# Guidance Law and Neural Control for Hypersonic Missile to Track Targets

Authors: Wenxing Fu; Binbin Yan; Xiaofei Chang; Jie Yan
Journal: Discrete Dynamics in Nature and Society
 (2016)

---

## Abstract

Hypersonic technology plays an important role in prompt global strike. Because the flight dynamics of a hypersonic vehicle is nonlinear, uncertain, and highly coupled, the controller design is challenging, especially to design its guidance and control law during the attack of a maneuvering target. In this paper, the sliding mode control (SMC) method is used to develop the guidance law from which the desired flight path angle is derived. With the desired information as control command, the adaptive neural control in discrete time is investigated ingeniously for the longitudinal dynamics of the hypersonic missile. The proposed guidance and control laws are validated by simulation of a hypersonic missile against a maneuvering target. It is demonstrated that the scheme has good robustness and high accuracy to attack a maneuvering target in the presence of external disturbance and missile model uncertainty.


---

## Body

## 1. Introduction

The hypersonic missile can propel the development of the prompt global strike. Controller design is crucial in making it feasible. To control the hypersonic vehicle whose flight dynamics is nonlinear, uncertain, and coupled, many researches were carried out. In [1], the development of flight dynamics and controller design is comprehensively reviewed.The SMC is used to control a generic hypersonic vehicle in [2]. The dynamics of the hypersonic vehicle is decoupled to velocity and attitude subsystems [3] that have been transformed into the linearly parameterized form. Considering the parametric model uncertainty and input saturations, the dynamic inverse control is presented via the back-stepping design in which the dynamic surface control is used. Furthermore, the dead-zone input is considered in [4] with Nussbaum gain based controller design. In [5], the aerothermoelastic effect is taken into consideration in the hypersonic vehicle model, and a robust controller is developed to achieve global exponential tracking of the reference model. Fixed-order μ controllers are designed to constrain the controller’s dimensions with the propulsion system perturbations and aeroelastic fuselage bending considered as uncertainties in [6]. A coupled linear parameter varying (LPV) and flatness design is proposed to solve the hypersonic guidance problem in [7]. In [8, 9], robust control using the genetic algorithm to search a design coefficient space is proposed.In these literatures, the control outputs are flight velocity and altitude which are driven to track the reference trajectory that increases from zero to constant value after a transient phase. When the hypersonic missile is used to attack targets, the dynamics will change greatly in the endgame phase due to the angle and angle rate changes. More problems exist in the control of the hypersonic missile in attack scenarios, and researchers began to pay more attention to them. The control of the hypersonic missile attacking a ground stationary target with a different vector-output and time-varying commanded output trajectories is using nonlinear control [10] and adaptive control [11] separately.In this paper, the guidance and control law of a hypersonic missile attacking a maneuvering target instead of a ground stationary target are discussed. It is a more challenging task owing to the uncertainties of target maneuvers. To solve this problem, the SMC guidance law and the back-stepping control law are designed to guarantee the robustness. The SMC method is one of the robust control design methods which is robust to parameter perturbations and external disturbances [12, 13]. Reference [14] proposed the switched bias proportional navigation guidance law which is derived by using the sliding mode control theory based on proportional navigation [15]. In [16], an adaptive sliding mode guidance law designed for intercepting a maneuvering target is proposed. It is robust in the presence of disturbances and parameter perturbations and is able to eliminate chattering. With target acceleration considered as bounded uncertainty, a sliding mode guidance law is derived based on the nonlinear planar engagement kinematics in [17]. In [18], the sliding mode control method with the zero-effort miss distance considered as the sliding surface is derived. In [19], a novel smooth second-order sliding mode (SSOSM) control was applied to guidance law design. Compared with augmented proportional navigation guidance law, the SSOSM can achieve better miss distance performance.Back-stepping design [20–22] is a powerful tool to design the controller for a nonlinear system in parameter strict-feedback form. Properties of strong stability and tracking are built into the nonlinear system in a number of steps, which are never higher than the system order. In [23], the sequential loop closure controller design decomposing equations into functional subsystems is adopted. Moreover, intelligent control receives increasing attention since the neural networks [24–28] or fuzzy logic system [29, 30] can learn the dynamics efficiently. In [31], the global design with neural approximation is proposed for both indirect and direct control of hypersonic dynamics.With the development of hardware, the research on discrete-time control has received a considerable attention [32–35]. As pointed out in [36], the use of digital computers and samplers in control circuitry has made the use of discrete-time system representation more justifiable than continuous-time system representation. For the control of a flight vehicle, a controller using continuous-time system is usually implemented by a digital computer with a certain sampling interval [37, 38]. As for the modeling, the discrete strict-feedback form of HFV model is obtained with the Euler approximation [39]. Focusing on the uncertainty estimation, the adaptive back-stepping discrete controller is investigated for hypersonic flight [40].In this paper, we first design the sliding mode guidance law from which the desired flight path angle is derived. Then, we design the controller with the back-stepping scheme recursively. We take into consideration the nominal nonlinearity for feedback design and the neural network (NN) is used to approximate the system uncertainty. In order to avoid the circular construction of control inputs, the upper bound is taken instead of the nominal value for coefficient design. The idea of the guidance and control design is shown in Figure2. This paper is organized as follows. Section 2 briefly presents the longitudinal dynamics of a generic HFV. Section 3 presents the guidance law design via sliding mode control. The system transformation of prediction model is given in Section 4. Section 5 presents the adaptive neural controller design and the stability analysis for the subsystems, respectively. The target interception simulation is included in Section 6. Section 7 presents several comments and final remarks.


## 2. Hypersonic Vehicle Modeling

This model comprises five state variablesX
h
=
V
,
h
,
α
,
γ
,
q
T and two control inputs U
c
=
δ
e
,
β
T, where V is the velocity, γ is the flight path angle, h is the altitude, α is the angle of attack, q is the pitch rate, δ
e is elevator deflection, and β is the throttle setting:(1)
V
˙
=
T
cos
⁡
α
-
D
m
-
μ
sin
⁡
γ
r
2
,
(2)
h
˙
=
V
sin
⁡
γ
,
(3)
γ
˙
=
L
+
T
sin
⁡
α
m
V
-
μ
-
V
2
r
cos
⁡
γ
V
r
2
,
(4)
α
˙
=
q
-
γ
˙
,
(5)
q
˙
=
M
y
y
I
y
y
.The definitions of the dynamics of hypersonic flight vehicle (HFV) in [41] are given as follows: r
=
h
+
R
E, q
¯
=
1
/
2
ρ
V
2, L
=
q
¯
S
C
L, D
=
q
¯
S
C
D, T
=
q
¯
S
C
T, M
y
y
=
q
¯
S
c
¯
C
M
α
+
C
M
δ
e
+
C
M
q, C
L
=
0.6203
α, C
D
=
0.6450
α
2
+
0.0043378
α
+
0.003772, C
M
(
α
)
=
-
0.035
α
2
+
0.036617
α
+
5.3261
×
1
0
-
6, and C
M
(
q
)
=
(
c
¯
/
2
V
)
q
(
-
6.796
α
2
+
0.3015
α
-
0.2289
).The control input is defined as(6)
C
T
=
0.02576
β
if
β
<
1
0.0224
+
0.00336
β
otherwise,
C
M
δ
e
=
0.0292
δ
e
-
α
, where ρ denotes the air density, S is the reference area, c
¯ is the reference length, and R
E is the radius of the Earth. C
x, x
=
L
,
D
,
T
,
M, are the force and moment coefficients.


## 3. Guidance Law Design

The geometry of an idealized interception in which the missile and the target are closing on each other at constant speed is shown in Figure1, where M is hypersonic missile, T is target, V
M and γ
M are the speed and path angle of the hypersonic missile, V
T and γ
T are the speed and path angle of the target, r is the distance between the hypersonic missile and the target, and λ is the line of sight (LOS). Then, the LOS rate can be represented as(7)
λ
˙
=
1
r
V
M
sin
⁡
λ
-
γ
M
-
V
T
sin
⁡
λ
-
γ
T
.The crucial part of the sliding mode controller design is the sliding surface design. To attain a collision triangle, the angle rate command of the missile is usually chosen to be proportional to the LOS rate when the speeds of the interceptor and the target are constants. So the switching function can be chosen as(8)
s
=
r
λ
˙
.As mentioned before, the missile and the target are closing on each other at constant speed. Then, the exponential approximation law can be used to achieve satisfactory dynamical performance. The adaptive approximation law can be designed as(9)
s
˙
=
-
ks
-
ε
sgn
⁡
s
, where k and ε are positive constants and sgn
⁡
(
) is defined as(10)
sgn
⁡
s
=
1
,
s
>
0
0
,
s
=
0
-
1
,
s
<
0
.To prevent the inertia of a constant approximation law from making the LOS angular rate diverge, the adaptive approximation law is designed as(11)
s
˙
=
-
k
r
˙
r
s
-
ε
sgn
⁡
s
.Equation (11) means that the sliding mode approximation speed is adjusted by the target-to-missile ranger r. When r is close to zero, the speed increases rapidly to prevent λ
˙ from diverging. To mitigate chattering, a continuous function λ
˙
/
λ
˙
+
δ can be a substitute for the variable structure item sgn
⁡
λ
˙, where δ is a small positive real number. Thus, the sliding mode guidance law is written as(12)
n
y
=
k
+
1
V
˙
R
λ
˙
+
ε
λ
˙
λ
˙
+
δ
.Substitute (12) into dynamic equation(13)
n
y
=
V
g
γ
˙
. By integrating γ
˙, we can obtain the desired path angle γ
d.Figure 1
Interception geometry.Figure 2
The control schematic diagram.


## 4. System Transformation

Equations (1)–(5) are applicable when the velocity is mainly related to throttle setting and the rate of altitude change is controlled by the elevator deflection. So the dynamics of the HFV can be decoupled into two functional subsystems. With the tracking references V
d and h
d, we design the velocity and altitude controller separately in Section 5. The schematic diagram is depicted in Figure 2.


### 4.1. Strict-Feedback Formulation

Assumption 1.
Sinceγ is small, we can take sin
⁡
γ
≈
γ in (2) for simplification. The thrust term T
sin
⁡
α in (3) can be neglected because it is generally much smaller than L.Assumption 2.
The velocity can be considered as constant during the hypersonic missile attacking the target.


#### 4.1.1. Velocity Subsystem

The velocity subsystem (1) can be rewritten into the following form:(14)
V
˙
=
f
V
+
g
V
u
V
,
u
V
=
β
,
y
V
=
V
,where f
V
=
-
(
D
/
m
+
μ
sin
⁡
γ
/
r
2
)
+
q
¯
S
×
0.0224
cos
⁡
α
/
m and g
V
=
q
¯
S
×
0.00336
cos
⁡
α
/
m if β
>
1. Otherwise, f
V
=
-
(
D
/
m
+
μ
sin
⁡
γ
/
r
2
) and g
V
=
q
¯
S
×
0.02576
cos
⁡
α
/
m.


#### 4.1.2. Altitude Subsystem

The altitude tracking error is defined ash
~
=
h
-
h
d and the flight path command is chosen as(15)
γ
d
=
arcsin
⁡
-
k
h
h
-
h
d
-
k
I
∫
h
-
h
d
d
t
+
h
˙
d
V
.If k
h
>
0 and k
I
>
0 are chosen and the flight path angle is controlled to follow γ
d, the altitude tracking error is regulated to zero exponentially [42].DefineX
A
=
x
1
,
x
2
,
x
3
T, x
1
=
γ, x
2
=
θ
p, and x
3
=
q, where θ
p
=
α
+
γ. Then, the strict-feedback form equations of the attitude subsystem (3)–(5) are written as(16)
x
˙
1
=
f
1
x
1
+
g
1
x
1
x
2
,
x
˙
2
=
f
2
x
1
,
x
2
+
g
2
x
1
,
x
2
x
3
,
x
˙
3
=
f
3
x
1
,
x
2
,
x
3
+
g
3
x
1
,
x
2
,
x
3
u
A
,
u
A
=
δ
e
,
y
=
x
1
, where f
1
=
-
(
μ
-
V
2
r
)
cos
⁡
γ
/
(
V
r
2
)
-
0.6203
q
¯
S
γ
/
(
m
V
), g
1
=
0.6203
q
¯
S
/
(
m
V
), f
2
=
0, g
2
=
1, f
3
=
q
¯
S
c
¯
[
C
M
(
α
)
+
C
M
(
q
)
-
0.0292
α
]
/
I
y
y, and g
3
=
0.0292
q
¯
S
c
¯
/
I
y
y.Assumption 3.
f
i and g
i are unknown smooth functions. There exist constants g
¯
i and g
_
i such that g
¯
i
≥
g
i
≥
g
_
i
>
0, i
=
1,3
,
V.


### 4.2. Discrete-Time Model

With the Euler expansion for the sampling periodT
s, systems (14) and (16) can be approximated as(17)
V
k
+
1
=
V
k
+
T
s
f
v
k
+
g
v
k
u
v
k
,
(18)
x
1
k
+
1
=
x
1
k
+
T
s
f
1
k
+
g
1
k
x
2
k
,
x
2
k
+
1
=
x
2
k
+
T
s
f
2
k
+
g
2
k
x
3
k
,
x
3
k
+
1
=
x
3
k
+
T
s
f
3
k
+
g
3
k
u
A
k
.


## 4.1. Strict-Feedback Formulation

Assumption 1.
Sinceγ is small, we can take sin
⁡
γ
≈
γ in (2) for simplification. The thrust term T
sin
⁡
α in (3) can be neglected because it is generally much smaller than L.Assumption 2.
The velocity can be considered as constant during the hypersonic missile attacking the target.


### 4.1.1. Velocity Subsystem

The velocity subsystem (1) can be rewritten into the following form:(14)
V
˙
=
f
V
+
g
V
u
V
,
u
V
=
β
,
y
V
=
V
,where f
V
=
-
(
D
/
m
+
μ
sin
⁡
γ
/
r
2
)
+
q
¯
S
×
0.0224
cos
⁡
α
/
m and g
V
=
q
¯
S
×
0.00336
cos
⁡
α
/
m if β
>
1. Otherwise, f
V
=
-
(
D
/
m
+
μ
sin
⁡
γ
/
r
2
) and g
V
=
q
¯
S
×
0.02576
cos
⁡
α
/
m.


### 4.1.2. Altitude Subsystem

The altitude tracking error is defined ash
~
=
h
-
h
d and the flight path command is chosen as(15)
γ
d
=
arcsin
⁡
-
k
h
h
-
h
d
-
k
I
∫
h
-
h
d
d
t
+
h
˙
d
V
.If k
h
>
0 and k
I
>
0 are chosen and the flight path angle is controlled to follow γ
d, the altitude tracking error is regulated to zero exponentially [42].DefineX
A
=
x
1
,
x
2
,
x
3
T, x
1
=
γ, x
2
=
θ
p, and x
3
=
q, where θ
p
=
α
+
γ. Then, the strict-feedback form equations of the attitude subsystem (3)–(5) are written as(16)
x
˙
1
=
f
1
x
1
+
g
1
x
1
x
2
,
x
˙
2
=
f
2
x
1
,
x
2
+
g
2
x
1
,
x
2
x
3
,
x
˙
3
=
f
3
x
1
,
x
2
,
x
3
+
g
3
x
1
,
x
2
,
x
3
u
A
,
u
A
=
δ
e
,
y
=
x
1
, where f
1
=
-
(
μ
-
V
2
r
)
cos
⁡
γ
/
(
V
r
2
)
-
0.6203
q
¯
S
γ
/
(
m
V
), g
1
=
0.6203
q
¯
S
/
(
m
V
), f
2
=
0, g
2
=
1, f
3
=
q
¯
S
c
¯
[
C
M
(
α
)
+
C
M
(
q
)
-
0.0292
α
]
/
I
y
y, and g
3
=
0.0292
q
¯
S
c
¯
/
I
y
y.Assumption 3.
f
i and g
i are unknown smooth functions. There exist constants g
¯
i and g
_
i such that g
¯
i
≥
g
i
≥
g
_
i
>
0, i
=
1,3
,
V.


## 4.1.1. Velocity Subsystem

The velocity subsystem (1) can be rewritten into the following form:(14)
V
˙
=
f
V
+
g
V
u
V
,
u
V
=
β
,
y
V
=
V
,where f
V
=
-
(
D
/
m
+
μ
sin
⁡
γ
/
r
2
)
+
q
¯
S
×
0.0224
cos
⁡
α
/
m and g
V
=
q
¯
S
×
0.00336
cos
⁡
α
/
m if β
>
1. Otherwise, f
V
=
-
(
D
/
m
+
μ
sin
⁡
γ
/
r
2
) and g
V
=
q
¯
S
×
0.02576
cos
⁡
α
/
m.


## 4.1.2. Altitude Subsystem

The altitude tracking error is defined ash
~
=
h
-
h
d and the flight path command is chosen as(15)
γ
d
=
arcsin
⁡
-
k
h
h
-
h
d
-
k
I
∫
h
-
h
d
d
t
+
h
˙
d
V
.If k
h
>
0 and k
I
>
0 are chosen and the flight path angle is controlled to follow γ
d, the altitude tracking error is regulated to zero exponentially [42].DefineX
A
=
x
1
,
x
2
,
x
3
T, x
1
=
γ, x
2
=
θ
p, and x
3
=
q, where θ
p
=
α
+
γ. Then, the strict-feedback form equations of the attitude subsystem (3)–(5) are written as(16)
x
˙
1
=
f
1
x
1
+
g
1
x
1
x
2
,
x
˙
2
=
f
2
x
1
,
x
2
+
g
2
x
1
,
x
2
x
3
,
x
˙
3
=
f
3
x
1
,
x
2
,
x
3
+
g
3
x
1
,
x
2
,
x
3
u
A
,
u
A
=
δ
e
,
y
=
x
1
, where f
1
=
-
(
μ
-
V
2
r
)
cos
⁡
γ
/
(
V
r
2
)
-
0.6203
q
¯
S
γ
/
(
m
V
), g
1
=
0.6203
q
¯
S
/
(
m
V
), f
2
=
0, g
2
=
1, f
3
=
q
¯
S
c
¯
[
C
M
(
α
)
+
C
M
(
q
)
-
0.0292
α
]
/
I
y
y, and g
3
=
0.0292
q
¯
S
c
¯
/
I
y
y.Assumption 3.
f
i and g
i are unknown smooth functions. There exist constants g
¯
i and g
_
i such that g
¯
i
≥
g
i
≥
g
_
i
>
0, i
=
1,3
,
V.


## 4.2. Discrete-Time Model

With the Euler expansion for the sampling periodT
s, systems (14) and (16) can be approximated as(17)
V
k
+
1
=
V
k
+
T
s
f
v
k
+
g
v
k
u
v
k
,
(18)
x
1
k
+
1
=
x
1
k
+
T
s
f
1
k
+
g
1
k
x
2
k
,
x
2
k
+
1
=
x
2
k
+
T
s
f
2
k
+
g
2
k
x
3
k
,
x
3
k
+
1
=
x
3
k
+
T
s
f
3
k
+
g
3
k
u
A
k
.


## 5. Discrete-Time Controller Design

(1) Definez
1
(
k
)
=
x
1
(
k
)
-
x
1
d
(
k
), where x
1
d
(
k
)
=
γ
d
(
k
) is derived from (15). Then,(19)
z
1
k
+
1
=
x
1
k
+
T
s
f
1
k
+
g
1
k
x
2
k
-
x
1
d
k
+
1
.The desired control input is(20)
x
2
d
∗
k
=
1
T
s
g
1
k
c
1
z
1
k
-
T
s
f
1
k
-
x
1
k
+
x
1
d
k
+
1
. However, it can be observed that functions f
1
(
k
) and x
1
d
(
k
+
1
) are unknown. Thus, the uncertainty is defined and approximated by NN:(21)
U
1
k
=
-
T
s
f
1
k
-
x
1
k
+
x
1
d
k
+
1
=
ω
1
∗
T
θ
1
X
1
k
+
ε
1
X
1
k
,where ω
1
∗ is the optimal NN weight vector to approximate U
1
(
k
) and X
1
(
k
)
=
V
k
,
x
1
k
,
x
2
k
,
x
3
(
k
)
,
x
d
k
T. Denoting θ
1
(
k
)
=
θ
1
(
X
1
(
k
)
), ε
1
(
k
)
=
ε
1
(
X
1
(
k
)
) is the NN constriction error with the upper bound ε
1
M.The virtual controllerx
2
d
(
k
) is designed with the following form:(22)
x
2
d
k
=
1
T
s
g
1
k
c
1
z
1
k
+
ω
^
1
T
θ
1
X
1
k
,where ω
^
1 is the estimation of ω
1
∗ and c
1 is the design constant.Definez
2
(
k
)
=
x
2
(
k
)
-
x
2
d
(
k
). From (19) and (22), we know(23)
z
1
k
+
1
=
c
1
z
1
k
+
T
s
g
1
k
z
2
k
+
ω
~
1
T
k
θ
1
k
-
ε
1
k
,where ω
~
1
=
ω
^
1
-
ω
1
∗.The NN updating law is proposed as(24)
ω
^
1
k
+
1
=
ω
^
1
k
-
λ
1
z
1
k
+
1
θ
1
k
-
δ
1
ω
^
1
k
.Define the Lyapunov function candidate:(25)
L
1
k
=
z
1
2
k
+
ω
~
1
T
k
ω
~
1
k
λ
1
. The first difference is calculated as(26)
Δ
L
1
k
=
L
1
k
+
1
-
L
1
k
=
z
1
2
k
+
1
-
z
1
2
k
+
ω
~
1
T
k
+
1
ω
~
1
k
+
1
-
ω
~
1
T
k
ω
~
1
k
λ
1
.From (24), it is obvious that(27)
ω
~
1
k
+
1
=
ω
~
1
k
-
λ
1
z
1
k
+
1
θ
1
k
-
δ
1
ω
^
1
k
.From (23), we have(28)
ω
~
1
T
k
θ
1
k
z
1
k
+
1
=
z
1
2
k
+
1
-
T
s
g
1
k
z
1
k
+
1
z
2
k
-
c
1
z
1
k
z
1
k
+
1
+
ε
1
k
z
1
k
+
1
.It is known that θ
1
(
k
)
2
≤
N
1, where N
1 is the number of NN nodes. Now, we have(29)
Δ
L
1
k
=
L
1
k
+
1
-
L
1
k
≤
z
1
2
k
+
1
-
z
1
2
k
+
λ
1
N
1
z
1
2
k
+
1
+
δ
1
2
λ
1
ω
^
1
k
2
-
2
z
1
2
k
+
1
+
2
T
s
g
1
k
z
1
k
+
1
z
2
k
+
2
c
1
z
1
k
z
1
k
+
1
-
2
ε
1
k
z
1
k
+
1
-
2
δ
1
λ
1
ω
~
1
T
k
ω
^
1
k
+
2
δ
1
ω
^
1
T
k
θ
1
k
z
1
k
+
1
.The following inequalities are obtained:(30)
2
ε
1
k
z
1
k
+
1
≤
ρ
11
ε
1
M
2
+
1
ρ
11
z
1
2
k
+
1
,
2
z
1
k
+
1
z
1
k
≤
ρ
12
z
1
2
k
+
1
+
1
ρ
12
z
1
2
k
,
2
z
2
k
z
1
k
+
1
≤
ρ
13
z
2
2
k
+
1
ρ
13
z
1
2
k
+
1
,
2
ω
~
1
T
k
ω
^
1
k
=
ω
~
1
T
k
ω
~
1
k
+
ω
^
1
k
2
-
ω
1
∗
2
,
2
ω
^
1
T
k
θ
1
k
z
1
k
+
1
≤
ρ
14
ω
^
1
k
2
+
1
ρ
14
N
1
z
1
2
k
+
1
,where ρ
1
,
i
,
i
=
1,2
,
3,4, are positive scalars.The first difference ofL
1
k is obtained as(31)
Δ
L
1
k
≤
z
1
2
k
+
1
-
z
1
2
k
+
λ
1
N
1
z
1
2
k
+
1
+
δ
1
2
λ
1
ω
^
1
k
2
-
2
z
1
2
k
+
1
+
ρ
13
T
s
g
¯
1
z
2
2
k
+
1
ρ
13
T
s
g
¯
1
z
1
2
k
+
1
+
c
1
ρ
12
z
1
2
k
+
1
+
c
1
ρ
12
z
1
2
k
+
ρ
11
ε
1
M
2
+
1
ρ
11
z
1
2
k
+
1
-
δ
1
λ
1
ω
~
1
T
k
ω
~
1
k
-
δ
1
λ
1
ω
^
1
k
2
+
δ
1
λ
1
ω
1
∗
2
+
δ
1
ρ
14
ω
^
1
k
2
+
δ
1
N
1
ρ
14
z
1
2
k
+
1
.Finally, we obtain(32)
Δ
L
1
k
≤
-
γ
11
z
1
2
k
+
1
-
γ
12
z
1
2
k
-
γ
13
ω
^
1
k
2
+
γ
14
z
2
2
k
+
γ
15
,where γ
11
=
1
-
λ
1
N
1
-
1
/
ρ
13
T
s
g
¯
1
-
k
1
ρ
12
-
1
/
ρ
11
-
δ
1
N
1
/
ρ
14, γ
12
=
1
-
c
1
/
ρ
12, γ
13
=
δ
1
/
λ
1
-
δ
1
2
/
λ
1
-
δ
1
ρ
14, γ
14
=
T
s
g
¯
1
ρ
13, and γ
15
=
ρ
11
ε
1
M
2
+
δ
1
/
λ
1
ω
1
∗
2.(2) Definez
2
(
k
)
=
x
2
(
k
)
-
x
2
d
(
k
), X
2
(
k
)
=
V
(
k
)
,
h
(
k
)
,
x
1
(
k
)
,
x
2
(
k
)
,
x
3
(
k
)
,
x
1
d
(
k
)
T. Since f
2
=
0 and g
2
=
1, we know(33)
z
2
k
+
1
=
x
2
k
+
T
s
x
3
k
-
x
2
d
k
+
1
.The uncertainty is defined and approximated with the NN:(34)
U
2
k
=
x
2
d
k
+
1
-
x
2
k
=
ω
2
∗
T
θ
2
X
2
k
+
ε
2
X
2
k
, where ω
2
∗ is the optimal NN weight vector to approximate U
2
(
k
). Denoting θ
2
k
=
θ
2
X
2
k, ε
2
k
=
ε
2
X
2
k is the NN construction error with upper bound ε
2
M.Design virtual controlx
3
d
(
k
) as(35)
x
3
d
k
=
c
2
z
2
k
+
ω
^
2
T
k
θ
2
k
T
s
,where ω
^
2 is the estimation of ω
2
∗ and c
2 is the design constant.Definez
3
(
k
)
=
x
3
(
k
)
-
x
3
d
(
k
). From (33) and (35), we have(36)
z
2
k
+
1
=
c
2
z
2
k
+
T
s
z
3
k
+
ω
~
2
T
k
θ
2
k
-
ε
2
k
,where ω
~
2
=
ω
^
2
-
ω
2
∗.The NN updating law is given as(37)
ω
^
2
k
+
1
=
ω
^
2
k
-
λ
2
z
2
k
+
1
θ
2
k
-
δ
2
ω
^
2
k
.Define the Lyapunov function candidate:(38)
L
2
k
=
z
2
2
k
+
ω
~
2
T
k
ω
~
2
k
λ
2
. Then,(39)
Δ
L
2
k
≤
-
γ
21
z
2
2
k
+
1
-
γ
22
z
2
2
k
-
γ
23
ω
^
2
k
2
+
γ
24
z
3
2
k
+
γ
25
,where γ
21
=
1
-
λ
2
N
2
-
1
/
ρ
23
T
s
-
c
2
ρ
22
-
1
/
ρ
21
-
δ
2
N
2
/
ρ
24, γ
22
=
1
/
T
s
-
c
2
/
ρ
22, γ
23
=
δ
2
/
λ
2
-
δ
2
2
/
λ
2
-
δ
2
ρ
24, γ
24
=
ρ
23
T
s, and γ
25
=
ρ
21
ε
2
M
2
+
δ
2
/
λ
2
ω
2
∗
2.(3) According to the definition ofz
3
(
k
), we have(40)
z
3
k
+
1
=
x
3
k
+
T
s
f
3
k
+
g
3
k
u
A
k
-
x
3
d
k
+
1
.Define X
3
(
k
)
=
V
(
k
)
,
h
(
k
)
,
x
1
(
k
)
,
x
2
(
k
)
,
x
3
(
k
)
,
x
1
d
(
k
)
T. The uncertainty is defined as(41)
U
3
k
=
-
T
s
f
3
k
-
x
3
k
+
x
3
d
k
+
1
=
ω
3
∗
T
θ
3
k
+
ε
3
k
,where ω
3
∗ is the optimal NN weight vector to approximate U
3
(
k
). Denoting θ
3
(
k
)
=
θ
3
(
X
3
(
k
)
), ε
3
(
k
)
=
ε
3
(
X
3
(
k
)
) is the NN construction error with the upper bound ε
3
M.The elevator deflectionu
A
(
k
) is designed as(42)
u
A
k
=
c
3
z
3
k
+
ω
^
3
T
k
θ
3
k
T
s
g
3
k
,where ω
^
3 is the estimation of ω
3
∗ and c
3 is a positive design constant.Combining (40) and (42), the error dynamics of z
3
k
+
1 is(43)
z
3
k
+
1
=
c
3
z
3
k
+
ω
~
3
T
k
θ
3
k
-
ε
3
k
,where ω
~
3
=
ω
^
3
-
ω
3
∗.The NN updating law is given as(44)
ω
^
3
k
+
1
=
ω
^
3
k
-
λ
3
z
3
k
+
1
θ
3
k
-
δ
3
ω
^
3
k
.Define the Lyapunov function candidate:(45)
L
3
k
=
z
3
2
k
+
ω
~
3
T
k
ω
~
3
k
λ
3
. Then,(46)
Δ
L
3
k
≤
-
γ
31
z
3
2
k
+
1
-
γ
32
z
3
2
k
-
γ
33
ω
^
3
k
2
+
γ
35
,where γ
31
=
1
-
λ
3
N
3
-
c
3
ρ
32
-
1
/
ρ
31
-
δ
3
N
3
/
ρ
34, γ
32
=
1
-
c
3
/
ρ
32, γ
33
=
δ
3
/
λ
3
-
δ
3
2
/
λ
3
-
δ
3
ρ
34, and γ
35
=
ρ
31
ε
3
M
2
+
δ
3
/
λ
3
ω
3
∗
2.By selectingρ
3
i, i
=
1
,
2
,
3
,
5, and k
3
,
λ
3
,
δ
3, we know that γ
3
i
>
0 can be guaranteed. From (46), if z
3
k
>
γ
35
/
γ
32, then Δ
L
3
≤
0. So z
3
k is bounded. This further implies that z
2
(
k
) is bounded from (39). Then, from (32), z
1
(
k
) is bounded.Now, the following theorem can be obtained.Theorem 4.
Consider system (18) with controllers (22), (35), and (42) and NN updating laws (24), (38), and (44). All the signals involved are bounded.


## 6. Simulation

For the velocity subsystem, the controller in [42] is simply employed. In this section, we verify the effectiveness and performance of the proposed guidance law and adaptive neural controller. The hypersonic missile starts from the state at h
=
118110
f
t, V
=
15060
f
t
/
s, α
=
0
d
e
g, γ
=
0
d
e
g, and q
=
0
d
e
g
/
s. The target starts from the state at h
=
119090
f
t and V
=
2230
f
t
/
s and has an upward maneuvering overload n
=
1.5
g in the interception phase.The parameters for the guidance law are selected ask
=
2.5, ε
=
0.1, and δ
=
0.5. The parameters for the controller are selected as T
s
=
0.005
s, c
1
=
0.9, c
2
=
0.95, c
3
=
0.95, c
V
=
0.95, λ
i
=
0.002, δ
i
=
0.05, i
=
1,2
,
3, λ
v
=
0.02, and ε
0
=
0.0001. The number of NN nodes is 50. The initial NN weights for g
^
v are set as 5, while for other NNs the weights are set as zero.The simulation results are shown in Figures3–12. From the target interception in Figures 3 and 4, it can be observed that the target (dotted line) maneuvering upward rapidly is impacted by the hypersonic missile (solid line) which tracks the trajectory (dash dot line) calculated precisely by the sliding mode guidance. From the flight path angle tracking in Figures 5 and 6, it can be concluded that the altitude control law tracks the flight path angle at the desired value calculated accurately by the guidance law. From the velocity tracking in Figures 7 through 10, it can be concluded that the velocity controller achieves good performance of keeping the velocity at constant value. The throttle setting changes with the value of angle of attack, indicating the coupling of aerodynamics and propulsion. Figures 7 through 10 depict the changes in the velocity, angle of attack, and control input which oscillate in the first second and then become stable. The reason for oscillation can be found in the tracking of the NN weights depicted in Figures 11 and 12. The NN has learned the dynamic characteristics of the hypersonic missile.Figure 3
Interception trajectory.Figure 4
Altitude tracking error.Figure 5
Flight path angle.Figure 6
Flight path angle tracking error.Figure 7
Velocity.Figure 8
Angle of attack.Figure 9
Elevator deflection.Figure 10
Throttle setting.Figure 11
NN weights response (1).Figure 12
NN weights response (2).


## 7. Conclusion and Future Work

Taking the target maneuver into consideration, this paper proposes the sliding mode guidance law and adaptive neural controller for the hypersonic missile via the back-stepping method. The flight path angle is designed as the guidance command instead of normal acceleration. The nominal part of the nonlinearity during each step is eliminated and the NN is used to approximate the system uncertainty. Simulation results show the effectiveness of the proposed guidance and control laws. For future work, the energy management control should be considered to derive more effective velocity control.In this paper, only one-step-ahead model is used for controller design. For future work, we will work on how to develop a new algorithm based on the equivalent prediction model developed in [43]. Also, the fault tolerant control [44] is of interest since it can guarantee flight safety. In case of time-varying disturbance, the disturbance observer based control [45] can be applied.



---

*Source: 6219609-2016-04-07.xml*