research article integrated guidance and control method
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Research ArticleIntegrated Guidance and Control Method forthe Interception of Maneuvering Hypersonic VehicleBased on High Order Sliding Mode Approach
Kang Chen1 Bin Fu1 Yuening Ding2 and Jie Yan1
1College of Astronautics Northwestern Polytechnical University Xirsquoan 710072 China2Flight Automation Control Research Institute Xirsquoan 710065 China
Correspondence should be addressed to Bin Fu binfumailnwpueducn
Received 27 April 2015 Revised 26 July 2015 Accepted 29 July 2015
Academic Editor Giuseppe Rega
Copyright copy 2015 Kang Chen et al This 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
This paper focuses on the integrated guidance and control (IGC) method applied in the interception of maneuvering near spacehypersonic vehicles using the homogeneous high order sliding mode (HOSM) approach The IGC model is derived by combiningthe target-missile relative motion and dynamic equations Then a fourth-order sliding mode controller is implemented in theaugmented IGC model To estimate the high order derivatives of the sliding manifold which is required in the HOSMmethod anArbitrary Order Robust Exact Differentiator is presented At last the idea of virtual control is introduced to alleviate the chatteringof the control input without using any saturation functions which may lead to a loss of the robustness And the stability of theclosed-loop system with presented fourth-order homogeneous HOSM controller is also proved theoretically Finally simulationresults are provided and analyzed to demonstrate the effectiveness of the proposed method in three typical engagement scenarios
1 Introduction
Because of its speed advantage and flexible maneuverabilitythe hypersonic vehicle may certainly become a severe threatin the future battlefield to deal with the threat the researchon the interception of the hypersonic vehicle is already onthe agenda Primarily the interception of the hypersonicvehicle faces the following issues (1) when the speed ofits target is much faster than the interception missile theeffective attack area of the traditional guidance law greatlyshrinks and it is impossible to accomplish the tail-chase orbackward interception (2) at the high altitude of 25 to 40 kmwhere the hypersonic vehicle flies the air is relatively thinthe aerodynamic efficiency of an interception missile is lowand there is a limited usable overload for the interceptor(3) it is difficult to destroy such a hypersonic target withthe traditional destructive means such as near explosionfragments requiring that the interception missile should useknock-on collision as much as possible to attack the targetnamely minimal target missing As the requirements for
guidance accuracy are higher researchers do massive workto advance the guidance and control theory
During the past decades the proportional navigation(PN) guidance law is a popular and widely used method inmissile interception missions for its ease of implementationand high efficiency The principle of PN guidance law in [1]is that the commanded normal acceleration of the missileis proportional to the line-of-sight (LOS) rate which issimple and effective under a wide range of engagementscenarios However the use of the PN guidance law is alsolimited for example as the distance between the missileand its target is closing or the target acts an unpredictablemaneuver the LOS rate may grow extremely fast andsubsequently the overload needed at the end phase alsodiverges at last As for the overload autopilot accountingfor a second-order dynamic the real acceleration responseof the missile to the fast-changing high-frequency overloadcommand lags behind and attenuates and eventually caused alarge miss distance Therefore many new guidance laws cropup
Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 648231 19 pageshttpdxdoiorg1011552015648231
2 Mathematical Problems in Engineering
Being famous for its good robustness to bounded distur-bances the sliding mode control method is introduced intothe study of guidance law In [2] Zhou et al gave the con-ditions for the sliding mode motion of a linear time-varyingsystem not to be disturbed by the disturbances and param-eter perturbations and proposed an adaptive sliding modeguidance law (ASMG) simulation results demonstrated thatthe ASMG is robust to uncertainties like target accelerationTo get finite time convergence modified SMC like terminalsliding mode control (TSMC) is proposed In [3] Zeng andHu combine the advantages of linear and terminal slidingmode controls which guarantee the convergence of trackingerrors in finite time And then nonsingular terminal slidingmode control (NTSMC) is introduced in the guidance lawdesign In [4] Kumar et al proposed a nonsingular terminalsliding mode guidance law with finite time convergencewhich avoids the singularity that may lead to the saturationof the control Integral SMC (ISMC) like that introduced in[5] is another revised sliding mode control which introducedthe integral sliding mode scheme to the guidance law designthe proposed ISMC guidance law provided a smaller controlmagnitude than the traditional sliding mode design Theadvance of the 119867
infincontroller greatly enriched the methods
to implement a robust control system In [6] Savkin etal suitably modified the 119867
infincontrol theory and provided
an effective framework for the precision missile guidanceproblem which showed much better performance than thelinear quadratic optimal guidance law in the computersimulation In [7] Chen and Yang proposed a mixed 119867
2119867infin
guidance design against maneuvering targets in which thecomplete nonlinear kinematics of the pursuit-evasionmotionwas considered
In addition to the separated guidance and control lawthe integrated guidance and control method is under heateddiscussion According to the relative motion between thetarget and the missile the traditional guidance law calculatesthe overload needed to hit the target and inputs the overloadcommand into the overload autopilot while the integratedguidance and control (IGC) method gives the rudder deflec-tion command directly to the missile according to therelative motion which evidently responds more quickly In[8] Menon et al designed an IGCmethod by using the linearquadratic optimal theory but the robustness of the system isnot satisfactory In [9] Vaddi et al provided a fully numericalmethodology for deriving state-dependent Riccati equationcontrollers for arbitrarily complex dynamic systems andapplied it in the IGC design of a missile Simulation resultsdemonstrated the effectiveness of the method In [10] Xinet al employed the theta-D method to give an approximateclosed-form suboptimal feedback controller to the nonlinearinfinite-horizon IGC problem Taking another look slidingmode approaches are also employed in the IGC design ofhoming missiles In [11] Shima et al proposed the slidingmode integrated guidance and control method based on theZEM Shtessel and Tournes doesmassive research on the highorder sliding mode controller design and proposed his newmethod In [12] he designed the high order sliding modeguidance law based on the smooth second-order slidingmode control (SSOSMC) which is smoother in high orders
than the traditional second-order sliding mode guidance lawIn [13] based on the geometrical homogeneity theory Donget al designed the tranquility control law for the integratedguidance and control model which however has rathermore parameters and is sensitive to parameters and theparameters must be carefully selected In [14] Mingzhe andGuangren used the sliding mode control theory to design theadaptive nonlinear feedback controller which is primarily forfixed ground target being unable to deal with the vehementperturbation caused by target maneuvering
Motivated by the aforementioned considerations thiswork will design an IGC scheme for the interception of thenear space maneuvering hypersonic vehicles Firstly a line-of-sight (LOS) rate feedback scheme is adopted to derive theIGC law and as the relative order of the control input to theLOS rate is higher than one a HOSM approach is introducedSecondly to implement the HOSM approach the 119899 timesderivations of the sliding manifold 119878 119878
119878 119878
(119899) must beknown However reconstruction of each 119878 119878
119878 119878
(119899) bytheir analytical expression could be rather difficult in prac-tice An alternative way adopted in this paper is to use theArbitrary-Order Robust Exact Differentiator (AORED) toestimate the derivations And next to alleviate the chatteringphenomenon caused by the HOSM controller the idea ofvirtual control is introducedThe virtual control V is designedand used as the control input of a system extended from theoriginal one and the real control 119906 acting on the real systemis obtained by integrating the virtual control V Benefittingfrom the integration element the real control input 119906 couldbe smooth enough for the implementation without reducingthe robustness of the HOSM method Finally the proposedmethod is implemented in a 3-dof model
The remaining part of this paper is organized as followsIn Section 2 the IGC model is derived in the longitudinalplane In Section 3 the quasi-continuous HOSM controlleras well as the AORED is designed In Section 4 the baselineseparated guidance laws and controller are given In Sec-tion 5 the numerical simulations are demonstrated in threetypical engagement scenarios And conclusions are made inSection 6
2 Integrated Guidance and Control Model
The traditional guidance and control algorithm usually usesthe guidance loop as its outer loop and is only responsible forgiving commanded overload then the control loop is onlyresponsible for tracking the overload command eventuallyachieving the missilersquos guidance toward its target Althoughit is always desirable to design the control loop or theautopilot to have better dynamic performance in actualitythe controller always has some delay and attenuation Asa result the missile always has some error in acting theoverload command That is one of the reasons why aninterceptor misses its target
The integrated guidance and control algorithm combinesthe guidance and control loops into one loop and avoidsthe delay and attenuation caused by them Its architecture isshown in Figure 1
Mathematical Problems in Engineering 3
Seeker
Guidancefiltering
Integratedguidance and control
ActuatorMissile
airframe
Missile position
Target position
Missile acceleration
Target acceleration
Guidancelaw
Autopilot(controller) Separated guidance and control
Integrated guidance and control +minus
120575CnC
q
120575120575C
q
q nm
Figure 1 Integrated guidance and control
O
Y1
X1
M
aM
xb
120572120579M
q
VM
T
VT
120579T
aT
120599
Figure 2 Engagement geometry
21 The Engagement Dynamics Without loss of generalitywe present hereinafter only the subsystems that govern themotion of an interceptor in its longitudinal plane The planarengagement between the interceptor and its target is shownin Figure 2
The study of guidance laws usually regards themissile andits target as mass points deliberates on the mass point of itsrigid motion and ignores the attitude of the missile airframeand changes in its attitude By contrast the integrated guid-ance and control method takes the dynamic characteristicsof the missile airframe into consideration therefore thederivation involves the pitch angle pitch rate and angle ofattack of the missile as shown in Figure 2
The motion equations of the missile in the longitudinalplane are as follows
119889119881119872
119889119905=
1
119898(119875 cos120572 minus 119883 minus 119898119892 sin 120579
119872) (1)
119889120579119872
119889119905=
1
119898119881119872
(119875 sin120572 + 119884 minus 119898119892 cos 120579119872
) (2)
119889119909119872
119889119905= 119881119872cos 120579119872
(3)
119889119910119872
119889119905= 119881119872sin 120579119872
(4)
119889120596119885
119889119905=
119872119885
119869119885
(5)
119889120599
119889119905= 120596119885 (6)
120572 = 120599 minus 120579 (7)
where (119909119872
119910119872
) is the position 119898 is the mass 119883 is the axialforce 119875 is the thrust force the missilersquos velocity is 119881
119872 and its
flight path angle is 120579119872 the LOS angle between themissile and
its target is 119902 the relative distance between the missile and itstarget is 119903 the velocity of the target is 119881
119879 and its flight path
angle is 120579119879 119909119887is the axle of the missilersquos airframe the pitch
angle is 120599 the angle of attack is 120572The motion equations of the target are as follows
119889119909119879
119889119905= minus119881119879cos 120579119879
119889119910119879
119889119905= 119881119879sin 120579119879
119889120579119879
119889119905=
119886119879119873
119881119879
(8)
The interception is characterized by two variablesnamely the target range and the LOS angle The kinematicequations are expressed by the following relations
119889119903
119889119905= minus119881119872cos (120579
119872minus 119902) minus 119881
119879cos (120579
119879+ 119902) (9)
119903119889119902
119889119905= minus119881119872sin (120579119872
minus 119902) + 119881119879sin (120579119879
+ 119902) (10)
22 The Model Simplification To simplify the model andmake further derivations wemake the following two assump-tions
Assumption 1 Within the terminal phase of the interceptionthe missile has no thrust and its gravity is not taken intoaccount
Assumption 2 Within the terminal phase of the interceptionthe missilersquos speed does not change
4 Mathematical Problems in Engineering
Then (1) and (2) can be reformed as below
119889119881119872
119889119905= 0
119889120579119872
119889119905=
119884
119898119881119872
(11)
Denote that 119886119872119873
= 119884119898 119884 is the normal force 119886119872119873
119886119879119873
are the normal acceleration of the missile and targetrespectively and then
120579119872
=119886119872119873
119881119872
120579119879
=119886119879119873
119881119879
(12)
The normal force 119884 and the pitch moment 119872119885acting on
the missile are usually expressed respectively as follows
119884 = 119862120572
119884120572119876119878ref + 119862
120575119885
119884120575119885119876119878ref (13)
119872119885
= 119898120575119885
119885120575119885119876119878ref119897 + 119898
120572
119885120572119876119878ref119897 (14)
where119862120572
119884is the coefficient of normal force caused by the angle
of attack 119862120575119885
119884is the coefficient of normal force caused by the
rudder deflection angle 120575119885 119876 is the dynamic pressure 119878ref is
the reference area 119897 is the reference length But the normalforce produced by the rudder deflection angle 120575
119885is orders of
magnitude smaller than that produced by the angle of attackso (13) is simplified as
119884 = 119862120572
119884120572119876119878ref (15)
Then
119886119872119873
=119884
119898=
119862120572
119884120572119876119878ref
119898 (16)
Then the dynamic equations can be simplified as
119889120579119872
119889119905=
119862120572
119884120572119876119878ref
119898119881119872
(17)
119889120596119885
119889119905=
119898120575119885
119885120575119885
119876119878ref119897 + 119898120572
119885120572119876119878ref119897
119869119885
(18)
119889120599
119889119905= 120596119885 (19)
120572 = 120599 minus 120579119872
(20)
23 The Integrated Guidance and Control Model Follow-ing the above derivation we select the state variables as
( 119902 120579119872
120572 120596119885)119879 and obtain the following nonlinear integrated
guidance and control model
119902 = minus2119903
119903sdot 119902 minus
119862120572
119884119876119878ref
119898119903cos (120579
119872minus 119902) sdot 120572
+119886119879119873
119903cos (120579
119879+ 119902)
119903 119902 = minus119881119872sin (120579119872
minus 119902) + 119881119879sin (120579119879
+ 119902)
120579119872
=1
119898119881119872
119862120572
119884119876119878ref sdot 120572
= 120596119885
minus1
119898119881119872
119862120572
119884119876119878ref sdot 120572
119885
=119898120575119885
119885119876119878ref119897
119869119885
sdot 120575119885
+119898120572
119885119876119878ref119897
119869119885
120572
(21)
24 The Relative Degree of Control Input To obtain therelative degree of the control input 120575 of the integratedguidance and control method we keep on deriving the LOSangular velocity 119902 until the explicit formula of its derivativeof a certain order contains the control input
The derivation of (10) produces119902
=1
119903[minus2 119903 119902 minus 119886
119872119873cos (120579
119872minus 119902) + 119886
119879119873cos (120579
119879+ 119902)]
(22)
The LOS angular velocity is expressed as the derivativeof the first order and does not contain the control input 120575
explicitly The continuous derivation of the above equationproduces
119902 =
1
119903minus2 119903 119902 minus 3 119903 119902 minus [119886
119872119873( 119902 minus 120579
119872) sin (120579
119872minus 119902)
+ 119886119872119873
cos (120579119872
minus 119902)] minus [119886119879119873
( 119902 + 120579119879) sin (120579
119879+ 119902)
minus 119886119879119873
cos (120579119879
+ 119902)]
(23)
where the LOS angular rate is expressed as the differentiatingof the second order and 119886
119872can be expressed as
119886119872119873
=119862120572
119884119876119878ref
119898 =
119862120572
119884119876119878ref
119898( 120599 minus 120579
119872)
=119862120572
119884119876119878ref
119898(120596119885
minus 120579119872
)
(24)
Although the control volume 120575119885does not appear in the
derivative of the second order (18) shows that 119885contains
120575119885 We continue to derive (23) and substitute 119886
119872as follows
119886119872119873
=119862120572
119884119876119878ref
119898(119885
minus119886119872
119881119872
)
=119862120572
119884119876119878ref
119898
119898120575119885
119885119876119878ref119897ref
119869119885
sdot 120575119885
+119862120572
119884119876119878ref
119898(
119898120572
119885119876119878ref119897ref
119869119885
120572 minus119886119872
119881119872
)
(25)
Mathematical Problems in Engineering 5
Thus
119902(4)
= 119891120575119885
sdot 120575119885
+1
119903(1198911
+ 1198912
+ 1198913
+ 1198914
+ 1198915
+ 1198916) (26)
where
119891120575119885
= minus1
119903
119862120572
119884119876119878ref cos 120578
119872
119898
119898120575119885
119885119876119878ref119897ref
119869119885
(27)
1198911
= (120596119885
minus 120579119872
) (minus2 120579119886119872119873
120572sin 120578119872
+ 119902119886119872119873
120572sin 120578119872
+ 120579119886119872119873
1205722cos 120578119872
) minus119886119872119873
cos 120578119872
120572
119898120572
119885120572119876119878ref119897ref
119869119885
(28)
1198912
= 120579119872
(( 120579119872
minus 119902) 119886119872119873
cos 120578119872
minus (120596119885
minus 120579119872
)119886119872119873
120572sin 120578119872
)
(29)
1198913
= 119902 (minus 120579119872
119886119872119873
cos 120578119872
+ 119902 (119886119872119873
cos 120578119872
minus 119886119879119873
cos 120578119879) minus 120579119879119886119879119873
cos 120578119879
+119886119872119873
120572(120596119885
minus 120579119872
) sin 120578119872
minus 119886119879sin 120578119879)
(30)
1198914
= 120579119879
(minus 119902119886119879cos 120578119879
minus 120579119879119886119879cos 120578119879
minus 119886119879sin 120578119879) (31)
1198915
= 119886119879
(minus 119902 sin 120578119879
minus 2 120579119879sin 120578119879) + 119886119879cos 120578119879 (32)
1198916
= 119902 (119886119872119873
sin 120578119872
minus 119886119879sin 120578119879
minus 2 119903) minus 4 119903 sdot119902 minus 3 119903 sdot 119902
minus 2119903 sdot 119902
(33)
120578119872
= 120579119872
minus 119902
120578119879
= 120579119879
+ 119902
(34)
In (26) it can be seen that the control input 120575119885appears
expressly in the third-order derivative of the control output119902 Therefore the relative degree of the control input 120575
119885is 3
3 The Quasi-Continuous High Order SlidingMode Controller
31 Sliding Mode Manifold Design To design the HOSMcontroller a sliding manifold must be chosen first In thisdesign we try to make the LOS rate converge to zero ora small neighbor domain near zero thus ensuring that themissile approaches its target in a quasi-parallel way whichwill lead to a minimal overload requirement So the slidingmanifold is chosen as follows
120590 = 119902 (35)
From the above discussion in Section 2 we know thatthe control input in relation to control output 119902 namelythe relative degree of sliding mode manifold 120590 is 3 So thefollowing design will be about a third-order sliding modecontroller
32 Design of the Quasi-Continuous HOSM Controller First(26) can be expressed as follows
120590 = ℎ (119905 119909) + 119892 (119905 119909) 119906 (36)
where ℎ(119905 119909) 119892(119905 119909) and 119906 are expressed as follows
ℎ (119905 119909) =1
119903(1198911
+ 1198912
+ 1198913
+ 1198914
+ 1198915
+ 1198916)
119892 (119905 119909) = 119891120575119885
119906 = 120575119885
(37)
According to the quasi-continuous high order slidingmode control method proposed by Levant in [15] the slidingmode manifold whose relative degree is 3 should be designedin the following form where 120573 is a control gain term
119906 = minus120573
+ 2 (|| + |120590|23
)minus12
( + |120590|23 sign120590)
|| + 2 (|| + |120590|23
)12
(38)
The conditions under which the LOS angular velocitymay converge are as follows
0 lt 119870119898
le 119892 (119905 119909) le 119870119872
|ℎ (119905 119909)| le 119862
(39)
where119870119898119870119872 and119862 are all larger than zeroThis is a proven
theorem by Levant in [15]The system we discussed meets the above requirements
and the proof is as followsEquation (27) shows the following
119892 (119905 119909) = 119891120575119885
= minus1
119903
119862120572
119884119876119878ref
119898
119898120575119885
119911119876119878ref119897ref
119869119885
cos (120579119872
minus 119902)
(40)
The dynamic pressure 119876 is 119876 = 1205881198812
1198722 where 120588 =
008803Kgm3 (altitude = 20Km) is the air density and119881119872
=
2000ms is the speed of the missile so 119876 is always positive119878ref = 026m2 and 119897ref = 365m denote the reference area
and the reference length of the missile they are both positiveconstant
119898 = 100Kg denotes the missile mass 119869119885
= 106m2 Kgdenotes the rotational inertia
119903 is the relative distance it is always a positive number119862120572
119884is the lift coefficient caused by the angle of attack it
varies from 018 to 037 and it is always a positive number119898120575119911
119911is the moment coefficient caused by the actuator
deflection In the normal layout (actuator lays behind thecenter of gravity) 119898
120575119911
119911is always negative
Meanwhile consider that the missile under guidance andcontrol is unlikely to fly away from its target namely the anglebetween themissilersquos velocity and its LOS direction cannot belarger than 90∘ then
1003816100381610038161003816120579119872 minus 1199021003816100381610038161003816 lt
120587
2
997904rArr cos (120579119872
minus 119902) gt 0
(41)
6 Mathematical Problems in Engineering
Summing up the above conditions then we can get
119892 (119905 119909) gt 0 (42)
In other words there is a positive real number 119870119898existing
that could satisfy the following condition
0 lt 119870119898
lt 119892 (119905 119909) (43)
Before the missile hits on the target the term will be positiveand limited then we can get
0 lt 119870119898
lt 119892 (119905 119909) lt 119870119872
(44)
With (37) then
ℎ (119905 119909) =1
119903(1198911
+ 1198912
+ 1198913
+ 1198914
+ 1198915
+ 1198916) (45)
In the practice sense the changes in both the LOS rateand the acceleration of the missile and the acceleration of thetarget are limited and continuous So the following variables119902 119902
119902 119886119872 119886119879 and 119886
119879are all bounded However because
ℎ(119905 119909) contains the item 1119903 when the relative distancebetween the missile and its target is zero the boundaryof ℎ(119905 119909) is not guaranteed In [15] Levant only requiresthat condition (39) should be locally valid not requiringthat it should be globally valid Therefore the integratedguidance and controlmethod is applicable here So the abovementioned condition is satisfied with a positive number 119862
|ℎ (119905 119909)| le 119862 (46)
33TheVirtual ControlDesign Whenusing the slidingmodecontrolmethod the avoidance of the chattering phenomenonhas always been a key issue being discussed In the tradi-tional method researchers in [16 17] have proposed severalsaturation functions to replace the sign functions to builda boundary layer to alleviate the chattering or to use fuzzylogic to displace the high-frequency switching term To ourknowledge none of these approaches has proven that therefined controller still retains their robustness against theuncertainties and disturbances In this work in order toalleviate the chattering phenomenon we do not directly usethe third-order controller but introduce the virtual control119906119894= 120575119885to perform the actual control
120575119885
= int 120575119885dt = int 119906
119894dt (47)
After the relative degree is increased to the fourth order weget the following expressions
120590(4)
= ℎlowast
(119905 119909) + 119892lowast
(119905 119909) 119906119894= ℎlowast
(119905 119909) + 119892 (119905 119909) 119906119894
ℎlowast
(119905 119909) = ℎ (119905 119909) + 119892 (119905 119909) 120575119885
119892 (119905 119909)
=119898120575119885
1199111198762
1198782
119897119862120572
119884
119869119885
1198981199032[( 119902 minus 120579
119872) 119903 sin 120578
119872+ 119903 cos 120578
119872]
(48)
Even though the expression of ℎ(119905 119909) is rather compli-cated it is still the function of 119902 119902 119902 119886
119872 119886119879 and 119886
119879 therefore
similar to ℎ(119905 119909) it has its boundary except themomentwhenthe missile hits on its target For the same reason 119892(119905 119909)
and the rudder deflection 120575119885also have their boundaries
Therefore we get the condition that |ℎlowast
(119905 119909)| le 119862 (119862 gt 0)Because 120575
119885is obtained through the derivation of 120575
119885 119892lowast(119905 119909)
is the same as 119892(119905 119909) thus 0 lt 119870119898
lt 119892lowast
(119905 119909) lt 119870119872
issatisfied
According to the formula of the fourth-order controllergiven by Levant in [15] we give the following formulae forthe virtual control 119906
119894
119906119894= minus120573
Φ34
11987334
Φ34
=120590 + 3 [||
+ (|| + 05 |120590|34
)minus13 10038161003816100381610038161003816
+ 05 |120590|34 sign120590
10038161003816100381610038161003816]
minus12
sdot [
+ (|| + 05 |120590|34
)minus13 10038161003816100381610038161003816
+ 05 |120590|34 sign120590
10038161003816100381610038161003816]
11987334
=1003816100381610038161003816
120590
1003816100381610038161003816 + 3 [||
+ (|| + 05 |120590|34
)minus13 10038161003816100381610038161003816
+ 05 |120590|34 sign120590
10038161003816100381610038161003816]
minus12
sdot
1003816100381610038161003816100381610038161003816
+ (|| + 05 |120590|34
)minus13
( + 05 |120590|34 sign120590)
1003816100381610038161003816100381610038161003816
(49)
The integral of the virtual control 120575119885produces the actual
control command 120575119885 120590 in the controller is obtained with the
Arbitrary-Order Robust Exact Differentiator presented in thefollowing section
34 The Arbitrary-Order Robust Exact Differentiator Thequasi-continuous HOSM control method needs to use thethird derivative of the sliding manifold namely 119902
(4) Howto calculate or accurately estimate 119902
(4) is one of the keyproblems to be solved We use the Arbitrary-Order RobustExact Differentiator designed by Levant to differentiate theLOS rate 119902 thus obtaining 119902 119902 and 119902
(4)According to (44) and (45) the following condition is
valid1003816100381610038161003816
120590
1003816100381610038161003816 le 119862 + 120573119870119872
(50)
The Arbitrary-Order Robust Exact Differentiator can beconstructed in accordance with high order sliding modesdifferentiation and output feedback control in [18]
If a certain signal 119891(119905) is a function consisting of abounded Lebesgue-measurable noise with unknown base
Mathematical Problems in Engineering 7
signal 1198910(119905) whose 119903th derivative has a known Lipschitz
constant 119871 gt 0 then the 119899th-order differentiator is definedas follows
0
= V0
V0
= minus12058201198711(119899+1) 10038161003816100381610038161199110 minus 119891 (119905)
1003816100381610038161003816
119899(119899+1) sign (1199110
minus 119891 (119905))
+ 1199111
1
= V1
V1
= minus12058211198712(119899+1) 10038161003816100381610038161199111 minus V
0
1003816100381610038161003816
(119899minus1)119899 sign (1199111
minus V0) + 1199112
119899minus1
= V119899minus1
V119899minus1
= minus120582119899minus1
11987112 1003816100381610038161003816119911119899minus1 minus V
119899minus2
1003816100381610038161003816
12 sign (119911119899minus1
minus V119899minus2
)
+ 119911119899
119899
= minus120582119899119871 sign (119911
119899minus V119899minus1
)
(51)
and if 120582119894
gt 0 is sufficiently large the convergence is guaran-teed
To obtain the third-order derivative of 119902 we constructthe third-order sliding mode differentiator and estimate thederivative of 119902 for each order In view of differential precisionwe configure the following fifth-order differentiator SeeAppendix A for comparison
0
= V0
V0
= minus120582011987116 10038161003816100381610038161199110 minus
1199021003816100381610038161003816
56 sign (1199110
minus 119902) + 1199111
1
= V1
V1
= minus120582111987115 10038161003816100381610038161199111 minus V
0
1003816100381610038161003816
45 sign (1199111
minus V0) + 1199112
2
= V2
V2
= minus120582211987114 10038161003816100381610038161199112 minus V
1
1003816100381610038161003816
34 sign (1199112
minus V1) + 1199113
3
= V3
V3
= minus120582311987113 10038161003816100381610038161199113 minus V
2
1003816100381610038161003816
23 sign (1199113
minus V2) + 1199114
4
= V4
V4
= minus120582411987112 10038161003816100381610038161199114 minus V
3
1003816100381610038161003816
12 sign (1199114
minus V3) + 1199115
5
= V5
V5
= minus1205825119871 sign (119911
5minus V4)
(52)
where 1199113 1199112 1199111 and 119911
0are the estimations of 119902
(4) 119902 119902 and 119902
4 Baseline Separated Guidance andControl Method
To verify the homing performance of the integrated guidanceand control method we compare it with the separatedguidance and control methodThe guidance laws we used arethe proportional navigation (PN) guidance law for attackinga nonmaneuvering target and the optimal sliding modeguidance law for attacking a maneuvering target
41 The Proportional Navigation (PN) Guidance Law Theguidance law has a simple formula and excellent perfor-mances for nonmaneuvering target Its formula is as follows
119899119888
= minus119873 119902119881119872
119892 (53)
119899119888is the overload command 119873 is the effective navigation
ratio 119902 is the LOS rate 119881119872
is the speed of the missile 119892 isthe acceleration of the gravity The guidance law gives theoverload command of the missile according to the LOS rateand then the controller gives the rudder deflection commandaccording to the overload command
42 The Optimal Sliding Mode Guidance Law The optimalsliding mode guidance law (OSMG) is a novel practicalguidance law proposed by D Zhou He combines the optimalguidance lawwith the slidingmode guidance law and designsthe new sliding mode guidance law that not only is robustto maneuvering target but also has the merits of the optimalguidance law such as good dynamic performance and energyconservation Its formula is as follows
119899119888
= minus3100381610038161003816100381610038160
10038161003816100381610038161003816119902 + 120576
119902
10038161003816100381610038161199021003816100381610038161003816 + 120575
(54)
where 119899119888is the overload command
0is the approach
velocity of the missile and its target 119902 is their LOS rate 120576 =
const is the compensatory gain 119902(| 119902| + 120575) is for substitutingfor sign( 119902) and for smoothing 120575 is a small quantity whichcould adjust the chattering
43 Separated Guidance and Control Design For simulationand comparisonwe use the conventional three-loop overloadautopilot as the controller which gives the rudder deflectioncommand according to the feedback of the three loops ofoverload pseudo-angle of attack and pitch rate The blockdiagram is as shown in Figure 3
As the figure shows the inner loop has the feedback onangular velocity which improves the damping characteristicsof the missile airframe
According to the aerodynamic coefficient of the missilewith selected working points we set 119870
119868= 019 119870
120572= 3 and
119870120596
= minus025 and the controller can well track the overloadcommand the rise time of its step response is 046 secondsand its settling time is 083 secondsThe step responses of themissile to overload command and the Bode diagram for openloop are shown in Figure 4
8 Mathematical Problems in Engineering
KIS
120596Z 120572 nY
nC K120596K120572 nY+minus+minus+minus dynamicsAirframe
modelServo
Figure 3 The working principles for three-loop overload autopilot
Step response
Time (s)0 02 04 06 08 1 12
0
02
04
06
08
1
Rise time (s) 0463
Settling time (s) 0831
Bode diagram
Frequency (rads)
To output pointFrom input pointTo output pointFrom input point
Gain margin (dB) 175 At frequency (rads) 15
Phase margin (deg) 709 At frequency (rads) 294
Am
plitu
de
Mag
nitu
de (d
B)Ph
ase (
deg)
0
minus180
minus360
minus54010410310210110010minus1
100
0
minus100
minus200
minus300
Figure 4 The autopilot performance step response and Bode diagram
5 Simulation Results
To verify the high order slidingmode integrated guidance andcontrol (HOSM-IGC) method we compare it with the base-line separated guidance and control Numerical simulationsare designed in three typical engagement scenarios
AORED parameters are as follows the initial value 1205820
=
1205821
= 1205822
= 1205823
= 50 1199110
= 01 1199111
= 1199112
= 1199113
= 0 119871 = 1400 thesimulation step is 00001 seconds
51 Scenario 1 Nonmaneuvering Target In the first scenariothe nonmaneuvering target does uniform rectilinear motionand PN guidance law with three-loop autopilot is introducedfor a comparison with the HOSM-IGCThe initial conditionsare set as shown in Table 1
The motion equations of the target are as follows
119910119879
= 119881119879cos 120579119879
119879
= minus119881119879sin 120579119879
120579119879
=119886119879
119881119879
119886119879
= 0
(55)
The simulation results are shown in Figures 5 and 6The missilersquos flight trajectory and overload curve show
that within the first 3 sec the HOSM-IGC method spendsmuch energy (overload) on changing the initial LOSdirection
Table 1 Initial conditions
Variable Description Value(119883119872 119884119872) Missile initial position (0 km 18 km)
(119883119879 119884119879) Target initial position (60 km 20 km)
119881119872
Missile initial velocity 1200 (ms)119881119879
Target initial velocity 2000 (ms)120579119872
Missile initial flight path angle 0 (deg)120579119879
Target initial flight path angle 0 (deg)120573 Parameter of the controller 10 or 30119873 Navigation ratio of PN 3
of the missile After the boresight adjustment 120590 reaches thedesired sliding manifold and the missile does not need anymaneuver to hit the target This is because the target doesnot maneuver any more which means no disturbance isintroduced in the engagement dynamic so the state (or thesliding mode) of the missile will stay on the manifold Incontrast the overload command given by the PN guidancelaw increases fast as the relative distance decreases Toexamine the performances of the two guidance laws furtherwe increase the target speed and analyze the commandchanges
Therefore we set the target speeds 3000ms 4000msand carry out simulations The simulation result is shown inthe overload curve in Figure 7
Mathematical Problems in Engineering 9
175
180
185
190
195
200
205
TargetHOSM-IGCPN
Y(k
m)
0 10 20 30 40 50 60X (km)
Figure 5 Target and missile trajectories
0
5
10
15
20
25
0 5 10 15 20Time (s)
HOSM-IGCPN
Initial boresight adjustment
Sliding manifold reached
Miss
ile ac
cele
ratio
n (G
)
minus5
Figure 6 Missile acceleration profile
It is evident that the speed of divergence of the overloadcommand given by the PN guidance law increases withthe target speed Specifically when the target speed reaches4000ms the commanded overload is almost 30 g which isobviously not ideal for the attack of a nonmaneuvering targetbut with the HOSM-IGC method the missile adjusts itsboresight very quickly then maintains it around 0 g and fliesto its target in the rectilinear ballistic trajectory not affectedby the increases of target speed still accomplishing the high-precision hit-on collision
Figure 8 shows that although the HOSM-IGC methodachieves a more effective overload command on the otherside it sees some chattering when the sliding mode reaches
0
5
10
15
20
25
30
0 5 10 15Time (s)
Miss
ile ac
cele
ratio
n (G
)
minus5
HOSM-IGC (Vt = minus3000)HOSM-IGC (Vt = minus4000)
PN (Vt = minus3000)PN (Vt = minus4000)
Figure 7 Missile acceleration profile
0
1
2
0 5 10 15 20Time (s)
4 45 5
0
02
04
minus02
minus04
Actu
ator
defl
ectio
n (d
eg)
minus1
minus2
minus3
minus4
HOSM-IGC (120573 = 30)
Figure 8 Actuator deflection
the desired manifold the commanded rudder deflectionangle chatters for around 045 deg at about 15Hz which iskind of harmful to the system To reduce the chatteringwe adjust the controllerrsquos parameter 120573 = 10 and carry outsimulations again Figure 9 shows that after 120573 decreasesthe command of rudder deflection angle converges slower(for about 7 sec) however the chattering weakens obviouslyits magnitude being only about 015 degrees The smallerchattering well enhances convergence precision eventuallyreducing the target missing The reason that minor 120573 resultsin an alleviative chattering can be seen from (47) Figure 10shows the target and missile trajectories
Table 2 compares the average miss distance of 50 simula-tions under the conditions discussed above the comparison
10 Mathematical Problems in Engineering
Table 2 Average miss distance of 50 simulations
Target speed PN HOSM-IGC(120573 = 30)
HOSM-IGC(120573 = 10)
119881119905= 2000ms 115m 086 073m
119881119905= 3000ms 261m 117 106
119881119905= 4000ms 542m 156 134
0
1
2
0 5 10 15 20Time (s)
72 74 76 78
0
02
minus02
minus04
Actu
ator
defl
ectio
n (d
eg)
minus1
minus2
minus3
minus4
HOSM-IGC (120573 = 10)HOSM-IGC (120573 = 30)
Figure 9 Actuator deflection
results show that when the missile intercepts a nonmaneu-vering target the hit precision of the HOSM-IGC methodis apparently higher than that of the PN guidance law andthat the low-gain HOSM-IGC method can effectively reducethe chattering magnitude thus enhancing the interceptionprecision
52 Scenario 2 Step Maneuvering Target At the first stagethe target flies at uniform speed and in a rectilinear way after10 seconds it maneuvers at the normal acceleration of 5 g Inthis scenario OSMG is introduced for a comparison with theHOSM-IGC
119910119879
= 119881119879cos 120579119879
119879
= minus119881119879sin 120579119879
120579119879
=119886119879
119881119879
119886119879
=
0 g 119905 lt 10 sec
5 g 119905 gt 10 sec
(56)
The initial simulation conditions are given in Table 3The overload curve in Figure 11 shows dearly that both
types of guidance laws can track the maneuvering targetDuring 0 to 10 seconds the target flies at uniform speed andin a rectilinear way and the missile converges its overload to
Table 3 Initial conditions
Variable Description Value(119883119872 119884119872) Missile initial position (0 km 22 km)
(119883119879 119884119879) Target initial position (60 km 20 km)
119881119872
Missile initial velocity 1200 (ms)119881119879
Target initial velocity 2000 (ms)120579119872
Missile initial flight path angle 0 (deg)120579119879
Target initial flight path angle 0 (deg)120576 Compensatory gain 3120575 Saturation parameter 0001
15
17
19
21
23
25
TargetHOSM-IGCOSMG
Y(k
m)
0 20 40 60X (km)
Figure 10 Target and missile trajectories
0 g and flies to its target after 10 seconds the target beginsto maneuver by 5 g Both types of guidance law respondby rapidly increasing the overload adjusting attitude andmaking the missilersquos boresight aim at its target
We can see that when both types of guidance law tracktheir targets the convergence speed of OSMG is almostthe same as the HOSM-IGC method But the HOSM-IGCmethod has higher convergence precision and needs loweroverload at the end phase
We can also see that after the target maneuvers if themissile is given enough time to track the targetrsquos maneuvernamely let the missilersquos overload command converge to theoverload of the target theremay not be largemiss distance Inother words for a certain period of time before the collisionthe targetmaneuver (it means only a limitedmaneuver whichdoes not include the condition that the maneuvering of thetarget for a long time may cause a change of the geometricalrelations between the missile and its target) has a small effecton both types of guidance law
But if themaneuver occurs rather late namely within oneto three seconds before collision when the overload com-mand of guidance law is not yet converged the approaching
Mathematical Problems in Engineering 11
0
5
10
0 5 10 15 20Time (s)
TargetOSMGHOSM-IGC
Miss
ile an
d ta
rget
acce
lera
tion
(G)
minus5
minus10
Figure 11 Missile and target acceleration profile
0
10
20
30
0 5 10 15 20Time (s)
tgo = 1 stgo = 2 stgo = 3 s
tgo = 3 s (diverge)
tgo = 2 s (diverge)
tgo = 1 s (diverge)minus10
minus20
minus30
Miss
ile ac
cele
ratio
n (G
)
HOSM-IGC (120573 = 10)
Figure 12 Missile acceleration profile
collision increases themiss distanceTherefore increasing theparameter 120573 can remarkably increase the convergence speedand effectively enhance guidance precision As Figure 12shows when 120573 = 10 the overload at the end phase convergesslowly even if the target maneuvers three seconds beforecollision the missilersquos overload still has no time to convergebeing unable to track themaneuvering target Figure 13 showsthat when 120573 increases to 30 and tgo = 3 seconds theoverload can converge to about 5 g but when tgo = 2 sec-onds the overload may continue to increase indicating that
0
10
20
30
0 5 10 15 20Time (s)
tgo = 1 stgo = 2 stgo = 3 s
tgo = 1 s (diverge)
tgo = 2 s (diverge)
tgo = 3 s (converge)
minus10
minus20
minus30
Miss
ile ac
cele
ratio
n (G
)
120573 = 30)HOSM-IGC (
Figure 13 Missile acceleration profile
0
10
20
30
0 5 10 15 20Time (s)
tgo = 1 s (diverge)
tgo = 2 s (converge)
tgo = 3 s (converge)
tgo = 1 stgo = 2 stgo = 3 s
minus10
minus20
minus30
Miss
ile ac
cele
ratio
n (G
)
HOSM-IGC (120573 = 50)
Figure 14 Missile acceleration profile
the convergence still needs to be quickened thus continuingto rise 120573 to 50 Figure 14 shows that the HOSM-IGC methodcan track the target that maneuvers when tgo = 2 secondsbut it causes the divergence of overload and the increase ofmiss distance if the target maneuvers when tgo = 1 secondFigure 15 gives the overload curve of the OSMG guidance lawand shows that the maneuver of the target before collisionmay cause the large-scale oscillation of the missilersquos overloadwhich may diverge to a large numerical value when thecollision occurs in the end
12 Mathematical Problems in Engineering
Table 4 Average miss distances of 50 simulations
Targetmaneuveringtiming
HOSM-IGC120573 = 30
HOSM-IGC120573 = 40
HOSM-IGC120573 = 50
OSMG
tgo = 1second 43539 3224 28936 36116
tgo = 2seconds 25424 18665 09322 35534
tgo = 3seconds 08124 08265 07538 11959
Time (s)0 5 10 15 20
0
10
20
30OSMG
tgo = 1 stgo = 2 stgo = 3 s
minus10
minus20
minus30
Miss
ile ac
cele
ratio
n (G
)
Figure 15 Missile acceleration profile
The analysis in Figures 12 13 14 and 15 shows that theoverload of the missile converges faster and its miss distanceis smaller with increasing 120573 To verify this finding we carryout 50 times Monte Carlo simulations in which the positionand speed of the target have 1 of random difference
The average miss distances are shown in Table 4 andFigure 16 Clearly the timing of the targetrsquos step maneuverdramatically affects the final interception precision morespecifically given a shorter reaction time for the guidanceand control system the missile seems more likely to missthe target To the OSMG guidance law in all three scenarioshardly does it show any advantages against theHOSM-IGC Itcan also be seen that with the increasing of the120573 theHOSM-IGC system responds even faster which leads to an obviousdecrease of the average miss distance The effect of 120573 on theresponse of the HOSM-IGC system is a valuable guidelinewhen implementing the proposed method into practice
53 Scenario 3 Weaving Target In this scenario the targetmaneuvers by 119886
119879= 40sin(1205871199052) OSMG with three-loop
Table 5 Initial conditions
Variable Description Value(119883119872 119884119872) Missile initial position (0 km 175 km)
(119883119879 119884119879) Target initial position (60 km 195 km)
119881119872
Missile initial velocity 1200 (ms)119881119879
Target initial velocity 2000 (ms)120579119872
Missile initial flight path angle 0 (deg)120579119879
Target initial flight path angle 0 (deg)120576 Compensatory gain 3120575 Saturation parameter 0001
autopilot is introduced for comparison The motion equa-tions of the target are as follows
119910119879
= 119881119879cos 120579119879
119879
= minus119881119879sin 120579119879
120579119879
=119886119879
119881119879
119886119879
= 40 sin(120587119905
2)
(57)
The initial simulation conditions are given in Table 5The missilersquos trajectory under the two guidance and
control methods given in Figure 17 does not see muchdifference
However the overload curve given in Figure 18 showsthat after the missile completes its initial attitude adjustmentwith the HOSM-IGC method it can almost ideally track itsmaneuvering target by contrast with the OSMG methodthe missile seems to have the tendency to track its targetrsquosmaneuver but has larger tracking errors Besides with theOSMGmethod the missilersquos overload increases rapidly at theend of attack primarily because of the divergence of its LOSrate On the other hand with the HOSM-IGC method themissile has no divergence even at the end of attack ensuringa smaller target missing quantity
The actuator deflection curve in Figure 19 shows thatin order to provide a rather big normal overload for theend phase the OSMG method produces a rather big rudderdeflection command however it may increase the missilersquostarget missing quantity once its rudder deflection saturatesand themissile does not have enough overloads or the ruddercannot respond that fast
As shown in Figure 20 because of the dramatic changein overload command the response of the missilersquos autopilotto high-frequency command sees an obvious phase lag andamplitude value attenuation its actual overload cannot trackthe command ideally this is a main reason why the missdistance increases However the controller in the HOSM-IGC method gives its rudder deflection command directlyand there is no lagging or attenuation caused by the autopilotthus enhancing the guidance precision effectively Further-more the fast convergence of the high order sliding modemakes the missile rapidly track its maneuvering target withthemost reasonable rudder deflection command reducing itsoverload effectively
Mathematical Problems in Engineering 13
0
05
1
15
2
25
3
35
4
45
5
0 10 20 30 40 50Simulation times
OSMG
Miss
(m)
0
05
1
15
2
25
3
35
4
45
5
0 10 20 30 40 50Simulation times
Miss
(m)
0
05
1
15
2
25
3
35
4
45
5
0 10 20 30 40 50Simulation times
Miss
(m)
tgo = 3 stgo = 2 stgo = 1 s
0
05
1
15
2
25
3
35
4
45
5
0 10 20 30 40 50Simulation times
Miss
(m)
tgo = 3 stgo = 2 stgo = 1 s
HOSM-IGC (120573 = 40)
HOSM-IGC (120573 = 50)
HOSM-IGC (120573 = 30)
Figure 16 Miss distances of HOSM-IGC (120573 = 30 40 50) and OSMG with target maneuvers at different time-to-go (tgo = 1 second 2seconds and 3 seconds)
The average miss distance of 50 simulations under theconditions is 073m for HOSM-IGC and 186m for OSMGWe can see that the HOSM-IGC method not only doesprovide a more reasonable actuator deflection command butalso achieves a higher interception precision
6 Conclusions
This paper proposes an LOS feedback integrated guidanceand control method using quasi-continuous high order
sliding mode guidance and control method With the fastand precise convergence of the quasi-continuous HOSMmethod the HOSM-IGCmethod performsmuch better thanthe traditional separated guidance and control method withless acceleration effort and less miss distance in all thethree simulation scenarios of nonmaneuvering target stepmaneuvering target and weaving target In addition the ideaof virtual control largely alleviates the chattering withoutany sacrifice of robustness As a result of the alleviationof the chattering the control input command 120575
119885becomes
14 Mathematical Problems in Engineering
175
180
185
190
195
200
TargetHOSM-IGCOSMG
Y(k
m)
0 20 40 60 0 20 40X (km)
Figure 17 The trajectories of the missile and its target
0
10
20
0 5 10 15 20Time (s)
TargetOSMGHOSM-IGC
Miss
ile an
d ta
rget
acce
lera
tion
(G)
minus10
minus20
Figure 18 Missile acceleration profile
smooth enough for the actuator to be implemented as aresult the interception accuracy is improved In designingthe quasi-continuous sliding mode controller the Arbitrary-Order Robust Exact Differentiator is introduced to estimatethe high order time derivative of the LOS rate 119902 which notonly ensures the guidance and control precision but alsosimplifies the calculation
0
10
20
30
0 5 10 15 20Time (s)
OSMGHOSM-IGC
minus10
minus20
minus30
minus40
Actu
ator
defl
ectio
n (d
eg)
Figure 19 Actuator deflection
0
50
100
150
200
15 16 17 18Time (s)
Commanded accelerationAchieved acceleration
Miss
ile ac
cele
ratio
n (G
)
minus50
Figure 20 Commanded acceleration and achieved acceleration
Appendices
A The Third-Order RobustExact Differentiator
The third-order robust exact differentiator built according tothe following parameters and equations is used to performthe simulation the initial value 120582
0= 1205821
= 1205822
= 1205823
= 50
Mathematical Problems in Engineering 15
0
0005
001
0015
002
0025
003
0035
004
0 02 04Time (s)
0
2
0 02 04Time (s)
0
100
200
300
0
100
200
300
400
0 02 04Time (s)
h(t)
z3
h(t) and z3
minus100
minus200
minus300
minus400
minus5000 02 04
Time (s)
g(t)
z2
g(t) and z2
minus100
minus200
minus300
minus400
f(t)
z1
f(t) and z1
minus2
minus4
minus6
minus8
minus10
minus12
minus14
F(t)
z0
F(t) and z0
minus0005
minus001
Figure 21 The outputs of the differentiator in the sampling interval of 0001 seconds
the initial values 1199110
= 01 1199111
= 1199112
= 1199113
= 0 119871 = 1400 thesimulation step is 0001 seconds Consider
0
= V0
V0
= minus120582011987114 10038161003816100381610038161199110 minus
1199021003816100381610038161003816
34 sign (1199110
minus 119902) + 1199111
1
= V1
V1
= minus120582111987113 10038161003816100381610038161199111 minus V
0
1003816100381610038161003816
23 sign (1199111
minus V0) + 1199112
2
= V2
V2
= minus120582211987112 10038161003816100381610038161199112 minus V
1
1003816100381610038161003816
12 sign (1199112
minus V1) + 1199113
3
= minus1205823119871 sign (119911
3minus V2)
(A1)
16 Mathematical Problems in Engineering
0
0005
001
0015
002
0025
003
0035
004
0 02 04Time (s)
0
2
0 02 04Time (s)
0
100
200
300
0 02 04Time (s)
0
200
400
0 02 04Time (s)
h(t) and z3
minus200
minus400
minus600
minus800
g(t) and z2
minus100
minus200
minus300
minus400
f(t) and z1
minus2
minus4
minus6
minus8
minus10
minus12
minus14
minus16
F(t) and z0
minus0005
minus001
h(t)
z3g(t)
z2
f(t)
z1F(t)
z0
Figure 22 The outputs of the differentiator in the sampling interval of 00001 seconds
If the designed signals are 119865(119905) = int 119891(119905)119889119905 119891(119905) =
int 119892(119905)119889119905 119892(119905) = int ℎ(119905)119889119905 and ℎ(119905) = sin(2120587119905) +
3cos(120587119905) then theoretically 119891(119905) is the first-order differ-ential of input signal 119892(119905) is the second-order differ-ential of input signal ℎ(119905) is the third-order differen-tial of input signal Through the simulation the differ-ential signal of actual output is compared with that of
theoretical output the comparison results are shown inFigure 21
Figure 21 shows that 1199110can effectively track the signal
119865(119905) but 1199112has obvious errors in tracking the second-order
differential 119892(119905) 1199113has greater errors in tracking the third-
order differential thus the overall differential effect is notideal
Mathematical Problems in Engineering 17
0 02 04
0
0005
001
0015
002
0025
003
0035
004
Time (s)0 02 04
0
01
02
03
04
05
Time (s)
0 02 04
0
05
1
15
2
Time (s)0 02 04
0
05
1
15
2
25
3
35
4
Time (s)
h(t) and z3g(t) and z2
minus04
minus05
f(t) and z1
minus2
minus15
minus1
minus05
minus1
minus05
minus01
minus02
minus03
F(t) and z0
minus0005
minus001
h(t)
z3g(t)
z2
f(t)
z1F(t)
z0
Figure 23 The outputs of the differentiator in the sampling interval of 00001 seconds
There are two causes for the errors in a usual differentia-tor the setting of sampling interval and the setting of numberof orders of the differentiator The following theorem applieshere
Theorem A1 (see [19]) If the sampling interval 120591 gt 0 theaccuracy of the controller is expressed as follows
10038161003816100381610038161003816120590(119894)
10038161003816100381610038161003816le 120583120591119903minus119894
119894 = 0 119903 minus 1 (A2)
That is to say the precision (understood as the differentialprecision of a differentiator) of the 119894th order control is in directproportion to the power of 119903 minus 119894 times of sampling interval120591 Therefore the decrease of iteration step can enhance thedifferential tracking precision
Under the conditions that the sampling interval is 10 timessmaller and that the setting is 00001 seconds the simulationresults are shown in Figure 22
18 Mathematical Problems in Engineering
Table 6 Tracking errors of the AORED with different orders andsampling interval
Orders of AORED Third order Fifth orderSample period 120591 = 0001 120591 = 00001 120591 = 00001
First-order differential error 001 Negligible NegligibleSecond-order differential error 15 002 NegligibleThird-order differential error 70 25 00005
As Figure 22 shows the signal tracking is rather idealexcept a peak in the beginning the tracking error in the first-order differential tracking can be neglected 119911
2has an obviously
smaller tracking error in the second-order differential 119892(119905)
tracking in the beginning 1199113has a rather sharp peak in the
third-order differential tracking but converges to a relativelyhigh precision after 01 seconds As we can see the differentialtracking errors for each number of orders are greatly reducedbut the third-order differential tracking error is still rather bigIn the following we will reduce the tracking errors by increasingthe number of orders of the differentiator
B The Fifth-Order Robust Exact Differentiator
The settings of the fifth-order differentiator are given asfollows
0
= V0
V0
= minus120582011987116 10038161003816100381610038161199110 minus
1199021003816100381610038161003816
56 sign (1199110
minus 119902) + 1199111
1
= V1
V1
= minus120582111987115 10038161003816100381610038161199111 minus V
0
1003816100381610038161003816
45 sign (1199111
minus V0) + 1199112
2
= V2
V2
= minus120582211987114 10038161003816100381610038161199112 minus V
1
1003816100381610038161003816
34 sign (1199112
minus V1) + 1199113
3
= V3
V3
= minus120582311987113 10038161003816100381610038161199113 minus V
2
1003816100381610038161003816
23 sign (1199113
minus V2) + 1199114
4
= V4
V4
= minus120582411987112 10038161003816100381610038161199114 minus V
3
1003816100381610038161003816
12 sign (1199114
minus V3) + 1199115
5
= V5
V5
= minus1205825119871 sign (119911
5minus V4)
(B1)
The initial value 1199110
= 01 1205820
= 1205821
= 1205822
= 1205823
= 1205824
=
1205825
= 50 the initial values 1199111 1199112 1199113 1199114 1199115
= 0 119871 = 1400the simulation step length is 00001 seconds The simulationresults are shown in Figure 23
As the figure shows the fifth-order differentiator hasa rather ideal tracking effect on the third-order differen-tial tracking the tracking error being only about 00005(see Table 6)
To sum up the Arbitrary-Order Robust Exact Differen-tiator can estimate rather ideal differential signals by selecting
Table 7
Length 365m119871 ref 365m119883119866
177m119878ref 0026m2
Diameter 0178mMass 1016 Kg119868119885
1063 Kgsdotm2
an appropriate sampling interval and using the differentiatorwith a relatively high number of orders
C Physical and Geometric Characteristics
See Table 7
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] P Zarchan Tactical and Strategic Missile Guidance 2002[2] D Zhou CMu andWXu ldquoAdaptive sliding-mode guidance of
a homing missilerdquo Journal of Guidance Control and Dynamicsvol 22 no 4 pp 589ndash594 1999
[3] G Zeng and M Hu ldquoFinite-time control for electromagneticsatellite formationsrdquo Acta Astronautica vol 74 pp 120ndash1302012
[4] S R Kumar S Rao and D Ghose ldquoNonsingular terminalsliding mode guidance with impact angle constraintsrdquo Journalof Guidance Control and Dynamics vol 37 no 4 pp 1114ndash11302014
[5] Y Liang C Chen D Liaw Y Feng C Cheng and CChen ldquoRobust guidance law via integral-sliding-mode schemerdquoJournal of Guidance Control and Dynamics vol 37 no 3 pp1038ndash1042 2014
[6] A V Savkin P N Pathirana and F Faruqi ldquoProblem ofprecision missile guidance LQR and 119867
infincontrol frameworksrdquo
IEEE Transactions on Aerospace and Electronic Systems vol 39no 3 pp 901ndash910 2003
[7] H Y Chen andC-C Yang ldquoNonlinear1198672119867infinguidance design
for homing missilesrdquo in Proceedings of the AIAA GuidanceNavigation and Control Conference and Exhibit vol 77 2001
[8] P K Menon G D Sweriduk and E J Ohlmeyer ldquoOptimalfixed-interval integrated guidance-control laws for hit-to-killmissilesrdquo in Proceedings of the AIAA Guidance Navigation andControl Conference and Exhibit AIAA 2003-5579 Austin TexUSA August 2003
[9] S S Vaddi P K Menon and E J Ohlmeyer ldquoNumericalSDRE approach for missile integrated guidance-controlrdquo inProceedings of the AIAA Guidance Navigation and ControlConference and Exhibit AIAA 2007-6672 Hilton Head SCUSA August 2007
[10] M Xin S N Balakrishnan and E J Ohlmeyer ldquoIntegratedguidance and control of missiles with Theta-D methodrdquo IEEE
Mathematical Problems in Engineering 19
Transactions on Control Systems Technology vol 14 no 6 pp981ndash992 2006
[11] T Shima M Idan and O M Golan ldquoSliding-mode controlfor integrated missile autopilot guidancerdquo Journal of GuidanceControl and Dynamics vol 29 no 2 pp 250ndash260 2006
[12] Y B Shtessel and C H Tournes ldquoIntegrated higher-ordersliding mode guidance and autopilot for dual control missilesrdquoJournal of Guidance Control and Dynamics vol 32 no 1 pp79ndash94 2009
[13] F Y Dong H M Lei and C J Zhou ldquoIntegrated robust highorder sliding mode guidance and control for missilesrdquo ActaAeronautica et Astronautica Sinica vol 34 no 9 pp 2212ndash22182013
[14] H Mingzhe and D Guangren ldquoIntegrated guidance and con-trol of homing missiles against ground fixed targetsrdquo ChineseJournal of Aeronautics vol 21 no 2 pp 162ndash168 2008
[15] A Levant ldquoHomogeneity approach to high-order sliding modedesignrdquo Automatica vol 41 no 5 pp 823ndash830 2005
[16] Y-W Liang C Chen D Liaw Y Feng C Cheng and CChen ldquoRobust guidance law via integral-sliding-mode schemerdquoJournal of Guidance Control and Dynamics vol 37 no 3 pp1038ndash1042 2014
[17] L Wang Y Sheng and X Liu ldquoContinuous time-varying slid-ingmode based attitude control for reentry vehiclerdquo Proceedingsof the Institution of Mechanical Engineers Part G Journal ofAerospace Engineering 2014
[18] A Levant ldquoHigher-order sliding modes differentiation andoutput-feedback controlrdquo International Journal of Control vol76 no 9-10 pp 924ndash941 2003
[19] A Levant ldquoQuasi-continuous high-order sliding-mode con-trollersrdquo IEEE Transactions on Automatic Control vol 50 no11 pp 1812ndash1816 2005
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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Algebra
Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
Discrete MathematicsJournal of
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
2 Mathematical Problems in Engineering
Being famous for its good robustness to bounded distur-bances the sliding mode control method is introduced intothe study of guidance law In [2] Zhou et al gave the con-ditions for the sliding mode motion of a linear time-varyingsystem not to be disturbed by the disturbances and param-eter perturbations and proposed an adaptive sliding modeguidance law (ASMG) simulation results demonstrated thatthe ASMG is robust to uncertainties like target accelerationTo get finite time convergence modified SMC like terminalsliding mode control (TSMC) is proposed In [3] Zeng andHu combine the advantages of linear and terminal slidingmode controls which guarantee the convergence of trackingerrors in finite time And then nonsingular terminal slidingmode control (NTSMC) is introduced in the guidance lawdesign In [4] Kumar et al proposed a nonsingular terminalsliding mode guidance law with finite time convergencewhich avoids the singularity that may lead to the saturationof the control Integral SMC (ISMC) like that introduced in[5] is another revised sliding mode control which introducedthe integral sliding mode scheme to the guidance law designthe proposed ISMC guidance law provided a smaller controlmagnitude than the traditional sliding mode design Theadvance of the 119867
infincontroller greatly enriched the methods
to implement a robust control system In [6] Savkin etal suitably modified the 119867
infincontrol theory and provided
an effective framework for the precision missile guidanceproblem which showed much better performance than thelinear quadratic optimal guidance law in the computersimulation In [7] Chen and Yang proposed a mixed 119867
2119867infin
guidance design against maneuvering targets in which thecomplete nonlinear kinematics of the pursuit-evasionmotionwas considered
In addition to the separated guidance and control lawthe integrated guidance and control method is under heateddiscussion According to the relative motion between thetarget and the missile the traditional guidance law calculatesthe overload needed to hit the target and inputs the overloadcommand into the overload autopilot while the integratedguidance and control (IGC) method gives the rudder deflec-tion command directly to the missile according to therelative motion which evidently responds more quickly In[8] Menon et al designed an IGCmethod by using the linearquadratic optimal theory but the robustness of the system isnot satisfactory In [9] Vaddi et al provided a fully numericalmethodology for deriving state-dependent Riccati equationcontrollers for arbitrarily complex dynamic systems andapplied it in the IGC design of a missile Simulation resultsdemonstrated the effectiveness of the method In [10] Xinet al employed the theta-D method to give an approximateclosed-form suboptimal feedback controller to the nonlinearinfinite-horizon IGC problem Taking another look slidingmode approaches are also employed in the IGC design ofhoming missiles In [11] Shima et al proposed the slidingmode integrated guidance and control method based on theZEM Shtessel and Tournes doesmassive research on the highorder sliding mode controller design and proposed his newmethod In [12] he designed the high order sliding modeguidance law based on the smooth second-order slidingmode control (SSOSMC) which is smoother in high orders
than the traditional second-order sliding mode guidance lawIn [13] based on the geometrical homogeneity theory Donget al designed the tranquility control law for the integratedguidance and control model which however has rathermore parameters and is sensitive to parameters and theparameters must be carefully selected In [14] Mingzhe andGuangren used the sliding mode control theory to design theadaptive nonlinear feedback controller which is primarily forfixed ground target being unable to deal with the vehementperturbation caused by target maneuvering
Motivated by the aforementioned considerations thiswork will design an IGC scheme for the interception of thenear space maneuvering hypersonic vehicles Firstly a line-of-sight (LOS) rate feedback scheme is adopted to derive theIGC law and as the relative order of the control input to theLOS rate is higher than one a HOSM approach is introducedSecondly to implement the HOSM approach the 119899 timesderivations of the sliding manifold 119878 119878
119878 119878
(119899) must beknown However reconstruction of each 119878 119878
119878 119878
(119899) bytheir analytical expression could be rather difficult in prac-tice An alternative way adopted in this paper is to use theArbitrary-Order Robust Exact Differentiator (AORED) toestimate the derivations And next to alleviate the chatteringphenomenon caused by the HOSM controller the idea ofvirtual control is introducedThe virtual control V is designedand used as the control input of a system extended from theoriginal one and the real control 119906 acting on the real systemis obtained by integrating the virtual control V Benefittingfrom the integration element the real control input 119906 couldbe smooth enough for the implementation without reducingthe robustness of the HOSM method Finally the proposedmethod is implemented in a 3-dof model
The remaining part of this paper is organized as followsIn Section 2 the IGC model is derived in the longitudinalplane In Section 3 the quasi-continuous HOSM controlleras well as the AORED is designed In Section 4 the baselineseparated guidance laws and controller are given In Sec-tion 5 the numerical simulations are demonstrated in threetypical engagement scenarios And conclusions are made inSection 6
2 Integrated Guidance and Control Model
The traditional guidance and control algorithm usually usesthe guidance loop as its outer loop and is only responsible forgiving commanded overload then the control loop is onlyresponsible for tracking the overload command eventuallyachieving the missilersquos guidance toward its target Althoughit is always desirable to design the control loop or theautopilot to have better dynamic performance in actualitythe controller always has some delay and attenuation Asa result the missile always has some error in acting theoverload command That is one of the reasons why aninterceptor misses its target
The integrated guidance and control algorithm combinesthe guidance and control loops into one loop and avoidsthe delay and attenuation caused by them Its architecture isshown in Figure 1
Mathematical Problems in Engineering 3
Seeker
Guidancefiltering
Integratedguidance and control
ActuatorMissile
airframe
Missile position
Target position
Missile acceleration
Target acceleration
Guidancelaw
Autopilot(controller) Separated guidance and control
Integrated guidance and control +minus
120575CnC
q
120575120575C
q
q nm
Figure 1 Integrated guidance and control
O
Y1
X1
M
aM
xb
120572120579M
q
VM
T
VT
120579T
aT
120599
Figure 2 Engagement geometry
21 The Engagement Dynamics Without loss of generalitywe present hereinafter only the subsystems that govern themotion of an interceptor in its longitudinal plane The planarengagement between the interceptor and its target is shownin Figure 2
The study of guidance laws usually regards themissile andits target as mass points deliberates on the mass point of itsrigid motion and ignores the attitude of the missile airframeand changes in its attitude By contrast the integrated guid-ance and control method takes the dynamic characteristicsof the missile airframe into consideration therefore thederivation involves the pitch angle pitch rate and angle ofattack of the missile as shown in Figure 2
The motion equations of the missile in the longitudinalplane are as follows
119889119881119872
119889119905=
1
119898(119875 cos120572 minus 119883 minus 119898119892 sin 120579
119872) (1)
119889120579119872
119889119905=
1
119898119881119872
(119875 sin120572 + 119884 minus 119898119892 cos 120579119872
) (2)
119889119909119872
119889119905= 119881119872cos 120579119872
(3)
119889119910119872
119889119905= 119881119872sin 120579119872
(4)
119889120596119885
119889119905=
119872119885
119869119885
(5)
119889120599
119889119905= 120596119885 (6)
120572 = 120599 minus 120579 (7)
where (119909119872
119910119872
) is the position 119898 is the mass 119883 is the axialforce 119875 is the thrust force the missilersquos velocity is 119881
119872 and its
flight path angle is 120579119872 the LOS angle between themissile and
its target is 119902 the relative distance between the missile and itstarget is 119903 the velocity of the target is 119881
119879 and its flight path
angle is 120579119879 119909119887is the axle of the missilersquos airframe the pitch
angle is 120599 the angle of attack is 120572The motion equations of the target are as follows
119889119909119879
119889119905= minus119881119879cos 120579119879
119889119910119879
119889119905= 119881119879sin 120579119879
119889120579119879
119889119905=
119886119879119873
119881119879
(8)
The interception is characterized by two variablesnamely the target range and the LOS angle The kinematicequations are expressed by the following relations
119889119903
119889119905= minus119881119872cos (120579
119872minus 119902) minus 119881
119879cos (120579
119879+ 119902) (9)
119903119889119902
119889119905= minus119881119872sin (120579119872
minus 119902) + 119881119879sin (120579119879
+ 119902) (10)
22 The Model Simplification To simplify the model andmake further derivations wemake the following two assump-tions
Assumption 1 Within the terminal phase of the interceptionthe missile has no thrust and its gravity is not taken intoaccount
Assumption 2 Within the terminal phase of the interceptionthe missilersquos speed does not change
4 Mathematical Problems in Engineering
Then (1) and (2) can be reformed as below
119889119881119872
119889119905= 0
119889120579119872
119889119905=
119884
119898119881119872
(11)
Denote that 119886119872119873
= 119884119898 119884 is the normal force 119886119872119873
119886119879119873
are the normal acceleration of the missile and targetrespectively and then
120579119872
=119886119872119873
119881119872
120579119879
=119886119879119873
119881119879
(12)
The normal force 119884 and the pitch moment 119872119885acting on
the missile are usually expressed respectively as follows
119884 = 119862120572
119884120572119876119878ref + 119862
120575119885
119884120575119885119876119878ref (13)
119872119885
= 119898120575119885
119885120575119885119876119878ref119897 + 119898
120572
119885120572119876119878ref119897 (14)
where119862120572
119884is the coefficient of normal force caused by the angle
of attack 119862120575119885
119884is the coefficient of normal force caused by the
rudder deflection angle 120575119885 119876 is the dynamic pressure 119878ref is
the reference area 119897 is the reference length But the normalforce produced by the rudder deflection angle 120575
119885is orders of
magnitude smaller than that produced by the angle of attackso (13) is simplified as
119884 = 119862120572
119884120572119876119878ref (15)
Then
119886119872119873
=119884
119898=
119862120572
119884120572119876119878ref
119898 (16)
Then the dynamic equations can be simplified as
119889120579119872
119889119905=
119862120572
119884120572119876119878ref
119898119881119872
(17)
119889120596119885
119889119905=
119898120575119885
119885120575119885
119876119878ref119897 + 119898120572
119885120572119876119878ref119897
119869119885
(18)
119889120599
119889119905= 120596119885 (19)
120572 = 120599 minus 120579119872
(20)
23 The Integrated Guidance and Control Model Follow-ing the above derivation we select the state variables as
( 119902 120579119872
120572 120596119885)119879 and obtain the following nonlinear integrated
guidance and control model
119902 = minus2119903
119903sdot 119902 minus
119862120572
119884119876119878ref
119898119903cos (120579
119872minus 119902) sdot 120572
+119886119879119873
119903cos (120579
119879+ 119902)
119903 119902 = minus119881119872sin (120579119872
minus 119902) + 119881119879sin (120579119879
+ 119902)
120579119872
=1
119898119881119872
119862120572
119884119876119878ref sdot 120572
= 120596119885
minus1
119898119881119872
119862120572
119884119876119878ref sdot 120572
119885
=119898120575119885
119885119876119878ref119897
119869119885
sdot 120575119885
+119898120572
119885119876119878ref119897
119869119885
120572
(21)
24 The Relative Degree of Control Input To obtain therelative degree of the control input 120575 of the integratedguidance and control method we keep on deriving the LOSangular velocity 119902 until the explicit formula of its derivativeof a certain order contains the control input
The derivation of (10) produces119902
=1
119903[minus2 119903 119902 minus 119886
119872119873cos (120579
119872minus 119902) + 119886
119879119873cos (120579
119879+ 119902)]
(22)
The LOS angular velocity is expressed as the derivativeof the first order and does not contain the control input 120575
explicitly The continuous derivation of the above equationproduces
119902 =
1
119903minus2 119903 119902 minus 3 119903 119902 minus [119886
119872119873( 119902 minus 120579
119872) sin (120579
119872minus 119902)
+ 119886119872119873
cos (120579119872
minus 119902)] minus [119886119879119873
( 119902 + 120579119879) sin (120579
119879+ 119902)
minus 119886119879119873
cos (120579119879
+ 119902)]
(23)
where the LOS angular rate is expressed as the differentiatingof the second order and 119886
119872can be expressed as
119886119872119873
=119862120572
119884119876119878ref
119898 =
119862120572
119884119876119878ref
119898( 120599 minus 120579
119872)
=119862120572
119884119876119878ref
119898(120596119885
minus 120579119872
)
(24)
Although the control volume 120575119885does not appear in the
derivative of the second order (18) shows that 119885contains
120575119885 We continue to derive (23) and substitute 119886
119872as follows
119886119872119873
=119862120572
119884119876119878ref
119898(119885
minus119886119872
119881119872
)
=119862120572
119884119876119878ref
119898
119898120575119885
119885119876119878ref119897ref
119869119885
sdot 120575119885
+119862120572
119884119876119878ref
119898(
119898120572
119885119876119878ref119897ref
119869119885
120572 minus119886119872
119881119872
)
(25)
Mathematical Problems in Engineering 5
Thus
119902(4)
= 119891120575119885
sdot 120575119885
+1
119903(1198911
+ 1198912
+ 1198913
+ 1198914
+ 1198915
+ 1198916) (26)
where
119891120575119885
= minus1
119903
119862120572
119884119876119878ref cos 120578
119872
119898
119898120575119885
119885119876119878ref119897ref
119869119885
(27)
1198911
= (120596119885
minus 120579119872
) (minus2 120579119886119872119873
120572sin 120578119872
+ 119902119886119872119873
120572sin 120578119872
+ 120579119886119872119873
1205722cos 120578119872
) minus119886119872119873
cos 120578119872
120572
119898120572
119885120572119876119878ref119897ref
119869119885
(28)
1198912
= 120579119872
(( 120579119872
minus 119902) 119886119872119873
cos 120578119872
minus (120596119885
minus 120579119872
)119886119872119873
120572sin 120578119872
)
(29)
1198913
= 119902 (minus 120579119872
119886119872119873
cos 120578119872
+ 119902 (119886119872119873
cos 120578119872
minus 119886119879119873
cos 120578119879) minus 120579119879119886119879119873
cos 120578119879
+119886119872119873
120572(120596119885
minus 120579119872
) sin 120578119872
minus 119886119879sin 120578119879)
(30)
1198914
= 120579119879
(minus 119902119886119879cos 120578119879
minus 120579119879119886119879cos 120578119879
minus 119886119879sin 120578119879) (31)
1198915
= 119886119879
(minus 119902 sin 120578119879
minus 2 120579119879sin 120578119879) + 119886119879cos 120578119879 (32)
1198916
= 119902 (119886119872119873
sin 120578119872
minus 119886119879sin 120578119879
minus 2 119903) minus 4 119903 sdot119902 minus 3 119903 sdot 119902
minus 2119903 sdot 119902
(33)
120578119872
= 120579119872
minus 119902
120578119879
= 120579119879
+ 119902
(34)
In (26) it can be seen that the control input 120575119885appears
expressly in the third-order derivative of the control output119902 Therefore the relative degree of the control input 120575
119885is 3
3 The Quasi-Continuous High Order SlidingMode Controller
31 Sliding Mode Manifold Design To design the HOSMcontroller a sliding manifold must be chosen first In thisdesign we try to make the LOS rate converge to zero ora small neighbor domain near zero thus ensuring that themissile approaches its target in a quasi-parallel way whichwill lead to a minimal overload requirement So the slidingmanifold is chosen as follows
120590 = 119902 (35)
From the above discussion in Section 2 we know thatthe control input in relation to control output 119902 namelythe relative degree of sliding mode manifold 120590 is 3 So thefollowing design will be about a third-order sliding modecontroller
32 Design of the Quasi-Continuous HOSM Controller First(26) can be expressed as follows
120590 = ℎ (119905 119909) + 119892 (119905 119909) 119906 (36)
where ℎ(119905 119909) 119892(119905 119909) and 119906 are expressed as follows
ℎ (119905 119909) =1
119903(1198911
+ 1198912
+ 1198913
+ 1198914
+ 1198915
+ 1198916)
119892 (119905 119909) = 119891120575119885
119906 = 120575119885
(37)
According to the quasi-continuous high order slidingmode control method proposed by Levant in [15] the slidingmode manifold whose relative degree is 3 should be designedin the following form where 120573 is a control gain term
119906 = minus120573
+ 2 (|| + |120590|23
)minus12
( + |120590|23 sign120590)
|| + 2 (|| + |120590|23
)12
(38)
The conditions under which the LOS angular velocitymay converge are as follows
0 lt 119870119898
le 119892 (119905 119909) le 119870119872
|ℎ (119905 119909)| le 119862
(39)
where119870119898119870119872 and119862 are all larger than zeroThis is a proven
theorem by Levant in [15]The system we discussed meets the above requirements
and the proof is as followsEquation (27) shows the following
119892 (119905 119909) = 119891120575119885
= minus1
119903
119862120572
119884119876119878ref
119898
119898120575119885
119911119876119878ref119897ref
119869119885
cos (120579119872
minus 119902)
(40)
The dynamic pressure 119876 is 119876 = 1205881198812
1198722 where 120588 =
008803Kgm3 (altitude = 20Km) is the air density and119881119872
=
2000ms is the speed of the missile so 119876 is always positive119878ref = 026m2 and 119897ref = 365m denote the reference area
and the reference length of the missile they are both positiveconstant
119898 = 100Kg denotes the missile mass 119869119885
= 106m2 Kgdenotes the rotational inertia
119903 is the relative distance it is always a positive number119862120572
119884is the lift coefficient caused by the angle of attack it
varies from 018 to 037 and it is always a positive number119898120575119911
119911is the moment coefficient caused by the actuator
deflection In the normal layout (actuator lays behind thecenter of gravity) 119898
120575119911
119911is always negative
Meanwhile consider that the missile under guidance andcontrol is unlikely to fly away from its target namely the anglebetween themissilersquos velocity and its LOS direction cannot belarger than 90∘ then
1003816100381610038161003816120579119872 minus 1199021003816100381610038161003816 lt
120587
2
997904rArr cos (120579119872
minus 119902) gt 0
(41)
6 Mathematical Problems in Engineering
Summing up the above conditions then we can get
119892 (119905 119909) gt 0 (42)
In other words there is a positive real number 119870119898existing
that could satisfy the following condition
0 lt 119870119898
lt 119892 (119905 119909) (43)
Before the missile hits on the target the term will be positiveand limited then we can get
0 lt 119870119898
lt 119892 (119905 119909) lt 119870119872
(44)
With (37) then
ℎ (119905 119909) =1
119903(1198911
+ 1198912
+ 1198913
+ 1198914
+ 1198915
+ 1198916) (45)
In the practice sense the changes in both the LOS rateand the acceleration of the missile and the acceleration of thetarget are limited and continuous So the following variables119902 119902
119902 119886119872 119886119879 and 119886
119879are all bounded However because
ℎ(119905 119909) contains the item 1119903 when the relative distancebetween the missile and its target is zero the boundaryof ℎ(119905 119909) is not guaranteed In [15] Levant only requiresthat condition (39) should be locally valid not requiringthat it should be globally valid Therefore the integratedguidance and controlmethod is applicable here So the abovementioned condition is satisfied with a positive number 119862
|ℎ (119905 119909)| le 119862 (46)
33TheVirtual ControlDesign Whenusing the slidingmodecontrolmethod the avoidance of the chattering phenomenonhas always been a key issue being discussed In the tradi-tional method researchers in [16 17] have proposed severalsaturation functions to replace the sign functions to builda boundary layer to alleviate the chattering or to use fuzzylogic to displace the high-frequency switching term To ourknowledge none of these approaches has proven that therefined controller still retains their robustness against theuncertainties and disturbances In this work in order toalleviate the chattering phenomenon we do not directly usethe third-order controller but introduce the virtual control119906119894= 120575119885to perform the actual control
120575119885
= int 120575119885dt = int 119906
119894dt (47)
After the relative degree is increased to the fourth order weget the following expressions
120590(4)
= ℎlowast
(119905 119909) + 119892lowast
(119905 119909) 119906119894= ℎlowast
(119905 119909) + 119892 (119905 119909) 119906119894
ℎlowast
(119905 119909) = ℎ (119905 119909) + 119892 (119905 119909) 120575119885
119892 (119905 119909)
=119898120575119885
1199111198762
1198782
119897119862120572
119884
119869119885
1198981199032[( 119902 minus 120579
119872) 119903 sin 120578
119872+ 119903 cos 120578
119872]
(48)
Even though the expression of ℎ(119905 119909) is rather compli-cated it is still the function of 119902 119902 119902 119886
119872 119886119879 and 119886
119879 therefore
similar to ℎ(119905 119909) it has its boundary except themomentwhenthe missile hits on its target For the same reason 119892(119905 119909)
and the rudder deflection 120575119885also have their boundaries
Therefore we get the condition that |ℎlowast
(119905 119909)| le 119862 (119862 gt 0)Because 120575
119885is obtained through the derivation of 120575
119885 119892lowast(119905 119909)
is the same as 119892(119905 119909) thus 0 lt 119870119898
lt 119892lowast
(119905 119909) lt 119870119872
issatisfied
According to the formula of the fourth-order controllergiven by Levant in [15] we give the following formulae forthe virtual control 119906
119894
119906119894= minus120573
Φ34
11987334
Φ34
=120590 + 3 [||
+ (|| + 05 |120590|34
)minus13 10038161003816100381610038161003816
+ 05 |120590|34 sign120590
10038161003816100381610038161003816]
minus12
sdot [
+ (|| + 05 |120590|34
)minus13 10038161003816100381610038161003816
+ 05 |120590|34 sign120590
10038161003816100381610038161003816]
11987334
=1003816100381610038161003816
120590
1003816100381610038161003816 + 3 [||
+ (|| + 05 |120590|34
)minus13 10038161003816100381610038161003816
+ 05 |120590|34 sign120590
10038161003816100381610038161003816]
minus12
sdot
1003816100381610038161003816100381610038161003816
+ (|| + 05 |120590|34
)minus13
( + 05 |120590|34 sign120590)
1003816100381610038161003816100381610038161003816
(49)
The integral of the virtual control 120575119885produces the actual
control command 120575119885 120590 in the controller is obtained with the
Arbitrary-Order Robust Exact Differentiator presented in thefollowing section
34 The Arbitrary-Order Robust Exact Differentiator Thequasi-continuous HOSM control method needs to use thethird derivative of the sliding manifold namely 119902
(4) Howto calculate or accurately estimate 119902
(4) is one of the keyproblems to be solved We use the Arbitrary-Order RobustExact Differentiator designed by Levant to differentiate theLOS rate 119902 thus obtaining 119902 119902 and 119902
(4)According to (44) and (45) the following condition is
valid1003816100381610038161003816
120590
1003816100381610038161003816 le 119862 + 120573119870119872
(50)
The Arbitrary-Order Robust Exact Differentiator can beconstructed in accordance with high order sliding modesdifferentiation and output feedback control in [18]
If a certain signal 119891(119905) is a function consisting of abounded Lebesgue-measurable noise with unknown base
Mathematical Problems in Engineering 7
signal 1198910(119905) whose 119903th derivative has a known Lipschitz
constant 119871 gt 0 then the 119899th-order differentiator is definedas follows
0
= V0
V0
= minus12058201198711(119899+1) 10038161003816100381610038161199110 minus 119891 (119905)
1003816100381610038161003816
119899(119899+1) sign (1199110
minus 119891 (119905))
+ 1199111
1
= V1
V1
= minus12058211198712(119899+1) 10038161003816100381610038161199111 minus V
0
1003816100381610038161003816
(119899minus1)119899 sign (1199111
minus V0) + 1199112
119899minus1
= V119899minus1
V119899minus1
= minus120582119899minus1
11987112 1003816100381610038161003816119911119899minus1 minus V
119899minus2
1003816100381610038161003816
12 sign (119911119899minus1
minus V119899minus2
)
+ 119911119899
119899
= minus120582119899119871 sign (119911
119899minus V119899minus1
)
(51)
and if 120582119894
gt 0 is sufficiently large the convergence is guaran-teed
To obtain the third-order derivative of 119902 we constructthe third-order sliding mode differentiator and estimate thederivative of 119902 for each order In view of differential precisionwe configure the following fifth-order differentiator SeeAppendix A for comparison
0
= V0
V0
= minus120582011987116 10038161003816100381610038161199110 minus
1199021003816100381610038161003816
56 sign (1199110
minus 119902) + 1199111
1
= V1
V1
= minus120582111987115 10038161003816100381610038161199111 minus V
0
1003816100381610038161003816
45 sign (1199111
minus V0) + 1199112
2
= V2
V2
= minus120582211987114 10038161003816100381610038161199112 minus V
1
1003816100381610038161003816
34 sign (1199112
minus V1) + 1199113
3
= V3
V3
= minus120582311987113 10038161003816100381610038161199113 minus V
2
1003816100381610038161003816
23 sign (1199113
minus V2) + 1199114
4
= V4
V4
= minus120582411987112 10038161003816100381610038161199114 minus V
3
1003816100381610038161003816
12 sign (1199114
minus V3) + 1199115
5
= V5
V5
= minus1205825119871 sign (119911
5minus V4)
(52)
where 1199113 1199112 1199111 and 119911
0are the estimations of 119902
(4) 119902 119902 and 119902
4 Baseline Separated Guidance andControl Method
To verify the homing performance of the integrated guidanceand control method we compare it with the separatedguidance and control methodThe guidance laws we used arethe proportional navigation (PN) guidance law for attackinga nonmaneuvering target and the optimal sliding modeguidance law for attacking a maneuvering target
41 The Proportional Navigation (PN) Guidance Law Theguidance law has a simple formula and excellent perfor-mances for nonmaneuvering target Its formula is as follows
119899119888
= minus119873 119902119881119872
119892 (53)
119899119888is the overload command 119873 is the effective navigation
ratio 119902 is the LOS rate 119881119872
is the speed of the missile 119892 isthe acceleration of the gravity The guidance law gives theoverload command of the missile according to the LOS rateand then the controller gives the rudder deflection commandaccording to the overload command
42 The Optimal Sliding Mode Guidance Law The optimalsliding mode guidance law (OSMG) is a novel practicalguidance law proposed by D Zhou He combines the optimalguidance lawwith the slidingmode guidance law and designsthe new sliding mode guidance law that not only is robustto maneuvering target but also has the merits of the optimalguidance law such as good dynamic performance and energyconservation Its formula is as follows
119899119888
= minus3100381610038161003816100381610038160
10038161003816100381610038161003816119902 + 120576
119902
10038161003816100381610038161199021003816100381610038161003816 + 120575
(54)
where 119899119888is the overload command
0is the approach
velocity of the missile and its target 119902 is their LOS rate 120576 =
const is the compensatory gain 119902(| 119902| + 120575) is for substitutingfor sign( 119902) and for smoothing 120575 is a small quantity whichcould adjust the chattering
43 Separated Guidance and Control Design For simulationand comparisonwe use the conventional three-loop overloadautopilot as the controller which gives the rudder deflectioncommand according to the feedback of the three loops ofoverload pseudo-angle of attack and pitch rate The blockdiagram is as shown in Figure 3
As the figure shows the inner loop has the feedback onangular velocity which improves the damping characteristicsof the missile airframe
According to the aerodynamic coefficient of the missilewith selected working points we set 119870
119868= 019 119870
120572= 3 and
119870120596
= minus025 and the controller can well track the overloadcommand the rise time of its step response is 046 secondsand its settling time is 083 secondsThe step responses of themissile to overload command and the Bode diagram for openloop are shown in Figure 4
8 Mathematical Problems in Engineering
KIS
120596Z 120572 nY
nC K120596K120572 nY+minus+minus+minus dynamicsAirframe
modelServo
Figure 3 The working principles for three-loop overload autopilot
Step response
Time (s)0 02 04 06 08 1 12
0
02
04
06
08
1
Rise time (s) 0463
Settling time (s) 0831
Bode diagram
Frequency (rads)
To output pointFrom input pointTo output pointFrom input point
Gain margin (dB) 175 At frequency (rads) 15
Phase margin (deg) 709 At frequency (rads) 294
Am
plitu
de
Mag
nitu
de (d
B)Ph
ase (
deg)
0
minus180
minus360
minus54010410310210110010minus1
100
0
minus100
minus200
minus300
Figure 4 The autopilot performance step response and Bode diagram
5 Simulation Results
To verify the high order slidingmode integrated guidance andcontrol (HOSM-IGC) method we compare it with the base-line separated guidance and control Numerical simulationsare designed in three typical engagement scenarios
AORED parameters are as follows the initial value 1205820
=
1205821
= 1205822
= 1205823
= 50 1199110
= 01 1199111
= 1199112
= 1199113
= 0 119871 = 1400 thesimulation step is 00001 seconds
51 Scenario 1 Nonmaneuvering Target In the first scenariothe nonmaneuvering target does uniform rectilinear motionand PN guidance law with three-loop autopilot is introducedfor a comparison with the HOSM-IGCThe initial conditionsare set as shown in Table 1
The motion equations of the target are as follows
119910119879
= 119881119879cos 120579119879
119879
= minus119881119879sin 120579119879
120579119879
=119886119879
119881119879
119886119879
= 0
(55)
The simulation results are shown in Figures 5 and 6The missilersquos flight trajectory and overload curve show
that within the first 3 sec the HOSM-IGC method spendsmuch energy (overload) on changing the initial LOSdirection
Table 1 Initial conditions
Variable Description Value(119883119872 119884119872) Missile initial position (0 km 18 km)
(119883119879 119884119879) Target initial position (60 km 20 km)
119881119872
Missile initial velocity 1200 (ms)119881119879
Target initial velocity 2000 (ms)120579119872
Missile initial flight path angle 0 (deg)120579119879
Target initial flight path angle 0 (deg)120573 Parameter of the controller 10 or 30119873 Navigation ratio of PN 3
of the missile After the boresight adjustment 120590 reaches thedesired sliding manifold and the missile does not need anymaneuver to hit the target This is because the target doesnot maneuver any more which means no disturbance isintroduced in the engagement dynamic so the state (or thesliding mode) of the missile will stay on the manifold Incontrast the overload command given by the PN guidancelaw increases fast as the relative distance decreases Toexamine the performances of the two guidance laws furtherwe increase the target speed and analyze the commandchanges
Therefore we set the target speeds 3000ms 4000msand carry out simulations The simulation result is shown inthe overload curve in Figure 7
Mathematical Problems in Engineering 9
175
180
185
190
195
200
205
TargetHOSM-IGCPN
Y(k
m)
0 10 20 30 40 50 60X (km)
Figure 5 Target and missile trajectories
0
5
10
15
20
25
0 5 10 15 20Time (s)
HOSM-IGCPN
Initial boresight adjustment
Sliding manifold reached
Miss
ile ac
cele
ratio
n (G
)
minus5
Figure 6 Missile acceleration profile
It is evident that the speed of divergence of the overloadcommand given by the PN guidance law increases withthe target speed Specifically when the target speed reaches4000ms the commanded overload is almost 30 g which isobviously not ideal for the attack of a nonmaneuvering targetbut with the HOSM-IGC method the missile adjusts itsboresight very quickly then maintains it around 0 g and fliesto its target in the rectilinear ballistic trajectory not affectedby the increases of target speed still accomplishing the high-precision hit-on collision
Figure 8 shows that although the HOSM-IGC methodachieves a more effective overload command on the otherside it sees some chattering when the sliding mode reaches
0
5
10
15
20
25
30
0 5 10 15Time (s)
Miss
ile ac
cele
ratio
n (G
)
minus5
HOSM-IGC (Vt = minus3000)HOSM-IGC (Vt = minus4000)
PN (Vt = minus3000)PN (Vt = minus4000)
Figure 7 Missile acceleration profile
0
1
2
0 5 10 15 20Time (s)
4 45 5
0
02
04
minus02
minus04
Actu
ator
defl
ectio
n (d
eg)
minus1
minus2
minus3
minus4
HOSM-IGC (120573 = 30)
Figure 8 Actuator deflection
the desired manifold the commanded rudder deflectionangle chatters for around 045 deg at about 15Hz which iskind of harmful to the system To reduce the chatteringwe adjust the controllerrsquos parameter 120573 = 10 and carry outsimulations again Figure 9 shows that after 120573 decreasesthe command of rudder deflection angle converges slower(for about 7 sec) however the chattering weakens obviouslyits magnitude being only about 015 degrees The smallerchattering well enhances convergence precision eventuallyreducing the target missing The reason that minor 120573 resultsin an alleviative chattering can be seen from (47) Figure 10shows the target and missile trajectories
Table 2 compares the average miss distance of 50 simula-tions under the conditions discussed above the comparison
10 Mathematical Problems in Engineering
Table 2 Average miss distance of 50 simulations
Target speed PN HOSM-IGC(120573 = 30)
HOSM-IGC(120573 = 10)
119881119905= 2000ms 115m 086 073m
119881119905= 3000ms 261m 117 106
119881119905= 4000ms 542m 156 134
0
1
2
0 5 10 15 20Time (s)
72 74 76 78
0
02
minus02
minus04
Actu
ator
defl
ectio
n (d
eg)
minus1
minus2
minus3
minus4
HOSM-IGC (120573 = 10)HOSM-IGC (120573 = 30)
Figure 9 Actuator deflection
results show that when the missile intercepts a nonmaneu-vering target the hit precision of the HOSM-IGC methodis apparently higher than that of the PN guidance law andthat the low-gain HOSM-IGC method can effectively reducethe chattering magnitude thus enhancing the interceptionprecision
52 Scenario 2 Step Maneuvering Target At the first stagethe target flies at uniform speed and in a rectilinear way after10 seconds it maneuvers at the normal acceleration of 5 g Inthis scenario OSMG is introduced for a comparison with theHOSM-IGC
119910119879
= 119881119879cos 120579119879
119879
= minus119881119879sin 120579119879
120579119879
=119886119879
119881119879
119886119879
=
0 g 119905 lt 10 sec
5 g 119905 gt 10 sec
(56)
The initial simulation conditions are given in Table 3The overload curve in Figure 11 shows dearly that both
types of guidance laws can track the maneuvering targetDuring 0 to 10 seconds the target flies at uniform speed andin a rectilinear way and the missile converges its overload to
Table 3 Initial conditions
Variable Description Value(119883119872 119884119872) Missile initial position (0 km 22 km)
(119883119879 119884119879) Target initial position (60 km 20 km)
119881119872
Missile initial velocity 1200 (ms)119881119879
Target initial velocity 2000 (ms)120579119872
Missile initial flight path angle 0 (deg)120579119879
Target initial flight path angle 0 (deg)120576 Compensatory gain 3120575 Saturation parameter 0001
15
17
19
21
23
25
TargetHOSM-IGCOSMG
Y(k
m)
0 20 40 60X (km)
Figure 10 Target and missile trajectories
0 g and flies to its target after 10 seconds the target beginsto maneuver by 5 g Both types of guidance law respondby rapidly increasing the overload adjusting attitude andmaking the missilersquos boresight aim at its target
We can see that when both types of guidance law tracktheir targets the convergence speed of OSMG is almostthe same as the HOSM-IGC method But the HOSM-IGCmethod has higher convergence precision and needs loweroverload at the end phase
We can also see that after the target maneuvers if themissile is given enough time to track the targetrsquos maneuvernamely let the missilersquos overload command converge to theoverload of the target theremay not be largemiss distance Inother words for a certain period of time before the collisionthe targetmaneuver (it means only a limitedmaneuver whichdoes not include the condition that the maneuvering of thetarget for a long time may cause a change of the geometricalrelations between the missile and its target) has a small effecton both types of guidance law
But if themaneuver occurs rather late namely within oneto three seconds before collision when the overload com-mand of guidance law is not yet converged the approaching
Mathematical Problems in Engineering 11
0
5
10
0 5 10 15 20Time (s)
TargetOSMGHOSM-IGC
Miss
ile an
d ta
rget
acce
lera
tion
(G)
minus5
minus10
Figure 11 Missile and target acceleration profile
0
10
20
30
0 5 10 15 20Time (s)
tgo = 1 stgo = 2 stgo = 3 s
tgo = 3 s (diverge)
tgo = 2 s (diverge)
tgo = 1 s (diverge)minus10
minus20
minus30
Miss
ile ac
cele
ratio
n (G
)
HOSM-IGC (120573 = 10)
Figure 12 Missile acceleration profile
collision increases themiss distanceTherefore increasing theparameter 120573 can remarkably increase the convergence speedand effectively enhance guidance precision As Figure 12shows when 120573 = 10 the overload at the end phase convergesslowly even if the target maneuvers three seconds beforecollision the missilersquos overload still has no time to convergebeing unable to track themaneuvering target Figure 13 showsthat when 120573 increases to 30 and tgo = 3 seconds theoverload can converge to about 5 g but when tgo = 2 sec-onds the overload may continue to increase indicating that
0
10
20
30
0 5 10 15 20Time (s)
tgo = 1 stgo = 2 stgo = 3 s
tgo = 1 s (diverge)
tgo = 2 s (diverge)
tgo = 3 s (converge)
minus10
minus20
minus30
Miss
ile ac
cele
ratio
n (G
)
120573 = 30)HOSM-IGC (
Figure 13 Missile acceleration profile
0
10
20
30
0 5 10 15 20Time (s)
tgo = 1 s (diverge)
tgo = 2 s (converge)
tgo = 3 s (converge)
tgo = 1 stgo = 2 stgo = 3 s
minus10
minus20
minus30
Miss
ile ac
cele
ratio
n (G
)
HOSM-IGC (120573 = 50)
Figure 14 Missile acceleration profile
the convergence still needs to be quickened thus continuingto rise 120573 to 50 Figure 14 shows that the HOSM-IGC methodcan track the target that maneuvers when tgo = 2 secondsbut it causes the divergence of overload and the increase ofmiss distance if the target maneuvers when tgo = 1 secondFigure 15 gives the overload curve of the OSMG guidance lawand shows that the maneuver of the target before collisionmay cause the large-scale oscillation of the missilersquos overloadwhich may diverge to a large numerical value when thecollision occurs in the end
12 Mathematical Problems in Engineering
Table 4 Average miss distances of 50 simulations
Targetmaneuveringtiming
HOSM-IGC120573 = 30
HOSM-IGC120573 = 40
HOSM-IGC120573 = 50
OSMG
tgo = 1second 43539 3224 28936 36116
tgo = 2seconds 25424 18665 09322 35534
tgo = 3seconds 08124 08265 07538 11959
Time (s)0 5 10 15 20
0
10
20
30OSMG
tgo = 1 stgo = 2 stgo = 3 s
minus10
minus20
minus30
Miss
ile ac
cele
ratio
n (G
)
Figure 15 Missile acceleration profile
The analysis in Figures 12 13 14 and 15 shows that theoverload of the missile converges faster and its miss distanceis smaller with increasing 120573 To verify this finding we carryout 50 times Monte Carlo simulations in which the positionand speed of the target have 1 of random difference
The average miss distances are shown in Table 4 andFigure 16 Clearly the timing of the targetrsquos step maneuverdramatically affects the final interception precision morespecifically given a shorter reaction time for the guidanceand control system the missile seems more likely to missthe target To the OSMG guidance law in all three scenarioshardly does it show any advantages against theHOSM-IGC Itcan also be seen that with the increasing of the120573 theHOSM-IGC system responds even faster which leads to an obviousdecrease of the average miss distance The effect of 120573 on theresponse of the HOSM-IGC system is a valuable guidelinewhen implementing the proposed method into practice
53 Scenario 3 Weaving Target In this scenario the targetmaneuvers by 119886
119879= 40sin(1205871199052) OSMG with three-loop
Table 5 Initial conditions
Variable Description Value(119883119872 119884119872) Missile initial position (0 km 175 km)
(119883119879 119884119879) Target initial position (60 km 195 km)
119881119872
Missile initial velocity 1200 (ms)119881119879
Target initial velocity 2000 (ms)120579119872
Missile initial flight path angle 0 (deg)120579119879
Target initial flight path angle 0 (deg)120576 Compensatory gain 3120575 Saturation parameter 0001
autopilot is introduced for comparison The motion equa-tions of the target are as follows
119910119879
= 119881119879cos 120579119879
119879
= minus119881119879sin 120579119879
120579119879
=119886119879
119881119879
119886119879
= 40 sin(120587119905
2)
(57)
The initial simulation conditions are given in Table 5The missilersquos trajectory under the two guidance and
control methods given in Figure 17 does not see muchdifference
However the overload curve given in Figure 18 showsthat after the missile completes its initial attitude adjustmentwith the HOSM-IGC method it can almost ideally track itsmaneuvering target by contrast with the OSMG methodthe missile seems to have the tendency to track its targetrsquosmaneuver but has larger tracking errors Besides with theOSMGmethod the missilersquos overload increases rapidly at theend of attack primarily because of the divergence of its LOSrate On the other hand with the HOSM-IGC method themissile has no divergence even at the end of attack ensuringa smaller target missing quantity
The actuator deflection curve in Figure 19 shows thatin order to provide a rather big normal overload for theend phase the OSMG method produces a rather big rudderdeflection command however it may increase the missilersquostarget missing quantity once its rudder deflection saturatesand themissile does not have enough overloads or the ruddercannot respond that fast
As shown in Figure 20 because of the dramatic changein overload command the response of the missilersquos autopilotto high-frequency command sees an obvious phase lag andamplitude value attenuation its actual overload cannot trackthe command ideally this is a main reason why the missdistance increases However the controller in the HOSM-IGC method gives its rudder deflection command directlyand there is no lagging or attenuation caused by the autopilotthus enhancing the guidance precision effectively Further-more the fast convergence of the high order sliding modemakes the missile rapidly track its maneuvering target withthemost reasonable rudder deflection command reducing itsoverload effectively
Mathematical Problems in Engineering 13
0
05
1
15
2
25
3
35
4
45
5
0 10 20 30 40 50Simulation times
OSMG
Miss
(m)
0
05
1
15
2
25
3
35
4
45
5
0 10 20 30 40 50Simulation times
Miss
(m)
0
05
1
15
2
25
3
35
4
45
5
0 10 20 30 40 50Simulation times
Miss
(m)
tgo = 3 stgo = 2 stgo = 1 s
0
05
1
15
2
25
3
35
4
45
5
0 10 20 30 40 50Simulation times
Miss
(m)
tgo = 3 stgo = 2 stgo = 1 s
HOSM-IGC (120573 = 40)
HOSM-IGC (120573 = 50)
HOSM-IGC (120573 = 30)
Figure 16 Miss distances of HOSM-IGC (120573 = 30 40 50) and OSMG with target maneuvers at different time-to-go (tgo = 1 second 2seconds and 3 seconds)
The average miss distance of 50 simulations under theconditions is 073m for HOSM-IGC and 186m for OSMGWe can see that the HOSM-IGC method not only doesprovide a more reasonable actuator deflection command butalso achieves a higher interception precision
6 Conclusions
This paper proposes an LOS feedback integrated guidanceand control method using quasi-continuous high order
sliding mode guidance and control method With the fastand precise convergence of the quasi-continuous HOSMmethod the HOSM-IGCmethod performsmuch better thanthe traditional separated guidance and control method withless acceleration effort and less miss distance in all thethree simulation scenarios of nonmaneuvering target stepmaneuvering target and weaving target In addition the ideaof virtual control largely alleviates the chattering withoutany sacrifice of robustness As a result of the alleviationof the chattering the control input command 120575
119885becomes
14 Mathematical Problems in Engineering
175
180
185
190
195
200
TargetHOSM-IGCOSMG
Y(k
m)
0 20 40 60 0 20 40X (km)
Figure 17 The trajectories of the missile and its target
0
10
20
0 5 10 15 20Time (s)
TargetOSMGHOSM-IGC
Miss
ile an
d ta
rget
acce
lera
tion
(G)
minus10
minus20
Figure 18 Missile acceleration profile
smooth enough for the actuator to be implemented as aresult the interception accuracy is improved In designingthe quasi-continuous sliding mode controller the Arbitrary-Order Robust Exact Differentiator is introduced to estimatethe high order time derivative of the LOS rate 119902 which notonly ensures the guidance and control precision but alsosimplifies the calculation
0
10
20
30
0 5 10 15 20Time (s)
OSMGHOSM-IGC
minus10
minus20
minus30
minus40
Actu
ator
defl
ectio
n (d
eg)
Figure 19 Actuator deflection
0
50
100
150
200
15 16 17 18Time (s)
Commanded accelerationAchieved acceleration
Miss
ile ac
cele
ratio
n (G
)
minus50
Figure 20 Commanded acceleration and achieved acceleration
Appendices
A The Third-Order RobustExact Differentiator
The third-order robust exact differentiator built according tothe following parameters and equations is used to performthe simulation the initial value 120582
0= 1205821
= 1205822
= 1205823
= 50
Mathematical Problems in Engineering 15
0
0005
001
0015
002
0025
003
0035
004
0 02 04Time (s)
0
2
0 02 04Time (s)
0
100
200
300
0
100
200
300
400
0 02 04Time (s)
h(t)
z3
h(t) and z3
minus100
minus200
minus300
minus400
minus5000 02 04
Time (s)
g(t)
z2
g(t) and z2
minus100
minus200
minus300
minus400
f(t)
z1
f(t) and z1
minus2
minus4
minus6
minus8
minus10
minus12
minus14
F(t)
z0
F(t) and z0
minus0005
minus001
Figure 21 The outputs of the differentiator in the sampling interval of 0001 seconds
the initial values 1199110
= 01 1199111
= 1199112
= 1199113
= 0 119871 = 1400 thesimulation step is 0001 seconds Consider
0
= V0
V0
= minus120582011987114 10038161003816100381610038161199110 minus
1199021003816100381610038161003816
34 sign (1199110
minus 119902) + 1199111
1
= V1
V1
= minus120582111987113 10038161003816100381610038161199111 minus V
0
1003816100381610038161003816
23 sign (1199111
minus V0) + 1199112
2
= V2
V2
= minus120582211987112 10038161003816100381610038161199112 minus V
1
1003816100381610038161003816
12 sign (1199112
minus V1) + 1199113
3
= minus1205823119871 sign (119911
3minus V2)
(A1)
16 Mathematical Problems in Engineering
0
0005
001
0015
002
0025
003
0035
004
0 02 04Time (s)
0
2
0 02 04Time (s)
0
100
200
300
0 02 04Time (s)
0
200
400
0 02 04Time (s)
h(t) and z3
minus200
minus400
minus600
minus800
g(t) and z2
minus100
minus200
minus300
minus400
f(t) and z1
minus2
minus4
minus6
minus8
minus10
minus12
minus14
minus16
F(t) and z0
minus0005
minus001
h(t)
z3g(t)
z2
f(t)
z1F(t)
z0
Figure 22 The outputs of the differentiator in the sampling interval of 00001 seconds
If the designed signals are 119865(119905) = int 119891(119905)119889119905 119891(119905) =
int 119892(119905)119889119905 119892(119905) = int ℎ(119905)119889119905 and ℎ(119905) = sin(2120587119905) +
3cos(120587119905) then theoretically 119891(119905) is the first-order differ-ential of input signal 119892(119905) is the second-order differ-ential of input signal ℎ(119905) is the third-order differen-tial of input signal Through the simulation the differ-ential signal of actual output is compared with that of
theoretical output the comparison results are shown inFigure 21
Figure 21 shows that 1199110can effectively track the signal
119865(119905) but 1199112has obvious errors in tracking the second-order
differential 119892(119905) 1199113has greater errors in tracking the third-
order differential thus the overall differential effect is notideal
Mathematical Problems in Engineering 17
0 02 04
0
0005
001
0015
002
0025
003
0035
004
Time (s)0 02 04
0
01
02
03
04
05
Time (s)
0 02 04
0
05
1
15
2
Time (s)0 02 04
0
05
1
15
2
25
3
35
4
Time (s)
h(t) and z3g(t) and z2
minus04
minus05
f(t) and z1
minus2
minus15
minus1
minus05
minus1
minus05
minus01
minus02
minus03
F(t) and z0
minus0005
minus001
h(t)
z3g(t)
z2
f(t)
z1F(t)
z0
Figure 23 The outputs of the differentiator in the sampling interval of 00001 seconds
There are two causes for the errors in a usual differentia-tor the setting of sampling interval and the setting of numberof orders of the differentiator The following theorem applieshere
Theorem A1 (see [19]) If the sampling interval 120591 gt 0 theaccuracy of the controller is expressed as follows
10038161003816100381610038161003816120590(119894)
10038161003816100381610038161003816le 120583120591119903minus119894
119894 = 0 119903 minus 1 (A2)
That is to say the precision (understood as the differentialprecision of a differentiator) of the 119894th order control is in directproportion to the power of 119903 minus 119894 times of sampling interval120591 Therefore the decrease of iteration step can enhance thedifferential tracking precision
Under the conditions that the sampling interval is 10 timessmaller and that the setting is 00001 seconds the simulationresults are shown in Figure 22
18 Mathematical Problems in Engineering
Table 6 Tracking errors of the AORED with different orders andsampling interval
Orders of AORED Third order Fifth orderSample period 120591 = 0001 120591 = 00001 120591 = 00001
First-order differential error 001 Negligible NegligibleSecond-order differential error 15 002 NegligibleThird-order differential error 70 25 00005
As Figure 22 shows the signal tracking is rather idealexcept a peak in the beginning the tracking error in the first-order differential tracking can be neglected 119911
2has an obviously
smaller tracking error in the second-order differential 119892(119905)
tracking in the beginning 1199113has a rather sharp peak in the
third-order differential tracking but converges to a relativelyhigh precision after 01 seconds As we can see the differentialtracking errors for each number of orders are greatly reducedbut the third-order differential tracking error is still rather bigIn the following we will reduce the tracking errors by increasingthe number of orders of the differentiator
B The Fifth-Order Robust Exact Differentiator
The settings of the fifth-order differentiator are given asfollows
0
= V0
V0
= minus120582011987116 10038161003816100381610038161199110 minus
1199021003816100381610038161003816
56 sign (1199110
minus 119902) + 1199111
1
= V1
V1
= minus120582111987115 10038161003816100381610038161199111 minus V
0
1003816100381610038161003816
45 sign (1199111
minus V0) + 1199112
2
= V2
V2
= minus120582211987114 10038161003816100381610038161199112 minus V
1
1003816100381610038161003816
34 sign (1199112
minus V1) + 1199113
3
= V3
V3
= minus120582311987113 10038161003816100381610038161199113 minus V
2
1003816100381610038161003816
23 sign (1199113
minus V2) + 1199114
4
= V4
V4
= minus120582411987112 10038161003816100381610038161199114 minus V
3
1003816100381610038161003816
12 sign (1199114
minus V3) + 1199115
5
= V5
V5
= minus1205825119871 sign (119911
5minus V4)
(B1)
The initial value 1199110
= 01 1205820
= 1205821
= 1205822
= 1205823
= 1205824
=
1205825
= 50 the initial values 1199111 1199112 1199113 1199114 1199115
= 0 119871 = 1400the simulation step length is 00001 seconds The simulationresults are shown in Figure 23
As the figure shows the fifth-order differentiator hasa rather ideal tracking effect on the third-order differen-tial tracking the tracking error being only about 00005(see Table 6)
To sum up the Arbitrary-Order Robust Exact Differen-tiator can estimate rather ideal differential signals by selecting
Table 7
Length 365m119871 ref 365m119883119866
177m119878ref 0026m2
Diameter 0178mMass 1016 Kg119868119885
1063 Kgsdotm2
an appropriate sampling interval and using the differentiatorwith a relatively high number of orders
C Physical and Geometric Characteristics
See Table 7
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] P Zarchan Tactical and Strategic Missile Guidance 2002[2] D Zhou CMu andWXu ldquoAdaptive sliding-mode guidance of
a homing missilerdquo Journal of Guidance Control and Dynamicsvol 22 no 4 pp 589ndash594 1999
[3] G Zeng and M Hu ldquoFinite-time control for electromagneticsatellite formationsrdquo Acta Astronautica vol 74 pp 120ndash1302012
[4] S R Kumar S Rao and D Ghose ldquoNonsingular terminalsliding mode guidance with impact angle constraintsrdquo Journalof Guidance Control and Dynamics vol 37 no 4 pp 1114ndash11302014
[5] Y Liang C Chen D Liaw Y Feng C Cheng and CChen ldquoRobust guidance law via integral-sliding-mode schemerdquoJournal of Guidance Control and Dynamics vol 37 no 3 pp1038ndash1042 2014
[6] A V Savkin P N Pathirana and F Faruqi ldquoProblem ofprecision missile guidance LQR and 119867
infincontrol frameworksrdquo
IEEE Transactions on Aerospace and Electronic Systems vol 39no 3 pp 901ndash910 2003
[7] H Y Chen andC-C Yang ldquoNonlinear1198672119867infinguidance design
for homing missilesrdquo in Proceedings of the AIAA GuidanceNavigation and Control Conference and Exhibit vol 77 2001
[8] P K Menon G D Sweriduk and E J Ohlmeyer ldquoOptimalfixed-interval integrated guidance-control laws for hit-to-killmissilesrdquo in Proceedings of the AIAA Guidance Navigation andControl Conference and Exhibit AIAA 2003-5579 Austin TexUSA August 2003
[9] S S Vaddi P K Menon and E J Ohlmeyer ldquoNumericalSDRE approach for missile integrated guidance-controlrdquo inProceedings of the AIAA Guidance Navigation and ControlConference and Exhibit AIAA 2007-6672 Hilton Head SCUSA August 2007
[10] M Xin S N Balakrishnan and E J Ohlmeyer ldquoIntegratedguidance and control of missiles with Theta-D methodrdquo IEEE
Mathematical Problems in Engineering 19
Transactions on Control Systems Technology vol 14 no 6 pp981ndash992 2006
[11] T Shima M Idan and O M Golan ldquoSliding-mode controlfor integrated missile autopilot guidancerdquo Journal of GuidanceControl and Dynamics vol 29 no 2 pp 250ndash260 2006
[12] Y B Shtessel and C H Tournes ldquoIntegrated higher-ordersliding mode guidance and autopilot for dual control missilesrdquoJournal of Guidance Control and Dynamics vol 32 no 1 pp79ndash94 2009
[13] F Y Dong H M Lei and C J Zhou ldquoIntegrated robust highorder sliding mode guidance and control for missilesrdquo ActaAeronautica et Astronautica Sinica vol 34 no 9 pp 2212ndash22182013
[14] H Mingzhe and D Guangren ldquoIntegrated guidance and con-trol of homing missiles against ground fixed targetsrdquo ChineseJournal of Aeronautics vol 21 no 2 pp 162ndash168 2008
[15] A Levant ldquoHomogeneity approach to high-order sliding modedesignrdquo Automatica vol 41 no 5 pp 823ndash830 2005
[16] Y-W Liang C Chen D Liaw Y Feng C Cheng and CChen ldquoRobust guidance law via integral-sliding-mode schemerdquoJournal of Guidance Control and Dynamics vol 37 no 3 pp1038ndash1042 2014
[17] L Wang Y Sheng and X Liu ldquoContinuous time-varying slid-ingmode based attitude control for reentry vehiclerdquo Proceedingsof the Institution of Mechanical Engineers Part G Journal ofAerospace Engineering 2014
[18] A Levant ldquoHigher-order sliding modes differentiation andoutput-feedback controlrdquo International Journal of Control vol76 no 9-10 pp 924ndash941 2003
[19] A Levant ldquoQuasi-continuous high-order sliding-mode con-trollersrdquo IEEE Transactions on Automatic Control vol 50 no11 pp 1812ndash1816 2005
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Mathematical Problems in Engineering
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OptimizationJournal of
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International Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
Discrete MathematicsJournal of
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 3
Seeker
Guidancefiltering
Integratedguidance and control
ActuatorMissile
airframe
Missile position
Target position
Missile acceleration
Target acceleration
Guidancelaw
Autopilot(controller) Separated guidance and control
Integrated guidance and control +minus
120575CnC
q
120575120575C
q
q nm
Figure 1 Integrated guidance and control
O
Y1
X1
M
aM
xb
120572120579M
q
VM
T
VT
120579T
aT
120599
Figure 2 Engagement geometry
21 The Engagement Dynamics Without loss of generalitywe present hereinafter only the subsystems that govern themotion of an interceptor in its longitudinal plane The planarengagement between the interceptor and its target is shownin Figure 2
The study of guidance laws usually regards themissile andits target as mass points deliberates on the mass point of itsrigid motion and ignores the attitude of the missile airframeand changes in its attitude By contrast the integrated guid-ance and control method takes the dynamic characteristicsof the missile airframe into consideration therefore thederivation involves the pitch angle pitch rate and angle ofattack of the missile as shown in Figure 2
The motion equations of the missile in the longitudinalplane are as follows
119889119881119872
119889119905=
1
119898(119875 cos120572 minus 119883 minus 119898119892 sin 120579
119872) (1)
119889120579119872
119889119905=
1
119898119881119872
(119875 sin120572 + 119884 minus 119898119892 cos 120579119872
) (2)
119889119909119872
119889119905= 119881119872cos 120579119872
(3)
119889119910119872
119889119905= 119881119872sin 120579119872
(4)
119889120596119885
119889119905=
119872119885
119869119885
(5)
119889120599
119889119905= 120596119885 (6)
120572 = 120599 minus 120579 (7)
where (119909119872
119910119872
) is the position 119898 is the mass 119883 is the axialforce 119875 is the thrust force the missilersquos velocity is 119881
119872 and its
flight path angle is 120579119872 the LOS angle between themissile and
its target is 119902 the relative distance between the missile and itstarget is 119903 the velocity of the target is 119881
119879 and its flight path
angle is 120579119879 119909119887is the axle of the missilersquos airframe the pitch
angle is 120599 the angle of attack is 120572The motion equations of the target are as follows
119889119909119879
119889119905= minus119881119879cos 120579119879
119889119910119879
119889119905= 119881119879sin 120579119879
119889120579119879
119889119905=
119886119879119873
119881119879
(8)
The interception is characterized by two variablesnamely the target range and the LOS angle The kinematicequations are expressed by the following relations
119889119903
119889119905= minus119881119872cos (120579
119872minus 119902) minus 119881
119879cos (120579
119879+ 119902) (9)
119903119889119902
119889119905= minus119881119872sin (120579119872
minus 119902) + 119881119879sin (120579119879
+ 119902) (10)
22 The Model Simplification To simplify the model andmake further derivations wemake the following two assump-tions
Assumption 1 Within the terminal phase of the interceptionthe missile has no thrust and its gravity is not taken intoaccount
Assumption 2 Within the terminal phase of the interceptionthe missilersquos speed does not change
4 Mathematical Problems in Engineering
Then (1) and (2) can be reformed as below
119889119881119872
119889119905= 0
119889120579119872
119889119905=
119884
119898119881119872
(11)
Denote that 119886119872119873
= 119884119898 119884 is the normal force 119886119872119873
119886119879119873
are the normal acceleration of the missile and targetrespectively and then
120579119872
=119886119872119873
119881119872
120579119879
=119886119879119873
119881119879
(12)
The normal force 119884 and the pitch moment 119872119885acting on
the missile are usually expressed respectively as follows
119884 = 119862120572
119884120572119876119878ref + 119862
120575119885
119884120575119885119876119878ref (13)
119872119885
= 119898120575119885
119885120575119885119876119878ref119897 + 119898
120572
119885120572119876119878ref119897 (14)
where119862120572
119884is the coefficient of normal force caused by the angle
of attack 119862120575119885
119884is the coefficient of normal force caused by the
rudder deflection angle 120575119885 119876 is the dynamic pressure 119878ref is
the reference area 119897 is the reference length But the normalforce produced by the rudder deflection angle 120575
119885is orders of
magnitude smaller than that produced by the angle of attackso (13) is simplified as
119884 = 119862120572
119884120572119876119878ref (15)
Then
119886119872119873
=119884
119898=
119862120572
119884120572119876119878ref
119898 (16)
Then the dynamic equations can be simplified as
119889120579119872
119889119905=
119862120572
119884120572119876119878ref
119898119881119872
(17)
119889120596119885
119889119905=
119898120575119885
119885120575119885
119876119878ref119897 + 119898120572
119885120572119876119878ref119897
119869119885
(18)
119889120599
119889119905= 120596119885 (19)
120572 = 120599 minus 120579119872
(20)
23 The Integrated Guidance and Control Model Follow-ing the above derivation we select the state variables as
( 119902 120579119872
120572 120596119885)119879 and obtain the following nonlinear integrated
guidance and control model
119902 = minus2119903
119903sdot 119902 minus
119862120572
119884119876119878ref
119898119903cos (120579
119872minus 119902) sdot 120572
+119886119879119873
119903cos (120579
119879+ 119902)
119903 119902 = minus119881119872sin (120579119872
minus 119902) + 119881119879sin (120579119879
+ 119902)
120579119872
=1
119898119881119872
119862120572
119884119876119878ref sdot 120572
= 120596119885
minus1
119898119881119872
119862120572
119884119876119878ref sdot 120572
119885
=119898120575119885
119885119876119878ref119897
119869119885
sdot 120575119885
+119898120572
119885119876119878ref119897
119869119885
120572
(21)
24 The Relative Degree of Control Input To obtain therelative degree of the control input 120575 of the integratedguidance and control method we keep on deriving the LOSangular velocity 119902 until the explicit formula of its derivativeof a certain order contains the control input
The derivation of (10) produces119902
=1
119903[minus2 119903 119902 minus 119886
119872119873cos (120579
119872minus 119902) + 119886
119879119873cos (120579
119879+ 119902)]
(22)
The LOS angular velocity is expressed as the derivativeof the first order and does not contain the control input 120575
explicitly The continuous derivation of the above equationproduces
119902 =
1
119903minus2 119903 119902 minus 3 119903 119902 minus [119886
119872119873( 119902 minus 120579
119872) sin (120579
119872minus 119902)
+ 119886119872119873
cos (120579119872
minus 119902)] minus [119886119879119873
( 119902 + 120579119879) sin (120579
119879+ 119902)
minus 119886119879119873
cos (120579119879
+ 119902)]
(23)
where the LOS angular rate is expressed as the differentiatingof the second order and 119886
119872can be expressed as
119886119872119873
=119862120572
119884119876119878ref
119898 =
119862120572
119884119876119878ref
119898( 120599 minus 120579
119872)
=119862120572
119884119876119878ref
119898(120596119885
minus 120579119872
)
(24)
Although the control volume 120575119885does not appear in the
derivative of the second order (18) shows that 119885contains
120575119885 We continue to derive (23) and substitute 119886
119872as follows
119886119872119873
=119862120572
119884119876119878ref
119898(119885
minus119886119872
119881119872
)
=119862120572
119884119876119878ref
119898
119898120575119885
119885119876119878ref119897ref
119869119885
sdot 120575119885
+119862120572
119884119876119878ref
119898(
119898120572
119885119876119878ref119897ref
119869119885
120572 minus119886119872
119881119872
)
(25)
Mathematical Problems in Engineering 5
Thus
119902(4)
= 119891120575119885
sdot 120575119885
+1
119903(1198911
+ 1198912
+ 1198913
+ 1198914
+ 1198915
+ 1198916) (26)
where
119891120575119885
= minus1
119903
119862120572
119884119876119878ref cos 120578
119872
119898
119898120575119885
119885119876119878ref119897ref
119869119885
(27)
1198911
= (120596119885
minus 120579119872
) (minus2 120579119886119872119873
120572sin 120578119872
+ 119902119886119872119873
120572sin 120578119872
+ 120579119886119872119873
1205722cos 120578119872
) minus119886119872119873
cos 120578119872
120572
119898120572
119885120572119876119878ref119897ref
119869119885
(28)
1198912
= 120579119872
(( 120579119872
minus 119902) 119886119872119873
cos 120578119872
minus (120596119885
minus 120579119872
)119886119872119873
120572sin 120578119872
)
(29)
1198913
= 119902 (minus 120579119872
119886119872119873
cos 120578119872
+ 119902 (119886119872119873
cos 120578119872
minus 119886119879119873
cos 120578119879) minus 120579119879119886119879119873
cos 120578119879
+119886119872119873
120572(120596119885
minus 120579119872
) sin 120578119872
minus 119886119879sin 120578119879)
(30)
1198914
= 120579119879
(minus 119902119886119879cos 120578119879
minus 120579119879119886119879cos 120578119879
minus 119886119879sin 120578119879) (31)
1198915
= 119886119879
(minus 119902 sin 120578119879
minus 2 120579119879sin 120578119879) + 119886119879cos 120578119879 (32)
1198916
= 119902 (119886119872119873
sin 120578119872
minus 119886119879sin 120578119879
minus 2 119903) minus 4 119903 sdot119902 minus 3 119903 sdot 119902
minus 2119903 sdot 119902
(33)
120578119872
= 120579119872
minus 119902
120578119879
= 120579119879
+ 119902
(34)
In (26) it can be seen that the control input 120575119885appears
expressly in the third-order derivative of the control output119902 Therefore the relative degree of the control input 120575
119885is 3
3 The Quasi-Continuous High Order SlidingMode Controller
31 Sliding Mode Manifold Design To design the HOSMcontroller a sliding manifold must be chosen first In thisdesign we try to make the LOS rate converge to zero ora small neighbor domain near zero thus ensuring that themissile approaches its target in a quasi-parallel way whichwill lead to a minimal overload requirement So the slidingmanifold is chosen as follows
120590 = 119902 (35)
From the above discussion in Section 2 we know thatthe control input in relation to control output 119902 namelythe relative degree of sliding mode manifold 120590 is 3 So thefollowing design will be about a third-order sliding modecontroller
32 Design of the Quasi-Continuous HOSM Controller First(26) can be expressed as follows
120590 = ℎ (119905 119909) + 119892 (119905 119909) 119906 (36)
where ℎ(119905 119909) 119892(119905 119909) and 119906 are expressed as follows
ℎ (119905 119909) =1
119903(1198911
+ 1198912
+ 1198913
+ 1198914
+ 1198915
+ 1198916)
119892 (119905 119909) = 119891120575119885
119906 = 120575119885
(37)
According to the quasi-continuous high order slidingmode control method proposed by Levant in [15] the slidingmode manifold whose relative degree is 3 should be designedin the following form where 120573 is a control gain term
119906 = minus120573
+ 2 (|| + |120590|23
)minus12
( + |120590|23 sign120590)
|| + 2 (|| + |120590|23
)12
(38)
The conditions under which the LOS angular velocitymay converge are as follows
0 lt 119870119898
le 119892 (119905 119909) le 119870119872
|ℎ (119905 119909)| le 119862
(39)
where119870119898119870119872 and119862 are all larger than zeroThis is a proven
theorem by Levant in [15]The system we discussed meets the above requirements
and the proof is as followsEquation (27) shows the following
119892 (119905 119909) = 119891120575119885
= minus1
119903
119862120572
119884119876119878ref
119898
119898120575119885
119911119876119878ref119897ref
119869119885
cos (120579119872
minus 119902)
(40)
The dynamic pressure 119876 is 119876 = 1205881198812
1198722 where 120588 =
008803Kgm3 (altitude = 20Km) is the air density and119881119872
=
2000ms is the speed of the missile so 119876 is always positive119878ref = 026m2 and 119897ref = 365m denote the reference area
and the reference length of the missile they are both positiveconstant
119898 = 100Kg denotes the missile mass 119869119885
= 106m2 Kgdenotes the rotational inertia
119903 is the relative distance it is always a positive number119862120572
119884is the lift coefficient caused by the angle of attack it
varies from 018 to 037 and it is always a positive number119898120575119911
119911is the moment coefficient caused by the actuator
deflection In the normal layout (actuator lays behind thecenter of gravity) 119898
120575119911
119911is always negative
Meanwhile consider that the missile under guidance andcontrol is unlikely to fly away from its target namely the anglebetween themissilersquos velocity and its LOS direction cannot belarger than 90∘ then
1003816100381610038161003816120579119872 minus 1199021003816100381610038161003816 lt
120587
2
997904rArr cos (120579119872
minus 119902) gt 0
(41)
6 Mathematical Problems in Engineering
Summing up the above conditions then we can get
119892 (119905 119909) gt 0 (42)
In other words there is a positive real number 119870119898existing
that could satisfy the following condition
0 lt 119870119898
lt 119892 (119905 119909) (43)
Before the missile hits on the target the term will be positiveand limited then we can get
0 lt 119870119898
lt 119892 (119905 119909) lt 119870119872
(44)
With (37) then
ℎ (119905 119909) =1
119903(1198911
+ 1198912
+ 1198913
+ 1198914
+ 1198915
+ 1198916) (45)
In the practice sense the changes in both the LOS rateand the acceleration of the missile and the acceleration of thetarget are limited and continuous So the following variables119902 119902
119902 119886119872 119886119879 and 119886
119879are all bounded However because
ℎ(119905 119909) contains the item 1119903 when the relative distancebetween the missile and its target is zero the boundaryof ℎ(119905 119909) is not guaranteed In [15] Levant only requiresthat condition (39) should be locally valid not requiringthat it should be globally valid Therefore the integratedguidance and controlmethod is applicable here So the abovementioned condition is satisfied with a positive number 119862
|ℎ (119905 119909)| le 119862 (46)
33TheVirtual ControlDesign Whenusing the slidingmodecontrolmethod the avoidance of the chattering phenomenonhas always been a key issue being discussed In the tradi-tional method researchers in [16 17] have proposed severalsaturation functions to replace the sign functions to builda boundary layer to alleviate the chattering or to use fuzzylogic to displace the high-frequency switching term To ourknowledge none of these approaches has proven that therefined controller still retains their robustness against theuncertainties and disturbances In this work in order toalleviate the chattering phenomenon we do not directly usethe third-order controller but introduce the virtual control119906119894= 120575119885to perform the actual control
120575119885
= int 120575119885dt = int 119906
119894dt (47)
After the relative degree is increased to the fourth order weget the following expressions
120590(4)
= ℎlowast
(119905 119909) + 119892lowast
(119905 119909) 119906119894= ℎlowast
(119905 119909) + 119892 (119905 119909) 119906119894
ℎlowast
(119905 119909) = ℎ (119905 119909) + 119892 (119905 119909) 120575119885
119892 (119905 119909)
=119898120575119885
1199111198762
1198782
119897119862120572
119884
119869119885
1198981199032[( 119902 minus 120579
119872) 119903 sin 120578
119872+ 119903 cos 120578
119872]
(48)
Even though the expression of ℎ(119905 119909) is rather compli-cated it is still the function of 119902 119902 119902 119886
119872 119886119879 and 119886
119879 therefore
similar to ℎ(119905 119909) it has its boundary except themomentwhenthe missile hits on its target For the same reason 119892(119905 119909)
and the rudder deflection 120575119885also have their boundaries
Therefore we get the condition that |ℎlowast
(119905 119909)| le 119862 (119862 gt 0)Because 120575
119885is obtained through the derivation of 120575
119885 119892lowast(119905 119909)
is the same as 119892(119905 119909) thus 0 lt 119870119898
lt 119892lowast
(119905 119909) lt 119870119872
issatisfied
According to the formula of the fourth-order controllergiven by Levant in [15] we give the following formulae forthe virtual control 119906
119894
119906119894= minus120573
Φ34
11987334
Φ34
=120590 + 3 [||
+ (|| + 05 |120590|34
)minus13 10038161003816100381610038161003816
+ 05 |120590|34 sign120590
10038161003816100381610038161003816]
minus12
sdot [
+ (|| + 05 |120590|34
)minus13 10038161003816100381610038161003816
+ 05 |120590|34 sign120590
10038161003816100381610038161003816]
11987334
=1003816100381610038161003816
120590
1003816100381610038161003816 + 3 [||
+ (|| + 05 |120590|34
)minus13 10038161003816100381610038161003816
+ 05 |120590|34 sign120590
10038161003816100381610038161003816]
minus12
sdot
1003816100381610038161003816100381610038161003816
+ (|| + 05 |120590|34
)minus13
( + 05 |120590|34 sign120590)
1003816100381610038161003816100381610038161003816
(49)
The integral of the virtual control 120575119885produces the actual
control command 120575119885 120590 in the controller is obtained with the
Arbitrary-Order Robust Exact Differentiator presented in thefollowing section
34 The Arbitrary-Order Robust Exact Differentiator Thequasi-continuous HOSM control method needs to use thethird derivative of the sliding manifold namely 119902
(4) Howto calculate or accurately estimate 119902
(4) is one of the keyproblems to be solved We use the Arbitrary-Order RobustExact Differentiator designed by Levant to differentiate theLOS rate 119902 thus obtaining 119902 119902 and 119902
(4)According to (44) and (45) the following condition is
valid1003816100381610038161003816
120590
1003816100381610038161003816 le 119862 + 120573119870119872
(50)
The Arbitrary-Order Robust Exact Differentiator can beconstructed in accordance with high order sliding modesdifferentiation and output feedback control in [18]
If a certain signal 119891(119905) is a function consisting of abounded Lebesgue-measurable noise with unknown base
Mathematical Problems in Engineering 7
signal 1198910(119905) whose 119903th derivative has a known Lipschitz
constant 119871 gt 0 then the 119899th-order differentiator is definedas follows
0
= V0
V0
= minus12058201198711(119899+1) 10038161003816100381610038161199110 minus 119891 (119905)
1003816100381610038161003816
119899(119899+1) sign (1199110
minus 119891 (119905))
+ 1199111
1
= V1
V1
= minus12058211198712(119899+1) 10038161003816100381610038161199111 minus V
0
1003816100381610038161003816
(119899minus1)119899 sign (1199111
minus V0) + 1199112
119899minus1
= V119899minus1
V119899minus1
= minus120582119899minus1
11987112 1003816100381610038161003816119911119899minus1 minus V
119899minus2
1003816100381610038161003816
12 sign (119911119899minus1
minus V119899minus2
)
+ 119911119899
119899
= minus120582119899119871 sign (119911
119899minus V119899minus1
)
(51)
and if 120582119894
gt 0 is sufficiently large the convergence is guaran-teed
To obtain the third-order derivative of 119902 we constructthe third-order sliding mode differentiator and estimate thederivative of 119902 for each order In view of differential precisionwe configure the following fifth-order differentiator SeeAppendix A for comparison
0
= V0
V0
= minus120582011987116 10038161003816100381610038161199110 minus
1199021003816100381610038161003816
56 sign (1199110
minus 119902) + 1199111
1
= V1
V1
= minus120582111987115 10038161003816100381610038161199111 minus V
0
1003816100381610038161003816
45 sign (1199111
minus V0) + 1199112
2
= V2
V2
= minus120582211987114 10038161003816100381610038161199112 minus V
1
1003816100381610038161003816
34 sign (1199112
minus V1) + 1199113
3
= V3
V3
= minus120582311987113 10038161003816100381610038161199113 minus V
2
1003816100381610038161003816
23 sign (1199113
minus V2) + 1199114
4
= V4
V4
= minus120582411987112 10038161003816100381610038161199114 minus V
3
1003816100381610038161003816
12 sign (1199114
minus V3) + 1199115
5
= V5
V5
= minus1205825119871 sign (119911
5minus V4)
(52)
where 1199113 1199112 1199111 and 119911
0are the estimations of 119902
(4) 119902 119902 and 119902
4 Baseline Separated Guidance andControl Method
To verify the homing performance of the integrated guidanceand control method we compare it with the separatedguidance and control methodThe guidance laws we used arethe proportional navigation (PN) guidance law for attackinga nonmaneuvering target and the optimal sliding modeguidance law for attacking a maneuvering target
41 The Proportional Navigation (PN) Guidance Law Theguidance law has a simple formula and excellent perfor-mances for nonmaneuvering target Its formula is as follows
119899119888
= minus119873 119902119881119872
119892 (53)
119899119888is the overload command 119873 is the effective navigation
ratio 119902 is the LOS rate 119881119872
is the speed of the missile 119892 isthe acceleration of the gravity The guidance law gives theoverload command of the missile according to the LOS rateand then the controller gives the rudder deflection commandaccording to the overload command
42 The Optimal Sliding Mode Guidance Law The optimalsliding mode guidance law (OSMG) is a novel practicalguidance law proposed by D Zhou He combines the optimalguidance lawwith the slidingmode guidance law and designsthe new sliding mode guidance law that not only is robustto maneuvering target but also has the merits of the optimalguidance law such as good dynamic performance and energyconservation Its formula is as follows
119899119888
= minus3100381610038161003816100381610038160
10038161003816100381610038161003816119902 + 120576
119902
10038161003816100381610038161199021003816100381610038161003816 + 120575
(54)
where 119899119888is the overload command
0is the approach
velocity of the missile and its target 119902 is their LOS rate 120576 =
const is the compensatory gain 119902(| 119902| + 120575) is for substitutingfor sign( 119902) and for smoothing 120575 is a small quantity whichcould adjust the chattering
43 Separated Guidance and Control Design For simulationand comparisonwe use the conventional three-loop overloadautopilot as the controller which gives the rudder deflectioncommand according to the feedback of the three loops ofoverload pseudo-angle of attack and pitch rate The blockdiagram is as shown in Figure 3
As the figure shows the inner loop has the feedback onangular velocity which improves the damping characteristicsof the missile airframe
According to the aerodynamic coefficient of the missilewith selected working points we set 119870
119868= 019 119870
120572= 3 and
119870120596
= minus025 and the controller can well track the overloadcommand the rise time of its step response is 046 secondsand its settling time is 083 secondsThe step responses of themissile to overload command and the Bode diagram for openloop are shown in Figure 4
8 Mathematical Problems in Engineering
KIS
120596Z 120572 nY
nC K120596K120572 nY+minus+minus+minus dynamicsAirframe
modelServo
Figure 3 The working principles for three-loop overload autopilot
Step response
Time (s)0 02 04 06 08 1 12
0
02
04
06
08
1
Rise time (s) 0463
Settling time (s) 0831
Bode diagram
Frequency (rads)
To output pointFrom input pointTo output pointFrom input point
Gain margin (dB) 175 At frequency (rads) 15
Phase margin (deg) 709 At frequency (rads) 294
Am
plitu
de
Mag
nitu
de (d
B)Ph
ase (
deg)
0
minus180
minus360
minus54010410310210110010minus1
100
0
minus100
minus200
minus300
Figure 4 The autopilot performance step response and Bode diagram
5 Simulation Results
To verify the high order slidingmode integrated guidance andcontrol (HOSM-IGC) method we compare it with the base-line separated guidance and control Numerical simulationsare designed in three typical engagement scenarios
AORED parameters are as follows the initial value 1205820
=
1205821
= 1205822
= 1205823
= 50 1199110
= 01 1199111
= 1199112
= 1199113
= 0 119871 = 1400 thesimulation step is 00001 seconds
51 Scenario 1 Nonmaneuvering Target In the first scenariothe nonmaneuvering target does uniform rectilinear motionand PN guidance law with three-loop autopilot is introducedfor a comparison with the HOSM-IGCThe initial conditionsare set as shown in Table 1
The motion equations of the target are as follows
119910119879
= 119881119879cos 120579119879
119879
= minus119881119879sin 120579119879
120579119879
=119886119879
119881119879
119886119879
= 0
(55)
The simulation results are shown in Figures 5 and 6The missilersquos flight trajectory and overload curve show
that within the first 3 sec the HOSM-IGC method spendsmuch energy (overload) on changing the initial LOSdirection
Table 1 Initial conditions
Variable Description Value(119883119872 119884119872) Missile initial position (0 km 18 km)
(119883119879 119884119879) Target initial position (60 km 20 km)
119881119872
Missile initial velocity 1200 (ms)119881119879
Target initial velocity 2000 (ms)120579119872
Missile initial flight path angle 0 (deg)120579119879
Target initial flight path angle 0 (deg)120573 Parameter of the controller 10 or 30119873 Navigation ratio of PN 3
of the missile After the boresight adjustment 120590 reaches thedesired sliding manifold and the missile does not need anymaneuver to hit the target This is because the target doesnot maneuver any more which means no disturbance isintroduced in the engagement dynamic so the state (or thesliding mode) of the missile will stay on the manifold Incontrast the overload command given by the PN guidancelaw increases fast as the relative distance decreases Toexamine the performances of the two guidance laws furtherwe increase the target speed and analyze the commandchanges
Therefore we set the target speeds 3000ms 4000msand carry out simulations The simulation result is shown inthe overload curve in Figure 7
Mathematical Problems in Engineering 9
175
180
185
190
195
200
205
TargetHOSM-IGCPN
Y(k
m)
0 10 20 30 40 50 60X (km)
Figure 5 Target and missile trajectories
0
5
10
15
20
25
0 5 10 15 20Time (s)
HOSM-IGCPN
Initial boresight adjustment
Sliding manifold reached
Miss
ile ac
cele
ratio
n (G
)
minus5
Figure 6 Missile acceleration profile
It is evident that the speed of divergence of the overloadcommand given by the PN guidance law increases withthe target speed Specifically when the target speed reaches4000ms the commanded overload is almost 30 g which isobviously not ideal for the attack of a nonmaneuvering targetbut with the HOSM-IGC method the missile adjusts itsboresight very quickly then maintains it around 0 g and fliesto its target in the rectilinear ballistic trajectory not affectedby the increases of target speed still accomplishing the high-precision hit-on collision
Figure 8 shows that although the HOSM-IGC methodachieves a more effective overload command on the otherside it sees some chattering when the sliding mode reaches
0
5
10
15
20
25
30
0 5 10 15Time (s)
Miss
ile ac
cele
ratio
n (G
)
minus5
HOSM-IGC (Vt = minus3000)HOSM-IGC (Vt = minus4000)
PN (Vt = minus3000)PN (Vt = minus4000)
Figure 7 Missile acceleration profile
0
1
2
0 5 10 15 20Time (s)
4 45 5
0
02
04
minus02
minus04
Actu
ator
defl
ectio
n (d
eg)
minus1
minus2
minus3
minus4
HOSM-IGC (120573 = 30)
Figure 8 Actuator deflection
the desired manifold the commanded rudder deflectionangle chatters for around 045 deg at about 15Hz which iskind of harmful to the system To reduce the chatteringwe adjust the controllerrsquos parameter 120573 = 10 and carry outsimulations again Figure 9 shows that after 120573 decreasesthe command of rudder deflection angle converges slower(for about 7 sec) however the chattering weakens obviouslyits magnitude being only about 015 degrees The smallerchattering well enhances convergence precision eventuallyreducing the target missing The reason that minor 120573 resultsin an alleviative chattering can be seen from (47) Figure 10shows the target and missile trajectories
Table 2 compares the average miss distance of 50 simula-tions under the conditions discussed above the comparison
10 Mathematical Problems in Engineering
Table 2 Average miss distance of 50 simulations
Target speed PN HOSM-IGC(120573 = 30)
HOSM-IGC(120573 = 10)
119881119905= 2000ms 115m 086 073m
119881119905= 3000ms 261m 117 106
119881119905= 4000ms 542m 156 134
0
1
2
0 5 10 15 20Time (s)
72 74 76 78
0
02
minus02
minus04
Actu
ator
defl
ectio
n (d
eg)
minus1
minus2
minus3
minus4
HOSM-IGC (120573 = 10)HOSM-IGC (120573 = 30)
Figure 9 Actuator deflection
results show that when the missile intercepts a nonmaneu-vering target the hit precision of the HOSM-IGC methodis apparently higher than that of the PN guidance law andthat the low-gain HOSM-IGC method can effectively reducethe chattering magnitude thus enhancing the interceptionprecision
52 Scenario 2 Step Maneuvering Target At the first stagethe target flies at uniform speed and in a rectilinear way after10 seconds it maneuvers at the normal acceleration of 5 g Inthis scenario OSMG is introduced for a comparison with theHOSM-IGC
119910119879
= 119881119879cos 120579119879
119879
= minus119881119879sin 120579119879
120579119879
=119886119879
119881119879
119886119879
=
0 g 119905 lt 10 sec
5 g 119905 gt 10 sec
(56)
The initial simulation conditions are given in Table 3The overload curve in Figure 11 shows dearly that both
types of guidance laws can track the maneuvering targetDuring 0 to 10 seconds the target flies at uniform speed andin a rectilinear way and the missile converges its overload to
Table 3 Initial conditions
Variable Description Value(119883119872 119884119872) Missile initial position (0 km 22 km)
(119883119879 119884119879) Target initial position (60 km 20 km)
119881119872
Missile initial velocity 1200 (ms)119881119879
Target initial velocity 2000 (ms)120579119872
Missile initial flight path angle 0 (deg)120579119879
Target initial flight path angle 0 (deg)120576 Compensatory gain 3120575 Saturation parameter 0001
15
17
19
21
23
25
TargetHOSM-IGCOSMG
Y(k
m)
0 20 40 60X (km)
Figure 10 Target and missile trajectories
0 g and flies to its target after 10 seconds the target beginsto maneuver by 5 g Both types of guidance law respondby rapidly increasing the overload adjusting attitude andmaking the missilersquos boresight aim at its target
We can see that when both types of guidance law tracktheir targets the convergence speed of OSMG is almostthe same as the HOSM-IGC method But the HOSM-IGCmethod has higher convergence precision and needs loweroverload at the end phase
We can also see that after the target maneuvers if themissile is given enough time to track the targetrsquos maneuvernamely let the missilersquos overload command converge to theoverload of the target theremay not be largemiss distance Inother words for a certain period of time before the collisionthe targetmaneuver (it means only a limitedmaneuver whichdoes not include the condition that the maneuvering of thetarget for a long time may cause a change of the geometricalrelations between the missile and its target) has a small effecton both types of guidance law
But if themaneuver occurs rather late namely within oneto three seconds before collision when the overload com-mand of guidance law is not yet converged the approaching
Mathematical Problems in Engineering 11
0
5
10
0 5 10 15 20Time (s)
TargetOSMGHOSM-IGC
Miss
ile an
d ta
rget
acce
lera
tion
(G)
minus5
minus10
Figure 11 Missile and target acceleration profile
0
10
20
30
0 5 10 15 20Time (s)
tgo = 1 stgo = 2 stgo = 3 s
tgo = 3 s (diverge)
tgo = 2 s (diverge)
tgo = 1 s (diverge)minus10
minus20
minus30
Miss
ile ac
cele
ratio
n (G
)
HOSM-IGC (120573 = 10)
Figure 12 Missile acceleration profile
collision increases themiss distanceTherefore increasing theparameter 120573 can remarkably increase the convergence speedand effectively enhance guidance precision As Figure 12shows when 120573 = 10 the overload at the end phase convergesslowly even if the target maneuvers three seconds beforecollision the missilersquos overload still has no time to convergebeing unable to track themaneuvering target Figure 13 showsthat when 120573 increases to 30 and tgo = 3 seconds theoverload can converge to about 5 g but when tgo = 2 sec-onds the overload may continue to increase indicating that
0
10
20
30
0 5 10 15 20Time (s)
tgo = 1 stgo = 2 stgo = 3 s
tgo = 1 s (diverge)
tgo = 2 s (diverge)
tgo = 3 s (converge)
minus10
minus20
minus30
Miss
ile ac
cele
ratio
n (G
)
120573 = 30)HOSM-IGC (
Figure 13 Missile acceleration profile
0
10
20
30
0 5 10 15 20Time (s)
tgo = 1 s (diverge)
tgo = 2 s (converge)
tgo = 3 s (converge)
tgo = 1 stgo = 2 stgo = 3 s
minus10
minus20
minus30
Miss
ile ac
cele
ratio
n (G
)
HOSM-IGC (120573 = 50)
Figure 14 Missile acceleration profile
the convergence still needs to be quickened thus continuingto rise 120573 to 50 Figure 14 shows that the HOSM-IGC methodcan track the target that maneuvers when tgo = 2 secondsbut it causes the divergence of overload and the increase ofmiss distance if the target maneuvers when tgo = 1 secondFigure 15 gives the overload curve of the OSMG guidance lawand shows that the maneuver of the target before collisionmay cause the large-scale oscillation of the missilersquos overloadwhich may diverge to a large numerical value when thecollision occurs in the end
12 Mathematical Problems in Engineering
Table 4 Average miss distances of 50 simulations
Targetmaneuveringtiming
HOSM-IGC120573 = 30
HOSM-IGC120573 = 40
HOSM-IGC120573 = 50
OSMG
tgo = 1second 43539 3224 28936 36116
tgo = 2seconds 25424 18665 09322 35534
tgo = 3seconds 08124 08265 07538 11959
Time (s)0 5 10 15 20
0
10
20
30OSMG
tgo = 1 stgo = 2 stgo = 3 s
minus10
minus20
minus30
Miss
ile ac
cele
ratio
n (G
)
Figure 15 Missile acceleration profile
The analysis in Figures 12 13 14 and 15 shows that theoverload of the missile converges faster and its miss distanceis smaller with increasing 120573 To verify this finding we carryout 50 times Monte Carlo simulations in which the positionand speed of the target have 1 of random difference
The average miss distances are shown in Table 4 andFigure 16 Clearly the timing of the targetrsquos step maneuverdramatically affects the final interception precision morespecifically given a shorter reaction time for the guidanceand control system the missile seems more likely to missthe target To the OSMG guidance law in all three scenarioshardly does it show any advantages against theHOSM-IGC Itcan also be seen that with the increasing of the120573 theHOSM-IGC system responds even faster which leads to an obviousdecrease of the average miss distance The effect of 120573 on theresponse of the HOSM-IGC system is a valuable guidelinewhen implementing the proposed method into practice
53 Scenario 3 Weaving Target In this scenario the targetmaneuvers by 119886
119879= 40sin(1205871199052) OSMG with three-loop
Table 5 Initial conditions
Variable Description Value(119883119872 119884119872) Missile initial position (0 km 175 km)
(119883119879 119884119879) Target initial position (60 km 195 km)
119881119872
Missile initial velocity 1200 (ms)119881119879
Target initial velocity 2000 (ms)120579119872
Missile initial flight path angle 0 (deg)120579119879
Target initial flight path angle 0 (deg)120576 Compensatory gain 3120575 Saturation parameter 0001
autopilot is introduced for comparison The motion equa-tions of the target are as follows
119910119879
= 119881119879cos 120579119879
119879
= minus119881119879sin 120579119879
120579119879
=119886119879
119881119879
119886119879
= 40 sin(120587119905
2)
(57)
The initial simulation conditions are given in Table 5The missilersquos trajectory under the two guidance and
control methods given in Figure 17 does not see muchdifference
However the overload curve given in Figure 18 showsthat after the missile completes its initial attitude adjustmentwith the HOSM-IGC method it can almost ideally track itsmaneuvering target by contrast with the OSMG methodthe missile seems to have the tendency to track its targetrsquosmaneuver but has larger tracking errors Besides with theOSMGmethod the missilersquos overload increases rapidly at theend of attack primarily because of the divergence of its LOSrate On the other hand with the HOSM-IGC method themissile has no divergence even at the end of attack ensuringa smaller target missing quantity
The actuator deflection curve in Figure 19 shows thatin order to provide a rather big normal overload for theend phase the OSMG method produces a rather big rudderdeflection command however it may increase the missilersquostarget missing quantity once its rudder deflection saturatesand themissile does not have enough overloads or the ruddercannot respond that fast
As shown in Figure 20 because of the dramatic changein overload command the response of the missilersquos autopilotto high-frequency command sees an obvious phase lag andamplitude value attenuation its actual overload cannot trackthe command ideally this is a main reason why the missdistance increases However the controller in the HOSM-IGC method gives its rudder deflection command directlyand there is no lagging or attenuation caused by the autopilotthus enhancing the guidance precision effectively Further-more the fast convergence of the high order sliding modemakes the missile rapidly track its maneuvering target withthemost reasonable rudder deflection command reducing itsoverload effectively
Mathematical Problems in Engineering 13
0
05
1
15
2
25
3
35
4
45
5
0 10 20 30 40 50Simulation times
OSMG
Miss
(m)
0
05
1
15
2
25
3
35
4
45
5
0 10 20 30 40 50Simulation times
Miss
(m)
0
05
1
15
2
25
3
35
4
45
5
0 10 20 30 40 50Simulation times
Miss
(m)
tgo = 3 stgo = 2 stgo = 1 s
0
05
1
15
2
25
3
35
4
45
5
0 10 20 30 40 50Simulation times
Miss
(m)
tgo = 3 stgo = 2 stgo = 1 s
HOSM-IGC (120573 = 40)
HOSM-IGC (120573 = 50)
HOSM-IGC (120573 = 30)
Figure 16 Miss distances of HOSM-IGC (120573 = 30 40 50) and OSMG with target maneuvers at different time-to-go (tgo = 1 second 2seconds and 3 seconds)
The average miss distance of 50 simulations under theconditions is 073m for HOSM-IGC and 186m for OSMGWe can see that the HOSM-IGC method not only doesprovide a more reasonable actuator deflection command butalso achieves a higher interception precision
6 Conclusions
This paper proposes an LOS feedback integrated guidanceand control method using quasi-continuous high order
sliding mode guidance and control method With the fastand precise convergence of the quasi-continuous HOSMmethod the HOSM-IGCmethod performsmuch better thanthe traditional separated guidance and control method withless acceleration effort and less miss distance in all thethree simulation scenarios of nonmaneuvering target stepmaneuvering target and weaving target In addition the ideaof virtual control largely alleviates the chattering withoutany sacrifice of robustness As a result of the alleviationof the chattering the control input command 120575
119885becomes
14 Mathematical Problems in Engineering
175
180
185
190
195
200
TargetHOSM-IGCOSMG
Y(k
m)
0 20 40 60 0 20 40X (km)
Figure 17 The trajectories of the missile and its target
0
10
20
0 5 10 15 20Time (s)
TargetOSMGHOSM-IGC
Miss
ile an
d ta
rget
acce
lera
tion
(G)
minus10
minus20
Figure 18 Missile acceleration profile
smooth enough for the actuator to be implemented as aresult the interception accuracy is improved In designingthe quasi-continuous sliding mode controller the Arbitrary-Order Robust Exact Differentiator is introduced to estimatethe high order time derivative of the LOS rate 119902 which notonly ensures the guidance and control precision but alsosimplifies the calculation
0
10
20
30
0 5 10 15 20Time (s)
OSMGHOSM-IGC
minus10
minus20
minus30
minus40
Actu
ator
defl
ectio
n (d
eg)
Figure 19 Actuator deflection
0
50
100
150
200
15 16 17 18Time (s)
Commanded accelerationAchieved acceleration
Miss
ile ac
cele
ratio
n (G
)
minus50
Figure 20 Commanded acceleration and achieved acceleration
Appendices
A The Third-Order RobustExact Differentiator
The third-order robust exact differentiator built according tothe following parameters and equations is used to performthe simulation the initial value 120582
0= 1205821
= 1205822
= 1205823
= 50
Mathematical Problems in Engineering 15
0
0005
001
0015
002
0025
003
0035
004
0 02 04Time (s)
0
2
0 02 04Time (s)
0
100
200
300
0
100
200
300
400
0 02 04Time (s)
h(t)
z3
h(t) and z3
minus100
minus200
minus300
minus400
minus5000 02 04
Time (s)
g(t)
z2
g(t) and z2
minus100
minus200
minus300
minus400
f(t)
z1
f(t) and z1
minus2
minus4
minus6
minus8
minus10
minus12
minus14
F(t)
z0
F(t) and z0
minus0005
minus001
Figure 21 The outputs of the differentiator in the sampling interval of 0001 seconds
the initial values 1199110
= 01 1199111
= 1199112
= 1199113
= 0 119871 = 1400 thesimulation step is 0001 seconds Consider
0
= V0
V0
= minus120582011987114 10038161003816100381610038161199110 minus
1199021003816100381610038161003816
34 sign (1199110
minus 119902) + 1199111
1
= V1
V1
= minus120582111987113 10038161003816100381610038161199111 minus V
0
1003816100381610038161003816
23 sign (1199111
minus V0) + 1199112
2
= V2
V2
= minus120582211987112 10038161003816100381610038161199112 minus V
1
1003816100381610038161003816
12 sign (1199112
minus V1) + 1199113
3
= minus1205823119871 sign (119911
3minus V2)
(A1)
16 Mathematical Problems in Engineering
0
0005
001
0015
002
0025
003
0035
004
0 02 04Time (s)
0
2
0 02 04Time (s)
0
100
200
300
0 02 04Time (s)
0
200
400
0 02 04Time (s)
h(t) and z3
minus200
minus400
minus600
minus800
g(t) and z2
minus100
minus200
minus300
minus400
f(t) and z1
minus2
minus4
minus6
minus8
minus10
minus12
minus14
minus16
F(t) and z0
minus0005
minus001
h(t)
z3g(t)
z2
f(t)
z1F(t)
z0
Figure 22 The outputs of the differentiator in the sampling interval of 00001 seconds
If the designed signals are 119865(119905) = int 119891(119905)119889119905 119891(119905) =
int 119892(119905)119889119905 119892(119905) = int ℎ(119905)119889119905 and ℎ(119905) = sin(2120587119905) +
3cos(120587119905) then theoretically 119891(119905) is the first-order differ-ential of input signal 119892(119905) is the second-order differ-ential of input signal ℎ(119905) is the third-order differen-tial of input signal Through the simulation the differ-ential signal of actual output is compared with that of
theoretical output the comparison results are shown inFigure 21
Figure 21 shows that 1199110can effectively track the signal
119865(119905) but 1199112has obvious errors in tracking the second-order
differential 119892(119905) 1199113has greater errors in tracking the third-
order differential thus the overall differential effect is notideal
Mathematical Problems in Engineering 17
0 02 04
0
0005
001
0015
002
0025
003
0035
004
Time (s)0 02 04
0
01
02
03
04
05
Time (s)
0 02 04
0
05
1
15
2
Time (s)0 02 04
0
05
1
15
2
25
3
35
4
Time (s)
h(t) and z3g(t) and z2
minus04
minus05
f(t) and z1
minus2
minus15
minus1
minus05
minus1
minus05
minus01
minus02
minus03
F(t) and z0
minus0005
minus001
h(t)
z3g(t)
z2
f(t)
z1F(t)
z0
Figure 23 The outputs of the differentiator in the sampling interval of 00001 seconds
There are two causes for the errors in a usual differentia-tor the setting of sampling interval and the setting of numberof orders of the differentiator The following theorem applieshere
Theorem A1 (see [19]) If the sampling interval 120591 gt 0 theaccuracy of the controller is expressed as follows
10038161003816100381610038161003816120590(119894)
10038161003816100381610038161003816le 120583120591119903minus119894
119894 = 0 119903 minus 1 (A2)
That is to say the precision (understood as the differentialprecision of a differentiator) of the 119894th order control is in directproportion to the power of 119903 minus 119894 times of sampling interval120591 Therefore the decrease of iteration step can enhance thedifferential tracking precision
Under the conditions that the sampling interval is 10 timessmaller and that the setting is 00001 seconds the simulationresults are shown in Figure 22
18 Mathematical Problems in Engineering
Table 6 Tracking errors of the AORED with different orders andsampling interval
Orders of AORED Third order Fifth orderSample period 120591 = 0001 120591 = 00001 120591 = 00001
First-order differential error 001 Negligible NegligibleSecond-order differential error 15 002 NegligibleThird-order differential error 70 25 00005
As Figure 22 shows the signal tracking is rather idealexcept a peak in the beginning the tracking error in the first-order differential tracking can be neglected 119911
2has an obviously
smaller tracking error in the second-order differential 119892(119905)
tracking in the beginning 1199113has a rather sharp peak in the
third-order differential tracking but converges to a relativelyhigh precision after 01 seconds As we can see the differentialtracking errors for each number of orders are greatly reducedbut the third-order differential tracking error is still rather bigIn the following we will reduce the tracking errors by increasingthe number of orders of the differentiator
B The Fifth-Order Robust Exact Differentiator
The settings of the fifth-order differentiator are given asfollows
0
= V0
V0
= minus120582011987116 10038161003816100381610038161199110 minus
1199021003816100381610038161003816
56 sign (1199110
minus 119902) + 1199111
1
= V1
V1
= minus120582111987115 10038161003816100381610038161199111 minus V
0
1003816100381610038161003816
45 sign (1199111
minus V0) + 1199112
2
= V2
V2
= minus120582211987114 10038161003816100381610038161199112 minus V
1
1003816100381610038161003816
34 sign (1199112
minus V1) + 1199113
3
= V3
V3
= minus120582311987113 10038161003816100381610038161199113 minus V
2
1003816100381610038161003816
23 sign (1199113
minus V2) + 1199114
4
= V4
V4
= minus120582411987112 10038161003816100381610038161199114 minus V
3
1003816100381610038161003816
12 sign (1199114
minus V3) + 1199115
5
= V5
V5
= minus1205825119871 sign (119911
5minus V4)
(B1)
The initial value 1199110
= 01 1205820
= 1205821
= 1205822
= 1205823
= 1205824
=
1205825
= 50 the initial values 1199111 1199112 1199113 1199114 1199115
= 0 119871 = 1400the simulation step length is 00001 seconds The simulationresults are shown in Figure 23
As the figure shows the fifth-order differentiator hasa rather ideal tracking effect on the third-order differen-tial tracking the tracking error being only about 00005(see Table 6)
To sum up the Arbitrary-Order Robust Exact Differen-tiator can estimate rather ideal differential signals by selecting
Table 7
Length 365m119871 ref 365m119883119866
177m119878ref 0026m2
Diameter 0178mMass 1016 Kg119868119885
1063 Kgsdotm2
an appropriate sampling interval and using the differentiatorwith a relatively high number of orders
C Physical and Geometric Characteristics
See Table 7
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] P Zarchan Tactical and Strategic Missile Guidance 2002[2] D Zhou CMu andWXu ldquoAdaptive sliding-mode guidance of
a homing missilerdquo Journal of Guidance Control and Dynamicsvol 22 no 4 pp 589ndash594 1999
[3] G Zeng and M Hu ldquoFinite-time control for electromagneticsatellite formationsrdquo Acta Astronautica vol 74 pp 120ndash1302012
[4] S R Kumar S Rao and D Ghose ldquoNonsingular terminalsliding mode guidance with impact angle constraintsrdquo Journalof Guidance Control and Dynamics vol 37 no 4 pp 1114ndash11302014
[5] Y Liang C Chen D Liaw Y Feng C Cheng and CChen ldquoRobust guidance law via integral-sliding-mode schemerdquoJournal of Guidance Control and Dynamics vol 37 no 3 pp1038ndash1042 2014
[6] A V Savkin P N Pathirana and F Faruqi ldquoProblem ofprecision missile guidance LQR and 119867
infincontrol frameworksrdquo
IEEE Transactions on Aerospace and Electronic Systems vol 39no 3 pp 901ndash910 2003
[7] H Y Chen andC-C Yang ldquoNonlinear1198672119867infinguidance design
for homing missilesrdquo in Proceedings of the AIAA GuidanceNavigation and Control Conference and Exhibit vol 77 2001
[8] P K Menon G D Sweriduk and E J Ohlmeyer ldquoOptimalfixed-interval integrated guidance-control laws for hit-to-killmissilesrdquo in Proceedings of the AIAA Guidance Navigation andControl Conference and Exhibit AIAA 2003-5579 Austin TexUSA August 2003
[9] S S Vaddi P K Menon and E J Ohlmeyer ldquoNumericalSDRE approach for missile integrated guidance-controlrdquo inProceedings of the AIAA Guidance Navigation and ControlConference and Exhibit AIAA 2007-6672 Hilton Head SCUSA August 2007
[10] M Xin S N Balakrishnan and E J Ohlmeyer ldquoIntegratedguidance and control of missiles with Theta-D methodrdquo IEEE
Mathematical Problems in Engineering 19
Transactions on Control Systems Technology vol 14 no 6 pp981ndash992 2006
[11] T Shima M Idan and O M Golan ldquoSliding-mode controlfor integrated missile autopilot guidancerdquo Journal of GuidanceControl and Dynamics vol 29 no 2 pp 250ndash260 2006
[12] Y B Shtessel and C H Tournes ldquoIntegrated higher-ordersliding mode guidance and autopilot for dual control missilesrdquoJournal of Guidance Control and Dynamics vol 32 no 1 pp79ndash94 2009
[13] F Y Dong H M Lei and C J Zhou ldquoIntegrated robust highorder sliding mode guidance and control for missilesrdquo ActaAeronautica et Astronautica Sinica vol 34 no 9 pp 2212ndash22182013
[14] H Mingzhe and D Guangren ldquoIntegrated guidance and con-trol of homing missiles against ground fixed targetsrdquo ChineseJournal of Aeronautics vol 21 no 2 pp 162ndash168 2008
[15] A Levant ldquoHomogeneity approach to high-order sliding modedesignrdquo Automatica vol 41 no 5 pp 823ndash830 2005
[16] Y-W Liang C Chen D Liaw Y Feng C Cheng and CChen ldquoRobust guidance law via integral-sliding-mode schemerdquoJournal of Guidance Control and Dynamics vol 37 no 3 pp1038ndash1042 2014
[17] L Wang Y Sheng and X Liu ldquoContinuous time-varying slid-ingmode based attitude control for reentry vehiclerdquo Proceedingsof the Institution of Mechanical Engineers Part G Journal ofAerospace Engineering 2014
[18] A Levant ldquoHigher-order sliding modes differentiation andoutput-feedback controlrdquo International Journal of Control vol76 no 9-10 pp 924ndash941 2003
[19] A Levant ldquoQuasi-continuous high-order sliding-mode con-trollersrdquo IEEE Transactions on Automatic Control vol 50 no11 pp 1812ndash1816 2005
Submit your manuscripts athttpwwwhindawicom
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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
4 Mathematical Problems in Engineering
Then (1) and (2) can be reformed as below
119889119881119872
119889119905= 0
119889120579119872
119889119905=
119884
119898119881119872
(11)
Denote that 119886119872119873
= 119884119898 119884 is the normal force 119886119872119873
119886119879119873
are the normal acceleration of the missile and targetrespectively and then
120579119872
=119886119872119873
119881119872
120579119879
=119886119879119873
119881119879
(12)
The normal force 119884 and the pitch moment 119872119885acting on
the missile are usually expressed respectively as follows
119884 = 119862120572
119884120572119876119878ref + 119862
120575119885
119884120575119885119876119878ref (13)
119872119885
= 119898120575119885
119885120575119885119876119878ref119897 + 119898
120572
119885120572119876119878ref119897 (14)
where119862120572
119884is the coefficient of normal force caused by the angle
of attack 119862120575119885
119884is the coefficient of normal force caused by the
rudder deflection angle 120575119885 119876 is the dynamic pressure 119878ref is
the reference area 119897 is the reference length But the normalforce produced by the rudder deflection angle 120575
119885is orders of
magnitude smaller than that produced by the angle of attackso (13) is simplified as
119884 = 119862120572
119884120572119876119878ref (15)
Then
119886119872119873
=119884
119898=
119862120572
119884120572119876119878ref
119898 (16)
Then the dynamic equations can be simplified as
119889120579119872
119889119905=
119862120572
119884120572119876119878ref
119898119881119872
(17)
119889120596119885
119889119905=
119898120575119885
119885120575119885
119876119878ref119897 + 119898120572
119885120572119876119878ref119897
119869119885
(18)
119889120599
119889119905= 120596119885 (19)
120572 = 120599 minus 120579119872
(20)
23 The Integrated Guidance and Control Model Follow-ing the above derivation we select the state variables as
( 119902 120579119872
120572 120596119885)119879 and obtain the following nonlinear integrated
guidance and control model
119902 = minus2119903
119903sdot 119902 minus
119862120572
119884119876119878ref
119898119903cos (120579
119872minus 119902) sdot 120572
+119886119879119873
119903cos (120579
119879+ 119902)
119903 119902 = minus119881119872sin (120579119872
minus 119902) + 119881119879sin (120579119879
+ 119902)
120579119872
=1
119898119881119872
119862120572
119884119876119878ref sdot 120572
= 120596119885
minus1
119898119881119872
119862120572
119884119876119878ref sdot 120572
119885
=119898120575119885
119885119876119878ref119897
119869119885
sdot 120575119885
+119898120572
119885119876119878ref119897
119869119885
120572
(21)
24 The Relative Degree of Control Input To obtain therelative degree of the control input 120575 of the integratedguidance and control method we keep on deriving the LOSangular velocity 119902 until the explicit formula of its derivativeof a certain order contains the control input
The derivation of (10) produces119902
=1
119903[minus2 119903 119902 minus 119886
119872119873cos (120579
119872minus 119902) + 119886
119879119873cos (120579
119879+ 119902)]
(22)
The LOS angular velocity is expressed as the derivativeof the first order and does not contain the control input 120575
explicitly The continuous derivation of the above equationproduces
119902 =
1
119903minus2 119903 119902 minus 3 119903 119902 minus [119886
119872119873( 119902 minus 120579
119872) sin (120579
119872minus 119902)
+ 119886119872119873
cos (120579119872
minus 119902)] minus [119886119879119873
( 119902 + 120579119879) sin (120579
119879+ 119902)
minus 119886119879119873
cos (120579119879
+ 119902)]
(23)
where the LOS angular rate is expressed as the differentiatingof the second order and 119886
119872can be expressed as
119886119872119873
=119862120572
119884119876119878ref
119898 =
119862120572
119884119876119878ref
119898( 120599 minus 120579
119872)
=119862120572
119884119876119878ref
119898(120596119885
minus 120579119872
)
(24)
Although the control volume 120575119885does not appear in the
derivative of the second order (18) shows that 119885contains
120575119885 We continue to derive (23) and substitute 119886
119872as follows
119886119872119873
=119862120572
119884119876119878ref
119898(119885
minus119886119872
119881119872
)
=119862120572
119884119876119878ref
119898
119898120575119885
119885119876119878ref119897ref
119869119885
sdot 120575119885
+119862120572
119884119876119878ref
119898(
119898120572
119885119876119878ref119897ref
119869119885
120572 minus119886119872
119881119872
)
(25)
Mathematical Problems in Engineering 5
Thus
119902(4)
= 119891120575119885
sdot 120575119885
+1
119903(1198911
+ 1198912
+ 1198913
+ 1198914
+ 1198915
+ 1198916) (26)
where
119891120575119885
= minus1
119903
119862120572
119884119876119878ref cos 120578
119872
119898
119898120575119885
119885119876119878ref119897ref
119869119885
(27)
1198911
= (120596119885
minus 120579119872
) (minus2 120579119886119872119873
120572sin 120578119872
+ 119902119886119872119873
120572sin 120578119872
+ 120579119886119872119873
1205722cos 120578119872
) minus119886119872119873
cos 120578119872
120572
119898120572
119885120572119876119878ref119897ref
119869119885
(28)
1198912
= 120579119872
(( 120579119872
minus 119902) 119886119872119873
cos 120578119872
minus (120596119885
minus 120579119872
)119886119872119873
120572sin 120578119872
)
(29)
1198913
= 119902 (minus 120579119872
119886119872119873
cos 120578119872
+ 119902 (119886119872119873
cos 120578119872
minus 119886119879119873
cos 120578119879) minus 120579119879119886119879119873
cos 120578119879
+119886119872119873
120572(120596119885
minus 120579119872
) sin 120578119872
minus 119886119879sin 120578119879)
(30)
1198914
= 120579119879
(minus 119902119886119879cos 120578119879
minus 120579119879119886119879cos 120578119879
minus 119886119879sin 120578119879) (31)
1198915
= 119886119879
(minus 119902 sin 120578119879
minus 2 120579119879sin 120578119879) + 119886119879cos 120578119879 (32)
1198916
= 119902 (119886119872119873
sin 120578119872
minus 119886119879sin 120578119879
minus 2 119903) minus 4 119903 sdot119902 minus 3 119903 sdot 119902
minus 2119903 sdot 119902
(33)
120578119872
= 120579119872
minus 119902
120578119879
= 120579119879
+ 119902
(34)
In (26) it can be seen that the control input 120575119885appears
expressly in the third-order derivative of the control output119902 Therefore the relative degree of the control input 120575
119885is 3
3 The Quasi-Continuous High Order SlidingMode Controller
31 Sliding Mode Manifold Design To design the HOSMcontroller a sliding manifold must be chosen first In thisdesign we try to make the LOS rate converge to zero ora small neighbor domain near zero thus ensuring that themissile approaches its target in a quasi-parallel way whichwill lead to a minimal overload requirement So the slidingmanifold is chosen as follows
120590 = 119902 (35)
From the above discussion in Section 2 we know thatthe control input in relation to control output 119902 namelythe relative degree of sliding mode manifold 120590 is 3 So thefollowing design will be about a third-order sliding modecontroller
32 Design of the Quasi-Continuous HOSM Controller First(26) can be expressed as follows
120590 = ℎ (119905 119909) + 119892 (119905 119909) 119906 (36)
where ℎ(119905 119909) 119892(119905 119909) and 119906 are expressed as follows
ℎ (119905 119909) =1
119903(1198911
+ 1198912
+ 1198913
+ 1198914
+ 1198915
+ 1198916)
119892 (119905 119909) = 119891120575119885
119906 = 120575119885
(37)
According to the quasi-continuous high order slidingmode control method proposed by Levant in [15] the slidingmode manifold whose relative degree is 3 should be designedin the following form where 120573 is a control gain term
119906 = minus120573
+ 2 (|| + |120590|23
)minus12
( + |120590|23 sign120590)
|| + 2 (|| + |120590|23
)12
(38)
The conditions under which the LOS angular velocitymay converge are as follows
0 lt 119870119898
le 119892 (119905 119909) le 119870119872
|ℎ (119905 119909)| le 119862
(39)
where119870119898119870119872 and119862 are all larger than zeroThis is a proven
theorem by Levant in [15]The system we discussed meets the above requirements
and the proof is as followsEquation (27) shows the following
119892 (119905 119909) = 119891120575119885
= minus1
119903
119862120572
119884119876119878ref
119898
119898120575119885
119911119876119878ref119897ref
119869119885
cos (120579119872
minus 119902)
(40)
The dynamic pressure 119876 is 119876 = 1205881198812
1198722 where 120588 =
008803Kgm3 (altitude = 20Km) is the air density and119881119872
=
2000ms is the speed of the missile so 119876 is always positive119878ref = 026m2 and 119897ref = 365m denote the reference area
and the reference length of the missile they are both positiveconstant
119898 = 100Kg denotes the missile mass 119869119885
= 106m2 Kgdenotes the rotational inertia
119903 is the relative distance it is always a positive number119862120572
119884is the lift coefficient caused by the angle of attack it
varies from 018 to 037 and it is always a positive number119898120575119911
119911is the moment coefficient caused by the actuator
deflection In the normal layout (actuator lays behind thecenter of gravity) 119898
120575119911
119911is always negative
Meanwhile consider that the missile under guidance andcontrol is unlikely to fly away from its target namely the anglebetween themissilersquos velocity and its LOS direction cannot belarger than 90∘ then
1003816100381610038161003816120579119872 minus 1199021003816100381610038161003816 lt
120587
2
997904rArr cos (120579119872
minus 119902) gt 0
(41)
6 Mathematical Problems in Engineering
Summing up the above conditions then we can get
119892 (119905 119909) gt 0 (42)
In other words there is a positive real number 119870119898existing
that could satisfy the following condition
0 lt 119870119898
lt 119892 (119905 119909) (43)
Before the missile hits on the target the term will be positiveand limited then we can get
0 lt 119870119898
lt 119892 (119905 119909) lt 119870119872
(44)
With (37) then
ℎ (119905 119909) =1
119903(1198911
+ 1198912
+ 1198913
+ 1198914
+ 1198915
+ 1198916) (45)
In the practice sense the changes in both the LOS rateand the acceleration of the missile and the acceleration of thetarget are limited and continuous So the following variables119902 119902
119902 119886119872 119886119879 and 119886
119879are all bounded However because
ℎ(119905 119909) contains the item 1119903 when the relative distancebetween the missile and its target is zero the boundaryof ℎ(119905 119909) is not guaranteed In [15] Levant only requiresthat condition (39) should be locally valid not requiringthat it should be globally valid Therefore the integratedguidance and controlmethod is applicable here So the abovementioned condition is satisfied with a positive number 119862
|ℎ (119905 119909)| le 119862 (46)
33TheVirtual ControlDesign Whenusing the slidingmodecontrolmethod the avoidance of the chattering phenomenonhas always been a key issue being discussed In the tradi-tional method researchers in [16 17] have proposed severalsaturation functions to replace the sign functions to builda boundary layer to alleviate the chattering or to use fuzzylogic to displace the high-frequency switching term To ourknowledge none of these approaches has proven that therefined controller still retains their robustness against theuncertainties and disturbances In this work in order toalleviate the chattering phenomenon we do not directly usethe third-order controller but introduce the virtual control119906119894= 120575119885to perform the actual control
120575119885
= int 120575119885dt = int 119906
119894dt (47)
After the relative degree is increased to the fourth order weget the following expressions
120590(4)
= ℎlowast
(119905 119909) + 119892lowast
(119905 119909) 119906119894= ℎlowast
(119905 119909) + 119892 (119905 119909) 119906119894
ℎlowast
(119905 119909) = ℎ (119905 119909) + 119892 (119905 119909) 120575119885
119892 (119905 119909)
=119898120575119885
1199111198762
1198782
119897119862120572
119884
119869119885
1198981199032[( 119902 minus 120579
119872) 119903 sin 120578
119872+ 119903 cos 120578
119872]
(48)
Even though the expression of ℎ(119905 119909) is rather compli-cated it is still the function of 119902 119902 119902 119886
119872 119886119879 and 119886
119879 therefore
similar to ℎ(119905 119909) it has its boundary except themomentwhenthe missile hits on its target For the same reason 119892(119905 119909)
and the rudder deflection 120575119885also have their boundaries
Therefore we get the condition that |ℎlowast
(119905 119909)| le 119862 (119862 gt 0)Because 120575
119885is obtained through the derivation of 120575
119885 119892lowast(119905 119909)
is the same as 119892(119905 119909) thus 0 lt 119870119898
lt 119892lowast
(119905 119909) lt 119870119872
issatisfied
According to the formula of the fourth-order controllergiven by Levant in [15] we give the following formulae forthe virtual control 119906
119894
119906119894= minus120573
Φ34
11987334
Φ34
=120590 + 3 [||
+ (|| + 05 |120590|34
)minus13 10038161003816100381610038161003816
+ 05 |120590|34 sign120590
10038161003816100381610038161003816]
minus12
sdot [
+ (|| + 05 |120590|34
)minus13 10038161003816100381610038161003816
+ 05 |120590|34 sign120590
10038161003816100381610038161003816]
11987334
=1003816100381610038161003816
120590
1003816100381610038161003816 + 3 [||
+ (|| + 05 |120590|34
)minus13 10038161003816100381610038161003816
+ 05 |120590|34 sign120590
10038161003816100381610038161003816]
minus12
sdot
1003816100381610038161003816100381610038161003816
+ (|| + 05 |120590|34
)minus13
( + 05 |120590|34 sign120590)
1003816100381610038161003816100381610038161003816
(49)
The integral of the virtual control 120575119885produces the actual
control command 120575119885 120590 in the controller is obtained with the
Arbitrary-Order Robust Exact Differentiator presented in thefollowing section
34 The Arbitrary-Order Robust Exact Differentiator Thequasi-continuous HOSM control method needs to use thethird derivative of the sliding manifold namely 119902
(4) Howto calculate or accurately estimate 119902
(4) is one of the keyproblems to be solved We use the Arbitrary-Order RobustExact Differentiator designed by Levant to differentiate theLOS rate 119902 thus obtaining 119902 119902 and 119902
(4)According to (44) and (45) the following condition is
valid1003816100381610038161003816
120590
1003816100381610038161003816 le 119862 + 120573119870119872
(50)
The Arbitrary-Order Robust Exact Differentiator can beconstructed in accordance with high order sliding modesdifferentiation and output feedback control in [18]
If a certain signal 119891(119905) is a function consisting of abounded Lebesgue-measurable noise with unknown base
Mathematical Problems in Engineering 7
signal 1198910(119905) whose 119903th derivative has a known Lipschitz
constant 119871 gt 0 then the 119899th-order differentiator is definedas follows
0
= V0
V0
= minus12058201198711(119899+1) 10038161003816100381610038161199110 minus 119891 (119905)
1003816100381610038161003816
119899(119899+1) sign (1199110
minus 119891 (119905))
+ 1199111
1
= V1
V1
= minus12058211198712(119899+1) 10038161003816100381610038161199111 minus V
0
1003816100381610038161003816
(119899minus1)119899 sign (1199111
minus V0) + 1199112
119899minus1
= V119899minus1
V119899minus1
= minus120582119899minus1
11987112 1003816100381610038161003816119911119899minus1 minus V
119899minus2
1003816100381610038161003816
12 sign (119911119899minus1
minus V119899minus2
)
+ 119911119899
119899
= minus120582119899119871 sign (119911
119899minus V119899minus1
)
(51)
and if 120582119894
gt 0 is sufficiently large the convergence is guaran-teed
To obtain the third-order derivative of 119902 we constructthe third-order sliding mode differentiator and estimate thederivative of 119902 for each order In view of differential precisionwe configure the following fifth-order differentiator SeeAppendix A for comparison
0
= V0
V0
= minus120582011987116 10038161003816100381610038161199110 minus
1199021003816100381610038161003816
56 sign (1199110
minus 119902) + 1199111
1
= V1
V1
= minus120582111987115 10038161003816100381610038161199111 minus V
0
1003816100381610038161003816
45 sign (1199111
minus V0) + 1199112
2
= V2
V2
= minus120582211987114 10038161003816100381610038161199112 minus V
1
1003816100381610038161003816
34 sign (1199112
minus V1) + 1199113
3
= V3
V3
= minus120582311987113 10038161003816100381610038161199113 minus V
2
1003816100381610038161003816
23 sign (1199113
minus V2) + 1199114
4
= V4
V4
= minus120582411987112 10038161003816100381610038161199114 minus V
3
1003816100381610038161003816
12 sign (1199114
minus V3) + 1199115
5
= V5
V5
= minus1205825119871 sign (119911
5minus V4)
(52)
where 1199113 1199112 1199111 and 119911
0are the estimations of 119902
(4) 119902 119902 and 119902
4 Baseline Separated Guidance andControl Method
To verify the homing performance of the integrated guidanceand control method we compare it with the separatedguidance and control methodThe guidance laws we used arethe proportional navigation (PN) guidance law for attackinga nonmaneuvering target and the optimal sliding modeguidance law for attacking a maneuvering target
41 The Proportional Navigation (PN) Guidance Law Theguidance law has a simple formula and excellent perfor-mances for nonmaneuvering target Its formula is as follows
119899119888
= minus119873 119902119881119872
119892 (53)
119899119888is the overload command 119873 is the effective navigation
ratio 119902 is the LOS rate 119881119872
is the speed of the missile 119892 isthe acceleration of the gravity The guidance law gives theoverload command of the missile according to the LOS rateand then the controller gives the rudder deflection commandaccording to the overload command
42 The Optimal Sliding Mode Guidance Law The optimalsliding mode guidance law (OSMG) is a novel practicalguidance law proposed by D Zhou He combines the optimalguidance lawwith the slidingmode guidance law and designsthe new sliding mode guidance law that not only is robustto maneuvering target but also has the merits of the optimalguidance law such as good dynamic performance and energyconservation Its formula is as follows
119899119888
= minus3100381610038161003816100381610038160
10038161003816100381610038161003816119902 + 120576
119902
10038161003816100381610038161199021003816100381610038161003816 + 120575
(54)
where 119899119888is the overload command
0is the approach
velocity of the missile and its target 119902 is their LOS rate 120576 =
const is the compensatory gain 119902(| 119902| + 120575) is for substitutingfor sign( 119902) and for smoothing 120575 is a small quantity whichcould adjust the chattering
43 Separated Guidance and Control Design For simulationand comparisonwe use the conventional three-loop overloadautopilot as the controller which gives the rudder deflectioncommand according to the feedback of the three loops ofoverload pseudo-angle of attack and pitch rate The blockdiagram is as shown in Figure 3
As the figure shows the inner loop has the feedback onangular velocity which improves the damping characteristicsof the missile airframe
According to the aerodynamic coefficient of the missilewith selected working points we set 119870
119868= 019 119870
120572= 3 and
119870120596
= minus025 and the controller can well track the overloadcommand the rise time of its step response is 046 secondsand its settling time is 083 secondsThe step responses of themissile to overload command and the Bode diagram for openloop are shown in Figure 4
8 Mathematical Problems in Engineering
KIS
120596Z 120572 nY
nC K120596K120572 nY+minus+minus+minus dynamicsAirframe
modelServo
Figure 3 The working principles for three-loop overload autopilot
Step response
Time (s)0 02 04 06 08 1 12
0
02
04
06
08
1
Rise time (s) 0463
Settling time (s) 0831
Bode diagram
Frequency (rads)
To output pointFrom input pointTo output pointFrom input point
Gain margin (dB) 175 At frequency (rads) 15
Phase margin (deg) 709 At frequency (rads) 294
Am
plitu
de
Mag
nitu
de (d
B)Ph
ase (
deg)
0
minus180
minus360
minus54010410310210110010minus1
100
0
minus100
minus200
minus300
Figure 4 The autopilot performance step response and Bode diagram
5 Simulation Results
To verify the high order slidingmode integrated guidance andcontrol (HOSM-IGC) method we compare it with the base-line separated guidance and control Numerical simulationsare designed in three typical engagement scenarios
AORED parameters are as follows the initial value 1205820
=
1205821
= 1205822
= 1205823
= 50 1199110
= 01 1199111
= 1199112
= 1199113
= 0 119871 = 1400 thesimulation step is 00001 seconds
51 Scenario 1 Nonmaneuvering Target In the first scenariothe nonmaneuvering target does uniform rectilinear motionand PN guidance law with three-loop autopilot is introducedfor a comparison with the HOSM-IGCThe initial conditionsare set as shown in Table 1
The motion equations of the target are as follows
119910119879
= 119881119879cos 120579119879
119879
= minus119881119879sin 120579119879
120579119879
=119886119879
119881119879
119886119879
= 0
(55)
The simulation results are shown in Figures 5 and 6The missilersquos flight trajectory and overload curve show
that within the first 3 sec the HOSM-IGC method spendsmuch energy (overload) on changing the initial LOSdirection
Table 1 Initial conditions
Variable Description Value(119883119872 119884119872) Missile initial position (0 km 18 km)
(119883119879 119884119879) Target initial position (60 km 20 km)
119881119872
Missile initial velocity 1200 (ms)119881119879
Target initial velocity 2000 (ms)120579119872
Missile initial flight path angle 0 (deg)120579119879
Target initial flight path angle 0 (deg)120573 Parameter of the controller 10 or 30119873 Navigation ratio of PN 3
of the missile After the boresight adjustment 120590 reaches thedesired sliding manifold and the missile does not need anymaneuver to hit the target This is because the target doesnot maneuver any more which means no disturbance isintroduced in the engagement dynamic so the state (or thesliding mode) of the missile will stay on the manifold Incontrast the overload command given by the PN guidancelaw increases fast as the relative distance decreases Toexamine the performances of the two guidance laws furtherwe increase the target speed and analyze the commandchanges
Therefore we set the target speeds 3000ms 4000msand carry out simulations The simulation result is shown inthe overload curve in Figure 7
Mathematical Problems in Engineering 9
175
180
185
190
195
200
205
TargetHOSM-IGCPN
Y(k
m)
0 10 20 30 40 50 60X (km)
Figure 5 Target and missile trajectories
0
5
10
15
20
25
0 5 10 15 20Time (s)
HOSM-IGCPN
Initial boresight adjustment
Sliding manifold reached
Miss
ile ac
cele
ratio
n (G
)
minus5
Figure 6 Missile acceleration profile
It is evident that the speed of divergence of the overloadcommand given by the PN guidance law increases withthe target speed Specifically when the target speed reaches4000ms the commanded overload is almost 30 g which isobviously not ideal for the attack of a nonmaneuvering targetbut with the HOSM-IGC method the missile adjusts itsboresight very quickly then maintains it around 0 g and fliesto its target in the rectilinear ballistic trajectory not affectedby the increases of target speed still accomplishing the high-precision hit-on collision
Figure 8 shows that although the HOSM-IGC methodachieves a more effective overload command on the otherside it sees some chattering when the sliding mode reaches
0
5
10
15
20
25
30
0 5 10 15Time (s)
Miss
ile ac
cele
ratio
n (G
)
minus5
HOSM-IGC (Vt = minus3000)HOSM-IGC (Vt = minus4000)
PN (Vt = minus3000)PN (Vt = minus4000)
Figure 7 Missile acceleration profile
0
1
2
0 5 10 15 20Time (s)
4 45 5
0
02
04
minus02
minus04
Actu
ator
defl
ectio
n (d
eg)
minus1
minus2
minus3
minus4
HOSM-IGC (120573 = 30)
Figure 8 Actuator deflection
the desired manifold the commanded rudder deflectionangle chatters for around 045 deg at about 15Hz which iskind of harmful to the system To reduce the chatteringwe adjust the controllerrsquos parameter 120573 = 10 and carry outsimulations again Figure 9 shows that after 120573 decreasesthe command of rudder deflection angle converges slower(for about 7 sec) however the chattering weakens obviouslyits magnitude being only about 015 degrees The smallerchattering well enhances convergence precision eventuallyreducing the target missing The reason that minor 120573 resultsin an alleviative chattering can be seen from (47) Figure 10shows the target and missile trajectories
Table 2 compares the average miss distance of 50 simula-tions under the conditions discussed above the comparison
10 Mathematical Problems in Engineering
Table 2 Average miss distance of 50 simulations
Target speed PN HOSM-IGC(120573 = 30)
HOSM-IGC(120573 = 10)
119881119905= 2000ms 115m 086 073m
119881119905= 3000ms 261m 117 106
119881119905= 4000ms 542m 156 134
0
1
2
0 5 10 15 20Time (s)
72 74 76 78
0
02
minus02
minus04
Actu
ator
defl
ectio
n (d
eg)
minus1
minus2
minus3
minus4
HOSM-IGC (120573 = 10)HOSM-IGC (120573 = 30)
Figure 9 Actuator deflection
results show that when the missile intercepts a nonmaneu-vering target the hit precision of the HOSM-IGC methodis apparently higher than that of the PN guidance law andthat the low-gain HOSM-IGC method can effectively reducethe chattering magnitude thus enhancing the interceptionprecision
52 Scenario 2 Step Maneuvering Target At the first stagethe target flies at uniform speed and in a rectilinear way after10 seconds it maneuvers at the normal acceleration of 5 g Inthis scenario OSMG is introduced for a comparison with theHOSM-IGC
119910119879
= 119881119879cos 120579119879
119879
= minus119881119879sin 120579119879
120579119879
=119886119879
119881119879
119886119879
=
0 g 119905 lt 10 sec
5 g 119905 gt 10 sec
(56)
The initial simulation conditions are given in Table 3The overload curve in Figure 11 shows dearly that both
types of guidance laws can track the maneuvering targetDuring 0 to 10 seconds the target flies at uniform speed andin a rectilinear way and the missile converges its overload to
Table 3 Initial conditions
Variable Description Value(119883119872 119884119872) Missile initial position (0 km 22 km)
(119883119879 119884119879) Target initial position (60 km 20 km)
119881119872
Missile initial velocity 1200 (ms)119881119879
Target initial velocity 2000 (ms)120579119872
Missile initial flight path angle 0 (deg)120579119879
Target initial flight path angle 0 (deg)120576 Compensatory gain 3120575 Saturation parameter 0001
15
17
19
21
23
25
TargetHOSM-IGCOSMG
Y(k
m)
0 20 40 60X (km)
Figure 10 Target and missile trajectories
0 g and flies to its target after 10 seconds the target beginsto maneuver by 5 g Both types of guidance law respondby rapidly increasing the overload adjusting attitude andmaking the missilersquos boresight aim at its target
We can see that when both types of guidance law tracktheir targets the convergence speed of OSMG is almostthe same as the HOSM-IGC method But the HOSM-IGCmethod has higher convergence precision and needs loweroverload at the end phase
We can also see that after the target maneuvers if themissile is given enough time to track the targetrsquos maneuvernamely let the missilersquos overload command converge to theoverload of the target theremay not be largemiss distance Inother words for a certain period of time before the collisionthe targetmaneuver (it means only a limitedmaneuver whichdoes not include the condition that the maneuvering of thetarget for a long time may cause a change of the geometricalrelations between the missile and its target) has a small effecton both types of guidance law
But if themaneuver occurs rather late namely within oneto three seconds before collision when the overload com-mand of guidance law is not yet converged the approaching
Mathematical Problems in Engineering 11
0
5
10
0 5 10 15 20Time (s)
TargetOSMGHOSM-IGC
Miss
ile an
d ta
rget
acce
lera
tion
(G)
minus5
minus10
Figure 11 Missile and target acceleration profile
0
10
20
30
0 5 10 15 20Time (s)
tgo = 1 stgo = 2 stgo = 3 s
tgo = 3 s (diverge)
tgo = 2 s (diverge)
tgo = 1 s (diverge)minus10
minus20
minus30
Miss
ile ac
cele
ratio
n (G
)
HOSM-IGC (120573 = 10)
Figure 12 Missile acceleration profile
collision increases themiss distanceTherefore increasing theparameter 120573 can remarkably increase the convergence speedand effectively enhance guidance precision As Figure 12shows when 120573 = 10 the overload at the end phase convergesslowly even if the target maneuvers three seconds beforecollision the missilersquos overload still has no time to convergebeing unable to track themaneuvering target Figure 13 showsthat when 120573 increases to 30 and tgo = 3 seconds theoverload can converge to about 5 g but when tgo = 2 sec-onds the overload may continue to increase indicating that
0
10
20
30
0 5 10 15 20Time (s)
tgo = 1 stgo = 2 stgo = 3 s
tgo = 1 s (diverge)
tgo = 2 s (diverge)
tgo = 3 s (converge)
minus10
minus20
minus30
Miss
ile ac
cele
ratio
n (G
)
120573 = 30)HOSM-IGC (
Figure 13 Missile acceleration profile
0
10
20
30
0 5 10 15 20Time (s)
tgo = 1 s (diverge)
tgo = 2 s (converge)
tgo = 3 s (converge)
tgo = 1 stgo = 2 stgo = 3 s
minus10
minus20
minus30
Miss
ile ac
cele
ratio
n (G
)
HOSM-IGC (120573 = 50)
Figure 14 Missile acceleration profile
the convergence still needs to be quickened thus continuingto rise 120573 to 50 Figure 14 shows that the HOSM-IGC methodcan track the target that maneuvers when tgo = 2 secondsbut it causes the divergence of overload and the increase ofmiss distance if the target maneuvers when tgo = 1 secondFigure 15 gives the overload curve of the OSMG guidance lawand shows that the maneuver of the target before collisionmay cause the large-scale oscillation of the missilersquos overloadwhich may diverge to a large numerical value when thecollision occurs in the end
12 Mathematical Problems in Engineering
Table 4 Average miss distances of 50 simulations
Targetmaneuveringtiming
HOSM-IGC120573 = 30
HOSM-IGC120573 = 40
HOSM-IGC120573 = 50
OSMG
tgo = 1second 43539 3224 28936 36116
tgo = 2seconds 25424 18665 09322 35534
tgo = 3seconds 08124 08265 07538 11959
Time (s)0 5 10 15 20
0
10
20
30OSMG
tgo = 1 stgo = 2 stgo = 3 s
minus10
minus20
minus30
Miss
ile ac
cele
ratio
n (G
)
Figure 15 Missile acceleration profile
The analysis in Figures 12 13 14 and 15 shows that theoverload of the missile converges faster and its miss distanceis smaller with increasing 120573 To verify this finding we carryout 50 times Monte Carlo simulations in which the positionand speed of the target have 1 of random difference
The average miss distances are shown in Table 4 andFigure 16 Clearly the timing of the targetrsquos step maneuverdramatically affects the final interception precision morespecifically given a shorter reaction time for the guidanceand control system the missile seems more likely to missthe target To the OSMG guidance law in all three scenarioshardly does it show any advantages against theHOSM-IGC Itcan also be seen that with the increasing of the120573 theHOSM-IGC system responds even faster which leads to an obviousdecrease of the average miss distance The effect of 120573 on theresponse of the HOSM-IGC system is a valuable guidelinewhen implementing the proposed method into practice
53 Scenario 3 Weaving Target In this scenario the targetmaneuvers by 119886
119879= 40sin(1205871199052) OSMG with three-loop
Table 5 Initial conditions
Variable Description Value(119883119872 119884119872) Missile initial position (0 km 175 km)
(119883119879 119884119879) Target initial position (60 km 195 km)
119881119872
Missile initial velocity 1200 (ms)119881119879
Target initial velocity 2000 (ms)120579119872
Missile initial flight path angle 0 (deg)120579119879
Target initial flight path angle 0 (deg)120576 Compensatory gain 3120575 Saturation parameter 0001
autopilot is introduced for comparison The motion equa-tions of the target are as follows
119910119879
= 119881119879cos 120579119879
119879
= minus119881119879sin 120579119879
120579119879
=119886119879
119881119879
119886119879
= 40 sin(120587119905
2)
(57)
The initial simulation conditions are given in Table 5The missilersquos trajectory under the two guidance and
control methods given in Figure 17 does not see muchdifference
However the overload curve given in Figure 18 showsthat after the missile completes its initial attitude adjustmentwith the HOSM-IGC method it can almost ideally track itsmaneuvering target by contrast with the OSMG methodthe missile seems to have the tendency to track its targetrsquosmaneuver but has larger tracking errors Besides with theOSMGmethod the missilersquos overload increases rapidly at theend of attack primarily because of the divergence of its LOSrate On the other hand with the HOSM-IGC method themissile has no divergence even at the end of attack ensuringa smaller target missing quantity
The actuator deflection curve in Figure 19 shows thatin order to provide a rather big normal overload for theend phase the OSMG method produces a rather big rudderdeflection command however it may increase the missilersquostarget missing quantity once its rudder deflection saturatesand themissile does not have enough overloads or the ruddercannot respond that fast
As shown in Figure 20 because of the dramatic changein overload command the response of the missilersquos autopilotto high-frequency command sees an obvious phase lag andamplitude value attenuation its actual overload cannot trackthe command ideally this is a main reason why the missdistance increases However the controller in the HOSM-IGC method gives its rudder deflection command directlyand there is no lagging or attenuation caused by the autopilotthus enhancing the guidance precision effectively Further-more the fast convergence of the high order sliding modemakes the missile rapidly track its maneuvering target withthemost reasonable rudder deflection command reducing itsoverload effectively
Mathematical Problems in Engineering 13
0
05
1
15
2
25
3
35
4
45
5
0 10 20 30 40 50Simulation times
OSMG
Miss
(m)
0
05
1
15
2
25
3
35
4
45
5
0 10 20 30 40 50Simulation times
Miss
(m)
0
05
1
15
2
25
3
35
4
45
5
0 10 20 30 40 50Simulation times
Miss
(m)
tgo = 3 stgo = 2 stgo = 1 s
0
05
1
15
2
25
3
35
4
45
5
0 10 20 30 40 50Simulation times
Miss
(m)
tgo = 3 stgo = 2 stgo = 1 s
HOSM-IGC (120573 = 40)
HOSM-IGC (120573 = 50)
HOSM-IGC (120573 = 30)
Figure 16 Miss distances of HOSM-IGC (120573 = 30 40 50) and OSMG with target maneuvers at different time-to-go (tgo = 1 second 2seconds and 3 seconds)
The average miss distance of 50 simulations under theconditions is 073m for HOSM-IGC and 186m for OSMGWe can see that the HOSM-IGC method not only doesprovide a more reasonable actuator deflection command butalso achieves a higher interception precision
6 Conclusions
This paper proposes an LOS feedback integrated guidanceand control method using quasi-continuous high order
sliding mode guidance and control method With the fastand precise convergence of the quasi-continuous HOSMmethod the HOSM-IGCmethod performsmuch better thanthe traditional separated guidance and control method withless acceleration effort and less miss distance in all thethree simulation scenarios of nonmaneuvering target stepmaneuvering target and weaving target In addition the ideaof virtual control largely alleviates the chattering withoutany sacrifice of robustness As a result of the alleviationof the chattering the control input command 120575
119885becomes
14 Mathematical Problems in Engineering
175
180
185
190
195
200
TargetHOSM-IGCOSMG
Y(k
m)
0 20 40 60 0 20 40X (km)
Figure 17 The trajectories of the missile and its target
0
10
20
0 5 10 15 20Time (s)
TargetOSMGHOSM-IGC
Miss
ile an
d ta
rget
acce
lera
tion
(G)
minus10
minus20
Figure 18 Missile acceleration profile
smooth enough for the actuator to be implemented as aresult the interception accuracy is improved In designingthe quasi-continuous sliding mode controller the Arbitrary-Order Robust Exact Differentiator is introduced to estimatethe high order time derivative of the LOS rate 119902 which notonly ensures the guidance and control precision but alsosimplifies the calculation
0
10
20
30
0 5 10 15 20Time (s)
OSMGHOSM-IGC
minus10
minus20
minus30
minus40
Actu
ator
defl
ectio
n (d
eg)
Figure 19 Actuator deflection
0
50
100
150
200
15 16 17 18Time (s)
Commanded accelerationAchieved acceleration
Miss
ile ac
cele
ratio
n (G
)
minus50
Figure 20 Commanded acceleration and achieved acceleration
Appendices
A The Third-Order RobustExact Differentiator
The third-order robust exact differentiator built according tothe following parameters and equations is used to performthe simulation the initial value 120582
0= 1205821
= 1205822
= 1205823
= 50
Mathematical Problems in Engineering 15
0
0005
001
0015
002
0025
003
0035
004
0 02 04Time (s)
0
2
0 02 04Time (s)
0
100
200
300
0
100
200
300
400
0 02 04Time (s)
h(t)
z3
h(t) and z3
minus100
minus200
minus300
minus400
minus5000 02 04
Time (s)
g(t)
z2
g(t) and z2
minus100
minus200
minus300
minus400
f(t)
z1
f(t) and z1
minus2
minus4
minus6
minus8
minus10
minus12
minus14
F(t)
z0
F(t) and z0
minus0005
minus001
Figure 21 The outputs of the differentiator in the sampling interval of 0001 seconds
the initial values 1199110
= 01 1199111
= 1199112
= 1199113
= 0 119871 = 1400 thesimulation step is 0001 seconds Consider
0
= V0
V0
= minus120582011987114 10038161003816100381610038161199110 minus
1199021003816100381610038161003816
34 sign (1199110
minus 119902) + 1199111
1
= V1
V1
= minus120582111987113 10038161003816100381610038161199111 minus V
0
1003816100381610038161003816
23 sign (1199111
minus V0) + 1199112
2
= V2
V2
= minus120582211987112 10038161003816100381610038161199112 minus V
1
1003816100381610038161003816
12 sign (1199112
minus V1) + 1199113
3
= minus1205823119871 sign (119911
3minus V2)
(A1)
16 Mathematical Problems in Engineering
0
0005
001
0015
002
0025
003
0035
004
0 02 04Time (s)
0
2
0 02 04Time (s)
0
100
200
300
0 02 04Time (s)
0
200
400
0 02 04Time (s)
h(t) and z3
minus200
minus400
minus600
minus800
g(t) and z2
minus100
minus200
minus300
minus400
f(t) and z1
minus2
minus4
minus6
minus8
minus10
minus12
minus14
minus16
F(t) and z0
minus0005
minus001
h(t)
z3g(t)
z2
f(t)
z1F(t)
z0
Figure 22 The outputs of the differentiator in the sampling interval of 00001 seconds
If the designed signals are 119865(119905) = int 119891(119905)119889119905 119891(119905) =
int 119892(119905)119889119905 119892(119905) = int ℎ(119905)119889119905 and ℎ(119905) = sin(2120587119905) +
3cos(120587119905) then theoretically 119891(119905) is the first-order differ-ential of input signal 119892(119905) is the second-order differ-ential of input signal ℎ(119905) is the third-order differen-tial of input signal Through the simulation the differ-ential signal of actual output is compared with that of
theoretical output the comparison results are shown inFigure 21
Figure 21 shows that 1199110can effectively track the signal
119865(119905) but 1199112has obvious errors in tracking the second-order
differential 119892(119905) 1199113has greater errors in tracking the third-
order differential thus the overall differential effect is notideal
Mathematical Problems in Engineering 17
0 02 04
0
0005
001
0015
002
0025
003
0035
004
Time (s)0 02 04
0
01
02
03
04
05
Time (s)
0 02 04
0
05
1
15
2
Time (s)0 02 04
0
05
1
15
2
25
3
35
4
Time (s)
h(t) and z3g(t) and z2
minus04
minus05
f(t) and z1
minus2
minus15
minus1
minus05
minus1
minus05
minus01
minus02
minus03
F(t) and z0
minus0005
minus001
h(t)
z3g(t)
z2
f(t)
z1F(t)
z0
Figure 23 The outputs of the differentiator in the sampling interval of 00001 seconds
There are two causes for the errors in a usual differentia-tor the setting of sampling interval and the setting of numberof orders of the differentiator The following theorem applieshere
Theorem A1 (see [19]) If the sampling interval 120591 gt 0 theaccuracy of the controller is expressed as follows
10038161003816100381610038161003816120590(119894)
10038161003816100381610038161003816le 120583120591119903minus119894
119894 = 0 119903 minus 1 (A2)
That is to say the precision (understood as the differentialprecision of a differentiator) of the 119894th order control is in directproportion to the power of 119903 minus 119894 times of sampling interval120591 Therefore the decrease of iteration step can enhance thedifferential tracking precision
Under the conditions that the sampling interval is 10 timessmaller and that the setting is 00001 seconds the simulationresults are shown in Figure 22
18 Mathematical Problems in Engineering
Table 6 Tracking errors of the AORED with different orders andsampling interval
Orders of AORED Third order Fifth orderSample period 120591 = 0001 120591 = 00001 120591 = 00001
First-order differential error 001 Negligible NegligibleSecond-order differential error 15 002 NegligibleThird-order differential error 70 25 00005
As Figure 22 shows the signal tracking is rather idealexcept a peak in the beginning the tracking error in the first-order differential tracking can be neglected 119911
2has an obviously
smaller tracking error in the second-order differential 119892(119905)
tracking in the beginning 1199113has a rather sharp peak in the
third-order differential tracking but converges to a relativelyhigh precision after 01 seconds As we can see the differentialtracking errors for each number of orders are greatly reducedbut the third-order differential tracking error is still rather bigIn the following we will reduce the tracking errors by increasingthe number of orders of the differentiator
B The Fifth-Order Robust Exact Differentiator
The settings of the fifth-order differentiator are given asfollows
0
= V0
V0
= minus120582011987116 10038161003816100381610038161199110 minus
1199021003816100381610038161003816
56 sign (1199110
minus 119902) + 1199111
1
= V1
V1
= minus120582111987115 10038161003816100381610038161199111 minus V
0
1003816100381610038161003816
45 sign (1199111
minus V0) + 1199112
2
= V2
V2
= minus120582211987114 10038161003816100381610038161199112 minus V
1
1003816100381610038161003816
34 sign (1199112
minus V1) + 1199113
3
= V3
V3
= minus120582311987113 10038161003816100381610038161199113 minus V
2
1003816100381610038161003816
23 sign (1199113
minus V2) + 1199114
4
= V4
V4
= minus120582411987112 10038161003816100381610038161199114 minus V
3
1003816100381610038161003816
12 sign (1199114
minus V3) + 1199115
5
= V5
V5
= minus1205825119871 sign (119911
5minus V4)
(B1)
The initial value 1199110
= 01 1205820
= 1205821
= 1205822
= 1205823
= 1205824
=
1205825
= 50 the initial values 1199111 1199112 1199113 1199114 1199115
= 0 119871 = 1400the simulation step length is 00001 seconds The simulationresults are shown in Figure 23
As the figure shows the fifth-order differentiator hasa rather ideal tracking effect on the third-order differen-tial tracking the tracking error being only about 00005(see Table 6)
To sum up the Arbitrary-Order Robust Exact Differen-tiator can estimate rather ideal differential signals by selecting
Table 7
Length 365m119871 ref 365m119883119866
177m119878ref 0026m2
Diameter 0178mMass 1016 Kg119868119885
1063 Kgsdotm2
an appropriate sampling interval and using the differentiatorwith a relatively high number of orders
C Physical and Geometric Characteristics
See Table 7
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] P Zarchan Tactical and Strategic Missile Guidance 2002[2] D Zhou CMu andWXu ldquoAdaptive sliding-mode guidance of
a homing missilerdquo Journal of Guidance Control and Dynamicsvol 22 no 4 pp 589ndash594 1999
[3] G Zeng and M Hu ldquoFinite-time control for electromagneticsatellite formationsrdquo Acta Astronautica vol 74 pp 120ndash1302012
[4] S R Kumar S Rao and D Ghose ldquoNonsingular terminalsliding mode guidance with impact angle constraintsrdquo Journalof Guidance Control and Dynamics vol 37 no 4 pp 1114ndash11302014
[5] Y Liang C Chen D Liaw Y Feng C Cheng and CChen ldquoRobust guidance law via integral-sliding-mode schemerdquoJournal of Guidance Control and Dynamics vol 37 no 3 pp1038ndash1042 2014
[6] A V Savkin P N Pathirana and F Faruqi ldquoProblem ofprecision missile guidance LQR and 119867
infincontrol frameworksrdquo
IEEE Transactions on Aerospace and Electronic Systems vol 39no 3 pp 901ndash910 2003
[7] H Y Chen andC-C Yang ldquoNonlinear1198672119867infinguidance design
for homing missilesrdquo in Proceedings of the AIAA GuidanceNavigation and Control Conference and Exhibit vol 77 2001
[8] P K Menon G D Sweriduk and E J Ohlmeyer ldquoOptimalfixed-interval integrated guidance-control laws for hit-to-killmissilesrdquo in Proceedings of the AIAA Guidance Navigation andControl Conference and Exhibit AIAA 2003-5579 Austin TexUSA August 2003
[9] S S Vaddi P K Menon and E J Ohlmeyer ldquoNumericalSDRE approach for missile integrated guidance-controlrdquo inProceedings of the AIAA Guidance Navigation and ControlConference and Exhibit AIAA 2007-6672 Hilton Head SCUSA August 2007
[10] M Xin S N Balakrishnan and E J Ohlmeyer ldquoIntegratedguidance and control of missiles with Theta-D methodrdquo IEEE
Mathematical Problems in Engineering 19
Transactions on Control Systems Technology vol 14 no 6 pp981ndash992 2006
[11] T Shima M Idan and O M Golan ldquoSliding-mode controlfor integrated missile autopilot guidancerdquo Journal of GuidanceControl and Dynamics vol 29 no 2 pp 250ndash260 2006
[12] Y B Shtessel and C H Tournes ldquoIntegrated higher-ordersliding mode guidance and autopilot for dual control missilesrdquoJournal of Guidance Control and Dynamics vol 32 no 1 pp79ndash94 2009
[13] F Y Dong H M Lei and C J Zhou ldquoIntegrated robust highorder sliding mode guidance and control for missilesrdquo ActaAeronautica et Astronautica Sinica vol 34 no 9 pp 2212ndash22182013
[14] H Mingzhe and D Guangren ldquoIntegrated guidance and con-trol of homing missiles against ground fixed targetsrdquo ChineseJournal of Aeronautics vol 21 no 2 pp 162ndash168 2008
[15] A Levant ldquoHomogeneity approach to high-order sliding modedesignrdquo Automatica vol 41 no 5 pp 823ndash830 2005
[16] Y-W Liang C Chen D Liaw Y Feng C Cheng and CChen ldquoRobust guidance law via integral-sliding-mode schemerdquoJournal of Guidance Control and Dynamics vol 37 no 3 pp1038ndash1042 2014
[17] L Wang Y Sheng and X Liu ldquoContinuous time-varying slid-ingmode based attitude control for reentry vehiclerdquo Proceedingsof the Institution of Mechanical Engineers Part G Journal ofAerospace Engineering 2014
[18] A Levant ldquoHigher-order sliding modes differentiation andoutput-feedback controlrdquo International Journal of Control vol76 no 9-10 pp 924ndash941 2003
[19] A Levant ldquoQuasi-continuous high-order sliding-mode con-trollersrdquo IEEE Transactions on Automatic Control vol 50 no11 pp 1812ndash1816 2005
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Mathematical PhysicsAdvances in
Complex AnalysisJournal of
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OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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Algebra
Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 5
Thus
119902(4)
= 119891120575119885
sdot 120575119885
+1
119903(1198911
+ 1198912
+ 1198913
+ 1198914
+ 1198915
+ 1198916) (26)
where
119891120575119885
= minus1
119903
119862120572
119884119876119878ref cos 120578
119872
119898
119898120575119885
119885119876119878ref119897ref
119869119885
(27)
1198911
= (120596119885
minus 120579119872
) (minus2 120579119886119872119873
120572sin 120578119872
+ 119902119886119872119873
120572sin 120578119872
+ 120579119886119872119873
1205722cos 120578119872
) minus119886119872119873
cos 120578119872
120572
119898120572
119885120572119876119878ref119897ref
119869119885
(28)
1198912
= 120579119872
(( 120579119872
minus 119902) 119886119872119873
cos 120578119872
minus (120596119885
minus 120579119872
)119886119872119873
120572sin 120578119872
)
(29)
1198913
= 119902 (minus 120579119872
119886119872119873
cos 120578119872
+ 119902 (119886119872119873
cos 120578119872
minus 119886119879119873
cos 120578119879) minus 120579119879119886119879119873
cos 120578119879
+119886119872119873
120572(120596119885
minus 120579119872
) sin 120578119872
minus 119886119879sin 120578119879)
(30)
1198914
= 120579119879
(minus 119902119886119879cos 120578119879
minus 120579119879119886119879cos 120578119879
minus 119886119879sin 120578119879) (31)
1198915
= 119886119879
(minus 119902 sin 120578119879
minus 2 120579119879sin 120578119879) + 119886119879cos 120578119879 (32)
1198916
= 119902 (119886119872119873
sin 120578119872
minus 119886119879sin 120578119879
minus 2 119903) minus 4 119903 sdot119902 minus 3 119903 sdot 119902
minus 2119903 sdot 119902
(33)
120578119872
= 120579119872
minus 119902
120578119879
= 120579119879
+ 119902
(34)
In (26) it can be seen that the control input 120575119885appears
expressly in the third-order derivative of the control output119902 Therefore the relative degree of the control input 120575
119885is 3
3 The Quasi-Continuous High Order SlidingMode Controller
31 Sliding Mode Manifold Design To design the HOSMcontroller a sliding manifold must be chosen first In thisdesign we try to make the LOS rate converge to zero ora small neighbor domain near zero thus ensuring that themissile approaches its target in a quasi-parallel way whichwill lead to a minimal overload requirement So the slidingmanifold is chosen as follows
120590 = 119902 (35)
From the above discussion in Section 2 we know thatthe control input in relation to control output 119902 namelythe relative degree of sliding mode manifold 120590 is 3 So thefollowing design will be about a third-order sliding modecontroller
32 Design of the Quasi-Continuous HOSM Controller First(26) can be expressed as follows
120590 = ℎ (119905 119909) + 119892 (119905 119909) 119906 (36)
where ℎ(119905 119909) 119892(119905 119909) and 119906 are expressed as follows
ℎ (119905 119909) =1
119903(1198911
+ 1198912
+ 1198913
+ 1198914
+ 1198915
+ 1198916)
119892 (119905 119909) = 119891120575119885
119906 = 120575119885
(37)
According to the quasi-continuous high order slidingmode control method proposed by Levant in [15] the slidingmode manifold whose relative degree is 3 should be designedin the following form where 120573 is a control gain term
119906 = minus120573
+ 2 (|| + |120590|23
)minus12
( + |120590|23 sign120590)
|| + 2 (|| + |120590|23
)12
(38)
The conditions under which the LOS angular velocitymay converge are as follows
0 lt 119870119898
le 119892 (119905 119909) le 119870119872
|ℎ (119905 119909)| le 119862
(39)
where119870119898119870119872 and119862 are all larger than zeroThis is a proven
theorem by Levant in [15]The system we discussed meets the above requirements
and the proof is as followsEquation (27) shows the following
119892 (119905 119909) = 119891120575119885
= minus1
119903
119862120572
119884119876119878ref
119898
119898120575119885
119911119876119878ref119897ref
119869119885
cos (120579119872
minus 119902)
(40)
The dynamic pressure 119876 is 119876 = 1205881198812
1198722 where 120588 =
008803Kgm3 (altitude = 20Km) is the air density and119881119872
=
2000ms is the speed of the missile so 119876 is always positive119878ref = 026m2 and 119897ref = 365m denote the reference area
and the reference length of the missile they are both positiveconstant
119898 = 100Kg denotes the missile mass 119869119885
= 106m2 Kgdenotes the rotational inertia
119903 is the relative distance it is always a positive number119862120572
119884is the lift coefficient caused by the angle of attack it
varies from 018 to 037 and it is always a positive number119898120575119911
119911is the moment coefficient caused by the actuator
deflection In the normal layout (actuator lays behind thecenter of gravity) 119898
120575119911
119911is always negative
Meanwhile consider that the missile under guidance andcontrol is unlikely to fly away from its target namely the anglebetween themissilersquos velocity and its LOS direction cannot belarger than 90∘ then
1003816100381610038161003816120579119872 minus 1199021003816100381610038161003816 lt
120587
2
997904rArr cos (120579119872
minus 119902) gt 0
(41)
6 Mathematical Problems in Engineering
Summing up the above conditions then we can get
119892 (119905 119909) gt 0 (42)
In other words there is a positive real number 119870119898existing
that could satisfy the following condition
0 lt 119870119898
lt 119892 (119905 119909) (43)
Before the missile hits on the target the term will be positiveand limited then we can get
0 lt 119870119898
lt 119892 (119905 119909) lt 119870119872
(44)
With (37) then
ℎ (119905 119909) =1
119903(1198911
+ 1198912
+ 1198913
+ 1198914
+ 1198915
+ 1198916) (45)
In the practice sense the changes in both the LOS rateand the acceleration of the missile and the acceleration of thetarget are limited and continuous So the following variables119902 119902
119902 119886119872 119886119879 and 119886
119879are all bounded However because
ℎ(119905 119909) contains the item 1119903 when the relative distancebetween the missile and its target is zero the boundaryof ℎ(119905 119909) is not guaranteed In [15] Levant only requiresthat condition (39) should be locally valid not requiringthat it should be globally valid Therefore the integratedguidance and controlmethod is applicable here So the abovementioned condition is satisfied with a positive number 119862
|ℎ (119905 119909)| le 119862 (46)
33TheVirtual ControlDesign Whenusing the slidingmodecontrolmethod the avoidance of the chattering phenomenonhas always been a key issue being discussed In the tradi-tional method researchers in [16 17] have proposed severalsaturation functions to replace the sign functions to builda boundary layer to alleviate the chattering or to use fuzzylogic to displace the high-frequency switching term To ourknowledge none of these approaches has proven that therefined controller still retains their robustness against theuncertainties and disturbances In this work in order toalleviate the chattering phenomenon we do not directly usethe third-order controller but introduce the virtual control119906119894= 120575119885to perform the actual control
120575119885
= int 120575119885dt = int 119906
119894dt (47)
After the relative degree is increased to the fourth order weget the following expressions
120590(4)
= ℎlowast
(119905 119909) + 119892lowast
(119905 119909) 119906119894= ℎlowast
(119905 119909) + 119892 (119905 119909) 119906119894
ℎlowast
(119905 119909) = ℎ (119905 119909) + 119892 (119905 119909) 120575119885
119892 (119905 119909)
=119898120575119885
1199111198762
1198782
119897119862120572
119884
119869119885
1198981199032[( 119902 minus 120579
119872) 119903 sin 120578
119872+ 119903 cos 120578
119872]
(48)
Even though the expression of ℎ(119905 119909) is rather compli-cated it is still the function of 119902 119902 119902 119886
119872 119886119879 and 119886
119879 therefore
similar to ℎ(119905 119909) it has its boundary except themomentwhenthe missile hits on its target For the same reason 119892(119905 119909)
and the rudder deflection 120575119885also have their boundaries
Therefore we get the condition that |ℎlowast
(119905 119909)| le 119862 (119862 gt 0)Because 120575
119885is obtained through the derivation of 120575
119885 119892lowast(119905 119909)
is the same as 119892(119905 119909) thus 0 lt 119870119898
lt 119892lowast
(119905 119909) lt 119870119872
issatisfied
According to the formula of the fourth-order controllergiven by Levant in [15] we give the following formulae forthe virtual control 119906
119894
119906119894= minus120573
Φ34
11987334
Φ34
=120590 + 3 [||
+ (|| + 05 |120590|34
)minus13 10038161003816100381610038161003816
+ 05 |120590|34 sign120590
10038161003816100381610038161003816]
minus12
sdot [
+ (|| + 05 |120590|34
)minus13 10038161003816100381610038161003816
+ 05 |120590|34 sign120590
10038161003816100381610038161003816]
11987334
=1003816100381610038161003816
120590
1003816100381610038161003816 + 3 [||
+ (|| + 05 |120590|34
)minus13 10038161003816100381610038161003816
+ 05 |120590|34 sign120590
10038161003816100381610038161003816]
minus12
sdot
1003816100381610038161003816100381610038161003816
+ (|| + 05 |120590|34
)minus13
( + 05 |120590|34 sign120590)
1003816100381610038161003816100381610038161003816
(49)
The integral of the virtual control 120575119885produces the actual
control command 120575119885 120590 in the controller is obtained with the
Arbitrary-Order Robust Exact Differentiator presented in thefollowing section
34 The Arbitrary-Order Robust Exact Differentiator Thequasi-continuous HOSM control method needs to use thethird derivative of the sliding manifold namely 119902
(4) Howto calculate or accurately estimate 119902
(4) is one of the keyproblems to be solved We use the Arbitrary-Order RobustExact Differentiator designed by Levant to differentiate theLOS rate 119902 thus obtaining 119902 119902 and 119902
(4)According to (44) and (45) the following condition is
valid1003816100381610038161003816
120590
1003816100381610038161003816 le 119862 + 120573119870119872
(50)
The Arbitrary-Order Robust Exact Differentiator can beconstructed in accordance with high order sliding modesdifferentiation and output feedback control in [18]
If a certain signal 119891(119905) is a function consisting of abounded Lebesgue-measurable noise with unknown base
Mathematical Problems in Engineering 7
signal 1198910(119905) whose 119903th derivative has a known Lipschitz
constant 119871 gt 0 then the 119899th-order differentiator is definedas follows
0
= V0
V0
= minus12058201198711(119899+1) 10038161003816100381610038161199110 minus 119891 (119905)
1003816100381610038161003816
119899(119899+1) sign (1199110
minus 119891 (119905))
+ 1199111
1
= V1
V1
= minus12058211198712(119899+1) 10038161003816100381610038161199111 minus V
0
1003816100381610038161003816
(119899minus1)119899 sign (1199111
minus V0) + 1199112
119899minus1
= V119899minus1
V119899minus1
= minus120582119899minus1
11987112 1003816100381610038161003816119911119899minus1 minus V
119899minus2
1003816100381610038161003816
12 sign (119911119899minus1
minus V119899minus2
)
+ 119911119899
119899
= minus120582119899119871 sign (119911
119899minus V119899minus1
)
(51)
and if 120582119894
gt 0 is sufficiently large the convergence is guaran-teed
To obtain the third-order derivative of 119902 we constructthe third-order sliding mode differentiator and estimate thederivative of 119902 for each order In view of differential precisionwe configure the following fifth-order differentiator SeeAppendix A for comparison
0
= V0
V0
= minus120582011987116 10038161003816100381610038161199110 minus
1199021003816100381610038161003816
56 sign (1199110
minus 119902) + 1199111
1
= V1
V1
= minus120582111987115 10038161003816100381610038161199111 minus V
0
1003816100381610038161003816
45 sign (1199111
minus V0) + 1199112
2
= V2
V2
= minus120582211987114 10038161003816100381610038161199112 minus V
1
1003816100381610038161003816
34 sign (1199112
minus V1) + 1199113
3
= V3
V3
= minus120582311987113 10038161003816100381610038161199113 minus V
2
1003816100381610038161003816
23 sign (1199113
minus V2) + 1199114
4
= V4
V4
= minus120582411987112 10038161003816100381610038161199114 minus V
3
1003816100381610038161003816
12 sign (1199114
minus V3) + 1199115
5
= V5
V5
= minus1205825119871 sign (119911
5minus V4)
(52)
where 1199113 1199112 1199111 and 119911
0are the estimations of 119902
(4) 119902 119902 and 119902
4 Baseline Separated Guidance andControl Method
To verify the homing performance of the integrated guidanceand control method we compare it with the separatedguidance and control methodThe guidance laws we used arethe proportional navigation (PN) guidance law for attackinga nonmaneuvering target and the optimal sliding modeguidance law for attacking a maneuvering target
41 The Proportional Navigation (PN) Guidance Law Theguidance law has a simple formula and excellent perfor-mances for nonmaneuvering target Its formula is as follows
119899119888
= minus119873 119902119881119872
119892 (53)
119899119888is the overload command 119873 is the effective navigation
ratio 119902 is the LOS rate 119881119872
is the speed of the missile 119892 isthe acceleration of the gravity The guidance law gives theoverload command of the missile according to the LOS rateand then the controller gives the rudder deflection commandaccording to the overload command
42 The Optimal Sliding Mode Guidance Law The optimalsliding mode guidance law (OSMG) is a novel practicalguidance law proposed by D Zhou He combines the optimalguidance lawwith the slidingmode guidance law and designsthe new sliding mode guidance law that not only is robustto maneuvering target but also has the merits of the optimalguidance law such as good dynamic performance and energyconservation Its formula is as follows
119899119888
= minus3100381610038161003816100381610038160
10038161003816100381610038161003816119902 + 120576
119902
10038161003816100381610038161199021003816100381610038161003816 + 120575
(54)
where 119899119888is the overload command
0is the approach
velocity of the missile and its target 119902 is their LOS rate 120576 =
const is the compensatory gain 119902(| 119902| + 120575) is for substitutingfor sign( 119902) and for smoothing 120575 is a small quantity whichcould adjust the chattering
43 Separated Guidance and Control Design For simulationand comparisonwe use the conventional three-loop overloadautopilot as the controller which gives the rudder deflectioncommand according to the feedback of the three loops ofoverload pseudo-angle of attack and pitch rate The blockdiagram is as shown in Figure 3
As the figure shows the inner loop has the feedback onangular velocity which improves the damping characteristicsof the missile airframe
According to the aerodynamic coefficient of the missilewith selected working points we set 119870
119868= 019 119870
120572= 3 and
119870120596
= minus025 and the controller can well track the overloadcommand the rise time of its step response is 046 secondsand its settling time is 083 secondsThe step responses of themissile to overload command and the Bode diagram for openloop are shown in Figure 4
8 Mathematical Problems in Engineering
KIS
120596Z 120572 nY
nC K120596K120572 nY+minus+minus+minus dynamicsAirframe
modelServo
Figure 3 The working principles for three-loop overload autopilot
Step response
Time (s)0 02 04 06 08 1 12
0
02
04
06
08
1
Rise time (s) 0463
Settling time (s) 0831
Bode diagram
Frequency (rads)
To output pointFrom input pointTo output pointFrom input point
Gain margin (dB) 175 At frequency (rads) 15
Phase margin (deg) 709 At frequency (rads) 294
Am
plitu
de
Mag
nitu
de (d
B)Ph
ase (
deg)
0
minus180
minus360
minus54010410310210110010minus1
100
0
minus100
minus200
minus300
Figure 4 The autopilot performance step response and Bode diagram
5 Simulation Results
To verify the high order slidingmode integrated guidance andcontrol (HOSM-IGC) method we compare it with the base-line separated guidance and control Numerical simulationsare designed in three typical engagement scenarios
AORED parameters are as follows the initial value 1205820
=
1205821
= 1205822
= 1205823
= 50 1199110
= 01 1199111
= 1199112
= 1199113
= 0 119871 = 1400 thesimulation step is 00001 seconds
51 Scenario 1 Nonmaneuvering Target In the first scenariothe nonmaneuvering target does uniform rectilinear motionand PN guidance law with three-loop autopilot is introducedfor a comparison with the HOSM-IGCThe initial conditionsare set as shown in Table 1
The motion equations of the target are as follows
119910119879
= 119881119879cos 120579119879
119879
= minus119881119879sin 120579119879
120579119879
=119886119879
119881119879
119886119879
= 0
(55)
The simulation results are shown in Figures 5 and 6The missilersquos flight trajectory and overload curve show
that within the first 3 sec the HOSM-IGC method spendsmuch energy (overload) on changing the initial LOSdirection
Table 1 Initial conditions
Variable Description Value(119883119872 119884119872) Missile initial position (0 km 18 km)
(119883119879 119884119879) Target initial position (60 km 20 km)
119881119872
Missile initial velocity 1200 (ms)119881119879
Target initial velocity 2000 (ms)120579119872
Missile initial flight path angle 0 (deg)120579119879
Target initial flight path angle 0 (deg)120573 Parameter of the controller 10 or 30119873 Navigation ratio of PN 3
of the missile After the boresight adjustment 120590 reaches thedesired sliding manifold and the missile does not need anymaneuver to hit the target This is because the target doesnot maneuver any more which means no disturbance isintroduced in the engagement dynamic so the state (or thesliding mode) of the missile will stay on the manifold Incontrast the overload command given by the PN guidancelaw increases fast as the relative distance decreases Toexamine the performances of the two guidance laws furtherwe increase the target speed and analyze the commandchanges
Therefore we set the target speeds 3000ms 4000msand carry out simulations The simulation result is shown inthe overload curve in Figure 7
Mathematical Problems in Engineering 9
175
180
185
190
195
200
205
TargetHOSM-IGCPN
Y(k
m)
0 10 20 30 40 50 60X (km)
Figure 5 Target and missile trajectories
0
5
10
15
20
25
0 5 10 15 20Time (s)
HOSM-IGCPN
Initial boresight adjustment
Sliding manifold reached
Miss
ile ac
cele
ratio
n (G
)
minus5
Figure 6 Missile acceleration profile
It is evident that the speed of divergence of the overloadcommand given by the PN guidance law increases withthe target speed Specifically when the target speed reaches4000ms the commanded overload is almost 30 g which isobviously not ideal for the attack of a nonmaneuvering targetbut with the HOSM-IGC method the missile adjusts itsboresight very quickly then maintains it around 0 g and fliesto its target in the rectilinear ballistic trajectory not affectedby the increases of target speed still accomplishing the high-precision hit-on collision
Figure 8 shows that although the HOSM-IGC methodachieves a more effective overload command on the otherside it sees some chattering when the sliding mode reaches
0
5
10
15
20
25
30
0 5 10 15Time (s)
Miss
ile ac
cele
ratio
n (G
)
minus5
HOSM-IGC (Vt = minus3000)HOSM-IGC (Vt = minus4000)
PN (Vt = minus3000)PN (Vt = minus4000)
Figure 7 Missile acceleration profile
0
1
2
0 5 10 15 20Time (s)
4 45 5
0
02
04
minus02
minus04
Actu
ator
defl
ectio
n (d
eg)
minus1
minus2
minus3
minus4
HOSM-IGC (120573 = 30)
Figure 8 Actuator deflection
the desired manifold the commanded rudder deflectionangle chatters for around 045 deg at about 15Hz which iskind of harmful to the system To reduce the chatteringwe adjust the controllerrsquos parameter 120573 = 10 and carry outsimulations again Figure 9 shows that after 120573 decreasesthe command of rudder deflection angle converges slower(for about 7 sec) however the chattering weakens obviouslyits magnitude being only about 015 degrees The smallerchattering well enhances convergence precision eventuallyreducing the target missing The reason that minor 120573 resultsin an alleviative chattering can be seen from (47) Figure 10shows the target and missile trajectories
Table 2 compares the average miss distance of 50 simula-tions under the conditions discussed above the comparison
10 Mathematical Problems in Engineering
Table 2 Average miss distance of 50 simulations
Target speed PN HOSM-IGC(120573 = 30)
HOSM-IGC(120573 = 10)
119881119905= 2000ms 115m 086 073m
119881119905= 3000ms 261m 117 106
119881119905= 4000ms 542m 156 134
0
1
2
0 5 10 15 20Time (s)
72 74 76 78
0
02
minus02
minus04
Actu
ator
defl
ectio
n (d
eg)
minus1
minus2
minus3
minus4
HOSM-IGC (120573 = 10)HOSM-IGC (120573 = 30)
Figure 9 Actuator deflection
results show that when the missile intercepts a nonmaneu-vering target the hit precision of the HOSM-IGC methodis apparently higher than that of the PN guidance law andthat the low-gain HOSM-IGC method can effectively reducethe chattering magnitude thus enhancing the interceptionprecision
52 Scenario 2 Step Maneuvering Target At the first stagethe target flies at uniform speed and in a rectilinear way after10 seconds it maneuvers at the normal acceleration of 5 g Inthis scenario OSMG is introduced for a comparison with theHOSM-IGC
119910119879
= 119881119879cos 120579119879
119879
= minus119881119879sin 120579119879
120579119879
=119886119879
119881119879
119886119879
=
0 g 119905 lt 10 sec
5 g 119905 gt 10 sec
(56)
The initial simulation conditions are given in Table 3The overload curve in Figure 11 shows dearly that both
types of guidance laws can track the maneuvering targetDuring 0 to 10 seconds the target flies at uniform speed andin a rectilinear way and the missile converges its overload to
Table 3 Initial conditions
Variable Description Value(119883119872 119884119872) Missile initial position (0 km 22 km)
(119883119879 119884119879) Target initial position (60 km 20 km)
119881119872
Missile initial velocity 1200 (ms)119881119879
Target initial velocity 2000 (ms)120579119872
Missile initial flight path angle 0 (deg)120579119879
Target initial flight path angle 0 (deg)120576 Compensatory gain 3120575 Saturation parameter 0001
15
17
19
21
23
25
TargetHOSM-IGCOSMG
Y(k
m)
0 20 40 60X (km)
Figure 10 Target and missile trajectories
0 g and flies to its target after 10 seconds the target beginsto maneuver by 5 g Both types of guidance law respondby rapidly increasing the overload adjusting attitude andmaking the missilersquos boresight aim at its target
We can see that when both types of guidance law tracktheir targets the convergence speed of OSMG is almostthe same as the HOSM-IGC method But the HOSM-IGCmethod has higher convergence precision and needs loweroverload at the end phase
We can also see that after the target maneuvers if themissile is given enough time to track the targetrsquos maneuvernamely let the missilersquos overload command converge to theoverload of the target theremay not be largemiss distance Inother words for a certain period of time before the collisionthe targetmaneuver (it means only a limitedmaneuver whichdoes not include the condition that the maneuvering of thetarget for a long time may cause a change of the geometricalrelations between the missile and its target) has a small effecton both types of guidance law
But if themaneuver occurs rather late namely within oneto three seconds before collision when the overload com-mand of guidance law is not yet converged the approaching
Mathematical Problems in Engineering 11
0
5
10
0 5 10 15 20Time (s)
TargetOSMGHOSM-IGC
Miss
ile an
d ta
rget
acce
lera
tion
(G)
minus5
minus10
Figure 11 Missile and target acceleration profile
0
10
20
30
0 5 10 15 20Time (s)
tgo = 1 stgo = 2 stgo = 3 s
tgo = 3 s (diverge)
tgo = 2 s (diverge)
tgo = 1 s (diverge)minus10
minus20
minus30
Miss
ile ac
cele
ratio
n (G
)
HOSM-IGC (120573 = 10)
Figure 12 Missile acceleration profile
collision increases themiss distanceTherefore increasing theparameter 120573 can remarkably increase the convergence speedand effectively enhance guidance precision As Figure 12shows when 120573 = 10 the overload at the end phase convergesslowly even if the target maneuvers three seconds beforecollision the missilersquos overload still has no time to convergebeing unable to track themaneuvering target Figure 13 showsthat when 120573 increases to 30 and tgo = 3 seconds theoverload can converge to about 5 g but when tgo = 2 sec-onds the overload may continue to increase indicating that
0
10
20
30
0 5 10 15 20Time (s)
tgo = 1 stgo = 2 stgo = 3 s
tgo = 1 s (diverge)
tgo = 2 s (diverge)
tgo = 3 s (converge)
minus10
minus20
minus30
Miss
ile ac
cele
ratio
n (G
)
120573 = 30)HOSM-IGC (
Figure 13 Missile acceleration profile
0
10
20
30
0 5 10 15 20Time (s)
tgo = 1 s (diverge)
tgo = 2 s (converge)
tgo = 3 s (converge)
tgo = 1 stgo = 2 stgo = 3 s
minus10
minus20
minus30
Miss
ile ac
cele
ratio
n (G
)
HOSM-IGC (120573 = 50)
Figure 14 Missile acceleration profile
the convergence still needs to be quickened thus continuingto rise 120573 to 50 Figure 14 shows that the HOSM-IGC methodcan track the target that maneuvers when tgo = 2 secondsbut it causes the divergence of overload and the increase ofmiss distance if the target maneuvers when tgo = 1 secondFigure 15 gives the overload curve of the OSMG guidance lawand shows that the maneuver of the target before collisionmay cause the large-scale oscillation of the missilersquos overloadwhich may diverge to a large numerical value when thecollision occurs in the end
12 Mathematical Problems in Engineering
Table 4 Average miss distances of 50 simulations
Targetmaneuveringtiming
HOSM-IGC120573 = 30
HOSM-IGC120573 = 40
HOSM-IGC120573 = 50
OSMG
tgo = 1second 43539 3224 28936 36116
tgo = 2seconds 25424 18665 09322 35534
tgo = 3seconds 08124 08265 07538 11959
Time (s)0 5 10 15 20
0
10
20
30OSMG
tgo = 1 stgo = 2 stgo = 3 s
minus10
minus20
minus30
Miss
ile ac
cele
ratio
n (G
)
Figure 15 Missile acceleration profile
The analysis in Figures 12 13 14 and 15 shows that theoverload of the missile converges faster and its miss distanceis smaller with increasing 120573 To verify this finding we carryout 50 times Monte Carlo simulations in which the positionand speed of the target have 1 of random difference
The average miss distances are shown in Table 4 andFigure 16 Clearly the timing of the targetrsquos step maneuverdramatically affects the final interception precision morespecifically given a shorter reaction time for the guidanceand control system the missile seems more likely to missthe target To the OSMG guidance law in all three scenarioshardly does it show any advantages against theHOSM-IGC Itcan also be seen that with the increasing of the120573 theHOSM-IGC system responds even faster which leads to an obviousdecrease of the average miss distance The effect of 120573 on theresponse of the HOSM-IGC system is a valuable guidelinewhen implementing the proposed method into practice
53 Scenario 3 Weaving Target In this scenario the targetmaneuvers by 119886
119879= 40sin(1205871199052) OSMG with three-loop
Table 5 Initial conditions
Variable Description Value(119883119872 119884119872) Missile initial position (0 km 175 km)
(119883119879 119884119879) Target initial position (60 km 195 km)
119881119872
Missile initial velocity 1200 (ms)119881119879
Target initial velocity 2000 (ms)120579119872
Missile initial flight path angle 0 (deg)120579119879
Target initial flight path angle 0 (deg)120576 Compensatory gain 3120575 Saturation parameter 0001
autopilot is introduced for comparison The motion equa-tions of the target are as follows
119910119879
= 119881119879cos 120579119879
119879
= minus119881119879sin 120579119879
120579119879
=119886119879
119881119879
119886119879
= 40 sin(120587119905
2)
(57)
The initial simulation conditions are given in Table 5The missilersquos trajectory under the two guidance and
control methods given in Figure 17 does not see muchdifference
However the overload curve given in Figure 18 showsthat after the missile completes its initial attitude adjustmentwith the HOSM-IGC method it can almost ideally track itsmaneuvering target by contrast with the OSMG methodthe missile seems to have the tendency to track its targetrsquosmaneuver but has larger tracking errors Besides with theOSMGmethod the missilersquos overload increases rapidly at theend of attack primarily because of the divergence of its LOSrate On the other hand with the HOSM-IGC method themissile has no divergence even at the end of attack ensuringa smaller target missing quantity
The actuator deflection curve in Figure 19 shows thatin order to provide a rather big normal overload for theend phase the OSMG method produces a rather big rudderdeflection command however it may increase the missilersquostarget missing quantity once its rudder deflection saturatesand themissile does not have enough overloads or the ruddercannot respond that fast
As shown in Figure 20 because of the dramatic changein overload command the response of the missilersquos autopilotto high-frequency command sees an obvious phase lag andamplitude value attenuation its actual overload cannot trackthe command ideally this is a main reason why the missdistance increases However the controller in the HOSM-IGC method gives its rudder deflection command directlyand there is no lagging or attenuation caused by the autopilotthus enhancing the guidance precision effectively Further-more the fast convergence of the high order sliding modemakes the missile rapidly track its maneuvering target withthemost reasonable rudder deflection command reducing itsoverload effectively
Mathematical Problems in Engineering 13
0
05
1
15
2
25
3
35
4
45
5
0 10 20 30 40 50Simulation times
OSMG
Miss
(m)
0
05
1
15
2
25
3
35
4
45
5
0 10 20 30 40 50Simulation times
Miss
(m)
0
05
1
15
2
25
3
35
4
45
5
0 10 20 30 40 50Simulation times
Miss
(m)
tgo = 3 stgo = 2 stgo = 1 s
0
05
1
15
2
25
3
35
4
45
5
0 10 20 30 40 50Simulation times
Miss
(m)
tgo = 3 stgo = 2 stgo = 1 s
HOSM-IGC (120573 = 40)
HOSM-IGC (120573 = 50)
HOSM-IGC (120573 = 30)
Figure 16 Miss distances of HOSM-IGC (120573 = 30 40 50) and OSMG with target maneuvers at different time-to-go (tgo = 1 second 2seconds and 3 seconds)
The average miss distance of 50 simulations under theconditions is 073m for HOSM-IGC and 186m for OSMGWe can see that the HOSM-IGC method not only doesprovide a more reasonable actuator deflection command butalso achieves a higher interception precision
6 Conclusions
This paper proposes an LOS feedback integrated guidanceand control method using quasi-continuous high order
sliding mode guidance and control method With the fastand precise convergence of the quasi-continuous HOSMmethod the HOSM-IGCmethod performsmuch better thanthe traditional separated guidance and control method withless acceleration effort and less miss distance in all thethree simulation scenarios of nonmaneuvering target stepmaneuvering target and weaving target In addition the ideaof virtual control largely alleviates the chattering withoutany sacrifice of robustness As a result of the alleviationof the chattering the control input command 120575
119885becomes
14 Mathematical Problems in Engineering
175
180
185
190
195
200
TargetHOSM-IGCOSMG
Y(k
m)
0 20 40 60 0 20 40X (km)
Figure 17 The trajectories of the missile and its target
0
10
20
0 5 10 15 20Time (s)
TargetOSMGHOSM-IGC
Miss
ile an
d ta
rget
acce
lera
tion
(G)
minus10
minus20
Figure 18 Missile acceleration profile
smooth enough for the actuator to be implemented as aresult the interception accuracy is improved In designingthe quasi-continuous sliding mode controller the Arbitrary-Order Robust Exact Differentiator is introduced to estimatethe high order time derivative of the LOS rate 119902 which notonly ensures the guidance and control precision but alsosimplifies the calculation
0
10
20
30
0 5 10 15 20Time (s)
OSMGHOSM-IGC
minus10
minus20
minus30
minus40
Actu
ator
defl
ectio
n (d
eg)
Figure 19 Actuator deflection
0
50
100
150
200
15 16 17 18Time (s)
Commanded accelerationAchieved acceleration
Miss
ile ac
cele
ratio
n (G
)
minus50
Figure 20 Commanded acceleration and achieved acceleration
Appendices
A The Third-Order RobustExact Differentiator
The third-order robust exact differentiator built according tothe following parameters and equations is used to performthe simulation the initial value 120582
0= 1205821
= 1205822
= 1205823
= 50
Mathematical Problems in Engineering 15
0
0005
001
0015
002
0025
003
0035
004
0 02 04Time (s)
0
2
0 02 04Time (s)
0
100
200
300
0
100
200
300
400
0 02 04Time (s)
h(t)
z3
h(t) and z3
minus100
minus200
minus300
minus400
minus5000 02 04
Time (s)
g(t)
z2
g(t) and z2
minus100
minus200
minus300
minus400
f(t)
z1
f(t) and z1
minus2
minus4
minus6
minus8
minus10
minus12
minus14
F(t)
z0
F(t) and z0
minus0005
minus001
Figure 21 The outputs of the differentiator in the sampling interval of 0001 seconds
the initial values 1199110
= 01 1199111
= 1199112
= 1199113
= 0 119871 = 1400 thesimulation step is 0001 seconds Consider
0
= V0
V0
= minus120582011987114 10038161003816100381610038161199110 minus
1199021003816100381610038161003816
34 sign (1199110
minus 119902) + 1199111
1
= V1
V1
= minus120582111987113 10038161003816100381610038161199111 minus V
0
1003816100381610038161003816
23 sign (1199111
minus V0) + 1199112
2
= V2
V2
= minus120582211987112 10038161003816100381610038161199112 minus V
1
1003816100381610038161003816
12 sign (1199112
minus V1) + 1199113
3
= minus1205823119871 sign (119911
3minus V2)
(A1)
16 Mathematical Problems in Engineering
0
0005
001
0015
002
0025
003
0035
004
0 02 04Time (s)
0
2
0 02 04Time (s)
0
100
200
300
0 02 04Time (s)
0
200
400
0 02 04Time (s)
h(t) and z3
minus200
minus400
minus600
minus800
g(t) and z2
minus100
minus200
minus300
minus400
f(t) and z1
minus2
minus4
minus6
minus8
minus10
minus12
minus14
minus16
F(t) and z0
minus0005
minus001
h(t)
z3g(t)
z2
f(t)
z1F(t)
z0
Figure 22 The outputs of the differentiator in the sampling interval of 00001 seconds
If the designed signals are 119865(119905) = int 119891(119905)119889119905 119891(119905) =
int 119892(119905)119889119905 119892(119905) = int ℎ(119905)119889119905 and ℎ(119905) = sin(2120587119905) +
3cos(120587119905) then theoretically 119891(119905) is the first-order differ-ential of input signal 119892(119905) is the second-order differ-ential of input signal ℎ(119905) is the third-order differen-tial of input signal Through the simulation the differ-ential signal of actual output is compared with that of
theoretical output the comparison results are shown inFigure 21
Figure 21 shows that 1199110can effectively track the signal
119865(119905) but 1199112has obvious errors in tracking the second-order
differential 119892(119905) 1199113has greater errors in tracking the third-
order differential thus the overall differential effect is notideal
Mathematical Problems in Engineering 17
0 02 04
0
0005
001
0015
002
0025
003
0035
004
Time (s)0 02 04
0
01
02
03
04
05
Time (s)
0 02 04
0
05
1
15
2
Time (s)0 02 04
0
05
1
15
2
25
3
35
4
Time (s)
h(t) and z3g(t) and z2
minus04
minus05
f(t) and z1
minus2
minus15
minus1
minus05
minus1
minus05
minus01
minus02
minus03
F(t) and z0
minus0005
minus001
h(t)
z3g(t)
z2
f(t)
z1F(t)
z0
Figure 23 The outputs of the differentiator in the sampling interval of 00001 seconds
There are two causes for the errors in a usual differentia-tor the setting of sampling interval and the setting of numberof orders of the differentiator The following theorem applieshere
Theorem A1 (see [19]) If the sampling interval 120591 gt 0 theaccuracy of the controller is expressed as follows
10038161003816100381610038161003816120590(119894)
10038161003816100381610038161003816le 120583120591119903minus119894
119894 = 0 119903 minus 1 (A2)
That is to say the precision (understood as the differentialprecision of a differentiator) of the 119894th order control is in directproportion to the power of 119903 minus 119894 times of sampling interval120591 Therefore the decrease of iteration step can enhance thedifferential tracking precision
Under the conditions that the sampling interval is 10 timessmaller and that the setting is 00001 seconds the simulationresults are shown in Figure 22
18 Mathematical Problems in Engineering
Table 6 Tracking errors of the AORED with different orders andsampling interval
Orders of AORED Third order Fifth orderSample period 120591 = 0001 120591 = 00001 120591 = 00001
First-order differential error 001 Negligible NegligibleSecond-order differential error 15 002 NegligibleThird-order differential error 70 25 00005
As Figure 22 shows the signal tracking is rather idealexcept a peak in the beginning the tracking error in the first-order differential tracking can be neglected 119911
2has an obviously
smaller tracking error in the second-order differential 119892(119905)
tracking in the beginning 1199113has a rather sharp peak in the
third-order differential tracking but converges to a relativelyhigh precision after 01 seconds As we can see the differentialtracking errors for each number of orders are greatly reducedbut the third-order differential tracking error is still rather bigIn the following we will reduce the tracking errors by increasingthe number of orders of the differentiator
B The Fifth-Order Robust Exact Differentiator
The settings of the fifth-order differentiator are given asfollows
0
= V0
V0
= minus120582011987116 10038161003816100381610038161199110 minus
1199021003816100381610038161003816
56 sign (1199110
minus 119902) + 1199111
1
= V1
V1
= minus120582111987115 10038161003816100381610038161199111 minus V
0
1003816100381610038161003816
45 sign (1199111
minus V0) + 1199112
2
= V2
V2
= minus120582211987114 10038161003816100381610038161199112 minus V
1
1003816100381610038161003816
34 sign (1199112
minus V1) + 1199113
3
= V3
V3
= minus120582311987113 10038161003816100381610038161199113 minus V
2
1003816100381610038161003816
23 sign (1199113
minus V2) + 1199114
4
= V4
V4
= minus120582411987112 10038161003816100381610038161199114 minus V
3
1003816100381610038161003816
12 sign (1199114
minus V3) + 1199115
5
= V5
V5
= minus1205825119871 sign (119911
5minus V4)
(B1)
The initial value 1199110
= 01 1205820
= 1205821
= 1205822
= 1205823
= 1205824
=
1205825
= 50 the initial values 1199111 1199112 1199113 1199114 1199115
= 0 119871 = 1400the simulation step length is 00001 seconds The simulationresults are shown in Figure 23
As the figure shows the fifth-order differentiator hasa rather ideal tracking effect on the third-order differen-tial tracking the tracking error being only about 00005(see Table 6)
To sum up the Arbitrary-Order Robust Exact Differen-tiator can estimate rather ideal differential signals by selecting
Table 7
Length 365m119871 ref 365m119883119866
177m119878ref 0026m2
Diameter 0178mMass 1016 Kg119868119885
1063 Kgsdotm2
an appropriate sampling interval and using the differentiatorwith a relatively high number of orders
C Physical and Geometric Characteristics
See Table 7
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] P Zarchan Tactical and Strategic Missile Guidance 2002[2] D Zhou CMu andWXu ldquoAdaptive sliding-mode guidance of
a homing missilerdquo Journal of Guidance Control and Dynamicsvol 22 no 4 pp 589ndash594 1999
[3] G Zeng and M Hu ldquoFinite-time control for electromagneticsatellite formationsrdquo Acta Astronautica vol 74 pp 120ndash1302012
[4] S R Kumar S Rao and D Ghose ldquoNonsingular terminalsliding mode guidance with impact angle constraintsrdquo Journalof Guidance Control and Dynamics vol 37 no 4 pp 1114ndash11302014
[5] Y Liang C Chen D Liaw Y Feng C Cheng and CChen ldquoRobust guidance law via integral-sliding-mode schemerdquoJournal of Guidance Control and Dynamics vol 37 no 3 pp1038ndash1042 2014
[6] A V Savkin P N Pathirana and F Faruqi ldquoProblem ofprecision missile guidance LQR and 119867
infincontrol frameworksrdquo
IEEE Transactions on Aerospace and Electronic Systems vol 39no 3 pp 901ndash910 2003
[7] H Y Chen andC-C Yang ldquoNonlinear1198672119867infinguidance design
for homing missilesrdquo in Proceedings of the AIAA GuidanceNavigation and Control Conference and Exhibit vol 77 2001
[8] P K Menon G D Sweriduk and E J Ohlmeyer ldquoOptimalfixed-interval integrated guidance-control laws for hit-to-killmissilesrdquo in Proceedings of the AIAA Guidance Navigation andControl Conference and Exhibit AIAA 2003-5579 Austin TexUSA August 2003
[9] S S Vaddi P K Menon and E J Ohlmeyer ldquoNumericalSDRE approach for missile integrated guidance-controlrdquo inProceedings of the AIAA Guidance Navigation and ControlConference and Exhibit AIAA 2007-6672 Hilton Head SCUSA August 2007
[10] M Xin S N Balakrishnan and E J Ohlmeyer ldquoIntegratedguidance and control of missiles with Theta-D methodrdquo IEEE
Mathematical Problems in Engineering 19
Transactions on Control Systems Technology vol 14 no 6 pp981ndash992 2006
[11] T Shima M Idan and O M Golan ldquoSliding-mode controlfor integrated missile autopilot guidancerdquo Journal of GuidanceControl and Dynamics vol 29 no 2 pp 250ndash260 2006
[12] Y B Shtessel and C H Tournes ldquoIntegrated higher-ordersliding mode guidance and autopilot for dual control missilesrdquoJournal of Guidance Control and Dynamics vol 32 no 1 pp79ndash94 2009
[13] F Y Dong H M Lei and C J Zhou ldquoIntegrated robust highorder sliding mode guidance and control for missilesrdquo ActaAeronautica et Astronautica Sinica vol 34 no 9 pp 2212ndash22182013
[14] H Mingzhe and D Guangren ldquoIntegrated guidance and con-trol of homing missiles against ground fixed targetsrdquo ChineseJournal of Aeronautics vol 21 no 2 pp 162ndash168 2008
[15] A Levant ldquoHomogeneity approach to high-order sliding modedesignrdquo Automatica vol 41 no 5 pp 823ndash830 2005
[16] Y-W Liang C Chen D Liaw Y Feng C Cheng and CChen ldquoRobust guidance law via integral-sliding-mode schemerdquoJournal of Guidance Control and Dynamics vol 37 no 3 pp1038ndash1042 2014
[17] L Wang Y Sheng and X Liu ldquoContinuous time-varying slid-ingmode based attitude control for reentry vehiclerdquo Proceedingsof the Institution of Mechanical Engineers Part G Journal ofAerospace Engineering 2014
[18] A Levant ldquoHigher-order sliding modes differentiation andoutput-feedback controlrdquo International Journal of Control vol76 no 9-10 pp 924ndash941 2003
[19] A Levant ldquoQuasi-continuous high-order sliding-mode con-trollersrdquo IEEE Transactions on Automatic Control vol 50 no11 pp 1812ndash1816 2005
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
6 Mathematical Problems in Engineering
Summing up the above conditions then we can get
119892 (119905 119909) gt 0 (42)
In other words there is a positive real number 119870119898existing
that could satisfy the following condition
0 lt 119870119898
lt 119892 (119905 119909) (43)
Before the missile hits on the target the term will be positiveand limited then we can get
0 lt 119870119898
lt 119892 (119905 119909) lt 119870119872
(44)
With (37) then
ℎ (119905 119909) =1
119903(1198911
+ 1198912
+ 1198913
+ 1198914
+ 1198915
+ 1198916) (45)
In the practice sense the changes in both the LOS rateand the acceleration of the missile and the acceleration of thetarget are limited and continuous So the following variables119902 119902
119902 119886119872 119886119879 and 119886
119879are all bounded However because
ℎ(119905 119909) contains the item 1119903 when the relative distancebetween the missile and its target is zero the boundaryof ℎ(119905 119909) is not guaranteed In [15] Levant only requiresthat condition (39) should be locally valid not requiringthat it should be globally valid Therefore the integratedguidance and controlmethod is applicable here So the abovementioned condition is satisfied with a positive number 119862
|ℎ (119905 119909)| le 119862 (46)
33TheVirtual ControlDesign Whenusing the slidingmodecontrolmethod the avoidance of the chattering phenomenonhas always been a key issue being discussed In the tradi-tional method researchers in [16 17] have proposed severalsaturation functions to replace the sign functions to builda boundary layer to alleviate the chattering or to use fuzzylogic to displace the high-frequency switching term To ourknowledge none of these approaches has proven that therefined controller still retains their robustness against theuncertainties and disturbances In this work in order toalleviate the chattering phenomenon we do not directly usethe third-order controller but introduce the virtual control119906119894= 120575119885to perform the actual control
120575119885
= int 120575119885dt = int 119906
119894dt (47)
After the relative degree is increased to the fourth order weget the following expressions
120590(4)
= ℎlowast
(119905 119909) + 119892lowast
(119905 119909) 119906119894= ℎlowast
(119905 119909) + 119892 (119905 119909) 119906119894
ℎlowast
(119905 119909) = ℎ (119905 119909) + 119892 (119905 119909) 120575119885
119892 (119905 119909)
=119898120575119885
1199111198762
1198782
119897119862120572
119884
119869119885
1198981199032[( 119902 minus 120579
119872) 119903 sin 120578
119872+ 119903 cos 120578
119872]
(48)
Even though the expression of ℎ(119905 119909) is rather compli-cated it is still the function of 119902 119902 119902 119886
119872 119886119879 and 119886
119879 therefore
similar to ℎ(119905 119909) it has its boundary except themomentwhenthe missile hits on its target For the same reason 119892(119905 119909)
and the rudder deflection 120575119885also have their boundaries
Therefore we get the condition that |ℎlowast
(119905 119909)| le 119862 (119862 gt 0)Because 120575
119885is obtained through the derivation of 120575
119885 119892lowast(119905 119909)
is the same as 119892(119905 119909) thus 0 lt 119870119898
lt 119892lowast
(119905 119909) lt 119870119872
issatisfied
According to the formula of the fourth-order controllergiven by Levant in [15] we give the following formulae forthe virtual control 119906
119894
119906119894= minus120573
Φ34
11987334
Φ34
=120590 + 3 [||
+ (|| + 05 |120590|34
)minus13 10038161003816100381610038161003816
+ 05 |120590|34 sign120590
10038161003816100381610038161003816]
minus12
sdot [
+ (|| + 05 |120590|34
)minus13 10038161003816100381610038161003816
+ 05 |120590|34 sign120590
10038161003816100381610038161003816]
11987334
=1003816100381610038161003816
120590
1003816100381610038161003816 + 3 [||
+ (|| + 05 |120590|34
)minus13 10038161003816100381610038161003816
+ 05 |120590|34 sign120590
10038161003816100381610038161003816]
minus12
sdot
1003816100381610038161003816100381610038161003816
+ (|| + 05 |120590|34
)minus13
( + 05 |120590|34 sign120590)
1003816100381610038161003816100381610038161003816
(49)
The integral of the virtual control 120575119885produces the actual
control command 120575119885 120590 in the controller is obtained with the
Arbitrary-Order Robust Exact Differentiator presented in thefollowing section
34 The Arbitrary-Order Robust Exact Differentiator Thequasi-continuous HOSM control method needs to use thethird derivative of the sliding manifold namely 119902
(4) Howto calculate or accurately estimate 119902
(4) is one of the keyproblems to be solved We use the Arbitrary-Order RobustExact Differentiator designed by Levant to differentiate theLOS rate 119902 thus obtaining 119902 119902 and 119902
(4)According to (44) and (45) the following condition is
valid1003816100381610038161003816
120590
1003816100381610038161003816 le 119862 + 120573119870119872
(50)
The Arbitrary-Order Robust Exact Differentiator can beconstructed in accordance with high order sliding modesdifferentiation and output feedback control in [18]
If a certain signal 119891(119905) is a function consisting of abounded Lebesgue-measurable noise with unknown base
Mathematical Problems in Engineering 7
signal 1198910(119905) whose 119903th derivative has a known Lipschitz
constant 119871 gt 0 then the 119899th-order differentiator is definedas follows
0
= V0
V0
= minus12058201198711(119899+1) 10038161003816100381610038161199110 minus 119891 (119905)
1003816100381610038161003816
119899(119899+1) sign (1199110
minus 119891 (119905))
+ 1199111
1
= V1
V1
= minus12058211198712(119899+1) 10038161003816100381610038161199111 minus V
0
1003816100381610038161003816
(119899minus1)119899 sign (1199111
minus V0) + 1199112
119899minus1
= V119899minus1
V119899minus1
= minus120582119899minus1
11987112 1003816100381610038161003816119911119899minus1 minus V
119899minus2
1003816100381610038161003816
12 sign (119911119899minus1
minus V119899minus2
)
+ 119911119899
119899
= minus120582119899119871 sign (119911
119899minus V119899minus1
)
(51)
and if 120582119894
gt 0 is sufficiently large the convergence is guaran-teed
To obtain the third-order derivative of 119902 we constructthe third-order sliding mode differentiator and estimate thederivative of 119902 for each order In view of differential precisionwe configure the following fifth-order differentiator SeeAppendix A for comparison
0
= V0
V0
= minus120582011987116 10038161003816100381610038161199110 minus
1199021003816100381610038161003816
56 sign (1199110
minus 119902) + 1199111
1
= V1
V1
= minus120582111987115 10038161003816100381610038161199111 minus V
0
1003816100381610038161003816
45 sign (1199111
minus V0) + 1199112
2
= V2
V2
= minus120582211987114 10038161003816100381610038161199112 minus V
1
1003816100381610038161003816
34 sign (1199112
minus V1) + 1199113
3
= V3
V3
= minus120582311987113 10038161003816100381610038161199113 minus V
2
1003816100381610038161003816
23 sign (1199113
minus V2) + 1199114
4
= V4
V4
= minus120582411987112 10038161003816100381610038161199114 minus V
3
1003816100381610038161003816
12 sign (1199114
minus V3) + 1199115
5
= V5
V5
= minus1205825119871 sign (119911
5minus V4)
(52)
where 1199113 1199112 1199111 and 119911
0are the estimations of 119902
(4) 119902 119902 and 119902
4 Baseline Separated Guidance andControl Method
To verify the homing performance of the integrated guidanceand control method we compare it with the separatedguidance and control methodThe guidance laws we used arethe proportional navigation (PN) guidance law for attackinga nonmaneuvering target and the optimal sliding modeguidance law for attacking a maneuvering target
41 The Proportional Navigation (PN) Guidance Law Theguidance law has a simple formula and excellent perfor-mances for nonmaneuvering target Its formula is as follows
119899119888
= minus119873 119902119881119872
119892 (53)
119899119888is the overload command 119873 is the effective navigation
ratio 119902 is the LOS rate 119881119872
is the speed of the missile 119892 isthe acceleration of the gravity The guidance law gives theoverload command of the missile according to the LOS rateand then the controller gives the rudder deflection commandaccording to the overload command
42 The Optimal Sliding Mode Guidance Law The optimalsliding mode guidance law (OSMG) is a novel practicalguidance law proposed by D Zhou He combines the optimalguidance lawwith the slidingmode guidance law and designsthe new sliding mode guidance law that not only is robustto maneuvering target but also has the merits of the optimalguidance law such as good dynamic performance and energyconservation Its formula is as follows
119899119888
= minus3100381610038161003816100381610038160
10038161003816100381610038161003816119902 + 120576
119902
10038161003816100381610038161199021003816100381610038161003816 + 120575
(54)
where 119899119888is the overload command
0is the approach
velocity of the missile and its target 119902 is their LOS rate 120576 =
const is the compensatory gain 119902(| 119902| + 120575) is for substitutingfor sign( 119902) and for smoothing 120575 is a small quantity whichcould adjust the chattering
43 Separated Guidance and Control Design For simulationand comparisonwe use the conventional three-loop overloadautopilot as the controller which gives the rudder deflectioncommand according to the feedback of the three loops ofoverload pseudo-angle of attack and pitch rate The blockdiagram is as shown in Figure 3
As the figure shows the inner loop has the feedback onangular velocity which improves the damping characteristicsof the missile airframe
According to the aerodynamic coefficient of the missilewith selected working points we set 119870
119868= 019 119870
120572= 3 and
119870120596
= minus025 and the controller can well track the overloadcommand the rise time of its step response is 046 secondsand its settling time is 083 secondsThe step responses of themissile to overload command and the Bode diagram for openloop are shown in Figure 4
8 Mathematical Problems in Engineering
KIS
120596Z 120572 nY
nC K120596K120572 nY+minus+minus+minus dynamicsAirframe
modelServo
Figure 3 The working principles for three-loop overload autopilot
Step response
Time (s)0 02 04 06 08 1 12
0
02
04
06
08
1
Rise time (s) 0463
Settling time (s) 0831
Bode diagram
Frequency (rads)
To output pointFrom input pointTo output pointFrom input point
Gain margin (dB) 175 At frequency (rads) 15
Phase margin (deg) 709 At frequency (rads) 294
Am
plitu
de
Mag
nitu
de (d
B)Ph
ase (
deg)
0
minus180
minus360
minus54010410310210110010minus1
100
0
minus100
minus200
minus300
Figure 4 The autopilot performance step response and Bode diagram
5 Simulation Results
To verify the high order slidingmode integrated guidance andcontrol (HOSM-IGC) method we compare it with the base-line separated guidance and control Numerical simulationsare designed in three typical engagement scenarios
AORED parameters are as follows the initial value 1205820
=
1205821
= 1205822
= 1205823
= 50 1199110
= 01 1199111
= 1199112
= 1199113
= 0 119871 = 1400 thesimulation step is 00001 seconds
51 Scenario 1 Nonmaneuvering Target In the first scenariothe nonmaneuvering target does uniform rectilinear motionand PN guidance law with three-loop autopilot is introducedfor a comparison with the HOSM-IGCThe initial conditionsare set as shown in Table 1
The motion equations of the target are as follows
119910119879
= 119881119879cos 120579119879
119879
= minus119881119879sin 120579119879
120579119879
=119886119879
119881119879
119886119879
= 0
(55)
The simulation results are shown in Figures 5 and 6The missilersquos flight trajectory and overload curve show
that within the first 3 sec the HOSM-IGC method spendsmuch energy (overload) on changing the initial LOSdirection
Table 1 Initial conditions
Variable Description Value(119883119872 119884119872) Missile initial position (0 km 18 km)
(119883119879 119884119879) Target initial position (60 km 20 km)
119881119872
Missile initial velocity 1200 (ms)119881119879
Target initial velocity 2000 (ms)120579119872
Missile initial flight path angle 0 (deg)120579119879
Target initial flight path angle 0 (deg)120573 Parameter of the controller 10 or 30119873 Navigation ratio of PN 3
of the missile After the boresight adjustment 120590 reaches thedesired sliding manifold and the missile does not need anymaneuver to hit the target This is because the target doesnot maneuver any more which means no disturbance isintroduced in the engagement dynamic so the state (or thesliding mode) of the missile will stay on the manifold Incontrast the overload command given by the PN guidancelaw increases fast as the relative distance decreases Toexamine the performances of the two guidance laws furtherwe increase the target speed and analyze the commandchanges
Therefore we set the target speeds 3000ms 4000msand carry out simulations The simulation result is shown inthe overload curve in Figure 7
Mathematical Problems in Engineering 9
175
180
185
190
195
200
205
TargetHOSM-IGCPN
Y(k
m)
0 10 20 30 40 50 60X (km)
Figure 5 Target and missile trajectories
0
5
10
15
20
25
0 5 10 15 20Time (s)
HOSM-IGCPN
Initial boresight adjustment
Sliding manifold reached
Miss
ile ac
cele
ratio
n (G
)
minus5
Figure 6 Missile acceleration profile
It is evident that the speed of divergence of the overloadcommand given by the PN guidance law increases withthe target speed Specifically when the target speed reaches4000ms the commanded overload is almost 30 g which isobviously not ideal for the attack of a nonmaneuvering targetbut with the HOSM-IGC method the missile adjusts itsboresight very quickly then maintains it around 0 g and fliesto its target in the rectilinear ballistic trajectory not affectedby the increases of target speed still accomplishing the high-precision hit-on collision
Figure 8 shows that although the HOSM-IGC methodachieves a more effective overload command on the otherside it sees some chattering when the sliding mode reaches
0
5
10
15
20
25
30
0 5 10 15Time (s)
Miss
ile ac
cele
ratio
n (G
)
minus5
HOSM-IGC (Vt = minus3000)HOSM-IGC (Vt = minus4000)
PN (Vt = minus3000)PN (Vt = minus4000)
Figure 7 Missile acceleration profile
0
1
2
0 5 10 15 20Time (s)
4 45 5
0
02
04
minus02
minus04
Actu
ator
defl
ectio
n (d
eg)
minus1
minus2
minus3
minus4
HOSM-IGC (120573 = 30)
Figure 8 Actuator deflection
the desired manifold the commanded rudder deflectionangle chatters for around 045 deg at about 15Hz which iskind of harmful to the system To reduce the chatteringwe adjust the controllerrsquos parameter 120573 = 10 and carry outsimulations again Figure 9 shows that after 120573 decreasesthe command of rudder deflection angle converges slower(for about 7 sec) however the chattering weakens obviouslyits magnitude being only about 015 degrees The smallerchattering well enhances convergence precision eventuallyreducing the target missing The reason that minor 120573 resultsin an alleviative chattering can be seen from (47) Figure 10shows the target and missile trajectories
Table 2 compares the average miss distance of 50 simula-tions under the conditions discussed above the comparison
10 Mathematical Problems in Engineering
Table 2 Average miss distance of 50 simulations
Target speed PN HOSM-IGC(120573 = 30)
HOSM-IGC(120573 = 10)
119881119905= 2000ms 115m 086 073m
119881119905= 3000ms 261m 117 106
119881119905= 4000ms 542m 156 134
0
1
2
0 5 10 15 20Time (s)
72 74 76 78
0
02
minus02
minus04
Actu
ator
defl
ectio
n (d
eg)
minus1
minus2
minus3
minus4
HOSM-IGC (120573 = 10)HOSM-IGC (120573 = 30)
Figure 9 Actuator deflection
results show that when the missile intercepts a nonmaneu-vering target the hit precision of the HOSM-IGC methodis apparently higher than that of the PN guidance law andthat the low-gain HOSM-IGC method can effectively reducethe chattering magnitude thus enhancing the interceptionprecision
52 Scenario 2 Step Maneuvering Target At the first stagethe target flies at uniform speed and in a rectilinear way after10 seconds it maneuvers at the normal acceleration of 5 g Inthis scenario OSMG is introduced for a comparison with theHOSM-IGC
119910119879
= 119881119879cos 120579119879
119879
= minus119881119879sin 120579119879
120579119879
=119886119879
119881119879
119886119879
=
0 g 119905 lt 10 sec
5 g 119905 gt 10 sec
(56)
The initial simulation conditions are given in Table 3The overload curve in Figure 11 shows dearly that both
types of guidance laws can track the maneuvering targetDuring 0 to 10 seconds the target flies at uniform speed andin a rectilinear way and the missile converges its overload to
Table 3 Initial conditions
Variable Description Value(119883119872 119884119872) Missile initial position (0 km 22 km)
(119883119879 119884119879) Target initial position (60 km 20 km)
119881119872
Missile initial velocity 1200 (ms)119881119879
Target initial velocity 2000 (ms)120579119872
Missile initial flight path angle 0 (deg)120579119879
Target initial flight path angle 0 (deg)120576 Compensatory gain 3120575 Saturation parameter 0001
15
17
19
21
23
25
TargetHOSM-IGCOSMG
Y(k
m)
0 20 40 60X (km)
Figure 10 Target and missile trajectories
0 g and flies to its target after 10 seconds the target beginsto maneuver by 5 g Both types of guidance law respondby rapidly increasing the overload adjusting attitude andmaking the missilersquos boresight aim at its target
We can see that when both types of guidance law tracktheir targets the convergence speed of OSMG is almostthe same as the HOSM-IGC method But the HOSM-IGCmethod has higher convergence precision and needs loweroverload at the end phase
We can also see that after the target maneuvers if themissile is given enough time to track the targetrsquos maneuvernamely let the missilersquos overload command converge to theoverload of the target theremay not be largemiss distance Inother words for a certain period of time before the collisionthe targetmaneuver (it means only a limitedmaneuver whichdoes not include the condition that the maneuvering of thetarget for a long time may cause a change of the geometricalrelations between the missile and its target) has a small effecton both types of guidance law
But if themaneuver occurs rather late namely within oneto three seconds before collision when the overload com-mand of guidance law is not yet converged the approaching
Mathematical Problems in Engineering 11
0
5
10
0 5 10 15 20Time (s)
TargetOSMGHOSM-IGC
Miss
ile an
d ta
rget
acce
lera
tion
(G)
minus5
minus10
Figure 11 Missile and target acceleration profile
0
10
20
30
0 5 10 15 20Time (s)
tgo = 1 stgo = 2 stgo = 3 s
tgo = 3 s (diverge)
tgo = 2 s (diverge)
tgo = 1 s (diverge)minus10
minus20
minus30
Miss
ile ac
cele
ratio
n (G
)
HOSM-IGC (120573 = 10)
Figure 12 Missile acceleration profile
collision increases themiss distanceTherefore increasing theparameter 120573 can remarkably increase the convergence speedand effectively enhance guidance precision As Figure 12shows when 120573 = 10 the overload at the end phase convergesslowly even if the target maneuvers three seconds beforecollision the missilersquos overload still has no time to convergebeing unable to track themaneuvering target Figure 13 showsthat when 120573 increases to 30 and tgo = 3 seconds theoverload can converge to about 5 g but when tgo = 2 sec-onds the overload may continue to increase indicating that
0
10
20
30
0 5 10 15 20Time (s)
tgo = 1 stgo = 2 stgo = 3 s
tgo = 1 s (diverge)
tgo = 2 s (diverge)
tgo = 3 s (converge)
minus10
minus20
minus30
Miss
ile ac
cele
ratio
n (G
)
120573 = 30)HOSM-IGC (
Figure 13 Missile acceleration profile
0
10
20
30
0 5 10 15 20Time (s)
tgo = 1 s (diverge)
tgo = 2 s (converge)
tgo = 3 s (converge)
tgo = 1 stgo = 2 stgo = 3 s
minus10
minus20
minus30
Miss
ile ac
cele
ratio
n (G
)
HOSM-IGC (120573 = 50)
Figure 14 Missile acceleration profile
the convergence still needs to be quickened thus continuingto rise 120573 to 50 Figure 14 shows that the HOSM-IGC methodcan track the target that maneuvers when tgo = 2 secondsbut it causes the divergence of overload and the increase ofmiss distance if the target maneuvers when tgo = 1 secondFigure 15 gives the overload curve of the OSMG guidance lawand shows that the maneuver of the target before collisionmay cause the large-scale oscillation of the missilersquos overloadwhich may diverge to a large numerical value when thecollision occurs in the end
12 Mathematical Problems in Engineering
Table 4 Average miss distances of 50 simulations
Targetmaneuveringtiming
HOSM-IGC120573 = 30
HOSM-IGC120573 = 40
HOSM-IGC120573 = 50
OSMG
tgo = 1second 43539 3224 28936 36116
tgo = 2seconds 25424 18665 09322 35534
tgo = 3seconds 08124 08265 07538 11959
Time (s)0 5 10 15 20
0
10
20
30OSMG
tgo = 1 stgo = 2 stgo = 3 s
minus10
minus20
minus30
Miss
ile ac
cele
ratio
n (G
)
Figure 15 Missile acceleration profile
The analysis in Figures 12 13 14 and 15 shows that theoverload of the missile converges faster and its miss distanceis smaller with increasing 120573 To verify this finding we carryout 50 times Monte Carlo simulations in which the positionand speed of the target have 1 of random difference
The average miss distances are shown in Table 4 andFigure 16 Clearly the timing of the targetrsquos step maneuverdramatically affects the final interception precision morespecifically given a shorter reaction time for the guidanceand control system the missile seems more likely to missthe target To the OSMG guidance law in all three scenarioshardly does it show any advantages against theHOSM-IGC Itcan also be seen that with the increasing of the120573 theHOSM-IGC system responds even faster which leads to an obviousdecrease of the average miss distance The effect of 120573 on theresponse of the HOSM-IGC system is a valuable guidelinewhen implementing the proposed method into practice
53 Scenario 3 Weaving Target In this scenario the targetmaneuvers by 119886
119879= 40sin(1205871199052) OSMG with three-loop
Table 5 Initial conditions
Variable Description Value(119883119872 119884119872) Missile initial position (0 km 175 km)
(119883119879 119884119879) Target initial position (60 km 195 km)
119881119872
Missile initial velocity 1200 (ms)119881119879
Target initial velocity 2000 (ms)120579119872
Missile initial flight path angle 0 (deg)120579119879
Target initial flight path angle 0 (deg)120576 Compensatory gain 3120575 Saturation parameter 0001
autopilot is introduced for comparison The motion equa-tions of the target are as follows
119910119879
= 119881119879cos 120579119879
119879
= minus119881119879sin 120579119879
120579119879
=119886119879
119881119879
119886119879
= 40 sin(120587119905
2)
(57)
The initial simulation conditions are given in Table 5The missilersquos trajectory under the two guidance and
control methods given in Figure 17 does not see muchdifference
However the overload curve given in Figure 18 showsthat after the missile completes its initial attitude adjustmentwith the HOSM-IGC method it can almost ideally track itsmaneuvering target by contrast with the OSMG methodthe missile seems to have the tendency to track its targetrsquosmaneuver but has larger tracking errors Besides with theOSMGmethod the missilersquos overload increases rapidly at theend of attack primarily because of the divergence of its LOSrate On the other hand with the HOSM-IGC method themissile has no divergence even at the end of attack ensuringa smaller target missing quantity
The actuator deflection curve in Figure 19 shows thatin order to provide a rather big normal overload for theend phase the OSMG method produces a rather big rudderdeflection command however it may increase the missilersquostarget missing quantity once its rudder deflection saturatesand themissile does not have enough overloads or the ruddercannot respond that fast
As shown in Figure 20 because of the dramatic changein overload command the response of the missilersquos autopilotto high-frequency command sees an obvious phase lag andamplitude value attenuation its actual overload cannot trackthe command ideally this is a main reason why the missdistance increases However the controller in the HOSM-IGC method gives its rudder deflection command directlyand there is no lagging or attenuation caused by the autopilotthus enhancing the guidance precision effectively Further-more the fast convergence of the high order sliding modemakes the missile rapidly track its maneuvering target withthemost reasonable rudder deflection command reducing itsoverload effectively
Mathematical Problems in Engineering 13
0
05
1
15
2
25
3
35
4
45
5
0 10 20 30 40 50Simulation times
OSMG
Miss
(m)
0
05
1
15
2
25
3
35
4
45
5
0 10 20 30 40 50Simulation times
Miss
(m)
0
05
1
15
2
25
3
35
4
45
5
0 10 20 30 40 50Simulation times
Miss
(m)
tgo = 3 stgo = 2 stgo = 1 s
0
05
1
15
2
25
3
35
4
45
5
0 10 20 30 40 50Simulation times
Miss
(m)
tgo = 3 stgo = 2 stgo = 1 s
HOSM-IGC (120573 = 40)
HOSM-IGC (120573 = 50)
HOSM-IGC (120573 = 30)
Figure 16 Miss distances of HOSM-IGC (120573 = 30 40 50) and OSMG with target maneuvers at different time-to-go (tgo = 1 second 2seconds and 3 seconds)
The average miss distance of 50 simulations under theconditions is 073m for HOSM-IGC and 186m for OSMGWe can see that the HOSM-IGC method not only doesprovide a more reasonable actuator deflection command butalso achieves a higher interception precision
6 Conclusions
This paper proposes an LOS feedback integrated guidanceand control method using quasi-continuous high order
sliding mode guidance and control method With the fastand precise convergence of the quasi-continuous HOSMmethod the HOSM-IGCmethod performsmuch better thanthe traditional separated guidance and control method withless acceleration effort and less miss distance in all thethree simulation scenarios of nonmaneuvering target stepmaneuvering target and weaving target In addition the ideaof virtual control largely alleviates the chattering withoutany sacrifice of robustness As a result of the alleviationof the chattering the control input command 120575
119885becomes
14 Mathematical Problems in Engineering
175
180
185
190
195
200
TargetHOSM-IGCOSMG
Y(k
m)
0 20 40 60 0 20 40X (km)
Figure 17 The trajectories of the missile and its target
0
10
20
0 5 10 15 20Time (s)
TargetOSMGHOSM-IGC
Miss
ile an
d ta
rget
acce
lera
tion
(G)
minus10
minus20
Figure 18 Missile acceleration profile
smooth enough for the actuator to be implemented as aresult the interception accuracy is improved In designingthe quasi-continuous sliding mode controller the Arbitrary-Order Robust Exact Differentiator is introduced to estimatethe high order time derivative of the LOS rate 119902 which notonly ensures the guidance and control precision but alsosimplifies the calculation
0
10
20
30
0 5 10 15 20Time (s)
OSMGHOSM-IGC
minus10
minus20
minus30
minus40
Actu
ator
defl
ectio
n (d
eg)
Figure 19 Actuator deflection
0
50
100
150
200
15 16 17 18Time (s)
Commanded accelerationAchieved acceleration
Miss
ile ac
cele
ratio
n (G
)
minus50
Figure 20 Commanded acceleration and achieved acceleration
Appendices
A The Third-Order RobustExact Differentiator
The third-order robust exact differentiator built according tothe following parameters and equations is used to performthe simulation the initial value 120582
0= 1205821
= 1205822
= 1205823
= 50
Mathematical Problems in Engineering 15
0
0005
001
0015
002
0025
003
0035
004
0 02 04Time (s)
0
2
0 02 04Time (s)
0
100
200
300
0
100
200
300
400
0 02 04Time (s)
h(t)
z3
h(t) and z3
minus100
minus200
minus300
minus400
minus5000 02 04
Time (s)
g(t)
z2
g(t) and z2
minus100
minus200
minus300
minus400
f(t)
z1
f(t) and z1
minus2
minus4
minus6
minus8
minus10
minus12
minus14
F(t)
z0
F(t) and z0
minus0005
minus001
Figure 21 The outputs of the differentiator in the sampling interval of 0001 seconds
the initial values 1199110
= 01 1199111
= 1199112
= 1199113
= 0 119871 = 1400 thesimulation step is 0001 seconds Consider
0
= V0
V0
= minus120582011987114 10038161003816100381610038161199110 minus
1199021003816100381610038161003816
34 sign (1199110
minus 119902) + 1199111
1
= V1
V1
= minus120582111987113 10038161003816100381610038161199111 minus V
0
1003816100381610038161003816
23 sign (1199111
minus V0) + 1199112
2
= V2
V2
= minus120582211987112 10038161003816100381610038161199112 minus V
1
1003816100381610038161003816
12 sign (1199112
minus V1) + 1199113
3
= minus1205823119871 sign (119911
3minus V2)
(A1)
16 Mathematical Problems in Engineering
0
0005
001
0015
002
0025
003
0035
004
0 02 04Time (s)
0
2
0 02 04Time (s)
0
100
200
300
0 02 04Time (s)
0
200
400
0 02 04Time (s)
h(t) and z3
minus200
minus400
minus600
minus800
g(t) and z2
minus100
minus200
minus300
minus400
f(t) and z1
minus2
minus4
minus6
minus8
minus10
minus12
minus14
minus16
F(t) and z0
minus0005
minus001
h(t)
z3g(t)
z2
f(t)
z1F(t)
z0
Figure 22 The outputs of the differentiator in the sampling interval of 00001 seconds
If the designed signals are 119865(119905) = int 119891(119905)119889119905 119891(119905) =
int 119892(119905)119889119905 119892(119905) = int ℎ(119905)119889119905 and ℎ(119905) = sin(2120587119905) +
3cos(120587119905) then theoretically 119891(119905) is the first-order differ-ential of input signal 119892(119905) is the second-order differ-ential of input signal ℎ(119905) is the third-order differen-tial of input signal Through the simulation the differ-ential signal of actual output is compared with that of
theoretical output the comparison results are shown inFigure 21
Figure 21 shows that 1199110can effectively track the signal
119865(119905) but 1199112has obvious errors in tracking the second-order
differential 119892(119905) 1199113has greater errors in tracking the third-
order differential thus the overall differential effect is notideal
Mathematical Problems in Engineering 17
0 02 04
0
0005
001
0015
002
0025
003
0035
004
Time (s)0 02 04
0
01
02
03
04
05
Time (s)
0 02 04
0
05
1
15
2
Time (s)0 02 04
0
05
1
15
2
25
3
35
4
Time (s)
h(t) and z3g(t) and z2
minus04
minus05
f(t) and z1
minus2
minus15
minus1
minus05
minus1
minus05
minus01
minus02
minus03
F(t) and z0
minus0005
minus001
h(t)
z3g(t)
z2
f(t)
z1F(t)
z0
Figure 23 The outputs of the differentiator in the sampling interval of 00001 seconds
There are two causes for the errors in a usual differentia-tor the setting of sampling interval and the setting of numberof orders of the differentiator The following theorem applieshere
Theorem A1 (see [19]) If the sampling interval 120591 gt 0 theaccuracy of the controller is expressed as follows
10038161003816100381610038161003816120590(119894)
10038161003816100381610038161003816le 120583120591119903minus119894
119894 = 0 119903 minus 1 (A2)
That is to say the precision (understood as the differentialprecision of a differentiator) of the 119894th order control is in directproportion to the power of 119903 minus 119894 times of sampling interval120591 Therefore the decrease of iteration step can enhance thedifferential tracking precision
Under the conditions that the sampling interval is 10 timessmaller and that the setting is 00001 seconds the simulationresults are shown in Figure 22
18 Mathematical Problems in Engineering
Table 6 Tracking errors of the AORED with different orders andsampling interval
Orders of AORED Third order Fifth orderSample period 120591 = 0001 120591 = 00001 120591 = 00001
First-order differential error 001 Negligible NegligibleSecond-order differential error 15 002 NegligibleThird-order differential error 70 25 00005
As Figure 22 shows the signal tracking is rather idealexcept a peak in the beginning the tracking error in the first-order differential tracking can be neglected 119911
2has an obviously
smaller tracking error in the second-order differential 119892(119905)
tracking in the beginning 1199113has a rather sharp peak in the
third-order differential tracking but converges to a relativelyhigh precision after 01 seconds As we can see the differentialtracking errors for each number of orders are greatly reducedbut the third-order differential tracking error is still rather bigIn the following we will reduce the tracking errors by increasingthe number of orders of the differentiator
B The Fifth-Order Robust Exact Differentiator
The settings of the fifth-order differentiator are given asfollows
0
= V0
V0
= minus120582011987116 10038161003816100381610038161199110 minus
1199021003816100381610038161003816
56 sign (1199110
minus 119902) + 1199111
1
= V1
V1
= minus120582111987115 10038161003816100381610038161199111 minus V
0
1003816100381610038161003816
45 sign (1199111
minus V0) + 1199112
2
= V2
V2
= minus120582211987114 10038161003816100381610038161199112 minus V
1
1003816100381610038161003816
34 sign (1199112
minus V1) + 1199113
3
= V3
V3
= minus120582311987113 10038161003816100381610038161199113 minus V
2
1003816100381610038161003816
23 sign (1199113
minus V2) + 1199114
4
= V4
V4
= minus120582411987112 10038161003816100381610038161199114 minus V
3
1003816100381610038161003816
12 sign (1199114
minus V3) + 1199115
5
= V5
V5
= minus1205825119871 sign (119911
5minus V4)
(B1)
The initial value 1199110
= 01 1205820
= 1205821
= 1205822
= 1205823
= 1205824
=
1205825
= 50 the initial values 1199111 1199112 1199113 1199114 1199115
= 0 119871 = 1400the simulation step length is 00001 seconds The simulationresults are shown in Figure 23
As the figure shows the fifth-order differentiator hasa rather ideal tracking effect on the third-order differen-tial tracking the tracking error being only about 00005(see Table 6)
To sum up the Arbitrary-Order Robust Exact Differen-tiator can estimate rather ideal differential signals by selecting
Table 7
Length 365m119871 ref 365m119883119866
177m119878ref 0026m2
Diameter 0178mMass 1016 Kg119868119885
1063 Kgsdotm2
an appropriate sampling interval and using the differentiatorwith a relatively high number of orders
C Physical and Geometric Characteristics
See Table 7
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] P Zarchan Tactical and Strategic Missile Guidance 2002[2] D Zhou CMu andWXu ldquoAdaptive sliding-mode guidance of
a homing missilerdquo Journal of Guidance Control and Dynamicsvol 22 no 4 pp 589ndash594 1999
[3] G Zeng and M Hu ldquoFinite-time control for electromagneticsatellite formationsrdquo Acta Astronautica vol 74 pp 120ndash1302012
[4] S R Kumar S Rao and D Ghose ldquoNonsingular terminalsliding mode guidance with impact angle constraintsrdquo Journalof Guidance Control and Dynamics vol 37 no 4 pp 1114ndash11302014
[5] Y Liang C Chen D Liaw Y Feng C Cheng and CChen ldquoRobust guidance law via integral-sliding-mode schemerdquoJournal of Guidance Control and Dynamics vol 37 no 3 pp1038ndash1042 2014
[6] A V Savkin P N Pathirana and F Faruqi ldquoProblem ofprecision missile guidance LQR and 119867
infincontrol frameworksrdquo
IEEE Transactions on Aerospace and Electronic Systems vol 39no 3 pp 901ndash910 2003
[7] H Y Chen andC-C Yang ldquoNonlinear1198672119867infinguidance design
for homing missilesrdquo in Proceedings of the AIAA GuidanceNavigation and Control Conference and Exhibit vol 77 2001
[8] P K Menon G D Sweriduk and E J Ohlmeyer ldquoOptimalfixed-interval integrated guidance-control laws for hit-to-killmissilesrdquo in Proceedings of the AIAA Guidance Navigation andControl Conference and Exhibit AIAA 2003-5579 Austin TexUSA August 2003
[9] S S Vaddi P K Menon and E J Ohlmeyer ldquoNumericalSDRE approach for missile integrated guidance-controlrdquo inProceedings of the AIAA Guidance Navigation and ControlConference and Exhibit AIAA 2007-6672 Hilton Head SCUSA August 2007
[10] M Xin S N Balakrishnan and E J Ohlmeyer ldquoIntegratedguidance and control of missiles with Theta-D methodrdquo IEEE
Mathematical Problems in Engineering 19
Transactions on Control Systems Technology vol 14 no 6 pp981ndash992 2006
[11] T Shima M Idan and O M Golan ldquoSliding-mode controlfor integrated missile autopilot guidancerdquo Journal of GuidanceControl and Dynamics vol 29 no 2 pp 250ndash260 2006
[12] Y B Shtessel and C H Tournes ldquoIntegrated higher-ordersliding mode guidance and autopilot for dual control missilesrdquoJournal of Guidance Control and Dynamics vol 32 no 1 pp79ndash94 2009
[13] F Y Dong H M Lei and C J Zhou ldquoIntegrated robust highorder sliding mode guidance and control for missilesrdquo ActaAeronautica et Astronautica Sinica vol 34 no 9 pp 2212ndash22182013
[14] H Mingzhe and D Guangren ldquoIntegrated guidance and con-trol of homing missiles against ground fixed targetsrdquo ChineseJournal of Aeronautics vol 21 no 2 pp 162ndash168 2008
[15] A Levant ldquoHomogeneity approach to high-order sliding modedesignrdquo Automatica vol 41 no 5 pp 823ndash830 2005
[16] Y-W Liang C Chen D Liaw Y Feng C Cheng and CChen ldquoRobust guidance law via integral-sliding-mode schemerdquoJournal of Guidance Control and Dynamics vol 37 no 3 pp1038ndash1042 2014
[17] L Wang Y Sheng and X Liu ldquoContinuous time-varying slid-ingmode based attitude control for reentry vehiclerdquo Proceedingsof the Institution of Mechanical Engineers Part G Journal ofAerospace Engineering 2014
[18] A Levant ldquoHigher-order sliding modes differentiation andoutput-feedback controlrdquo International Journal of Control vol76 no 9-10 pp 924ndash941 2003
[19] A Levant ldquoQuasi-continuous high-order sliding-mode con-trollersrdquo IEEE Transactions on Automatic Control vol 50 no11 pp 1812ndash1816 2005
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MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
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Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 7
signal 1198910(119905) whose 119903th derivative has a known Lipschitz
constant 119871 gt 0 then the 119899th-order differentiator is definedas follows
0
= V0
V0
= minus12058201198711(119899+1) 10038161003816100381610038161199110 minus 119891 (119905)
1003816100381610038161003816
119899(119899+1) sign (1199110
minus 119891 (119905))
+ 1199111
1
= V1
V1
= minus12058211198712(119899+1) 10038161003816100381610038161199111 minus V
0
1003816100381610038161003816
(119899minus1)119899 sign (1199111
minus V0) + 1199112
119899minus1
= V119899minus1
V119899minus1
= minus120582119899minus1
11987112 1003816100381610038161003816119911119899minus1 minus V
119899minus2
1003816100381610038161003816
12 sign (119911119899minus1
minus V119899minus2
)
+ 119911119899
119899
= minus120582119899119871 sign (119911
119899minus V119899minus1
)
(51)
and if 120582119894
gt 0 is sufficiently large the convergence is guaran-teed
To obtain the third-order derivative of 119902 we constructthe third-order sliding mode differentiator and estimate thederivative of 119902 for each order In view of differential precisionwe configure the following fifth-order differentiator SeeAppendix A for comparison
0
= V0
V0
= minus120582011987116 10038161003816100381610038161199110 minus
1199021003816100381610038161003816
56 sign (1199110
minus 119902) + 1199111
1
= V1
V1
= minus120582111987115 10038161003816100381610038161199111 minus V
0
1003816100381610038161003816
45 sign (1199111
minus V0) + 1199112
2
= V2
V2
= minus120582211987114 10038161003816100381610038161199112 minus V
1
1003816100381610038161003816
34 sign (1199112
minus V1) + 1199113
3
= V3
V3
= minus120582311987113 10038161003816100381610038161199113 minus V
2
1003816100381610038161003816
23 sign (1199113
minus V2) + 1199114
4
= V4
V4
= minus120582411987112 10038161003816100381610038161199114 minus V
3
1003816100381610038161003816
12 sign (1199114
minus V3) + 1199115
5
= V5
V5
= minus1205825119871 sign (119911
5minus V4)
(52)
where 1199113 1199112 1199111 and 119911
0are the estimations of 119902
(4) 119902 119902 and 119902
4 Baseline Separated Guidance andControl Method
To verify the homing performance of the integrated guidanceand control method we compare it with the separatedguidance and control methodThe guidance laws we used arethe proportional navigation (PN) guidance law for attackinga nonmaneuvering target and the optimal sliding modeguidance law for attacking a maneuvering target
41 The Proportional Navigation (PN) Guidance Law Theguidance law has a simple formula and excellent perfor-mances for nonmaneuvering target Its formula is as follows
119899119888
= minus119873 119902119881119872
119892 (53)
119899119888is the overload command 119873 is the effective navigation
ratio 119902 is the LOS rate 119881119872
is the speed of the missile 119892 isthe acceleration of the gravity The guidance law gives theoverload command of the missile according to the LOS rateand then the controller gives the rudder deflection commandaccording to the overload command
42 The Optimal Sliding Mode Guidance Law The optimalsliding mode guidance law (OSMG) is a novel practicalguidance law proposed by D Zhou He combines the optimalguidance lawwith the slidingmode guidance law and designsthe new sliding mode guidance law that not only is robustto maneuvering target but also has the merits of the optimalguidance law such as good dynamic performance and energyconservation Its formula is as follows
119899119888
= minus3100381610038161003816100381610038160
10038161003816100381610038161003816119902 + 120576
119902
10038161003816100381610038161199021003816100381610038161003816 + 120575
(54)
where 119899119888is the overload command
0is the approach
velocity of the missile and its target 119902 is their LOS rate 120576 =
const is the compensatory gain 119902(| 119902| + 120575) is for substitutingfor sign( 119902) and for smoothing 120575 is a small quantity whichcould adjust the chattering
43 Separated Guidance and Control Design For simulationand comparisonwe use the conventional three-loop overloadautopilot as the controller which gives the rudder deflectioncommand according to the feedback of the three loops ofoverload pseudo-angle of attack and pitch rate The blockdiagram is as shown in Figure 3
As the figure shows the inner loop has the feedback onangular velocity which improves the damping characteristicsof the missile airframe
According to the aerodynamic coefficient of the missilewith selected working points we set 119870
119868= 019 119870
120572= 3 and
119870120596
= minus025 and the controller can well track the overloadcommand the rise time of its step response is 046 secondsand its settling time is 083 secondsThe step responses of themissile to overload command and the Bode diagram for openloop are shown in Figure 4
8 Mathematical Problems in Engineering
KIS
120596Z 120572 nY
nC K120596K120572 nY+minus+minus+minus dynamicsAirframe
modelServo
Figure 3 The working principles for three-loop overload autopilot
Step response
Time (s)0 02 04 06 08 1 12
0
02
04
06
08
1
Rise time (s) 0463
Settling time (s) 0831
Bode diagram
Frequency (rads)
To output pointFrom input pointTo output pointFrom input point
Gain margin (dB) 175 At frequency (rads) 15
Phase margin (deg) 709 At frequency (rads) 294
Am
plitu
de
Mag
nitu
de (d
B)Ph
ase (
deg)
0
minus180
minus360
minus54010410310210110010minus1
100
0
minus100
minus200
minus300
Figure 4 The autopilot performance step response and Bode diagram
5 Simulation Results
To verify the high order slidingmode integrated guidance andcontrol (HOSM-IGC) method we compare it with the base-line separated guidance and control Numerical simulationsare designed in three typical engagement scenarios
AORED parameters are as follows the initial value 1205820
=
1205821
= 1205822
= 1205823
= 50 1199110
= 01 1199111
= 1199112
= 1199113
= 0 119871 = 1400 thesimulation step is 00001 seconds
51 Scenario 1 Nonmaneuvering Target In the first scenariothe nonmaneuvering target does uniform rectilinear motionand PN guidance law with three-loop autopilot is introducedfor a comparison with the HOSM-IGCThe initial conditionsare set as shown in Table 1
The motion equations of the target are as follows
119910119879
= 119881119879cos 120579119879
119879
= minus119881119879sin 120579119879
120579119879
=119886119879
119881119879
119886119879
= 0
(55)
The simulation results are shown in Figures 5 and 6The missilersquos flight trajectory and overload curve show
that within the first 3 sec the HOSM-IGC method spendsmuch energy (overload) on changing the initial LOSdirection
Table 1 Initial conditions
Variable Description Value(119883119872 119884119872) Missile initial position (0 km 18 km)
(119883119879 119884119879) Target initial position (60 km 20 km)
119881119872
Missile initial velocity 1200 (ms)119881119879
Target initial velocity 2000 (ms)120579119872
Missile initial flight path angle 0 (deg)120579119879
Target initial flight path angle 0 (deg)120573 Parameter of the controller 10 or 30119873 Navigation ratio of PN 3
of the missile After the boresight adjustment 120590 reaches thedesired sliding manifold and the missile does not need anymaneuver to hit the target This is because the target doesnot maneuver any more which means no disturbance isintroduced in the engagement dynamic so the state (or thesliding mode) of the missile will stay on the manifold Incontrast the overload command given by the PN guidancelaw increases fast as the relative distance decreases Toexamine the performances of the two guidance laws furtherwe increase the target speed and analyze the commandchanges
Therefore we set the target speeds 3000ms 4000msand carry out simulations The simulation result is shown inthe overload curve in Figure 7
Mathematical Problems in Engineering 9
175
180
185
190
195
200
205
TargetHOSM-IGCPN
Y(k
m)
0 10 20 30 40 50 60X (km)
Figure 5 Target and missile trajectories
0
5
10
15
20
25
0 5 10 15 20Time (s)
HOSM-IGCPN
Initial boresight adjustment
Sliding manifold reached
Miss
ile ac
cele
ratio
n (G
)
minus5
Figure 6 Missile acceleration profile
It is evident that the speed of divergence of the overloadcommand given by the PN guidance law increases withthe target speed Specifically when the target speed reaches4000ms the commanded overload is almost 30 g which isobviously not ideal for the attack of a nonmaneuvering targetbut with the HOSM-IGC method the missile adjusts itsboresight very quickly then maintains it around 0 g and fliesto its target in the rectilinear ballistic trajectory not affectedby the increases of target speed still accomplishing the high-precision hit-on collision
Figure 8 shows that although the HOSM-IGC methodachieves a more effective overload command on the otherside it sees some chattering when the sliding mode reaches
0
5
10
15
20
25
30
0 5 10 15Time (s)
Miss
ile ac
cele
ratio
n (G
)
minus5
HOSM-IGC (Vt = minus3000)HOSM-IGC (Vt = minus4000)
PN (Vt = minus3000)PN (Vt = minus4000)
Figure 7 Missile acceleration profile
0
1
2
0 5 10 15 20Time (s)
4 45 5
0
02
04
minus02
minus04
Actu
ator
defl
ectio
n (d
eg)
minus1
minus2
minus3
minus4
HOSM-IGC (120573 = 30)
Figure 8 Actuator deflection
the desired manifold the commanded rudder deflectionangle chatters for around 045 deg at about 15Hz which iskind of harmful to the system To reduce the chatteringwe adjust the controllerrsquos parameter 120573 = 10 and carry outsimulations again Figure 9 shows that after 120573 decreasesthe command of rudder deflection angle converges slower(for about 7 sec) however the chattering weakens obviouslyits magnitude being only about 015 degrees The smallerchattering well enhances convergence precision eventuallyreducing the target missing The reason that minor 120573 resultsin an alleviative chattering can be seen from (47) Figure 10shows the target and missile trajectories
Table 2 compares the average miss distance of 50 simula-tions under the conditions discussed above the comparison
10 Mathematical Problems in Engineering
Table 2 Average miss distance of 50 simulations
Target speed PN HOSM-IGC(120573 = 30)
HOSM-IGC(120573 = 10)
119881119905= 2000ms 115m 086 073m
119881119905= 3000ms 261m 117 106
119881119905= 4000ms 542m 156 134
0
1
2
0 5 10 15 20Time (s)
72 74 76 78
0
02
minus02
minus04
Actu
ator
defl
ectio
n (d
eg)
minus1
minus2
minus3
minus4
HOSM-IGC (120573 = 10)HOSM-IGC (120573 = 30)
Figure 9 Actuator deflection
results show that when the missile intercepts a nonmaneu-vering target the hit precision of the HOSM-IGC methodis apparently higher than that of the PN guidance law andthat the low-gain HOSM-IGC method can effectively reducethe chattering magnitude thus enhancing the interceptionprecision
52 Scenario 2 Step Maneuvering Target At the first stagethe target flies at uniform speed and in a rectilinear way after10 seconds it maneuvers at the normal acceleration of 5 g Inthis scenario OSMG is introduced for a comparison with theHOSM-IGC
119910119879
= 119881119879cos 120579119879
119879
= minus119881119879sin 120579119879
120579119879
=119886119879
119881119879
119886119879
=
0 g 119905 lt 10 sec
5 g 119905 gt 10 sec
(56)
The initial simulation conditions are given in Table 3The overload curve in Figure 11 shows dearly that both
types of guidance laws can track the maneuvering targetDuring 0 to 10 seconds the target flies at uniform speed andin a rectilinear way and the missile converges its overload to
Table 3 Initial conditions
Variable Description Value(119883119872 119884119872) Missile initial position (0 km 22 km)
(119883119879 119884119879) Target initial position (60 km 20 km)
119881119872
Missile initial velocity 1200 (ms)119881119879
Target initial velocity 2000 (ms)120579119872
Missile initial flight path angle 0 (deg)120579119879
Target initial flight path angle 0 (deg)120576 Compensatory gain 3120575 Saturation parameter 0001
15
17
19
21
23
25
TargetHOSM-IGCOSMG
Y(k
m)
0 20 40 60X (km)
Figure 10 Target and missile trajectories
0 g and flies to its target after 10 seconds the target beginsto maneuver by 5 g Both types of guidance law respondby rapidly increasing the overload adjusting attitude andmaking the missilersquos boresight aim at its target
We can see that when both types of guidance law tracktheir targets the convergence speed of OSMG is almostthe same as the HOSM-IGC method But the HOSM-IGCmethod has higher convergence precision and needs loweroverload at the end phase
We can also see that after the target maneuvers if themissile is given enough time to track the targetrsquos maneuvernamely let the missilersquos overload command converge to theoverload of the target theremay not be largemiss distance Inother words for a certain period of time before the collisionthe targetmaneuver (it means only a limitedmaneuver whichdoes not include the condition that the maneuvering of thetarget for a long time may cause a change of the geometricalrelations between the missile and its target) has a small effecton both types of guidance law
But if themaneuver occurs rather late namely within oneto three seconds before collision when the overload com-mand of guidance law is not yet converged the approaching
Mathematical Problems in Engineering 11
0
5
10
0 5 10 15 20Time (s)
TargetOSMGHOSM-IGC
Miss
ile an
d ta
rget
acce
lera
tion
(G)
minus5
minus10
Figure 11 Missile and target acceleration profile
0
10
20
30
0 5 10 15 20Time (s)
tgo = 1 stgo = 2 stgo = 3 s
tgo = 3 s (diverge)
tgo = 2 s (diverge)
tgo = 1 s (diverge)minus10
minus20
minus30
Miss
ile ac
cele
ratio
n (G
)
HOSM-IGC (120573 = 10)
Figure 12 Missile acceleration profile
collision increases themiss distanceTherefore increasing theparameter 120573 can remarkably increase the convergence speedand effectively enhance guidance precision As Figure 12shows when 120573 = 10 the overload at the end phase convergesslowly even if the target maneuvers three seconds beforecollision the missilersquos overload still has no time to convergebeing unable to track themaneuvering target Figure 13 showsthat when 120573 increases to 30 and tgo = 3 seconds theoverload can converge to about 5 g but when tgo = 2 sec-onds the overload may continue to increase indicating that
0
10
20
30
0 5 10 15 20Time (s)
tgo = 1 stgo = 2 stgo = 3 s
tgo = 1 s (diverge)
tgo = 2 s (diverge)
tgo = 3 s (converge)
minus10
minus20
minus30
Miss
ile ac
cele
ratio
n (G
)
120573 = 30)HOSM-IGC (
Figure 13 Missile acceleration profile
0
10
20
30
0 5 10 15 20Time (s)
tgo = 1 s (diverge)
tgo = 2 s (converge)
tgo = 3 s (converge)
tgo = 1 stgo = 2 stgo = 3 s
minus10
minus20
minus30
Miss
ile ac
cele
ratio
n (G
)
HOSM-IGC (120573 = 50)
Figure 14 Missile acceleration profile
the convergence still needs to be quickened thus continuingto rise 120573 to 50 Figure 14 shows that the HOSM-IGC methodcan track the target that maneuvers when tgo = 2 secondsbut it causes the divergence of overload and the increase ofmiss distance if the target maneuvers when tgo = 1 secondFigure 15 gives the overload curve of the OSMG guidance lawand shows that the maneuver of the target before collisionmay cause the large-scale oscillation of the missilersquos overloadwhich may diverge to a large numerical value when thecollision occurs in the end
12 Mathematical Problems in Engineering
Table 4 Average miss distances of 50 simulations
Targetmaneuveringtiming
HOSM-IGC120573 = 30
HOSM-IGC120573 = 40
HOSM-IGC120573 = 50
OSMG
tgo = 1second 43539 3224 28936 36116
tgo = 2seconds 25424 18665 09322 35534
tgo = 3seconds 08124 08265 07538 11959
Time (s)0 5 10 15 20
0
10
20
30OSMG
tgo = 1 stgo = 2 stgo = 3 s
minus10
minus20
minus30
Miss
ile ac
cele
ratio
n (G
)
Figure 15 Missile acceleration profile
The analysis in Figures 12 13 14 and 15 shows that theoverload of the missile converges faster and its miss distanceis smaller with increasing 120573 To verify this finding we carryout 50 times Monte Carlo simulations in which the positionand speed of the target have 1 of random difference
The average miss distances are shown in Table 4 andFigure 16 Clearly the timing of the targetrsquos step maneuverdramatically affects the final interception precision morespecifically given a shorter reaction time for the guidanceand control system the missile seems more likely to missthe target To the OSMG guidance law in all three scenarioshardly does it show any advantages against theHOSM-IGC Itcan also be seen that with the increasing of the120573 theHOSM-IGC system responds even faster which leads to an obviousdecrease of the average miss distance The effect of 120573 on theresponse of the HOSM-IGC system is a valuable guidelinewhen implementing the proposed method into practice
53 Scenario 3 Weaving Target In this scenario the targetmaneuvers by 119886
119879= 40sin(1205871199052) OSMG with three-loop
Table 5 Initial conditions
Variable Description Value(119883119872 119884119872) Missile initial position (0 km 175 km)
(119883119879 119884119879) Target initial position (60 km 195 km)
119881119872
Missile initial velocity 1200 (ms)119881119879
Target initial velocity 2000 (ms)120579119872
Missile initial flight path angle 0 (deg)120579119879
Target initial flight path angle 0 (deg)120576 Compensatory gain 3120575 Saturation parameter 0001
autopilot is introduced for comparison The motion equa-tions of the target are as follows
119910119879
= 119881119879cos 120579119879
119879
= minus119881119879sin 120579119879
120579119879
=119886119879
119881119879
119886119879
= 40 sin(120587119905
2)
(57)
The initial simulation conditions are given in Table 5The missilersquos trajectory under the two guidance and
control methods given in Figure 17 does not see muchdifference
However the overload curve given in Figure 18 showsthat after the missile completes its initial attitude adjustmentwith the HOSM-IGC method it can almost ideally track itsmaneuvering target by contrast with the OSMG methodthe missile seems to have the tendency to track its targetrsquosmaneuver but has larger tracking errors Besides with theOSMGmethod the missilersquos overload increases rapidly at theend of attack primarily because of the divergence of its LOSrate On the other hand with the HOSM-IGC method themissile has no divergence even at the end of attack ensuringa smaller target missing quantity
The actuator deflection curve in Figure 19 shows thatin order to provide a rather big normal overload for theend phase the OSMG method produces a rather big rudderdeflection command however it may increase the missilersquostarget missing quantity once its rudder deflection saturatesand themissile does not have enough overloads or the ruddercannot respond that fast
As shown in Figure 20 because of the dramatic changein overload command the response of the missilersquos autopilotto high-frequency command sees an obvious phase lag andamplitude value attenuation its actual overload cannot trackthe command ideally this is a main reason why the missdistance increases However the controller in the HOSM-IGC method gives its rudder deflection command directlyand there is no lagging or attenuation caused by the autopilotthus enhancing the guidance precision effectively Further-more the fast convergence of the high order sliding modemakes the missile rapidly track its maneuvering target withthemost reasonable rudder deflection command reducing itsoverload effectively
Mathematical Problems in Engineering 13
0
05
1
15
2
25
3
35
4
45
5
0 10 20 30 40 50Simulation times
OSMG
Miss
(m)
0
05
1
15
2
25
3
35
4
45
5
0 10 20 30 40 50Simulation times
Miss
(m)
0
05
1
15
2
25
3
35
4
45
5
0 10 20 30 40 50Simulation times
Miss
(m)
tgo = 3 stgo = 2 stgo = 1 s
0
05
1
15
2
25
3
35
4
45
5
0 10 20 30 40 50Simulation times
Miss
(m)
tgo = 3 stgo = 2 stgo = 1 s
HOSM-IGC (120573 = 40)
HOSM-IGC (120573 = 50)
HOSM-IGC (120573 = 30)
Figure 16 Miss distances of HOSM-IGC (120573 = 30 40 50) and OSMG with target maneuvers at different time-to-go (tgo = 1 second 2seconds and 3 seconds)
The average miss distance of 50 simulations under theconditions is 073m for HOSM-IGC and 186m for OSMGWe can see that the HOSM-IGC method not only doesprovide a more reasonable actuator deflection command butalso achieves a higher interception precision
6 Conclusions
This paper proposes an LOS feedback integrated guidanceand control method using quasi-continuous high order
sliding mode guidance and control method With the fastand precise convergence of the quasi-continuous HOSMmethod the HOSM-IGCmethod performsmuch better thanthe traditional separated guidance and control method withless acceleration effort and less miss distance in all thethree simulation scenarios of nonmaneuvering target stepmaneuvering target and weaving target In addition the ideaof virtual control largely alleviates the chattering withoutany sacrifice of robustness As a result of the alleviationof the chattering the control input command 120575
119885becomes
14 Mathematical Problems in Engineering
175
180
185
190
195
200
TargetHOSM-IGCOSMG
Y(k
m)
0 20 40 60 0 20 40X (km)
Figure 17 The trajectories of the missile and its target
0
10
20
0 5 10 15 20Time (s)
TargetOSMGHOSM-IGC
Miss
ile an
d ta
rget
acce
lera
tion
(G)
minus10
minus20
Figure 18 Missile acceleration profile
smooth enough for the actuator to be implemented as aresult the interception accuracy is improved In designingthe quasi-continuous sliding mode controller the Arbitrary-Order Robust Exact Differentiator is introduced to estimatethe high order time derivative of the LOS rate 119902 which notonly ensures the guidance and control precision but alsosimplifies the calculation
0
10
20
30
0 5 10 15 20Time (s)
OSMGHOSM-IGC
minus10
minus20
minus30
minus40
Actu
ator
defl
ectio
n (d
eg)
Figure 19 Actuator deflection
0
50
100
150
200
15 16 17 18Time (s)
Commanded accelerationAchieved acceleration
Miss
ile ac
cele
ratio
n (G
)
minus50
Figure 20 Commanded acceleration and achieved acceleration
Appendices
A The Third-Order RobustExact Differentiator
The third-order robust exact differentiator built according tothe following parameters and equations is used to performthe simulation the initial value 120582
0= 1205821
= 1205822
= 1205823
= 50
Mathematical Problems in Engineering 15
0
0005
001
0015
002
0025
003
0035
004
0 02 04Time (s)
0
2
0 02 04Time (s)
0
100
200
300
0
100
200
300
400
0 02 04Time (s)
h(t)
z3
h(t) and z3
minus100
minus200
minus300
minus400
minus5000 02 04
Time (s)
g(t)
z2
g(t) and z2
minus100
minus200
minus300
minus400
f(t)
z1
f(t) and z1
minus2
minus4
minus6
minus8
minus10
minus12
minus14
F(t)
z0
F(t) and z0
minus0005
minus001
Figure 21 The outputs of the differentiator in the sampling interval of 0001 seconds
the initial values 1199110
= 01 1199111
= 1199112
= 1199113
= 0 119871 = 1400 thesimulation step is 0001 seconds Consider
0
= V0
V0
= minus120582011987114 10038161003816100381610038161199110 minus
1199021003816100381610038161003816
34 sign (1199110
minus 119902) + 1199111
1
= V1
V1
= minus120582111987113 10038161003816100381610038161199111 minus V
0
1003816100381610038161003816
23 sign (1199111
minus V0) + 1199112
2
= V2
V2
= minus120582211987112 10038161003816100381610038161199112 minus V
1
1003816100381610038161003816
12 sign (1199112
minus V1) + 1199113
3
= minus1205823119871 sign (119911
3minus V2)
(A1)
16 Mathematical Problems in Engineering
0
0005
001
0015
002
0025
003
0035
004
0 02 04Time (s)
0
2
0 02 04Time (s)
0
100
200
300
0 02 04Time (s)
0
200
400
0 02 04Time (s)
h(t) and z3
minus200
minus400
minus600
minus800
g(t) and z2
minus100
minus200
minus300
minus400
f(t) and z1
minus2
minus4
minus6
minus8
minus10
minus12
minus14
minus16
F(t) and z0
minus0005
minus001
h(t)
z3g(t)
z2
f(t)
z1F(t)
z0
Figure 22 The outputs of the differentiator in the sampling interval of 00001 seconds
If the designed signals are 119865(119905) = int 119891(119905)119889119905 119891(119905) =
int 119892(119905)119889119905 119892(119905) = int ℎ(119905)119889119905 and ℎ(119905) = sin(2120587119905) +
3cos(120587119905) then theoretically 119891(119905) is the first-order differ-ential of input signal 119892(119905) is the second-order differ-ential of input signal ℎ(119905) is the third-order differen-tial of input signal Through the simulation the differ-ential signal of actual output is compared with that of
theoretical output the comparison results are shown inFigure 21
Figure 21 shows that 1199110can effectively track the signal
119865(119905) but 1199112has obvious errors in tracking the second-order
differential 119892(119905) 1199113has greater errors in tracking the third-
order differential thus the overall differential effect is notideal
Mathematical Problems in Engineering 17
0 02 04
0
0005
001
0015
002
0025
003
0035
004
Time (s)0 02 04
0
01
02
03
04
05
Time (s)
0 02 04
0
05
1
15
2
Time (s)0 02 04
0
05
1
15
2
25
3
35
4
Time (s)
h(t) and z3g(t) and z2
minus04
minus05
f(t) and z1
minus2
minus15
minus1
minus05
minus1
minus05
minus01
minus02
minus03
F(t) and z0
minus0005
minus001
h(t)
z3g(t)
z2
f(t)
z1F(t)
z0
Figure 23 The outputs of the differentiator in the sampling interval of 00001 seconds
There are two causes for the errors in a usual differentia-tor the setting of sampling interval and the setting of numberof orders of the differentiator The following theorem applieshere
Theorem A1 (see [19]) If the sampling interval 120591 gt 0 theaccuracy of the controller is expressed as follows
10038161003816100381610038161003816120590(119894)
10038161003816100381610038161003816le 120583120591119903minus119894
119894 = 0 119903 minus 1 (A2)
That is to say the precision (understood as the differentialprecision of a differentiator) of the 119894th order control is in directproportion to the power of 119903 minus 119894 times of sampling interval120591 Therefore the decrease of iteration step can enhance thedifferential tracking precision
Under the conditions that the sampling interval is 10 timessmaller and that the setting is 00001 seconds the simulationresults are shown in Figure 22
18 Mathematical Problems in Engineering
Table 6 Tracking errors of the AORED with different orders andsampling interval
Orders of AORED Third order Fifth orderSample period 120591 = 0001 120591 = 00001 120591 = 00001
First-order differential error 001 Negligible NegligibleSecond-order differential error 15 002 NegligibleThird-order differential error 70 25 00005
As Figure 22 shows the signal tracking is rather idealexcept a peak in the beginning the tracking error in the first-order differential tracking can be neglected 119911
2has an obviously
smaller tracking error in the second-order differential 119892(119905)
tracking in the beginning 1199113has a rather sharp peak in the
third-order differential tracking but converges to a relativelyhigh precision after 01 seconds As we can see the differentialtracking errors for each number of orders are greatly reducedbut the third-order differential tracking error is still rather bigIn the following we will reduce the tracking errors by increasingthe number of orders of the differentiator
B The Fifth-Order Robust Exact Differentiator
The settings of the fifth-order differentiator are given asfollows
0
= V0
V0
= minus120582011987116 10038161003816100381610038161199110 minus
1199021003816100381610038161003816
56 sign (1199110
minus 119902) + 1199111
1
= V1
V1
= minus120582111987115 10038161003816100381610038161199111 minus V
0
1003816100381610038161003816
45 sign (1199111
minus V0) + 1199112
2
= V2
V2
= minus120582211987114 10038161003816100381610038161199112 minus V
1
1003816100381610038161003816
34 sign (1199112
minus V1) + 1199113
3
= V3
V3
= minus120582311987113 10038161003816100381610038161199113 minus V
2
1003816100381610038161003816
23 sign (1199113
minus V2) + 1199114
4
= V4
V4
= minus120582411987112 10038161003816100381610038161199114 minus V
3
1003816100381610038161003816
12 sign (1199114
minus V3) + 1199115
5
= V5
V5
= minus1205825119871 sign (119911
5minus V4)
(B1)
The initial value 1199110
= 01 1205820
= 1205821
= 1205822
= 1205823
= 1205824
=
1205825
= 50 the initial values 1199111 1199112 1199113 1199114 1199115
= 0 119871 = 1400the simulation step length is 00001 seconds The simulationresults are shown in Figure 23
As the figure shows the fifth-order differentiator hasa rather ideal tracking effect on the third-order differen-tial tracking the tracking error being only about 00005(see Table 6)
To sum up the Arbitrary-Order Robust Exact Differen-tiator can estimate rather ideal differential signals by selecting
Table 7
Length 365m119871 ref 365m119883119866
177m119878ref 0026m2
Diameter 0178mMass 1016 Kg119868119885
1063 Kgsdotm2
an appropriate sampling interval and using the differentiatorwith a relatively high number of orders
C Physical and Geometric Characteristics
See Table 7
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] P Zarchan Tactical and Strategic Missile Guidance 2002[2] D Zhou CMu andWXu ldquoAdaptive sliding-mode guidance of
a homing missilerdquo Journal of Guidance Control and Dynamicsvol 22 no 4 pp 589ndash594 1999
[3] G Zeng and M Hu ldquoFinite-time control for electromagneticsatellite formationsrdquo Acta Astronautica vol 74 pp 120ndash1302012
[4] S R Kumar S Rao and D Ghose ldquoNonsingular terminalsliding mode guidance with impact angle constraintsrdquo Journalof Guidance Control and Dynamics vol 37 no 4 pp 1114ndash11302014
[5] Y Liang C Chen D Liaw Y Feng C Cheng and CChen ldquoRobust guidance law via integral-sliding-mode schemerdquoJournal of Guidance Control and Dynamics vol 37 no 3 pp1038ndash1042 2014
[6] A V Savkin P N Pathirana and F Faruqi ldquoProblem ofprecision missile guidance LQR and 119867
infincontrol frameworksrdquo
IEEE Transactions on Aerospace and Electronic Systems vol 39no 3 pp 901ndash910 2003
[7] H Y Chen andC-C Yang ldquoNonlinear1198672119867infinguidance design
for homing missilesrdquo in Proceedings of the AIAA GuidanceNavigation and Control Conference and Exhibit vol 77 2001
[8] P K Menon G D Sweriduk and E J Ohlmeyer ldquoOptimalfixed-interval integrated guidance-control laws for hit-to-killmissilesrdquo in Proceedings of the AIAA Guidance Navigation andControl Conference and Exhibit AIAA 2003-5579 Austin TexUSA August 2003
[9] S S Vaddi P K Menon and E J Ohlmeyer ldquoNumericalSDRE approach for missile integrated guidance-controlrdquo inProceedings of the AIAA Guidance Navigation and ControlConference and Exhibit AIAA 2007-6672 Hilton Head SCUSA August 2007
[10] M Xin S N Balakrishnan and E J Ohlmeyer ldquoIntegratedguidance and control of missiles with Theta-D methodrdquo IEEE
Mathematical Problems in Engineering 19
Transactions on Control Systems Technology vol 14 no 6 pp981ndash992 2006
[11] T Shima M Idan and O M Golan ldquoSliding-mode controlfor integrated missile autopilot guidancerdquo Journal of GuidanceControl and Dynamics vol 29 no 2 pp 250ndash260 2006
[12] Y B Shtessel and C H Tournes ldquoIntegrated higher-ordersliding mode guidance and autopilot for dual control missilesrdquoJournal of Guidance Control and Dynamics vol 32 no 1 pp79ndash94 2009
[13] F Y Dong H M Lei and C J Zhou ldquoIntegrated robust highorder sliding mode guidance and control for missilesrdquo ActaAeronautica et Astronautica Sinica vol 34 no 9 pp 2212ndash22182013
[14] H Mingzhe and D Guangren ldquoIntegrated guidance and con-trol of homing missiles against ground fixed targetsrdquo ChineseJournal of Aeronautics vol 21 no 2 pp 162ndash168 2008
[15] A Levant ldquoHomogeneity approach to high-order sliding modedesignrdquo Automatica vol 41 no 5 pp 823ndash830 2005
[16] Y-W Liang C Chen D Liaw Y Feng C Cheng and CChen ldquoRobust guidance law via integral-sliding-mode schemerdquoJournal of Guidance Control and Dynamics vol 37 no 3 pp1038ndash1042 2014
[17] L Wang Y Sheng and X Liu ldquoContinuous time-varying slid-ingmode based attitude control for reentry vehiclerdquo Proceedingsof the Institution of Mechanical Engineers Part G Journal ofAerospace Engineering 2014
[18] A Levant ldquoHigher-order sliding modes differentiation andoutput-feedback controlrdquo International Journal of Control vol76 no 9-10 pp 924ndash941 2003
[19] A Levant ldquoQuasi-continuous high-order sliding-mode con-trollersrdquo IEEE Transactions on Automatic Control vol 50 no11 pp 1812ndash1816 2005
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
Volume 2014
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Complex AnalysisJournal of
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OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
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Algebra
Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
Discrete MathematicsJournal of
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
8 Mathematical Problems in Engineering
KIS
120596Z 120572 nY
nC K120596K120572 nY+minus+minus+minus dynamicsAirframe
modelServo
Figure 3 The working principles for three-loop overload autopilot
Step response
Time (s)0 02 04 06 08 1 12
0
02
04
06
08
1
Rise time (s) 0463
Settling time (s) 0831
Bode diagram
Frequency (rads)
To output pointFrom input pointTo output pointFrom input point
Gain margin (dB) 175 At frequency (rads) 15
Phase margin (deg) 709 At frequency (rads) 294
Am
plitu
de
Mag
nitu
de (d
B)Ph
ase (
deg)
0
minus180
minus360
minus54010410310210110010minus1
100
0
minus100
minus200
minus300
Figure 4 The autopilot performance step response and Bode diagram
5 Simulation Results
To verify the high order slidingmode integrated guidance andcontrol (HOSM-IGC) method we compare it with the base-line separated guidance and control Numerical simulationsare designed in three typical engagement scenarios
AORED parameters are as follows the initial value 1205820
=
1205821
= 1205822
= 1205823
= 50 1199110
= 01 1199111
= 1199112
= 1199113
= 0 119871 = 1400 thesimulation step is 00001 seconds
51 Scenario 1 Nonmaneuvering Target In the first scenariothe nonmaneuvering target does uniform rectilinear motionand PN guidance law with three-loop autopilot is introducedfor a comparison with the HOSM-IGCThe initial conditionsare set as shown in Table 1
The motion equations of the target are as follows
119910119879
= 119881119879cos 120579119879
119879
= minus119881119879sin 120579119879
120579119879
=119886119879
119881119879
119886119879
= 0
(55)
The simulation results are shown in Figures 5 and 6The missilersquos flight trajectory and overload curve show
that within the first 3 sec the HOSM-IGC method spendsmuch energy (overload) on changing the initial LOSdirection
Table 1 Initial conditions
Variable Description Value(119883119872 119884119872) Missile initial position (0 km 18 km)
(119883119879 119884119879) Target initial position (60 km 20 km)
119881119872
Missile initial velocity 1200 (ms)119881119879
Target initial velocity 2000 (ms)120579119872
Missile initial flight path angle 0 (deg)120579119879
Target initial flight path angle 0 (deg)120573 Parameter of the controller 10 or 30119873 Navigation ratio of PN 3
of the missile After the boresight adjustment 120590 reaches thedesired sliding manifold and the missile does not need anymaneuver to hit the target This is because the target doesnot maneuver any more which means no disturbance isintroduced in the engagement dynamic so the state (or thesliding mode) of the missile will stay on the manifold Incontrast the overload command given by the PN guidancelaw increases fast as the relative distance decreases Toexamine the performances of the two guidance laws furtherwe increase the target speed and analyze the commandchanges
Therefore we set the target speeds 3000ms 4000msand carry out simulations The simulation result is shown inthe overload curve in Figure 7
Mathematical Problems in Engineering 9
175
180
185
190
195
200
205
TargetHOSM-IGCPN
Y(k
m)
0 10 20 30 40 50 60X (km)
Figure 5 Target and missile trajectories
0
5
10
15
20
25
0 5 10 15 20Time (s)
HOSM-IGCPN
Initial boresight adjustment
Sliding manifold reached
Miss
ile ac
cele
ratio
n (G
)
minus5
Figure 6 Missile acceleration profile
It is evident that the speed of divergence of the overloadcommand given by the PN guidance law increases withthe target speed Specifically when the target speed reaches4000ms the commanded overload is almost 30 g which isobviously not ideal for the attack of a nonmaneuvering targetbut with the HOSM-IGC method the missile adjusts itsboresight very quickly then maintains it around 0 g and fliesto its target in the rectilinear ballistic trajectory not affectedby the increases of target speed still accomplishing the high-precision hit-on collision
Figure 8 shows that although the HOSM-IGC methodachieves a more effective overload command on the otherside it sees some chattering when the sliding mode reaches
0
5
10
15
20
25
30
0 5 10 15Time (s)
Miss
ile ac
cele
ratio
n (G
)
minus5
HOSM-IGC (Vt = minus3000)HOSM-IGC (Vt = minus4000)
PN (Vt = minus3000)PN (Vt = minus4000)
Figure 7 Missile acceleration profile
0
1
2
0 5 10 15 20Time (s)
4 45 5
0
02
04
minus02
minus04
Actu
ator
defl
ectio
n (d
eg)
minus1
minus2
minus3
minus4
HOSM-IGC (120573 = 30)
Figure 8 Actuator deflection
the desired manifold the commanded rudder deflectionangle chatters for around 045 deg at about 15Hz which iskind of harmful to the system To reduce the chatteringwe adjust the controllerrsquos parameter 120573 = 10 and carry outsimulations again Figure 9 shows that after 120573 decreasesthe command of rudder deflection angle converges slower(for about 7 sec) however the chattering weakens obviouslyits magnitude being only about 015 degrees The smallerchattering well enhances convergence precision eventuallyreducing the target missing The reason that minor 120573 resultsin an alleviative chattering can be seen from (47) Figure 10shows the target and missile trajectories
Table 2 compares the average miss distance of 50 simula-tions under the conditions discussed above the comparison
10 Mathematical Problems in Engineering
Table 2 Average miss distance of 50 simulations
Target speed PN HOSM-IGC(120573 = 30)
HOSM-IGC(120573 = 10)
119881119905= 2000ms 115m 086 073m
119881119905= 3000ms 261m 117 106
119881119905= 4000ms 542m 156 134
0
1
2
0 5 10 15 20Time (s)
72 74 76 78
0
02
minus02
minus04
Actu
ator
defl
ectio
n (d
eg)
minus1
minus2
minus3
minus4
HOSM-IGC (120573 = 10)HOSM-IGC (120573 = 30)
Figure 9 Actuator deflection
results show that when the missile intercepts a nonmaneu-vering target the hit precision of the HOSM-IGC methodis apparently higher than that of the PN guidance law andthat the low-gain HOSM-IGC method can effectively reducethe chattering magnitude thus enhancing the interceptionprecision
52 Scenario 2 Step Maneuvering Target At the first stagethe target flies at uniform speed and in a rectilinear way after10 seconds it maneuvers at the normal acceleration of 5 g Inthis scenario OSMG is introduced for a comparison with theHOSM-IGC
119910119879
= 119881119879cos 120579119879
119879
= minus119881119879sin 120579119879
120579119879
=119886119879
119881119879
119886119879
=
0 g 119905 lt 10 sec
5 g 119905 gt 10 sec
(56)
The initial simulation conditions are given in Table 3The overload curve in Figure 11 shows dearly that both
types of guidance laws can track the maneuvering targetDuring 0 to 10 seconds the target flies at uniform speed andin a rectilinear way and the missile converges its overload to
Table 3 Initial conditions
Variable Description Value(119883119872 119884119872) Missile initial position (0 km 22 km)
(119883119879 119884119879) Target initial position (60 km 20 km)
119881119872
Missile initial velocity 1200 (ms)119881119879
Target initial velocity 2000 (ms)120579119872
Missile initial flight path angle 0 (deg)120579119879
Target initial flight path angle 0 (deg)120576 Compensatory gain 3120575 Saturation parameter 0001
15
17
19
21
23
25
TargetHOSM-IGCOSMG
Y(k
m)
0 20 40 60X (km)
Figure 10 Target and missile trajectories
0 g and flies to its target after 10 seconds the target beginsto maneuver by 5 g Both types of guidance law respondby rapidly increasing the overload adjusting attitude andmaking the missilersquos boresight aim at its target
We can see that when both types of guidance law tracktheir targets the convergence speed of OSMG is almostthe same as the HOSM-IGC method But the HOSM-IGCmethod has higher convergence precision and needs loweroverload at the end phase
We can also see that after the target maneuvers if themissile is given enough time to track the targetrsquos maneuvernamely let the missilersquos overload command converge to theoverload of the target theremay not be largemiss distance Inother words for a certain period of time before the collisionthe targetmaneuver (it means only a limitedmaneuver whichdoes not include the condition that the maneuvering of thetarget for a long time may cause a change of the geometricalrelations between the missile and its target) has a small effecton both types of guidance law
But if themaneuver occurs rather late namely within oneto three seconds before collision when the overload com-mand of guidance law is not yet converged the approaching
Mathematical Problems in Engineering 11
0
5
10
0 5 10 15 20Time (s)
TargetOSMGHOSM-IGC
Miss
ile an
d ta
rget
acce
lera
tion
(G)
minus5
minus10
Figure 11 Missile and target acceleration profile
0
10
20
30
0 5 10 15 20Time (s)
tgo = 1 stgo = 2 stgo = 3 s
tgo = 3 s (diverge)
tgo = 2 s (diverge)
tgo = 1 s (diverge)minus10
minus20
minus30
Miss
ile ac
cele
ratio
n (G
)
HOSM-IGC (120573 = 10)
Figure 12 Missile acceleration profile
collision increases themiss distanceTherefore increasing theparameter 120573 can remarkably increase the convergence speedand effectively enhance guidance precision As Figure 12shows when 120573 = 10 the overload at the end phase convergesslowly even if the target maneuvers three seconds beforecollision the missilersquos overload still has no time to convergebeing unable to track themaneuvering target Figure 13 showsthat when 120573 increases to 30 and tgo = 3 seconds theoverload can converge to about 5 g but when tgo = 2 sec-onds the overload may continue to increase indicating that
0
10
20
30
0 5 10 15 20Time (s)
tgo = 1 stgo = 2 stgo = 3 s
tgo = 1 s (diverge)
tgo = 2 s (diverge)
tgo = 3 s (converge)
minus10
minus20
minus30
Miss
ile ac
cele
ratio
n (G
)
120573 = 30)HOSM-IGC (
Figure 13 Missile acceleration profile
0
10
20
30
0 5 10 15 20Time (s)
tgo = 1 s (diverge)
tgo = 2 s (converge)
tgo = 3 s (converge)
tgo = 1 stgo = 2 stgo = 3 s
minus10
minus20
minus30
Miss
ile ac
cele
ratio
n (G
)
HOSM-IGC (120573 = 50)
Figure 14 Missile acceleration profile
the convergence still needs to be quickened thus continuingto rise 120573 to 50 Figure 14 shows that the HOSM-IGC methodcan track the target that maneuvers when tgo = 2 secondsbut it causes the divergence of overload and the increase ofmiss distance if the target maneuvers when tgo = 1 secondFigure 15 gives the overload curve of the OSMG guidance lawand shows that the maneuver of the target before collisionmay cause the large-scale oscillation of the missilersquos overloadwhich may diverge to a large numerical value when thecollision occurs in the end
12 Mathematical Problems in Engineering
Table 4 Average miss distances of 50 simulations
Targetmaneuveringtiming
HOSM-IGC120573 = 30
HOSM-IGC120573 = 40
HOSM-IGC120573 = 50
OSMG
tgo = 1second 43539 3224 28936 36116
tgo = 2seconds 25424 18665 09322 35534
tgo = 3seconds 08124 08265 07538 11959
Time (s)0 5 10 15 20
0
10
20
30OSMG
tgo = 1 stgo = 2 stgo = 3 s
minus10
minus20
minus30
Miss
ile ac
cele
ratio
n (G
)
Figure 15 Missile acceleration profile
The analysis in Figures 12 13 14 and 15 shows that theoverload of the missile converges faster and its miss distanceis smaller with increasing 120573 To verify this finding we carryout 50 times Monte Carlo simulations in which the positionand speed of the target have 1 of random difference
The average miss distances are shown in Table 4 andFigure 16 Clearly the timing of the targetrsquos step maneuverdramatically affects the final interception precision morespecifically given a shorter reaction time for the guidanceand control system the missile seems more likely to missthe target To the OSMG guidance law in all three scenarioshardly does it show any advantages against theHOSM-IGC Itcan also be seen that with the increasing of the120573 theHOSM-IGC system responds even faster which leads to an obviousdecrease of the average miss distance The effect of 120573 on theresponse of the HOSM-IGC system is a valuable guidelinewhen implementing the proposed method into practice
53 Scenario 3 Weaving Target In this scenario the targetmaneuvers by 119886
119879= 40sin(1205871199052) OSMG with three-loop
Table 5 Initial conditions
Variable Description Value(119883119872 119884119872) Missile initial position (0 km 175 km)
(119883119879 119884119879) Target initial position (60 km 195 km)
119881119872
Missile initial velocity 1200 (ms)119881119879
Target initial velocity 2000 (ms)120579119872
Missile initial flight path angle 0 (deg)120579119879
Target initial flight path angle 0 (deg)120576 Compensatory gain 3120575 Saturation parameter 0001
autopilot is introduced for comparison The motion equa-tions of the target are as follows
119910119879
= 119881119879cos 120579119879
119879
= minus119881119879sin 120579119879
120579119879
=119886119879
119881119879
119886119879
= 40 sin(120587119905
2)
(57)
The initial simulation conditions are given in Table 5The missilersquos trajectory under the two guidance and
control methods given in Figure 17 does not see muchdifference
However the overload curve given in Figure 18 showsthat after the missile completes its initial attitude adjustmentwith the HOSM-IGC method it can almost ideally track itsmaneuvering target by contrast with the OSMG methodthe missile seems to have the tendency to track its targetrsquosmaneuver but has larger tracking errors Besides with theOSMGmethod the missilersquos overload increases rapidly at theend of attack primarily because of the divergence of its LOSrate On the other hand with the HOSM-IGC method themissile has no divergence even at the end of attack ensuringa smaller target missing quantity
The actuator deflection curve in Figure 19 shows thatin order to provide a rather big normal overload for theend phase the OSMG method produces a rather big rudderdeflection command however it may increase the missilersquostarget missing quantity once its rudder deflection saturatesand themissile does not have enough overloads or the ruddercannot respond that fast
As shown in Figure 20 because of the dramatic changein overload command the response of the missilersquos autopilotto high-frequency command sees an obvious phase lag andamplitude value attenuation its actual overload cannot trackthe command ideally this is a main reason why the missdistance increases However the controller in the HOSM-IGC method gives its rudder deflection command directlyand there is no lagging or attenuation caused by the autopilotthus enhancing the guidance precision effectively Further-more the fast convergence of the high order sliding modemakes the missile rapidly track its maneuvering target withthemost reasonable rudder deflection command reducing itsoverload effectively
Mathematical Problems in Engineering 13
0
05
1
15
2
25
3
35
4
45
5
0 10 20 30 40 50Simulation times
OSMG
Miss
(m)
0
05
1
15
2
25
3
35
4
45
5
0 10 20 30 40 50Simulation times
Miss
(m)
0
05
1
15
2
25
3
35
4
45
5
0 10 20 30 40 50Simulation times
Miss
(m)
tgo = 3 stgo = 2 stgo = 1 s
0
05
1
15
2
25
3
35
4
45
5
0 10 20 30 40 50Simulation times
Miss
(m)
tgo = 3 stgo = 2 stgo = 1 s
HOSM-IGC (120573 = 40)
HOSM-IGC (120573 = 50)
HOSM-IGC (120573 = 30)
Figure 16 Miss distances of HOSM-IGC (120573 = 30 40 50) and OSMG with target maneuvers at different time-to-go (tgo = 1 second 2seconds and 3 seconds)
The average miss distance of 50 simulations under theconditions is 073m for HOSM-IGC and 186m for OSMGWe can see that the HOSM-IGC method not only doesprovide a more reasonable actuator deflection command butalso achieves a higher interception precision
6 Conclusions
This paper proposes an LOS feedback integrated guidanceand control method using quasi-continuous high order
sliding mode guidance and control method With the fastand precise convergence of the quasi-continuous HOSMmethod the HOSM-IGCmethod performsmuch better thanthe traditional separated guidance and control method withless acceleration effort and less miss distance in all thethree simulation scenarios of nonmaneuvering target stepmaneuvering target and weaving target In addition the ideaof virtual control largely alleviates the chattering withoutany sacrifice of robustness As a result of the alleviationof the chattering the control input command 120575
119885becomes
14 Mathematical Problems in Engineering
175
180
185
190
195
200
TargetHOSM-IGCOSMG
Y(k
m)
0 20 40 60 0 20 40X (km)
Figure 17 The trajectories of the missile and its target
0
10
20
0 5 10 15 20Time (s)
TargetOSMGHOSM-IGC
Miss
ile an
d ta
rget
acce
lera
tion
(G)
minus10
minus20
Figure 18 Missile acceleration profile
smooth enough for the actuator to be implemented as aresult the interception accuracy is improved In designingthe quasi-continuous sliding mode controller the Arbitrary-Order Robust Exact Differentiator is introduced to estimatethe high order time derivative of the LOS rate 119902 which notonly ensures the guidance and control precision but alsosimplifies the calculation
0
10
20
30
0 5 10 15 20Time (s)
OSMGHOSM-IGC
minus10
minus20
minus30
minus40
Actu
ator
defl
ectio
n (d
eg)
Figure 19 Actuator deflection
0
50
100
150
200
15 16 17 18Time (s)
Commanded accelerationAchieved acceleration
Miss
ile ac
cele
ratio
n (G
)
minus50
Figure 20 Commanded acceleration and achieved acceleration
Appendices
A The Third-Order RobustExact Differentiator
The third-order robust exact differentiator built according tothe following parameters and equations is used to performthe simulation the initial value 120582
0= 1205821
= 1205822
= 1205823
= 50
Mathematical Problems in Engineering 15
0
0005
001
0015
002
0025
003
0035
004
0 02 04Time (s)
0
2
0 02 04Time (s)
0
100
200
300
0
100
200
300
400
0 02 04Time (s)
h(t)
z3
h(t) and z3
minus100
minus200
minus300
minus400
minus5000 02 04
Time (s)
g(t)
z2
g(t) and z2
minus100
minus200
minus300
minus400
f(t)
z1
f(t) and z1
minus2
minus4
minus6
minus8
minus10
minus12
minus14
F(t)
z0
F(t) and z0
minus0005
minus001
Figure 21 The outputs of the differentiator in the sampling interval of 0001 seconds
the initial values 1199110
= 01 1199111
= 1199112
= 1199113
= 0 119871 = 1400 thesimulation step is 0001 seconds Consider
0
= V0
V0
= minus120582011987114 10038161003816100381610038161199110 minus
1199021003816100381610038161003816
34 sign (1199110
minus 119902) + 1199111
1
= V1
V1
= minus120582111987113 10038161003816100381610038161199111 minus V
0
1003816100381610038161003816
23 sign (1199111
minus V0) + 1199112
2
= V2
V2
= minus120582211987112 10038161003816100381610038161199112 minus V
1
1003816100381610038161003816
12 sign (1199112
minus V1) + 1199113
3
= minus1205823119871 sign (119911
3minus V2)
(A1)
16 Mathematical Problems in Engineering
0
0005
001
0015
002
0025
003
0035
004
0 02 04Time (s)
0
2
0 02 04Time (s)
0
100
200
300
0 02 04Time (s)
0
200
400
0 02 04Time (s)
h(t) and z3
minus200
minus400
minus600
minus800
g(t) and z2
minus100
minus200
minus300
minus400
f(t) and z1
minus2
minus4
minus6
minus8
minus10
minus12
minus14
minus16
F(t) and z0
minus0005
minus001
h(t)
z3g(t)
z2
f(t)
z1F(t)
z0
Figure 22 The outputs of the differentiator in the sampling interval of 00001 seconds
If the designed signals are 119865(119905) = int 119891(119905)119889119905 119891(119905) =
int 119892(119905)119889119905 119892(119905) = int ℎ(119905)119889119905 and ℎ(119905) = sin(2120587119905) +
3cos(120587119905) then theoretically 119891(119905) is the first-order differ-ential of input signal 119892(119905) is the second-order differ-ential of input signal ℎ(119905) is the third-order differen-tial of input signal Through the simulation the differ-ential signal of actual output is compared with that of
theoretical output the comparison results are shown inFigure 21
Figure 21 shows that 1199110can effectively track the signal
119865(119905) but 1199112has obvious errors in tracking the second-order
differential 119892(119905) 1199113has greater errors in tracking the third-
order differential thus the overall differential effect is notideal
Mathematical Problems in Engineering 17
0 02 04
0
0005
001
0015
002
0025
003
0035
004
Time (s)0 02 04
0
01
02
03
04
05
Time (s)
0 02 04
0
05
1
15
2
Time (s)0 02 04
0
05
1
15
2
25
3
35
4
Time (s)
h(t) and z3g(t) and z2
minus04
minus05
f(t) and z1
minus2
minus15
minus1
minus05
minus1
minus05
minus01
minus02
minus03
F(t) and z0
minus0005
minus001
h(t)
z3g(t)
z2
f(t)
z1F(t)
z0
Figure 23 The outputs of the differentiator in the sampling interval of 00001 seconds
There are two causes for the errors in a usual differentia-tor the setting of sampling interval and the setting of numberof orders of the differentiator The following theorem applieshere
Theorem A1 (see [19]) If the sampling interval 120591 gt 0 theaccuracy of the controller is expressed as follows
10038161003816100381610038161003816120590(119894)
10038161003816100381610038161003816le 120583120591119903minus119894
119894 = 0 119903 minus 1 (A2)
That is to say the precision (understood as the differentialprecision of a differentiator) of the 119894th order control is in directproportion to the power of 119903 minus 119894 times of sampling interval120591 Therefore the decrease of iteration step can enhance thedifferential tracking precision
Under the conditions that the sampling interval is 10 timessmaller and that the setting is 00001 seconds the simulationresults are shown in Figure 22
18 Mathematical Problems in Engineering
Table 6 Tracking errors of the AORED with different orders andsampling interval
Orders of AORED Third order Fifth orderSample period 120591 = 0001 120591 = 00001 120591 = 00001
First-order differential error 001 Negligible NegligibleSecond-order differential error 15 002 NegligibleThird-order differential error 70 25 00005
As Figure 22 shows the signal tracking is rather idealexcept a peak in the beginning the tracking error in the first-order differential tracking can be neglected 119911
2has an obviously
smaller tracking error in the second-order differential 119892(119905)
tracking in the beginning 1199113has a rather sharp peak in the
third-order differential tracking but converges to a relativelyhigh precision after 01 seconds As we can see the differentialtracking errors for each number of orders are greatly reducedbut the third-order differential tracking error is still rather bigIn the following we will reduce the tracking errors by increasingthe number of orders of the differentiator
B The Fifth-Order Robust Exact Differentiator
The settings of the fifth-order differentiator are given asfollows
0
= V0
V0
= minus120582011987116 10038161003816100381610038161199110 minus
1199021003816100381610038161003816
56 sign (1199110
minus 119902) + 1199111
1
= V1
V1
= minus120582111987115 10038161003816100381610038161199111 minus V
0
1003816100381610038161003816
45 sign (1199111
minus V0) + 1199112
2
= V2
V2
= minus120582211987114 10038161003816100381610038161199112 minus V
1
1003816100381610038161003816
34 sign (1199112
minus V1) + 1199113
3
= V3
V3
= minus120582311987113 10038161003816100381610038161199113 minus V
2
1003816100381610038161003816
23 sign (1199113
minus V2) + 1199114
4
= V4
V4
= minus120582411987112 10038161003816100381610038161199114 minus V
3
1003816100381610038161003816
12 sign (1199114
minus V3) + 1199115
5
= V5
V5
= minus1205825119871 sign (119911
5minus V4)
(B1)
The initial value 1199110
= 01 1205820
= 1205821
= 1205822
= 1205823
= 1205824
=
1205825
= 50 the initial values 1199111 1199112 1199113 1199114 1199115
= 0 119871 = 1400the simulation step length is 00001 seconds The simulationresults are shown in Figure 23
As the figure shows the fifth-order differentiator hasa rather ideal tracking effect on the third-order differen-tial tracking the tracking error being only about 00005(see Table 6)
To sum up the Arbitrary-Order Robust Exact Differen-tiator can estimate rather ideal differential signals by selecting
Table 7
Length 365m119871 ref 365m119883119866
177m119878ref 0026m2
Diameter 0178mMass 1016 Kg119868119885
1063 Kgsdotm2
an appropriate sampling interval and using the differentiatorwith a relatively high number of orders
C Physical and Geometric Characteristics
See Table 7
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] P Zarchan Tactical and Strategic Missile Guidance 2002[2] D Zhou CMu andWXu ldquoAdaptive sliding-mode guidance of
a homing missilerdquo Journal of Guidance Control and Dynamicsvol 22 no 4 pp 589ndash594 1999
[3] G Zeng and M Hu ldquoFinite-time control for electromagneticsatellite formationsrdquo Acta Astronautica vol 74 pp 120ndash1302012
[4] S R Kumar S Rao and D Ghose ldquoNonsingular terminalsliding mode guidance with impact angle constraintsrdquo Journalof Guidance Control and Dynamics vol 37 no 4 pp 1114ndash11302014
[5] Y Liang C Chen D Liaw Y Feng C Cheng and CChen ldquoRobust guidance law via integral-sliding-mode schemerdquoJournal of Guidance Control and Dynamics vol 37 no 3 pp1038ndash1042 2014
[6] A V Savkin P N Pathirana and F Faruqi ldquoProblem ofprecision missile guidance LQR and 119867
infincontrol frameworksrdquo
IEEE Transactions on Aerospace and Electronic Systems vol 39no 3 pp 901ndash910 2003
[7] H Y Chen andC-C Yang ldquoNonlinear1198672119867infinguidance design
for homing missilesrdquo in Proceedings of the AIAA GuidanceNavigation and Control Conference and Exhibit vol 77 2001
[8] P K Menon G D Sweriduk and E J Ohlmeyer ldquoOptimalfixed-interval integrated guidance-control laws for hit-to-killmissilesrdquo in Proceedings of the AIAA Guidance Navigation andControl Conference and Exhibit AIAA 2003-5579 Austin TexUSA August 2003
[9] S S Vaddi P K Menon and E J Ohlmeyer ldquoNumericalSDRE approach for missile integrated guidance-controlrdquo inProceedings of the AIAA Guidance Navigation and ControlConference and Exhibit AIAA 2007-6672 Hilton Head SCUSA August 2007
[10] M Xin S N Balakrishnan and E J Ohlmeyer ldquoIntegratedguidance and control of missiles with Theta-D methodrdquo IEEE
Mathematical Problems in Engineering 19
Transactions on Control Systems Technology vol 14 no 6 pp981ndash992 2006
[11] T Shima M Idan and O M Golan ldquoSliding-mode controlfor integrated missile autopilot guidancerdquo Journal of GuidanceControl and Dynamics vol 29 no 2 pp 250ndash260 2006
[12] Y B Shtessel and C H Tournes ldquoIntegrated higher-ordersliding mode guidance and autopilot for dual control missilesrdquoJournal of Guidance Control and Dynamics vol 32 no 1 pp79ndash94 2009
[13] F Y Dong H M Lei and C J Zhou ldquoIntegrated robust highorder sliding mode guidance and control for missilesrdquo ActaAeronautica et Astronautica Sinica vol 34 no 9 pp 2212ndash22182013
[14] H Mingzhe and D Guangren ldquoIntegrated guidance and con-trol of homing missiles against ground fixed targetsrdquo ChineseJournal of Aeronautics vol 21 no 2 pp 162ndash168 2008
[15] A Levant ldquoHomogeneity approach to high-order sliding modedesignrdquo Automatica vol 41 no 5 pp 823ndash830 2005
[16] Y-W Liang C Chen D Liaw Y Feng C Cheng and CChen ldquoRobust guidance law via integral-sliding-mode schemerdquoJournal of Guidance Control and Dynamics vol 37 no 3 pp1038ndash1042 2014
[17] L Wang Y Sheng and X Liu ldquoContinuous time-varying slid-ingmode based attitude control for reentry vehiclerdquo Proceedingsof the Institution of Mechanical Engineers Part G Journal ofAerospace Engineering 2014
[18] A Levant ldquoHigher-order sliding modes differentiation andoutput-feedback controlrdquo International Journal of Control vol76 no 9-10 pp 924ndash941 2003
[19] A Levant ldquoQuasi-continuous high-order sliding-mode con-trollersrdquo IEEE Transactions on Automatic Control vol 50 no11 pp 1812ndash1816 2005
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
Volume 2014
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OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 9
175
180
185
190
195
200
205
TargetHOSM-IGCPN
Y(k
m)
0 10 20 30 40 50 60X (km)
Figure 5 Target and missile trajectories
0
5
10
15
20
25
0 5 10 15 20Time (s)
HOSM-IGCPN
Initial boresight adjustment
Sliding manifold reached
Miss
ile ac
cele
ratio
n (G
)
minus5
Figure 6 Missile acceleration profile
It is evident that the speed of divergence of the overloadcommand given by the PN guidance law increases withthe target speed Specifically when the target speed reaches4000ms the commanded overload is almost 30 g which isobviously not ideal for the attack of a nonmaneuvering targetbut with the HOSM-IGC method the missile adjusts itsboresight very quickly then maintains it around 0 g and fliesto its target in the rectilinear ballistic trajectory not affectedby the increases of target speed still accomplishing the high-precision hit-on collision
Figure 8 shows that although the HOSM-IGC methodachieves a more effective overload command on the otherside it sees some chattering when the sliding mode reaches
0
5
10
15
20
25
30
0 5 10 15Time (s)
Miss
ile ac
cele
ratio
n (G
)
minus5
HOSM-IGC (Vt = minus3000)HOSM-IGC (Vt = minus4000)
PN (Vt = minus3000)PN (Vt = minus4000)
Figure 7 Missile acceleration profile
0
1
2
0 5 10 15 20Time (s)
4 45 5
0
02
04
minus02
minus04
Actu
ator
defl
ectio
n (d
eg)
minus1
minus2
minus3
minus4
HOSM-IGC (120573 = 30)
Figure 8 Actuator deflection
the desired manifold the commanded rudder deflectionangle chatters for around 045 deg at about 15Hz which iskind of harmful to the system To reduce the chatteringwe adjust the controllerrsquos parameter 120573 = 10 and carry outsimulations again Figure 9 shows that after 120573 decreasesthe command of rudder deflection angle converges slower(for about 7 sec) however the chattering weakens obviouslyits magnitude being only about 015 degrees The smallerchattering well enhances convergence precision eventuallyreducing the target missing The reason that minor 120573 resultsin an alleviative chattering can be seen from (47) Figure 10shows the target and missile trajectories
Table 2 compares the average miss distance of 50 simula-tions under the conditions discussed above the comparison
10 Mathematical Problems in Engineering
Table 2 Average miss distance of 50 simulations
Target speed PN HOSM-IGC(120573 = 30)
HOSM-IGC(120573 = 10)
119881119905= 2000ms 115m 086 073m
119881119905= 3000ms 261m 117 106
119881119905= 4000ms 542m 156 134
0
1
2
0 5 10 15 20Time (s)
72 74 76 78
0
02
minus02
minus04
Actu
ator
defl
ectio
n (d
eg)
minus1
minus2
minus3
minus4
HOSM-IGC (120573 = 10)HOSM-IGC (120573 = 30)
Figure 9 Actuator deflection
results show that when the missile intercepts a nonmaneu-vering target the hit precision of the HOSM-IGC methodis apparently higher than that of the PN guidance law andthat the low-gain HOSM-IGC method can effectively reducethe chattering magnitude thus enhancing the interceptionprecision
52 Scenario 2 Step Maneuvering Target At the first stagethe target flies at uniform speed and in a rectilinear way after10 seconds it maneuvers at the normal acceleration of 5 g Inthis scenario OSMG is introduced for a comparison with theHOSM-IGC
119910119879
= 119881119879cos 120579119879
119879
= minus119881119879sin 120579119879
120579119879
=119886119879
119881119879
119886119879
=
0 g 119905 lt 10 sec
5 g 119905 gt 10 sec
(56)
The initial simulation conditions are given in Table 3The overload curve in Figure 11 shows dearly that both
types of guidance laws can track the maneuvering targetDuring 0 to 10 seconds the target flies at uniform speed andin a rectilinear way and the missile converges its overload to
Table 3 Initial conditions
Variable Description Value(119883119872 119884119872) Missile initial position (0 km 22 km)
(119883119879 119884119879) Target initial position (60 km 20 km)
119881119872
Missile initial velocity 1200 (ms)119881119879
Target initial velocity 2000 (ms)120579119872
Missile initial flight path angle 0 (deg)120579119879
Target initial flight path angle 0 (deg)120576 Compensatory gain 3120575 Saturation parameter 0001
15
17
19
21
23
25
TargetHOSM-IGCOSMG
Y(k
m)
0 20 40 60X (km)
Figure 10 Target and missile trajectories
0 g and flies to its target after 10 seconds the target beginsto maneuver by 5 g Both types of guidance law respondby rapidly increasing the overload adjusting attitude andmaking the missilersquos boresight aim at its target
We can see that when both types of guidance law tracktheir targets the convergence speed of OSMG is almostthe same as the HOSM-IGC method But the HOSM-IGCmethod has higher convergence precision and needs loweroverload at the end phase
We can also see that after the target maneuvers if themissile is given enough time to track the targetrsquos maneuvernamely let the missilersquos overload command converge to theoverload of the target theremay not be largemiss distance Inother words for a certain period of time before the collisionthe targetmaneuver (it means only a limitedmaneuver whichdoes not include the condition that the maneuvering of thetarget for a long time may cause a change of the geometricalrelations between the missile and its target) has a small effecton both types of guidance law
But if themaneuver occurs rather late namely within oneto three seconds before collision when the overload com-mand of guidance law is not yet converged the approaching
Mathematical Problems in Engineering 11
0
5
10
0 5 10 15 20Time (s)
TargetOSMGHOSM-IGC
Miss
ile an
d ta
rget
acce
lera
tion
(G)
minus5
minus10
Figure 11 Missile and target acceleration profile
0
10
20
30
0 5 10 15 20Time (s)
tgo = 1 stgo = 2 stgo = 3 s
tgo = 3 s (diverge)
tgo = 2 s (diverge)
tgo = 1 s (diverge)minus10
minus20
minus30
Miss
ile ac
cele
ratio
n (G
)
HOSM-IGC (120573 = 10)
Figure 12 Missile acceleration profile
collision increases themiss distanceTherefore increasing theparameter 120573 can remarkably increase the convergence speedand effectively enhance guidance precision As Figure 12shows when 120573 = 10 the overload at the end phase convergesslowly even if the target maneuvers three seconds beforecollision the missilersquos overload still has no time to convergebeing unable to track themaneuvering target Figure 13 showsthat when 120573 increases to 30 and tgo = 3 seconds theoverload can converge to about 5 g but when tgo = 2 sec-onds the overload may continue to increase indicating that
0
10
20
30
0 5 10 15 20Time (s)
tgo = 1 stgo = 2 stgo = 3 s
tgo = 1 s (diverge)
tgo = 2 s (diverge)
tgo = 3 s (converge)
minus10
minus20
minus30
Miss
ile ac
cele
ratio
n (G
)
120573 = 30)HOSM-IGC (
Figure 13 Missile acceleration profile
0
10
20
30
0 5 10 15 20Time (s)
tgo = 1 s (diverge)
tgo = 2 s (converge)
tgo = 3 s (converge)
tgo = 1 stgo = 2 stgo = 3 s
minus10
minus20
minus30
Miss
ile ac
cele
ratio
n (G
)
HOSM-IGC (120573 = 50)
Figure 14 Missile acceleration profile
the convergence still needs to be quickened thus continuingto rise 120573 to 50 Figure 14 shows that the HOSM-IGC methodcan track the target that maneuvers when tgo = 2 secondsbut it causes the divergence of overload and the increase ofmiss distance if the target maneuvers when tgo = 1 secondFigure 15 gives the overload curve of the OSMG guidance lawand shows that the maneuver of the target before collisionmay cause the large-scale oscillation of the missilersquos overloadwhich may diverge to a large numerical value when thecollision occurs in the end
12 Mathematical Problems in Engineering
Table 4 Average miss distances of 50 simulations
Targetmaneuveringtiming
HOSM-IGC120573 = 30
HOSM-IGC120573 = 40
HOSM-IGC120573 = 50
OSMG
tgo = 1second 43539 3224 28936 36116
tgo = 2seconds 25424 18665 09322 35534
tgo = 3seconds 08124 08265 07538 11959
Time (s)0 5 10 15 20
0
10
20
30OSMG
tgo = 1 stgo = 2 stgo = 3 s
minus10
minus20
minus30
Miss
ile ac
cele
ratio
n (G
)
Figure 15 Missile acceleration profile
The analysis in Figures 12 13 14 and 15 shows that theoverload of the missile converges faster and its miss distanceis smaller with increasing 120573 To verify this finding we carryout 50 times Monte Carlo simulations in which the positionand speed of the target have 1 of random difference
The average miss distances are shown in Table 4 andFigure 16 Clearly the timing of the targetrsquos step maneuverdramatically affects the final interception precision morespecifically given a shorter reaction time for the guidanceand control system the missile seems more likely to missthe target To the OSMG guidance law in all three scenarioshardly does it show any advantages against theHOSM-IGC Itcan also be seen that with the increasing of the120573 theHOSM-IGC system responds even faster which leads to an obviousdecrease of the average miss distance The effect of 120573 on theresponse of the HOSM-IGC system is a valuable guidelinewhen implementing the proposed method into practice
53 Scenario 3 Weaving Target In this scenario the targetmaneuvers by 119886
119879= 40sin(1205871199052) OSMG with three-loop
Table 5 Initial conditions
Variable Description Value(119883119872 119884119872) Missile initial position (0 km 175 km)
(119883119879 119884119879) Target initial position (60 km 195 km)
119881119872
Missile initial velocity 1200 (ms)119881119879
Target initial velocity 2000 (ms)120579119872
Missile initial flight path angle 0 (deg)120579119879
Target initial flight path angle 0 (deg)120576 Compensatory gain 3120575 Saturation parameter 0001
autopilot is introduced for comparison The motion equa-tions of the target are as follows
119910119879
= 119881119879cos 120579119879
119879
= minus119881119879sin 120579119879
120579119879
=119886119879
119881119879
119886119879
= 40 sin(120587119905
2)
(57)
The initial simulation conditions are given in Table 5The missilersquos trajectory under the two guidance and
control methods given in Figure 17 does not see muchdifference
However the overload curve given in Figure 18 showsthat after the missile completes its initial attitude adjustmentwith the HOSM-IGC method it can almost ideally track itsmaneuvering target by contrast with the OSMG methodthe missile seems to have the tendency to track its targetrsquosmaneuver but has larger tracking errors Besides with theOSMGmethod the missilersquos overload increases rapidly at theend of attack primarily because of the divergence of its LOSrate On the other hand with the HOSM-IGC method themissile has no divergence even at the end of attack ensuringa smaller target missing quantity
The actuator deflection curve in Figure 19 shows thatin order to provide a rather big normal overload for theend phase the OSMG method produces a rather big rudderdeflection command however it may increase the missilersquostarget missing quantity once its rudder deflection saturatesand themissile does not have enough overloads or the ruddercannot respond that fast
As shown in Figure 20 because of the dramatic changein overload command the response of the missilersquos autopilotto high-frequency command sees an obvious phase lag andamplitude value attenuation its actual overload cannot trackthe command ideally this is a main reason why the missdistance increases However the controller in the HOSM-IGC method gives its rudder deflection command directlyand there is no lagging or attenuation caused by the autopilotthus enhancing the guidance precision effectively Further-more the fast convergence of the high order sliding modemakes the missile rapidly track its maneuvering target withthemost reasonable rudder deflection command reducing itsoverload effectively
Mathematical Problems in Engineering 13
0
05
1
15
2
25
3
35
4
45
5
0 10 20 30 40 50Simulation times
OSMG
Miss
(m)
0
05
1
15
2
25
3
35
4
45
5
0 10 20 30 40 50Simulation times
Miss
(m)
0
05
1
15
2
25
3
35
4
45
5
0 10 20 30 40 50Simulation times
Miss
(m)
tgo = 3 stgo = 2 stgo = 1 s
0
05
1
15
2
25
3
35
4
45
5
0 10 20 30 40 50Simulation times
Miss
(m)
tgo = 3 stgo = 2 stgo = 1 s
HOSM-IGC (120573 = 40)
HOSM-IGC (120573 = 50)
HOSM-IGC (120573 = 30)
Figure 16 Miss distances of HOSM-IGC (120573 = 30 40 50) and OSMG with target maneuvers at different time-to-go (tgo = 1 second 2seconds and 3 seconds)
The average miss distance of 50 simulations under theconditions is 073m for HOSM-IGC and 186m for OSMGWe can see that the HOSM-IGC method not only doesprovide a more reasonable actuator deflection command butalso achieves a higher interception precision
6 Conclusions
This paper proposes an LOS feedback integrated guidanceand control method using quasi-continuous high order
sliding mode guidance and control method With the fastand precise convergence of the quasi-continuous HOSMmethod the HOSM-IGCmethod performsmuch better thanthe traditional separated guidance and control method withless acceleration effort and less miss distance in all thethree simulation scenarios of nonmaneuvering target stepmaneuvering target and weaving target In addition the ideaof virtual control largely alleviates the chattering withoutany sacrifice of robustness As a result of the alleviationof the chattering the control input command 120575
119885becomes
14 Mathematical Problems in Engineering
175
180
185
190
195
200
TargetHOSM-IGCOSMG
Y(k
m)
0 20 40 60 0 20 40X (km)
Figure 17 The trajectories of the missile and its target
0
10
20
0 5 10 15 20Time (s)
TargetOSMGHOSM-IGC
Miss
ile an
d ta
rget
acce
lera
tion
(G)
minus10
minus20
Figure 18 Missile acceleration profile
smooth enough for the actuator to be implemented as aresult the interception accuracy is improved In designingthe quasi-continuous sliding mode controller the Arbitrary-Order Robust Exact Differentiator is introduced to estimatethe high order time derivative of the LOS rate 119902 which notonly ensures the guidance and control precision but alsosimplifies the calculation
0
10
20
30
0 5 10 15 20Time (s)
OSMGHOSM-IGC
minus10
minus20
minus30
minus40
Actu
ator
defl
ectio
n (d
eg)
Figure 19 Actuator deflection
0
50
100
150
200
15 16 17 18Time (s)
Commanded accelerationAchieved acceleration
Miss
ile ac
cele
ratio
n (G
)
minus50
Figure 20 Commanded acceleration and achieved acceleration
Appendices
A The Third-Order RobustExact Differentiator
The third-order robust exact differentiator built according tothe following parameters and equations is used to performthe simulation the initial value 120582
0= 1205821
= 1205822
= 1205823
= 50
Mathematical Problems in Engineering 15
0
0005
001
0015
002
0025
003
0035
004
0 02 04Time (s)
0
2
0 02 04Time (s)
0
100
200
300
0
100
200
300
400
0 02 04Time (s)
h(t)
z3
h(t) and z3
minus100
minus200
minus300
minus400
minus5000 02 04
Time (s)
g(t)
z2
g(t) and z2
minus100
minus200
minus300
minus400
f(t)
z1
f(t) and z1
minus2
minus4
minus6
minus8
minus10
minus12
minus14
F(t)
z0
F(t) and z0
minus0005
minus001
Figure 21 The outputs of the differentiator in the sampling interval of 0001 seconds
the initial values 1199110
= 01 1199111
= 1199112
= 1199113
= 0 119871 = 1400 thesimulation step is 0001 seconds Consider
0
= V0
V0
= minus120582011987114 10038161003816100381610038161199110 minus
1199021003816100381610038161003816
34 sign (1199110
minus 119902) + 1199111
1
= V1
V1
= minus120582111987113 10038161003816100381610038161199111 minus V
0
1003816100381610038161003816
23 sign (1199111
minus V0) + 1199112
2
= V2
V2
= minus120582211987112 10038161003816100381610038161199112 minus V
1
1003816100381610038161003816
12 sign (1199112
minus V1) + 1199113
3
= minus1205823119871 sign (119911
3minus V2)
(A1)
16 Mathematical Problems in Engineering
0
0005
001
0015
002
0025
003
0035
004
0 02 04Time (s)
0
2
0 02 04Time (s)
0
100
200
300
0 02 04Time (s)
0
200
400
0 02 04Time (s)
h(t) and z3
minus200
minus400
minus600
minus800
g(t) and z2
minus100
minus200
minus300
minus400
f(t) and z1
minus2
minus4
minus6
minus8
minus10
minus12
minus14
minus16
F(t) and z0
minus0005
minus001
h(t)
z3g(t)
z2
f(t)
z1F(t)
z0
Figure 22 The outputs of the differentiator in the sampling interval of 00001 seconds
If the designed signals are 119865(119905) = int 119891(119905)119889119905 119891(119905) =
int 119892(119905)119889119905 119892(119905) = int ℎ(119905)119889119905 and ℎ(119905) = sin(2120587119905) +
3cos(120587119905) then theoretically 119891(119905) is the first-order differ-ential of input signal 119892(119905) is the second-order differ-ential of input signal ℎ(119905) is the third-order differen-tial of input signal Through the simulation the differ-ential signal of actual output is compared with that of
theoretical output the comparison results are shown inFigure 21
Figure 21 shows that 1199110can effectively track the signal
119865(119905) but 1199112has obvious errors in tracking the second-order
differential 119892(119905) 1199113has greater errors in tracking the third-
order differential thus the overall differential effect is notideal
Mathematical Problems in Engineering 17
0 02 04
0
0005
001
0015
002
0025
003
0035
004
Time (s)0 02 04
0
01
02
03
04
05
Time (s)
0 02 04
0
05
1
15
2
Time (s)0 02 04
0
05
1
15
2
25
3
35
4
Time (s)
h(t) and z3g(t) and z2
minus04
minus05
f(t) and z1
minus2
minus15
minus1
minus05
minus1
minus05
minus01
minus02
minus03
F(t) and z0
minus0005
minus001
h(t)
z3g(t)
z2
f(t)
z1F(t)
z0
Figure 23 The outputs of the differentiator in the sampling interval of 00001 seconds
There are two causes for the errors in a usual differentia-tor the setting of sampling interval and the setting of numberof orders of the differentiator The following theorem applieshere
Theorem A1 (see [19]) If the sampling interval 120591 gt 0 theaccuracy of the controller is expressed as follows
10038161003816100381610038161003816120590(119894)
10038161003816100381610038161003816le 120583120591119903minus119894
119894 = 0 119903 minus 1 (A2)
That is to say the precision (understood as the differentialprecision of a differentiator) of the 119894th order control is in directproportion to the power of 119903 minus 119894 times of sampling interval120591 Therefore the decrease of iteration step can enhance thedifferential tracking precision
Under the conditions that the sampling interval is 10 timessmaller and that the setting is 00001 seconds the simulationresults are shown in Figure 22
18 Mathematical Problems in Engineering
Table 6 Tracking errors of the AORED with different orders andsampling interval
Orders of AORED Third order Fifth orderSample period 120591 = 0001 120591 = 00001 120591 = 00001
First-order differential error 001 Negligible NegligibleSecond-order differential error 15 002 NegligibleThird-order differential error 70 25 00005
As Figure 22 shows the signal tracking is rather idealexcept a peak in the beginning the tracking error in the first-order differential tracking can be neglected 119911
2has an obviously
smaller tracking error in the second-order differential 119892(119905)
tracking in the beginning 1199113has a rather sharp peak in the
third-order differential tracking but converges to a relativelyhigh precision after 01 seconds As we can see the differentialtracking errors for each number of orders are greatly reducedbut the third-order differential tracking error is still rather bigIn the following we will reduce the tracking errors by increasingthe number of orders of the differentiator
B The Fifth-Order Robust Exact Differentiator
The settings of the fifth-order differentiator are given asfollows
0
= V0
V0
= minus120582011987116 10038161003816100381610038161199110 minus
1199021003816100381610038161003816
56 sign (1199110
minus 119902) + 1199111
1
= V1
V1
= minus120582111987115 10038161003816100381610038161199111 minus V
0
1003816100381610038161003816
45 sign (1199111
minus V0) + 1199112
2
= V2
V2
= minus120582211987114 10038161003816100381610038161199112 minus V
1
1003816100381610038161003816
34 sign (1199112
minus V1) + 1199113
3
= V3
V3
= minus120582311987113 10038161003816100381610038161199113 minus V
2
1003816100381610038161003816
23 sign (1199113
minus V2) + 1199114
4
= V4
V4
= minus120582411987112 10038161003816100381610038161199114 minus V
3
1003816100381610038161003816
12 sign (1199114
minus V3) + 1199115
5
= V5
V5
= minus1205825119871 sign (119911
5minus V4)
(B1)
The initial value 1199110
= 01 1205820
= 1205821
= 1205822
= 1205823
= 1205824
=
1205825
= 50 the initial values 1199111 1199112 1199113 1199114 1199115
= 0 119871 = 1400the simulation step length is 00001 seconds The simulationresults are shown in Figure 23
As the figure shows the fifth-order differentiator hasa rather ideal tracking effect on the third-order differen-tial tracking the tracking error being only about 00005(see Table 6)
To sum up the Arbitrary-Order Robust Exact Differen-tiator can estimate rather ideal differential signals by selecting
Table 7
Length 365m119871 ref 365m119883119866
177m119878ref 0026m2
Diameter 0178mMass 1016 Kg119868119885
1063 Kgsdotm2
an appropriate sampling interval and using the differentiatorwith a relatively high number of orders
C Physical and Geometric Characteristics
See Table 7
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] P Zarchan Tactical and Strategic Missile Guidance 2002[2] D Zhou CMu andWXu ldquoAdaptive sliding-mode guidance of
a homing missilerdquo Journal of Guidance Control and Dynamicsvol 22 no 4 pp 589ndash594 1999
[3] G Zeng and M Hu ldquoFinite-time control for electromagneticsatellite formationsrdquo Acta Astronautica vol 74 pp 120ndash1302012
[4] S R Kumar S Rao and D Ghose ldquoNonsingular terminalsliding mode guidance with impact angle constraintsrdquo Journalof Guidance Control and Dynamics vol 37 no 4 pp 1114ndash11302014
[5] Y Liang C Chen D Liaw Y Feng C Cheng and CChen ldquoRobust guidance law via integral-sliding-mode schemerdquoJournal of Guidance Control and Dynamics vol 37 no 3 pp1038ndash1042 2014
[6] A V Savkin P N Pathirana and F Faruqi ldquoProblem ofprecision missile guidance LQR and 119867
infincontrol frameworksrdquo
IEEE Transactions on Aerospace and Electronic Systems vol 39no 3 pp 901ndash910 2003
[7] H Y Chen andC-C Yang ldquoNonlinear1198672119867infinguidance design
for homing missilesrdquo in Proceedings of the AIAA GuidanceNavigation and Control Conference and Exhibit vol 77 2001
[8] P K Menon G D Sweriduk and E J Ohlmeyer ldquoOptimalfixed-interval integrated guidance-control laws for hit-to-killmissilesrdquo in Proceedings of the AIAA Guidance Navigation andControl Conference and Exhibit AIAA 2003-5579 Austin TexUSA August 2003
[9] S S Vaddi P K Menon and E J Ohlmeyer ldquoNumericalSDRE approach for missile integrated guidance-controlrdquo inProceedings of the AIAA Guidance Navigation and ControlConference and Exhibit AIAA 2007-6672 Hilton Head SCUSA August 2007
[10] M Xin S N Balakrishnan and E J Ohlmeyer ldquoIntegratedguidance and control of missiles with Theta-D methodrdquo IEEE
Mathematical Problems in Engineering 19
Transactions on Control Systems Technology vol 14 no 6 pp981ndash992 2006
[11] T Shima M Idan and O M Golan ldquoSliding-mode controlfor integrated missile autopilot guidancerdquo Journal of GuidanceControl and Dynamics vol 29 no 2 pp 250ndash260 2006
[12] Y B Shtessel and C H Tournes ldquoIntegrated higher-ordersliding mode guidance and autopilot for dual control missilesrdquoJournal of Guidance Control and Dynamics vol 32 no 1 pp79ndash94 2009
[13] F Y Dong H M Lei and C J Zhou ldquoIntegrated robust highorder sliding mode guidance and control for missilesrdquo ActaAeronautica et Astronautica Sinica vol 34 no 9 pp 2212ndash22182013
[14] H Mingzhe and D Guangren ldquoIntegrated guidance and con-trol of homing missiles against ground fixed targetsrdquo ChineseJournal of Aeronautics vol 21 no 2 pp 162ndash168 2008
[15] A Levant ldquoHomogeneity approach to high-order sliding modedesignrdquo Automatica vol 41 no 5 pp 823ndash830 2005
[16] Y-W Liang C Chen D Liaw Y Feng C Cheng and CChen ldquoRobust guidance law via integral-sliding-mode schemerdquoJournal of Guidance Control and Dynamics vol 37 no 3 pp1038ndash1042 2014
[17] L Wang Y Sheng and X Liu ldquoContinuous time-varying slid-ingmode based attitude control for reentry vehiclerdquo Proceedingsof the Institution of Mechanical Engineers Part G Journal ofAerospace Engineering 2014
[18] A Levant ldquoHigher-order sliding modes differentiation andoutput-feedback controlrdquo International Journal of Control vol76 no 9-10 pp 924ndash941 2003
[19] A Levant ldquoQuasi-continuous high-order sliding-mode con-trollersrdquo IEEE Transactions on Automatic Control vol 50 no11 pp 1812ndash1816 2005
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
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OptimizationJournal of
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International Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
10 Mathematical Problems in Engineering
Table 2 Average miss distance of 50 simulations
Target speed PN HOSM-IGC(120573 = 30)
HOSM-IGC(120573 = 10)
119881119905= 2000ms 115m 086 073m
119881119905= 3000ms 261m 117 106
119881119905= 4000ms 542m 156 134
0
1
2
0 5 10 15 20Time (s)
72 74 76 78
0
02
minus02
minus04
Actu
ator
defl
ectio
n (d
eg)
minus1
minus2
minus3
minus4
HOSM-IGC (120573 = 10)HOSM-IGC (120573 = 30)
Figure 9 Actuator deflection
results show that when the missile intercepts a nonmaneu-vering target the hit precision of the HOSM-IGC methodis apparently higher than that of the PN guidance law andthat the low-gain HOSM-IGC method can effectively reducethe chattering magnitude thus enhancing the interceptionprecision
52 Scenario 2 Step Maneuvering Target At the first stagethe target flies at uniform speed and in a rectilinear way after10 seconds it maneuvers at the normal acceleration of 5 g Inthis scenario OSMG is introduced for a comparison with theHOSM-IGC
119910119879
= 119881119879cos 120579119879
119879
= minus119881119879sin 120579119879
120579119879
=119886119879
119881119879
119886119879
=
0 g 119905 lt 10 sec
5 g 119905 gt 10 sec
(56)
The initial simulation conditions are given in Table 3The overload curve in Figure 11 shows dearly that both
types of guidance laws can track the maneuvering targetDuring 0 to 10 seconds the target flies at uniform speed andin a rectilinear way and the missile converges its overload to
Table 3 Initial conditions
Variable Description Value(119883119872 119884119872) Missile initial position (0 km 22 km)
(119883119879 119884119879) Target initial position (60 km 20 km)
119881119872
Missile initial velocity 1200 (ms)119881119879
Target initial velocity 2000 (ms)120579119872
Missile initial flight path angle 0 (deg)120579119879
Target initial flight path angle 0 (deg)120576 Compensatory gain 3120575 Saturation parameter 0001
15
17
19
21
23
25
TargetHOSM-IGCOSMG
Y(k
m)
0 20 40 60X (km)
Figure 10 Target and missile trajectories
0 g and flies to its target after 10 seconds the target beginsto maneuver by 5 g Both types of guidance law respondby rapidly increasing the overload adjusting attitude andmaking the missilersquos boresight aim at its target
We can see that when both types of guidance law tracktheir targets the convergence speed of OSMG is almostthe same as the HOSM-IGC method But the HOSM-IGCmethod has higher convergence precision and needs loweroverload at the end phase
We can also see that after the target maneuvers if themissile is given enough time to track the targetrsquos maneuvernamely let the missilersquos overload command converge to theoverload of the target theremay not be largemiss distance Inother words for a certain period of time before the collisionthe targetmaneuver (it means only a limitedmaneuver whichdoes not include the condition that the maneuvering of thetarget for a long time may cause a change of the geometricalrelations between the missile and its target) has a small effecton both types of guidance law
But if themaneuver occurs rather late namely within oneto three seconds before collision when the overload com-mand of guidance law is not yet converged the approaching
Mathematical Problems in Engineering 11
0
5
10
0 5 10 15 20Time (s)
TargetOSMGHOSM-IGC
Miss
ile an
d ta
rget
acce
lera
tion
(G)
minus5
minus10
Figure 11 Missile and target acceleration profile
0
10
20
30
0 5 10 15 20Time (s)
tgo = 1 stgo = 2 stgo = 3 s
tgo = 3 s (diverge)
tgo = 2 s (diverge)
tgo = 1 s (diverge)minus10
minus20
minus30
Miss
ile ac
cele
ratio
n (G
)
HOSM-IGC (120573 = 10)
Figure 12 Missile acceleration profile
collision increases themiss distanceTherefore increasing theparameter 120573 can remarkably increase the convergence speedand effectively enhance guidance precision As Figure 12shows when 120573 = 10 the overload at the end phase convergesslowly even if the target maneuvers three seconds beforecollision the missilersquos overload still has no time to convergebeing unable to track themaneuvering target Figure 13 showsthat when 120573 increases to 30 and tgo = 3 seconds theoverload can converge to about 5 g but when tgo = 2 sec-onds the overload may continue to increase indicating that
0
10
20
30
0 5 10 15 20Time (s)
tgo = 1 stgo = 2 stgo = 3 s
tgo = 1 s (diverge)
tgo = 2 s (diverge)
tgo = 3 s (converge)
minus10
minus20
minus30
Miss
ile ac
cele
ratio
n (G
)
120573 = 30)HOSM-IGC (
Figure 13 Missile acceleration profile
0
10
20
30
0 5 10 15 20Time (s)
tgo = 1 s (diverge)
tgo = 2 s (converge)
tgo = 3 s (converge)
tgo = 1 stgo = 2 stgo = 3 s
minus10
minus20
minus30
Miss
ile ac
cele
ratio
n (G
)
HOSM-IGC (120573 = 50)
Figure 14 Missile acceleration profile
the convergence still needs to be quickened thus continuingto rise 120573 to 50 Figure 14 shows that the HOSM-IGC methodcan track the target that maneuvers when tgo = 2 secondsbut it causes the divergence of overload and the increase ofmiss distance if the target maneuvers when tgo = 1 secondFigure 15 gives the overload curve of the OSMG guidance lawand shows that the maneuver of the target before collisionmay cause the large-scale oscillation of the missilersquos overloadwhich may diverge to a large numerical value when thecollision occurs in the end
12 Mathematical Problems in Engineering
Table 4 Average miss distances of 50 simulations
Targetmaneuveringtiming
HOSM-IGC120573 = 30
HOSM-IGC120573 = 40
HOSM-IGC120573 = 50
OSMG
tgo = 1second 43539 3224 28936 36116
tgo = 2seconds 25424 18665 09322 35534
tgo = 3seconds 08124 08265 07538 11959
Time (s)0 5 10 15 20
0
10
20
30OSMG
tgo = 1 stgo = 2 stgo = 3 s
minus10
minus20
minus30
Miss
ile ac
cele
ratio
n (G
)
Figure 15 Missile acceleration profile
The analysis in Figures 12 13 14 and 15 shows that theoverload of the missile converges faster and its miss distanceis smaller with increasing 120573 To verify this finding we carryout 50 times Monte Carlo simulations in which the positionand speed of the target have 1 of random difference
The average miss distances are shown in Table 4 andFigure 16 Clearly the timing of the targetrsquos step maneuverdramatically affects the final interception precision morespecifically given a shorter reaction time for the guidanceand control system the missile seems more likely to missthe target To the OSMG guidance law in all three scenarioshardly does it show any advantages against theHOSM-IGC Itcan also be seen that with the increasing of the120573 theHOSM-IGC system responds even faster which leads to an obviousdecrease of the average miss distance The effect of 120573 on theresponse of the HOSM-IGC system is a valuable guidelinewhen implementing the proposed method into practice
53 Scenario 3 Weaving Target In this scenario the targetmaneuvers by 119886
119879= 40sin(1205871199052) OSMG with three-loop
Table 5 Initial conditions
Variable Description Value(119883119872 119884119872) Missile initial position (0 km 175 km)
(119883119879 119884119879) Target initial position (60 km 195 km)
119881119872
Missile initial velocity 1200 (ms)119881119879
Target initial velocity 2000 (ms)120579119872
Missile initial flight path angle 0 (deg)120579119879
Target initial flight path angle 0 (deg)120576 Compensatory gain 3120575 Saturation parameter 0001
autopilot is introduced for comparison The motion equa-tions of the target are as follows
119910119879
= 119881119879cos 120579119879
119879
= minus119881119879sin 120579119879
120579119879
=119886119879
119881119879
119886119879
= 40 sin(120587119905
2)
(57)
The initial simulation conditions are given in Table 5The missilersquos trajectory under the two guidance and
control methods given in Figure 17 does not see muchdifference
However the overload curve given in Figure 18 showsthat after the missile completes its initial attitude adjustmentwith the HOSM-IGC method it can almost ideally track itsmaneuvering target by contrast with the OSMG methodthe missile seems to have the tendency to track its targetrsquosmaneuver but has larger tracking errors Besides with theOSMGmethod the missilersquos overload increases rapidly at theend of attack primarily because of the divergence of its LOSrate On the other hand with the HOSM-IGC method themissile has no divergence even at the end of attack ensuringa smaller target missing quantity
The actuator deflection curve in Figure 19 shows thatin order to provide a rather big normal overload for theend phase the OSMG method produces a rather big rudderdeflection command however it may increase the missilersquostarget missing quantity once its rudder deflection saturatesand themissile does not have enough overloads or the ruddercannot respond that fast
As shown in Figure 20 because of the dramatic changein overload command the response of the missilersquos autopilotto high-frequency command sees an obvious phase lag andamplitude value attenuation its actual overload cannot trackthe command ideally this is a main reason why the missdistance increases However the controller in the HOSM-IGC method gives its rudder deflection command directlyand there is no lagging or attenuation caused by the autopilotthus enhancing the guidance precision effectively Further-more the fast convergence of the high order sliding modemakes the missile rapidly track its maneuvering target withthemost reasonable rudder deflection command reducing itsoverload effectively
Mathematical Problems in Engineering 13
0
05
1
15
2
25
3
35
4
45
5
0 10 20 30 40 50Simulation times
OSMG
Miss
(m)
0
05
1
15
2
25
3
35
4
45
5
0 10 20 30 40 50Simulation times
Miss
(m)
0
05
1
15
2
25
3
35
4
45
5
0 10 20 30 40 50Simulation times
Miss
(m)
tgo = 3 stgo = 2 stgo = 1 s
0
05
1
15
2
25
3
35
4
45
5
0 10 20 30 40 50Simulation times
Miss
(m)
tgo = 3 stgo = 2 stgo = 1 s
HOSM-IGC (120573 = 40)
HOSM-IGC (120573 = 50)
HOSM-IGC (120573 = 30)
Figure 16 Miss distances of HOSM-IGC (120573 = 30 40 50) and OSMG with target maneuvers at different time-to-go (tgo = 1 second 2seconds and 3 seconds)
The average miss distance of 50 simulations under theconditions is 073m for HOSM-IGC and 186m for OSMGWe can see that the HOSM-IGC method not only doesprovide a more reasonable actuator deflection command butalso achieves a higher interception precision
6 Conclusions
This paper proposes an LOS feedback integrated guidanceand control method using quasi-continuous high order
sliding mode guidance and control method With the fastand precise convergence of the quasi-continuous HOSMmethod the HOSM-IGCmethod performsmuch better thanthe traditional separated guidance and control method withless acceleration effort and less miss distance in all thethree simulation scenarios of nonmaneuvering target stepmaneuvering target and weaving target In addition the ideaof virtual control largely alleviates the chattering withoutany sacrifice of robustness As a result of the alleviationof the chattering the control input command 120575
119885becomes
14 Mathematical Problems in Engineering
175
180
185
190
195
200
TargetHOSM-IGCOSMG
Y(k
m)
0 20 40 60 0 20 40X (km)
Figure 17 The trajectories of the missile and its target
0
10
20
0 5 10 15 20Time (s)
TargetOSMGHOSM-IGC
Miss
ile an
d ta
rget
acce
lera
tion
(G)
minus10
minus20
Figure 18 Missile acceleration profile
smooth enough for the actuator to be implemented as aresult the interception accuracy is improved In designingthe quasi-continuous sliding mode controller the Arbitrary-Order Robust Exact Differentiator is introduced to estimatethe high order time derivative of the LOS rate 119902 which notonly ensures the guidance and control precision but alsosimplifies the calculation
0
10
20
30
0 5 10 15 20Time (s)
OSMGHOSM-IGC
minus10
minus20
minus30
minus40
Actu
ator
defl
ectio
n (d
eg)
Figure 19 Actuator deflection
0
50
100
150
200
15 16 17 18Time (s)
Commanded accelerationAchieved acceleration
Miss
ile ac
cele
ratio
n (G
)
minus50
Figure 20 Commanded acceleration and achieved acceleration
Appendices
A The Third-Order RobustExact Differentiator
The third-order robust exact differentiator built according tothe following parameters and equations is used to performthe simulation the initial value 120582
0= 1205821
= 1205822
= 1205823
= 50
Mathematical Problems in Engineering 15
0
0005
001
0015
002
0025
003
0035
004
0 02 04Time (s)
0
2
0 02 04Time (s)
0
100
200
300
0
100
200
300
400
0 02 04Time (s)
h(t)
z3
h(t) and z3
minus100
minus200
minus300
minus400
minus5000 02 04
Time (s)
g(t)
z2
g(t) and z2
minus100
minus200
minus300
minus400
f(t)
z1
f(t) and z1
minus2
minus4
minus6
minus8
minus10
minus12
minus14
F(t)
z0
F(t) and z0
minus0005
minus001
Figure 21 The outputs of the differentiator in the sampling interval of 0001 seconds
the initial values 1199110
= 01 1199111
= 1199112
= 1199113
= 0 119871 = 1400 thesimulation step is 0001 seconds Consider
0
= V0
V0
= minus120582011987114 10038161003816100381610038161199110 minus
1199021003816100381610038161003816
34 sign (1199110
minus 119902) + 1199111
1
= V1
V1
= minus120582111987113 10038161003816100381610038161199111 minus V
0
1003816100381610038161003816
23 sign (1199111
minus V0) + 1199112
2
= V2
V2
= minus120582211987112 10038161003816100381610038161199112 minus V
1
1003816100381610038161003816
12 sign (1199112
minus V1) + 1199113
3
= minus1205823119871 sign (119911
3minus V2)
(A1)
16 Mathematical Problems in Engineering
0
0005
001
0015
002
0025
003
0035
004
0 02 04Time (s)
0
2
0 02 04Time (s)
0
100
200
300
0 02 04Time (s)
0
200
400
0 02 04Time (s)
h(t) and z3
minus200
minus400
minus600
minus800
g(t) and z2
minus100
minus200
minus300
minus400
f(t) and z1
minus2
minus4
minus6
minus8
minus10
minus12
minus14
minus16
F(t) and z0
minus0005
minus001
h(t)
z3g(t)
z2
f(t)
z1F(t)
z0
Figure 22 The outputs of the differentiator in the sampling interval of 00001 seconds
If the designed signals are 119865(119905) = int 119891(119905)119889119905 119891(119905) =
int 119892(119905)119889119905 119892(119905) = int ℎ(119905)119889119905 and ℎ(119905) = sin(2120587119905) +
3cos(120587119905) then theoretically 119891(119905) is the first-order differ-ential of input signal 119892(119905) is the second-order differ-ential of input signal ℎ(119905) is the third-order differen-tial of input signal Through the simulation the differ-ential signal of actual output is compared with that of
theoretical output the comparison results are shown inFigure 21
Figure 21 shows that 1199110can effectively track the signal
119865(119905) but 1199112has obvious errors in tracking the second-order
differential 119892(119905) 1199113has greater errors in tracking the third-
order differential thus the overall differential effect is notideal
Mathematical Problems in Engineering 17
0 02 04
0
0005
001
0015
002
0025
003
0035
004
Time (s)0 02 04
0
01
02
03
04
05
Time (s)
0 02 04
0
05
1
15
2
Time (s)0 02 04
0
05
1
15
2
25
3
35
4
Time (s)
h(t) and z3g(t) and z2
minus04
minus05
f(t) and z1
minus2
minus15
minus1
minus05
minus1
minus05
minus01
minus02
minus03
F(t) and z0
minus0005
minus001
h(t)
z3g(t)
z2
f(t)
z1F(t)
z0
Figure 23 The outputs of the differentiator in the sampling interval of 00001 seconds
There are two causes for the errors in a usual differentia-tor the setting of sampling interval and the setting of numberof orders of the differentiator The following theorem applieshere
Theorem A1 (see [19]) If the sampling interval 120591 gt 0 theaccuracy of the controller is expressed as follows
10038161003816100381610038161003816120590(119894)
10038161003816100381610038161003816le 120583120591119903minus119894
119894 = 0 119903 minus 1 (A2)
That is to say the precision (understood as the differentialprecision of a differentiator) of the 119894th order control is in directproportion to the power of 119903 minus 119894 times of sampling interval120591 Therefore the decrease of iteration step can enhance thedifferential tracking precision
Under the conditions that the sampling interval is 10 timessmaller and that the setting is 00001 seconds the simulationresults are shown in Figure 22
18 Mathematical Problems in Engineering
Table 6 Tracking errors of the AORED with different orders andsampling interval
Orders of AORED Third order Fifth orderSample period 120591 = 0001 120591 = 00001 120591 = 00001
First-order differential error 001 Negligible NegligibleSecond-order differential error 15 002 NegligibleThird-order differential error 70 25 00005
As Figure 22 shows the signal tracking is rather idealexcept a peak in the beginning the tracking error in the first-order differential tracking can be neglected 119911
2has an obviously
smaller tracking error in the second-order differential 119892(119905)
tracking in the beginning 1199113has a rather sharp peak in the
third-order differential tracking but converges to a relativelyhigh precision after 01 seconds As we can see the differentialtracking errors for each number of orders are greatly reducedbut the third-order differential tracking error is still rather bigIn the following we will reduce the tracking errors by increasingthe number of orders of the differentiator
B The Fifth-Order Robust Exact Differentiator
The settings of the fifth-order differentiator are given asfollows
0
= V0
V0
= minus120582011987116 10038161003816100381610038161199110 minus
1199021003816100381610038161003816
56 sign (1199110
minus 119902) + 1199111
1
= V1
V1
= minus120582111987115 10038161003816100381610038161199111 minus V
0
1003816100381610038161003816
45 sign (1199111
minus V0) + 1199112
2
= V2
V2
= minus120582211987114 10038161003816100381610038161199112 minus V
1
1003816100381610038161003816
34 sign (1199112
minus V1) + 1199113
3
= V3
V3
= minus120582311987113 10038161003816100381610038161199113 minus V
2
1003816100381610038161003816
23 sign (1199113
minus V2) + 1199114
4
= V4
V4
= minus120582411987112 10038161003816100381610038161199114 minus V
3
1003816100381610038161003816
12 sign (1199114
minus V3) + 1199115
5
= V5
V5
= minus1205825119871 sign (119911
5minus V4)
(B1)
The initial value 1199110
= 01 1205820
= 1205821
= 1205822
= 1205823
= 1205824
=
1205825
= 50 the initial values 1199111 1199112 1199113 1199114 1199115
= 0 119871 = 1400the simulation step length is 00001 seconds The simulationresults are shown in Figure 23
As the figure shows the fifth-order differentiator hasa rather ideal tracking effect on the third-order differen-tial tracking the tracking error being only about 00005(see Table 6)
To sum up the Arbitrary-Order Robust Exact Differen-tiator can estimate rather ideal differential signals by selecting
Table 7
Length 365m119871 ref 365m119883119866
177m119878ref 0026m2
Diameter 0178mMass 1016 Kg119868119885
1063 Kgsdotm2
an appropriate sampling interval and using the differentiatorwith a relatively high number of orders
C Physical and Geometric Characteristics
See Table 7
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] P Zarchan Tactical and Strategic Missile Guidance 2002[2] D Zhou CMu andWXu ldquoAdaptive sliding-mode guidance of
a homing missilerdquo Journal of Guidance Control and Dynamicsvol 22 no 4 pp 589ndash594 1999
[3] G Zeng and M Hu ldquoFinite-time control for electromagneticsatellite formationsrdquo Acta Astronautica vol 74 pp 120ndash1302012
[4] S R Kumar S Rao and D Ghose ldquoNonsingular terminalsliding mode guidance with impact angle constraintsrdquo Journalof Guidance Control and Dynamics vol 37 no 4 pp 1114ndash11302014
[5] Y Liang C Chen D Liaw Y Feng C Cheng and CChen ldquoRobust guidance law via integral-sliding-mode schemerdquoJournal of Guidance Control and Dynamics vol 37 no 3 pp1038ndash1042 2014
[6] A V Savkin P N Pathirana and F Faruqi ldquoProblem ofprecision missile guidance LQR and 119867
infincontrol frameworksrdquo
IEEE Transactions on Aerospace and Electronic Systems vol 39no 3 pp 901ndash910 2003
[7] H Y Chen andC-C Yang ldquoNonlinear1198672119867infinguidance design
for homing missilesrdquo in Proceedings of the AIAA GuidanceNavigation and Control Conference and Exhibit vol 77 2001
[8] P K Menon G D Sweriduk and E J Ohlmeyer ldquoOptimalfixed-interval integrated guidance-control laws for hit-to-killmissilesrdquo in Proceedings of the AIAA Guidance Navigation andControl Conference and Exhibit AIAA 2003-5579 Austin TexUSA August 2003
[9] S S Vaddi P K Menon and E J Ohlmeyer ldquoNumericalSDRE approach for missile integrated guidance-controlrdquo inProceedings of the AIAA Guidance Navigation and ControlConference and Exhibit AIAA 2007-6672 Hilton Head SCUSA August 2007
[10] M Xin S N Balakrishnan and E J Ohlmeyer ldquoIntegratedguidance and control of missiles with Theta-D methodrdquo IEEE
Mathematical Problems in Engineering 19
Transactions on Control Systems Technology vol 14 no 6 pp981ndash992 2006
[11] T Shima M Idan and O M Golan ldquoSliding-mode controlfor integrated missile autopilot guidancerdquo Journal of GuidanceControl and Dynamics vol 29 no 2 pp 250ndash260 2006
[12] Y B Shtessel and C H Tournes ldquoIntegrated higher-ordersliding mode guidance and autopilot for dual control missilesrdquoJournal of Guidance Control and Dynamics vol 32 no 1 pp79ndash94 2009
[13] F Y Dong H M Lei and C J Zhou ldquoIntegrated robust highorder sliding mode guidance and control for missilesrdquo ActaAeronautica et Astronautica Sinica vol 34 no 9 pp 2212ndash22182013
[14] H Mingzhe and D Guangren ldquoIntegrated guidance and con-trol of homing missiles against ground fixed targetsrdquo ChineseJournal of Aeronautics vol 21 no 2 pp 162ndash168 2008
[15] A Levant ldquoHomogeneity approach to high-order sliding modedesignrdquo Automatica vol 41 no 5 pp 823ndash830 2005
[16] Y-W Liang C Chen D Liaw Y Feng C Cheng and CChen ldquoRobust guidance law via integral-sliding-mode schemerdquoJournal of Guidance Control and Dynamics vol 37 no 3 pp1038ndash1042 2014
[17] L Wang Y Sheng and X Liu ldquoContinuous time-varying slid-ingmode based attitude control for reentry vehiclerdquo Proceedingsof the Institution of Mechanical Engineers Part G Journal ofAerospace Engineering 2014
[18] A Levant ldquoHigher-order sliding modes differentiation andoutput-feedback controlrdquo International Journal of Control vol76 no 9-10 pp 924ndash941 2003
[19] A Levant ldquoQuasi-continuous high-order sliding-mode con-trollersrdquo IEEE Transactions on Automatic Control vol 50 no11 pp 1812ndash1816 2005
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 11
0
5
10
0 5 10 15 20Time (s)
TargetOSMGHOSM-IGC
Miss
ile an
d ta
rget
acce
lera
tion
(G)
minus5
minus10
Figure 11 Missile and target acceleration profile
0
10
20
30
0 5 10 15 20Time (s)
tgo = 1 stgo = 2 stgo = 3 s
tgo = 3 s (diverge)
tgo = 2 s (diverge)
tgo = 1 s (diverge)minus10
minus20
minus30
Miss
ile ac
cele
ratio
n (G
)
HOSM-IGC (120573 = 10)
Figure 12 Missile acceleration profile
collision increases themiss distanceTherefore increasing theparameter 120573 can remarkably increase the convergence speedand effectively enhance guidance precision As Figure 12shows when 120573 = 10 the overload at the end phase convergesslowly even if the target maneuvers three seconds beforecollision the missilersquos overload still has no time to convergebeing unable to track themaneuvering target Figure 13 showsthat when 120573 increases to 30 and tgo = 3 seconds theoverload can converge to about 5 g but when tgo = 2 sec-onds the overload may continue to increase indicating that
0
10
20
30
0 5 10 15 20Time (s)
tgo = 1 stgo = 2 stgo = 3 s
tgo = 1 s (diverge)
tgo = 2 s (diverge)
tgo = 3 s (converge)
minus10
minus20
minus30
Miss
ile ac
cele
ratio
n (G
)
120573 = 30)HOSM-IGC (
Figure 13 Missile acceleration profile
0
10
20
30
0 5 10 15 20Time (s)
tgo = 1 s (diverge)
tgo = 2 s (converge)
tgo = 3 s (converge)
tgo = 1 stgo = 2 stgo = 3 s
minus10
minus20
minus30
Miss
ile ac
cele
ratio
n (G
)
HOSM-IGC (120573 = 50)
Figure 14 Missile acceleration profile
the convergence still needs to be quickened thus continuingto rise 120573 to 50 Figure 14 shows that the HOSM-IGC methodcan track the target that maneuvers when tgo = 2 secondsbut it causes the divergence of overload and the increase ofmiss distance if the target maneuvers when tgo = 1 secondFigure 15 gives the overload curve of the OSMG guidance lawand shows that the maneuver of the target before collisionmay cause the large-scale oscillation of the missilersquos overloadwhich may diverge to a large numerical value when thecollision occurs in the end
12 Mathematical Problems in Engineering
Table 4 Average miss distances of 50 simulations
Targetmaneuveringtiming
HOSM-IGC120573 = 30
HOSM-IGC120573 = 40
HOSM-IGC120573 = 50
OSMG
tgo = 1second 43539 3224 28936 36116
tgo = 2seconds 25424 18665 09322 35534
tgo = 3seconds 08124 08265 07538 11959
Time (s)0 5 10 15 20
0
10
20
30OSMG
tgo = 1 stgo = 2 stgo = 3 s
minus10
minus20
minus30
Miss
ile ac
cele
ratio
n (G
)
Figure 15 Missile acceleration profile
The analysis in Figures 12 13 14 and 15 shows that theoverload of the missile converges faster and its miss distanceis smaller with increasing 120573 To verify this finding we carryout 50 times Monte Carlo simulations in which the positionand speed of the target have 1 of random difference
The average miss distances are shown in Table 4 andFigure 16 Clearly the timing of the targetrsquos step maneuverdramatically affects the final interception precision morespecifically given a shorter reaction time for the guidanceand control system the missile seems more likely to missthe target To the OSMG guidance law in all three scenarioshardly does it show any advantages against theHOSM-IGC Itcan also be seen that with the increasing of the120573 theHOSM-IGC system responds even faster which leads to an obviousdecrease of the average miss distance The effect of 120573 on theresponse of the HOSM-IGC system is a valuable guidelinewhen implementing the proposed method into practice
53 Scenario 3 Weaving Target In this scenario the targetmaneuvers by 119886
119879= 40sin(1205871199052) OSMG with three-loop
Table 5 Initial conditions
Variable Description Value(119883119872 119884119872) Missile initial position (0 km 175 km)
(119883119879 119884119879) Target initial position (60 km 195 km)
119881119872
Missile initial velocity 1200 (ms)119881119879
Target initial velocity 2000 (ms)120579119872
Missile initial flight path angle 0 (deg)120579119879
Target initial flight path angle 0 (deg)120576 Compensatory gain 3120575 Saturation parameter 0001
autopilot is introduced for comparison The motion equa-tions of the target are as follows
119910119879
= 119881119879cos 120579119879
119879
= minus119881119879sin 120579119879
120579119879
=119886119879
119881119879
119886119879
= 40 sin(120587119905
2)
(57)
The initial simulation conditions are given in Table 5The missilersquos trajectory under the two guidance and
control methods given in Figure 17 does not see muchdifference
However the overload curve given in Figure 18 showsthat after the missile completes its initial attitude adjustmentwith the HOSM-IGC method it can almost ideally track itsmaneuvering target by contrast with the OSMG methodthe missile seems to have the tendency to track its targetrsquosmaneuver but has larger tracking errors Besides with theOSMGmethod the missilersquos overload increases rapidly at theend of attack primarily because of the divergence of its LOSrate On the other hand with the HOSM-IGC method themissile has no divergence even at the end of attack ensuringa smaller target missing quantity
The actuator deflection curve in Figure 19 shows thatin order to provide a rather big normal overload for theend phase the OSMG method produces a rather big rudderdeflection command however it may increase the missilersquostarget missing quantity once its rudder deflection saturatesand themissile does not have enough overloads or the ruddercannot respond that fast
As shown in Figure 20 because of the dramatic changein overload command the response of the missilersquos autopilotto high-frequency command sees an obvious phase lag andamplitude value attenuation its actual overload cannot trackthe command ideally this is a main reason why the missdistance increases However the controller in the HOSM-IGC method gives its rudder deflection command directlyand there is no lagging or attenuation caused by the autopilotthus enhancing the guidance precision effectively Further-more the fast convergence of the high order sliding modemakes the missile rapidly track its maneuvering target withthemost reasonable rudder deflection command reducing itsoverload effectively
Mathematical Problems in Engineering 13
0
05
1
15
2
25
3
35
4
45
5
0 10 20 30 40 50Simulation times
OSMG
Miss
(m)
0
05
1
15
2
25
3
35
4
45
5
0 10 20 30 40 50Simulation times
Miss
(m)
0
05
1
15
2
25
3
35
4
45
5
0 10 20 30 40 50Simulation times
Miss
(m)
tgo = 3 stgo = 2 stgo = 1 s
0
05
1
15
2
25
3
35
4
45
5
0 10 20 30 40 50Simulation times
Miss
(m)
tgo = 3 stgo = 2 stgo = 1 s
HOSM-IGC (120573 = 40)
HOSM-IGC (120573 = 50)
HOSM-IGC (120573 = 30)
Figure 16 Miss distances of HOSM-IGC (120573 = 30 40 50) and OSMG with target maneuvers at different time-to-go (tgo = 1 second 2seconds and 3 seconds)
The average miss distance of 50 simulations under theconditions is 073m for HOSM-IGC and 186m for OSMGWe can see that the HOSM-IGC method not only doesprovide a more reasonable actuator deflection command butalso achieves a higher interception precision
6 Conclusions
This paper proposes an LOS feedback integrated guidanceand control method using quasi-continuous high order
sliding mode guidance and control method With the fastand precise convergence of the quasi-continuous HOSMmethod the HOSM-IGCmethod performsmuch better thanthe traditional separated guidance and control method withless acceleration effort and less miss distance in all thethree simulation scenarios of nonmaneuvering target stepmaneuvering target and weaving target In addition the ideaof virtual control largely alleviates the chattering withoutany sacrifice of robustness As a result of the alleviationof the chattering the control input command 120575
119885becomes
14 Mathematical Problems in Engineering
175
180
185
190
195
200
TargetHOSM-IGCOSMG
Y(k
m)
0 20 40 60 0 20 40X (km)
Figure 17 The trajectories of the missile and its target
0
10
20
0 5 10 15 20Time (s)
TargetOSMGHOSM-IGC
Miss
ile an
d ta
rget
acce
lera
tion
(G)
minus10
minus20
Figure 18 Missile acceleration profile
smooth enough for the actuator to be implemented as aresult the interception accuracy is improved In designingthe quasi-continuous sliding mode controller the Arbitrary-Order Robust Exact Differentiator is introduced to estimatethe high order time derivative of the LOS rate 119902 which notonly ensures the guidance and control precision but alsosimplifies the calculation
0
10
20
30
0 5 10 15 20Time (s)
OSMGHOSM-IGC
minus10
minus20
minus30
minus40
Actu
ator
defl
ectio
n (d
eg)
Figure 19 Actuator deflection
0
50
100
150
200
15 16 17 18Time (s)
Commanded accelerationAchieved acceleration
Miss
ile ac
cele
ratio
n (G
)
minus50
Figure 20 Commanded acceleration and achieved acceleration
Appendices
A The Third-Order RobustExact Differentiator
The third-order robust exact differentiator built according tothe following parameters and equations is used to performthe simulation the initial value 120582
0= 1205821
= 1205822
= 1205823
= 50
Mathematical Problems in Engineering 15
0
0005
001
0015
002
0025
003
0035
004
0 02 04Time (s)
0
2
0 02 04Time (s)
0
100
200
300
0
100
200
300
400
0 02 04Time (s)
h(t)
z3
h(t) and z3
minus100
minus200
minus300
minus400
minus5000 02 04
Time (s)
g(t)
z2
g(t) and z2
minus100
minus200
minus300
minus400
f(t)
z1
f(t) and z1
minus2
minus4
minus6
minus8
minus10
minus12
minus14
F(t)
z0
F(t) and z0
minus0005
minus001
Figure 21 The outputs of the differentiator in the sampling interval of 0001 seconds
the initial values 1199110
= 01 1199111
= 1199112
= 1199113
= 0 119871 = 1400 thesimulation step is 0001 seconds Consider
0
= V0
V0
= minus120582011987114 10038161003816100381610038161199110 minus
1199021003816100381610038161003816
34 sign (1199110
minus 119902) + 1199111
1
= V1
V1
= minus120582111987113 10038161003816100381610038161199111 minus V
0
1003816100381610038161003816
23 sign (1199111
minus V0) + 1199112
2
= V2
V2
= minus120582211987112 10038161003816100381610038161199112 minus V
1
1003816100381610038161003816
12 sign (1199112
minus V1) + 1199113
3
= minus1205823119871 sign (119911
3minus V2)
(A1)
16 Mathematical Problems in Engineering
0
0005
001
0015
002
0025
003
0035
004
0 02 04Time (s)
0
2
0 02 04Time (s)
0
100
200
300
0 02 04Time (s)
0
200
400
0 02 04Time (s)
h(t) and z3
minus200
minus400
minus600
minus800
g(t) and z2
minus100
minus200
minus300
minus400
f(t) and z1
minus2
minus4
minus6
minus8
minus10
minus12
minus14
minus16
F(t) and z0
minus0005
minus001
h(t)
z3g(t)
z2
f(t)
z1F(t)
z0
Figure 22 The outputs of the differentiator in the sampling interval of 00001 seconds
If the designed signals are 119865(119905) = int 119891(119905)119889119905 119891(119905) =
int 119892(119905)119889119905 119892(119905) = int ℎ(119905)119889119905 and ℎ(119905) = sin(2120587119905) +
3cos(120587119905) then theoretically 119891(119905) is the first-order differ-ential of input signal 119892(119905) is the second-order differ-ential of input signal ℎ(119905) is the third-order differen-tial of input signal Through the simulation the differ-ential signal of actual output is compared with that of
theoretical output the comparison results are shown inFigure 21
Figure 21 shows that 1199110can effectively track the signal
119865(119905) but 1199112has obvious errors in tracking the second-order
differential 119892(119905) 1199113has greater errors in tracking the third-
order differential thus the overall differential effect is notideal
Mathematical Problems in Engineering 17
0 02 04
0
0005
001
0015
002
0025
003
0035
004
Time (s)0 02 04
0
01
02
03
04
05
Time (s)
0 02 04
0
05
1
15
2
Time (s)0 02 04
0
05
1
15
2
25
3
35
4
Time (s)
h(t) and z3g(t) and z2
minus04
minus05
f(t) and z1
minus2
minus15
minus1
minus05
minus1
minus05
minus01
minus02
minus03
F(t) and z0
minus0005
minus001
h(t)
z3g(t)
z2
f(t)
z1F(t)
z0
Figure 23 The outputs of the differentiator in the sampling interval of 00001 seconds
There are two causes for the errors in a usual differentia-tor the setting of sampling interval and the setting of numberof orders of the differentiator The following theorem applieshere
Theorem A1 (see [19]) If the sampling interval 120591 gt 0 theaccuracy of the controller is expressed as follows
10038161003816100381610038161003816120590(119894)
10038161003816100381610038161003816le 120583120591119903minus119894
119894 = 0 119903 minus 1 (A2)
That is to say the precision (understood as the differentialprecision of a differentiator) of the 119894th order control is in directproportion to the power of 119903 minus 119894 times of sampling interval120591 Therefore the decrease of iteration step can enhance thedifferential tracking precision
Under the conditions that the sampling interval is 10 timessmaller and that the setting is 00001 seconds the simulationresults are shown in Figure 22
18 Mathematical Problems in Engineering
Table 6 Tracking errors of the AORED with different orders andsampling interval
Orders of AORED Third order Fifth orderSample period 120591 = 0001 120591 = 00001 120591 = 00001
First-order differential error 001 Negligible NegligibleSecond-order differential error 15 002 NegligibleThird-order differential error 70 25 00005
As Figure 22 shows the signal tracking is rather idealexcept a peak in the beginning the tracking error in the first-order differential tracking can be neglected 119911
2has an obviously
smaller tracking error in the second-order differential 119892(119905)
tracking in the beginning 1199113has a rather sharp peak in the
third-order differential tracking but converges to a relativelyhigh precision after 01 seconds As we can see the differentialtracking errors for each number of orders are greatly reducedbut the third-order differential tracking error is still rather bigIn the following we will reduce the tracking errors by increasingthe number of orders of the differentiator
B The Fifth-Order Robust Exact Differentiator
The settings of the fifth-order differentiator are given asfollows
0
= V0
V0
= minus120582011987116 10038161003816100381610038161199110 minus
1199021003816100381610038161003816
56 sign (1199110
minus 119902) + 1199111
1
= V1
V1
= minus120582111987115 10038161003816100381610038161199111 minus V
0
1003816100381610038161003816
45 sign (1199111
minus V0) + 1199112
2
= V2
V2
= minus120582211987114 10038161003816100381610038161199112 minus V
1
1003816100381610038161003816
34 sign (1199112
minus V1) + 1199113
3
= V3
V3
= minus120582311987113 10038161003816100381610038161199113 minus V
2
1003816100381610038161003816
23 sign (1199113
minus V2) + 1199114
4
= V4
V4
= minus120582411987112 10038161003816100381610038161199114 minus V
3
1003816100381610038161003816
12 sign (1199114
minus V3) + 1199115
5
= V5
V5
= minus1205825119871 sign (119911
5minus V4)
(B1)
The initial value 1199110
= 01 1205820
= 1205821
= 1205822
= 1205823
= 1205824
=
1205825
= 50 the initial values 1199111 1199112 1199113 1199114 1199115
= 0 119871 = 1400the simulation step length is 00001 seconds The simulationresults are shown in Figure 23
As the figure shows the fifth-order differentiator hasa rather ideal tracking effect on the third-order differen-tial tracking the tracking error being only about 00005(see Table 6)
To sum up the Arbitrary-Order Robust Exact Differen-tiator can estimate rather ideal differential signals by selecting
Table 7
Length 365m119871 ref 365m119883119866
177m119878ref 0026m2
Diameter 0178mMass 1016 Kg119868119885
1063 Kgsdotm2
an appropriate sampling interval and using the differentiatorwith a relatively high number of orders
C Physical and Geometric Characteristics
See Table 7
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] P Zarchan Tactical and Strategic Missile Guidance 2002[2] D Zhou CMu andWXu ldquoAdaptive sliding-mode guidance of
a homing missilerdquo Journal of Guidance Control and Dynamicsvol 22 no 4 pp 589ndash594 1999
[3] G Zeng and M Hu ldquoFinite-time control for electromagneticsatellite formationsrdquo Acta Astronautica vol 74 pp 120ndash1302012
[4] S R Kumar S Rao and D Ghose ldquoNonsingular terminalsliding mode guidance with impact angle constraintsrdquo Journalof Guidance Control and Dynamics vol 37 no 4 pp 1114ndash11302014
[5] Y Liang C Chen D Liaw Y Feng C Cheng and CChen ldquoRobust guidance law via integral-sliding-mode schemerdquoJournal of Guidance Control and Dynamics vol 37 no 3 pp1038ndash1042 2014
[6] A V Savkin P N Pathirana and F Faruqi ldquoProblem ofprecision missile guidance LQR and 119867
infincontrol frameworksrdquo
IEEE Transactions on Aerospace and Electronic Systems vol 39no 3 pp 901ndash910 2003
[7] H Y Chen andC-C Yang ldquoNonlinear1198672119867infinguidance design
for homing missilesrdquo in Proceedings of the AIAA GuidanceNavigation and Control Conference and Exhibit vol 77 2001
[8] P K Menon G D Sweriduk and E J Ohlmeyer ldquoOptimalfixed-interval integrated guidance-control laws for hit-to-killmissilesrdquo in Proceedings of the AIAA Guidance Navigation andControl Conference and Exhibit AIAA 2003-5579 Austin TexUSA August 2003
[9] S S Vaddi P K Menon and E J Ohlmeyer ldquoNumericalSDRE approach for missile integrated guidance-controlrdquo inProceedings of the AIAA Guidance Navigation and ControlConference and Exhibit AIAA 2007-6672 Hilton Head SCUSA August 2007
[10] M Xin S N Balakrishnan and E J Ohlmeyer ldquoIntegratedguidance and control of missiles with Theta-D methodrdquo IEEE
Mathematical Problems in Engineering 19
Transactions on Control Systems Technology vol 14 no 6 pp981ndash992 2006
[11] T Shima M Idan and O M Golan ldquoSliding-mode controlfor integrated missile autopilot guidancerdquo Journal of GuidanceControl and Dynamics vol 29 no 2 pp 250ndash260 2006
[12] Y B Shtessel and C H Tournes ldquoIntegrated higher-ordersliding mode guidance and autopilot for dual control missilesrdquoJournal of Guidance Control and Dynamics vol 32 no 1 pp79ndash94 2009
[13] F Y Dong H M Lei and C J Zhou ldquoIntegrated robust highorder sliding mode guidance and control for missilesrdquo ActaAeronautica et Astronautica Sinica vol 34 no 9 pp 2212ndash22182013
[14] H Mingzhe and D Guangren ldquoIntegrated guidance and con-trol of homing missiles against ground fixed targetsrdquo ChineseJournal of Aeronautics vol 21 no 2 pp 162ndash168 2008
[15] A Levant ldquoHomogeneity approach to high-order sliding modedesignrdquo Automatica vol 41 no 5 pp 823ndash830 2005
[16] Y-W Liang C Chen D Liaw Y Feng C Cheng and CChen ldquoRobust guidance law via integral-sliding-mode schemerdquoJournal of Guidance Control and Dynamics vol 37 no 3 pp1038ndash1042 2014
[17] L Wang Y Sheng and X Liu ldquoContinuous time-varying slid-ingmode based attitude control for reentry vehiclerdquo Proceedingsof the Institution of Mechanical Engineers Part G Journal ofAerospace Engineering 2014
[18] A Levant ldquoHigher-order sliding modes differentiation andoutput-feedback controlrdquo International Journal of Control vol76 no 9-10 pp 924ndash941 2003
[19] A Levant ldquoQuasi-continuous high-order sliding-mode con-trollersrdquo IEEE Transactions on Automatic Control vol 50 no11 pp 1812ndash1816 2005
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
12 Mathematical Problems in Engineering
Table 4 Average miss distances of 50 simulations
Targetmaneuveringtiming
HOSM-IGC120573 = 30
HOSM-IGC120573 = 40
HOSM-IGC120573 = 50
OSMG
tgo = 1second 43539 3224 28936 36116
tgo = 2seconds 25424 18665 09322 35534
tgo = 3seconds 08124 08265 07538 11959
Time (s)0 5 10 15 20
0
10
20
30OSMG
tgo = 1 stgo = 2 stgo = 3 s
minus10
minus20
minus30
Miss
ile ac
cele
ratio
n (G
)
Figure 15 Missile acceleration profile
The analysis in Figures 12 13 14 and 15 shows that theoverload of the missile converges faster and its miss distanceis smaller with increasing 120573 To verify this finding we carryout 50 times Monte Carlo simulations in which the positionand speed of the target have 1 of random difference
The average miss distances are shown in Table 4 andFigure 16 Clearly the timing of the targetrsquos step maneuverdramatically affects the final interception precision morespecifically given a shorter reaction time for the guidanceand control system the missile seems more likely to missthe target To the OSMG guidance law in all three scenarioshardly does it show any advantages against theHOSM-IGC Itcan also be seen that with the increasing of the120573 theHOSM-IGC system responds even faster which leads to an obviousdecrease of the average miss distance The effect of 120573 on theresponse of the HOSM-IGC system is a valuable guidelinewhen implementing the proposed method into practice
53 Scenario 3 Weaving Target In this scenario the targetmaneuvers by 119886
119879= 40sin(1205871199052) OSMG with three-loop
Table 5 Initial conditions
Variable Description Value(119883119872 119884119872) Missile initial position (0 km 175 km)
(119883119879 119884119879) Target initial position (60 km 195 km)
119881119872
Missile initial velocity 1200 (ms)119881119879
Target initial velocity 2000 (ms)120579119872
Missile initial flight path angle 0 (deg)120579119879
Target initial flight path angle 0 (deg)120576 Compensatory gain 3120575 Saturation parameter 0001
autopilot is introduced for comparison The motion equa-tions of the target are as follows
119910119879
= 119881119879cos 120579119879
119879
= minus119881119879sin 120579119879
120579119879
=119886119879
119881119879
119886119879
= 40 sin(120587119905
2)
(57)
The initial simulation conditions are given in Table 5The missilersquos trajectory under the two guidance and
control methods given in Figure 17 does not see muchdifference
However the overload curve given in Figure 18 showsthat after the missile completes its initial attitude adjustmentwith the HOSM-IGC method it can almost ideally track itsmaneuvering target by contrast with the OSMG methodthe missile seems to have the tendency to track its targetrsquosmaneuver but has larger tracking errors Besides with theOSMGmethod the missilersquos overload increases rapidly at theend of attack primarily because of the divergence of its LOSrate On the other hand with the HOSM-IGC method themissile has no divergence even at the end of attack ensuringa smaller target missing quantity
The actuator deflection curve in Figure 19 shows thatin order to provide a rather big normal overload for theend phase the OSMG method produces a rather big rudderdeflection command however it may increase the missilersquostarget missing quantity once its rudder deflection saturatesand themissile does not have enough overloads or the ruddercannot respond that fast
As shown in Figure 20 because of the dramatic changein overload command the response of the missilersquos autopilotto high-frequency command sees an obvious phase lag andamplitude value attenuation its actual overload cannot trackthe command ideally this is a main reason why the missdistance increases However the controller in the HOSM-IGC method gives its rudder deflection command directlyand there is no lagging or attenuation caused by the autopilotthus enhancing the guidance precision effectively Further-more the fast convergence of the high order sliding modemakes the missile rapidly track its maneuvering target withthemost reasonable rudder deflection command reducing itsoverload effectively
Mathematical Problems in Engineering 13
0
05
1
15
2
25
3
35
4
45
5
0 10 20 30 40 50Simulation times
OSMG
Miss
(m)
0
05
1
15
2
25
3
35
4
45
5
0 10 20 30 40 50Simulation times
Miss
(m)
0
05
1
15
2
25
3
35
4
45
5
0 10 20 30 40 50Simulation times
Miss
(m)
tgo = 3 stgo = 2 stgo = 1 s
0
05
1
15
2
25
3
35
4
45
5
0 10 20 30 40 50Simulation times
Miss
(m)
tgo = 3 stgo = 2 stgo = 1 s
HOSM-IGC (120573 = 40)
HOSM-IGC (120573 = 50)
HOSM-IGC (120573 = 30)
Figure 16 Miss distances of HOSM-IGC (120573 = 30 40 50) and OSMG with target maneuvers at different time-to-go (tgo = 1 second 2seconds and 3 seconds)
The average miss distance of 50 simulations under theconditions is 073m for HOSM-IGC and 186m for OSMGWe can see that the HOSM-IGC method not only doesprovide a more reasonable actuator deflection command butalso achieves a higher interception precision
6 Conclusions
This paper proposes an LOS feedback integrated guidanceand control method using quasi-continuous high order
sliding mode guidance and control method With the fastand precise convergence of the quasi-continuous HOSMmethod the HOSM-IGCmethod performsmuch better thanthe traditional separated guidance and control method withless acceleration effort and less miss distance in all thethree simulation scenarios of nonmaneuvering target stepmaneuvering target and weaving target In addition the ideaof virtual control largely alleviates the chattering withoutany sacrifice of robustness As a result of the alleviationof the chattering the control input command 120575
119885becomes
14 Mathematical Problems in Engineering
175
180
185
190
195
200
TargetHOSM-IGCOSMG
Y(k
m)
0 20 40 60 0 20 40X (km)
Figure 17 The trajectories of the missile and its target
0
10
20
0 5 10 15 20Time (s)
TargetOSMGHOSM-IGC
Miss
ile an
d ta
rget
acce
lera
tion
(G)
minus10
minus20
Figure 18 Missile acceleration profile
smooth enough for the actuator to be implemented as aresult the interception accuracy is improved In designingthe quasi-continuous sliding mode controller the Arbitrary-Order Robust Exact Differentiator is introduced to estimatethe high order time derivative of the LOS rate 119902 which notonly ensures the guidance and control precision but alsosimplifies the calculation
0
10
20
30
0 5 10 15 20Time (s)
OSMGHOSM-IGC
minus10
minus20
minus30
minus40
Actu
ator
defl
ectio
n (d
eg)
Figure 19 Actuator deflection
0
50
100
150
200
15 16 17 18Time (s)
Commanded accelerationAchieved acceleration
Miss
ile ac
cele
ratio
n (G
)
minus50
Figure 20 Commanded acceleration and achieved acceleration
Appendices
A The Third-Order RobustExact Differentiator
The third-order robust exact differentiator built according tothe following parameters and equations is used to performthe simulation the initial value 120582
0= 1205821
= 1205822
= 1205823
= 50
Mathematical Problems in Engineering 15
0
0005
001
0015
002
0025
003
0035
004
0 02 04Time (s)
0
2
0 02 04Time (s)
0
100
200
300
0
100
200
300
400
0 02 04Time (s)
h(t)
z3
h(t) and z3
minus100
minus200
minus300
minus400
minus5000 02 04
Time (s)
g(t)
z2
g(t) and z2
minus100
minus200
minus300
minus400
f(t)
z1
f(t) and z1
minus2
minus4
minus6
minus8
minus10
minus12
minus14
F(t)
z0
F(t) and z0
minus0005
minus001
Figure 21 The outputs of the differentiator in the sampling interval of 0001 seconds
the initial values 1199110
= 01 1199111
= 1199112
= 1199113
= 0 119871 = 1400 thesimulation step is 0001 seconds Consider
0
= V0
V0
= minus120582011987114 10038161003816100381610038161199110 minus
1199021003816100381610038161003816
34 sign (1199110
minus 119902) + 1199111
1
= V1
V1
= minus120582111987113 10038161003816100381610038161199111 minus V
0
1003816100381610038161003816
23 sign (1199111
minus V0) + 1199112
2
= V2
V2
= minus120582211987112 10038161003816100381610038161199112 minus V
1
1003816100381610038161003816
12 sign (1199112
minus V1) + 1199113
3
= minus1205823119871 sign (119911
3minus V2)
(A1)
16 Mathematical Problems in Engineering
0
0005
001
0015
002
0025
003
0035
004
0 02 04Time (s)
0
2
0 02 04Time (s)
0
100
200
300
0 02 04Time (s)
0
200
400
0 02 04Time (s)
h(t) and z3
minus200
minus400
minus600
minus800
g(t) and z2
minus100
minus200
minus300
minus400
f(t) and z1
minus2
minus4
minus6
minus8
minus10
minus12
minus14
minus16
F(t) and z0
minus0005
minus001
h(t)
z3g(t)
z2
f(t)
z1F(t)
z0
Figure 22 The outputs of the differentiator in the sampling interval of 00001 seconds
If the designed signals are 119865(119905) = int 119891(119905)119889119905 119891(119905) =
int 119892(119905)119889119905 119892(119905) = int ℎ(119905)119889119905 and ℎ(119905) = sin(2120587119905) +
3cos(120587119905) then theoretically 119891(119905) is the first-order differ-ential of input signal 119892(119905) is the second-order differ-ential of input signal ℎ(119905) is the third-order differen-tial of input signal Through the simulation the differ-ential signal of actual output is compared with that of
theoretical output the comparison results are shown inFigure 21
Figure 21 shows that 1199110can effectively track the signal
119865(119905) but 1199112has obvious errors in tracking the second-order
differential 119892(119905) 1199113has greater errors in tracking the third-
order differential thus the overall differential effect is notideal
Mathematical Problems in Engineering 17
0 02 04
0
0005
001
0015
002
0025
003
0035
004
Time (s)0 02 04
0
01
02
03
04
05
Time (s)
0 02 04
0
05
1
15
2
Time (s)0 02 04
0
05
1
15
2
25
3
35
4
Time (s)
h(t) and z3g(t) and z2
minus04
minus05
f(t) and z1
minus2
minus15
minus1
minus05
minus1
minus05
minus01
minus02
minus03
F(t) and z0
minus0005
minus001
h(t)
z3g(t)
z2
f(t)
z1F(t)
z0
Figure 23 The outputs of the differentiator in the sampling interval of 00001 seconds
There are two causes for the errors in a usual differentia-tor the setting of sampling interval and the setting of numberof orders of the differentiator The following theorem applieshere
Theorem A1 (see [19]) If the sampling interval 120591 gt 0 theaccuracy of the controller is expressed as follows
10038161003816100381610038161003816120590(119894)
10038161003816100381610038161003816le 120583120591119903minus119894
119894 = 0 119903 minus 1 (A2)
That is to say the precision (understood as the differentialprecision of a differentiator) of the 119894th order control is in directproportion to the power of 119903 minus 119894 times of sampling interval120591 Therefore the decrease of iteration step can enhance thedifferential tracking precision
Under the conditions that the sampling interval is 10 timessmaller and that the setting is 00001 seconds the simulationresults are shown in Figure 22
18 Mathematical Problems in Engineering
Table 6 Tracking errors of the AORED with different orders andsampling interval
Orders of AORED Third order Fifth orderSample period 120591 = 0001 120591 = 00001 120591 = 00001
First-order differential error 001 Negligible NegligibleSecond-order differential error 15 002 NegligibleThird-order differential error 70 25 00005
As Figure 22 shows the signal tracking is rather idealexcept a peak in the beginning the tracking error in the first-order differential tracking can be neglected 119911
2has an obviously
smaller tracking error in the second-order differential 119892(119905)
tracking in the beginning 1199113has a rather sharp peak in the
third-order differential tracking but converges to a relativelyhigh precision after 01 seconds As we can see the differentialtracking errors for each number of orders are greatly reducedbut the third-order differential tracking error is still rather bigIn the following we will reduce the tracking errors by increasingthe number of orders of the differentiator
B The Fifth-Order Robust Exact Differentiator
The settings of the fifth-order differentiator are given asfollows
0
= V0
V0
= minus120582011987116 10038161003816100381610038161199110 minus
1199021003816100381610038161003816
56 sign (1199110
minus 119902) + 1199111
1
= V1
V1
= minus120582111987115 10038161003816100381610038161199111 minus V
0
1003816100381610038161003816
45 sign (1199111
minus V0) + 1199112
2
= V2
V2
= minus120582211987114 10038161003816100381610038161199112 minus V
1
1003816100381610038161003816
34 sign (1199112
minus V1) + 1199113
3
= V3
V3
= minus120582311987113 10038161003816100381610038161199113 minus V
2
1003816100381610038161003816
23 sign (1199113
minus V2) + 1199114
4
= V4
V4
= minus120582411987112 10038161003816100381610038161199114 minus V
3
1003816100381610038161003816
12 sign (1199114
minus V3) + 1199115
5
= V5
V5
= minus1205825119871 sign (119911
5minus V4)
(B1)
The initial value 1199110
= 01 1205820
= 1205821
= 1205822
= 1205823
= 1205824
=
1205825
= 50 the initial values 1199111 1199112 1199113 1199114 1199115
= 0 119871 = 1400the simulation step length is 00001 seconds The simulationresults are shown in Figure 23
As the figure shows the fifth-order differentiator hasa rather ideal tracking effect on the third-order differen-tial tracking the tracking error being only about 00005(see Table 6)
To sum up the Arbitrary-Order Robust Exact Differen-tiator can estimate rather ideal differential signals by selecting
Table 7
Length 365m119871 ref 365m119883119866
177m119878ref 0026m2
Diameter 0178mMass 1016 Kg119868119885
1063 Kgsdotm2
an appropriate sampling interval and using the differentiatorwith a relatively high number of orders
C Physical and Geometric Characteristics
See Table 7
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] P Zarchan Tactical and Strategic Missile Guidance 2002[2] D Zhou CMu andWXu ldquoAdaptive sliding-mode guidance of
a homing missilerdquo Journal of Guidance Control and Dynamicsvol 22 no 4 pp 589ndash594 1999
[3] G Zeng and M Hu ldquoFinite-time control for electromagneticsatellite formationsrdquo Acta Astronautica vol 74 pp 120ndash1302012
[4] S R Kumar S Rao and D Ghose ldquoNonsingular terminalsliding mode guidance with impact angle constraintsrdquo Journalof Guidance Control and Dynamics vol 37 no 4 pp 1114ndash11302014
[5] Y Liang C Chen D Liaw Y Feng C Cheng and CChen ldquoRobust guidance law via integral-sliding-mode schemerdquoJournal of Guidance Control and Dynamics vol 37 no 3 pp1038ndash1042 2014
[6] A V Savkin P N Pathirana and F Faruqi ldquoProblem ofprecision missile guidance LQR and 119867
infincontrol frameworksrdquo
IEEE Transactions on Aerospace and Electronic Systems vol 39no 3 pp 901ndash910 2003
[7] H Y Chen andC-C Yang ldquoNonlinear1198672119867infinguidance design
for homing missilesrdquo in Proceedings of the AIAA GuidanceNavigation and Control Conference and Exhibit vol 77 2001
[8] P K Menon G D Sweriduk and E J Ohlmeyer ldquoOptimalfixed-interval integrated guidance-control laws for hit-to-killmissilesrdquo in Proceedings of the AIAA Guidance Navigation andControl Conference and Exhibit AIAA 2003-5579 Austin TexUSA August 2003
[9] S S Vaddi P K Menon and E J Ohlmeyer ldquoNumericalSDRE approach for missile integrated guidance-controlrdquo inProceedings of the AIAA Guidance Navigation and ControlConference and Exhibit AIAA 2007-6672 Hilton Head SCUSA August 2007
[10] M Xin S N Balakrishnan and E J Ohlmeyer ldquoIntegratedguidance and control of missiles with Theta-D methodrdquo IEEE
Mathematical Problems in Engineering 19
Transactions on Control Systems Technology vol 14 no 6 pp981ndash992 2006
[11] T Shima M Idan and O M Golan ldquoSliding-mode controlfor integrated missile autopilot guidancerdquo Journal of GuidanceControl and Dynamics vol 29 no 2 pp 250ndash260 2006
[12] Y B Shtessel and C H Tournes ldquoIntegrated higher-ordersliding mode guidance and autopilot for dual control missilesrdquoJournal of Guidance Control and Dynamics vol 32 no 1 pp79ndash94 2009
[13] F Y Dong H M Lei and C J Zhou ldquoIntegrated robust highorder sliding mode guidance and control for missilesrdquo ActaAeronautica et Astronautica Sinica vol 34 no 9 pp 2212ndash22182013
[14] H Mingzhe and D Guangren ldquoIntegrated guidance and con-trol of homing missiles against ground fixed targetsrdquo ChineseJournal of Aeronautics vol 21 no 2 pp 162ndash168 2008
[15] A Levant ldquoHomogeneity approach to high-order sliding modedesignrdquo Automatica vol 41 no 5 pp 823ndash830 2005
[16] Y-W Liang C Chen D Liaw Y Feng C Cheng and CChen ldquoRobust guidance law via integral-sliding-mode schemerdquoJournal of Guidance Control and Dynamics vol 37 no 3 pp1038ndash1042 2014
[17] L Wang Y Sheng and X Liu ldquoContinuous time-varying slid-ingmode based attitude control for reentry vehiclerdquo Proceedingsof the Institution of Mechanical Engineers Part G Journal ofAerospace Engineering 2014
[18] A Levant ldquoHigher-order sliding modes differentiation andoutput-feedback controlrdquo International Journal of Control vol76 no 9-10 pp 924ndash941 2003
[19] A Levant ldquoQuasi-continuous high-order sliding-mode con-trollersrdquo IEEE Transactions on Automatic Control vol 50 no11 pp 1812ndash1816 2005
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 13
0
05
1
15
2
25
3
35
4
45
5
0 10 20 30 40 50Simulation times
OSMG
Miss
(m)
0
05
1
15
2
25
3
35
4
45
5
0 10 20 30 40 50Simulation times
Miss
(m)
0
05
1
15
2
25
3
35
4
45
5
0 10 20 30 40 50Simulation times
Miss
(m)
tgo = 3 stgo = 2 stgo = 1 s
0
05
1
15
2
25
3
35
4
45
5
0 10 20 30 40 50Simulation times
Miss
(m)
tgo = 3 stgo = 2 stgo = 1 s
HOSM-IGC (120573 = 40)
HOSM-IGC (120573 = 50)
HOSM-IGC (120573 = 30)
Figure 16 Miss distances of HOSM-IGC (120573 = 30 40 50) and OSMG with target maneuvers at different time-to-go (tgo = 1 second 2seconds and 3 seconds)
The average miss distance of 50 simulations under theconditions is 073m for HOSM-IGC and 186m for OSMGWe can see that the HOSM-IGC method not only doesprovide a more reasonable actuator deflection command butalso achieves a higher interception precision
6 Conclusions
This paper proposes an LOS feedback integrated guidanceand control method using quasi-continuous high order
sliding mode guidance and control method With the fastand precise convergence of the quasi-continuous HOSMmethod the HOSM-IGCmethod performsmuch better thanthe traditional separated guidance and control method withless acceleration effort and less miss distance in all thethree simulation scenarios of nonmaneuvering target stepmaneuvering target and weaving target In addition the ideaof virtual control largely alleviates the chattering withoutany sacrifice of robustness As a result of the alleviationof the chattering the control input command 120575
119885becomes
14 Mathematical Problems in Engineering
175
180
185
190
195
200
TargetHOSM-IGCOSMG
Y(k
m)
0 20 40 60 0 20 40X (km)
Figure 17 The trajectories of the missile and its target
0
10
20
0 5 10 15 20Time (s)
TargetOSMGHOSM-IGC
Miss
ile an
d ta
rget
acce
lera
tion
(G)
minus10
minus20
Figure 18 Missile acceleration profile
smooth enough for the actuator to be implemented as aresult the interception accuracy is improved In designingthe quasi-continuous sliding mode controller the Arbitrary-Order Robust Exact Differentiator is introduced to estimatethe high order time derivative of the LOS rate 119902 which notonly ensures the guidance and control precision but alsosimplifies the calculation
0
10
20
30
0 5 10 15 20Time (s)
OSMGHOSM-IGC
minus10
minus20
minus30
minus40
Actu
ator
defl
ectio
n (d
eg)
Figure 19 Actuator deflection
0
50
100
150
200
15 16 17 18Time (s)
Commanded accelerationAchieved acceleration
Miss
ile ac
cele
ratio
n (G
)
minus50
Figure 20 Commanded acceleration and achieved acceleration
Appendices
A The Third-Order RobustExact Differentiator
The third-order robust exact differentiator built according tothe following parameters and equations is used to performthe simulation the initial value 120582
0= 1205821
= 1205822
= 1205823
= 50
Mathematical Problems in Engineering 15
0
0005
001
0015
002
0025
003
0035
004
0 02 04Time (s)
0
2
0 02 04Time (s)
0
100
200
300
0
100
200
300
400
0 02 04Time (s)
h(t)
z3
h(t) and z3
minus100
minus200
minus300
minus400
minus5000 02 04
Time (s)
g(t)
z2
g(t) and z2
minus100
minus200
minus300
minus400
f(t)
z1
f(t) and z1
minus2
minus4
minus6
minus8
minus10
minus12
minus14
F(t)
z0
F(t) and z0
minus0005
minus001
Figure 21 The outputs of the differentiator in the sampling interval of 0001 seconds
the initial values 1199110
= 01 1199111
= 1199112
= 1199113
= 0 119871 = 1400 thesimulation step is 0001 seconds Consider
0
= V0
V0
= minus120582011987114 10038161003816100381610038161199110 minus
1199021003816100381610038161003816
34 sign (1199110
minus 119902) + 1199111
1
= V1
V1
= minus120582111987113 10038161003816100381610038161199111 minus V
0
1003816100381610038161003816
23 sign (1199111
minus V0) + 1199112
2
= V2
V2
= minus120582211987112 10038161003816100381610038161199112 minus V
1
1003816100381610038161003816
12 sign (1199112
minus V1) + 1199113
3
= minus1205823119871 sign (119911
3minus V2)
(A1)
16 Mathematical Problems in Engineering
0
0005
001
0015
002
0025
003
0035
004
0 02 04Time (s)
0
2
0 02 04Time (s)
0
100
200
300
0 02 04Time (s)
0
200
400
0 02 04Time (s)
h(t) and z3
minus200
minus400
minus600
minus800
g(t) and z2
minus100
minus200
minus300
minus400
f(t) and z1
minus2
minus4
minus6
minus8
minus10
minus12
minus14
minus16
F(t) and z0
minus0005
minus001
h(t)
z3g(t)
z2
f(t)
z1F(t)
z0
Figure 22 The outputs of the differentiator in the sampling interval of 00001 seconds
If the designed signals are 119865(119905) = int 119891(119905)119889119905 119891(119905) =
int 119892(119905)119889119905 119892(119905) = int ℎ(119905)119889119905 and ℎ(119905) = sin(2120587119905) +
3cos(120587119905) then theoretically 119891(119905) is the first-order differ-ential of input signal 119892(119905) is the second-order differ-ential of input signal ℎ(119905) is the third-order differen-tial of input signal Through the simulation the differ-ential signal of actual output is compared with that of
theoretical output the comparison results are shown inFigure 21
Figure 21 shows that 1199110can effectively track the signal
119865(119905) but 1199112has obvious errors in tracking the second-order
differential 119892(119905) 1199113has greater errors in tracking the third-
order differential thus the overall differential effect is notideal
Mathematical Problems in Engineering 17
0 02 04
0
0005
001
0015
002
0025
003
0035
004
Time (s)0 02 04
0
01
02
03
04
05
Time (s)
0 02 04
0
05
1
15
2
Time (s)0 02 04
0
05
1
15
2
25
3
35
4
Time (s)
h(t) and z3g(t) and z2
minus04
minus05
f(t) and z1
minus2
minus15
minus1
minus05
minus1
minus05
minus01
minus02
minus03
F(t) and z0
minus0005
minus001
h(t)
z3g(t)
z2
f(t)
z1F(t)
z0
Figure 23 The outputs of the differentiator in the sampling interval of 00001 seconds
There are two causes for the errors in a usual differentia-tor the setting of sampling interval and the setting of numberof orders of the differentiator The following theorem applieshere
Theorem A1 (see [19]) If the sampling interval 120591 gt 0 theaccuracy of the controller is expressed as follows
10038161003816100381610038161003816120590(119894)
10038161003816100381610038161003816le 120583120591119903minus119894
119894 = 0 119903 minus 1 (A2)
That is to say the precision (understood as the differentialprecision of a differentiator) of the 119894th order control is in directproportion to the power of 119903 minus 119894 times of sampling interval120591 Therefore the decrease of iteration step can enhance thedifferential tracking precision
Under the conditions that the sampling interval is 10 timessmaller and that the setting is 00001 seconds the simulationresults are shown in Figure 22
18 Mathematical Problems in Engineering
Table 6 Tracking errors of the AORED with different orders andsampling interval
Orders of AORED Third order Fifth orderSample period 120591 = 0001 120591 = 00001 120591 = 00001
First-order differential error 001 Negligible NegligibleSecond-order differential error 15 002 NegligibleThird-order differential error 70 25 00005
As Figure 22 shows the signal tracking is rather idealexcept a peak in the beginning the tracking error in the first-order differential tracking can be neglected 119911
2has an obviously
smaller tracking error in the second-order differential 119892(119905)
tracking in the beginning 1199113has a rather sharp peak in the
third-order differential tracking but converges to a relativelyhigh precision after 01 seconds As we can see the differentialtracking errors for each number of orders are greatly reducedbut the third-order differential tracking error is still rather bigIn the following we will reduce the tracking errors by increasingthe number of orders of the differentiator
B The Fifth-Order Robust Exact Differentiator
The settings of the fifth-order differentiator are given asfollows
0
= V0
V0
= minus120582011987116 10038161003816100381610038161199110 minus
1199021003816100381610038161003816
56 sign (1199110
minus 119902) + 1199111
1
= V1
V1
= minus120582111987115 10038161003816100381610038161199111 minus V
0
1003816100381610038161003816
45 sign (1199111
minus V0) + 1199112
2
= V2
V2
= minus120582211987114 10038161003816100381610038161199112 minus V
1
1003816100381610038161003816
34 sign (1199112
minus V1) + 1199113
3
= V3
V3
= minus120582311987113 10038161003816100381610038161199113 minus V
2
1003816100381610038161003816
23 sign (1199113
minus V2) + 1199114
4
= V4
V4
= minus120582411987112 10038161003816100381610038161199114 minus V
3
1003816100381610038161003816
12 sign (1199114
minus V3) + 1199115
5
= V5
V5
= minus1205825119871 sign (119911
5minus V4)
(B1)
The initial value 1199110
= 01 1205820
= 1205821
= 1205822
= 1205823
= 1205824
=
1205825
= 50 the initial values 1199111 1199112 1199113 1199114 1199115
= 0 119871 = 1400the simulation step length is 00001 seconds The simulationresults are shown in Figure 23
As the figure shows the fifth-order differentiator hasa rather ideal tracking effect on the third-order differen-tial tracking the tracking error being only about 00005(see Table 6)
To sum up the Arbitrary-Order Robust Exact Differen-tiator can estimate rather ideal differential signals by selecting
Table 7
Length 365m119871 ref 365m119883119866
177m119878ref 0026m2
Diameter 0178mMass 1016 Kg119868119885
1063 Kgsdotm2
an appropriate sampling interval and using the differentiatorwith a relatively high number of orders
C Physical and Geometric Characteristics
See Table 7
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] P Zarchan Tactical and Strategic Missile Guidance 2002[2] D Zhou CMu andWXu ldquoAdaptive sliding-mode guidance of
a homing missilerdquo Journal of Guidance Control and Dynamicsvol 22 no 4 pp 589ndash594 1999
[3] G Zeng and M Hu ldquoFinite-time control for electromagneticsatellite formationsrdquo Acta Astronautica vol 74 pp 120ndash1302012
[4] S R Kumar S Rao and D Ghose ldquoNonsingular terminalsliding mode guidance with impact angle constraintsrdquo Journalof Guidance Control and Dynamics vol 37 no 4 pp 1114ndash11302014
[5] Y Liang C Chen D Liaw Y Feng C Cheng and CChen ldquoRobust guidance law via integral-sliding-mode schemerdquoJournal of Guidance Control and Dynamics vol 37 no 3 pp1038ndash1042 2014
[6] A V Savkin P N Pathirana and F Faruqi ldquoProblem ofprecision missile guidance LQR and 119867
infincontrol frameworksrdquo
IEEE Transactions on Aerospace and Electronic Systems vol 39no 3 pp 901ndash910 2003
[7] H Y Chen andC-C Yang ldquoNonlinear1198672119867infinguidance design
for homing missilesrdquo in Proceedings of the AIAA GuidanceNavigation and Control Conference and Exhibit vol 77 2001
[8] P K Menon G D Sweriduk and E J Ohlmeyer ldquoOptimalfixed-interval integrated guidance-control laws for hit-to-killmissilesrdquo in Proceedings of the AIAA Guidance Navigation andControl Conference and Exhibit AIAA 2003-5579 Austin TexUSA August 2003
[9] S S Vaddi P K Menon and E J Ohlmeyer ldquoNumericalSDRE approach for missile integrated guidance-controlrdquo inProceedings of the AIAA Guidance Navigation and ControlConference and Exhibit AIAA 2007-6672 Hilton Head SCUSA August 2007
[10] M Xin S N Balakrishnan and E J Ohlmeyer ldquoIntegratedguidance and control of missiles with Theta-D methodrdquo IEEE
Mathematical Problems in Engineering 19
Transactions on Control Systems Technology vol 14 no 6 pp981ndash992 2006
[11] T Shima M Idan and O M Golan ldquoSliding-mode controlfor integrated missile autopilot guidancerdquo Journal of GuidanceControl and Dynamics vol 29 no 2 pp 250ndash260 2006
[12] Y B Shtessel and C H Tournes ldquoIntegrated higher-ordersliding mode guidance and autopilot for dual control missilesrdquoJournal of Guidance Control and Dynamics vol 32 no 1 pp79ndash94 2009
[13] F Y Dong H M Lei and C J Zhou ldquoIntegrated robust highorder sliding mode guidance and control for missilesrdquo ActaAeronautica et Astronautica Sinica vol 34 no 9 pp 2212ndash22182013
[14] H Mingzhe and D Guangren ldquoIntegrated guidance and con-trol of homing missiles against ground fixed targetsrdquo ChineseJournal of Aeronautics vol 21 no 2 pp 162ndash168 2008
[15] A Levant ldquoHomogeneity approach to high-order sliding modedesignrdquo Automatica vol 41 no 5 pp 823ndash830 2005
[16] Y-W Liang C Chen D Liaw Y Feng C Cheng and CChen ldquoRobust guidance law via integral-sliding-mode schemerdquoJournal of Guidance Control and Dynamics vol 37 no 3 pp1038ndash1042 2014
[17] L Wang Y Sheng and X Liu ldquoContinuous time-varying slid-ingmode based attitude control for reentry vehiclerdquo Proceedingsof the Institution of Mechanical Engineers Part G Journal ofAerospace Engineering 2014
[18] A Levant ldquoHigher-order sliding modes differentiation andoutput-feedback controlrdquo International Journal of Control vol76 no 9-10 pp 924ndash941 2003
[19] A Levant ldquoQuasi-continuous high-order sliding-mode con-trollersrdquo IEEE Transactions on Automatic Control vol 50 no11 pp 1812ndash1816 2005
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
14 Mathematical Problems in Engineering
175
180
185
190
195
200
TargetHOSM-IGCOSMG
Y(k
m)
0 20 40 60 0 20 40X (km)
Figure 17 The trajectories of the missile and its target
0
10
20
0 5 10 15 20Time (s)
TargetOSMGHOSM-IGC
Miss
ile an
d ta
rget
acce
lera
tion
(G)
minus10
minus20
Figure 18 Missile acceleration profile
smooth enough for the actuator to be implemented as aresult the interception accuracy is improved In designingthe quasi-continuous sliding mode controller the Arbitrary-Order Robust Exact Differentiator is introduced to estimatethe high order time derivative of the LOS rate 119902 which notonly ensures the guidance and control precision but alsosimplifies the calculation
0
10
20
30
0 5 10 15 20Time (s)
OSMGHOSM-IGC
minus10
minus20
minus30
minus40
Actu
ator
defl
ectio
n (d
eg)
Figure 19 Actuator deflection
0
50
100
150
200
15 16 17 18Time (s)
Commanded accelerationAchieved acceleration
Miss
ile ac
cele
ratio
n (G
)
minus50
Figure 20 Commanded acceleration and achieved acceleration
Appendices
A The Third-Order RobustExact Differentiator
The third-order robust exact differentiator built according tothe following parameters and equations is used to performthe simulation the initial value 120582
0= 1205821
= 1205822
= 1205823
= 50
Mathematical Problems in Engineering 15
0
0005
001
0015
002
0025
003
0035
004
0 02 04Time (s)
0
2
0 02 04Time (s)
0
100
200
300
0
100
200
300
400
0 02 04Time (s)
h(t)
z3
h(t) and z3
minus100
minus200
minus300
minus400
minus5000 02 04
Time (s)
g(t)
z2
g(t) and z2
minus100
minus200
minus300
minus400
f(t)
z1
f(t) and z1
minus2
minus4
minus6
minus8
minus10
minus12
minus14
F(t)
z0
F(t) and z0
minus0005
minus001
Figure 21 The outputs of the differentiator in the sampling interval of 0001 seconds
the initial values 1199110
= 01 1199111
= 1199112
= 1199113
= 0 119871 = 1400 thesimulation step is 0001 seconds Consider
0
= V0
V0
= minus120582011987114 10038161003816100381610038161199110 minus
1199021003816100381610038161003816
34 sign (1199110
minus 119902) + 1199111
1
= V1
V1
= minus120582111987113 10038161003816100381610038161199111 minus V
0
1003816100381610038161003816
23 sign (1199111
minus V0) + 1199112
2
= V2
V2
= minus120582211987112 10038161003816100381610038161199112 minus V
1
1003816100381610038161003816
12 sign (1199112
minus V1) + 1199113
3
= minus1205823119871 sign (119911
3minus V2)
(A1)
16 Mathematical Problems in Engineering
0
0005
001
0015
002
0025
003
0035
004
0 02 04Time (s)
0
2
0 02 04Time (s)
0
100
200
300
0 02 04Time (s)
0
200
400
0 02 04Time (s)
h(t) and z3
minus200
minus400
minus600
minus800
g(t) and z2
minus100
minus200
minus300
minus400
f(t) and z1
minus2
minus4
minus6
minus8
minus10
minus12
minus14
minus16
F(t) and z0
minus0005
minus001
h(t)
z3g(t)
z2
f(t)
z1F(t)
z0
Figure 22 The outputs of the differentiator in the sampling interval of 00001 seconds
If the designed signals are 119865(119905) = int 119891(119905)119889119905 119891(119905) =
int 119892(119905)119889119905 119892(119905) = int ℎ(119905)119889119905 and ℎ(119905) = sin(2120587119905) +
3cos(120587119905) then theoretically 119891(119905) is the first-order differ-ential of input signal 119892(119905) is the second-order differ-ential of input signal ℎ(119905) is the third-order differen-tial of input signal Through the simulation the differ-ential signal of actual output is compared with that of
theoretical output the comparison results are shown inFigure 21
Figure 21 shows that 1199110can effectively track the signal
119865(119905) but 1199112has obvious errors in tracking the second-order
differential 119892(119905) 1199113has greater errors in tracking the third-
order differential thus the overall differential effect is notideal
Mathematical Problems in Engineering 17
0 02 04
0
0005
001
0015
002
0025
003
0035
004
Time (s)0 02 04
0
01
02
03
04
05
Time (s)
0 02 04
0
05
1
15
2
Time (s)0 02 04
0
05
1
15
2
25
3
35
4
Time (s)
h(t) and z3g(t) and z2
minus04
minus05
f(t) and z1
minus2
minus15
minus1
minus05
minus1
minus05
minus01
minus02
minus03
F(t) and z0
minus0005
minus001
h(t)
z3g(t)
z2
f(t)
z1F(t)
z0
Figure 23 The outputs of the differentiator in the sampling interval of 00001 seconds
There are two causes for the errors in a usual differentia-tor the setting of sampling interval and the setting of numberof orders of the differentiator The following theorem applieshere
Theorem A1 (see [19]) If the sampling interval 120591 gt 0 theaccuracy of the controller is expressed as follows
10038161003816100381610038161003816120590(119894)
10038161003816100381610038161003816le 120583120591119903minus119894
119894 = 0 119903 minus 1 (A2)
That is to say the precision (understood as the differentialprecision of a differentiator) of the 119894th order control is in directproportion to the power of 119903 minus 119894 times of sampling interval120591 Therefore the decrease of iteration step can enhance thedifferential tracking precision
Under the conditions that the sampling interval is 10 timessmaller and that the setting is 00001 seconds the simulationresults are shown in Figure 22
18 Mathematical Problems in Engineering
Table 6 Tracking errors of the AORED with different orders andsampling interval
Orders of AORED Third order Fifth orderSample period 120591 = 0001 120591 = 00001 120591 = 00001
First-order differential error 001 Negligible NegligibleSecond-order differential error 15 002 NegligibleThird-order differential error 70 25 00005
As Figure 22 shows the signal tracking is rather idealexcept a peak in the beginning the tracking error in the first-order differential tracking can be neglected 119911
2has an obviously
smaller tracking error in the second-order differential 119892(119905)
tracking in the beginning 1199113has a rather sharp peak in the
third-order differential tracking but converges to a relativelyhigh precision after 01 seconds As we can see the differentialtracking errors for each number of orders are greatly reducedbut the third-order differential tracking error is still rather bigIn the following we will reduce the tracking errors by increasingthe number of orders of the differentiator
B The Fifth-Order Robust Exact Differentiator
The settings of the fifth-order differentiator are given asfollows
0
= V0
V0
= minus120582011987116 10038161003816100381610038161199110 minus
1199021003816100381610038161003816
56 sign (1199110
minus 119902) + 1199111
1
= V1
V1
= minus120582111987115 10038161003816100381610038161199111 minus V
0
1003816100381610038161003816
45 sign (1199111
minus V0) + 1199112
2
= V2
V2
= minus120582211987114 10038161003816100381610038161199112 minus V
1
1003816100381610038161003816
34 sign (1199112
minus V1) + 1199113
3
= V3
V3
= minus120582311987113 10038161003816100381610038161199113 minus V
2
1003816100381610038161003816
23 sign (1199113
minus V2) + 1199114
4
= V4
V4
= minus120582411987112 10038161003816100381610038161199114 minus V
3
1003816100381610038161003816
12 sign (1199114
minus V3) + 1199115
5
= V5
V5
= minus1205825119871 sign (119911
5minus V4)
(B1)
The initial value 1199110
= 01 1205820
= 1205821
= 1205822
= 1205823
= 1205824
=
1205825
= 50 the initial values 1199111 1199112 1199113 1199114 1199115
= 0 119871 = 1400the simulation step length is 00001 seconds The simulationresults are shown in Figure 23
As the figure shows the fifth-order differentiator hasa rather ideal tracking effect on the third-order differen-tial tracking the tracking error being only about 00005(see Table 6)
To sum up the Arbitrary-Order Robust Exact Differen-tiator can estimate rather ideal differential signals by selecting
Table 7
Length 365m119871 ref 365m119883119866
177m119878ref 0026m2
Diameter 0178mMass 1016 Kg119868119885
1063 Kgsdotm2
an appropriate sampling interval and using the differentiatorwith a relatively high number of orders
C Physical and Geometric Characteristics
See Table 7
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] P Zarchan Tactical and Strategic Missile Guidance 2002[2] D Zhou CMu andWXu ldquoAdaptive sliding-mode guidance of
a homing missilerdquo Journal of Guidance Control and Dynamicsvol 22 no 4 pp 589ndash594 1999
[3] G Zeng and M Hu ldquoFinite-time control for electromagneticsatellite formationsrdquo Acta Astronautica vol 74 pp 120ndash1302012
[4] S R Kumar S Rao and D Ghose ldquoNonsingular terminalsliding mode guidance with impact angle constraintsrdquo Journalof Guidance Control and Dynamics vol 37 no 4 pp 1114ndash11302014
[5] Y Liang C Chen D Liaw Y Feng C Cheng and CChen ldquoRobust guidance law via integral-sliding-mode schemerdquoJournal of Guidance Control and Dynamics vol 37 no 3 pp1038ndash1042 2014
[6] A V Savkin P N Pathirana and F Faruqi ldquoProblem ofprecision missile guidance LQR and 119867
infincontrol frameworksrdquo
IEEE Transactions on Aerospace and Electronic Systems vol 39no 3 pp 901ndash910 2003
[7] H Y Chen andC-C Yang ldquoNonlinear1198672119867infinguidance design
for homing missilesrdquo in Proceedings of the AIAA GuidanceNavigation and Control Conference and Exhibit vol 77 2001
[8] P K Menon G D Sweriduk and E J Ohlmeyer ldquoOptimalfixed-interval integrated guidance-control laws for hit-to-killmissilesrdquo in Proceedings of the AIAA Guidance Navigation andControl Conference and Exhibit AIAA 2003-5579 Austin TexUSA August 2003
[9] S S Vaddi P K Menon and E J Ohlmeyer ldquoNumericalSDRE approach for missile integrated guidance-controlrdquo inProceedings of the AIAA Guidance Navigation and ControlConference and Exhibit AIAA 2007-6672 Hilton Head SCUSA August 2007
[10] M Xin S N Balakrishnan and E J Ohlmeyer ldquoIntegratedguidance and control of missiles with Theta-D methodrdquo IEEE
Mathematical Problems in Engineering 19
Transactions on Control Systems Technology vol 14 no 6 pp981ndash992 2006
[11] T Shima M Idan and O M Golan ldquoSliding-mode controlfor integrated missile autopilot guidancerdquo Journal of GuidanceControl and Dynamics vol 29 no 2 pp 250ndash260 2006
[12] Y B Shtessel and C H Tournes ldquoIntegrated higher-ordersliding mode guidance and autopilot for dual control missilesrdquoJournal of Guidance Control and Dynamics vol 32 no 1 pp79ndash94 2009
[13] F Y Dong H M Lei and C J Zhou ldquoIntegrated robust highorder sliding mode guidance and control for missilesrdquo ActaAeronautica et Astronautica Sinica vol 34 no 9 pp 2212ndash22182013
[14] H Mingzhe and D Guangren ldquoIntegrated guidance and con-trol of homing missiles against ground fixed targetsrdquo ChineseJournal of Aeronautics vol 21 no 2 pp 162ndash168 2008
[15] A Levant ldquoHomogeneity approach to high-order sliding modedesignrdquo Automatica vol 41 no 5 pp 823ndash830 2005
[16] Y-W Liang C Chen D Liaw Y Feng C Cheng and CChen ldquoRobust guidance law via integral-sliding-mode schemerdquoJournal of Guidance Control and Dynamics vol 37 no 3 pp1038ndash1042 2014
[17] L Wang Y Sheng and X Liu ldquoContinuous time-varying slid-ingmode based attitude control for reentry vehiclerdquo Proceedingsof the Institution of Mechanical Engineers Part G Journal ofAerospace Engineering 2014
[18] A Levant ldquoHigher-order sliding modes differentiation andoutput-feedback controlrdquo International Journal of Control vol76 no 9-10 pp 924ndash941 2003
[19] A Levant ldquoQuasi-continuous high-order sliding-mode con-trollersrdquo IEEE Transactions on Automatic Control vol 50 no11 pp 1812ndash1816 2005
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 15
0
0005
001
0015
002
0025
003
0035
004
0 02 04Time (s)
0
2
0 02 04Time (s)
0
100
200
300
0
100
200
300
400
0 02 04Time (s)
h(t)
z3
h(t) and z3
minus100
minus200
minus300
minus400
minus5000 02 04
Time (s)
g(t)
z2
g(t) and z2
minus100
minus200
minus300
minus400
f(t)
z1
f(t) and z1
minus2
minus4
minus6
minus8
minus10
minus12
minus14
F(t)
z0
F(t) and z0
minus0005
minus001
Figure 21 The outputs of the differentiator in the sampling interval of 0001 seconds
the initial values 1199110
= 01 1199111
= 1199112
= 1199113
= 0 119871 = 1400 thesimulation step is 0001 seconds Consider
0
= V0
V0
= minus120582011987114 10038161003816100381610038161199110 minus
1199021003816100381610038161003816
34 sign (1199110
minus 119902) + 1199111
1
= V1
V1
= minus120582111987113 10038161003816100381610038161199111 minus V
0
1003816100381610038161003816
23 sign (1199111
minus V0) + 1199112
2
= V2
V2
= minus120582211987112 10038161003816100381610038161199112 minus V
1
1003816100381610038161003816
12 sign (1199112
minus V1) + 1199113
3
= minus1205823119871 sign (119911
3minus V2)
(A1)
16 Mathematical Problems in Engineering
0
0005
001
0015
002
0025
003
0035
004
0 02 04Time (s)
0
2
0 02 04Time (s)
0
100
200
300
0 02 04Time (s)
0
200
400
0 02 04Time (s)
h(t) and z3
minus200
minus400
minus600
minus800
g(t) and z2
minus100
minus200
minus300
minus400
f(t) and z1
minus2
minus4
minus6
minus8
minus10
minus12
minus14
minus16
F(t) and z0
minus0005
minus001
h(t)
z3g(t)
z2
f(t)
z1F(t)
z0
Figure 22 The outputs of the differentiator in the sampling interval of 00001 seconds
If the designed signals are 119865(119905) = int 119891(119905)119889119905 119891(119905) =
int 119892(119905)119889119905 119892(119905) = int ℎ(119905)119889119905 and ℎ(119905) = sin(2120587119905) +
3cos(120587119905) then theoretically 119891(119905) is the first-order differ-ential of input signal 119892(119905) is the second-order differ-ential of input signal ℎ(119905) is the third-order differen-tial of input signal Through the simulation the differ-ential signal of actual output is compared with that of
theoretical output the comparison results are shown inFigure 21
Figure 21 shows that 1199110can effectively track the signal
119865(119905) but 1199112has obvious errors in tracking the second-order
differential 119892(119905) 1199113has greater errors in tracking the third-
order differential thus the overall differential effect is notideal
Mathematical Problems in Engineering 17
0 02 04
0
0005
001
0015
002
0025
003
0035
004
Time (s)0 02 04
0
01
02
03
04
05
Time (s)
0 02 04
0
05
1
15
2
Time (s)0 02 04
0
05
1
15
2
25
3
35
4
Time (s)
h(t) and z3g(t) and z2
minus04
minus05
f(t) and z1
minus2
minus15
minus1
minus05
minus1
minus05
minus01
minus02
minus03
F(t) and z0
minus0005
minus001
h(t)
z3g(t)
z2
f(t)
z1F(t)
z0
Figure 23 The outputs of the differentiator in the sampling interval of 00001 seconds
There are two causes for the errors in a usual differentia-tor the setting of sampling interval and the setting of numberof orders of the differentiator The following theorem applieshere
Theorem A1 (see [19]) If the sampling interval 120591 gt 0 theaccuracy of the controller is expressed as follows
10038161003816100381610038161003816120590(119894)
10038161003816100381610038161003816le 120583120591119903minus119894
119894 = 0 119903 minus 1 (A2)
That is to say the precision (understood as the differentialprecision of a differentiator) of the 119894th order control is in directproportion to the power of 119903 minus 119894 times of sampling interval120591 Therefore the decrease of iteration step can enhance thedifferential tracking precision
Under the conditions that the sampling interval is 10 timessmaller and that the setting is 00001 seconds the simulationresults are shown in Figure 22
18 Mathematical Problems in Engineering
Table 6 Tracking errors of the AORED with different orders andsampling interval
Orders of AORED Third order Fifth orderSample period 120591 = 0001 120591 = 00001 120591 = 00001
First-order differential error 001 Negligible NegligibleSecond-order differential error 15 002 NegligibleThird-order differential error 70 25 00005
As Figure 22 shows the signal tracking is rather idealexcept a peak in the beginning the tracking error in the first-order differential tracking can be neglected 119911
2has an obviously
smaller tracking error in the second-order differential 119892(119905)
tracking in the beginning 1199113has a rather sharp peak in the
third-order differential tracking but converges to a relativelyhigh precision after 01 seconds As we can see the differentialtracking errors for each number of orders are greatly reducedbut the third-order differential tracking error is still rather bigIn the following we will reduce the tracking errors by increasingthe number of orders of the differentiator
B The Fifth-Order Robust Exact Differentiator
The settings of the fifth-order differentiator are given asfollows
0
= V0
V0
= minus120582011987116 10038161003816100381610038161199110 minus
1199021003816100381610038161003816
56 sign (1199110
minus 119902) + 1199111
1
= V1
V1
= minus120582111987115 10038161003816100381610038161199111 minus V
0
1003816100381610038161003816
45 sign (1199111
minus V0) + 1199112
2
= V2
V2
= minus120582211987114 10038161003816100381610038161199112 minus V
1
1003816100381610038161003816
34 sign (1199112
minus V1) + 1199113
3
= V3
V3
= minus120582311987113 10038161003816100381610038161199113 minus V
2
1003816100381610038161003816
23 sign (1199113
minus V2) + 1199114
4
= V4
V4
= minus120582411987112 10038161003816100381610038161199114 minus V
3
1003816100381610038161003816
12 sign (1199114
minus V3) + 1199115
5
= V5
V5
= minus1205825119871 sign (119911
5minus V4)
(B1)
The initial value 1199110
= 01 1205820
= 1205821
= 1205822
= 1205823
= 1205824
=
1205825
= 50 the initial values 1199111 1199112 1199113 1199114 1199115
= 0 119871 = 1400the simulation step length is 00001 seconds The simulationresults are shown in Figure 23
As the figure shows the fifth-order differentiator hasa rather ideal tracking effect on the third-order differen-tial tracking the tracking error being only about 00005(see Table 6)
To sum up the Arbitrary-Order Robust Exact Differen-tiator can estimate rather ideal differential signals by selecting
Table 7
Length 365m119871 ref 365m119883119866
177m119878ref 0026m2
Diameter 0178mMass 1016 Kg119868119885
1063 Kgsdotm2
an appropriate sampling interval and using the differentiatorwith a relatively high number of orders
C Physical and Geometric Characteristics
See Table 7
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] P Zarchan Tactical and Strategic Missile Guidance 2002[2] D Zhou CMu andWXu ldquoAdaptive sliding-mode guidance of
a homing missilerdquo Journal of Guidance Control and Dynamicsvol 22 no 4 pp 589ndash594 1999
[3] G Zeng and M Hu ldquoFinite-time control for electromagneticsatellite formationsrdquo Acta Astronautica vol 74 pp 120ndash1302012
[4] S R Kumar S Rao and D Ghose ldquoNonsingular terminalsliding mode guidance with impact angle constraintsrdquo Journalof Guidance Control and Dynamics vol 37 no 4 pp 1114ndash11302014
[5] Y Liang C Chen D Liaw Y Feng C Cheng and CChen ldquoRobust guidance law via integral-sliding-mode schemerdquoJournal of Guidance Control and Dynamics vol 37 no 3 pp1038ndash1042 2014
[6] A V Savkin P N Pathirana and F Faruqi ldquoProblem ofprecision missile guidance LQR and 119867
infincontrol frameworksrdquo
IEEE Transactions on Aerospace and Electronic Systems vol 39no 3 pp 901ndash910 2003
[7] H Y Chen andC-C Yang ldquoNonlinear1198672119867infinguidance design
for homing missilesrdquo in Proceedings of the AIAA GuidanceNavigation and Control Conference and Exhibit vol 77 2001
[8] P K Menon G D Sweriduk and E J Ohlmeyer ldquoOptimalfixed-interval integrated guidance-control laws for hit-to-killmissilesrdquo in Proceedings of the AIAA Guidance Navigation andControl Conference and Exhibit AIAA 2003-5579 Austin TexUSA August 2003
[9] S S Vaddi P K Menon and E J Ohlmeyer ldquoNumericalSDRE approach for missile integrated guidance-controlrdquo inProceedings of the AIAA Guidance Navigation and ControlConference and Exhibit AIAA 2007-6672 Hilton Head SCUSA August 2007
[10] M Xin S N Balakrishnan and E J Ohlmeyer ldquoIntegratedguidance and control of missiles with Theta-D methodrdquo IEEE
Mathematical Problems in Engineering 19
Transactions on Control Systems Technology vol 14 no 6 pp981ndash992 2006
[11] T Shima M Idan and O M Golan ldquoSliding-mode controlfor integrated missile autopilot guidancerdquo Journal of GuidanceControl and Dynamics vol 29 no 2 pp 250ndash260 2006
[12] Y B Shtessel and C H Tournes ldquoIntegrated higher-ordersliding mode guidance and autopilot for dual control missilesrdquoJournal of Guidance Control and Dynamics vol 32 no 1 pp79ndash94 2009
[13] F Y Dong H M Lei and C J Zhou ldquoIntegrated robust highorder sliding mode guidance and control for missilesrdquo ActaAeronautica et Astronautica Sinica vol 34 no 9 pp 2212ndash22182013
[14] H Mingzhe and D Guangren ldquoIntegrated guidance and con-trol of homing missiles against ground fixed targetsrdquo ChineseJournal of Aeronautics vol 21 no 2 pp 162ndash168 2008
[15] A Levant ldquoHomogeneity approach to high-order sliding modedesignrdquo Automatica vol 41 no 5 pp 823ndash830 2005
[16] Y-W Liang C Chen D Liaw Y Feng C Cheng and CChen ldquoRobust guidance law via integral-sliding-mode schemerdquoJournal of Guidance Control and Dynamics vol 37 no 3 pp1038ndash1042 2014
[17] L Wang Y Sheng and X Liu ldquoContinuous time-varying slid-ingmode based attitude control for reentry vehiclerdquo Proceedingsof the Institution of Mechanical Engineers Part G Journal ofAerospace Engineering 2014
[18] A Levant ldquoHigher-order sliding modes differentiation andoutput-feedback controlrdquo International Journal of Control vol76 no 9-10 pp 924ndash941 2003
[19] A Levant ldquoQuasi-continuous high-order sliding-mode con-trollersrdquo IEEE Transactions on Automatic Control vol 50 no11 pp 1812ndash1816 2005
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
16 Mathematical Problems in Engineering
0
0005
001
0015
002
0025
003
0035
004
0 02 04Time (s)
0
2
0 02 04Time (s)
0
100
200
300
0 02 04Time (s)
0
200
400
0 02 04Time (s)
h(t) and z3
minus200
minus400
minus600
minus800
g(t) and z2
minus100
minus200
minus300
minus400
f(t) and z1
minus2
minus4
minus6
minus8
minus10
minus12
minus14
minus16
F(t) and z0
minus0005
minus001
h(t)
z3g(t)
z2
f(t)
z1F(t)
z0
Figure 22 The outputs of the differentiator in the sampling interval of 00001 seconds
If the designed signals are 119865(119905) = int 119891(119905)119889119905 119891(119905) =
int 119892(119905)119889119905 119892(119905) = int ℎ(119905)119889119905 and ℎ(119905) = sin(2120587119905) +
3cos(120587119905) then theoretically 119891(119905) is the first-order differ-ential of input signal 119892(119905) is the second-order differ-ential of input signal ℎ(119905) is the third-order differen-tial of input signal Through the simulation the differ-ential signal of actual output is compared with that of
theoretical output the comparison results are shown inFigure 21
Figure 21 shows that 1199110can effectively track the signal
119865(119905) but 1199112has obvious errors in tracking the second-order
differential 119892(119905) 1199113has greater errors in tracking the third-
order differential thus the overall differential effect is notideal
Mathematical Problems in Engineering 17
0 02 04
0
0005
001
0015
002
0025
003
0035
004
Time (s)0 02 04
0
01
02
03
04
05
Time (s)
0 02 04
0
05
1
15
2
Time (s)0 02 04
0
05
1
15
2
25
3
35
4
Time (s)
h(t) and z3g(t) and z2
minus04
minus05
f(t) and z1
minus2
minus15
minus1
minus05
minus1
minus05
minus01
minus02
minus03
F(t) and z0
minus0005
minus001
h(t)
z3g(t)
z2
f(t)
z1F(t)
z0
Figure 23 The outputs of the differentiator in the sampling interval of 00001 seconds
There are two causes for the errors in a usual differentia-tor the setting of sampling interval and the setting of numberof orders of the differentiator The following theorem applieshere
Theorem A1 (see [19]) If the sampling interval 120591 gt 0 theaccuracy of the controller is expressed as follows
10038161003816100381610038161003816120590(119894)
10038161003816100381610038161003816le 120583120591119903minus119894
119894 = 0 119903 minus 1 (A2)
That is to say the precision (understood as the differentialprecision of a differentiator) of the 119894th order control is in directproportion to the power of 119903 minus 119894 times of sampling interval120591 Therefore the decrease of iteration step can enhance thedifferential tracking precision
Under the conditions that the sampling interval is 10 timessmaller and that the setting is 00001 seconds the simulationresults are shown in Figure 22
18 Mathematical Problems in Engineering
Table 6 Tracking errors of the AORED with different orders andsampling interval
Orders of AORED Third order Fifth orderSample period 120591 = 0001 120591 = 00001 120591 = 00001
First-order differential error 001 Negligible NegligibleSecond-order differential error 15 002 NegligibleThird-order differential error 70 25 00005
As Figure 22 shows the signal tracking is rather idealexcept a peak in the beginning the tracking error in the first-order differential tracking can be neglected 119911
2has an obviously
smaller tracking error in the second-order differential 119892(119905)
tracking in the beginning 1199113has a rather sharp peak in the
third-order differential tracking but converges to a relativelyhigh precision after 01 seconds As we can see the differentialtracking errors for each number of orders are greatly reducedbut the third-order differential tracking error is still rather bigIn the following we will reduce the tracking errors by increasingthe number of orders of the differentiator
B The Fifth-Order Robust Exact Differentiator
The settings of the fifth-order differentiator are given asfollows
0
= V0
V0
= minus120582011987116 10038161003816100381610038161199110 minus
1199021003816100381610038161003816
56 sign (1199110
minus 119902) + 1199111
1
= V1
V1
= minus120582111987115 10038161003816100381610038161199111 minus V
0
1003816100381610038161003816
45 sign (1199111
minus V0) + 1199112
2
= V2
V2
= minus120582211987114 10038161003816100381610038161199112 minus V
1
1003816100381610038161003816
34 sign (1199112
minus V1) + 1199113
3
= V3
V3
= minus120582311987113 10038161003816100381610038161199113 minus V
2
1003816100381610038161003816
23 sign (1199113
minus V2) + 1199114
4
= V4
V4
= minus120582411987112 10038161003816100381610038161199114 minus V
3
1003816100381610038161003816
12 sign (1199114
minus V3) + 1199115
5
= V5
V5
= minus1205825119871 sign (119911
5minus V4)
(B1)
The initial value 1199110
= 01 1205820
= 1205821
= 1205822
= 1205823
= 1205824
=
1205825
= 50 the initial values 1199111 1199112 1199113 1199114 1199115
= 0 119871 = 1400the simulation step length is 00001 seconds The simulationresults are shown in Figure 23
As the figure shows the fifth-order differentiator hasa rather ideal tracking effect on the third-order differen-tial tracking the tracking error being only about 00005(see Table 6)
To sum up the Arbitrary-Order Robust Exact Differen-tiator can estimate rather ideal differential signals by selecting
Table 7
Length 365m119871 ref 365m119883119866
177m119878ref 0026m2
Diameter 0178mMass 1016 Kg119868119885
1063 Kgsdotm2
an appropriate sampling interval and using the differentiatorwith a relatively high number of orders
C Physical and Geometric Characteristics
See Table 7
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] P Zarchan Tactical and Strategic Missile Guidance 2002[2] D Zhou CMu andWXu ldquoAdaptive sliding-mode guidance of
a homing missilerdquo Journal of Guidance Control and Dynamicsvol 22 no 4 pp 589ndash594 1999
[3] G Zeng and M Hu ldquoFinite-time control for electromagneticsatellite formationsrdquo Acta Astronautica vol 74 pp 120ndash1302012
[4] S R Kumar S Rao and D Ghose ldquoNonsingular terminalsliding mode guidance with impact angle constraintsrdquo Journalof Guidance Control and Dynamics vol 37 no 4 pp 1114ndash11302014
[5] Y Liang C Chen D Liaw Y Feng C Cheng and CChen ldquoRobust guidance law via integral-sliding-mode schemerdquoJournal of Guidance Control and Dynamics vol 37 no 3 pp1038ndash1042 2014
[6] A V Savkin P N Pathirana and F Faruqi ldquoProblem ofprecision missile guidance LQR and 119867
infincontrol frameworksrdquo
IEEE Transactions on Aerospace and Electronic Systems vol 39no 3 pp 901ndash910 2003
[7] H Y Chen andC-C Yang ldquoNonlinear1198672119867infinguidance design
for homing missilesrdquo in Proceedings of the AIAA GuidanceNavigation and Control Conference and Exhibit vol 77 2001
[8] P K Menon G D Sweriduk and E J Ohlmeyer ldquoOptimalfixed-interval integrated guidance-control laws for hit-to-killmissilesrdquo in Proceedings of the AIAA Guidance Navigation andControl Conference and Exhibit AIAA 2003-5579 Austin TexUSA August 2003
[9] S S Vaddi P K Menon and E J Ohlmeyer ldquoNumericalSDRE approach for missile integrated guidance-controlrdquo inProceedings of the AIAA Guidance Navigation and ControlConference and Exhibit AIAA 2007-6672 Hilton Head SCUSA August 2007
[10] M Xin S N Balakrishnan and E J Ohlmeyer ldquoIntegratedguidance and control of missiles with Theta-D methodrdquo IEEE
Mathematical Problems in Engineering 19
Transactions on Control Systems Technology vol 14 no 6 pp981ndash992 2006
[11] T Shima M Idan and O M Golan ldquoSliding-mode controlfor integrated missile autopilot guidancerdquo Journal of GuidanceControl and Dynamics vol 29 no 2 pp 250ndash260 2006
[12] Y B Shtessel and C H Tournes ldquoIntegrated higher-ordersliding mode guidance and autopilot for dual control missilesrdquoJournal of Guidance Control and Dynamics vol 32 no 1 pp79ndash94 2009
[13] F Y Dong H M Lei and C J Zhou ldquoIntegrated robust highorder sliding mode guidance and control for missilesrdquo ActaAeronautica et Astronautica Sinica vol 34 no 9 pp 2212ndash22182013
[14] H Mingzhe and D Guangren ldquoIntegrated guidance and con-trol of homing missiles against ground fixed targetsrdquo ChineseJournal of Aeronautics vol 21 no 2 pp 162ndash168 2008
[15] A Levant ldquoHomogeneity approach to high-order sliding modedesignrdquo Automatica vol 41 no 5 pp 823ndash830 2005
[16] Y-W Liang C Chen D Liaw Y Feng C Cheng and CChen ldquoRobust guidance law via integral-sliding-mode schemerdquoJournal of Guidance Control and Dynamics vol 37 no 3 pp1038ndash1042 2014
[17] L Wang Y Sheng and X Liu ldquoContinuous time-varying slid-ingmode based attitude control for reentry vehiclerdquo Proceedingsof the Institution of Mechanical Engineers Part G Journal ofAerospace Engineering 2014
[18] A Levant ldquoHigher-order sliding modes differentiation andoutput-feedback controlrdquo International Journal of Control vol76 no 9-10 pp 924ndash941 2003
[19] A Levant ldquoQuasi-continuous high-order sliding-mode con-trollersrdquo IEEE Transactions on Automatic Control vol 50 no11 pp 1812ndash1816 2005
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 17
0 02 04
0
0005
001
0015
002
0025
003
0035
004
Time (s)0 02 04
0
01
02
03
04
05
Time (s)
0 02 04
0
05
1
15
2
Time (s)0 02 04
0
05
1
15
2
25
3
35
4
Time (s)
h(t) and z3g(t) and z2
minus04
minus05
f(t) and z1
minus2
minus15
minus1
minus05
minus1
minus05
minus01
minus02
minus03
F(t) and z0
minus0005
minus001
h(t)
z3g(t)
z2
f(t)
z1F(t)
z0
Figure 23 The outputs of the differentiator in the sampling interval of 00001 seconds
There are two causes for the errors in a usual differentia-tor the setting of sampling interval and the setting of numberof orders of the differentiator The following theorem applieshere
Theorem A1 (see [19]) If the sampling interval 120591 gt 0 theaccuracy of the controller is expressed as follows
10038161003816100381610038161003816120590(119894)
10038161003816100381610038161003816le 120583120591119903minus119894
119894 = 0 119903 minus 1 (A2)
That is to say the precision (understood as the differentialprecision of a differentiator) of the 119894th order control is in directproportion to the power of 119903 minus 119894 times of sampling interval120591 Therefore the decrease of iteration step can enhance thedifferential tracking precision
Under the conditions that the sampling interval is 10 timessmaller and that the setting is 00001 seconds the simulationresults are shown in Figure 22
18 Mathematical Problems in Engineering
Table 6 Tracking errors of the AORED with different orders andsampling interval
Orders of AORED Third order Fifth orderSample period 120591 = 0001 120591 = 00001 120591 = 00001
First-order differential error 001 Negligible NegligibleSecond-order differential error 15 002 NegligibleThird-order differential error 70 25 00005
As Figure 22 shows the signal tracking is rather idealexcept a peak in the beginning the tracking error in the first-order differential tracking can be neglected 119911
2has an obviously
smaller tracking error in the second-order differential 119892(119905)
tracking in the beginning 1199113has a rather sharp peak in the
third-order differential tracking but converges to a relativelyhigh precision after 01 seconds As we can see the differentialtracking errors for each number of orders are greatly reducedbut the third-order differential tracking error is still rather bigIn the following we will reduce the tracking errors by increasingthe number of orders of the differentiator
B The Fifth-Order Robust Exact Differentiator
The settings of the fifth-order differentiator are given asfollows
0
= V0
V0
= minus120582011987116 10038161003816100381610038161199110 minus
1199021003816100381610038161003816
56 sign (1199110
minus 119902) + 1199111
1
= V1
V1
= minus120582111987115 10038161003816100381610038161199111 minus V
0
1003816100381610038161003816
45 sign (1199111
minus V0) + 1199112
2
= V2
V2
= minus120582211987114 10038161003816100381610038161199112 minus V
1
1003816100381610038161003816
34 sign (1199112
minus V1) + 1199113
3
= V3
V3
= minus120582311987113 10038161003816100381610038161199113 minus V
2
1003816100381610038161003816
23 sign (1199113
minus V2) + 1199114
4
= V4
V4
= minus120582411987112 10038161003816100381610038161199114 minus V
3
1003816100381610038161003816
12 sign (1199114
minus V3) + 1199115
5
= V5
V5
= minus1205825119871 sign (119911
5minus V4)
(B1)
The initial value 1199110
= 01 1205820
= 1205821
= 1205822
= 1205823
= 1205824
=
1205825
= 50 the initial values 1199111 1199112 1199113 1199114 1199115
= 0 119871 = 1400the simulation step length is 00001 seconds The simulationresults are shown in Figure 23
As the figure shows the fifth-order differentiator hasa rather ideal tracking effect on the third-order differen-tial tracking the tracking error being only about 00005(see Table 6)
To sum up the Arbitrary-Order Robust Exact Differen-tiator can estimate rather ideal differential signals by selecting
Table 7
Length 365m119871 ref 365m119883119866
177m119878ref 0026m2
Diameter 0178mMass 1016 Kg119868119885
1063 Kgsdotm2
an appropriate sampling interval and using the differentiatorwith a relatively high number of orders
C Physical and Geometric Characteristics
See Table 7
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] P Zarchan Tactical and Strategic Missile Guidance 2002[2] D Zhou CMu andWXu ldquoAdaptive sliding-mode guidance of
a homing missilerdquo Journal of Guidance Control and Dynamicsvol 22 no 4 pp 589ndash594 1999
[3] G Zeng and M Hu ldquoFinite-time control for electromagneticsatellite formationsrdquo Acta Astronautica vol 74 pp 120ndash1302012
[4] S R Kumar S Rao and D Ghose ldquoNonsingular terminalsliding mode guidance with impact angle constraintsrdquo Journalof Guidance Control and Dynamics vol 37 no 4 pp 1114ndash11302014
[5] Y Liang C Chen D Liaw Y Feng C Cheng and CChen ldquoRobust guidance law via integral-sliding-mode schemerdquoJournal of Guidance Control and Dynamics vol 37 no 3 pp1038ndash1042 2014
[6] A V Savkin P N Pathirana and F Faruqi ldquoProblem ofprecision missile guidance LQR and 119867
infincontrol frameworksrdquo
IEEE Transactions on Aerospace and Electronic Systems vol 39no 3 pp 901ndash910 2003
[7] H Y Chen andC-C Yang ldquoNonlinear1198672119867infinguidance design
for homing missilesrdquo in Proceedings of the AIAA GuidanceNavigation and Control Conference and Exhibit vol 77 2001
[8] P K Menon G D Sweriduk and E J Ohlmeyer ldquoOptimalfixed-interval integrated guidance-control laws for hit-to-killmissilesrdquo in Proceedings of the AIAA Guidance Navigation andControl Conference and Exhibit AIAA 2003-5579 Austin TexUSA August 2003
[9] S S Vaddi P K Menon and E J Ohlmeyer ldquoNumericalSDRE approach for missile integrated guidance-controlrdquo inProceedings of the AIAA Guidance Navigation and ControlConference and Exhibit AIAA 2007-6672 Hilton Head SCUSA August 2007
[10] M Xin S N Balakrishnan and E J Ohlmeyer ldquoIntegratedguidance and control of missiles with Theta-D methodrdquo IEEE
Mathematical Problems in Engineering 19
Transactions on Control Systems Technology vol 14 no 6 pp981ndash992 2006
[11] T Shima M Idan and O M Golan ldquoSliding-mode controlfor integrated missile autopilot guidancerdquo Journal of GuidanceControl and Dynamics vol 29 no 2 pp 250ndash260 2006
[12] Y B Shtessel and C H Tournes ldquoIntegrated higher-ordersliding mode guidance and autopilot for dual control missilesrdquoJournal of Guidance Control and Dynamics vol 32 no 1 pp79ndash94 2009
[13] F Y Dong H M Lei and C J Zhou ldquoIntegrated robust highorder sliding mode guidance and control for missilesrdquo ActaAeronautica et Astronautica Sinica vol 34 no 9 pp 2212ndash22182013
[14] H Mingzhe and D Guangren ldquoIntegrated guidance and con-trol of homing missiles against ground fixed targetsrdquo ChineseJournal of Aeronautics vol 21 no 2 pp 162ndash168 2008
[15] A Levant ldquoHomogeneity approach to high-order sliding modedesignrdquo Automatica vol 41 no 5 pp 823ndash830 2005
[16] Y-W Liang C Chen D Liaw Y Feng C Cheng and CChen ldquoRobust guidance law via integral-sliding-mode schemerdquoJournal of Guidance Control and Dynamics vol 37 no 3 pp1038ndash1042 2014
[17] L Wang Y Sheng and X Liu ldquoContinuous time-varying slid-ingmode based attitude control for reentry vehiclerdquo Proceedingsof the Institution of Mechanical Engineers Part G Journal ofAerospace Engineering 2014
[18] A Levant ldquoHigher-order sliding modes differentiation andoutput-feedback controlrdquo International Journal of Control vol76 no 9-10 pp 924ndash941 2003
[19] A Levant ldquoQuasi-continuous high-order sliding-mode con-trollersrdquo IEEE Transactions on Automatic Control vol 50 no11 pp 1812ndash1816 2005
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
18 Mathematical Problems in Engineering
Table 6 Tracking errors of the AORED with different orders andsampling interval
Orders of AORED Third order Fifth orderSample period 120591 = 0001 120591 = 00001 120591 = 00001
First-order differential error 001 Negligible NegligibleSecond-order differential error 15 002 NegligibleThird-order differential error 70 25 00005
As Figure 22 shows the signal tracking is rather idealexcept a peak in the beginning the tracking error in the first-order differential tracking can be neglected 119911
2has an obviously
smaller tracking error in the second-order differential 119892(119905)
tracking in the beginning 1199113has a rather sharp peak in the
third-order differential tracking but converges to a relativelyhigh precision after 01 seconds As we can see the differentialtracking errors for each number of orders are greatly reducedbut the third-order differential tracking error is still rather bigIn the following we will reduce the tracking errors by increasingthe number of orders of the differentiator
B The Fifth-Order Robust Exact Differentiator
The settings of the fifth-order differentiator are given asfollows
0
= V0
V0
= minus120582011987116 10038161003816100381610038161199110 minus
1199021003816100381610038161003816
56 sign (1199110
minus 119902) + 1199111
1
= V1
V1
= minus120582111987115 10038161003816100381610038161199111 minus V
0
1003816100381610038161003816
45 sign (1199111
minus V0) + 1199112
2
= V2
V2
= minus120582211987114 10038161003816100381610038161199112 minus V
1
1003816100381610038161003816
34 sign (1199112
minus V1) + 1199113
3
= V3
V3
= minus120582311987113 10038161003816100381610038161199113 minus V
2
1003816100381610038161003816
23 sign (1199113
minus V2) + 1199114
4
= V4
V4
= minus120582411987112 10038161003816100381610038161199114 minus V
3
1003816100381610038161003816
12 sign (1199114
minus V3) + 1199115
5
= V5
V5
= minus1205825119871 sign (119911
5minus V4)
(B1)
The initial value 1199110
= 01 1205820
= 1205821
= 1205822
= 1205823
= 1205824
=
1205825
= 50 the initial values 1199111 1199112 1199113 1199114 1199115
= 0 119871 = 1400the simulation step length is 00001 seconds The simulationresults are shown in Figure 23
As the figure shows the fifth-order differentiator hasa rather ideal tracking effect on the third-order differen-tial tracking the tracking error being only about 00005(see Table 6)
To sum up the Arbitrary-Order Robust Exact Differen-tiator can estimate rather ideal differential signals by selecting
Table 7
Length 365m119871 ref 365m119883119866
177m119878ref 0026m2
Diameter 0178mMass 1016 Kg119868119885
1063 Kgsdotm2
an appropriate sampling interval and using the differentiatorwith a relatively high number of orders
C Physical and Geometric Characteristics
See Table 7
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] P Zarchan Tactical and Strategic Missile Guidance 2002[2] D Zhou CMu andWXu ldquoAdaptive sliding-mode guidance of
a homing missilerdquo Journal of Guidance Control and Dynamicsvol 22 no 4 pp 589ndash594 1999
[3] G Zeng and M Hu ldquoFinite-time control for electromagneticsatellite formationsrdquo Acta Astronautica vol 74 pp 120ndash1302012
[4] S R Kumar S Rao and D Ghose ldquoNonsingular terminalsliding mode guidance with impact angle constraintsrdquo Journalof Guidance Control and Dynamics vol 37 no 4 pp 1114ndash11302014
[5] Y Liang C Chen D Liaw Y Feng C Cheng and CChen ldquoRobust guidance law via integral-sliding-mode schemerdquoJournal of Guidance Control and Dynamics vol 37 no 3 pp1038ndash1042 2014
[6] A V Savkin P N Pathirana and F Faruqi ldquoProblem ofprecision missile guidance LQR and 119867
infincontrol frameworksrdquo
IEEE Transactions on Aerospace and Electronic Systems vol 39no 3 pp 901ndash910 2003
[7] H Y Chen andC-C Yang ldquoNonlinear1198672119867infinguidance design
for homing missilesrdquo in Proceedings of the AIAA GuidanceNavigation and Control Conference and Exhibit vol 77 2001
[8] P K Menon G D Sweriduk and E J Ohlmeyer ldquoOptimalfixed-interval integrated guidance-control laws for hit-to-killmissilesrdquo in Proceedings of the AIAA Guidance Navigation andControl Conference and Exhibit AIAA 2003-5579 Austin TexUSA August 2003
[9] S S Vaddi P K Menon and E J Ohlmeyer ldquoNumericalSDRE approach for missile integrated guidance-controlrdquo inProceedings of the AIAA Guidance Navigation and ControlConference and Exhibit AIAA 2007-6672 Hilton Head SCUSA August 2007
[10] M Xin S N Balakrishnan and E J Ohlmeyer ldquoIntegratedguidance and control of missiles with Theta-D methodrdquo IEEE
Mathematical Problems in Engineering 19
Transactions on Control Systems Technology vol 14 no 6 pp981ndash992 2006
[11] T Shima M Idan and O M Golan ldquoSliding-mode controlfor integrated missile autopilot guidancerdquo Journal of GuidanceControl and Dynamics vol 29 no 2 pp 250ndash260 2006
[12] Y B Shtessel and C H Tournes ldquoIntegrated higher-ordersliding mode guidance and autopilot for dual control missilesrdquoJournal of Guidance Control and Dynamics vol 32 no 1 pp79ndash94 2009
[13] F Y Dong H M Lei and C J Zhou ldquoIntegrated robust highorder sliding mode guidance and control for missilesrdquo ActaAeronautica et Astronautica Sinica vol 34 no 9 pp 2212ndash22182013
[14] H Mingzhe and D Guangren ldquoIntegrated guidance and con-trol of homing missiles against ground fixed targetsrdquo ChineseJournal of Aeronautics vol 21 no 2 pp 162ndash168 2008
[15] A Levant ldquoHomogeneity approach to high-order sliding modedesignrdquo Automatica vol 41 no 5 pp 823ndash830 2005
[16] Y-W Liang C Chen D Liaw Y Feng C Cheng and CChen ldquoRobust guidance law via integral-sliding-mode schemerdquoJournal of Guidance Control and Dynamics vol 37 no 3 pp1038ndash1042 2014
[17] L Wang Y Sheng and X Liu ldquoContinuous time-varying slid-ingmode based attitude control for reentry vehiclerdquo Proceedingsof the Institution of Mechanical Engineers Part G Journal ofAerospace Engineering 2014
[18] A Levant ldquoHigher-order sliding modes differentiation andoutput-feedback controlrdquo International Journal of Control vol76 no 9-10 pp 924ndash941 2003
[19] A Levant ldquoQuasi-continuous high-order sliding-mode con-trollersrdquo IEEE Transactions on Automatic Control vol 50 no11 pp 1812ndash1816 2005
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 19
Transactions on Control Systems Technology vol 14 no 6 pp981ndash992 2006
[11] T Shima M Idan and O M Golan ldquoSliding-mode controlfor integrated missile autopilot guidancerdquo Journal of GuidanceControl and Dynamics vol 29 no 2 pp 250ndash260 2006
[12] Y B Shtessel and C H Tournes ldquoIntegrated higher-ordersliding mode guidance and autopilot for dual control missilesrdquoJournal of Guidance Control and Dynamics vol 32 no 1 pp79ndash94 2009
[13] F Y Dong H M Lei and C J Zhou ldquoIntegrated robust highorder sliding mode guidance and control for missilesrdquo ActaAeronautica et Astronautica Sinica vol 34 no 9 pp 2212ndash22182013
[14] H Mingzhe and D Guangren ldquoIntegrated guidance and con-trol of homing missiles against ground fixed targetsrdquo ChineseJournal of Aeronautics vol 21 no 2 pp 162ndash168 2008
[15] A Levant ldquoHomogeneity approach to high-order sliding modedesignrdquo Automatica vol 41 no 5 pp 823ndash830 2005
[16] Y-W Liang C Chen D Liaw Y Feng C Cheng and CChen ldquoRobust guidance law via integral-sliding-mode schemerdquoJournal of Guidance Control and Dynamics vol 37 no 3 pp1038ndash1042 2014
[17] L Wang Y Sheng and X Liu ldquoContinuous time-varying slid-ingmode based attitude control for reentry vehiclerdquo Proceedingsof the Institution of Mechanical Engineers Part G Journal ofAerospace Engineering 2014
[18] A Levant ldquoHigher-order sliding modes differentiation andoutput-feedback controlrdquo International Journal of Control vol76 no 9-10 pp 924ndash941 2003
[19] A Levant ldquoQuasi-continuous high-order sliding-mode con-trollersrdquo IEEE Transactions on Automatic Control vol 50 no11 pp 1812ndash1816 2005
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of