# Concise Neural Nonaffine Control of Air-Breathing Hypersonic Vehicles Subject to Parametric Uncertainties

Authors: Xiangwei Bu; Qiong Wang; Yan Zhao; Guangjun He
Journal: International Journal of Aerospace Engineering
 (2017)

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## Abstract

In this paper, a novel simplified neural control strategy is proposed for the longitudinal dynamics of an air-breathing hypersonic vehicle (AHV) directly using nonaffine models instead of affine ones. For the velocity dynamics, an adaptive neural controller is devised based on a minimal-learning parameter (MLP) technique for the sake of decreasing computational loads. The altitude dynamics is rewritten as a pure feedback nonaffine formulation, for which a novel concise neural control approach is achieved without backstepping. The special contributions are that the control architecture is concise and the computational cost is low. Moreover, the exploited controller possesses good practicability since there is no need for affine models. The semiglobally uniformly ultimate boundedness of all the closed-loop system signals is guaranteed via Lyapunov stability theory. Finally, simulation results are presented to validate the effectiveness of the investigated control methodology in the presence of parametric uncertainties.


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## Body

## 1. Introduction

Air-breathing hypersonic vehicles (AHVs) are regarded as a promising technology for cost-efficient and reliable access to space. One of the major issues with respect to the developments of AHVs is to maintain controllability of the vehicle during various phases of flight [1–3]. However, flight control design for such vehicles is very challenging due to significant uncertainties, strong couplings, nonminimum phase characteristics, and significant flexible effects [4–7].Recently, many classic control methodologies have been proposed for the longitudinal dynamics of AHVs. To guarantee the robustness with respect to parametric uncertainties, an adaptive sliding mode controller is addressed for an AHV [8, 9]. The starting point of that strategy [8, 9] is to linearize the vehicle model as an affine one, which is required for the control design. In [10], a continuous high-order sliding mode control scheme is investigated using an affine AHV model. In that study, the undesired chattering phenomenon is avoided owing to the continuous control inputs. For an AHV with actuator faults, an adaptive fault-tolerant controller is devised on the basis of a Takagi-Sugeno (T-S) fuzzy system [11]. In [12, 13], a linear parameter-varying (LPV) affine model is first constructed to describe the complex longitudinal dynamics of an AHV. Then, a nonfragile switching tracking control methodology is proposed [12, 13]. In [14], a compound control approach-combined trajectory linearization control (TLC) with active disturbance rejection control (ADRC) is presented for an AHV. Moreover, the robustness of that strategy is enhanced via linear-extended state observers [14]. For the longitudinal dynamics of an AHV, a new tracking control scheme is exploited by solving a system of linear algebraic equations [15]. In [16], a robust controller is explored for an AHV, and moreover, a new nonlinear disturbance observer (NDO) is developed for robustness enhancement.It is noted that the altitude dynamics of AHVs can be rewritten as a strict feedback affine model based on strict assumptions, which makes backstepping control achievable [17–20]. In [1], a robust adaptive control method is studied via backstepping. However, the smooth parameter projection utilized in that paper may result in high computational loads because of too many adaptive parameters. Meanwhile, the problem of “explosion of terms” is ignored [1]. In [19], a new tracking differentiator is exploited to provide estimations of the time derivatives of virtual controllers while the problem of “explosion of terms” is eliminated. Based on the results of [21], a new NDO is constructed, which guarantees that the proposed backstepping controller can provide robust tracking of velocity and altitude reference trajectories in the presence of parametric uncertainties and external disturbances [19]. It has been demonstrated that neural networks (NNs) have the ability of global approximation [22–25]. Thus, improved backstepping control methodologies with robust performance can be achieved if the unknown nonlinearities are approximated by NNs [26–29]. To reduce the computational cost of neural approximation, the NNs are employed to approach the devised backstepping controllers rather than unknown functions while a nonsingular direct neural control strategy is addressed for an AHV [30]. Furthermore, a novel control scheme with low computational loads is exploited for an AHV based on a pure feedback affine model [31]. Meanwhile, the learning parameters are decreased via a minimal-learning parameter (MLP) method [31].It is worth pointing out that all the above results are obtained based on affine models. That is, it requires that the AHV model has to be linear in the control inputs. However, in fact, the AHV model has no linear appearance of the control inputs [32, 33]. On the basis of the previous approaches, we have to first transform the AHV model into an affine formulation and then exploit controllers utilizing linearized or affine AHV models. In this paper, we propose a novel adaptive neural controller for the longitudinal dynamics of an AHV without using affine models. The vehicle dynamics is decomposed as the velocity subsystem and the altitude subsystem to be controlled separately. For both subsystems, simplified neural controllers with low computational loads are addressed via an MLP technique. By using a novel model transformation method, the altitude dynamics is rewritten as a pure feedback formulation. Hence, a simplified design is achieved. Finally, the effectiveness of the studied methodology is verified by simulation results. The special contributions of this paper are summarized as follows:
(1)
Different from the existing studies, the exploited controller is directly derived from nonaffine models. Moreover, the strict requirements [30] that the unknown functions have to be strictly positive and bounded are released in this paper. Thereby, the proposed control scheme possesses excellent practicability and reliability.(2)
Compared with backstepping control strategies [18–20, 30], the explored controller exhibits a more concise structure and a lower computational load. In this study, there is no need of a complicated recursive design process of backstepping. In comparison with the neural backstepping control approach [30], much less controllers, NNs, and learning parameters are needed in this study.The remainder of this paper is structured as follows. The vehicle model and preliminaries are shown in Section2. Section 3 presents the control design process. The simulation studies are made in Section 4, and the conclusions are proposed in Section 5.


## 2. AHV Model and Preliminaries

### 2.1. The AHV Model and Control Goal

In this paper, we consider a longitudinal dynamic model of an AHV developed by Bolender and Doman [32]. The AHV’s longitudinal sketch and force map are shown in Figure 1. The equations of motion are constructed by applying Lagrange’s equations. Moreover, the vehicle is viewed as a single flexible structure with mass-normalized mode shapes. Thus, the vehicle model contains flexible effects. The equations of motion are expressed as follows [33]:
(1)V̇=Tcosθ−γ−Dm−gsinγ,(2)ḣ=Vsinγ,(3)γ̇=L+Tsinθ−γmV−gcosγV,(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 the approximations of T,D,L,M,N1, and N2 are designed as follows [33]:
(8)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,q¯=12ρV2,ρ=ρ0exp−h−h0hs,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¯.Figure 1
The geometry and force map of an AHV.The above model is composed of five rigid body states (i.e.,V,h,γ,θ, and Q), four flexible states (i.e., η1, η̇1, η2, and η̇2), and two control inputs (i.e., Φ and δe). It is noticed that the control inputs do not occur explicitly in (1), (2), (3), (4), (5), (6), and (7). However, they appear through T,D,L,M,N1, and N2. For more detailed definitions of the model parameters and coefficients, the reader could refer to [33] or the Nomenclature.The control goal pursued in this study is to devise adaptive neural controllers for an AHV without using affine modes such that velocityV and altitude h track their reference trajectories Vref and href, respectively. Meanwhile, all the closed-loop system signals are bounded and the proposed controller is robust to parametric uncertainties.


### 2.2. NN Description

To guarantee the controller’s robustness, we employ the radial basis function NN (RBFNN) to approximate the lumped uncertainties of both subsystems of AHVs. The RBFNN is defined as the mapping relationship between the input vectorX=ℓ1,ℓ2,…,ℓnT∈Rn and the output y∈R [34].
(10)y=WThX,where W=w1,w2,…,wpT∈Rp denotes a weight vector; p and n represent the node number and input number, respectively; hX=h1X,h2X,…,hpXT∈Rp with hjX is defined as follows:
(11)hjX=exp−X−c22bj2,j=1,2,…,p,where c=c1,c2,…,cnT∈Rn and b=bj1,bj2,…,bjnT∈Rn mean a center vector and a width vector of hjX, respectively.For an arbitrary continuous unknown functionFX, it has be proved that there exists an ideal weight vector W∗=w1∗,w2∗,…,wp∗T∈Rp such that [34]
(12)FX=W∗ThX+ε,ε≤εM,where ε and εM are the approximation error and its upper bound, respectively. Noting that W∗ is completely unknown, its elements w1∗,w2∗,…,wp∗ are required to be adjusted adaptively. In what follows, we define φ=W∗2 and regulate φ rather than w1∗,w2∗,…,wp∗ to achieve satisfactory neural approximation. In this way, the learning parameters are decreased greatly such that the computational burden is quite low. This is the so-called MLP scheme.


### 2.3. High-Order Tracking Differentiator

In the subsequent developments, we will employ a high-order tracking differentiator (HTD) [35] to estimate the newly defined states. This HTD is formulated as follows:
(13)χ̇1=χ2,χ̇2=χ3,χ̇3=χ4,χ̇4=−R4a1χ1−υ+a2χ2R+a3χ3R2+a4χ4R3,where R,a1,a2,a3,anda4 are positive constants to be chosen; υ is the input signal; χ1,χ2,χ3, and χ4 denote the states of the HTD and they stand for the estimations of υ, υ̇, υ¨, and υ⃛, respectively. The relative estimation errors are defined as follows: e1=χ1−υ, e2=χ2−υ̇, e3=χ3−υ¨, and e4=χ4−υ⃛. By choosing an infinitely large R for the HTD, we have
(14)limR→+∞χ1−υ=limR→+∞e1=0,dlimR→+∞χ1−υdt=limR→+∞χ̇1−υ̇=limR→+∞χ2−υ̇=limR→+∞e2=0,dlimR→+∞χ2−υ̇dt=limR→+∞χ̇2−υ¨=limR→+∞χ3−υ¨=limR→+∞e3=0,dlimR→+∞χ3−υ¨dt=limR→+∞χ̇3−υ⃛=limR→+∞χ4−υ⃛=limR→+∞e4=0.Remark 1.
It is obvious that the estimation errorseii=1,2,3,4 can converge to zero by designing an adequately large R. In choosing an infinitely large but bounded R for the THD in simulation study, we know that there exist positive constants e¯ii=1,2,3,4 such that e¯i≥eii=1,2,3,4 [23].


### 2.4. Preliminaries

To start the control design, the following preliminaries are required.Definition 1 (see [36]).
For any compact setΩ0, if there exist a positive parameter Σ and I=IΣ,Y0t0 such that Y0t≤Σ for all Yt0∈Ω0 and t≥t0+I, then Y0t is semiglobally uniformly ultimately bounded.Lemma 1 (implicit function theorem, see [36, 37]).
The implicit function
H
:
R
l
×
R
r
→
R
l
is continuously differentiable at each
ϖ
,
σ
of an open set
Y
⊂
R
l
×
R
r
. Define
ϖ
0
,
σ
0
as a point in Y for which
H
ϖ
0
,
σ
0
=
0
and the Jacobian matrix
∂
H
/
∂
ϖ
ϖ
0
,
σ
0
is nonsingular. Then, there are neighborhoods
U
⊂
R
l
of
ϖ
0
and
G
⊂
R
r
of
σ
0
such that for each
σ
∈
G
, the equation
H
ϖ
,
σ
=
0
has a unique solution
ϖ
∈
U
. Furthermore, the solution can be expressed as
ϖ
=
g
0
σ
, where
g
0
·
is a continuously differentiable function at
σ
=
σ
0.Remark 2.
For the detailed proof ofLemma 1, the reader could refer to [36]. It is noted that if the implicit function Hϖ,σ satisfies Lemma 1, Hϖ,σ can be viewed as a function of σ since there exists a function g0· such that ϖ=g0σ.


## 2.1. The AHV Model and Control Goal

In this paper, we consider a longitudinal dynamic model of an AHV developed by Bolender and Doman [32]. The AHV’s longitudinal sketch and force map are shown in Figure 1. The equations of motion are constructed by applying Lagrange’s equations. Moreover, the vehicle is viewed as a single flexible structure with mass-normalized mode shapes. Thus, the vehicle model contains flexible effects. The equations of motion are expressed as follows [33]:
(1)V̇=Tcosθ−γ−Dm−gsinγ,(2)ḣ=Vsinγ,(3)γ̇=L+Tsinθ−γmV−gcosγV,(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 the approximations of T,D,L,M,N1, and N2 are designed as follows [33]:
(8)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,q¯=12ρV2,ρ=ρ0exp−h−h0hs,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¯.Figure 1
The geometry and force map of an AHV.The above model is composed of five rigid body states (i.e.,V,h,γ,θ, and Q), four flexible states (i.e., η1, η̇1, η2, and η̇2), and two control inputs (i.e., Φ and δe). It is noticed that the control inputs do not occur explicitly in (1), (2), (3), (4), (5), (6), and (7). However, they appear through T,D,L,M,N1, and N2. For more detailed definitions of the model parameters and coefficients, the reader could refer to [33] or the Nomenclature.The control goal pursued in this study is to devise adaptive neural controllers for an AHV without using affine modes such that velocityV and altitude h track their reference trajectories Vref and href, respectively. Meanwhile, all the closed-loop system signals are bounded and the proposed controller is robust to parametric uncertainties.


## 2.2. NN Description

To guarantee the controller’s robustness, we employ the radial basis function NN (RBFNN) to approximate the lumped uncertainties of both subsystems of AHVs. The RBFNN is defined as the mapping relationship between the input vectorX=ℓ1,ℓ2,…,ℓnT∈Rn and the output y∈R [34].
(10)y=WThX,where W=w1,w2,…,wpT∈Rp denotes a weight vector; p and n represent the node number and input number, respectively; hX=h1X,h2X,…,hpXT∈Rp with hjX is defined as follows:
(11)hjX=exp−X−c22bj2,j=1,2,…,p,where c=c1,c2,…,cnT∈Rn and b=bj1,bj2,…,bjnT∈Rn mean a center vector and a width vector of hjX, respectively.For an arbitrary continuous unknown functionFX, it has be proved that there exists an ideal weight vector W∗=w1∗,w2∗,…,wp∗T∈Rp such that [34]
(12)FX=W∗ThX+ε,ε≤εM,where ε and εM are the approximation error and its upper bound, respectively. Noting that W∗ is completely unknown, its elements w1∗,w2∗,…,wp∗ are required to be adjusted adaptively. In what follows, we define φ=W∗2 and regulate φ rather than w1∗,w2∗,…,wp∗ to achieve satisfactory neural approximation. In this way, the learning parameters are decreased greatly such that the computational burden is quite low. This is the so-called MLP scheme.


## 2.3. High-Order Tracking Differentiator

In the subsequent developments, we will employ a high-order tracking differentiator (HTD) [35] to estimate the newly defined states. This HTD is formulated as follows:
(13)χ̇1=χ2,χ̇2=χ3,χ̇3=χ4,χ̇4=−R4a1χ1−υ+a2χ2R+a3χ3R2+a4χ4R3,where R,a1,a2,a3,anda4 are positive constants to be chosen; υ is the input signal; χ1,χ2,χ3, and χ4 denote the states of the HTD and they stand for the estimations of υ, υ̇, υ¨, and υ⃛, respectively. The relative estimation errors are defined as follows: e1=χ1−υ, e2=χ2−υ̇, e3=χ3−υ¨, and e4=χ4−υ⃛. By choosing an infinitely large R for the HTD, we have
(14)limR→+∞χ1−υ=limR→+∞e1=0,dlimR→+∞χ1−υdt=limR→+∞χ̇1−υ̇=limR→+∞χ2−υ̇=limR→+∞e2=0,dlimR→+∞χ2−υ̇dt=limR→+∞χ̇2−υ¨=limR→+∞χ3−υ¨=limR→+∞e3=0,dlimR→+∞χ3−υ¨dt=limR→+∞χ̇3−υ⃛=limR→+∞χ4−υ⃛=limR→+∞e4=0.Remark 1.
It is obvious that the estimation errorseii=1,2,3,4 can converge to zero by designing an adequately large R. In choosing an infinitely large but bounded R for the THD in simulation study, we know that there exist positive constants e¯ii=1,2,3,4 such that e¯i≥eii=1,2,3,4 [23].


## 2.4. Preliminaries

To start the control design, the following preliminaries are required.Definition 1 (see [36]).
For any compact setΩ0, if there exist a positive parameter Σ and I=IΣ,Y0t0 such that Y0t≤Σ for all Yt0∈Ω0 and t≥t0+I, then Y0t is semiglobally uniformly ultimately bounded.Lemma 1 (implicit function theorem, see [36, 37]).
The implicit function
H
:
R
l
×
R
r
→
R
l
is continuously differentiable at each
ϖ
,
σ
of an open set
Y
⊂
R
l
×
R
r
. Define
ϖ
0
,
σ
0
as a point in Y for which
H
ϖ
0
,
σ
0
=
0
and the Jacobian matrix
∂
H
/
∂
ϖ
ϖ
0
,
σ
0
is nonsingular. Then, there are neighborhoods
U
⊂
R
l
of
ϖ
0
and
G
⊂
R
r
of
σ
0
such that for each
σ
∈
G
, the equation
H
ϖ
,
σ
=
0
has a unique solution
ϖ
∈
U
. Furthermore, the solution can be expressed as
ϖ
=
g
0
σ
, where
g
0
·
is a continuously differentiable function at
σ
=
σ
0.Remark 2.
For the detailed proof ofLemma 1, the reader could refer to [36]. It is noted that if the implicit function Hϖ,σ satisfies Lemma 1, Hϖ,σ can be viewed as a function of σ since there exists a function g0· such that ϖ=g0σ.


## 3. Controller Design

According to the discussions presented in [1, 7, 19, 20], we know that the vehicle dynamics can be reasonably decomposed as the velocity subsystem (i.e., (1)) and the altitude subsystem (i.e., (2), (3), (4), and (5)).


### 3.1. Velocity Controller Design

In this subsection, the objective is to devise a neural controllerΦ for the velocity subsystem by applying a nonaffine model such that velocity V follows its reference command Vref.The velocity subsystem can be expressed as the following nonaffine model:(15)V̇=fVV,Φ,where fVV,Φ is a completely unknown and continuously differentiable function.Assumption 1.
The following inequality(16)∂fVV,Φ∂Φ>0holds for all V,Φ∈ΩV×R with a controllability region ΩV.Remark 3.
According to [33] and the response bounds of the rigid-body states, we conclude that Assumption 1 is satisfied.Define velocity tracking error as follows:(17)V˜=V−Vref.Taking time derivative along (17) and substituting (15) lead to
(18)V˜̇=V̇−V̇ref=kVΦ+FVV,Φ−V̇ref,where kV>0 is a design parameter; FVV,Φ=fVV,Φ−kVΦ denotes the lumped uncertainty of the velocity subsystem.The control effortΦ is chosen as follows:
(19)Φ=kV−1Φ0−Φ1,where Φ0=−kV1V˜−kV2∫0tV˜dτ+V̇ref with kV1>0 and kV2>0; Φ1 is a neural control signal designed to cancel FVV,Φ by employing an RBFNN.InvokingLemma 1, we have that there exists Φ1∗ such that
(20)H1V,Φ0,Φ1∗≜FVV,kV−1Φ0−Φ1∗−Φ1∗=0.Then, we obtain the following theorem.Theorem 1.
Define(21)kV>12∂fVV,Φ∂Φ.Then there exist a setΩV⊂R and a unique Φ1∗ that is a function of V and Φ0 such that Φ1∗V,Φ0 satisfies (20) for all V,Φ0∈ΩV×R.Proof.
The sufficient condition for the existence ofΦ1∗ is that the following inequality holds [37, 38]:
(22)∂FVV,Φ∂Φ1∗<1.
Considering (16), (19), and (21) and the fact that FVV,Φ=fVV,Φ−kVΦ, we obtain
(23)∂FVV,Φ∂Φ1∗=∂∂Φ1∗fVV,Φ−kVΦ=∂fVV,Φ−kVΦ∂Φ∂Φ∂Φ1∗=∂fVV,Φ∂Φ−kV1kV=1kV∂fVV,Φ∂Φ−1<1.
It is clear thatΦ1∗ exists.
Furthermore, notice that(24)∂∂Φ1∗H1V,Φ0,Φ1∗=∂∂Φ1∗FVV,Φ∗−Φ1∗=∂∂Φ∗fVV,Φ∗−kVΦ∗∂Φ∗∂Φ1∗−1=∂∂Φ∗fVV,Φ∗−kV−1kV−1=−1kV∂fVV,Φ∗∂Φ∗,with Φ∗=kV−1Φ0−Φ1∗.
By (16) and (21), we know that ∂H1V,Φ0,Φ1∗/∂Φ1∗ is nonsingular. Thus, Theorem 1 holds. This is the end of proof.Theorem1 reveals that Φ is a function of V and Φ0 for all V,Φ0∈ΩV×R due to Φ=kV−1Φ0−Φ1. Thereby, FVV,Φ=fVV,Φ−kVΦ also is a function of V and Φ0. Define X1=V,Φ0T∈R2 as an input vector. Then we introduce an RBFNN to approximate FVV,Φ.
(25)FVV,Φ=W1∗Th1X1+ι1,ι1≤ι1M,where W1∗=w11∗,w12∗,…,w1p1∗T∈Rp1 is an ideal weight vector; ι1 and ι1M are the approximation error and its upper bound, respectively; h1X1=h11X1,h12X1,…,h1p1X1T∈Rp1, h1jX1j=1,2,…,p1 have similar formulations to (11).Defineφ1=W1∗2 and choose
(26)Φ1=12V˜φ^1h1TX1h1X1,where φ^1 is the estimation of φ1 with the following adaptive law:
(27)φ^̇1=λ12V˜2h1TX1h1X1−2φ^1,where λ1>0 is a design parameter.Theorem 2.
Consider the closed-loop system consisting of plant (
15
) under Assumption
1
with controller (
19
) and adaptive law (
27
). Then all the signals involved are semiglobally uniformly ultimately bounded.Proof.
Define(28)φ˜1=φ^1−φ1.
Substituting (19), (25), and (26) into (18) yields
(29)V˜̇=Φ0−Φ1+W1∗Th1X1+ι1−V̇ref=−kV1V˜−kV2∫0tV˜dτ−12V˜φ^1h1TX1h1X1+W1∗Th1X1+ι1.
The Lyapunov function candidate is chosen as follows:(30)WV=V˜22+kV22∫0tV˜dτ2+φ˜122λ1.
Invoking (27), (28), and (29), the time derivative of WV is derived as follows:
(31)ẆV=V˜V˜̇+kV2V˜∫0tV˜dτ+φ˜1φ^̇1λ1=−kV1V˜2−12V˜2φ1h1TX1h1X1+V˜W1∗Th1X1+V˜ι1−2φ˜1φ^1λ1.
Due toφ˜12+2φ˜1φ^1−φ˜1+φ12=φ˜12+2φ˜1φ1+φ12=φ˜1+φ12≥0, it follows that 2φ˜1φ^1≥φ˜12−φ12. Thus, the following inequality holds:
(32)ẆV≤−kV1V˜2−φ˜12λ1−12V˜2φ1h1TX1h1X1+V˜W1∗Th1X1+V˜ι1+φ12λ1.
Notice that(33)V˜W1∗Th1X1≤V˜22W1∗Th1X12+12=V˜22W1∗2h1X12+12=V˜22φ1h1TX1h1X1+12,V˜ι1≤12V˜2ι1M2+12.
Inequality (32) becomes
(34)ẆV≤−kV1−12ι1M2V˜2−φ˜12λ1+1+φ12λ1.
LetkV1>ι1M2/2 and define the following compact sets:
(35)ΩV˜=V˜V˜≤1+φ12/λ1kV1−1/2ι1M2,Ωφ˜1=φ˜1φ˜1≤1+φ12/λ11/λ1.
It can be seen thatẆV will be negative if V˜∉ΩV˜ or φ˜1∉Ωφ˜1. Hence, V˜ and φ˜1 are semiglobally uniformly ultimately bounded. This completes the proof.


### 3.2. Altitude Controller Design

In this subsection, we will explore a simplified neural controllerδe for the altitude subsystem without using backstepping such that altitude h tracks its command href.Define altitude tracking error ash˜=h−href. The command of γ is chosen as follows:
(36)γd=arcsin−kγh˜+ḣrefV,where kγ>0 is a design parameter. If γ converges to γd, then we have that h converges to href [26]. Hence, the subsequent control goal is to let γ converge to γd.Definex1=γ,x2=θ,x3=Q. Then, a nonaffine model can be derived from the rest of the altitude subsystem (i.e., (3), (4), and (5)).
(37)ẋ1=f1x1,x2,ẋ2=x3,ẋ3=f3x,δe,with x=x1,x2,x3T. In (37), f1x1,x2 and f3x,δe are completely unknown and continuously differentiable functions.Assumption 2.
The following are inequalities:(38)∂f1x1,x2∂x2>0,∂f3x,δe∂δe>0.The inequalities above are satisfied for all x,δe∈Ωx×R with a controllability region Ωx.Remark 4.
Taking into account [33] and the response bounds of the rigid-body states, we know that Assumption 2 holds.To avoid the complex design process of backstepping, the following model transformation is made.Step 1.
Definez1=x1=γ and z2=ż1=f1x1,x2. Considering (37), the time derivative of z2 is given by the following:
(39)ż2=∂f1x1,x2∂x1ẋ1+∂f1x1,x2∂x2ẋ2=∂f1x1,x2∂x1f1x1,x2+∂f1x1,x2∂x2x3≜fh1x.Step 2.
Definez3=ż2=fh1x. Invoking (37), the time derivative of z3 is derived as follows:
(40)ż3=∂fh1x∂x1ẋ1+∂fh1x∂x2ẋ2+∂fh1x∂x3ẋ3=∂fh1x∂x1f1x1,x2+∂fh1x∂x2x3+∂fh1x∂x3f3x,δe≜fh2x,δe.As a result, a pure feedback model with a nonaffine formulation is obtained:(41)ż1=z2,ż2=z3,ż3=fh2x,δe,where fh2x,δe is an unknown continuous function.Remark 5.
From (37), (38), (39), and (40), we obtain that
(42)∂fh2x,δe∂δe=∂fh1x∂x3∂f3x,δe∂δe=∂f1x1,x2∂x2∂f3x,δe∂δe>0.Remark 6.
Compared with (37), the expression of (41) is simpler. On the basis of (41), the complicated design process of backstepping can be eliminated.Define flight-path angle tracking errore0 and error function E as follows:
(43)e0=γ−γd=z1−γd,E=ddt+μ3∫0te0dτ=e¨0+3μė0+3μ2e0+μ3∫0te0dτ,where μ is a positive constant to be chosen. Moreover, because the polynomial s+μ3 is Hurwitz, the boundedness of e0 can be guaranteed if E is bounded.Considering (41), the first three order time derivatives of e0 are derived as follows:
(44)ė0=ż1−γ̇d=z2−γ̇d,e¨0=ż2−γ¨d=z3−γ¨d,e⃛0=ż3−γ⃛d=khδe+Fhx,δe−γ⃛d,where kh>0 is a design parameter; Fhx,δe=fh2x,δe−khδe is a completely unknown function.Thus, the time derivative ofE is given as follows:
(45)Ė=e⃛0+3μe¨0+3μ2ė0+μ3e0=khδe+Fhx,δe−γ⃛d+3μe¨0+3μ2ė0+μ3e0.Note that the newly defined statesz2 and z3 are unknown. From the above model transformation process, we have that z2=γ̇ and z3=γ¨. Thus, let γ pass through the HTD presented in Section 2.3. We can get the estimations of z2 and z3, denoted by z^2 and z^3, respectively. Meanwhile, let γd be the input signal of the HTD. We can obtain the estimations of γ̇d, γ¨d, and γ⃛d, represented by γ^̇d, γ^¨d, and γ^⃛d, respectively. The relative estimation errors are defined as follows:
(46)s1=z^2−z2,s2=z^3−z3,s3=γ^̇d−γ̇d,s4=γ^¨d−γ¨d,s5=γ^⃛d−γ⃛d.By Remark1, it is concluded that there exist positive constants s¯ii=1−5 such that s¯i≥sii=1−5. Thus, (44) and (45) can be modified as follows:
(47)e^̇0=z^2−γ^̇d,e^¨0=z^3−γ^¨d,e^⃛0=khδe+Fhx,δe−γ^⃛d,(48)E^̇=khδe+Fhx,δe−γ^⃛d+3μe^¨0+3μ2e^̇0+μ3e0.The control law is selected as follows:(49)δe=kh−1δe0−δe1,where δe0=−kh1E^+γ^⃛d−3μe^¨0−3μ2e^̇0−μ3e0; kh>0, kh1>0 are design parameters; δe1 is a neural control signal devised to cancel Fhx,δe by applying an RBFNN.ConsideringLemma 1, we know that there exists a δe1∗ such that
(50)H2x,δe0,δe1∗≜Fhx,kh−1δe0−δe1∗−δe1∗=0.Theorem 3.
Define(51)kh>12∂fh2x,δe∂δe.Then, there exist a setΩx⊂R3 and a unique δe1∗ that is a function of x and δe such that δe1∗x,δe0 satisfies (50) for all x,δe0∈Ωx×R.Proof.
The sufficient condition for the existence ofδe1∗ is that the following inequality holds [37, 38]:
(52)∂Fhx,δe∂δe1∗<1.
Considering (42), (49), and (51), it follows that
(53)∂Fhx,δe∂δe1∗=∂fh2x,δe−khδe∂δe1∗=∂fh2x,δe−khδe∂δe∂δe∂δe1∗=∂fh2x,δe∂δe−kh1kh=1kh∂fh2x,δe∂δe−1<1.
Thereby,δe1∗ exists.
Furthermore,(54)∂∂δe1∗H2x,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∗,with δe∗=kh−1δe0−δe1∗. Obviously, ∂H2x,δe0,δe1∗/∂δe1∗ is nonsingular. Hence, Theorem 3 is satisfied. The proof is completed.It is observed fromTheorem 3 and Lemma 1 that δe can be treated as a function of x and δe0 because of δe=kh−1δe0−δe1. Hence, Fhx,δe=fh2x,δe−khδe can also be viewed as a function of x and δe0.DefineX2=xT,δe0T∈R4 as an input vector. Then, we employ an RBFNN to approximate Fhx,δe.
(55)Fhx,δe=W2∗Th2X2+ι2,ι2≤ι2M,where W2∗=w21∗,w22∗,…,w2p2∗T∈Rp2 means an ideal weight vector; ι2 and ι2M represent the approximation error and its upper bound, respectively; h2X2=h21X2,h22X2,…,h2p2X2T∈Rp2, h2jX2j=1,2,…,p2 have similar formulations to (11).δ
e
1 is chosen as follows:
(56)δe1=12E^φ^2h2TX2h2X2,where φ2=W2∗2, φ^2 is the estimation of φ2 with the following adaptive law:
(57)φ^̇2=λ22E^2h2TX2h2X2−2φ^2,with λ2>0.Theorem 4.
Consider the closed-loop system consisting of plant (41) under Assumption2with controller (50) and adaptive law (58). Then all the signals involved are semi-globally uniformly ultimately bounded.Proof.
Define(58)φ˜2=φ^2−φ2.
Substituting (49), (55), and (56) into (48), we have
(59)E^̇=−kh1E^−12E^φ^2h2TX2h2X2+W2∗Th2X2+ι2.
Design the following Lyapunov function candidate:(60)Wh=12E^2+φ˜222λ2.
Differentiating along (60) with respect to time and substituting (57), (58), and (59), we obtain
(61)Ẇh=E^E^̇+φ˜2φ^̇2λ2=−kh1E^2−12E^2φ2h2TX2h2X2+E^W2∗Th2X2+ι2E^−2φ˜2φ^2λ2.
Notice that(62)E^W2∗Th2X2≤E^22W2∗Th2X22+12=E^22W2∗2h2X22+12=E^22φ2h2TX2h2X2+122φ˜2φ^2λ2≥φ˜22λ2−φ22λ2,E^ι2≤12E^2ι2M2+12.
Thus, the following inequality is satisfied:(63)Ẇh=−kh1−12ι2M2E^2−1λ2φ˜22+1+φ22λ2.
Letkh1>ι2M2/2 and define the following compact sets:
(64)ΩE^=E^E^≤1+φ22/λ2kh1−1/2ι2M2,Ωφ˜2=φ˜2φ˜2≤1+φ22/λ21/λ2.
It is clear thatẆh will be negative if E^∉ΩE^ or φ˜2∉Ωφ˜2. Therefore, E^ and φ˜2 are semiglobally uniformly ultimately bounded. The proof is completed.Remark 7.
From (43), (44), (45), (46), (47), and (48), we get that
(65)E^=e^¨0+3μe^̇0+3μ2e0+μ3∫0te0dτ=e¨0+s2−s4+3μė0+s1−s3+3μ2e0+μ3∫0te0dτ=E+s2−s4+3μs1−s3.
It further follows that(66)E=E^−s2+s4−3μs1−s3≤E^+s¯2+s¯4+3μs¯1+s¯3.
Thus,E and e0 are also bounded. That is, γ can converge to γd.Remark 8.
Different from the affine control strategies [26–30], the proposed controller is derived from a nonaffine model such that it exhibits better practicability. Moreover, the prior information [30] that the unknown control gains approximated by NNs have to be strictly positive is not needed in this paper.Remark 9.
It is obvious that there is no need of the complicated design process of backstepping. Compared with [26, 30], the architecture of the developed controller is more concise. Meanwhile, the problem of “explosion of terms” is successfully eliminated.Remark 10.
Throughout this paper, only two NNs are applied to approach the lumped uncertainties of both subsystems. Furthermore, by the merit of the MLP scheme, there are totally only two learning parameters (i.e.,φ^1 and φ^2) required for neural approximation. In comparison with [30], a lower computational burden design is achieved in this study.Remark 11.
If the input vectors of RBFNNs are chosen asX1=V,ΦT and X2=xT,δeT, it will lead to a fixed-point problem, which will result in a computational burden. In this paper, such problem is avoided.


## 3.1. Velocity Controller Design

In this subsection, the objective is to devise a neural controllerΦ for the velocity subsystem by applying a nonaffine model such that velocity V follows its reference command Vref.The velocity subsystem can be expressed as the following nonaffine model:(15)V̇=fVV,Φ,where fVV,Φ is a completely unknown and continuously differentiable function.Assumption 1.
The following inequality(16)∂fVV,Φ∂Φ>0holds for all V,Φ∈ΩV×R with a controllability region ΩV.Remark 3.
According to [33] and the response bounds of the rigid-body states, we conclude that Assumption 1 is satisfied.Define velocity tracking error as follows:(17)V˜=V−Vref.Taking time derivative along (17) and substituting (15) lead to
(18)V˜̇=V̇−V̇ref=kVΦ+FVV,Φ−V̇ref,where kV>0 is a design parameter; FVV,Φ=fVV,Φ−kVΦ denotes the lumped uncertainty of the velocity subsystem.The control effortΦ is chosen as follows:
(19)Φ=kV−1Φ0−Φ1,where Φ0=−kV1V˜−kV2∫0tV˜dτ+V̇ref with kV1>0 and kV2>0; Φ1 is a neural control signal designed to cancel FVV,Φ by employing an RBFNN.InvokingLemma 1, we have that there exists Φ1∗ such that
(20)H1V,Φ0,Φ1∗≜FVV,kV−1Φ0−Φ1∗−Φ1∗=0.Then, we obtain the following theorem.Theorem 1.
Define(21)kV>12∂fVV,Φ∂Φ.Then there exist a setΩV⊂R and a unique Φ1∗ that is a function of V and Φ0 such that Φ1∗V,Φ0 satisfies (20) for all V,Φ0∈ΩV×R.Proof.
The sufficient condition for the existence ofΦ1∗ is that the following inequality holds [37, 38]:
(22)∂FVV,Φ∂Φ1∗<1.
Considering (16), (19), and (21) and the fact that FVV,Φ=fVV,Φ−kVΦ, we obtain
(23)∂FVV,Φ∂Φ1∗=∂∂Φ1∗fVV,Φ−kVΦ=∂fVV,Φ−kVΦ∂Φ∂Φ∂Φ1∗=∂fVV,Φ∂Φ−kV1kV=1kV∂fVV,Φ∂Φ−1<1.
It is clear thatΦ1∗ exists.
Furthermore, notice that(24)∂∂Φ1∗H1V,Φ0,Φ1∗=∂∂Φ1∗FVV,Φ∗−Φ1∗=∂∂Φ∗fVV,Φ∗−kVΦ∗∂Φ∗∂Φ1∗−1=∂∂Φ∗fVV,Φ∗−kV−1kV−1=−1kV∂fVV,Φ∗∂Φ∗,with Φ∗=kV−1Φ0−Φ1∗.
By (16) and (21), we know that ∂H1V,Φ0,Φ1∗/∂Φ1∗ is nonsingular. Thus, Theorem 1 holds. This is the end of proof.Theorem1 reveals that Φ is a function of V and Φ0 for all V,Φ0∈ΩV×R due to Φ=kV−1Φ0−Φ1. Thereby, FVV,Φ=fVV,Φ−kVΦ also is a function of V and Φ0. Define X1=V,Φ0T∈R2 as an input vector. Then we introduce an RBFNN to approximate FVV,Φ.
(25)FVV,Φ=W1∗Th1X1+ι1,ι1≤ι1M,where W1∗=w11∗,w12∗,…,w1p1∗T∈Rp1 is an ideal weight vector; ι1 and ι1M are the approximation error and its upper bound, respectively; h1X1=h11X1,h12X1,…,h1p1X1T∈Rp1, h1jX1j=1,2,…,p1 have similar formulations to (11).Defineφ1=W1∗2 and choose
(26)Φ1=12V˜φ^1h1TX1h1X1,where φ^1 is the estimation of φ1 with the following adaptive law:
(27)φ^̇1=λ12V˜2h1TX1h1X1−2φ^1,where λ1>0 is a design parameter.Theorem 2.
Consider the closed-loop system consisting of plant (
15
) under Assumption
1
with controller (
19
) and adaptive law (
27
). Then all the signals involved are semiglobally uniformly ultimately bounded.Proof.
Define(28)φ˜1=φ^1−φ1.
Substituting (19), (25), and (26) into (18) yields
(29)V˜̇=Φ0−Φ1+W1∗Th1X1+ι1−V̇ref=−kV1V˜−kV2∫0tV˜dτ−12V˜φ^1h1TX1h1X1+W1∗Th1X1+ι1.
The Lyapunov function candidate is chosen as follows:(30)WV=V˜22+kV22∫0tV˜dτ2+φ˜122λ1.
Invoking (27), (28), and (29), the time derivative of WV is derived as follows:
(31)ẆV=V˜V˜̇+kV2V˜∫0tV˜dτ+φ˜1φ^̇1λ1=−kV1V˜2−12V˜2φ1h1TX1h1X1+V˜W1∗Th1X1+V˜ι1−2φ˜1φ^1λ1.
Due toφ˜12+2φ˜1φ^1−φ˜1+φ12=φ˜12+2φ˜1φ1+φ12=φ˜1+φ12≥0, it follows that 2φ˜1φ^1≥φ˜12−φ12. Thus, the following inequality holds:
(32)ẆV≤−kV1V˜2−φ˜12λ1−12V˜2φ1h1TX1h1X1+V˜W1∗Th1X1+V˜ι1+φ12λ1.
Notice that(33)V˜W1∗Th1X1≤V˜22W1∗Th1X12+12=V˜22W1∗2h1X12+12=V˜22φ1h1TX1h1X1+12,V˜ι1≤12V˜2ι1M2+12.
Inequality (32) becomes
(34)ẆV≤−kV1−12ι1M2V˜2−φ˜12λ1+1+φ12λ1.
LetkV1>ι1M2/2 and define the following compact sets:
(35)ΩV˜=V˜V˜≤1+φ12/λ1kV1−1/2ι1M2,Ωφ˜1=φ˜1φ˜1≤1+φ12/λ11/λ1.
It can be seen thatẆV will be negative if V˜∉ΩV˜ or φ˜1∉Ωφ˜1. Hence, V˜ and φ˜1 are semiglobally uniformly ultimately bounded. This completes the proof.


## 3.2. Altitude Controller Design

In this subsection, we will explore a simplified neural controllerδe for the altitude subsystem without using backstepping such that altitude h tracks its command href.Define altitude tracking error ash˜=h−href. The command of γ is chosen as follows:
(36)γd=arcsin−kγh˜+ḣrefV,where kγ>0 is a design parameter. If γ converges to γd, then we have that h converges to href [26]. Hence, the subsequent control goal is to let γ converge to γd.Definex1=γ,x2=θ,x3=Q. Then, a nonaffine model can be derived from the rest of the altitude subsystem (i.e., (3), (4), and (5)).
(37)ẋ1=f1x1,x2,ẋ2=x3,ẋ3=f3x,δe,with x=x1,x2,x3T. In (37), f1x1,x2 and f3x,δe are completely unknown and continuously differentiable functions.Assumption 2.
The following are inequalities:(38)∂f1x1,x2∂x2>0,∂f3x,δe∂δe>0.The inequalities above are satisfied for all x,δe∈Ωx×R with a controllability region Ωx.Remark 4.
Taking into account [33] and the response bounds of the rigid-body states, we know that Assumption 2 holds.To avoid the complex design process of backstepping, the following model transformation is made.Step 1.
Definez1=x1=γ and z2=ż1=f1x1,x2. Considering (37), the time derivative of z2 is given by the following:
(39)ż2=∂f1x1,x2∂x1ẋ1+∂f1x1,x2∂x2ẋ2=∂f1x1,x2∂x1f1x1,x2+∂f1x1,x2∂x2x3≜fh1x.Step 2.
Definez3=ż2=fh1x. Invoking (37), the time derivative of z3 is derived as follows:
(40)ż3=∂fh1x∂x1ẋ1+∂fh1x∂x2ẋ2+∂fh1x∂x3ẋ3=∂fh1x∂x1f1x1,x2+∂fh1x∂x2x3+∂fh1x∂x3f3x,δe≜fh2x,δe.As a result, a pure feedback model with a nonaffine formulation is obtained:(41)ż1=z2,ż2=z3,ż3=fh2x,δe,where fh2x,δe is an unknown continuous function.Remark 5.
From (37), (38), (39), and (40), we obtain that
(42)∂fh2x,δe∂δe=∂fh1x∂x3∂f3x,δe∂δe=∂f1x1,x2∂x2∂f3x,δe∂δe>0.Remark 6.
Compared with (37), the expression of (41) is simpler. On the basis of (41), the complicated design process of backstepping can be eliminated.Define flight-path angle tracking errore0 and error function E as follows:
(43)e0=γ−γd=z1−γd,E=ddt+μ3∫0te0dτ=e¨0+3μė0+3μ2e0+μ3∫0te0dτ,where μ is a positive constant to be chosen. Moreover, because the polynomial s+μ3 is Hurwitz, the boundedness of e0 can be guaranteed if E is bounded.Considering (41), the first three order time derivatives of e0 are derived as follows:
(44)ė0=ż1−γ̇d=z2−γ̇d,e¨0=ż2−γ¨d=z3−γ¨d,e⃛0=ż3−γ⃛d=khδe+Fhx,δe−γ⃛d,where kh>0 is a design parameter; Fhx,δe=fh2x,δe−khδe is a completely unknown function.Thus, the time derivative ofE is given as follows:
(45)Ė=e⃛0+3μe¨0+3μ2ė0+μ3e0=khδe+Fhx,δe−γ⃛d+3μe¨0+3μ2ė0+μ3e0.Note that the newly defined statesz2 and z3 are unknown. From the above model transformation process, we have that z2=γ̇ and z3=γ¨. Thus, let γ pass through the HTD presented in Section 2.3. We can get the estimations of z2 and z3, denoted by z^2 and z^3, respectively. Meanwhile, let γd be the input signal of the HTD. We can obtain the estimations of γ̇d, γ¨d, and γ⃛d, represented by γ^̇d, γ^¨d, and γ^⃛d, respectively. The relative estimation errors are defined as follows:
(46)s1=z^2−z2,s2=z^3−z3,s3=γ^̇d−γ̇d,s4=γ^¨d−γ¨d,s5=γ^⃛d−γ⃛d.By Remark1, it is concluded that there exist positive constants s¯ii=1−5 such that s¯i≥sii=1−5. Thus, (44) and (45) can be modified as follows:
(47)e^̇0=z^2−γ^̇d,e^¨0=z^3−γ^¨d,e^⃛0=khδe+Fhx,δe−γ^⃛d,(48)E^̇=khδe+Fhx,δe−γ^⃛d+3μe^¨0+3μ2e^̇0+μ3e0.The control law is selected as follows:(49)δe=kh−1δe0−δe1,where δe0=−kh1E^+γ^⃛d−3μe^¨0−3μ2e^̇0−μ3e0; kh>0, kh1>0 are design parameters; δe1 is a neural control signal devised to cancel Fhx,δe by applying an RBFNN.ConsideringLemma 1, we know that there exists a δe1∗ such that
(50)H2x,δe0,δe1∗≜Fhx,kh−1δe0−δe1∗−δe1∗=0.Theorem 3.
Define(51)kh>12∂fh2x,δe∂δe.Then, there exist a setΩx⊂R3 and a unique δe1∗ that is a function of x and δe such that δe1∗x,δe0 satisfies (50) for all x,δe0∈Ωx×R.Proof.
The sufficient condition for the existence ofδe1∗ is that the following inequality holds [37, 38]:
(52)∂Fhx,δe∂δe1∗<1.
Considering (42), (49), and (51), it follows that
(53)∂Fhx,δe∂δe1∗=∂fh2x,δe−khδe∂δe1∗=∂fh2x,δe−khδe∂δe∂δe∂δe1∗=∂fh2x,δe∂δe−kh1kh=1kh∂fh2x,δe∂δe−1<1.
Thereby,δe1∗ exists.
Furthermore,(54)∂∂δe1∗H2x,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∗,with δe∗=kh−1δe0−δe1∗. Obviously, ∂H2x,δe0,δe1∗/∂δe1∗ is nonsingular. Hence, Theorem 3 is satisfied. The proof is completed.It is observed fromTheorem 3 and Lemma 1 that δe can be treated as a function of x and δe0 because of δe=kh−1δe0−δe1. Hence, Fhx,δe=fh2x,δe−khδe can also be viewed as a function of x and δe0.DefineX2=xT,δe0T∈R4 as an input vector. Then, we employ an RBFNN to approximate Fhx,δe.
(55)Fhx,δe=W2∗Th2X2+ι2,ι2≤ι2M,where W2∗=w21∗,w22∗,…,w2p2∗T∈Rp2 means an ideal weight vector; ι2 and ι2M represent the approximation error and its upper bound, respectively; h2X2=h21X2,h22X2,…,h2p2X2T∈Rp2, h2jX2j=1,2,…,p2 have similar formulations to (11).δ
e
1 is chosen as follows:
(56)δe1=12E^φ^2h2TX2h2X2,where φ2=W2∗2, φ^2 is the estimation of φ2 with the following adaptive law:
(57)φ^̇2=λ22E^2h2TX2h2X2−2φ^2,with λ2>0.Theorem 4.
Consider the closed-loop system consisting of plant (41) under Assumption2with controller (50) and adaptive law (58). Then all the signals involved are semi-globally uniformly ultimately bounded.Proof.
Define(58)φ˜2=φ^2−φ2.
Substituting (49), (55), and (56) into (48), we have
(59)E^̇=−kh1E^−12E^φ^2h2TX2h2X2+W2∗Th2X2+ι2.
Design the following Lyapunov function candidate:(60)Wh=12E^2+φ˜222λ2.
Differentiating along (60) with respect to time and substituting (57), (58), and (59), we obtain
(61)Ẇh=E^E^̇+φ˜2φ^̇2λ2=−kh1E^2−12E^2φ2h2TX2h2X2+E^W2∗Th2X2+ι2E^−2φ˜2φ^2λ2.
Notice that(62)E^W2∗Th2X2≤E^22W2∗Th2X22+12=E^22W2∗2h2X22+12=E^22φ2h2TX2h2X2+122φ˜2φ^2λ2≥φ˜22λ2−φ22λ2,E^ι2≤12E^2ι2M2+12.
Thus, the following inequality is satisfied:(63)Ẇh=−kh1−12ι2M2E^2−1λ2φ˜22+1+φ22λ2.
Letkh1>ι2M2/2 and define the following compact sets:
(64)ΩE^=E^E^≤1+φ22/λ2kh1−1/2ι2M2,Ωφ˜2=φ˜2φ˜2≤1+φ22/λ21/λ2.
It is clear thatẆh will be negative if E^∉ΩE^ or φ˜2∉Ωφ˜2. Therefore, E^ and φ˜2 are semiglobally uniformly ultimately bounded. The proof is completed.Remark 7.
From (43), (44), (45), (46), (47), and (48), we get that
(65)E^=e^¨0+3μe^̇0+3μ2e0+μ3∫0te0dτ=e¨0+s2−s4+3μė0+s1−s3+3μ2e0+μ3∫0te0dτ=E+s2−s4+3μs1−s3.
It further follows that(66)E=E^−s2+s4−3μs1−s3≤E^+s¯2+s¯4+3μs¯1+s¯3.
Thus,E and e0 are also bounded. That is, γ can converge to γd.Remark 8.
Different from the affine control strategies [26–30], the proposed controller is derived from a nonaffine model such that it exhibits better practicability. Moreover, the prior information [30] that the unknown control gains approximated by NNs have to be strictly positive is not needed in this paper.Remark 9.
It is obvious that there is no need of the complicated design process of backstepping. Compared with [26, 30], the architecture of the developed controller is more concise. Meanwhile, the problem of “explosion of terms” is successfully eliminated.Remark 10.
Throughout this paper, only two NNs are applied to approach the lumped uncertainties of both subsystems. Furthermore, by the merit of the MLP scheme, there are totally only two learning parameters (i.e.,φ^1 and φ^2) required for neural approximation. In comparison with [30], a lower computational burden design is achieved in this study.Remark 11.
If the input vectors of RBFNNs are chosen asX1=V,ΦT and X2=xT,δeT, it will lead to a fixed-point problem, which will result in a computational burden. In this paper, such problem is avoided.


## 4. Simulation Results

In this section, we will make simulation studies to validate the tracking performance of the proposed control methodology. All the aerodynamic coefficients and parameters in (1), (2), (3), (4), (5), (6), and (7) are the same as [33]. The initial trim conditions of the vehicle are shown in Table 1. The reference trajectories of velocity and altitude are smoothed via the following filter:
(67)0.12s2+2×0.9×0.1×s+0.12.Table 1
Initial trim conditions.
Item
Value
Units
V
7700
ft/s
h
85,000
ft
γ
0
deg
θ
1.6325
deg
Q
0
deg/s
η
1
0.9700
—
η
̇
1
0
—
η
2
0.7967
—
η
̇
2
0
—The centers of RBFNNs are evenly spaced inc1∈7700ft/s,8700ft/s×−0.1,1,c2∈−1deg,1deg×0deg,5deg×−5deg/s,5deg/s×0rad,0.35rad. The weight vectors are chosen as b1=33.33ft/s,0.0367T,b2=0.0012rad,0.0029rad,0.0058rad/s,0.0117radT. The node number is set as p1=p2=20. The design parameters are chosen as follows: kV=0.9,kV1=0.2,kV2=0.8,kγ=2,kh=0.9,kh1=50,μ=7,λ1=λ2=0.05,R=0.05,a1=a3=0.5,a2=a4=0.1. To show the superiority, the proposed controller is compared with a traditional backstepping control strategy presented in [19]. The following two cases are considered.Case 1.
In this case, the AHV is assumed to climb a maneuver from the initial trim conditions, listed in Table1, to the final values h=85800 ft and V=7900 ft/s. To test the robustness of the designed controller, we define
(68)C=C01+0.4sin0.05πt,where C means the value of uncertain coefficient and C0 stands for the nominal value of C. With this definition, a parameter uncertainty up to 40% of the nominal value is considered.The simulation results are shown in Figures2, 3, 4, 5, 6, and 7. From Figures 2 and 3, it can be seen that the obtained velocity and altitude tracking performance is better than the one provided by the backstepping controller. Meanwhile, the attitude angles, the control inputs, and the flexible states, achieved by both control schemes, are smooth and without high-frequency chattering, as depicted in Figures 4, 5, and 6. Figure 7 reveals that the estimations of φ1 and φ2 are bounded.Case 2.
In this case, we take into consideration a more practical scenario. The altitude follows a square command with period 200 s and magnitude 800 ft. The velocity tracks the step command with 333.33 ft/s for every 100 seconds. The following condition of parametric uncertainty is considered.(69)C=C0,0s≤t<50s,C01+0.4sin0.1πt,50s≤t<100s,C0,100s≤t<150s,C01+0.4sin0.1πt,150s≤t<200s,C0,200s≤t<250s,C01+0.4sin0.1πt,250s≤t≤300s..Figure 2
Velocity tracking performance.Figure 3
Altitude tracking performance.Figure 4
The attitude angles.Figure 5
The control inputs.Figure 6
The flexible states.Figure 7
The estimations ofφ1 and φ2.The obtained simulation results, presented in Figures8, 9, 10, 11, 12, and 13, indicate that the exploited control methodology can provide stable tracking of velocity and altitude commands in the presence of severe uncertainties. Moreover, it is observed from Figures 8 and 9 that compared with backstepping control, much smaller tracking errors are achieved by the explored control algorithm when the model parameters are uncertain. The attitude angles, the control inputs, and the flexible states, shown in Figures 10, 11, and 12, are even and can converge to stable values. Finally, the boundedness of φ^1 and φ^2 is presented in Figure 13.Figure 8
Velocity tracking and tracking error.Figure 9
Altitude tracking and tracking error.Figure 10
The flight-path angle, pitch angle, and pitch rate.Figure 11
The control inputs.Figure 12
The flexible states.Figure 13
The estimations ofφ1 and φ2.


## 5. Conclusions

This paper investigates the design of a concise neural controller for an AHV with parametric uncertainties. The vehicle dynamics is decomposed into the respective velocity and altitude subsystems that are controlled separately. For both subsystems, simplified neural controllers are devised directly employing nonaffine models. Specially, a novel control scheme with a concise structure is proposed for the altitude dynamics without utilizing backstepping. Throughout this paper, only two NNs are applied and only two learning parameters are required for neural approximation. It is proved that all the closed-loop system signals are semiglobally uniformly ultimately bounded. Finally, the superiority of the design in command tracking over the traditional backstepping control is well validated by simulation results in the presence of model uncertainties.



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*Source: 1374932-2017-10-01.xml*