗=Fhx,kh−1δe0∗−δe1∗−δe1∗.Similar to velocity subsystem, to illustrate thatHx,δe0∗,δe1∗ satisfies the implicit function theorem, the following theorem is given.Theorem 5.
If(61)kh>12∂fh2x,δe∗∂δe∗,then there are a controllable area Ωh⊂R3 and a unique δe1∗ for any x,δe1∗∈Ωh×R; δe1∗ satisfies
(62)Hx,δe0∗,δe1∗=0.Proof 4.
From [23], a sufficient condition for the existence of δe1∗ is that the following inequality holds:
(63)∂Fhx,δe∗∂δe1∗<1.
Consider (52), (59) and (61), there are
(64)∂Fhx,δe∗∂δe1∗=∂fh2x,δe∗−khδe∗∂δe1∗=∂fh2x,δe∗−khδe∗∂δe∗∂δe∗∂δe1∗=∂fh2x,δe∗∂δe∗−kh1kh=1kh∂fh2x,δe∗∂δe∗−1<1.
Therefore,δe1∗ exists.Further,(65)∂∂δe1∗Hx,δe0∗,δe1∗=∂∂δe1∗Fhx,δe∗−δe1∗=∂∂δe1∗fh2x,δe∗−khδe∗−1=∂∂δe∗fh2x,δe∗−khδe∗∂δe∗∂δe1∗−1=∂fh2x,δe∗∂δe∗−kh−1kh−1=−1kh∂fh2x,δe∗∂δe∗.Combine (52) and (65)
(66)∂∂δe1∗Hx,δe0∗,δe1∗<0According to Theorem5 and (66), Hx,δe0∗,δe1∗ satisfies the implicit function theorem. Therefore, δe1∗ can be regarded as a function of δe0∗ and x, and further Fhx,δe∗ can be regarded as a function of δe0∗ and x.DefineX2=xT,δe0∗T∈R4 as the input vector of RBFNN and introduce RBFNN to approximate Fhx,δe∗
(67)Fhx,δe∗=Wh∗ThX2+ι,ι≤ιM,where Wh∗T=wh1∗,wh2∗,…,whN2∗T∈RN2 is the weight vector, N2 is the number of nodes, and ι and ιM stand for approximation errors and its upper bounds, respectively. hX2=h1X2,h2X2,…,hN1X2T∈RN2, hiX2, is the activation function which is chosen as the Gaussian function here.Defineωh=Wh∗2 and design δe1∗ as
(68)δe1∗=12E^ω^hhTX2hX2,where ω^h is the estimation of ωh, and its adaptive law is designed as
(69)ω^̇h=μh2E^2hTX2hX2−2ω^h,where μh∈R+ is the design parameter.


### 3.4. Finite-Time Convergence Differentiator (FD)

Theorem 6.
Consider the following FD [26]:
(70)ξ̇1=ξ2,ξ̇2=ξ3,⋮ξ̇n−1=ξn,ξ̇n=Rn−a1arctanξ1−υt−a2arctanξ2R−⋯−anarctanξnRn−1,where R,aii=1,2,…,n∈R+ stand for the design parameters. There are ϕ>0 and ιϕ>n so that
(71)ξi−υi−1t=O1Rιϕ−i+1,i=1,2,…,n,where O1/Rιϕ−i+1 denotes the approximation of 1/Rτϕ−i+1 order between ξi and υi−1t, ϕ=1−ϑ/ϑ, ϑ∈0,minι/ι+n,1/2, n≥2.Proof 5.
The detailed proof process of Theorem6 can be found in [26].Remark 8.
In (70), ξii=1,2,…,n is the state variable of the system, ξ1 is the estimation of υt, and ξii=2,3,…,n is the estimation of the i−1th derivative of υt. At the same time, (71) shows that the estimation error is a high-order infinitesimal of 1/Rτϕ−i+1.Use the above FD to estimatez2, z3, γ̇d, γ¨d, and γ⃛d of (54)
(72)z^̇1=z^2,z^̇2=z^3,z^̇3=z^4,z^̇4=R14−a11arctanz^1−γ−a12arctanz^2R1−a13arctanz^3R12−a14arctanz^4R13,γ^d′=γ^̇d,γ^̇d′=γ^¨d,γ^¨d′=γ^⃛d,γ^⃛d′=R24−a21arctanγ^d−γd−a22arctanγ^̇dR2−a23arctanγ^¨dR12−a24arctanγ^⃛dR23,where Ri,aiji=1,2;j=1,2,3,4∈R+ are design parameters; z^1, z^2, z^3, and z^4 represent the estimations of z1γ, z2γ̇, z3γ¨, and z4γ⃛, respectively; and γ^d, γ^̇d, γ^¨d, and γ^⃛d stand for the estimations of γd, γ̇d, γ¨d, and γ⃛d.


## 3.1. RBFNN Approximation

RBFNN has the characteristics of simple structure, strong learning, and fault tolerance and has the ability of globally approximating nonlinear continuous functions [21]. It can be expressed as a mapping from input to output.
(12)y=WThX,where X=X1,X2,…,XnT∈Rn is the input vector (n represents the dimension of the input vector), W=w1,w2,…,wNT∈RN stands for the weight vector (N is the number of hidden layer nodes), hX=h1X,h2X,…,hNXT∈RN, hiX, denotes the activation function. Here, hiX is chosen as the following Gaussian function:
(13)hiX=exp−X−ci22bi2,i=1,2,…,N,where b=b1,b2,…,bNT∈RN, bi, is the width of the ith Gaussian function; c=c1,c2,…,cN∈Rn×N (ci represents the center of the ith Gaussian function), which can be expressed as follows:
(14)c=c11⋯c1N⋮⋱⋮cn1⋯cnN.For any unknown nonlinear continuous functionFX, using the RBFNN and by selecting enough nodes (selecting a sufficiently large N), there must be an ideal weight vector W∗=w1∗,w2∗,…,wN∗T∈RN that satisfies [21]
(15)FX=W∗ThX+μ,μ≤μM,where μ∈R is the approximation error and μM∈R+ stands for the upper bound of approximation error. When taking N large enough, μM can be arbitrarily small.Remark 3.
Sinceexp⋅ is a strictly monotonically increasing and positive function and −X−ci2/2bi2≤0, there is 0<hiX≤hi0=1. Therefore, there must be a bounded constant h¯∈R+ such that hX≤h¯.


## 3.2. Controller Design for Velocity Subsystem

Based on the research conclusion of [7], here velocity subsystem is further expressed as a more general nonaffine form of control input
(16)V̇=FVV,Φ.Combined with (11), the above formula can be rewritten as
(17)V̇=FVV,Φ∗,where FVV,Φ∗ is a completely unknown continuously differentiable function.In order to design the control lawΦ∗, the following theorem is given firstly.Theorem 1.
For anyV,Φ∗∈ΩV×R, the following inequality is established:
(18)∂FVV,Φ∗∂Φ∗>0,where ΩV is a controllable area.Proof 1.
From (11), (16), and (17), we can know
(19)∂FVV,Φ∗∂Φ∗=∂FVV,Φ∗∂Φ·∂Φ∂Φ∗=∂FVV,Φ/∂Φcosh22Φ∗/Φmax−Φmin−1.
According to [17], there is
(20)∂FVV,Φ∂Φ>0.
Therefore,(21)∂FVV,Φ∗∂Φ∗>0.Define velocity tracking error(22)V~=V−Vref.Taking time derivative along (22) and using (17) yield
(23)V~̇=V̇−V̇ref=FVV,Φ∗−V̇ref.Let(24)FV∗V,Φ∗=FVV,Φ∗−kVΦ∗.where kV∈R+ is the design parameter.Combine (23) and (24)
(25)V~̇=V̇−V̇ref=FV∗V,Φ∗+kVΦ∗−V̇ref.Design the control lawΦ∗ as
(26)Φ∗=kV−1Φ0∗−Φ1∗,where
(27)Φ0∗=−kV1V~−kV2∫0tV~dτ+V̇ref.kV1 and kV2∈R+ stand for design parameters and Φ1∗ is the neural control law to be designed to counteract the influence of uncertain term FV∗V,Φ∗.In order to facilitate the subsequent design process, the implicit function theorem is introduced here [22].Theorem 2.
Assume that the implicit functionΨ:Rl×Rr→Rl is continuously differentiable at each point ς,σ on the open set Y⊂Rl×Rr. ς0,σ0 is the point in Y; Ψς0,σ0=0 and the Jacobian matrix ∂Ψ/∂ςς0,σ0 is nonsingular. Then, for any σ∈G, a neighborhood U⊂Rl of ς0 and a neighborhood G⊂Rr of σ0 can make the equation Ψς,σ=0 which has a unique solution ς∈U, and the solution can be expressed as ς=g0σ, where g0⋅ is a continuously differentiable function on σ=σ0.Remark 4.
Theorem2 shows that once the implicit function Ψς,σ satisfies all conditions in the theorem, ς can be expressed as a continuously differentiable function of σ, that is, ς=g0σ. At this time, using RBFNN to approximate Ψς,σ, only σ, instead of ς, is used as the input signal of the neural network to obtain a satisfactory approximation effect. This is where the special meaning of the implicit function theorem lies.Let(28)ΓV,Φ0∗,Φ1∗≜FV∗V,Φ∗−Φ1∗=FV∗V,kV−1Φ0∗−Φ1∗−Φ1∗.To illustrate thatΓV,Φ0∗,Φ1∗ satisfies the implicit function theorem, the following theorem is given.Theorem 3.
Define(29)kV>12∂FVV,Φ∗∂Φ∗.
Then there are a controllable areaΩV⊂R and a unique Φ1∗ for any V,Φ1∗∈ΩV×R; Φ1∗ satisfies
(30)ΓV,Φ0∗,Φ1∗=0.Proof 2.
According to [23], a sufficient condition for the existence of Φ1∗ is that the following inequality holds:
(31)∂FV∗V,Φ∗∂Φ1∗<1.
Consider (18), (24), (26), and (29), there are
(32)∂FV∗V,Φ∗∂Φ1∗=∂FVV,Φ∗−kVΦ∗∂Φ1∗=∂FVV,Φ∗−kVΦ∗∂Φ∗∂Φ∗∂Φ1∗=∂FVV,Φ∗∂Φ∗−kV1kV=1kV∂FVV,Φ∗∂Φ∗−1<1.
Therefore,Φ1∗ exists.Further,(33)∂ΓV,Φ0∗,Φ1∗∂Φ1∗=∂FV∗V,Φ∗−Φ1∗∂Φ1∗=∂FVV,Φ∗−kVΦ∗∂Φ1∗−1=∂FVV,Φ∗−kVΦ∗∂Φ∗∂Φ∗∂Φ1∗−1=∂FVV,Φ∗∂Φ∗−kV−1kV−1=−1kV∂FVV,Φ∗∂Φ∗,where Φ∗=kV−1Φ0∗−Φ1∗.Combine (18) and (33)
(34)∂ΓV,Φ0∗,Φ1∗∂Φ1∗<0.According to Theorem3 and (34), ΓV,Φ0∗,Φ1∗ satisfies the implicit function theorem. Therefore, Φ1∗ can be regarded as a function of Φ0∗ and V, and further FV∗V,Φ∗ can be regarded as a function of Φ0∗ and V.DefineX1=V,Φ0∗T∈R2 as the input vector of RBFNN and introduce RBFNN to approximate FV∗V,Φ∗
(35)FV∗V,Φ∗=WV∗ThX1+ε,ε≤εM,where WV∗T=wV1∗,wV2∗,…,wVN1∗T∈RN1 is the weight vector, N1 is the number of nodes, and ε and εM are approximation errors and its upper bounds, respectively. And hX1=h1X1,h2X1,…,hN1X1T∈RN1, where hiX1 is the activation function; here, hiX1 is chosen as the Gaussian function.The following is based on the MLP technique to adaptively adjust the norm of the RBFNN weight vector.DefineωV=WV∗2 and design Φ1∗ as
(36)Φ1∗=12V~ω^VhTX1hX1,where ω^V is the estimation of ωV, and its adaptive law is designed as
(37)ω^̇V=μV2V~2hTX1hX1−2ω^V,where μV∈R+ is the design parameter.Substitute (27) and (36) into (26) and finally get the control law Φ∗
(38)Φ∗=kV−1−kV1V~−kV2∫0tV~dτ+V̇ref−12V~ω^VhTX1hX1.Remark 5.
Different from [24] in which the weight vector of the neural network is directly adjusted online, this paper regards WV∗ as a whole based on the MLP technique. Adaptive adjustment of ωV requires only one online learning parameter ω^V, and the computational complexity of the approximation algorithm is significantly reduced.


## 3.3. Controller Design for Altitude Subsystem

Define altitude tracking error(39)h~=h−href.Taking time derivative along (39) and using (2) yield
(40)h~̇=Vsinγ−ḣref.The reference trajectory ofγ is chosen as
(41)γd=arcsin−kγ1h~−kγ2∫0th~dτ+ḣrefV,where kγ1, and kγ2∈R+ are design parameters. If γ→γd, known by (40) and (41), the dynamic response of h~ is
(42)h~¨+kγ1h~̇+kγ2h~=0.Performing Laplace transformation on both sides of (42) can get its characteristic equation
(43)s2+kγ1s+kγ2=0.The two characteristic roots−kγ1−kγ12−4kγ2/2 and −kγ1+kγ12−4kγ2/2 of (43) are negative real numbers, so the system (42) is stable and h~ is exponentially convergent. Therefore, as long as γ→γd, h can track href stably. In this way, the control objective of the altitude subsystem turns to ensure that γ tracks γd.Definex1=γ, x2=θ, and x3=Q. Based on the research conclusion of [7], the remaining part of the altitude subsystem ((3), (4), and (5)) is further expressed as the following more general nonaffine form:
(44)ẋ1=f1x1,x2,ẋ2=x3,ẋ3=f3x,δe.Considering (11), the above formula can be rewritten as
(45)ẋ1=f1x1,x2,ẋ2=x3,ẋ3=f3x,δe∗,where x=x1,x2,x3T. f1x1,x2 and f3x,δe∗ are completely unknown continuously differentiable functions.Remark 6.
Due to the strong coupling between the rigid body states and the flexible modes in the AHV model ((1), (2), (3), (4), (5), (6), and (7)), here the dynamic characteristics of strong nonlinearity and strong coupling in the original model are regarded as completely unknown continuous differentiable functions ((16) and (44)) by referring to the method of [25]. The proposed method in this paper will use RBFNN to accurately approximate these continuous differentiable functions and then complete the design of the control law and ensure the stability of the closed-loop control system.In order to design the control lawδe∗, the following theorem is given firstly.Theorem 4.
For anyx,δe∗∈Ωx×R, the following inequalities are established:
(46)∂f1x1,x2∂x2>0,∂f3x,δe∗∂δe∗>0,where Ωx is a controllable area.Proof 3.
According to the results in [17], we can know
(47)∂f1x1,x2∂x2>0,∂f3x,δe∂δe>0.
Consider (11), (44) and (45), there is
(48)∂f3x,δe∗∂δe∗=∂f3x,δe∗∂δe·∂δe∂δe∗=∂f3x,δe/∂δecosh2δe∗/δemax.
Combining (47) and (48), we can see that (46) is true.In order to avoid the cumbersome and complicated design process of the traditional backstepping method in designing the control law, the following equivalent transformation is performed on the system (45).Step 1.
Letz1=x1=γ and z2=ż1=f1x1,x2. From (45), the time derivative of z2 is derived as
(49)ż2=∂f1x1,x2∂x1ẋ1+∂f1x1,x2∂x2ẋ2=∂f1x1,x2∂x1f1x1,x2+∂f1x1,x2∂x2x3≜fh1x.Step 2.
Letz3=ż2=fh1x. From (45), the time derivative of z3 is derived as
(50)ż3=∂fh1x∂x1ẋ1+∂fh1x∂x2ẋ2+∂fh1x∂x3ẋ3=∂fh1x∂x1f1x1,x2+∂fh1x∂x2x3+∂fh1x∂x3f3x,δe∗≜fh2x,δe∗.
After the above transformation, system (45) is as follows:
(51)ż1=z2,ż2=z3,ż3=fh2x,δe∗,where fh2x,δe∗ is a completely unknown continuously differentiable function.Remark 7.
From (46), (49) and (50), there is
(52)∂fh2x,δe∗∂δe∗=∂fh1x∂x3∂f3x,δe∗∂δe∗=∂f1x1,x2∂x2∂f3x,δe∗∂δe∗>0.
Define flight-path tracking errore and error function E
(53)e=γ−γd=z1−γd,E=ddt+λ3∫0tedτ=e¨+3λė+3λ2e+λ3∫0tedτ,where λ∈R+ stands for the design parameter. Since s+λ3 is a Hurwitz polynomial, when E is bounded, e must be bounded.According to (51) and (53), we obtain
(54)ė=ż1−γ̇d=z2−γ̇d,e¨=ż2−γ¨d=z3−γ¨d,e⃛=ż3−γ⃛d=fh2x,δe∗−γ⃛d.Considering thatz2 and z3 are unknown, we can know that z2=γ̇ and z3=γ¨ from the previous model transformation process. A new finite-time convergence differentiator (FD) will be used below to accurately estimate differential signal. By taking γ as the input signal of FD (take n=4), we can get the estimated values of z2 and z3, which are expressed as z^2 and z^3, respectively. Similarly, the estimations of γ̇d, γ¨d, and γ⃛d can be obtained via taking γd as the input signal of FD (take n=4), which are expressed as γ^̇d, γ^¨d, and γ^⃛d.Therefore, the estimations of the first three derivatives ofe can be expressed as
(55)e^̇=z^2−γ^̇d,e^¨=z^3−γ^¨d,e^⃛=fh2x,δe∗−γ^⃛d.From (53) and (55), we obtain the estimation of E
(56)E^=e^¨+3λe^̇+3λ2e+λ3∫0tedτ.Let(57)Fhx,δe∗=fh2x,δe∗−khδe∗,where kh∈R+ is the design parameter.Combine (55), (56), and (57)
(58)E^̇=khδe∗+Fhx,δe∗−γ^⃛d+3λe^¨+3λ2e^̇+λ3e.Design the control lawδe∗ as
(59)δe∗=kh−1δe0∗−δe1∗,where δe0∗=−kh1E^+γ^⃛d−3λe^¨−3λ2e^̇−λ3e, kh1∈R+ represents the design parameter, and δe1∗ is the neural control law to be designed to counteract the influence of the uncertain term Fhx,δe∗.Let(60)Hx,δe0∗,δe1∗≜Fhx,δe∗−δe1∗=Fhx,kh−1δe0∗−δe1∗−δe1∗.Similar to velocity subsystem, to illustrate thatHx,δe0∗,δe1∗ satisfies the implicit function theorem, the following theorem is given.Theorem 5.
If(61)kh>12∂fh2x,δe∗∂δe∗,then there are a controllable area Ωh⊂R3 and a unique δe1∗ for any x,δe1∗∈Ωh×R; δe1∗ satisfies
(62)Hx,δe0∗,δe1∗=0.Proof 4.
From [23], a sufficient condition for the existence of δe1∗ is that the following inequality holds:
(63)∂Fhx,δe∗∂δe1∗<1.
Consider (52), (59) and (61), there are
(64)∂Fhx,δe∗∂δe1∗=∂fh2x,δe∗−khδe∗∂δe1∗=∂fh2x,δe∗−khδe∗∂δe∗∂δe∗∂δe1∗=∂fh2x,δe∗∂δe∗−kh1kh=1kh∂fh2x,δe∗∂δe∗−1<1.
Therefore,δe1∗ exists.Further,(65)∂∂δe1∗Hx,δe0∗,δe1∗=∂∂δe1∗Fhx,δe∗−δe1∗=∂∂δe1∗fh2x,δe∗−khδe∗−1=∂∂δe∗fh2x,δe∗−khδe∗∂δe∗∂δe1∗−1=∂fh2x,δe∗∂δe∗−kh−1kh−1=−1kh∂fh2x,δe∗∂δe∗.Combine (52) and (65)
(66)∂∂δe1∗Hx,δe0∗,δe1∗<0According to Theorem5 and (66), Hx,δe0∗,δe1∗ satisfies the implicit function theorem. Therefore, δe1∗ can be regarded as a function of δe0∗ and x, and further Fhx,δe∗ can be regarded as a function of δe0∗ and x.DefineX2=xT,δe0∗T∈R4 as the input vector of RBFNN and introduce RBFNN to approximate Fhx,δe∗
(67)Fhx,δe∗=Wh∗ThX2+ι,ι≤ιM,where Wh∗T=wh1∗,wh2∗,…,whN2∗T∈RN2 is the weight vector, N2 is the number of nodes, and ι and ιM stand for approximation errors and its upper bounds, respectively. hX2=h1X2,h2X2,…,hN1X2T∈RN2, hiX2, is the activation function which is chosen as the Gaussian function here.Defineωh=Wh∗2 and design δe1∗ as
(68)δe1∗=12E^ω^hhTX2hX2,where ω^h is the estimation of ωh, and its adaptive law is designed as
(69)ω^̇h=μh2E^2hTX2hX2−2ω^h,where μh∈R+ is the design parameter.


## 3.4. Finite-Time Convergence Differentiator (FD)

Theorem 6.
Consider the following FD [26]:
(70)ξ̇1=ξ2,ξ̇2=ξ3,⋮ξ̇n−1=ξn,ξ̇n=Rn−a1arctanξ1−υt−a2arctanξ2R−⋯−anarctanξnRn−1,where R,aii=1,2,…,n∈R+ stand for the design parameters. There are ϕ>0 and ιϕ>n so that
(71)ξi−υi−1t=O1Rιϕ−i+1,i=1,2,…,n,where O1/Rιϕ−i+1 denotes the approximation of 1/Rτϕ−i+1 order between ξi and υi−1t, ϕ=1−ϑ/ϑ, ϑ∈0,minι/ι+n,1/2, n≥2.Proof 5.
The detailed proof process of Theorem6 can be found in [26].Remark 8.
In (70), ξii=1,2,…,n is the state variable of the system, ξ1 is the estimation of υt, and ξii=2,3,…,n is the estimation of the i−1th derivative of υt. At the same time, (71) shows that the estimation error is a high-order infinitesimal of 1/Rτϕ−i+1.Use the above FD to estimatez2, z3, γ̇d, γ¨d, and γ⃛d of (54)
(72)z^̇1=z^2,z^̇2=z^3,z^̇3=z^4,z^̇4=R14−a11arctanz^1−γ−a12arctanz^2R1−a13arctanz^3R12−a14arctanz^4R13,γ^d′=γ^̇d,γ^̇d′=γ^¨d,γ^¨d′=γ^⃛d,γ^⃛d′=R24−a21arctanγ^d−γd−a22arctanγ^̇dR2−a23arctanγ^¨dR12−a24arctanγ^⃛dR23,where Ri,aiji=1,2;j=1,2,3,4∈R+ are design parameters; z^1, z^2, z^3, and z^4 represent the estimations of z1γ, z2γ̇, z3γ¨, and z4γ⃛, respectively; and γ^d, γ^̇d, γ^¨d, and γ^⃛d stand for the estimations of γd, γ̇d, γ¨d, and γ⃛d.


## 4. Stability Analysis

Theorem 7.
For the velocity subsystem of AHV (17), considering the saturation characteristic of Φ (11), and adopting the control law (26) and the adaptive law (37) under the premise of Theorem 1, the closed-loop control system is semiglobally uniformly ultimately bounded.Proof 6.
Define the estimation error ofωV
(73)ω~V=ω^V−ωV.
Substitute (26), (27), (35), and (36) into (25)
(74)V~̇=Φ0∗−Φ1∗+FV∗V,Φ∗−V̇ref=−kV1V~−kV2∫0tV~dτ−12V~ω^VhTX1hX1+WV∗ThX1+ε.
Choose the following Lyapunov function candidate:(75)LV=V~22+kV22∫0tV~dτ2+ω~V22μV.
Taking time derivative of (75) and invoking (37), (73) and (74) yield
(76)L̇V=V~V~̇+kV2V~∫0tV~dτ+ω~Vω^̇VμV=V~−kV1V~−kV2∫0tV~dτ−12V~ω^VhTX1hX1+WV∗ThX1+ε+kV2V~∫0tV~dτ+ω~VμVμV2V~2hTX1hX1−2ω^V=−kV1V~2−12V~2ωVhTX1hX1+V~WV∗ThX1+V~ε−2ω~Vω^VμV.
Notice that(77)V~WV∗ThX1≤V~22WV∗ThX12+12,V~ε≤V~24+εM2,2ω~Vω^V≥ω~V2−ωV2.
Also from Cauchy-Schwarz inequality(78)WV∗ThX1≤WV∗hX1.
Further,(79)V~WV∗ThX1≤V~22WV∗2hX12+12=V~22ωVhTX1hX1+12.
Then (76) becomes
(80)L̇V≤−kV1−14V~2−ω~V2μV+12+εM2+ωV2μV.
LetkV1>1/4, and define the following compact sets:
(81)ΩV~=V~∣V~≤1/2+εM2+ωV2/μVkV1−1/4,Ωω~V=ω~V∣ω~V≤1/2+εM2+ωV2/μV1/μV.
IfV~∉ΩV~ or ω~V∉Ωω~V, then L̇V<0. Therefore, the closed-loop control system is semiglobally uniformly ultimately bounded. Further, these error signals V~ and ω~V are semiglobally uniformly ultimately bounded and can be invariant to the following sets ΩV~ and Ωω~V. The radiuses of ΩV~ and Ωω~V can be made arbitrarily small by choosing kV1 that is big enough and μV that is small enough, and the tracking errors V~ and ω~V can also be arbitrarily small.Theorem 8.
For the altitude subsystem of AHV (51), considering the saturation characteristic of δe (11) and adopting the control law (59), the adaptive law (69) and FD (72) under the premise of Theorem 4, the closed-loop control system is semiglobally uniformly ultimately bounded.Proof 7.
Define the estimation error ofωh
(82)ω~h=ω^h−ωh.
Then, define the estimation error of FD(83)χ1=z^2−z2,χ2=z^3−z3,χ3=γ^̇d−γ̇d,χ4=γ^¨d−γ¨d,χ5=γ^⃛d−γ⃛d.
According to Theorem6, there is χiM∈R+i=1,2,…,5
(84)χi≤χiM.
Substitute (59), (67), and (68) into (58)
(85)E^̇=−kh1E^−12E^ω^hhTX2hX2+Wh∗ThX2+ι.
Choose the following Lyapunov function candidate:(86)Lh=12E^2+ω~h22μh.
Taking time derivative of (86) and invoke (69), (82), and (85) yield
(87)L̇h=E^E^̇+ω~hω^̇hμh=E^−kh1E^−12E^ω^hhTX2hX2+Wh∗ThX2+ι+ω~hμhμh2E^2hTX2hX2−2ω^h=−kh1E^2−12E^2ωhhTX2hX2+E^Wh∗ThX2+E^ι−2ω~hω^hμh.
Since(88)E^Wh∗ThX2≤E^22Wh∗ThX22+12,2ω~hω^hμh≥ω~h2μh−ωh2μh,E^ι≤E^ι≤E^24+ιM2.
According to Cauchy-Schwarz inequality,(89)Wh∗ThX2≤Wh∗hX2.
Further,(90)E^Wh∗ThX2≤E^22Wh∗2hX22+12=E^22ωhhTX2hX2+12.
Then (87) becomes
(91)L̇h≤−kh1−14E^2−ω~h2μh+12+ιM2+ωh2μh.
Letkh1>1/4, and define the following compact sets:
(92)ΩE^=E^∣E^≤1/2+ιM2+ωh2/μhkh1−1/4,Ωω~h=ω~h∣ω~h≤1/2+ιM2+ωh2/μh1/μh.
IfE^∉ΩE^ or ω~h∉Ωω~h, then L̇h<0. Thus, these error signals E^ and ω~h are semiglobally uniformly ultimately bounded and can be invariant to the following sets ΩE^ and Ωω~h. The radiuses of ΩE^ and Ωω~h can be made arbitrarily small by choosing big enough kh1 and small enough μh, and the tracking errors E^ and ω~h can also be arbitrarily small.
Combine (53), (54), (55) and (56) and (83)
(93)E=E^+z3−z^3+γ^¨d−γ¨d+3λz2−z^2+γ^̇d−γ̇d=E^+χ4−χ2+3λχ3−χ1.
Consider (84), then (93) can become
(94)E≤E^+χ4M+χ2M+3λχ3M+χ1M.
Therefore,E and e are also bounded, then the closed-loop control system is semiglobally uniformly ultimately bounded.


## 5. Simulation Results

With AHV longitudinal motion model as the controlled object, the tracking simulation of velocity and altitude reference commands is performed. The initial velocity and altitude of AHV are taken asV=2500 m/s and h=27000 m. Velocity step and altitude step are chosen as ΔV=200 m/s and Δh=200 m. Both the velocity and altitude reference inputs are given by a second-order reference model with a damping ratio of 0.9 and a natural frequency of 0.1 rad/s. The design parameters of the controller are chosen as kV=5, kV1=8, kV2=0.01, kγ1=2, kγ2=0.01, kh=0.9, kh1=30, and λ=7. The design parameters of the adaptive law are taken as μV=μh=0.05. The design parameters of FD are chosen as R1=R2=0.05, a11=a13=a21=a23=0.5, and a12=a14=a22=a24=0.1. The node number of RBFNN is chosen as N1=N2=20. In the velocity subsystem, the center c1 of the Gaussian function is evenly spaced in [2500 m/s, 3100 m/s] × [−0.1, 1], and the width b1 is selected as 6.56. In altitude subsystem, the center c2 of the Gaussian function is evenly spaced in [−1°, 1°] × [0°, 5°] × [−5°/s, 5°/s] × [0 rad, 0.35 rad]; the width b2 is selected as 0.01. Three cases are considered in the simulation study.Remark 9.
In the above stability analysis of the control law, in order to ensure the stability of the closed-loop control system, the range of values of related parameters (such askV1>1/4, kh1>1/4) is given. At the same time, in order to ensure tracking accuracy, the selection principle of related parameters is also analyzed (e.g., parameters kV1 and kh1 should be taken large enough, and parameters μV and μh should be taken small enough). The selection of parameters in the simulation is based on the above principles and is determined by repeated debugging.Case 1.
Considering Remark1, we assume that actuators are constrained as Φ∈0.05,1.5 and δe∈−20∘,20∘. Simultaneously, to verify the robustness of the controller, it is assumed that there is a perturbation of ±40% for the aerodynamic coefficient of the AHV model, which is expressed as C=C01+0.4sin0.1πt, where C0 represents the nominal value and C stands for the simulation value. And after 50 s, add external disturbances d1=2sin0.1πt and d2=0.02sin0.1πt to (1) and (5) in the AHV model, respectively. The subscript “1” in the figures shows the result obtained by the proposed method, and the subscript “2” shows the result obtained by the method of [27].Case 2.
Combining Remark2, we further suppose that actuators are subject to stricter constraints, resulting in a smaller executable range than the theoretical value, which are taken as Φ∈0.05,1.2 and δe∈−18∘,18∘. Other simulation conditions are the same as Case 1.Case 3.
The control problem of nonlinear systems with actuator constraints has been considered extensively in the existing results, such as [28, 29]. In order to compare with the method in this paper, the adaptive dynamic surface control method considering actuator constraints proposed in [29] is applied to the AHV model. All conditions in the simulation are the same as in Case 2. The subscript “1” represents the result of the proposed method. The subscript “2” represents the result of the method proposed in [29].When the control inputs are constrained as described in Case1 (see Figures 2(h) and 2(i)), both the proposed method and method of [26] can guarantee that V and h track their reference commands Vref and href (see Figures 2(a) and 2(d)). However, the tracking accuracy and anti-interference ability of the proposed method in this paper are significantly improved than those of the method in [26] (see Figures 2(b) and 2(e)). At the same time, the changes in the flight-path angle and flexible states are also smoother than [26] (see Figures 2(c), 2(f), and 2(g)). When actuators of AHV are subject to stricter constraints (see Figures 3(h) and 3(i)), the stability of the closed-loop system cannot be guaranteed with the method of [26]. But through using the proposed method in this paper, the control objective can still be achieved (see Figures 3(a), 3(b), 3(d), and 3(e)). And the proposed method can effectively inhibit flexible states (see Figures 3(c) and 3(f)). In Case 3, the method proposed in [29] can guarantee that V and h track their reference commands when the actuators are constrained (see Figures 4(a) and 4(d)). However, since the method does not use the relevant means to estimate the uncertain dynamics of the model, the tracking error is significantly larger than the method proposed in this paper (see Figures 4(b) and 4(e)). At the same time, compared with the method in [29], the control curve obtained by the method is smoother (see Figures 4(h) and 4(i)), which is more suitable for engineering applications.Figure 2
Simulation result of Case1.
(a)
Velocity tracking
(b)
Velocity tracking error
(c)
The responses of flexible statesη1
(d)
Altitude tracking
(e)
Altitude tracking error
(f)
The responses of flexible statesη2
(g)
Flight-path angle
(h)
Fuel equivalence ratio
(i)
Elevator angular deflectionFigure 3
Simulation result of Case2.
(a)
Velocity tracking
(b)
Velocity tracking error
(c)
The responses of flexible statesη1
(d)
Altitude tracking
(e)
Altitude tracking error
(f)
The responses of flexible statesη2
(g)
Flight-path angle
(h)
Fuel equivalence ratio
(i)
Elevator angular deflectionFigure 4
Simulation result of Case3.
(a)
Velocity tracking
(b)
Velocity tracking error
(c)
The responses of flexible statesη1
(d)
Altitude tracking
(e)
Altitude tracking error
(f)
The responses of flexible statesη2
(g)
Flight-path angle
(h)
Fuel equivalence ratio
(i)
Elevator angular deflection


## 6. Conclusion

(1)
An adaptive neural control law based on input saturation function is designed in this paper, which guarantees the stability of the closed-loop system and the boundedness of tracking error when the actuator reaches saturation.
(2)
When the AHV model encounters perturbation of aerodynamic coefficient and external disturbances, the proposed method can still ensure good tracking accuracy, which proves that the algorithm has strong robustness.
(3)
The controller designed in this paper can still effectively suppress the flexible states when the actuator is constrained. Simulation examples verify the effectiveness and superiority of the design method.
(4)
In this paper, only the most common amplitude saturation problem of the actuator is considered, and the rate and bandwidth constraint problem is not discussed. These will serve as a further research direction.



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*Source: 1284753-2018-09-09.xml*