On Alternative ormulation for Linearied Mi Ditance Analyi Domenic Bucco and Rick Gorecki Weapon Sytem Diviion Defence Science and Technology Organiation DSTO-TR-845 ABSTRACT In thi report, technique generally employed in the analyi of intercept guidance problem are reviewed. rom the governing non-linear equation decribing uch problem, two baic linear model are derived. Traditionally, thee linear model are utilied a a bai for preliminary intercept engagement tudie. Under certain input condition, the two model are mathematically equivalent and, hence, have been ued interchangeably by weapon analyt to yield appropriate deign and performance data in upport of their program. However, for a pecific et of initial condition, which include a very important cla of practical problem that may be aeed with the ue of thee model, it i noted herein that one of thee linear model produce incorrect performance data when compared to a non-linear imulation of the engagement. In contrat, the other model produce conitent reult with thoe generated by the non-linear imulation regardle of the initial condition conidered. To remedy thi dicrepancy, the neceary mathematic are derived to bring the two formulation into alignment for any form of the initial condition and input to the ytem. Conequently, thi lead to a conitency in the correponding adjoint model which are contructed from thee linear model, thu enuring the generation of correct output data regardle of which model i employed by the analyt. RELEASE LIMITATION Approved for public releae
Publihed by Weapon Sytem Diviion DSTO Defence Science and Technology Organiation PO Box 500 Edinburgh South Autralia 5 Autralia Telephone: 300 DEENCE ax: (08) 7389 6567 Commonwealth of Autralia 03 AR-05-6 May 03 APPROVED OR PUBLIC RELEASE
On Alternative ormulation for Linearied Mi Ditance Analyi Executive Summary In thi report, the miile-target engagement problem i analyed. A part of the analyi, the non-linear governing equation are reviewed for motion in a ingle plane. Thee nonlinear equation are employed for two purpoe. irtly, the equation form the bai of a non-linear imulation program developed in Simulink. Thi Simulink model i controlled and executed via a graphical uer interface pecifically developed a an aid for the weapon analyt to tudy the engagement problem. Secondly, the non-linear equation are utilied a a bai for lineariation and, hence, the derivation of an approximate linear model of the engagement. Two different formulation of the linear model are derived. Thee are deignated a Model A and Model B in the report. Although both model are generally ued interchangeably in the literature for guided miile homing loop analyi, it i demontrated herein that, under certain input condition, care need to be exercied when uing one of thee model, Model A, for performance analyi. In order to enure that both model yield the ame performance data for all input condition conidered, a correction factor i derived. Thi correction factor need to be included in the form of an added initial condition on one of the tate in the tate pace repreentation of Model A. Simulation reult how that the two model are in agreement when thi correction factor i applied. Knowledge of thi fact i important to enure that analyt generate correct performance data when uing linear technique uch a the adjoint method. The adjoint model i contructed from a knowledge of the forward linear model, that i, Model A or Model B, and i traditionally employed by analyt a part of the olution proce. To gain further inight into the nature of the miile-target engagement problem and the parameter that influence performance, alo included in thi report i an analytical treatment of the problem. It i well known that, when the miile guidance dynamic i repreented by a firt order lag (ingle time contant ytem), then the linear differential equation decribing the intercept problem are readily amenable to analytical treatment. Conequently, an analytical invetigation of each model (A and B) i carried out for the range of input condition conidered herein. The reulting cloed form olution from each model are compared and are hown to be mathematically equivalent for all input condition conidered provided that Model A ha the propoed correction factor implemented. or imulation purpoe, the correponding adjoint model of Model A and Model B are contructed and then implemented in Simulink. A with the analytical reult, output from the two model are hown to be in agreement. inally, the model are extended to repreent more realitic guidance ytem dynamic (fifth order ytem) and imulation reult are generated and compared.
inally, a pecial relationhip linking two of the derived mi ditance formula i noted and explored further. Thi relationhip highlight a connection between the mi ditance due to an initial target diplacement and that due to an initial heading error in the context of the linear analyi. ollowing verification uing linear imulation, a formula baed on thi relationhip i derived and propoed a a mean for predicting performance data of more complex linear ytem. Thi formula may alo be ued in connection with the nonlinear imulation model for generating approximate mi ditance profile due to the effect of a tep in target poition prior to intercept. A tep in target poition typically arie in problem aociated with the eeker reolution of the target in a multi-target cenario. It may alo arie in the cae of a ingle target cenario. or thi cae, the miile may be on a colliion coure with the predicted intercept point (PIP) of the target. However, at the time of eeker turn-on, the PIP may not necearily coincide with the actual target poition.
DSTO-TR-845 Author Domenic Bucco Weapon Sytem Diviion Domenic Bucco received hi PhD in Applied Mathematic from the Univerity of Adelaide, South Autralia. Since 980, he ha been working at the Defence Science and Technology Organization (DSTO) in Autralia a a reearch cientit in the Weapon Sytem Diviion. He ha worked in variou area of the diviion including guidance and control, modelling and imulation, hardware in the loop and weapon robotic. During 987/988, he wa poted to the Naval Air Warfare Centre, China Lake, California, a an exchange cientit for 8 month. While there, he worked on autopilot deign and hardware-in-the-loop imulation. During 008/009, he wa awarded a Defence ellowhip to conduct reearch on the aeropace application of adjoint theory. He i currently working a a principal reearch cientit upporting Guidance and Control reearch in the Diviion. Rick Gorecki DSTO Mr Richard Gorecki wa awarded a B Tech in Electronic Engineering at Adelaide Univerity and a Graduate Diploma in Mathematic at the Univerity of South Autralia. During hi time at the Defence Science and Technology Organiation (DSTO) in Autralia he worked motly on miile modelling and imulation including ix degree of freedom (6DO) model of mot of the miile in the AD inventory. He alo pent ome time in hip Radar ignature meaurement and modelling and Optical and IR enor modelling and analyi of C3I ytem. In later year he developed a model of a hardware in the loop (HWIL) facility. Since hi retirement from DSTO he ha worked for Daintree Sytem on contract involving towed target attrition, HWIL imulation of two target cenario and 6DO miile imulation model.
DSTO-TR-845
DSTO-TR-845 Content. INTRODUCTION.... NON-LINEAR ENGAGEMENT EQUATIONS.... Planar Engagement Model.... Simulation Reult... 6.. Step in target manoeuvre... 6.. Heading error... 6..3 Step in target poition... 9 3. LINEARISED ENGAGEMENT EQUATIONS... 4. COMPARISON O THE LINEARISED MODELS... 4 4.. Step in target manoeuvre... 5 4.. Heading error... 7 4..3 Step in target poition... 7 5. DERIVATION O A CORRECTION OR MODEL A... 0 6. ANALYTICAL ORMULAS... 6. Cloed-orm Solution for Model A... 3 6.. Adjoint of Model A in Simulink... 7 6.. Comparion with non-linear reult... 9 6. Cloed-orm Solution for Model B... 3 6.. Cloed orm Solution for N =3... 36 7. HIGHER ORDER GUIDANCE SYSTEM DYNAMICS... 39 8. CONCLUSIONS... 44 9. REERENCES... 45 APPENDIX A: APPENDIX B: NON-LINEAR ENGAGEMENT MODEL IN MATLAB/SIMULINK... 46 A.. Detail of the Simulink Model... 46 A.. The Simulink Subytem Model... 46 A.. The Aociated MATLAB Code... 49 CLOSED ORM SOLUTIONS OR SINGLE TIME CONSTANT GUIDANCE SYSTEM... 59 B.. Solution for N = 4... 59
DSTO-TR-845 B.. Solution for N = 5... 59 B.3. Solution for N = 6... 60 APPENDIX C: ALTERNATIVE APPROACH TO PERORMANCE PREDICTION O MISS DISTANCE DUE TO A STEP IN TARGET DISPLACEMENT... 6 C.. Linear Sytem Invetigation... 6 C.. Non-linear Sytem Invetigation... 64
DSTO-TR-845. Introduction The equation decribing the dynamic and geometrical interaction between an interceptor and it target are generally non-linear and complex and are uually beyond the cope of analytical invetigation. Therefore, they are olved approximately uing computer imulation. Simulation provide the miile analyt with a baic capability for aeing the performance and behaviour of the interceptor under varying condition of the engagement and can provide anwer to quetion uch a, What i the overall effect on mi ditance due to a udden hift in target direction? However, to gain critical inight into the nature of uch interaction and to identify key parameter that may affect the performance and behaviour of the interceptor, engineer often make ue of implifying aumption in order to linearie the governing equation. Once in linear form, the engagement equation are more eaily tackled uing etablihed linear technique. Two uch technique are the adjoint imulation method and the covariance analyi method. In thi report, we firt review the engagement model baed on two dimenional (D) non-linear dynamic. ollowing thi, we make ue of lineariation to reduce the equation to a form amenable to linear analyi. Uing tandard block diagram algebra, thee linear equation are then ued to derive two baic model, referred to a Model A and Model B, which may be ued a the bai for the linear analyi of planar miile/target engagement. In the literature, thee model are often ued interchangeably for preliminary analyi of the performance of guided miile ytem [-6]. However, a i demontrated herein, caution mut be ued when adopting one of thee model under certain initial condition of the tate variable. It i hown that the mathematical equivalence of the two block diagram topologie repreenting the model i dependent on the nature of the initial condition impoed on the tate of the ytem. or example, if the initial condition i baed on an initial heading error in the miile, then both model yield the ame mi ditance reult. However, if the initial condition tem from a tep in target diplacement (thi condition i ued to tudy miile performance againt multiple target), then the two block diagram topologie, a currently employed in the open literature, will lead to mi ditance reult that differ ignificantly. Conequently, applying the adjoint method to thee two ytem will alo give different reult. To correct thi inconitency, the neceary mathematic i derived to bring the two formulation into alignment for any of the initial tate condition. Thi then lead to conitency in the aociated adjoint model regardle of which linear formulation i purued by the analyt. In addition, cloed form olution of the mi ditance variation in term of flight time have been derived for the cae when the miile guidance loop may be approximated dynamically by a firt order lag term. inally, imulation reult and deign curve are generated for more realitic guidance ytem.
DSTO-TR-845. Non-linear Engagement Equation. Planar Engagement Model Conider a miile-target engagement under the following aumption; the engagement i confined to the X-Y plane, the force due to gravity i ignored, and the miile velocity and the target velocity remain contant. The geometry of the engagement i depicted in igure. The non-linear differential equation decribing the motion of the target are [], () X T V T co( ) () Y V in( ) T T n V T (3) where ( X, Y T T ) define the target poition, VT i the contant target peed, nt i the target acceleration while refer to the target flight path angle a hown in igure. The dot over the variable denote differentiation with time. The initial condition are given by ( ) X, ( ) Y and ( 0). X T 0 T 0 YT 0 T 0 0 Y T V T n T T n C V M R M L HE X igure. Engagement Geometry
DSTO-TR-845 Similarly, the non-linear differential equation decribing the motion of the miile are, (4) X M V MX V Y V (5) M MY (6) V MX n c MY n c in() (7) co() Here, ( X M, Y ) define the intantaneou poition of the miile while ( V, V are the M MX MY ) component of it velocity vector. urthermore, n c denote the miile commanded acceleration and i the line of ight angle a hown in igure. The initial condition are given by X ( 0),, M X M 0 YM ( 0) YM 0 VMX ( 0) VM co( L HE ) and VMY ( 0) VM in( L HE ) where VM denote the contant miile peed, L i the miile lead angle aociated with the colliion triangle and HE refer to the initial deviation of the miile from the colliion triangle commonly known a the heading error. The lead angle L can be found by application of the law of ine on the colliion triangle yielding the formula L in { V T in( ) / V M }. (8) To find the miile acceleration component, it i neceary to determine the component of the relative miile-target eparation. Let the component of the relative miile-target eparation be expreed a X X, (9) X T M Y Y. (0) Conequently, the range R between miile and target i given by Y T M R ( ), () X Y while the line of ight angle i given by tan Y. () X The line of ight rate can be eaily derived from thi expreion by differentiating with repect to time, giving 3
DSTO-TR-845 R X Y X Y (3) The cloing velocity i defined a the negative rate of change of the miile target eparation, that i, R, which, after uing eq () above, lead to the relation, V c. V C ( X X R Y ) Y. (4) Conequently, the magnitude of the miile guidance command definition for proportional navigation guidance, n C can be found from the ' n N V, C c (5) where ' N i a given contant. If we model the actual acceleration of the miile n L by a firt order lag term, then n n L C. (6) A D imulation model i eaily developed uing the above equation. or thi tudy, MATLAB/Simulink wa ued to develop the imulation model. The top level Simulink model i preented in igure. The detail of each ubytem of the model are given in Appendix A. igure 3 preent a creen hot of the Graphical Uer Interface (GUI) ued in thi tudy. The GUI wa pecifically deigned a an aid in running the imulation. The MATLAB oftware underpinning the contruction of the GUI i alo preented in the Appendix. At the GUI level, the miile and target parameter may be entered by the analyt. On the left of the GUI, the analyt can enter initial target poition, target peed and flight path angle. On the right of the GUI, the analyt can enter initial miile poition, miile peed, effective navigation contant and guidance loop time contant. Input for the imulation include target manoeuvre (NT), error in the initial heading angle (HE) and jump in the target poition (Diplace) at ome time (THOM) prior to intercept. 4
DSTO-TR-845 9.4 NT Nt Tgo XT YT VTX VTY 3 XT 4 YT XM YM XT YT VTX lamda Nc 0.3+ AP 5 NL lamda XM YM XM YM Target Dynamic VTY Vc VMX NL VMX Tgo VMY VMY Vc Relative Dynamic Stop Condition Miile Dynamic igure Two Dimenional Engagement Model in Simulink igure 3. Graphical Uer Interface to run the D Engagement Model 5
DSTO-TR-845. Simulation Reult The Simulink model given above may now be ued to conduct imulation tudie of the miile/target engagement problem. The imulation tudie will be baed on the following three imulation input; (a) (b) (c) a tep in target manoeuvre an error in initial heading angle a tep in target poition Interet in (c) above tem from the analyi of multi-target cenario a i decribed in Section..3... Step in target manoeuvre Conider the cae in which the only diturbance i a 3g target manoeuvre tarting at time t 0. The nominal value for target and miile parameter may be eaily identified in the GUI given in igure 3. In thi cenario, the miile and target are initially on a colliion coure and flying along the downrange component of the earth fixed co-ordinate ytem. Thu, the target velocity vector i initially along the line of ight and, at firt, all 3g of the target acceleration are perpendicular to the line of ight. However, a the target manoeuvre, the magnitude of the target acceleration perpendicular to the line of ight reduce due to the turning of the target. Sample miile/target trajectorie for thi cae with effective navigation ratio of 3 and 5 are hown in igure 4. It i clear from the figure that the higher effective navigation ratio caue the miile to lead the target lightly more than the lower navigation ratio cae. igure 5 preent the repective miile acceleration profile obtained from the imulation. Note that, although both acceleration profile are monotonically increaing for mot of the flight, the higher effective navigation ratio cae lead to le acceleration requirement of the miile toward the end of flight. Alo noteworthy from the plot i the obervation that the peak acceleration required by the miile to hit the target i ignificantly higher than the manoeuvre level of the target (3g)... Heading error Next, conider the cae in which the only diturbance i a 0 degree error in the initial heading angle, that i, HE = -0 deg. Again the imulation wa run for two value of the effective navigation ratio. Sample trajectorie for effective navigation ratio of 3 and 5 are preented in igure 6. rom the figure, it i apparent that initially the miile i flying in the wrong direction becaue of the heading error. Gradually the guidance law force the miile to head toward the target. It i intereting to note from the figure that the larger effective navigation ratio enable the miile to remove the initial heading error more rapidly, thu leading to a tighter miile trajectory in thi cae. 6
DSTO-TR-845 In igure 7 i plotted the reultant miile acceleration profile for each cae. rom the figure, it i oberved that the fater removal of heading error in the higher effective navigation ratio cae i aociated with larger miile acceleration at the beginning of flight 750 700 3g Target Manoeuvre 650 Crorange (m) 600 550 500 Miile, N'=5 450 Target 400 Miile, N'=3 350 0 000 000 3000 4000 5000 6000 Downrange (m) igure 4: Miile performance againt manoeuvring target 9 8 7 N' = 3 n L (g) 6 5 4 3 0 N' = 5-0 3 4 5 time () igure 5: Acceleration profile for Target Manoeuvre Cae 7
DSTO-TR-845 400 350-0 deg Heading Error 300 Crorange (m) 50 00 50 00 Miile, N'=5 Target 50 0 Miile, N'=3-50 0 000 000 3000 4000 5000 6000 Downrange (m) igure 6: Miile performance when initial heading angle i in error 30 5 N' = 5 0 n L (g) 5 0 N'=3 5 0-5 0 3 4 5 6 time () igure 7: Miile performance when initial heading angle i in error 8
DSTO-TR-845..3 Step in target poition The non-linear engagement imulation program may be ued to analye the miile repone to a udden jump in target poition at a certain time prior to intercept. Thi i ueful in upporting imulation tudie concerning the effect of eeker reolution on the performance of the miile when confronted with multiple target. Conider the cenario hown in igure 8 []. On the left i the miile engaging two target which are flying in formation. Both target have the ame peed and are eparated by a pacing of d metre. It i aumed here that the power centroid i located half way between the target. In thi cae, the input parameter, Diplace in the GUI of igure 3 i equivalent to d / metre. or the imulation, it i aumed that the miile i initially on a colliion coure with the power centroid. At a certain time to go before intercept with the power centroid, eeker reolution occur and the miile i preented with the true target, that i, Target. At thi time, it will appear to the miile a if the target poition jump from the power centroid to Target. rom a imulation perpective, it i only neceary to model the target currently een by the miile. Therefore, for mot of the flight, the miile will be guiding on the power centroid and for the ret of the flight, following eeker reolution, the miile will be guiding on Target. Thi i repreented in the imulation by the target poition being updated by a tep in target diplacement at a given time to go prior to intercept (THOM). The non-linear D engagement model i ued to generate ample imulation reult of thi type of engagement. Relevant imulation reult are preented in igure 9 and 0 below. Y V T Target Miile V Power Centroid V T M V T Target d d igure 8: Miile engaging two target flying in formation X 9
DSTO-TR-845 430 40 Target 40 Crorange (m) 400 390 380 370 360 N'=5 350 340 N'=3 Power Centroid 330 0 000 000 3000 4000 5000 6000 7000 Downrange (m) igure 9: Miile performance in multi-target cae 40 30 N'=3 0 N'=5 n L (g) 0 0-0 -0 0 3 4 5 time () igure 0: Acceleration profile in multi-target cae 0
DSTO-TR-845 3. Linearied Engagement Equation In thi ection, analytical tool are employed to gain further inight into the performance of the interceptor under different engagement condition and to better undertand it dynamic repone. In particular, the adjoint method i applied to a implified verion of the engagement equation. The implified equation of motion are derived uing linearization about the line of ight angle between miile and target. The procedure follow cloely that outlined in Zarchan []. irt, define the relative eparation between the miile and target perpendicular to the fixed reference a defined in igure, y Y T Y M. (7) The relative acceleration i expreed, by inpection of the figure, a Y y n T co n C co. (8) V T n T T n C M V M L HE R y Y T Y M igure : Engagement Geometry for Lineariation or mall flight path angle, that i, near head-on or tail chae cae, the coine term are approximately unity, and the previou equation (8), reduce to y n T n C. (9) X
DSTO-TR-845 Similarly, the line of ight angle, which i given by, in y, R (0) may alo be linearied, uing the mall angle approximation, yielding the expreion y. R or a head-on cae, the cloing velocity i approximated by () V C V V, () M T and in a tail chae ituation, the cloing velocity i approximated by V C V V. (3) M T Conequently, for contant miile and target peed, the cloing velocity may be treated a a poitive contant. However, the cloing velocity ha previouly been defined a the negative derivative of the range from miile to target, that i, V c R, and ince the range mut go to zero at the end of flight, we can approximate the range equation with the time varying relationhip, R V ( t t). (4) C In the above expreion, t denote the current time and engagement. Note that t i a contant here. t i the total flight time of the The linearied mi ditance i defined to be the relative eparation between miile and target at the end of flight, namely, MD y t ). (5) ( Uing equation (9), (), (4) and (5), and adding the information given by equation (5) and (6), we are able to build a block diagram model for the linearied verion of the homing equation. Thi model i diplayed in igure below. In the diagram, the ymbol refer to differentiation in the frequency domain uing Laplace Tranform terminology. In order to implement thi block diagram model in Simulink, it i prudent to combine the ingle block with the tranfer function block given by /( ). Thi i poible in thi cae a N ' V C i a contant. Thu, following thi proce, the revied block diagram model for the linearied equation i preented in igure 3. Thi model i deignated here a Model A. Note
DSTO-TR-845 that the block diagram diplayed in igure 3 now contain an inner cloed loop block tructure. Thi alternative block tructure i mathematically equivalent to the ingle block with tranfer function /( ). The proce i known a reducing the block to it fundamental cloed loop form and ha been achieved here uing tandard block diagram algebra. Thi alternative repreentation i important in the equel. MD y( t ) n T y y y V ( t t) C n L n C N ' V C igure : Linearied Geometry Model MD y( t ) n T n L y y y V ( t t) C N ' V C igure 3: Linear Model A Model A i typically employed in upport of analytical tudie of the homing loop problem in the literature [,,3]. Thi model i alo ued a the bai for adjoint analyi of the guidance loop. It i hown in the equel that while thi model i ueful for performance tudie aociated with target manoeuvre and heading error effect, caution need to be exercied when uing the model a a bai for analyi of multi-target problem. An alternative block diagram tructure typically employed in the literature for homing loop tudie [,4,5,6] may be derived uing the analytical expreion for the rate of change of line of ight angle. or the non-linear model, the rate of change of the line of ight angle i given in equation (3). After applying the lineariation condition tipulated above, the expreion for the line of ight rate become 3
DSTO-TR-845 y V ( t t) V C C y ( t t), (6) or equivalently, ( t y y) go, V t C go (7) where the term t denote the time to go and i defined a t t t. go go After incorporating the analytical expreion for the line of ight rate into the linearied et of equation, an alternative block diagram for the homing loop dynamic may be derived. Thi alternative block diagram, deignated a Model B, i diplayed in igure 4. MD y( t ) n T y y y V C t go n L t go N ' V C igure 4: Linear Model B In the literature, thee two model (Model A and Model B) are often ued interchangeably a the bai for a preliminary analyi of the performance of the generic guided miile homing loop [-6]. The next ection cover a comparion of the two model through the ue of imulation and highlight input condition under which the model yield different reult. or thee input condition, caution hould be exercied by the analyt when adopting thee model, particularly when conidering Model A. 4. Comparion of the Linearied Model The two linear model have been implemented in Simulink in preparation for a comparative imulation tudy. The Simulink implementation of Model A and B are preented in igure 5 and 6, repectively. or the preent work, the imulation of the linear model are carried out under the ame input condition conidered in Section. above. Thi allow a comparion of the imulation reult generated by the linear model againt thoe obtained by the non-linear model. To do thi, the time of intercept diplayed in the GUI after running the non-linear 4
DSTO-TR-845 model i recorded a the approximate flight time,, for the engagement. Thi flight time i then ued in the linear model. Additionally, the ame numerical value for the parameter are ued, a hown in the GUI in igure 3. t MD Nt In Out Nt Integ Integ /VcTgo NL Np Np Vc Vc /Tau Gain Integ igure 5: Simulink Implementation of Linear Model A MD Nt In Out Nt Integ Integ /VcTgo In Out Tgo NL Integ /Tau Gain Np Np Vc Vc igure 6: Simulink Implementation of Linear Model B 4.. Step in target manoeuvre A tep in target acceleration of magnitude 3g i applied to both linear model and all other input are et to zero. The imulation reult are preented in igure 7 and 8 below. igure 7 how the time hitory of the relative diplacement profile while igure 8 how the achieved miile acceleration profile. It i clear from the figure that both Model A and 5
DSTO-TR-845 Model B imulation reult agree. urthermore, the linear model provide a good approximation to the non-linear engagement model in thi cae. 00 90 80 Model A Non-Linear Model B 70 Rel Dipl y (m) 60 50 40 30 0 0 0 0 3 4 5 6 Time () igure 7: Comparion of relative diplacement profile in the cae of a 3g tep in target man oeuvre 5 0 Non-Linear Model A Model B Accel N L (g) 5 0-5 0 3 4 5 6 Time () igure 8: Comparion of miile acceleration profile in the cae of a 3g tep in target manoeuvre 6
DSTO-TR-845 4.. Heading error In thi cae, an error in initial heading angle of -0 degree i conidered with all other input et to zero. Again, the imulation reult from the linear model are compared with thoe generated by the non-linear engagement model. The heading error i introduced in the linear model a an appropriate initial condition on the firt integrator in the Simulink model of igure 5 and 6, repectively. The reult of the imulation are preented in igure 9 and 0 below. The figure indicate that both linear model agree in thi cae and that they repreent a good approximation to the dynamic of the non-linear engagement model. 50 0-50 Non-Linear Model A Model B -00 Rel Dipl y (m) -50-00 -50-300 -350-400 -450 0 3 4 5 6 Time () igure 9: Comparion of relative diplacement profile in the cae of a heading error of -0 deg 4..3 Step in target poition Here, interet lie in the repone of the miile to a udden tep in target poition at a certain time to go prior to intercept. Thi cenario imulate the effect of target reolution by the onboard eeker in a multi-target cenario. or the linearied model, thi condition tranlate into an appropriate initial condition placed on the econd integrator (Integ) in igure 5 and 6. Simulink imulation were carried out for both linear model and then compared to the correponding non-linear reult, a hown in igure 0 and below. Thee plot how that Model A reult do not agree with Model B reult. urthermore, the plot demontrate that Model B reult are a good approximation to the imulation reult generated by the nonlinear model. However, the ame cannot be aid of Model A in thi cae. 7
DSTO-TR-845 0 5 Non-Linear Model A Model B 0 Accel N L (g) -5-0 -5-0 0 3 4 5 6 Time () igure 0: Comparion of miile acceleration profile in the cae of a heading error of -0 deg Conequently, the obervation gleaned from the foregoing imulation ugget that Model B i a ufficiently accurate linear model when conidering the miile target engagement problem. And thi i o regardle of the three input condition conidered. However, caution need to be exercied when uing Model A for thi purpoe a it ha been hown via imulation to produce erroneou reult under one of the tipulated input condition (ee igure and ). 8
DSTO-TR-845 70 60 50 Non-Linear Model A Model B 40 Rel Dipl y (m) 30 0 0 0-0 igure : -0 4 4. 4.4 4.6 4.8 5 Time () Comparion of relative diplacement profile in the cae of a jump in target poition of 60 m 00 80 60 Non-Linear Model A Model B 40 Accel N L (g) 0 0-0 -40-60 -80-00 4 4. 4.4 4.6 4.8 5 Time () igure : Comparion of miile acceleration profile in the cae of a jump in target poition of 60 m 9
DSTO-TR-845 5. Derivation of a Correction for Model A In thi ection, an analyi i carried out to explore whether Model A can be brought into alignment with Model B when the input condition i a tep in target diplacement. Conider the block diagram of Model A given in igure 3 and reproduced in igure 3 for convenience. In igure 3, the tate of the ytem have been included in the block diagram, deignated by x, a ha the deired input condition y. i ic MD y( t ) n T n L y y y x x y ic V ( t t) C N ' V C x 3 igure 3: Linear Model A with ytem tate diplayed By inpection of the diagram, the tate pace equation may be eaily written down. They are, x ' n N V x T C 3, (8) (9) x x, x x ( V t x 3 3 C go ). (30) In thi cae, the initial condition are, x (0) 0, zero, equation (30) yield x (0) y ic and x (0) 0. Thu, for time 3 yic x (0) 3 V t C. (3) Conequently, the initial value of the relative acceleration y(0) will have the magnitude 0
DSTO-TR-845 ' N yic (3) y(0) x (0) n. T t Thi i phyically impoible a it implie a jump in miile achieved acceleration whenever the initial condition on y i non-zero. To counteract thi effect, a requirement i placed on the filter initial condition a follow, yic x (0) 3 V t C. (33) When thi correction, in the form of a non-zero initial condition on the filter, i applied to Model A, the imulation reult generated by both linear model agree a hown in igure 4 and 5 below. Thu, in concluion, any of the linear model, Model A or Model B, derived previouly may be ued for linear analyi of the miile target engagement problem provided that Model A i ued in conjunction with the correction factor derived above. It i noteworthy to point out that in the cae when i zero, no initial correction of the filter i neceary. y ic 70 60 Non-Linear Model B Model A with IC 50 Rel Dipl y (m) 40 30 0 0 0 igure 4: -0 4 4. 4.4 4.6 4.8 5 Time () The relative diplacement profile generated by the linear model now agree for the initial diplacement cae
DSTO-TR-845 00 80 60 Non-Linear Model B Model A with IC 40 Accel N L (g) 0 0-0 -40-60 -80-00 4 4. 4.4 4.6 4.8 5 Time () igure 5: The miile acceleration profile generated by the linear model now agree for the initial diplacement cae 6. Analytical ormula The lineariation of the engagement model i important for two reaon. irtly, with a linear model, powerful computeried technique, uch a the adjoint method, may be utilied to analye the miile guidance ytem both tatitically and determinitically in one computer imulation. Moreover, with thi technique, error budget are automatically generated o that key ytem driver may be identified and a balanced ytem deign can be achieved. Secondly, under pecial circumtance, the linear engagement model i mathematically amenable to analytical olution. Thee olution can ait, during the preliminary phae of a miile deign, in gaining inight for ytem izing. urthermore, the form of the analytical olution will provide clue on how key parameter may influence ytem performance. In thi ection, cloed-form olution are derived for the three important cae that were conidered above for the engagement imulation. The aim i to derive cloed-form olution aociated with Model A and Model B independently and then to how that thee olution are mathematically equivalent provided that the filter initial condition i included in Model A. It ha been hown in the literature [] that cloed form olution for the linear engagement model may be obtained more eaily uing the adjoint method. Therefore, thi i the method employed in the derivation proce.
DSTO-TR-845 6. Cloed-orm Solution for Model A The adjoint block diagram of Model A i preented in igure 6. Thi model wa contructed from the forward loop model given in igure 3 uing the uual rule of adjoint contruction given in reference [, 3]. Note that x in the diagram refer to the filter initial condition ic derived in the lat ection and i expreed, in thi cae, a yic x ic V t C. (34) Alo in the diagram, the initial condition for the heading angle error i defined a HA V HE. (35) M MD HA HA y ic MD yic V C t n T MD xic x ic N ' V C MD NT igure 6: Adjoint Model A In thi form, the tate equation aociated with the adjoint block diagram (igure 6) are mathematically challenging for analytical work. However, uing block diagram algebra [], the tructure of thi block diagram may be conveniently reduced to a form uch a to allow eay mathematical treatment of the problem. The reduced form of the block diagram i preented in igure 7 below. In the diagram, h (t) i deignated a an adjoint output ignal while the weighting function w (t), after frequency domain tranformation, i defined a W ( ) ' N. ( ) (36) 3
DSTO-TR-845 MD yic n T M D NT y ic HA MD HA W () t V x C ic h(t) MD xic igure 7: Reduced orm of Adjoint Model A Uing the convolution integral, the adjoint output i related to the input by h( t) t t 0 w( )[ ( t ) h( t )] d. After taking the Laplace Tranform of the preceding equation, one obtain (37) However, ince dh W ( )[ H ( )]. (38) d dh d [ H ( )], (39) d d the technique of eparation of variable i invoked to arrive at d ( H ) ( ). W d (40) H The olution to equation (40) i given by H ( ) C exp( W ( ) d), (4) where C i a contant of integration. Thi contant can be evaluated by firt noting from igure 7 that the mi ditance due to a tep in target diplacement of magnitude i given by (uing frequency domain notation) y ic 4
DSTO-TR-845 yic L{ MD } [ H ( )]. y ic (4) L In the above expreion, the term refer to the Laplace tranform of it argument. Let the magnitude of the tep be unity. Then, in the time domain, the mi ditance due to a unit tep in target diplacement at time zero will be unity. Thu, invoking the initial value theorem lead to MDyic lim H ( ) ( 0) Therefore, C i choen to atify the following relation,. (43) lim C exp( ( ) ) W d. (44) In thi cae, ince () i reduced to W i given by equation (36), then it follow that C, and equation (4) H ( ) exp( W ( ) d). (45) Conequently, ' d W ( ) d N, ( ) ' N [ ] d ( / ), (46) (47) N ' ln /. (48) Now, after ubtitution of thi expreion into equation (45) above, one obtain the fundamental relationhip H ( ) / ' N. (49) Conequently, from inpection of the adjoint block diagram given in igure 7 above, it i eay to write down expreion (in frequency domain notation) for each of the deired output a follow. The mi ditance due to a tep manoeuvre of magnitude i given by n T 5
DSTO-TR-845 MD L{ n T NT H ( ) } 3 3 / The mi ditance due to an initial heading error i given by ' N. (50) MD L{ HE HE } V M [ H ( )] V M / ' N. (5) or a tep in target diplacement, the mi ditance formula i compoed of two part, namely, that due to y and that due to x. The part due to y i ic ic ic while the part due to MD L{ y ic yic H ( ) } / x i, according to igure 7, ic ' N, (5) MD L{ V x C ic [ H ( )] W ( ) N } ' / ' N, / xic (53) Therefore, for an effective navigation ratio of 3, the invere Laplace Tranform of the above expreion i taken, yielding the analytical expreion in equation (54) and (55). Note that the time variable appearing in the expreion i adjoint time which may be interpreted a total flight time t for the engagement problem. MD n T NT ' N 3 0.5t e t /, (54) and MD HE t / V t e. (55) M HE ' N 3 t urthermore, the mi ditance due to a tep in target diplacement may be determined by the addition of the invere Laplace Tranform of equation (5) and (53). The cloed form expreion for thi i MD yic yic xic (56) y ic Tot ' N 3 MD y ic ' N 3 MD y ic ' N 3, 6
DSTO-TR-845 where MD y ic yic ' N 3 t t [ 0.5 ] e t /, (57) and MD y ic xic ' N 3 t t [.5 0.5 ] e t /. (58) In deriving equation (58) above, ue wa made of equation (34). Conequently, adding the above expreion yield the mi ditance due to a tep in target diplacement of magnitude, namely y ic MD yic yic / (59) Tot ' N 3 t [ 0.5 ] e t. 6.. Adjoint of Model A in Simulink A Simulink model of the adjoint ytem of Model A (igure 6) i hown in igure 8. Thi model i ued to generate imulation data for comparion with the cloed form olution given previouly. The comparion are diplayed in the following figure. MDHE HE HE yic MDyic 3 MDNT Nt Nt Int Int Int Out /Vc/t In - /Tau Int3 4 MDxic yic Out In /Vc/T - Minu Np Np Vc Vc igure 8: Adjoint Model A in Simulink 7
DSTO-TR-845 0.8 0.7 0.6 Simulation ormula Mi Ditance (m) 0.5 0.4 0.3 3g Target Man N'=3, Tau=.3 0. 0. 0 0 4 6 8 0 light Time () igure 9: Adjoint Simulation and ormula agree for MD NT 0 5 Simulation ormula 0 Mi Ditance (m) -5-0 -5-0 Deg Heading Error N'=3, Tau=0.3-0 -5 0 4 6 8 0 light Time () igure 30: Adjoint Simulation and ormula agree for MD HE 8
DSTO-TR-845 60 50 Simulation ormula 40 Mi Ditance (m) 30 0 0 60 m Target Diplacement N'=3, Tau=0.3 0-0 0 4 6 8 0 light Time () igure 3: Adjoint Simulation and ormula agree for MDyic The plot confirm that the cloed form olution and the Simulink imulation reult baed on the linear adjoint of Model A agree for the particular cae conidered here. Thu, either approach may be ued for preliminary ytem deign. Recall that the cloed form olution are baed on a ingle time contant guidance loop. However, for higher order ytem, the equation become mathematically intractable. Thu, uing the imulation model allow the analyt added flexibility for tudying more complex homing loop ytem with minimum extra effort due to the block diagram feature of Simulink. Thi apect hall be conidered further in a later ection of the report. 6.. Comparion with non-linear reult Here, the non-linear engagement model, a preented in Section, i ued in a multi-run mode to generate the mi ditance profile a a function of flight time or time to go. Thi will enable a comparion of the non-linear imulation reult with thoe obtained uing either the linear adjoint model in Simulink or the cloed form olution derived earlier. or the non-linear model, the approximate mi ditance calculation method, a employed by Zarchan [], i utilied. urthermore, in order to obtain the mi ditance profile a a function of flight time uing the non-linear imulation, a range to go parameter i defined and utilied in the proce. The mi ditance generated uing the non-linear model for a tep in target manoeuvre of magnitude 3g i hown in igure 3 for different flight time. Superimpoed on thi plot i the mi ditance profile computed from the cloed form olution for thi cenario. It i clear from thi plot that the cloed form olution provide ufficiently accurate reult for low value of flight time. A the flight time increae, the plot how that the non-linear reult tend to 9
DSTO-TR-845 diverge. According to linear theory, the mi ditance tend to zero for flight time that are approximately greater than ten time the miile repone time contant. Thi i clearly evident in the figure a the miile time contant i 0.3 in thi cae. Conequently, beyond a flight time of 3 econd, linear theory i le accurate in thi cae. In igure 33, the non-linear imulation reult are compared with thoe obtained uing the cloed form olution for the cae of an error in initial miile heading angle. In thi cae, good agreement i obtained for mot of the flight time conidered. Large dicrepancie occur when the total flight time i very mall. Thi tem from the fact that under uch condition, the engagement cenario i ignificantly non-linear. A a final comparion with the non-linear imulation, the cae of a tep in target diplacement i conidered. In thi cae, the non-linear imulation reult were obtained by looping over different value of homing time, or THOM in igure 3. Again, the cloed form olution, baed on the linear analyi, agree reaonably well with the non-linear imulation reult generated by the Simulink model (ee igure 34). 0.8 0.7 Non-Linear Simulation Cloed orm Solution 0.6 n T =3g, N'=3, =.3 0.5 MD (m) 0.4 0.3 0. 0. 0-0. 0 3 4 5 6 light Time () igure 3: ormula agree cloely with non-linear imulation for MD NT 30
DSTO-TR-845 35 30 Non-Linear Simulation Cloed orm Solution 5 HE=0 deg, N'=3, =.3 0 5 MD (m) 0 5 0-5 -0 0 3 4 5 6 light Time () igure 33: ormula agree cloely with non-linear imulation for MD HE 60 50 Non-Linear Simulation Cloed orm Solution Tgt Step = 60 m N' = 3, =.3 40 30 MD (m) 0 0 0-0 0 0.5.5.5 3 3.5 4 4.5 5 light Time () igure 34: ormula agree cloely with non-linear imulation for MDyic 3
DSTO-TR-845 6. Cloed-orm Solution for Model B In thi ection, linear Model B i invetigated uing analytical mean. The forward loop block diagram of Model B i given in igure 4. Applying the adjoint contruction rule to thi block diagram yield the adjoint block diagram of Model B a preented in igure 35. In igure 35, the output of two of the integrator in the block diagram are deignated a z and z. Thee are clearly indicated in the figure. rom inpection of the block diagram, the following expreion may be deduced, namely z z tz, z (0). (60) Conequently, after integrating thi equation, one obtain the expreion z tz, (6) C where C i a contant. The value of thi contant can be determined by noting from igure 35 that z ( t ) i directly linked to the mi ditance due to a heading angle error. Recall that the time variable in thi cae (adjoint model) i interpreted a flight time. Thu, for zero flight time, the following mut be true, namely, z ( 0) C 0. Conequently, equation (6) reduce to z. (6) tz MD HA MD NT n T HA YT z z MD YT t N ' V C V C t igure 35: Adjoint Model B Hence, making ue of equation (6) above, and after introducing the expreion for the weighting function defined earlier in equation (5), the block diagram diplayed in igure 35 may be judiciouly manipulated into the block tructure preented in igure 36. 3
DSTO-TR-845 or eae of analyi, the block diagram diplayed in igure 36 i re-cat into the tandard input-output form hown in igure 37 below. Note that care mut be exercied when doing thi ince the block diagram contain variable from different domain, that i, from the time domain (t) and the frequency domain (). In the block diagram of igure 37, two extra variable have been included a adjoint ignal of interet, namely, z ( ) and z ( ). 3 t 4 t MD HA HA YT t z z MD YT t n T MD NT W () igure 36: Equivalent block diagram of adjoint Model B HA M D HA MD YT YT n T MD NT z 3 z t z W () t z 4 igure 37: Equivalent block diagram in tandard form rom igure 37, the following time domain expreion may be deduced, t z4 ( t) ( ) ( ), f z t d (63) t 0 33
DSTO-TR-845 z ( t) t [ ( t ) z4 ( t )] d 0, (64) where f (t) i defined a f ( t) L { W( )}. (65) Taking the Laplace Tranform of equation (63) and (64) yield the following relationhip, Z d Z ( ) 4 W ( ) Z ( ), (66) d Z ( ) Z ( ) 4 ). (67) 3 ( urthermore, taking the Laplace Tranform of equation (6), yield the relation dz ( ) Z ( ) d (68) After ome algebra, the following differential equation for Z ( ) i derived, W ( ) Z 0 d.. dz (69) Thi differential equation may be readily olved uing eparation of variable to yield C C ( ) W ( ) d e Z (70) where i a contant. The value of the contant may be determined a follow. rom the block diagram in igure 37, the expreion for the mi ditance due to target diplacement may be derived. Thi ha the form, MD YT YT * z ( t). (7) Therefore, for a unit tep in target diplacement at zero flight time, the following condition mut be true, namely, z (0). (7) 34
DSTO-TR-845 Note the conitency with equation (60) above, where the ame condition aroe a a reult of the impulive input. After ubtituting the expreion for W (), a defined in equation (5), into equation (70) above, and after ome algebra, the following expreion i obtained, C Z ( ) / ' N. (73) rom equation (6) Thu at zero flight time, the following expreion i valid, z ( t) z ( t). (74) t lim z ( t) z ( 0) t 0 t (75) Uing L Hopital rule and making ue of equation (7) yield the expreion, lim dz t 0 dt.. (76) Now, the Laplace Tranform of dz / dt i given by dz L Z( ) z (0) dt (77) However, a wa hown earlier, z (0) 0. Hence, the above expreion reduce to dz L Z( dt ).. (78) Conequently, returning to equation (76) and invoking the initial value theorem lead to the following expreion lim Z ( ). (79) Hence, after ubtituting equation (70) into equation (79), and making ue of equation (5), the above expreion reduce to 35
DSTO-TR-845 lim C / ' N. (80) Thi implie that C. Conequently, the expreion for Z ( ) i Z ( ) / ' N. (8) Thi expreion may now be ued to deduce the Laplace Tranform of the mi ditance output of interet highlighted in igure 37 above, namely, the mi ditance due to target manoeuvre, MD NT n ( ) T 3 / ' N, (8) and the mi ditance due to an initial heading error, MD HA V HE M ( ) / ' N. (83) inally, the mi ditance due to target diplacement i found by making ue of equation (74). After time domain converion, thi may be expreed a where t denote flight time. 6.. Cloed orm Solution for N =3 MD t ) YT * z ( t ) / t, (84) ( YT Here cloed form olution are derived for a particular value of the navigation ratio, namely, ' N 3. Subtitution of thi value into the above expreion, and after taking the invere Laplace Tranform, yield the following cloed form olution, MD n T NT N ' 3 0.5t e t /, (85) MD HA t / t V t e ( ) M HE N ' 3, (86) MD YT t / t e ( ) YT N ' 3. (87) 36
DSTO-TR-845 Thee agree exactly with the correponding formula derived earlier and baed on Model A. Cloed form olution may alo be derived for other value of N provided they are integer. In fact, everal of thee cloed form olution have been derived and are reproduced in Appendix B. rom the cloed form olution given above (and in Appendix B), it i clear that the mi ditance due to a tep in target diplacement i related to the mi ditance due to a heading error. Thi relationhip i invetigated further in Appendix C. However, for non-integer value of N, the mathematic i complex. In thi cae, the bet approach for a olution would be to ue imulation. A Simulink repreentation of Model B i diplayed in igure 38 below. A the imulation reult agree exactly with thoe previouly produced uing Model A (ee igure 9-3), only thoe reult generated uing Model B will be preented here. MDHE HE HE yic yic MDYT 3 MNT Nt Nt Integrator Integrator Integrator Out /VcTgo In Out In Tgo - minu Tau.+ Tranfer cn Np Np Vc Vc igure 38: Adjoint Model B in Simulink The mi ditance due to a target manoeuvre of magnitude 3g (that i, output 3 in igure 38) i preented in igure 39 for variou value of the proportional navigation ratio. rom the ame imulation, it i poible to extract imilar information relating to the mi ditance due to a heading error (output in the figure) and the mi ditance due to a tep in target poition (output in igure 38). Thee are preented in igure 40 and 4 below. It i clear from thee figure that the mi ditance i generally maller a the navigation ratio increae. 37
DSTO-TR-845 Mi due to n T (m). 0.8 0.6 0.4 Tgt Man, n T =3g =0.3 N'=.5 N'=3 N'=3.5 N'=4 N'=4.5 N'=5 0. 0-0. 0 4 6 8 0 light Time () igure 39: Mi due to Target Manoeuvre for Variou N 5 0 Heading Error -0 deg = 0.3 V M = 000 m/ 5 Mi due to HE (m) 0-5 -0-5 -0-5 N'=.5 N'=3 N'=3.5 N'=4 N'=4.5 N'=5-30 0 3 4 5 light Time () igure 40: Mi due to Heading Error for Variou N 38
DSTO-TR-845 60 Mi due to Tgt Dipl (m) 50 40 30 0 0 Tgt Dipl = 60 m = 0.3 N'=.5 N'=3 N'=3.5 N'=4 N'=4.5 N'=5 0-0 0 0.5.5 light Time () igure 4: Mi due to Target Diplacement for Variou N 7. Higher Order Guidance Sytem Dynamic It ha been hown that when the guidance ytem dynamic may be characteried by a ingle time contant, then it i poible to derive cloed form olution for the mi ditance. However, it i known from reference [] that the ingle time contant repreentation of the guidance ytem eriouly underetimate the mi ditance. or a more realitic repreentation, reference [] recommend the ue of a fifth order tranfer function repreentation. In thi cae, no cloed form olution i poible. Thu the only option i to ue numerical imulation. In thi ection, the fifth order binomial model, a recommended by Zarchan [], ha been adopted. The model may be expreed in the form, ' nl N VC, (88) 5 ( / 5) Uing the above model, it i hown that both linear model, namely, Model A and Model B, till produce equivalent imulation reult provided that the correction term i applied to Model A a previouly propoed for the ingle time contant cae. 39
DSTO-TR-845 irtly, Model A (ee igure 3) i modified to reflect the implementation of the fifth order guidance ytem repreentation expreed in equation (88) above. After a block diagram retructure, thi new model i deignated a Model A5 and i hown in igure 4 below. Note that, a propoed earlier in the ingle time contant cae, an initial condition x on the firt ic integrator downtream of the entry point ha been included in the model to reflect the cae of a tep in target diplacement y ic. A previouly, the value of thi initial condition i given by the expreion, yic x ic V t C. (89) MD y( t ) n T n L y y y HA y ic V ( t t) C 5 x ic ' N V C ( / 5) 4 igure 4: Block Diagram of Model A5 Now, following the adjoint contruction rule given in [, 3], it i an eay tak to contruct the adjoint of Model A5. Thi i hown in igure 43. In a imilar fahion, the alternative adjoint model baed on Model B i contructed. or the fifth order ytem, Model B i deignated a Model B5. Thi i hown in igure 44 and the correponding adjoint model i hown in igure 45. Both model were then programmed in Simulink for ubequent adjoint imulation. The reult are preented in igure 46 through to igure 48 incluively. It i clear that both model yield the ame reult in thi cae and hence any of the model may be ued interchangeably for the guidance loop analyi provided that the propoed correction trategy derived here i adopted when uing Model A. 40
DSTO-TR-845 MD yic MD HA HA y ic V ( t t) C 5 n T MD NT MD xic x ic ' N V C ( / 5) 4 igure 43: Adjoint Block Diagram of Model A5 n T YT MD y( t ) y y y V C t go n L HA t go ' N V C ( / 5) 5 igure 44: Block Diagram of Model B5 4
DSTO-TR-845 MD HA n T HA YT MD YT t V C t MD NT ' N V C ( / 5) 5 igure 45: Adjoint Block Diagram of Model B5.5 ifth Order Binomial 3g Tgt Man = 0.3, N' = 5 Mi due to n T (m) 0.5 0-0.5 - -.5 Model A5 & Model B5 - -.5 0 3 4 5 light Time () igure 46: Mi Ditance due to Target Manoeuvre for 5 th Order Sytem 4
DSTO-TR-845 0 00 80 60 ifth Order Binomial HE = -0 deg, V m = 000 m/ = 0.3, N' = 5 Mi due to HE (m) 40 0 0-0 -40-60 Model A5 & Model B5-80 0 3 4 5 light Time () igure 47: Mi Ditance due to Heading Error for 5 th Order Sytem 60 Mi due to YT (m) 50 40 30 0 0 0 ifth Order Binomial YT = 60 m = 0.3, N' = 5-0 -0 Model A5 & Model B5-30 0 3 4 5 light Time () igure 48: Mi Ditance due to Target Diplacement for 5 th Order Sytem 43
DSTO-TR-845 8. Concluion In thi report, attention i focued on the model and tool often adopted for the linear analyi of miile-target engagement in two dimenion. ollowing a review of the non-linear dynamic equation decribing the problem, two linear model were developed for the linear analyi. Thee model are often ued interchangeably in the literature for miile guidance loop analyi. However, it i hown here that under certain initial condition of the tate variable, the two model yield ignificantly different reult. Hence, caution need to be exercied by the analyt when adopting thee model for miile homing loop analyi. ollowing cloer analyi, a correction factor wa derived which may be ued to render the two model equivalent. Thi correction factor wa then ued a the bai for deriving cloed form olution to the problem in the cae when the guidance ytem may be approximately repreented by a ingle time contant ytem. After carrying out the mathematic, it wa demontrated that both model yield the ame cloed form olution for the particular input conidered. The model were then implemented in Simulink and it wa hown how they could be ued for aement of other problem in which the cloed form olution are not available. In addition to inight gained on ytem performance, another benefit of having derived the cloed form olution to thi problem wa the obervation that two of the analytical relationhip were indeed connected. Conequently, it wa hown how thi fact could be utilied for the generation of performance data aociated with the tudy of multiple target uing the ytem repone to a heading error input. inally, the power and flexibility of Simulink wa utilied to demontrate how eaily thee linearied engagement model and their adjoint could be extended to accommodate higher order and more complex dynamic ytem repreentation of the guidance loop. 44
DSTO-TR-845 9. Reference. Zarchan, P., Tactical and Strategic Miile Guidance, 5 th Ed. Progre in Atronautic and Aeronautic, Vol. 4, AIAA Inc. Wahington DC, 007.. Shneydor, N.A., Miile Guidance and Puruit, Kinematic, Dynamic and Control, Horwood Publihing Ltd, Chicheter, UK, 998. 3. Bucco, D., Aeropace Application of Adjoint Theory, Monograph, DSTO ellowhip Program, Autralia, Jan 00. 4. Gutman, S., Applied Min-Max Approach to Miile Guidance and Control, Progre in Atronautic and Aeronautic, Vol. 09, AIAA Inc. Wahington DC, 005. 5. Wei, M., Adjoint Method for Miile Performance Analyi on State-Space Model, Journal of Guidance, Control and Dynamic, Vol. 8, No., pp. 36-48, 005. 6. Taylor, J.H., Price, C.., Siegel, J. & Gelb, A., Covariance Analyi of Nonlinear Stochatic Sytem via Statitical Linearization, ASME, Nonlinear Sytem and Synthei, Vol., Technique and Application, eb 98. 45
DSTO-TR-845 Appendix A: Non-linear Engagement Model in MATLAB/Simulink A.. Detail of the Simulink Model A.. The Simulink Subytem Model The miile dynamic equation and the target dynamic equation are implemented in Simulink a ubytem, a hown in the figure below. 3 VMX lamda i n i n - minu NL Prod Xdot X XM co co Prod Ydot Y YM 4 VMY igure A. Miile Dynamic Subytem The Miile Dynamic Subytem (igure A) contain all of the block required to imulate the miile dynamic a repreented by the non-linear equation given in the text. In a imilar fahion, the Target Dynamic Subytem (igure A) contain all of the block required to imulate the target dynamic in accordance with the target equation given in the text. In order to accommodate a udden change in target poition, a witch block ha been included in the Simulink model. The witch, in thi cae, i ued to imulate the different value of the target poition to be repreented in the imulation at the moment of eeker reolution in a multiple target cenario. It wa neceary to inert a unit delay in the loop in order to avoid an inherent algebraic loop condition cauing the imulation to fail. 46
DSTO-TR-845 3 VTX co - X XT 300 VT0 Beta Divide 4 VTY Nt in Y YT 350 YT0 Tgo >= Thom 350 >= 0.5 z Unit Delay (To avoid algebraic loop) YT Switch igure A. Target Dynamic The implementation of the relative dynamic i hown in igure A3. XM YM 3 XT XM YM XT DelX DelY 4 YT YT DelR DelX Tgo 4 DelY Tgo DelVX DelVY Vc 3 Vc 7 VMX 8 VMX VMY DelVX Cloing Vel Prod Np Np Nc VMY 5 VTX DelX VTX DelVY 6 VTY VTY DelV DelY DelVX LamDot lamda Int DelVY Lamda Dot igure A3. Relative Dynamic Subytem 47
DSTO-TR-845 XM 3 XT DelX VMX 3 VTX DelVX YM 4 YT DelY VMY 4 VTY DelVY igure A4. DelR Subytem igure A5. DelV Subytem DelX u SQ qrt DelY 3 DelVX Prod Prod u SQ 4 DelVY Add RTM Divide Tgo Vc < 0 Compare To Zero STOP Stop Simulation - Gain Divide Vc igure A6. Cloing Vel Subytem igure A7. Stop Condition DelX u SQ DelY u SQ Add 3 DelVX Prod Prod 4 DelVY Divide LamDot igure A8. Lamda Dot Subytem 48
DSTO-TR-845 Note that the Relative Dynamic ubytem block hown in igure A3 i compoed of the ubytem given in igure A4, A5, A6 and A8. The imulation topping condition model i hown in igure A7. A.. The Aociated MATLAB Code The miile-target engagement model repreented in Simulink i controlled by a Graphical Uer Interface underpinned by MATLAB code. The complete MATLAB code i lited over the following page. igure A9. MATLAB Code 49
DSTO-TR-845 igure A0. MATLAB Code (continued) igure A. MATLAB Code (continued) 50
DSTO-TR-845 igure A. MATLAB Code (continued) igure A3. MATLAB Code (continued) 5
DSTO-TR-845 igure A4. MATLAB Code (continued) igure A5. MATLAB Code (continued) 5
DSTO-TR-845 igure A6. MATLAB Code (continued) igure A7. MATLAB Code (continued) 53
DSTO-TR-845 igure A8. MATLAB Code (continued) igure A9. MATLAB Code (continued) 54
DSTO-TR-845 igure A0. MATLAB Code (continued) igure A. MATLAB Code (continued) 55
DSTO-TR-845 igure A. MATLAB Code (continued) igure A3. MATLAB Code (continued) 56
DSTO-TR-845 igure A4. MATLAB Code (continued) igure A5. MATLAB Code (continued) 57
DSTO-TR-845 igure A6. MATLAB Code (continued) 58