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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.

2506 J. GUIDANCE, VOL. 41, NO. 11: ENGINEERING NOTES

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

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

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

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

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.

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