5th International Conference on Advanced Materials and Computer Science (ICAMCS 2016)

Transcription

1 5th Intenational Confeence on Advanced Mateials and Copute Science (ICAMCS 016) Design and Siulation of Finite-tie Convegence with Ipact angle constaint Sliding-ode Guidance Law Kai Zhang, Suo-chang Yang, Kuan-iao Zhang, Yun-zhi Yao Peng Chen Depatent of Missile Engineeing Odnance Engineeing College Shijiazhuang China Keywods: Ipact angle constaint Finite-tie convegence; Sliding-ode guidance law Abstact. Fo issile falling angle constaint tagets to pevent taget evasive aneuves, issile taget elative otion elationship is established, based on the sliding ode vaiable stuctue theoy, the linea sliding suface, select adaptive powe eaching law is poposed with the dop angle constaint of sliding ode guidance law. By using Lyapunov theoy guidance law of stability analysis, the analysis of the guidance law with finite tie convegence. Siulation esults show that the designed guidance law is copaed to the taditional guidance law, the pefoance is oe supeio, has stong obustness, eet the design euieents. Intoduction In oden wa, as the issile weapons, with pecise taget effect, if at a cetain angle to hit the taget, will ealize the killing effect of wahead oe. Since the scholas Ki and Gide fo the fist tie in issile eenty guidance of angula constaint pobles, oe and oe scholas and expets in diffeent backgound, accoding to the the design of a vaiety of diffeent theoetical ethods guidance law with ipact angle constaint [1-3]. Above guidance law put fowad in the design of the angula constaint euieents, siulation studies have shown it to eet the euieents in tes of pefoance, but the guidance syste fo the issue of convegence ove tie liteatue eally did not ention. Thus, the angula constaint guidance law with finite peiod of tie will be cobined with the new diection of eseach [4]. The theoetical knowledge Definition1 Conside the following nonlinea syste [4]: x f ( x, t ), f (0, t ) 0, x R n (1) Aong the, f : U 0 R R in U 0 R is continuous, and U 0 is the oigin of x 0 an open neighbohood. In the definition of a nonlinea syste (1) the euilibiu point x 0 (local) finite tie convegence, is that of abitay initial tie t0 and the given initial state x(t0 ) x0 U and in the n pesence of a dependency x0 on the stopped tie T 0, akes the syste of euations (1) x0 to the initial state solution x(t ) ϕ (t ; t0, x0 ) is defined (and possibly not only), and li ϕ (t ; t0, x0 ) 0 t T ( x0 ) 0 if t > T ( x0 ), than ϕ (t ; t0, x0 ) () When t [t0, T ( x0 )], ϕ (t ; t0, x0 ) U /{0} in addition, the euilibiu point x 0 (local) finite tie stability of this syste is that it is Lyapunov stable and finite tie convegence in a 016. The authos - Published by Atlantis Pess 818

2 n neighbohood U U0 of the oigin. If the oiginu R is the euilibiu point of the global finite tie stability. Based on the finite tie contol theoy, Thee ae the following lea: Lea 1 conside the nonlinea syste (1), assue the existence of a definition in the neighbohood ˆ n U U0 R of the oigin of the sooth function V( x ), and thee ae ealc > 0 and0< λ < 1 the V( x ) in Uˆ is positive and V ( x) + cv λ ( x) inu ˆ is negative, the oigin of the syste is finite tie stable. Stable tie is deteined by the initial value x(0) x0 and the uppe bound fo V ( x0) Tx ( x0) (3) c(1 λ) Poof: V( x) in Uˆ is positive definite and V ( x) + cv λ ( x) in Uˆ is uppe sei negative definite V λ ( x) cv ( x), t 0 (4) On the integal solution can be got V ( x) V ( x0) c(1 λ) t, (5) 0 t Tx ( x0) when t T ( ) x x0, V( x ) 0, The finite convegence tie of the syste can be obtained V ( x0) Tx ( x0) c(1 λ) The poof is coplete. Relative otion odel of issile and taget Befoe the design of the guidance law, the need to establish a two-diensional odel of the elative otion of the issile and the taget. Fo the convenience of analysis, the assuption that the issile and the taget ae paticles, and the velocity of the issile is geate than the taget velocity, as a constant. T v t θ t v M θ Fig. 1 Scheatic diaga of the elative otion of pojectile Aong the, issile M and T ae consideed as a paticle; v is the issile speed, vt as speed of the taget; θ as issile tajectoy angle, θt as the taget heading angle; between the issile taget elative distance, as angle of issile taget line of sight (LOS), the povisions of all angles is a counteclockwise diection, othewise it is negative. Relative otion euation of pojectile: 819

3 (6) (7) vcos( θ) + vtcos( θt) vsin( θ) vtsin( θt) a θ v (8) a θ t t v t (9) Aong the, a and at as the issile and the taget of the noal acceleation is expessed, espectively. The foula deivation, available: v [ tsin( θt) vsin( θ)] + v tcos( θt) (10) + vθ sin( θ ) v cos( θ ) v θ sin( θ ) t t t + [ vcos( θ) vtcos( θt)] + vsin( θ) v cos( θ ) θ v sin( θ ) + v cos( θ ) θ t t t t t (11) Put The type of substitution (6) in (11) available + u + w (1) u w and espectively fo the issile and the taget acceleation in the line of sight coponent on the noal vecto. Cobining with (1) w u + (13) In the teinal guidance, t () < 0, 0 t (), t > 0 u u select x w w, w u x x+ (14) Guidance law design analysis Guidance law design In ode to accuately hit the taget, and obtain the desied Angle, accoding to the guidance weapon teinal guidance poble, can know the line of sight angula ate to zeo on behalf of the t () issile hit the taget unde ideal condition, the constaint conditions d 0 to be fulfilled ae given in the end, so take sliding ode switching function fo λv s x+ ( t () d) Aong the, λ > 0 and d as constaint Angle. (15) 80

4 Based on sliding ode vaiable stuctue contol theoy, in ode to ake the syste state speeding and eaching the sliding along the sliding ode suface, selecting the adaptive exponential ε α eaching law fo s s sgn( s), type: ε > 0, 0< α < 1. By the selection of eaching law ε α s s sgn( s) s (16) Due to the ss, > 0 sgn( s) s ss, < 0 Naely sgn( s) s s So thee ae ε α + 1 ss s Consideing the guidance phase α > 0 Theefoe, ε α + 1 ss s < 0 (17) Theefoe, eaching slide condition syste can each the sliding suface. Fo selection of the sliding ode suface, such as type (15), the deivative λv λvt s ( x + ) (18) Will type (16) into type (18), available λv λvt ε α x + s sgn( s) s (19) By the type available w u ε α λv λvt 0 x+ + s sgn( s) s+ (0) To sot out the α λvt u x + w + ε s sgn( s) + λv (1) The guidance law expession shows that the guidance law fo vaiable stuctue contol and the paaete petubation and extenal distubance obust sgn( s ), switching function is contained in the vaiable stuctue guidance law, the need to constantly switch contol uantity, and in the actual syste, thee is a cetain aount of tie lag, can cause a shaking of the actuato, a shaking of the issile is the jitte is actually, if the wobble is too big, can affect the accuacy of its hit taget. Vaiable stuctue guidance law and jitte is the inheent shotcoings, can't eliinate, only to decease it. So in this guidance law, sybolic function s s + δ is eplaced by a satuation function sgn( s) with high gain, the effect of educing jitte [5]. Such as type () α s λvt u x + w + ε s + λv () s + δ 81

5 Fo play yad distance t () and distance ate t (), active hoing issile can be easued o calculated by, while the passive hoing issile can not diectly get, need fo eal-tie estiation. If the taget is aneuveing taget, then its acceleation w geneally cannot get accuately, theefoe, it can be seen as intefeence estiate the uppe bound. Mobility poo taget fo sea and on the gound, can be diectly hypothesis w 0, this guidance law was α s λvt u x + ε s + λv (3) s + δ Finite tie convegence analysis Syste otion state can be divided into phases and eaching slide along the sliding ode otion stage, because the selection of guidance law of sliding ode suface fo linea sliding ode suface, so you just need to aive in stage of sliding ode fo finite tie convegence analysis. Fo eaching slide stage, the Lyapunov functionv1 s is selected, its deivation, too α + 1 V ss ss k s k s < 0 (4) 1 1 In the teinal guidance, t () < 0,0 t (), t > 0 select 1 1 ( α+ 1) ( α+ 1) 0 (5) By the lea 1, syste status, convege to the sliding suface in finite tie fo tie V kv k V kv, t > ( α + 1) 1 x0 V (,0) T1 k (1 α) 1 In conclusion shows that the designed guidance law can ake guidance syste state of convegence in a liited tie. The esults of siulation In this pape, design of finite tie convegence (FTCG) guidance law with fall Angle constaint is ainly used in fixed taget, to veify the pecision guidance pefoance, using MATLAB siulink odeling capabilities to validate the designed guidance law, and with the ipoveent of the taditional popotional guidance law biased popotional guidance law (BPNG) wee copaed. Selection of the initial position of issile (X0 Y0) (0, 0 ), taget initial position fo (Xt0 Yt0) (0, 000 ), the speed of the issile is 00 /s, incident Angle of constaint of 50º, the siulation esults ae shown in figue -5. (6) Fig. Missile tajectoy cuve Fig. 3 Elevation Angle diaga 8

6 Fig.4 Angle of the line of sight Fig. 5 Missile oveload scheatic diaga Can be seen fo figue, issile guidance in the last stage, copaed FTCG BPNG has a staighte flight tajectoy, the size of the issile pecision and speed is vey beneficial, to attack the gound fixed taget o the passive stapdown hoing issile gound fotifications, flat tajectoy fo pecise egulation of teinal guidance. By figue 3 shows two guidance law of convegence than expected Angle, elevation Angle ae basic to ealize the function of the guidance law is designed. Fo figue 4, you can see that the guidance law is designed in this pape FTCG in convegence condition when a issile hit the taget, and at the tie of the teinal guidance BPNG divegence Angle of the line of sight, lead to the stability of the issile, geatly influenced the pefoance of the guidance, the seious influence issile accuacy. Can be seen fo figue 5, FTCG oveload ae uch salle than actual application oveload, and the actual oveload BPNG oveload is geate than the set of available. The designed guidance law fo the pespective of a issile oto oveload is easy to ipleent, ensue the issile in the pocess of flying to the taget to aintain the stability of the stong. Based on analysis, this pape design the guidance law of pefoance due to the taditional offset popotional guidance law, has stonge obustness and bette. Suay Fall fo issile teinal guidance Angle constaint poble, this pape selected the linea sliding ode suface, designed a kind of finite tie convegence with fall Angle constaint sliding-ode guidance law, analyze the finite tie and poves that the and to weaken the chatteing phenoenon analysis, though siulation copaison, in this pape, the guidance law designed siple fo, has stong obustness and good guidance pefoance, has wide eseach space and pospect. Refeences [1] Ki M, Gide k. Teinal guidance fo ipact attitude Angle constained flight tajectoies [J]. IEEE the Tansactions on Aeospace and Electonic Systes, 1973, 9 (6) : [], scene, tang jiao, etc. A biased popotional guidance law based on fall Angle constaint [J]. Jounal of Beijing institute of technology, 014, 34 (3) : [3] zhang xu, hu-in lei, Ceng Hua XiaoZengBo, Ye Jikun, etc. With fall Angle constaint adaptive popotional guidance law [J]. J solid ocket technology, 011, 34 (6) : [4] Xiong Shaofeng, wei-hong wang, liu xd, sen, wang wu. Consideing the dynaic chaacteistics of issile autopilot with attack Angle constaint guidance law [J]. Contol and decision, 015, 30 (4) :

7 [5] Zhang Yunxi, Sun Mingwei, Zengiang vaiable stuctue guidance law fo finite tie convegence [J] Contol Theoy and Applications, 01, 9 (11):