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Research ArticleCooperative Intercept Guidance of Multiple Aircraft with a LureRole Included
Shuai Zhang ,1 Yang Guo ,1,2 and Shicheng Wang1
1Xi’an Research Institute of Hi-Tech, Xi’an 710025, China2Science and Technology on Electro-optical Control Laboratory, Luoyang 741000, China
Correspondence should be addressed to Yang Guo; [email protected]
Received 4 September 2017; Revised 17 January 2018; Accepted 21 January 2018; Published 10 April 2018
Academic Editor: Franco Bernelli-Zazzera
Copyright © 2018 Shuai Zhang et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
When threatened by a pursuer, an evading aircraft launches two defenders to accomplish a cooperative evasion, constituting a four-aircraft interception engagement. Under the assumption that the pursuing aircraft adopts the augmented proportional guidancelaw and first-order dynamics, a cooperative intercept mathematical model with an intercept angle constraint is established,allowing for the cooperative maneuvering of the evader. Based on the differential linear matrix inequality (DLMI), a controllerdesign method of input and output finite time stability (IO-FTS) is proposed and applied to the aircraft’s cooperative interceptscenario. A cooperation performance analysis is carried out for two cases: (1) two defenders intercept the pursuer with variousintercept angle constraints, and (2) the evader acts in a lure role to cooperate with two defenders. The simulation resultsindicate that the proposed method of controller design has the ability to guarantee that the two defenders intercept the pursuerat the preassigned intercept angles. The cooperative intercept scenario with a lure role is shown to be a very effective method forreducing the maximum required acceleration for defenders, which confirms the availability and advantage of cooperation. Thestrong adaptability and robustness of the cooperative guidance law with respect to various initial launch conditions is also verified.
1. Introduction
A lone wolf cannot defeat a lion, but a group of wolves caneasily contend against a lion, the reason for which is thatthe explicit division of labor, coordinated in cooperation,makes wolves much more powerful. There is an analogouscase in an interception confrontation between multiple air-craft in which cooperation is also particularly important.The multiple-aircraft cooperative intercept scenario hasbecome a subject of great interest in recent years. This paperaddresses the tactics of cooperative defense in an engagementbetween four aircraft: an evader, two defenders, and a pur-suer. For simplicity, the participants in this scenario aredescribed by the terms Evader, Defender 1, Defender 2, andPursuer, as shown in Figure 1. In this scenario, Evader maybe a manned or unmanned aircraft of high value, such as aballistic missile, aerial aircraft, or rocket. When confrontedwith a homing Pursuer, Evader launches two Defendersto intercept the incoming threat, which forms the four-aircraft interception engagement. It is expected that Evader,
Defender 1, and Defender 2 can present superior intercept per-formance and improve the evader’s probability of survivalthrough mutual cooperation. The survivability of Evader ismainly contingent on two pivotal components of the four-aircraft engagement: the cooperative intercept strategy ofDefender 1 and Defender 2 with intercept angle constraints,and the maneuver strategy of Evader in cooperation with theDefenders. The Defenders are expected to achieve a betterintercept performance (e.g., all-around intercept) by pursuingdifferent intercept angles in cooperation with the maneuversof Evader, in which Evader takes on the role of a lure [1, 2].
According to the different strategies and roles inmultiple-aircraft cooperative engagement, the existing coop-erative scenario generally can be classified into two modes:one is the cooperative scenario of different collaborative tasksand roles and the other is the cooperative scenario of thesame collaborative tasks and roles. In the first category, thecurrent research focuses on the evader-attacker-defender sce-nario (an evader, a pursuer, and a defender), which meansthat the evading aircraft carries a defensive missile (defender)
HindawiInternational Journal of Aerospace EngineeringVolume 2018, Article ID 4126807, 15 pageshttps://doi.org/10.1155/2018/4126807
by itself, and the evader acts as a lure and the defender acts asan interception to cope with the incoming interceptortogether. This scenario was proposed by Boyell [3, 4]. Inrecent years, this subject had been widely investigated. Shima[5] presented the optimal cooperative evasion and pursuitstrategies for the evader and its defender missile on the con-dition that the pursuer use proportional navigation (PN),augmented proportional navigation (APN), and optimalguidance law (OGL). Prokopov and Shima [6] derived theoptimal cooperation scheme for the evader and its defendermissile based on a contrast analysis of three different linearquadratic cooperative strategies. Weiss et al. [7] proposedtwo minimum-effort guidance algorithms based on therequired maximum miss distance for the attacker and theminimum required effort for target evasion. Additionally,Yamasaki and Balakrishnan [8] detailed a novel line-of-sight (LOS) geometry guidance law (referred to as triangleguidance) using a geometrical approach. The proposed guid-ance law ensures that the defender is located along the linesegment between the evader and the pursuer. Ratnoo andShima [9] derived a LOS geometry guidance law using a sim-ilar method. Guo et al. [10] proposed a cooperative guidancelaw for a three-aircraft engagement based on finite time theoryin which the feedback controller was derived for cooperativeevasion. In later research, Guo et al. [11] also derived maneu-ver control strategies for the evader based on finite time theoryin the ballistic middle phase. This scenario involved the coop-erative defense of evader and defender. The current researchfocuses on the optimal maneuvering strategy of the evaderand defender to enable effective cooperative evasion.
The second cooperative scenario is developed on thebasis of the traditional one-to-one interception. The assign-ment and role of each aircraft in the interception processare basically the same. The core idea of this scenario is theconsensus of flight status or impact time under differentflight conditions (e.g., communication delay, different topol-ogies, and salvo attack with intercept angle constraint).Recently, the research focus lies in the many-to-one cooper-ative intercept scenario based on two-to-one (namely,attacker-attacker-target) scenario, which is very useful andcritical for the cluster operations. For the attacker-attacker-target scenario, Shaferman and Shima [1, 2, 12] derived the
optimal geometrical guidance law for explicit cooperationbased on the optimal control theory. This two-to-one inter-ception engagement was conducted by imposing a defenderintercept angle without the cooperation of the evader in alure role. Recently, Fonod and Shima [13, 14] proposed acooperative guidance law via sharing the LOS angle betweena team of aircraft. This method could enhance the interceptestimation performance by imposing an intercept angle,which relaxed the assumptions of constant acceleration andperfect information of the pursuer’s actions considered in[1, 2, 12]. This cooperative scenario was an explicit extensionof the traditional one-to-one intercept scenario, the focus ofwhich was the design of a cooperative guidance law forattackers by improving the estimation of relative states. Forthe multiple-aircraft cooperative scenario, Du and Li [15]and Li et al. [16] propose a distributed delayed attitude con-trol algorithm to ensure the attitude synchronization for flex-ible spacecraft when communication delay exists based on amultiagent system’s theory. Zhang et al. [17] presented a dis-tributed cooperative guidance law for the consensus ofimpact time with different communication networks. Insummary, many scholars have conducted in-depth researchon three-aircraft engagements and achieved some importantresults. However, the aforementioned research pays moreattention to the optimal maneuvering of the attacker withthe same role and performance. The cooperation betweendifferent roles in the engagement with an imposed interceptangle constraint in this scenario is rarely addressed [18–20].
A multiple-aircraft engagement is a typical finite timeprocess: the intercept process must be completed within afinite time interval rather than over an infinite time period.The guidance command is also meaningful in a finite timeinterval [21–23]. Recently, many scholars have conductedrelated research to obtain a finite time convergent guidancelaw based on finite time theory in the terminal guidancephase [24–26]. The concept of a finite time theory was firstproposed by Dorato in 1961 [27]. Recently, with the intro-duction of the effective solution of differential linear matrixinequality (DLMI), many scholars have turned their atten-tion to the design of a finite time feedback controller. At thistime, there are two main types of finite time stability con-cepts. The first is a theory proposed by Bernstein and Bhat[28] and extended to address nonautonomous systems byHong et al. [29]. Du et al. [30, 31] and Xi et al. [32, 33]derived a finite-time cooperative control method that guaran-tees the state consensus in a finite time for a multiagent sys-tem. This finite time theory mainly relies on the Lyapunovstability analysis of nonlinear systems whose state trajectoriesconverge to an equilibrium point in the finite time interval.The second finite time concept was proposed by Amato et al.[34], proposing the input and output finite time stability(IO-FTS) concept, which assumes that the output variablesdo not exceed a given threshold if the input variables arebounded in a finite time interval. Amato et al. [35–38]proposed sufficient conditions for the analysis of linear time-invariant systems in the IO-FTS concept via a feedback con-troller. The current paper focuses on the IO-FTS finite timetheory proposed by Amato et al. because it is more practicalin some cases of state saturation and input boundedness.
Evader
Pursuer
Defender 1
qPE
qPD1
�훼P
rPE
rPD1
�훼E
�훼D1
Defender 2
�훼D2
rPD2
qPD2
Figure 1: Schematic diagram of a four-aircraft engagement.
2 International Journal of Aerospace Engineering
Inspired by the first scenario, we carried out the explor-atory research based on the different roles in cooperation.As mentioned above, the aforementioned research focuseson the design of the optimal cooperative guidance law for atwo-aircraft cooperative scenario based on optimal controltheory, under the assumption that either the evader ordefender possesses certain perfect information. Research intocooperative strategies between an evader and multipledefenders under intercept angles constraints is relatively lack-ing. An intercept scenario different than scenarios addressedin [12–14] is considered in this study based on the finite timetheory: a pursuer, an evader, andmultiple defenders, focusingon cooperative defense with different intercept angle con-straints and evader lure role cooperation. The three-aircraftcooperative defense scenario (Evader, Defender 1, andDefender 2) is expected to achieve superior intercept perfor-mance. As the multiple-aircraft cooperative intercept issue isa typical finite time process, the current paper attempts toadopt the finite time theory to design a cooperative guidancelaw for different interception situations. Compared with theprevious research, the current paper makes two main contri-butions: (1) the proposal of a cooperative intercept guidancelaw for two defenders with intercept angle constraints basedon finite time theory and (2) the derivation of a multiple-aircraft cooperative interception strategy with an evader lurerole to improve defender intercept performance.
The remainder of this paper is organized as follows. InSection 2, three types of multiple-aircraft cooperative inter-cept mathematical models are formulated, namely, (1) acooperative interception model without the participation ofthe evader, (2) a cooperative interception model with thecooperation of the evader, and (3) a cooperative interceptionmodel with intercept angle constraints. In Section 3, weintroduce the theory of input and output finite time stability(IO-FTS) and provide sufficient conditions for the input andoutput finite time stability of the linear time-variant (LTV)systems. A state feedback controller design method for theLTV system is proposed in the form of a differential linearmatrix inequality based on finite time theory. In Section 4,a cooperative intercept performance analysis is carried out.Four intercept cases are simulated to verify the cooperativeperformance under intercept angle constraints and lure rolecooperation. Section 5 presents concluding remarks.
2. Mathematical Model
A planar engagement between the four aircraft is considered.A schematic view of this engagement is shown in Figure 2.Variables associated with Evader, Defender 1, Defender 2,and Pursuer are denoted by the subscripts E, D1, D2, andP. Aircraft velocity is denoted by V , and normal accelerationis denoted by a. The ranges between Evader and Pursuer,Defender 1 and Pursuer, and Defender 2 and Pursuer aredenoted by rPE, rPD1, and rPD2, respectively. The LOS anglesfor Evader-to-Pursuer, Defender 1-to-Pursuer, and Defender2-to-Pursuer are denoted by qPE, qPD1, and qPD2, respectively.The rotation rate of LOS angles for Evader-to-Pursuer,Defender 1-to-Pursuer, and Defender 2-to-Pursuer isdenoted by qPE, qPD1, and qPD2, respectively.
2.1. Kinematics and Dynamics
Assumption A. The four aircraft have first-order dynamics.
Assumption B. Pursuer adopts augmented proportionalnavigation.
Assumption C. Evader, Defender 1, and Defender 2 can shareinformation with each other, and any delay is ignored.
According to the relative motion of the particles and thecoordinate transformation, the relative motion equationsbetween Evader and Pursuer can be derived as
qPE = −2rPErPE
qPE +aE − aPrPE
1
Similarly, the LOS rotation motions of Defender 1-Pursuer and Defender 2-Pursuer satisfy
qPD1 = −2rPD1rPD1
qPD1 +aD1 − aPrPD1
,
qPD2 = −2rPD2rPD2
qPD2 +aD2 − aP
rD2
2
During the engagement, the velocities of the four agentsare assumed to be constant, and intercept time can beapproximated by
t fPE = rPE0
VPE,
t fPD1 = rPD10
VPD1,
t fPD2 = rPD20
VPD2
3
where rPE 0 , rPD1 0 , and rPD2 0 denote the initial range;and t fPE, t fPD1, and t fPD2 denote the total flight time of thePursuer, Defender 1, and Defender 2, respectively.
E
P
x
y
0
VD2
VD1
rPD1
Vp
�훼p
qPD2
qPD1
qPErPEVE�훼E
�훼D1
D1
rPD2
�훼D2
D2
Figure 2: Schematic planar geometry of the four-aircraftengagement.
3International Journal of Aerospace Engineering
Then the corresponding times-to-go tgPi can be derived,where i = E, D1, D2 .
tgPE = t fPE − t,
tgPD1 = t fPD1 − t,
tgPD2 = t fPD2 − t
4
Obviously, the critical condition to ensure cooperativeevasion is tgPE > tgPD1, tgPE > tgPD2. While the relativeapproaching velocities are positive values, the LOS anglerotation rate q needs to converge to zero in the terminal guid-ance stage to achieve the objective of accurate interception.
The dynamic equation can be expressed by (5), (6), (7),and (8) according to Assumption A.
aP =aPC − aP
τP, 5
aE =aEC − aE
τE, 6
aD1 =aD1C − aD1
τD1, 7
aD2 =aD2C − aD2
τD2, 8
where the variables τi denote the response time constant andthe variables aiC denote the normal acceleration command ofvehicle i, where i = E, D1, D2, P .
According to Assumption B, the guidance law of thePursuer is derived.
aPC =NVPEqPE +K2aE 9
By (5) and (9), we have
aP =1τP
aPC −1τP
aP =1τP
NVPEqPE +K2aE −
1τP
aP
=NVPEτP
qPE +K2τP
aE −1τP
aP,10
where N and K denote navigation parameters.The relative movement of the four aircraft constitutes a
system in which we describe the cooperative guidance lawof Evader, Defender 1, and Defender 2. Therefore, the designof the acceleration commands aiC forms the essence of thecollaborative guidance law for i = E, D1, D2 .
2.2. Evader Noncooperation with Defenders. In this case,Evader is considered to have a special mission that precludesits ability to freely maneuver in cooperation with Defender 1and Defender 2 against Pursuer. The acceleration commandof Evader is known by its defenders and is regarded as theexternal input vector w t of the four-aircraft engagementsystem. The acceleration commands of Defender 1 andDefender 2 are regarded as the control input vector u t ,while x t is regarded as the set of internal state variables.
Because we are concerned with the normal accelerationai, i = E, D1, D2, P for each aircraft in the cooperativeintercept process, as well as the relative LOS angle qi qi ,i = PE, PD1, PD2 among the four aircraft, the system statevector x t can be taken as
x t = qPE qPD1 qPD2 qPE qPD1 qPD2 aP aE aD1 aD2T
11
According to (2), (3), (4), (5), (6), (7), (8), (9), and (10),the state equations of the system are deduced as
x t =A t x t + B t u t +G t w t , 12
y t =C t x t , 13
where
A t =
0 0 0 1 0 0 0 0 0 0
0 0 0 0 1 0 0 0 0 0
0 0 0 0 0 1 0 0 0 0
0 0 0−2tgPE
0 0−1
VPEtgPE
1VPEtgPE
0 0
0 0 0 0−2tgPD1
0−1
VPD1tgPD10
1VPD1tgPD1
0
0 0 0 0 0−2tgPD2
−1VPD2tgPD2
0 0−1
VPD2tgPD2
0 0 0NVPEτP
0 0−1τP
K2τP
0 0
0 0 0 0 0 0 0−1τE
0 0
0 0 0 0 0 0 0 0−1τD1
0
0 0 0 0 0 0 0 0 0−1τD2
, 14
4 International Journal of Aerospace Engineering
in which G t denotes the corresponding transfer matrix ofthe external input.
Owing to the noncooperation between Evader andDefenders, we can only control the Defenders to interceptPursuer, meaning the input vector u t can be expressed as
u t = aD1CaD2CT 15
The corresponding vectors B t and G t are expressedas
B t =0 0 0 0 0 0 0 0
1τD1
0
0 0 0 0 0 0 0 0 01τD2
T
,
G t = 0 0 0 0 0 0 01τE
0 0T
16
Setting y t as the evaluation output vector, and y t =C t x t , the corresponding vector C t is expressed as
C t =
0 1 0 0 0 0 0 0 0 0
0 0 1 0 0 0 0 0 0 0
0 0 0 0 1 0 0 0 0 0
0 0 0 0 0 1 0 0 0 0
17
That is,
y t = qPD1 qPD2 qPD1 qPD2T 18
The system described by (12) and (13) is a typical time-varying system. In the four-aircraft interception engagement,we are concerned that the LOS angles qPD1 and qPD2 reach adesignated intercept angle and that the LOS angle rotationrates qPD1 and qPD2 converge to zero in the finite time inter-val. In other words, the system evaluation output y t mustmeet the preassigned threshold in the finite time interval toprovide accurate intercept.
2.3. Evader Cooperation with Defenders. In this case, theEvader is able to freely maneuver with Defender 1 andDefender 2 against Pursuer for improved intercept per-formance. It is then necessary to design the accelerationcommands of Evader, Defender 1, and Defender 2 for coop-erative evasion. At this point, according to (2), (3), (4), (5),(6), (7), (8), (9), and (10), the state equations of the systemare deduced as
x t =A t x t + B t u t +G t w t , 19
y t = C t x t , 20
where the acceleration commands of Evader, Defender 1, andDefender 2 constitute the input vector u t together, whichimplies that u t can be expressed as
u t = aEC aD1C aD2CT 21
The corresponding vector B t is expressed as
B t =
0 0 0 0 0 0 01τE
0 0
0 0 0 0 0 0 0 01τD1
0
0 0 0 0 0 0 0 0 01τD2
T
22
The external disturbance of the system can be regarded asthe external input vectorw t . MatricesA t andC t are thesame as those given in Section 2.2.
2.4. Intercept Model with Intercept Angle Constraint.When itis desired that Defender 1 and Defender 2 intercept Pursuerat designated intercept angles, an intercept model with angleconstraints is considered. The state equations with an inter-cept angle constraint for the Defenders can be derived basedon state equations (12), (13), (19), and (20), where the aug-mented state vector is denoted by x t , and x t = x t + δ.The variable δ can be regarded as the augmented vector ofthe system state, and δ ∈ℝ10. The augmented system stateequations can then be rewritten as
x t =A1 t x t + B t u t +G t w t , 23
y t = C t x t , 24
where A1 t denotes the augmented state matrix. By systemequations (22) and (23), the cooperative guidance systemequation with intercept angle constraint can be formulated.
For system equations (23) and (24), if the controller u tis designed in the form of state feedback, and u t =K t x t ,the following system equations (25) and (26) can be derivedin which K t denotes the state feedback matrix.
x t = A1 t + B t K t x t +G t w t , 25
y t =C t x t 26
Remark 1. According to the state equations in Sections 2.2,2.3, and 2.4, the essential role of the cooperative interceptguidance law is to design a state feedback controller u t inorder to guarantee that the system output vector y t meetsthe preassigned threshold in the finite time interval whenthe input vector w t is bounded. Specifically, one needs todesign a state feedback controller u t that ensures that theLOS angles qPD1 and qPD2 converge to a preassigned interceptangle, and the LOS angle rotation rates qPD1 and qPD2 con-verge to zero in the finite time interval.
3. Cooperative Intercept Guidance Law
As mentioned above, the four-aircraft cooperative guidanceprocess is a typical finite time process, and the cooperativeintercept model established in Section 2 is also appliedin the finite time interval. Therefore, in this section, we
5International Journal of Aerospace Engineering
introduce the input-output finite time stability theory fordesigning the controller u t .
3.1. Input-Output Finite Time Stability Theory
Definition 1 (IO-FTS). Assume that the linear time-varying(LTV) system equations (27) and (28) are in the zero initialcondition, namely, x 0 = 0.
x t =A t x t +G t w t , 27
y t = C t x t 28
If the LTV system equations (27) and (28) satisfy the follow-ing condition,
T
0wT t Sww t dt ≤ 1⇒yT t Sy t y t ≤ 1, ∀t ∈ 0, T ,
29
then the system equations (27) and (28) are said to possessinput-output finite time stability with respect to T , Sw, Sy ·. In Definition 1, w t denotes the external input vector, yt denotes the system output vector, Sw and Sy t denote agiven measurement matrix and a measurement matrix func-tion, respectively, and T denotes a given positive time scalar.
Remark 2. IO-FTS and Lyapunov bounded input-boundedoutput (BIBO) stability are two different concepts. The IO-FTS theory focuses on signals defined over a finite time inter-val. The input output signals of IO-FTS are all norm boundedby preassigned quantitative bounds. However, the LyapunovBIBO stability focuses on the existence of the bounds of inputand output signals during an infinite time period.
Remark 3. IO-FTS is usually defined in the zero initialcondition, which is due to the fact that the initial conditionsand input signals will affect both the system states and outputsimultaneously. However, the effects of system states andoutput can be superimposed for linear systems, such as thesystem described in (27) and (28). Moreover, controllersdesigned with zero initial conditions and nonzero initialconditions have a similar control effect.
Remark 4. The multiple-aircraft cooperative interceptionproblem given above can also be described as follows: Givena positive time scalar t f , a class of external input signals w tdefined over 0, t f , and a positive definite matrix-valuedfunction Sy t , design the state feedback control law u tand u t =K t x t , where K t is a piecewise continuousmatrix-valued function, such that the closed-loop systemgiven in (25) and (26) satisfies output vector y t stabilityin the finite time interval.
Theorem 1 (sufficient condition of IO-FTS) [35]. If thereexists a symmetric positive definite matrix-valued functionP t ∈ Rn×n, such that the inequality equations (30), (31),and (32) hold,
P t +AT t P t + P t A t + P t G t S−1w GT t P t < 0,30
P t ≥CT t Sy t C t , 31
t ∈ 0, T , 32
then the system given by (27) and (28) is described by IO-FTSwith respect to T , Sw, Sy t in the finite time interval 0, T .
3.2. Controller Design. In order to design the cooperativeguidance law for the four-aircraft engagement, we need toseek an appropriate state feedback controller u t , such thatthe system given in (25) and (26) satisfies the input and out-put finite time stability. In this section, on the basis of Section3, we propose a controller design method in the form of a dif-ferential linear matrix inequality.
Theorem 2 (IO-FTS state feedback controller design method).If there exists a symmetric positive definite matrix-valuedfunction X t ∈ Rn×n and L t ∈ Rr×n, such that the inequalityequations (33), (34), and (35) hold,
−X t +X t AT t +A t X t + B t L t + LT t BT t G t
GT t −Sw< 0,
33
X t ≤ S−1 t , 34
S t =CT t Sy t C t 35
Then the system given by (26) and (27) is described by IO-FTS with respect to T , Sw, Sy t in the finite time interval0, T . The state feedback controller u t =K t x t , andK t = L t X−1 t .
Proof. Let V x t = xT t P t x t , and then one has
V =ddt
xTPx = xTPx + xTPx + xTPx
= xT P +ATP + PA x + wTGTPx + xTPGw36
(In the following proofs, one omits variable t for brevity.)By (30) and (36), the inequality equation (37) can be
derived.
ddt
xTPx <wTGTPx + xTPGw − xTPGS−1w GTPx 37
Assume v t as the middle variable, and let v t =S1/2w w − S−1/2w GTPx .
Then (38) can be obtained.
vTv =wTSww + xTPGS−1w GTPx −wTGTPx − xTPGw 38
According to (37) and (38), one can derive
ddt
xTPx <wTSww − vTv <wTSww 39
Integrating (39) from 0 to t, t ∈ 0, T , and one has
6 International Journal of Aerospace Engineering
x t TP t x t <t
0w τ TSww τ dτ ≤ 1 40
By (28), (31), (32), and (40), one can obtain
y t TSy t y t = x t TCT t Sy t C t x t
< x t TP t x t < 1, for all t ∈ 0, T41
By using Schur complements [39], inequality equation(42) can be derived according to (30) and (41).
P t +AT t P t + P t A t P t G t
GT t P t −Sw< 0 42
Let X t = P−1 t , and one has X t = −P−1 t P t P−1 t .Multiply matrix inequality equation (42) by diag P−1 t , Ion the right- and left-hand sides, and then the matrixinequality equation (43) can be derived.
−X t +X t AT t +A t X t G t
GT t −Sw< 0 43
By system equations (27) and (28), replace matrixA t byA t in matrix inequality equation (43), and A t =A1 t +B t K t .
Let L t =K t X t , and then matrix inequality equation(33) can be deduced by simple replacement. Then one provesthat matrix inequality equations (33), (34), and (35) can beobtained by transformation of matrix inequality (43), (31),and (32). Therefore, Theorem 2 is proven.
Remark 5. To determine the state feedback controller u t , itis necessary to solve the differential linear matrix inequality(DLMI) equation, such as matrix inequality equation (33).The DLMI can be recast in term of several criteria linearmatrix inequality equations, which is an appropriate methodfor solving a DLMI.
Remark 6. One discretizes DLMI into several standard LMIs,and the issue of controller design is transformed into theproblem of solving LMIs. Based on the Matlab control tool-box and the in-point method, the numerical solution of LMIscan be calculated after several iteration loops. Therefore, thefeasibility of the solution can be guaranteed by the existingmethod and software. The details for this procedure can bereferred to in [39].
4. Simulation Analysis
In this section, a simulation analysis is conducted to evaluatethe behavior of the multiple-aircraft cooperative interceptscenario. The specific process can be divided into two steps:(1) Based on the system state equations (25) and (26), thestate feedback controller u t is designed based on the finitetime controller design theory; (2) multiple-aircraft coopera-tive performance is examined using the closed-loop system.
Owing to the fact that the intercept angles of theDefenders and cooperative maneuvering of Evader can affectflight performance simultaneously, in Sections 4.1–4.3, weconduct three different simulation experiments to analyzetheir cooperative performance. The validity and stability ofthe cooperative guidance law are then verified for differentinitial launch conditions in Section 4.4.
4.1. Two Defenders Intercept Pursuer from One Side withoutCooperation of Evader. In this case, Evader launchesDefender 1 and Defender 2 for cooperative intercept, andEvader does not cooperate with Defender 1 and Defender 2.We assume that the two Defenders hit the Pursuer from thetop for a better attack effect; therefore, the intercept anglesof Defender 1 and Defender 2 are set to −15° and −30°,respectively. Based on the system equations derived in Sec-tion 2.2 and Section 2.4, the controller u t is solved basedon Theorem 2. The simulation parameters are shown inTable 1.
Figure 3 presents the flight trajectory of the four aircraftwithout the cooperative maneuvering of Evader. It is appar-ent that the trajectory of Evader is relatively flat, and thatEvader does not maneuver with Defenders for cooperativeevasion. Pursuer attempts to attack Evader according to itsguidance command. Evader launches Defender 1 andDefender 2 for cooperative intercept at the time 0 s. Defender1 and Defender 2 cooperate with each other to intercept Pur-suer from the same side.
Figure 4 shows the variation of the interception angles,with the intercept angles of Defender 1 and Defender 2 con-trolled at −15° and −30° during the terminal guidance period.
Figure 5 shows the variation of the LOS angle rotationrate. The simulation results show that the LOS angle rotationrates of Defender 1 and Defender 2 finally converge to zero,indicating that the Defenders can intercept Pursuer in thecase of positive approaching speed.
The above simulation results support the conclusion thatthe Defenders accomplish the cooperative interception in thefinite time interval, showing that the designed controller canguarantee the ability of the Defenders to cooperatively inter-cept at preassigned angles. Thus, the theoretical methodsunderpinning the controller design are verified.
In order to illustrate the effectiveness and advantages ofthe guidance law proposed in this paper, we conduct a com-parative simulation experiment. Two defenders use the pro-posed guidance law and the augmented proportionalnavigation (APN, N = 5, K = 2) with angle restraints in [17],respectively. Evader does not cooperate with Defender 1and Defender 2. The intercept angles of Defender 1 andDefender 2 are set to −15° and −30°, respectively. Theremaining parameters are the same as those given in Table 1.
The simulation results are shown in Figures 6–8. ByFigures 6–8, the trajectory of the proposed guidance law ismore gentle than that of the APN with the same interceptionangle constraints and initial conditions, which indicates thatthe proposed guidance law can effectively reduce the requiredacceleration of the defenders. The acceleration requirementshown in Figures 7 and 8 also verifies the conclusion inFigure 6. These simulation results confirm the usefulness
7International Journal of Aerospace Engineering
and advantages of the proposed guidance law in this paperwhen compared with other guidance law.
4.2. Two Defenders Intercept Pursuer from One Side with theCooperation of Evader. In this case, based on the system stateequations derived in Section 2.3 and Section 2.4, the control-ler u t is solved by Theorem 2. Unlike in Section 4.1, Evadercooperates with the Defenders to improve intercept perfor-mance. The interception angles of Defender 1 and Defender2 are set to −15° and −30°, respectively. The remainingparameters are the same as those given in Table 1.
Figure 9 presents the flight trajectories of the four air-craft for the condition of cooperative maneuver of Evader.It is apparent that Evader maneuvers with Defenders to per-form a collaborative defense, allowing Defender 1 and
Defender 2 to intercept Pursuer together at their preassignedintercept angles.
Figure 10 indicates that intercept angles of Defender 1and Defender 2 converge to −15° and −30° in the finalphase. Figure 11 shows the change of the LOS angle rota-tion rate with the cooperation of Evader. By Figures 9–11,it can be concluded that the Defenders successfully inter-cept Pursuer at a preassigned intercept angle with thecooperation of Evader.
Comparing the accelerations of Defender 1 andDefender 2 in the cooperation and noncooperation scenariospresented in Cases A and B, respectively, we can obtain thecomparative accelerations illustrated in Figures 12 and 13.The simulation results prove that for the one-side intercept
0 5000 10000 150000
1000
2000
3000
4000
5000
6000
7000
x (m)
y (m
)
EvaderPursuer
Defender 1Defender 2
Figure 3: Four aircraft flight trajectories (one-side intercept withoutcooperation of Evader).
Table 1: Simulation parameters for cooperative intercept.
Parameter Symbol Initial value
Initial range rPE 0 15,000m
Initial range rPD1 0 , rPD2 0 15,000m
Velocity of Evader VE 250m/s
Velocity of Pursuer VP 500m/s
Velocity of Defenders VD1, VD2 500m/s
Response time of Evader τE 0.1 s
Response time of Defenders τD1, τD2 0.05 s
Response time of Pursuer τP 0.05 s
Navigation parameter N 4
Navigation parameter K 3
Initial system state vector x 0 0 0 0 0 05 −0 02 −0 03 0 0 0 0 T
Measurement matrix function Sy · 950t ⋅ I4×4Measurement matrix Sw I1×1
PursuerDefender 1Defender 2
0 5 10 15−35
−30
−25
−20
−15
−10
−5
0
5
10
t (s)
q (d
egre
e)
Figure 4: Intercept angle variation of the aircraft.
8 International Journal of Aerospace Engineering
scenarios, cooperative guidance with the cooperation ofEvader can significantly reduce the maximum requiredacceleration for both Defenders. The Defenders achieve abetter intercept performance by pursuing different interceptangles in cooperation with the maneuvers of Evader. Thesesimulation results confirm the usefulness and advantages ofcooperation for both Defenders.
4.3. Two Defenders Intercept Pursuer from Two Sides withCooperation of Evader. From the point of view of probability,if the Defenders attempt to intercept Pursuer from two sides,it will be more difficult for Pursuer to avoid the interception[2, 12]. The two-side cooperative intercept case is expected topresent a more effective collaborative defense for Evader.Therefore, in this case, the Defenders use a two-side interceptmode to increase the intercept probability, setting the
intercept angles of the Defenders to 30° and −30°. Addi-tionally, Evader cooperates with its defense in the cooper-ative interception. The initial value of the state vector isx 0 = 0 0 0 0 05 0 03 −0 03 0 0 0 0 T. Theremaining parameters are the same as those given inTable 1. Based on the system state equations derived inSection 2.3 and Section 2.4, we can design the cooperativeguidance law u t .
Figure 14 shows the flight trajectory of the four aircraftwithout the cooperation of Evader. Figure 15 shows theflight trajectory of the four aircraft with the cooperationof Evader. The comparison of Figures 14 and 15 indicatesthat Evader substantially cooperates with the Defenders incooperative evasion.
Figures 16 and 17 reveal the variation of intercept anglesand LOS angle rotation rates with the cooperation of Evader.
PursuerDefender 1Defender 2
0 5 10 15t (s)
−0.14
−0.12
−0.1
−0.08
−0.06
−0.04
−0.02
0
0.02
0.04
0.06
dq/dt (
rad/
s)
Figure 5: The variation of LOS angle rotation rate.
0 5000 10000 15000−1000
0
1000
2000
3000
4000
5000
6000
7000
8000
x (m)
y (m
)
EvaderPursuer
Defender 1Defender 2
APN
Proposed guidance law
Figure 6: Four aircraft flight trajectories.
0 5 10 15t (s)
−150
−100
−50
0
50
100
150
200
250
300
350
�훼 (m
/s2 )
Proposed guidance lawAPN
Figure 7: The acceleration variation of Defender 1.
0 5 10 15t (s)
Proposed guidance lawAPN
−200
−100
0
100
200
300
400
500
600�훼
(m/s
2 )
Figure 8: The acceleration variation of Defender 2.
9International Journal of Aerospace Engineering
By Figures 15–17, it can be concluded that the Defenders cancollaboratively intercept Pursuer at predetermined intercep-tion angles from both sides.
Figures 18 and 19 show the compared results of accelera-tion in the two-side intercept case. In the case of the nonco-operation of Evader, the maximum required accelerationfor the Defender 2 reaches 403.5m/s2, and that of Defender1 reaches 338.4m/s2, indicating that compared withDefender 1, Defender 2 is more difficult to maneuver for suc-cessful intercept. Consequently, the simulation results indi-cate that for the two-side cooperative intercept case, Evadertends to cooperate more with the defender who demands agreater required acceleration, illustrating that the cooperative
intercept scenario can reduce the required acceleration of thedefender with whom Evader cooperates.
4.4. Performance Analysis under Different Initial LaunchConditions. The initial launch conditions of the Defendersmay be affected by the various flying situations of Evader.Additionally, while engaging with the different requirementsof the combat mission, the Defenders must confront differentbattlefield launch conditions in which the validity and stabil-ity of the guidance law is critical to ensure accurate attack.Based on the cooperative guidance law proposed in Section3, in this section, the performance analysis of the cooperativeguidance law is conducted under different initial launch
0 5000 10000 15000x (m)
0
1000
2000
3000
4000
5000
6000
7000
y (m
)
EvaderPursuer
Defender 1Defender 2
Figure 9: Four aircraft flight trajectories (one-side intercept withcooperation of Evader).
0 5 10 15t (s)
−35
−30
−25
−20
−15
−10
−5
0
5
10
q (d
egre
e)
PursuerDefender 1Defender 2
Figure 10: Intercept angle variation of the aircraft.
0 5 10 15t (s)
PursuerDefender 1Defender 2
−0.14
−0.12
−0.1
−0.08
−0.06
−0.04
−0.02
0
0.02
0.04
0.06
dq/dt (
rad/
s)Figure 11: LOS angle rotation rate variation.
0 5 10 15t (s)
−150
−100
−50
0
50
100
150
200
250
300
�훼 (m
/s2 )
CooperationNoncooperation
Figure 12: Acceleration variation of Defender 1 (one-sideintercept).
10 International Journal of Aerospace Engineering
conditions. When the Defenders are separated from Evader,the different launch conditions will ultimately affect the ini-tial LOS angular rate qi, i = PD1, PD2 .
The simulation scenario is set up as follows: Evader coop-erates with the two Defenders for cooperative evasion, andDefender 1 and Defender 2 intercept the Pursuer from differ-ent sides. The intercept angle of Defender 1 is set to 30°, andthe intercept angle of Defender 2 is set to −30°. The initialvalue of the LOS angle rotation rate q is set differently for fivecases, shown in Table 2. The remaining parameters are thesame as those given in Table 1.
Figures 20 and 21 show the variation of the interceptangles of Defender 1 and Defender 2, respectively, in the five
considered cases. Figures 22 and 23 show the variation of theLOS angle rotation rates qPD1 and qPD2 for the two Defendersin the five considered cases. As can be seen from these figures,Defender 1 and Defender 2 have the ability to interceptPursuer at preassigned intercept angles under all five dif-ferent initial launch conditions. The intercept angles eventu-ally converge to 30° and −30°, and the LOS angle rotationrates qPD1 and qPD2 eventually converge to 0 rad/s, despitethe variation in initial launch conditions. These simulationresults verify the effectiveness of the proposed cooperativeguidance law. Moreover, these simulations also demonstratethe strong adaptability and robustness of the proposed guid-ance law under various initial launch conditions.
�훼 (m
/s2 )
0 5 10 15t (s)
CooperationNoncooperation
−200
−100
0
100
200
300
400
500
600
Figure 13: Acceleration variation of Defender 2 (one-sideintercept).
0 5000 10000 15000x (m)
EvaderPursuer
Defender 1Defender 2
−3000
−2000
−1000
0
1000
2000
3000
4000
5000
6000
7000
y (m
)
Figure 14: Four aircraft flight trajectories (two-side interceptwithout cooperation of Evader).
0 5000 10000 15000x (m)
EvaderPursuer
Defender 1Defender 2
−3000
−2000
−1000
0
1000
2000
3000
4000
5000
6000
7000
y (m
)
Figure 15: Four aircraft flight trajectories (two-side intercept withcooperation of Evader).
0 5 10 15t (s)
PursuerDefender 1Defender 2
−40
−30
−20
−10
0
10
20
30
40
q (d
egre
e)
Figure 16: Intercept angle variation of the aircraft.
11International Journal of Aerospace Engineering
5. Conclusions
This study focuses on a four-aircraft intercept engagementscenario. Compared with previous research, this study pri-marily addresses two novel aspects of the intercept engage-ment scenario: (1) the cooperative intercept performanceanalysis of multiple defenders under intercept angle con-straints and (2) the cooperative intercept performance analy-sis of the aircraft with the cooperation of the Evader in a lurerole. Based on the relative motion and kinetic equations ofthe four-aircraft cooperative intercept models, three differentcases of engagement are established. The design method forthe state feedback controller is proposed based on the inputand output finite time stability theory and is applied in the
controller solution for the four-aircraft engagement. Thesimulation results show that the proposed method can guar-antee that Defender 1 and Defender 2 intercept Pursuer at
0 5 10 15t (s)
PursuerDefender 1Defender 2
−0.2
−0.15
−0.1
−0.05
0
0.05
0.1
0.15
dq/dt (
rad/
s)
Figure 17: LOS angle rotation rate variation.
0 5 10 15t (s)
CooperationNoncooperation
−500
−400
−300
−200
−100
0
100
�훼 (m
/s2 )
Figure 18: Acceleration variation of Defender 1 (two-sideintercept).
0 5 10 15t (s)
CooperationNoncooperation
−200
−100
0
100
200
300
400
500
�훼 (m
/s2 )
Figure 19: Acceleration variation of Defender 2 (two-sideintercept).
Table 2: Initial launch parameters of the two Defenders.
Case Initial value qPD1 (rad/s) Initial value qPD2 (rad/s)
1 0.01 −0.012 0.03 −0.033 0.06 −0.064 0.09 −0.095 0.1 −0.1
0
5
10
15
20
25
30
35
q (d
egre
e)
Case 1Case 2Case 3
Case 4Case 5
0 5 10 15t (s)
Figure 20: Intercept angle variation of Defender 1.
12 International Journal of Aerospace Engineering
the preassigned intercept angles, and that the Defenders canachieve improved interception performance with the cooper-ation of Evader. It was also shown that the two Defenders canintercept Pursuer from either the same side or both sides, as afunction of the desired intercept angle. The one-side cooper-ative intercept scenario reduces the maximum requiredacceleration of both defenders, while the two-side coopera-tive intercept scenario increases the intercept probability.When cooperating with the Defenders in a two-sided inter-ception scenario, Evader tends to cooperate more with thedefender who demands a greater required acceleration.Finally, the simulations demonstrate that the proposed coop-erative guidance law possesses strong adaptability androbustness in the face of different initial launch conditions.
Nomenclature
ai: Normal acceleration of vehicle i (m/s2)aiC: Normal acceleration command (m/s2)A, B, C,G: State-space representations of the dynamicsA1 t : The augmented system state matrixD1, D2: Defender 1 and Defender 2E: EvaderK t : The state feedback matrixN , K : The augmented proportional navigation
parametersP: PursuerP,X, L: Symmetric positive definite matrix-valued
functionqij: LOS angle from the vehicle i to the vehicle j (rad)qij: LOS angle rotation rate from the vehicle i to the
vehicle j (rad/s)rij t : Range between the vehicle i to the vehicle j (m)rij 0 : Initial range between the vehicle i to the vehicle
j (m)Sw: Measurement matrixSy t : Measurement matrix functiont f : The total flight time (s)T : Positive time scalaru t : System control input vectorV : Velocity (m/s)w t : System external input vectorx t : System state variablesx 0 : The initial system state variablesx t : The augmented system state variablesy t : System evaluation output vectory t : The augmented system evaluation output vectorτi: Dynamic response time constants of the vehicle
i (s)δ: The augmented vector of system state, δ ∈ℝ10.
−40
−35
−30
−25
−20
−15
−10
−5
0
q (d
egre
e)
Case 1Case 2Case 3
Case 4Case 5
0 5 10 15t (s)
Figure 21: Intercept angle variation of Defender 2.
−0.02
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
dq/dt (
rad/
s)
Case 1Case 2Case 3
Case 4Case 5
0 5 10 15t (s)
Figure 22: LOS angle rotation rate variation of Defender 1.
−0.14
−0.12
−0.1
−0.08
−0.06
−0.04
−0.02
0
0.02
0.04
dq/dt (
rad/
s)
Case 1Case 2Case 3
Case 4Case 5
0 5 10 15t (s)
Figure 23: LOS angle rotation rate variation of Defender 2.
13International Journal of Aerospace Engineering
Conflicts of Interest
The authors declare that there is no conflict of interestsregarding the publication of this paper.
Acknowledgments
This work was supported by the National Natural Sci-ence Foundation (NNSF) of China under grant number61673386 and the Aviation Foundation of China under grantnumber 201651U8006.
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