Research ArticleAdaptive Super-Twisting Algorithm-Based Nonsingular TerminalSliding Mode Guidance Law
Fang Yang1 Kuanqiao Zhang 2 and Lei Yu2
1Xirsquoan Aeronautical University Xirsquoan 710077 China2Luoyang Electronic Equipment Test Center of China Luoyang 471003 China
Correspondence should be addressed to Kuanqiao Zhang zkuanqiao163com
Received 22 January 2020 Revised 28 May 2020 Accepted 11 June 2020 Published 4 July 2020
Academic Editor Yu Wang
Copyright copy 2020 Fang Yang et al -is is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
A nonsingular fast terminal sliding mode guidance law with an impact angle constraint is proposed to solve the problem of missileguidance accuracy and impact angle constraint for maneuvering targets Aiming at the singularity problem of the terminal slidingmode a fast terminal sliding mode surface with finite-time convergence and impact angle constraint is designed based on fixed-time convergence and piecewise sliding mode theory To weaken chattering and suppress interference a second-order slidingmode supertwisting algorithm is improved By designing the parameter adaptive law an adaptive smooth supertwisting algorithmis designed-is algorithm can effectively weaken chattering without knowing the upper bound information of interference and itconverges faster Based on the proposed adaptive supertwisting algorithm and the sliding mode surface a guidance law with theimpact angle constraint is designed -e global finite-time convergence of the guidance law is proved by constructing theLyapunov function -e simulation results verify the effectiveness of the proposed guidance law and compared with the existingterminal sliding mode guidance laws the proposed guidance law has higher guidance precision and angle constraint accuracy
1 Introduction
In modern warfare many missiles (such as some antishipmissiles antitank missiles and air defense missiles) need tohit the target with certain impact angles to increase thedamage effectiveness of the warheads -erefore the impactangle constraint is a problem that needs to be considered inthe design of the guidance law [1]
Sliding mode control is widely used in the design of theguidance law because of its invariability to interference in thesliding mode In [2] the line-of-sight (LOS) angular velocityand impact angle constraint have been used as the slidingsurface and the sliding mode control is applied to design theguidance law with the impact angle constraint In [3] theadaptive exponential approach law has been used to designthe sliding mode guidance law which increases the adapt-ability and dynamic performance of the guidance lawHowever none of these methods has finite-time conver-gence For the finite-time control problem a finite-timeconvergence guidance law with the impact angle constraint
is designed based on the terminal sliding mode control in[4ndash6] In [7 8] linear terms are added to the terminal slidingmode surface to further speed up the convergence of systemstates But the negative exponential term of the state quantityin the terminal sliding mode control law will cause thesingular problem For the singular problem a nonsingularterminal sliding surface is improved to avoid the singularityproblem in [9ndash11] and the corresponding guidance law isdesigned However the proposed guidance laws cannotguarantee strictly finite-time convergence of the slidingmode surface and there are nonconvergence factors so theconvergence rate will be reduced In [12] the problem ofnonstrict convergence of the sliding surface is studied and anonsingular terminal sliding surface with strictly finite-timeconvergence is proposed However the sliding surfacefunction is not smooth the system can only converge to abounded region and the specific range cannot be given
For the disturbance problems such as target maneu-vering and system disturbance there are currently threemethods for processing most documents (1) designing the
HindawiJournal of Control Science and EngineeringVolume 2020 Article ID 1058347 11 pageshttpsdoiorg10115520201058347
disturbance observer to estimate the disturbance in real timeand online [13 14] (2) designing the adaptive law to esti-mate the upper bound of disturbance [15 16] and (3) usingthe robustness of the sliding mode control to resist inter-ference -ese methods need to introduce symbolic functionterms which will make the control quantity discontinuousand easy to cause the chattering phenomenon Most liter-ature studies smooth the symbol terms to reduce chatteringHowever at the same time they also change the inherentstructure of the sliding mode control and weaken the ro-bustness of the sliding mode control system For the chat-tering problem a second-order sliding mode supertwistingalgorithm is proposed in [17] It has the advantages of simpleform avoiding chattering and strong robustness Howeverthe control law of the supertwisting algorithm is not smooththe parameter selection needs to know the upper boundaryinformation of system disturbance and the convergencespeed is slow when the system states are far from theequilibrium point
In order to solve the above problems this paper im-proves an adaptive smooth supertwisting algorithm whichsolves the problems of slow convergence speed and theunsmooth control law of the traditional supertwisting al-gorithm and greatly weakens the chattering problem of thesliding mode control At the same time the parameteradaptive law is designed against the disturbance withoutknowing the upper bound information of the disturbanceBased on the idea of fixed-time convergence and piecewisesliding surface a nonsingular fast terminal sliding surfacewith the impact angle constraint is designed A nonsingularfast terminal sliding mode guidance law with the impactangle constraint is proposed based on the adaptive super-twisting algorithm -e global finite-time convergence isproved by constructing the Lyapunov function Finally theeffectiveness and superiority of the guidance law are verifiedby simulation experiments
2 Preparatory Knowledge
21 Relative Dynamics between the Missile and the TargetIn the inertial coordinate system the relative motion rela-tionship between the missile and the target is established asshown in Figure 1 M and T represent the missile and thetarget respectively r is the relative distance between themissile and the target and q is the LOS angle vm and vt arevelocities of the missile and the target respectively and θm
and θt are track angles of the missile and the targetrespectively
According to the relative motion relationship of themissile and the target the relative motion equations of themissile and the target can be obtained as follows
_r minus vm cos q minus θm( 1113857 + vt cos q minus θt( 1113857
r _q vm sin q minus θm( 1113857 minus vt sin q minus θt( 1113857
am vm_θm
at vt_θt
⎧⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎩
(1)
Differentiating _q with respect to time gives
euroq minus 2 _r _q
rminus
am cos q minus θm( 1113857
r+
at cos q minus θt( 1113857
r (2)
-e impact angle is the angle between the missile and thetarget velocity vector at the time of guidance terminal andthe impact angle constraint problem can be transformed intothe terminal LOS angle constraint problem [3ndash8] -ereforethe state equation of the guidance system with the impactangle constraint can be obtained based on (2) as follows
_x1 x2
_x2 f1x2 + f2u + d1113896 (3)
withx1 q minus qd
x2 _q
u am
f1 minus2 _r
r1113888 1113889
f2 minus1r
1113874 1113875
d am minus am cos q minus θm( 1113857 + at cos q minus θt( 1113857( 1113857
r
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
(4)
where qd is the desired terminal LOS angle d can beregarded as the total disturbance of the system
22 Related Lemma For the convenience of analysis andproof the following lemmas are introduced
Lemma 1 (see [18]) Assume that there is a smooth functionV(x) defined on the neighborhood 1113954U sub U0 sub Rn of the originand a1gt 0 and 0 lt b1lt 1 _V(x) + a1V
b1(x)le 0 then theorigin of the system is finite-time stable and the convergencetime satisfies
vm
vt
am θm
θtat
xo
y
M
T
r
q
Figure 1 Relationship of missile-to-target motion
2 Journal of Control Science and Engineering
T1 leV1minus b1 x0( 1113857
a1 1 minus b1( 1113857 (5)
Lemma 2 (see [19]) Assume that Lyapunov function V(x)
satisfies _V(x) le minus a1Vb1(x) minus a2V(x) and a1gt 0 a2gt 0 and
0lt b1lt 1 then the system can converge to the origin in finitetime and the convergence time satisfies
T2 le1
a2 1 minus b1( 1113857ln 1 +
a2V1minus b1 x0( 1113857
a11113888 1113889 (6)
Lemma 3 (see [20 21]) For the nonlinear system_y minus a1|y|b1 sgn(y) minus a2|y|b2 sgn(y) if a1gt 0 a2gt 00lt b1lt 1 and b2gt 1 the system is stable in finite time andthe convergence time satisfies
T3 lt1
a1 1 minus b1( 1113857+
1a2 b2 minus 1( 1113857
(7)
In addition if the system has a small disturbance that is_y minus a1|y|b1 sgn(y) minus a2|y|b2 sgn(y) + ς and ς is a smallpositive number the system can converge to the neigh-borhood Ω |y|le 2ϑ | a1ϑ
b1 + a2ϑb2 ς1113966 1113967 of the origin in
finite time and the convergence time satisfies
T4 lt1
a1 2b1 minus 1( 1113857 1 minus b1( 1113857+
1a2 b2 minus 1( 1113857
(8)
3 Adaptive Fast Supertwisting Algorithm
For the following first-order system_y u + ξ (9)
where y is the system state u is the input and ξ is thedisturbance the supertwisting algorithm can be expressed asfollows
u minus m1|y|(12)sgn(y) + u1
_u1 minus m2sgn(y)
⎧⎨
⎩ (10)
-e supertwisting algorithm can greatly reducechattering and has strong robustness and high precisioncontrol performance [17] However the supertwistingalgorithm has the following disadvantages (1) the controllaw is a continuous function but not a smooth functionwhich will affect the control performance (2) the selectionof control parameters needs to know the upper boundinformation of the system disturbances and (3) when thesystem states are far from the equilibrium point theconvergence speed is slow In view of the above short-comings this paper speeds up the convergence of thealgorithm by adding linear terms to the algorithm Andthe adaptive law does not need the information of theinterference -e improved adaptive supertwisting algo-rithm can be expressed as follows
u minus m1φ1(y) + u1
_u1 minus m2φ2(y)
φ1(y) 1β
|y|βsgn(y) + y
φ2(y) y +y|2βminus 1sgn(y)
1113868111386811138681113868
β+
1β
+ 11113888 1113889|y|βsgn(y)
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
(11)
where (12)le βlt 1-e parameter adaptive law is designed as follows
_m1 (a + c)φ1(y)φ2(y)sgn(|y| minus ε)
m2 bm1
m1(0)gt 0
⎧⎪⎪⎨
⎪⎪⎩(12)
where agt 0 bgt 0 cgt 0 and εgt 0Substituting (11) into (9)
_y minus m1φ1(y) + u1 + ξ
_u1 minus m2φ2(y)1113896 (13)
Remark 1 It can be seen from (13) that when the systemstate is far away from the equilibrium point the linear term yin (11) will accelerate the convergence rate of the systemWhen the system is close to the equilibrium point thenonlinear term |y|βsgn(y) plays an important role in ac-celerating the convergence rate of the system -ereforecompared with the traditional supertwisting algorithm theimproved adaptive supertwisting algorithm (11) has a fasterconvergence speed
Remark 2 Due to the measurement noise of the system thestate of the system cannot reach the equilibrium pointcompletely In order to avoid the parameter increasing toinfinity the term sgn(|y| minus ε) is added to the adaptive law toavoid the problem of overestimation [22]
Remark 3 It is obvious that φ1(s)φ2(s)ge 0 So it can be seenfrom (12) that when |y|gt ε m1 and m2 will gradually in-crease making the system state to converge When thesystem state converges to |y|lt ε m1 and m2 will decreasegradually If m1 and m2 decrease to the point where theinterference cannot be eliminated the system state willdeviate from |y|lt ε At this point m1 and m2 will graduallyincrease under the effect of the adaptive law making thesystem state converge to |y|lt ε Repeat the previous processm1 andm2 will gradually decrease -ereforem1 andm2 areglobally bounded
For the total disturbance ξ of the system the followingassumption can be made
Journal of Control Science and Engineering 3
Assumption 1 ξ is bounded ξ ξ1 + ξ2 ξ1 is non-differentiable and ξ2 is differentiable they satisfy
minus cφ1(y)φ2(y) 1113957m11113868111386811138681113868
1113868111386811138681113868
minus2
radiccφ1(y)φ2(y)V
(12)k
_V2 le minusbλ(1minus β2)
min (P)V(1+β2)z
λmax(P)minus
b
λmax(P)Vz
minus2
radiccφ1(y)φ2(y)V
(12)k
minus μVc2
(31)
with
μ μbλ(1minus β2)
min (P)V(1+β2)z
λmax(P)
b
λmax(P)
2
radiccφ1(y)φ2(y)gt 0
(32)
c c1 + β2
112
1113888 1113889 (33)
It is known that (12)lt clt 1 from (12)le βlt 1 so_V2 le minus μV
c2 le 0 (34)
According to Lemma 1 V2 can converge in finite time ycan converge to |y|le ε and the convergence time satisfies
t1 leV
1minus c2 (0)
μ(1 minus c) (35)
When |y|le ε if m1 and m2 decrease to the point wherethe interference cannot be eliminated the system state willdeviate from |y|le ε In this case m1 and m2 will increaseagain under the effect of the adaptive law making the systemstate converge to |y|le ε
4 Guidance Law Design
-e terminal sliding mode control adopts the nonlinearfunction as the sliding mode surface which can make thesystem states converge in finite time but the method hassingular problems In order to avoid singular problems
based on the piecewise sliding surface [12] and Lemma 3 anonsingular fast terminal sliding surface is designed as
Remark 4 Condition (1) can ensure that gprime(x1) is a con-tinuous bounded function and eliminates singular problemsand when the system reaches the sliding surface s 0 x1 andx2 are always with different signs ensuring that the systemstate is convergent Condition (2) ensures that the slidingsurface s is a continuous function Condition (3) guaranteesthat g(x1) is bounded in x1 isin (minus δ δ) and ψprime(x1) is acontinuous function so s is a smooth function
According to the above conditions this paper selectsfunction g(x1) as follows
g x1( 1113857 λ1x1 + λ2x31 (40)
where λ1 (3 minus α22)εα2minus 1 and λ2 (α2 minus 12)εα2minus 3Substituting (3) into (38) yields
_s f1x2 + f2u + d + k1α1 x11113868111386811138681113868
In order to counteract the disturbance suppress chat-tering and accelerate the convergence speed of the slidingsurface based on the adaptive smooth fast supertwistingalgorithm proposed in the second section an auxiliaryguidance law is designed as
Journal of Control Science and Engineering 5
uaux minus fminus 12 k3φ1(s) + k4 1113946
t
0φ2(s)dt (44)
-e parameter adaptive law is designed as_k3 (a + c)φ1(s)φ2(s)sgn(|s| minus ε)
k4 bk3
k3(0)gt 0
⎧⎪⎪⎪⎨
⎪⎪⎪⎩
(45)
Combining with (42) and (44) we can design a non-singular terminal sliding mode guidance law with the impactangle constraint based on the adaptive supertwisting algo-rithm as
For the convenience of description the design guidancelaw (46) is abbreviated as ASNTSMG
5 Simulation Analysis
In order to test the performance of the designed guidance lawASNTSMG this section conducts simulation analysis basedon ballistic simulation in different scenarios -e initial po-sition of themissile is (0m 0m) and the initial position of thetarget is (1000m 5000m) -e missilersquos velocity isvm 500ms and the targetrsquos velocity is vt 250ms -eacceleration of gravity is g 98ms2 and the maximumavailable overload of the missile is 20 g -e parameters ofASNTSMG are set as follows k1 k2 2 a1 3 α2 (57)ε 001 δ 0001 a 05 and b c 1
In order to verify the superiority of the designed guidancelaw this section also carries out the nonsingular fast terminalsliding mode guidance law (NFTSMG) proposed in [23] andthe second-order nonsingular terminal sliding mode guidancelaw (SONTSMG) proposed in [24] to perform a comparativesimulation -e expression of NFTSMG is
u r
cos q minus θm( 1113857
1k2a2
x2minus a22 1 + k1a1x
a1minus 111113872 1113873 minus
2 _r
rx21113890
+αs + β|s|csgn(s)
r1113891
(47)
-e parameters are set as follows α 600 β 500α1 (75) α2 (57) k1 k2 2 and c 05
-e expression of SONTSMG is
u minus2 _r
rx2 +
rβα
x2minus α2 + z1 + k1|s|
1minus (1c)sgn(s)
+ k2 1113946t
0
xαminus 12r
|s|1minus (2c)sgn(s)dt
(48)
-e parameters are set as follows k1 600 k2 100α (75) β 05 and c 21
-e average overload Nme (unit g) is introduced toevaluate the energy consumption in the process of guidancewhich is defined as follows
Case 1 Attack moving target with different impact angleconstraints set qd as 20deg 30deg 40deg and 50deg respectively andθm0 45deg -e target makes sinusoidal maneuver and itsacceleration is at 30sin (πt5)ms2 and θt0 150deg -esimulation results are shown in Figure 2
It can be seen from Figures 2(a) and 2(b) that ASNTSMGcan effectively intercept the target with different impactangle constraints -e miss distances are 0374m 0428m0408m and 0479m respectively -is method can hit thetarget accurately It can be seen from Figures 2(c)ndash2(e) thatthe sliding surface and the LOS angular rate can converge tozero in finite time and LOS angle can effectively converge tothe expected impact angle With the increase of qd theconvergence time increases -is is because the larger qd isthe larger the initial LOS angle deviation will be and theconvergence time is related to the initial value which leads tothe corresponding growth of the convergence time
Figure 2(f) shows the overload curve of the missilewhich is saturated in the early stage and the larger θm0 thelonger the saturation time which is mainly due to the largeroverload needed in the earlier stage which makes the missilemeet the requirements of angle constraint and guidanceaccuracy When q and _q approach the expected values theoverload gradually approaches zero which ensures that themissile has sufficient overload margin to deal with otherunknown disturbances in the later stage of guidance
Case 2 Comparative simulation of ASNTSMG NFTSMGand SONTSMG the relevant initial parameters are set toθm0 45deg qd 45deg and θt0 180deg -e movement of thetarget is set as follows
(1) Cosine motion at 30cos (πt5)ms2
(2) Square wave motion at 30sgn (sin (πt5))ms2
-e simulation results are shown in Figures 3ndash6 andTable 1
Figure 3 shows the trajectories of the missile and thetarget It can be seen that the missile can track and interceptthe target under the three guidance laws Compared withNFTSMG and SONTSMG the trajectory of ASNTSMG isrelatively smooth indicating that its attack time is relativelyshort which can be verified by Table 1 Figure 4 shows theLOS angle curve All three guidance laws can make the LOSangle gradually converge to the expected angle ASNTSMGcanmake the LOS angle converge to the expected angle morequickly NFTSMG adopts the robustness of the sliding modecontrol to cancel the disturbance of target maneuver so itcan only make the system states converge to the neigh-borhood of origin in finite time NFTSMG and SONTSMGadopt the traditional nonsingular terminal sliding surface
Figure 2 Simulation results of Case 1 (a) Missile and target motion trajectory (b) Relative distance between the missile and the target(c) Sliding mode (d) LOS angle (e) LOS angular rate (f ) Missile overload
Journal of Control Science and Engineering 7
which has the nonstrict finite-time convergence problemand the nonconvergence factor slows down the convergencespeed Figure 5 shows the LOS angular rate curve Under thethree guidance laws the LOS angular rates converge to zeroin finite time ASNTSMG has smaller convergence error andfaster convergence speed Figure 6 shows the overload curveof the missile Due to the finite-time convergence of LOSangle and angular rate the missile needs a large overload inthe early stage of guidance -erefore the overloads of threeguidance laws are all saturated in the early stage When theLOS angle and the angular rate converge the missile
overload gradually converges to zero in the later stage Andthe convergence speed of ASNTSMG is faster
Table 1 shows the simulation results of attack timemiss distance LOS angle error and average overloadunder the three guidance laws It can be seen that com-pared with NFTSMG and SONTSMG ASNTSMG hassmaller attack time miss distance terminal LOS angleerror and average overload so ASNTSMG has betterguidance performance
According to the analysis of the simulation results of twocases ASNTSMG can hit the target precisely with the
expected impact angle under different expected LOS anglesand target maneuvering conditions Compared with theexisting guidance laws NFTSMG and SONTSMGASNTSMG can effectively attack the target with less impactangle error miss distance and energy consumption whichverifies the effectiveness and superiority of ASNTSMG
6 Conclusion
In this paper a nonsingular fast terminal sliding modeguidance law is proposed to solve the problem of guidanceaccuracy and impact angle constraint -rough theoreticalanalysis and simulation verification the following conclu-sions can be obtained
(1) -e proposed adaptive smooth supertwisting algo-rithm can effectively counteract the disturbance ofthe system and accelerate the convergence speed ofthe system without knowing the upper bound of thedisturbance
(2) -e designed nonsingular terminal sliding modesurface can realize the fast finite-time convergence ofthe system states and ensure the impact angleconstraint and guidance accuracy requirements
(3) -is guidance law can attack the target preciselyunder the conditions of different expected LOSangles and target maneuvers Compared with theexisting nonsingular fast terminal sliding modeguidance law and second-order nonsingular termi-nal sliding mode guidance law this law has higherguidance accuracy and angle constraint accuracy andconsumes less energy
Data Availability
-e data used to support the findings of this study are in-cluded within the article
Conflicts of Interest
-e authors declare that there are no conflicts of interestregarding the publication of this paper
References
[1] H Cai Z D Hu and Y Cao ldquoA survey of guidance law withterminal impact angle constraintsrdquo Journal of Astronauticsvol 31 no 2 pp 315ndash323 2010
[2] J M Song and T Q Zhang ldquoPassive homingmissilersquos variablestructure proportional navigation with terminal angularconstraintrdquo Chinese Journal of Aeronautics vol 14 no 2pp 83ndash87 2001
[3] P Wu and M Yang ldquoVariable structure guidance law withterminal attack angle constraintrdquo Journal of Solid RocketTechnology vol 31 no 2 pp 116ndash120 2002
[4] S R Kumar S Rao and D Ghose ldquoSliding-mode guidanceand control for all-aspect interceptors with terminal angleconstraintsrdquo Journal of Guidance Control and Dynamicsvol 35 no 4 pp 1230ndash1246 2012
[5] H B Zhou S M Song and M Y Xu ldquoDesign of terminalsliding-mode guidance law with attack angle constraintsrdquo in
Proceedings of the 2013 25th Chinese Control and DecisionConference (CCDC) Guiyang China May 2013
[6] Y X Zhang M W Sun and Z Q Chen ldquoFinite-timeconvergent guidance law with impact angle constraint basedon sliding-mode controlrdquo Nonlinear Dynamic vol 7 no 3pp 619ndash625 2012
[7] X H Yu and Z H Man ldquoFast terminal sliding-mode controldesign for nonlinear dynamical systemsrdquo IEEE Transactionson Circuits and Systems I Fundamental Jeory and Appli-cations vol 49 no 2 pp 261ndash264 2002
[8] J M Song S M Song and Y Guo ldquoNonlinear disturbanceobserver based fast terminal sliding mode guidance withimpact angle constraintsrdquo International Journal of InnovativeComputing Information and Control vol 11 no 3pp 787ndash802 2015
[9] S R Kumar S Rao and D Ghose ldquoNonsingular terminalslidingmode guidance with impact angle constraintsrdquo Journalof Guidance Control and Dynamics vol 37 no 4pp 1114ndash1130 2014
[10] S F Xiong W H Wang and S Wang ldquoNonsingular fastterminal sliding-mode guidance with intercept angle con-straintrdquo Control Jeory amp Applications vol 31 no 3pp 269ndash278 2014
[11] S C Yang K Q Zhang and P Chen ldquoAdaptive terminalsliding mode guidance law with impact angle constraintrdquoJournal of Beijing University of Aeronautics and Astronauticsvol 42 no 8 pp 1566ndash1574 2016
[12] Z L Zhang and J Zhou ldquoStrictly convergent non-singularterminal sliding mode guidance law with impact angle con-straintsrdquo Optik vol 127 no 22 pp 10971ndash10980 2016
[13] Z Zhang C Man S Li and S Jin ldquoFinite-time guidance lawsfor three-dimensional missile-target interceptionrdquo Journal ofAerospace Engineering vol 230 no 2 pp 392ndash403 2015
[14] N Zhang W Gai M Zhong and J Zhang ldquoA fast finite-timeconvergent guidance law with nonlinear disturbance observerfor unmanned aerial vehicles collision avoidancerdquo AerospaceScience and Technology vol 86 pp 204ndash214 2019
[15] J Song S Song and H Zhou ldquoAdaptive nonsingular fastterminal sliding mode guidance law with impact angle con-straintsrdquo International Journal of Control Automation andSystems vol 14 no 1 pp 99ndash114 2016
[16] J Song and S Song ldquoRobust impact angle constraintsguidance law with autopilot lag and acceleration saturationconsiderationrdquo Transactions of the Institute of Measurementand Control vol 41 no 1 pp 182ndash192 2019
[17] A Levant ldquoPrinciples of 2-sliding mode designrdquo Automaticavol 43 no 4 pp 576ndash586 2007
[18] A M Zhou ldquoFinite-time output feedback attitude trackingcontrol for rigid spacecraftrdquo IEEE Transactions on ControlSystems Technology vol 22 no 1 pp 338ndash345 2014
[19] B Li Q Hu Y Yu and G Ma ldquoObserver-based fault-tolerantattitude control for rigid spacecraftrdquo IEEE Transactions onAerospace and Electronic Systems vol 53 no 5 pp 2572ndash2582 2017
[20] H Li and Y Cai ldquoOn SFTSM control with fixed-time con-vergencerdquo IET Control Jeory amp Applications vol 11 no 6pp 766ndash773 2017
[21] J Ni L Liu C Liu X Hu and T Shen ldquoFixed-time dynamicsurface high-order slidingmode control for chaotic oscillationin power systemrdquo Nonlinear Dynamics vol 86 no 1pp 401ndash420 2016
[22] Q Zong Z S Zhao and J Zhang ldquoHigher order sliding modecontrol with self-tuning law based on integral sliding moderdquo
10 Journal of Control Science and Engineering
IET Control Jeory and Application vol 4 no 7 pp 1282ndash1289 2008
[23] S He D Lin and J Wang ldquoContinuous second-order slidingmode based impact angle guidance lawrdquo Aerospace Scienceand Technology vol 41 pp 199ndash208 2015
[24] B L Cong Z Chen and X D Liu ldquoOn adaptive sliding modecontrol without switching gain overestimationrdquo InternationalJournal of Robust and Nonlinear Control vol 24 no 3pp 515ndash531 2014
Journal of Control Science and Engineering 11
disturbance observer to estimate the disturbance in real timeand online [13 14] (2) designing the adaptive law to esti-mate the upper bound of disturbance [15 16] and (3) usingthe robustness of the sliding mode control to resist inter-ference -ese methods need to introduce symbolic functionterms which will make the control quantity discontinuousand easy to cause the chattering phenomenon Most liter-ature studies smooth the symbol terms to reduce chatteringHowever at the same time they also change the inherentstructure of the sliding mode control and weaken the ro-bustness of the sliding mode control system For the chat-tering problem a second-order sliding mode supertwistingalgorithm is proposed in [17] It has the advantages of simpleform avoiding chattering and strong robustness Howeverthe control law of the supertwisting algorithm is not smooththe parameter selection needs to know the upper boundaryinformation of system disturbance and the convergencespeed is slow when the system states are far from theequilibrium point
In order to solve the above problems this paper im-proves an adaptive smooth supertwisting algorithm whichsolves the problems of slow convergence speed and theunsmooth control law of the traditional supertwisting al-gorithm and greatly weakens the chattering problem of thesliding mode control At the same time the parameteradaptive law is designed against the disturbance withoutknowing the upper bound information of the disturbanceBased on the idea of fixed-time convergence and piecewisesliding surface a nonsingular fast terminal sliding surfacewith the impact angle constraint is designed A nonsingularfast terminal sliding mode guidance law with the impactangle constraint is proposed based on the adaptive super-twisting algorithm -e global finite-time convergence isproved by constructing the Lyapunov function Finally theeffectiveness and superiority of the guidance law are verifiedby simulation experiments
2 Preparatory Knowledge
21 Relative Dynamics between the Missile and the TargetIn the inertial coordinate system the relative motion rela-tionship between the missile and the target is established asshown in Figure 1 M and T represent the missile and thetarget respectively r is the relative distance between themissile and the target and q is the LOS angle vm and vt arevelocities of the missile and the target respectively and θm
and θt are track angles of the missile and the targetrespectively
According to the relative motion relationship of themissile and the target the relative motion equations of themissile and the target can be obtained as follows
_r minus vm cos q minus θm( 1113857 + vt cos q minus θt( 1113857
r _q vm sin q minus θm( 1113857 minus vt sin q minus θt( 1113857
am vm_θm
at vt_θt
⎧⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎩
(1)
Differentiating _q with respect to time gives
euroq minus 2 _r _q
rminus
am cos q minus θm( 1113857
r+
at cos q minus θt( 1113857
r (2)
-e impact angle is the angle between the missile and thetarget velocity vector at the time of guidance terminal andthe impact angle constraint problem can be transformed intothe terminal LOS angle constraint problem [3ndash8] -ereforethe state equation of the guidance system with the impactangle constraint can be obtained based on (2) as follows
_x1 x2
_x2 f1x2 + f2u + d1113896 (3)
withx1 q minus qd
x2 _q
u am
f1 minus2 _r
r1113888 1113889
f2 minus1r
1113874 1113875
d am minus am cos q minus θm( 1113857 + at cos q minus θt( 1113857( 1113857
r
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
(4)
where qd is the desired terminal LOS angle d can beregarded as the total disturbance of the system
22 Related Lemma For the convenience of analysis andproof the following lemmas are introduced
Lemma 1 (see [18]) Assume that there is a smooth functionV(x) defined on the neighborhood 1113954U sub U0 sub Rn of the originand a1gt 0 and 0 lt b1lt 1 _V(x) + a1V
b1(x)le 0 then theorigin of the system is finite-time stable and the convergencetime satisfies
vm
vt
am θm
θtat
xo
y
M
T
r
q
Figure 1 Relationship of missile-to-target motion
2 Journal of Control Science and Engineering
T1 leV1minus b1 x0( 1113857
a1 1 minus b1( 1113857 (5)
Lemma 2 (see [19]) Assume that Lyapunov function V(x)
satisfies _V(x) le minus a1Vb1(x) minus a2V(x) and a1gt 0 a2gt 0 and
0lt b1lt 1 then the system can converge to the origin in finitetime and the convergence time satisfies
T2 le1
a2 1 minus b1( 1113857ln 1 +
a2V1minus b1 x0( 1113857
a11113888 1113889 (6)
Lemma 3 (see [20 21]) For the nonlinear system_y minus a1|y|b1 sgn(y) minus a2|y|b2 sgn(y) if a1gt 0 a2gt 00lt b1lt 1 and b2gt 1 the system is stable in finite time andthe convergence time satisfies
T3 lt1
a1 1 minus b1( 1113857+
1a2 b2 minus 1( 1113857
(7)
In addition if the system has a small disturbance that is_y minus a1|y|b1 sgn(y) minus a2|y|b2 sgn(y) + ς and ς is a smallpositive number the system can converge to the neigh-borhood Ω |y|le 2ϑ | a1ϑ
b1 + a2ϑb2 ς1113966 1113967 of the origin in
finite time and the convergence time satisfies
T4 lt1
a1 2b1 minus 1( 1113857 1 minus b1( 1113857+
1a2 b2 minus 1( 1113857
(8)
3 Adaptive Fast Supertwisting Algorithm
For the following first-order system_y u + ξ (9)
where y is the system state u is the input and ξ is thedisturbance the supertwisting algorithm can be expressed asfollows
u minus m1|y|(12)sgn(y) + u1
_u1 minus m2sgn(y)
⎧⎨
⎩ (10)
-e supertwisting algorithm can greatly reducechattering and has strong robustness and high precisioncontrol performance [17] However the supertwistingalgorithm has the following disadvantages (1) the controllaw is a continuous function but not a smooth functionwhich will affect the control performance (2) the selectionof control parameters needs to know the upper boundinformation of the system disturbances and (3) when thesystem states are far from the equilibrium point theconvergence speed is slow In view of the above short-comings this paper speeds up the convergence of thealgorithm by adding linear terms to the algorithm Andthe adaptive law does not need the information of theinterference -e improved adaptive supertwisting algo-rithm can be expressed as follows
u minus m1φ1(y) + u1
_u1 minus m2φ2(y)
φ1(y) 1β
|y|βsgn(y) + y
φ2(y) y +y|2βminus 1sgn(y)
1113868111386811138681113868
β+
1β
+ 11113888 1113889|y|βsgn(y)
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
(11)
where (12)le βlt 1-e parameter adaptive law is designed as follows
_m1 (a + c)φ1(y)φ2(y)sgn(|y| minus ε)
m2 bm1
m1(0)gt 0
⎧⎪⎪⎨
⎪⎪⎩(12)
where agt 0 bgt 0 cgt 0 and εgt 0Substituting (11) into (9)
_y minus m1φ1(y) + u1 + ξ
_u1 minus m2φ2(y)1113896 (13)
Remark 1 It can be seen from (13) that when the systemstate is far away from the equilibrium point the linear term yin (11) will accelerate the convergence rate of the systemWhen the system is close to the equilibrium point thenonlinear term |y|βsgn(y) plays an important role in ac-celerating the convergence rate of the system -ereforecompared with the traditional supertwisting algorithm theimproved adaptive supertwisting algorithm (11) has a fasterconvergence speed
Remark 2 Due to the measurement noise of the system thestate of the system cannot reach the equilibrium pointcompletely In order to avoid the parameter increasing toinfinity the term sgn(|y| minus ε) is added to the adaptive law toavoid the problem of overestimation [22]
Remark 3 It is obvious that φ1(s)φ2(s)ge 0 So it can be seenfrom (12) that when |y|gt ε m1 and m2 will gradually in-crease making the system state to converge When thesystem state converges to |y|lt ε m1 and m2 will decreasegradually If m1 and m2 decrease to the point where theinterference cannot be eliminated the system state willdeviate from |y|lt ε At this point m1 and m2 will graduallyincrease under the effect of the adaptive law making thesystem state converge to |y|lt ε Repeat the previous processm1 andm2 will gradually decrease -ereforem1 andm2 areglobally bounded
For the total disturbance ξ of the system the followingassumption can be made
Journal of Control Science and Engineering 3
Assumption 1 ξ is bounded ξ ξ1 + ξ2 ξ1 is non-differentiable and ξ2 is differentiable they satisfy
minus cφ1(y)φ2(y) 1113957m11113868111386811138681113868
1113868111386811138681113868
minus2
radiccφ1(y)φ2(y)V
(12)k
_V2 le minusbλ(1minus β2)
min (P)V(1+β2)z
λmax(P)minus
b
λmax(P)Vz
minus2
radiccφ1(y)φ2(y)V
(12)k
minus μVc2
(31)
with
μ μbλ(1minus β2)
min (P)V(1+β2)z
λmax(P)
b
λmax(P)
2
radiccφ1(y)φ2(y)gt 0
(32)
c c1 + β2
112
1113888 1113889 (33)
It is known that (12)lt clt 1 from (12)le βlt 1 so_V2 le minus μV
c2 le 0 (34)
According to Lemma 1 V2 can converge in finite time ycan converge to |y|le ε and the convergence time satisfies
t1 leV
1minus c2 (0)
μ(1 minus c) (35)
When |y|le ε if m1 and m2 decrease to the point wherethe interference cannot be eliminated the system state willdeviate from |y|le ε In this case m1 and m2 will increaseagain under the effect of the adaptive law making the systemstate converge to |y|le ε
4 Guidance Law Design
-e terminal sliding mode control adopts the nonlinearfunction as the sliding mode surface which can make thesystem states converge in finite time but the method hassingular problems In order to avoid singular problems
based on the piecewise sliding surface [12] and Lemma 3 anonsingular fast terminal sliding surface is designed as
Remark 4 Condition (1) can ensure that gprime(x1) is a con-tinuous bounded function and eliminates singular problemsand when the system reaches the sliding surface s 0 x1 andx2 are always with different signs ensuring that the systemstate is convergent Condition (2) ensures that the slidingsurface s is a continuous function Condition (3) guaranteesthat g(x1) is bounded in x1 isin (minus δ δ) and ψprime(x1) is acontinuous function so s is a smooth function
According to the above conditions this paper selectsfunction g(x1) as follows
g x1( 1113857 λ1x1 + λ2x31 (40)
where λ1 (3 minus α22)εα2minus 1 and λ2 (α2 minus 12)εα2minus 3Substituting (3) into (38) yields
_s f1x2 + f2u + d + k1α1 x11113868111386811138681113868
In order to counteract the disturbance suppress chat-tering and accelerate the convergence speed of the slidingsurface based on the adaptive smooth fast supertwistingalgorithm proposed in the second section an auxiliaryguidance law is designed as
Journal of Control Science and Engineering 5
uaux minus fminus 12 k3φ1(s) + k4 1113946
t
0φ2(s)dt (44)
-e parameter adaptive law is designed as_k3 (a + c)φ1(s)φ2(s)sgn(|s| minus ε)
k4 bk3
k3(0)gt 0
⎧⎪⎪⎪⎨
⎪⎪⎪⎩
(45)
Combining with (42) and (44) we can design a non-singular terminal sliding mode guidance law with the impactangle constraint based on the adaptive supertwisting algo-rithm as
For the convenience of description the design guidancelaw (46) is abbreviated as ASNTSMG
5 Simulation Analysis
In order to test the performance of the designed guidance lawASNTSMG this section conducts simulation analysis basedon ballistic simulation in different scenarios -e initial po-sition of themissile is (0m 0m) and the initial position of thetarget is (1000m 5000m) -e missilersquos velocity isvm 500ms and the targetrsquos velocity is vt 250ms -eacceleration of gravity is g 98ms2 and the maximumavailable overload of the missile is 20 g -e parameters ofASNTSMG are set as follows k1 k2 2 a1 3 α2 (57)ε 001 δ 0001 a 05 and b c 1
In order to verify the superiority of the designed guidancelaw this section also carries out the nonsingular fast terminalsliding mode guidance law (NFTSMG) proposed in [23] andthe second-order nonsingular terminal sliding mode guidancelaw (SONTSMG) proposed in [24] to perform a comparativesimulation -e expression of NFTSMG is
u r
cos q minus θm( 1113857
1k2a2
x2minus a22 1 + k1a1x
a1minus 111113872 1113873 minus
2 _r
rx21113890
+αs + β|s|csgn(s)
r1113891
(47)
-e parameters are set as follows α 600 β 500α1 (75) α2 (57) k1 k2 2 and c 05
-e expression of SONTSMG is
u minus2 _r
rx2 +
rβα
x2minus α2 + z1 + k1|s|
1minus (1c)sgn(s)
+ k2 1113946t
0
xαminus 12r
|s|1minus (2c)sgn(s)dt
(48)
-e parameters are set as follows k1 600 k2 100α (75) β 05 and c 21
-e average overload Nme (unit g) is introduced toevaluate the energy consumption in the process of guidancewhich is defined as follows
Case 1 Attack moving target with different impact angleconstraints set qd as 20deg 30deg 40deg and 50deg respectively andθm0 45deg -e target makes sinusoidal maneuver and itsacceleration is at 30sin (πt5)ms2 and θt0 150deg -esimulation results are shown in Figure 2
It can be seen from Figures 2(a) and 2(b) that ASNTSMGcan effectively intercept the target with different impactangle constraints -e miss distances are 0374m 0428m0408m and 0479m respectively -is method can hit thetarget accurately It can be seen from Figures 2(c)ndash2(e) thatthe sliding surface and the LOS angular rate can converge tozero in finite time and LOS angle can effectively converge tothe expected impact angle With the increase of qd theconvergence time increases -is is because the larger qd isthe larger the initial LOS angle deviation will be and theconvergence time is related to the initial value which leads tothe corresponding growth of the convergence time
Figure 2(f) shows the overload curve of the missilewhich is saturated in the early stage and the larger θm0 thelonger the saturation time which is mainly due to the largeroverload needed in the earlier stage which makes the missilemeet the requirements of angle constraint and guidanceaccuracy When q and _q approach the expected values theoverload gradually approaches zero which ensures that themissile has sufficient overload margin to deal with otherunknown disturbances in the later stage of guidance
Case 2 Comparative simulation of ASNTSMG NFTSMGand SONTSMG the relevant initial parameters are set toθm0 45deg qd 45deg and θt0 180deg -e movement of thetarget is set as follows
(1) Cosine motion at 30cos (πt5)ms2
(2) Square wave motion at 30sgn (sin (πt5))ms2
-e simulation results are shown in Figures 3ndash6 andTable 1
Figure 3 shows the trajectories of the missile and thetarget It can be seen that the missile can track and interceptthe target under the three guidance laws Compared withNFTSMG and SONTSMG the trajectory of ASNTSMG isrelatively smooth indicating that its attack time is relativelyshort which can be verified by Table 1 Figure 4 shows theLOS angle curve All three guidance laws can make the LOSangle gradually converge to the expected angle ASNTSMGcanmake the LOS angle converge to the expected angle morequickly NFTSMG adopts the robustness of the sliding modecontrol to cancel the disturbance of target maneuver so itcan only make the system states converge to the neigh-borhood of origin in finite time NFTSMG and SONTSMGadopt the traditional nonsingular terminal sliding surface
Figure 2 Simulation results of Case 1 (a) Missile and target motion trajectory (b) Relative distance between the missile and the target(c) Sliding mode (d) LOS angle (e) LOS angular rate (f ) Missile overload
Journal of Control Science and Engineering 7
which has the nonstrict finite-time convergence problemand the nonconvergence factor slows down the convergencespeed Figure 5 shows the LOS angular rate curve Under thethree guidance laws the LOS angular rates converge to zeroin finite time ASNTSMG has smaller convergence error andfaster convergence speed Figure 6 shows the overload curveof the missile Due to the finite-time convergence of LOSangle and angular rate the missile needs a large overload inthe early stage of guidance -erefore the overloads of threeguidance laws are all saturated in the early stage When theLOS angle and the angular rate converge the missile
overload gradually converges to zero in the later stage Andthe convergence speed of ASNTSMG is faster
Table 1 shows the simulation results of attack timemiss distance LOS angle error and average overloadunder the three guidance laws It can be seen that com-pared with NFTSMG and SONTSMG ASNTSMG hassmaller attack time miss distance terminal LOS angleerror and average overload so ASNTSMG has betterguidance performance
According to the analysis of the simulation results of twocases ASNTSMG can hit the target precisely with the
expected impact angle under different expected LOS anglesand target maneuvering conditions Compared with theexisting guidance laws NFTSMG and SONTSMGASNTSMG can effectively attack the target with less impactangle error miss distance and energy consumption whichverifies the effectiveness and superiority of ASNTSMG
6 Conclusion
In this paper a nonsingular fast terminal sliding modeguidance law is proposed to solve the problem of guidanceaccuracy and impact angle constraint -rough theoreticalanalysis and simulation verification the following conclu-sions can be obtained
(1) -e proposed adaptive smooth supertwisting algo-rithm can effectively counteract the disturbance ofthe system and accelerate the convergence speed ofthe system without knowing the upper bound of thedisturbance
(2) -e designed nonsingular terminal sliding modesurface can realize the fast finite-time convergence ofthe system states and ensure the impact angleconstraint and guidance accuracy requirements
(3) -is guidance law can attack the target preciselyunder the conditions of different expected LOSangles and target maneuvers Compared with theexisting nonsingular fast terminal sliding modeguidance law and second-order nonsingular termi-nal sliding mode guidance law this law has higherguidance accuracy and angle constraint accuracy andconsumes less energy
Data Availability
-e data used to support the findings of this study are in-cluded within the article
Conflicts of Interest
-e authors declare that there are no conflicts of interestregarding the publication of this paper
References
[1] H Cai Z D Hu and Y Cao ldquoA survey of guidance law withterminal impact angle constraintsrdquo Journal of Astronauticsvol 31 no 2 pp 315ndash323 2010
[2] J M Song and T Q Zhang ldquoPassive homingmissilersquos variablestructure proportional navigation with terminal angularconstraintrdquo Chinese Journal of Aeronautics vol 14 no 2pp 83ndash87 2001
[3] P Wu and M Yang ldquoVariable structure guidance law withterminal attack angle constraintrdquo Journal of Solid RocketTechnology vol 31 no 2 pp 116ndash120 2002
[4] S R Kumar S Rao and D Ghose ldquoSliding-mode guidanceand control for all-aspect interceptors with terminal angleconstraintsrdquo Journal of Guidance Control and Dynamicsvol 35 no 4 pp 1230ndash1246 2012
[5] H B Zhou S M Song and M Y Xu ldquoDesign of terminalsliding-mode guidance law with attack angle constraintsrdquo in
Proceedings of the 2013 25th Chinese Control and DecisionConference (CCDC) Guiyang China May 2013
[6] Y X Zhang M W Sun and Z Q Chen ldquoFinite-timeconvergent guidance law with impact angle constraint basedon sliding-mode controlrdquo Nonlinear Dynamic vol 7 no 3pp 619ndash625 2012
[7] X H Yu and Z H Man ldquoFast terminal sliding-mode controldesign for nonlinear dynamical systemsrdquo IEEE Transactionson Circuits and Systems I Fundamental Jeory and Appli-cations vol 49 no 2 pp 261ndash264 2002
[8] J M Song S M Song and Y Guo ldquoNonlinear disturbanceobserver based fast terminal sliding mode guidance withimpact angle constraintsrdquo International Journal of InnovativeComputing Information and Control vol 11 no 3pp 787ndash802 2015
[9] S R Kumar S Rao and D Ghose ldquoNonsingular terminalslidingmode guidance with impact angle constraintsrdquo Journalof Guidance Control and Dynamics vol 37 no 4pp 1114ndash1130 2014
[10] S F Xiong W H Wang and S Wang ldquoNonsingular fastterminal sliding-mode guidance with intercept angle con-straintrdquo Control Jeory amp Applications vol 31 no 3pp 269ndash278 2014
[11] S C Yang K Q Zhang and P Chen ldquoAdaptive terminalsliding mode guidance law with impact angle constraintrdquoJournal of Beijing University of Aeronautics and Astronauticsvol 42 no 8 pp 1566ndash1574 2016
[12] Z L Zhang and J Zhou ldquoStrictly convergent non-singularterminal sliding mode guidance law with impact angle con-straintsrdquo Optik vol 127 no 22 pp 10971ndash10980 2016
[13] Z Zhang C Man S Li and S Jin ldquoFinite-time guidance lawsfor three-dimensional missile-target interceptionrdquo Journal ofAerospace Engineering vol 230 no 2 pp 392ndash403 2015
[14] N Zhang W Gai M Zhong and J Zhang ldquoA fast finite-timeconvergent guidance law with nonlinear disturbance observerfor unmanned aerial vehicles collision avoidancerdquo AerospaceScience and Technology vol 86 pp 204ndash214 2019
[15] J Song S Song and H Zhou ldquoAdaptive nonsingular fastterminal sliding mode guidance law with impact angle con-straintsrdquo International Journal of Control Automation andSystems vol 14 no 1 pp 99ndash114 2016
[16] J Song and S Song ldquoRobust impact angle constraintsguidance law with autopilot lag and acceleration saturationconsiderationrdquo Transactions of the Institute of Measurementand Control vol 41 no 1 pp 182ndash192 2019
[17] A Levant ldquoPrinciples of 2-sliding mode designrdquo Automaticavol 43 no 4 pp 576ndash586 2007
[18] A M Zhou ldquoFinite-time output feedback attitude trackingcontrol for rigid spacecraftrdquo IEEE Transactions on ControlSystems Technology vol 22 no 1 pp 338ndash345 2014
[19] B Li Q Hu Y Yu and G Ma ldquoObserver-based fault-tolerantattitude control for rigid spacecraftrdquo IEEE Transactions onAerospace and Electronic Systems vol 53 no 5 pp 2572ndash2582 2017
[20] H Li and Y Cai ldquoOn SFTSM control with fixed-time con-vergencerdquo IET Control Jeory amp Applications vol 11 no 6pp 766ndash773 2017
[21] J Ni L Liu C Liu X Hu and T Shen ldquoFixed-time dynamicsurface high-order slidingmode control for chaotic oscillationin power systemrdquo Nonlinear Dynamics vol 86 no 1pp 401ndash420 2016
[22] Q Zong Z S Zhao and J Zhang ldquoHigher order sliding modecontrol with self-tuning law based on integral sliding moderdquo
10 Journal of Control Science and Engineering
IET Control Jeory and Application vol 4 no 7 pp 1282ndash1289 2008
[23] S He D Lin and J Wang ldquoContinuous second-order slidingmode based impact angle guidance lawrdquo Aerospace Scienceand Technology vol 41 pp 199ndash208 2015
[24] B L Cong Z Chen and X D Liu ldquoOn adaptive sliding modecontrol without switching gain overestimationrdquo InternationalJournal of Robust and Nonlinear Control vol 24 no 3pp 515ndash531 2014
Journal of Control Science and Engineering 11
T1 leV1minus b1 x0( 1113857
a1 1 minus b1( 1113857 (5)
Lemma 2 (see [19]) Assume that Lyapunov function V(x)
satisfies _V(x) le minus a1Vb1(x) minus a2V(x) and a1gt 0 a2gt 0 and
0lt b1lt 1 then the system can converge to the origin in finitetime and the convergence time satisfies
T2 le1
a2 1 minus b1( 1113857ln 1 +
a2V1minus b1 x0( 1113857
a11113888 1113889 (6)
Lemma 3 (see [20 21]) For the nonlinear system_y minus a1|y|b1 sgn(y) minus a2|y|b2 sgn(y) if a1gt 0 a2gt 00lt b1lt 1 and b2gt 1 the system is stable in finite time andthe convergence time satisfies
T3 lt1
a1 1 minus b1( 1113857+
1a2 b2 minus 1( 1113857
(7)
In addition if the system has a small disturbance that is_y minus a1|y|b1 sgn(y) minus a2|y|b2 sgn(y) + ς and ς is a smallpositive number the system can converge to the neigh-borhood Ω |y|le 2ϑ | a1ϑ
b1 + a2ϑb2 ς1113966 1113967 of the origin in
finite time and the convergence time satisfies
T4 lt1
a1 2b1 minus 1( 1113857 1 minus b1( 1113857+
1a2 b2 minus 1( 1113857
(8)
3 Adaptive Fast Supertwisting Algorithm
For the following first-order system_y u + ξ (9)
where y is the system state u is the input and ξ is thedisturbance the supertwisting algorithm can be expressed asfollows
u minus m1|y|(12)sgn(y) + u1
_u1 minus m2sgn(y)
⎧⎨
⎩ (10)
-e supertwisting algorithm can greatly reducechattering and has strong robustness and high precisioncontrol performance [17] However the supertwistingalgorithm has the following disadvantages (1) the controllaw is a continuous function but not a smooth functionwhich will affect the control performance (2) the selectionof control parameters needs to know the upper boundinformation of the system disturbances and (3) when thesystem states are far from the equilibrium point theconvergence speed is slow In view of the above short-comings this paper speeds up the convergence of thealgorithm by adding linear terms to the algorithm Andthe adaptive law does not need the information of theinterference -e improved adaptive supertwisting algo-rithm can be expressed as follows
u minus m1φ1(y) + u1
_u1 minus m2φ2(y)
φ1(y) 1β
|y|βsgn(y) + y
φ2(y) y +y|2βminus 1sgn(y)
1113868111386811138681113868
β+
1β
+ 11113888 1113889|y|βsgn(y)
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
(11)
where (12)le βlt 1-e parameter adaptive law is designed as follows
_m1 (a + c)φ1(y)φ2(y)sgn(|y| minus ε)
m2 bm1
m1(0)gt 0
⎧⎪⎪⎨
⎪⎪⎩(12)
where agt 0 bgt 0 cgt 0 and εgt 0Substituting (11) into (9)
_y minus m1φ1(y) + u1 + ξ
_u1 minus m2φ2(y)1113896 (13)
Remark 1 It can be seen from (13) that when the systemstate is far away from the equilibrium point the linear term yin (11) will accelerate the convergence rate of the systemWhen the system is close to the equilibrium point thenonlinear term |y|βsgn(y) plays an important role in ac-celerating the convergence rate of the system -ereforecompared with the traditional supertwisting algorithm theimproved adaptive supertwisting algorithm (11) has a fasterconvergence speed
Remark 2 Due to the measurement noise of the system thestate of the system cannot reach the equilibrium pointcompletely In order to avoid the parameter increasing toinfinity the term sgn(|y| minus ε) is added to the adaptive law toavoid the problem of overestimation [22]
Remark 3 It is obvious that φ1(s)φ2(s)ge 0 So it can be seenfrom (12) that when |y|gt ε m1 and m2 will gradually in-crease making the system state to converge When thesystem state converges to |y|lt ε m1 and m2 will decreasegradually If m1 and m2 decrease to the point where theinterference cannot be eliminated the system state willdeviate from |y|lt ε At this point m1 and m2 will graduallyincrease under the effect of the adaptive law making thesystem state converge to |y|lt ε Repeat the previous processm1 andm2 will gradually decrease -ereforem1 andm2 areglobally bounded
For the total disturbance ξ of the system the followingassumption can be made
Journal of Control Science and Engineering 3
Assumption 1 ξ is bounded ξ ξ1 + ξ2 ξ1 is non-differentiable and ξ2 is differentiable they satisfy
minus cφ1(y)φ2(y) 1113957m11113868111386811138681113868
1113868111386811138681113868
minus2
radiccφ1(y)φ2(y)V
(12)k
_V2 le minusbλ(1minus β2)
min (P)V(1+β2)z
λmax(P)minus
b
λmax(P)Vz
minus2
radiccφ1(y)φ2(y)V
(12)k
minus μVc2
(31)
with
μ μbλ(1minus β2)
min (P)V(1+β2)z
λmax(P)
b
λmax(P)
2
radiccφ1(y)φ2(y)gt 0
(32)
c c1 + β2
112
1113888 1113889 (33)
It is known that (12)lt clt 1 from (12)le βlt 1 so_V2 le minus μV
c2 le 0 (34)
According to Lemma 1 V2 can converge in finite time ycan converge to |y|le ε and the convergence time satisfies
t1 leV
1minus c2 (0)
μ(1 minus c) (35)
When |y|le ε if m1 and m2 decrease to the point wherethe interference cannot be eliminated the system state willdeviate from |y|le ε In this case m1 and m2 will increaseagain under the effect of the adaptive law making the systemstate converge to |y|le ε
4 Guidance Law Design
-e terminal sliding mode control adopts the nonlinearfunction as the sliding mode surface which can make thesystem states converge in finite time but the method hassingular problems In order to avoid singular problems
based on the piecewise sliding surface [12] and Lemma 3 anonsingular fast terminal sliding surface is designed as
Remark 4 Condition (1) can ensure that gprime(x1) is a con-tinuous bounded function and eliminates singular problemsand when the system reaches the sliding surface s 0 x1 andx2 are always with different signs ensuring that the systemstate is convergent Condition (2) ensures that the slidingsurface s is a continuous function Condition (3) guaranteesthat g(x1) is bounded in x1 isin (minus δ δ) and ψprime(x1) is acontinuous function so s is a smooth function
According to the above conditions this paper selectsfunction g(x1) as follows
g x1( 1113857 λ1x1 + λ2x31 (40)
where λ1 (3 minus α22)εα2minus 1 and λ2 (α2 minus 12)εα2minus 3Substituting (3) into (38) yields
_s f1x2 + f2u + d + k1α1 x11113868111386811138681113868
In order to counteract the disturbance suppress chat-tering and accelerate the convergence speed of the slidingsurface based on the adaptive smooth fast supertwistingalgorithm proposed in the second section an auxiliaryguidance law is designed as
Journal of Control Science and Engineering 5
uaux minus fminus 12 k3φ1(s) + k4 1113946
t
0φ2(s)dt (44)
-e parameter adaptive law is designed as_k3 (a + c)φ1(s)φ2(s)sgn(|s| minus ε)
k4 bk3
k3(0)gt 0
⎧⎪⎪⎪⎨
⎪⎪⎪⎩
(45)
Combining with (42) and (44) we can design a non-singular terminal sliding mode guidance law with the impactangle constraint based on the adaptive supertwisting algo-rithm as
For the convenience of description the design guidancelaw (46) is abbreviated as ASNTSMG
5 Simulation Analysis
In order to test the performance of the designed guidance lawASNTSMG this section conducts simulation analysis basedon ballistic simulation in different scenarios -e initial po-sition of themissile is (0m 0m) and the initial position of thetarget is (1000m 5000m) -e missilersquos velocity isvm 500ms and the targetrsquos velocity is vt 250ms -eacceleration of gravity is g 98ms2 and the maximumavailable overload of the missile is 20 g -e parameters ofASNTSMG are set as follows k1 k2 2 a1 3 α2 (57)ε 001 δ 0001 a 05 and b c 1
In order to verify the superiority of the designed guidancelaw this section also carries out the nonsingular fast terminalsliding mode guidance law (NFTSMG) proposed in [23] andthe second-order nonsingular terminal sliding mode guidancelaw (SONTSMG) proposed in [24] to perform a comparativesimulation -e expression of NFTSMG is
u r
cos q minus θm( 1113857
1k2a2
x2minus a22 1 + k1a1x
a1minus 111113872 1113873 minus
2 _r
rx21113890
+αs + β|s|csgn(s)
r1113891
(47)
-e parameters are set as follows α 600 β 500α1 (75) α2 (57) k1 k2 2 and c 05
-e expression of SONTSMG is
u minus2 _r
rx2 +
rβα
x2minus α2 + z1 + k1|s|
1minus (1c)sgn(s)
+ k2 1113946t
0
xαminus 12r
|s|1minus (2c)sgn(s)dt
(48)
-e parameters are set as follows k1 600 k2 100α (75) β 05 and c 21
-e average overload Nme (unit g) is introduced toevaluate the energy consumption in the process of guidancewhich is defined as follows
Case 1 Attack moving target with different impact angleconstraints set qd as 20deg 30deg 40deg and 50deg respectively andθm0 45deg -e target makes sinusoidal maneuver and itsacceleration is at 30sin (πt5)ms2 and θt0 150deg -esimulation results are shown in Figure 2
It can be seen from Figures 2(a) and 2(b) that ASNTSMGcan effectively intercept the target with different impactangle constraints -e miss distances are 0374m 0428m0408m and 0479m respectively -is method can hit thetarget accurately It can be seen from Figures 2(c)ndash2(e) thatthe sliding surface and the LOS angular rate can converge tozero in finite time and LOS angle can effectively converge tothe expected impact angle With the increase of qd theconvergence time increases -is is because the larger qd isthe larger the initial LOS angle deviation will be and theconvergence time is related to the initial value which leads tothe corresponding growth of the convergence time
Figure 2(f) shows the overload curve of the missilewhich is saturated in the early stage and the larger θm0 thelonger the saturation time which is mainly due to the largeroverload needed in the earlier stage which makes the missilemeet the requirements of angle constraint and guidanceaccuracy When q and _q approach the expected values theoverload gradually approaches zero which ensures that themissile has sufficient overload margin to deal with otherunknown disturbances in the later stage of guidance
Case 2 Comparative simulation of ASNTSMG NFTSMGand SONTSMG the relevant initial parameters are set toθm0 45deg qd 45deg and θt0 180deg -e movement of thetarget is set as follows
(1) Cosine motion at 30cos (πt5)ms2
(2) Square wave motion at 30sgn (sin (πt5))ms2
-e simulation results are shown in Figures 3ndash6 andTable 1
Figure 3 shows the trajectories of the missile and thetarget It can be seen that the missile can track and interceptthe target under the three guidance laws Compared withNFTSMG and SONTSMG the trajectory of ASNTSMG isrelatively smooth indicating that its attack time is relativelyshort which can be verified by Table 1 Figure 4 shows theLOS angle curve All three guidance laws can make the LOSangle gradually converge to the expected angle ASNTSMGcanmake the LOS angle converge to the expected angle morequickly NFTSMG adopts the robustness of the sliding modecontrol to cancel the disturbance of target maneuver so itcan only make the system states converge to the neigh-borhood of origin in finite time NFTSMG and SONTSMGadopt the traditional nonsingular terminal sliding surface
Figure 2 Simulation results of Case 1 (a) Missile and target motion trajectory (b) Relative distance between the missile and the target(c) Sliding mode (d) LOS angle (e) LOS angular rate (f ) Missile overload
Journal of Control Science and Engineering 7
which has the nonstrict finite-time convergence problemand the nonconvergence factor slows down the convergencespeed Figure 5 shows the LOS angular rate curve Under thethree guidance laws the LOS angular rates converge to zeroin finite time ASNTSMG has smaller convergence error andfaster convergence speed Figure 6 shows the overload curveof the missile Due to the finite-time convergence of LOSangle and angular rate the missile needs a large overload inthe early stage of guidance -erefore the overloads of threeguidance laws are all saturated in the early stage When theLOS angle and the angular rate converge the missile
overload gradually converges to zero in the later stage Andthe convergence speed of ASNTSMG is faster
Table 1 shows the simulation results of attack timemiss distance LOS angle error and average overloadunder the three guidance laws It can be seen that com-pared with NFTSMG and SONTSMG ASNTSMG hassmaller attack time miss distance terminal LOS angleerror and average overload so ASNTSMG has betterguidance performance
According to the analysis of the simulation results of twocases ASNTSMG can hit the target precisely with the
expected impact angle under different expected LOS anglesand target maneuvering conditions Compared with theexisting guidance laws NFTSMG and SONTSMGASNTSMG can effectively attack the target with less impactangle error miss distance and energy consumption whichverifies the effectiveness and superiority of ASNTSMG
6 Conclusion
In this paper a nonsingular fast terminal sliding modeguidance law is proposed to solve the problem of guidanceaccuracy and impact angle constraint -rough theoreticalanalysis and simulation verification the following conclu-sions can be obtained
(1) -e proposed adaptive smooth supertwisting algo-rithm can effectively counteract the disturbance ofthe system and accelerate the convergence speed ofthe system without knowing the upper bound of thedisturbance
(2) -e designed nonsingular terminal sliding modesurface can realize the fast finite-time convergence ofthe system states and ensure the impact angleconstraint and guidance accuracy requirements
(3) -is guidance law can attack the target preciselyunder the conditions of different expected LOSangles and target maneuvers Compared with theexisting nonsingular fast terminal sliding modeguidance law and second-order nonsingular termi-nal sliding mode guidance law this law has higherguidance accuracy and angle constraint accuracy andconsumes less energy
Data Availability
-e data used to support the findings of this study are in-cluded within the article
Conflicts of Interest
-e authors declare that there are no conflicts of interestregarding the publication of this paper
References
[1] H Cai Z D Hu and Y Cao ldquoA survey of guidance law withterminal impact angle constraintsrdquo Journal of Astronauticsvol 31 no 2 pp 315ndash323 2010
[2] J M Song and T Q Zhang ldquoPassive homingmissilersquos variablestructure proportional navigation with terminal angularconstraintrdquo Chinese Journal of Aeronautics vol 14 no 2pp 83ndash87 2001
[3] P Wu and M Yang ldquoVariable structure guidance law withterminal attack angle constraintrdquo Journal of Solid RocketTechnology vol 31 no 2 pp 116ndash120 2002
[4] S R Kumar S Rao and D Ghose ldquoSliding-mode guidanceand control for all-aspect interceptors with terminal angleconstraintsrdquo Journal of Guidance Control and Dynamicsvol 35 no 4 pp 1230ndash1246 2012
[5] H B Zhou S M Song and M Y Xu ldquoDesign of terminalsliding-mode guidance law with attack angle constraintsrdquo in
Proceedings of the 2013 25th Chinese Control and DecisionConference (CCDC) Guiyang China May 2013
[6] Y X Zhang M W Sun and Z Q Chen ldquoFinite-timeconvergent guidance law with impact angle constraint basedon sliding-mode controlrdquo Nonlinear Dynamic vol 7 no 3pp 619ndash625 2012
[7] X H Yu and Z H Man ldquoFast terminal sliding-mode controldesign for nonlinear dynamical systemsrdquo IEEE Transactionson Circuits and Systems I Fundamental Jeory and Appli-cations vol 49 no 2 pp 261ndash264 2002
[8] J M Song S M Song and Y Guo ldquoNonlinear disturbanceobserver based fast terminal sliding mode guidance withimpact angle constraintsrdquo International Journal of InnovativeComputing Information and Control vol 11 no 3pp 787ndash802 2015
[9] S R Kumar S Rao and D Ghose ldquoNonsingular terminalslidingmode guidance with impact angle constraintsrdquo Journalof Guidance Control and Dynamics vol 37 no 4pp 1114ndash1130 2014
[10] S F Xiong W H Wang and S Wang ldquoNonsingular fastterminal sliding-mode guidance with intercept angle con-straintrdquo Control Jeory amp Applications vol 31 no 3pp 269ndash278 2014
[11] S C Yang K Q Zhang and P Chen ldquoAdaptive terminalsliding mode guidance law with impact angle constraintrdquoJournal of Beijing University of Aeronautics and Astronauticsvol 42 no 8 pp 1566ndash1574 2016
[12] Z L Zhang and J Zhou ldquoStrictly convergent non-singularterminal sliding mode guidance law with impact angle con-straintsrdquo Optik vol 127 no 22 pp 10971ndash10980 2016
[13] Z Zhang C Man S Li and S Jin ldquoFinite-time guidance lawsfor three-dimensional missile-target interceptionrdquo Journal ofAerospace Engineering vol 230 no 2 pp 392ndash403 2015
[14] N Zhang W Gai M Zhong and J Zhang ldquoA fast finite-timeconvergent guidance law with nonlinear disturbance observerfor unmanned aerial vehicles collision avoidancerdquo AerospaceScience and Technology vol 86 pp 204ndash214 2019
[15] J Song S Song and H Zhou ldquoAdaptive nonsingular fastterminal sliding mode guidance law with impact angle con-straintsrdquo International Journal of Control Automation andSystems vol 14 no 1 pp 99ndash114 2016
[16] J Song and S Song ldquoRobust impact angle constraintsguidance law with autopilot lag and acceleration saturationconsiderationrdquo Transactions of the Institute of Measurementand Control vol 41 no 1 pp 182ndash192 2019
[17] A Levant ldquoPrinciples of 2-sliding mode designrdquo Automaticavol 43 no 4 pp 576ndash586 2007
[18] A M Zhou ldquoFinite-time output feedback attitude trackingcontrol for rigid spacecraftrdquo IEEE Transactions on ControlSystems Technology vol 22 no 1 pp 338ndash345 2014
[19] B Li Q Hu Y Yu and G Ma ldquoObserver-based fault-tolerantattitude control for rigid spacecraftrdquo IEEE Transactions onAerospace and Electronic Systems vol 53 no 5 pp 2572ndash2582 2017
[20] H Li and Y Cai ldquoOn SFTSM control with fixed-time con-vergencerdquo IET Control Jeory amp Applications vol 11 no 6pp 766ndash773 2017
[21] J Ni L Liu C Liu X Hu and T Shen ldquoFixed-time dynamicsurface high-order slidingmode control for chaotic oscillationin power systemrdquo Nonlinear Dynamics vol 86 no 1pp 401ndash420 2016
[22] Q Zong Z S Zhao and J Zhang ldquoHigher order sliding modecontrol with self-tuning law based on integral sliding moderdquo
10 Journal of Control Science and Engineering
IET Control Jeory and Application vol 4 no 7 pp 1282ndash1289 2008
[23] S He D Lin and J Wang ldquoContinuous second-order slidingmode based impact angle guidance lawrdquo Aerospace Scienceand Technology vol 41 pp 199ndash208 2015
[24] B L Cong Z Chen and X D Liu ldquoOn adaptive sliding modecontrol without switching gain overestimationrdquo InternationalJournal of Robust and Nonlinear Control vol 24 no 3pp 515ndash531 2014
Journal of Control Science and Engineering 11
Assumption 1 ξ is bounded ξ ξ1 + ξ2 ξ1 is non-differentiable and ξ2 is differentiable they satisfy
minus cφ1(y)φ2(y) 1113957m11113868111386811138681113868
1113868111386811138681113868
minus2
radiccφ1(y)φ2(y)V
(12)k
_V2 le minusbλ(1minus β2)
min (P)V(1+β2)z
λmax(P)minus
b
λmax(P)Vz
minus2
radiccφ1(y)φ2(y)V
(12)k
minus μVc2
(31)
with
μ μbλ(1minus β2)
min (P)V(1+β2)z
λmax(P)
b
λmax(P)
2
radiccφ1(y)φ2(y)gt 0
(32)
c c1 + β2
112
1113888 1113889 (33)
It is known that (12)lt clt 1 from (12)le βlt 1 so_V2 le minus μV
c2 le 0 (34)
According to Lemma 1 V2 can converge in finite time ycan converge to |y|le ε and the convergence time satisfies
t1 leV
1minus c2 (0)
μ(1 minus c) (35)
When |y|le ε if m1 and m2 decrease to the point wherethe interference cannot be eliminated the system state willdeviate from |y|le ε In this case m1 and m2 will increaseagain under the effect of the adaptive law making the systemstate converge to |y|le ε
4 Guidance Law Design
-e terminal sliding mode control adopts the nonlinearfunction as the sliding mode surface which can make thesystem states converge in finite time but the method hassingular problems In order to avoid singular problems
based on the piecewise sliding surface [12] and Lemma 3 anonsingular fast terminal sliding surface is designed as
Remark 4 Condition (1) can ensure that gprime(x1) is a con-tinuous bounded function and eliminates singular problemsand when the system reaches the sliding surface s 0 x1 andx2 are always with different signs ensuring that the systemstate is convergent Condition (2) ensures that the slidingsurface s is a continuous function Condition (3) guaranteesthat g(x1) is bounded in x1 isin (minus δ δ) and ψprime(x1) is acontinuous function so s is a smooth function
According to the above conditions this paper selectsfunction g(x1) as follows
g x1( 1113857 λ1x1 + λ2x31 (40)
where λ1 (3 minus α22)εα2minus 1 and λ2 (α2 minus 12)εα2minus 3Substituting (3) into (38) yields
_s f1x2 + f2u + d + k1α1 x11113868111386811138681113868
In order to counteract the disturbance suppress chat-tering and accelerate the convergence speed of the slidingsurface based on the adaptive smooth fast supertwistingalgorithm proposed in the second section an auxiliaryguidance law is designed as
Journal of Control Science and Engineering 5
uaux minus fminus 12 k3φ1(s) + k4 1113946
t
0φ2(s)dt (44)
-e parameter adaptive law is designed as_k3 (a + c)φ1(s)φ2(s)sgn(|s| minus ε)
k4 bk3
k3(0)gt 0
⎧⎪⎪⎪⎨
⎪⎪⎪⎩
(45)
Combining with (42) and (44) we can design a non-singular terminal sliding mode guidance law with the impactangle constraint based on the adaptive supertwisting algo-rithm as
For the convenience of description the design guidancelaw (46) is abbreviated as ASNTSMG
5 Simulation Analysis
In order to test the performance of the designed guidance lawASNTSMG this section conducts simulation analysis basedon ballistic simulation in different scenarios -e initial po-sition of themissile is (0m 0m) and the initial position of thetarget is (1000m 5000m) -e missilersquos velocity isvm 500ms and the targetrsquos velocity is vt 250ms -eacceleration of gravity is g 98ms2 and the maximumavailable overload of the missile is 20 g -e parameters ofASNTSMG are set as follows k1 k2 2 a1 3 α2 (57)ε 001 δ 0001 a 05 and b c 1
In order to verify the superiority of the designed guidancelaw this section also carries out the nonsingular fast terminalsliding mode guidance law (NFTSMG) proposed in [23] andthe second-order nonsingular terminal sliding mode guidancelaw (SONTSMG) proposed in [24] to perform a comparativesimulation -e expression of NFTSMG is
u r
cos q minus θm( 1113857
1k2a2
x2minus a22 1 + k1a1x
a1minus 111113872 1113873 minus
2 _r
rx21113890
+αs + β|s|csgn(s)
r1113891
(47)
-e parameters are set as follows α 600 β 500α1 (75) α2 (57) k1 k2 2 and c 05
-e expression of SONTSMG is
u minus2 _r
rx2 +
rβα
x2minus α2 + z1 + k1|s|
1minus (1c)sgn(s)
+ k2 1113946t
0
xαminus 12r
|s|1minus (2c)sgn(s)dt
(48)
-e parameters are set as follows k1 600 k2 100α (75) β 05 and c 21
-e average overload Nme (unit g) is introduced toevaluate the energy consumption in the process of guidancewhich is defined as follows
Case 1 Attack moving target with different impact angleconstraints set qd as 20deg 30deg 40deg and 50deg respectively andθm0 45deg -e target makes sinusoidal maneuver and itsacceleration is at 30sin (πt5)ms2 and θt0 150deg -esimulation results are shown in Figure 2
It can be seen from Figures 2(a) and 2(b) that ASNTSMGcan effectively intercept the target with different impactangle constraints -e miss distances are 0374m 0428m0408m and 0479m respectively -is method can hit thetarget accurately It can be seen from Figures 2(c)ndash2(e) thatthe sliding surface and the LOS angular rate can converge tozero in finite time and LOS angle can effectively converge tothe expected impact angle With the increase of qd theconvergence time increases -is is because the larger qd isthe larger the initial LOS angle deviation will be and theconvergence time is related to the initial value which leads tothe corresponding growth of the convergence time
Figure 2(f) shows the overload curve of the missilewhich is saturated in the early stage and the larger θm0 thelonger the saturation time which is mainly due to the largeroverload needed in the earlier stage which makes the missilemeet the requirements of angle constraint and guidanceaccuracy When q and _q approach the expected values theoverload gradually approaches zero which ensures that themissile has sufficient overload margin to deal with otherunknown disturbances in the later stage of guidance
Case 2 Comparative simulation of ASNTSMG NFTSMGand SONTSMG the relevant initial parameters are set toθm0 45deg qd 45deg and θt0 180deg -e movement of thetarget is set as follows
(1) Cosine motion at 30cos (πt5)ms2
(2) Square wave motion at 30sgn (sin (πt5))ms2
-e simulation results are shown in Figures 3ndash6 andTable 1
Figure 3 shows the trajectories of the missile and thetarget It can be seen that the missile can track and interceptthe target under the three guidance laws Compared withNFTSMG and SONTSMG the trajectory of ASNTSMG isrelatively smooth indicating that its attack time is relativelyshort which can be verified by Table 1 Figure 4 shows theLOS angle curve All three guidance laws can make the LOSangle gradually converge to the expected angle ASNTSMGcanmake the LOS angle converge to the expected angle morequickly NFTSMG adopts the robustness of the sliding modecontrol to cancel the disturbance of target maneuver so itcan only make the system states converge to the neigh-borhood of origin in finite time NFTSMG and SONTSMGadopt the traditional nonsingular terminal sliding surface
Figure 2 Simulation results of Case 1 (a) Missile and target motion trajectory (b) Relative distance between the missile and the target(c) Sliding mode (d) LOS angle (e) LOS angular rate (f ) Missile overload
Journal of Control Science and Engineering 7
which has the nonstrict finite-time convergence problemand the nonconvergence factor slows down the convergencespeed Figure 5 shows the LOS angular rate curve Under thethree guidance laws the LOS angular rates converge to zeroin finite time ASNTSMG has smaller convergence error andfaster convergence speed Figure 6 shows the overload curveof the missile Due to the finite-time convergence of LOSangle and angular rate the missile needs a large overload inthe early stage of guidance -erefore the overloads of threeguidance laws are all saturated in the early stage When theLOS angle and the angular rate converge the missile
overload gradually converges to zero in the later stage Andthe convergence speed of ASNTSMG is faster
Table 1 shows the simulation results of attack timemiss distance LOS angle error and average overloadunder the three guidance laws It can be seen that com-pared with NFTSMG and SONTSMG ASNTSMG hassmaller attack time miss distance terminal LOS angleerror and average overload so ASNTSMG has betterguidance performance
According to the analysis of the simulation results of twocases ASNTSMG can hit the target precisely with the
expected impact angle under different expected LOS anglesand target maneuvering conditions Compared with theexisting guidance laws NFTSMG and SONTSMGASNTSMG can effectively attack the target with less impactangle error miss distance and energy consumption whichverifies the effectiveness and superiority of ASNTSMG
6 Conclusion
In this paper a nonsingular fast terminal sliding modeguidance law is proposed to solve the problem of guidanceaccuracy and impact angle constraint -rough theoreticalanalysis and simulation verification the following conclu-sions can be obtained
(1) -e proposed adaptive smooth supertwisting algo-rithm can effectively counteract the disturbance ofthe system and accelerate the convergence speed ofthe system without knowing the upper bound of thedisturbance
(2) -e designed nonsingular terminal sliding modesurface can realize the fast finite-time convergence ofthe system states and ensure the impact angleconstraint and guidance accuracy requirements
(3) -is guidance law can attack the target preciselyunder the conditions of different expected LOSangles and target maneuvers Compared with theexisting nonsingular fast terminal sliding modeguidance law and second-order nonsingular termi-nal sliding mode guidance law this law has higherguidance accuracy and angle constraint accuracy andconsumes less energy
Data Availability
-e data used to support the findings of this study are in-cluded within the article
Conflicts of Interest
-e authors declare that there are no conflicts of interestregarding the publication of this paper
References
[1] H Cai Z D Hu and Y Cao ldquoA survey of guidance law withterminal impact angle constraintsrdquo Journal of Astronauticsvol 31 no 2 pp 315ndash323 2010
[2] J M Song and T Q Zhang ldquoPassive homingmissilersquos variablestructure proportional navigation with terminal angularconstraintrdquo Chinese Journal of Aeronautics vol 14 no 2pp 83ndash87 2001
[3] P Wu and M Yang ldquoVariable structure guidance law withterminal attack angle constraintrdquo Journal of Solid RocketTechnology vol 31 no 2 pp 116ndash120 2002
[4] S R Kumar S Rao and D Ghose ldquoSliding-mode guidanceand control for all-aspect interceptors with terminal angleconstraintsrdquo Journal of Guidance Control and Dynamicsvol 35 no 4 pp 1230ndash1246 2012
[5] H B Zhou S M Song and M Y Xu ldquoDesign of terminalsliding-mode guidance law with attack angle constraintsrdquo in
Proceedings of the 2013 25th Chinese Control and DecisionConference (CCDC) Guiyang China May 2013
[6] Y X Zhang M W Sun and Z Q Chen ldquoFinite-timeconvergent guidance law with impact angle constraint basedon sliding-mode controlrdquo Nonlinear Dynamic vol 7 no 3pp 619ndash625 2012
[7] X H Yu and Z H Man ldquoFast terminal sliding-mode controldesign for nonlinear dynamical systemsrdquo IEEE Transactionson Circuits and Systems I Fundamental Jeory and Appli-cations vol 49 no 2 pp 261ndash264 2002
[8] J M Song S M Song and Y Guo ldquoNonlinear disturbanceobserver based fast terminal sliding mode guidance withimpact angle constraintsrdquo International Journal of InnovativeComputing Information and Control vol 11 no 3pp 787ndash802 2015
[9] S R Kumar S Rao and D Ghose ldquoNonsingular terminalslidingmode guidance with impact angle constraintsrdquo Journalof Guidance Control and Dynamics vol 37 no 4pp 1114ndash1130 2014
[10] S F Xiong W H Wang and S Wang ldquoNonsingular fastterminal sliding-mode guidance with intercept angle con-straintrdquo Control Jeory amp Applications vol 31 no 3pp 269ndash278 2014
[11] S C Yang K Q Zhang and P Chen ldquoAdaptive terminalsliding mode guidance law with impact angle constraintrdquoJournal of Beijing University of Aeronautics and Astronauticsvol 42 no 8 pp 1566ndash1574 2016
[12] Z L Zhang and J Zhou ldquoStrictly convergent non-singularterminal sliding mode guidance law with impact angle con-straintsrdquo Optik vol 127 no 22 pp 10971ndash10980 2016
[13] Z Zhang C Man S Li and S Jin ldquoFinite-time guidance lawsfor three-dimensional missile-target interceptionrdquo Journal ofAerospace Engineering vol 230 no 2 pp 392ndash403 2015
[14] N Zhang W Gai M Zhong and J Zhang ldquoA fast finite-timeconvergent guidance law with nonlinear disturbance observerfor unmanned aerial vehicles collision avoidancerdquo AerospaceScience and Technology vol 86 pp 204ndash214 2019
[15] J Song S Song and H Zhou ldquoAdaptive nonsingular fastterminal sliding mode guidance law with impact angle con-straintsrdquo International Journal of Control Automation andSystems vol 14 no 1 pp 99ndash114 2016
[16] J Song and S Song ldquoRobust impact angle constraintsguidance law with autopilot lag and acceleration saturationconsiderationrdquo Transactions of the Institute of Measurementand Control vol 41 no 1 pp 182ndash192 2019
[17] A Levant ldquoPrinciples of 2-sliding mode designrdquo Automaticavol 43 no 4 pp 576ndash586 2007
[18] A M Zhou ldquoFinite-time output feedback attitude trackingcontrol for rigid spacecraftrdquo IEEE Transactions on ControlSystems Technology vol 22 no 1 pp 338ndash345 2014
[19] B Li Q Hu Y Yu and G Ma ldquoObserver-based fault-tolerantattitude control for rigid spacecraftrdquo IEEE Transactions onAerospace and Electronic Systems vol 53 no 5 pp 2572ndash2582 2017
[20] H Li and Y Cai ldquoOn SFTSM control with fixed-time con-vergencerdquo IET Control Jeory amp Applications vol 11 no 6pp 766ndash773 2017
[21] J Ni L Liu C Liu X Hu and T Shen ldquoFixed-time dynamicsurface high-order slidingmode control for chaotic oscillationin power systemrdquo Nonlinear Dynamics vol 86 no 1pp 401ndash420 2016
[22] Q Zong Z S Zhao and J Zhang ldquoHigher order sliding modecontrol with self-tuning law based on integral sliding moderdquo
10 Journal of Control Science and Engineering
IET Control Jeory and Application vol 4 no 7 pp 1282ndash1289 2008
[23] S He D Lin and J Wang ldquoContinuous second-order slidingmode based impact angle guidance lawrdquo Aerospace Scienceand Technology vol 41 pp 199ndash208 2015
[24] B L Cong Z Chen and X D Liu ldquoOn adaptive sliding modecontrol without switching gain overestimationrdquo InternationalJournal of Robust and Nonlinear Control vol 24 no 3pp 515ndash531 2014
Journal of Control Science and Engineering 11
A minus m1 + ρ1 1
minus bm1 + ρ2 01113890 1113891
1113957A A minus A minus 1113957m1 0
minus b 1113957m1 01113890 1113891
(29)
According to (26) we can get
_Vz le minusbλ(1minus β2)
min (P)V(1+β2)z
λmax(P)minus
b
λmax(P)Vz (30)
When |y|gt ε combining with (12) gives
_Vm1 |y|βminus 1
+ 11113872 1113873zT 1113957A
TP + P1113957A1113874 1113875z + _m1 1113957m1
minus aφ1(y)φ2(y) +(a + c)φ1(y)φ2(y) 1113957m1
minus cφ1(y)φ2(y) 1113957m11113868111386811138681113868
1113868111386811138681113868
minus2
radiccφ1(y)φ2(y)V
(12)k
_V2 le minusbλ(1minus β2)
min (P)V(1+β2)z
λmax(P)minus
b
λmax(P)Vz
minus2
radiccφ1(y)φ2(y)V
(12)k
minus μVc2
(31)
with
μ μbλ(1minus β2)
min (P)V(1+β2)z
λmax(P)
b
λmax(P)
2
radiccφ1(y)φ2(y)gt 0
(32)
c c1 + β2
112
1113888 1113889 (33)
It is known that (12)lt clt 1 from (12)le βlt 1 so_V2 le minus μV
c2 le 0 (34)
According to Lemma 1 V2 can converge in finite time ycan converge to |y|le ε and the convergence time satisfies
t1 leV
1minus c2 (0)
μ(1 minus c) (35)
When |y|le ε if m1 and m2 decrease to the point wherethe interference cannot be eliminated the system state willdeviate from |y|le ε In this case m1 and m2 will increaseagain under the effect of the adaptive law making the systemstate converge to |y|le ε
4 Guidance Law Design
-e terminal sliding mode control adopts the nonlinearfunction as the sliding mode surface which can make thesystem states converge in finite time but the method hassingular problems In order to avoid singular problems
based on the piecewise sliding surface [12] and Lemma 3 anonsingular fast terminal sliding surface is designed as
Remark 4 Condition (1) can ensure that gprime(x1) is a con-tinuous bounded function and eliminates singular problemsand when the system reaches the sliding surface s 0 x1 andx2 are always with different signs ensuring that the systemstate is convergent Condition (2) ensures that the slidingsurface s is a continuous function Condition (3) guaranteesthat g(x1) is bounded in x1 isin (minus δ δ) and ψprime(x1) is acontinuous function so s is a smooth function
According to the above conditions this paper selectsfunction g(x1) as follows
g x1( 1113857 λ1x1 + λ2x31 (40)
where λ1 (3 minus α22)εα2minus 1 and λ2 (α2 minus 12)εα2minus 3Substituting (3) into (38) yields
_s f1x2 + f2u + d + k1α1 x11113868111386811138681113868
In order to counteract the disturbance suppress chat-tering and accelerate the convergence speed of the slidingsurface based on the adaptive smooth fast supertwistingalgorithm proposed in the second section an auxiliaryguidance law is designed as
Journal of Control Science and Engineering 5
uaux minus fminus 12 k3φ1(s) + k4 1113946
t
0φ2(s)dt (44)
-e parameter adaptive law is designed as_k3 (a + c)φ1(s)φ2(s)sgn(|s| minus ε)
k4 bk3
k3(0)gt 0
⎧⎪⎪⎪⎨
⎪⎪⎪⎩
(45)
Combining with (42) and (44) we can design a non-singular terminal sliding mode guidance law with the impactangle constraint based on the adaptive supertwisting algo-rithm as
For the convenience of description the design guidancelaw (46) is abbreviated as ASNTSMG
5 Simulation Analysis
In order to test the performance of the designed guidance lawASNTSMG this section conducts simulation analysis basedon ballistic simulation in different scenarios -e initial po-sition of themissile is (0m 0m) and the initial position of thetarget is (1000m 5000m) -e missilersquos velocity isvm 500ms and the targetrsquos velocity is vt 250ms -eacceleration of gravity is g 98ms2 and the maximumavailable overload of the missile is 20 g -e parameters ofASNTSMG are set as follows k1 k2 2 a1 3 α2 (57)ε 001 δ 0001 a 05 and b c 1
In order to verify the superiority of the designed guidancelaw this section also carries out the nonsingular fast terminalsliding mode guidance law (NFTSMG) proposed in [23] andthe second-order nonsingular terminal sliding mode guidancelaw (SONTSMG) proposed in [24] to perform a comparativesimulation -e expression of NFTSMG is
u r
cos q minus θm( 1113857
1k2a2
x2minus a22 1 + k1a1x
a1minus 111113872 1113873 minus
2 _r
rx21113890
+αs + β|s|csgn(s)
r1113891
(47)
-e parameters are set as follows α 600 β 500α1 (75) α2 (57) k1 k2 2 and c 05
-e expression of SONTSMG is
u minus2 _r
rx2 +
rβα
x2minus α2 + z1 + k1|s|
1minus (1c)sgn(s)
+ k2 1113946t
0
xαminus 12r
|s|1minus (2c)sgn(s)dt
(48)
-e parameters are set as follows k1 600 k2 100α (75) β 05 and c 21
-e average overload Nme (unit g) is introduced toevaluate the energy consumption in the process of guidancewhich is defined as follows
Case 1 Attack moving target with different impact angleconstraints set qd as 20deg 30deg 40deg and 50deg respectively andθm0 45deg -e target makes sinusoidal maneuver and itsacceleration is at 30sin (πt5)ms2 and θt0 150deg -esimulation results are shown in Figure 2
It can be seen from Figures 2(a) and 2(b) that ASNTSMGcan effectively intercept the target with different impactangle constraints -e miss distances are 0374m 0428m0408m and 0479m respectively -is method can hit thetarget accurately It can be seen from Figures 2(c)ndash2(e) thatthe sliding surface and the LOS angular rate can converge tozero in finite time and LOS angle can effectively converge tothe expected impact angle With the increase of qd theconvergence time increases -is is because the larger qd isthe larger the initial LOS angle deviation will be and theconvergence time is related to the initial value which leads tothe corresponding growth of the convergence time
Figure 2(f) shows the overload curve of the missilewhich is saturated in the early stage and the larger θm0 thelonger the saturation time which is mainly due to the largeroverload needed in the earlier stage which makes the missilemeet the requirements of angle constraint and guidanceaccuracy When q and _q approach the expected values theoverload gradually approaches zero which ensures that themissile has sufficient overload margin to deal with otherunknown disturbances in the later stage of guidance
Case 2 Comparative simulation of ASNTSMG NFTSMGand SONTSMG the relevant initial parameters are set toθm0 45deg qd 45deg and θt0 180deg -e movement of thetarget is set as follows
(1) Cosine motion at 30cos (πt5)ms2
(2) Square wave motion at 30sgn (sin (πt5))ms2
-e simulation results are shown in Figures 3ndash6 andTable 1
Figure 3 shows the trajectories of the missile and thetarget It can be seen that the missile can track and interceptthe target under the three guidance laws Compared withNFTSMG and SONTSMG the trajectory of ASNTSMG isrelatively smooth indicating that its attack time is relativelyshort which can be verified by Table 1 Figure 4 shows theLOS angle curve All three guidance laws can make the LOSangle gradually converge to the expected angle ASNTSMGcanmake the LOS angle converge to the expected angle morequickly NFTSMG adopts the robustness of the sliding modecontrol to cancel the disturbance of target maneuver so itcan only make the system states converge to the neigh-borhood of origin in finite time NFTSMG and SONTSMGadopt the traditional nonsingular terminal sliding surface
Figure 2 Simulation results of Case 1 (a) Missile and target motion trajectory (b) Relative distance between the missile and the target(c) Sliding mode (d) LOS angle (e) LOS angular rate (f ) Missile overload
Journal of Control Science and Engineering 7
which has the nonstrict finite-time convergence problemand the nonconvergence factor slows down the convergencespeed Figure 5 shows the LOS angular rate curve Under thethree guidance laws the LOS angular rates converge to zeroin finite time ASNTSMG has smaller convergence error andfaster convergence speed Figure 6 shows the overload curveof the missile Due to the finite-time convergence of LOSangle and angular rate the missile needs a large overload inthe early stage of guidance -erefore the overloads of threeguidance laws are all saturated in the early stage When theLOS angle and the angular rate converge the missile
overload gradually converges to zero in the later stage Andthe convergence speed of ASNTSMG is faster
Table 1 shows the simulation results of attack timemiss distance LOS angle error and average overloadunder the three guidance laws It can be seen that com-pared with NFTSMG and SONTSMG ASNTSMG hassmaller attack time miss distance terminal LOS angleerror and average overload so ASNTSMG has betterguidance performance
According to the analysis of the simulation results of twocases ASNTSMG can hit the target precisely with the
expected impact angle under different expected LOS anglesand target maneuvering conditions Compared with theexisting guidance laws NFTSMG and SONTSMGASNTSMG can effectively attack the target with less impactangle error miss distance and energy consumption whichverifies the effectiveness and superiority of ASNTSMG
6 Conclusion
In this paper a nonsingular fast terminal sliding modeguidance law is proposed to solve the problem of guidanceaccuracy and impact angle constraint -rough theoreticalanalysis and simulation verification the following conclu-sions can be obtained
(1) -e proposed adaptive smooth supertwisting algo-rithm can effectively counteract the disturbance ofthe system and accelerate the convergence speed ofthe system without knowing the upper bound of thedisturbance
(2) -e designed nonsingular terminal sliding modesurface can realize the fast finite-time convergence ofthe system states and ensure the impact angleconstraint and guidance accuracy requirements
(3) -is guidance law can attack the target preciselyunder the conditions of different expected LOSangles and target maneuvers Compared with theexisting nonsingular fast terminal sliding modeguidance law and second-order nonsingular termi-nal sliding mode guidance law this law has higherguidance accuracy and angle constraint accuracy andconsumes less energy
Data Availability
-e data used to support the findings of this study are in-cluded within the article
Conflicts of Interest
-e authors declare that there are no conflicts of interestregarding the publication of this paper
References
[1] H Cai Z D Hu and Y Cao ldquoA survey of guidance law withterminal impact angle constraintsrdquo Journal of Astronauticsvol 31 no 2 pp 315ndash323 2010
[2] J M Song and T Q Zhang ldquoPassive homingmissilersquos variablestructure proportional navigation with terminal angularconstraintrdquo Chinese Journal of Aeronautics vol 14 no 2pp 83ndash87 2001
[3] P Wu and M Yang ldquoVariable structure guidance law withterminal attack angle constraintrdquo Journal of Solid RocketTechnology vol 31 no 2 pp 116ndash120 2002
[4] S R Kumar S Rao and D Ghose ldquoSliding-mode guidanceand control for all-aspect interceptors with terminal angleconstraintsrdquo Journal of Guidance Control and Dynamicsvol 35 no 4 pp 1230ndash1246 2012
[5] H B Zhou S M Song and M Y Xu ldquoDesign of terminalsliding-mode guidance law with attack angle constraintsrdquo in
Proceedings of the 2013 25th Chinese Control and DecisionConference (CCDC) Guiyang China May 2013
[6] Y X Zhang M W Sun and Z Q Chen ldquoFinite-timeconvergent guidance law with impact angle constraint basedon sliding-mode controlrdquo Nonlinear Dynamic vol 7 no 3pp 619ndash625 2012
[7] X H Yu and Z H Man ldquoFast terminal sliding-mode controldesign for nonlinear dynamical systemsrdquo IEEE Transactionson Circuits and Systems I Fundamental Jeory and Appli-cations vol 49 no 2 pp 261ndash264 2002
[8] J M Song S M Song and Y Guo ldquoNonlinear disturbanceobserver based fast terminal sliding mode guidance withimpact angle constraintsrdquo International Journal of InnovativeComputing Information and Control vol 11 no 3pp 787ndash802 2015
[9] S R Kumar S Rao and D Ghose ldquoNonsingular terminalslidingmode guidance with impact angle constraintsrdquo Journalof Guidance Control and Dynamics vol 37 no 4pp 1114ndash1130 2014
[10] S F Xiong W H Wang and S Wang ldquoNonsingular fastterminal sliding-mode guidance with intercept angle con-straintrdquo Control Jeory amp Applications vol 31 no 3pp 269ndash278 2014
[11] S C Yang K Q Zhang and P Chen ldquoAdaptive terminalsliding mode guidance law with impact angle constraintrdquoJournal of Beijing University of Aeronautics and Astronauticsvol 42 no 8 pp 1566ndash1574 2016
[12] Z L Zhang and J Zhou ldquoStrictly convergent non-singularterminal sliding mode guidance law with impact angle con-straintsrdquo Optik vol 127 no 22 pp 10971ndash10980 2016
[13] Z Zhang C Man S Li and S Jin ldquoFinite-time guidance lawsfor three-dimensional missile-target interceptionrdquo Journal ofAerospace Engineering vol 230 no 2 pp 392ndash403 2015
[14] N Zhang W Gai M Zhong and J Zhang ldquoA fast finite-timeconvergent guidance law with nonlinear disturbance observerfor unmanned aerial vehicles collision avoidancerdquo AerospaceScience and Technology vol 86 pp 204ndash214 2019
[15] J Song S Song and H Zhou ldquoAdaptive nonsingular fastterminal sliding mode guidance law with impact angle con-straintsrdquo International Journal of Control Automation andSystems vol 14 no 1 pp 99ndash114 2016
[16] J Song and S Song ldquoRobust impact angle constraintsguidance law with autopilot lag and acceleration saturationconsiderationrdquo Transactions of the Institute of Measurementand Control vol 41 no 1 pp 182ndash192 2019
[17] A Levant ldquoPrinciples of 2-sliding mode designrdquo Automaticavol 43 no 4 pp 576ndash586 2007
[18] A M Zhou ldquoFinite-time output feedback attitude trackingcontrol for rigid spacecraftrdquo IEEE Transactions on ControlSystems Technology vol 22 no 1 pp 338ndash345 2014
[19] B Li Q Hu Y Yu and G Ma ldquoObserver-based fault-tolerantattitude control for rigid spacecraftrdquo IEEE Transactions onAerospace and Electronic Systems vol 53 no 5 pp 2572ndash2582 2017
[20] H Li and Y Cai ldquoOn SFTSM control with fixed-time con-vergencerdquo IET Control Jeory amp Applications vol 11 no 6pp 766ndash773 2017
[21] J Ni L Liu C Liu X Hu and T Shen ldquoFixed-time dynamicsurface high-order slidingmode control for chaotic oscillationin power systemrdquo Nonlinear Dynamics vol 86 no 1pp 401ndash420 2016
[22] Q Zong Z S Zhao and J Zhang ldquoHigher order sliding modecontrol with self-tuning law based on integral sliding moderdquo
10 Journal of Control Science and Engineering
IET Control Jeory and Application vol 4 no 7 pp 1282ndash1289 2008
[23] S He D Lin and J Wang ldquoContinuous second-order slidingmode based impact angle guidance lawrdquo Aerospace Scienceand Technology vol 41 pp 199ndash208 2015
[24] B L Cong Z Chen and X D Liu ldquoOn adaptive sliding modecontrol without switching gain overestimationrdquo InternationalJournal of Robust and Nonlinear Control vol 24 no 3pp 515ndash531 2014
Journal of Control Science and Engineering 11
uaux minus fminus 12 k3φ1(s) + k4 1113946
t
0φ2(s)dt (44)
-e parameter adaptive law is designed as_k3 (a + c)φ1(s)φ2(s)sgn(|s| minus ε)
k4 bk3
k3(0)gt 0
⎧⎪⎪⎪⎨
⎪⎪⎪⎩
(45)
Combining with (42) and (44) we can design a non-singular terminal sliding mode guidance law with the impactangle constraint based on the adaptive supertwisting algo-rithm as
For the convenience of description the design guidancelaw (46) is abbreviated as ASNTSMG
5 Simulation Analysis
In order to test the performance of the designed guidance lawASNTSMG this section conducts simulation analysis basedon ballistic simulation in different scenarios -e initial po-sition of themissile is (0m 0m) and the initial position of thetarget is (1000m 5000m) -e missilersquos velocity isvm 500ms and the targetrsquos velocity is vt 250ms -eacceleration of gravity is g 98ms2 and the maximumavailable overload of the missile is 20 g -e parameters ofASNTSMG are set as follows k1 k2 2 a1 3 α2 (57)ε 001 δ 0001 a 05 and b c 1
In order to verify the superiority of the designed guidancelaw this section also carries out the nonsingular fast terminalsliding mode guidance law (NFTSMG) proposed in [23] andthe second-order nonsingular terminal sliding mode guidancelaw (SONTSMG) proposed in [24] to perform a comparativesimulation -e expression of NFTSMG is
u r
cos q minus θm( 1113857
1k2a2
x2minus a22 1 + k1a1x
a1minus 111113872 1113873 minus
2 _r
rx21113890
+αs + β|s|csgn(s)
r1113891
(47)
-e parameters are set as follows α 600 β 500α1 (75) α2 (57) k1 k2 2 and c 05
-e expression of SONTSMG is
u minus2 _r
rx2 +
rβα
x2minus α2 + z1 + k1|s|
1minus (1c)sgn(s)
+ k2 1113946t
0
xαminus 12r
|s|1minus (2c)sgn(s)dt
(48)
-e parameters are set as follows k1 600 k2 100α (75) β 05 and c 21
-e average overload Nme (unit g) is introduced toevaluate the energy consumption in the process of guidancewhich is defined as follows
Case 1 Attack moving target with different impact angleconstraints set qd as 20deg 30deg 40deg and 50deg respectively andθm0 45deg -e target makes sinusoidal maneuver and itsacceleration is at 30sin (πt5)ms2 and θt0 150deg -esimulation results are shown in Figure 2
It can be seen from Figures 2(a) and 2(b) that ASNTSMGcan effectively intercept the target with different impactangle constraints -e miss distances are 0374m 0428m0408m and 0479m respectively -is method can hit thetarget accurately It can be seen from Figures 2(c)ndash2(e) thatthe sliding surface and the LOS angular rate can converge tozero in finite time and LOS angle can effectively converge tothe expected impact angle With the increase of qd theconvergence time increases -is is because the larger qd isthe larger the initial LOS angle deviation will be and theconvergence time is related to the initial value which leads tothe corresponding growth of the convergence time
Figure 2(f) shows the overload curve of the missilewhich is saturated in the early stage and the larger θm0 thelonger the saturation time which is mainly due to the largeroverload needed in the earlier stage which makes the missilemeet the requirements of angle constraint and guidanceaccuracy When q and _q approach the expected values theoverload gradually approaches zero which ensures that themissile has sufficient overload margin to deal with otherunknown disturbances in the later stage of guidance
Case 2 Comparative simulation of ASNTSMG NFTSMGand SONTSMG the relevant initial parameters are set toθm0 45deg qd 45deg and θt0 180deg -e movement of thetarget is set as follows
(1) Cosine motion at 30cos (πt5)ms2
(2) Square wave motion at 30sgn (sin (πt5))ms2
-e simulation results are shown in Figures 3ndash6 andTable 1
Figure 3 shows the trajectories of the missile and thetarget It can be seen that the missile can track and interceptthe target under the three guidance laws Compared withNFTSMG and SONTSMG the trajectory of ASNTSMG isrelatively smooth indicating that its attack time is relativelyshort which can be verified by Table 1 Figure 4 shows theLOS angle curve All three guidance laws can make the LOSangle gradually converge to the expected angle ASNTSMGcanmake the LOS angle converge to the expected angle morequickly NFTSMG adopts the robustness of the sliding modecontrol to cancel the disturbance of target maneuver so itcan only make the system states converge to the neigh-borhood of origin in finite time NFTSMG and SONTSMGadopt the traditional nonsingular terminal sliding surface
Figure 2 Simulation results of Case 1 (a) Missile and target motion trajectory (b) Relative distance between the missile and the target(c) Sliding mode (d) LOS angle (e) LOS angular rate (f ) Missile overload
Journal of Control Science and Engineering 7
which has the nonstrict finite-time convergence problemand the nonconvergence factor slows down the convergencespeed Figure 5 shows the LOS angular rate curve Under thethree guidance laws the LOS angular rates converge to zeroin finite time ASNTSMG has smaller convergence error andfaster convergence speed Figure 6 shows the overload curveof the missile Due to the finite-time convergence of LOSangle and angular rate the missile needs a large overload inthe early stage of guidance -erefore the overloads of threeguidance laws are all saturated in the early stage When theLOS angle and the angular rate converge the missile
overload gradually converges to zero in the later stage Andthe convergence speed of ASNTSMG is faster
Table 1 shows the simulation results of attack timemiss distance LOS angle error and average overloadunder the three guidance laws It can be seen that com-pared with NFTSMG and SONTSMG ASNTSMG hassmaller attack time miss distance terminal LOS angleerror and average overload so ASNTSMG has betterguidance performance
According to the analysis of the simulation results of twocases ASNTSMG can hit the target precisely with the
expected impact angle under different expected LOS anglesand target maneuvering conditions Compared with theexisting guidance laws NFTSMG and SONTSMGASNTSMG can effectively attack the target with less impactangle error miss distance and energy consumption whichverifies the effectiveness and superiority of ASNTSMG
6 Conclusion
In this paper a nonsingular fast terminal sliding modeguidance law is proposed to solve the problem of guidanceaccuracy and impact angle constraint -rough theoreticalanalysis and simulation verification the following conclu-sions can be obtained
(1) -e proposed adaptive smooth supertwisting algo-rithm can effectively counteract the disturbance ofthe system and accelerate the convergence speed ofthe system without knowing the upper bound of thedisturbance
(2) -e designed nonsingular terminal sliding modesurface can realize the fast finite-time convergence ofthe system states and ensure the impact angleconstraint and guidance accuracy requirements
(3) -is guidance law can attack the target preciselyunder the conditions of different expected LOSangles and target maneuvers Compared with theexisting nonsingular fast terminal sliding modeguidance law and second-order nonsingular termi-nal sliding mode guidance law this law has higherguidance accuracy and angle constraint accuracy andconsumes less energy
Data Availability
-e data used to support the findings of this study are in-cluded within the article
Conflicts of Interest
-e authors declare that there are no conflicts of interestregarding the publication of this paper
References
[1] H Cai Z D Hu and Y Cao ldquoA survey of guidance law withterminal impact angle constraintsrdquo Journal of Astronauticsvol 31 no 2 pp 315ndash323 2010
[2] J M Song and T Q Zhang ldquoPassive homingmissilersquos variablestructure proportional navigation with terminal angularconstraintrdquo Chinese Journal of Aeronautics vol 14 no 2pp 83ndash87 2001
[3] P Wu and M Yang ldquoVariable structure guidance law withterminal attack angle constraintrdquo Journal of Solid RocketTechnology vol 31 no 2 pp 116ndash120 2002
[4] S R Kumar S Rao and D Ghose ldquoSliding-mode guidanceand control for all-aspect interceptors with terminal angleconstraintsrdquo Journal of Guidance Control and Dynamicsvol 35 no 4 pp 1230ndash1246 2012
[5] H B Zhou S M Song and M Y Xu ldquoDesign of terminalsliding-mode guidance law with attack angle constraintsrdquo in
Proceedings of the 2013 25th Chinese Control and DecisionConference (CCDC) Guiyang China May 2013
[6] Y X Zhang M W Sun and Z Q Chen ldquoFinite-timeconvergent guidance law with impact angle constraint basedon sliding-mode controlrdquo Nonlinear Dynamic vol 7 no 3pp 619ndash625 2012
[7] X H Yu and Z H Man ldquoFast terminal sliding-mode controldesign for nonlinear dynamical systemsrdquo IEEE Transactionson Circuits and Systems I Fundamental Jeory and Appli-cations vol 49 no 2 pp 261ndash264 2002
[8] J M Song S M Song and Y Guo ldquoNonlinear disturbanceobserver based fast terminal sliding mode guidance withimpact angle constraintsrdquo International Journal of InnovativeComputing Information and Control vol 11 no 3pp 787ndash802 2015
[9] S R Kumar S Rao and D Ghose ldquoNonsingular terminalslidingmode guidance with impact angle constraintsrdquo Journalof Guidance Control and Dynamics vol 37 no 4pp 1114ndash1130 2014
[10] S F Xiong W H Wang and S Wang ldquoNonsingular fastterminal sliding-mode guidance with intercept angle con-straintrdquo Control Jeory amp Applications vol 31 no 3pp 269ndash278 2014
[11] S C Yang K Q Zhang and P Chen ldquoAdaptive terminalsliding mode guidance law with impact angle constraintrdquoJournal of Beijing University of Aeronautics and Astronauticsvol 42 no 8 pp 1566ndash1574 2016
[12] Z L Zhang and J Zhou ldquoStrictly convergent non-singularterminal sliding mode guidance law with impact angle con-straintsrdquo Optik vol 127 no 22 pp 10971ndash10980 2016
[13] Z Zhang C Man S Li and S Jin ldquoFinite-time guidance lawsfor three-dimensional missile-target interceptionrdquo Journal ofAerospace Engineering vol 230 no 2 pp 392ndash403 2015
[14] N Zhang W Gai M Zhong and J Zhang ldquoA fast finite-timeconvergent guidance law with nonlinear disturbance observerfor unmanned aerial vehicles collision avoidancerdquo AerospaceScience and Technology vol 86 pp 204ndash214 2019
[15] J Song S Song and H Zhou ldquoAdaptive nonsingular fastterminal sliding mode guidance law with impact angle con-straintsrdquo International Journal of Control Automation andSystems vol 14 no 1 pp 99ndash114 2016
[16] J Song and S Song ldquoRobust impact angle constraintsguidance law with autopilot lag and acceleration saturationconsiderationrdquo Transactions of the Institute of Measurementand Control vol 41 no 1 pp 182ndash192 2019
[17] A Levant ldquoPrinciples of 2-sliding mode designrdquo Automaticavol 43 no 4 pp 576ndash586 2007
[18] A M Zhou ldquoFinite-time output feedback attitude trackingcontrol for rigid spacecraftrdquo IEEE Transactions on ControlSystems Technology vol 22 no 1 pp 338ndash345 2014
[19] B Li Q Hu Y Yu and G Ma ldquoObserver-based fault-tolerantattitude control for rigid spacecraftrdquo IEEE Transactions onAerospace and Electronic Systems vol 53 no 5 pp 2572ndash2582 2017
[20] H Li and Y Cai ldquoOn SFTSM control with fixed-time con-vergencerdquo IET Control Jeory amp Applications vol 11 no 6pp 766ndash773 2017
[21] J Ni L Liu C Liu X Hu and T Shen ldquoFixed-time dynamicsurface high-order slidingmode control for chaotic oscillationin power systemrdquo Nonlinear Dynamics vol 86 no 1pp 401ndash420 2016
[22] Q Zong Z S Zhao and J Zhang ldquoHigher order sliding modecontrol with self-tuning law based on integral sliding moderdquo
10 Journal of Control Science and Engineering
IET Control Jeory and Application vol 4 no 7 pp 1282ndash1289 2008
[23] S He D Lin and J Wang ldquoContinuous second-order slidingmode based impact angle guidance lawrdquo Aerospace Scienceand Technology vol 41 pp 199ndash208 2015
[24] B L Cong Z Chen and X D Liu ldquoOn adaptive sliding modecontrol without switching gain overestimationrdquo InternationalJournal of Robust and Nonlinear Control vol 24 no 3pp 515ndash531 2014
Figure 2 Simulation results of Case 1 (a) Missile and target motion trajectory (b) Relative distance between the missile and the target(c) Sliding mode (d) LOS angle (e) LOS angular rate (f ) Missile overload
Journal of Control Science and Engineering 7
which has the nonstrict finite-time convergence problemand the nonconvergence factor slows down the convergencespeed Figure 5 shows the LOS angular rate curve Under thethree guidance laws the LOS angular rates converge to zeroin finite time ASNTSMG has smaller convergence error andfaster convergence speed Figure 6 shows the overload curveof the missile Due to the finite-time convergence of LOSangle and angular rate the missile needs a large overload inthe early stage of guidance -erefore the overloads of threeguidance laws are all saturated in the early stage When theLOS angle and the angular rate converge the missile
overload gradually converges to zero in the later stage Andthe convergence speed of ASNTSMG is faster
Table 1 shows the simulation results of attack timemiss distance LOS angle error and average overloadunder the three guidance laws It can be seen that com-pared with NFTSMG and SONTSMG ASNTSMG hassmaller attack time miss distance terminal LOS angleerror and average overload so ASNTSMG has betterguidance performance
According to the analysis of the simulation results of twocases ASNTSMG can hit the target precisely with the
expected impact angle under different expected LOS anglesand target maneuvering conditions Compared with theexisting guidance laws NFTSMG and SONTSMGASNTSMG can effectively attack the target with less impactangle error miss distance and energy consumption whichverifies the effectiveness and superiority of ASNTSMG
6 Conclusion
In this paper a nonsingular fast terminal sliding modeguidance law is proposed to solve the problem of guidanceaccuracy and impact angle constraint -rough theoreticalanalysis and simulation verification the following conclu-sions can be obtained
(1) -e proposed adaptive smooth supertwisting algo-rithm can effectively counteract the disturbance ofthe system and accelerate the convergence speed ofthe system without knowing the upper bound of thedisturbance
(2) -e designed nonsingular terminal sliding modesurface can realize the fast finite-time convergence ofthe system states and ensure the impact angleconstraint and guidance accuracy requirements
(3) -is guidance law can attack the target preciselyunder the conditions of different expected LOSangles and target maneuvers Compared with theexisting nonsingular fast terminal sliding modeguidance law and second-order nonsingular termi-nal sliding mode guidance law this law has higherguidance accuracy and angle constraint accuracy andconsumes less energy
Data Availability
-e data used to support the findings of this study are in-cluded within the article
Conflicts of Interest
-e authors declare that there are no conflicts of interestregarding the publication of this paper
References
[1] H Cai Z D Hu and Y Cao ldquoA survey of guidance law withterminal impact angle constraintsrdquo Journal of Astronauticsvol 31 no 2 pp 315ndash323 2010
[2] J M Song and T Q Zhang ldquoPassive homingmissilersquos variablestructure proportional navigation with terminal angularconstraintrdquo Chinese Journal of Aeronautics vol 14 no 2pp 83ndash87 2001
[3] P Wu and M Yang ldquoVariable structure guidance law withterminal attack angle constraintrdquo Journal of Solid RocketTechnology vol 31 no 2 pp 116ndash120 2002
[4] S R Kumar S Rao and D Ghose ldquoSliding-mode guidanceand control for all-aspect interceptors with terminal angleconstraintsrdquo Journal of Guidance Control and Dynamicsvol 35 no 4 pp 1230ndash1246 2012
[5] H B Zhou S M Song and M Y Xu ldquoDesign of terminalsliding-mode guidance law with attack angle constraintsrdquo in
Proceedings of the 2013 25th Chinese Control and DecisionConference (CCDC) Guiyang China May 2013
[6] Y X Zhang M W Sun and Z Q Chen ldquoFinite-timeconvergent guidance law with impact angle constraint basedon sliding-mode controlrdquo Nonlinear Dynamic vol 7 no 3pp 619ndash625 2012
[7] X H Yu and Z H Man ldquoFast terminal sliding-mode controldesign for nonlinear dynamical systemsrdquo IEEE Transactionson Circuits and Systems I Fundamental Jeory and Appli-cations vol 49 no 2 pp 261ndash264 2002
[8] J M Song S M Song and Y Guo ldquoNonlinear disturbanceobserver based fast terminal sliding mode guidance withimpact angle constraintsrdquo International Journal of InnovativeComputing Information and Control vol 11 no 3pp 787ndash802 2015
[9] S R Kumar S Rao and D Ghose ldquoNonsingular terminalslidingmode guidance with impact angle constraintsrdquo Journalof Guidance Control and Dynamics vol 37 no 4pp 1114ndash1130 2014
[10] S F Xiong W H Wang and S Wang ldquoNonsingular fastterminal sliding-mode guidance with intercept angle con-straintrdquo Control Jeory amp Applications vol 31 no 3pp 269ndash278 2014
[11] S C Yang K Q Zhang and P Chen ldquoAdaptive terminalsliding mode guidance law with impact angle constraintrdquoJournal of Beijing University of Aeronautics and Astronauticsvol 42 no 8 pp 1566ndash1574 2016
[12] Z L Zhang and J Zhou ldquoStrictly convergent non-singularterminal sliding mode guidance law with impact angle con-straintsrdquo Optik vol 127 no 22 pp 10971ndash10980 2016
[13] Z Zhang C Man S Li and S Jin ldquoFinite-time guidance lawsfor three-dimensional missile-target interceptionrdquo Journal ofAerospace Engineering vol 230 no 2 pp 392ndash403 2015
[14] N Zhang W Gai M Zhong and J Zhang ldquoA fast finite-timeconvergent guidance law with nonlinear disturbance observerfor unmanned aerial vehicles collision avoidancerdquo AerospaceScience and Technology vol 86 pp 204ndash214 2019
[15] J Song S Song and H Zhou ldquoAdaptive nonsingular fastterminal sliding mode guidance law with impact angle con-straintsrdquo International Journal of Control Automation andSystems vol 14 no 1 pp 99ndash114 2016
[16] J Song and S Song ldquoRobust impact angle constraintsguidance law with autopilot lag and acceleration saturationconsiderationrdquo Transactions of the Institute of Measurementand Control vol 41 no 1 pp 182ndash192 2019
[17] A Levant ldquoPrinciples of 2-sliding mode designrdquo Automaticavol 43 no 4 pp 576ndash586 2007
[18] A M Zhou ldquoFinite-time output feedback attitude trackingcontrol for rigid spacecraftrdquo IEEE Transactions on ControlSystems Technology vol 22 no 1 pp 338ndash345 2014
[19] B Li Q Hu Y Yu and G Ma ldquoObserver-based fault-tolerantattitude control for rigid spacecraftrdquo IEEE Transactions onAerospace and Electronic Systems vol 53 no 5 pp 2572ndash2582 2017
[20] H Li and Y Cai ldquoOn SFTSM control with fixed-time con-vergencerdquo IET Control Jeory amp Applications vol 11 no 6pp 766ndash773 2017
[21] J Ni L Liu C Liu X Hu and T Shen ldquoFixed-time dynamicsurface high-order slidingmode control for chaotic oscillationin power systemrdquo Nonlinear Dynamics vol 86 no 1pp 401ndash420 2016
[22] Q Zong Z S Zhao and J Zhang ldquoHigher order sliding modecontrol with self-tuning law based on integral sliding moderdquo
10 Journal of Control Science and Engineering
IET Control Jeory and Application vol 4 no 7 pp 1282ndash1289 2008
[23] S He D Lin and J Wang ldquoContinuous second-order slidingmode based impact angle guidance lawrdquo Aerospace Scienceand Technology vol 41 pp 199ndash208 2015
[24] B L Cong Z Chen and X D Liu ldquoOn adaptive sliding modecontrol without switching gain overestimationrdquo InternationalJournal of Robust and Nonlinear Control vol 24 no 3pp 515ndash531 2014
Journal of Control Science and Engineering 11
which has the nonstrict finite-time convergence problemand the nonconvergence factor slows down the convergencespeed Figure 5 shows the LOS angular rate curve Under thethree guidance laws the LOS angular rates converge to zeroin finite time ASNTSMG has smaller convergence error andfaster convergence speed Figure 6 shows the overload curveof the missile Due to the finite-time convergence of LOSangle and angular rate the missile needs a large overload inthe early stage of guidance -erefore the overloads of threeguidance laws are all saturated in the early stage When theLOS angle and the angular rate converge the missile
overload gradually converges to zero in the later stage Andthe convergence speed of ASNTSMG is faster
Table 1 shows the simulation results of attack timemiss distance LOS angle error and average overloadunder the three guidance laws It can be seen that com-pared with NFTSMG and SONTSMG ASNTSMG hassmaller attack time miss distance terminal LOS angleerror and average overload so ASNTSMG has betterguidance performance
According to the analysis of the simulation results of twocases ASNTSMG can hit the target precisely with the
expected impact angle under different expected LOS anglesand target maneuvering conditions Compared with theexisting guidance laws NFTSMG and SONTSMGASNTSMG can effectively attack the target with less impactangle error miss distance and energy consumption whichverifies the effectiveness and superiority of ASNTSMG
6 Conclusion
In this paper a nonsingular fast terminal sliding modeguidance law is proposed to solve the problem of guidanceaccuracy and impact angle constraint -rough theoreticalanalysis and simulation verification the following conclu-sions can be obtained
(1) -e proposed adaptive smooth supertwisting algo-rithm can effectively counteract the disturbance ofthe system and accelerate the convergence speed ofthe system without knowing the upper bound of thedisturbance
(2) -e designed nonsingular terminal sliding modesurface can realize the fast finite-time convergence ofthe system states and ensure the impact angleconstraint and guidance accuracy requirements
(3) -is guidance law can attack the target preciselyunder the conditions of different expected LOSangles and target maneuvers Compared with theexisting nonsingular fast terminal sliding modeguidance law and second-order nonsingular termi-nal sliding mode guidance law this law has higherguidance accuracy and angle constraint accuracy andconsumes less energy
Data Availability
-e data used to support the findings of this study are in-cluded within the article
Conflicts of Interest
-e authors declare that there are no conflicts of interestregarding the publication of this paper
References
[1] H Cai Z D Hu and Y Cao ldquoA survey of guidance law withterminal impact angle constraintsrdquo Journal of Astronauticsvol 31 no 2 pp 315ndash323 2010
[2] J M Song and T Q Zhang ldquoPassive homingmissilersquos variablestructure proportional navigation with terminal angularconstraintrdquo Chinese Journal of Aeronautics vol 14 no 2pp 83ndash87 2001
[3] P Wu and M Yang ldquoVariable structure guidance law withterminal attack angle constraintrdquo Journal of Solid RocketTechnology vol 31 no 2 pp 116ndash120 2002
[4] S R Kumar S Rao and D Ghose ldquoSliding-mode guidanceand control for all-aspect interceptors with terminal angleconstraintsrdquo Journal of Guidance Control and Dynamicsvol 35 no 4 pp 1230ndash1246 2012
[5] H B Zhou S M Song and M Y Xu ldquoDesign of terminalsliding-mode guidance law with attack angle constraintsrdquo in
Proceedings of the 2013 25th Chinese Control and DecisionConference (CCDC) Guiyang China May 2013
[6] Y X Zhang M W Sun and Z Q Chen ldquoFinite-timeconvergent guidance law with impact angle constraint basedon sliding-mode controlrdquo Nonlinear Dynamic vol 7 no 3pp 619ndash625 2012
[7] X H Yu and Z H Man ldquoFast terminal sliding-mode controldesign for nonlinear dynamical systemsrdquo IEEE Transactionson Circuits and Systems I Fundamental Jeory and Appli-cations vol 49 no 2 pp 261ndash264 2002
[8] J M Song S M Song and Y Guo ldquoNonlinear disturbanceobserver based fast terminal sliding mode guidance withimpact angle constraintsrdquo International Journal of InnovativeComputing Information and Control vol 11 no 3pp 787ndash802 2015
[9] S R Kumar S Rao and D Ghose ldquoNonsingular terminalslidingmode guidance with impact angle constraintsrdquo Journalof Guidance Control and Dynamics vol 37 no 4pp 1114ndash1130 2014
[10] S F Xiong W H Wang and S Wang ldquoNonsingular fastterminal sliding-mode guidance with intercept angle con-straintrdquo Control Jeory amp Applications vol 31 no 3pp 269ndash278 2014
[11] S C Yang K Q Zhang and P Chen ldquoAdaptive terminalsliding mode guidance law with impact angle constraintrdquoJournal of Beijing University of Aeronautics and Astronauticsvol 42 no 8 pp 1566ndash1574 2016
[12] Z L Zhang and J Zhou ldquoStrictly convergent non-singularterminal sliding mode guidance law with impact angle con-straintsrdquo Optik vol 127 no 22 pp 10971ndash10980 2016
[13] Z Zhang C Man S Li and S Jin ldquoFinite-time guidance lawsfor three-dimensional missile-target interceptionrdquo Journal ofAerospace Engineering vol 230 no 2 pp 392ndash403 2015
[14] N Zhang W Gai M Zhong and J Zhang ldquoA fast finite-timeconvergent guidance law with nonlinear disturbance observerfor unmanned aerial vehicles collision avoidancerdquo AerospaceScience and Technology vol 86 pp 204ndash214 2019
[15] J Song S Song and H Zhou ldquoAdaptive nonsingular fastterminal sliding mode guidance law with impact angle con-straintsrdquo International Journal of Control Automation andSystems vol 14 no 1 pp 99ndash114 2016
[16] J Song and S Song ldquoRobust impact angle constraintsguidance law with autopilot lag and acceleration saturationconsiderationrdquo Transactions of the Institute of Measurementand Control vol 41 no 1 pp 182ndash192 2019
[17] A Levant ldquoPrinciples of 2-sliding mode designrdquo Automaticavol 43 no 4 pp 576ndash586 2007
[18] A M Zhou ldquoFinite-time output feedback attitude trackingcontrol for rigid spacecraftrdquo IEEE Transactions on ControlSystems Technology vol 22 no 1 pp 338ndash345 2014
[19] B Li Q Hu Y Yu and G Ma ldquoObserver-based fault-tolerantattitude control for rigid spacecraftrdquo IEEE Transactions onAerospace and Electronic Systems vol 53 no 5 pp 2572ndash2582 2017
[20] H Li and Y Cai ldquoOn SFTSM control with fixed-time con-vergencerdquo IET Control Jeory amp Applications vol 11 no 6pp 766ndash773 2017
[21] J Ni L Liu C Liu X Hu and T Shen ldquoFixed-time dynamicsurface high-order slidingmode control for chaotic oscillationin power systemrdquo Nonlinear Dynamics vol 86 no 1pp 401ndash420 2016
[22] Q Zong Z S Zhao and J Zhang ldquoHigher order sliding modecontrol with self-tuning law based on integral sliding moderdquo
10 Journal of Control Science and Engineering
IET Control Jeory and Application vol 4 no 7 pp 1282ndash1289 2008
[23] S He D Lin and J Wang ldquoContinuous second-order slidingmode based impact angle guidance lawrdquo Aerospace Scienceand Technology vol 41 pp 199ndash208 2015
[24] B L Cong Z Chen and X D Liu ldquoOn adaptive sliding modecontrol without switching gain overestimationrdquo InternationalJournal of Robust and Nonlinear Control vol 24 no 3pp 515ndash531 2014
expected impact angle under different expected LOS anglesand target maneuvering conditions Compared with theexisting guidance laws NFTSMG and SONTSMGASNTSMG can effectively attack the target with less impactangle error miss distance and energy consumption whichverifies the effectiveness and superiority of ASNTSMG
6 Conclusion
In this paper a nonsingular fast terminal sliding modeguidance law is proposed to solve the problem of guidanceaccuracy and impact angle constraint -rough theoreticalanalysis and simulation verification the following conclu-sions can be obtained
(1) -e proposed adaptive smooth supertwisting algo-rithm can effectively counteract the disturbance ofthe system and accelerate the convergence speed ofthe system without knowing the upper bound of thedisturbance
(2) -e designed nonsingular terminal sliding modesurface can realize the fast finite-time convergence ofthe system states and ensure the impact angleconstraint and guidance accuracy requirements
(3) -is guidance law can attack the target preciselyunder the conditions of different expected LOSangles and target maneuvers Compared with theexisting nonsingular fast terminal sliding modeguidance law and second-order nonsingular termi-nal sliding mode guidance law this law has higherguidance accuracy and angle constraint accuracy andconsumes less energy
Data Availability
-e data used to support the findings of this study are in-cluded within the article
Conflicts of Interest
-e authors declare that there are no conflicts of interestregarding the publication of this paper
References
[1] H Cai Z D Hu and Y Cao ldquoA survey of guidance law withterminal impact angle constraintsrdquo Journal of Astronauticsvol 31 no 2 pp 315ndash323 2010
[2] J M Song and T Q Zhang ldquoPassive homingmissilersquos variablestructure proportional navigation with terminal angularconstraintrdquo Chinese Journal of Aeronautics vol 14 no 2pp 83ndash87 2001
[3] P Wu and M Yang ldquoVariable structure guidance law withterminal attack angle constraintrdquo Journal of Solid RocketTechnology vol 31 no 2 pp 116ndash120 2002
[4] S R Kumar S Rao and D Ghose ldquoSliding-mode guidanceand control for all-aspect interceptors with terminal angleconstraintsrdquo Journal of Guidance Control and Dynamicsvol 35 no 4 pp 1230ndash1246 2012
[5] H B Zhou S M Song and M Y Xu ldquoDesign of terminalsliding-mode guidance law with attack angle constraintsrdquo in
Proceedings of the 2013 25th Chinese Control and DecisionConference (CCDC) Guiyang China May 2013
[6] Y X Zhang M W Sun and Z Q Chen ldquoFinite-timeconvergent guidance law with impact angle constraint basedon sliding-mode controlrdquo Nonlinear Dynamic vol 7 no 3pp 619ndash625 2012
[7] X H Yu and Z H Man ldquoFast terminal sliding-mode controldesign for nonlinear dynamical systemsrdquo IEEE Transactionson Circuits and Systems I Fundamental Jeory and Appli-cations vol 49 no 2 pp 261ndash264 2002
[8] J M Song S M Song and Y Guo ldquoNonlinear disturbanceobserver based fast terminal sliding mode guidance withimpact angle constraintsrdquo International Journal of InnovativeComputing Information and Control vol 11 no 3pp 787ndash802 2015
[9] S R Kumar S Rao and D Ghose ldquoNonsingular terminalslidingmode guidance with impact angle constraintsrdquo Journalof Guidance Control and Dynamics vol 37 no 4pp 1114ndash1130 2014
[10] S F Xiong W H Wang and S Wang ldquoNonsingular fastterminal sliding-mode guidance with intercept angle con-straintrdquo Control Jeory amp Applications vol 31 no 3pp 269ndash278 2014
[11] S C Yang K Q Zhang and P Chen ldquoAdaptive terminalsliding mode guidance law with impact angle constraintrdquoJournal of Beijing University of Aeronautics and Astronauticsvol 42 no 8 pp 1566ndash1574 2016
[12] Z L Zhang and J Zhou ldquoStrictly convergent non-singularterminal sliding mode guidance law with impact angle con-straintsrdquo Optik vol 127 no 22 pp 10971ndash10980 2016
[13] Z Zhang C Man S Li and S Jin ldquoFinite-time guidance lawsfor three-dimensional missile-target interceptionrdquo Journal ofAerospace Engineering vol 230 no 2 pp 392ndash403 2015
[14] N Zhang W Gai M Zhong and J Zhang ldquoA fast finite-timeconvergent guidance law with nonlinear disturbance observerfor unmanned aerial vehicles collision avoidancerdquo AerospaceScience and Technology vol 86 pp 204ndash214 2019
[15] J Song S Song and H Zhou ldquoAdaptive nonsingular fastterminal sliding mode guidance law with impact angle con-straintsrdquo International Journal of Control Automation andSystems vol 14 no 1 pp 99ndash114 2016
[16] J Song and S Song ldquoRobust impact angle constraintsguidance law with autopilot lag and acceleration saturationconsiderationrdquo Transactions of the Institute of Measurementand Control vol 41 no 1 pp 182ndash192 2019
[17] A Levant ldquoPrinciples of 2-sliding mode designrdquo Automaticavol 43 no 4 pp 576ndash586 2007
[18] A M Zhou ldquoFinite-time output feedback attitude trackingcontrol for rigid spacecraftrdquo IEEE Transactions on ControlSystems Technology vol 22 no 1 pp 338ndash345 2014
[19] B Li Q Hu Y Yu and G Ma ldquoObserver-based fault-tolerantattitude control for rigid spacecraftrdquo IEEE Transactions onAerospace and Electronic Systems vol 53 no 5 pp 2572ndash2582 2017
[20] H Li and Y Cai ldquoOn SFTSM control with fixed-time con-vergencerdquo IET Control Jeory amp Applications vol 11 no 6pp 766ndash773 2017
[21] J Ni L Liu C Liu X Hu and T Shen ldquoFixed-time dynamicsurface high-order slidingmode control for chaotic oscillationin power systemrdquo Nonlinear Dynamics vol 86 no 1pp 401ndash420 2016
[22] Q Zong Z S Zhao and J Zhang ldquoHigher order sliding modecontrol with self-tuning law based on integral sliding moderdquo
10 Journal of Control Science and Engineering
IET Control Jeory and Application vol 4 no 7 pp 1282ndash1289 2008
[23] S He D Lin and J Wang ldquoContinuous second-order slidingmode based impact angle guidance lawrdquo Aerospace Scienceand Technology vol 41 pp 199ndash208 2015
[24] B L Cong Z Chen and X D Liu ldquoOn adaptive sliding modecontrol without switching gain overestimationrdquo InternationalJournal of Robust and Nonlinear Control vol 24 no 3pp 515ndash531 2014
Journal of Control Science and Engineering 11
expected impact angle under different expected LOS anglesand target maneuvering conditions Compared with theexisting guidance laws NFTSMG and SONTSMGASNTSMG can effectively attack the target with less impactangle error miss distance and energy consumption whichverifies the effectiveness and superiority of ASNTSMG
6 Conclusion
In this paper a nonsingular fast terminal sliding modeguidance law is proposed to solve the problem of guidanceaccuracy and impact angle constraint -rough theoreticalanalysis and simulation verification the following conclu-sions can be obtained
(1) -e proposed adaptive smooth supertwisting algo-rithm can effectively counteract the disturbance ofthe system and accelerate the convergence speed ofthe system without knowing the upper bound of thedisturbance
(2) -e designed nonsingular terminal sliding modesurface can realize the fast finite-time convergence ofthe system states and ensure the impact angleconstraint and guidance accuracy requirements
(3) -is guidance law can attack the target preciselyunder the conditions of different expected LOSangles and target maneuvers Compared with theexisting nonsingular fast terminal sliding modeguidance law and second-order nonsingular termi-nal sliding mode guidance law this law has higherguidance accuracy and angle constraint accuracy andconsumes less energy
Data Availability
-e data used to support the findings of this study are in-cluded within the article
Conflicts of Interest
-e authors declare that there are no conflicts of interestregarding the publication of this paper
References
[1] H Cai Z D Hu and Y Cao ldquoA survey of guidance law withterminal impact angle constraintsrdquo Journal of Astronauticsvol 31 no 2 pp 315ndash323 2010
[2] J M Song and T Q Zhang ldquoPassive homingmissilersquos variablestructure proportional navigation with terminal angularconstraintrdquo Chinese Journal of Aeronautics vol 14 no 2pp 83ndash87 2001
[3] P Wu and M Yang ldquoVariable structure guidance law withterminal attack angle constraintrdquo Journal of Solid RocketTechnology vol 31 no 2 pp 116ndash120 2002
[4] S R Kumar S Rao and D Ghose ldquoSliding-mode guidanceand control for all-aspect interceptors with terminal angleconstraintsrdquo Journal of Guidance Control and Dynamicsvol 35 no 4 pp 1230ndash1246 2012
[5] H B Zhou S M Song and M Y Xu ldquoDesign of terminalsliding-mode guidance law with attack angle constraintsrdquo in
Proceedings of the 2013 25th Chinese Control and DecisionConference (CCDC) Guiyang China May 2013
[6] Y X Zhang M W Sun and Z Q Chen ldquoFinite-timeconvergent guidance law with impact angle constraint basedon sliding-mode controlrdquo Nonlinear Dynamic vol 7 no 3pp 619ndash625 2012
[7] X H Yu and Z H Man ldquoFast terminal sliding-mode controldesign for nonlinear dynamical systemsrdquo IEEE Transactionson Circuits and Systems I Fundamental Jeory and Appli-cations vol 49 no 2 pp 261ndash264 2002
[8] J M Song S M Song and Y Guo ldquoNonlinear disturbanceobserver based fast terminal sliding mode guidance withimpact angle constraintsrdquo International Journal of InnovativeComputing Information and Control vol 11 no 3pp 787ndash802 2015
[9] S R Kumar S Rao and D Ghose ldquoNonsingular terminalslidingmode guidance with impact angle constraintsrdquo Journalof Guidance Control and Dynamics vol 37 no 4pp 1114ndash1130 2014
[10] S F Xiong W H Wang and S Wang ldquoNonsingular fastterminal sliding-mode guidance with intercept angle con-straintrdquo Control Jeory amp Applications vol 31 no 3pp 269ndash278 2014
[11] S C Yang K Q Zhang and P Chen ldquoAdaptive terminalsliding mode guidance law with impact angle constraintrdquoJournal of Beijing University of Aeronautics and Astronauticsvol 42 no 8 pp 1566ndash1574 2016
[12] Z L Zhang and J Zhou ldquoStrictly convergent non-singularterminal sliding mode guidance law with impact angle con-straintsrdquo Optik vol 127 no 22 pp 10971ndash10980 2016
[13] Z Zhang C Man S Li and S Jin ldquoFinite-time guidance lawsfor three-dimensional missile-target interceptionrdquo Journal ofAerospace Engineering vol 230 no 2 pp 392ndash403 2015
[14] N Zhang W Gai M Zhong and J Zhang ldquoA fast finite-timeconvergent guidance law with nonlinear disturbance observerfor unmanned aerial vehicles collision avoidancerdquo AerospaceScience and Technology vol 86 pp 204ndash214 2019
[15] J Song S Song and H Zhou ldquoAdaptive nonsingular fastterminal sliding mode guidance law with impact angle con-straintsrdquo International Journal of Control Automation andSystems vol 14 no 1 pp 99ndash114 2016
[16] J Song and S Song ldquoRobust impact angle constraintsguidance law with autopilot lag and acceleration saturationconsiderationrdquo Transactions of the Institute of Measurementand Control vol 41 no 1 pp 182ndash192 2019
[17] A Levant ldquoPrinciples of 2-sliding mode designrdquo Automaticavol 43 no 4 pp 576ndash586 2007
[18] A M Zhou ldquoFinite-time output feedback attitude trackingcontrol for rigid spacecraftrdquo IEEE Transactions on ControlSystems Technology vol 22 no 1 pp 338ndash345 2014
[19] B Li Q Hu Y Yu and G Ma ldquoObserver-based fault-tolerantattitude control for rigid spacecraftrdquo IEEE Transactions onAerospace and Electronic Systems vol 53 no 5 pp 2572ndash2582 2017
[20] H Li and Y Cai ldquoOn SFTSM control with fixed-time con-vergencerdquo IET Control Jeory amp Applications vol 11 no 6pp 766ndash773 2017
[21] J Ni L Liu C Liu X Hu and T Shen ldquoFixed-time dynamicsurface high-order slidingmode control for chaotic oscillationin power systemrdquo Nonlinear Dynamics vol 86 no 1pp 401ndash420 2016
[22] Q Zong Z S Zhao and J Zhang ldquoHigher order sliding modecontrol with self-tuning law based on integral sliding moderdquo
10 Journal of Control Science and Engineering
IET Control Jeory and Application vol 4 no 7 pp 1282ndash1289 2008
[23] S He D Lin and J Wang ldquoContinuous second-order slidingmode based impact angle guidance lawrdquo Aerospace Scienceand Technology vol 41 pp 199ndash208 2015
[24] B L Cong Z Chen and X D Liu ldquoOn adaptive sliding modecontrol without switching gain overestimationrdquo InternationalJournal of Robust and Nonlinear Control vol 24 no 3pp 515ndash531 2014
Journal of Control Science and Engineering 11
IET Control Jeory and Application vol 4 no 7 pp 1282ndash1289 2008
[23] S He D Lin and J Wang ldquoContinuous second-order slidingmode based impact angle guidance lawrdquo Aerospace Scienceand Technology vol 41 pp 199ndash208 2015
[24] B L Cong Z Chen and X D Liu ldquoOn adaptive sliding modecontrol without switching gain overestimationrdquo InternationalJournal of Robust and Nonlinear Control vol 24 no 3pp 515ndash531 2014
