# Constrained Adaptive Neural Control for Air-Breathing Hypersonic Vehicles without Backstepping

Authors: Shili Tan; Humin Lei; Pengfei Wang
Journal: Mathematical Problems in Engineering
 (2018)

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## Abstract

An adaptive neural control scheme without backstepping is proposed for the air-breathing hypersonic vehicle subject to input constraints. To estimate the unknown nonlinearity of velocity subsystem and altitude subsystem, two radial basis function neural networks (RBFNNs) are constructed. Since the complex backstepping design steps are not needed, the proposed control structure is quite concise and the problem of “explosion of terms” is avoided. Moreover, a novel nonlinear auxiliary system is constructed to solve the problem of input constraints. The advantage of the proposed auxiliary system is that its high-order form has good performance and the parameter tuning is relatively easy. Simulation results show that the designed controllers achieve stable tracking of reference commands with good performance.


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## Body

## 1. Introduction

Air-breathing hypersonic vehicle (AHV) has attracted wide concern, because it offers a promising and economic access to near space [1, 2]. The design of the flight control system is an important issue to ensure that the AHV can accomplish its intended task. Due to the complex flight environment, the highly nonlinear dynamic behavior, and the large uncertainty [3, 4], it is a huge challenge to design the controllers with satisfactory performance.Recently, there are already numerous publications on the design of the flight control system for AHV. A robust linear output-control strategy is investigated for AHV, achieving the stable tracking of reference trajectories [5]. Since the feedback linearization (FL) remains based on the accurate model which is difficult to be obtained, the robustness of FL is weak when there is the model uncertainty. To enhance the robustness with respect to model parameters, a nonfragile control scheme is studied based on the linear parameter-varying (LPV) model [6, 7]. Also the sliding mode control is an effective nonlinear control method and is widely studied for AHV subject to the uncertainty. A key technique for sliding mode control is how to solve the chattering problem [8]. To reduce the chattering, the quasi-continuous high-order mode controllers are exploited, effectively reducing the flutter of the AHV [9]. By introducing the nonlinear disturbance observers (NDO), the total uncertainty is estimated online, and the robustness of the controller is guaranteed [10, 11]. Noticing that the robustness of the designed controllers is closely related to the NDO, an novel NDO based on hyperbolic sine function is investigated and the robustness is enhanced [12]. Moreover, much concern has been on the intelligent controls such as the fuzzy logical system (FLS)[13–15] and the neural network (NN)[16, 17]. The minimal learning parameter (MLP) scheme [13, 14, 16] and the composite learning-based parameter adaptive law [15, 17] are constructed to update the FLS/NN weights. By estimating the model uncertainty accurately and compensating for the controllers, the robustness of the proposed control scheme is ensured.Backstepping control is regarded as an effective tool to solve the problem of unmatched uncertainty [18, 19]. Noting that the traditional backstepping control needs the tedious analytical calculation of derivatives, the problem of the “explosion of terms” is inevitable. Therefore, the dynamic surface control [20, 21], command filtered approaches [22], and tracking differentiator (TD)[12] are proposed to solve the problem of “explosion of terms”. Though good results in terms of command tracking can be gained by the above method, the whole control system becomes cumbersome and it is unfavorable to the design of the controller. To simplify the design process of the controllers, the controllers without the virtual control laws are creatively designed [16]. However, the expression of the final actual controller becomes complex with a large number of state variables and design parameters. Motivated by the above studies [13, 14, 16, 17], the radial basis function neural networks (RBFNNs) with MLP algorithm are constructed to approximate the model uncertainty. Different from the previous studies, the control scheme without backstepping is proposed by introducing the high-order differentiator. Thus there is no need of complex strict-feedback form and virtual controllers. Also only two RBFNNs are needed to estimate the unknown nonlinearity of velocity subsystem and altitude subsystem, respectively.From a practical perspective, the control inputs are limited to a certain range. The designed controller without considering the input constraints may suffer performance degradation or even instability appearance. Therefore, it is necessary to take the input constraints into consideration. The Nussbaum function based auxiliary design system is employed for the strict-feedback systems and the nonstrict-feedback systems, preventing the outputs from violating the constraints [23, 24]. Also a linear auxiliary system is adopted for the backstepping scheme [25] and the disturbance observer-based feedback linearization control [26]. Moreover, an novel hybrid auxiliary system is designed to solve the problem of the nonsymmetric nonlinear input constraints [27], and it has been applied to the AHV systems [28, 29] and the flexible spacecraft systems [30]. However, these antiwindup methods can only be applied in some specific control scheme such as the backstepping control. Regardless of the linear auxiliary system [25, 26] or the hybrid auxiliary system [27–30], it is not suitable to be extended to the high-order forms. Since a high-order auxiliary system is required for the altitude subsystem under the proposed control scheme, a novel nonlinear auxiliary system is constructed.The remainder of the paper is organized as follows: the inputs constrained AHV model and the description of RBFNN are presented in Section2. Then, the main design process of the controllers is provided in Section 3. Next, Sections 4 and 5 show the stability analysis and simulation, respectively. Finally, the conclusions are summarized in Section 6. The main contributions of this paper are summarized as follows.( 1 ) Different from the adaptive neural control schemes in [16, 17, 28], there is no need of complex backstepping design steps in this paper. Thus the proposed control structure is quite concise and the problem of “explosion of terms” is avoided.( 2 ) Compared with the linear auxiliary system [25, 26] and the hybrid auxiliary system [27–30], the nonlinear auxiliary system proposed in this paper can be extended to the high-order forms and applied to the altitude subsystem under the proposed control scheme. Moreover, the parameter tuning for the proposed auxiliary system is relatively easy.


## 2. Inputs Constrained AHV Model and RBFNN Description

### 2.1. AHV Model

Based on the X43A aircraft developed by NASA, M. Bolender et al. presented an AHV model [31]. This mode has been widely used in the design of flight control systems. Then, Parker simplified the model by ignoring the weak couplings and slow dynamics [32]. The dynamic equations are formulated as (1)V˙=1mTcos⁡θ-γ-D-gsin⁡γ,(2)h˙=Vsin⁡γ,(3)γ˙=1mVL+Tsin⁡θ-γ-gVcos⁡γ,(4)θ˙=Q,(5)IyyQ˙=M+ψ~1η¨1+ψ~2η¨2,(6)k1η¨1=-2ζ1ω1η˙1-ω12η1+N1-ψ~1MIyy-ψ~1ψ~2η¨2Iyy,(7)k2η¨2=-2ζ2ω2η˙2-ω22η2+N2-ψ~2MIyy-ψ~2ψ~1η¨1Iyy,where V, h, γ, θ, and Q denote the velocity, altitude, flight path angle, pitch angle, and pitch rate, respectively. The flexible states η=[η1,η˙1,η2,η˙2] are the first two vibrational modes. m is the mass, Iyy the moment of inertia, ξi the damping ratio, and ωi the natural frequency. ki=1+ψ~i/Iyy, with ψ~i the constrained beam coupling constant for ηi. The thrust T, lift L, drag D, pitching moment M, and generalized forces Ni(i=1,2) are given as follows [17]:(8)L≈q-SCLαα+CTδeδe+CL0,D≈q-SCDα2α2+CDαα+CDδe2δe2+CDδeδe+CD0,T≈CTα3α3+CTα2α2+CTαα+CT0,M≈zTT+q-Sc-CM,αα2α2+CM,ααα+CM,α0+ceδe,Ni=Niα2α2+Niαα+Niδeδe+Ni0,i=1,2,with(9)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,(10)ρ-=ρ-0exp-h-h0hs,where ϕ and δe are the fuel equivalence ratio and the elevator deflection, respectively; S and c- denote the reference area and the aerodynamic chord, respectively; q- is the dynamic pressure, ρ- the air density, and zT the thrust to moment coupling coefficient.


### 2.2. Input Constraints

Two inputs of the AHV areu=[ϕ,δe]T. In engineering practice, the inputs cannot be any desired values, but must be limited within a certain range. ϕ must be limited to a certain range to ensure the normal operation of the scramjet. Similarly, the limitation of δe is determined by the physical structure of the elevator. The above input constraints can be described as(11)ϕ=ϕmax,ϕc≥ϕmaxϕc,ϕmin≤ϕc≤ϕmaxϕmin,ϕc≤ϕmin(12)δe=δemax,δec≥δemaxδec,δemin≤δec≤δemaxδemin,δec≤δeminwhere ϕc and δec are the inputs to be designed; the “max” and “min” denote the maximum and minimum value in the constrained range, respectively.


### 2.3. Description of RBFNN

The RBFNN has been widely used because it can approximate any continuous function to an arbitrary precision [16, 33]. Assume that the number of RBFNN nodes is l≥1. The mapping from the input layer X∈Ω⊂Rn to the output layer y∈R is (13)y=WTφX,where W=[w1,w2,…,wl]T∈Rl denotes the weight vector; φ(X)=[φ1(X),φ2(X),…,φl(X)]T∈Rl, φj(X) is chosen as(14)φjX=exp-X-μjTX-μjσj2,j=1,2,…,lwhere μj=[μj1,μj2,…,μjn]T denotes the center vector and σj is the width vector.Assume thatF(X):Rn→R is the function to be approximated. There exists W∗ such that(15)FX=W∗TφX+ε,ε≤ε-,∀X∈ΩX,where W∗denotes the ideal weight vector; ε is the approximation error.


## 2.1. AHV Model

Based on the X43A aircraft developed by NASA, M. Bolender et al. presented an AHV model [31]. This mode has been widely used in the design of flight control systems. Then, Parker simplified the model by ignoring the weak couplings and slow dynamics [32]. The dynamic equations are formulated as (1)V˙=1mTcos⁡θ-γ-D-gsin⁡γ,(2)h˙=Vsin⁡γ,(3)γ˙=1mVL+Tsin⁡θ-γ-gVcos⁡γ,(4)θ˙=Q,(5)IyyQ˙=M+ψ~1η¨1+ψ~2η¨2,(6)k1η¨1=-2ζ1ω1η˙1-ω12η1+N1-ψ~1MIyy-ψ~1ψ~2η¨2Iyy,(7)k2η¨2=-2ζ2ω2η˙2-ω22η2+N2-ψ~2MIyy-ψ~2ψ~1η¨1Iyy,where V, h, γ, θ, and Q denote the velocity, altitude, flight path angle, pitch angle, and pitch rate, respectively. The flexible states η=[η1,η˙1,η2,η˙2] are the first two vibrational modes. m is the mass, Iyy the moment of inertia, ξi the damping ratio, and ωi the natural frequency. ki=1+ψ~i/Iyy, with ψ~i the constrained beam coupling constant for ηi. The thrust T, lift L, drag D, pitching moment M, and generalized forces Ni(i=1,2) are given as follows [17]:(8)L≈q-SCLαα+CTδeδe+CL0,D≈q-SCDα2α2+CDαα+CDδe2δe2+CDδeδe+CD0,T≈CTα3α3+CTα2α2+CTαα+CT0,M≈zTT+q-Sc-CM,αα2α2+CM,ααα+CM,α0+ceδe,Ni=Niα2α2+Niαα+Niδeδe+Ni0,i=1,2,with(9)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,(10)ρ-=ρ-0exp-h-h0hs,where ϕ and δe are the fuel equivalence ratio and the elevator deflection, respectively; S and c- denote the reference area and the aerodynamic chord, respectively; q- is the dynamic pressure, ρ- the air density, and zT the thrust to moment coupling coefficient.


## 2.2. Input Constraints

Two inputs of the AHV areu=[ϕ,δe]T. In engineering practice, the inputs cannot be any desired values, but must be limited within a certain range. ϕ must be limited to a certain range to ensure the normal operation of the scramjet. Similarly, the limitation of δe is determined by the physical structure of the elevator. The above input constraints can be described as(11)ϕ=ϕmax,ϕc≥ϕmaxϕc,ϕmin≤ϕc≤ϕmaxϕmin,ϕc≤ϕmin(12)δe=δemax,δec≥δemaxδec,δemin≤δec≤δemaxδemin,δec≤δeminwhere ϕc and δec are the inputs to be designed; the “max” and “min” denote the maximum and minimum value in the constrained range, respectively.


## 2.3. Description of RBFNN

The RBFNN has been widely used because it can approximate any continuous function to an arbitrary precision [16, 33]. Assume that the number of RBFNN nodes is l≥1. The mapping from the input layer X∈Ω⊂Rn to the output layer y∈R is (13)y=WTφX,where W=[w1,w2,…,wl]T∈Rl denotes the weight vector; φ(X)=[φ1(X),φ2(X),…,φl(X)]T∈Rl, φj(X) is chosen as(14)φjX=exp-X-μjTX-μjσj2,j=1,2,…,lwhere μj=[μj1,μj2,…,μjn]T denotes the center vector and σj is the width vector.Assume thatF(X):Rn→R is the function to be approximated. There exists W∗ such that(15)FX=W∗TφX+ε,ε≤ε-,∀X∈ΩX,where W∗denotes the ideal weight vector; ε is the approximation error.


## 3. Controller Design

The control target is selected as developing a controlleru=[ϕ,δe]T so that the outputs y=[V,h]T can follow the reference commands yd=[Vd,hd]T. Considering that V is steered by ϕ to a large degree and h is mainly affected by δe, we naturally decompose the AHV model into the V-subsystem and the h-subsystem. In what follows, two constrained adaptive neural controllers will be separately designed for the two subsystems such that V→Vd and h→hd.Step 1 (designing control law for theV-subsystem).
The dynamic equation ofV-subsystem is described as(16)V˙=fV+ϕ,where fV=[Tcos⁡(θ-γ)-D]/m-gsin⁡γ-ϕ contains large uncertainty and is approximated by one RBFNN:(17)fV=WV∗TφVXV+εV,εV≤ε-V,where WV∗=[wV1∗,wV2∗,…,wVl1∗]T∈Rl1 is the ideal weight vector; φV(XV)=[φV1(XV),φV2(XV),…,φVl1(XV)]T∈Rl1; φVj(XV), j=1,2,…,l1 has same formulation to (14); εV is the approximation error.
The error betweenV and Vd is expressed by(18)zV=V-Vd.
DifferentiatingzV by time t, we have (19)z˙V=fV+ϕ-V˙d.
Inspired by the MLP method [13, 14, 16], the neural controller ϕ is designed as(20)ϕ=-kV1zV-kV2∫0tzVτdτ-12zVλ^VφVTXVφVXV+V˙d,where kV1>0, kV2>0 are design parameters; λV=WV∗2, with the estimation λ^V. λ^V is decided by the adaptive law as follows:(21)λ^˙V=κV2zV′2φVTXVφVXV-2λ^V,where κV>0 is a design parameter.
Noting thatϕ is constrained as (11), a novel auxiliary system is designed as(22)χ˙V=-lVχVα1·sign⁡χVχV+δV+ϕ-ϕc,where lV>0, δV>0, α1>0 are design parameters; χV is the state variable.
Define the compensated velocity tracking error as(23)zV′=zV-χV
DifferentiatingzV′ by time t and invoking (19) and (22), we have (24)z˙V′=fV-V˙d+lVχVα1·sign⁡χVχV+δV+ϕc.
The desired control inputϕc is determined by the chosen control law: (25)ϕc=-kV1zV′-kV2∫0tzV′τdτ-12zV′λ^VφVTXVφVXV+V˙d-lVχVα1·sign⁡χVχV+δV.
The controller structure of the V-subsystem is presented in Figure1.Figure 1
The controller structure of the V-subsystem.Remark 1.
As shown in Figure1, if the designed control law ϕc directly contains the variable ϕ and fV, or the RBFNN input layer XV contains the element ϕ, the “algebraic loop” problem will appear. Thus, it can be concluded from (25) that the “algebraic loop” problem will not exist as long as ϕ∉XV.Step 2 (designing control law for theh-subsystem).
The error betweenh and hd is defined as (26)zh=h-hd.
Choose the command ofγ as(27)γd=arcsin⁡-kh1zh-kh2∫0tzhτdτ+h˙dV.where kh1>0, kh2>0 are design parameters.
According to [33], zh can converge to zero exponentially, if we choose γ→γd. Hence, a constrained adaptive neural controller is developed to steer γ→γd.
The dynamic equations ofh-subsystem are described as(28)γ˙=fγ+θ,θ˙=Q,Q˙=fQ+δe,where fγ=[L+Tsin⁡(θ-γ)]/(mV)-gcos⁡(γ)/V-θ, fQ=(M+ψ~1η¨1+ψ~2η¨2)/Iyy-δe.
The flight path angle tracking error is given by(29)Zγ=ddt+μ3∫0tzγτdτ,where zγ=γ-γd, μ>0 is a chosen constant. Obviously, the tracking error zγ is bounded only if Zγ is bounded, because the polynomial (s+μ)3 is Hurwitz.
DifferentiatingZγ by time t and invoking (28) yields(30)Z˙γ=fh+δe+3μz¨γ+3μ2z˙γ+μ3zγ-γ⃛d,where fh=f¨γ+fQ is highly uncertain and is approached via the following RBFNN:(31)fh=Wh∗TφhXh+εh,εh≤ε-h,where Wh∗=[wh1∗,wh2∗,…,whl2∗]T∈Rl2 is the ideal weight vector; φh(Xh)=[φh1(Xh),φh2(Xh),…,φhl2(Xh)]T∈Rl2; φhj(Xh), j=1,2,…,l2 has same formulation to (14); εh is the approximation error.
Noting thatδe is constrained as (12), a novel auxiliary system is designed as(32)χ˙h,1=χh,2χ˙h,2=χh,3χ˙h,3=-lh,1χh,1α2·sign⁡χh,1χh,1+δh,1-lh,2χh,2α2·sign⁡χh,2χh,2+δh,2-lh,3χh,3α2·sign⁡χh,3χh,3+δh,3+δe-δecwhere lh,i>0, δh,i>0(i=1,2,3), and α2∈(0,1) are the design parameters. χh,i(i=1,2,3) are the state variables.
The compensated flight path angle tracking error is given by(33)zγ′=zγ-χh,1Zγ′=ddt+μ3∫0tzγ′τdτ
since(34)z˙γ′=γ˙-γ˙d-χh,2z¨γ′=γ¨-γ¨d-χh,3z⃛γ′=fh+δec-γ⃛d+lh,1χh,1α2·sign⁡χh,1χh,1+δh,1+lh,2χh,2α2·sign⁡χh,1χh,2+δh,2+lh,3χh,3α2·sign⁡χh,1χh,3+δh,3
DifferentiatingZγ′ by time t, we have(35)Z˙γ′=fh+δec-γ⃛d+3μz¨γ′+3μ2z˙γ′+μ3zγ′+lh,1χh,1α2·sign⁡χh,1χh,1+δh,1+lh,2χh,2α2·sign⁡χh,2χh,2+δh,2+lh,3χh,3α2·sign⁡χh,3χh,3+δh,3
Considering thatγ˙, γ¨, γ˙d, γ¨d, and γ⃛d in (34) cannot be obtained directly, we get their estimations by introducing the finite-time-convergent differentiator [34]. Assume that the estimations are γ^˙, γ^¨, γ^˙d, γ^¨d, and γ^⃛d. The estimation errors are defined as(36)e1=γ^˙-γ˙,e2=γ^¨-γ¨e3=γ^˙d-γ˙d,e4=γ^¨d-γ¨d,e5=γ^⃛d-γ⃛d
According to [34], there exist positive constants e-i such that(37)ei≤e-i,i=1,2,…,5.
The desired control inputδec is determined by the chosen control law:(38)δec=-kγZγ′-12Zγ′λ^hφhTXhφhXh-3μz¨γ′-3μ2z˙γ′-μ3zγ′+γ^⃛d-lh,1χh,1α2·sign⁡χh,1χh,1+δh,1-lh,2χh,2α2·sign⁡χh,2χh,2+δh,2-lh,3χh,3α2·sign⁡χh,3χh,3+δh,3,where kγ>0 is design parameter; λ^h is the approximation of λh=Wh∗2 with the following adaptive law: (39)λ^˙h=κh2Z^γ′2φhTXhφhXh-2λ^h,where κh>0 is design parameter.Remark 2.
(a) Different from the backstepping design, by applying the finite-time-convergent differentiator to obtain γ˙, γ¨, γ˙d, γ¨d, and γ⃛d in the chosen control law δec, there is no need of complex strict-feedback form and virtual controller.(b) Similarly, the RBFNN input layer Xh must meet the requirement that δe∉Xh.


## 4. Stability Analysis

The stability of the designed auxiliary systems (22) and (32) is illustrated first (i.e., Step 1). Assume that ϕm>0, δem>0 meet ϕ-ϕc≤ϕm, δe-δec≤δem. Choose the Lyapunov function as(40)Y1=12χV2Y2=lh,2∫0χh,1τ1α2·sign⁡τ1τ1+δh,1dτ1+lh,2∫χh,1χh,2τ2α2·sign⁡τ2τ2+δh,2dτ2+χh,322DifferentiatingY1 and Y2 by time t and invoking (22) and (32) yield(41)Y˙1=χVχ˙V=-lVχVα1+1χV+δV+χVϕ-ϕc≤-lVχVα1χV+δV-ϕmχV.(42)Y˙2=lh,2χh,1α2·sign⁡χh,1χh,1+δh,1χ˙h,1+lh,2χh,2α2·sign⁡χh,2χh,2+δh,2χ˙h,2-χh,1α2·sign⁡χh,1χh,1+δh,1χ˙h,1+χh,3χ˙h,3=χh,3δe-δec-lh,1χh,1α2·sign⁡χh,1χh,1+δh,1+lh,3χh,3α2·sign⁡χh,3χh,3+δh,3χh,3≤χh,3δem-lh,1χh,1α2·sign⁡χh,1χh,1+δh,1χh,3-lh,3χh,3α2·sign⁡χh,3χh,3+δh,3χh,3≤-lh,1χh,1α2·sign⁡χh,1χh,1+δh,1-δemχh,3Note that the following inequality holds:(43)χVα1χV+δV≤ηV,α1∈0,1,(44)lh,1χh,1α2·sign⁡χh,1χh,1+δh,1≤ηh,1,α2∈0,1,where ηV>0 and ηh,1>0 are constants.There existlV>ϕm/ηV and lh,1>δem/ηh,1such that Y˙1<0 and Y˙2<0. According to Lyapunov theory, states χV, χh,1, χh,2, and χh,3 are bounded and can converge to zero in finite time by setting bounded χV(0), χh,1(0), χh,2(0), and χh,3(0).Then, we illustrate the stability of the closed-loop system consisting of the plant (1)~(7) and controllers (25) and (38), (i.e., Step 2). For the V-subsystem, the Lyapunov function is selected as(45)YV=zV′22+kV22∫0tzV′τdτ2+λ~V22κV,where λ~V=λ^V-λV.Substituting (17) and (25) into (24), we have(46)z˙V′=-kV1zV′-kV2∫0tzV′τdτ+WV∗TφVXV-12zV′λ^VφVTXVφVXV+εV.Invoking (21) and (46), the time derivative of YV is(47)Y˙V=zV′z˙V′+kV2zV′∫0tzV′τdτ+1κVλ~Vλ^˙V=-kV1zV′2+zV′WV∗TφVXV+zV′εV-2λ~Vλ^VκV-12zV′2λVφVTXVφVXV.Since(48)zV′WV∗TφVXV≤zV′22WV∗TφVXV2+12=zV′22WV∗2φVXV2+12,(49)zV′εV≤12zV′2ε-V2+12,(50)-2λ~Vλ^VκV≤λV2-λ~V2κV≤λV2κV,we obtain(51)Y˙V≤-kV1-12ε-V2zV′2+1κVλV2+1.For theh-subsystem, the Lyapunov function is selected as(52)Yh=12Zγ′2+λ~h22κh.where λ~h=λ^h-λh.Combining (31), (35) and (38), we obtain(53)Z^˙γ′=-kγZγ′+Wh∗TφhXh-12Zγ′λ^hφhTXhφhXh+εh.Invoking (39) and (53), the time derivative of Yh is (54)Y˙h=Zγ′Z˙γ′+λ~hλ^˙h=-kγZγ′2+Zγ′Wh∗TφhXh+Zγ′εh-2λ~hλ^hκh-Zγ′22λhφhTXhφhXh.Since(55)Zγ′Wh∗TφhXh≤Zγ′22Wh∗2φhXh2+12,(56)Zγ′εh≤12Zγ′2ε-h2+12,(57)-2λ~hλ^hκh≤λh2-λ~h2κh≤λh2κh,we obtain(58)Y˙h≤-kγ-12ε-h2Zγ′2+1κhλh2+1.For the closed-loop control system consisting of the plant (1)~(7) and controllers (25) and (38), the Lyapunov function candidate is selected as follows:(59)Y3=YV+YhTaking time derivative along (59), we have(60)Y˙3=Y˙V+Y˙h≤-kV1-12ε-V2zV′2-kγ-12ε-h2Zγ′2+Σwhere Σ=(1/κV)λV2+(1/κh)λh2+2.ChoosingkV1>ε-V2/2 and kγ>ε-h2/2, the compact sets are expressed by(61)ΩzV′=zV′zV′≤ΣkV1-1/2ε-V2.(62)ΩZ^γ′=Zγ′Zγ′≤Σkγ-1/2ε-h2.The radiuses ofΩzV′ and ΩZ^γ′ can be arbitrarily small by choosing enough large kV1, kγ. Besides, Y˙3 is negative when zV′∉ΩzV′, Z^γ′∉ΩZ^γ′. Thus, zV′ and Zγ′ can be arbitrarily small. According to (33), the compensated tracking error zγ′ can be arbitrarily small since μ>0.Assume that the upper bounds ofχV, χh,1 are positive constants χVM, χh,1M, respectively. From (23) and (33), we have(63)zV=zV′+χV≤zV′+χVMzγ=+χh,1≤zγ′+χh,1MThe tracking errorszV and zγ are bounded. Therefore, the proposed control scheme can make V→Vd and γ→γd. Furthermore, we obtain h→hd.


## 5. Simulation Results

The aerodynamic parameters and physical properties of the AHV model are given in Tables2–6. To characterize model uncertainty (including parameter uncertainties and external disturbances), we add the perturbation into the nominal aerodynamic coefficient. Define the actual coefficient as(64)C=C01+ΔCsin⁡0.05πt,where C0 represents the nominal value and ΔC denotes the amplitude of coefficient perturbation.The states initialization of the AHV are given in Table1.Table 1
States initialization.
Item Value Unit V 8200 f t / s h 88600 f t γ 0 d e g θ 1.5295 d e g Q 0 d e g / s η 1 0.9374 - η ˙ 1 0 - η 2 0.7709 - η ˙ 2 0 -Table 2
Miscellaneous coefficient values.
Coefficient Value Units S 1.7000 × 1 0 1 f t 2 ∙ f t - 1 ρ 0 6.7429 × 1 0 - 5 s l u g s ∙ f t - 3 h 0 8.5000 × 1 0 4 f t h s 2.1358 × 1 0 4 f t m 3.0000 × 1 0 2 l b ∙ f t - 1 g 3.2000 × 1 0 1 f t ∙ s - 2 I y y 5.0000 × 1 0 5 l b ∙ f tTable 3
Lift and drag coefficient values.
Coefficient Value Units C L α 4.6773 × 1 0 0 r a d - 1 C T δ e 7.6224 × 1 0 - 1 r a d - 1 C L 0 - 1.8714 × 1 0 - 2 - C D α 2 5.8224 × 1 0 0 r a d - 2 C D α - 4.5315 × 1 0 - 2 r a d - 1 C D δ e 2 8.1993 × 1 0 - 1 r a d - 2 C D δ e 2.7669 × 1 0 - 4 r a d - 1 C D 0 1.0131 × 1 0 - 2 -Table 4
Moment coefficient values.
Coefficient Value Units z T 8.3600 × 1 0 0 f t c - 1.7000 × 1 0 1 f t C M , α α 2 6.2926 × 1 0 0 r a d - 2 C M , α α 2.1335 × 1 0 0 r a d - 1 C M , α 0 1.8979 × 1 0 - 1 - c e - 1.2897 × 1 0 0 r a d - 1Table 5
Thrust coefficient values.
Coefficient Value Units β 1 - 3.7693 × 1 0 5 l b ∙ f t - 1 ∙ r a d - 3 β 2 - 3.7225 × 1 0 4 l b ∙ f t - 1 ∙ r a d - 3 β 3 2.6814 × 1 0 4 l b ∙ f t - 1 ∙ r a d - 2 β 4 - 1.7277 × 1 0 4 l b ∙ f t - 1 ∙ r a d - 2 β 5 3.5542 × 1 0 4 l b ∙ f t - 1 ∙ r a d - 1 β 6 - 2.4216 × 1 0 3 l b ∙ f t - 1 ∙ r a d - 1 β 7 6.3785 × 1 0 3 l b ∙ f t - 1 β 8 - 1.0090 × 1 0 2 l b ∙ f t - 1Table 6
N 1 and N2 coefficient values.
Coefficient Value Units N 1 α 2 1.4013 × 1 0 3 l b ∙ f t - 1 ∙ s l u g - 0.5 ∙ r a d - 2 N 1 α 4.5737 × 1 0 3 l b ∙ f t - 1 ∙ s l u g - 0.5 ∙ r a d - 1 N 1 δ e 1.1752 × 1 0 3 l b ∙ f t - 1 ∙ s l u g - 0.5 ∙ r a d - 1 N 1 0 0.0000 × 1 0 1 l b ∙ f t - 1 ∙ s l u g - 0.5 N 2 α 2 - 5.0227 × 1 0 3 l b ∙ f t - 1 ∙ s l u g - 0.5 ∙ r a d - 2 N 2 α 2.8633 × 1 0 3 l b ∙ f t - 1 ∙ s l u g - 0.5 ∙ r a d - 1 N 2 δ e 1.2465 × 1 0 3 l b ∙ f t - 1 ∙ s l u g - 0.5 ∙ r a d - 1 N 2 0 - 4.4201 × 1 0 1 l b ∙ f t - 1 ∙ s l u g - 0.5The reference commandsVd and hd are obtained through a second-order system with a natural frequency ωn=0.1 and a damping coefficient ς=0.9. The velocity steps are from 8200 ft/s to 8800 ft/s and the altitude steps from 88600 ft to 89200 ft.Choose the design parameters as follows:kV1=0.3, kV2=0.8, kh1=2, kh2=0.1, kγ=50, μ=1, α1=α2=0.9, lV=1, δV=0.1, lh,1=0.5, lh,2=lh,3=1,, δh,i=0.1(i=1,2,3), κV=κh=0.05. The inputs of RBFNNs ((17), (31)) are chosen as XV=[V,V˙d]T, Xh=[γ,θ,Q,γ˙d]T, and the node numbers are selected as l1=l2=20. The center vectors μV, μh are evenly distributed and the width vectors are set as σV=σf1/l1VV-μV2, σh=σf1/l2Vh-μh2 with the regulating parameter σf=0.5.Scenario 1.
To illustrate the effect of the designed auxiliary systems, the comparative simulations are conducted as follows.
Case I. The input constraints are ϕ∈[0.05,1.1], δe∈[-18∘,18∘], and the aerodynamic coefficient perturbation is ΔC=40%. Simulation results are presented in Figures 2–9.
Case II. All simulation conditions remain the same except that the auxiliary systems ( (22) and (32)) do not work. For the designed controllers without the auxiliary systems, simulation results are shown in Figures 10–14.
Figures2 and 3 depict the tracking performance that V→Vd and h→hd. It is shown that the outputs V and h steadily track the reference commands with the satisfactory tracking error. The states γ, θ, Q, and the flexible states, as shown in Figures 4 and 5, change smoothly without the chattering and mutation during the entire simulation process. Figures 6–8 depict the inputs and the states of the auxiliary systems. It is obvious that the auxiliary systems compensate for the inputs when the actuators are saturated. Figure 9 shows that the estimations of λV and λh are bounded and smooth.
As shown in Figures10–12, the velocity and altitude become divergent because the control inputs hit their saturation for a long time. It can be observed from Figure 13 that the angle of attack oscillates near zero when the elevator deflection is constrained. Negative angle of attack will directly lead to engine flameout. Figure 14 reveals that the flexible states become divergent, and the aircraft will disintegrate by violent vibration. In conclusion, the designed controllers without the auxiliary systems fail to achieve the tracking of reference commands.Figure 2
Velocity tracking.Figure 3
Altitude tracking.Figure 4
Responses ofγ, θ, and Q.Figure 5
Flexible states.Figure 6
Control inputs.Figure 7
Response ofχV.Figure 8
Response ofχh,1, χh,2, and χh,3.Figure 9
Estimations ofλV and λh.Figure 10
Velocity tracking.Figure 11
Altitude tracking.Figure 12
Control inputs.Figure 13
Angle of attack.Figure 14
Flexible states.Scenario 2.
To show the robustness and effectiveness of the proposed algorithm, the cases are considered as follows.
Case I. Simulations with and without parameter uncertainties and external disturbances are included and compared. The aerodynamic coefficient perturbations are selected as ΔC=40% and ΔC=0, respectively.
Case II. To show how the proposal works under a noisy environment, the quantization errors in the state variables are described as Verror=0.6×sin(0.1πt), herror=0.2×sin(0.1πt), Qerror=0.1×π/180×sin(0.1πt). In addition, the control input signals are added with the white noise whose mean is 0 and variance is 0.01. Also the aerodynamic coefficient perturbations are selected as ΔC=40%.
Case III. The backstepping control scheme based on novel NDO in [12] is selected for comparison with the proposed controller in this article. All simulation conditions remain the same as Case II.
In order to ensure the fairness of the comparison, the input constraints are not considered. Simulation results inCase I and Case II are shown in Figures 15 and 16 and simulation results in CaseIII are presented in Figures 17–20.
From Figures15-16, it can be seen that the proposed control scheme achieved robust tracking of the velocity and altitude under the environment of parameter uncertainties and additive noise. Figures 17 and 18 reveal that the control method proposed in this article has higher accuracy in the velocity and altitude tracking, compared with the method in [12]. It can be observed from Figures 19 and 20 that the inputs of the method in [12] cannot be adjusted in time, especially in the initial time period, resulting in the flight-path angle, pitch angle, and pitch rate not meeting the requirements of the tracking task.Figure 15
Velocity tracking.Figure 16
Altitude tracking.Figure 17
Velocity tracking.Figure 18
Altitude tracking.Figure 19
Responses ofγ, θ, and Q.Figure 20
Control inputs.


## 6. Conclusions

A constrained adaptive neural control without backstepping is proposed for the AHV in this article. It is shown that the proposed control structure is quite concise and the problem of “explosion of terms” is avoided. By conducting the comparative simulations, the effectiveness of the proposed nonlinear auxiliary systems is verified. Compared with the backstepping control scheme based on novel NDO, the proposed algorithm has stronger robustness. Our further works will concentrate on designing better learning algorithms for RBFNN.



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*Source: 6968526-2018-08-06.xml*