Research ArticleAdaptive Fuzzy Sliding Mode Guidance Law considering AvailableAcceleration and Autopilot Dynamics

Yulin Wang ,1 Shengjing Tang ,1 Wei Shang,2 and Jie Guo 1

1School of Aerospace Engineering, Beijing Institute of Technology, Beijing 100081, China2System Design Institute of Hubei Aerospace Technology Academy, Hubei, 430040, China

Correspondence should be addressed to Jie Guo; guojie1981@bit.edu.cn

Received 21 November 2017; Accepted 8 April 2018; Published 29 April 2018

Academic Editor: Vaios Lappas

Copyright © 2018 Yulin Wang et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Terminal guidance law for missiles intercepting high maneuvering targets considering the limited available acceleration andautopilot dynamics of interceptor is investigated. Conventional guidance laws based on adaptive sliding mode control theorywere designed to intercept a maneuvering target. However, they demand a large acceleration for interceptor at the end of theterminal guidance, which may have acceleration saturation especially when the target acceleration is close to the availableacceleration of interceptor. In this paper, a terminal guidance law considering the available acceleration and autopilotdynamics of interceptor is proposed. Then, a fuzzy system is utilized to approximate and replace the variable structureterm, which can handle the unknown target acceleration. And an adaptive neural network system is adopted to compensatethe effects caused by the designed overlarge acceleration of interceptor such that the interceptor with small availableacceleration can intercept the high maneuvering target. Simulation results show that the guidance law with available accelerationand autopilot dynamics (AAADG) is highly effective for reducing the acceleration command and achieving a small finalmiss distance.

1. Introduction

In recent years, the hypersonic flight vehicles with highmaneuverabilities have been widely developed [1]. It is neces-sary to design an appropriate guidance law to intercept thehypersonic flight vehicles for its high strike ability. However,the widely used PNG law [2, 3] and its variants [4–8] wouldfail to satisfy the requirement of high accuracy for the task ofintercepting the hypersonic flight vehicles with high maneu-verabilities [9]. An effective approach to intercept maneuver-able targets is applying robust control methods, such assliding mode control (SMC) [10], adaptive control [11], non-linear H∞ control [12], and optimal control [13].

The guidance law based on SMC has become an activeresearch area because of its strong robustness and easyimplementation [10, 14, 15]. In the guidance law design,the zero-effort miss distance is usually used to define slidingmode surface. For example, Shima et al. [16] proposed asliding mode guidance law for integrated missile autopilotwith the zero-effort miss distance as sliding mode surface.

The sliding mode guidance laws are also designed by guid-ing the line-of-sight (LOS) angular rate converge to zero.Then, Zhang et al. [17] developed a novel guidance lawbased on integral sliding mode control and employed a non-linear disturbance observer to estimate the target accelera-tion. By estimating the target acceleration via an extendedstate observer, Zhu et al.[18] designed a guidance law with-out requiring the information on the target acceleration.However, a compromise should be made between the track-ing accuracy and control input since the high tracking accu-racy will lead to a very large control input. Thus, the propervalues for the control parameters, such as the amplitude ofvariable structure term, are hard to choose. Consideringfinite-time convergence, Zhou et al. [19] proposed a variablestructure finite-time-convergent guidance law and provedthat the LOS angular rate would converge to zero beforethe ending of the terminal guidance. To attack the targetwith a predefined impact angle, the guidance laws withimpact angle constraints were proposed in [20–23]. Leeet al. [24] developed a guidance law using the high-

HindawiInternational Journal of Aerospace EngineeringVolume 2018, Article ID 6081801, 10 pageshttps://doi.org/10.1155/2018/6081801

performance sliding mode control theory to intercept a sta-tionary or slowly moving target. Kumar et al. [20] presenteda guidance law to intercept the targets from any initial head-ing angle without exhibiting any singularity. Furthermore,some guidance laws take autopilot dynamics into consider-ation [25, 26]. Sun et al. [27] studied a second order slidingmode guidance law with autopilot dynamics. Li et al. [28]took both the autopilot dynamics and uncertainties intoconsideration and proposed a finite-time-convergent guid-ance law to intercept a maneuvering target.

To the best of our knowledge, the existing sliding modeguidance laws may require a greater guidance command thanthe available acceleration of interceptor when the maximumacceleration of target is equal to or larger than the availableacceleration of interceptor. To solve this problem, an acceler-ation limiter is adopted to reduce the saturated accelerationdemand [29, 30]. Xiong et al. [31] proposed a strategy mak-ing the coefficients of the sliding mode guidance law varyaccording to the fuzzy rule to reduce the acceleration demandat the beginning of the terminal guidance. However, it cannotguarantee the stability of the guidance system when thedesigned acceleration is significantly reduced at the end ofthe homing guidance. In this paper, a neural network systemis employed to compensate the effects of the designed over-large acceleration and the stability of the guidance system isproved. However, to guarantee a good performance of theguidance system when the missile intercepts the hypersonicflight vehicle with a high maneuverability, the gain of switch-ing term needs to be chosen larger than the upper bound oftarget acceleration, which is hard to choose in advance andwill cause the chattering problem. Then, with the universalapproximation property of fuzzy systems, a continuousadaptive fuzzy system is used to approximate and replacethe variable structure term, which can handle the unknowntarget maneuver and avoid the chattering phenomenon.The input information of the adaptive fuzzy system is therelative distance and the LOS angular rate which can be mea-sured by the interceptor seeker easily. Besides, before design-ing a guidance law, it is necessary to consider the autopilotdynamics because the autopilot lag of a homing missile willdecrease the precision of guidance. Then, a sliding modeguidance law considering the available acceleration and auto-pilot dynamics of missile for intercepting a high maneuver-ing target is proposed.

The main contribution of this paper is that, by adoptingneural network system to reduce the acceleration command,so that the interceptor is able to intercept the hypersonicflight vehicles whose maximum acceleration is equal to orlarger than the available acceleration of interceptor. In addi-tion, the information needed to estimate the upper bound oftarget acceleration is limited and easy to obtain, so the pro-posed guidance law is easy to be implemented in practice.The rest of this paper is organized as follows. Section 2describes the formulations of the guidance system consider-ing the available acceleration and autopilot dynamics ofinterceptor. In Section 3, the control methods for the pro-posed guidance law are introduced. In Section 4, the stabilityof the guidance law is verified. At last, compared with pro-portional navigation guidance (PNG) and finite-time

convergence guidance (FTCG), simulation results demon-strate the effectiveness of AAADG.

2. The Dynamics of the RelativeMotion between the Interceptor and Target

This section presents the mathematic model of the guidancesystem for intercepting. The planar relative motion of theinterceptor and target is shown in Figure 1. The interceptorand the target are denoted by the subscripts M and T, respec-tively. In order to simplify the guidance law design, the inter-ceptor and target are assumed to have two mass points. Thecorresponding equations are given by as follows:

r = −vMcos σM + vTcos σT, 1

rq = vMsin σM − vTsin σT, 2

σM = q − θM, 3

σT = q − θT, 4

θM =aMvM

, 5

θT =aTvT

, 6

where r is the relative distance and r is the relative velocitybetween the interceptor and the target. vM is the velocity ofthe interceptor. aM is the normal acceleration of the intercep-tor. θM is the flight-path angle of the interceptor. vT is the tar-get velocity. aT is the normal acceleration of the target. θT isthe flight-path angle of the target. q and q are the light-of-sight (LOS) angle and LOS angular rate between the intercep-tor and target, respectively.

Assumption 1: [31]. In the terminal guidance, the magnitudesof the speeds of the interceptor and target are assumed as twoconstants such that vM = 0 and vT = 0.

In practice, because of the physical limitation of the inter-ceptor actuator, the interceptor cannot achieve the requiredacceleration when the target acceleration is equal to or largerthan the available acceleration of the interceptor. It may leadto a large final miss distance. Therefore, the required acceler-ation of the interceptor should be limited to a certain level.From (5), it can be seen that the acceleration of interceptorvaries with the change of θM. So, we can limit θM as follows:

θM = sat θM =−θMmax θM > −θMmax

θM −θMmax ≤ θM ≤ θMmax

θMmax θM > θMmax

, 7

θM − θM = δ, 8

where θMmax = const > 0 is the maximum flight-path angularrate of the interceptor, which is influenced by the flight envi-

ronment of the interceptor, and θM is the actual flight-pathangular rate. δ is a nonlinear function caused by saturation.

2 International Journal of Aerospace Engineering

The normal acceleration command generated by theguidance law is controlled by the autopilot of the interceptor,which tracks the expected acceleration command by modu-lating fins or thrusters. However, there is a lag between theexpected acceleration and the achieved acceleration. There-fore, the performance of the interceptor is expected to beimproved by using a guidance law considering the autopilotdynamics. The autopilot dynamics can be well modeled andapproximated as the following first-order system:

vMθM = −1T1

vMθM +1T1

vMθC, 9

where θC is the expected flight-path angular rate command ofthe interceptor. θM is the achieved flight-path angular rate.T1 is the time constant of the autopilot, which is easy toobtain in advance.

The designing purpose of the guidance law is adjustingaM so that the LOS angular rate will converge to zero. Basedon this idea, we define three state variables as follows:

x1 = q,

x2 = q,

x3 = θM

10

Then, we obtain the integrated dynamics equations forthe guidance system with the autopilot dynamics of the inter-ceptor as follows:

x1 = x2,

x2 = −2rrx2 −

vMcos σMr

x3 + δ +1rω,

x3 = −1T1

x3 +1T1

u,

11

where ω = vTcos σTθT is the target motion information. u isthe actual control input.

3. Guidance Law Design consideringAvailable Acceleration and AutopilotDynamics of the Interceptor

In this section, the guidance law considering the availableacceleration and autopilot dynamics of the interceptor isdesigned. It can be seen that the guidance system describedby (11) is in a strict feedback form. However, the guidancelaw designed using backstepping method contains the high-order time derivatives of LOS angle and interceptor-to-target distance. To avoid this problem, the dynamic surfacecontrol method is proposed by introducing a first-orderlow-pass filter. The first-order low-pass filter allows thedesign where the model is not differentiable. For the guidancesystem described by (11), the goal of the guidance law is tomake the LOS angular rate converge to zero. The design ofthe guidance law is given as follows:

s1 = x2,

x3d =k1 + 2 r x2 + ε sgn s1 − vM sin σM

vM cos σM− δ,

x3cx3d

=1

T2s + 1, x3 0 = x3c 0 ,

s2 = x3 − x3c,

u = x3 +T1T2

x3d − x3c − k2T1s2,

12

where s1 and s2 are designed as the dynamic surfaces. x3d isthe virtual control obtained from the first step. k1 and k2are the dynamic surface gains. x3c is the command input ofthe second step, which is obtained by x3d passing throughthe low-pass filter. T2 is the time constant of the filter. ε sgns1 is the variable structure term to enhance the robustnessof the system against the unknown target movement, wherethe switching term amplitude ε satisfies ε > ω . δ is the adap-tive acceleration compensator, which will be designed later.

Remark 1. The choice of the switching term amplitude ε isconnected to the maximum acceleration of the target.Because of the uncertain maneuver of the target, the boundof target acceleration cannot be estimated in advance. It isdifficult to determine the value of switching term amplitude,since the small value of ε makes the q diverge and cannotconverge to zero, and the large value of ε will cause heavychattering. Moreover, as ε sgn s1 is not a continuous func-tion, the interceptor actuator cannot response quickly to real-ize ε sgn s1 in practice, which will cause large miss distance.

In this paper, the adaptive fuzzy function f q, r ∣ η is cho-sen to approximate the switching term ε sgn s1. The inputs ofadaptive fuzzy function are the relative distance r betweenthe interceptor and target and LOS angular rate q. Theswitching term is replaced by the continuous function whichis the output of adaptive fuzzy system, so that the impropervalue of ε cannot influence the precision of the guidancelaw and the chattering phenomenon is avoided. The adaptivefuzzy system is designed as follows:

vM

q

x

y

r

qvT

aM

M

T

aT

O

�휎T

�휃T

�휎M

�휃M

Figure 1: Relative motion between the interceptor and target.

3International Journal of Aerospace Engineering

Define b1 = q and b2 = r/r0, where r0 is the initial relativedistance between the interceptor and target. The knowledgebased on adaptive fuzzy systems comprises a collection offuzzy if-then rules as the following form:

(i) R1: if b1 is F11 and b2 is F

12, then y is G1

.

(ii) R2: if b2 is F21 and b2 is F

22, then y is G2

.

(iii) R3: if b3 is F31 and b2 is F

32, then y is G3

.

(iv) R4: if b4 is F41 and b2 is F

42, then y is G4

.

(v) R5: if b5 is F51 and b2 is F

52, then y is G5

.

In order to approximate the switching term ε sgn s1properly, the fuzzy rules should be symmetric and uniformand cover all the potential regions of b1 and b2. Thus, theadaptive fuzzy membership functions of the fuzzy rules inthis paper are chosen as follows:

u1 b1, b2 =1

1 + e 25b1+1×

11 + e0 5b2

,

u2 b1, b2 = e− 5b1+0 2 2× e− 0 1b2+0 2 2

,

u3 b1, b2 = e− 5b12× e− 0 1b2

2,

u4 b1, b2 = e− 5b1−0 2 2× e− 0 1b2−0 2

2,

u5 b1, b2 = 11 + e −25b1+1

× 11 + e−0 5b2

13

Define the fuzzy basis functions as follows:

ξm q, r =um b1, b2

〠5i=1u

i b1, b2, m = 1, 2,… , 5 14

The switching term can be approximated and replaced bythe adaptive fuzzy system. Using singleton function, centeraverage defuzzification, and product inference, the adaptivefuzzy logic system can be expressed as follows:

f q, r η = ηTξ q, r , 15

where η is the adaptive parameter vector of the fuzzy system.If the designed acceleration exceeds the available acceler-

ation of interceptor too much during the terminal guidance,the control law may cause the system instable and lead to alarge final miss distance. In order to solve this problem, theadaptive radial basis function neural network δ is designedas the acceleration compensator which can compensate theinfluence of designed overlarge acceleration. Using the uni-versal approximation property of neural network function,there exists an ideal neural network that approximates thenonlinear function δ in (8) closely, such that

δ =WTh x3 + ς, 16

where ς is the approximation error which is very small andcan be ignored.W is the ideal constant weight vector of neu-ral network. And the adaptive radial basis function neuralnetwork is as follows:

δ = WTh x3 , 17

where h x3 is the basis vector and hj x3 = exp − x3 − cj2/

2b2j , j = 1, 2,… , 5. c = c1, c2,… , c5 is the center of the recep-tive field and b = b1, b2,… , b5 is the width of the Gaussianfunction. W is the adaptive weight vector of neural network.

Neural network system is designed to compensate theinfluence caused by acceleration saturation of the interceptor.Therefore, the system can eliminate the limit of the availableacceleration.

4. Stability Analysis

Lemma 1: [32] Consider the system

x = g x, t 18

Suppose that there is a continuously differentiable Lya-punov function V x, t defined in a neighborhood of theorigin. For the equation V x, t ≤ −αV x, t + C and condi-tions V 0 = p, α > C/p Then, the zero solution of the sys-tem described by (18) is stable.

Theorem 1. Employing the guidance law described in (12),whose adaptive control parameters for the fuzzy systemin (15) and neural network system in (17) are designed as

η = Tηr0/r ξs1 and W = − TWvMcos σM/r hs1, respectively,the guidance system described by (11) considering the avail-able acceleration and autopilot dynamics of the interceptoris stable.

Proof 1. Define the state error vector of the low-pass filteras follows:

x3 = x3c − x3d 19

The approximation error of the radial basis function neu-ral network weights is defined as follows:

W =W − W 20

Furthermore, define the optimal parameter vector η∗ forthe fuzzy system in (15) as follow:

η∗ = arg minη∈Ω

supq,r∈R

ηTξ q, r − ε sgn s1 21

Then, the estimated error of the adaptive parameter vec-tor for the fuzzy system is defined as follows:

η = η∗ − η 22

With the guidance law designed in (12), the time deriva-tive of the state error vector in (19) is calculated as follows:

x3 = x3c − x3d =1T2

x3d −1T2

x3c − x3d 23

As the optimal weight vector η∗ of the fuzzy system in(21) and the ideal weight vectorW of the neural network sys-tem in (16) are two constant vectors, the time derivative of

4 International Journal of Aerospace Engineering

the adaptive parameter error vectors in (20) and (22) arecalculated as follows:

W = −W =TWvM cos σM

rhs1, 24

η = −η = −Tηr0r

ξs1 25

With the guidance law designed in (12), the time deriva-tive of s1 is as follows:

s1 = −k1 rr

s1 −ηTξ

r−vM cos σM

rδ − δ

+1rω −

vM cos σMr

x3 + s2

26

Similar to the process of (23), with the guidance law in(12), the time derivative of s2 is calculated as follows:

s2 = −k2s2 27

Define the Lyapunov function candidate as follows:

V =12s21 +

12s22 +

12x23 +

12Tηr0

ηTη + 12TW

WTW, 28

where Tη and TW are two positive constant. The time deriv-ative of the Lyapunov function is computed as follows:

V = s1s1 + s2s2 + x3x3 +1

Tηr0ηTη + 1

TWWTW 29

With the results calculated in (23), (26), and (27), itholds that

s1s1 = −k1 rr

s21 −ηθ

rs1 −

vM cos σMr

δ − δ s1

−vM cos σM

rx3 + s2 s1 +

1rωs1,

30

s2s2 = −k2s22, 31

x3x3 = x3c − x3d1T2

x3d −1T2

x3c − x3d ≤ −1T2

x3c − x3d2

+12x23d x3d − x3c

2 +12= −

1T2

−12x23d x23 +

1232

Substituting the results of (30), (31), and (32) into (29),the time derivative of the Lyapunov function can be reformu-lated as follows:

V ≤ −k1 rr

s21 −ηξrs1 −

vM cos σMr

δ − δ s1

−vM cos σM

rx3 + s2 s1 +

s1rω − k2s

22 −

1T2

−12x23d x23

+12+

1Tηr0

ηTη + 1TW

WTW ≤ −k1 rr

s21 − k2s22

−1T2

−12x23d x23 +

s1rω −

η∗ Tξ

rs1

+vM cos σM

2rx23 + 2s21 + s22 + ηT 1

rξs1 −

1Tηr0

η

−WT vM cos σMr

hs1 +1TW

W +12

33

With the time derivatives of the estimated parametererror vectors in (24) and (25), and considering the universalapproximation property of the fuzzy system as shown in (21),(33) can be rewritten as follows:

V ≤ −k1 rr

−vM cos σM

rs21 − k2 −

vM cos σM2r

s22

−1T2

−12x23d −

vM cos σM2r

x23 +12+s1rω

−η∗ Tξ

rs1 ≤ −cV +M,

34

where c and M are two positive constants which satisfy thefollowing:

c ≤min 2k1 rr

−vM cos σM

r, 2k2 −

vM cos σMr

,

· 21T2

−12x23d −

vM cos σM2r

,

35

M ≥max1

2Tηr0cηTη +max

12TWc

WTW +12

36

With Lemma 1, it can be concluded from (34) that thesystem is stable. The proof is finished.

Remark 1. From the guidance law designed in (12), we canget that the main parameters affecting the precisions of guid-ance are k1, k2, and T2. k1 + 2 is similar to the navigationratio of the traditional PNG law, which is usually createdfrom 2 to 6. And considering the choice of the constant c in(35), k1 can take a value from 1 to 4. According to someempirical tests, k2 can be chosen from 1 to 5. T2 takes a valuefrom 0.01 to 0.5. The adaptive parameters W and η can beadjusted on line using the states of the interceptor and target,and their initial values are chosen as zero.

Remark 2. Although AAADG needs to know more informa-tion than PNG guidance law, it is not difficult to realize the

5International Journal of Aerospace Engineering

guidance law. In the proposed guidance law in (12), all theparameters needed can be measured by a target seeker easily.

Remark 3. Compared with the existing guidance laws,AAADG neither needs to estimate the exact maneuver infor-mation of the target nor simply treats the unknown targetacceleration as zero. Meanwhile, this guidance law ensuressmall final miss distance under the challenging situation,where the acceleration of target is equal to or larger thanthe available acceleration of the interceptor.

5. Simulation Results

In this section, the effectiveness of AAADG is verified bynumerical simulations. For the comparison purpose, weapply the proposed guidance law, the PNG and the FTCGdesigned in [19] to each simulation case. The proportionalparameter of PNG is chosen as kp = 2. The parameters ofFTCG are chosen as N = 3, η = 0 5, β = 2, and f = 100. InFTCG, the sign function sgn x is replaced with a saturationfunction sat x which is expressed as follows:

sat x =

1, x > ϑ,xϑ,  x ≤ ϑ,

−1, x < −ϑ,

37

where ϑ is the small positive constant, which is chosen as0.001 in simulation.

Case 1. Intercepting the target with periodic maneuver andthe available acceleration of interceptor is less than the max-imum acceleration of the target.

In this case, we use the proposed guidance law to inter-cept a target with constant velocity and sinusoidal normalacceleration. The initial position of the target is xT0 = 2300m and yT0 = 3000m. The initial flight-path angle is θT0 = 0deg. The velocity of the target is vT = 1000m/s. And thetarget maneuvers with a normal acceleration is aT = 200cos tm/s2.

The initial position of the interceptor is set as xM0 = 0mand yM0 = 0m, and the initial flight-path angle is θM0 = 30deg. The velocity of the interceptor is vM = 1200m/s. The ini-tial relative distance between the interceptor and target is cal-culated as r0 = 3780m. The initial light-of-sight angle iscalculated as q0 = 52 5 deg. And the available acceleration ofthe interceptor is 160m/s2, which is smaller than the maxi-mum acceleration of the target. The parameters of theAAADG are chosen as k1 = 2, k2 = 4, T1 = 0 05 s, T2 = 0 06 s,TW = 1, b = 1 1 1 1 1 , c = −2, −1, 0, 1, 2 , and Tη = 100.

For Case 1, the final miss distances and impact timesunder three guidance laws are listed in Table 1. The acceler-ations of the interceptor and the LOS angle rates under threeguidance laws are plotted in Figures 2 and 3, respectively. Thetrajectories of the interceptor and target under three guid-ance laws are plotted in Figure 4. The relative distances

between the interceptor and target under three guidance lawsare plotted in Figure 5.

Figure 2 shows that the accelerations under three guid-ance laws are limited to 160m/s2. And at the end of the ter-minal guidance, the acceleration command under PNGreaches to 160m/s2 earlier than those under other two guid-ance laws. Moreover, from Figure 2, the acceleration com-mands under FTCG and AAADG reached to 160m/s2 at

Table 1: Final miss distance and impact time.

Guidance law PNG FTCG AAADG

Final miss distance (m) 35.2539 8.7406 0.6663

Impact time (s) 13.17 12.97 12.97

0 2 4 6 8 10 12 14−200

−150

−100

−50

0

50

100

150

200

250

Time (s)Ac

cele

ratio

n co

mm

and

(m/s

2 )

PNGFTCGAAADG

Figure 2: Accelerations under three guidance laws.

0 2 4 6 8 10 12 14−10

−8

−6

−4

−2

0

2

4

6

8

10

Time (s)

LOS

angu

lar r

ate (

deg/

s)

PNGFTCGAAADG

Figure 3: LOS angular rates under three guidance laws.

6 International Journal of Aerospace Engineering

the end of the terminal guidance, but the acceleration com-mand under FTCG has the chattering phenomenon at 11 s.

Figure 3 shows that the LOS angular rate at the beginningof the guidance is extremely large because of the initial stateerrors between the interceptor and target. The LOS angularrate under PNG becomes divergent too early and does notconverge to zero in the terminal guidance, so the final missdistance under PNG is very large. The LOS angular ratesunder FTCG and AAADG converge to zero in 2 s. However,the LOS angular rate under FTCG diverges earlier than thatunder AAADG, such that the final miss distance underPNG is larger than that under AAADG.

Figure 4 shows the trajectories of the interceptor and tar-get under three guidance laws. Figure 5 shows the relative

distance between the interceptor and target as well as theimpact time. The impact time under AAADG is earlier thanthat under other two guidance laws, which can be seen fromFigures 4 and 5 and Table 1. And the final miss distanceunder AAADG is the smallest.

Case 2. Intercepting a target with nonperiodic maneuver andthe available acceleration of interceptor is equal to the maxi-mum acceleration of the target.

In this case, the initial positions and velocities of the tar-get and interceptor are the same as those in Case 1. The non-periodic acceleration of the target is shown in Figure 6. Themaximum acceleration of the target is 200m/s2. And theavailable acceleration of the interceptor is 200m/s2 which isequal to the maximum acceleration of the target.

For Case 2, the final miss distances and impact timeunder three guidance laws are listed in Table 2. The acceler-ations of the interceptor and the LOS angular rates underthree guidance laws are plotted in Figures 7 and 8, respec-tively. The trajectories of the interceptor and target underthree guidance laws are plotted in Figure 9. The relative dis-tances between the interceptor and target under three guid-ance laws are plotted in Figure 10.

Figure 7 shows that the accelerations under PNG andFTCG reached to 200m/s2 too early and the accelerationunder AAADG dis not reach to 200m/s2 at the end of the ter-minal guidance. Moreover, from Figure 8, the LOS angularrates under PNG and FTCG are divergent at the end of theterminal guidance. However, the LOS angular rate underAAADG will converge to zero at the end of the terminalguidance. Figures 9 and 10 and Table 2 show that the finalmiss distance under AAADG is smaller than that under othertwo guidance laws.

Consequently, considering the available acceleration andautopilot dynamics of the interceptor, AAADG is able tointercept the periodic and nonperiodic high maneuveringtargets with small final miss distance.

0 2000 4000 6000 8000 10000 12000 14000 160000

500

1000

1500

2000

2500

3000

3500

x (m)

y (m

)

PNGFTCG

AAADGTarget

1.48 1.5 1.52 1.54×104

2950

3000

3050

Figure 4: Trajectories of the interceptor and target under threeguidance laws.

0 2 4 6 8 10 12 140

500

1000

1500

2000

2500

3000

3500

4000

Time (s)

Relat

ive d

istan

ce o

f the

miss

ile an

d ta

rget

(m)

PNGFTCGAAADG

12.8 13 13.20

20

40

60

Figure 5: Relative distances between the interceptor and targetunder three guidance laws.

0 2 4 6 8 10 12 14−250

−200

−150

−100

−50

0

50

100

150

200

250

Time (s)

Acce

lera

tion

com

man

d (m

/s2 )

Figure 6: Target acceleration.

7International Journal of Aerospace Engineering

6. Conclusion

In this paper, the terminal guidance law considering theavailable acceleration and autopilot dynamics of the inter-ceptor has been designed to intercept a strong maneuverabletarget. In the proposed guidance law, the adaptive fuzzysystem is employed to approximate and replace the variablestructure term, which can handle the unknown targetmaneuver and eliminate the chattering phenomenon.

Moreover, the adaptive radial basis function neural networksystem is designed to compensate the influence caused bythe acceleration saturation of the interceptor. Therefore, theinterceptor under AAADG can intercept the hypersonicflight vehicle whose maximum acceleration is greater thanor equal to the available acceleration of the interceptor.Moreover, the target information needed for AAADG is easyto obtain, so that the proposed AAADG is highly practical.

Conflicts of Interest

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

Table 2: Final miss distance and impact time.

Guidance law PNG FTCG AAADG

Final miss distance (m) 51.5315 42.9289 0.8256

Impact time (s) 13.52 13.44 13.31

0 2 4 6 8 10 12 14−200

−150

−100

−50

0

50

100

150

200

250

Time (s)

Acce

lera

tion

com

man

d (m

/s2 )

PNGFTCGAAADG

Figure 7: Accelerations under three guidance laws.

0 2 4 6 8 10 12 14−6

−4

−2

0

2

4

6

8

10

Time (s)

LOS

angu

lar r

ate (

deg/

s)

PNGFTCGAAADG

Figure 8: LOS angular rates under three guidance laws.

0 2000 4000 6000 8000 10000 12000 14000 160000

500

1000

1500

2000

2500

3000

3500

x (m)

y (m

)

PNGFTCG

AAADGTarget

1.45 1.5 1.55 1.6×104

31003200330034003500

Figure 9: Trajectories of the interceptor and target under threeguidance laws.

0 2 4 6 8 10 12 140

500

1000

1500

2000

2500

3000

3500

4000

Time (s)

Relat

ive d

istan

ce o

f the

miss

ile an

d ta

rget

(m)

PNGFTCGAAADG

12 12.5 13 13.50

100

200

300

Figure 10: Relative distances between the interceptor and targetunder three guidance laws.

8 International Journal of Aerospace Engineering

Acknowledgments

This paper was sponsored by the National Natural ScienceFoundation of China 11202024.

References

[1] E. N. Johnson, A. J. Calise, M. D. Curry, K. D. Mease, and J. E.Corban, “Adaptive guidance and control for autonomoushypersonic vehicles,” Journal of Guidance, Control, andDynamics, vol. 29, no. 3, pp. 725–737, 2006.

[2] P. J. Yuan and J. S. Chern, “Solutions of true proportionalnavigation for maneuvering and nonmaneuvering targets,”Journal of Guidance, Control, and Dynamics, vol. 15, no. 1,pp. 268–271, 1992.

[3] C.-D. Yang, F.-B. Hsiao, and F.-B. Yeh, “Generalized guidancelaw for homing missiles,” IEEE Transactions on Aerospace andElectronic Systems, vol. 25, no. 2, pp. 197–212, 1989.

[4] M. Guelman and J. Shinar, “Optimal guidance law in theplane,” Journal of Guidance, Control, and Dynamics, vol. 7,no. 4, pp. 471–476, 1984.

[5] S. E. Talole and R. N. Banavar, “Proportional navigationthrough predictive control,” Journal of Guidance, Control,and Dynamics, vol. 21, no. 6, pp. 1004–1006, 1998.

[6] T.-H. Kim, B.-G. Park, and M.-J. Tahk, “Bias-shaping methodfor biased proportional navigation with terminal-angle con-straint,” Journal of Guidance, Control, and Dynamics, vol. 36,no. 6, pp. 1810–1816, 2013.

[7] Y. A. Zhang, G. X. Ma, and H. L. Wu, “A biased proportionalnavigation guidance law with large impact angle constraintand the time-to-go estimation,” Proceedings of the Institutionof Mechanical Engineers, Part G: Journal of Aerospace Engi-neering, vol. 228, no. 10, pp. 1725–1734, 2014.

[8] B. S. Kim, J. G. Lee, and H. S. Han, “Biased PNG law forimpact with angular constraint,” IEEE Transactions on Aero-space and Electronic Systems, vol. 34, no. 1, pp. 277–288,1998.

[9] C.-D. Yang and C.-C. Yang, “A unified approach to propor-tional navigation,” IEEE Transactions on Aerospace and Elec-tronic Systems, vol. 33, no. 2, pp. 557–567, 1997.

[10] D. Zhou, C. Mu, andW. Xu, “Adaptive sliding-mode guidanceof a homing missile,” Journal of Guidance, Control, andDynamics, vol. 22, no. 4, pp. 589–594, 1999.

[11] D. Chwa and J. Y. Choi, “Adaptive nonlinear guidance lawconsidering control loop dynamics,” IEEE Transactions onAerospace and Electronic Systems, vol. 39, no. 4, pp. 1134–1143, 2003.

[12] C. D. Yang and H. Y. Chen, “Nonlinear H robust guidance lawfor homing missiles,” Journal of Guidance, Control, andDynamics, vol. 21, no. 6, pp. 882–890, 1998.

[13] B.-G. Park, T.-H. Kim, and M.-J. Tahk, “Optimal impact anglecontrol guidance law considering the seeker’s field-of-viewlimits,” Proceedings of the Institution of Mechanical Engineers,Part G: Journal of Aerospace Engineering, vol. 227, no. 8,pp. 1347–1364, 2013.

[14] J. Moon, K. Kim, and Y. Kim, “Design of missile guidance lawvia variable structure control,” Journal of Guidance, Control,and Dynamics, vol. 24, no. 4, pp. 659–664, 2001.

[15] S. D. Brierley and R. Longchamp, “Application of sliding-modecontrol to air-air interception problem,” IEEE Transactions

on Aerospace and Electronic Systems, vol. 26, no. 2, pp. 306–325, 1990.

[16] T. Shima, M. Idan, and O. M. Golan, “Sliding–mode controlfor integrated missile autopilot guidance,” Journal of Guid-ance, Control, and Dynamics, vol. 29, no. 2, pp. 250–260, 2006.

[17] Z. Zhang, S. Li, and S. Luo, “Terminal guidance laws of missilebased on ISMC and NDOB with impact angle constraint,”Aerospace Science and Technology, vol. 31, no. 1, pp. 30–41,2013.

[18] Z. Zhu, D. Xu, J. Liu, and Y. Xia, “Missile guidance law basedon extended state observer,” IEEE Transactions on IndustrialElectronics, vol. 60, no. 12, pp. 5882–5891, 2013.

[19] D. Zhou, S. Sun, and K. L. Teo, “Guidance laws with finite timeconvergence,” Journal of Guidance, Control, and Dynamics,vol. 32, no. 6, pp. 1838–1846, 2009.

[20] S. R. Kumar, S. Rao, and D. Ghose, “Nonsingular terminal slid-ing mode guidance with impact angle constraints,” Journal ofGuidance, Control, and Dynamics, vol. 37, no. 4, pp. 1114–1130, 2014.

[21] Y. Zhao, Y. Sheng, and X. Liu, “Sliding mode control basedguidance law with impact angle constraint,” Chinese Journalof Aeronautics, vol. 27, no. 1, pp. 145–152, 2014.

[22] Y. Zhang, M. Sun, and Z. Chen, “Finite-time convergent guid-ance law with impact angle constraint based on sliding-modecontrol,” Nonlinear Dynamics, vol. 70, no. 1, pp. 619–625,2012.

[23] Z. Zhang, S. Li, and S. Luo, “Composite guidance laws based onsliding mode control with impact angle constraint and autopi-lot lag,” Transactions of the Institute of Measurement and Con-trol, vol. 35, no. 6, pp. 764–776, 2013.

[24] C.-H. Lee, T.-H. Kim, and M.-J. Tahk, “Design of impact anglecontrol guidance laws via high-performance sliding mode con-trol,” Proceedings of the Institution of Mechanical Engineers,Part G: Journal of Aerospace Engineering, vol. 227, no. 2,pp. 235–253, 2013.

[25] P. Qu and D. Zhou, “A dimension reduction observer-basedguidance law accounting for dynamics of missile autopilot,”Proceedings of the Institution of Mechanical Engineers, Part G:Journal of Aerospace Engineering, vol. 227, no. 7, pp. 1114–1121, 2013.

[26] S. S. Moosapour, G. Alizadeh, S. Khanmohammadi, andH. Moosapour, “A novel robust proportional navigation guid-ance law design for missile considering autopilot dynamic,”Transactions of the Institute of Measurement and Control,vol. 35, no. 5, pp. 703–710, 2013.

[27] S. Sun, D. Zhou, andW. Hou, “A guidance law with finite timeconvergence accounting for autopilot lag,” Aerospace Scienceand Technology, vol. 25, no. 1, pp. 132–137, 2013.

[28] G.-L. Li, H. Yan, and H.-B. Ji, “A guidance law with finite timeconvergence considering autopilot dynamics and uncer-tainties,” International Journal of Control Automation andSystems, vol. 12, no. 5, pp. 1011–1017, 2014.

[29] W. Wang, S. Xiong, X. Liu, S. Wang, and L. Ma, “Adaptivenonsingular terminal sliding mode guidance law againstmaneuvering targets with impact angle constraint,” Proceed-ings of the Institution of Mechanical Engineers, Part G: Jour-nal of Aerospace Engineering, vol. 229, no. 5, pp. 867–890,2014.

[30] C.-M. Lin and C.-F. Hsu, “Guidance law design by adaptivefuzzy sliding-mode control,” Journal of Guidance, Control,and Dynamics, vol. 25, no. 2, pp. 248–256, 2002.

9International Journal of Aerospace Engineering

[31] J. H. Xiong, S. J. Tang, J. Guo, and T. N. Wang, “Improvedsliding mode guidance law based on fuzzy variable coefficientsstrategy,” The Aeronautical Journal, vol. 118, no. 1202,pp. 435–451, 2014.

[32] M. Hou, X. Liang, and G. Duan, “Adaptive block dynamic sur-face control for integrated missile guidance and autopilot,”Chinese Journal of Aeronautics, vol. 26, no. 3, pp. 741–750,2013.

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