The Design of Finite-time Convergence Guidance Law for Head
Pursuit based on Adaptive Sliding Mode Control
Cheng Zhang
1,2
, Ke Zhang
1,2
and Jingyu Wang
1,2
1
School of Astronautics, Northwestern Polytechnical University, Xi’an 710072, China
2
National Key Laboratory of Aerospace Flight Dynamics, Northwestern Polytechnical University, Xi’an 710072, China
Keywords: Finite-time Convergence, Head Pursuit, Terminal Sliding Mode, Sliding Mode Control.
Abstract: The high-speed target interception plays a vital role in the modern industry with many application scenarios.
Due to the difficulties of direct interception in high speed, the head pursuit intercept is frequently considered
for the target of re-entering flight vehicle. In this paper, a novel terminal sliding mode control method is
proposed for the interception of high-speed maneuvering target, in which the finite convergence guidance
law is initially designed under the constraint of intercept angle. By introducing the mathematical model of
correlative motion between interceptor and target, the sliding surface is researched and designed to meet the
critical conditions of head pursuit intercepting. Meanwhile, considering the dynamic characteristics of both
interceptor and target, an adaptive guidance law is therefore proposed to compensate the modelling errors,
which goal is to improve the accuracy of interception. The stability analysis is theoretically proved in terms
of the Lyapunov method. Numerical simulations are presented to validate the robustness and effectiveness
of the proposed guidance law, by which good intercepting performance can be supported.
1 INTRODUCTION
In recent years, high-speed targets such as re-
entering flight vehicle have brought severe
challenges to the traditional interception system.
Hypersonic vehicle target requires higher accuracy
of the interception system, such as long distance
target detection, the quick response of each
subsystem in the system, the ability to withstand
high temperature of seeker and the ability of the
guidance system to adapt to harsh aerodynamic
environment interference and so on.
Generally speaking, the requirement for the
speed of interceptor is higher than the target in the
terminal interception process, it is therefore not very
suitable for the traditional rear-end interception issue.
However, considering the head-on interception, due
to the extreme high speed of the target, it will cause
the missile - target relative speed become very large.
Therefore, it will cause more stringent requirements
to the computation capacity of the computer as well
as other hardware devices on-board. Since all these
requirements cannot be achieved in a very short time,
the designing of appropriate control method will be
a practically realizable way.
To solve the problem of interception in the high
speed situation, the head pursuit(HP) interception is
proposed (Shima, 2007). By using the HP
interception, not only the seeker can be protected
from the vital influence of high temperature and bad
aerodynamic interference, but also the energy
consumption of the interceptor can be reduced.
Therefore, the research of HP interception has
attracted huge attention in recent years.
As a frequently used design method in control
system, the flexibility of sliding mode control (SMC)
is strong. As the dynamic characteristic of a control
system is related to the designed sliding surface, so
when the control system is running on a specifically
designed sliding surface, the quality of which will
significantly affect the performance of the system.
In fact, as long as the system is assured to finally
approach the sliding surface and satisfy the stability
condition, there will be no limit to the form of used
sliding surface. Therefore, how to design a sliding
surface with better control performance has become
the focus of research, from which also derived a lot
of new SMC methods (Goyal, 2015). However,
traditional SMC approaches generally use the linear
sliding surface, thus the system is only asymptotic
convergence in this case. If the target is with
complex non-linearity property, good control
110
Zhang, C., Zhang, K. and Wang, J.
The Design of Finite-time Convergence Guidance Law for Head Pursuit based on Adaptive Sliding Mode Control.
DOI: 10.5220/0005987701100118
In Proceedings of the 13th International Conference on Informatics in Control, Automation and Robotics (ICINCO 2016) - Volume 2, pages 110-118
ISBN: 978-989-758-198-4
Copyright
c
2016 by SCITEPRESS – Science and Technology Publications, Lda. All rights reserved
performance cannot be achieved by applying the
traditional method or even worse, the stability of
system cannot be guaranteed. However, a terminal
sliding mode (TSM) control method that can
converge to zero in finite time is initially proposed
in (Zhihong, 1994), which has opened up a new
research field of SMC. Thereafter, the TSM based
control method has become one of the hot topics in
the research of SMC (Feng, 2013 and Kumar, 2014).
2 RELATED WORKS
In the research of sliding mode control method, the
essential key is to design the guidance law. Due to
the fact that relative motion equations of target and
interceptor have strong nonlinear cross coupling,
parameter uncertainty and external disturbance
inevitably exist, so the target interception system is a
nonlinear uncertain system.
To deal with the control issue of this complex
system, many research methods have been proposed
in recent years. In the research works of (Liu, 2010
and Li, 2011), two control methods are presented to
solve the control problem of the classic SISO and
MIMO nonlinear system, in which the variable
structure control and fuzzy control theory are used
respectively. Though the achieved results have
shown that good outputs can be obtained by
applying the proposed methods, the target systems
that considered in these works are simple.
Regarding the control problem of the complex
system in real world, a fuzzy terminal sliding mode
control strategy is proposed in (Gu, 2015) for the
control of marine current power system. The
proposed control strategy mainly consists of two
major parts, which are a fuzzy logic controller for
deriving the reference generator current and a non-
singular terminal sliding mode controller to track the
aforementioned derived current, respectively.
Meanwhile, in the work of (Zhao, 2015), an output
feedback terminal sliding mode control framework
for the continuous stirred tank reactor is presented.
In this work, two novel control algorithms of TSM
are studied, of which the theoretical stability
analysis is given. The experimental results of above-
mentioned works show that, if appropriate designed,
the TSM based control approaches can be effective
solutions to deal with the nonlinear system, while its
robustness to external disturbance is superior to the
traditional research works. However, the high-speed
target is a highly nonlinear system which far more
complex than the system considered in the industrial
application scenario. Therefore, the presented works
have obvious limitations and thus cannot be applied
on the interception, which bring out the requirement
of novel control method.
In the work of (Behera, 2015), the steady-state
behaviour of discretized TSM control is
comprehensively studied, in which the continuous-
time system is discretized and the analysis of steady-
state behaviour is given in the terms of periodic
orbits. By finding all possible orbits, the research
reveals that only period-2 orbit is satisfied by the
state evolution and thus retained in steady-state.
Though the analytic result is achieved by only
considering a second-order system, it can be
developed for a nth order SISO system as well. This
work shows that the continuous-time system can be
theoretically researched by conducting a discretized
processing, which supports promising application in
the design of TSM control method for higher order
nonlinear complex system. Moreover, a non-singular
TSM controller is proposed for actuated exoskeleton
in (Madani, 2015), of which the target system is
actually an active orthosis used for rehabilitation
purpose. Since the system has a complex dynamical
model with no prior knowledge, the proposed
controller uses non-singular TSM technique to
obtain the expected good performance in terms of
convergence in a finite time. Supported by the
theoretical stability proof in closed loop, the
proposed approach is ensured to be stable according
to the Lyapunov formalism. Experimental results
have shown that smooth movements could be
conducted for flexion and extension, which indicates
the potential application in real world. However, it
should be pointed out that, the system considered in
this work is still not as complex as the target system
considered in the interception scenario, which shows
that improvement of previous proposed approaches
should be researched.
Considering the specific control target system of
spacecraft, a SMC approach is designed in (Lu,
2012), in which the existence of external
disturbances and inertia uncertainties are both taken
into account. The simulation results show that, the
proposed approach is capable to achieve the goal of
effectively tracking on the attitude of a spacecraft.
Meanwhile, a novel adaptive sliding mode control
scheme for spacecraft body-fixed hovering in the
proximity of an asteroid is proposed in (Lee, 2015).
Since the bounds of parametric uncertainties as well
as the disturbances are not required in advance, the
scheme is proposed to guarantee the asymptotic
stability of states for both position and attitude
stabilization. The achieved results of numerical
simulation have demonstrated both low control
The Design of Finite-time Convergence Guidance Law for Head Pursuit based on Adaptive Sliding Mode Control
111
inputs and effectiveness of control strategy. Though
both of these works have shown that sliding mode
control method is a practically useful tool for the
control issue of spacecraft, the targets considered in
these systems are regular spacecraft with normal
speed, which cause the limitations when apply them
on the HP interception problem.
Particularly, a HP sliding mode variable structure
guidance law is introduced in the work of (Hua,
2011), in which the deviation and related derivative
are used as switching function by using the
Lyapunov method. Although it can obtain better
results for the case of high-speed interception, much
more information of the system is required, which is
difficult to achieve. The research work accomplished
in (Song, 2015) is similar to our work, in which the
design of a guidance law for non-decoupling 3D
engagement geometry is studied. By using finite
time integral sliding mode manifold, an improved
control law is developed to estimate the upper bound
of the target acceleration. Experiment results show
that, the proposed control method can control the
target in both constant speed and varying speed
situations effectively. However, despite that the
target can be precisely intercepted by applying the
proposed method, the designed control laws are
specifically developed regarding the classic way of
interception which is different from HP interception.
Consequently, novel control method for the HP
interception system should be theoretically studied.
In this paper, by considering the target maneuvering
as bounded uncertain item of the system, an adaptive
sliding mode guidance law suitable for head pursuit
interception scenario is proposed. The derivation of
the designed guidance law is based on the dynamic
characteristics of both interceptor and target, while
their model errors are taken into account as well.
The stability analysis of the proposed TSM adaptive
control method is studied by providing the detailed
theoretically proof. Finally, simulation experiments
show that robustness and convergence in finite time
could be achieved for the HP interception.
3 OVERVIEW OF HEAD
PURSUIT INTERCEPTION
3.1 HP Working Process
The schematic diagram is shown in Figure 1. In
general, the head pursuit intercept can be divided
into three phases: a) approach; b) control the orbit
direction of interceptor to enter the preset orbit at a
predetermined time; c) terminal guidance.
The major task of interception in the terminal
process is to adjust the position as well as direction
of the interceptor flight errors, while making the tail
of interceptor as close as to the target, then intercept
the target head.
Figure 1: Head pursuit interception.
3.2 HP Kinematic Equation
In the stage of the terminal guidance, the interceptor
will fly in the same general direction as the target,
while ahead of it at a lower speed. Because of the
relatively short interception time and the overload
restriction, the interceptor fight direction will rarely
change. As a result, the interception process can be
decoupled into two orthogonal planes, which are
vertical plane and horizontal plane respectively.
Correspondingly, the expected guidance law can be
designed in the two planes.
To simplify the problem of HP interception, we
assume that the interceptor and the target can be
treated as a particle. The acceleration vectors of both
interceptor and target are vertical with their velocity
vector, thus the acceleration vector will only change
the velocity of the interceptor and target, while the
direction will not change.
Considering the case in the longitudinal plane as
an example, the planar geometry is shown in Figure
2. It can be seen that, the target
T
is located behind
the interceptor
I
which has lower speed. Here, the
speed, maneuvring acceleration and flight-path angle
are respectively denoted by
V
, a and
. The range
between the target and interceptor is defined as
r
,
while
q
is the line of sight (LOS) angle relative to a
fixed reference. Meanwhile, the angles
and
are
the instantaneous target and interceptor direction of
flight relative to the LOS.
In Figure 2, the
r
V
denotes the relative speed
between the interceptor and target, which is defined
as
ICINCO 2016 - 13th International Conference on Informatics in Control, Automation and Robotics
112
cos cos
rI T
VV V
(1)
and the speed perpendicular to the LOS is
sin sin
qI T
VV V
(2)
I
V
T
V
T
a
I
a
T
I
q
q
T
q
I
q
Figure 2: HP geometry.
Therefore, we can have the equations of HP
interception as follows
cos cos
rI T
TT q
II q
q
rV V V
aV Vr
aV V r
qVr
(3)
Meanwhile, the speeds of interceptor and target
are represented by
I
V
and
T
V
, respectively. Thus,
the speeds should be constant and non-dimensional
parameter
K
can be defined as the speed ratio as
1
IT
KVV
(4)
Let the running time is
t
, the terminal guidance
initials at
0t
with
(t 0) 0r
and terminates at
f
tt
where
arg 0
f
t
trtrt
(5)
Considering the goal of successfully intercept,
the conditions that must be met are as follow
lim 0
f
tt
rt
(6)
lim 0
lim 0
f
f
tt
tt
(7)
The purpose of HP guidance control is to guide
the interceptor to the point that satisfies the
conditions of Eq.(6) and Eq.(7). Therefore, the
angles
and
need to be proportionally changed
during the design process
n
(8)
where n is guidance coefficient.
Nevertheless, if in the real interception scenario,
the interceptor might need to hit the target with a
certain angle which brings more requirements to the
designed control law. Therefore, it is quite necessary
to put forward a more objective terminal interception
geometric condition, thus the Eq. (8) is amended as
rr
n
(9)
where 2
r
is selected purposely and
1
sin sin
rr
K
(10)
4 PROPOSED APPROACH
At the very beginning of the design of guidance law,
we should determine the dynamic of the interceptor
and target, and then we use the TSM control to
design the guidance law, finally prove the stability.
In order to compensate the interference of both
model errors and external disturbances, the finite
time convergence adaptive reaching law is designed,
the design method is based on the following three
steps: First, select the proper sliding surface. Second,
design the adaptive guidance law. Finally, prove the
stability of both processes in either reaching phase
or in sliding phase.
4.1 Dynamic
Though Eq.(9) describes the geometric rule between
the interceptor and target, the explicit control value
is however not included. Therefore, it is necessary to
derive relationship between this rule and interceptor
acceleration command
I
a
.
We assume that closed-loop lateral acceleration
of both interceptor and target can be represented by
equivalent first-order transfer function, by ignoring
model error we have
I
Ic I I
aaa
(11)
TTcTT
aaa
(12)
where
Tc
a
,
I
c
a
are the guidance commands of target
and interceptor,
T
and
I
are their time constants.
Notably, the acceleration command is restricted to
max
c
TT
aM
(13)
4.2 Sliding Surface Design
In contrast to the existing methods of HP intercept
guidance law, this paper uses the terminal sliding
mode surface which can converge in a finite time.
We define the HP interception deviation as
The Design of Finite-time Convergence Guidance Law for Head Pursuit based on Adaptive Sliding Mode Control
113
rr
en
(14)
where
n is the time constant. Since interceptor and
target have first order characteristics, the derivation
of
e is required to have the accelerate command
I
c
a
.
Lemma 1. (Yu, 2005) For any real numbers
1
0
,
2
0
,
01k
, an extended Lyapunov condition of
finite-time stability can be given in the form of TSM
as
12
0
k
Vx Vx V x
(15)
where the settling time can be estimated by
1
102
12
1
ln
1
k
r
Vx
T
k
(16)
Define the terminal sliding variable as
s
eesigne
(17)
where
0
,
01
. Substituting Eqs (1), (2),
(3), and (14), we can write the sliding variable as
1
rr
IT
IT
snnPsignP
V
aa
nn PsignP
VV r
(18)
where
Pe
, and differentiating the Eq. (18) as
1
cos cos
1
2
11
Ic I Tc T
IT
II TT
rIT
IT
aa aa
aa
snn
VV r
Vaa
nqe nnq
rVV
(19)
Substituting Eqs.(1), (2) and (3), into Eq.(19) as
I
ITT IcIcTcTc
s
Ga Ga Gq G a G a
(20)
where
1
1
1
1cos
1
cos
1
21 1
1
I
I
II
TT
TT
r
Ic
II
Tc
TT
Gn eV
Vr
n
GnneV
Vr
V
Gn n e
r
G
V
n
G
V
(21)
Therefore, the first-order dynamic characteristic
of sliding variable can be simplified to
I
I T T q Ic Ic Tc Tc
s
Ga Ga Gq G a G a
(22)
4.3 Guidance Law Design
Based on the terminal sliding variable given in
Eq.(17), the sliding mode controller
I
c
a
can be
composed of two parts:
eq
a
and
uc
a
.
To be specific,
eq
a
is called the equivalent part,
which indicates that if there is no model error and
target maneuvering, the system could continue to
approach within the sliding surface. Meanwhile,
uc
a
is called the uncertainty part, which means that if the
system has modelling errors and target maneuvering,
this part can drive the system to the designed sliding
surface
s
, so that the robustness of the system can be
achieved. Therefore,
I
c
a
can be written as
I
cequc
aaa
(23)
eq I I T T Ic
at Ga Ga GqG
(24)
In this paper, we design the uncertainty part of
uc
a
based on the reaching law as
12k
s
ks k sig s
(25)
where
1
0k
,
2
0k
, 01
. According to Eq.
(22) and Eq.(25), it can be written as
12
ˆˆ
0
Ic
uc
ks ksig s f s
at
s
(26)
where
1
ˆ
k
,
2
ˆ
k
are the estimated gain values. For the
research objective in this paper, it is better to
eliminate the influence of external disturbance and
accelerate the convergence speed of the sliding
surface, thus the adaptive law can be designed as
2
1
1
ˆ
0
ss
k
s
(27)
1
2
2
ˆ
0
ss
k
s
(28)
where
1
,
2
,
are positive design parameters.
4.4 Stability Proof
Proof. We select the Lyapunov function as
22
12
12
111
22 2
T
Vss k k
(29)
By using the time derivative of the Lyapunov
function, we have
ICINCO 2016 - 13th International Conference on Informatics in Control, Automation and Robotics
114
1
1
1
1
12
2
11 2 2
2
2
12 11
22 max
2
12 max
ˆˆ
ˆˆ
ˆ
ˆˆ
ˆ
T
Tc Tc
Tc T
Tc T
Vs ksksigsGa
kks kks
ks k s k k s
kks Ga s
ks k s Ga s
(30)
In order to make
0V
, Eq.(30) can be rewritten
into the following two forms as
11
max
22
12
-2 - 2
Tc T
Ga
Vk V kV
s
(31)
11
max
1
22
2
2
2
Tc T
Ga
k
VV k V
s
(32)
If we want to make
0V
, the initial values of
parameters
1
k
and
2
k
should meet the following
conditions as
max
1
Tc T
Ga
k
s
(33)
max
2
Tc T
Ga
k
s
(34)
Consequently, both Eq.(31) and Eq.(32) can be
simplified as
1
2
12
0VVV
(35)
where
11
2k
,
1
2
22
2 k
.
According to the Lemma 1, we can figure out
that the sliding surface defined in Eq.(17) will be
driven onto the plane of
0s
in finite time, which
can be mathematically described as
1
1
2
10 2
2
1
1
ln
1
1
2
r
V
T
(36)
From Eq.(14) and Eq.(17), it can be seen that the
plane
0s
is not qualified to meet the condition of
head pursuit intercept, which is
0e
. Therefore, we
prove that the system can reach the origin from the
sliding surface.
When
0s
, if
0e
, then
0esigne
.
As a result,
0e
and the parameter e will increase
along with the increase of time, and finally tends to
zero. If
0e
, then
0esigne
, so
0e
, and
e will decreases until finally tends to 0 too. This
shows that the states of the system will convergence
to the origin in finite time.
5 SIMULATION EXPERIMENT
AND ANALYSIS
In this section, we apply the proposed approach in
experimental scenario by using numerical simulation
to validate the effectiveness.
Table 1: Simulation parameters setup.
Interceptor Target Guidance
V
I
= 1600m/s V
T
= 2000m/s n = 2
τ
I
= 0.2s
τ
T
= 0.2s
τ
= 0.2s
Notably, the targets with constant velocity and
constant maneuvering (20g,-20g) are considered to
test the designed method. Moreover, we analyze the
change of interceptor trajectory, overload and sliding
surface in multiple attack angles. The simulation
parameters are given in Table 1.
According to Table 1, when the initial LOS angle
q is 20 deg, the initial distance between the target
and interceptor is
(0)r =3000m, [20,0,20]g
T
a
and
0
r
. 3 scenarios are tested: non-maneuvering
target and two constant maneuvering targets. In
Figure 3, the trajectories of both interceptor and
target are graphically demonstrated.
0 2000 4000 6000 8000 10000 12000 14000
-6000
-4000
-2000
0
2000
4000
6000
8000
X( m )
Y(m)
target
interceptor
aT=20g
aT=0g
aT=-20g
Figure 3: Response curves of intercept trajectory.
-25 -20 -15 -10 -5 0 5 10
0
500
1000
1500
2000
2500
3000
(deg)
r(m)
aT=20g
aT=0g
aT=-20g
Figure 4: Response curves of trjectories in target
coordinate system;
0deg
r
.
The Design of Finite-time Convergence Guidance Law for Head Pursuit based on Adaptive Sliding Mode Control
115
From Figure 3 we can see that, whether if the
target has maneuvering or not, the trajectory of
interceptor will gradually approach to the trajectory
of target and track it until it is caught up. In Figure 4
and 5, the relative trajectories of interceptor in non-
rotating polar coordinate attached to the target are
given. It can be seen from the figure that, the
interception is successfully achieved in the presence
of target maneuvers and the initial heading errors
(
(0) 20deg
), while the angle error is small. To
be specific, as we set
0
r
in Figure 4, the angle
errors are 0.34deg and 0.05deg while the
T
a
of target
maneuvering are 20g and -20g, respectively. In
Figure 5,
r
is set to be -10deg.
-30 -25 -20 -15 -10 -5
0
0
500
1000
1500
2000
2500
3000
(deg)
r(m )
aT=20g
aT=0g
aT=-20g
Figure 5: Response curves of trjectories in target
coordinate system;
10deg
r
.
From the experimental results we can see that,
the angle errors are 0.12deg and 0.29deg separately,
while the corresponding
T
a of target maneuvering
are 20g and -20g. Moreover, both Figure 4 and 5
show that the angle error is almost zero if the target
has no maneuver.
0 1 2 3 4 5 6 7 8
-100
-80
-60
-40
-20
0
20
40
t(s)
aI(g)
aT=0g
aT=20g
aT=-20g
Figure 6: Response curves of interceptor acceleration;
0deg
r
.
0 1 2 3 4 5 6 7 8
-80
-60
-40
-20
0
20
40
t
(
s
)
aI(g)
aT=20g
aT=0g
aT=-20g
Figure 7: Response curves of interceptor acceleration
10deg
r
.
Considering that both target and interceptor are
set to be the typical high-speed target, the results
show a good performance. As shown in Figure 6 and
7, the response curves of interceptor acceleration for
different target maneuvers are presented respectively,
where
0
r
and
10deg
r
.
0 0.5 1 1.5 2 2.5 3 3.5 4
-2
-1
0
1
2
3
4
5
t(s)
s(deg)
aT=20g
aT=0g
aT=-20g
Figure 8: Response curves of sliding surface
value; 0deg
r
.
0 0.5 1 1.5 2 2.5 3 3.5 4
-4
-2
0
2
4
6
8
10
12
t(s)
s(deg)
aT=20g
aT=0g
aT=-20g
Figure 9: Response curves of sliding surface
value; 10deg
r
.
We can see that the magnitude of the interceptor
acceleration is adjusted dynamically over time. The
change of interceptor acceleration decreases after a
short period of time. Finally, the tracking of target is
realized. Compared with the head pursuit guidance
laws that proposed in recent years, the maneuvering
acceleration of interceptor is greatly reduced.
ICINCO 2016 - 13th International Conference on Informatics in Control, Automation and Robotics
116
The value of the sliding variable for
0
r
and
10deg
r
are graphically showed in Figure 8 and
9, respectively. Obviously, it also can be seen from
Figure 8 and 9 that, the sliding variable
s
could
convergence to the sliding surface rapidly, then
reach steady state.
In order to further illustrate the effectiveness and
the advance of the guidance law, we compared it
wih the guidance law proposed in paper (Dan, 2013).
Table 2 shows the intercept time under different
target maneuvers.
Moreover, in order to evaluate the convergence
rate of the proposed method, the comparison of
interception time between our method and the
guidance law proposed in (Dan, 2013) are presented.
The comparison results are given in the Table 2, in
which 4 target acceleration values are applied.
Table 2: Intercept time of the two guidance law.
a
T
10g 15g 20g 25g
Our
8.413s 7.982s 7.523s 7.051 s
Dan 13.827s 13.304s 12.506s 11.563s
We can see that the guidance law proposed in
this paper has shorter intercept time and has better
convergence performance.
6 CONCLUSIONS
In this paper, a finite time convergent guidance law
for head pursuit is proposed, in which the terminal
sliding mode control theory is used. By considering
the dynamics characteristics of target and interceptor,
an adaptive law is theoretically designed based on
the interference factors of target maneuvering and
model errors. The results of numerical simulation
show that the guidance law can be implemented on
the head pursuit intercept of high-speed targets and
achieve successful interception in different attack
angles and target maneuvering, while having smaller
intercept error compared with other methods. The
proposed method has lower requirement on the way
of maneuvering, which show a potential application
in real interception of high-speed vehicle. Moreover,
the sliding variable can also convergence to sliding
surface more quickly with strong robustness.
ACKNOWLEDGEMENTS
This work was supported by the National Natural
Science Foundation of China under Grant 61502391,
the Foundation of National Key Laboratory of
Aerospace Flight Dynamics and the China Space
Foundation under Grant 2015KC020121.
REFERENCES
Shima, T., & Golan, O. M. (2007). Head pursuit
guidance. Journal of Guidance, Control, and
Dynamics, 30(5), 1437-1444.
Goyal, V., Deolia, V. K., & Sharma, T. N. (2015). Robust
Sliding Mode Control for Nonlinear Discrete-Time
Delayed Systems Based on Neural Network.
Intelligent Control and Automation, 6(01), 75.
Zhihong, M., Paplinski, A. P., & Wu, H. R. (1994). A
robust MIMO terminal sliding mode control scheme
for rigid robotic manipulators. Automatic Control,
IEEE Transactions on, 39(12), 2464-2469.
Feng, Y., Yu, X., & Han, F. (2013). High-order terminal
sliding-mode observer for parameter estimation of a
permanent-magnet synchronous motor. Industrial
Electronics, IEEE Transactions on, 60(10), 4272-
4280.
Kumar, S. R., Rao, S., & Ghose, D. (2014). Nonsingular
terminal sliding mode guidance with impact angle
constraints. Journal of Guidance, Control, and
Dynamics, 37(4), 1114-1130.
Liu, Y. J., & Li, Y. X. (2010). Adaptive fuzzy output-
feedback control of uncertain SISO nonlinear systems.
Nonlinear Dynamics, 61(4), 749-761.
Li, P., & Yang, G. (2011). An adaptive fuzzy design for
fault-tolerant control of MIMO nonlinear uncertain
systems. Journal of Control Theory and Applications,
9(2), 244-250.
Gu, Y. J., Yin, X. X., Liu, H. W., Li, W., & Lin, Y. G.
(2015). Fuzzy terminal sliding mode control for
extracting maximum marine current energy. Energy,90,
258-265.
Zhao, D., Zhu, Q., & Dubbeldam, J. (2015). Terminal
sliding mode control for continuous stirred tank
reactor. Chemical Engineering Research and Design,
94, 266-274.
Behera, A. K., & Bandyopadhyay, B. (2015). Steady-state
behaviour of discretized terminal sliding mode.
Automatica, 54, 176-181.
Madani, T., Daachi, B., & Djouani, K. (2015). Non-
singular terminal sliding mode controller: Application
to an actuated exoskeleton. Mechatronics, 33, 136-145.
Lu, K., Xia, Y., Zhu, Z., & Basin, M. V. (2012). Sliding
mode attitude tracking of rigid spacecraft with
disturbances. Journal of the Franklin Institute, 349(2),
413-440.
Lee, D., & Vukovich, G. (2015). Adaptive sliding mode
control for spacecraft body-fixed hovering in the
proximity of an asteroid. Aerospace Science and
Technology, 46, 471-483.
Hua, W., & Chen, X. (2011). Adaptive sliding-mode
guidance law for head pursuit. Journal of Harbin
Institute of Technology, 43(11), 30-33.
The Design of Finite-time Convergence Guidance Law for Head Pursuit based on Adaptive Sliding Mode Control
117
Song, J., & Song, S. (2015). Three-dimensional guidance
law based on adaptive integral sliding mode
control. Chinese Journal of Aeronautics, 29(1), 202-
214.
Yu, S., Yu, X., Shirinzadeh, B., & Man, Z. (2005).
Continuous finite-time control for robotic
manipulators with terminal sliding mode. Automatica,
41(11), 1957-1964.
Dan Yang. (2013). Research on variable structure terminal
guidance law of space interception. Shanghai Jiao
Tong University.
ICINCO 2016 - 13th International Conference on Informatics in Control, Automation and Robotics
118
