# Research on Integrated Guidance and Control of Distributed Cooperation of Multi-Interceptor with State Coupling

Authors: Xiang Liu; Xiaogeng Liang
Journal: Journal of Control Science and Engineering
 (2018)

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## Abstract

With the aim of achieving cooperative target interception by using multi-interceptor, a distributed cooperative control algorithm of the multi-interceptor with state coupling is proposed based on the IGC (integrated guidance and control) method. Considering the coupling relationship between the pitch and ya w channels, a state coupling “leader” IGC model is established, an FTDO (finite-time disturbance observer) is designed for estimating the unknown interference of the model, and the “leader” controller is designed according to the adaptive dynamic surface sliding-mode control law. Secondly, the cooperative control strategy of the multi-interceptor is designed with the “leader-follower” distributed network mode to obtain the speed in the three directions of the interceptor in air and transform them to the general flight speed, trajectory inclination angle, and trajectory deflection instruction by using the transformational relation of kinematics. Finally, the “follower” controller is designed with the FTDO and dynamic surface sliding-mode control. The designed multi-interceptor distributed cooperative IGC algorithm with state coupling has good stability according to the simulation results of two different communication topologies.


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## Body

## 1. Introduction

With the rapid development of antimissile technology, it is getting difficult for a single interceptor to break through the defense and intercept targets efficiently, thus making it hard for an interceptor to adapt to the demands of future battlefield scenarios. Therefore, an interceptor with cooperative target interception capability would be more suitable for the future. Multi-interceptor can realize functional complementation through information interaction and sharing, which can not only considerably enhance defense penetration and counterforce of the interceptor but also finish tasks that cannot be achieved by a single interceptor [1].With respect to cooperative guidance and control of multi-interceptors, the authors in [2, 3] proposed a guidance law with controllable attack time and angle-of-attack constraint and applied it to the salvo attack of anti-ship missiles. Based on this idea, researchers subsequently introduced some other guidance and control methods, including sliding-mode control [4, 5], optimal control [6], differential game [7], and dynamic surface control [8]. This group of methods relies on specifying the attack time before launching to achieve coordination. No information exchange occurs between missiles during flight; hence, these methods apparently have temporal limitations. With the progress in consensus of multiagent systems, researchers have begun to use the consensus theory to study the cooperative guidance and control of multi-interceptors. Using the coordination strategy under the cooperative guidance framework, the authors in [9] adjusted missile trajectories, such that the coordination variable of each missile can approach the expected coordination variable and realize cooperative guidance. The authors in [10] applied the “leader-follower” formation control to cooperative guidance of multi-interceptors by putting forward an analogous “leader-follower” cooperative guidance framework. The authors in [11, 12] explored the guidance and control law of the “leader-follower” topology with angle constraint and topology switch present. By constructing an integrated cost function for multiple missiles, the authors in [13] designed a cooperative guidance law for multiple missiles intercepting a maneuvering target. However, the application of this integrated cost function was faced with multiple constrains because each missile required the global information of all the participating partners.The interceptor guidance and control system is a highly dynamic, strong-coupling, varying, and uncertain multivariant system featuring complicated dynamic characteristics. Therefore, the integrated guidance and control (IGC) design method can allocate the control ability of interceptors more properly. It mainly generates control power according to the relative motive between the interception targets and the interceptors and then drives the interceptors to chase the targets. Moreover, it cannot only stabilize flight attitude but also enhance guidance precision [14]. In recent years, many researchers worldwide conducted studies focusing on the design method of IGC. The authors in [15, 16] designed an IGC control law with the sliding control mode and back-stepping control algorithm. The sliding-mode control method has been widely used in the design of IGC of aircraft [17], missile [18], and unmanned helicopter [19, 20]. In existing papers, most of them were designed in a single channel [21, 22], regardless of the coupling between channels. The authors in [23, 24] designed IGC algorithms in three dimensions, but designing the controller is difficult when establishing the model with a high order.According to the above literature review, the guidance loop and control loop of the multi-interceptor cooperative guidance and control have been studied separately by experts. However, external disturbance and its coupling relation during the multi-interceptor flight cannot be ignored. In the meantime, multi-interceptor needs to communicate during its flight to finish cooperative control. Thus, unsmooth local communication should be considered while designing the controller. In light of this, a distributed cooperative control strategy was introduced on top of the integrated guidance and control (IGC) method by considering the coupling between the interception pitch and yaw channels, a design method of cooperative IGC of the multi-interceptor with state coupling of the “leader-follower” distributed topology structure is proposed. The design of the “leader” and “follower” control algorithm using the dynamic surface sliding-mode and finite-time disturbance observer can effectively enhance the stability of the interceptor during the flight and furthermore ensure the target to be hit by the “leader” and “follower” simultaneously following the distributed cooperative controlling strategy. The proposed method can enhance the stability of cooperative guidance and control of the multi-interceptor.


## 2. “Leader” IGC Model of Interceptor with State Coupling

According to the relative motion relation between the interceptor “leader” and target [25, 26], the relative motion model of the interceptor “leader” and target is established as follows: (1)rq¨ε=-2r˙q˙ε-rq˙β2sin⁡qεcos⁡qε-atε-am4εr¨=rq˙ε2+rq˙β2cos2qε+atr-am4r-rq¨βcos⁡qε=2r˙q˙βcos⁡qε-2rq˙εq˙βsin⁡qε+atβ-am4βIn the equation,qβ and qε denote the elevation angle and horizontal sight angle of the “leader” and target, respectively; am4ε and am4β denote the longitudinal and lateral motion acceleration of the “leader,” respectively; atε and atβ denote the longitudinal and lateral motion acceleration of the target, respectively; r represents the relative distance between the “leader” and target.The kinetic model of the interceptor “leader” can be indicated as follows:(2)α˙=ωz+ωytan⁡βsin⁡α-qSCyααcos⁡αmVmcos⁡β+dαβ˙=ωycos⁡α+qSCyααsin⁡αsin⁡βmVm+qSCzββcos⁡βmVm+dβω˙z=qSL-mzααJz+qSL-mzωzωzJz+MzJz+dωzω˙y=qSL-myββJy+qSL-myωyωyJy+MyJy+dωy(3)am3ε=qSCyααmam3β=qSCzββmIn the equation,S is the reference area of the “leader;” L- is the reference length of the “leader;” m is the mass of the “leader;” α and β are the attack angle and sideslip angle, respectively; ωz and ωy are the pitch angular velocity and yaw rate, respectively; and dα, dωz, dβ, and dωy are the disturbance and uncertain disturbance of the various links of the system. Jz and Jy are the rotational inertia; Cyα, Czβ, mzα, mzωz, myβ, and myωy are the related aerodynamic force and torque coefficient, respectively; Mz and My are the pitch moment and yawing moment of the “leader,” respectively; and am3ε and am3β are the longitudinal and lateral acceleration of the “leader.”Assuming that the sight angle of the interceptor in the terminal guidance stage changes slightly and the included angle of the sight angle and velocity direction of the interceptor are relatively small, letam3ε=am4ε and am3β=am4β. By defining x1=q˙ε,q˙βT,  x2=α,βT, and x3=ωz,ωyT, the nonlinear model of the “leader” IGC of the interceptor with state coupling can be obtained, according to (1), (2), and (3). (4)x˙1=f1x1+g11x1x2+g12x1atx˙2=f2x2+g2x2x3+d2x˙3=f3x2,x3+g3u+d3y=x1In the equation,(5)f1x1=-2r˙rq˙ε-q˙β2sin⁡qεcos⁡qε,-2r˙rq˙β+2q˙εq˙βtan⁡qεT,f2x2=-qSCyααcos⁡αmVmcos⁡β,qSCyααsin⁡αsin⁡βmVm+qSCzββcos⁡βmVmT,f3x2,x3=qSL-mzααJz+qSL-mzωzωzJz,qSL-myββJy+qSL-myωyωyJyTg11x1=-qSCyαmr00qSCzβmrcos⁡qε,g12x1=1r001rcos⁡qε,g2x2=1tan⁡βsin⁡α0cos⁡α,g3=diagJz-1,Jy-1,u=Mz,MyT,at=atε,-atβT,d2=dα,dβT,d3=dωz,dωyTThe unknown disturbancesd2 and d3 in the “leader” IGC model are assumed to be continuously differentiable and the first-order derivative is bounded. di<N,i=2,3,N is a positive constant.


## 3. Design of Interceptor “Leader” Controller

### 3.1. Design of Finite-Time Disturbance Observer

Aiming at the uncertaintyat, d2, and d3 included in the system model (4), an FTDO (finite-time disturbance observer) is designed for estimating unknown disturbances, in order to eliminate the impact of unknown disturbances on the leader system of the interceptor. Define vr=r˙,  vε=rq˙ε,  vβ=rq˙βcos⁡qε, and(6)v˙ε=r˙q˙ε+rq¨ε=-r˙q˙ε-rq˙β2sin⁡qεcos⁡qε+atε-am4ε=-vrvεr-vβ2tan⁡qεr+atε-am4ε(7)v˙β=r˙q˙βcos⁡qε+rq¨βcos⁡qε-rq˙εq˙βsin⁡qε=-r˙q˙βcos⁡qε+rq˙εq˙βsin⁡qε-atβ+am4β=-vrvβr+vεvβtan⁡qεr-atβ+am4βDefinew1=vε,vβT, and according to (6) and (7),(8)w˙1=h1w1+h2w1x2+atIn the equation,(9)
h
1
w
1
=
-
v
r
v
ε
r
-
v
β
2
tan
⁡
q
ε
r
-
v
r
v
β
r
+
v
ε
v
β
tan
⁡
q
ε
r
,
h
2
w
1
=
-
q
S
C
y
α
α
m
0
0
q
S
C
z
β
β
mThe following FTDO is designed to estimate the accelerationat of the target, and(10)z˙10=v10+h1w1+h2w1x2,z˙11=v11,z˙12=v12v10=-λ10L11/3z10-w12/3sgnz10-w1+z11v11=-λ11L11/2z11-v101/2sgnz11-v10+z12v12=-λ12L1z12-v11q1/p1sgnz12-v11z10=w^1,z11=a^t,z12=a˙^tIn the equationw1=w11,w12T,  z10=z101,z102T,  z11=z111,z112T,  z12=z121,z122T,  v10=v101,v102T,  v11=v111,v112T,  v12=v121,v122T,  L1=L11,L12T,  a^t is the estimated acceleration at of the target, the estimated value w1 and a˙t is w^1 and a˙^t, λ10, λ11, and λ12 are the coefficients to be designed for the disturbance observer, q1 and p1 are the terminal coefficients, respectively, and 0<p1<q1.It can be learnt according to [27] that appropriate parameters can guarantee that the FTDO error system is steady in finite time. The estimated error of the acceleration at of the target is defined as e11=z11-at.Similarly, the disturbanced2 and d3 of the second subsystem and third subsystem is estimated, and(11)z˙20=v20+f2x2+g2x2x3,z˙21=v21,z˙22=v22v20=-λ20L21/3z20-x22/3sgnz20-x2+z21v21=-λ21L21/2z21-v201/2sgnz21-v20+z22v22=-λ22L2z22-v21q2/p2sgnz22-v21z20=x^2,z21=d^2,z22=d^˙2(12)z˙30=v30+f3x2,x3+g3u,z˙31=v31,z˙32=v32v30=-λ30L31/3z30-x32/3sgnz30-x3+z31v31=-λ31L31/2z31-v301/2sgnz31-v30+z32v32=-λ32L3z32-v31q3/p3sgnz32-v31z30=x^3,z31=d^3,z32=d^˙3In the equation, the estimated value of disturbance d2 and d3 is d^2 and d^3, respectively, and the estimated error is e21=z21-d2 and e31=z31-d3, respectively.


### 3.2. Design of Adaptive Dynamic Sliding-Mode Controller

Because the interceptor IGC model is an unmatched and uncertain system, and aiming at the state coupling IGC model (4) and FTDO estimated value (10)–(12), the “leader” control algorithm is designed by taking advantage of the adaptive dynamic sliding-mode control law.( 1 ) The command signal of the first subsystem of (4) is defined as x1d. In order to realize the guidance goal, the sight angular velocity should be removed. According to the design method of dynamic surface sliding-mode control, the first dynamic error surface is defined as follows: (13)s1=-g11-1x1x1-x1dTaking the derivative ofs1, the dynamic equation of error is given by(14)s˙1=g11-2x1g˙11x1x1-x1d-g11-1x1f1x1+g12x1at-x˙1d-x2According to the dynamic surface design method and FTDO estimated valuea^t in (10), the virtual control amount of the first dynamic surface can be obtained as follows: (15)x2∗=g11-2x1g˙11x1x1-x1d-g11-1x1f1x1+g12x1a^t-x˙1d+k1s1In the equation,k1=diagk11,k12 is the positive definite matrix. In the design process, differential blast would occur, while the differential of the virtual control amount x2∗ is taken. In order to avoid the complicated computation process owing to item inflation, x2∗ must be obtained through the first-order low-pass filter, and the virtual control amount of the filter can be obtained as follows: (16)τ2x-˙2∗+x-2∗=x2∗,x-2∗0=x2∗0In the equation,τ2=diagτ21,τ22 is the time constant of the filter, and the differential of the virtual control after the error surface filter can be obtained.(17)x-˙2∗=-τ2-1x-2∗-x2∗( 2 ) The second dynamic error surface is defined as(18)s2=x2-x-2∗Taking the derivative ofs2, the dynamic equation of error can be obtained as follows: (19)s˙2=x˙2-x-˙2∗=f2x2+g2x2x3+d2-x-˙2∗Similar to the first dynamic surface design method, the estimated FTDO valued^2 is substituted in (11). Thus, the virtual control of the second dynamic surface can be obtained as follows: (20)x3∗=g2-1x2-f2x2-d^2+x-˙2∗-k2s2In the equation,k2=diagk21,k22 is the positive definite matrix. Similarly, by obtaining x3∗ through the first-order low-pass filter, the virtual control amount of the filter can be obtained as follows: (21)τ3x-˙3∗+x-3∗=x3∗,x-3∗0=x3∗0In the equation,τ3=diagτ31,τ32 is the time constant of the filter. The differential of virtual control after the error surface filter can be obtained as follows:(22)x-˙3∗=-τ3-1x-3∗-x3∗( 3 ) The third dynamic error surface is defined as follows:(23)s3=x3-x-3∗Taking the derivative ofs3, the dynamic equation of error can be obtained as follows:(24)s˙3=x˙3-x-˙3∗=f3x2,x3+g3u+d3-x-˙3∗To guarantee the convergence velocity of the interceptor “leader,” an adaptive sliding-mode reaching law is designed:(25)s˙=-kar˙rs-kbs∂sgnsIn the equation,ka>0,  kb>0, and r˙ denotes the change in relative distance between the “leader” and target.According to (24) and (25) and estimated FTDO d^3 of (12), the adaptive dynamic surface sliding-mode control law of the interceptor “leader” is given by(26)u=g3-1-f3x2,x3-d^3+x-˙3∗-k3r˙rs3-k4s3∂sgns3In the equation,k3=diagk31,k32,  k4=diagk41,k42, and ∂=diag∂11,∂12 are positive definite matrices, and 0<∂<1.


### 3.3. Stability Analysis

Theorem 1.
Consider the integrated guidance and control system for the “leader” (equation (4)). If the convergence rate is calculated using (25), the disturbance values of the system (see (4)) are estimated using (10)-(12), and filter equations (16) and (21) are implemented; then finally under the dynamic surface sliding-mode control law (see (26)), imposing the constraint for ensuring the system (see (4)) output error converging into the adjacent area of the origin, an arbitrary adjacent area of the origin can be obtained with the appropriate design parameter determined.To prove:Let us assume that the estimated error of FTDO system meets(27)e11<N1,e21<N2,e31<N3In the equation,N1, N2, and N3 are positive constants.The filter error is defined as follows:(28)y2=x-2∗-x2∗,y3=x-3∗-x3∗Taking the derivative ofy2 and y3, the dynamic error of the filter can be obtained as follows:(29)y˙2=-τ2-1y2-x˙2∗,y˙3=-τ3-1y3-x˙3∗According to (13)–(23) and (28),(30)x1=-g11x1s1+x1dx2=s2+x-2∗=s2+y2+x2∗x3=s4+x-3∗=s3+y3+x3∗According to (4), (11)–(21), and (26)–(28),(31)s˙1=g11-2x1g˙11x1x1-x1d-g11-1x1f1x1+g12x1at-x˙1d-x2=-s2-y2-k1s1+e~11In the equation,e~11=g11-1(x1)g12(x1)e11. Let us assume that e~11<N~1, where N~1 is a positive constant. (32)s˙2=x˙2-x-˙2∗=f2x2+g2x2x3+d2-x-˙2∗=g2x2s3+y3-k2s2-e21(33)s˙3=x˙3-x-˙3∗=f3x2,x3+g3u+d3-x-˙3∗=-k3r˙rs3-k4s3∂sgns3-e31According to Young’s equation and (30)–(33),(34)s1Ts˙1=s1T-s2-y2-k1s1+e~11≤s1T32I-k1s1+12s2Ts2+12y2Ty2+12N~12(35)s2Ts˙2=s2Tg2x2s3+y3-k2s2-e21≤s2T12I+g22-k2s2+12s3Ts3+12y3Ty3+12N22(36)s3Ts˙3=s3T-k3r˙rs3-k4s3∂sgns3-e31≤s3T12I-k3+32k4s3+12N32+12k4It can be learnt that variables and their differential in the system model are bounded, and there are continuous functionsz~2 and z~3, where z~2>0 and z~3>0, enabling variables x˙2∗ and x˙3∗ to meet(37)x˙2∗≤z~2,x˙3∗≤z~3According to Young’s equation and (28)–(29) and (37): (38)y2Ty˙2=y2T-τ2-1y2-x˙2∗≤y2T12I-τ2-1y2+12z~22(39)y3Ty˙3=y3T-τ3-1y3-x˙3∗≤y3T12I-τ3-1y3+12z~32According to the state coupling IGC nonlinear system model (4), a Lyapunov function is selected: (40)V=12s1Ts1+s2Ts2+s3Ts3+y2Ty2+y3Ty3Taking the derivative of (40),(41)V˙=s1Ts˙1+s2Ts˙2+s3Ts˙3+y2Ty˙2+y3Ty˙3≤s1T32I-k1s1+12s2Ts2+12y2Ty2+12N~12+s2T12I+g22-k2s2+12s3Ts3+12y3Ty3+12N22+s3T12I-k3+32k4s3+12N32+12k4+y2T12I-τ2-1y2+12z~22+y3T12I-τ3-1y3+12z~32The design parameters meet the following rules:(42)k1≥32I+12κI,k2≥I+g22+12κI,k3≥I+32k4+12κIτ2-1≥I+12κI,τ3-1≥I+12κIIn the equation,κ is a constant, and κ>0. Therefore,(43)V˙≤-κV+A⌢In the equation,A⌢=1/2N~12+1/2N22+1/2N32+1/2z~22+1/2z~32+1/2k4.According to (43),(44)Vt≤κV0-A⌢e-κt+A⌢κs 1, s2, s3, y2, and y3 are consistent and eventually bounded. Thus, large parameters k1, k2, k3, and k4, as well as small parameters τ2 and τ3 are selected, to make the value of κ sufficiently large and A⌢/κ sufficiently small to ensure control precision.Remark 2.
Theoretically, the final boundaries of error surfacess1, s2, and s3 and filter errors y2 and y3 will become smaller with the increasing design parameters k1, k2, k3, and k4 and the decreasing τ2 and τ3. This change leads to a higher controlling precision. However, in reality, using too large parameters (k1, k2, k3, and k4) and too small parameters (τ2 and τ3) will result in an input saturation for the interceptor control system. The nonlinear behavior of the saturated system results in a higher requirement of overload exceeding the available overload. Therefore, the angle of attack and the sideslip angle of the interceptor exceed the allowable range leading to a reduced controlling performance of the system. Furthermore, the physical constraints of the low-pass filter prohibit parameters τ2 and τ3 from being too small. Therefore, the parameters of the control algorithm should be properly selected by combining practical situations.


## 3.1. Design of Finite-Time Disturbance Observer

Aiming at the uncertaintyat, d2, and d3 included in the system model (4), an FTDO (finite-time disturbance observer) is designed for estimating unknown disturbances, in order to eliminate the impact of unknown disturbances on the leader system of the interceptor. Define vr=r˙,  vε=rq˙ε,  vβ=rq˙βcos⁡qε, and(6)v˙ε=r˙q˙ε+rq¨ε=-r˙q˙ε-rq˙β2sin⁡qεcos⁡qε+atε-am4ε=-vrvεr-vβ2tan⁡qεr+atε-am4ε(7)v˙β=r˙q˙βcos⁡qε+rq¨βcos⁡qε-rq˙εq˙βsin⁡qε=-r˙q˙βcos⁡qε+rq˙εq˙βsin⁡qε-atβ+am4β=-vrvβr+vεvβtan⁡qεr-atβ+am4βDefinew1=vε,vβT, and according to (6) and (7),(8)w˙1=h1w1+h2w1x2+atIn the equation,(9)
h
1
w
1
=
-
v
r
v
ε
r
-
v
β
2
tan
⁡
q
ε
r
-
v
r
v
β
r
+
v
ε
v
β
tan
⁡
q
ε
r
,
h
2
w
1
=
-
q
S
C
y
α
α
m
0
0
q
S
C
z
β
β
mThe following FTDO is designed to estimate the accelerationat of the target, and(10)z˙10=v10+h1w1+h2w1x2,z˙11=v11,z˙12=v12v10=-λ10L11/3z10-w12/3sgnz10-w1+z11v11=-λ11L11/2z11-v101/2sgnz11-v10+z12v12=-λ12L1z12-v11q1/p1sgnz12-v11z10=w^1,z11=a^t,z12=a˙^tIn the equationw1=w11,w12T,  z10=z101,z102T,  z11=z111,z112T,  z12=z121,z122T,  v10=v101,v102T,  v11=v111,v112T,  v12=v121,v122T,  L1=L11,L12T,  a^t is the estimated acceleration at of the target, the estimated value w1 and a˙t is w^1 and a˙^t, λ10, λ11, and λ12 are the coefficients to be designed for the disturbance observer, q1 and p1 are the terminal coefficients, respectively, and 0<p1<q1.It can be learnt according to [27] that appropriate parameters can guarantee that the FTDO error system is steady in finite time. The estimated error of the acceleration at of the target is defined as e11=z11-at.Similarly, the disturbanced2 and d3 of the second subsystem and third subsystem is estimated, and(11)z˙20=v20+f2x2+g2x2x3,z˙21=v21,z˙22=v22v20=-λ20L21/3z20-x22/3sgnz20-x2+z21v21=-λ21L21/2z21-v201/2sgnz21-v20+z22v22=-λ22L2z22-v21q2/p2sgnz22-v21z20=x^2,z21=d^2,z22=d^˙2(12)z˙30=v30+f3x2,x3+g3u,z˙31=v31,z˙32=v32v30=-λ30L31/3z30-x32/3sgnz30-x3+z31v31=-λ31L31/2z31-v301/2sgnz31-v30+z32v32=-λ32L3z32-v31q3/p3sgnz32-v31z30=x^3,z31=d^3,z32=d^˙3In the equation, the estimated value of disturbance d2 and d3 is d^2 and d^3, respectively, and the estimated error is e21=z21-d2 and e31=z31-d3, respectively.


## 3.2. Design of Adaptive Dynamic Sliding-Mode Controller

Because the interceptor IGC model is an unmatched and uncertain system, and aiming at the state coupling IGC model (4) and FTDO estimated value (10)–(12), the “leader” control algorithm is designed by taking advantage of the adaptive dynamic sliding-mode control law.( 1 ) The command signal of the first subsystem of (4) is defined as x1d. In order to realize the guidance goal, the sight angular velocity should be removed. According to the design method of dynamic surface sliding-mode control, the first dynamic error surface is defined as follows: (13)s1=-g11-1x1x1-x1dTaking the derivative ofs1, the dynamic equation of error is given by(14)s˙1=g11-2x1g˙11x1x1-x1d-g11-1x1f1x1+g12x1at-x˙1d-x2According to the dynamic surface design method and FTDO estimated valuea^t in (10), the virtual control amount of the first dynamic surface can be obtained as follows: (15)x2∗=g11-2x1g˙11x1x1-x1d-g11-1x1f1x1+g12x1a^t-x˙1d+k1s1In the equation,k1=diagk11,k12 is the positive definite matrix. In the design process, differential blast would occur, while the differential of the virtual control amount x2∗ is taken. In order to avoid the complicated computation process owing to item inflation, x2∗ must be obtained through the first-order low-pass filter, and the virtual control amount of the filter can be obtained as follows: (16)τ2x-˙2∗+x-2∗=x2∗,x-2∗0=x2∗0In the equation,τ2=diagτ21,τ22 is the time constant of the filter, and the differential of the virtual control after the error surface filter can be obtained.(17)x-˙2∗=-τ2-1x-2∗-x2∗( 2 ) The second dynamic error surface is defined as(18)s2=x2-x-2∗Taking the derivative ofs2, the dynamic equation of error can be obtained as follows: (19)s˙2=x˙2-x-˙2∗=f2x2+g2x2x3+d2-x-˙2∗Similar to the first dynamic surface design method, the estimated FTDO valued^2 is substituted in (11). Thus, the virtual control of the second dynamic surface can be obtained as follows: (20)x3∗=g2-1x2-f2x2-d^2+x-˙2∗-k2s2In the equation,k2=diagk21,k22 is the positive definite matrix. Similarly, by obtaining x3∗ through the first-order low-pass filter, the virtual control amount of the filter can be obtained as follows: (21)τ3x-˙3∗+x-3∗=x3∗,x-3∗0=x3∗0In the equation,τ3=diagτ31,τ32 is the time constant of the filter. The differential of virtual control after the error surface filter can be obtained as follows:(22)x-˙3∗=-τ3-1x-3∗-x3∗( 3 ) The third dynamic error surface is defined as follows:(23)s3=x3-x-3∗Taking the derivative ofs3, the dynamic equation of error can be obtained as follows:(24)s˙3=x˙3-x-˙3∗=f3x2,x3+g3u+d3-x-˙3∗To guarantee the convergence velocity of the interceptor “leader,” an adaptive sliding-mode reaching law is designed:(25)s˙=-kar˙rs-kbs∂sgnsIn the equation,ka>0,  kb>0, and r˙ denotes the change in relative distance between the “leader” and target.According to (24) and (25) and estimated FTDO d^3 of (12), the adaptive dynamic surface sliding-mode control law of the interceptor “leader” is given by(26)u=g3-1-f3x2,x3-d^3+x-˙3∗-k3r˙rs3-k4s3∂sgns3In the equation,k3=diagk31,k32,  k4=diagk41,k42, and ∂=diag∂11,∂12 are positive definite matrices, and 0<∂<1.


## 3.3. Stability Analysis

Theorem 1.
Consider the integrated guidance and control system for the “leader” (equation (4)). If the convergence rate is calculated using (25), the disturbance values of the system (see (4)) are estimated using (10)-(12), and filter equations (16) and (21) are implemented; then finally under the dynamic surface sliding-mode control law (see (26)), imposing the constraint for ensuring the system (see (4)) output error converging into the adjacent area of the origin, an arbitrary adjacent area of the origin can be obtained with the appropriate design parameter determined.To prove:Let us assume that the estimated error of FTDO system meets(27)e11<N1,e21<N2,e31<N3In the equation,N1, N2, and N3 are positive constants.The filter error is defined as follows:(28)y2=x-2∗-x2∗,y3=x-3∗-x3∗Taking the derivative ofy2 and y3, the dynamic error of the filter can be obtained as follows:(29)y˙2=-τ2-1y2-x˙2∗,y˙3=-τ3-1y3-x˙3∗According to (13)–(23) and (28),(30)x1=-g11x1s1+x1dx2=s2+x-2∗=s2+y2+x2∗x3=s4+x-3∗=s3+y3+x3∗According to (4), (11)–(21), and (26)–(28),(31)s˙1=g11-2x1g˙11x1x1-x1d-g11-1x1f1x1+g12x1at-x˙1d-x2=-s2-y2-k1s1+e~11In the equation,e~11=g11-1(x1)g12(x1)e11. Let us assume that e~11<N~1, where N~1 is a positive constant. (32)s˙2=x˙2-x-˙2∗=f2x2+g2x2x3+d2-x-˙2∗=g2x2s3+y3-k2s2-e21(33)s˙3=x˙3-x-˙3∗=f3x2,x3+g3u+d3-x-˙3∗=-k3r˙rs3-k4s3∂sgns3-e31According to Young’s equation and (30)–(33),(34)s1Ts˙1=s1T-s2-y2-k1s1+e~11≤s1T32I-k1s1+12s2Ts2+12y2Ty2+12N~12(35)s2Ts˙2=s2Tg2x2s3+y3-k2s2-e21≤s2T12I+g22-k2s2+12s3Ts3+12y3Ty3+12N22(36)s3Ts˙3=s3T-k3r˙rs3-k4s3∂sgns3-e31≤s3T12I-k3+32k4s3+12N32+12k4It can be learnt that variables and their differential in the system model are bounded, and there are continuous functionsz~2 and z~3, where z~2>0 and z~3>0, enabling variables x˙2∗ and x˙3∗ to meet(37)x˙2∗≤z~2,x˙3∗≤z~3According to Young’s equation and (28)–(29) and (37): (38)y2Ty˙2=y2T-τ2-1y2-x˙2∗≤y2T12I-τ2-1y2+12z~22(39)y3Ty˙3=y3T-τ3-1y3-x˙3∗≤y3T12I-τ3-1y3+12z~32According to the state coupling IGC nonlinear system model (4), a Lyapunov function is selected: (40)V=12s1Ts1+s2Ts2+s3Ts3+y2Ty2+y3Ty3Taking the derivative of (40),(41)V˙=s1Ts˙1+s2Ts˙2+s3Ts˙3+y2Ty˙2+y3Ty˙3≤s1T32I-k1s1+12s2Ts2+12y2Ty2+12N~12+s2T12I+g22-k2s2+12s3Ts3+12y3Ty3+12N22+s3T12I-k3+32k4s3+12N32+12k4+y2T12I-τ2-1y2+12z~22+y3T12I-τ3-1y3+12z~32The design parameters meet the following rules:(42)k1≥32I+12κI,k2≥I+g22+12κI,k3≥I+32k4+12κIτ2-1≥I+12κI,τ3-1≥I+12κIIn the equation,κ is a constant, and κ>0. Therefore,(43)V˙≤-κV+A⌢In the equation,A⌢=1/2N~12+1/2N22+1/2N32+1/2z~22+1/2z~32+1/2k4.According to (43),(44)Vt≤κV0-A⌢e-κt+A⌢κs 1, s2, s3, y2, and y3 are consistent and eventually bounded. Thus, large parameters k1, k2, k3, and k4, as well as small parameters τ2 and τ3 are selected, to make the value of κ sufficiently large and A⌢/κ sufficiently small to ensure control precision.Remark 2.
Theoretically, the final boundaries of error surfacess1, s2, and s3 and filter errors y2 and y3 will become smaller with the increasing design parameters k1, k2, k3, and k4 and the decreasing τ2 and τ3. This change leads to a higher controlling precision. However, in reality, using too large parameters (k1, k2, k3, and k4) and too small parameters (τ2 and τ3) will result in an input saturation for the interceptor control system. The nonlinear behavior of the saturated system results in a higher requirement of overload exceeding the available overload. Therefore, the angle of attack and the sideslip angle of the interceptor exceed the allowable range leading to a reduced controlling performance of the system. Furthermore, the physical constraints of the low-pass filter prohibit parameters τ2 and τ3 from being too small. Therefore, the parameters of the control algorithm should be properly selected by combining practical situations.


## 4. Distributed Network Synchronization Strategy

### 4.1. Design of Cooperative Control Strategy Based on the Distributed Network

Based on the principle of time consistency of a multiagent system, the multi-interceptor cooperative control strategy is designed to ensure that all interceptors hit the targets at the same time. In the cooperative system of the multi-interceptor, the state information of other interceptors can be obtained through information interaction, for realizing time consistency, and such information interaction can be described using the graph theory. Assuming that each interceptor is a communication node, the information exchange among interceptors can be indicated asG-={V-,E-,A-}, where V-={v-i,i=1,2,…,n} denotes the set of interceptor nodes and E- denotes the lines between the interceptor nodes. The weighted coefficient matrix is indicated as A-=[a-ij]∈Rn×n; a-ij>0 implies that the interceptor node i and node j can exchange information. However, if a-ij=0, information cannot be exchanged. L- denotes the Laplace matrix of the undirected graph G-, among which the elements satisfy(45)l-ii=∑j=1,j≠ina-ijl-ij=-a-ij,j≠iB - = d i a g { b - 1 , b - 2 , … , b - j } denotes whether the interceptors can obtain the state information of the leader, b-i>0,i∈{1,2,3,…,j} indicates that the interceptors can obtain the state information of the leader, and b-i=0,i∈{1,2,3,…,j} indicates that the interceptors cannot obtain the state information of the leader.Based on the “leader-follower” topology structure, the distributed cooperative control strategy of the multi-interceptor is designed as follows:(46)vi=k-i1∑j=1na-ijςj-ςi+b-iς0-ςi+ς˙0In the equation,ς0=x0,y0,z0T denotes the position of the “leader,” ςi=xi,yi,ziT,i∈{1,2,3,…,j} denotes the position of the interceptors, vi=ς˙i,i∈{1,2,3,…,j} denotes the velocity of the interceptors, k-i1=diagk-i11,k-i12,k-i13 is a constant, and k-i1>0.Theorem 3.
If the “follower” state of the interceptor can converge to the “leader” state following the cooperative control strategy (see (46)), such cooperative strategy is then considered to be successful.To prove:Lemma 4.
Laplace matrixM- has the following properties:
(1) If G- is connected, the characteristic value of M- is λmin(M-)>0, and it is called the algebraic connectivity of the network-connected graph. The larger the value of λmin(M-), the more connected the network.
(2) One of the characteristic values of M- is 0, and its corresponding characteristic vector is1.The error variable is defined asei=ςi-ς0, and(47)e˙i=ς˙i-ς˙0=ki1∑j=1na-ijej-ei+b-ieiThe Lyapunov function is defined as(48)V=12eTM-+B-eIn the equation,e=[e1,e2,…,en]T.k - i = min ⁡ { k - i 1 }, and taking the derivative of the above equation,(49)V˙=e˙TM-+B-e=-∑i=1n∑j=1naijej-ei-b-ieie˙i=-∑i=1nk-i1∑j=1naijej-ei-b-iei∑j=1na-ijej-ei+b-iei=-∑i=1nk-i1∑j=1naijej-ei-b-iei2≤-k-i∑i=1n∑j=1naijej-ei-b-iei2LetV(e)≠0. Then, according to the above equation,(50)∑i=1n∑j=1naijej-ei-b-iei2Ve=eTM-+B-TM-+B-e1/2eTM-+B-Te≥2λminM-+B-According to (49)–(50),(51)V˙t≤-k-i2λminM-+B-Therefore,V(t) is convergent in finite time, namely, the convergence state from the “follower” to the “leader”; it can realize cooperative guidance and control of the multi-interceptor.


### 4.2. Implementation of Distributed Network Cooperative Control Strategy

In order to implement the distributed network synchronization strategy, instructions provided to the synchronization strategy should be traced for each interceptor “follower.” The motion relation of the interceptors involved in cooperative interception is given as follows:(52)x˙i=Vmicos⁡θmicos⁡φmviy˙i=Vmisin⁡θmiz˙i=-Vmicos⁡θmisin⁡φmviIn the equation,x˙i, y˙i, and z˙i are the velocity components of the ith interceptor in the inertial frame, and θmi and φmvi are the trajectory inclination angle and trajectory deflection angle of the ith interceptor.According to the distributed network synchronization strategy (46), the velocity reference instruction of the interceptor is given by(53)V-mxi=k-i1∑j=1na-ijxj-xi+b-ixm-xi+x˙mV-myi=k-i1∑j=1na-ijyj-yi+b-iym-yi+y˙mV-mzi=k-i1∑j=1na-ijzj-zi+b-izm-zi+z˙mAccording to (52), the total velocity, trajectory inclination angle, and trajectory deflection angle of the interceptor can be obtained as follows:(54)V-mi∗=V-mxi2+V-myi2+V-mzi2θ-mi∗=arcsin⁡V-myiV-miφ-vmi∗=-arctan⁡V-mziV-mxiTo obtain the differential of the total velocity and trajectory inclination angle of the tractorx-˙, the signal is obtained through the filter. Let x- and x-1∗ be the postfiltering instruction and prefiltering instruction, respectively. Then,(55)x-¨=-2ζnωnx-˙-ωn2x-+ωn2x-1∗In the equation,ζn and ωn are the damping and bandwidth of the filter, respectively.


## 4.1. Design of Cooperative Control Strategy Based on the Distributed Network

Based on the principle of time consistency of a multiagent system, the multi-interceptor cooperative control strategy is designed to ensure that all interceptors hit the targets at the same time. In the cooperative system of the multi-interceptor, the state information of other interceptors can be obtained through information interaction, for realizing time consistency, and such information interaction can be described using the graph theory. Assuming that each interceptor is a communication node, the information exchange among interceptors can be indicated asG-={V-,E-,A-}, where V-={v-i,i=1,2,…,n} denotes the set of interceptor nodes and E- denotes the lines between the interceptor nodes. The weighted coefficient matrix is indicated as A-=[a-ij]∈Rn×n; a-ij>0 implies that the interceptor node i and node j can exchange information. However, if a-ij=0, information cannot be exchanged. L- denotes the Laplace matrix of the undirected graph G-, among which the elements satisfy(45)l-ii=∑j=1,j≠ina-ijl-ij=-a-ij,j≠iB - = d i a g { b - 1 , b - 2 , … , b - j } denotes whether the interceptors can obtain the state information of the leader, b-i>0,i∈{1,2,3,…,j} indicates that the interceptors can obtain the state information of the leader, and b-i=0,i∈{1,2,3,…,j} indicates that the interceptors cannot obtain the state information of the leader.Based on the “leader-follower” topology structure, the distributed cooperative control strategy of the multi-interceptor is designed as follows:(46)vi=k-i1∑j=1na-ijςj-ςi+b-iς0-ςi+ς˙0In the equation,ς0=x0,y0,z0T denotes the position of the “leader,” ςi=xi,yi,ziT,i∈{1,2,3,…,j} denotes the position of the interceptors, vi=ς˙i,i∈{1,2,3,…,j} denotes the velocity of the interceptors, k-i1=diagk-i11,k-i12,k-i13 is a constant, and k-i1>0.Theorem 3.
If the “follower” state of the interceptor can converge to the “leader” state following the cooperative control strategy (see (46)), such cooperative strategy is then considered to be successful.To prove:Lemma 4.
Laplace matrixM- has the following properties:
(1) If G- is connected, the characteristic value of M- is λmin(M-)>0, and it is called the algebraic connectivity of the network-connected graph. The larger the value of λmin(M-), the more connected the network.
(2) One of the characteristic values of M- is 0, and its corresponding characteristic vector is1.The error variable is defined asei=ςi-ς0, and(47)e˙i=ς˙i-ς˙0=ki1∑j=1na-ijej-ei+b-ieiThe Lyapunov function is defined as(48)V=12eTM-+B-eIn the equation,e=[e1,e2,…,en]T.k - i = min ⁡ { k - i 1 }, and taking the derivative of the above equation,(49)V˙=e˙TM-+B-e=-∑i=1n∑j=1naijej-ei-b-ieie˙i=-∑i=1nk-i1∑j=1naijej-ei-b-iei∑j=1na-ijej-ei+b-iei=-∑i=1nk-i1∑j=1naijej-ei-b-iei2≤-k-i∑i=1n∑j=1naijej-ei-b-iei2LetV(e)≠0. Then, according to the above equation,(50)∑i=1n∑j=1naijej-ei-b-iei2Ve=eTM-+B-TM-+B-e1/2eTM-+B-Te≥2λminM-+B-According to (49)–(50),(51)V˙t≤-k-i2λminM-+B-Therefore,V(t) is convergent in finite time, namely, the convergence state from the “follower” to the “leader”; it can realize cooperative guidance and control of the multi-interceptor.


## 4.2. Implementation of Distributed Network Cooperative Control Strategy

In order to implement the distributed network synchronization strategy, instructions provided to the synchronization strategy should be traced for each interceptor “follower.” The motion relation of the interceptors involved in cooperative interception is given as follows:(52)x˙i=Vmicos⁡θmicos⁡φmviy˙i=Vmisin⁡θmiz˙i=-Vmicos⁡θmisin⁡φmviIn the equation,x˙i, y˙i, and z˙i are the velocity components of the ith interceptor in the inertial frame, and θmi and φmvi are the trajectory inclination angle and trajectory deflection angle of the ith interceptor.According to the distributed network synchronization strategy (46), the velocity reference instruction of the interceptor is given by(53)V-mxi=k-i1∑j=1na-ijxj-xi+b-ixm-xi+x˙mV-myi=k-i1∑j=1na-ijyj-yi+b-iym-yi+y˙mV-mzi=k-i1∑j=1na-ijzj-zi+b-izm-zi+z˙mAccording to (52), the total velocity, trajectory inclination angle, and trajectory deflection angle of the interceptor can be obtained as follows:(54)V-mi∗=V-mxi2+V-myi2+V-mzi2θ-mi∗=arcsin⁡V-myiV-miφ-vmi∗=-arctan⁡V-mziV-mxiTo obtain the differential of the total velocity and trajectory inclination angle of the tractorx-˙, the signal is obtained through the filter. Let x- and x-1∗ be the postfiltering instruction and prefiltering instruction, respectively. Then,(55)x-¨=-2ζnωnx-˙-ωn2x-+ωn2x-1∗In the equation,ζn and ωn are the damping and bandwidth of the filter, respectively.


## 5. Design of Interceptor “Follower” Controller

The instructions provided by the cooperative control strategy can be transformed into velocity, trajectory inclination angle, and trajectory deflection angle instruction. In order to track the command signal of interceptor “follower” in the cooperative network, the “follower” controller adopts the dynamic surface sliding-mode control algorithm. Assuming that the velocity of the interceptor “follower” is controllable, the flight velocity can be indicated as follows:(56)V˙m=cos⁡αicos⁡βimPi-gsin⁡θmiIn the equation,Pi is the motor power, m is the quality of “follower”, αi and βi are the attack angle and sideslip angle, respectively, θmi is the trajectory inclination angle, and g is the gravitational acceleration.According to (56), the error surface is defined as follows:(57)sv=Vm-V-miIn the equation,V-mi is the reference velocity command of the “follower” after filtering. Taking the derivative of sv,(58)s˙v=cos⁡αicos⁡βimPi-gsin⁡θmi-V-˙miIn the equation,V-˙mi is the differential of total velocity after filtering.To ensure that the velocity of the “follower” can track the system command rapidly, the following sliding-mode reaching law is adopted:(59)s˙=-kas-kbs∂sgnsAccording to (56)–(59), the thrust of the “follower” can be obtained as follows: (60)Pi=mcos⁡αicos⁡βiV-˙mi+gsin⁡θmi-kv1sv-kv2sv∂vsgnsvIn the equation,kv1>0,  kv2>0,  0<∂v<1.By definingxi1=θmi,φvmiT,  xi2=αi,βiT, and xi3=ωzi,ωyiT, the state coupling kinetic equations of the ith “follower” can be indicated as follows:(61)x˙i1=gi1xi2+di1x˙i2=fi2xi2+gi2xi2xi3+di2x˙i3=fi3xi2,xi3+gi3ui+di3In the equation,(62)fi2xi2=-qSCyαiαicos⁡αimVmicos⁡βi,qSCyαiαisin⁡αisin⁡βimVmi+qSCzβiβicos⁡βimVmiT,fi3xi2,xi3=qSL-mzαiαiJz+qSL-mzωziωziJz,qSL-myβiβiJy+qSL-myωyiωyiJyTgi1=-qSCyαim00qSCzβim,gi2xi2=1tan⁡βisin⁡αi0cos⁡αi,gi3=diagJz-1,Jy-1,ui=Mzi,MyiT,di1=dθi,dφviT,di2=dαi,dβiT,di3=dωzi,dωyiTAccording to (10)–(12), the following form of FTDO is designed for evaluating the disturbance di1, di2, and di3 to (61),(63)z˙i10=vi10+gi1xi2,z˙i11=vi11,z˙i12=vi12vi10=-λi10Li11/3zi10-xi12/3sgnzi10-xi1+zi11vi11=-λi11Li11/2zi11-vi101/2sgnzi11-vi10+zi12vi12=-λi12Li1zi12-vi11qi1/pi1sgnzi12-vi11zi10=x^i1,zi11=d^i1,zi12=d^˙i1In the equation, the estimated value ofdi1 is d^i1, and the estimated error is ei11=zi11-di1.Similarly, the estimated values ofdi2 and di3 are d^i2 and d^i3, respectively, and the estimated errors are ei12=zi12-di2 and ei13=zi13-di3.To ensure that the “follower” can track the command signal of the cooperative control strategy rapidly and guarantee steady flight attitude, the “follower” controller is designed with the dynamic surface sliding-mode control law, and according to the FTDO estimated value and state coupling kinetic equation (61).( 1 ) The first dynamic error surface is defined as follows:(64)si1=gi1-1xi1-xi1dIn the equation,xi1d=θ-mi,φ-vmiT is the instruction of the trajectory inclination angle and trajectory deflection angle after filtering. Taking the derivative of si1,(65)s˙i1=xi2+gi1-1di1-x˙i1dAccording to the dynamic surface sliding-mode control method and FTDO estimated valued^i1, the virtual control of the first dynamic surface is selected as(66)xi2∗=gi1-1-d^i1+x˙i1d-ki1si1In the equation,x˙i1d is the differential of the trajectory inclination angle and trajectory deflection angle after filtering, and ki1=diagki11,ki12 is the positive definite matrix. The value of xi2∗ is obtained through the first-order low-pass filter, and the virtual control after filtering and its differential are given by(67)τi2x-˙i2∗+x-i2∗=xi2∗,x-i2∗0=xi2∗0x-˙i2∗=-τi2-1x-i2∗-xi2∗In the equation,τi2=diagτi21,τi22 is the time constant of the filter.( 2 ) The second dynamic error surface is defined by(68)si2=xi2-x-i2∗In the equation,x-i2∗ is the command signal after filtering. Taking the derivative of si2,(69)s˙i2=fi2xi2+gi2xi2xi3+di2-x-˙i2∗According to the dynamic surface sliding-mode control method and FTDO estimated valued^i2, the virtual control of the first dynamic surface is given by(70)xi3∗=gi2-1xi2-fi2xi2-d^i2+x-˙i2∗-ki2si2In the equation,x-˙i2∗ is the instruction differential after filtering, and ki2=diagki21,ki22 is the positive definite matrix. The value of xi3∗ is obtained through the first-order low-pass filter, and the virtual control after filtering and its differential can be obtained as follows:(71)τi3x-˙i3∗+x-i3∗=xi3∗,x-i3∗0=xi3∗0x-˙i3∗=-τi3-1x-i3∗-xi3∗In the equation,τi3=diagτi31,τi32 is the time constant of the filter.( 3 ) The third dynamic error surface is defined as follows:(72)si3=xi3-x-i3∗In the equation,x-i3∗ is the command signal after filtering. Taking the derivative of si3,(73)s˙i3=fi3xi2,xi3+gi3ui+di3-x-˙i3∗According to the sliding-mode reaching law (see (59)), and FTDO estimated value d^i3, the “follower” dynamic surface sliding-mode control law is designed as follows: (74)ui=gi3-1-fi3xi2,xi3-d^i3+x-˙i3∗-ki3si3-ki4si3∂i3sgnsi3In the equationx-˙i3∗ is the instruction differential after filtering, ki3=diagki31,ki32 and ki4=diagki41,ki42 are the positive definite matrices, ∂i3=diag∂i31,∂i32, and 0<∂i3<1.It can be learnt by referring to (27)–(44) that the stability of the control algorithm of the interceptor “follower” can be guaranteed by selecting appropriate parameters.


## 6. Simulation Verification

To verify the effectiveness of the distributed cooperative IGC algorithm of the multi-interceptor with state coupling designed in this study, it is assumed that the flight velocity of the interceptor “leader” remains the same. According to the global communication topology structure shown in Figure1 and the local communication topology structure shown in Figure 2, Figure 1 assumes that the “leader” can communicate with the remaining three “followers,” while the “followers” can communicate with each other. Figure 2 assumes that the “leader” can only communicate with “follower 1,” while “followers” can communicate with each other. The initial conditions of interceptor “leader,” “follower,” and target are listed in Table 1.Table 1
Initial conditions of the leader, follower, and target.
No.  Parameter Value Parameter Value Parameter Value Parameter Value 1 Leader x m 0 0m y m 0 0m z m 0 0m V m 0 800 m/s 2 Follower1 x m 10 600m y m 10 0m z m 10 500m V m 10 800 m/s 3 Follower2 x m 20 800m y m 20 0m z m 20 1000m V m 20 800 m/s 4 Follower3 x m 30 400m y m 30 0m z m 30 -500m V m 30 800 m/s 5 Target x t 0 4000m y t 0 5000m z m 40 3000m V t 0 400 m/sFigure 1
Global communication topology structure of interceptor “leader” and “follower”.Figure 2
Local communication topology structure of the interceptor “leader” and “follower”.Focusing on the two communication topology structures shown in Figures1 and 2, a simulation study is conducted for the cooperative IGC algorithm of the multi-interceptor with state coupling designed in this study. It is assumed that the disturbance of the system is d2=d3=di1=di2=di3=0.02sin⁡(t), and the intercepted target has a linear acceleration of atε=atβ=5m/s2. The comparison of the simulation results of the cooperative IGC algorithm of the multi-interceptor with state coupling in the global and local communication topology is given, as shown in Figures 3–11.Figure 3
Motion trails of the interceptor “leader,” “follower,” and target in the global communication topology: X denotes the horizontal motion trail of the interceptor and target, Y denotes the longitudinal motion distance of the interceptor and target, and Z denotes the lateral motion trail of the interceptor and target.Figure 4
Motion trails of the interceptor “leader,” “follower,” and target in the local communication topology. X denotes the horizontal motion trail of the interceptor and target, Y denotes the longitudinal motion distance of the interceptor and target, and Z denotes the lateral motion trail of the interceptor and target.Figure 5
Interceptor velocity curve in the global and local communication topologies: (a) velocity curve in the global communication topology; (b) velocity curve in the local communication topology.
(a) (b)Figure 6
Trajectory inclination angle curve in the global and local communication topologies: (a) trajectory inclination angle curve in the global communication topology; (b) trajectory inclination angle curve in the local communication topology.
(a) (b)Figure 7
Trajectory deflection angle curve in the global and local communication topology: (a) trajectory deflection angle curve in global communication topology; (b) trajectory deflection angle curve in local communication topology.
(a) (b)Figure 8
Attack angle curve in the global and local communication topologies: (a) attack angle curve in the global communication topology; (b) attack angle curve in the local communication topology.
(a) (b)Figure 9
Sideslip angle curve in the global and local communication topologies: (a) sideslip angle curve in the global communication topology; (b) sideslip angle curve in the local communication topology.
(a) (b)Figure 10
Pitch angular velocities curve in the global and local communication topologies: (a) pitch angular velocities curve in the global communication topology; (b) pitch angular velocities curve in the local communication topology.
(a) (b)Figure 11
Yaw rate curve in the global and local communication topologies: (a) yaw rate curve in the global communication topology; (b) yaw rate curve in the local communication topology.
(a) (b)Figures3 and 4 show the motion trail of the interceptor “leader,” “follower,” and target in the local and global communication topologies. It can be seen that the motion trail of the interceptor “follower” in two different communication topologies is gradually consistent with that of the “leader.” Eventually, the “leader” and “follower” hit the target at the same time. The motion trail curve is smooth, showing short interception duration, fast convergence speed, and good stability.Figures5(a) and 5(b) show the velocity curve of the interceptor in the global and local topologies, from which it can be seen that the interceptor “follower” features a considerable overstriking property in the initial stage due to the lack of “leader.” The convergence rate is slower than that in the global topology, but it can eventually reach the steady state of the “leader.” The convergence process changes smoothly, and it shows good robustness to external disturbance. Similarly, it is clear from Figures 6–11 that the interceptor “follower” in the global and local communication topologies realizes the tracking of control instruction of the “leader.” The convergence process is relatively smooth, and it also shows good robustness to external disturbance.It can be seen from the simulation results of the global and local communication topologies that the distributed cooperative IGC algorithm of the multi-interceptor with state coupling designed in this study completes the tracking the instructions of the cooperative control strategy in the two different topologies and eventually achieves cooperative target interception.The cooperative IGC algorithm of the multi-interceptor with state coupling proposed in this study and the traditional method for multi-interceptor cooperation without state coupling are compared through numerical simulations. Figures12 and 13 show the comparison results.Figure 12
Comparison of the interceptor’s attack angle curves in the design methods with and without state coupling: (a) attack angle curve in the design method with state coupling; (b) attack angle curve in the design method without state coupling.
(a) (b)Figure 13
Comparison of the interceptor’s sideslip angle curves in the design methods with and without state coupling: (a) sideslip angle curve in the design method with state coupling; (b) sideslip angle curve in the design method without state coupling.
(a) (b)Figures12 and 13 present the comparison of the attack and sideslip angle curves associated with the cooperative IGC algorithm of the multi-interceptor with state coupling and the traditional method for the multi-interceptor cooperation without state coupling. The convergence process associated with the cooperative IGC algorithm of the multi-interceptor with state coupling appeared to be smoother compared to that associated with the traditional method for the multi-interceptor cooperation without state coupling. The section of the curve after t = 4s particularly revealed more stable angle and sideslip angle curves using the state-coupled design methods. In contrast, fluctuations in the attack and sideslip angle curves were observed for the traditional design methods without state coupling. In other words, the cooperative IGC algorithm of the multi-interceptor with state coupling exhibited better resistance to interference and allowed for a more stable control of the interceptor compared to the traditional method without state coupling.


## 7. Conclusions

This study focused on the cooperative target interception by multi-interceptor and designed cooperative IGC algorithm of the multi-interceptor with state coupling “leader-follower” structure. The algorithm is designed by considering the coupling relation between the pitch and yaw channels of the interceptor. Further, this study combines the IGC method and introduces the distributed cooperative control strategy. The interceptor “leader” and “follower” control algorithm is designed separately by employing the dynamic surface sliding-mode control law and FTDO. The distributed cooperative control strategy guarantees that the “leader” and “follower” can hit the targets at the same time. The algorithm displays ideal trajectory characteristics in the simulation verification, and it can realize the cooperative interception of targets in both the global and local communication topologies. Furthermore, the study provides a design method for the cooperative target interception of the multi-interceptor, with certain engineering values.



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*Source: 3193405-2018-08-01.xml*