Trajectory estimation based on extended state observer with Fal-filter

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1 The Aeonauical Jounal Augus 05 Volue 9 No 8 07 Tajecoy esiaion based on exended sae obseve wih Fal-file C- in chunlin@dagon.nchu.edu.w S- Hsieh and Y-P in Depaen of Elecical Engineeing Naional Chung Hsing Univesiy Taichung Taiwan ABSTRACT This pape inends o develop a age ajecoy esiaion algoih wih applicaion o he ballisic age esiaion in he einal phase. The poposed design is based on he applicaion of a second-ode exended sae obseve (ESO echnique using age infoaion acquied fo he seeke o esiae he ajecoy of anoeuveable ballisic ages. Saisfacoy esuls have been eceived while applying he design in esiaion of eihe wo-dienional o heedienional age evasive acceleaion via copue siulaion..0 INTRODUCTION Tajecoy esiaions of ballisic issiles and he conol syse design fo ani-issiles have been eceiving a gea deal of aenion as pa of he defense indusy (-4 in he pas wo decades. Recenly, advanced acical ballisic issiles (TBMs aleady possess he capabiliy of evasion duing he eeny phase which akes he ajecoy esiaion eihe fo on-boad adas o fo gound-based adas oe difficul han befoe. In he aspec of ajecoy esiaions, α-β file (5 is useful fo ada easueen. Kalan file (KF and exended Kalan file (EKF (6,7 have he abiliies of ajecoy esiaion of aicaf wih noise by using ecusive algoihs which have been widely applied in navigaion, fixed sona, and indusial auoaions. In Ref. 8, he auhos have pesened he esuls of copaison fo esiaion pefoance of e-eny ages using he EKF, saisical lineaizaion, paicle fileing, and unscened Kalan file (UKF. The esuls favoed he EKF. Pape No Manuscip eceived 3 May 03, s evised vesion eceived 0 Ocobe 03, nd evised vesion eceived 7 June 04, 3d evised vesion eceived 9 Decebe 04, acceped 6 Januay 05.

2 08 The Aeonauical Jounal Augus 05 Figue. Geoeic elaionship beween TBM and seeke. Howeve, in eal-wold syses, hee ae odelling eos which ay lead o esiaion eo in Kalan fileing design. In addiion, coplicaed aix-based copuaion ay equie uch copuaion ie. Besides, he infoaion eceived fo he seeke is liied, one hus needs a oe siple and efficien way o esiae he evasive age ajecoy. Owing o he less dependence on odel infoaion, song capabiliies fo disubance ejecion and siple copuaion, he use of exended sae obseve (ESO fo on-line esiaing he oal unceainies, which lups he inenal nonlinea and unceain dynaics and he exenal disubance, has been eceived uch aenion. In he lieaue, he auhos in Ref. 9 designed a hee-dienional guidance uilized in hi-o-kill inecepo based on ESO. In Refs 0,, he ESO was used o esiae he evasive acceleaion of TBM and design he guidance law of inecepo. Accuacy of ESO is affeced by he obseve gain, because of he disubance, unceainy, and easueen noise (. Recenly, he ehod of expanding fis-ode sae o ESO was consideed ( while he auhos of Ref. 3 used he ehod of expanding he file sae o ESO. In Refs 4, 5, he poble was solved based on Fal file and feedback conol. The appoaches pesened in Refs,3 need o incease he ode of ESO, which esul in he incease of seing paaees. The ehod in Ref. 4 doesn need o incease he ode of ESO ha siplifies paaee adjusen while ainaining esiaion pefoance. Howeve, he case sudy given in Ref. 4 fo ESO design was acually conduced in he plana envionen, which eans ha pich and yaw dynaics ae assued o be fully decoupled. Howeve, ignoing he coupling effec in he hee-diensional space ay induce significan pefoance degadaion of sae esiaion when he elaive disance beween age and inecepo deceases gadually. This pape inends o develop an ipoved design of ajecoy esiaion based on he secondode ESO wih Fal file in he hee-diensional space. In addiion o conside he coupling effec of he pich and yaw dynaics, his eseach also invesigaes he effec of paaee changes in ESO design o esiaion pefoance. The esuls povide useful guidelines while designing ESOs.

3 Hsieh e al Tajecoy esiaion based on exended sae obseve wih fal-file 09.0 SYSTEM DESCRIPTION. TBM odel in e-eny phase Assuing a TBM wih he co-odinaes (X R, Y R, Z R, and he issile seeke is locaed a he oigin O of he ineial co-odinae syse as illusaed in Fig.. e he TBM be egaded as a poin ass which has a consan weigh W influenced by dag and gaviy in eeny phase so ha he dynaic equaion can be expessed as v x v y v g Cos Cos a x v g CosSin a y v z v g Sin g a z... ( whee V x, V y and V z ae, especively, he coponens of age s velociy along X, Y and Z axes; a x, a y and a z ae he especive coponens of evasive acceleaion; pich angle γ, yaw angle ψ, and ballisic coefficien β ae defined, especively, as Tan ψ Tan W S C ef v v x v y D0 v z v x y... ( whee S ef, W and C D0 denoe he efeence aea, weigh, and zeo-lif dag coefficien of he ballisic age, especively.. Deivaion of he equaions of oion Deivaion of equaions of oion fo he pich and yaw dynaics in OS co-odinaes is sandad. One is efeed o Fig. fo illusaion of he ineial co-odinae syse and OS co-odinae syse, whee he diecion cosine aix (DCM elaing he ineial co-odinaes (O l X l Y l Z l o he OS co-odinaes (O X Y Z is given by X CosCos CosSin Sin X I Y Sin Cos 0 Y I Z SinCos SinSin Cos ZI... (3 X I C I (, YI Z I The disance veco beween issile and age is denoed as = [ 0 0] T whee is he elaive disance in he OS co-odinaes. The acceleaion vecos of he issile and age ae expessed as a = [ ax, ay, az ] and T a, especively. The angula velociy veco along X, Y, and Z = [ ax, ay, az ] axes is denoed as ω = [ω X ω X ω X ] T and he aix of oaion (8 is given by

4 00 The Aeonauical Jounal Augus 05 Figue. Illusaion of ineial co-odinae syse and OS co-odinae syse. 0 z y 0 z x y x 0... (4 Consideing he age kineaics one can obain i.e. a a ( a a x ax z y ay ay z x y z az az y ω x z y... (5 Afe conducing soe calculaions, one can ge ay a y z z x y, az a z y y x z... (6 Clealy he wo es ae coupled. Fo Equaion (3, he angula velociy veco ω can be descibed in es of OS angles as x ω y z Cos 0 Sin0 CosCos CosSin Sin Sin Cos 0 0 Sin 0 Cos 0 SinCos SinSin Cos Sin Cos... (7

5 Hsieh e al Tajecoy esiaion based on exended sae obseve wih fal-file 0 Subsiuing Equaion (7 ino Equaion (6 yields he equaions of oion in he OS co-odinaes as follows a y a y Tan Cos... (8... (9 I is seen ha he pich and yaw dynaics ae closely coupled due o he exisence of wo coupling es ψ. θ. Tan θ. and ψ. Sin θ. Cos θ NOISE ANAYSIS az a z Sin Cos Conside he issiles deecing age posiion via an acive seeke. Duing engageen, an acive seeke igh be disubed by abien noises including glin noise, fading noise, and eceive noise duing fligh. Fading noise is independen on he elaive disance beween issile and age which is usually a high fequencies. I can be assued o be a Gaussian whie noise, wih is sandad deviaion denoed by σ fade (ad, passing hough a low-pass file. Effec of eceive noise is popoional o he elaive disance beween issile and age. The eceive noise of an acive seeke is popoional o he squae of he elaive disance beween issile (inecepo and age. I appeas a a highe fequency han he opeaional fequency of he guiding syse. Suppose ha he sandad deviaion is denoed as σ eceive (ad. Effec of glin noise inceases wih deceen of he elaive disance beween issile and age. Is fequency is usually highe han he opeaional fequency of he guiding syse. Usually, i is geneaed by a Gaussian whie noise souce passing hough a low-pass file wih is sandad deviaion denoed σ glin (ad. Suaion of he sandad deviaions of noises deeced by he acive seeke can be expessed as Ref. 3 4 glin eceive angle fade 4... (0 whee 0 is a efeence disance elaed o he signal inensiy of he eceive. Fo he ballisic age, assue ha is RCS is copaaively salle while copaing o egula je fighes, he effec due o glin noise is hus ignoable. Theefoe, Equaion (0 is siplified o 0 angle fade 4 eceive ( 4.0 ESO AND ESO FITER ESO is used o linealy appoxiae and asceain a non-linea and unceain plane by wo channel copensao. 4. Deivaion of ESO equaions ESO is a new ype of obseve ( which is designed fo siulaneously obseving he saes, unceainy of he syse, and he exenal disubance.

6 0 The Aeonauical Jounal Augus 05 Conside an nh-ode nonlinea SISO syse wih exenal disubance and unceainy, as shown below ( n ( n x f( xx,&,, x, w (... ( whee f (x, ẋ,..., x (n, is an unknown funcion of he plan and w( is unknown disubance inpu. e he saes of he syse be x ( x ( x( x &( M ( n xn ( x ( ( n xn( f( x, x&,, x, w(... (3 whee x (n+, he n+h sae, is he exended sae of he syse. Equaion ( can be ewien as x& ( x( x& ( x3( M x & n( xn( x& n( xn ( x& n ( h(... (4 whee h(, he deivaive of f (x, ẋ,..., x (n, + w(, is an unknown funcion. Based on Equaion (4, he coesponding sae-space odel can be expessed as whee x( Ax( Eh ( y Cx... ( A M M M O M , T 0 C 0 0 and E = [ ] T. Fo which, we can design a sae feedback esiao fo he closed-loop syse as follows x ˆ ( Ax ˆ( ( ycx ( y Cx( whee he obseve gain = [β... β n+ ] T. Tha is... (6 xˆ xˆ ( xˆ x xˆ ˆ ˆ x3 ( x x xˆ ˆ ˆ n xn n( x x xˆ ˆ n n ( x x... (7

7 Hsieh e al Tajecoy esiaion based on exended sae obseve wih fal-file 03 e he sae esiae z i ( = x^i ( and esiaion eo e ( = z ( x (, i =,..., n+ so ha i i i Equaion (7 can be conveed ino he following eo dynaic equaions: e& & & zx z x ( zx e ( e ( e& z& x& z3 x3 ( z x e3 ( e ( M e& n z& n x& n zn ( xn b0u n( z x en n( e( e & n z& n x& n x& nn( zx h( n ( e (... (8 I s easy o see ha if he es β (e i... β n+ (e i saisfy e ( e 0, e 0, ( 0 0, i,, n i i... (9 hen he syse descibed in Equaion (8 would be sable a he oigin wih espec o he bounded h(. I ean ha as long as one chooses β (e i... β n+ (e i appopiaely, he saes in Equaion (4 can be well acked by he saes in Equaion (7, i.e. z ( x (,, z ( x (, z ( x ( n n n n... (0 Refeing o Equaion (8 one can obain he geneal fo of ESO as below e zx z y z& z ( e( z& z3 ( e( M z& n zn n(e ( z& n h( n ( e (... ( 4. ESO wih Fal file The accuacy of ESO is affeced by he obseve gain, because of he disubance, unceainy, and easueen noise (8. Incopoaion of a low-pass file can help o pe-file easueen noise. Howeve, i also bings exa phase lag which ay deeioae esiaion pefoance. The poble can be esolved by eliinaing easueen noise using he ESO wih Fal file (4. Conside a syse wih easueen noise given as below ( n ( n x f( xx,&,, x, w ( y xvy... ( whee he plan odel funcion f (x, ẋ,..., x (n, is unknown, w is an unknown disubance inpu, and v y is easueen noise. Fo he syse of Equaion (, he ESO is designed as

8 04 The Aeonauical Jounal Augus 05 y xvy y0 fal _ file( y, kf, af, f e z( y0 z& ( z( ge z& ( z3( g fal( e, a, M z& ( z ( fal( e, a, n n gn z& n g( n fal e an ( (,, n... (3 and he file fal_file (y, k f, a f, δ f is defined as x kf fal(, ea f, f y0 x e yx... (4 whee a e sign (, e if e fal(, ea, e, if e a wih 0 a and 0 < δ. The funcion fal exhibis vial fileing chaaceisics on noise inpu. When a = i acs as a linea gain of and i becoes a sauaion funcion when a = 0. Fo he lae, when e > δ he nonlinea funcion fal fixes a + o, wheeas fo e δ, i acs like a linea gain of /δ. Siilaly, fo a (0, i acs as a linea gain /δ a fo e δ which deceases wih he deceen of a iplying a bee noise fileing effec. On he ohe hand, he equivalen gain wihin e δ inceases wih he deceasing value of δ. This gives ise o bee esiaion pefoance. 4.3 Esiaion of evasive acceleaion The equaion of ineial oion has been pesened in Equaion (8 fo he OS yaw plane and in Equaion (9 fo he pich plane. And he esiaed OS ae θ.^ = θ. = σ. angle and ψ.^ = ψ. = σ. angle whee θ. and ψ. ae he noinal angle aes. e θ.^ and ψ.^ pass hough a especive Fal file. Tha is ˆ ( ˆ FalFile, k, a, 0 f f f... (5 and ˆ FalFile( ˆ, k, a, 0 f f f... (6 The evasive acceleaions a and yl a ae egaded as unceain es and expanded as wo new saes. xl e he saes x = θ.^0 and x = ψ.^ 3 0 Cos, hen Equaion (8 and Equaion (9 can be wien as θ^ x Tanˆ x3 x az a z... (7

9 Hsieh e al Tajecoy esiaion based on exended sae obseve wih fal-file 05 and x x x Tanˆ 3 3 x3 ay a y... (8 Since,., θ^, θ^0 and ψ.^ 0 ae deecable by he seeke, we le he known coponens xtanˆ x3 f x3 x ˆ x3tan 3 Tanˆ x 3 and f whee 3 and xxtanˆ ae he coupling es and unknown coponens f = a zl and f 4 = a yl. Equaions (7 and (8 can be fuhe expessed as x f f a z... (9 and x f f a y (30 Expand he unknown e f o a fis-ode sae, and assue x = w p whee w p is unknown. Then Equaion (9 becoes x f f az x wp yp x... (3 On he basis of he above developen, we ae able o synhesise a second-ode ESO as follows ep z yp z Tanˆ z3 z z ep az z fal( ep, ap, p In Equaion (30, le he exended sae ẋ 4 = w y whee x 4 = f 4. We obain... (3 and ESO is designed as x 3 f3 f4 a x 4 wy yy x3 y ey z3 yy z z z Tanˆ β z 4 4fal( ey, ay, y 3 3 z3 z4 3ey ay... (33... (34 whee (a p, δ p, β, β and (a y, δ y, β 3, β 4 ae paaees of ESO, y p = x θ^0 and y = x = y 3 ψ.^0 Cos θ^ ae deecable by he seeke, he issile laeal acceleaion coand a and yl a ae deeined zl by he einal guidance of he issile. In Equaion (3 and Equaion (34, z, z, z 3 and z 4 ae he esiaed values of x, x, x 3 and x 4, especively. Once e p and e y convege, he age evasive acceleaions a and zl a ae obained. yl Due o he pesence of disubance, unceainy, and easueen noise, he accuacy of ESO is affeced by he paaees ses (a p, δ p, β, β and (a y, δ y, β 3, β 4. The funcion and effec of α

10 06 The Aeonauical Jounal Augus 05 and δ have been depiced peviously. In pacice, he noisy age infoaion easued by he seeke is pos pocessed via a low-pass file. Howeve, he guidance file canno file he noise hooughly. I will cause an exa dynaic lag and ay deeioae esiaion pefoance of ESO. Fo ESO design, he following ules can be efeeed: he value of β i deeines he esponse speed of he file. The lage i is, he bee acking pefoance will be; howeve, his will also ake he syse o be easily couped by he exenal noises, and vice vesa. Theefoe, one can se he values of β i invesely popoional o he inecepo-age elaive ange. Fo Equaion (3, Equaion (3 and Equaion (34, i is seen ha wihin he linea opeaional egion, he undaped naual fequency of he lineaised second dynaics of he ESO is appoxiaely deeined by β wih he daping aio β. Theefoe, one ay selec β > β and β 4 > β β 3 and wih β = β 4 fo he consisen pich and yaw dynaics. Design of δ depends on he esiaion eo in acual siuaion. Refeing o Equaion (3 and Equaion (34, seing δ elaes o he elaive ange fo an appopiae seing of he gain. A paaee opiisaion algoih can be used o seach fo he opial ESO paaees by iniising he oveall esiaion eo such as geneic algoih (6. Fo an advanced design of ESO, he vaiable obsevaion gains can be consideed. Fo exaple, when he inecepo is sill fa away fo he age, a salle β i will yield bee fileing effec o noise coupion. On he ohe hand, inceasing is values when he inecepo gadually appoaches he evasive age will bing bee esiaion o he age evasive acceleaion. The design guideline would be beneficial o guidance design fo he appopiae saegy of age engageen. One is also efeed o Ref. 6 fo he sabiliy analysis of he esiaion eo dynaics unde vaiable esiaion gains. 5.0 CASE STUDY The siulaion sudy consiss of hee pas. Fo he fis case, he esiaion accuacy wih coupling es and wihou coupling es ae copaed. In his case, he noise effec is ignoed, hus i is needless o use a Fal file. The second case discusses he effec of paaees of Fal file and ESO. Finally, on he basis of Cases and, he siulaion sudy wih he coupled pich and yaw dynaics is consideed. Case : Copaison of he esiaion esuls wih and wihou coupling es The siulaed condiions ae se as follows: he issile is a he oigin (0,0,0(, he issile velociy (V x,v y,v z = (,000,,000,,000 (/sec, he acceleaion of he issile (A x,a y,a z = (,3, (G, he age iniial posiion (X,Y,Z = = (0,000, 0,000, 0,000 (, he age velociy (V x,v y,v z = ( 00, 00, 50 (/sec, and he age evasive acceleaion (A x,a y = ( 5, (G fo one o five seconds; ESO paaees ae se as follows: a = 0.5, δ = 0.0, β 0 = 50, β 0 = 500, β 03 = 50, and β 04 = 500. By he appoach given in Ref. 4, if one ignoes he coupled dynaics, he esiaed and eal value of he evasive acceleaion in OS co-odinae will behave as he geen lines appeaed in Figs. 3 and 4. Howeve, involving he coupling es as ha consideed in his eseach and wih he poposed appoach, he esuls ae diffeen as shown wih he ed lines in Figs. 3 and 4. I s easily seen ha, a he beginning of esiaion, he effec of coupling es is ignoable, howeve, as wih he inceasing angula changing aes, he coupling effec becoe significan and should no be negleced. Alhough he incease of OS angle aes in pich and yaw planes ae no lage, as shown in Figs 5 and 6, he esiaed saes x = θ.^ and x = ϕ^. 3 Cos, as shown in Figs 7 and 8, becoe significan θ^ when uliplying he by he elaive disance, hus he coupling effec should no be eaed loosely.

11 Hsieh e al Tajecoy esiaion based on exended sae obseve wih fal-file 07 Figue 3. Esiaed and eal value of he evasive acceleaion in pich plane. Figue 4. Esiaed and eal value of he evasive acceleaion in yaw plane. Figue 5. OS angle ae in pich plane. Figue 6. OS angle ae in yaw plane. Case : Effec of Fal file and ESO paaees We esiae he evasive acceleaion wih easueen noise σ angle and o = 5k. The ohe siulaed condiions ae he sae as hose in Case. ESO and Fal file paaees a = 0.5, and δ = 0.0 ae consan. We use diffeen ESO paaees β, β and Fal file paaees k f in pich plane, and hen obseve he influence of easueen noise on hese paaees. As he fal funcion exhibi highly nonlinea chaaceisics, we exaine sabiliy of he esuling esiao design via siulaion. Fis, we conside he effec of Fal file. Se β = 00, β = 500, β 3 = 00, β 4 =500 and k f = 0. Figs 9 and 0 show, especively, he ESO design wihou and wih Fal file when he noise deviaion σ angle = ad wih he daa easueen ae 0.s (3 (i.e. he equivalen powe specal densiy Φ σ angle = 5x0 8 ad /Hz. One is efeed o Ref. 3 fo chaaceisaion of he noise inensiy. I s eviden ha he ESO design wih Fal file pefos bee in he acking esponse. Nex, we conside he effec of he gain k f in Fal file. Reain he sae noise inensiy. Se β =

12 08 The Aeonauical Jounal Augus 05 Figue 7. Esiaed sae x in pich plane. Figue 8. Esiaed sae x 3 in yaw plane. Figue 9. Resuls of ESO design in pich plane (a wihou Fal file (b wih Fal file. 00, β = 500 and adjus k f. While choosing k f =, 0, 40, 50 he siulaion esuls ae illusaed especively, in Figs -4. While k f =, he oscillaion of esiaed values is sall, bu hee is a sligh phase lag. When k f is inceased o 0, he phase lag is ipoved and ansien behavio of he esiaed esul is copaaively accepable. Inceasing k f o 40 leads o explici oscillaion in he esul. Keeping inceen of k f unil 50 leads o an unsable esponse. Which eveals ha he adissible k f would be less han 50. Nex, we incease he easueen noise inensiy Φ σ angle = 5x0 6 ad /Hz and eain he sae Fal file paaee seings. While k f = 0, he easueen noise has a lage effec on he esiaed esul, as shown in Fig. 5. When choosing k f =, he noise effec deceases, see Fig. 6; he esiaion pefoance is bee han k f = 0. In Equaion (4, a lage k f eans he fase vaiaion of he oupu y 0. In Equaion (3, a lage β eans he fase vaiaion of z, and a lage β eans he slowe vaiaion of z.

13 Hsieh e al Tajecoy esiaion based on exended sae obseve wih fal-file 09 (a Figue 0. Resuls of ESO design in yaw plane (a wihou Fal file (b wih Fal file. (b Figue. Esiaed and eal value of he evasive acceleaion while k f =. Figue. Esiaed and eal value of he evasive acceleaion while k f = 0. Figue 3. Esiaed and eal value of he evasive acceleaion while k f = 40. Figue 4. Esiaed and eal value of he evasive acceleaion while k f = 50.

14 030 The Aeonauical Jounal Augus 05 Figue 5. Esiaed and eal value of he evasive acceleaion while k f = 0. Figue 6. Esiaed and eal value of he evasive acceleaion while k f =. Figue 7. Esiaed and eal value of he evasive acceleaion using he accuae dynaics in pich plan. Figue 8. Esiaed and eal value of he evasive acceleaion using he accuae dynaics in yaw plan. In bief, he fileing effec will be noable when he Fal file paaee k f is chosen o be sall. Howeve, choosing a salle k f ay lead o a lage phase lag. The funcion of he ESO paaee β is siila o ha of k f, wheeas, he effec of selecion of β is conay o he ha of k f. Case 3: Siulaion esul based on Cases and The siulaed condiions of inecepo and age eain invaian as hose in Case wih excepion σ fade = ad and σ eceive = 0.000ad. When he inecepo is fa away fo he age, he easueen noise effec is significan. As he elaive disance beween boh objecs deceases, he easueen noise effec is copaaively salle. Theefoe, one has o choose he appopiae paaees of ESO and Fal file o guaanee esiaion pefoance in he einal engageen phase. Hee, he ESO and Fal file paaee ae chosen as follows: a = 0.5, δ = 0.0, β = 00, β = 500, β 3 = 00, β 4 = 500 and k f =0. Esiaion of he age evasive acceleaion in pich and yaw planes ae illusaed, especively, in Figs 7 and 8 which show saisfacoy pefoance of he pesen appoach.

15 Hsieh e al Tajecoy esiaion based on exended sae obseve wih fal-file CONCUSIONS This eseach has developed an evasive acceleaion esiaion schee fo he ediu- and highe-ie ballisic ages in he einal phase. The design is based on a second-ode exended saes obseve fo he coupled OS angle ae dynaics. I is shown ha he coupling effec o esiaion accuacy would be significan when he inecepo appoaches o he age in he einal phase. Effecs of he paaees in ESO and Fal file have been discussed via a vaiey of case sudies. The case sudies have veified applicabiliy of he poposed design. REFERENCES. in, C.F. Moden Navigaion. Guidance and Conol Pocessing, 99, Penice Hall, NJ, USA.. in, C.F. Advanced Conol Syse Design, 994, Penice Hall, Englewood Cliffs, NJ, USA. 3. Zachan, P. Tacical and Saegic Missile Guidance, Fifh Ediion, 007, AIAA, Washingon, DC, USA. 4. Kalan, R.E. A new appoach o linea fileing and pedicion pobles, ASME J Basic Eng, 960, Seies 8d, pp Bookne, E. Tacking and Kalan Fileing Made Easy, 998, Wiley-Inescience. 6. Soenson, H.W. Paaee Esiaion: Pinciples and Poble, 985, Maiel Dekke, NY, USA. 7. Huang, C.W., in, C.. and in, Y.P. Esiao design fo e-eny ages, ISA Tansacions, 04, 53, pp Faina, A., Risic, B. and Benvenui, D. Tacking a ballisic age: Copaison of seveal nonlinea files, IEEE Tansacions on Aeospace and Eleconic Syses, 00, 38, pp Yuan, T., Jian, C. and Zhang, R. Design of hee-diensional guidance law based on exended sae obseve fo hi-o-kill inecepos, 009, Poceedings of he IEEE Inenaional Confeence on Auoaion and ogisics, Shenyang, China, pp Yao, Y. and Wang, Y.H. Acceleaion esiaion of aneuveing ages based on exended sae obseve, 009, Conol and Siulaion Cene, Habin Ins of Technology, Habin, China.. Wang, Y.H. Design of laeal hus and aeodynaics blended conol syse based on auo disubance ejecion conolle, 009, Conol and Siulaion Cene, Habin Ins of Technology, Habin, China.. Song, J.., Gan, Z.X. and Han, J.Q. Sudy of acive disubance ejecion conolle on fileing, Conol and Design, 003, 8, (. 3. in, F., Sun, H., Zheng, Q.. and Xia, Y.F. Novel exended sae obseve fo unceain syse wih easueen noise, Conol Theoy and Applicaions, 005,, (. 4. Wang, Y.H., Yao, Y. and Ma, K.M. A new ype exended sae obseve fo syse wih easueen noise, 008, Poceedings of he IEEE Inenaional Confeence on Auoaion and ogisics, Qingdao, pp Wang, Y.H., Yao, Y. and Ma, K.M. Analysis and applicaion of Fal funcion file, Elecic Machines and Conol, 00, 4, (. 6. in, Y.P. Esiaion and Engageen of Ballisic Tage wih Vaiable Tajecoy, 03, Docoal disseaion, Naional Chung Hsing Univesiy, Taichung, Taiwan.

ÖRNEK 1: THE LINEAR IMPULSE-MOMENTUM RELATION Calculate the linear momentum of a particle of mass m=10 kg which has a. kg m s

ÖRNEK 1: THE LINEAR IMPULSE-MOMENTUM RELATION Calculate the linear momentum of a particle of mass m=10 kg which has a. kg m s MÜHENDİSLİK MEKANİĞİ. HAFTA İMPULS- MMENTUM-ÇARPIŞMA Linea oenu of a paicle: The sybol L denoes he linea oenu and is defined as he ass ies he elociy of a paicle. L ÖRNEK : THE LINEAR IMPULSE-MMENTUM RELATIN