integrated guidance and control of interceptor missile

Research ArticleIntegrated Guidance and Control of Interceptor Missile Based onAsymmetric Barrier Lyapunov Function

Xiang Liu and Xiaogeng Liang

School of Automation, Northwestern Polytechnical University, Xi’an 710072, China

Correspondence should be addressed to Xiang Liu; [email protected]

Received 29 July 2018; Accepted 13 February 2019; Published 28 April 2019

Academic Editor: Zhiguang Song

Copyright © 2019 Xiang Liu and Xiaogeng Liang. This is an open access article distributed under the Creative CommonsAttribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original workis properly cited.

In this study, a novel integrated guidance and control (IGC) algorithm based on an IGC method and the asymmetric barrierLyapunov function is designed; this algorithm is designed for the interceptor missile which uses a direct-force/aerodynamic-force control scheme. First, by considering the coupling between the pitch and the yaw channels of the interceptor missile, anIGC model of these channels is established, and a time-varying gain extended state observer (TVGESO) is designed to estimateunknown interferences in the model. Second, by considering the system output constraint problem, an asymmetric barrierLyapunov function and a dynamic surface sliding-mode control method are employed to design the control law of the pitchand yaw channels to obtain the desired control moments. Finally, in light of redundancy in such actuators as aerodynamicrudders and jet devices, a dynamic control allocation algorithm is designed to assign the desired control moments to theactuators. Moreover, the results of simulations show that the IGC algorithm based on the asymmetric barrier Lyapunovfunction for the interceptor missile allows the outputs to meet the constraints and improves the stability of the control systemof the interceptor missile.

1. Introduction

The interceptor missile plays an important role in modernanti-missile systems. Given the rapid development of hyper-sonic aircraft, demands on accurate guidance and control ofthe interceptor missile has become increasingly stringent.To cope with high-speed aircraft with a strong penetrationcapability, the interceptor missile generally adopts a direct-force/aerodynamic-force control scheme that can acceleratethe speed of command response. The guidance and controlsystem of the interceptor missile is a highly dynamic andmultivariate system of strong coupling, rapid temporalchange, and uncertainty, thereby making it critical to theexamination of the guidance and control of interceptor mis-sile at high speed with strict control requirements.

Integrated guidance and control (IGC) refers to using therelative interceptor–target motion information and thedynamic information of the interceptor missile to generatea control force that drives it to strike the target. IGC allowsfor a rational allocation of interceptor missile control

capability to not only maintain the interceptor missile’s flightattitude but also improve the precision of its guidance [1, 2].The authors in [3, 4] explored the IGC control law by usinghigh-order sliding-mode control methods and backsteppingcontrol algorithms. On the basis of backstepping controlalgorithms, the authors in [5, 6] considered saturation factorsand introduced dynamic models of actuators models todesign algorithms for longitudinal and anti-saturation IGCfor aircraft. An adaptive sliding-mode control method wasused to implement the design of an IGC system on the pitchplane to configure missiles to attack ground targets [7], butthe problem of target mobility was not considered. Basedon a backstepping method, the missile control loop wasregarded as a second-order element in the study [8], whichintroduced a first-order integral filter to estimate the deriva-tive of the input to the virtual control and design the controllaw. Based on a three-dimensional (3D) ICG model, a robustadaptive backstepping method was implemented to designan IGC algorithm for a missile [9]. Methods of IGC designhave been widely used in guidance and control systems of

HindawiInternational Journal of Aerospace EngineeringVolume 2019, Article ID 8531584, 17 pageshttps://doi.org/10.1155/2019/8531584

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https://doi.org/10.1155/2019/8531584

aircraft [10], missiles [11–13], and unmanned aerial vehicles[14, 15]. In recent years, researchers have combined thismethod with modern control theories, such as dynamic sur-face control [16], optimal control [17], and predictive control[18], to generate methods of IGC design for aircrafts. How-ever, most prevalent IGC design methods do not considerthe coupling relationship between the pitch and the yawchannels. Moreover, to improve the stability of the guidanceprocess, constraints concerning the angular velocities of theline of sight during the guidance of the interceptor missileshould be considered.

Nonlinearity commonly exists in the IGC system of theinterceptor missile. In recent years, with practical engineer-ing problems’ increasing demand on the control perfor-mance, there has been considerable progress in thedevelopment of nonlinear control theory, especially in adap-tive control [19], neural network control [20], fuzzy control[21], etc. All of these have laid a solid foundation for thein-depth research of nonlinear control theory. However, inactual engineering applications, the use of nonlinear systemsis always subject to input, output, and state constraints,among others, and violation of these constraints can resultin the control system’s downgraded performance. Therefore,it has now become an important research direction, whenconstructing the control system to consider the effect of theseconstraints and to properly handle them in the controllerdesign process. The authors in [22] proposed an adaptiveneural network constrained control algorithm for single-input/single-output nonlinear stochastic switching systems;this algorithm constructed traditional Lyapunov function tohandle constraint control, which achieved good results. Asbarrier Lyapunov function does not require an exact solutionof the system, the constrained control method based on bar-rier Lyapunov function has been widely used in state con-straint and output constraint problems in recent years.Barrier Lyapunov function is a special type of continuousfunction, unlike traditional Lyapunov function which is radi-ally unbounded, in barrier Lyapunov function, when theparameters approach the limit value, the function value willtend to infinity to ensure that the control system satisfiesthe constraints [23]. Barrier Lyapunov function can be uti-lized to satisfy the constraints of both symmetric and asym-metric constraint controls, even when the constraint is atime-varying asymmetric one. According to the authors in[24], barrier Lyapunov function has been used to solve con-straint problems in a hybrid PDE-ODE system that describesa nonuniform gantry crane system. The authors in [25] pro-posed an adaptive fuzzy neural network control methodusing impedance learning for a constrained robot systembased on barrier Lyapunov function. According to theauthors in [26, 27], barrier Lyapunov functions have beenused to solve constraint problems in nonlinear and uncertainsystems and to expand the definitions of the constraints.Methods based on these functions can effectively solve prob-lems of symmetrically and asymmetrically constrained con-trol. The authors in [28, 29] have expanded output controlconstraints to include time-varying outputs while relaxingthe limitations on the initial values of control systems. Bycombining barrier Lyapunov functions with dynamic surface

control technologies, some studies [30, 31] have proposedbarrier Lyapunov function-based methods suitable for con-strained dynamic surface control to solve the computationalinflation problem caused by backstepping control.Researchers subsequently applied this method to brushlessDC motors [32], plane braking systems [33], and hypersonicaircraft [34, 35] to achieve satisfactory results in terms ofconstrained control. However, few studies have investigatedthe application of this method to interceptor missile control.Design methods based on barrier Lyapunov functions areadvantageous because they can solve the output constraintproblem of interceptor missile guidance control systemsand improve their stability.

In view of the above analysis, to aim at the interceptormissile which uses a direct-force/aerodynamic-force controlscheme, and to consider the coupling relationship betweenthe pitch and the yaw channels as well as the constraints onthe system’s output, this study proposes an IGC algorithmbased on the asymmetric barrier Lyapunov function. First, atime-varying gain extended state observer (TVGESO) isdesigned to estimate interferences in the system. Second, anasymmetric barrier Lyapunov function and a dynamic sur-face sliding-mode control method, respectively, are used todesign control laws for the interceptor missile to obtain thedesired moments. Finally, a dynamic control allocation algo-rithm is designed to allocate the desired control moments.The results of simulations show that the proposed algorithmenables the outputs to meet the constraints and improves thestability of the interceptor missile control systems.

2. IGC Model of Interceptor Missile

The relationship of relative motion between the interceptormissile and its target in 3D space is shown in Figure 1.

In the figure, Oxyz refers to an inertial coordinate systemand Ox4y4z4 refers to a line-of-sight coordinate system,respectively; M and T refer to the interceptor missile andthe target, respectively; qε and qβ are the vertical and hori-zontal angles of the line of sight with the interceptor missileand the target, respectively, and r denotes the relative dis-tance between the interceptor missile and the target. Themodel of the relative motion of the interceptor missile andtarget is as follows:

rqε = −2rqε − rq2β sin qε cos qε − atε − am4ε,

r = rq2ε + rq2β cos2qε + atr − am4r ,−rqβ cos qε = 2rqβ cos qε − 2rqεqβ sin qε + atβ − am4β,

1

where qε and qβ denote the vertical and horizontal angularvelocities of the line of sight with the interceptor missileand the target, respectively; am4ε and am4β denote the longitu-dinal and lateral accelerations of the interceptor missile,respectively; and atε and atβ denote the longitudinal and lat-eral accelerations of the target, respectively.

2 International Journal of Aerospace Engineering

The interceptor missile uses a direct-force/aerodynamic-force control scheme. Assuming that the direct force isadjustable and continuous, the angles of deflection of therudder equivalent to the direct force in the pitch and theyaw channels are, respectively, defined as

δsy =Fy

Fs max,

δsz =Fz

Fs max,

2

where Fz and Fy denote the thrust generated by the jetdevice, and Fs max is the maximum steady-state thrust ofthe jet devices.

In light of the coupling relationship between the pitchand yaw channels of the interceptor missile, its dynamicmodel is expressed as follows:

α = ωz + ωy tan β sin α −qSCα

yα cos αmVm cos β + dα,

β = ωy cos α +qSCα

yα sin α sin β

mVm+ qSCβ

zβ cos βmVm

+ dβ,

ωz =qSLmα

zα

Jz+ qSLmωz

z ωz

Jz+ qSLmδz

z δzJz

+1 + Kz FyLm

Jz+ dωz

,

ωy =qSLmβ

yβ

Jy+qSLm

ωyy ωy

Jy+qSLm

δyy δy

Jy+

1 + Ky FzLmJy

+ dωy,

3

am3ε =qSCα

yα

m,

am3β =qSCβ

zβ

m,

4

where S is the reference area of the interceptor missile; q is thedynamic pressure; Vm is the speed of the interceptor missile;L is its reference length; Lm is the average distance between

the jet device and its center of mass; m is the mass of theinterceptor missile; α and β denote the attack angle and thesideslip angle, respectively; ωz and ωy denote the angularvelocities of the pitch and the yaw, respectively; δz and δydenote the angles of deflection of the aerodynamic rudder;Kz and Ky denote the amplification factors of moment (usedto describe the effect of the mutual interference between lat-eral jets and incoming flow on the aerodynamic moment ofthe interceptor missile); dα, dωz

, dβ, and dωyrefer to the dis-

turbances and uncertain interferences at each link of the sys-tem; Jz and Jy refer to the moments of inertia; Cα

y , Cβz , m

αz ,

mωzz , mδz

z , mβy , m

ωyy , and m

δyy refer to the relevant aerody-

namic forces and coefficients of moment; and am3ε andam3β refer to the vertical and lateral overloads of the inter-ceptor missile, respectively.

Assuming that the angle of the line of sight of the inter-ceptor missile in the terminal guidance stage changes slightlyand that the angle of line of sight and direction of velocity ofthe interceptor missile are relatively small, let am3ε = am4ε andam3β = am4β. According to Equations (1)–(4), by definingxz1 = qε, xz2 = qε, xz3 = α, xz4 = ωz , xy1 = qβ, xy2 = qβ, xy3 = β,and xy4 = ωy, one can have the nonlinear IGC model in thepitch channel for the interceptor missile:

xz2 = f z2xz2 + gz2xz3 − x2y2 sin xz1 cos xz1 +atεr,

xz3 = f z3xz3 + xz4 + dz3,xz4 = f z4xz3 + f z5xz4 + gz3uz + dz4,yz = xz2,

5

where f z2 = − 2r/r , f z3 = − qSCαy /mVm , f z4 = qSLmα

z /Jz ,f z5 = qSLmωz

z /Jz , gz2 = − qSCαy /mr , gz3 = 1/Jz , dz3 = ωy tan

β sin α + dα, dz4 = dωz, and uz = qSLmδz

z δz + 1 + Kz FyLm,with uz representing the moment generated jointly by boththe aerodynamic rudders and the jet devices in the pitchchannels.

Similarly, the nonlinear IGC model for the interceptormissile in its yaw channel is as follows:

xy2 = f y2xy2 + gy2xy3 + 2xz2xy2 tan xz1 −atβ

r cos xz1,

xy3 = f y3xy3 + xy4 + dy3,xy4 = f y4xy3 + f y5xy4 + gy3uy + dy4,yy = xy2,

6

where f y2 = − 2r/r , f y3 = qSCβy /mVm, f y4 = qSLmβ

y /Jy, f y5 =qSLm

ωyy /Jy, gy2 = qSCβ

y /mr cos xz1, gy3 = 1/Jy, dy3 = qSCαyα

sin α sin β/mVm + dβ, dy4 = dωy, and uy = qSLm

δyy δy + 1 +

Ky FzLm, with uy representing the moment generated jointly

q�휀O(M)

yy4

x4

x

r

x

T

q�훽

z

z4

Figure 1: The relative relationship between the interceptor missileand its target in 3D space.

3International Journal of Aerospace Engineering

by both the aerodynamic rudders and the jet devices in theyaw channel.

Assumption 1. The unknown interferences dz3, dz4, dy3, anddy4 in the IGC models in Equations (5) and (6) of the inter-ceptor missile are continuously differentiable, and the deriv-atives are bounded.

3. Design of TVGESO

A TVGESO can estimate nonlinear uncertainties in themodel of the system and feed the estimates back into the con-trol system for compensation. To eliminate the effects ofunknown uncertain interferences atε, atβ, dz3, dz4, dy3, anddy4 in the system models in Equations (5) and (6) on the con-trol system of the interceptor missile, a TVGESO is designedto estimate these interferences.

By defining vr = r, vε = rqε, and vβ = rqβ cos qε, oneobtains

vε = rqε − 2rqε − rq2β sin qε cos qε + atε−am4ε

= −vrvεr

−v2β tan qε

r− am4ε + atε,

7

vβ = rqβ cos qε + rqβ cos qε−rqεqβ sin qε

= −vrvβr

+vεvβ tan qε

r+ am4β − atβ,

8

Considering the system in Equation (5) and Equation (7),one can design the following TVGESO to estimate accelera-tion atε of the target:

ez21 = vz21 − vε, ez22 = vz22 − atε,vz21 = vz22 − λz21ez21 + gz2,vz22 = −λz22ez21,

9

where gz2 = − vrvε/r − v2β tan qε/r − qSCαy /m α; vz21 and

vz22, respectively, are the estimated values of vε and atε; ez21and ez22 are the estimated errors; and λz21 and λz22 are thetime-varying gain coefficients designed for the state observer.They are defined as λz21 = 2L t and λz22 = L2 t , respec-tively. Function L t is defined as

L t =γ ez21 , ez21 ≠ 0,0, ez21 = 0,L 0 = λz0,

10

where γ is the adaptive coefficient and is greater than zero. Asindicated in the literature [36], appropriate values of the coef-ficient can ensure that the error system of the TVGESO isstable for a limited time.

Similarly, by estimating interferences dz3 and dz4 of theangle-of-attack loop and the pitch angular velocity loop,respectively, in Equation (5), one can obtain the following:

ez31 = vz31 − xz3, ez32 = vz32 − dz3,vz31 = vz32 − λz31ez31 + gz3,vz32 = −λz32ez31,

11

ez41 = vz41 − xz4, ez42 = vz42 − dz4,vz41 = vz42 − λz41ez41 + gz4,vz42 = −λz42ez41,

12

where gz3 = f z3xz3 + xz4 and gz4 = f z4xz3 + f z5xz4 + gz3uz , theinterferences dz3 and dz4 are estimated as vz32 and vz42,respectively, and the estimation errors for them are denotedby ez32 and ez42, respectively.

According to Equations (9)–(12), the interferences atβ,dy3, and dy4 in the system in Equation (6) have estimatedvalues of vy22, vy32, and vy42, respectively, with the estimationerrors of ey22, ey32, and ey42, respectively.

4. Design of the Dynamic Surface Sliding-ModeControl Law Based on Asymmetric BarrierLyapunov Function

Let us define ζ as an open region containing the origin andthe barrier Lyapunov function V x as a scalar functiondefined in ζ for the system x = f x, t . It also has the follow-ing characteristics: (1) smooth and positive definite, (2) has afirst-order continuous partial derivative at each point in ζ, (3)tends to infinity when x approaches the edge of ζ, and (4) sat-isfies the expression V x t ≤ b for ∀t ≥ 0 if x 0 ∈ ζ, whereb > 0.

Assumption 2. For any t > 0, there exist constants Kci andKci i = 0, 1, 2 that satisfy kc1 ≥ Kc0 and kc1 ≤ Kc0, with their

derivatives satisfying k ici t ≥ Kci, k

ici t ≤ Kci, i = 1, 2,

and ∀t ≥ 0.

Assumption 3. For any kc1 和 kc1, there exist functions Y 0 tand Y0 t as well as positive constants Y1 and Y2 satisfyingY 0 t > kc1 t and kc1 t < Y0 t , such that they make thesystem track command x2d = diag xz2d , xy2d , and its time

derivative satisfies kc1 t ≤ x2d t ≤ kc1 t and x2d t ≤ Y1as well as x2d t ≤ Y2 for ∀t ≥ 0. There exists a continuousset satisfying Mx2d

= x2d ∈ℝ x2d + x2d + x2d ≤ ϕx2d ⊂ℝ3.

Given that the IGC model of interceptor missile is a mis-matching and uncertain system, to enable the guidance andcontrol system to accurately pursue the target, and not violat-ing the constraints on the control system, the control lawallows using a dynamic surface sliding mode algorithm basedon the asymmetric barrier Lyapunov function. It can enablethe control system to pursue the target highly precisely,meanwhile ensuring that the closed-loop system is consistentand ultimately bounded, and the tracking error converges toa small set.


4.1. Design of Control Law for Pitch Channel. For the sys-tem in Equation (5), define xz2d as the system’s track com-mand signal.

(1) Define the first dynamic error surface:

sz2 = xz2 − xz2d 13

Taking the derivative of sz2, one obtains the errordynamic equation:

sz2 = f z2xz2 − x2y2 sin xz1 cos xz1 +atεr

− xz2d + gz2xz3 14

Because the backstepping method does not have a perfectsolution to the expansion of items and the problems causedby the expansion of items in the derivation process of the vir-tual control, this shortcoming is particularly prominent inthe higher-order system. By using the dynamic surface con-trol method and using the first-order filter to calculate thederivative of the virtual control, the expansion of the differ-ential items can be eliminated and the controller and param-eters can be designed simply [37]. Introduce virtual controlvariables x∗z3 and x∗z3. To avoid complicated calculations ofthe expansion of the number of items during the derivationof the virtual control variables, the virtual control variablex∗z3 before filtering is passed through a first-order low-passfilter to become the virtual control variable x∗z3:

τz3x∗z3 + x∗z3 = x∗z3, x∗z3 0 = x∗z3 0 15

In the above expression, τz3 is the filter’s time constant,τz3 > 0, and the filtering error is defined as yz3 = x∗z3 − x∗z3.

Considering boundary layer errors of the dynamic sur-face, one can construct the following asymmetric barrier Lya-punov function:

Vz1 =1 − p1 sz2

2 log k2a11k2a11 − s2z2

+ p1 sz22 log k2b11

k2b11 − s2z2+ 12 y

2z3,

16

where

p1 sz2 =1 0 < sz2,0 sz2 ≤ 0

17

ka11 = xz2d − kc1, kb11 = kc1 − xz2d , log • represents a naturallogarithm, and kc1 and kc1 represent output constraints.

Given the independence characteristic of the output con-straints kc1 and kc1, the tracking error constraints ka11 andkb11 can be designed independently. When constraints ka11and kb11 are constant, and ka11 ≠ kb11, the output constrainedcontrol can be extended to a static asymmetric constraint,whereas the output constraint becomes a symmetric con-straint when ka11 = kb11. This means that the initial output

can be changed depending on the setting of the constraint.It is evident that the asymmetric barrier Lyapunov functionrelaxes the constraint on the initial condition of the output.

As shown by Equation (16), the expression Vz1 ≥ 0 sim-plifies to Vz1 = 0 if and only if sz2 = 0 and x∗z3 0 = x∗z3 0simultaneously. Therefore, Vz1 is a positive-definite functionin the range −ka11 < sz2 < kb11. Moreover, given lim

sz2→0+dVz1/

dsz2 = limsz2→0−

dVz1/dsz2 = 0, Vz1 is a piecewise continuously

differentiable function in the ranges sz2 ∈ −ka11, 0 and sz2∈ 0, kb11 . Therefore, Vz1 is a valid Lyapunov function thatcan ensure that the system’s output error is constrained inthe ranges sz2 ∈ −ka11, 0 and sz2 ∈ 0, kb11 .

Substituting the estimated values of TVGESO in Equation(9), one can design the virtual control variable for the firstdynamic surface as shown in Equation (18):

x∗z3 = −g−1z2 f z2xz2 − x2y2 sin xz1 cos xz1 +vz22r

− xz2d

− kz2 + kz2 sz2,18

where kz2 > 0, gz2 > 0, and r > 0, and the control coefficientkz2 is defined as

kz2 = 1 − p1 k2a11 + p1k2b11 19

According to Equation (15), the derivative of the virtualcontrol variable for the error surface after filtering is

x∗z3 = −τ−1z3 x∗z3 − x∗z3 20

(2) Define the second dynamic error surface as

sz3 = xz3 − x∗z3 21

By taking the derivation of sz3, one can obtain the equa-tion for the dynamic error as follows:

sz3 = xz3 − x∗z3 = f z3xz3 + xz4 + dz3 − x∗z3 22

Introduce two virtual control variables x∗z4 and x∗z4. Bypassing the virtual control variable x∗z4 through a first-orderlow-pass filter, one obtains the virtual control variable x∗z4:

τz4x∗z4 + x∗z4 = x∗z4, x∗z4 0 = x∗z4 0 23

In Equation (23), τz4 denotes the filter time constant,τz4 > 0, and the filtering error is defined as yz4 = x∗z4 − x∗z4.

Construct the following Lyapunov function:

Vz2 =12 s

2z3 +

12 y

2z4 24


By substituting the estimated values of TVGESO intoEquation (11), one can design the virtual control variablefor the second dynamic surface as shown in Equation (25):

x∗z4 = −f z3xz3 − vz32 + x∗z3 − kz3sz3 − μgz2sz2, 25

where kz3 > 0 and μ = 1 − p1/k2a11 − s2z2 + p1/k2b11 − s2z2 .According to Equation (23), the derivative of the virtual

control variable for the error surface after filtering is

x∗z4 = −τ−1z4 x∗z4 − x∗z4 26

(3) Define the third dynamic error surface:

sz4 = xz4 − x∗z4 27

By taking the derivative of sz4, one can obtain the equa-tion of error dynamic as follows:

sz4 = xz4 − x∗z4 = f z4xz3 + f z5x4 + gz3uz + dz4 − x∗z4 28

Construct the following Lyapunov function:

Vz3 =12 s

2z4 29

By substituting vz42 with the estimated values ofTVGESO in Equation (12), one can design the dynamicsliding-mode control law of the interceptor missile as shownin Equation (30):

uz = g−1z3 −f z4xz3 − f z5xz4 − vz42

+ x∗z4 − kz4sz4 − kz5 sz4∂z sgn sz4 ,

30

where kz4 > 0, kz5 > 0, 0 < ∂z < 1, and gz3 > 0.

4.2. Stability Analysis of Control Law for Pitch Channel.

V1 S1 ⟶∞,γ1 ϖ ≤U ϖ ≤ γ2 ϖ ,

31

where γ1 and γ2 are K∞ functions. Suppose V η ≔V1 S1 +U ϖ and S1 0 ∈ S1. If the inequality,

V = ∂V∂η

h ≤ −cV + υ, 32

is valid in the domain θ ∈N , and c and υ are positive con-stants, then S1 t is still in the set for ∀t ∈ 0,∞ .

Lemma 1 (see [20]). For any two functions ka t and kb t ,suppose S1 ≔ S1 ∈ℝ −ka t < S1 < kb t ⊂ℝ and N ≔ℝl

× S1 ⊂ℝl+1 to be open sets. For system η = h t, η with η≔

ϖ, S1 T ∈N as state, function h ℝ+ ×N →ℝl+1 is piecewisecontinuous with respect to t, and S1 satisfies the Lipschitz con-dition in ℝ+ ×N . If there exist two functions U ℝl →ℝ+and V1 S1 →ℝ+ that are continuous and positively definitein their domains, respectively, the following two expressionsare valid when S1 → −ka t or S1 → kb t :

Theorem 1. For the closed-loop system Equation (5), if thevirtual control variables and the control law satisfy Equations(18), (25), and (30), then for any set M0

z to which the initialcondition yz 0 belongs, there always exists a sufficiently largeset Mz ≔ kc1 t <−ka11 + xz2d t ≤ yz t ≤ kb11 + xz2d t < kc1t that can prevent the system’s output constraints frombeing violated. Moreover, by selecting appropriate designparameters, it is possible to bound all closed-loop signals ofthe closed-loop system Equation (5) and make the output errorconverge in the neighborhood of the origin.

Proof 1. According to Equations (16), (24), and (29), one canconstruct a Lyapunov function that is a combination ofasymmetric barrier Lyapunov functions and traditional Lya-punov functions as follows:

Vz =Vz1 +Vz2 + Vz3 =1 − p1 sz2

2 log k2a11k2a11 − s2z2

+ p1 sz22 log k2b11

k2b11 − s2z2+ 12 y

2z3 +

12 s

2z3 +

12 y

2z4 +

12 s

2z4

33

By taking derivatives on both sides of the Lyapunov func-tion in Equation (33), one obtains Equation (34):

Vz = μsz2sz2 + yz3yz3 + sz3sz3 + yz4yz4 + sz4sz4, 34

where μ = 1 − p1/k2a11 − s2z2 + p1/k2b11 − s2z2 .Define the estimated error of the TVGESO system to

satisfy Equation (35):

∣ez22∣ <Nz2,∣ez32∣ <Nz3,∣ez42∣ <Nz4,

35

where Nz2, Nz3, and Nz4 are positive constants.It can be seen that the filtering errors are

yz3 = x∗z3 − x∗z3,yz4 = x∗z4 − x∗z4

36

By taking derivatives of yz3 and yz4, one obtains thedynamic filtering errors:

yz3 = −τ−1z3 yz3 − x∗z3,yz4 = −τ−1z4 yz4 − x∗z4

37


From Equations (13)–(30) and (36), one has

xz2 = sz2 + xz2d ,xz3 = sz3 + x∗z3 = sz3 + yz3 + x∗z3,xz4 = sz4 + x∗z4 = sz4 + yz4 + x∗z4

38

From Equations (13)–(28) and (36)–(38), one has

sz2 = f z2xz2 − x2y2 sin xz1 cos xz1 +atεr

− xz2d

+ gz2 sz3 + yz3 + x∗z3= gz2sz3 + gz2yz3 − gz2 kz2 + kz2 sz2 − ez22,

39

where ez22 = vz22 − atε /r, and it is assumed that ez22 <Nz2,Nz2 is a positive constant, and μNz2 > 0.

sz3 = f z3xz3 + sz4 + yz4 + x∗z4 + dz3 − x∗z3= sz4 + yz4 − kz3sz3 − ez31 − μgz2sz2,

40

sz4 = f z4xz3 + f z5xz4 + gz3uz + dz4−x∗z4

= −kz4sz4 − kz5 sz4∂z sgn sz4 − ez41

41

According to Young’s inequality and Equations(39)–(41), one has

sz2sz2 = sz2 gz2sz3 + gz2yz3 − gz2 kz2 + kz2 sz2 − ez22

≤12 + 1

2gz2 − gz2kz2 − gz2kz2 s2z2 +12gz2y

2z3

+ 12N

2z2 + gz2sz2sz3,

42

sz3sz3 = sz3 sz4 + yz4 − kz3sz3 − ez31 − μgz2sz2 ≤32 − kz3 s2z3

+ 12 s

2z4 +

12 y

2z4 +

12N

2z3 − μgz2sz2sz3,

43

sz4sz4 = sz4 −kz4sz4 − kz5 sz4∂z sgn sz4 − ez41

≤12 − kz4 +

32 kz5 s2z4 +

12N

2z4 +

12 kz5

44

It is clear that the variables in the system model and theirderivatives are all bounded. If there exist continuous func-tions zz3 and zz4 that satisfy zz3 > 0 and zz4 > 0, the variablesx∗z3 and x∗z4 satisfy

x∗z3 ≤ zz3,x∗z4 ≤ zz4

45

According to Young’s inequality and Equations(36)–(37) and (45), one has

yz3yz3 = yz3 −τ−1z3 yz3−x∗z3 ≤ −τ−1z3 y

2z3 +

12 y2z3 + z2z3 ,

46

yz4yz4 = yz4 −τ−1z4 yz4−x∗z4 ≤ −τ−1z4 y

2z4 +

12 y2z4 + z2z4

47

By substituting Equations (42)–(47) into Equation (34)and arranging the terms, one has

Vz = μsz2sz2 + yz3yz3 + sz3sz3 + yz4yz4 + sz4sz4

≤ −μ gz2kz2 + gz2kz2 −12gz2 −

12 s2z2

− kz3 −32 −

12 μ s2z3 − kz4 − 1 − 3

2 kz5 s2z4

− τ−1z3 −12 −

12 μgz2 y2z3 − τ−1z4 − 1 y2z4 +

12 μN

2z2

+ 12N

2z3 +

12N

2z4 +

12 z

2z3 +

12 z

2z4 +

12 kz5

48

For convenience of description, define the followingparameters:

Kz2 = gz2kz2 + gz2kz2 −12gz2 −

12 ,

Kz3 = kz3 −32 −

12 μ,

Kz4 = kz4 − 1 − 32 kz5,

σz3 = τ−1z3 −12 −

12 μgz2,

σz4 = τ−1z4 − 1,

A = 12 μN

2z2 +

12N

2z3 +

12N

2z4 +

12 z

2z3 +

12 z

2z4 +

12 kz5

49

Then, Equation (48) can be rewritten as

Vz ≤ −μKz2s2z2 − Kz3s

2z3 − Kz4s

24 − σz3y

2z3 − σz4y

2z4 + A 50

In the set −ka11 t < sz2 < kb11 t , one has

−1 − p1

k2a11 − s2z2+ p1k2b11 − s2z2

s2z2

≤ − 1 − p1 log k2a11k2a11 − s2z2

+ p1 logk2b11

k2b11 − s2z2

51


Then, Equation (50) can be further rewritten as

Vz ≤ − 1 − p1 log k2a11k2a11 − s2z2

+ p1 logk2b11

k2b11 − s2z2Kz2

− Kz3s2z3 − Kz4s

2z4 − σz3y

2z3 − σz4y

2z4 + A

52

Define positive-definite matrix Q as

Q =Kz2 0 00 Kz3 00 0 Kz4

53

By selecting κ =min 2λmin Q , 2σz3, 2σz4 , where κ is apositive constant and λmin Q denotes the minimum eigen-value of matrix Q, one has

Vz ≤ −κVz + A 54

Define two sets M0z ≔ yz ∈ℝ kc0 ≤ yz 0 ≤ kc0 ⊂ℝ

and Mz ≔ yz ∈ℝ kc1 ≤ yz t ≤ kc1 ⊂ℝ that satisfy kc1 t

> kc0 + Y0 + xz2d 0 ⊂ℝ and kc1 t < kc0 + Y 0 + xz2d 0⊂ℝ. From sz2 0 = yz 0 − xz2d 0 , one has

−ka11 ≤ sz2 0 ≤ kb11 55

Therefore, set Mz is an invariant set. According toLemma 1, it can be seen that given Vz≤−κVz + A, the follow-ing expression is valid for ∀t ≥ 0:

−ka11 ≤ sz2 t ≤ kb11 56

According to Equations (13) and (56), one has the follow-ing expression for ∀t ≥ 0:

kc1 t < −ka11 + xz2d t ≤ yz t ≤ kb11 + xz2d t < kc1 t

57

Therefore, it can be concluded that for any initial com-pact set M0

z defined by yz 0 ∈M0z , there is always a suffi-

ciently large compact set Mz to make yz ∈Mz for ∀t ≥ 0.Define parameters according to Equation (48) to satisfy

the following rules:

Multiply both sides of Equation (54) by eκt and arrangethe terms to obtain the following expression:

Vz t ≤κVz 0 − A e−κt + A

κ59

Therefore, it is clear that by designing parameters kz2, kz3,kz4, and kz5 and parameters τz3 and τz4, it is possible toensure that all closed-loop signals of the system are bounded.Opting to increase kz2, kz3, kz4, and kz5 and decrease τz3 andτz4 can ensure that κ is sufficiently large to make the filteringerrors and the error surface small enough to control accu-racy. Q.E.D.

Remark 1. Theoretically, larger values of the designedparameters kz2, kz3, kz4, and kz5; smaller ones of τz3 andτz4; the final boundary of the resulting error surfaces sz2,sz3, and sz4; and filtering errors yz3 and yz4 indicate an

increase in the precision of control. However, in practice,very large values of kz2, kz3, kz4, and kz5, and very smallones of τz3 and τz4 can easily lead to input saturation ininterceptor missile control systems, which triggers the sat-uration nonlinearity of the system such that the requiredoverloads of the interceptor missile are beyond the avail-able capacity, resulting in a reduction in the system’s con-trol performance. Moreover, given the physical limitationsof the low-pass filter, τz3 and τz4 should not be chosen tobe arbitrarily small. Therefore, the parameters of the con-trol algorithm should be properly selected in light of prac-tical considerations.

4.3. Design of Control Law for Yaw Channel. Based on thedesign of the control law for the pitch channel, one can sub-stitute vy22, vy32, and vy42—the estimated interferences of theTVGESO—into the system of Equation (6) and define xy2d asthe system’s track command sign for the following form ofcontrol law for the yaw channel:

kz2 ≥12g

−1z2 +

12 − kz2 +

12 κ, kz3 ≥

32 + 1

2 μ +12 κ, kz4 ≥

32 kz5 + 1 + 1

2 κ,

τ−1z3 ≥12 + 1

2μgz2 +12 κ, τ−1z4 ≥ 1 + 1

2 κ58


where ky2 = 1 − p2 k2a12 + p2k2b12,

p2 sy2 =1 0 < sy2,0 sy2 ≤ 0,

61

ka12 = xy2d − kc1, kb12 = kc1 − xy2d , μ = 1 − p2/k2a12 − s2y2 +p2/k2b12 − s2y2 , ky2 > 0, ky3 > 0, ky4 > 0, ky5 > 0, 0 < ∂y < 1,gy2 > 0, gy3 > 0, and r > 0. The parameters sy2, sy3, and sy4are error surfaces; τy3 and τy4 are filter time constants,τy3 > 0 and τy4 > 0; x∗y3 and x∗y4 are virtual control variablesof the system Equation (6); and x∗y3 and x∗y4, respectively,are virtual control variables after filtering.

According to the stability analysis method in Equations(33)–(59), it can be proven that the closed-loop system ofEquation (6) is stable and the system’s control precisioncan be achieved by designing appropriate parameters.

5. Dynamic Control Allocation Algorithm

Based on Equations (30) and (60), one can obtain the controlmoments jointly generated by the aerodynamic rudders andthe jet devices using control inputs uz and uy as the desiredcontrol moments. Therefore, it is necessary to use a dynamiccontrol allocation technique to distribute the desired controlmoments to the actuators.

Define virtual control moment u as

u = Bδ, 62

where u = uz , uy T , δ = δz , δsy , δy, δsz T ,

B =Bz1 Bz2 0 00 0 By1 By2

T

, 63

Bz1 = qSLmδzz , Bz2 = 1 + Kz Fs maxLm, By1 = qSLm

δyy , and

By2 = 1 + Ky Fs maxLm.

Given that actuators are subject to physical constraints,such as structural and load-related constraints, the range ofdeflection and speed are limited. Define the control momentas in the feasible range δmin t ≤ δ t ≤ δmax t . To achievestable control performance, define the rate of change of thecontrol moment to satisfy δmin t ≤ δ t ≤ δmax t , whereδmin and δmax are the minimum and maximum positionalconstraints of the actuators, respectively, with δmin and δmaxbeing, respectively, the minimum and maximum speed con-straints. Using T to denote sampling time, one can rewritethe feasible range of the actuators as

δ t ≤ δ t ≤ δ t , 64

where δ t =max δmin, δ t − T + δminT and δ t =minδmax, δ t − T + δmaxT .To achieve stable and smooth actuator trajectories to

suppress the impacts of noise and interference on the con-troller, a dynamic control allocation algorithm is designedto solve the allocation problem between the aerodynamicrudders and the jet devices.

The dynamic control allocation problem can beexpressed as the following hybrid optimization problem:

minδ t

J0 + J1 + J2,

subject to u = Bδ, δ t ≤ δ t ≤ δ t ,65

where J0 = W0δ t 2, J2 = W2 δ t − δ t − T 2 +W2 δ t − δ t − 2T 2, J1 = W1 δ t − δs t

2, W0 =WT

0 > 0, W1 =WT1 > 0, and W2 =WT

2 > 0.· represents a norm defined as x = xTx, δs t

denotes the desired control command of the actuators, J0denotes the minimum energy principle, J1 denotes the min-imum error principle for the desired control commands ofthe actuators, and J2 denotes the minimum transition rateprinciple for the relative sampling time of the expected con-trol commands.

sy2 = xy2 − xy2d ,

τy3x∗y3 + x∗y3 = x∗y3, x∗y3 0 = x∗y3 0 ,

x∗y3 = −g−1y2 f y2xy2 + 2xz2xy2 tan xz1 −vy22

r cos xz1− xy2d − ky2 + ky2 sy2,

sy3 = xy3 − x∗y3,

τy4x∗y4 + x∗y4 = x∗y4, x∗y4 0 = x∗y4 0 ,

x∗y4 = −f y3xy3 − vy32 + x∗y3 − ky3sy3 − μgy2sy2,

sy4 = xy4 − x∗y4,

uy = g−1y3 −f y4xy3 − f y5xy4 − vy42 + x

∗y4 − ky4sy4 − ky5 sy4

∂y sgn sy4 ,

60


Define diagonal matrices R0, R1, and R2 with correspond-ing dimensions

R0 = diag R01, R02, R03, R04 ,R1 = diag R11, R12, R13, R14 ,R2 = diag R21, R22, R23, R24 ,

66

where

R0i =

δi tδi t

, δi t > δi t ,

1, δi t ≤ δi t ≤ δi t

δi tδi t

, δi t < δi t ,

,

R1i =

δsi tδsi t

, δsi t > δsi t ,

1, δsi t ≤ δsi t ≤ δsi t

δsi tδsi t

, δsi t < δsi t ,

,

67

and

R2i =

δi t − Tδi t − T

, δi t − T > δi t − T ,

1, δi t − T ≤ δi t − T ≤ δi t − T ,δi t − Tδi t − T

, δi t − T < δi t − T ,

68

with δi t , δi t , δi t , δsi t , δsi t , δsi t , δi t − T , δi t − T ,and δi t − T the elements of the corresponding vectors i ∈1, 2, 3, 4 .

Theorem 2. The hybrid optimization problem in Equation(65) is equivalent to

minδ t

J0 + J1 + J2,

subject to u = Bδ, δ t ≤ δ t ≤ δ t ,69

where J0 = R0W0δ t 2, J2 = R2W2 δ t − δ t − T 2 +W2 δ t − δ t − 2T 2, and J1 = R1W1 δ t − δs t

2,and the general solution is given in Equation (70):

δ t = Eδs t + F1δ t − T + F2δ t − 2T + Gu t , 70

where E = I −GB W−2R21W

21, F1 = I −GB W−2R2

2W22,

F2 = I − GB W−2W22, G =W−2BT BW−2BT −1

, and W =R20W

20 + R2

1W21 + R2

2W22 +W2

21/2.

Proof 2. From Equation (69), one has

J = J0 + J1 + J2 = δT t R20W

20δ + δ t − δs t

TR21W

21 δ t

− δs t + δ t − δ t − T TR22W

22 δ t − δ t − T

+ δ t − δ t − 2T TW22 δ t − δ t − 2T

= W δ t − δ∗0 t 2,71

where δ∗0 t =W−2 R21W

21δs t + R2

2W22δ t − T +W2

2δ t − 2T . Therefore, the optimization problem in Equation (69)is equivalent to

minδ t

= W δ t − δ∗0 t 2,

subject to u = Bδ, δ t ≤ δ t ≤ δ t72

According to the weighted pseudo-inverse theorem [38],the general solution of the optimization problem in Equation(69) is

δ t = I −GB δ∗0 t +Gu t 73

By substituting Equation (71) into Equation (73), one canfind that Theorem 2 is valid. Q.E.D.

Remark 2. By including the minimum error term related tothe error between the command of the given step and thoseone or two steps prior, the purpose is to reduce chatteringin the system and smoothen the control commands. Byincreasing the coefficient matrices R0, R1, and R2, one cansolve the constraint problem of control moment to solvethe optimization problem.

6. Simulation Analysis

To verify the effectiveness of the asymmetric barrier Lyapu-nov function-based IGC algorithm designed in this studyfor the interceptor missile, it was assumed that given an inter-ceptor missile, its initial position was Xm, Ym, Zm = 0m,0m, 0m , its initial velocity was Vm0 = 1600m/s, and theinitial position of the target was Xt, Yt, Zt = 5000m,6000m, 4000m , and the initial velocity was Vt0 = 1200m/s.The rudder deflection angle of the interceptor missile waslimited in the range -30° to +30°, the angular velocity of therudder’s deflection was limited in the range -300°/s to+300°/s, the maximum steady-state thrust of the jet deviceswas Fs max = 4800N, and the their moment amplification fac-tor was Kz = Ky = 0 3. The asymmetric output constraints

were kc1 = −4 and kc1 = 2, and the time-varying asymmetricoutput constraints were kc1 = −4 + 0 2 cos t and kc1 = 2 +0 1 sin t . The aerodynamic parameters of the interceptormissile are shown in Table 1.

The method designed in this study (i.e., ABLFIGC+DCA) was compared with the combined use of IGC andthe traditional dynamic surface sliding-mode control law(i.e., IGC+DCA). Assuming that the interferences in the


system were dz3 = dz4 = dy3 = dy4 = 0 02 sin t , target inter-ception was maneuvered in the following two scenarios:

Scenario 1. The target was in uniform motion: atε = atβ = 0m/s2.

Scenario 2. The target was in accelerated motion: atε = atβ =10m/s2.

When the asymmetric output constraints were kc1 = −4and kc1 = 2, the simulation comparison plots of Scenario 1are shown in Figures 2–7.

Figures 2 and 3, respectively, show the curves of angularvelocities of the vertical and horizontal lines of sight of theinterceptor missile in Scenario 1. It is clear that a certaindegree of chattering occurred during convergence whenthe traditional IGC algorithm was used, and the two angularvelocities exhibited large jumps in the late stage of guidance.By contrast, the ABLFIGC+DCA algorithm designed in thisstudy caused the two angular velocities to converge morequickly to steady-state values in a smoother convergenceprocess that met the asymmetric output constraints kc1and kc1, thus improving the stability of the guidance andcontrol system and showing good robustness against exter-nal interferences.

Figures 4–7 show the curves of the attack angle, sideslipangle, and angular velocities of the pitch and yaw of the inter-ceptor missile in Scenario 1, respectively. A certain degree ofchattering appeared due to interference when the traditionalIGC algorithm was used, and the above four parameters allexhibited large jumps in the late stage of guidance. By con-trast, the ABLFIGC+DCA algorithm designed in this studyrenders the entire convergence process smoother, with theabove four parameters stable and not recording large jumpsin the late stage of guidance. Moreover, this algorithmshowed good robustness against external interference.

The simulation comparison plots of Scenario 2 are shownin Figures 8–13.

Figures 8–13 show the simulation comparison plots ofthe target motion Scenario 2. From Figures 8 and 9, it is clearthat the ABLFIGC+DCA algorithm smoothened conver-gence and was free of large jumps in the guidance process,met the asymmetric output constraints kc1 and kc1, and

showed good robustness against external interference. Asshown in Figures 10–13, the ABLFIGC+DCA algorithmensured that the flight attitude of the interceptor missilewas stable and changed smoothly.

The results of comparison of the two scenarios in termsof the target miss distance and interception time of the inter-ceptor missile are shown in Table 2.

As shown in Table 2, the algorithm designed in this studyhad a smaller miss distance and interception time than theIGC algorithm subject to the traditional dynamic sliding-mode control law.

For Scenario 1, the simulation comparison plots of δz , δsy ,δy, and δsz of the actuators obtained by the dynamic controlallocation algorithm are shown in Figures 14–17.

Figures 14–17 show the curves of the elevator deflectionangle δz , the rudder deflection angle rudder angle δy , and theequivalent rudder deflection angles δsz and δsy, respectively,all of which were obtained via the dynamic control allocation

Time (s)1086420 12 14 16

Ver

tical

line

-of-s

ight

angu

lar v

eloc

ities

(°/s

)

–6

–4–2

0246

ABLFIGC+DCAIGC+DCA

Figure 2: Curves of angular velocities of vertical line of sight of theinterceptor missile in Scenario 1: ABLFIGC+DCA refers to acombination of the proposed asymmetric barrier Lyapunovfunction-based IGC algorithm and the dynamic control allocationalgorithm; IGC+DCA refers to a combination of the IGCalgorithm subject to the traditional dynamic surface sliding-modecontrol law and the dynamic control allocation algorithm.

Time (s)1086420 12 14 16

Hor

izon

tal l

ine-

of-s

ight

angu

lar v

eloc

ities

(°/s

)

−6−4−2

0246

ABLFIGC+DCAIGC+DCA

Figure 3: Curves of angular velocities of horizontal line of sight ofthe interceptor missile in Scenario 1: ABLFIGC+DCA refers to acombination of the proposed asymmetric barrier Lyapunovfunction-based IGC algorithm and the dynamic control allocationalgorithm; IGC+DCA refers to a combination of the IGCalgorithm subject to the traditional dynamic surface sliding-modecontrol law and the dynamic control allocation algorithm.

Table 1: Aerodynamic parameters of the interceptor missile.

Parameter Value Parameter Value

Jz 247.43 kg·m2 Cβz 19.79

Jy 247.43 kg·m2 mαz -15.26

m 204 kg mβy -15.26

L 0.22m mωzz -0.16

Lm 2m mωyy -0.16

S 0.0409m2 mδzz -11.81

Cαy -19.79 m

δyy -11.81


algorithm. As shown in these figures, the ABLFIGC+DCAalgorithm rendered the actuator outputs bounded andcaused them to alter smoothly, had good robustness againstexternal interferences, and did not incur a large jump inthe late stage of guidance, indicating a good control alloca-tion performance.

When the time-varying asymmetric output constraintswere kc1 = −4 + 0 2 cos t and kc1 = 2 + 0 1 sin t , due tothe attack angle, sideslip angle, and angular velocities of thepitch and yaw of the interceptor missile which are similarto the above, therefore, only the angular velocities of the ver-tical and horizontal lines of sight of the interceptor missileare compared. The simulation comparison plots of Scenario1 are shown in Figures 18 and 19.

The simulation comparison plots of Scenario 2 are shownin Figures 20 and 21.

Figures 18–21, respectively, show the curves of angularvelocities of the vertical and horizontal lines of sight of theinterceptor missile in Scenario 1 and Scenario 2. FromFigures 18 and 21, it is clear that the ABLFIGC+DCA algo-rithm smoothened convergence and was free of large jumpsin the guidance process, met the asymmetric output con-straints kc1 = −4 + 0 2 cos t and kc1 = 2 + 0 1 sin t , andshowed good robustness against external interference.

As shown by the above simulation results, the asymmet-ric barrier Lyapunov function-based IGC algorithm pro-posed in this study for the interceptor missile, whencompared with the conventional IGC algorithm, can enablethe outputs to meet constraints and the system to have bettercontrol performance and anti-interference ability, thusimproving the stability of the guidance and control systemsfor the interceptor missile.

Time (s)1086420 12 14 16

Side

slip

angl

e (°)

–16

–12

–8

–4

0

4

8

ABLFIGC+DCAIGC+DCA

Figure 5: Sideslip angle curves of the interceptor missile in Scenario1: ABLFIGC+DCA refers to a combination of the proposedasymmetric barrier Lyapunov function-based IGC algorithm andthe dynamic control allocation algorithm; IGC+DCA refers to acombination of the IGC algorithm subject to the traditionaldynamic surface sliding-mode control law and the dynamiccontrol allocation algorithm.

Time (s)1086420 12 14 16

Pitc

h an

gula

r vel

ociti

es (°

/s)

−50

−25

0

25

50

ABLFIGC+DCAIGC+DCA

Figure 6: Pitch angular velocity curves of the interceptor missile inScenario 1: ABLFIGC+DCA refers to a combination of the proposedasymmetric barrier Lyapunov function-based IGC algorithm andthe dynamic control allocation algorithm; IGC+DCA refers to acombination of the IGC algorithm subject to the traditionaldynamic surface sliding-mode control law and the dynamiccontrol allocation algorithm.

Time (s)1086420 12 14 16

Yaw

angu

lar v

eloc

ities

(°/s

)

−80−60−40−20

020406080

ABLFIGC+DCAIGC+DCA

Figure 7: Yaw angular velocity curves of the interceptor missile inScenario 1: ABLFIGC+DCA refers to a combination of theproposed asymmetric barrier Lyapunov function-based IGCalgorithm and the dynamic control allocation algorithm; IGC+DCA refers to a combination of the IGC algorithm subject to thetraditional dynamic surface sliding-mode control law and thedynamic control allocation algorithm.

Time (s)1086420 12 14 16

Atta

ck an

gle (

°)

-6

-4-202468

ABLFIGC+DCADSC

Figure 4: Attack angle curves of the interceptor missile in Scenario1: ABLFIGC+DCA refers to a combination of the proposedasymmetric barrier Lyapunov function-based IGC algorithm andthe dynamic control allocation algorithm; IGC+DCA refers to acombination of the IGC algorithm subject to the traditionaldynamic surface sliding-mode control law and the dynamiccontrol allocation algorithm.


Time (s)1086420 12 14 16 18 20 22 24

Ver

tical

line

-of-s

ight

angu

lar v

eloc

ities

(°/s

)

−6−4−2

0246

ABLFIGC+DCAIGC+DCA


Time (s)1086420 12 14 16 18 20 22 24

Hor

izon

tal l

ine-

of-s

ight

angu

lar v

eloc

ities

(°/s

)

−6

−4

−2

0

2

4

6

ABLFIGC+DCAIGC+DCA

Figure 9: Curves of angular velocities of the horizontal line of sightof the interceptor missile in Scenario 2: ABLFIGC+DCA refers to acombination of the proposed asymmetric barrier Lyapunovfunction-based IGC algorithm and the dynamic control allocationalgorithm; IGC+DCA refers to a combination of the IGCalgorithm subject to the traditional dynamic surface sliding-modecontrol law and the dynamic control allocation algorithm.

Time (s)1086420 12 14 16 18 20 22 24

Atta

ck an

gle (

°)

-6-4-202468

ABLFIGC+DCADSC

Figure 10: Attack angle curves of the interceptor missile in Scenario2: ABLFIGC+DCA refers to a combination of the proposedasymmetric barrier Lyapunov function-based IGC algorithm andthe dynamic control allocation algorithm; IGC+DCA refers to acombination of the IGC algorithm subject to the traditionaldynamic surface sliding-mode control law and the dynamiccontrol allocation algorithm.

Time (s)1086420 12 14 16 18 20 22 24

Side

slip

angl

e (°)

−16

−12

−8

−4

0

4

8

ABLFIGC+DCAIGC+DCA

Figure 11: Sideslip angle curves of the interceptor missile inScenario 2: ABLFIGC+DCA refers to a combination of theproposed asymmetric barrier Lyapunov function-based IGCalgorithm and the dynamic control allocation algorithm; IGC+DCA refers to a combination of the IGC algorithm subject to thetraditional dynamic surface sliding-mode control law and thedynamic control allocation algorithm.

Time (s)1086420 12 14 16 18 20 22 24

Pitc

h an

gula

r vel

ocity

(°/s

)

−50

−25

0

25

50

ABLFIGC+DCADSC

Figure 12: Pitch angular velocity curves of the interceptor missile inScenario 2: ABLFIGC+DCA refers to a combination of the proposedasymmetric barrier Lyapunov function-based IGC algorithm andthe dynamic control allocation algorithm; IGC+DCA refers to acombination of the IGC algorithm subject to the traditionaldynamic surface sliding-mode control law and the dynamiccontrol allocation algorithm.

Time (s)1086420 12 14 16 18 20 22 24

Yaw

angu

lar v

eloc

ity (°

/s)

−80−60−40−20

020406080

ABLFIGC+DCAIGC+DCA

Figure 13: Yaw angular velocity curves of the interceptor missile inScenario 2: ABLFIGC+DCA refers to a combination of the proposedasymmetric barrier Lyapunov function-based IGC algorithm andthe dynamic control allocation algorithm; IGC+DCA refers to acombination of the IGC algorithm subject to the traditionaldynamic surface sliding-mode control law and the dynamiccontrol allocation algorithm.


7. Conclusion

In the context of the interceptor missile which uses a direct-force/aerodynamic-force control scheme, by considering thecoupling relationship between the pitch and yaw channels ofthe interceptor missile as well as the output constraint

Table 2: Comparison of simulation results between the twoscenarios.

Scenario Control modeMiss

distance (m)Interceptiontime (s)

1ABLFIGC+DCA 0.608 15.21

IGC+DCA 0.792 15.26

2ABLFIGC+DCA 0.854 21.42

IGC+DCA 0.992 21.48

Time (s)1086420 12 14 16

Elev

ator

defl

ectio

n δs

z (°)

–15

–10

–5

0

5

10

15

ABLFIGC+DCAIGC+DCA

Figure 14: Curves of elevator deflection angle δz : ABLFIGC+DCArefers to a combination of the proposed asymmetric barrierLyapunov function-based IGC algorithm and the dynamic controlallocation algorithm; IGC+DCA refers to a combination of theIGC algorithm subject to the traditional dynamic surface sliding-mode control law and the dynamic control allocation algorithm.

Time (s)1086420 12 14 16

Rudd

er d

efle

ctio

n δs

z (°)

–15

–10

–5

0

5

10

15

ABLFIGC+DCAIGC+DCA

Figure 15: Curves of rudder deflection angle δy : ABLFIGC+DCArefers to a combination of the proposed asymmetric barrierLyapunov function-based IGC algorithm and the dynamic controlallocation algorithm; IGC+DCA refers to a combination of theIGC algorithm subject to the traditional dynamic surface sliding-mode control law and the dynamic control allocation algorithm.

Time (s)1086420 12 14 16

Equi

vale

nt d

efle

ctio

n δs

z (°)

−1

−0.5

0

0.5

1

ABLFIGC+DCAIGC+DCA

Figure 16: Curves of equivalent rudder deflection angle δsz :ABLFIGC+DCA refers to a combination of the proposedasymmetric barrier Lyapunov function-based IGC algorithm andthe dynamic control allocation algorithm; IGC+DCA refers to acombination of the IGC algorithm subject to the traditionaldynamic surface sliding-mode control law and the dynamiccontrol allocation algorithm.

Time (s)1086420 12 14 16

Equi

vale

nt d

efle

ctio

n δs

y (°)

–2–1.5

–1–0.5

00.5

11.5

2

ABLFIGC+DCAIGC+DCA

Figure 17: Curves of equivalent rudder deflection angle δsy :ABLFIGC+DCA refers to a combination of the proposedasymmetric barrier Lyapunov function-based IGC algorithm andthe dynamic control allocation algorithm; IGC+DCA refers to acombination of the IGC algorithm subject to the traditionaldynamic surface sliding-mode control law and the dynamiccontrol allocation algorithm.

Time (s)1086420 12 14 16

Ver

tical

line

-of-s

ight

angu

lar v

eloc

ities

(°/s

)

−6

−4−2

0246

ABLFIGC+DCAIGC+DCA

−4+0.2cos(t)2+0.1sin(t)



problems of the system. This study combined the IGCmethod with an asymmetric barrier Lyapunov function todesign an asymmetric barrier Lyapunov function-basedIGC algorithm for the interceptor missile. Compared withtraditional algorithms, by adopting the asymmetric barrierLyapunov function, the proposed algorithm relaxed the con-straints on the initial conditions of the system. In addition,the study utilized the dynamic surface sliding-mode controllaw and a TVGESO to design an IGC algorithm for intercep-tor missile, which not only reduced the system’s require-ments for high-order differentiability of the stabilityfunction but also enhanced the control system’s resistanceto unknown interferences. The study also considered theexisting redundancy of the aerodynamic rudder and the reac-tion jet device in the actuator and consequently designed adynamic control allocation algorithm to allocate the desiredcontrol moments to the actuator, thereby improving its effi-ciency. Finally, the simulation results showed that the pro-posed algorithm demonstrated relatively good dynamiccharacteristics and was able to satisfy the interceptor missileIGC system’s requirements for accuracy and stability withoutbreaking the constraints.

Data Availability

The simulation data used to support the findings of this studywere supplied by Northwestern Polytechnical Universityunder license and so cannot be made freely available.Requests for access to these data should be made to Prof.Xiaogeng Liang, E-mail: [email protected].

Conflicts of Interest

The authors declare that there is no conflict of interestregarding the publication of this paper.

Acknowledgments

This work was supported by the Aeronautical ScienceFoundation of China (grant number 2016ZC12005). Theauthors gratefully acknowledge the suggestions and helpby the Prof. Xiaogeng Liang and Northwestern Polytechni-cal University.

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