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Engineering Notes
Active Defense Guidance Law via
Cooperative Identification and
Estimation
Xinguang Zou∗and Di Zhou†
Harbin Institute of Technology, 150001 Harbin,
People’s Republic of China
and
Runle Du‡and Jiaqi Liu§
National Key Laboratory of Science and Technology on Test
Physics and Numerical Mathematics, 100076 Beijing,
People’s Republic of China
DOI: 10.2514/1.G003372
I. Introduction
IT IS a challenge to protect a vehicle such as an aircraft or a ballistic
missile against the interception of an enemy missile. Some passive
approaches have been proposed to address this problem, such as
electronic jamming, stealthy technology, deploying decoys, multi-
warheads, orbit varying for ballistic missiles, etc. In recent years,
some active defense approaches have gained considerable attention,
such as the scenario in which an aircraft launches a defensive missile
to intercept the incoming enemy missile.
For the sake of conciseness, in the following content, the incoming
enemy missile is called the pursuer; the vehicle intercepted by the
pursuer is called the evader; and the defensivemissile that protects the
evader against the pursuer is called the defender.
The earliest research on the active defense was done in [1], in
which a closed-form interception condition for defending an aircraft
or ship was obtained with the assumption of a constant collision
course between the target and the interceptor. In [2], a cooperative
guidance law between the evader and the defender was proposed
with the assumption that the pursuer used a linear guidance law.
Furthermore, in [3], three cooperative schemes with different
communication directions between the evader and the defender were
proposed, which were also under the assumption that the pursuer
employed a linear guidance law. However, this assumption limited
the applications of these guidance schemes in real scenarios. To
address this issue, a multiple-model adaptive guidance law was
proposed in [4], in which a multiple model adaptive estimator
(MMAE) was used to identify the pursuer’s guidance law. In the
study, it was assumed that the pursuer’s guidance law was in a set of
finite guidance laws known by the evader in advance.
The engagement of the evader, the pursuer, and the defender can
also be seen as a three-person game problem, in which the defender
tries to minimize its distance to the pursuer while the pursuer tries to
maximize its distance to the defender and, at the same time, minimize
its distance to the evader. Some active defense guidance laws based
on the multiple objective optimization and the differential game
theory were derived in [5]. Another differential game guidance law
was derived in [6], in which an optimal cooperative evasion strategy
for the evader and an optimal cooperative interception strategy for the
defender were obtained simultaneously. It was shown in [6] that this
optimal cooperative interception guidance law for the defender could
significantly reduce the requirements for defender’s maneuverability.
To derive the differential game guidance laws, the pursuer’s
acceleration boundary has to be known in advance by the defender.
With regard to the problem of the active defense guidance, two
guidance strategies are often adopted to design guidance laws, i.e.,
the triangle guidance strategy and the strategy of nullifying the line-
of-sight (LOS) angular rate. With the triangle guidance strategy, the
defender tries to keep itself on the LOS connecting the evader and the
pursuer. In this case, if the pursuer tries to intercept the evader along
the LOS, it eventually collides with the defender before intercepting
the evader. Based on the triangle guidance strategy, theguidance laws
presented in [7,8] found their foundation in the optimal control
framework.
When the optimal control theory is used to derive the active
defense cooperative guidance laws, the guidance strategy of the
pursuer has to be known by the defender in advance and the time to go
has to be estimated online. Therefore, it is difficult to implement an
optimal cooperative guidance law in practical applications. By
contrast, a cooperative guidance law based on the nonsingular
terminal sliding mode control was proposed for the ballistic missile
protection in [9]. This cooperative guidance law is relatively easy to
implement because it does not rely on the guidance strategy of the
pursuer or the estimation of the time to go. To use this sliding mode
guidance law, the boundary of the system uncertainty should be
known a priori in practical applications. Commonly, a conservative
value is chosen or an adaptive law is designed to estimate the
boundary. These methods usually get an estimation of boundary
larger than its actual value, resulting in a higher requirement on
the maneuverability of defender and possible chattering in the
guidance system.
In real guidance scenarios, not all the required information needed
by the guidance law can be directly measured, and the measurements
are corrupted by noises. Thus, to guide the defender with accuracy,
the estimation of the state of the pursuer is essential. Some Kalman-
type filters such as the Kalman filter (KF), the extended Kalman filter
(EKF), the unscented Kalman filter (UKF), etc., have been widely
used to track the states of various kinds of vehicles for decades. The
Kalman filters are a kind of predictor–corrector approach, for which
the performance heavily depends upon the correctness of the model.
Thus, the variations of the motion mode of the pursuer during the
tracking process will affect the performance of the Kalman filters
remarkably.
The proportional navigation guidance (PNG) law is a popular
guidance law widely used because it is easy to implement. In many
scenarios, it is practical to assume that the pursuer’s guidance law is
the PNG law. The navigation ratio of the PNG law is commonly set to
a constant. However, to cope with the guidance problems in some
complex scenarios or to fulfill some specific objectives, the pursuer
may switch its navigation ratio during the guidance process. For
example, when the range between the pursuer and the evader is large,
the measurement noise is relatively high. In this case, the navigation
ratio should be a small value; otherwise, the measurement noise will
have a great impact on the guidance command. With the range
Received 7 October 2017; revision received 7 April 2018; accepted for
publication 25 June 2018; published online 30 August 2018. Copyright
© 2018 by the American Institute of Aeronautics and Astronautics, Inc. All
rights reserved. All requests for copying and permission to reprint should be
submitted to CCC at www.copyright.com; employ the ISSN 0731-5090
(print) or 1533-3884 (online) to initiate your request. See also AIAA Rights
and Permissions www.aiaa.org/randp.
*Associate Professor, School of Electrical Engineering and Automation,
Mailbox 327; xgzou@hit.edu.cn.
†Professor, Department of Control Science and Engineering, Mailbox 327;
zhoud@hit.edu.cn (Corresponding Author).
‡Staff Engineer, P.O. Box 9200-76-7; jenniferdu@126.com.
§Staff Engineer, P.O. Box 9200-76-7; liujiaqi_business@163.com.
2503
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Vol. 41, No. 11, November 2018
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becoming shorter, the measurement noise is decreasing, and the
navigation ratio should be a relatively large value to nullify the LOS
angular rate and achieve a small miss distance. In another example, a
two-stage pure proportional navigation guidance law is proposed to
intercept a target with a desired impact angle and without violating
the field-of-view limit [10]. In its first stage, a small navigation ratio
(i.e., N1) is selected to orient its trajectory. This helps to achieve
the desired impact angle in the next stage. Then, in the second stage,
the navigation ratio is switched to a larger value (e.g., N≥2)to
achieve the desired impact angle. In these scenarios, the navigation
ratio switching results in the change of the pursuer’s motion model.
Due to the sensitivity to the assumed motion models, the Kalman
filters’performances degenerate when dealing with the switching of
models. To our best knowledge, no open literature has studied the
identification of the PNG law with the switching guidance ratio. In
this Note, we investigate this practical issue and design the new active
defense cooperative guidance laws via the cooperative estimation
between the evader and the defender.
Recently, the multiple model adaptive estimators [11] have seen
usage in missile guidance scenarios to address the target acceleration
estimation problem. A multiple model-based guidance law (MMGL)
was proposed and compared against the shaping filter in [12]. With
regard to the root mean square of the miss distance, the MMGL
outperformed the shaping filter-based guidance law. In [13], an
MMAE approach was used in an air-to-air missile interception
scenario, in which the target was a highly maneuverable aircraft
with an electronic countermeasure device. Several combinations of
maneuvers and electronic countermeasures were taken to form the
MMAE’s model set. In [14], to address the target tracking problem in
a ballistic missile interception scenario, a modified MMAE filter
focusing on reducing the computational effort was proposed for the
interceptor. In [15], an MMAE approach was proposed to track a
pursuer with a guidance law in a set of guidance laws. In this study,
the navigation ratio of the guidance law remains time invariant. The
MMAE approach is a superior choice if the pursuer’s motion model is
time invariant. However, with the navigation ratio varying during the
guidance process, such as the scenarios already mentioned, the
performance of the MMAE will degenerate dramatically due to its
static structure. Compared with the MMAE approach, the interactive
multiple model (IMM) filtering [16] issuitable to copewith the model
switching problem. With the MMAE, all elemental filters works
independently; whereas with the IMM filtering, all elemental filters
interact with each other via a mixture procedure, which makes the
IMM filtering more flexible. The models’posterior probabilities,
which denote the extent of correctness of the models, are updated in
each filtering cycle. Due to its ability to cope with the model
switching, the IMM filtering is adopted in this work to identify the
pursuer’s guidance law and to estimate the states of the pursuer.
Owing to its robustness, the sliding mode control has gained much
popularityin the past decades. The traditional sliding mode controllers
suffered from the chattering problem. To attenuate the chattering,
some smooth saturation functions (e.g., the sigmoid functions) were
commonly used to approximate the discontinuous sign function.
However, such approximation sacrificed the control performance
because the sliding variables could only converge to the vicinity of the
equilibrium point.
Higher-order sliding mode controllers are effective to attenuate
the chattering [17]. With regard to the system of relative degree one,
a second-order sliding mode controller can effectively attenuate
the chattering. One of the most popular second-order sliding mode
controllers is the supertwisting sliding mode controller [18].
Although the supertwisting sliding mode control can attenuate the
chattering, the large switching gain or the noise in the sliding variable
still possibly induces the remarkable chattering. As the magnitude of
the switching gain has a solid relationship with the boundary of the
system uncertainty, an effective way to attenuate the chattering is to
reduce the system uncertainty. The idea in this study is to estimate
the system uncertainty and compensate it in the guidance law via the
cooperative estimation between the evaderand the defender. After the
estimation and compensation, the system uncertainty is reduced to
the estimation error, which has a much smaller boundary due to the
effectiveness of the cooperative estimation. With the much smaller
boundary of the system uncertainty, a much smaller switching gain
can be chosen, which in turn dramatically reduces the chattering.
Furthermore, the noise-corrupted sliding variable can also be
estimated with accuracy via this cooperative estimation approach to
attenuate the chattering further.
The contributions of our work are as follows:
1) In the guidance law design, the idea of the system uncertainty
estimation and compensation is used. To our knowledge, this is the
first attempt at applying a cooperative estimation approach in
the sliding-mode-based guidance law between the evader and the
defender.
2) In the estimation design, a cooperative estimation approach
between the evader and the defender is proposed. An IMM filter on
the evader is proposed to identify the guidance law of the pursuer and
to estimate the states of the pursuer. A Kalman filter on the defender is
proposed to estimate the states of the pursuer. This cooperation can
notably improve the accuracy of the estimation. And, it can still work
with very good performance if the pursuer switches its guidance ratio
during the guidance process. However, the performance of Kalman
filters or MMAEs degenerates under such conditions.
3) Two supertwisting sliding mode guidance laws via the
cooperative estimation are proposed. One exploits the triangle guidance
strategy, and the other exploits the strategy of nullifying the LOS
angular rate. They can notably reduce the miss distance, dramatically
reduce the maximum acceleration requirement, and effectively
attenuate the control chattering for the defender.
II. Problem Formulation
A. Engagement of an Evader, a Pursuer, and a Defender
In this Note, as shown in Fig. 1, the planar engagement of an
evader, a pursuer, and a defender is investigated. The initial LOS
coordinate system of the evader to the pursuer is taken as the scene
inertial coordinate system OXY, for which the xaxis is along the LOS
and the yaxis is perpendicular to the LOS and points upward. The
evader, the pursuer, and the defender are denoted by e,p, and d,
respectively. The evader’s and the defender’s inertial coordinate
systems are both set in the same direction with the scene inertial
coordinate system. The yaxis of the pursuer’s inertial coordinate
system is in the same direction with the yaxis of the scene coordinate
system, and the direction of the xaxis of the pursuer’s inertial
coordinate system is opposite to the xaxis of the scene inertial
coordinate system. In Fig. 1, Veis the evader’s velocity vector, aeis
the evader’s acceleration vector, and qep is the evader’s LOS angle to
the pursuer; Vdis the defender’s velocity vector, adis the defender’s
acceleration vector, and qdp is the defender’s LOS angle to the
pursuer; Vpis the pursuer’s velocity vector, apis the pursuer’s
acceleration vector, and qpe is the pursuer’s LOS angle to the evader;
e
p
d
qdp
q
ep
OX
Yap1
ap2
ad
Ve
V
p
Vd
qpe
rpe rdp
a
e
ap
Fig. 1 Planar engagement of an evader, a pursuer, and a defender.
2504 J. GUIDANCE, VOL. 41, NO. 11: ENGINEERING NOTES
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and rpe is the range between the pursuer and the evader, and andrdp is
the range between the defender and the pursuer.
The engagement equations of the evader, the pursuer, and the
defender can be expressed as follows:
qpe −2_
rpe
rpe
_
qpe 1
rpe
ae4−1
rpe
ap1(1)
qdp −2_
rdp
rdp
_
qdp 1
rdp
ap2−1
rdp
ad4(2)
where _qpe is the pursuer’s LOS angular rate to the evader, _qdp is
the defender’s LOS angular rate to the pursuer, ae4is the y-axis
acceleration of the evader in the pursuer’s LOS coordinate system,
ad4is the y-axis acceleration of the defender in the defender’sLOS
coordinate system, ap1is the y-axis acceleration of the pursuer
in the pursuer’s LOS coordinate system, and ap2is the y-axis
acceleration of the pursuer in the defender’sLOScoordinate
system.
B. Pursuer’s Motion Model in the View of Evader
As the PNG law is the most popular guidance law, here, we
reasonably assume that the pursuer intercepts the evader with the
PNG law and names the pursuer’s motion model the proportional
navigation (PN) model in this circumstance. To estimate the states of
the pursuer with regard to the evader, the state vector of the pursuer is
chosen as xerx;e;r
y;e;v
x;e;v
y;e;a
px;a
pyT, where rx;e ;r
y;eTis
the relative position to the pursuer, vx;e;v
y;eTis the relative velocity
to the pursuer, and apx;a
pyTis the pursuer’s acceleration. These
vectors are all in the scene inertial coordinate system. The motion
model for the pursuer is given by
_
rx;e vx;e
_
ry;e vy;e
_
vx;e apx −aex
_
vy;e apy −aey
_
apx apxc −apx∕τ
_
apy apyc −apy∕τ(3)
where aex;a
eyTis the acceleration of the evader in the scene inertial
coordinate system, apxc;a
pycTare the projections of the guidance
command of the pursuer in the scene inertial coordinate system, and τ
is a time constant with a typical value of 0.1.
The PNG law of the pursuer can be
ap4x0;a
p4y−N_
rpe _
qpe (4)
where ap4xand ap4yare the x-axis and y-axis guidance commands in
the pursuer’s LOS coordinate system, respectively. Nis the navigation
ratio. The relationship between ap4x;a
p4yTand apxc;a
pycTis
given by
apxc
apyc CISCLI ap4x
ap4y(5)
where CLI is the transformation matrix from the pursuer’sLOS
coordinate system to the pursuer’s inertial coordinate system, i.e.,
CLI cos qpe −sin qpe
sin qpe cos qpe (6)
and CIS is the transformation matrix from the pursuer’sinertial
coordinate system to the scene inertial coordinate system. According
to the aforementioned relationship between the pursuer’s inertial
coordinate system and the scene inertial coordinate system,
CIS diag−1;1. Substituting the expressions of CLI and CIS
into Eq. (5) yields
apxc
apyc −sin qpe
−cos qpe N_
rpe _
qpe (7)
It is easy to find in the engagement geometry for which
qep −qpe _
qep −_
qpe _
rep _
rpe (8)
where _
qep is the evader’s LOS angular rates to the pursuer, _
rep is the
closing velocity from the evader to the pursuer, and _
rpe is the closing
velocity from the pursuer to the evader. The formulations of the LOS
angle and range are
qep arctan ry;e
rx;e
rep
r2
x;e r2
y;e
q(9)
sin qep ry;e
r2
x;e r2
y;e
qcos qep rx;e
r2
x;e r2
y;e
q(10)
Differentiating Eq. (9) with respect to time, we obtain
_
qep rx;evy;e −ry;e vx;e
r2
x;e r2
y;e
_
rep rx;evx;e ry;e vy;e
r2
x;e r2
y;e
q(11)
Substituting Eqs. (7), (8), (10), and (11) into Eq. (3) yields
_
rx;e vx;e
_
ry;e vy;e
_
vx;e apx −aex
_
vy;e apy −aey
_
apx −Nry;erx;evx;e ry;e vy;erx;e vy;e −ry;evx;e
r2
x;e r2
y;e2τ−apx
τ
_
apy Nrx;erx;evx;e ry;e vy;erx;e vy;e −ry;evx;e
r2
x;e r2
y;e2τ−apy
τ(12)
Equation (12) is the PN model of the pursuer in the view of
the evader. The evader is assumed to have the ability to get the
measurements of its relative position to the pursuer in the scene inertial
coordinate system. Thus, the pursuer’s measurement matrix is
He100000
010000
(13)
Different from the fixed multiple mode filter (e.g., MMAE), in each
iteration of an IMM filter, the inputs of the element filters are mixed
values of the previous step’s estimations, namely, interaction. The
IMM filterhas a model set MfM1;···;M
ng,whereMjdenotesthe
jth model. Each model corresponds to a filter (e.g., KF, EKF, or UKF),
namely, the element filter. Thus, a model set corresponds to a bank of
element filters. The IMM filter uses the model probability weighted
sum of the estimations from all the element filters as its estimation
output. In this design, each model is a PN model with a specific
navigation ratio, which corresponds to a UKF.
The a priori probability of model Mjis defined as μj
0PfMj
0g.
Let Pij PfMj
kjMi
k−1gdenote a known mode transition probability.
The state equation and measurement equation corresponding to
model Mjis given as
xkfjk; xk−1wj
k−1
zkhjk; xkvj
k(14)
J. GUIDANCE, VOL. 41, NO. 11: ENGINEERING NOTES 2505
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where fjk; xk−1,hjk; xk,wj
k−1, and vj
kare the state transition
function, measurement function, process noise, and measurement
noise, respectively. The IMM algorithm is a recursive algorithm with
each iteration divided into three steps: mixing, filtering, and
combination. The process in the iteration at time kis described as
follows:
1. Step 1: Mixing
In each iteration, an element filter uses a combination of last
model-conditioned estimation as its state estimation and covariance
inputs. Denote μijj
kas the mixing probability from models Mito Mj.It
is calculated by
μijj
k1
cj
pijμi
k−1
cjX
n
i1
pijμi
k−1(15)
where μi
k−1is the probability of model Miat time k−1, and
cjis the
normalizing factor. The mixing input is given by
xj;0
k−1X
n
i1
μijj
k^
xi
k−1;k−1
Pj;0
k−1X
n
i1
μijj
khPi
k−1jk−1^
xi
k−1jk−1−xj;0
k−1 ^
xi
k−1jk−1−xj;0
k−1Ti
(16)
where xj;0
k−1and Pj;0
k−1denote the mixing input and covariance,
respectively.
2. Step 2: Filtering
Using xj;0
k−1and Pj;0
k−1in Eq. (16) as inputs, each UKF in the filter
bank executes a filtering iteration. The filtering is divided into two
substeps, including the state prediction and state update:
h^
xj
k;k−1;P
j
k;k−1iUKFpxj;0
k−1;P
j;0
k−1;fj;Q
j
k−1
h^
xj
k;P
j
k;νj
k;Sj
kiUKFu^
xj
k;k−1;P
j
k;k−1;zk;hj;R
j
k(17)
where the function UKFp⋅is the state prediction function, and
UKFu⋅is the state update function. For model Mj,Qj
k−1is the process
noise matrix at time k−1,and ^
xj
k;k−1and Pj
k;k−1are the updated state
mean and updated state covariance at time k−1, respectively. Note that
zkis the measurement at time k;Rj
kis the measurement noise matrix at
time k;and ^
xj
kand Pj
kare the state estimation and state covariance,
respectively. Also, νj
kand Sj
kare the innovation and the innovation
covariance at time k, respectively. The likelihood of model Mjat time k
is Λj
kNνj
k;0;Sj
k. Denote μj
kas the probability of model Mjat time
k, and it is calculated as follows:
μj
k1
cΛj
k
cjcX
n
j1
Λj
k
cj(18)
3. Step 3: Combination
At last, the state estimation and covariance are given by
^
xkX
N
j1
μj
k^
xj
k
PkX
N
j1hPj
k^
xj
k−^
xk ^
xj
k−^
xkTi(19)
C. Pursuer’s Motion Model in the View of the Defender
Some parameters required by the guidance law for the defender
cannot be measured directly, such as the LOS angles, LOS angular
rates, etc. They need to be estimated via an estimator mounted on the
defender. Moreover, it is assumed that the estimated accelerations of
the pursuer by the IMM filter on the evader (i.e., ^
apx;^
apyT) can be
shared with the defender via some communication link.
The state of the pursuer is chosen as xdrx;d ;r
y;d;v
x;d;v
y;dT,
where rx;d:ry;d Tis the relative position to the pursuer and vx;d;v
y;dT
is the relative velocity to the pursuer. The acceleration of the pursuer
apx;a
pyTis replaced with its estimation ^
apx;^
apyT. Thus, the
motion model of the pursuer in the view of the defender is
_
rx;d vx;d
_
ry;d vy;d
_
vx;d ^
apx −adx
_
vy;d ^
apy −ady (20)
For such a linear time-invariant system, a linear Kalman filter
works well. The defender is able to get the measurements of its
relative position to the pursuer in the scene inertial coordinate system.
Thus, the defender’s measurement matrix is
Hd1000
0100
(21)
III. Design of Active Defense Guidance Law
The supertwisting sliding mode control is well known for its ability
to attenuate the chattering, and so it is used to design two active
defense guidance laws: one exploiting the triangle guidance strategy
by which the defender keeps itself on the evader–pursuer LOS, and
the other one trying to nullify the defender’s LOS angular rate to the
pursuer.
A. Supertwisting Control
Consider a single-input/single-output dynamical system with the
following form:
_
xfx;thx;tu(22)
where x∈Rnis a state vector; uis a scalar control input; and the term
fx;tis an unknown function, which is taken as the system
uncertainty. Let σbe the sliding variable such that the system’s
input/output dynamics (u→σ) has a relative degree of one, i.e.,
_σφx;tbx;tu(23)
where bx;t≠0for any xand t. Let
ubx;tu; then, the
preceding equation can be rewritten as
_σφx;t
u(24)
The function φx;tis assumed to be bounded (i.e., jφx;tj ≤
δjσj1∕2), where δis a known positive constant. The aim is to design a
sliding mode control law to drive the sliding variableσto zero in finite
time and attenuate the control chattering as far as possible in the
presence of the bounded system uncertainty. And, a supertwisting
controller has the following form [17,18]:
u−αjσj1∕2signσw
_
w−βsignσ(25)
where the amplitudes of switching functions αand βsatisfy
α>λ−1εδ2λ4ε22εδε
and β2εα λ4ε2to ensure the stability of the system, and
meanwhile attenuate the chattering. Note that εand λare arbitrary
positive constants.
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In Eq. (19), the boundary of the system uncertainty φx;tis
usually unknown. To ensure the stability of the system, commonly, a
conservative boundary (i.e., a larger one than it needs to be) is chosen,
resulting in a larger δ. A larger δin turn produces larger αand β.
Eventually, larger δand βinduce the remarkable chattering and then
the larger control inputs. If we want to keep the chattering as low as
possible and ensure reasonably low control inputs, we should keep
the boundary of the system uncertainty φx;tas small as possible.
To address this problem, we propose a system uncertainty
estimation and compensation approach. In this approach, we estimate
φx;twith a cooperative estimation method that will be discussed
later, and then we compensate it in the controller. Let ^
φx;tdenote
the estimate of φx;t; then, Eq. (24) can be rewritten in the form of
_σεx;t ^φx;t u
εx;tφx;t−^
φx;t(26)
Now, the estimation error εx;tis the system uncertainty in
Eq. (26), satisfying jεx;tj ≤δ0jσj1∕2. Due to the effectiveness of
the cooperative estimation method, the system uncertainty boundary
δ0can be kept much smaller than δin Eq. (24).
Then, a modified supertwisting sliding mode controller is
designed as
u−α0jσj1∕2signσ−^
φx;tw
_
w−β0signσ(27)
where
α0>λ−1
0ε0δ2
0λ04ε2
02ε0δ0ε0
β02ε0α0λ04ε2
0, and ε0and λ0are arbitrary positive
constants. As δ0is much less than δ, the chattering is further
attenuated and the control input can be remarkably reduced.
B. Triangle Guidance Law
In this section, the supertwisting sliding mode control-based
triangle guidance law (STTGL) is designed. And, the system
uncertainty estimation and compensation approach is used. With this
guidance law, the defender tries to keep itself on the LOS connecting
the evader and the pursuer. To achieve this purpose, the LOS angles
qpe and qdp should satisfy
qpe qdp 0 (28)
Let eqep qdp and the sliding variable σ1be
σ1_
eke (29)
where kis a positive constant. By differentiating σ1with respect to
time, we get
_σ1φ1−1
rdp
ad(30)
where
φ1−2_rpe
rpe
_
qpe −2_rdp
rdp
_
qdp k_
qpe k_
qdp 1
rpe
ae4−1
rpe
ap11
rdp
ap2
The parameters rdp,_
rdp, and _
qdp can be estimated by the defender’s
Kalman filter; and rpe,_
rpe,_
qpe, and ap1can be estimated by the
evader’s IMM filter. Because the defender tries to be on the LOS
connecting the evader and the pursuer, the evader’s LOS coordinate
system is almost parallel to the defender’s LOS coordinate system.
Then, the accelerations ap1and ap2are almost the same. Here, both
ap1and ap2are replaced with the estimation ^
ap1. At last, Eq. (30) can
be rewritten as
_σ1^
φ1Δe1Δd1−1
^
rdp
ad(31)
where
^
φ1−2
^
_
rpe
^
rpe
^
_
qpe −2
^
_
rdp
^
rdp
^
_
qdp k^
_
qpe k^
_
qdp 1
^
rpe
ae4−1
^
rpe
^
ap11
^
rdp
^
ap1
can be calculated with the estimations of the filters. And, the
estimation error Δe1is
Δe1−2_
rpe
rpe
_
qpe 2
^
_
rpe
^
rpe
^
_
qpe k_
qpe −k^
_
qpe 1
rpe
ae4−1
^
rpe
ae4
−1
rpe
ap11
rdp
ap21
^
rpe
^
ap1−1
^
rdp
^
ap1(32)
The estimation error Δd1is
Δd1−2_
rdp
rdp
_
qdp 2
^
_
rdp
^
rdp
^
_
qdp k_
qdp −k^
_
qdp −1
rdp
ad1
^
rdp
ad(33)
The sum of the aforementioned two errors is the system
uncertainty in Eq. (31). Due to the effectiveness of the cooperation
between the IMM filter and Kalman filter, the system uncertainty
(i.e., the estimation error) can be kept to such a small value that a
much smaller system uncertainty boundary can be chosen. Thus, the
chattering can be effectively attenuated.
Let ε1Δe1Δd1, where jε1j≤δ1jσ1j1∕2with constant δ1>0,
and
u−ad∕^
rdp. With Eq. (23), the guidance law is designed as
ad^
rdpα1jσ1j1∕2signσ1−^
rdpw^
rdp ^
φ1
_
w−β1signσ1(34)
where
α1>λ−1
1ε1δ2
1λ14ε2
12ε1δ1ε1
β12ε1α1λ14ε2
1, and ε1and λ1are arbitrary positive
constants.
Substituting Eq. (34) into Eq. (31) yields
_σ1−α1jσ1j1∕2signσ1wε1
_
w−β1signσ1(35)
According to [17], the stability analysis is divided into two steps.
First, select a pair of new states to change system (35) into a form that
is convenient for Lyapunov stability analysis. That is,
z1z1;1;z
1;2Tjσ1j1∕2signσ1;wT(36)
As jz1;1jjσ1j1∕2,ifz1;1and z1;2converge to zero in finite time,
then σ1and _σ1also converge to zero in finite time. A Lyapunov
function is selected as
V1z1zT
1P1z1(37)
where P1is a positive definite matrix with the form
P1λ14ε2
1−2ε1
2ε11(38)
In [17], it has been proved that
_
V1≤−r1V1∕2
1(39)
J. GUIDANCE, VOL. 41, NO. 11: ENGINEERING NOTES 2507
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where r12ε1λ1∕2
minP1∕λmax P1; and λminP1and λmaxP1are
the minimal and maximal eigenvalues of P1, respectively. And,
according to [19], the settling time is
ts1≤λmaxP1V1∕2
1x0∕ε1λ1∕2
minP1
Thus, according to Eq. (36), σ1and _σ1also converge to zero in
finite time. From Eq. (29), _
eke 0holds in finite time with σ1
converging to zero in finite time. Solving this equation gives
ee0exp−kt, meaning the STTGL asymptotically drives the
defender onto the LOS connecting the evader and the pursuer.
C. LOS Angular Rate Nullifying Guidance Law
In this section, the supertwisting sliding mode control-based LOS
angular rate nullifying guidance law (STLGL) is designed. Like
the STTGL, the system uncertainty estimation and compensation
approach is used.
Let the LOS angular rate be the sliding variable
σ2_
qdp (40)
From the Eq. (2), the derivative of σ2with respect to time is
_σ2φ2−1
rdp
ad(41)
where φ2−2_
rdp _
qdp∕rdp up2∕rdp . The parameters rdp,_
rdp, and
_
qdp can be estimated by the defender’s filter. Also, ^
ap2, which is the
estimate of ap2, can be calculated from ^
ap1via coordinate
transformations. Thus, Eq. (41) can be rewritten as
_σ2^
φ2Δe2Δd2−1
^
rdp
ad(42)
where ^
φ2−2_
^
rdp _
^
qdp∕^
rdp ^
ap1∕^
rdp,Δe2ap2∕rdp −^
ap2∕^
rdp is
the estimation error of the evader’s filter, and
Δd2−2_
rdpx2∕rdp 2_
^
rdp _
^
qdp∕^
rdp −ad∕rdp ad∕^
rdp
is the estimation error of the defender’s filter. The sum of the two
errors is the system uncertainty in Eq. (42), which is much smaller
than φ2due to the effectiveness of the filters.
Let ε2Δe2Δd2, where jε2j≤δ2jσ2j1∕2with constant δ2>0,
and u−ad∕^rdp . With Eq. (42), the guidance law is designed as
ad^
rdpα2jσ2j1∕2signσ2−^
rdpw^
rdp ^
φ2
_
w−β2signσ2(43)
where
α2>λ−1
2ε2δ2
2λ24ε2
22ε2δ2ε2
β22ε2α2λ24ε2
2, and ε2and λ2are arbitrary positive
constants. Note that the two guidance laws [Eqs. (34) and (43)] have
the same form.
Substituting Eq. (43) into Eq. (42) yields
_σ2−α2jσ2j1∕2signσ2wε2
_
w−β2signσ2(44)
Similarly, according to [17], first select a pair of new states to
change system (44) in a form as follows:
z2z2;1;z
2;2Tjσ2j1∕2signσ2;wT(45)
As jz2;1jjσ2j1∕2,ifz2;1and z2;2converge to zero in finite time,
then σ2and _σ2also converge to zero in finite time. A Lyapunov
function is selected as
V2z2zT
2P2z2(46)
where P2is a positive definite matrix with the form
P2λ24ε2
2−2ε2
2ε21(47)
Similarly, the work in [17] proved that
_
V2≤−r2V1∕2
2(48)
where r22ε2λ1∕2
minP2∕λmax P2; and λminP2and λmaxP2are
the minimal and maximal eigenvalues of P2, respectively. And,
according to the work in [19], the settling time is
ts2≤λmaxP2V1∕2
2x0ε2λ1∕2
minP2
Thus, according to Eq. (45), σ2and _σ2also converge to zero in finite
time. With Eq. (40), _
qdp converges to zero in finite time.
IV. Simulation Results
The simulation is performed in an active defense scenario that
involves three vehicles: an evader, a pursuer, and a defender. First, we
present the initial conditions of the scenarios. Then, we compare
the performance of the two proposed guidance laws against a
supertwisting sliding mode guidance law that does not use the
approach of the system uncertainty estimation and compensation.
A. Simulation Scenario
In a 200-run Monte Carlo simulation, the initial position of the
evader is set to be the origin of the coordinate system, i.e., 0;0Tm.
The initialposition of the pursueris set to be 105;0Tδm,whereδis
a position drift, which is a two-dimensional uniform distributed
random variablewith each component in therange of [−1000,1000] m.
The initial position of the defender is 5000;0Tm. The initial speeds
of the evader and defender are both 6000 m∕s, and the initial speed of
the pursuer is 2500 m∕s. The initial flight-path angle of the evader
is γe−5deg, the initial flight-path angle of the pursuer is
γp−10.23 deg, and the initial flight-path angle of the defender
is γd−0.74 deg. The maximum accelerations of the evader and
pursuer are set to be 2 and 6g, respectively.
It is assumed that the evader and defender can measure the relative
position to the pursuer with the sampling rate of 100 Hz. The
measurements are contaminated by a Gaussian noise with zero mean
and a variance of 25 m2. The pursuer uses the PN guidance law to
intercept the evader. The initial navigation ratio is set to three, which
is a typical value in many scenarios; after the range between the
pursuer and the evader reduces to half of the initial range, the ratio
switches to five, which is a relatively larger one.
B. Performance of Guidance Law
To evaluate the performance of the proposed guidance laws, a
benchmark guidance law is implemented for comparison. Like the
STLGL, this guidance law also adopts the strategy of nullifying the
LOS angular rate. However, it does not use the approach of system
uncertainty estimation and compensation. Here, we named it as
STTGL0, which has the following form:
ud^
rdpα3jσ3j1∕2signσ3−^
rdpw
_
w−β3signσ3(49)
where jφ2j≤δ3jσ3j1∕2with the positive constant δ3,
α3>λ−1
3ε3δ2
3λ34ε2
32ε3δ3ε3
β32ε3α3λ34ε2
3, and ε3and λ3are arbitrary positive
constants. Due to the absence of the system uncertainty estimation
2508 J. GUIDANCE, VOL. 41, NO. 11: ENGINEERING NOTES
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and compensation, compared with δ1and δ2, the constant δ3in
Eq. (49) must be set to a much larger value to keep the system stable.
The larger boundary δ3results in the larger switching gains α3and β3,
which in turn result in the larger acceleration requirement and more
intensive chattering. Thus, the major advantage of the proposed
guidance laws over the benchmark guidance law is the notably
smaller acceleration requirement and significantly more attenuated
chattering.
In Fig. 2, the defender’s cumulative distributions functions of the
miss distances are given. In simulations, the evader conducts a
sinusoidal maneuver or does not maneuver. As to the cumulative
distribution function of the miss distance, the steeper the profile is, the
better the guidance law performs. When the evader conducts the
sinusoidal maneuver, the profiles with the STTGL and STLGL are
steeper than that with the STLGL0, meaning their mean miss
distances have a substantial reduction as compared with that of the
STLGL0. When the evader does not maneuver, the three guidance
laws have almost the same cumulative distribution profiles.
Meanwhile, it can be found that the performances of all the three
guidance laws in this case are better than their performances in the
case in which the evader conducts the sinusoidal maneuver. This is
because, when the evader does not maneuver, it is easy to be
intercepted by the pursuer with less acceleration requirements;
whereas a pursuer with less acceleration requirements is, in turn, easy
to intercept by the defender.
The defender’sy-axis accelerations in the scene inertial coordinate
system are shown in Fig. 3. It can be found that, at the initial stage, the
guidance command in the STLGL0 is dramatically larger than those
of the STTGL and STLGL. As the STLGL0 does not estimate and
compensate the system uncertainty, a much higher uncertainty
boundary has to be chosen to make the system stable. This, in turn,
results in the much larger acceleration requirement. The larger
acceleration helps to rapidly reduce the LOS angular rate to zero;
thus, the needed acceleration by the STLGL0 reduces fast after
successfully nullifying the LOS angular rate. However, due to the
larger system uncertainty boundary, the chattering in the STLGL0 is
much more notable than that under the STTGL and STLGL.
C. Performance of the IMM Filter
As mentioned in the guidance law design, many parameters
needed by the guidance laws are not available directly, and they need
to be estimated via the cooperation between the evader’s IMM filter
and the defender’s Kalman filter. In this research, the evader knows
that the pursuer is intercepting itself using a PN guidance law.
However, the evader only knows the range of the navigation ratio
(e.g., in this research, it is assumed to be in the range of three to five)
instead of the exact value. Considering that the pursuer also switches
its guidance ratio during the guidance, an IMM filter is used to
address this problem. The IMM filter uses five PN models of PN3,
PN3.5,PN4,PN4.5 , and PN5with navigation ratios of 3, 3.5, 4, 4.5,
and 5, respectively. The IMM filter estimates the relative position,
relative velocity, and acceleration of the pursuer and communicates
all these estimations to the defender. With the pursuer’s acceleration
estimation from the evader, the defender uses a Kalman filter to
estimate the states of the pursuer. All the estimation values are
eventually provided to the guidance law.Thus, theperformance ofthe
IMM filter has a great influence on the guidance law. Here, we focus
on the performance of the IMM filter.
There are five element filters in the IMM filter, with each for a PN
model. For each element filter, the relative position components of the
initial states are set to the initial measurements. The initial relative
velocitycomponents are assumedto be available by some methodsuch
as time differentiating of the relative position components. The
acceleration of the pursuers can hardly be calculated with sufficient
precisionvia differentiating the velocity, which is already an estimation
value from the differentiation of the position measurement corrupted
by noise. Thus, the initial acceleration of the pursuer is set to zero.
The element filters of the IMM filter’s process noise matrix is
Qe10−3diag1;1;1;1;1;1, and the measurement noise matrix
is R25diag1;1. In the scenarios, the evader has no prior
information on the navigation ratio of the pursuer. Thus, all the prior
model probabilities are set to the same value, i.e., the initial model
probabilities of μ0.2 0.2 0.2 0.2 0.2 .
The IMM filter’s model probabilities are shown in Fig. 4. In both
figures, from the initial time, model PN3’s probability increases and
converges to one very fast, meaning that the IMM filter effectively
Fig. 2 Miss distance cumulative distribution function of the defender:
a) evader conducts sinusoidal maneuver; and b) evader does not
maneuver.
Fig. 3 Accelerations of the defender: a) evader conducts sinusoidal
maneuver; and b) evader does not maneuver.
Fig. 4 Model probabilities: a) evader conducts a sinusoidal maneuver;
and b) evader does not maneuver.
J. GUIDANCE, VOL. 41, NO. 11: ENGINEERING NOTES 2509
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identifies the pursuer’s navigation ratio, which is three, in a very short
time. As mentioned earlier, the pursuer switches its navigation ratio
from three to five when the range reduces to half of the initial range. In
these two simulations, the switching of the guidance ratio happens at
about 5.9 s. Due to the time lag effect of the filter, it takes about 0.5 s
for the IMM filter to respond to the pursuer switching its navigation
ratio such that the model probabilities are changed accordingly. It can
be found that the probability of model PN3drops immediately at
6.4 s. Because the current navigation ratio of the pursuer is five, the
probability of model PN5should increase and converge to one very
soon. In the simulations, the probability of model PN5does increase,
and it is interesting to find that the probabilities of some other models
also increase. In Fig. 4a, they are models PN4and PN4.5 (with
navigation ratios of 4 and 4.5, respectively); in Fig. 4b, it is model
PN4.5. In Fig. 4a, after about 2 s, model PN5’s probability begins to
show its dominance, whereas in Fig. 4b, model PN5’s probability
fails to show its dominance and converges to 0.5. The reason for the
phenomenon is twofold. One is the time lag effect of the filter. The
second reason is that, with the LOS angular rate converging to zero
due to the long-time effort of the pursuer’s guidance law, the
observability of the pursuer becomes weak. The weak observability,
in turn, makes the time lag effect even more evident. In Fig. 4a,
the evader conducts a sinusoidal maneuver, and this makes the
observability a bit stronger. It appears that this improvement of the
observability helps the filter discriminate the real model (i.e., PN5)
from the others. However, in Fig. 4b, as the evader does not conduct
any maneuver, the observability is very weak due to the LOS angular
rate nullification. And, at the same time, the navigation ratios of
models PN4.5 and PN5are so close that the filter fails to discriminate
these two models under such a weak observability situation.
Fortunately, because the estimation of the IMM filter is actually the
probabilistically weighted sum of the estimations of all the element
filters, the slow convergence or even nonconvergence of the model
probabilities has little influence on the miss distance of the defender
(see Fig. 2) and the estimation of the pursuer’s acceleration
(see Fig. 5).
The estimation errors of the relative position and relative velocity
between the evader and the pursuer, as well as the estimation errors of
the pursuer’s acceleration in the same Monte Carlo simulation, are
shown in Figs. 6, 7, and 5, respectively. It is shown that the estimation
root-mean-square errors (RMSEs) are close to zero, except in the
initial phase or the middle phase of the guidance process. As
mentioned before, the initial state of the IMM filter has notable errors.
In the middle phase of the guidance process, the pursuer switches its
navigation ratio, resulting in the model mismatch in the IMM filter.
Due to the time lag effect, the filter needs some time to adjust itself.
Thus, the errors deviate from zero over this period. Fortunately, as the
guidance ratio switching happens in the middle of the guidance
process, it leaves enough time for the IMM filter to converge again.
In light of this, it seems that one of the pursuer’s strategies against the
tracking is to switch its navigation ratio abruptly at the later phase of
the guidance process by leaving a very narrow time window for the
IMM filter to converge. Interestingly, this strategy has little impact on
the IMM filter. The reason is as follows. At the terminal phase of the
guidance process, the LOS angular rate is very close to zero. By
Eq. (4), now, the guidance command is also very small and is
nearly unchanged, whatever the guidance ratio (varying in a finite
range), due to the fact that the guidance command is proportional to
the product of the navigation ratio and the LOS angular rate.
Additionally, the remaining time is so little that the pursuer’s motion
model can hardly have a distinct change. Thus, the IMM filter can still
effectively estimate the states of the pursuer in this circumstance.
In conclusion, with the assumption that the pursuer is intercepting
the evader via the PNG law, the IMM filter can effectively estimate
the states of the pursuer.
V. Conclusions
To address the active defense guidance problem, two supertwisting
sliding mode guidance laws are proposed. One adopts the triangle
guidance strategy, and the other one adopts the strategy of nullifying
the LOS angular rate. To provide the guidance laws with the estimated
parameters with accuracy, a cooperative estimation approach between
the evader’s IMM filter and the defender’s Kalman filter is proposed.
By assuming the pursuer uses the PNG guidance law, the IMM
filter can effectively estimate the states of the pursuer, even if the
pursuer switches its guidance ratio during the guidance process.
Fig. 5 IMM filter’s RMSE of the pursuer’s acceleration in the scene
inertial coordinate system.
Fig. 6 IMM filter’s RMSE of the relative position in the scene inertial
coordinate system.
Fig. 7 IMM filter’s RMSE of the relative velocity in the scene inertial
coordinate system.
2510 J. GUIDANCE, VOL. 41, NO. 11: ENGINEERING NOTES
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The cooperative estimation and compensation of the system
uncertainty significantly attenuate the chattering under active defense
guidance laws designed using the supertwisting sliding mode control.
Numerical simulations demonstrate that the proposed guidance
laws can notably improve the accuracy of the active defense
guidance, dramatically reduce the maximum acceleration require-
ment, and effectively attenuate the chattering.
Acknowledgments
This work is supported by the National Natural Science Foundation
of China ( grant nos. 61773142 a nd 61471023). The a uthors would like
to thank X. R. Li, Huimin Chen, and especially the three anonymous
reviewers for their many useful suggestions that substantially
improved the quality of this Note.
References
[1] Boyell, R. L., “Defending a Moving Target Against Missile or Torpedo
Attack,”IEEE Transactions on Aerospace and Electronic Systems,
Vol. AES-12, No. 4, 1976, pp. 522–526.
doi:10.1109/TAES.1976.308338
[2] Shima, T., “Optimal Cooperative Pursuit and Evasion Strategies Against
a Homing Missile,”Journal of Guidance, Control, and Dynamics,
Vol. 34, No. 2, 2011, pp. 414–425.
doi:10.2514/1.51765
[3] Prokopov, O., and Shima, T., “Linear Quadratic Optimal Cooperative
Strategies for Active Aircraft Protection,”Journalof Guidance, Control,
and Dynamics, Vol. 36, No. 3, 2013, pp. 753–764.
doi:10.2514/1.58531
[4] Shaferman, V., and Shima, T., “Cooperative Multiple-Model
Adaptive Guidance for an Aircraft Defending Missile,”Journal
of Guidance, Control, and Dynamics, Vol. 33, No. 6, 2010,
pp. 1801–1813.
doi:10.2514/1.49515
[5] Rusnak, I., Weiss, H., and Hexner, G., “Guidance Laws in Target-
Missile-Defender Scenario with an Aggressive Defender,”18th IFAC
World Congress in Italy, International Federation of Automatic Control,
Laxenburg, Austria, 2011, pp. 9349–9354.
doi:10.3182/20110828-6-IT-1002.02587
[6] Perelman, A., Shima, T., and Rusnak, I., “Cooperative Differential
Games Strategies for Active Aircraft Protection from a Homing
Missile,”Journal of Guidance, Control, and Dynamics, Vol. 34, No. 3,
2011, pp. 761–773.
doi:10.2514/1.51611
[7] Ratnoo, A., and Shima, T., “Line-of-Sight Interceptor Guidance for
Defending an Aircraft,”Journal of Guidance, Control, and Dynamics,
Vol. 34, No. 2, 2011, pp. 522–532.
doi:10.2514/1.50572
[8] Yamasaki, T., Balakrishnan, S. N., and Takano, H., “Geometrical
Approach-Based Defense-Missile Intercept Guidance for Aircraft
Protection Against Missile Attack,”Journal of Aerospace Engineering,
Vol. 226, No. 8, 2012, pp. 1014–1028.
doi:10.1177/0954410011416053
[9] Zou, X. G., Zhou, D., Du, R. L., and Liu, J. Q., “Adaptive Nonsingular
Terminal Sliding Mode Cooperative Guidance Law in Active Defense
Scenario,”Journal of Aerospace Engineering, Vol. 230, No. 2, 2016,
pp. 307–320.
doi:10.1177/0954410015591613
[10] Ratnoo, A., “Analysis of Two-Stage Proportional Navigation with
Heading Constraints,”Journal of Guidance, Control, and Dynamics,
Vol. 39, No. 1, 2016, pp. 156–164.
doi:10.2514/1.G001262
[11] Magill, D. T., “Optimal Adaptive Estimation of Sampled Stochastic
Processes,”IEEE Transactions on Automatic Control, Vol. 10, No. 4,
1965, pp. 434–439.
doi:10.1109/TAC.1965.1098191
[12] Rusnak, I., “Multiple Model-Bas ed Terminal Guidance Law,”Journal
of Guidance, Control, and Dynamics, Vol. 23, No. 4, 2000,
pp. 742–746.
doi:10.2514/2.4593
[13] Oshman, Y., Shinar, J., and Weizman, S. A., “Using a Multiple-Model
Adaptive Estimator in a Random Evasion Missile/Aircraft Encounter,”
Journal of Guidance, Control, and Dynamics, Vol. 24, No. 6, 2001,
pp. 1176–1186.
doi:10.2514/2.4833
[14] Shima, T., Oshman, Y., and Shinar, J., “Efficient Multiple Model
Adaptive Estimation in Ballistic Missile Interception Scenarios,”
Journal of Guidance, Control, and Dynamics, Vol. 25, No. 4, 2002,
pp. 667–675.
doi:10.2514/2.4961
[15] Shaferman, V., and Shima, T., “Cooperative Multiple-Model Adaptive
Guidance for an Aircraft Defending Missile,”Journal of Guidance,
Control, and Dynamics, Vol. 33, No. 6, 2010, pp. 1801–1813.
doi:10.2514/1.49515
[16] Li, X. R., Zhi, X. R., and Zhang, X. M., “Multiple-Model Estimation
with Variable Structure-Part III: Model-Group Switching Algorithm,”
IEEE Transactions on Aerospace and Electronic Systems, Vol. 35,
No. 1, 1999, pp. 225–241.
doi:10.1109/7.745694
[17] Shtessel, Y. B., Moreno, J. A., Plestan, F., and Poznyak, A. S., “Super-
Twisting Adaptive Sliding Mode Control: A Lyapunov Design,”49th
IEEE Conference on Decision and Control, IEEE Publ., Piscataway, NJ,
Dec. 2010, pp. 5109–5113.
doi:10.1109/CDC.2010.5717908
[18] Levant, A., “Sliding Order and Sliding Accuracy in Sliding Mode
Control,”International Journal of Control, Vol. 58, No. 6, 1993,
pp. 1247–1263.
doi:10.1080/00207179308923053
[19] Zhou, D., Sun, S., and Teo, K. L., “Guidance Laws with Finite Time
Convergence,”Journal of Guidance, Control, and Dynamics, Vol. 32,
No. 6, 2009, pp. 1838–1846.
doi:10.2514/1.42976
J. GUIDANCE, VOL. 41, NO. 11: ENGINEERING NOTES 2511
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