# Adaptive Neural Back-Stepping Control with Constrains for a Flexible Air-Breathing Hypersonic Vehicle

Authors: Pengfei Wang; Jie Wang; Jianming Shi; Chang Luo; Shili Tan; Qiyun Xu
Journal: Mathematical Problems in Engineering
 (2015)

---

## Abstract

The design of an adaptive neural back-stepping control for a flexible air-breathing hypersonic vehicle (AHV) in the presence of input constraint and aerodynamic uncertainty is discussed. Based on functional decomposition, the dynamics can be decomposed into the velocity subsystem and the altitude subsystem. To guarantee the exploited controller’s robustness with respect to parametric uncertainties, neural network (NN) is applied to approximate the lumped uncertainty of each subsystem of AHV model. The exceptional contribution is that novel auxiliary systems are introduced to compensate both the tracking errors and desired control laws, based on which the explored controller can still provide effective tracking of velocity and altitude commands when the actuators are saturated. Finally, simulation studies are made to illustrate the effectiveness of the proposed control approach in spite of the flexible effects, system uncertainties, and varying disturbances.


---

## Body

## 1. Introduction

Air-breathing hypersonic vehicles (AHV) are crucial because they may represent a more efficient way to make access to space routine or even make the space travel routine and intercontinental travel as easy as intercity travel. A key issue in making AHV feasible and efficient is the flight control design [1]. However, the flight control of AHV is still an open challenge due to its peculiarity of flight dynamics. It is worth noting that there exists strong coupling between the propulsive and the aerodynamic forces, which makes the aerodynamic characteristics of AHV very difficult to be estimated and measured [2, 3].Recently, lots of efforts have been put into flight control for AHV. Linear control theory was widely employed for flight control design based on linearized model. As shown in [4], the robust control of AHV is studied by introducing a linear quadratic regulator (LQR) with stochastic robustness analysis. In [5], LQR controller is proposed for the linearization model based on any equilibrium points of flight envelope.Nonlinear control scheme is also employed for AHV. Back-stepping has been proved to be a powerful tool for the tracking control of a large class of strict-feedback systems or pure-feedback ones [6–12]. Back-stepping design for nonlinear systems consists of a recursive design procedure, which breaks down the full system control problem into a sequence of designs for lower-order subsystems. A back-stepping controller is designed with multilayer online adaptive neural networks, which can provide good tracking performance [13]. In [14], a combination of novelty command filtered back-stepping technology and dynamic inversion methodology is adopted for designing a dynamic state-feedback controller that provides stable tracking of the altitude and velocity reference commands. However, it is well known that there exists a problem of “explosion of terms” in the traditional back-stepping design, which is caused by the repeated differentiations of virtual control laws. To solve this problem, dynamic surface control [15] and tracking differentiators [16] should be applied.Since the aerodynamic characteristics of AHV are sensitive to the flight condition changes, they are difficult to be measured. Thus, the aerodynamic uncertainty needs to be dealt with at the control design as the main issues. Conventionally, the flight control is accomplished by feedback linearization with neural networks to deal with the uncertainties. Efforts in [17] approximate the unknown nonlinear functions by radial basis function networks and incorporating the dynamic surface technique into a neural network based adaptive control design framework. Reference [18] investigates the discrete time controller for the longitudinal dynamics of the hypersonic flight vehicle with throttle setting constraint. The controller is proposed by estimating the system uncertainty and unknown control gain separately with neural networks. The auxiliary error signal is designed to compensate the effect of throttle setting constraint.Moreover, input constraint also cannot be ignored in practice since the outputs of actuators are constrained for physical limitations. If the input constraint is ignored, control systems may suffer from performance limitations or even lose stability [19–21]. Much literature theoretically focuses on the control problem with input constraint [22–26]. In practice, when input constraint occurs, aircraft body may change seriously even disintegrating. So it is necessary to research the control design problem with input constraint.Motivated by the results of the previous studies, an adaptive neural back-stepping control approach is addressed for the longitudinal dynamical model of AHV. By viewing the flexible effects as system uncertainties, the longitudinal dynamics of AHV are decomposed into two functional subsystems, namely, the respective velocity subsystem and altitude subsystem. To ensure the controller’s robustness, NNs are applied to estimate the lumped uncertainty of each subsystem. Particularly, novel auxiliary systems are exploited to deal with the problem of control input constraint. Finally, simulation results are presented to demonstrate the efficacy of the proposed control methodology. The special advantages of the approach proposed herein include the following:(1)
The novel auxiliary systems are employed to eliminate the error between the desired control laws and actual control laws, which makes sure that the semi-globally uniformly bounded stability of closed-loop system can be still achieved even when the physical limitations are in effect.
(2)
The second-order reference model is designed for the precise estimation of the derivatives of virtual control laws, which predigests the design of controller.The organization of paper is outlined as follows. Firstly, the longitudinal motion model and control-oriented model of an AHV are described in Section2. Section 3 presents the design procedures of adaptive neural back-stepping controller. Then, simulation results are given in Section 4. Finally, brief concluding remarks end the paper in Section 5.


## 2. Problem Formulation

### 2.1. Longitudinal Dynamic Model of AHV

The model taken into consideration in this paper is developed by Bolender and Doman [27] for the longitudinal dynamics of an AHV. The flexible effects are included in these equations by modeling the vehicle as a single flexible structure with mass-normalized mode shapes.Assuming a flat Earth and normalizing the span of an AHV to unit depth, the nonlinear motion equations are written as [27](1)
V
˙
=
T
cos
⁡
α
-
D
m
-
μ
sin
⁡
γ
R
E
+
h
2
,
h
˙
=
V
sin
⁡
γ
,
γ
˙
=
L
+
T
sin
⁡
α
m
V
-
μ
-
V
2
R
E
+
h
cos
⁡
γ
V
R
E
+
h
2
,
α
˙
=
-
γ
˙
+
Q
,
Q
˙
=
z
T
T
+
M
I
y
y
,
η
¨
i
=
-
2
ζ
i
ω
i
η
˙
i
-
ω
i
2
η
i
+
N
i
,
i
=
1,2
,
3
,where the rigid body states V, h, γ, θ, and Q represent velocity, altitude, flight path angle, pitch angle, and pitch rate, respectively; μ and R
E are gravity constant and the radial distance from center of the earth; m and I
y
y are mass of vehicle and moment of inertia about pitch axis; T, D, L, and M represent thrust, drag, lift, and pitch moment, respectively; z
T denotes the thrust moment arm; the flexible states η
=
η
1
η
˙
1
η
2
η
˙
2
η
3
η
˙
3
T denote the first three bending modes of the fuselage; 0
<
ζ
i
<
1 and ω
i
>
0 (i
=
1,2
,
3) mean the damping ratio and natural frequency of the mass-normalized generalized coordinates of the flexible structure.The dynamic system of an AHV can be decomposed as velocity subsystem and altitude-related subsystem. Define the control inputsu
=
[
Φ
,
δ
e
], which are fuel equivalence ratio and elevator angular deflection, respectively.Remark 1.
In [28], the controller design requires an auxiliary actuator as the canard. However, the presence of a canard is quite problematic for the vehicle configuration, as this control surface must withstand the expected high temperatures at hypersonic speed. Therefore, it is assumed that only the fuel-to-air ratio and elevator are the actuators available for controlling the vehicle.Let(2)
L
-
=
L
+
T
sin
⁡
α
.The thrustT, drag D, lift L
-, pitch moment M, and generalized forces N
i (i
=
1,2
,
3) are defined as(3)
T
=
q
-
S
C
T
α
,
M
a
,
Φ
+
ε
T
x
,
η
,
u
,
D
=
q
-
S
C
D
α
,
M
a
,
δ
e
+
ε
D
x
,
η
,
u
,
L
-
=
q
-
S
C
L
-
α
,
M
a
,
Φ
+
ε
L
-
x
,
η
,
u
,
M
=
q
-
S
c
-
C
M
α
,
M
a
,
δ
e
+
ε
M
x
,
η
,
u
,
C
T
α
,
M
a
,
Φ
=
C
T
0
α
,
M
a
+
C
T
Φ
α
,
M
a
,
Φ
Φ
,
C
M
α
,
M
a
,
δ
e
=
C
M
0
α
,
M
a
+
C
M
δ
e
α
,
M
a
,
δ
e
δ
e
,
C
L
-
α
,
M
a
,
Φ
=
C
L
-
0
M
a
,
Φ
+
C
L
-
α
α
,
M
a
,
Φ
α
,
N
i
≈
q
-
S
N
i
α
2
α
2
+
N
i
α
α
+
N
i
δ
e
δ
e
+
N
i
0
+
N
i
η
η
,where q
-, S, and c
- stand for the respective dynamic pressure, reference area, and aerodynamic chord. ε
T, ε
L
-, ε
D, and ε
M are denoted as the fitting error of T, L
-, D
, and M, respectively.To be convenient for design of back-stepping, (1) can be rewritten as(4)
V
˙
=
f
V
q
-
,
M
a
,
α
,
δ
e
+
g
V
q
-
,
α
,
M
a
,
Φ
Φ
+
F
V
+
Δ
V
x
,
η
,
u
,
(5)
h
˙
=
V
γ
+
Δ
h
V
,
γ
,
(6)
γ
˙
=
f
γ
q
-
,
M
a
,
Φ
+
g
γ
q
-
,
α
,
M
a
,
Φ
α
+
F
γ
+
Δ
γ
x
,
η
,
u
,
(7)
α
˙
=
f
α
q
-
,
α
,
M
a
,
Φ
+
Q
+
F
α
+
Δ
α
x
,
η
,
u
,
(8)
Q
˙
=
f
Q
q
-
,
α
,
M
a
,
Φ
+
g
Q
q
-
,
α
,
M
a
,
Φ
,
δ
e
δ
e
+
Δ
Q
x
,
η
,
uwith(9)
f
V
=
q
-
S
C
T
0
cos
⁡
α
-
C
D
m
,
g
V
=
q
-
S
C
T
Φ
cos
⁡
α
m
,
f
γ
=
q
-
S
C
L
0
m
V
,
g
γ
=
q
-
S
C
L
α
m
V
,
f
α
=
-
f
γ
+
g
γ
α
,
f
Q
=
q
-
S
z
T
C
T
0
+
C
T
Φ
Φ
+
c
-
C
M
0
I
y
y
,
g
Q
=
q
-
S
c
-
C
M
δ
e
I
y
y
,
F
V
=
-
μ
sin
⁡
γ
R
E
+
h
2
,
F
γ
=
-
F
α
=
-
μ
-
V
2
R
E
+
h
cos
⁡
γ
V
R
E
+
h
2
.The model errors are considered as(10)
Δ
V
x
,
η
,
u
=
ε
T
cos
⁡
α
-
ε
D
m
,
Δ
h
V
,
γ
=
V
γ
-
2
sin
⁡
γ
-
2
,
Δ
γ
x
,
η
,
u
=
ε
L
-
m
V
,
Δ
α
x
,
η
,
u
=
-
ε
L
-
m
V
,
Δ
Q
x
,
η
,
u
=
z
T
ε
T
+
ε
M
I
y
y
.Remark 2.
The flexible dynamics are taken as perturbations on the rigid body system, and their effects are evaluated in simulation. According to (1) and (4), η
i would be asymptotic stability if Φ, δ
e, and α are bounded.


### 2.2. Control Objective

In practice, due to physical limitations, the outputs of the actuator are constrained. Input constraints studied in the paper include the constraint on fuel equivalence ratio, elevator deflection. The constraint on fuel equivalence ratio is imposed by the very nature of the propulsion system, which is required to maintain the conditions that sustain scramjet operation [29]. If the limit is violated, the thermal choking will occur. It could induce that engine unstarts which could jeopardize mission, vehicle, and its contents [30]. The constraints on elevator deflection and canard deflection are mainly imposed by the limits on control surface displacement. The above input constraint can be expressed as(11)
Φ
=
Φ
m
a
x
,
Φ
c
≥
Φ
m
a
x
,
Φ
=
Φ
c
,
Φ
m
i
n
≤
Φ
c
≤
Φ
m
a
x
,
Φ
=
Φ
m
i
n
,
Φ
c
≤
Φ
m
i
n
,
δ
e
=
δ
e
m
a
x
,
δ
e
c
≥
δ
e
m
a
x
,
δ
e
=
δ
e
c
,
δ
e
m
i
n
≤
δ
e
c
≤
δ
e
m
a
x
,
δ
e
=
δ
e
m
i
n
,
δ
e
c
≤
δ
e
m
i
n
,where Φ
m
a
x and Φ
m
i
n denote the upper and lower bound of Φ, respectively; δ
e
m
a
x and δ
e
m
i
n stand for the upper and lower bound of δ
e, respectively; Φ
c and δ
e
c are the desired control inputs to be designed in the subsequent section.A second-order reference model with amplitude, rate, and bandwidth limitation is introduced to deal with the input constraint. The structure of second-order reference model is given in Figure1. Φ
0, δ
e
0, and γ
c
0, α
c
0, and Q
c
0 are the designed actual control variables and virtual control variables. Define Φ, δ
e, and γ
c, α
c, and Q
c as the executable control laws and virtual control laws with the second-order reference model.Figure 1
Second-order reference model.


### 2.3. Radial Basis Function NN Approximation

The radial basis function NN (RBFNN) will be introduced to approach the unknown functionsf
x
z and g
x
(
z
)
x
=
V
,
γ
,
Q owing to its excellent performance and global approximation. It has been proved that RBFNN can approximate an arbitrary continuous function over a compact set Ω
z
⊆
R
n to an arbitrary accuracy. RBFNN is formulated as θ
T
ξ
z with the input vector z
⊆
R
n, the weight vector θ
⊆
R
m, the basis function vector ξ
z
⊆
R
m, the node number m, and the input number n. The basis function is selected as the following Gaussian function:(12)
ξ
j
z
=
exp
⁡
-
z
-
c
i
2
2
b
i
2
,
i
=
1,2
,
…
,
m
,where c
i
=
c
i
1
,
c
i
2
,
…
,
c
i
m
T and b
i stand for the center and width of Gaussian function, respectively.Assumption 3.
f
x
z and g
x
z
x
=
V
,
γ
,
Q are continuous real function. The unknown functions f
x
(
z
) and g
x
(
z
) can be designed as(13)
f
x
z
=
θ
f
x
∗
T
ξ
f
x
z
+
ε
f
x
∗
,
g
x
z
=
θ
g
x
∗
T
ξ
g
x
z
+
ε
g
x
∗
,where ξ
f
x
(
z
) and ξ
g
x
(
z
) are radial basis function vector; ε
f
x
∗ and ε
g
x
∗ denote the approximation error, and |
ε
f
x
∗
|
≤
ε
f
x
M and |
ε
g
x
∗
|
≤
ε
g
x
M; θ
f
x
∗ and θ
g
x
∗ represent optimal weight vector, θ
^
f
x and θ
^
g
x are the estimation of θ
f
x
∗ and θ
g
x
∗, respectively, and θ
~
f
x
=
θ
f
x
∗
-
θ
^
f
x and θ
~
g
x
=
θ
g
x
∗
-
θ
^
g
x are estimation error of θ
f
x
∗ and θ
g
x
∗, respectively.Lemma 4.
According to(14)
θ
~
T
θ
^
=
1
2
θ
~
T
θ
∗
-
θ
~
+
θ
∗
-
θ
^
T
θ
^
=
1
2
-
θ
~
2
-
θ
^
2
+
θ
∗
2there exist(15)
θ
~
T
θ
^
≤
1
2
-
θ
^
2
+
θ
∗
2
,
θ
~
T
θ
^
≤
1
2
-
θ
~
2
+
θ
∗
2
.Lemma 5 (see [31]).
The adaptive law ofθ
^ is designed as (16)
θ
^
˙
=
Γ
Proj
⁡
θ
^
,
ν
,
β
,
Γ
>
0
,
β
>
0
,where(17)
Proj
⁡
θ
^
,
ν
,
β
=
ν
-
β
θ
^
,
i
f
θ
^
<
θ
M
o
r
θ
^
=
θ
M
,
θ
^
T
ν
≤
0
ν
-
β
θ
^
-
θ
^
θ
^
T
ν
θ
^
2
,
i
f
θ
^
=
θ
M
,
θ
^
T
ν
>
0
;then θ
^
(
t
)
≤
θ
M; θ
M is positive constant.


## 2.1. Longitudinal Dynamic Model of AHV

The model taken into consideration in this paper is developed by Bolender and Doman [27] for the longitudinal dynamics of an AHV. The flexible effects are included in these equations by modeling the vehicle as a single flexible structure with mass-normalized mode shapes.Assuming a flat Earth and normalizing the span of an AHV to unit depth, the nonlinear motion equations are written as [27](1)
V
˙
=
T
cos
⁡
α
-
D
m
-
μ
sin
⁡
γ
R
E
+
h
2
,
h
˙
=
V
sin
⁡
γ
,
γ
˙
=
L
+
T
sin
⁡
α
m
V
-
μ
-
V
2
R
E
+
h
cos
⁡
γ
V
R
E
+
h
2
,
α
˙
=
-
γ
˙
+
Q
,
Q
˙
=
z
T
T
+
M
I
y
y
,
η
¨
i
=
-
2
ζ
i
ω
i
η
˙
i
-
ω
i
2
η
i
+
N
i
,
i
=
1,2
,
3
,where the rigid body states V, h, γ, θ, and Q represent velocity, altitude, flight path angle, pitch angle, and pitch rate, respectively; μ and R
E are gravity constant and the radial distance from center of the earth; m and I
y
y are mass of vehicle and moment of inertia about pitch axis; T, D, L, and M represent thrust, drag, lift, and pitch moment, respectively; z
T denotes the thrust moment arm; the flexible states η
=
η
1
η
˙
1
η
2
η
˙
2
η
3
η
˙
3
T denote the first three bending modes of the fuselage; 0
<
ζ
i
<
1 and ω
i
>
0 (i
=
1,2
,
3) mean the damping ratio and natural frequency of the mass-normalized generalized coordinates of the flexible structure.The dynamic system of an AHV can be decomposed as velocity subsystem and altitude-related subsystem. Define the control inputsu
=
[
Φ
,
δ
e
], which are fuel equivalence ratio and elevator angular deflection, respectively.Remark 1.
In [28], the controller design requires an auxiliary actuator as the canard. However, the presence of a canard is quite problematic for the vehicle configuration, as this control surface must withstand the expected high temperatures at hypersonic speed. Therefore, it is assumed that only the fuel-to-air ratio and elevator are the actuators available for controlling the vehicle.Let(2)
L
-
=
L
+
T
sin
⁡
α
.The thrustT, drag D, lift L
-, pitch moment M, and generalized forces N
i (i
=
1,2
,
3) are defined as(3)
T
=
q
-
S
C
T
α
,
M
a
,
Φ
+
ε
T
x
,
η
,
u
,
D
=
q
-
S
C
D
α
,
M
a
,
δ
e
+
ε
D
x
,
η
,
u
,
L
-
=
q
-
S
C
L
-
α
,
M
a
,
Φ
+
ε
L
-
x
,
η
,
u
,
M
=
q
-
S
c
-
C
M
α
,
M
a
,
δ
e
+
ε
M
x
,
η
,
u
,
C
T
α
,
M
a
,
Φ
=
C
T
0
α
,
M
a
+
C
T
Φ
α
,
M
a
,
Φ
Φ
,
C
M
α
,
M
a
,
δ
e
=
C
M
0
α
,
M
a
+
C
M
δ
e
α
,
M
a
,
δ
e
δ
e
,
C
L
-
α
,
M
a
,
Φ
=
C
L
-
0
M
a
,
Φ
+
C
L
-
α
α
,
M
a
,
Φ
α
,
N
i
≈
q
-
S
N
i
α
2
α
2
+
N
i
α
α
+
N
i
δ
e
δ
e
+
N
i
0
+
N
i
η
η
,where q
-, S, and c
- stand for the respective dynamic pressure, reference area, and aerodynamic chord. ε
T, ε
L
-, ε
D, and ε
M are denoted as the fitting error of T, L
-, D
, and M, respectively.To be convenient for design of back-stepping, (1) can be rewritten as(4)
V
˙
=
f
V
q
-
,
M
a
,
α
,
δ
e
+
g
V
q
-
,
α
,
M
a
,
Φ
Φ
+
F
V
+
Δ
V
x
,
η
,
u
,
(5)
h
˙
=
V
γ
+
Δ
h
V
,
γ
,
(6)
γ
˙
=
f
γ
q
-
,
M
a
,
Φ
+
g
γ
q
-
,
α
,
M
a
,
Φ
α
+
F
γ
+
Δ
γ
x
,
η
,
u
,
(7)
α
˙
=
f
α
q
-
,
α
,
M
a
,
Φ
+
Q
+
F
α
+
Δ
α
x
,
η
,
u
,
(8)
Q
˙
=
f
Q
q
-
,
α
,
M
a
,
Φ
+
g
Q
q
-
,
α
,
M
a
,
Φ
,
δ
e
δ
e
+
Δ
Q
x
,
η
,
uwith(9)
f
V
=
q
-
S
C
T
0
cos
⁡
α
-
C
D
m
,
g
V
=
q
-
S
C
T
Φ
cos
⁡
α
m
,
f
γ
=
q
-
S
C
L
0
m
V
,
g
γ
=
q
-
S
C
L
α
m
V
,
f
α
=
-
f
γ
+
g
γ
α
,
f
Q
=
q
-
S
z
T
C
T
0
+
C
T
Φ
Φ
+
c
-
C
M
0
I
y
y
,
g
Q
=
q
-
S
c
-
C
M
δ
e
I
y
y
,
F
V
=
-
μ
sin
⁡
γ
R
E
+
h
2
,
F
γ
=
-
F
α
=
-
μ
-
V
2
R
E
+
h
cos
⁡
γ
V
R
E
+
h
2
.The model errors are considered as(10)
Δ
V
x
,
η
,
u
=
ε
T
cos
⁡
α
-
ε
D
m
,
Δ
h
V
,
γ
=
V
γ
-
2
sin
⁡
γ
-
2
,
Δ
γ
x
,
η
,
u
=
ε
L
-
m
V
,
Δ
α
x
,
η
,
u
=
-
ε
L
-
m
V
,
Δ
Q
x
,
η
,
u
=
z
T
ε
T
+
ε
M
I
y
y
.Remark 2.
The flexible dynamics are taken as perturbations on the rigid body system, and their effects are evaluated in simulation. According to (1) and (4), η
i would be asymptotic stability if Φ, δ
e, and α are bounded.


## 2.2. Control Objective

In practice, due to physical limitations, the outputs of the actuator are constrained. Input constraints studied in the paper include the constraint on fuel equivalence ratio, elevator deflection. The constraint on fuel equivalence ratio is imposed by the very nature of the propulsion system, which is required to maintain the conditions that sustain scramjet operation [29]. If the limit is violated, the thermal choking will occur. It could induce that engine unstarts which could jeopardize mission, vehicle, and its contents [30]. The constraints on elevator deflection and canard deflection are mainly imposed by the limits on control surface displacement. The above input constraint can be expressed as(11)
Φ
=
Φ
m
a
x
,
Φ
c
≥
Φ
m
a
x
,
Φ
=
Φ
c
,
Φ
m
i
n
≤
Φ
c
≤
Φ
m
a
x
,
Φ
=
Φ
m
i
n
,
Φ
c
≤
Φ
m
i
n
,
δ
e
=
δ
e
m
a
x
,
δ
e
c
≥
δ
e
m
a
x
,
δ
e
=
δ
e
c
,
δ
e
m
i
n
≤
δ
e
c
≤
δ
e
m
a
x
,
δ
e
=
δ
e
m
i
n
,
δ
e
c
≤
δ
e
m
i
n
,where Φ
m
a
x and Φ
m
i
n denote the upper and lower bound of Φ, respectively; δ
e
m
a
x and δ
e
m
i
n stand for the upper and lower bound of δ
e, respectively; Φ
c and δ
e
c are the desired control inputs to be designed in the subsequent section.A second-order reference model with amplitude, rate, and bandwidth limitation is introduced to deal with the input constraint. The structure of second-order reference model is given in Figure1. Φ
0, δ
e
0, and γ
c
0, α
c
0, and Q
c
0 are the designed actual control variables and virtual control variables. Define Φ, δ
e, and γ
c, α
c, and Q
c as the executable control laws and virtual control laws with the second-order reference model.Figure 1
Second-order reference model.


## 2.3. Radial Basis Function NN Approximation

The radial basis function NN (RBFNN) will be introduced to approach the unknown functionsf
x
z and g
x
(
z
)
x
=
V
,
γ
,
Q owing to its excellent performance and global approximation. It has been proved that RBFNN can approximate an arbitrary continuous function over a compact set Ω
z
⊆
R
n to an arbitrary accuracy. RBFNN is formulated as θ
T
ξ
z with the input vector z
⊆
R
n, the weight vector θ
⊆
R
m, the basis function vector ξ
z
⊆
R
m, the node number m, and the input number n. The basis function is selected as the following Gaussian function:(12)
ξ
j
z
=
exp
⁡
-
z
-
c
i
2
2
b
i
2
,
i
=
1,2
,
…
,
m
,where c
i
=
c
i
1
,
c
i
2
,
…
,
c
i
m
T and b
i stand for the center and width of Gaussian function, respectively.Assumption 3.
f
x
z and g
x
z
x
=
V
,
γ
,
Q are continuous real function. The unknown functions f
x
(
z
) and g
x
(
z
) can be designed as(13)
f
x
z
=
θ
f
x
∗
T
ξ
f
x
z
+
ε
f
x
∗
,
g
x
z
=
θ
g
x
∗
T
ξ
g
x
z
+
ε
g
x
∗
,where ξ
f
x
(
z
) and ξ
g
x
(
z
) are radial basis function vector; ε
f
x
∗ and ε
g
x
∗ denote the approximation error, and |
ε
f
x
∗
|
≤
ε
f
x
M and |
ε
g
x
∗
|
≤
ε
g
x
M; θ
f
x
∗ and θ
g
x
∗ represent optimal weight vector, θ
^
f
x and θ
^
g
x are the estimation of θ
f
x
∗ and θ
g
x
∗, respectively, and θ
~
f
x
=
θ
f
x
∗
-
θ
^
f
x and θ
~
g
x
=
θ
g
x
∗
-
θ
^
g
x are estimation error of θ
f
x
∗ and θ
g
x
∗, respectively.Lemma 4.
According to(14)
θ
~
T
θ
^
=
1
2
θ
~
T
θ
∗
-
θ
~
+
θ
∗
-
θ
^
T
θ
^
=
1
2
-
θ
~
2
-
θ
^
2
+
θ
∗
2there exist(15)
θ
~
T
θ
^
≤
1
2
-
θ
^
2
+
θ
∗
2
,
θ
~
T
θ
^
≤
1
2
-
θ
~
2
+
θ
∗
2
.Lemma 5 (see [31]).
The adaptive law ofθ
^ is designed as (16)
θ
^
˙
=
Γ
Proj
⁡
θ
^
,
ν
,
β
,
Γ
>
0
,
β
>
0
,where(17)
Proj
⁡
θ
^
,
ν
,
β
=
ν
-
β
θ
^
,
i
f
θ
^
<
θ
M
o
r
θ
^
=
θ
M
,
θ
^
T
ν
≤
0
ν
-
β
θ
^
-
θ
^
θ
^
T
ν
θ
^
2
,
i
f
θ
^
=
θ
M
,
θ
^
T
ν
>
0
;then θ
^
(
t
)
≤
θ
M; θ
M is positive constant.


## 3. Controller Design

The control objective pursued in this section is to develop an adaptive neural back-stepping controller for an AHV to provide robust tracking of velocity and altitude commandsV
r
e
f and h
r
e
f. It is assumed that the rigid body states V, h, γ, α, and Q are available for measurement. It is easy to note that the velocity V is mainly related to Φ and the altitude h is mainly affected by δ
e since the thrust T affects V and δ
e has a dominant contribution to h change in (1). In what follows, the respective control laws Φ and δ
e will be designed to make V
→
V
r
e
f and h
→
h
r
e
f.Assumption 6.
Considersup
⁡
|
Δ
x
|
≤
Δ
x
M; Δ
x
M is positive constant. Define Δ
~
x
M
=
Δ
x
M
-
Δ
^
x
M which stand for the estimation error of Δ
x
M.Lemma 7 (see [32]).
Ifx
∈
R and ε
>
0, then 0
≤
x
-
x
tanh
⁡
x
/
ε
≤
k
ε, k
=
0.2785
.


### 3.1. Controller Design for Velocity Subsystem

Differentiating the velocity track errorV
~
=
V
-
V
r
e
f with respect to time results in(18)
V
~
˙
=
f
V
+
g
V
Φ
+
F
V
+
Δ
V
-
V
˙
r
e
f
=
f
^
V
+
g
^
V
Φ
0
+
f
V
-
f
^
V
+
g
V
-
g
^
V
Φ
+
g
^
V
Φ
-
Φ
0
+
F
V
+
Δ
V
-
V
˙
refwith(19)
f
V
-
f
^
V
+
g
V
-
g
^
V
Φ
=
θ
f
V
∗
-
θ
^
f
V
T
ξ
f
V
+
θ
g
V
∗
-
θ
^
g
V
T
ξ
g
V
Φ
+
ε
f
V
∗
+
ε
g
V
∗
Φ
.To eliminate the error betweenΦ and Φ
0, a novel auxiliary system is introduced as(20)
λ
˙
V
=
-
κ
V
λ
V
-
V
~
g
^
V
Φ
-
Φ
0
λ
V
,where κ
V
>
0 is a design parameter.The adaptive law of weight vector can be designed as(21)
θ
^
˙
f
V
=
Γ
f
V
ξ
f
V
V
~
-
β
f
V
θ
^
f
V
,
θ
^
˙
g
V
=
Γ
g
V
P
r
o
j
θ
^
g
V
,
ξ
g
V
Φ
V
~
,
β
g
V
,where Γ
f
V
=
Γ
f
V
T, Γ
g
V
=
Γ
g
V
T, and β
f
V and β
g
V are design parameters.The adaptive law ofΔ
V
M is given as follows:(22)
Δ
^
˙
V
M
=
σ
V
V
~
-
β
V
Δ
^
V
M
,where σ
V and β
V are design parameters.The desired controllerΦ
0 is introduced as(23)
Φ
0
=
g
^
V
-
1
-
k
V
,
1
V
~
-
k
V
,
2
∫
0
t
V
~
d
τ
+
k
V
,
3
λ
V
-
F
V
+
V
˙
ref
-
f
^
V
-
Δ
^
V
M
tanh
⁡
V
~
ε
V
,where k
V
,
1 and k
V
,
2 are positive constants to be designed.Plugging (23) into (18) results in(24)
V
~
˙
=
-
k
V
,
1
V
~
-
k
V
,
2
∫
0
t
V
~
d
τ
+
k
V
,
3
λ
V
+
θ
~
f
V
T
ξ
f
V
+
θ
~
g
V
T
ξ
g
V
Φ
+
ε
f
V
∗
+
ε
g
V
∗
Φ
+
g
^
V
Φ
-
Φ
0
+
Δ
V
-
Δ
^
V
M
tanh
⁡
V
~
ε
V
.The Lyapunov function can be designed as(25)
W
V
=
1
2
V
~
2
+
k
V
,
2
2
∫
0
t
V
~
d
τ
2
+
1
2
λ
V
2
+
1
2
σ
V
Δ
~
V
M
2
+
1
2
θ
~
f
V
T
Γ
f
V
-
1
θ
~
f
V
+
1
2
θ
~
g
V
T
Γ
g
V
-
1
θ
~
g
V
.DifferentiatingW
V with respect to time results in(26)
W
˙
V
=
-
k
V
,
1
V
~
2
+
k
V
,
3
V
~
λ
V
-
κ
V
λ
V
2
+
V
~
ε
f
V
∗
+
ε
g
V
∗
Φ
+
V
~
Δ
V
-
Δ
^
V
M
tanh
⁡
V
~
ε
V
+
V
~
Δ
~
V
M
-
β
V
Δ
~
V
M
Δ
^
V
M
-
β
f
V
θ
~
f
V
T
θ
^
f
V
-
β
g
V
θ
~
g
V
T
θ
^
g
V
-
υ
f
V
θ
~
f
V
T
θ
^
f
V
θ
^
f
V
T
ξ
f
V
V
~
θ
^
f
V
2
-
υ
g
V
θ
~
g
V
T
θ
^
g
V
θ
^
g
V
T
ξ
g
V
Φ
V
~
θ
^
g
V
2
.When θ
^
g
V
≤
θ
g
V
M and θ
^
g
V
T
≤
0, υ
f
V
=
υ
g
V
=
0; when θ
^
g
V
=
θ
g
V
M and θ
^
g
V
T
>
0, υ
f
V
=
υ
g
V
=
1.Invoking Lemma7, we have(27)
V
~
Δ
V
-
Δ
^
V
M
tanh
⁡
V
~
ε
V
≤
V
~
Δ
~
V
M
+
Δ
^
V
M
V
~
-
V
~
tanh
⁡
V
~
ε
V
≤
-
V
~
Δ
~
V
M
+
k
V
Δ
^
V
M
ε
V
.Substituting (27) into (26) results in(28)
W
˙
V
≤
-
μ
V
,
1
V
~
-
μ
V
,
2
λ
V
2
-
μ
V
,
3
V
~
-
μ
V
,
4
ε
f
V
∗
+
ε
g
V
∗
Φ
2
+
μ
V
,
5
ε
f
V
∗
+
ε
g
V
∗
Φ
2
+
k
V
Δ
^
V
M
ε
V
-
β
V
Δ
~
V
M
Δ
^
V
M
-
β
f
V
θ
~
f
V
T
θ
^
f
V
-
β
g
V
θ
~
g
V
T
θ
^
g
V
,where(29)
μ
V
,
1
=
k
V
,
3
2
4
κ
V
,
μ
V
,
2
=
2
κ
V
k
V
,
3
,
μ
V
,
3
=
k
V
,
1
-
μ
V
,
1
,
μ
V
,
4
=
1
2
μ
V
,
3
,
μ
V
,
5
=
μ
V
,
3
μ
V
,
4
2
.According to2
x
y
≤
c
x
2
+
(
1
/
c
)
y
2, we obtain(30)
k
V
Δ
^
V
M
ε
V
≤
β
V
Δ
^
V
M
2
2
+
k
V
ε
V
2
2
β
V
.Invoking Lemma4, it is deduced as(31)
β
V
Δ
~
V
M
Δ
^
V
M
=
β
V
2
-
Δ
~
V
M
2
-
Δ
^
V
M
2
+
Δ
V
M
2
,
(32)
β
f
V
θ
~
f
V
T
θ
^
f
V
≤
β
f
V
2
-
θ
~
f
V
2
+
θ
f
V
∗
2
,
(33)
β
g
V
θ
~
g
V
T
θ
^
g
V
≤
β
g
V
2
-
θ
~
g
V
2
+
θ
g
V
∗
2
.Combining (30)–(33), we have the expression of(34)
W
˙
V
≤
-
μ
V
,
1
V
~
-
μ
V
,
2
λ
V
2
-
μ
V
,
3
V
~
-
μ
V
,
4
ε
f
V
∗
+
ε
g
V
∗
Φ
2
-
β
V
2
Δ
~
V
M
2
-
β
f
V
2
θ
~
f
V
2
-
β
g
V
2
θ
~
g
V
2
+
μ
V
,
5
ε
f
V
∗
+
ε
g
V
∗
Φ
2
+
k
V
2
2
β
V
ε
V
2
+
β
V
2
Δ
V
M
2
+
β
f
V
2
θ
f
V
∗
2
+
β
g
V
2
θ
g
V
∗
2
.


### 3.2. Controller Design for Altitude Subsystem

Define the tracking errors ofh, γ
,
α, and Q, respectively, as (35)
h
~
=
h
-
h
r
e
f
,
(36)
γ
~
=
γ
-
γ
c
,
(37)
α
~
=
α
-
α
c
,
(38)
Q
~
=
Q
-
Q
c
.Differentiating (35)–(38) and invoking (5)–(8) yield(39)
h
~
˙
=
V
γ
+
Δ
h
-
h
˙
ref
=
V
γ
c
0
+
V
γ
~
+
V
γ
c
-
γ
c
0
+
Δ
h
-
h
˙
ref
,
(40)
γ
~
˙
=
f
γ
+
g
γ
α
+
F
γ
+
Δ
γ
-
γ
˙
c
=
f
^
γ
+
g
^
γ
α
c
0
+
f
γ
-
f
^
γ
+
g
γ
-
g
^
γ
α
+
g
^
γ
α
c
-
α
c
0
+
g
^
γ
α
~
+
F
γ
+
Δ
γ
-
γ
˙
c
,
(41)
α
~
˙
=
-
f
γ
+
g
γ
α
+
Q
+
F
α
+
Δ
α
-
α
˙
c
=
-
f
^
γ
+
g
^
γ
α
+
Q
c
0
-
f
γ
-
f
^
γ
+
g
γ
-
g
^
γ
α
+
Q
c
-
Q
c
0
+
Q
~
+
F
α
+
Δ
α
-
α
˙
c
,
(42)
Q
~
˙
=
f
Q
+
g
Q
δ
e
+
Δ
Q
-
Q
˙
c
=
f
^
Q
+
g
^
Q
δ
e
0
+
f
Q
-
f
^
Q
+
g
Q
-
g
^
Q
δ
e
+
g
^
Q
δ
e
-
δ
e
0
+
Δ
Q
-
Q
˙
c
,where(43)
f
γ
-
f
^
γ
+
g
γ
-
g
^
γ
α
=
θ
f
γ
∗
-
θ
^
f
γ
T
ξ
f
γ
+
θ
g
γ
∗
-
θ
^
g
γ
T
ξ
g
γ
α
+
ε
f
γ
∗
+
ε
g
γ
∗
α
f
Q
-
f
^
Q
+
g
Q
-
g
^
Q
δ
e
=
θ
f
Q
∗
-
θ
^
f
Q
T
ξ
f
Q
+
θ
g
Q
∗
-
θ
^
g
Q
T
ξ
g
Q
δ
e
+
ε
f
Q
∗
+
ε
g
Q
∗
δ
e
.The auxiliary systems are introduced as(44)
λ
˙
h
=
-
κ
h
λ
h
-
h
~
V
γ
c
-
γ
c
0
λ
h
,
λ
˙
γ
=
-
κ
γ
λ
γ
-
γ
~
g
^
γ
α
c
-
α
c
0
λ
γ
,
λ
˙
α
=
-
κ
α
λ
α
-
α
~
Q
c
-
Q
c
0
λ
α
,
λ
˙
Q
=
-
κ
Q
λ
Q
-
Q
~
g
^
Q
δ
e
-
δ
e
0
λ
Q
,where κ
x
x
=
h
,
γ
,
α
,
Q stand for design parameters.The adaptive laws of weight vector can be designed as(45)
θ
^
˙
f
γ
=
Γ
f
γ
ξ
f
γ
γ
~
-
α
~
-
β
f
γ
θ
^
f
γ
,
θ
^
˙
g
γ
=
Γ
g
γ
Proj
⁡
θ
^
g
γ
,
ξ
g
γ
α
γ
~
-
α
~
,
β
g
γ
,
θ
^
˙
f
Q
=
Γ
f
Q
ξ
f
Q
Q
~
-
β
f
Q
θ
^
f
Q
,
θ
^
˙
g
Q
=
Γ
g
Q
Proj
⁡
θ
^
g
Q
,
ξ
g
Q
δ
e
Q
~
,
β
g
Q
,where Γ
f
x
=
Γ
f
x
T and Γ
g
x
=
Γ
g
x
T; β
f
x and β
g
x
x
=
γ
,
Q are design parameters.The adaptive laws ofΔ
x
M (x
=
h
,
γ
,
α
,
Q) are given as follows:(46)
Δ
^
˙
h
M
=
σ
h
h
~
-
β
h
Δ
^
h
M
Δ
^
˙
γ
M
=
σ
γ
γ
~
-
β
γ
Δ
^
γ
M
Δ
^
˙
α
M
=
σ
α
α
~
-
β
α
Δ
^
α
M
Δ
^
˙
Q
M
=
σ
Q
Q
~
-
β
Q
Δ
^
Q
M
,where σ
x and β
x
x
=
h
,
γ
,
α
,
Q are design parameters.The desired control lawsγ
c
0, α
c
0, Q
c
0, and δ
e
0 are designed as(47)
γ
c
0
=
V
-
1
-
k
h
,
1
h
~
-
k
h
,
2
∫
0
t
h
~
d
τ
+
k
h
,
3
λ
h
+
h
˙
ref
-
Δ
^
h
M
tanh
⁡
h
~
ε
h
,
(48)
α
c
0
=
g
^
γ
-
1
-
k
γ
,
1
γ
~
-
k
γ
,
2
∫
0
t
γ
~
d
τ
+
k
γ
,
3
λ
γ
-
f
^
γ
-
F
γ
-
V
h
~
+
γ
˙
c
-
Δ
^
γ
M
·
tanh
⁡
γ
~
ε
γ
-
c
γ
γ
~
α
2
2
,
(49)
Q
c
0
=
-
k
α
,
1
α
~
-
k
α
,
2
∫
0
t
α
~
d
τ
+
k
α
,
3
λ
α
+
f
^
γ
+
g
^
γ
α
-
F
α
-
g
^
γ
γ
~
+
α
˙
c
-
Δ
^
α
M
tanh
⁡
α
~
ε
α
-
c
α
,
1
+
c
α
,
2
α
2
α
~
2
,
(50)
δ
e
0
=
g
^
Q
-
1
-
k
Q
,
1
Q
~
-
k
Q
,
2
∫
0
t
Q
~
d
τ
+
k
Q
,
3
λ
Q
-
f
^
Q
-
α
~
+
Q
˙
c
-
Δ
^
Q
tanh
⁡
Q
~
ε
Q
,where k
x
,
1, k
x
,
2  x
=
h
,
γ
,
α
,
Q are positive constants to be designed.Substituting (47)–(50) into (39)–(42) gives(51)
h
~
˙
=
-
k
h
,
1
h
~
-
k
h
,
2
∫
0
t
h
~
d
τ
+
k
h
,
3
λ
h
+
V
γ
~
+
V
γ
c
-
γ
c
0
+
Δ
h
-
Δ
^
h
M
tanh
⁡
h
~
ε
h
,
(52)
γ
~
˙
=
-
k
γ
,
1
γ
~
-
k
γ
,
2
∫
0
t
γ
~
d
τ
+
k
γ
,
3
λ
γ
-
V
h
~
+
g
^
γ
α
~
-
c
γ
γ
~
α
2
2
+
θ
~
f
γ
T
ξ
f
γ
+
θ
~
g
γ
T
ξ
g
γ
α
+
ε
f
γ
∗
+
ε
g
γ
∗
α
+
g
^
γ
α
c
-
α
c
0
+
Δ
γ
-
Δ
^
γ
M
tanh
⁡
γ
~
ε
γ
,
(53)
α
~
˙
=
-
k
α
,
1
α
~
-
k
α
,
2
∫
0
t
α
~
d
τ
+
k
α
,
3
λ
α
-
g
^
γ
γ
~
+
c
α
,
1
+
c
α
,
2
α
2
α
~
2
+
Q
~
-
θ
~
f
γ
T
ξ
f
γ
+
θ
~
g
γ
T
ξ
g
γ
α
-
ε
f
γ
∗
+
ε
g
γ
∗
α
+
Q
c
-
Q
c
0
+
Δ
α
-
Δ
^
α
M
tanh
⁡
α
~
ε
α
,
(54)
Q
~
˙
=
-
k
Q
,
1
Q
~
-
k
Q
,
2
∫
0
t
Q
~
d
τ
+
k
Q
,
3
λ
Q
-
α
~
+
θ
~
f
Q
T
ξ
f
Q
+
θ
~
g
Q
T
ξ
g
Q
δ
e
+
ε
f
Q
∗
+
ε
g
Q
∗
δ
e
+
g
^
Q
δ
e
-
δ
e
0
+
Δ
Q
-
Δ
^
Q
M
tanh
⁡
Q
~
ε
Q
.The Lyapunov function can be designed as(55)
W
h
=
1
2
h
~
2
+
k
h
,
2
2
∫
0
t
h
~
d
τ
2
+
1
2
λ
h
2
+
1
2
σ
h
Δ
~
h
M
2
,
W
γ
=
1
2
γ
~
2
+
k
γ
,
2
2
∫
0
t
γ
~
d
τ
2
+
1
2
λ
γ
2
+
1
2
σ
γ
Δ
~
γ
M
2
+
1
2
θ
~
f
γ
T
Γ
f
γ
-
1
θ
~
f
γ
+
1
2
θ
~
g
γ
T
Γ
g
γ
-
1
θ
~
g
γ
,
W
α
=
1
2
α
~
2
+
k
α
,
2
2
∫
0
t
α
~
d
τ
2
+
1
2
λ
α
2
+
1
2
σ
α
Δ
~
α
M
2
,
W
Q
=
1
2
Q
~
2
+
k
Q
,
2
2
∫
0
t
Q
~
d
τ
2
+
1
2
λ
Q
2
+
1
2
σ
Q
Δ
~
Q
M
2
+
1
2
θ
~
f
Q
T
Γ
f
Q
-
1
θ
~
f
Q
+
1
2
θ
~
g
Q
T
Γ
g
Q
-
1
θ
~
g
Q
.DifferentiatingW
x
x
=
h
,
γ
,
α
,
Q and invoking (49)–(52) yield(56)
W
˙
h
=
-
k
h
,
1
h
~
2
+
k
h
,
3
h
~
λ
h
-
κ
h
λ
h
2
+
V
h
~
γ
~
+
h
~
Δ
h
-
Δ
^
h
M
tanh
⁡
h
~
ε
h
-
Δ
~
h
M
Δ
^
˙
h
M
σ
h
,
W
˙
γ
=
-
k
γ
,
1
γ
~
2
+
k
γ
,
3
γ
~
λ
γ
-
κ
γ
λ
γ
2
-
V
γ
~
h
~
+
g
^
γ
γ
~
α
~
-
c
γ
γ
~
2
α
2
2
+
γ
~
θ
~
f
γ
T
ξ
f
γ
+
θ
~
g
γ
T
ξ
g
γ
α
+
γ
~
ε
f
γ
∗
+
ε
g
γ
∗
α
+
γ
~
Δ
γ
-
Δ
^
γ
M
tanh
⁡
γ
~
ε
γ
-
Δ
~
γ
M
Δ
^
˙
γ
M
σ
γ
+
θ
~
f
γ
T
Γ
f
γ
-
1
θ
^
˙
f
γ
+
θ
~
g
γ
T
Γ
g
γ
-
1
θ
^
˙
g
γ
,
W
˙
α
=
-
k
α
,
1
α
~
2
+
k
α
,
3
α
~
λ
α
-
κ
α
λ
α
2
-
g
^
γ
α
~
γ
~
+
α
~
Q
~
-
c
α
,
1
+
c
α
,
2
α
2
α
~
2
2
-
α
~
θ
~
f
γ
T
ξ
f
γ
+
θ
~
g
γ
T
ξ
g
γ
α
-
α
~
ε
f
γ
∗
+
ε
g
γ
∗
α
+
α
~
Δ
α
-
Δ
^
α
M
tanh
⁡
α
~
ε
α
-
Δ
~
α
M
Δ
^
˙
α
M
σ
α
,
W
˙
Q
=
-
k
Q
,
1
Q
~
2
+
k
Q
,
3
Q
~
λ
Q
-
κ
Q
λ
Q
2
-
Q
~
α
~
+
Q
~
θ
~
f
Q
T
ξ
f
Q
+
θ
~
g
Q
T
ξ
g
Q
δ
e
+
Q
~
ε
f
Q
∗
+
ε
g
Q
∗
δ
e
+
Q
~
Δ
Q
-
Δ
^
Q
M
tanh
⁡
Q
~
ε
Q
-
Δ
~
Q
M
Δ
^
˙
Q
M
σ
Q
+
θ
~
f
Q
T
Γ
f
Q
-
1
θ
^
˙
f
Q
+
θ
~
g
Q
T
Γ
g
Q
-
1
θ
^
˙
g
Q
.Invoking Lemma5, we can derive(57)
x
~
Δ
x
-
Δ
^
x
M
tanh
⁡
x
~
ε
x
≤
x
~
Δ
~
x
M
+
Δ
^
x
M
x
~
-
x
~
tanh
⁡
x
~
ε
x
≤
x
~
Δ
~
x
M
+
k
x
Δ
^
x
M
ε
x
,where x
∈
{
h
,
γ
,
α
,
Q
}.The following inequalities hold:(58)
γ
~
ε
g
γ
∗
α
≤
c
γ
γ
~
2
α
2
2
+
ε
g
γ
∗
2
2
c
γ
-
α
~
ε
f
γ
∗
+
ε
g
γ
∗
α
=
-
α
~
ε
f
γ
∗
-
α
~
α
ε
g
γ
∗
≤
c
α
,
1
α
~
2
2
+
ε
f
γ
∗
2
2
c
α
,
1
+
c
α
,
2
α
~
2
α
2
2
+
ε
g
γ
∗
2
2
c
α
,
2
=
α
~
2
c
α
,
1
2
+
c
α
,
2
α
2
2
+
ε
f
γ
∗
2
2
c
α
,
1
+
ε
g
γ
∗
2
2
c
α
,
2
,where c
γ
>
0, c
α
,
1
>
0, and c
α
,
2
>
0.Plugging (44)–(46) and (57)–(58) into (56) results in(59)
W
˙
h
=
-
μ
h
,
1
h
~
-
μ
h
,
2
λ
h
2
-
μ
h
,
3
h
~
2
+
V
h
~
γ
~
+
k
h
Δ
^
h
M
ε
h
-
β
h
Δ
~
h
M
Δ
^
h
M
,
W
˙
γ
+
W
˙
α
≤
-
μ
γ
,
1
γ
~
-
μ
γ
,
2
λ
γ
2
-
μ
α
,
1
α
~
-
μ
α
,
2
λ
α
2
-
μ
γ
,
3
γ
~
-
μ
γ
,
4
ε
f
γ
∗
2
-
μ
α
,
3
α
~
2
+
μ
γ
,
5
+
1
2
c
α
,
1
ε
f
γ
∗
2
+
1
2
c
γ
+
1
2
c
α
,
2
ε
g
γ
∗
2
-
V
γ
~
h
~
+
α
~
Q
~
+
k
γ
Δ
^
γ
M
ε
γ
+
k
α
Δ
^
α
M
ε
α
-
β
γ
Δ
~
γ
M
Δ
^
γ
M
-
β
α
Δ
~
α
M
Δ
^
α
M
-
β
f
γ
θ
~
f
γ
T
θ
^
f
γ
-
β
g
γ
θ
~
g
γ
T
θ
^
g
γ
-
υ
f
γ
θ
~
f
γ
T
θ
^
f
γ
θ
^
f
γ
T
ξ
f
γ
γ
~
-
α
~
θ
^
f
γ
2
-
υ
g
γ
θ
~
g
γ
T
θ
^
g
γ
θ
^
g
γ
T
ξ
g
γ
α
γ
~
-
α
~
θ
^
g
γ
2
,
W
˙
Q
≤
-
μ
Q
,
1
Q
~
-
μ
Q
,
2
λ
Q
2
-
μ
Q
,
3
Q
~
-
μ
Q
,
4
ε
f
Q
∗
+
ε
g
Q
∗
δ
e
2
+
μ
Q
,
5
ε
f
Q
∗
+
ε
g
Q
∗
δ
e
2
-
Q
~
α
~
+
Q
~
Δ
Q
-
Δ
^
Q
-
β
f
Q
θ
~
f
Q
T
θ
^
f
Q
-
β
g
Q
θ
~
g
Q
T
θ
^
g
Q
-
υ
f
Q
θ
~
f
Q
T
θ
^
f
Q
θ
^
f
Q
T
ξ
f
Q
Q
~
θ
^
f
Q
2
-
υ
g
Q
θ
~
g
Q
T
θ
^
g
Q
θ
^
g
Q
T
ξ
g
Q
δ
e
Q
~
θ
^
g
Q
2
,where μ
x
,
1
=
k
x
,
3
2
/
(
4
κ
x
), μ
x
,
2
=
2
κ
x
/
k
x
,
3, μ
x
,
3
=
k
x
,
1
-
μ
x
,
1, μ
x
,
4
=
1
/
(
2
μ
x
,
3
), and μ
x
,
5
=
μ
x
,
3
μ
x
,
4
2, x
∈
{
h
,
γ
,
α
,
Q
}.Whenθ
^
g
x
≤
θ
g
x
M and θ
^
g
x
T
≤
0, υ
f
x
=
υ
g
x
=
0
x
=
h
,
γ
,
α
,
Q; when θ
^
g
x
=
θ
g
x
M and θ
^
g
x
T
>
0, υ
f
x
=
υ
g
x
=
1
x
=
h
,
γ
,
α
,
Q.Invoking Lemma4, we can derive(60)
k
x
Δ
^
x
M
ε
x
≤
β
x
Δ
^
x
M
2
2
+
k
x
ε
x
2
2
β
x
,
(61)
β
x
Δ
~
x
M
Δ
^
x
M
=
β
x
2
-
Δ
~
x
M
2
-
Δ
^
x
M
2
+
Δ
x
M
2
,
(62)
β
f
x
θ
~
f
x
T
θ
^
f
x
≤
β
f
x
2
-
θ
~
f
x
2
+
θ
f
x
∗
2
,
(63)
β
g
x
θ
~
g
x
T
θ
^
g
x
≤
β
g
x
2
-
θ
~
g
x
2
+
θ
g
x
∗
2
.Combining (60)–(63), we have the expression of(64)
W
˙
h
=
-
μ
h
,
1
h
~
-
μ
h
,
2
λ
h
2
-
μ
h
,
3
h
~
2
-
β
V
2
Δ
~
h
M
2
+
V
h
~
γ
~
+
k
h
2
2
β
h
ε
h
2
+
β
h
2
Δ
h
M
2
,
(65)
W
˙
γ
+
W
˙
α
≤
-
μ
γ
,
1
γ
~
-
μ
γ
,
2
λ
γ
2
-
μ
α
,
1
α
~
-
μ
α
,
2
λ
α
2
-
μ
γ
,
3
γ
~
-
μ
γ
,
4
ε
f
γ
∗
2
-
μ
α
,
3
α
~
2
-
β
γ
2
Δ
~
γ
M
2
-
β
α
2
Δ
~
α
M
2
-
β
f
γ
2
θ
~
f
γ
2
-
β
g
γ
2
θ
~
g
γ
2
-
V
γ
~
h
~
+
α
~
Q
~
+
μ
γ
,
5
+
1
2
c
α
,
1
ε
f
γ
∗
2
+
1
2
c
γ
+
1
2
c
α
,
2
ε
g
γ
∗
2
+
k
γ
2
2
β
γ
ε
γ
2
+
k
α
2
2
β
α
ε
α
2
+
β
γ
2
Δ
γ
M
2
+
β
α
2
Δ
α
M
2
+
β
f
γ
2
θ
f
γ
∗
2
+
β
g
γ
2
θ
g
γ
∗
2
(66)
W
˙
Q
≤
-
μ
Q
,
1
Q
~
-
μ
Q
,
2
λ
Q
2
-
μ
Q
,
3
Q
~
-
μ
Q
,
4
ε
f
Q
∗
+
ε
g
Q
∗
δ
e
2
-
β
Q
2
Δ
~
Q
M
2
-
β
f
Q
2
θ
~
f
Q
2
-
β
g
Q
2
θ
~
g
Q
2
+
μ
Q
,
5
ε
f
Q
∗
+
ε
g
Q
∗
δ
e
2
+
k
Q
2
2
β
Q
ε
Q
2
+
β
Q
2
Δ
Q
M
2
+
β
f
Q
2
θ
f
γ
∗
2
+
β
g
Q
2
θ
g
γ
∗
2
.


### 3.3. Stability Analysis

The Lyapunov function can be designed as(67)
W
=
W
V
+
W
h
+
W
γ
+
W
α
+
W
Q
.Combining (34) and (60)–(63) and differentiating W with respect to time result in (68)
W
˙
≤
-
μ
V
,
3
V
~
-
μ
V
,
4
ε
f
V
∗
+
ε
g
V
∗
Φ
2
-
μ
h
,
3
h
~
2
-
μ
γ
,
3
γ
~
-
μ
γ
,
4
ε
f
γ
∗
2
-
μ
α
,
3
α
~
2
-
μ
Q
,
3
Q
~
-
μ
Q
,
4
ε
f
Q
∗
+
ε
g
Q
∗
δ
e
2
-
∑
x
μ
x
,
1
x
~
-
μ
x
,
2
λ
x
2
-
1
2
∑
x
β
x
Δ
~
x
M
2
-
1
2
∑
x
β
f
x
θ
~
f
x
2
-
1
2
∑
x
β
g
x
θ
~
g
x
2
+
μ
V
,
5
ε
f
V
∗
+
ε
g
V
∗
Φ
2
+
μ
γ
,
5
+
1
2
a
α
,
1
ε
f
γ
∗
2
+
1
2
c
γ
+
1
2
c
α
,
2
·
ε
g
γ
∗
2
+
μ
Q
,
5
ε
f
Q
∗
+
ε
g
Q
∗
δ
e
2
+
1
2
∑
x
β
f
x
θ
f
x
∗
2
+
1
2
∑
x
β
g
x
θ
g
x
∗
2
+
1
2
∑
x
k
x
ε
x
2
β
x
+
1
2
∑
x
β
x
Δ
x
M
2
,where x
∈
{
V
,
h
,
γ
,
α
,
Q
}.Let(69)
Σ
=
μ
V
,
5
ε
f
V
M
+
ε
g
V
M
Φ
M
2
+
μ
γ
,
5
+
1
2
c
α
,
1
·
ε
f
γ
M
2
+
1
2
c
γ
+
1
2
c
α
,
2
ε
g
γ
M
2
+
μ
Q
,
5
ε
f
Q
M
+
ε
g
Q
M
δ
e
M
2
+
1
2
∑
x
k
x
ε
x
2
β
x
+
β
x
Δ
x
M
2
+
β
f
x
θ
f
x
M
2
+
β
g
x
θ
g
x
M
2
.Define(70)
Ω
V
~
=
V
~
∣
V
~
≤
μ
V
,
4
ε
f
V
M
+
ε
g
V
M
Φ
M
+
Σ
μ
V
,
3
,
Ω
h
~
=
h
~
∣
h
~
≤
Σ
μ
h
,
3
,
Ω
γ
~
=
γ
~
∣
γ
~
≤
μ
γ
,
4
ε
f
γ
M
+
Σ
μ
γ
,
3
,
Ω
α
~
=
α
~
∣
α
~
≤
Σ
μ
α
,
3
,
Ω
Q
~
=
Q
~
∣
Q
~
≤
μ
Q
,
4
ε
f
Q
M
+
ε
g
Q
M
δ
e
M
+
Σ
μ
Q
,
3
,
Ω
Δ
~
x
M
=
Δ
~
x
M
∣
Δ
~
x
M
≤
Σ
β
x
,
Ω
θ
~
f
x
=
θ
~
f
x
∣
θ
~
f
x
≤
Σ
β
f
x
,
Ω
θ
~
g
x
=
θ
~
g
x
∣
θ
~
g
x
≤
Σ
β
g
x
, where Φ
M
=
max
⁡
{
|
Φ
L
|
,
|
Φ
U
|
}; δ
e
L
=
max
⁡
{
|
δ
e
L
|
,
|
δ
e
U
|
}.Let(71)
k
x
,
1
>
k
x
,
3
2
4
κ
x
,
k
x
,
2
>
0
,
k
x
,
3
>
0
,
κ
x
>
0
,
Γ
f
x
=
Γ
f
x
T
>
0
,
Γ
g
x
=
Γ
g
x
T
>
0
,
β
f
x
>
0
,
β
g
x
>
0
,
σ
x
>
0
,
β
x
>
0
;If x
~, θ
~
f
x, θ
~
g
x and Δ
~
x
M, x
~
∈
{
V
~
,
h
~
,
γ
~
,
α
~
,
Q
~
}, do not belong to the compact set Ω
x
~, Ω
θ
~
f
x, Ω
θ
~
g
x and Ω
Δ
~
x
M, respectively, W
˙
≤
0. x
~, θ
~
f
x, θ
~
g
x, and Δ
~
x
M are bounded. Further, λ
x also is bounded according to (66). By regulating the designed parameters, the radius of Ω
x
~, Ω
W
~
f
x, Ω
W
~
g
x, and Ω
Δ
~
x
M can be made arbitrarily small.Theorem 8.
Consider the system of (1) under Assumptions 3 and 6, auxiliary analytical system (20) and (44), controllers (23) and (47)–(50), adaptive laws (21) and (45), and upper bound estimation of system error (22) and (46). If (71) is satisfied, all the signals involved are bounded.


## 3.1. Controller Design for Velocity Subsystem

Differentiating the velocity track errorV
~
=
V
-
V
r
e
f with respect to time results in(18)
V
~
˙
=
f
V
+
g
V
Φ
+
F
V
+
Δ
V
-
V
˙
r
e
f
=
f
^
V
+
g
^
V
Φ
0
+
f
V
-
f
^
V
+
g
V
-
g
^
V
Φ
+
g
^
V
Φ
-
Φ
0
+
F
V
+
Δ
V
-
V
˙
refwith(19)
f
V
-
f
^
V
+
g
V
-
g
^
V
Φ
=
θ
f
V
∗
-
θ
^
f
V
T
ξ
f
V
+
θ
g
V
∗
-
θ
^
g
V
T
ξ
g
V
Φ
+
ε
f
V
∗
+
ε
g
V
∗
Φ
.To eliminate the error betweenΦ and Φ
0, a novel auxiliary system is introduced as(20)
λ
˙
V
=
-
κ
V
λ
V
-
V
~
g
^
V
Φ
-
Φ
0
λ
V
,where κ
V
>
0 is a design parameter.The adaptive law of weight vector can be designed as(21)
θ
^
˙
f
V
=
Γ
f
V
ξ
f
V
V
~
-
β
f
V
θ
^
f
V
,
θ
^
˙
g
V
=
Γ
g
V
P
r
o
j
θ
^
g
V
,
ξ
g
V
Φ
V
~
,
β
g
V
,where Γ
f
V
=
Γ
f
V
T, Γ
g
V
=
Γ
g
V
T, and β
f
V and β
g
V are design parameters.The adaptive law ofΔ
V
M is given as follows:(22)
Δ
^
˙
V
M
=
σ
V
V
~
-
β
V
Δ
^
V
M
,where σ
V and β
V are design parameters.The desired controllerΦ
0 is introduced as(23)
Φ
0
=
g
^
V
-
1
-
k
V
,
1
V
~
-
k
V
,
2
∫
0
t
V
~
d
τ
+
k
V
,
3
λ
V
-
F
V
+
V
˙
ref
-
f
^
V
-
Δ
^
V
M
tanh
⁡
V
~
ε
V
,where k
V
,
1 and k
V
,
2 are positive constants to be designed.Plugging (23) into (18) results in(24)
V
~
˙
=
-
k
V
,
1
V
~
-
k
V
,
2
∫
0
t
V
~
d
τ
+
k
V
,
3
λ
V
+
θ
~
f
V
T
ξ
f
V
+
θ
~
g
V
T
ξ
g
V
Φ
+
ε
f
V
∗
+
ε
g
V
∗
Φ
+
g
^
V
Φ
-
Φ
0
+
Δ
V
-
Δ
^
V
M
tanh
⁡
V
~
ε
V
.The Lyapunov function can be designed as(25)
W
V
=
1
2
V
~
2
+
k
V
,
2
2
∫
0
t
V
~
d
τ
2
+
1
2
λ
V
2
+
1
2
σ
V
Δ
~
V
M
2
+
1
2
θ
~
f
V
T
Γ
f
V
-
1
θ
~
f
V
+
1
2
θ
~
g
V
T
Γ
g
V
-
1
θ
~
g
V
.DifferentiatingW
V with respect to time results in(26)
W
˙
V
=
-
k
V
,
1
V
~
2
+
k
V
,
3
V
~
λ
V
-
κ
V
λ
V
2
+
V
~
ε
f
V
∗
+
ε
g
V
∗
Φ
+
V
~
Δ
V
-
Δ
^
V
M
tanh
⁡
V
~
ε
V
+
V
~
Δ
~
V
M
-
β
V
Δ
~
V
M
Δ
^
V
M
-
β
f
V
θ
~
f
V
T
θ
^
f
V
-
β
g
V
θ
~
g
V
T
θ
^
g
V
-
υ
f
V
θ
~
f
V
T
θ
^
f
V
θ
^
f
V
T
ξ
f
V
V
~
θ
^
f
V
2
-
υ
g
V
θ
~
g
V
T
θ
^
g
V
θ
^
g
V
T
ξ
g
V
Φ
V
~
θ
^
g
V
2
.When θ
^
g
V
≤
θ
g
V
M and θ
^
g
V
T
≤
0, υ
f
V
=
υ
g
V
=
0; when θ
^
g
V
=
θ
g
V
M and θ
^
g
V
T
>
0, υ
f
V
=
υ
g
V
=
1.Invoking Lemma7, we have(27)
V
~
Δ
V
-
Δ
^
V
M
tanh
⁡
V
~
ε
V
≤
V
~
Δ
~
V
M
+
Δ
^
V
M
V
~
-
V
~
tanh
⁡
V
~
ε
V
≤
-
V
~
Δ
~
V
M
+
k
V
Δ
^
V
M
ε
V
.Substituting (27) into (26) results in(28)
W
˙
V
≤
-
μ
V
,
1
V
~
-
μ
V
,
2
λ
V
2
-
μ
V
,
3
V
~
-
μ
V
,
4
ε
f
V
∗
+
ε
g
V
∗
Φ
2
+
μ
V
,
5
ε
f
V
∗
+
ε
g
V
∗
Φ
2
+
k
V
Δ
^
V
M
ε
V
-
β
V
Δ
~
V
M
Δ
^
V
M
-
β
f
V
θ
~
f
V
T
θ
^
f
V
-
β
g
V
θ
~
g
V
T
θ
^
g
V
,where(29)
μ
V
,
1
=
k
V
,
3
2
4
κ
V
,
μ
V
,
2
=
2
κ
V
k
V
,
3
,
μ
V
,
3
=
k
V
,
1
-
μ
V
,
1
,
μ
V
,
4
=
1
2
μ
V
,
3
,
μ
V
,
5
=
μ
V
,
3
μ
V
,
4
2
.According to2
x
y
≤
c
x
2
+
(
1
/
c
)
y
2, we obtain(30)
k
V
Δ
^
V
M
ε
V
≤
β
V
Δ
^
V
M
2
2
+
k
V
ε
V
2
2
β
V
.Invoking Lemma4, it is deduced as(31)
β
V
Δ
~
V
M
Δ
^
V
M
=
β
V
2
-
Δ
~
V
M
2
-
Δ
^
V
M
2
+
Δ
V
M
2
,
(32)
β
f
V
θ
~
f
V
T
θ
^
f
V
≤
β
f
V
2
-
θ
~
f
V
2
+
θ
f
V
∗
2
,
(33)
β
g
V
θ
~
g
V
T
θ
^
g
V
≤
β
g
V
2
-
θ
~
g
V
2
+
θ
g
V
∗
2
.Combining (30)–(33), we have the expression of(34)
W
˙
V
≤
-
μ
V
,
1
V
~
-
μ
V
,
2
λ
V
2
-
μ
V
,
3
V
~
-
μ
V
,
4
ε
f
V
∗
+
ε
g
V
∗
Φ
2
-
β
V
2
Δ
~
V
M
2
-
β
f
V
2
θ
~
f
V
2
-
β
g
V
2
θ
~
g
V
2
+
μ
V
,
5
ε
f
V
∗
+
ε
g
V
∗
Φ
2
+
k
V
2
2
β
V
ε
V
2
+
β
V
2
Δ
V
M
2
+
β
f
V
2
θ
f
V
∗
2
+
β
g
V
2
θ
g
V
∗
2
.


## 3.2. Controller Design for Altitude Subsystem

Define the tracking errors ofh, γ
,
α, and Q, respectively, as (35)
h
~
=
h
-
h
r
e
f
,
(36)
γ
~
=
γ
-
γ
c
,
(37)
α
~
=
α
-
α
c
,
(38)
Q
~
=
Q
-
Q
c
.Differentiating (35)–(38) and invoking (5)–(8) yield(39)
h
~
˙
=
V
γ
+
Δ
h
-
h
˙
ref
=
V
γ
c
0
+
V
γ
~
+
V
γ
c
-
γ
c
0
+
Δ
h
-
h
˙
ref
,
(40)
γ
~
˙
=
f
γ
+
g
γ
α
+
F
γ
+
Δ
γ
-
γ
˙
c
=
f
^
γ
+
g
^
γ
α
c
0
+
f
γ
-
f
^
γ
+
g
γ
-
g
^
γ
α
+
g
^
γ
α
c
-
α
c
0
+
g
^
γ
α
~
+
F
γ
+
Δ
γ
-
γ
˙
c
,
(41)
α
~
˙
=
-
f
γ
+
g
γ
α
+
Q
+
F
α
+
Δ
α
-
α
˙
c
=
-
f
^
γ
+
g
^
γ
α
+
Q
c
0
-
f
γ
-
f
^
γ
+
g
γ
-
g
^
γ
α
+
Q
c
-
Q
c
0
+
Q
~
+
F
α
+
Δ
α
-
α
˙
c
,
(42)
Q
~
˙
=
f
Q
+
g
Q
δ
e
+
Δ
Q
-
Q
˙
c
=
f
^
Q
+
g
^
Q
δ
e
0
+
f
Q
-
f
^
Q
+
g
Q
-
g
^
Q
δ
e
+
g
^
Q
δ
e
-
δ
e
0
+
Δ
Q
-
Q
˙
c
,where(43)
f
γ
-
f
^
γ
+
g
γ
-
g
^
γ
α
=
θ
f
γ
∗
-
θ
^
f
γ
T
ξ
f
γ
+
θ
g
γ
∗
-
θ
^
g
γ
T
ξ
g
γ
α
+
ε
f
γ
∗
+
ε
g
γ
∗
α
f
Q
-
f
^
Q
+
g
Q
-
g
^
Q
δ
e
=
θ
f
Q
∗
-
θ
^
f
Q
T
ξ
f
Q
+
θ
g
Q
∗
-
θ
^
g
Q
T
ξ
g
Q
δ
e
+
ε
f
Q
∗
+
ε
g
Q
∗
δ
e
.The auxiliary systems are introduced as(44)
λ
˙
h
=
-
κ
h
λ
h
-
h
~
V
γ
c
-
γ
c
0
λ
h
,
λ
˙
γ
=
-
κ
γ
λ
γ
-
γ
~
g
^
γ
α
c
-
α
c
0
λ
γ
,
λ
˙
α
=
-
κ
α
λ
α
-
α
~
Q
c
-
Q
c
0
λ
α
,
λ
˙
Q
=
-
κ
Q
λ
Q
-
Q
~
g
^
Q
δ
e
-
δ
e
0
λ
Q
,where κ
x
x
=
h
,
γ
,
α
,
Q stand for design parameters.The adaptive laws of weight vector can be designed as(45)
θ
^
˙
f
γ
=
Γ
f
γ
ξ
f
γ
γ
~
-
α
~
-
β
f
γ
θ
^
f
γ
,
θ
^
˙
g
γ
=
Γ
g
γ
Proj
⁡
θ
^
g
γ
,
ξ
g
γ
α
γ
~
-
α
~
,
β
g
γ
,
θ
^
˙
f
Q
=
Γ
f
Q
ξ
f
Q
Q
~
-
β
f
Q
θ
^
f
Q
,
θ
^
˙
g
Q
=
Γ
g
Q
Proj
⁡
θ
^
g
Q
,
ξ
g
Q
δ
e
Q
~
,
β
g
Q
,where Γ
f
x
=
Γ
f
x
T and Γ
g
x
=
Γ
g
x
T; β
f
x and β
g
x
x
=
γ
,
Q are design parameters.The adaptive laws ofΔ
x
M (x
=
h
,
γ
,
α
,
Q) are given as follows:(46)
Δ
^
˙
h
M
=
σ
h
h
~
-
β
h
Δ
^
h
M
Δ
^
˙
γ
M
=
σ
γ
γ
~
-
β
γ
Δ
^
γ
M
Δ
^
˙
α
M
=
σ
α
α
~
-
β
α
Δ
^
α
M
Δ
^
˙
Q
M
=
σ
Q
Q
~
-
β
Q
Δ
^
Q
M
,where σ
x and β
x
x
=
h
,
γ
,
α
,
Q are design parameters.The desired control lawsγ
c
0, α
c
0, Q
c
0, and δ
e
0 are designed as(47)
γ
c
0
=
V
-
1
-
k
h
,
1
h
~
-
k
h
,
2
∫
0
t
h
~
d
τ
+
k
h
,
3
λ
h
+
h
˙
ref
-
Δ
^
h
M
tanh
⁡
h
~
ε
h
,
(48)
α
c
0
=
g
^
γ
-
1
-
k
γ
,
1
γ
~
-
k
γ
,
2
∫
0
t
γ
~
d
τ
+
k
γ
,
3
λ
γ
-
f
^
γ
-
F
γ
-
V
h
~
+
γ
˙
c
-
Δ
^
γ
M
·
tanh
⁡
γ
~
ε
γ
-
c
γ
γ
~
α
2
2
,
(49)
Q
c
0
=
-
k
α
,
1
α
~
-
k
α
,
2
∫
0
t
α
~
d
τ
+
k
α
,
3
λ
α
+
f
^
γ
+
g
^
γ
α
-
F
α
-
g
^
γ
γ
~
+
α
˙
c
-
Δ
^
α
M
tanh
⁡
α
~
ε
α
-
c
α
,
1
+
c
α
,
2
α
2
α
~
2
,
(50)
δ
e
0
=
g
^
Q
-
1
-
k
Q
,
1
Q
~
-
k
Q
,
2
∫
0
t
Q
~
d
τ
+
k
Q
,
3
λ
Q
-
f
^
Q
-
α
~
+
Q
˙
c
-
Δ
^
Q
tanh
⁡
Q
~
ε
Q
,where k
x
,
1, k
x
,
2  x
=
h
,
γ
,
α
,
Q are positive constants to be designed.Substituting (47)–(50) into (39)–(42) gives(51)
h
~
˙
=
-
k
h
,
1
h
~
-
k
h
,
2
∫
0
t
h
~
d
τ
+
k
h
,
3
λ
h
+
V
γ
~
+
V
γ
c
-
γ
c
0
+
Δ
h
-
Δ
^
h
M
tanh
⁡
h
~
ε
h
,
(52)
γ
~
˙
=
-
k
γ
,
1
γ
~
-
k
γ
,
2
∫
0
t
γ
~
d
τ
+
k
γ
,
3
λ
γ
-
V
h
~
+
g
^
γ
α
~
-
c
γ
γ
~
α
2
2
+
θ
~
f
γ
T
ξ
f
γ
+
θ
~
g
γ
T
ξ
g
γ
α
+
ε
f
γ
∗
+
ε
g
γ
∗
α
+
g
^
γ
α
c
-
α
c
0
+
Δ
γ
-
Δ
^
γ
M
tanh
⁡
γ
~
ε
γ
,
(53)
α
~
˙
=
-
k
α
,
1
α
~
-
k
α
,
2
∫
0
t
α
~
d
τ
+
k
α
,
3
λ
α
-
g
^
γ
γ
~
+
c
α
,
1
+
c
α
,
2
α
2
α
~
2
+
Q
~
-
θ
~
f
γ
T
ξ
f
γ
+
θ
~
g
γ
T
ξ
g
γ
α
-
ε
f
γ
∗
+
ε
g
γ
∗
α
+
Q
c
-
Q
c
0
+
Δ
α
-
Δ
^
α
M
tanh
⁡
α
~
ε
α
,
(54)
Q
~
˙
=
-
k
Q
,
1
Q
~
-
k
Q
,
2
∫
0
t
Q
~
d
τ
+
k
Q
,
3
λ
Q
-
α
~
+
θ
~
f
Q
T
ξ
f
Q
+
θ
~
g
Q
T
ξ
g
Q
δ
e
+
ε
f
Q
∗
+
ε
g
Q
∗
δ
e
+
g
^
Q
δ
e
-
δ
e
0
+
Δ
Q
-
Δ
^
Q
M
tanh
⁡
Q
~
ε
Q
.The Lyapunov function can be designed as(55)
W
h
=
1
2
h
~
2
+
k
h
,
2
2
∫
0
t
h
~
d
τ
2
+
1
2
λ
h
2
+
1
2
σ
h
Δ
~
h
M
2
,
W
γ
=
1
2
γ
~
2
+
k
γ
,
2
2
∫
0
t
γ
~
d
τ
2
+
1
2
λ
γ
2
+
1
2
σ
γ
Δ
~
γ
M
2
+
1
2
θ
~
f
γ
T
Γ
f
γ
-
1
θ
~
f
γ
+
1
2
θ
~
g
γ
T
Γ
g
γ
-
1
θ
~
g
γ
,
W
α
=
1
2
α
~
2
+
k
α
,
2
2
∫
0
t
α
~
d
τ
2
+
1
2
λ
α
2
+
1
2
σ
α
Δ
~
α
M
2
,
W
Q
=
1
2
Q
~
2
+
k
Q
,
2
2
∫
0
t
Q
~
d
τ
2
+
1
2
λ
Q
2
+
1
2
σ
Q
Δ
~
Q
M
2
+
1
2
θ
~
f
Q
T
Γ
f
Q
-
1
θ
~
f
Q
+
1
2
θ
~
g
Q
T
Γ
g
Q
-
1
θ
~
g
Q
.DifferentiatingW
x
x
=
h
,
γ
,
α
,
Q and invoking (49)–(52) yield(56)
W
˙
h
=
-
k
h
,
1
h
~
2
+
k
h
,
3
h
~
λ
h
-
κ
h
λ
h
2
+
V
h
~
γ
~
+
h
~
Δ
h
-
Δ
^
h
M
tanh
⁡
h
~
ε
h
-
Δ
~
h
M
Δ
^
˙
h
M
σ
h
,
W
˙
γ
=
-
k
γ
,
1
γ
~
2
+
k
γ
,
3
γ
~
λ
γ
-
κ
γ
λ
γ
2
-
V
γ
~
h
~
+
g
^
γ
γ
~
α
~
-
c
γ
γ
~
2
α
2
2
+
γ
~
θ
~
f
γ
T
ξ
f
γ
+
θ
~
g
γ
T
ξ
g
γ
α
+
γ
~
ε
f
γ
∗
+
ε
g
γ
∗
α
+
γ
~
Δ
γ
-
Δ
^
γ
M
tanh
⁡
γ
~
ε
γ
-
Δ
~
γ
M
Δ
^
˙
γ
M
σ
γ
+
θ
~
f
γ
T
Γ
f
γ
-
1
θ
^
˙
f
γ
+
θ
~
g
γ
T
Γ
g
γ
-
1
θ
^
˙
g
γ
,
W
˙
α
=
-
k
α
,
1
α
~
2
+
k
α
,
3
α
~
λ
α
-
κ
α
λ
α
2
-
g
^
γ
α
~
γ
~
+
α
~
Q
~
-
c
α
,
1
+
c
α
,
2
α
2
α
~
2
2
-
α
~
θ
~
f
γ
T
ξ
f
γ
+
θ
~
g
γ
T
ξ
g
γ
α
-
α
~
ε
f
γ
∗
+
ε
g
γ
∗
α
+
α
~
Δ
α
-
Δ
^
α
M
tanh
⁡
α
~
ε
α
-
Δ
~
α
M
Δ
^
˙
α
M
σ
α
,
W
˙
Q
=
-
k
Q
,
1
Q
~
2
+
k
Q
,
3
Q
~
λ
Q
-
κ
Q
λ
Q
2
-
Q
~
α
~
+
Q
~
θ
~
f
Q
T
ξ
f
Q
+
θ
~
g
Q
T
ξ
g
Q
δ
e
+
Q
~
ε
f
Q
∗
+
ε
g
Q
∗
δ
e
+
Q
~
Δ
Q
-
Δ
^
Q
M
tanh
⁡
Q
~
ε
Q
-
Δ
~
Q
M
Δ
^
˙
Q
M
σ
Q
+
θ
~
f
Q
T
Γ
f
Q
-
1
θ
^
˙
f
Q
+
θ
~
g
Q
T
Γ
g
Q
-
1
θ
^
˙
g
Q
.Invoking Lemma5, we can derive(57)
x
~
Δ
x
-
Δ
^
x
M
tanh
⁡
x
~
ε
x
≤
x
~
Δ
~
x
M
+
Δ
^
x
M
x
~
-
x
~
tanh
⁡
x
~
ε
x
≤
x
~
Δ
~
x
M
+
k
x
Δ
^
x
M
ε
x
,where x
∈
{
h
,
γ
,
α
,
Q
}.The following inequalities hold:(58)
γ
~
ε
g
γ
∗
α
≤
c
γ
γ
~
2
α
2
2
+
ε
g
γ
∗
2
2
c
γ
-
α
~
ε
f
γ
∗
+
ε
g
γ
∗
α
=
-
α
~
ε
f
γ
∗
-
α
~
α
ε
g
γ
∗
≤
c
α
,
1
α
~
2
2
+
ε
f
γ
∗
2
2
c
α
,
1
+
c
α
,
2
α
~
2
α
2
2
+
ε
g
γ
∗
2
2
c
α
,
2
=
α
~
2
c
α
,
1
2
+
c
α
,
2
α
2
2
+
ε
f
γ
∗
2
2
c
α
,
1
+
ε
g
γ
∗
2
2
c
α
,
2
,where c
γ
>
0, c
α
,
1
>
0, and c
α
,
2
>
0.Plugging (44)–(46) and (57)–(58) into (56) results in(59)
W
˙
h
=
-
μ
h
,
1
h
~
-
μ
h
,
2
λ
h
2
-
μ
h
,
3
h
~
2
+
V
h
~
γ
~
+
k
h
Δ
^
h
M
ε
h
-
β
h
Δ
~
h
M
Δ
^
h
M
,
W
˙
γ
+
W
˙
α
≤
-
μ
γ
,
1
γ
~
-
μ
γ
,
2
λ
γ
2
-
μ
α
,
1
α
~
-
μ
α
,
2
λ
α
2
-
μ
γ
,
3
γ
~
-
μ
γ
,
4
ε
f
γ
∗
2
-
μ
α
,
3
α
~
2
+
μ
γ
,
5
+
1
2
c
α
,
1
ε
f
γ
∗
2
+
1
2
c
γ
+
1
2
c
α
,
2
ε
g
γ
∗
2
-
V
γ
~
h
~
+
α
~
Q
~
+
k
γ
Δ
^
γ
M
ε
γ
+
k
α
Δ
^
α
M
ε
α
-
β
γ
Δ
~
γ
M
Δ
^
γ
M
-
β
α
Δ
~
α
M
Δ
^
α
M
-
β
f
γ
θ
~
f
γ
T
θ
^
f
γ
-
β
g
γ
θ
~
g
γ
T
θ
^
g
γ
-
υ
f
γ
θ
~
f
γ
T
θ
^
f
γ
θ
^
f
γ
T
ξ
f
γ
γ
~
-
α
~
θ
^
f
γ
2
-
υ
g
γ
θ
~
g
γ
T
θ
^
g
γ
θ
^
g
γ
T
ξ
g
γ
α
γ
~
-
α
~
θ
^
g
γ
2
,
W
˙
Q
≤
-
μ
Q
,
1
Q
~
-
μ
Q
,
2
λ
Q
2
-
μ
Q
,
3
Q
~
-
μ
Q
,
4
ε
f
Q
∗
+
ε
g
Q
∗
δ
e
2
+
μ
Q
,
5
ε
f
Q
∗
+
ε
g
Q
∗
δ
e
2
-
Q
~
α
~
+
Q
~
Δ
Q
-
Δ
^
Q
-
β
f
Q
θ
~
f
Q
T
θ
^
f
Q
-
β
g
Q
θ
~
g
Q
T
θ
^
g
Q
-
υ
f
Q
θ
~
f
Q
T
θ
^
f
Q
θ
^
f
Q
T
ξ
f
Q
Q
~
θ
^
f
Q
2
-
υ
g
Q
θ
~
g
Q
T
θ
^
g
Q
θ
^
g
Q
T
ξ
g
Q
δ
e
Q
~
θ
^
g
Q
2
,where μ
x
,
1
=
k
x
,
3
2
/
(
4
κ
x
), μ
x
,
2
=
2
κ
x
/
k
x
,
3, μ
x
,
3
=
k
x
,
1
-
μ
x
,
1, μ
x
,
4
=
1
/
(
2
μ
x
,
3
), and μ
x
,
5
=
μ
x
,
3
μ
x
,
4
2, x
∈
{
h
,
γ
,
α
,
Q
}.Whenθ
^
g
x
≤
θ
g
x
M and θ
^
g
x
T
≤
0, υ
f
x
=
υ
g
x
=
0
x
=
h
,
γ
,
α
,
Q; when θ
^
g
x
=
θ
g
x
M and θ
^
g
x
T
>
0, υ
f
x
=
υ
g
x
=
1
x
=
h
,
γ
,
α
,
Q.Invoking Lemma4, we can derive(60)
k
x
Δ
^
x
M
ε
x
≤
β
x
Δ
^
x
M
2
2
+
k
x
ε
x
2
2
β
x
,
(61)
β
x
Δ
~
x
M
Δ
^
x
M
=
β
x
2
-
Δ
~
x
M
2
-
Δ
^
x
M
2
+
Δ
x
M
2
,
(62)
β
f
x
θ
~
f
x
T
θ
^
f
x
≤
β
f
x
2
-
θ
~
f
x
2
+
θ
f
x
∗
2
,
(63)
β
g
x
θ
~
g
x
T
θ
^
g
x
≤
β
g
x
2
-
θ
~
g
x
2
+
θ
g
x
∗
2
.Combining (60)–(63), we have the expression of(64)
W
˙
h
=
-
μ
h
,
1
h
~
-
μ
h
,
2
λ
h
2
-
μ
h
,
3
h
~
2
-
β
V
2
Δ
~
h
M
2
+
V
h
~
γ
~
+
k
h
2
2
β
h
ε
h
2
+
β
h
2
Δ
h
M
2
,
(65)
W
˙
γ
+
W
˙
α
≤
-
μ
γ
,
1
γ
~
-
μ
γ
,
2
λ
γ
2
-
μ
α
,
1
α
~
-
μ
α
,
2
λ
α
2
-
μ
γ
,
3
γ
~
-
μ
γ
,
4
ε
f
γ
∗
2
-
μ
α
,
3
α
~
2
-
β
γ
2
Δ
~
γ
M
2
-
β
α
2
Δ
~
α
M
2
-
β
f
γ
2
θ
~
f
γ
2
-
β
g
γ
2
θ
~
g
γ
2
-
V
γ
~
h
~
+
α
~
Q
~
+
μ
γ
,
5
+
1
2
c
α
,
1
ε
f
γ
∗
2
+
1
2
c
γ
+
1
2
c
α
,
2
ε
g
γ
∗
2
+
k
γ
2
2
β
γ
ε
γ
2
+
k
α
2
2
β
α
ε
α
2
+
β
γ
2
Δ
γ
M
2
+
β
α
2
Δ
α
M
2
+
β
f
γ
2
θ
f
γ
∗
2
+
β
g
γ
2
θ
g
γ
∗
2
(66)
W
˙
Q
≤
-
μ
Q
,
1
Q
~
-
μ
Q
,
2
λ
Q
2
-
μ
Q
,
3
Q
~
-
μ
Q
,
4
ε
f
Q
∗
+
ε
g
Q
∗
δ
e
2
-
β
Q
2
Δ
~
Q
M
2
-
β
f
Q
2
θ
~
f
Q
2
-
β
g
Q
2
θ
~
g
Q
2
+
μ
Q
,
5
ε
f
Q
∗
+
ε
g
Q
∗
δ
e
2
+
k
Q
2
2
β
Q
ε
Q
2
+
β
Q
2
Δ
Q
M
2
+
β
f
Q
2
θ
f
γ
∗
2
+
β
g
Q
2
θ
g
γ
∗
2
.


## 3.3. Stability Analysis

The Lyapunov function can be designed as(67)
W
=
W
V
+
W
h
+
W
γ
+
W
α
+
W
Q
.Combining (34) and (60)–(63) and differentiating W with respect to time result in (68)
W
˙
≤
-
μ
V
,
3
V
~
-
μ
V
,
4
ε
f
V
∗
+
ε
g
V
∗
Φ
2
-
μ
h
,
3
h
~
2
-
μ
γ
,
3
γ
~
-
μ
γ
,
4
ε
f
γ
∗
2
-
μ
α
,
3
α
~
2
-
μ
Q
,
3
Q
~
-
μ
Q
,
4
ε
f
Q
∗
+
ε
g
Q
∗
δ
e
2
-
∑
x
μ
x
,
1
x
~
-
μ
x
,
2
λ
x
2
-
1
2
∑
x
β
x
Δ
~
x
M
2
-
1
2
∑
x
β
f
x
θ
~
f
x
2
-
1
2
∑
x
β
g
x
θ
~
g
x
2
+
μ
V
,
5
ε
f
V
∗
+
ε
g
V
∗
Φ
2
+
μ
γ
,
5
+
1
2
a
α
,
1
ε
f
γ
∗
2
+
1
2
c
γ
+
1
2
c
α
,
2
·
ε
g
γ
∗
2
+
μ
Q
,
5
ε
f
Q
∗
+
ε
g
Q
∗
δ
e
2
+
1
2
∑
x
β
f
x
θ
f
x
∗
2
+
1
2
∑
x
β
g
x
θ
g
x
∗
2
+
1
2
∑
x
k
x
ε
x
2
β
x
+
1
2
∑
x
β
x
Δ
x
M
2
,where x
∈
{
V
,
h
,
γ
,
α
,
Q
}.Let(69)
Σ
=
μ
V
,
5
ε
f
V
M
+
ε
g
V
M
Φ
M
2
+
μ
γ
,
5
+
1
2
c
α
,
1
·
ε
f
γ
M
2
+
1
2
c
γ
+
1
2
c
α
,
2
ε
g
γ
M
2
+
μ
Q
,
5
ε
f
Q
M
+
ε
g
Q
M
δ
e
M
2
+
1
2
∑
x
k
x
ε
x
2
β
x
+
β
x
Δ
x
M
2
+
β
f
x
θ
f
x
M
2
+
β
g
x
θ
g
x
M
2
.Define(70)
Ω
V
~
=
V
~
∣
V
~
≤
μ
V
,
4
ε
f
V
M
+
ε
g
V
M
Φ
M
+
Σ
μ
V
,
3
,
Ω
h
~
=
h
~
∣
h
~
≤
Σ
μ
h
,
3
,
Ω
γ
~
=
γ
~
∣
γ
~
≤
μ
γ
,
4
ε
f
γ
M
+
Σ
μ
γ
,
3
,
Ω
α
~
=
α
~
∣
α
~
≤
Σ
μ
α
,
3
,
Ω
Q
~
=
Q
~
∣
Q
~
≤
μ
Q
,
4
ε
f
Q
M
+
ε
g
Q
M
δ
e
M
+
Σ
μ
Q
,
3
,
Ω
Δ
~
x
M
=
Δ
~
x
M
∣
Δ
~
x
M
≤
Σ
β
x
,
Ω
θ
~
f
x
=
θ
~
f
x
∣
θ
~
f
x
≤
Σ
β
f
x
,
Ω
θ
~
g
x
=
θ
~
g
x
∣
θ
~
g
x
≤
Σ
β
g
x
, where Φ
M
=
max
⁡
{
|
Φ
L
|
,
|
Φ
U
|
}; δ
e
L
=
max
⁡
{
|
δ
e
L
|
,
|
δ
e
U
|
}.Let(71)
k
x
,
1
>
k
x
,
3
2
4
κ
x
,
k
x
,
2
>
0
,
k
x
,
3
>
0
,
κ
x
>
0
,
Γ
f
x
=
Γ
f
x
T
>
0
,
Γ
g
x
=
Γ
g
x
T
>
0
,
β
f
x
>
0
,
β
g
x
>
0
,
σ
x
>
0
,
β
x
>
0
;If x
~, θ
~
f
x, θ
~
g
x and Δ
~
x
M, x
~
∈
{
V
~
,
h
~
,
γ
~
,
α
~
,
Q
~
}, do not belong to the compact set Ω
x
~, Ω
θ
~
f
x, Ω
θ
~
g
x and Ω
Δ
~
x
M, respectively, W
˙
≤
0. x
~, θ
~
f
x, θ
~
g
x, and Δ
~
x
M are bounded. Further, λ
x also is bounded according to (66). By regulating the designed parameters, the radius of Ω
x
~, Ω
W
~
f
x, Ω
W
~
g
x, and Ω
Δ
~
x
M can be made arbitrarily small.Theorem 8.
Consider the system of (1) under Assumptions 3 and 6, auxiliary analytical system (20) and (44), controllers (23) and (47)–(50), adaptive laws (21) and (45), and upper bound estimation of system error (22) and (46). If (71) is satisfied, all the signals involved are bounded.


## 4. Simulation Results

Simulations are done to illustrate the effectiveness of the robust adaptive approximate back-stepping control scheme proposed in the previous section. The vehicle model parameters are referred to in [27]. The damping of flexibility is ζ
i
=
0.02, i
=
1,2
,
3, and vibrational frequencies are ω
1
=
27.8944 rad/s, ω
2
=
76.8933 rad/s, and ω
3
=
150.752
6 rad/s.The aircraft is commanded to perform a cruising flight mission that the initial height is 25908 m and dynamic pressure is 95760 Pa. The initial trim conditions are designed as follows:γ
=
0
°, α
=
0.6132
°, Q
=
0
°
/
s, Ф
=
0.132
9, δ
e
=
9.0081
°, η
1
=
1.889 m·kg0.5/m0.5, η
2
=
-
0.028 m·kg0.5/m0.5, η
3
=
-
0.007
323 m·kg0.5/m0.5, and η
˙
i
=
0 m/s·kg0.5/m0.5, i
=
1,2
,
3.The reference commands of velocity and altitude are generated via the following filters:(72)
V
ref
s
V
c
s
=
0.0
3
2
s
2
+
2
×
0.95
×
0.03
×
s
+
0.0
3
2
,
h
ref
s
h
c
s
=
0.0
3
2
s
2
+
2
×
0.95
×
0.03
×
s
+
0.0
3
2
.The control laws are limited byΦ
∈
[
0.1,1.2
], δ
e
∈
[
-
15
°
,
15
°
], α
c
∈
[
-
5
°
,
5
°
], γ
c
∈
[
-
5
°
,
5
°
], and Q
c
∈
[
-
10
°
/
s
,
10
°
/
s
]. The bandwidth of Φ, δ
e, γ
c, α
c, and Q
c is limited by 20 rad/s, 20 rad/s, 30 rad/s, 40 rad/s, and 60 rad/s, respectively.The design parameters of controller are proposed as follows:κ
x
=
0.1, ε
x
=
0.01, k
V
,
1
=
k
h
,
1
=
0.4, k
γ
·
1
=
k
α
·
1
=
k
Q
·
1
=
1, k
x
·
2
=
0.01, and k
x
·
3
=
0.1. The adaptive law of RBF is considered as follows: Γ
f
V
=
0.01
×
I
625
×
625, Γ
g
V
=
0.1
×
I
625
×
625, Γ
f
γ
=
0.01
×
I
125
×
125, Γ
g
γ
=
0.1
×
I
625
×
625, Γ
f
Q
=
0.01
×
I
625
×
625, Γ
g
Q
=
0.1
×
I
3125
×
3125, β
f
x
=
β
g
x
=
0.1, β
x
=
0.1, σ
x
=
1, and the upper bound of weight vectors θ
f
V
M
=
20, θ
g
V
M
=
50, θ
f
γ
M
=
0.01, θ
g
γ
M
=
0.2, θ
f
Q
M
=
5, and θ
g
Q
M
=
10.


### 4.1. Trajectory Tracking Simulation without External Disturbance

To show efficiency of the proposed control law, the following two cases are proposed.Case 1.
The track height command is a square wave signal, whose amplitudeΔ
h
c
=
1524 m and period is 600 s.Case 2.
The track height command is a square wave signal, whose amplitudeΔ
h
c
=
3048 m and period is 600 s.Time responses of velocity, altitude, flight path angle, attack angle, pitch rate, fuel equivalence ratio, elevator deflection, and flexible states in two cases are given in Figures2–11.Figure 2
Response curves of velocity and velocity tracking error (Case1).Figure 3
Response curves of altitude and altitude tracking error (Case1).Figure 4
Time history of fuel equivalence ratio and elevator deflection (Case1)Figure 5
Response curves of flight path angle, attack angle, and pitch rate (Case1).Figure 6
The flexible states (Case1).Figure 7
Response curves of velocity and velocity tracking error (Case2).Figure 8
Response curves of altitude and altitude tracking error (Case2).Figure 9
Time history of fuel equivalence ratio and elevator deflection (Case2).Figure 10
Response curves of flight path angle, attack angle, and pitch rate (Case2).Figure 11
The flexible states (Case2).Time responses to altitude step change in Case1 are presented in Figures 2–6. As shown in Figures 2 and 3, the velocity and altitude can track their commands timely and without any steady tracking errors. The control inputs and attitude angles are smooth and within their bounds, just as shown in Figures 4 and 5. The flexible states can reach equilibrium state, which can be observed from Figure 6.Time responses to altitude step change in Case2 are presented in Figures 7–11. It can be seen from Figure 9 that the saturation occurs in elevator deflection during simulation, but the controller still provides a good track characteristic according to Figures 7–10. Furthermore, the flexible states also can maintain stability, which can be observed from Figure 11.


### 4.2. Trajectory Tracking Simulation with External Disturbance

To illustrate the robustness to external disturbance of the control laws developed herein, two vertical wind disturbances whose amplitudes are 17 m/s and 14 m/s are added to the control system in 100 s and 300 s, respectively, and the sustaining time of wind disturbance is 10 s. The additional attack angle with wind disturbance can be expressed as(73)
α
u
=
arctan
⁡
u
sin
⁡
γ
V
+
u
cos
⁡
γ
,
α
w
=
arctan
⁡
w
cos
⁡
γ
V
-
w
sin
⁡
γ
,where α
u and α
w are horizontal additional attack angle and vertical additional attack angle, respectively; u and w are level wind speed and vertical wind speed, respectively.The aircraft is commanded to perform a cruising flight mission that the initial height is 25908 m and dynamic pressure is 95760 Pa. When the simulation timet
≤
215
s, altitude follows the step command with 50 ft/s; when the simulation time 215
s
≤
t
≤
355
s, altitude follows the step command with 139 ft/s.The simulation results are illustrated in Figures12–15. It can be seen from Figure 14 that the saturation occurs in elevator deflection with wind disturbance in 100 s and 300 s, but the controller still provides a good track characteristic according to Figures 12–15, which indicate that the proposed adaptive neural back-stepping controller has a good performance on disposing input constraint under external disturbance.Figure 12
Response curves of velocity and velocity tracking error.Figure 13
Response curves of altitude and altitude tracking error.Figure 14
Time history of fuel equivalence ratio and elevator deflection.Figure 15
Response curves of flight path angle, attack angle, and pitch rate.


## 4.1. Trajectory Tracking Simulation without External Disturbance

To show efficiency of the proposed control law, the following two cases are proposed.Case 1.
The track height command is a square wave signal, whose amplitudeΔ
h
c
=
1524 m and period is 600 s.Case 2.
The track height command is a square wave signal, whose amplitudeΔ
h
c
=
3048 m and period is 600 s.Time responses of velocity, altitude, flight path angle, attack angle, pitch rate, fuel equivalence ratio, elevator deflection, and flexible states in two cases are given in Figures2–11.Figure 2
Response curves of velocity and velocity tracking error (Case1).Figure 3
Response curves of altitude and altitude tracking error (Case1).Figure 4
Time history of fuel equivalence ratio and elevator deflection (Case1)Figure 5
Response curves of flight path angle, attack angle, and pitch rate (Case1).Figure 6
The flexible states (Case1).Figure 7
Response curves of velocity and velocity tracking error (Case2).Figure 8
Response curves of altitude and altitude tracking error (Case2).Figure 9
Time history of fuel equivalence ratio and elevator deflection (Case2).Figure 10
Response curves of flight path angle, attack angle, and pitch rate (Case2).Figure 11
The flexible states (Case2).Time responses to altitude step change in Case1 are presented in Figures 2–6. As shown in Figures 2 and 3, the velocity and altitude can track their commands timely and without any steady tracking errors. The control inputs and attitude angles are smooth and within their bounds, just as shown in Figures 4 and 5. The flexible states can reach equilibrium state, which can be observed from Figure 6.Time responses to altitude step change in Case2 are presented in Figures 7–11. It can be seen from Figure 9 that the saturation occurs in elevator deflection during simulation, but the controller still provides a good track characteristic according to Figures 7–10. Furthermore, the flexible states also can maintain stability, which can be observed from Figure 11.


## 4.2. Trajectory Tracking Simulation with External Disturbance

To illustrate the robustness to external disturbance of the control laws developed herein, two vertical wind disturbances whose amplitudes are 17 m/s and 14 m/s are added to the control system in 100 s and 300 s, respectively, and the sustaining time of wind disturbance is 10 s. The additional attack angle with wind disturbance can be expressed as(73)
α
u
=
arctan
⁡
u
sin
⁡
γ
V
+
u
cos
⁡
γ
,
α
w
=
arctan
⁡
w
cos
⁡
γ
V
-
w
sin
⁡
γ
,where α
u and α
w are horizontal additional attack angle and vertical additional attack angle, respectively; u and w are level wind speed and vertical wind speed, respectively.The aircraft is commanded to perform a cruising flight mission that the initial height is 25908 m and dynamic pressure is 95760 Pa. When the simulation timet
≤
215
s, altitude follows the step command with 50 ft/s; when the simulation time 215
s
≤
t
≤
355
s, altitude follows the step command with 139 ft/s.The simulation results are illustrated in Figures12–15. It can be seen from Figure 14 that the saturation occurs in elevator deflection with wind disturbance in 100 s and 300 s, but the controller still provides a good track characteristic according to Figures 12–15, which indicate that the proposed adaptive neural back-stepping controller has a good performance on disposing input constraint under external disturbance.Figure 12
Response curves of velocity and velocity tracking error.Figure 13
Response curves of altitude and altitude tracking error.Figure 14
Time history of fuel equivalence ratio and elevator deflection.Figure 15
Response curves of flight path angle, attack angle, and pitch rate.


## 5. Conclusions

An adaptive neural back-stepping control design is presented in this paper for hypersonic vehicle with input constraint and uncertain aerodynamic parameters. To reduce the complexity of the controller design, the nonlinear model is divided into two subsystems, and the adaptive neural back-stepping controllers are developed for them, respectively. The neural network based radial basis function is designed to approximate the unknown system nonlinearity, which needs less knowledge of dynamic model. In order to eliminate the problem of “explosion of term,” a second-order reference model is designed for the precise estimation of the derivatives of virtual control laws. Expressly, the auxiliary systems are constructed to eliminate the saturation effect, where a novel auxiliary error compensation scheme is proposed. A comprehensive Lyapunov based stability analysis is provided to show that the control design achieves ultimate boundedness tracking for the hypersonic vehicle. Simulation results demonstrate that the proposed control strategy can provide satisfactory velocity and altitude tracking performance with input constraint and uncertain aerodynamic parameters.



---

*Source: 374838-2015-11-30.xml*