# Design of Head-Pursuit Guidance Law Based on Backstepping Sliding Mode Control

Authors: Chenqi Zhu; Zhengyu Guo
Journal: International Journal of Aerospace Engineering
 (2019)

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## Abstract

In order to meet the needs of high-precision guidance for intercepting hypersonic targets, a novel head-pursuit guidance law considering the dynamic characteristics of a missile control system and the target mobility is presented via combining a fast power reaching law with backstepping sliding mode control in this paper. Initially, a three-dimensional head-pursuit system model of the missile and target is established. Subsequently, the system model is decomposed into a pitch plane system and lateral plane system, the control system dynamics are equivalent to second-order systems, and finite-time disturbance observers are introduced to estimate the target accelerations. On the basis of the previous work, the head-pursuit guidance laws of the vertical system and the lateral system which can stabilize the closed-loop system are designed separately and strict proofs of the methods are given. Finally, simulations are carried out to verify the effectiveness of this head-pursuit guidance law.


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## Body

## 1. Introduction

Hypersonic vehicles have the characteristics of long flight distance, high flight speed, and high maneuverability, which has great military value and also presents great challenges to the interception of such aircraft. As the first condition to affect the interception accuracy, the guidance law is one of the most important things in the design of the antiproximity space defense system [1–3]. Traditionally, there are two ways to intercept the targets, namely, tail-chase interception and head-on interception. Tail-chase interception requires the interceptor’s speed to be higher than the target’s, but it is hard to achieve when the target is a hypersonic vehicle. Head-on interception makes the missile fly to the target from the front of the target; the relative speed of the missile and target is so large that the response time of the interceptor will be shortened seriously, and this will put forward great challenges to the overload capability of the missile and the guidance law.In order to solve the problems above, literatures [4, 5] put forward the head-pursuit interception for the first time; they proposed the conditions which must be met for head-pursuit interception and also designed a two-dimensional head-pursuit guidance law under the ideal condition of ignoring the dynamic characteristics of autopilot. The head-pursuit guidance law not only reduced the requirement of overload capacity but also avoided the problem of aerodynamic ablation of interceptor warhead cover. On this basis, literature [6] designed the head-pursuit guidance law based on sliding mode control theory and verified the advantages of this method. Similarly, combined with the optimization theory, head-pursuit guidance laws based on Zero-Effort Miss (ZEM) and energy optimization are designed, respectively, in literatures [7, 8]. However, the guidance laws designed above only consider the two-dimensional case, which is quite different from the actual interception.Therefore, literatures [9, 10] designed three-dimensional head-pursuit guidance laws based on sliding mode control severally via considering the coupling between the pitch channel and the lateral channel. Besides, regarding the coupling between the pitch channel and the lateral channel as uncertain interference, adaptive sliding mode guidance laws of the pitch channel and the lateral channel are designed, respectively, in literature [11], which has strong robustness to the target acceleration; at the same time, a smooth second-order sliding mode guidance law which can effectively solve the discontinuous control problem caused by the target unknown acceleration and weaken the chattering is presented, too. In addition, adaptive sliding mode guidance laws are designed by Si and Song [12] to intercept the hypersonic vehicle in the form of head-pursuit interception. But these guidance laws ignore the dynamic characteristics of the control system. Even though literatures [13, 14] designed two-dimensional and three-dimensional head-pursuit guidance laws, respectively, they regard target maneuver as the precise known information, which is difficult to obtain accurately.In order to meet the actual requirements of intercepting hypersonic vehicles, this paper considers the dynamic characteristics of the control system, estimates the unknown target maneuver with a finite-time disturbance observer (FTDO), and designs three-dimensional head-pursuit guidance laws based on the backstepping sliding mode control theory. Meanwhile, the fast reaching law is introduced to weaken chattering while ensuring the convergence speed of the closed-loop system.The rest of this paper is organized as follows. Firstly, the principle of head-pursuit interception is introduced, and a three-dimensional head-pursuit system model of the missile and target is established. Secondly, the target maneuver is estimated by the FTDO and head-pursuit guidance laws which consider that the dynamic characteristics of the control system are designed and the stability of the closed-loop system is verified. Finally, the effectiveness and superiorities of the guidance laws designed in this paper are verified through the digital simulations.


## 2. Problem Formulation

As shown in Figure1, the head-pursuit interception against the hypersonic vehicle can be divided into three stages: approximation stage, orbit change stage, and terminal guidance stage. First, the missile approaches the target head-on after detecting it. Then, in order to make the interceptor consistent with the target’s flight direction and reach the expected orbit, the rail control engine of the interceptor is used to change the trajectory at the predetermined time and position. Finally, the interceptor enters the terminal guidance stage, and the guidance law adjusts the trajectory to make the target collide with the interceptor from the back.Figure 1
Head-pursuit interception schematic.For the sake of simplicity, the missile and target are assumed to be point masses, and supposing that the acceleration just changes the direction of the velocity, the target acceleration and its rate of change are bounded. Now consider a three-dimensional interceptor and target engagement as shown in Figure2.Figure 2
Three-dimensional engagement geometry.In Figure2, T is the target, M is the missile, T‐XYZ is the inertial coordinate system (ICS), T‐XtYtZt is the target velocity coordinate system (TVCS), T‐XmYmZm is the missile velocity coordinate system (MVCS), T‐XlYlZl is the line-of-sight coordinate system (LCS), φl and θl are the direction angles of the LCS relative to the ICS, and φm,θm and φt,θt are the direction angles of the missile and target velocity vectors relative to the LCS, respectively. The moment when the missile enters the terminal guidance stage is defined as the initial moment, and the target position at this moment is taken as the coordinate origin to establish the ICS so as to obtain the three-dimensional relative motion model of head-pursuit interception in the form of Equations (1), (2), (3), (4), (5), (6) and (7).
(1)ṙ=vmcosθmcosφm−vtcosθtcosφt,(2)θ̇l=vmsinθm−vtsinθtr,(3)φ̇l=vmcosθmsinφm−vtcosθtsinφtrcosθl,(4)θ̇t=aytvt−φ̇lsinθlsinφt−θ̇lcosφt,(5)φ̇t=−aztvtcosθt+φ̇lsinθlcosφttanθt−θ̇lsinφttanθt−φ̇lcosθl,(6)θ̇m=aymvm−φ̇lsinθlsinφm−θ̇lcosφm,(7)φ̇m=−azmvmcosθm+φ̇lsinθlcosφmtanθm−θ̇lsinφmtanθm−φ̇lcosθl,where ayt,azt and aym,azm are the components of the missile and target acceleration vectors in the vertical direction of their respective velocity directions and vt and vm are the magnitude of missile and target velocities, respectively. According to the guidance conditions of the head-pursuit interception discussed in literature [9], if the missile’s variables meet Equations (8) and (12), the interceptor can achieve head-pursuit interception of the target:
(8)limr→0φm=0;(9)limr→0φt=0;(10)limr→0θm=0;(11)limr→0θt=0,(12)φm=n1φt;(13)θm=n2θt,where n1 and n2 are the lead factors and need to meet the following conditions:
(14)n1>vtvm,n2>vtvm.It has been proved that in the head-pursuit interception, if the system shown in Equations (1), (2), (3), (4), (5), (6) and (7) meets the guidance relationship corresponding to Equation (12), the target can be successfully intercepted. Therefore, the key of designing the head-pursuit guidance law is to design the controller to make the system satisfy Equation (12).


## 3. Introduction to Basic Knowledge

Literature [15] presented that the dynamic characteristics of the missile can be described by a second-order model approximately, which can be expressed as follows:
(15)a¨ym=−2ξωnȧym−ωn2aym+ωn2aymc,a¨zm=−2ξωnȧzm−ωn2azm+ωn2azmc,where ξ is the damping coefficient, ωn is oscillation frequency, and aymc and azmc are the acceleration instruction.Lemma 1.
[16]. Consider a nonlinear system in the form of ṡ=−k1s−k2sigρs, where k1,k2>0,ρ∈0,1. Then, s and ṡ stabilize within a limited time T which depends on initial conditions and satisfies the following equality:
(16)Ts0=ln1+k1/k2s01−ρk11−ρ.Note 1.
k1 and k2 reflect the speed of the system state approaching 0; when the system state is far from the stable state, k1 plays a major role, but when the system state is near the stable state, k2 plays a major role. The reaching law which has the same form as this system is defined as the fast power reaching law; the convergence speed of the system state can be accelerated, and the chattering can be weakened via selecting proper k1 and k2.Lemma 2.
[17]. Consider a nonlinear system in the form of ẋ=gt+ut, x∈R, u is a continuous control input and satisfies ut∈R, gt is a smooth uncertain function and satisfies ġt≤L. Then, there is a disturbance observer in the form of Equation (17), which makes the system satisfy Equation (18) after a transient process:
(17)ż0=v0+ut,v0=−λ2L1/3z0−x12/3signz0−x1−μ2z0−x1+z1,ż1=v1,v1=−λ1L1/2z1−v01/2signz1−v0−μ1z1−v0+z2,ż2=v2,v2=−λ0Lsignz2−v1−μ0z2−v1,(18)z10t=x11t,z11t=d1t,z12t=ḋ1t,where λ0,λ1,λ2 and μ0,μ1,μ2 are the design parameters of the disturbance observer.Note 2.
Zhou [17] gives a valid set of values for λ0,λ1,λ2 and μ0,μ1,μ2 as follows:
(19)λ0=8,λ1=5,λ2=3,μ0=12,μ1=11,μ2=10.


### 3.1. Design of the Head-Pursuit Guidance Law

The design of SMG law can be divided into two phases which are the reaching phase (RP) and the sliding mode phase (SMP). Literature [4] has proved the convergence of SMP; therefore, only the convergence of the first stage is studied here.Based on the previous description, the following variables are defined:(20)x11=θm−nθt,x12=aym,x13=ȧym,x21=φm−nφt,x22=azm,x23=ȧzm,uy=aymc,uz=azmc.Subsequently, the system model is decomposed into the pitch plane system in the form of Equation (21) and the lateral plane system in the form of Equation (22):
(21)ẋ11=x12vm+f1−nf3+d1,ẋ12=x13,ẋ13=−2ξωnx13−ωn2x12+ωn2uy,(22)ẋ21=−x22vmcosθm+f2−nf4+d2,ẋ22=x23,ẋ23=−2ξωnx23−ωn2x22+ωn2uz,where
(23)f1=−φ̇lsinθlsinφm−θ̇lcosφm,f2=φ̇lsinθlcosφmtanθm−θ̇lsinφmtanθm−φ̇lcosθl,f3=−φ̇lsinθlsinφt−θ̇lcosφt,f4=φ̇lsinθlcosφttanθt−θ̇lsinφttanθt−φ̇lcosθl,d1=−naytvt,d2=naztvtcosθt.


#### 3.1.1. Guidance Law Design for the Pitch Plane System

Firstly, defineL1 and h1 as follows:
(24)ḋ1t≤L1,h1=x12vm+f1−nf3.Secondly, a FTDO is designed to estimate the disturbanced1 in the pitch plane system via combining Lemma 2 with the initial angle conditions of head-pursuit interception discussed by Ge et al. [10]:
(25)ż10=v10+h1t,v10=−λ12L11/3z10−x112/3signz10−x11−μ12z10−x11+z11,ż11=v11,v11=−λ11L11/2z11−v101/2signz11−v10−μ11z11−v10+z12,ż12=v12,v12=−λ10L1signz12−v11−μ10z12−v11,where λ10,λ11,λ12 and μ10,μ11,μ12 are the design parameters of the disturbance observer.After a transient process of finite time, the following equation can be obtained:(26)z10t=x11t,(27)z11t=d1t,(28)z12t=ḋ1t.Then, guidance based on backstepping sliding mode control theory is designed according to the following steps.(1)
The 1st dynamic error surface is defined as
(29)s11=x11.Taking the derivative ofs11 and obtaining the error dynamic equation
(30)ṡ11=x12vm+f1−nf3+d1.Defining a fast power reaching law based on Lemma1 as follows:
(31)ṡ11=−k11s11−k12sigρ11s11,where k11>0, k12>0, and 0<ρ11<1. Substitute Equation (26) into Equation (30) and combine Equation (30) with Equation (31), the following virtual control input of the 1st dynamic surface can be obtained:
(32)x12c=−vmk11s11+k12sigρ11s11+f1−nf3+z11.A 1st-order low-pass filter is designed in the following form for x12c to avoid the increase of calculating complexity caused by the “explosion of terms” while finding the derivative of the virtual control input:
(33)ẋ12c∗=τ11−1x12c−x12c∗,x12c∗0=x12c0,where x12c∗ is filtered virtual control input and τ11 is the time constant of the filter
(2)
The 2nd dynamic error surface is defined as
(34)s12=x12−x12c∗.Taking the derivative ofs12 and obtaining the error dynamic equation
(35)ṡ12=x13−ẋ12c∗.Defining a fast power reaching law as follows:(36)ṡ12=−k13s12−k14sigρ12s12,where k13>0, k14>0, and 0<ρ12<1. Combine Equation (35) with Equation (36), the following virtual control input of the 2nd dynamic surface can be obtained:
(37)x13c=−k13s12−k14sigρ12s12+ẋ12c∗.Similarly,x13c is fed through a low-pass filter in the following form to obtain the filtered virtual control input:
(38)ẋ13c∗=τ12−1x13c−x13c∗,x13c∗0=x13c0,where x13c∗ is filtered virtual control input and τ12 is the time constant of the filter
(3)
The 3rd dynamic error surface is defined as
(39)s13=x13−x13c∗.Taking the derivative ofs13 and obtaining the error dynamic equation
(40)ṡ13=ẋ13−ẋ13c∗=−2ξωnx13−ωn2x12+ωn2uy−ẋ13c∗.Defining a fast power reaching law as follows:(41)ṡ13=−k15s13−k16sigρ13s13,where k15>0, k16>0, and 0<ρ13<1. Combine Equation (40) with Equation (41), the following control input of the 3rd dynamic surface can be obtained:
(42)uy=ωn−2−k15s13−k16sigρ13s13+2ξωnx13+ωn2x12+ẋ13c∗.Therefore, the head-pursuit guidance law for the pitch plane system has the following form:(43)s11=x11,x12c=−vmk11s11+k12sigρ11s11+f1−nf3+z11,ẋ12c∗=τ11−1x12c−x12c∗,x12c∗0=x12c0,s12=x12−x12c∗,x13c=−k13s12−k14sigρ12s12+ẋ12c∗,ẋ13c∗=τ12−1x13c−x13c∗,x13c∗0=x13c0,s13=x13−x13c∗,uy=ωn−2−k15s13−k16sigρ13s13+2ξωnx13+ωn2x12+ẋ13c∗.Finally, the stability of the guidance law in the form of Equation (43) is analyzed in the coming section.Defining the Lyapunov function as follows:(44)V1=12s112+12s122+12s132+12e132+12e142,where
(45)e13=x12c∗−x12c,e14=x13c∗−x13c.According to Equations (21) and (43), the following equations can be obtained:
(46)ṡ11=x12vm+f1−nf3+d1=s12+e13+x12cvm+f1−nf3+d1=s12+e13vm−k11s11+k12sigρ11s11,ṡ12=x13−ẋ12c∗=s13+x13c∗−ẋ12c∗=s13+e14+x13c−ẋ12c∗=s13+e14−k13s12−k14sigρ12s12,ṡ13=−2ξωnx13−ωn2x12+ωn2uy−ẋ13c∗=−k15s13−k16sigρ13s13,where
(47)e12=d1−z11.Then,(48)s11ṡ11=s11s12+e13vm−k11s11+k12sigρ11s11≤12vms112+s122+12vms112+e132−k11s112+k1212+32s112=1vm−k11+32k12s112+12vms122+12vme132+12k12,s12ṡ12=s12s13+e14−k13s12−k14sigρ12s12≤12s122+s132+s122+e142−k13s122+k1412+32s122=1−k13+32k14s122+12s132+12e142+12k14,s13ṡ13=−k15s132−k16s13ρ13+1≤−k15s132+k1612+32s132=−k15+32k16s132+12k16,e13ė13=e13−τ11−1e13−ẋ12c≤−τ11−1e132+12e132+12ẋ12c2=−τ11−1+12e13212ẋ12c2,e14ė14=e14−τ12−1e14−ẋ13c≤−τ12−1e142+12e142+12ẋ13c2=−τ12−1+12e14212ẋ13c2.Taking the derivative of Equation (44):
(49)V̇1=s11ṡ11+s12ṡ12+s13ṡ13+e13ė13+e14ė14≤1vm−k11+32k12s112+12vm+1−k13+32k14s122+12−k15+32k16s132+−τ11−1+12+12vme132+−τ12−1+1e142+12ẋ12c2+12ẋ13c2+12k12+12k14+12k16.Since all variables and their derivatives in the system are bounded, there exist nonnegative continuous functionsg1 and g2 as follows:
(50)ẋ12c∗≤g1s11,s12,e12,e13,k11,k12,ẋ13c∗≤g2s11,s12,s13,e12,e13,e14,k11,k12,k13,k14.There are positive constantR1 and compact set B1 which satisfy the following equation:
(51)B1=s11,s12,s13,e12,e13,e14T,V1<R1.Therefore, we have(52)ẋ12c≤M11,ẋ13c≤M12,where M11 and M12 are the maximum values of g1 and g2 on the compact set B1, respectively. Rewrite Equation (49), and we get
(53)V̇1≤1vm−k11+32k12s112+12vm+1−k13+32k14s122+12−k15+32k16s132+−τ11−1+12+12vme132+−τ12−1+1e142+12M112+12M122+12k12+12k14+12k16.LetN be a positive integer. If design parameters satisfy the following equation:
(54)k11−32k12≥1vm+0.5N,k13−32k14≥12vm+1+0.5N,k15−32k16≥12+0.5N,τ11−1≥12+12vm+0.5N,τ12−1≥1+0.5N.We get the following inequality:(55)V̇1≤−NV+E,where
(56)E=12M112+12M122+12k12+12k14+12k16.According to Equation (55), we can get
(57)V1≤NV0−Ee−Nt+EN.Based on the literature proposed by Liu and Liang [18], if N is large enough, it is guaranteed that filtering errors and error surfaces are sufficiently small. Therefore, just make sure that the design parameters k11,k13,k15 are large enough; the design parameters k12,k14,k16 and the time constant of the filter τ11,τ12 are small enough.


#### 3.1.2. Guidance Law Design for the Lateral Plane System

The design of the guidance law for the lateral plane system is similar to the pitch plane system. Firstly, a FTDO is designed to estimate the disturbanced2 in the lateral plane system:
(58)ż20=v20+h2t,v20=−λ22L21/3z20−x212/3signz20−x21−μ22z20−x21+z21,ż21=v21,v21=−λ21L21/2z21−v201/2signz21−v20−μ21z21−v20+z22,ż22=v22,v22=−λ20L2signz22−v21−μ20z22−v21,where λ20,λ21,λ22 and μ20,μ21,μ22 are the design parameters of the disturbance observer, z20,z21,z22 are corresponding estimates, and L2 and h2 satisfy the following relations, respectively:
(59)L2≥ḋ2t,h2=−x22vmcosθm+f2−nf4.The head-pursuit guidance law for the lateral plane system is designed as follows:(60)s21=x21,x22c=vmcosθmk21s21+k22sigρ21s21+f2−nf4+z21,ẋ22c∗=τ21−1x22c−x22c∗,x22c∗0=x22c0,s22=x22−x22c∗,x23c=−k23s22−k24sigρ22s22+ẋ22c∗,ẋ23c∗=τ22−1x23c−x23c∗,x23c∗0=x23c0,s23=x23−x23c∗,uz=ωn−2−k25s23−k26sigρ23s23+2ξωnx23+ωn2x22+ẋ23c∗,where s21,s22,s23 are the corresponding error surfaces, k21,k22,k23,k24,k25,k26 are the corresponding design parameters, x22c,x23c,x22c∗,x23c∗ are the corresponding virtual control inputs and filtered virtual control inputs, and τ21,τ22 are the corresponding time constants of the filters. If the design parameters k21,k23,k25 are large enough, the design parameters k22,k24,k26 and the time constant of the filter τ21,τ22 are small enough; it can be ensured that filtering errors and error surfaces are sufficiently small.The stability of the guidance law for the lateral plane system is similar to the pitch plane system; therefore, there is no specific elaboration here.


## 3.1. Design of the Head-Pursuit Guidance Law

The design of SMG law can be divided into two phases which are the reaching phase (RP) and the sliding mode phase (SMP). Literature [4] has proved the convergence of SMP; therefore, only the convergence of the first stage is studied here.Based on the previous description, the following variables are defined:(20)x11=θm−nθt,x12=aym,x13=ȧym,x21=φm−nφt,x22=azm,x23=ȧzm,uy=aymc,uz=azmc.Subsequently, the system model is decomposed into the pitch plane system in the form of Equation (21) and the lateral plane system in the form of Equation (22):
(21)ẋ11=x12vm+f1−nf3+d1,ẋ12=x13,ẋ13=−2ξωnx13−ωn2x12+ωn2uy,(22)ẋ21=−x22vmcosθm+f2−nf4+d2,ẋ22=x23,ẋ23=−2ξωnx23−ωn2x22+ωn2uz,where
(23)f1=−φ̇lsinθlsinφm−θ̇lcosφm,f2=φ̇lsinθlcosφmtanθm−θ̇lsinφmtanθm−φ̇lcosθl,f3=−φ̇lsinθlsinφt−θ̇lcosφt,f4=φ̇lsinθlcosφttanθt−θ̇lsinφttanθt−φ̇lcosθl,d1=−naytvt,d2=naztvtcosθt.


### 3.1.1. Guidance Law Design for the Pitch Plane System

Firstly, defineL1 and h1 as follows:
(24)ḋ1t≤L1,h1=x12vm+f1−nf3.Secondly, a FTDO is designed to estimate the disturbanced1 in the pitch plane system via combining Lemma 2 with the initial angle conditions of head-pursuit interception discussed by Ge et al. [10]:
(25)ż10=v10+h1t,v10=−λ12L11/3z10−x112/3signz10−x11−μ12z10−x11+z11,ż11=v11,v11=−λ11L11/2z11−v101/2signz11−v10−μ11z11−v10+z12,ż12=v12,v12=−λ10L1signz12−v11−μ10z12−v11,where λ10,λ11,λ12 and μ10,μ11,μ12 are the design parameters of the disturbance observer.After a transient process of finite time, the following equation can be obtained:(26)z10t=x11t,(27)z11t=d1t,(28)z12t=ḋ1t.Then, guidance based on backstepping sliding mode control theory is designed according to the following steps.(1)
The 1st dynamic error surface is defined as
(29)s11=x11.Taking the derivative ofs11 and obtaining the error dynamic equation
(30)ṡ11=x12vm+f1−nf3+d1.Defining a fast power reaching law based on Lemma1 as follows:
(31)ṡ11=−k11s11−k12sigρ11s11,where k11>0, k12>0, and 0<ρ11<1. Substitute Equation (26) into Equation (30) and combine Equation (30) with Equation (31), the following virtual control input of the 1st dynamic surface can be obtained:
(32)x12c=−vmk11s11+k12sigρ11s11+f1−nf3+z11.A 1st-order low-pass filter is designed in the following form for x12c to avoid the increase of calculating complexity caused by the “explosion of terms” while finding the derivative of the virtual control input:
(33)ẋ12c∗=τ11−1x12c−x12c∗,x12c∗0=x12c0,where x12c∗ is filtered virtual control input and τ11 is the time constant of the filter
(2)
The 2nd dynamic error surface is defined as
(34)s12=x12−x12c∗.Taking the derivative ofs12 and obtaining the error dynamic equation
(35)ṡ12=x13−ẋ12c∗.Defining a fast power reaching law as follows:(36)ṡ12=−k13s12−k14sigρ12s12,where k13>0, k14>0, and 0<ρ12<1. Combine Equation (35) with Equation (36), the following virtual control input of the 2nd dynamic surface can be obtained:
(37)x13c=−k13s12−k14sigρ12s12+ẋ12c∗.Similarly,x13c is fed through a low-pass filter in the following form to obtain the filtered virtual control input:
(38)ẋ13c∗=τ12−1x13c−x13c∗,x13c∗0=x13c0,where x13c∗ is filtered virtual control input and τ12 is the time constant of the filter
(3)
The 3rd dynamic error surface is defined as
(39)s13=x13−x13c∗.Taking the derivative ofs13 and obtaining the error dynamic equation
(40)ṡ13=ẋ13−ẋ13c∗=−2ξωnx13−ωn2x12+ωn2uy−ẋ13c∗.Defining a fast power reaching law as follows:(41)ṡ13=−k15s13−k16sigρ13s13,where k15>0, k16>0, and 0<ρ13<1. Combine Equation (40) with Equation (41), the following control input of the 3rd dynamic surface can be obtained:
(42)uy=ωn−2−k15s13−k16sigρ13s13+2ξωnx13+ωn2x12+ẋ13c∗.Therefore, the head-pursuit guidance law for the pitch plane system has the following form:(43)s11=x11,x12c=−vmk11s11+k12sigρ11s11+f1−nf3+z11,ẋ12c∗=τ11−1x12c−x12c∗,x12c∗0=x12c0,s12=x12−x12c∗,x13c=−k13s12−k14sigρ12s12+ẋ12c∗,ẋ13c∗=τ12−1x13c−x13c∗,x13c∗0=x13c0,s13=x13−x13c∗,uy=ωn−2−k15s13−k16sigρ13s13+2ξωnx13+ωn2x12+ẋ13c∗.Finally, the stability of the guidance law in the form of Equation (43) is analyzed in the coming section.Defining the Lyapunov function as follows:(44)V1=12s112+12s122+12s132+12e132+12e142,where
(45)e13=x12c∗−x12c,e14=x13c∗−x13c.According to Equations (21) and (43), the following equations can be obtained:
(46)ṡ11=x12vm+f1−nf3+d1=s12+e13+x12cvm+f1−nf3+d1=s12+e13vm−k11s11+k12sigρ11s11,ṡ12=x13−ẋ12c∗=s13+x13c∗−ẋ12c∗=s13+e14+x13c−ẋ12c∗=s13+e14−k13s12−k14sigρ12s12,ṡ13=−2ξωnx13−ωn2x12+ωn2uy−ẋ13c∗=−k15s13−k16sigρ13s13,where
(47)e12=d1−z11.Then,(48)s11ṡ11=s11s12+e13vm−k11s11+k12sigρ11s11≤12vms112+s122+12vms112+e132−k11s112+k1212+32s112=1vm−k11+32k12s112+12vms122+12vme132+12k12,s12ṡ12=s12s13+e14−k13s12−k14sigρ12s12≤12s122+s132+s122+e142−k13s122+k1412+32s122=1−k13+32k14s122+12s132+12e142+12k14,s13ṡ13=−k15s132−k16s13ρ13+1≤−k15s132+k1612+32s132=−k15+32k16s132+12k16,e13ė13=e13−τ11−1e13−ẋ12c≤−τ11−1e132+12e132+12ẋ12c2=−τ11−1+12e13212ẋ12c2,e14ė14=e14−τ12−1e14−ẋ13c≤−τ12−1e142+12e142+12ẋ13c2=−τ12−1+12e14212ẋ13c2.Taking the derivative of Equation (44):
(49)V̇1=s11ṡ11+s12ṡ12+s13ṡ13+e13ė13+e14ė14≤1vm−k11+32k12s112+12vm+1−k13+32k14s122+12−k15+32k16s132+−τ11−1+12+12vme132+−τ12−1+1e142+12ẋ12c2+12ẋ13c2+12k12+12k14+12k16.Since all variables and their derivatives in the system are bounded, there exist nonnegative continuous functionsg1 and g2 as follows:
(50)ẋ12c∗≤g1s11,s12,e12,e13,k11,k12,ẋ13c∗≤g2s11,s12,s13,e12,e13,e14,k11,k12,k13,k14.There are positive constantR1 and compact set B1 which satisfy the following equation:
(51)B1=s11,s12,s13,e12,e13,e14T,V1<R1.Therefore, we have(52)ẋ12c≤M11,ẋ13c≤M12,where M11 and M12 are the maximum values of g1 and g2 on the compact set B1, respectively. Rewrite Equation (49), and we get
(53)V̇1≤1vm−k11+32k12s112+12vm+1−k13+32k14s122+12−k15+32k16s132+−τ11−1+12+12vme132+−τ12−1+1e142+12M112+12M122+12k12+12k14+12k16.LetN be a positive integer. If design parameters satisfy the following equation:
(54)k11−32k12≥1vm+0.5N,k13−32k14≥12vm+1+0.5N,k15−32k16≥12+0.5N,τ11−1≥12+12vm+0.5N,τ12−1≥1+0.5N.We get the following inequality:(55)V̇1≤−NV+E,where
(56)E=12M112+12M122+12k12+12k14+12k16.According to Equation (55), we can get
(57)V1≤NV0−Ee−Nt+EN.Based on the literature proposed by Liu and Liang [18], if N is large enough, it is guaranteed that filtering errors and error surfaces are sufficiently small. Therefore, just make sure that the design parameters k11,k13,k15 are large enough; the design parameters k12,k14,k16 and the time constant of the filter τ11,τ12 are small enough.


### 3.1.2. Guidance Law Design for the Lateral Plane System

The design of the guidance law for the lateral plane system is similar to the pitch plane system. Firstly, a FTDO is designed to estimate the disturbanced2 in the lateral plane system:
(58)ż20=v20+h2t,v20=−λ22L21/3z20−x212/3signz20−x21−μ22z20−x21+z21,ż21=v21,v21=−λ21L21/2z21−v201/2signz21−v20−μ21z21−v20+z22,ż22=v22,v22=−λ20L2signz22−v21−μ20z22−v21,where λ20,λ21,λ22 and μ20,μ21,μ22 are the design parameters of the disturbance observer, z20,z21,z22 are corresponding estimates, and L2 and h2 satisfy the following relations, respectively:
(59)L2≥ḋ2t,h2=−x22vmcosθm+f2−nf4.The head-pursuit guidance law for the lateral plane system is designed as follows:(60)s21=x21,x22c=vmcosθmk21s21+k22sigρ21s21+f2−nf4+z21,ẋ22c∗=τ21−1x22c−x22c∗,x22c∗0=x22c0,s22=x22−x22c∗,x23c=−k23s22−k24sigρ22s22+ẋ22c∗,ẋ23c∗=τ22−1x23c−x23c∗,x23c∗0=x23c0,s23=x23−x23c∗,uz=ωn−2−k25s23−k26sigρ23s23+2ξωnx23+ωn2x22+ẋ23c∗,where s21,s22,s23 are the corresponding error surfaces, k21,k22,k23,k24,k25,k26 are the corresponding design parameters, x22c,x23c,x22c∗,x23c∗ are the corresponding virtual control inputs and filtered virtual control inputs, and τ21,τ22 are the corresponding time constants of the filters. If the design parameters k21,k23,k25 are large enough, the design parameters k22,k24,k26 and the time constant of the filter τ21,τ22 are small enough; it can be ensured that filtering errors and error surfaces are sufficiently small.The stability of the guidance law for the lateral plane system is similar to the pitch plane system; therefore, there is no specific elaboration here.


## 3.1.1. Guidance Law Design for the Pitch Plane System

Firstly, defineL1 and h1 as follows:
(24)ḋ1t≤L1,h1=x12vm+f1−nf3.Secondly, a FTDO is designed to estimate the disturbanced1 in the pitch plane system via combining Lemma 2 with the initial angle conditions of head-pursuit interception discussed by Ge et al. [10]:
(25)ż10=v10+h1t,v10=−λ12L11/3z10−x112/3signz10−x11−μ12z10−x11+z11,ż11=v11,v11=−λ11L11/2z11−v101/2signz11−v10−μ11z11−v10+z12,ż12=v12,v12=−λ10L1signz12−v11−μ10z12−v11,where λ10,λ11,λ12 and μ10,μ11,μ12 are the design parameters of the disturbance observer.After a transient process of finite time, the following equation can be obtained:(26)z10t=x11t,(27)z11t=d1t,(28)z12t=ḋ1t.Then, guidance based on backstepping sliding mode control theory is designed according to the following steps.(1)
The 1st dynamic error surface is defined as
(29)s11=x11.Taking the derivative ofs11 and obtaining the error dynamic equation
(30)ṡ11=x12vm+f1−nf3+d1.Defining a fast power reaching law based on Lemma1 as follows:
(31)ṡ11=−k11s11−k12sigρ11s11,where k11>0, k12>0, and 0<ρ11<1. Substitute Equation (26) into Equation (30) and combine Equation (30) with Equation (31), the following virtual control input of the 1st dynamic surface can be obtained:
(32)x12c=−vmk11s11+k12sigρ11s11+f1−nf3+z11.A 1st-order low-pass filter is designed in the following form for x12c to avoid the increase of calculating complexity caused by the “explosion of terms” while finding the derivative of the virtual control input:
(33)ẋ12c∗=τ11−1x12c−x12c∗,x12c∗0=x12c0,where x12c∗ is filtered virtual control input and τ11 is the time constant of the filter
(2)
The 2nd dynamic error surface is defined as
(34)s12=x12−x12c∗.Taking the derivative ofs12 and obtaining the error dynamic equation
(35)ṡ12=x13−ẋ12c∗.Defining a fast power reaching law as follows:(36)ṡ12=−k13s12−k14sigρ12s12,where k13>0, k14>0, and 0<ρ12<1. Combine Equation (35) with Equation (36), the following virtual control input of the 2nd dynamic surface can be obtained:
(37)x13c=−k13s12−k14sigρ12s12+ẋ12c∗.Similarly,x13c is fed through a low-pass filter in the following form to obtain the filtered virtual control input:
(38)ẋ13c∗=τ12−1x13c−x13c∗,x13c∗0=x13c0,where x13c∗ is filtered virtual control input and τ12 is the time constant of the filter
(3)
The 3rd dynamic error surface is defined as
(39)s13=x13−x13c∗.Taking the derivative ofs13 and obtaining the error dynamic equation
(40)ṡ13=ẋ13−ẋ13c∗=−2ξωnx13−ωn2x12+ωn2uy−ẋ13c∗.Defining a fast power reaching law as follows:(41)ṡ13=−k15s13−k16sigρ13s13,where k15>0, k16>0, and 0<ρ13<1. Combine Equation (40) with Equation (41), the following control input of the 3rd dynamic surface can be obtained:
(42)uy=ωn−2−k15s13−k16sigρ13s13+2ξωnx13+ωn2x12+ẋ13c∗.Therefore, the head-pursuit guidance law for the pitch plane system has the following form:(43)s11=x11,x12c=−vmk11s11+k12sigρ11s11+f1−nf3+z11,ẋ12c∗=τ11−1x12c−x12c∗,x12c∗0=x12c0,s12=x12−x12c∗,x13c=−k13s12−k14sigρ12s12+ẋ12c∗,ẋ13c∗=τ12−1x13c−x13c∗,x13c∗0=x13c0,s13=x13−x13c∗,uy=ωn−2−k15s13−k16sigρ13s13+2ξωnx13+ωn2x12+ẋ13c∗.Finally, the stability of the guidance law in the form of Equation (43) is analyzed in the coming section.Defining the Lyapunov function as follows:(44)V1=12s112+12s122+12s132+12e132+12e142,where
(45)e13=x12c∗−x12c,e14=x13c∗−x13c.According to Equations (21) and (43), the following equations can be obtained:
(46)ṡ11=x12vm+f1−nf3+d1=s12+e13+x12cvm+f1−nf3+d1=s12+e13vm−k11s11+k12sigρ11s11,ṡ12=x13−ẋ12c∗=s13+x13c∗−ẋ12c∗=s13+e14+x13c−ẋ12c∗=s13+e14−k13s12−k14sigρ12s12,ṡ13=−2ξωnx13−ωn2x12+ωn2uy−ẋ13c∗=−k15s13−k16sigρ13s13,where
(47)e12=d1−z11.Then,(48)s11ṡ11=s11s12+e13vm−k11s11+k12sigρ11s11≤12vms112+s122+12vms112+e132−k11s112+k1212+32s112=1vm−k11+32k12s112+12vms122+12vme132+12k12,s12ṡ12=s12s13+e14−k13s12−k14sigρ12s12≤12s122+s132+s122+e142−k13s122+k1412+32s122=1−k13+32k14s122+12s132+12e142+12k14,s13ṡ13=−k15s132−k16s13ρ13+1≤−k15s132+k1612+32s132=−k15+32k16s132+12k16,e13ė13=e13−τ11−1e13−ẋ12c≤−τ11−1e132+12e132+12ẋ12c2=−τ11−1+12e13212ẋ12c2,e14ė14=e14−τ12−1e14−ẋ13c≤−τ12−1e142+12e142+12ẋ13c2=−τ12−1+12e14212ẋ13c2.Taking the derivative of Equation (44):
(49)V̇1=s11ṡ11+s12ṡ12+s13ṡ13+e13ė13+e14ė14≤1vm−k11+32k12s112+12vm+1−k13+32k14s122+12−k15+32k16s132+−τ11−1+12+12vme132+−τ12−1+1e142+12ẋ12c2+12ẋ13c2+12k12+12k14+12k16.Since all variables and their derivatives in the system are bounded, there exist nonnegative continuous functionsg1 and g2 as follows:
(50)ẋ12c∗≤g1s11,s12,e12,e13,k11,k12,ẋ13c∗≤g2s11,s12,s13,e12,e13,e14,k11,k12,k13,k14.There are positive constantR1 and compact set B1 which satisfy the following equation:
(51)B1=s11,s12,s13,e12,e13,e14T,V1<R1.Therefore, we have(52)ẋ12c≤M11,ẋ13c≤M12,where M11 and M12 are the maximum values of g1 and g2 on the compact set B1, respectively. Rewrite Equation (49), and we get
(53)V̇1≤1vm−k11+32k12s112+12vm+1−k13+32k14s122+12−k15+32k16s132+−τ11−1+12+12vme132+−τ12−1+1e142+12M112+12M122+12k12+12k14+12k16.LetN be a positive integer. If design parameters satisfy the following equation:
(54)k11−32k12≥1vm+0.5N,k13−32k14≥12vm+1+0.5N,k15−32k16≥12+0.5N,τ11−1≥12+12vm+0.5N,τ12−1≥1+0.5N.We get the following inequality:(55)V̇1≤−NV+E,where
(56)E=12M112+12M122+12k12+12k14+12k16.According to Equation (55), we can get
(57)V1≤NV0−Ee−Nt+EN.Based on the literature proposed by Liu and Liang [18], if N is large enough, it is guaranteed that filtering errors and error surfaces are sufficiently small. Therefore, just make sure that the design parameters k11,k13,k15 are large enough; the design parameters k12,k14,k16 and the time constant of the filter τ11,τ12 are small enough.


## 3.1.2. Guidance Law Design for the Lateral Plane System

The design of the guidance law for the lateral plane system is similar to the pitch plane system. Firstly, a FTDO is designed to estimate the disturbanced2 in the lateral plane system:
(58)ż20=v20+h2t,v20=−λ22L21/3z20−x212/3signz20−x21−μ22z20−x21+z21,ż21=v21,v21=−λ21L21/2z21−v201/2signz21−v20−μ21z21−v20+z22,ż22=v22,v22=−λ20L2signz22−v21−μ20z22−v21,where λ20,λ21,λ22 and μ20,μ21,μ22 are the design parameters of the disturbance observer, z20,z21,z22 are corresponding estimates, and L2 and h2 satisfy the following relations, respectively:
(59)L2≥ḋ2t,h2=−x22vmcosθm+f2−nf4.The head-pursuit guidance law for the lateral plane system is designed as follows:(60)s21=x21,x22c=vmcosθmk21s21+k22sigρ21s21+f2−nf4+z21,ẋ22c∗=τ21−1x22c−x22c∗,x22c∗0=x22c0,s22=x22−x22c∗,x23c=−k23s22−k24sigρ22s22+ẋ22c∗,ẋ23c∗=τ22−1x23c−x23c∗,x23c∗0=x23c0,s23=x23−x23c∗,uz=ωn−2−k25s23−k26sigρ23s23+2ξωnx23+ωn2x22+ẋ23c∗,where s21,s22,s23 are the corresponding error surfaces, k21,k22,k23,k24,k25,k26 are the corresponding design parameters, x22c,x23c,x22c∗,x23c∗ are the corresponding virtual control inputs and filtered virtual control inputs, and τ21,τ22 are the corresponding time constants of the filters. If the design parameters k21,k23,k25 are large enough, the design parameters k22,k24,k26 and the time constant of the filter τ21,τ22 are small enough; it can be ensured that filtering errors and error surfaces are sufficiently small.The stability of the guidance law for the lateral plane system is similar to the pitch plane system; therefore, there is no specific elaboration here.


## 4. Numerical Simulation

In order to verify the effectiveness of the guidance law designed in this paper, the guidance laws based on the power reaching law and exponential approach law are selected for comparisons.The guidance law based on the power reaching law is given as Equation (61), and the guidance law based on the exponential approach law is given as Equation (62):
(61)U1=u11u12=−vmk31sigρ31s11+f1−nf3vmcosθmk32sigρ32s21+f2−nf4,(62)U2=u21u22=−vmk41s11+σ41signs11+f1−nf3vmcosθmk42s21+σ42signs21+f2−nf4.The guidance law proposed in this paper which considers the dynamic characteristics is rewritten as follows:(63)U3=uyuz=ωn−2−k15s13−k16sigρ13s13+2ξωnx13+ωn2x12+ẋ13c∗ωn−2−k25s23−k26sigρ23s23+2ξωnx23+ωn2x22+ẋ23c∗.We consider the situations in which the initial relative range between the missile and the target is about 6000 m, the initial missile coordinate is (4500, 3000, -2600), and the target is (0, 0, 0), the initial missile and target velocities areVm=1500m/s and Vt=2100m/s, respectively, the initial line-of-sight angles are θl0=30° and φl0=30°, the initial missile lead angles are θm0=−20° and φm0=20° and the target θt0=−20° and φt0=20°, the damping coefficient ξ=0.8, and the oscillation frequency ωn=20. These values are all the same in the following numerical simulation cases. According to the different target accelerations, the comparisons of guidance laws are divided into three cases.Case 1.
The target has no maneuver.
The simulation results of the three guidance laws are compared in Figures3–10.
The simulation results show that when the target flies with no maneuver, the three guidance laws have similar results. They can make the error surfaces tend to stable states steadily, and the control input has no obvious chattering. But the convergence speeds of the error surfaces are faster under the guidance lawsU2 and U3, which are determined by the proportion term introduced in these two guidance laws. Therefore, the guidance law designed in this paper has no clear advantage when the target flies with no maneuver.Case 2.
The target has small maneuvers.(a)
The target has constant maneuvers,ayt=azt=9.8m/s2. The simulation results of the three guidance laws are compared in Figures 11–20
(b)
The target maneuvers in cosine form,ayt=azt=9.8cosπt/4m/s2. The simulation results of the three guidance laws are compared in Figures 21–30
The simulation results show that when the target flies with small maneuvers, there are different manifestations in some respect even though the guidance laws can complete interception missions with similar trajectories. When the target flies with small constant maneuvers, the error surfaces have steady-state errors under the guidance lawU1, which can be shown in Figures 13 and 14, and at the moment of collision, the missile and target have direction angles under the guidance law U1, which can be shown in Figures 15 and 16. When the target flies with small cosine maneuvers, the error surfaces fluctuate with the peak of disturbance under guidance law U1, which can be shown in Figures 23 and 24. And at the moment of collision, the missile and target direction angles have results similar to small constant maneuvers, which can be shown in Figures 25 and 26. Under guidance law U2, the acceleration instructions have obvious chattering when the target flies with small constant maneuvers, which can be shown in Figures 27 and 28. The simulation results also show that there are no obvious defects in the interception process under guidance law U3 when the target flies with small maneuvers, and the FTDO designed in this paper can track target maneuvers quickly and accurately.Case 3.
The target has large maneuvers.(a)
The target has large constant maneuvers,ayt=azt=49m/s2. The simulation results of three guidance laws are compared in Figures 31–40
(b)
The target maneuvers in cosine form,ayt=azt=49cosπt/4m/s2. The simulation results of three guidance laws are compared in Figures 41–50
The simulation results show that missiles guided by three guidance laws, respectively, can intercept the targets when the target flies with large maneuvers, but there are significant differences in their performance. When the target flies with large constant maneuvers, the error surfaces have obvious steady-state errors under guidance lawsU1 and U2, which can be shown in Figures 33 and 34, and at the moment of collision, the missile and target have direction angles under guidance laws U1 and U2, which can be shown in Figures 35 and 36. When the target flies with large cosine maneuvers, the error surfaces fluctuate obviously with the peak of disturbance under guidance laws U1 and U2, which can be shown in Figures 43 and 44. Besides, Figures 45 and 46 show that the missile and target direction angles fail to converge to zero at the moment of collision. The acceleration instructions under guidance U2 have significant chattering compared to guidance laws U1 and U2, which can be shown in Figures 47 and 48. At the same time, the simulation results also show that there are no obvious defects in the interception process under the guidance law U3. Therefore, the guidance law U3 has better performances when the target flies with large maneuvers.Figure 3
Relative trajectories.Figure 4
Ranges between the missile and the target.Figure 5
Dynamic error surfacess11.Figure 6
Dynamic error surfacess21.Figure 7
Direction anglesθ.Figure 8
Direction anglesφ.Figure 9
Acceleration instructionuy.Figure 10
Acceleration instructionuz.Figure 11
Relative trajectories.Figure 12
Ranges between the missile and the target.Figure 13
Dynamic error surfacess11.Figure 14
Dynamic error surfacess21.Figure 15
Direction anglesθ.Figure 16
Direction anglesφ.Figure 17
Acceleration instructionuy.Figure 18
Acceleration instructionuz.Figure 19
Target acceleration estimation in the pitch plane.Figure 20
Target acceleration estimation in the lateral plane.Figure 21
Relative trajectories.Figure 22
Ranges between the missile and the target.Figure 23
Dynamic error surfacess11.Figure 24
Dynamic error surfacess21.Figure 25
Direction anglesθ.Figure 26
Direction anglesφ.Figure 27
Acceleration instructionuy.Figure 28
Acceleration instructionuz.Figure 29
Target acceleration estimation in the pitch plane.Figure 30
Target acceleration estimation in the lateral plane.Figure 31
Relative trajectories.Figure 32
Ranges between the missile and the target.Figure 33
Dynamic error surfacess11.Figure 34
Dynamic error surfacess21.Figure 35
Direction anglesθ.Figure 36
Direction anglesφ.Figure 37
Acceleration instructionuy.Figure 38
Acceleration instructionuz.Figure 39
Target acceleration estimation in the pitch plane.Figure 40
Target acceleration estimation in the lateral plane.Figure 41
Relative trajectories.Figure 42
Ranges between the missile and the target.Figure 43
Dynamic error surfacess11.Figure 44
Dynamic error surfacess21.Figure 45
Direction anglesθ.Figure 46
Direction anglesφ.Figure 47
Acceleration instructionuy.Figure 48
Acceleration instructionuz.Figure 49
Target acceleration estimation in the pitch plane.Figure 50
Target acceleration estimation in the lateral plane.It can be found that there are many differences between the three guidance laws in intercepting hypersonic targets by comparing the simulation results under different conditions. The specific differences and reasons are as follows.Under the guidance lawU1, the error surface cannot converge to zero but can only converge to a small neighborhood when the target flies with maneuvers, and this neighborhood expands gradually as the target maneuvers increase. This is because the guidance law U1 adopts the power reaching law which can only ensure that the system converges to a small neighborhood if the system has an uncertain external disturbance, and at the same time, the range of the neighborhood is related to the parameters of the power reaching law and the upper bound of the external disturbance. Since the missile and target direction angles cannot converge to zero until the error plane remains zero, when the target is maneuvering, guidance law U1 cannot meet the head-pursuit guidance conditions as shown in Equation (12), which can be solved by designing a disturbance observer to estimate the target maneuver online. Meanwhile, the convergence speed of guidance law U1 can be solved by adding proportional terms.Under the guidance lawU2, although the error surface has a fast convergence speed, the control input has a significant chattering when the target is a maneuvering flight with acceleration for the cosine. In addition, the error surface cannot converge to zero when the amplitude of the cosine maneuver is large. This is because the guidance law U2 adopts the exponential approach law, in which the coefficient of the sign function determines the chattering and robustness of the system, and this coefficient needs to match the external disturbance. If the coefficient is relatively large, the chattering of the system will increase, while if the coefficient is relatively small, the system will be unstable. When the target maneuver is small and the coefficient of the sign function is appropriate, the error surface can quickly converge to zero. When the amplitude of the cosine maneuver is large and the coefficient of the sign function is relatively small, the system will be unstable. This problem can be solved by adjusting the symbol function coefficients online or estimating the uncertainty disturbance online and compensating for it.


## 5. Conclusion

In view of the interception problems of hypersonic targets, this paper establishes the 3D head-pursuit model. Then, a head-pursuit guidance law based on the backstepping sliding model control theory and FTDO is proposed considering the autopilot dynamic characteristics, and its stability is proved using the Lyapunov stability theorem. Finally, the guidance law proposed in this paper is compared with other guidance laws in different scenes.We can get the following conclusions by comparing the simulation results in three cases: the guidance law proposed in this paper has better performance in the interception of hypersonic targets, especially in the face of targets with high maneuverability. This is because the guidance law proposed in this paper not only considers the convergence speed and chattering requirements of the error surface comprehensively but also combines the proportional term and the power term to accelerate the convergence speed of the error surface and weaken the buffeting. In addition, considering the characteristics of the target’s large maneuver capability, the FTDO is introduced to estimate and compensate the target’s maneuver online so that the interceptor missile can complete the interception according to the head-pursuit guidance conditions strictly even if the target has a large maneuver. Besides, the guidance law designed in this paper takes into account the dynamic characteristics of autopilot, which is closer to the actual guidance environment. Therefore, the effectiveness and superiorities of this head-pursuit guidance law are verified.



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*Source: 8214042-2019-12-03.xml*