End-game guidance laws for dual-control missiles

End-game guidance laws for dual-control missilesT Shima� and O M Golan

RAFAEL, Haifa, Israel

The manuscript was received on 6 February 2004 and was accepted after revision for publication on 16 June 2004.

DOI: 10.1243/095441005X7249

Abstract: New guidance laws derived for a dual-control missile are studied. Their performanceis compared with that of a conventional guidance system with flight control that distributesthe commands among the two control channels. A Monte Carlo simulation study is performedwhere the target executes random evasive manoeuvres. The interceptor, represented by linearhigh-order manoeuvring dynamics with bounded forward and aft controls, acquires noisymeasurements and uses a state estimator. Using the single-shot kill probability as a perform-ance criterion, the superiority of the new guidance and control architecture is shown. It isalso shown that the best performance is obtained by the bounded controls differential gameguidance law. Sensitivity to the selection of the interceptor first-order time constant, a designparameter in the guidance law derivation, is also investigated.

Keywords: missile, guidance, canard, tail, differential games, optimal control

1 INTRODUCTION

The design of an interceptor missile system requirestrade-offs between conflicting requirements. Forexample, it has been shown that canard controlprovides better homing performance during theend game [1]. However, if the missile is to performsharp initial turns, canard control may limit itsperformance due to aerodynamic saturation atlarge angles of attack, and tail control may be pre-ferred. By using both canard and tail controls areasonable design compromise can be obtained toprovide overall enhanced performance.

The additional degree of freedom offered by thedual-control system requires special considerationin the guidance and control design. A variable-structure approach with a linear strategy for blendingthe two control actions was suggested in references[2] and [3] while in reference [4] adaptive techniquesfor multiple actuators blending using fuzzy controlwere proposed. Neural networks were used inreference [5] for the design of adaptive non-linearcontrol for an agile missile with forward and aft

reaction control systems (RCSs) and aerodynamictail control surfaces. In reference [6] the controlproblem of a tail aero-fin and forward RCS configur-ation was treated, using the coefficient diagrammethod. A similar approach was used in reference[7] for an all-aerodynamic tail, fins, and canardconfiguration. In all these studies the prime focuswas on methods for controlling the airframe.

In recent papers by the present authors [8, 9] thefocus was different; it was on designing the end-game guidance strategy for a missile with dual con-trols. The approach is based on the assumptionthat the additional degree of freedom can be bestutilized by providing the guidance law with thecapability of optimally imparting the commands tothe two controls. The flight control and guidance ofthe interceptor missile were assumed to be spectrallyseparated and the closed-loop dynamics wererepresented by two first-order bi proper transferfunctions (TFs) (the zero in the canard and tail TFis in the left and right half S planes respectively).The target manoeuvring dynamics were representedby a first-order strictly proper TF. In reference [8]the missile control limits were treated indirectly, byincorporating penalties on the use of the controlsin the linear quadratic formulation of the problem.In reality, such an approach may lead to a conserva-tive design, if the weights are selected high enoughto avoid saturation. Hence in reference [9] a different

�Corresponding author: Currently visiting scientist at the Air

Vehicles Directorate, Air Force Research Laboratories, Building

146, Room 304, 2210 Eighth Street, Wright-Patterson AFB, OH

45433, USA. email: [email protected]

SPECIAL ISSUE PAPER 157

G00904 # IMechE 2005 Proc. IMechE Vol. 219 Part G: J. Aerospace Engineering

approach, where the control bounds are explicitlytaken into account, was studied.

In this paper, the new dual-control guidancescheme is examined, using a simulation of a genericinterceptor, having linear high-order manoeuvringdynamics with bounded forward and aft controls.In order to assess the advantage of the proposedguidance scheme, an alternative approach, inwhich the distribution of the commands is deter-mined by the flight control system, is examinedas well. In this case the guidance laws are reducedto the conventional form for a single controlinterceptor.

The guidance and control architecture of an inter-ceptor missile having two control systems, as wellas the interception scenario of an aerial target, isdescribed in the next section. In the following thedifferent guidance laws, tailored for the dual-controlconfiguration, are presented together with a modi-fied version of the classical proportional naviga-tion (PN) guidance law that serves as a reference.Section 4 is devoted to a performance analysis.Conclusions are offered in the last section. Themodel for the generic missile used in the analysis isdeveloped in the Appendix.

2 GUIDANCE AND CONTROL ARCHITECTURE

The role of flight control is to stabilize the airframeand to shape the response to achieve the desiredmanoeuvres required by the guidance law; this is,of course, while taking into account the controllimitations and other system requirements. For thepurpose of demonstrating the guidance law perform-ance, a simple example of linear dynamics is con-sidered. The flight control system is designed toimprove the stability by increasing the low naturaldamping of the airframe. This is achieved by feedingback a command proportional to the angular rate ofthe missile.

Two different architectures are considered: inthe first, the manoeuvring commands are issuedto the forward and aft controls directly from theguidance law, which is designed specifically forthe dual-control interceptor. In the second case,the manoeuvring commands are equally dividedbetween the two controls, thus enabling the inter-ceptor to receive conventional guidance commands.

2.1 Closed-loop dynamics

The interceptor airframe example used in thisstudy is based on a tutorial example from reference[10] which is modified to represent a dual-controlmissile. The obtained system is stable withundamped dynamic behaviour. The undesired

damping behaviour of the open-loop airframe canbe modified by feeding back a correction commandproportional to the measured pitch rate. If thecommand is equally split between the canard andtail, the fins will deflect in opposite directions to gen-erate the restraining moment; in the current modelthe net force due to this correction will be zero inthe steady state (details are given in the Appendix).Thus, the following closed-loop TFs are obtained

aPc

Kcdcc

¼ GcðsÞ

¼ð1þ0:0228sÞðs2=55:122þs=496:73þ1Þ

ð1þ0:0404sÞð1þ0:0216sÞðs2=15:772þs=11:19þ1Þ

ð1Þ

aPt

Ktdct

¼ GtðsÞ

¼ð1�0:0214sÞðs2=45:912þs=24:29þ1Þ

ð1þ0:0404sÞð1þ0:0216sÞðs2=15:772þs=11:19þ1Þ

ð2Þ

where dcc and dc

t are the control commands obtainedfrom the guidance law and aPc and aPt are theacceleration components due to the canard and tailactions respectively. The gains Kc and Kt in theabove TFs are obtained from the steady stateconditions and are used as scale factors. The non-minimum phase (NMP) nature of the tail control isevident.

If the manoeuvring command is equally dividedbetween the canard and tail control fins (so thatconventional guidance laws can be used), the missiledynamics is represented by the TF

aP

Kdc ¼ GðsÞ

¼ð1�0:007sÞð1þ0:0096sÞð1þ0:0197sÞ

ð1þ0:0404sÞð1þ0:0216sÞðs2=15:772þs=11:19þ1Þ

ð3Þ

where here also K is obtained from the steady stateconditions.

For the development of the guidance laws, simpli-fied closed-loop dynamics are assumed. The closed-loop dynamics of the interceptor is representedby two first order biproper TFs

aPi

ui

¼ di þ1� di

1þ stP

i ¼ c; t ð4Þ

where aPc and aPt are the missile acceleration com-ponents due to the control actions of the canardand tail respectively; uc and ut are the commandedaccelerations of the canard and tail channels res-pectively. The constants dc and dt are the direct lift

158 T Shima and O M Golan

Proc. IMechE Vol. 219 Part G: J. Aerospace Engineering G00904 # IMechE 2005

parts of the controls; note that for the canard controldc . 0 and hence the TF is minimum phase and forthe tail control dt , 0 and hence the TF is NMP.The second term on the right-hand side of equation(4) represents the response of the missile airframeto the command. The acceleration of the interceptormissile is

aP ¼ aPc þ aPt ¼ dcuc þ dtut þ abP ð5Þ

where abP is the specific force acting on the missile

airframe, excluding the direct lift contributions ofthe control fins.

The maximum admissible acceleration of theinterceptor missile is denoted amax

P . In order toavoid saturation it is assume that

uij j4 biamaxP ; 0 4 bi 4 1; i ¼ c; t ð6Þ

such that

bc þ bt ¼ 1 ð7Þ

The target is represented by a first-order strictlyproper TF

aE

v¼

1

1þ stE

ð8Þ

where v and aE are the commanded and realizedaccelerations respectively. It is also assumed thatthe target acceleration command is bounded

vj j4 amaxE ð9Þ

where amaxE is the target maximum admissible

acceleration.

2.2 Guidance dynamics

The guidance problem is treated in this paper asplanar. It is assumed that, during the end game,deviations from the collision triangle are small,

justifying linearization. The engagement geometryis plotted in Fig. 1 where the X axis is along the initialline of sight and Y is perpendicular to it. Note that thesubscript P denotes the pursuer (the interceptor)while the subscript E denotes the evader (the target).

The state vector used to represent the relativemotion normal to the reference line (the initial lineof sight) is

XT ¼ y _y aE abP

� �ð10Þ

where y is the relative displacement between thetarget and the interceptor. The correspondingequations of motion are

_x1 ¼ x2

_x2 ¼ x3 � x4 � dcuc � dtut

_x3 ¼v � x3

tE

_x4 ¼ð1� dcÞuc þ ð1� dtÞut � x4

tP

ð11Þ

Figure 2 shows the classical guidance and controlarchitecture where the guidance law issues a singlemanoeuvring command. The flight control systemis responsible for the distribution of the commandsto the two actuators following a heuristic rule (e.g.equal command to the forward and aft controls).In Fig. 3 the new architecture is shown, where theguidance law is designed for the dual-control con-figuration and is therefore providing the optimalmix of controls for the particular scenario. Figures 2and 3 also show the simplified first-order TFs repre-senting the closed-loop dynamics of the interceptorand the target in the guidance law derivation.

3 GUIDANCE LAWS

For the classical guidance and control architectureof Fig. 2, many well-known guidance laws can beimplemented. We consider three such guidancelaws that were developed using different for-mulations: optimal control [11], linear quadraticdifferential game (LQDG) [12] and bounded controlsdifferential game [13]. For the guidance lawdynamics of Fig. 3, new guidance laws have recentlybeen derived [8, 9]. These guidance laws that dis-tribute commands to the two different channels(canard and tail), thus utilizing the additionaldegree of freedom provided by the dual-controlconfiguration, are discussed next. It is also shownthat these laws degenerate to the guidance laws ofreferences [11–13] for the classical single-controlinterceptor configuration.Fig. 1 Engagement geometry

End-game guidance laws for dual-control missiles 159


3.1 Scenario parameters

There are several parameters that characterize theinterception scenario. The relative performance ofthe pursuer and the evader are described by thetarget–interceptor time-constant ratio

1 ¼D tE

tP

ð12Þ

and the interceptor–target maximum-accelerationsratio

m ¼D amax

P

amaxE

ð13Þ

Another parameter is the interception time, whichdepends on the scenario’s initial conditions and therelative speed of the two players. For the linearizedconstant-speed model the interception time can beapproximated by

tf ¼r0

Vcl

ð14Þ

where r0 is the initial range and Vcl is the closingspeed given by

Vcl ¼ VP cosðfPÞ þ VE cosðfEÞ ð15Þ

The time to go is defined as

tgo ¼ tf � t ð16Þ

and its normalized version is

u ¼tgo

tP

ð17Þ

3.2 Optimal guidance law

The quadratic cost function selected for the intercep-tion problem is

J ¼g

2y2ðtfÞ þ

1

2

ðtf

t0

½bu2c ðtÞ þ cu2

t ðtÞ�dt ð18Þ

The first term is the cost for the final miss and theintegral term represents the control effort. Theweights g . 0;b . 0 and c . 0 are the control lawdesign parameters. In order to ensure a perfect inter-cept, i.e. y(tf) ¼ 0, it is required that g! 1. In thiscase the weights b and c can be used only to balancethe relative control loads of the canard and tail. Inthe case where the missile has only a forward control,let c! 1 and similarly, for tail-only control, b! 1.

Fig. 2 Simplified conventional guidance dynamics block diagram

Fig. 3 Simplified new guidance dynamics block diagram



The optimal guidance law (OGL) derived in refer-ence [8] is

u�i ¼N 0Oi

t2go

zOðtÞ; i ¼ c; t ð19Þ

where zO, the well-known zero effort miss distanceterm, is given by

zOðtÞ ¼ Vc_lt2

go � abPt

2Pf ð0; uÞ þ 0:5aEt2

go ð20Þ

NOc0 and N 0Ot are the effective navigation gains of

the forward canard and aft tail controls respectivelygiven by

N 0Oc ¼u2f ðdc; uÞ

bgðuÞð21Þ

N 0Ot ¼u2f ðdt; uÞ

cgðuÞð22Þ

where

f ðd; zÞ ¼ dzþ ð1� dÞðe�z þ z� 1Þ ð23Þ

gðuÞ ¼1

b

ðu

0

f 2ðdc; zÞ dzþ1

c

ðu

0

f 2ðdt; zÞ dz ð24Þ

ðf

0

f 2ðd; zÞ dz ¼ �2ð1� dÞ½ð1þ fÞ e�f þ 0:5f2 � 1�

þ ð1� dÞ2½2 e�f � 0:5 e�2f þ f� 1:5�

þf3

3ð25Þ

Note that f (di, u) . 0 8di [ (0, 1), u . 0. If di [(�1, 0), then f (di, u) , 0 8 u , ut and f (di, u) . 0 8 u .

ut where

ut ¼ argu.0½ f ðdt; uÞ ¼ 0� ð26Þ

The critical value ut, which causes the navigationgain of the tail to change its sign near interceptiondue to its NMP nature, is plotted in Fig. 4. Notethat as the magnitude of the direct lift part ofthe tail control increases, so does the timing of thetail’s control direction change.

Remark 1

This guidance law degenerates to that of reference[11] by letting b! 1, dt ¼ 0, or c! 1, dc ¼ 0.

3.3 Linear quadratic differential game

The quadratic cost function of the game is chosen as

J ¼g

2y2ðtfÞ þ

1

2

ðtf

t0

½bu2c ðtÞ þ cu2

t ðtÞ � ev2ðtÞ�dt ð27Þ

Here, compared with equation (18), an additionalterm representing the target control effort isincluded, with a negative sign indicating the conflictof interest between the pursuer (the interceptor)and the evader (the target). The weights g, b, c ande are all positive and the controls uc, ut and v areL2. For given interceptor weight parameters band c, the weight parameter e is related to theinterceptor–target manoeuvring ratio m. As in theoptimal control formulation, b! 1 representsa tail-controlled missile and c! 1 represents aforward-controlled missile. Letting g! 1 yields aperfect interception game. If a conjugate pointappears in the solution, a finite weight g should beused, resulting in a larger-than-zero finite miss.

The LQDG guidance law derived in reference [8] is

u�i ¼N 0Li

t2go

zLðtÞ; i ¼ c; t ð28Þ

where

zLðtÞ ¼ Vc_lt2

go þ aEt2Ef 0;

u

1

� �� ab

Pt2Pf ð0; uÞ ð29Þ

N 0Lc ¼u2f ðdc; uÞ

b½1=ðat3PÞ þ gðuÞ � hðu; 1Þ�

ð30Þ

N 0Lt ¼u2f ðdt; uÞ

c½1=ðat3PÞ þ gðuÞ � hðu; 1Þ�

ð31Þ

Fig. 4 The influence of the tail’s direct lift on ut



and

hðu; 1Þ ¼1

e13

ðu=1

0

f 2ð0; zÞ dz . 0 8 u . 0 ð32Þ

Remark 2

This guidance law degenerates to that of reference[12] by letting b! 1, dt ¼ 0, or c! 1, dc ¼ 0.

3.4 Bounded control differential game law

The cost function of this game is chosen as

J ¼ yðtfÞ�� ð33Þ

Here, there is no term for the control effort as in theOGL and LQDG cases, since the control bounds arerepresented explicitly in the players’ models. Thiscost function is to be minimized by the interceptorand maximized by the target subject to equations(6) and (9).

The differential game law (DGL) derived in refer-ence [9] is

u�c ¼ bcamaxP sgn zLðtÞ½ � ð34Þ

u�t ¼ btamaxP sgn zLðtÞ½ � sgn f ðdt; uÞ

� �ð35Þ

Note that the zero effort miss zL(t) is identical withthat of the LQDG formulation given in equation(29). Assuming a scenario where interception canbe obtained 8u . 0 the following strategy, which islinear in part of the game space, can be used

u�c ¼ bcamaxP sat

zLðtÞ

z�LðtÞ

� ð36Þ

u�t ¼ btamaxP sgn f ðdt; uÞ

� �sat

zLðtÞ

z�LðtÞ

� ð37Þ

where sat(.) is the standard saturation function and

z�LðtÞ ¼ amaxE t2

P

mbc½xðdc; uÞ � f ðdc; uÞ� � 12

x 0;u

1

� �� f 0;

u

1

� �� þ G

�ð38Þ

G ¼�mbt½xðdt; uÞ � f ðdt; uÞ� u , ut

mbt½xðdt; uÞ � f ðdt; uÞ � 2xðdt; utÞ� u � ut

ð39Þ

xðdt; zÞ ¼ 0:5z2 þ dtz ð40Þ

Remark 3

This guidance law degenerates to that of reference[13] by letting bc ¼ 0, dt ¼ 0, or bt ¼ 0, dc ¼ 0.

3.5 Proportional navigation

The classical PN guidance law is

u ¼ N0Vc_l ð41Þ

where N0 is the well-known navigation constantchosen here with the value equal to three. Theadditional degree of freedom in the controls can betreated by applying an arbitrary rule that the controlis evenly divided between the canard and the tailcontrol channels. Hence

ui ¼3

2Vc

_l; i ¼ c; t ð42Þ

Remark 4

If a conventional guidance law (such as PN) is used,generating only one commanded input to the flightcontrol system (missile normal acceleration), thenthe additional degree of freedom provided by thedual-control configuration can be used to addressanother control task.

4 PERFORMANCE ANALYSIS

4.1 Tools

A general block diagram representing the guidancedynamics chosen for the guidance law evaluationis given in Fig. 5. Note that, unlike the simplifiedguidance dynamics of Figs 2 and 3 used for the gui-dance law development, here high-order interceptordynamics and an estimator are used.

It is assumed that the interceptor acquires themeasurement

Y ¼ lþ v �y

rþ v ð43Þ

at a given rate w, where r is the range (see Fig. 1) andv is the measurement error, modelled as a whiteGaussian noise

v � N ð0; s 2l Þ ð44Þ

For the estimator design, the well-known expo-nentially correlated acceleration shaping filter,associated with the Singer model [14], has beenused. The state vector of the filter is

W ¼ ½y; _y; aE�T

ð45Þ

The equation of the filter model is

_W ¼ AW W þ BW aP þ CWn ð46Þ



where

AW ¼

0 1 0

0 0 1

0 0 �1=ta

2

664

3

775; BW ¼

0

�1

0

2

664

3

775;

CW ¼

0

0

1

2

664

3

775 ð47Þ

and

n � N ð0; QÞ ð48Þ

The parameter ta is the assumed average time bet-ween the changes in the piecewise constant targetacceleration levels. This parameter and the varianceQ are design parameters.

Remark 5

A discrete time version of the above estimator wasimplemented with time steps of Dts.

For the homing performance assessment it isassumed that the target performs the random evasivemanoeuvre

v ¼þamax

E ; t , ðtgoÞsw

�amaxE ; t 5 ðtgoÞsw

( )

;

ðtgoÞsw � Uð0; tfÞ

ð49Þ

A possible objective of the interceptor designer isto guarantee the interception of the target with apredetermined probability of success, using thesmallest possible warhead lethal radius Rk. Hence,the selected performance criterion in this study isthe single-shot kill probability (SSKP) [15] defined by

SSKP ¼ E PdðRkÞ�

ð50Þ

where E is the expectation taken over the entire setof noise samples against any given feasible targetmanoeuvre. The simplified lethality function is

Pd ¼1; yðtfÞ4 Rk

0; yðtfÞ . Rk

ð51Þ

The nominal parameters of the interceptor modelfor the guidance law evaluation may be used asdesign parameters. Their initial values are selectedon the basis of a higher order, more precise, modeland the expected scenario. During the design processthese values must be tuned, using the simulation asa design tool. For this comparative analysis of thevarious guidance laws, the fixed set of parameterspresented in Table 1 is selected.

The only design parameter whose variations effectson the homing performance are examined in thisstudy is the interceptor time constant tP. As will beshown in the following, the sensitivity of the perform-ance to this parameter cannot be neglected, and thereexists an optimal value of tP that depends on the typeof guidance law and on the scenario parameters.

Note that tP used in equation (4) can be approxi-mated on the basis of the bandwidth of the TFs.This parameter, which is almost the same forboth TFs, will be denoted t

eqP and used to define the

equivalent dynamics ratio

1eq ¼tE

teqP

ð52Þ

The weights b, c and e are also design parameters inthe LQDG case. Their values are selected as b ¼ c ¼ 1and e ¼ 10 such that no conjugate point appears inthe solution. These values are not unique and the

Fig. 5 Simulated guidance dynamics block diagram

Table 1 Design parameters

Interceptor Estimator

dc ¼ 0:2 ta ¼ 1:5 sdt ¼ �0:2 q ¼ (amax

E )2=4



parameters can be tuned for a given scenario toachieve superior performance.

The simulations parameters are given in Table 2.For each test point, representing a combination ofthe parameters m, 1eq and tP, a set of 100 Monte Carlosimulation runs, with random target manoeuvreswitching times and different noise samples wereperformed.

4.2 Sample run results

In Figs 6 and 7 the evolution of the zero effort miss zL

and the interceptor’s acceleration, respectively,are plotted. The guidance law used is the dual-control DGL. Note that, when the value of the esti-mated zL reaches the border trajectory �z�L (at tgo ¼

0:5 and 0.3 s), the interceptor’s control commandssaturate, following the guidance law in equations(36) and (37).

From Fig. 7 the change in the sign of the tail con-trol near the end of the scenario at tgo ¼ uttP �

0:034 s is evident. This behaviour is common to allthe guidance laws tailored for the dual-controlconfiguration (DGL, LQDG and OGL) and is due tothe NMP nature of the tail control.

4.3 Tuning parameter

In Fig. 8 the required kill radius Rk for an SSKP of 0.95 isplotted for the various guidance laws as a function of

the design parameter tP. It can be observed that thedifferential game laws are quite sensitive to thisparameter. PN, which is analysed here for reference,is, as expected, not dependent on this parameter.Note also that, for the given scenario parameters,DGL provides superior homing performance.

Figures 9 and 10 depict t�P, the value of tP providingthe optimal homing performance. Figure 9 showsthat, for both OGL and LQDG, t�P is sensitive to themanoeuvre ratio m. In contrast, as shown in bothfigures, t�P for DGL is insensitive to either themanoeuvre ratio m or the time-constant ratio 1eq.Note that, for all the advanced guidance laws con-sidered, t�P is larger than the value of the equivalenttime-constant t

eqP . Adopting this value in the realiz-

ation increases the guidance law gains in order tocompensate for the assumed interceptor slower

Table 2 Simulation parameters

Interceptor Estimator Scenario

ac ¼ at ¼ 0:5 w ¼ 100 Hz tf ¼ 3 s

amaxP ¼ 30 g sl ¼ 0:2 mrad Dt ¼ 10�3 s

teqP ¼ 0:075 s

Fig. 6 Time history of the zero effort miss of DGL

½m ¼ 2:5(amaxP ¼ 30 g); 1eq ¼ 2:25(tP ¼ 0:15 s)�

Fig. 7 Time history of the interceptor’s acceleration

and commands for DGL ½m ¼ 2:5(amaxP ¼ 30 g);

1eq ¼ 2:25(tP ¼ 0:15 s)�

Fig. 8 The effect of the design parameter tP on the

homing performance (m ¼ 2:5; 1eq ¼ 1:5)



response. In the presence of saturations andmeasurement noise, a design trade-off leads to theoptimal value.

In Fig. 11 the normalized control effort depen-dence on the design parameter tP is shown forDGL, LQDG and OGL. The control efforts havebeen computed for the entire duration of the endgame and normalized by the control effort of PN,which is independent of tP. The graph shows thatOGL requires the smallest effort while DGL requiresthe highest. Note that for DGL an optimal tP exists,which minimizes the control effort. Its value isapproximately 0.25 s, the same as the optimal tP ofthe previous figures.

4.4 Homing accuracy

In Figs 12 and 13 the homing performance of the var-ious guidance laws are examined. Figure 12 shows

contours in the ð1eq;mÞ parameter plane for a requiredRk ¼ 10 m with an SSKP of 0.95. These contourswere drawn for t�P. It can be observed that the orderof preference of the guidance laws (from best toworst) is DGL, LQDG, OGL and PN. This result con-firms the superiority of the differential game lawsin general and that of DGL in particular. Note alsothat for all guidance laws the performance is sensi-tive to changes in m but much less to changes in 1eq.

Figure 13 presents a comparison of the perform-ance of the new guidance-control architectureversus the classical architecture, via contours of Rk ¼

5 and 10 m for DGL. The superiority of the newguidance-control architecture is established. Notethat, as the scenario is more difficult from the view-point of the interceptor, i.e. m1eq is smaller, thenthe advantage of using the new guidance-controlarchitecture is greater.

Fig. 9 The optimal tuning parameter t�P for the

different guidance laws (1eq ¼ 1) Fig. 11 The effect of the tuning parameter tP on the

normalized control effort (m ¼ 2:5 and 1eq ¼ 1)

Fig. 10 The insensitivity of the optimal tuning

parameter t�P of DGL Fig. 12 Contours of Rk ¼ 10 m



As expected, the plots show that smaller missrequirements demand higher interceptor/targetperformance capabilities. Also it can be observedthat, as the warhead lethal radius decreases, thesensitivity to changes in 1eq increases.

Remark 6

As mentioned in section 3 the performance of theLQDG law can be improved by carefully tailoringthe weight parameters to the particular scenario.As a result, the performance can come closer tothat of DGL. However, it is not likely that the LQDGlaw will generate superior results to DGL since, inthe latter case, the interceptor bounds are takeninto account explicitly.

Remark 7

A different application of DGL has also been exam-ined. In that application a bang–bang admissiblestrategy [equations (34) and (35)] has been imple-mented in the entire game space instead of thelinear strategy [equations (36) and (37)] imple-mented in some part of the game space. Such anapplication provided inferior homing performanceeven compared with OGL. Hence it is asserted thatin an actual implementation the linear variant ofDGL should be used.

Remark 8

As is customary in missile guidance design, the gui-dance laws presented in this paper have been derivedby implicitly assuming the certainty equivalenceproperty. This property states that the optimalcontrol law for a stochastic control problem is the

optimal control law for the associated deterministic(certainty equivalent) problem [16]. As shown in arecent paper [17], the performance of such guidancelaws can be improved by compensating for the delayin estimating the target acceleration.

Remark 9

The sliding mode control (SMC) methodology canalso be used to address errors in estimating thetarget acceleration. Such a missile guidance law, inthe class of PN, was proposed in reference [18]where the target manoeuvres were considered asbounded uncertainties. Using numerical simu-lations, the superiority of the proposed guidancelaw over the conventional PN was advocated. Ina recent paper [19], SMC has been used to derive aguidance law for a canard-controlled missile usingthe zero effort miss of equation (29) as the slidingsurface. However, a difficulty arises when applyingSMC to NMP systems due to the implied plantinversion of the method. Different ways weresuggested to treat NMP plants, e.g. in reference [20]by introducing dynamic sliding manifolds. Thestudy of SMC guidance law for the dual-controlmissile is deferred to future work.

5 CONCLUSIONS

The advantage of the proposed dual-control gui-dance scheme over the classical guidance scheme,in which the distribution of the commands isdetermined by the flight control system, was estab-lished. The performance of advanced guidancelaws tailored for the dual-control configuration wasanalysed in a noise-corrupted scenario where theinterceptor has linear high-order manoeuvringdynamics. Among the guidance laws examined inthis paper, DGL – implemented with a linear strategyin the singular part of the game space – provides thebest homing performance. However, OGL requiresthe smallest control effort. Since DGL provides amuch-improved homing performance comparedwith OGL and requires a marginal additionalcontrol effort for the relevant parameters, it is therecommended guidance law for the investigatedscenario.

It was shown that the performance of the advan-ced guidance laws is sensitive to the value of theinterceptor time constant used as a design par-ameter. It is possible and recommended to findan optimal value for this time constant which, forthe investigated parameters, was greater than thenominal value. For DGL this optimal value is virtu-ally insensitive to variations in the parametersinvestigated.

Fig. 13 Contours of Rk ¼ 5 and 10 m for DGL, using

the classical and new guidance-control

architecture



REFERENCES

1 Gutman, S. The superiority of canards in homing mis-siles. IEEE Transactions on Aerospace and ElectronicSystems, 2003, 39(3), 740–746.

2 Thukral, A. and Innocenti, M. A sliding mode missilepitch autopilot synthesis for high angle of attackmaneuvering. IEEE Trans. Control Systems Technol.,1998, 6(3), 359–371.

3 Bhat, M. S., Bai, D. S., Powly, A. A., Swami, K. N., andGhose, D. Variable structure controller design withapplication to missile tracking. J. Guidance, Control,Dynamics, 2001, 24(4), 859–862.

4 Menon, P. K. and Iragavarapu, V. R. Adaptive tech-niques for multiple actuator blending. In Proceedingsof the AIAA Guidance, Navigation, and Control Confer-ence, Conference Publication CP-1998-4494, Boston,Massachusetts, 1998.

5 McFarland, M. B. and Calise, A. J. Adaptive nonlinearcontrol of agile anti-air missiles using neural networks.IEEE Trans. Control Systems Technol., 2000, 8(5), 749–756.

6 Hirokawa, R., Sato, K., and Manabe, S. Autopilotdesign for a missile with reaction-jet using coefficientdiagram method. In Proceedings of the AIAA Guidance,Navigation, and Control Conference, ConferencePublication CP-2001-4162, Montreal, Canada, 2001.

7 Manabe, S. Application of coefficient diagram methodto dual-control-surface missile. In Proceedings ofthe 15th IFAC Symposium on Automatic Control inAerospace, Bologna, Italy, 2001, pp. 499–504.

8 Shima, T. and Golan, M. O. Linear quadratic guidancelaws for a dual controlled missile. In Proceedings of theAIAA Guidance Navigation and Control Conference,Conference Publication CP-2002-4841, Monterey,California, 2002.

9 Shima, T. and Golan, M. O. Bounded differential gameguidance law for a dual controlled missile. In Proceed-ings of the American Control Conference, Denver,Colorado, June 2003, pp. 390–395.

10 Friedland, B. Control System Design: An Introduction toState Space Methods, 1996, pp. 152–155 (McGraw-Hill,New York).

11 Cottrell, R. G. Optimal intercept guidance for short-range tactical missiles. Am. Inst. Aeronaut. Astronaut.J., 1971, 9(7), 1414–1415.

12 Ben-Asher, J. and Yaesh, I. Advances in MissileGuidance Theory, Progress in Astronautics and Aero-nautics, Vol. 180, 1998, pp. 89–126 (American Instituteof Aeronautics and Astronautics Washington, DC).

13 Shinar, J. Solution techniques for realistic pursuit—eva-sion games, Advances in Control and Dynamic Systems,Vol. 17, 1981, pp. 63–124 (Academic Press, New York).

14 Singer, R. A. Estimating optimal tracking filterperformance for manned maneuvering targets.IEEE Trans. Aerospace Electronic Systems, 1970, 6,473–483.

15 Forte, I. and Shinar, J. Improved guidance law designbased on mixed strategy concept. J. Guidance, ControlDynamics, 1989, 12(5), 739.

16 Witsenhausen, H. S. Separation of estimation andcontrol for discrete time systems. Proc. IEEE, November1971, 59(11), 1557–1566.

17 Shinar, J. and Shima, T. Non-orthodox guidance lawdevelopment approach for intercepting maneuveringtargets. J. Guidance, Control, Dynamics, 2002, 25(4),658–666.

18 Moon, J. and Kim, Y.. Design of missile guidance lawvia variable structure control. In Proceedings of theAIAA Guidance Navigation and Control Conference,Denver, Colorado, 2000, paper 4068.

19 Shima, T., Idan, M., and Golan O. M. Sliding mode con-trol for integrated missile autopilot-guidance. Presentedat the AIAA Guidance Navigation and Control Confer-ence, Providence, Rhode Island, August 2004.

20 Shkolnikov, I. A. and Shtessel, Y. B. Aircraft nonmini-mum phase control in dynamic sliding manifolds.J. Guidance Control Dynamics, 2001, 24(3), 566–572.

APPENDIX 1

Notation

a normal accelerationA dynamics matrixb weight on canard controlB control matrixc weight on tail controlC observation vectorCG centre of gravityDGL differential game lawe weight on target controlJ cost functionkq damping gainK steady state scale factorsLQDG linear quadratic differential gameM linear moment coefficientsMP minimum phaseN effective navigation gainNMP non-minimum phaseOGL optimal guidance lawPd lethality functionq pitch rateQ process noise covariance matrixr rangeRk warhead lethal radiusSSKP single-shot kill probabilityt timetgo time to goTF transfer functionu missile acceleration commandv target acceleration commandV speedW filter state vectory relative perpendicular displacementY measurementz zero effort missZ linear force coefficients

a angle of attackb control channel effectiveness



g weight on miss distanced fin angle deflectionDt time step1 target–interceptor time-constant ratiou normalized time-to-gol line of sightm interceptor–target maximum

accelerations ratiosl measurement standard deviationt time constanty process noisew measurement ratev measurement noise

Superscripts

b bodycom control command to fineq equivalentmax maximum� optimal

Subscripts

B bodyc canardcl closingd direct lifteq equivalentE evader/targetf finalL LQDGN normalO OGLP pursuer/missilesw switcht tailW filtera angle of attackdc canard angledt tail angle

APPENDIX 2

The generic interceptor model used in this study isbased on Friedland’s tail control missile exampleintroduced in reference [10]. The model assumesplanar and linear dynamics, with no thrust or dragforce and constant speed. Furthermore, it relies ona small-angles assumption. These assumptions arereasonable during the end-game period, the time ofinterest in this study, which is typically small forhigh-performance interceptors.

The governing equations of the missile configur-ation are modified in order to incorporate additionalcanard controls (Fig. 14)

_a ¼Za

Vaþ qþ

Zdc

Vdc þ

Zdt

Vdt ð53Þ

_q ¼ MaaþMqqþMdcdc þMdt

dt ð54Þ

The states a and q are the angle of attack and thepitch rate respectively. Za, Zdc

and Zdtare the linear

force coefficients of the angle of attack and thecanard and tail fin angles dc and dt. Similarly Ma,Mdc

and Mdtare the linear moment coefficients

about the centre of gravity (CG). Mq is the aero-dynamic damping coefficient and V is the missilespeed. In this representation the forces andmoments are normalized by the missile massand inertia respectively. The signs follow the right-hand convention, therefore, in this model, positiveangles generate negative force (in the 2Z direction).

The missile normal acceleration is obtained by

aN ¼ Zaaþ Zdcdc þ Zdt

dt ð55Þ

The missile manoeuvring acceleration is the accel-eration normal to the velocity vector. By the smallangles assumption

aP � aN ð56Þ

The canard and tail fin angles are controlled byactuators, the dynamics of which are representedby a first-order lag

_dc ¼dcom

c � dc

tc

ð57Þ

_dt ¼dcom

t � dt

tt

ð58Þ

where dcomc and dcom

t are the inputs to the actuators.

Fig. 14 Dual-control missile configuration



This model can be summarized in the vectorform

_x ¼ Axþ BU; y ¼ Cx

x ¼ ½a; q; dc; dt�T; U ¼ ½dcom

c ; dcomt �

Tð59Þ

where

A ¼

Za

V1

Zdc

V

Zdt

V

Ma 0 MdcMdt

0 0 �1

tc

0

0 0 0 �1

tt

2

6666666664

3

7777777775

;

B ¼

0 0

0 0

1tc

0

01

tt

2

66666664

3

77777775

; C ¼

Za

0

Zdc

Zdt

2

6666664

3

7777775

T

ð60Þ

and the output of the system is the interceptormanoeuvring acceleration aP.

The original values of Friedland’s [10] configur-ation are

V ¼ 1253 ft/s; Ma ¼ �248 rad/s2

Za ¼ �4170 ft/s2; Md ¼ �662 rad/s2

Zd ¼ �1115 ft/s2; Mq � 0 (61)

Neglecting aerodynamic coupling between thecontrol fins and the missile fuselage (the ‘body’),the body force and moment coefficients areobtained by

ZBa ¼ Za � Zd ¼ �3055 ft/s2ð62Þ

MBa ¼ Ma �Md ¼ 414 rad/s2ð63Þ

Hence, the missile fuselage without the tail fins isunstable.

In modifying the above missile to a dual-controlconfiguration the following requirements will beintroduced:

(R1) stable missile body (missile without controlfins);

(R2) equal force and moment contribution of thecanard and tail fins;

(R3) same overall manoeuvring capability as in theoriginal model (same Za).

Requirement (R1) can be achieved by splitting theoriginal tail surface into fixed and movable parts,such that the fixed part is stabilizing the airframe.Requirement (R2) dictates that the canard and tailcontrol fins have the same aerodynamic forcecoefficient and their centres of pressure be locatedat equal distance from both sides of the CG. FromRequirements (R3) the force, and hence themoment coefficients of the new control surfaces,can be determined.

Following these rules, the dual-control configur-ation will possess the following parameters

V ¼ 1253 ft/s; Ma ¼ �100 rad/s2

Za ¼ �4170 ft/s2; Mdc¼ 80 rad/s2

Zdc¼ �130 ft/s2; Mdt

¼ �80 rad/s2

Zdt¼ �130 ft/s2; Mq � 0 (64)

Introducing these values into the dynamic modelgiven in equations (53) to (58) gives a stable air-frame and hence A is Hurwitz; also, an undampeddynamic behaviour is obtained, due to the twoundamped poles of the aerodynamic model. Thisundesired behaviour can be modified by feedingback a correction command proportional to themeasured pitch rate. If the command is equallysplit between the canard and tail, the fins willdeflect in opposite direction to generate therestraining moment. In the current model the netforce due to this correction will be zero in thesteady state. Hence the control command is splitinto two parts

dcomc ¼ dc

c þ kqq ð65Þ

dcomt ¼ dc

t � kqq ð66Þ

where dcc and dc

t are the control commandsobtained from the guidance law, in the canardand tail channels respectively. Selecting kq ¼ 0:08will move the aerodynamic poles to the desiredlocation with damping coefficient of 0.7 and natu-ral frequency of 16 rad/s.

The new system model is obtained by introducingequations (65) and (66) into equation (59). The new



system matrix is

Ad ¼

Za

V1

Zdc

V

Zdt

VMa 0 Mdc

Mdt

0 �kq

tc

�1

tc

0

0kq

tt

0 �1

tt

2

666666664

3

777777775

ð67Þ

The missile’s TFs are obtained by

aP ¼ C sI� Adð Þ�1B

dcc

dct

" #

ð68Þ

In the steady state, the missile gains are

aP ¼ CA�1d B

dcc

dct

" #

¼ Kcdcc þ Ktd

ct ð69Þ

where Kc ¼ �2430 and Kt ¼ 2248 are scaling factors.Introducing the parameters of the configurationyields the TFs of equations (1) and (2) used in thisstudy.



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End-game guidance laws for dual-control missiles