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Robust finite-time guidance against maneuverable targets with unpredictable evasive strategies

01 Jun 2018-Aerospace Science and Technology (Elsevier BV)-Vol. 77, pp 534-544

TL;DR: Based on homogeneity technique, the local finite-time input-to-state stability is established for the closed-loop guidance system, thus implying the proposed RFTG law can quickly render the LOS rate within a bounded error throughout intercept.
Abstract: This paper presents a robust finite-time guidance (RFTG) law to a short-range interception problem. The main challenge is that the evasive strategy of the target is unpredictable because it is determined not only by the states of both the interceptor and the target, but also by external un-modeled factors. By robustly stabilizing a line-of-sight rate, this paper proposes an integrated continuous finite-time disturbance observer/bounded continuous finite-time stabilizer strategy. The design of this integrated strategy has two points: 1) effect of a target maneuver is modeled as disturbance and then is estimated by the second-order homogeneous observer; 2) the finite-time stabilizer is actively coupled with the observer. Based on homogeneity technique, the local finite-time input-to-state stability is established for the closed-loop guidance system, thus implying the proposed RFTG law can quickly render the LOS rate within a bounded error throughout intercept. Moreover, convergence properties of the LOS rate in the presence of control saturation are discussed. Numerical comparison studies demonstrate the guidance performance.

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The University of Manchester Research
Robust finite-time guidance against maneuverable targets
with unpredictable evasive strategies
DOI:
10.1016/j.ast.2018.04.004
Document Version
Accepted author manuscript
Link to publication record in Manchester Research Explorer
Citation for published version (APA):
Zhang, R., Wang, J., Li, H., Li, Z., & Ding, Z. (2018). Robust finite-time guidance against maneuverable targets
with unpredictable evasive strategies. Aerospace Science and Technology.
https://doi.org/10.1016/j.ast.2018.04.004
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Aerospace Science and Technology
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Download date:09. Aug. 2022

Robust finite-time guidance against maneuverable targets with unpredictable
evasive strategies
Ran Zhang
a
, Jiawei Wang
a
, Huifeng Li
a,
, Zhenhong Li
b
, Zhengtao Ding
b
a
School of Astronautics, Beihang University, Beijing 100191, P. R. China
b
School of Electrical and Electronic Engineering, University of Manchester, Manchester M13 9PL, UK
Abstract
This paper presents a robust finite-time guidance (RFTG) law to a short-range interception problem. The main challenge
is that the evasive strategy of the target is unpredictable because it is determined not only by the states of both the
interceptor and the target, but also by external un-modeled factors . By robustly stabilizing a line-of-si ght rate, this paper
proposes an integrated continuous finite-time disturbance observer/b oun de d continuous finite-time stabilizer strategy.
The design of this integrated strategy has two points: 1) eect of a target maneuver is modeled as disturbance and then
is estimated by the second-order homogeneous obs er ver; 2) the finite-time stabilizer is actively coupled with the observer.
Based on homogeneity technique, the local ni t e- t im e input-to-state stability is established for the closed-loop guidance
system, thus implying the propose d RFTG law can quickly render the LOS rate within a bounded error throughout
intercept. Moreover, convergence properties of the LOS rate in the presence of control saturation are di scu ss ed . Numerical
comparison studies demonstrate the guidance performance.
Keywords: Robust finite-time guidance, Maneuverable target, Unpredictable evasive strategy, Finite-time
input-to-state stability
1. Introduction
Interception of a maneuverable target is one of essential
questions in the study of homing guidance. With t he ad-
vancement of artificial intelligence, propulsion, composite
material, etc, evasive capability of the target has been in-
creasing at a rapid pace, and poses a challenging problem.
This new guidance problem is referred to as Unpredictable
Maneuverable Target Interception (UMTI) herein; that is,
the interceptor cannot predict th e evasive strategy of the
target. The existing evasive strategies consist of conven-
tional maneuver models [ 1, 2] and optimal evasive strate-
gies [3, 4, 5]. The conventional maneuver models, such as
a step maneuver, depend on prescribed maneuvers. The
optimal evasive strategies are completely deter mi n ed by
the relat i ve motion information of the interceptor and the
target. These two types cannot be applied t o the UMTI
since both of them do not explicitly take account of the
external un-modeled factors, particularly involving the de-
terministic yet unknown. In addition, conside ri n g that ma-
neuve r in g capability of the target is comparable to that of
the interceptor, the interceptor should be capable of ex-
ploiting its limit maneuverable capability, that is, control
saturation.
Corresponding author
Email addresses: zhangran@buaa.edu.cn (Ran Zhang),
j.wang@buaa.edu.cn (Jiawei Wang), lihuifeng@buaa.edu.cn
(Huifeng Li), zhenhong.li@postgrad.manchester.ac.uk
(Zhenhong Li), zhengtao.ding@manchester.ac.uk (Zhengtao Ding)
This paper is concerned wit h designing a robust guid-
ance law to achieve the UMTI. In the literature, a number
of guidance laws have been invest igat e d to deal with the
problems of the maneuverable target interception. These
guidance l aws can be roughly classified into two categories:
relative-motion-prediction-based guidance law (RMP-GL)
and manifold-stabilization-based guidance law (MS -G L) .
The RMP-GL usually obtains an optimal solution subject
to a predetermined interception engagement (such as an
ideal collision triangle scenario) using prediction of th e rel-
ative motion between the interceptor and the tar get . The
design tools of the RMP-GLs mainly include optimal con-
trol theory and dierential game theory. For example, in
[6], assuming an expli ci t model of the tar get maneuver and
a first-order missile dynami cs , the interception problem is
formul at ed as a linear quad rat i c control problem; in [4],
a nonlinear 3D-vector guidance law is designed, which is
an optimal strategy pair in the sense of the saddle-point
inequality. The RMP-GLs, however, cannot be applied
to the UMTI since th e prediction of the i ntercept motion
(including a time-to-go and an intercept point) i s dicult
to carry out due to significant uncertainties of the target
motion.
With res pect to the MS-GLs, they render the interceptor-
target relative motion around a prescribed manifold by
compensating for (or suppressing) adverse eect of vari-
ous disturbances and uncertainties related to the target
maneuver. The design tools mainly consist of adaptive
contr ol, sliding mode control, high-gain control, etc. In [7],
Preprint submitted to Aerospace Science and Technology April 3, 2018
To appear in Aerospace Science and Technology

the authors parameterize upper bounds of the target accel-
eration and then develop the resulting parameter adaptive
laws, t hus handling the target maneuver. As such, the con-
trol saturation an d the speed of the parameter adaptation
are two fundamental concerns in adaptive control design
[8, 9, 10]. To address these two concerns, in [11], a set of
relative-state-dependent basis functions is selected to rep-
resent the target acceleration , and then a weight vector
adaptive law is derived using optimal modification tech-
nique. Alth ough th e adapti ve guid anc e laws work well in
the above-mentioned sce nar i os, they have three drawbacks
herein: 1) the guidance performanc e heavily depends on
the convergence of adapti ve parameters, while these pa-
rameters are easily prone to diverge wh en the jerk of the
maneuverable target is considerable; 2) the robustness to
the target maneuver relies on the parameterization of the
target maneuver, while it is very h ar d to accurately pa-
rameterize the target maneuver under study; 3) the order
of the closed-loop guidance system will inevitably increase
as the dimension of the adaptive parameters grows.
The sliding mode control is used to suppress the ef-
fect of t h e target maneuver in [12, 13]. In [14], the high-
order sliding mode control is adopted to estimate and to
compensate for t he eect of the target maneuver. In this
regard, the sl i di n g mode guidance laws are upper-bound-
dependent. For the UMTI, the upper bounds of the target
acceleration are required to be chosen relatively big for
the sake of completely canceling the adverse eects of the
target maneuver. Consequently, the resulting control is
conservati ve and may induce severe chattering due t o var-
ious kinds of modeling imperfections [15]. As another way,
a smooth sliding mo d e guidanc e law is proposed in [16], in
which an adaptive law is us ed to estimate th e upper bound
of the target maneuver. However, the adaptive parameter
may di verge under the situation of control s at ur at i on.
In [17], the high-gain control is used to ensure input-to-
state stability for a closed-l oop LOS rate dynamics, and
thereby the residual error of the LOS rate can be made
suciently smal l in the presence of the target maneuver.
In [18], a finite-time guidance is proposed to nullify the
LOS rate, and the convergence boundary layer is theoret-
ically analyzed. The guidance pe rf or manc e of the high-
gain control, however, may det e ri or at e due to measure-
ment noise when the system bandwidth is excessively en-
larged. In fact, it is dicult to make a reason abl e trade-o
between the disturbance suppression and the noise attenu-
ation without the priori information of the evasive strategy
of the target.
Following the technical route of the MS-GLs, a ro-
bust finite-time guidan ce (RFTG) law is presented in this
paper. An integrated continuous finite-time disturbance
observer (CFTDO)/bounded continuous finite-time stabi-
lizer (BCFTS) strategy is proposed. The CFTDO is de-
signed to estimate the disturbances that are the eects
of the target maneuvers by making the observation-error
dynamics behave as a second-order homogeneous system.
The BCF TS uses a typical first-order homogeneous system
to specify the finite-time stability of the nominal guid-
ance system in consideration. Both the CFTDO and the
BCFTS are based on non-smooth yet continuous feedback
contr ol. Such non-smo ot h feedback control has a favor-
able characteristic: its equivalent control gain increases
as the feedback error decreases. In contrast to existing
smooth feedback controls, the non -s mooth feedback con-
trol may have better convergence and robustness [19, 20]
and is suitable to be applied to the UMTI. Given these
virtues, a fe w guidance laws based on the non-smooth
feedback control have been designed. For example, the
finite-time convergence of the LOS rate is realized in [12],
but the ee ct of the t ar get maneuver is suppressed using
the convent i on al sliding mode control; in [21], using the
high-gain control, the authors present a guidance law with
finite-time input-to-state stability (FTISS). Note that the
robustness of these two guidance laws is routinely guar an -
teed by the sliding mode control or the high-gain control.
As discussed previously, these two control methods have
their own drawbacks when adopted in the UMTI.
To address the preceding drawbacks, the int egr at ed
strategy is put forward with two key ideas: 1) the eect of
the target maneuver is modeled as the disturbance and is
estimated using the CFTDO; 2) the design of the BCFTS
is coupled wi t h that of the CFTDO. An unique charac-
teristic of this integrate d strategy is that it can explic-
itly d eal with the interplay between the CFTDO and the
BCFTS. This characterist i c is dierent from the existing
finite-time disturbance observer-based control methodolo-
gies (e.g., [22, 23, 24, 25, 26]), which d es ign the observer
and the finite-time controller in a decoupled fashion: they
are based on a zero-observation-error assumption. How-
ever, the observation errors cannot be perfectly canceled
in practi c e, especial l y concerning the UMT I. Nevertheless,
the proposed RFTG law can avoid such a restricted as-
sumption bec aus e of the integrated design adopted herein.
In addition, the proposed integrated strategy has sev-
eral appreciable advantages in the current scenari o. The
BCFTS, which uses the control saturation to quickly sta-
bilize the LOS rate, can take advantage of the limit of
the maneuver capability of the interceptor. Regarding the
CFTDO, it can rapidly estimate the disturbance of i n-
terest without suering from the parameter convergence
issues. Through cooperation between the BCFTS and the
CFTDO, the resultant RFTG law is independent of both
the explicit maneuver models and the upper bounds of
the target maneuver s. Further, the RFTG law can guar-
antee locally FTISS in the presence of bounded derivative
of the disturbance, implying the LOS rate can b e rendered
within a bounded error as q u ickly as possible. Numerical
comparison results demonstrate the proposed RFTG law
can guarant ee the high-precision miss-distance.
This paper is organized as follows. We begin by formu-
lating a new short-range interception problem. In partic-
ular, a notion of the unpredictable evasive strategies is in-
troduced. The next section pre sents the robust finite-time
guidance law. Then, the convergence analysis of the LO S
2

rate is conducted in non-saturation and saturation cases.
Finally, numerical comparison st u di es are performed t o as-
sess the guidance performance.
2. Problem formulation
The design and analysis of th e RFTG law is based on
the following assumptions:
Assumption 1. The motion of the interceptor and the
target is in a planar plane.
Assumption 2. Both the interceptor and the target per-
form maneuvers orthogonal to their velocity vectors .
Assumption 3. The speeds of the interceptor and the tar-
get, V
M
and V
T
,areconstant,andV
M
>V
T
.
The equations of the relative motion between the in-
terceptor and the target are given as follows [27]:
˙r = V
M
cos(
M
)+V
T
cos(
T
), (1)
˙
=
V
M
sin(
M
)
r
V
T
sin(
T
)
r
, (2)
˙! =
r!
r
cos(
M
)a
M
r
+
cos(
T
)a
T
r
, (3)
˙
M
=
a
M
V
M
, (4)
˙
T
=
a
T
V
T
, (5)
where r is the interceptor-target range. is the LOS angle.
! i s the LOS rate: ! =
˙
.
M
and
T
are flight-path
angles of the interceptor and the target. a
M
and a
T
are
acceleration of the interceptor and the target, respectively.
Herein, it is assumed that the variables, rr, , !, V
M
,
and
M
, are available to the interceptor, while the states
of the target, V
T
,
T
, and a
T
are unknown.
The target acceleration in the UMTI can be described
by
a
T
= (x, , $, t), (6)
where x represents the states of the interceptor and the
target, such as r, , V
M
, and V
T
. represents the states
of the prescribed maneuver models sat isfying
˙ = f(, & ,t), (7)
where & represents the parameter vector of the prescribed
target maneuvers, such as frequency in a sinusoidal fu n c-
tion. $ represents the determin is t ic yet unknown maneu-
vers, such as an artificial-intell i gen ce model and an event-
trigger-based model. It is $ that makes the target accel-
eration unpredictable.
Assumption 4. The target acceleration (x, , $, t) is
continuously dierentiable with respect to t,andisbounded,
i.e., |(x, , $, t)|U
T
. The target jerk
d(x,,$,t)
dt
is also
bounded.
It is wor t h mentioning that the target m ane uver model
(6) almost encompasses existing target maneuver models
used so far in the homing guidance literature. To be spe-
cific,
1. when a
T
= (x, t), the resulting maneuvers are de-
termined by the information of the interceptor and
the target. The optimal evasive strategies fall into
this class;
2. when a
T
= (, t), the resulting maneuvers repre-
sent the predictable maneuvers with the prescr i bed
forms defined by (7), such as step maneuver, vertical-
S maneuver, spiraling maneuver, etc;
3. when a
T
= (x, , $, t), the resulting maneuvers are
the model of the UMTI, which can not be predicted
onboard due to unavailable mathematical descrip-
tion for $.
Remark 1. From the view of the target maneuver mod-
els, the above three cases clarify the main dierences be-
tween the UMTI and the exis t ing interception problems.
A specific target maneuver model concerning (6) is gi ven
in Section 5.1 to assess the guida nce performance of the
proposed gu idan ce law.
At this point, we end this section by the objective of
the present study: design a guidance law to intercept the
target with acceptable small miss-distance in the presence
of the target maneuvers descr i bed by (6).
3. Robust finite-time guidance design
In th is section, the RFTG law is p r es ented: first, th e
CFTDO is designed to estimate the eect of the target ma-
neuve r s; second, the integrated CFTDO/BCFTS strategy
is proposed to robustly stabil iz e the LOS r at e. Note that
the design of both the CFTDO and the BCFTS is based on
homogeneity technology, which is able to construct homo-
geneous dynamic systems with favorable finite-time sta-
bility. The fundamental notions used herein are briefly
intr oduce d i n Append i x 7. 1, and Appendix 7.2.
3.1. Continuous finite-time disturbance observer (CFTDO)
To estimate the eect of the target maneuver, the CFTD O
is designed as follows:
˙
ˆ! =
r!
r
cos(
M
)a
M
r
+
ˆ
d
T
+ k
1
|! ˆ!|
sgn(! ˆ!),
˙
ˆ
d
T
= k
2
|! ˆ!|
21
sgn(! ˆ!),
(8)
where ˆ! is the estimated LOS rate.
ˆ
d
T
is the es t i mat ed
eect of the target maneuver d
T
=
cos(
T
)a
T
r
.
1
2
<<
1. k
1
> 0. k
2
> 0. ˆ!(0) = ˆ!
0
.
ˆ
d
T
(0) =
ˆ
d
T 0
. sgn(·)isa
signum function.
3

Correspondingly, the observation-error dynamics are
given by
˙e
!
= e
d
k
1
|e
!
|
sgn(e
!
),
˙e
d
= k
2
|e
!
|
21
sgn(e
!
)+(t),
(9)
where e
!
= !ˆ!. e
d
= d
T
ˆ
d
T
. (t)=
˙
d
T
. e
!
(0) = !(0)
ˆ!
0
. e
d
(0) = d
T
(0) d
T 0
.When(t) = 0, the obtained
observation -e rr or dynamics are reduced to a second-order
homogeneous syst e m whose origin is finite-time stable [20].
The stability analysis is carried out i n Section 4.1. When
(t) 6= 0 and |(t)| is bounded, the observati on errors can
be made suciently small, which is proved in Section 4.2.
Remark 2. In light of the terms involving
1
r
in (8), their
value will be singu lar when r =0. Nevertheless, as dis-
cussed in the guidance literature, such as [28], this kind of
singularity has little impact on the guidance performa nce.
Besides, in practice, the enga gemen t ends before r =0.
Remark 3. If =1, th e observer (8) is a conventional
Luenberger observer [29]. Compared with the Luenberger
observer, the observer (8) has better performance in con-
vergence and disturbance rejectio n by virtue of the non-
smooth feedback. In the case of =
1
2
,itisasecond-
order sliding mode observer. Both the second-order slid-
ing mode observer and the observer (8) can guarantee the
finite-time convergence. Unlike the second-order s li di ng
mode observer, the observer ( 8) does not require the up-
per bound of the target jerk, which is hard to be properly
determined in the UMTI .
Remark 4. Another favorable property of the CFTDO is
that it works well in the presen ce of the control saturation,
which is observed in the numerical simul ati ons .
3.2. Integrated CFTDO/BCFTS strategy
To stabilize the LOS rate with finite-time convergence,
the robus t finite-time guidance (RFTG) law is designed
based on the integrated CFTDO /B CF TS s tr at egy :
a
M
= U
M
sat
(, !,
ˆ
d
T
)
U
M
,
(, !,
ˆ
d
T
)=
1
cos(
M
)
(r! + c
1
r|!|
sgn(!)+r
ˆ
d
T
),
(10)
where U
M
is the upper bound of the interceptor accelera-
tion. c
1
> 0. 0 <<1. sat(·) is a saturation function:
sat(y)=min{1, |y|}s gn( y).
As shown in (10), the RFTG law is composed by the
CFTDO and the BCFTS. The task of the BCFTS is to
stabilize the LOS rate within a small enough error in finite
time. In particular, when |(, !,
ˆ
d
T
)| <U
M
,theresulting
closed-loop dynamics of the LOS rate is
˙! = c
1
|!|
sgn(!)+e
d
. (11)
If e
d
= 0, the resu lt i n g dynami cs r ed uc e to a typical first-
order homogeneous system with the finite-time stability,
specifying the nominal performance of the BCFTS.
The following lemma states a particular homogeneous
property of the integrated CFTDO/BCF TS strategy.
Lemma 1. Suppose that (t)=0,and|(, !,
ˆ
d
T
)| <
U
M
. Then, by = , the obtained closed-loop LOS rate
dynamics are homogeneous of degree 1
1
with dilation
(
1
,
1
, 1).
Proof. Consider (t) = 0 and |(, !,
ˆ
d
T
)| <U
M
. Using
the RFTG law (10), together with = , the closed-loop
closed-loop LOS rate dynamics are given by
˙! = c
1
|!|
sgn(!)+e
d
,
˙e
!
= e
d
k
1
|e
!
|
sgn(e
!
),
˙e
d
= k
2
|e
!
|
21
sgn(e
!
).
(12)
By definition of the homogeneous vector field, it is straight-
forward to check that t h e dynamic system (12) is homo-
geneous of degree 1
1
with dilation (
1
,
1
, 1).
If = , this integrated CFTDO/BCFTS strategy has
two advantages. First, it can explicitly cope with the inter-
play between the CFTDO an d the BCF TS . Using th e inte-
grated strategy, the cl osed -l oop system is constructed as a
third-order homogeneous system with the finite-time sta-
bility. In the literature, the existing finite-time disturbance
observer-based control methodologies (e.g., [23, 25, 26])
design t h e observer and the finite-time controller inde-
pendent l y. In this case, the di s tu r ban ce s must be com-
pletely cancele d by the observer in finite time, and then
the finite-time controller is designed b ase d on this zero-
observation -e rr or ass ump t i on. However, such convenient
design methodologies are not acceptable for the UMTI
since the zero-observation-error assumption is not valid.
Second, it can eectively reduce the so-called chattering
eects in implement at ion . As pointed out by [15], the
finite-time convergence is in conflict with the chattering
suppression demand. A small power in the non-smooth
feedback is the main reason. In this respect, inc re asi n g
the power is a reasonable solution to relieving chattering.
By virtue of the integrated design, the power can be in-
creased without sacrifici n g the robustness, thereby reduc-
ing the chattering eects.
3.3. Guidance scheme of RFTG law
The RMP-GLs and the MS-GLs employ dist in ct guid-
ance schemes. The scheme of the former is to first p r ed ic t
the relative motion between the interceptor and the tar-
get through out the engagement, and then to correct the
guidance command, so that the guided relat ive motion
meets the prescribed guidance requir em ents. The MS-
GLs’s guidance scheme is to directly maintain a desired
relative motion relationship, which is a suc ie nt condi-
tion that can guarantee a successful interception. From
4

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Journal ArticleDOI
TL;DR: Interceptor using norm differential game guidance laws is able to pursue the spacecraft without being intercepted by defender, even though the maneuverability of both interceptor and protector is lower than defender.
Abstract: This paper investigates the guidance problem for interceptors against spacecraft with active defense in a two-on-two engagement. There are four adversaries in the engagement which are interceptor, protector, spacecraft and active defender. The interceptor is required to capture the spacecraft and evade the defender with the assist of the protector. Two classes of guidance laws are proposed based on norm differential game strategy and linear quadratic differential game strategy, respectively. Interceptor using norm differential game guidance laws is able to pursue the spacecraft without being intercepted by defender, even though the maneuverability of both interceptor and protector is lower than defender. Additionally, linear quadratic differential game guidance laws derived by numerical solution of Riccati differential equation take into account the control saturation, fuel cost and chattering phenomenon simultaneously. Finally, the effectiveness and performance of the developed guidance approaches are demonstrated by nonlinear numerical simulations.

10 citations


Proceedings ArticleDOI
Zhongguo Li1, Zhengtao Ding1Institutions (1)
10 Jul 2019-
TL;DR: It is proved that, for any communication network with a connected graph, a fixed-time consensus is achieved, independent of the initial conditions, based on which an NE can be obtained asymptotically by the gradient descent term for the fixed- time consensus-based algorithm.
Abstract: In this paper, distributed algorithms are designed to search the Nash equilibrium (NE) for an $N$ -player game in continuous-time. The agents are not assumed to have direct access of other agents' states, and instead, they estimate other agents' states by communicating with their neighbours. Advanced consensus algorithms are implemented for such purposes, and consequently the game is decentralised into $N$ subsystems interacting over a communication network. It is proved that, for any communication network with a connected graph, a fixed-time consensus is achieved, independent of the initial conditions, based on which an NE can be obtained asymptotically by the gradient descent term for the fixed-time consensus-based algorithm. Then, the results are extended to a fixed-time NE seeking with modifications of the gradient terms, where both the consensus and the optimisation can be obtained in fixed time, and the upper bound of the settling time is established by the Lyapunov theory. A simulation example is presented to verify the effectiveness of the theoretical development, where some comparisons with other works are studied to demonstrate the advantages of the proposed algorithms.

6 citations


Journal ArticleDOI
Feng Yang1, Guangqing Xia1Institutions (1)
TL;DR: A relative Line-of-Sight (LOS) velocity based finite-time three-dimensional guidance law design framework is presented, and the application of fixed-time convergence disturbance observer in this framework is discussed.
Abstract: In order to achieve accurate interception of high-speed maneuvering targets, this paper presents a relative Line-of-Sight (LOS) velocity based finite-time three-dimensional guidance law design framework, and discusses the application of fixed-time convergence disturbance observer in this framework. Firstly, a simple Lyapunov function is provided to show that the coupled terms in the relative kinematics can be ignored in the proposed guidance law design framework. Secondly, the realizations of several classical guidance laws are analyzed with the proposed framework, including TPN guidance law, finite-time Input-to-State Stability (ISS) guidance law, and sliding mode guidance law. Thirdly, fixed-time convergence disturbance observers are introduced to design the composite finite-time 3D guidance law, and Lyapunov method is employed to show the stability of the guidance system. Numerical simulations with different scenarios show that the proposed generalized guidance law performs high interception accuracy.

5 citations


References
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S.R. Liberty1Institutions (1)
01 Nov 1981-
Abstract: Most of the signal processing that we will study in this course involves local operations on a signal, namely transforming the signal by applying linear combinations of values in the neighborhood of each sample point. You are familiar with such operations from Calculus, namely, taking derivatives and you are also familiar with this from optics namely blurring a signal. We will be looking at sampled signals only. Let's start with a few basic examples. Local difference Suppose we have a 1D image and we take the local difference of intensities, DI(x) = 1 2 (I(x + 1) − I(x − 1)) which give a discrete approximation to a partial derivative. (We compute this for each x in the image.) What is the effect of such a transformation? One key idea is that such a derivative would be useful for marking positions where the intensity changes. Such a change is called an edge. It is important to detect edges in images because they often mark locations at which object properties change. These can include changes in illumination along a surface due to a shadow boundary, or a material (pigment) change, or a change in depth as when one object ends and another begins. The computational problem of finding intensity edges in images is called edge detection. We could look for positions at which DI(x) has a large negative or positive value. Large positive values indicate an edge that goes from low to high intensity, and large negative values indicate an edge that goes from high to low intensity. Example Suppose the image consists of a single (slightly sloped) edge:

1,684 citations


"Robust finite-time guidance against..." refers background in this paper

  • ...If ↵ = 1, the observer (8) is a conventional Luenberger observer [29]....

    [...]


Book
01 Jan 1990-
Abstract: Numerical Techniques Fundamentals of Tactical Missile Guidance Method of Adjoints and the Homing Loop Noise Analysis Convariance Analysis and the Homing Loop Proportional Navigation and Miss Distance Digital Fading Memory Noise Filters in the Homing Loop Advanced Guidance Laws Kalman Filters and the Homing Loop Other Forms of Tactical Guidance Tactical Zones Strategic Considerations Boosters Lambert Guidance Strategic Intercepts Miscellaneous Topics Ballistic Target Properties Extended Kalman Filtering and Ballistic Coefficient Estimation Ballistic Target Challenges Multiple Targets Weaving Targets Representing Missile Airframe with Transfer Functions Introduction to Flight Control Design Three-Loop Autopilot. Appendices: Tactical and Strategic Missile Guidance Software Converting Programmes to C Converting Programmes to MATLAB Units.

1,487 citations


"Robust finite-time guidance against..." refers background in this paper

  • ...The equations of the relative motion between the interceptor and the target are given as follows [27]: ṙ = V M cos( M ) + V T cos( T ), (1) ̇ = V M sin( M )...

    [...]


Journal ArticleDOI
Lionel Rosier1Institutions (1)
Abstract: The goal of this article is to provide a construction of a homogeneous Lyapunov function P associated with a system of differential equations J = f(x), x ~ R ~ (n > 1), under the hypotheses: (1) f ~ C(R n, ~) vanishes at x = 0 and is homogeneous; (2) the zero solution of this system is locally asymptotically stable. Moreover, the Lyapunov function V(x) tends to infinity with 1( x (I, and belongs to C=(R~\{0}, R)n CP(~ ~, ~), with p E (~* as large as wanted. As application to the theory of homogeneous systems, we present two well known results of robustness, in a slightly extended form, and with simpler proofs.

643 citations


Journal ArticleDOI
01 Aug 2007-Automatica
TL;DR: The smooth second-order sliding mode control-based guidance law is designed and compared with augmented proportional navigation guidance law via computer simulations of a guided missile intercepting a maneuvering ballistic target.
Abstract: A new smooth second-order sliding mode control is proposed and proved using homogeneity-based technique for a system driven by sufficiently smooth uncertain disturbances. The main target application of this technique-the missile-interceptor guidance system against targets performing evasive maneuvers is considered. The smooth second-order sliding mode control-based guidance law is designed and compared with augmented proportional navigation guidance law via computer simulations of a guided missile intercepting a maneuvering ballistic target.

504 citations


"Robust finite-time guidance against..." refers methods in this paper

  • ...The sliding mode guidance law (SMGL) consists of the second-order sliding mode observer (SOSMO) and the second-order sliding mode stabilizer (SOSMS) [14]:...

    [...]

  • ...The sliding mode guidance law (SMGL) consists of the second-order sliding mode observer (SOSMO) and the second-order sliding mode stabilizer (SOSMS) [14]: a M = 1 cos( M ) (⇣ 1 |⌦|2/3 sgn(⌦) + ⇣ 2 Z |⌦|1/3 sgn⌦dt N s ṙ! c s ṙ/(2 p r) + z 1 ), (56) where z 1 is obtained by the following observer: ż 0 = v 0 cos( M )a M ṙ! csṙ/(2 p r), ż 1 = v 1 , ż 2 = 1.1L sgn(z 2 v 1 ), (57) in which v 0 = 2L1/3|z 0 ⌦|2/3 sgn(z 0 ⌦) + z 1 , v 1 = 1.5L1/2|z 1 v 0 |1/2 sgn(z 1 v 0 ) + z 2 ....

    [...]

  • ...In [14], the highorder sliding mode control is adopted to estimate and to compensate for the e↵ect of the target maneuver....

    [...]


Proceedings ArticleDOI
Sanjay P. Bhat1, Dennis S. Bernstein1Institutions (1)
04 Jun 1997-
Abstract: Examines finite-time stability of homogeneous systems. The main result is that a homogeneous system is finite-time stable if and only if it is asymptotically stable and has a negative degree of homogeneity.

406 citations


"Robust finite-time guidance against..." refers background or methods in this paper

  • ...In contrast to existing smooth feedback controls, the non-smooth feedback control may have better convergence and robustness [19, 20] and is suitable to be applied to the UMTI....

    [...]

  • ...Correspondingly, the finite-time stability of the zero solution can be established by the following lemma, which has been proved in [20] and is given below for the convenience of the reader....

    [...]

  • ...When (t) = 0, the obtained observation-error dynamics are reduced to a second-order homogeneous system whose origin is finite-time stable [20]....

    [...]