Chinese Journal of Aeronautics, (2022), 35(4): 320–331

Chinese Society of Aeronautics and Astronautics
& Beihang University

Chinese Journal of Aeronautics
cja@buaa.edu.cn
www.sciencedirect.com

Optimal predictive sliding-mode guidance law for
intercepting near-space hypersonic maneuvering
target
Bolun ZHANG, Di ZHOU *
School of Astronautics, Harbin Institute of Technology, Harbin 150001, China
Received 17 November 2020; revised 9 January 2021; accepted 8 April 2021
Available online 5 July 2021

KEYWORDS
Adaptive sliding-mode control;
Near-space hypersonic aircraft;
Neural networks;
Optimal control;
Predictive guidance law

Abstract The design of optimal guidance law for intercepting a near-space hypersonic maneuvering target with bounded inputs is considered. Firstly, a maneuvering model for near-space hypersonic aircraft is given. Then, the aircraft acceleration prediction can be obtained using this model
with two neural networks. By using the target acceleration prediction, which is taken into account
when calculating the Zero Effort Miss (ZEM), an optimal sliding-mode guidance law is proposed to
fulfill the guidance task. An adaptive sliding-mode switch term is designed to deal with actuator saturation and prediction errors. Finally, numerical simulations show that the proposed guidance law
can reduce the energy consumption and the terminal acceleration command of the interceptor effectively.
Ó 2021 Chinese Society of Aeronautics and Astronautics. Production and hosting by Elsevier Ltd. This is
an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

1. Introduction
In recent years, the technique of near-space hypersonic vehicle
has been developed rapidly.1,2 The strong maneuvering nearspace vehicle has brought great challenges to defense technology. For example, the design of a terminal guidance law for
intercepting the near-space hypersonic maneuvering target
whose maximum acceleration is very close to that of the inter* Corresponding author.
E-mail address: zhoud@hit.edu.cn (D. ZHOU).
Peer review under responsibility of Editorial Committee of CJA.

Production and hosting by Elsevier

ceptor missile has become an attractive topic. In order to intercept high-speed and high-maneuverability aircraft, some
researchers studied the cooperative guidance schemes of multiple missiles. For example, Wang et al.3 and Shaferman et al.4,5
used a cooperative guidance strategy, where multiple interceptors attack the target from different directions simultaneously
to reduce the impact of the target’s maneuvering. Shaferman
and Oshman6 proposed a strategy that two interceptors attack
the target at a certain distance apart. The key idea of the strategy is to improve the interception performance of the trailing
missile by the leading missile collecting the superior information. The aforementioned cooperative guidance schemes based
on multiple missile interceptors must cost much more than a
guidance scheme based on a single missile interceptor. Moreover, it is hard to realize the real time communications between
multiple missiles.

https://doi.org/10.1016/j.cja.2021.05.021
1000-9361 Ó 2021 Chinese Society of Aeronautics and Astronautics. Production and hosting by Elsevier Ltd.
This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

Optimal predictive sliding-mode guidance law
The cost of a near space interceptor missile must be much
more than an endo-atmospheric interceptor missile. Therefore,
herein we study the design of a terminal guidance law for a single missile intercepting near-space hypersonic maneuvering
targets. In the terminal guidance process of a near space interceptor, its flight path is controlled by reaction control thrusters. So it is necessary to consider the energy optimization of
the interceptor. However, there are two main difficulties to
design an optimal guidance law in this case. The first one is
that traditional optimal guidance laws will become suboptimal when the target is maneuvering.7,8 Especially when
the maneuver is strong enough, those guidance laws do not
show any advantage over the Proportional Navigation guidance law. The second difficulty is that, in near-space, the maximum value of the acceleration of a near-space hypersonic
flight vehicle is even close to that of an interceptor missile. This
usually causes the saturation of the control input of the interceptor and then the failure of an interception. In addition, The
guidance strategies based on game theory was proposed using
energy consumption as an indicator of a game.9–11 However,
the above game guidance strategies are so complicated that
their heavy calculation burden is hard to meet the requirement
of real time applications.
If the target acceleration can be well predicted and the prediction is applied to the design of a guidance law, the above
two difficulties can be overcome. Guo et al.12 introduced the
ZEM and Zero Effort Velocity (ZEF) into the design of a guidance law and obtained an optimal solution in a uniform gravitational environment. Tournes et al.13 and Kumar et al.14
designed a predictive guidance law for intercepting ballistic
missiles according to the characteristics of ballistic missile in
different flight segments. For the near-space interception problem, the above methods are not applicable because the target
aircraft is also subject to aerodynamic forces. Therefore, how
to predict the acceleration of the near-space aircraft has
become the first problem to be solved in this work. The traditional Singer model15 and multi-model filtering methods16,17
can only be used to estimate the acceleration of target, but cannot be used to predict the target acceleration. Aiming at estimating the acceleration of a periodically maneuvering nearspace vehicle, Li and Xiong18 proposed an Adaptive Nonzero Mean Sine Wave (ANM-SW) model. This model is based
on the Constant Jerk model, but it needs the prior information
on the target maneuvering period and maximum maneuver
acceleration. In addition, the model does not propose a
method to predict the target states. Maeder et al.19 used
Interacting-Multiple-Model (IMM) method to estimate the
target states and then extrapolate. However, in the models
used in the above two methods, the aerodynamic force, an
important factor that affects the flight status of the nearspace vehicle, is not considered. Li et al.20 proposed a trajectory prediction method based on aerodynamic acceleration
empirical mode decomposition Li and Jilkov21 introduced a
target maneuvering model that includes aerodynamic force
to estimate the target accelerations generated by lift and drag,
respectively. In order to obtain the prediction of the acceleration of a near-space hypersonic target, we first propose a
maneuvering model related to aerodynamic parameters
inspired by Li and Jilkov.21 Then, we can get the predicted
value by using the aerodynamic parameters to train two neural
networks. Recurrent Neural Network (RNN)22 and General
Regression Neural Network (GRNN),23 which have good per-

321
formance in learning nonlinear features on time series, are used
to complete this task. However, using this method to obtain a
more accurate prediction requires a sufficiently long observation time. Therefore, an auxiliary observer is necessary to track
the target before the target seeker on the interceptor finds the
target.
When the predicted value of the target acceleration is
obtained, it can be applied into the calculation of the ZEM to
obtain the predicted ZEM. Using the predicted ZEM, we can
get the analytical expression of the optimal guidance law
according to the optimal theory.8 However, due to the errors
between the prediction of acceleration and the actual value,
the guidance law is required to have strong robustness. A
purely optimal guidance law does not have this characteristic.
Pang and Wang24 and Dong et al.25 introduced the optimal
sliding mode control of uncertain systems. To improve the
robustness of an optimal guidance law, Zhou et al.26 combined
optimal control with sliding-mode control and proposed an
optimal sliding-mode guidance law. Wang et al.27 proposed a
stochastic sliding mode guidance law based on optimal control
theory considering measurement noise. Refs.26,27 had ignored a
very important feature (actuator saturation), which always
causes performance deterioration and even system instability
in practical systems. Goebel28 and Lebedev29 considered the
saturation problem of optimal control and sliding mode control
respectively. There is also a problem worth noting that the disturbance, caused by the prediction error in the ZEM dynamic
equation, is a function related to the acceleration prediction
error and time to go. Therefore, it is difficult to have priori
knowledge on its upper bound. Zhu et al.30 applied an adaptive
sliding mode control method to aircraft attitude control system
where the adaptive sliding-mode term solves the problems of
parameter uncertainty and unknown disturbance. In this work,
we consider designing an adaptive sliding mode switch term to
deal with the prediction error and actuator saturation.
This paper is organized as follows. In Section 2, a new
maneuver model for the near-space vehicle is given. Based
on the model, a method for predicting the target acceleration
is introduced. In Section 3, by decoupling the 3-dimensional
nonlinear guidance model, an optimal sliding mode guidance
(OSMG) law is introduced. By designing an adaptive sliding
mode switch term, the stability of the guidance law is ensured
in the presence of prediction errors and actuator saturation.
Then, some numerical simulations in Section 4 demonstrates
the effectiveness of OSMG law. Finally, some concluding
remarks will be made in Section 5.
2. Prediction of target acceleration
The acceleration of a near-space hypersonic flight vehicle are
generated by aerodynamic force. It has various forms of
maneuver with long periods. Some traditional maneuver models, such as Singer model,15 Constant Acceleration (CA)
model,19 and Constant Jerk (CJ) model,18 are difficult to exhibit the property of the maneuver of a near-space hypersonic
vehicle. This makes it difficult to predict the future accelerations of a near-space hypersonic vehicle by using a traditional
maneuver model even if the acceleration can be estimated via
using this model. In this section, we introduce a maneuver
model in terms of aerodynamics to predict the accelerations
of a near-space aircraft by iterative calculations.

322

B. ZHANG, D. ZHOU

For a near-space vehicle, the acceleration is generated by
the aerodynamic force, which is projected into the flight path
coordinate system. The origin of the flight path coordinate system is located at the target mass center. The Xfli-axis points to
the direction of target velocity, the Yfli-axis is perpendicular to
the Xfli-axis and located in the plane which contains the Xfliaxis and is vertical to the horizontal plane, and the Zfli-axis
is determined according to the right-handed rule. In addition,
the prediction of the target acceleration needs to be completed
by a filtering and predicting algorithm. The value of the predicted acceleration is related to the observation inertial coordinate system, which is presented in Fig. 1. The origin ‘‘O” of the
inertial coordinate system is located at the position of the
observer (radar). The Y-axis is the connection between the
earth center and the observer while the positive direction
pointing to the sky. The X-axis is perpendicular to the Y-axis
and located in the plane containing the earth center, the observer and the initial target position. The Z-axis is determined
according to the right-handed rule.
Firstly, three state variables are introduced as
8
Zx ¼  CmxTS
>
>
<
C S
Zy ¼ myT
ð1Þ
>
>
: Z ¼ Cz S
z

mT

where Cx ; Cy ; and Cz are three coefficients of the components
of aerodynamic force along the flight path coordinate system,
mT is the mass of the target, and S is the equivalent reference
area of the target.
Choose
the
state
variable
as

T
X ¼ x; y; z; vx ; vy ; vz ; Zx ; Zy ; Zz , where ½x; y; zT is the
target position vector under the inertial coordinate system,

T
and vx ; vy ; vz is the velocity vector. Suppose Zx ; Zy ; and
Zz are slow time-varying random variables. The target model
under the inertial coordinate system can be written as
8
x_ ¼ vx
>
>
>
>
>
y_ ¼ vy
>
>
>
>
>
_ ¼ vz
z
>
>
>
>
>
>
< v_x ¼ aTx ¼ ax  gx
ð2Þ
v_y ¼ aTy ¼ ay  gy
>
>
>
v_z ¼ aTz ¼ az  gz
>
>
>
>
>
>
Z_ x ¼ kZx þ wx
>
>
>
>
>
> Z_ y ¼ kZy þ wy
>
: _
Zz ¼ kZz þ wz
In Eq. (2), k is a positive constant; wx ; wy ; and wz are white
noises, representing slow time-varying random processes;
2 3
2 3
Zx
ax
1
6 7
7
i 6
ð3Þ
4 ay 5 ¼ qv2 Cfli 4 Zy 5
2
az
Zz
and
2

3

2

lx
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ffi
3

3

7
6
gx
7
6
lðyþRe Þ
7
6 7 6 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ffi
g ¼ 4 gy 5 ¼ 6
3 7
2
2
2
6 ðx þðyþRe Þ þz Þ 7
5
4
lz
gz
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ffi
ðx2 þðyþRe Þ2 þz2 Þ

3

ðx2 þðyþRe Þ2 þz2 Þ

ð4Þ

Fig. 1 Inertial coordinate system and line of sight coordinate
system.

represent the acceleration vectors generated by aerodynamic
forces and gravity, respectively, where q is the density of air,
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
v ¼ v2x þ v2y þ v2z
is the velocity of the target,
l ¼ 3:98199  1014 ; Re is the earth radius, and Cifli is the transform matrix from the flight path coordinate system to the inertial coordinate system, which can be calculated by the
following equation:
2
3
coshcoswv sinhcoswv sinwv
6
7
Cifli ¼ 4
sinh
cosh
0 5
ð5Þ
coshsinwv
sinhsinwv
coswv
and
(
vy
ffi
h ¼ arctan pffiffiffiffiffiffiffiffi
2
2
vx þvz

wv ¼ p þ arctan vvxz

ð6Þ

In Eq. (3), the target acceleration is in terms of parameters
q; v; Cifli ; and the gravity g. Substituting Eqs. (3) and (6) into
Eq. (2), the maneuvering target model can be written as
8
x_ ¼ vx
>
>
>
>
>
y_ ¼ vy
>
>
>
>
>
>
_ ¼ vz
z
>
>

>
>
>
_ ¼ 1 qv2  ðZx cosh  coswv  Zy sinh  coswv þ Zz sinwv
v
>
> x 2
>
lx
>
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ffi
>
>
>
3
>
2 þðyþR Þ2 þz2
x
>
ð
Þ
e
>
>
>

>
<
lðyþRe Þ
ffi
v_y ¼ 12 qv2  ðZx sinh  Zy cosh  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
3
2
x þðyþRe Þ2 þz2 Þ
ð
>
>
>

>
>
> v_z ¼ 12 qv2  ðZx cosh  sinwv þ Zy sinh  sinwv þ Zz coswv
>
>
>
>
lx
>
ffi
 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
>
>
3
>
2
>
x þðyþRe Þ2 þz2 Þ
ð
>
>
>
>
>
>
Z_ x ¼ kZx þ wx
>
>
>
>
> Z_ y ¼ kZy þ wy
>
>
>
: _
Zz ¼ kZz þ wz
ð7Þ
Eq. (7) is a nonlinear model. It is natural to use the
Extended Kalman Filtering (EKF) method to estimate X.

Optimal predictive sliding-mode guidance law
State equations and measurement equations can be described
as
(
X_ ¼ fðXÞ þ W
ð8Þ
Y ¼ hðXÞ þ V
In Eq. (8), X 2 R9 ; Y 2 R3 is the measurements of observer, W and V are white noise matrixes, fðXÞ is the state equation matrix determined by Eq. (7), and hðXÞ is the
measurement matrix given by
3
2
x 0 0 0 0 0 0 0 0
7
6
hðXÞ ¼ 4 0 y 0 0 0 0 0 0 0 5
0 0 z 0 0 0 0 0 0


^ ¼ x^; y^; z^; v^x ; v^y ; v^z ; Z^x ; Z^y ; Z^z T for the
The estimate X
state X in the target tracking phase can be obtained by the
EKF. According to Eq. (7), if the predicted values Zxp ; Zyp ;
and Zzp of Zx ; Zy ; and Zz can be obtained, then the predictions
of target position, target velocity, and target acceleration can
be calculated step by step. In this paper, the subscript ‘‘p” represents the prediction of the corresponding variable.
Without loss of generality, take the calculation of Zzp as an
example to introduce the process of obtaining Zxp ; Zyp ; and


Zzp . Firstly, use the estimate sequence Z^z ðtÞ; p0 P t P k0 ,
which has been obtained in the target tracking phase, as the
input of a neural network, where k0 is a reliable instant used
to judge whether the filter has reached a steady state and to
prevent the unreasonable estimated values affecting the neural
network, and p0 is the instant when the prediction starts. Then,
the neural network will output a time-related function Zzf ðtÞ.
When t > p0 , Zzf ðtÞ is regarded as the predicted value of Zz ,
i.e., Zzp ¼ Zzf ðtÞ; t > p0 ;.
Definition 1. k0 is the reliable instant, if the following equations
8 Z^ ðk þ1ÞZ^ ðk Þ
>
 z 0Z^z ðk0 Þ z 0  < d
>
>
>

>
>
>
ÞZ^z ðk0 þ1Þ
< Z^z ðk0 þ2
<d
^
Z ðk þ1Þ
z

323
Z^z ðk þ 1Þ is the residual, i.e., E(k), which is the input of
GRNN. The output of the GRNN Ep(t), is the prediction of
the residual EðkÞ. The prediction Zzf ðtÞ contains the preliminary prediction and residual prediction.
Similarly, we can get Zxp and Zyp . Applying Zxp , Zyp and
Zzp into Eq. (7), we can get the predictions of the target accel
T 
T
erations aTp ¼ aTxp ; aTyp ; aTzp ¼ v_xp ; v_yp ; v_zp .
This prediction method requires just the target position
measurements continuing for a long enough period of time.
Simulations under different scenarios show that when using
this model to estimate the states of a near-space target, it takes
about 30 s for each axial acceleration estimation error converging to a bound within 1 m=s2 . Details about this part will be
explained in Section 4. In the terminal guidance stage, it is difficult for the interceptor to obtain the acceleration prediction
of the target by itself, and so an auxiliary radar is required
to observe the target and then send the target acceleration predictions to the interceptor.
Remarks 1. For a near-space hypersonic vehicle in the cruising
phase, its maneuverability is provided by aerodynamics. According to Eqs. (1) and (7), it can be seen that the aerodynamic
acceleration is related to the aircraft’s position, velocity,
aerodynamic coefficient, mass, and equivalent reference area.
The mass and the equivalent reference area can be regarded as
constants and the aerodynamic coefficient is determined by the
aircraft’s velocity, attack angle, and sideslip angle. Compared
with Singer model, the model proposed in this paper takes into
account the specific factors that affect the acceleration. So, the
estimation for target acceleration is more accurate. In addition,
it is more reasonable to regard Zx ; Zy ; and Zz instead of
acceleration as slow time-varying random processes because in
the terminal stage of its cruising phase the aircraft is not liable to
change its maneuver scheme due to the constraint on its landing
conditions.
3. Optimal sliding-mode guidance law

0

>
..
>
>
>
>

> . ^
>
: Zz ðk0 þiÞZ^z ðk0 þi1Þ < d
Z^z ðk0 þi1Þ

hold, where i is a positive integer and d is a positive constant.
Herein we use a neural network which combines RNN and
GRNN to obtain Zzf ðtÞ. The specific process is shown in
Fig. 2. The RNN is used to predict the value of Zz where
Zzp1 is a preliminary prediction of Zz . The difference between
the preliminary prediction Zzp1 ðk þ 1Þ and the estimated value

The line of sight coordinate system (Xlos-Ylos-Zlos), as shown in
Fig. 1, depends on the position of the interceptor and the target at the current time. The guidance dynamics in the line of
sight coordinate system is represented by
8
>
R€ ¼ Rq_ 2e þ Rq_ 2b cos2 qe þ aTR  aIR
>
>
>
>
2
>
_
>
< q€e ¼ 2Rq_ e =R  q_ b cosqe sinqe  ðaIe  aTe Þ=R
ð9Þ
q€b ¼ 2R_ q_ b =R þ 2q_ e q_ b tanqe þ ðaIb  aTb Þ=Rcosqe
>
>
>
> a_ Ie ¼ ðaIe  ue Þ=se
>
>
>
:
a_ Ib ¼ ðaIb  ub Þ=sb
where R is the relative range between the interceptor and the
target; qe and qb are the line of sight angles; aTR ; aTe ; and aTb
are the target accelerations along the line-of-sight axes;
aIR ; aIe ; and aIb are the interceptor accelerations along the
line-of-sight axes; ue and ub are the control inputs; se and sb
are time constants of the interceptor.
In this problem, aIR is not controllable. We convert the
nonlinear model (9) to a linear model. Then, the 3dimensional guidance problem becomes two uncoupled 2dimensional problems. Taking the longitudinal plane as an

Fig. 2

Process to get the prediction of Zz

324

B. ZHANG, D. ZHOU

example, the optimal sliding mode guidance law considering
the target acceleration prediction will be introduced.
3.1. Optimal guidance law considering the acceleration
prediction

t

ð11Þ

where tf is the final interception time,
ð12Þ

and R_ can be estimated from the model in Section 3. That is
because the velocities of the interceptor and the target are
mainly in the line-of-sight direction. Additionally, it is necessary to get the prediction of target acceleration aTep . In Section 2, we obtain the prediction of target acceleration aTp ,
which is defined under the inertial coordinate system, and
aTep can be calculated easily as follows:
2
3
aTRp
6
7
l
ð13Þ
4 aTep 5 ¼ Ci  aTp
aTbp
where
2

cosqe cosqb
6 sinq cosq
l
Ci ¼ 4
e
b
sinqb

sinqe
cosqe
0

3
cosqe sinqb
sinqe sinqb 7
5

ð15Þ

wðtgo =se Þ ¼ expðtgo =se Þ þ ðtgo =se Þ  1

The relative dynamics between the interceptor and the target in the longitudinal plane are shown in Fig. 3, where qe0 is
the initial line-of-sight angle and Ze is the component of the
current relative range perpendicular to the initial line of sight.
To apply the theory of optimal guidance law, the purpose of
control is to make the predictive ZEM converge to 0.
The predictive ZEM is defined as
Z tf Z s
ZEMep ðtÞ ¼ Ze þ Z_ e  tgo  aIe s2e wðtgo =se Þ þ
aTep ðs1 Þds1 ds
t
t
Z tf Z s
¼ R_ q_ e t2go  aIe s2e wðtgo =se Þ þ
aTep ðs1 Þds1 ds

tgo ¼ R=R_

_ ep ðtÞ ¼ ue se wðtgo =se Þ
ZEM
where

When qe and qb are small, we can get the linearized dynamic
model of Eq. (9) in the longitudinal plane as follows:
(
q€e ¼ 2R_ q_ e =R  ðaIe  aTe Þ=R
ð10Þ
a_ Ie ¼ ðaIe  ue Þ=se

t

Then, we can get

ð14Þ

cosqb

is the transform matrix from the inertial coordinate system to
the line of sight coordinate system

ð16Þ

8

According to Ref. , the energy optimal guidance law ueopt
can be obtained as
ueopt ¼ Nopt

ZEMep
t2go

ð17Þ

where
Nopt ¼

6ðtgo =se Þ2 wðtgo =se Þ
2

3

3 þ 6ðtgo =se Þ  6ðtgo =se Þ þ 2ðtgo =se Þ  3expð2tgo =se Þ  12ðtgo =se Þexpðtgo =se Þ
ð18Þ

Remarks 2. The existing optimal guidance laws for intercepting
a maneuvering target usually ignore the target acceleration
during the design process, i.e., the ZEM is defined as
ZEMðtÞ ¼ R_ q_ e t2go  aIe s2e wðtgo =se Þ

ð19Þ

To compensate for the target acceleration, an additional
compensation term is introduced into the guidance law, which
makes the guidance law suboptimal. The suboptimal guidance
law is designed as
uesopt ¼ Nopt

ZEM
þ a^Te
t2go

ð20Þ

where a^Te is the estimated value of the target current acceleration. But in this work, considering the characteristics of the
near-space vehicle, we directly use the target acceleration predictions to design the optimal guidance law.
Remarks 3. The third term in Eq. (11) is the integral of the
target acceleration prediction, which is difficult to be calculated.
If qe  qe0 holds, the integral term can be obtained through the
acceleration prediction model (7). Firstly, we can get the
predicted target position at the final time, i.e.,
h
iT
xp ðtf Þ; yp ðtf Þ; zp ðtf Þ
which has been calculated when we
predict the target acceleration. Then, let the target acceleration

T
prediction in system (7) becomes 0, i.e., v_x ; v_y ; v_z ¼ 0; we
can get the non-maneuvering target position at the final time, i.e.,
Rt Rs
½x0 ðtf Þ; y0 ðtf Þ; z0 ðtf ÞT . The integral term t f t aTep ðs1 Þds1 ds
can be approximated as
3
3
2
2

xp ðtf Þ  x0 ðtf Þ
7
6
6 R tf R s a ðs Þds ds 7
l
ð21Þ
4 t t Tep 1 1 5 ¼ Ci  4 yp ðtf Þ  y0 ðtf Þ 5
R tf R s
ðt
Þ

z
ðt
Þ
z
a
ðs
Þds
ds
p
f
0
f
Tbp
1
1
t
t
3.2. An adaptive sliding-mode switch term to deal with the
prediction errors and saturation

Fig. 3

2-dimensional guidance geometry.

Since the predicted values are used in the optimal guidance
law, strong robustness of the guidance law is necessary to deal
with prediction errors, estimation errors of the line-of-sight
angular rate and the actuator saturation. The sliding-mode
control method is a well-established robust control algorithm

Optimal predictive sliding-mode guidance law

325

that can handle bounded matched disturbances with different
forms. So, in this work, we design a sliding-mode switch term
to deal with the prediction errors and saturation.
When there exist prediction errors, the true value of ZEM is
Rt Rs
ZEMe ðtÞ ¼ Ze þ Z_ e  tgo  aIe s2e wðtgo =se Þ þ t f t aTe ðs1 Þds1 ds
R
t Rs
¼ R_ q_ e t2  aIe s2 wðtgo =se Þ þ f
aTe ðs1 Þds1 ds
go

e

t

But we can only get the measurement of ZEMe ðtÞ with prediction errors, i.e.,
ð23Þ
ZEMemea ðtÞ ¼ ZEMep ðtÞ ¼ ZEMe ðtÞ þ d2e ðtÞ
R tf R s
where d2e ðtÞ ¼ t t ðaTep ðs1 Þ  aTe ðs1 ÞÞds1 ds. The ideal objective is to make the ZEMe ðtÞ converges to 0. However, due to
the inability to obtain the value of ZEMe ðtÞ, the above objective cannot be achieved. When the distribution characteristics
^ e ðtÞ can be
of d2e ðtÞ are known, the estimated value ZEM
obtained by designing an observer. Furthermore, the stability
^ e ðtÞ.
theory of stochastic systems can be used to control ZEM
However, since d2e ðtÞ contains the information of the target’s
future acceleration, it is difficult to obtain its distribution characteristics. Observing the expression of d2e ðtÞ, we can find that
it has a special property, i.e., d2e ðtf Þ ¼ 0; which means
ZEMep ðtf Þ ¼ ZEMe ðtf Þ. Therefore, the control objective can
be converted to make ZEMep ðtÞ converge to 0.
When there exist prediction errors and input saturation, the
dynamic of ZEMep ðtÞ becomes
ð24Þ

where d1e ðtÞ ¼ ðaTep ðtÞ  aTe ðtÞÞ  tgo ; satðue Þ ¼ jðue Þ  ue ; where
jðue Þ is defined as
8
ue > umax
>
< umax =ue ;
umax 6 ue 6 umax
jðue Þ ¼ 1;
ð25Þ
>
:
umax =ue ;
ue < umax
where umax represents the maximum acceleration generated by
the interceptor.
There exists a constant d that makes the following
inequality
0 < d 6 minðjðuÞÞ 6 1

1
1 2 2
ZEM2ep þ d þ c
2
p


ð29Þ



^ c ¼ d  ^c1 , and d is defined in (26). Its time
where d ¼ d  d;
derivation is


^ þ c ^c2^c
_ ep  1 d d
V_ ¼ ZEMep  ZEM
p
¼ ZEMep  ðsatðueopt þ ues Þ  se wðtgo =se Þ þ d1 Þ





^ZEMep 
 d ZEMep  þ l c ^cd
6 d  ZEMep  ðueopt se wðtgo =se ÞÞ þ ZEMep d1

 



^ZEMep   d ZEMep  þ l c ^cd
^ZEMep 
 d  l^cd
6 d  ZEMep  ðueopt se wðtgo =se ÞÞ
|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
1




 



^ZEMep   d ZEMep  þ l c ^cd
^ZEMep 
þ dZEMep   d  l^cd
|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
2

ð30Þ

According to the optimal control theory, the part 1 in Eq.

(30) is less than 0. Applying c ¼ d  ^c1 into the part 2, we
have

 






dZEMep   d ZEMep   d  l^cd^ZEMep  þ l c ^cd^ZEMep 








¼ d^ZEMep   d  l^cd^ZEMep  þ d  l^cd^ZEMep   ld^ZEMep 


ð31Þ
¼ ð1  lÞd^ZEMep 


Since l > 1; d^ > 0; we have ð1  lÞd^ZEMep  < 0 if
ZEMep –0. The state ZEMep ðtÞ is uniformly asymptotically
stable. Above all, ZEMep will converge to 0 with the OSMG
law (27). h
Similarly, we can design the optimal sliding-mode guidance
law in the horizontal guidance plane.
Remarks 4. The OSMG law does not avoid the saturation of the
control input, but guarantees the stability under saturation,
making it possible to take full advantage of the interceptor’s
maximum maneuverability.

ð26Þ

holds.
The OSMG law has the following form:
ueOSMG ¼ ueopt þ ues

V¼

t

ð22Þ

_ ep ðtÞ ¼ satðue Þ  se wðtgo =se Þ þ d1e ðtÞ
ZEM

Proof of Theorem 1. Consider the following Lyapunov function:

ð27Þ

where ueopt is calculated from Eq. (17) and ues is the switch
term. Without loss of generality, suppose jd1e ðtÞj has an upper
bound d, i.e., jd1e ðtÞj 6 d. Then, the switch term ues can be
designed as
8
u ¼ l^cd^  sgnðZEMep Þ=ðse wðtgo =se ÞÞ
>
>
< es 

_
ð28Þ
d^ ¼ pZEMep 
>
>


:_
3 ^
^c ¼ l^c d ZEMep 
where l > 1; p > 0;^c is the adaptive gain, and d^ is the adaptive
^ > 0 and ^cð0Þ > 0.
estimate of d. The initial values dð0Þ
Theorem 1. For the ZEM system consisting of Eqs. (11) and
(24), ZEMep will converge to 0 with the OSMG law (27).

Remarks 5. In this work, the essence of OSMG law is to design
the optimal guidance law according to the target acceleration
prediction and to compensate for the prediction errors via using
the sliding mode switch term. However, when the target acceleration is predicted for a long period of time, a large acceleration
prediction error may be generated, making the energy consumption even greater than the case where no prediction is used.
Therefore, we can adjust the ZEMep to the following form:
ZEMep ðtÞ ¼ R_ q_ e t2go  aIe s2e wðtgo =se Þ
Z tf Z s
t  s1
 aep ðs1 Þds1 ds
exp
þ
c
t
t

ð32Þ

where c is a positive constant. If we have confidence in the target acceleration prediction, then we may assign a large number
to c; otherwise, assign a small number to it. When c ! 1, Eq.
(32) is equal to Eq. (11). When c ! 0, Eq. (32) is equal to Eq.
(19), meaning the guidance law doesn’t consider the predicted
information.

326

B. ZHANG, D. ZHOU

4. Numerical simulations
In this section, we select a hypersonic aircraft gliding in nearspace as the target. The lift-to-drag ratio of the aircraft is
about 3:1, and the velocity of the aircraft is about 12Mach.
The flying altitude is about 30 km to 40 km. The acceleration
estimation results for the target are given firstly. Then use
the estimated value of the target acceleration and the measurements of line-of-sight provided by the interceptor seeker to
estimate the line-of-sight angle rate required by the OSMG
law. The simulation results of target state estimation are all
compared with the results of using Singer model, ANM-SW
model,16 and IMM filter,17 respectively. The target state prediction results will be compared with the IMM prediction
method.17 Then, the interceptor uses OSMG law to intercept
the target. We consider a more complicated situation, that is,
assuming that the target discovers the interceptor and makes
an abrupt escaping maneuver with a maximum constant acceleration at ðtf  1Þ s. The interception results under the OSMG
law are compared with that under the Augmented Proportion
Navigation (APN) law and the Optimal Guidance Law
(OGL). Finally, we will discuss the convergence time of the
proposed model for state estimation of near-space targets,
and analyze the influence of large tracking errors within short
tracking time on the predictive guidance law.
The maximum value of the interceptor is set to be 25 m=s2 ,
and the time constants se ¼ sb ¼ 0:1. The starting time of the
interceptor’s terminal guidance is taken as t = 0 s. We start
estimating the target acceleration from t =  96 s and predicting it from t = 0 s to obtain the ZEMep ðtÞ and
ZEMbp ðtÞ. The information of interceptor and target at
t = 0 s is shown in Table 1.
For simplicity, we use ‘‘new model” to refer to the nearspace vehicle maneuver model proposed in Eq. (7). Fig. 4
shows the estimated values of the target acceleration where it
is seen that when Singer model, ANM-SW model, and IMM
method are used to estimate the target acceleration, although
the initial values are better than the new model, their results
are much worse than the new model. Since the aerodynamic
force is not introduced in the models, the variances of the estimated value calculated by Singer model, ANM-SW model,
and IMM method respectively will be much larger than that
of the new model. When using the new model to estimate
the target acceleration for 30 seconds, the estimation errors
are always less than 1 m/s2. To illustrate this specifically, we
regard the value of the target acceleration, i.e.,
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
aT ð0Þ ¼ a2Tx ð0Þ þ a2Ty ð0Þ þ a2Tz ð0Þ; as the independent vari-

Fig. 4

get acceleration is less than 20 m/s2, the acceleration estimation
error converges to a bound less than 1 m/s2 in 40 s. According
to Eq. (9), the dynamics of the line-of-sight angular rate
includes the target acceleration. Therefore, the more accurate
the estimated values of the target acceleration, the more accurate the estimated values of the line-of-sight angular rate can
be obtained. Eqs. (11) and (17) indicate that the realization

able, and the time for the acceleration estimation error converging to a required bound, i.e., tes ; as an indicator to
measure the convergence time of the new model. Fig. 5 shows
the relationship between at ð0Þ and tes while Z^x ð0Þ; Z^y ð0Þ; and
Z^z ð0Þ equal to 0. The sampling interval of aT ð0Þ in Fig. 5 is
0.5 m/s2. As shown in Fig. 5, when the initial value of the tar-

Table 1

Estimations of target acceleration.

Initial information of interceptor and target.

Aircraft

x(m)

y(m)

z(m)

vx(m/s)

vy(m/s)

vz(m/s)

Interceptor
Target

0
75000

0
2500

0
400

1900
3700

30
200

550
500

Optimal predictive sliding-mode guidance law

327

Fig. 5 Time when the acceleration estimation error is stable
within 1 m/s2.

of OSMG law requires line-of-sight angular rate, but only the
line-of-sight can be measured by the seeker of the interceptor.
Therefore, it is necessary to estimate the line-of-sight angular
rate based on the estimated value of the target acceleration.
Fig. 6 shows the effects of using different models to estimate
target acceleration on the estimation of line-of-sight angular
rate. It can be seen that the effect of using the acceleration estimated by the new model to estimate the line-of-sight angular
rate is also better.
Since the OSMG law also requires target prediction information, it is necessary to predict the target acceleration at
the beginning of the terminal guidance process. To study the
impact of large estimation errors caused by short state estimation time on prediction and the OSMG law, the following simulations consider the situation that the state estimation is run
at the instant 10 s before the interceptor’s terminal guidance
process starts. The prediction results of the target acceleration
during the terminal guidance phase at t = 0 s is shown in
Fig. 7. In the simulations, in addition to the use of neural networks, another method is also used to obtain the acceleration
predictions. This method regards the Zxp ; Zyp ; and Zzp as constants during the prediction process, which equals to the last
Z^x ; Z^y ; and Z^z obtained before the start of the prediction,
respectively. Since only the last information is used and all

Fig. 6

the previous information is discarded, this method is greatly
affected by the estimation errors at the last moment. In addition, it is difficult to obtain the trend of Zx ; Zy ; and Zz by simply using the estimated values at the last moment. When the
prediction horizon is longer, the prediction errors will increase
more rapidly. For the situation where the target is tracked
throughout the simulation process, it is obvious that the prediction error generated by the IMM method diverges significantly faster than that generated by the new model proposed
in this paper, regardless of whether the new model uses a neural network. In the case of tracking the target only 10 s before
the start of the terminal guidance process, the acceleration prediction errors will become larger. However, the trend of the
acceleration prediction change is closer to the real situation
than the acceleration prediction value obtained by the IMM
method. Eq. (21) indicates that the prediction errors of the target position at t ¼ tf determine the errors between the predictive ZEM and the true value of ZEM. Denote the position of
target at t ¼ tf as ½xðtf Þ; yðtf Þ; zðtf Þ. Fig. 8 shows the predic

tion errors xðtf Þ  xp ðtf Þ; yðtf Þ  yp ðtf Þ; zðtf Þ  zp ðtf Þ as the
terminal guidance process proceeds. Since the estimations of
the target states and the predictions of the target acceleration
are constantly updated, the prediction errors of the ZEM are
converging.
Next, we use OSMG law, APN law, and OGL in interception simulations. Fig. 9 shows the control inputs under different guidance laws. The trajectories of the interceptor using
different guidance laws are shown in Fig. 10. In fact, the
OSMG law has a great advantage in energy consumption.
To illustrate this specifically, an integral function
Rt
Je ¼ 0 f k u k2 dt is selected as an index of energy consumption.
The energy consumption and final miss distances of the above
three guidance laws are shown in Table 2 where the final Miss
Distance (MD) under the OSMG law is only 0.16 m while that
under the APN law and OGL is 0.69 m and 0.56 m, respectively. Even if the target tracking is carried out 10 s before
the start of terminal guidance, the final miss distance is only
0.49 m. Thus, the OSMG law is more accurate than the two
other guidance laws. Moreover, the energy consumption is less
than the other two guidance laws significantly. However, the
larger state estimation errors caused by the shorter state estimation time will significantly increase the final miss distance
and energy consumption.

Estimations of the line-of-sight angular rate.

328

B. ZHANG, D. ZHOU

Fig. 7

Predictions of the target acceleration.

For the interceptor working in the near-space, the maneuverability of the interceptor does not have an apparent advantage over the maneuverability of target. From Fig. 9, it is
obvious that in the most time of terminal guidance process,
the magnitude of acceleration under the OSMG law is less
than that under the APN law and OGL, and so the energy consumption in the terminal guidance process under the OSMG
law is much less than that under the APN law and OGL.
Next, consider a more complicated situation that the target
finds the interceptor at ðtf  1Þ s and makes an evasive maneuver. Fig. 11 shows the control inputs of the three guidance laws
in this case where the target lateral acceleration aTb after the

Fig. 8 Prediction errors of the target final position as the
terminal guidance proceeds.

evasion maneuver is close to the interceptor’s maximum
maneuverability. That makes it difficult to intercept the target.
The simulation results of the three guidance laws when the target makes an evasive maneuver are shown in Table 3 where the
final miss distance under the OSMG law is only 0.79 m while

Optimal predictive sliding-mode guidance law

Fig. 9

329

Control inputs of interceptor.

that under the APN and OGL is 3.99 m and 3.07 m, respectively. Thus, the advantage of OSMG law in accuracy is apparent compared with the two other guidance laws. Moreover, the
energy consumption is still less than the other two guidance
laws in this case with abrupt target maneuver. However, the
energy consumption of OSMG law in this case with abrupt target maneuver is obviously larger than that in the previous case
without abrupt target maneuver, because the abrupt target
maneuver had not been predicted.
We have carried out 100 runs Monte Carlo simulation and
counted the energy consumptions and the final miss distances
of the three guidance laws. The results are shown in Table 4.
The subscripts ‘‘min”, ‘‘max” and ‘‘ave” represent the maximum, minimum, and average values, respectively. It is shown
in Table 4 that the maximum miss distance under the OSMG

Table 2

Simulation results.

Guidance law

MD (m)

Je

APN
OGL
OSMG
OSMG (10 s)

0.6966
0.5655
0.1619
0.4917

1781.8
1683.6
1179.9
1588.6

Fig. 11

Table 3 Simulation results when the target makes an evasive
maneuver.
Guidance law

MD(m)

Je

APN
OGL
OSMG
OSMG(10 s)

3.9959
3.0774
0.7973
2.2616

1881.8
1805.1
1285.7
1744.6

Control inputs of the interceptor when the target makes an evasive maneuver.

330
Table 4

B. ZHANG, D. ZHOU
100 Monte Carlo simulation results.

Guidance law

MDave(m)

MDmax(m)

MDmin(m)

Jeave

Jemax

Jemin

APN
OGL
OSMG
OSMG(10 s)

4.0018
3.2878
0.6859
2.6178

4.5361
3.9816
0.8271
4.8369

3.1228
2.6652
0.5871
1.6897

1850.9
1788.6
1316.1
1721.1

1902.1
1832.2
1351.4
1840.5

1796.4
1733.7
1235.3
1664.6

law is 0.82 m, while the minimum miss distance under the APN
law and that under the OGL is 3.12 m and 2.66 m, respectively.
The OSMG law avoids some unnecessary maneuvers so that
its control inputs are smaller than that of the other two guidance laws, making the interceptor be more capable of dealing
with model errors and abrupt target maneuvers at the final
stage of terminal guidance. However, when the time for state
estimation is short, a large final miss distance may occur. In
addition, the advantage of the OSMG law in energy is no
longer obvious.
Through the above simulation results, we can conclude that
the OSMG law can effectively reduce the energy consumption
of the interceptor if the tracking of the target can be carried
out at least 30 s before the start of terminal guidance. In addition, it has a great advantage in intercepting maneuvering targets whose maneuverability is comparable to that of the
interceptor.
5. Conclusions
In this paper, we propose an optimal sliding-mode guidance
law based on a target acceleration prediction model. By using
the proposed target acceleration prediction model, more accurate estimations for the target acceleration can be obtained,
and this is also beneficial to obtain more accurate line-ofsight angular rate estimations. By using the neural networks
to predict the trend of variations of aerodynamic parameters,
the prediction of target acceleration is obtained and applied
to designing the optimal sliding-mode guidance law. There
are three main advantages for the designed guidance law. (A)
The stability of the guidance system in the presence of prediction errors and actuator saturation is guaranteed by introducing an adaptive sliding-mode switch term. The switch term also
improves the robustness of optimal control. As a result, for
intercepting a maneuvering target, especially a target with an
abrupt target maneuver, the OSMG law has a smaller final
miss distance than the APN law and OGL. (B) This guidance
law makes the interceptor avoid unnecessary maneuvers by
using the predicted information, and so the energy consumption is much less than other guidance laws without using the
predicted information. (C) In most time of the terminal guidance process, the control inputs under OSMG law is smaller,
making the interceptor be more capable of dealing with model
errors and the abrupt target maneuver at the final stage of terminal guidance.
Declaration of Competing Interest
The authors declare that they have no known competing
financial interests or personal relationships that could have
appeared to influence the work reported in this paper.

Acknowledgement
This study was supported by the National Natural Science
Foundation of China (No. 61773142).
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