Ijeas0304034

International Journal of Engineering and Applied Sciences (IJEAS) ISSN: 2394-3661, Volume-3, Issue-4, April 2016

Missile Guidance Law Design via Backstepping Technique Hamza Zaidi, Panlong Wu, Ali Bellahcene 

Abstract — In this paper a Back-stepping Control technique is proposed for command to line-of-sight missile guidance law design. In this design, the three-dimensional (3-D) CLOS guidance problem is formulated as a tracking problem of a time-varying nonlinear system. Simulation results for different engagement scenarios illustrate the validity of the proposed Backstepping-based Guidance Law.

Table I: Definition of symbols

Symbol

Description Yaw angle of target

t t m m mc t t m m  

Index Terms—Command line-of-sight (CLOS), Backstepping Control system, missile guidance law.

I. INTRODUCTION The Concept of command to line-of-sight (CLOS) guidance is to oblige (force) a missile to fly as nearly as possible along the instantaneous line-of-sight (LOS) between the land tracker and the target. If the missile can continuously stay on the LOS, missile will intercept the target. To set demanded accelerations for the missile, a guidance controller is used at the ground station to take computation of tracker information about the missile and target position, angular velocity and acceleration of the LOS. These acceleration commands can then be transmitted to the missile by a radio link. The CLOS guidance has been identify as a low-cost guidance concept because it conformance placement of avionics on the launch platform, as opposed to mounting on the expendable weapons [1], [2]. Theoretically, the missile-target model is nonlinear and time-varying. Many different guidance laws have been developed over the years, and with the advent of highly maneuverable targets, research on improved guidance laws is continuing [3]–[5].

Pitch angle of target.

Yaw angle of missile.

Pitch angle of missile Roll angle command.

Azimuth angle of LOS to target.

Elevation angle of LOS to target.

Azimuth angle of LOS to missile Elevation angle of LOS to missile m t

m t

g

Gravity acceleration.

Axial acceleration of missile

ax a yc

Yaw acceleration command

a zc a yt

Pitch acceleration command.

atz Rm Rt

Pitch acceleration of target.

Yaw acceleration of target.

Missile range from ground tracker.

Target range from ground tracker.

II. PROBLEMATIQUE OF THE THREE-DIMENSIONAL CLOS GUIDANCE The three-dimensional CLOS guidance problem shown in Fig. 1 is a well-known guidance model [2], which involves guiding the missile along the LOS to the target. The three-dimensional CLOS guidance model in [5, 7] will be repeated here for convenience. The following description in Table 1 will be adopted to derive the dynamic equations of missile.

In this study, a Backstepping control system is proposed for commanding line-of-sight CLOS. The Lyapunov stability theorem is used to ensure the stability of the control system.

Simulations results demonstrate the effectiveness of the proposed control.

Zm

ZI

This paper is organized as follows. Formulation of missile-target engagement is described in Section II. The design procedures of the proposed Backstepping guidance system are constructed in Section III. Simulation results are set to confirm the effectiveness of the proposed control system in Section IV. Conclusions are drawn in Section V.

Ym

a zc ayc Missile

m

X m

ax

m

Target

RM



Ground tracker



m



RT

y t

m

yt

YI

m



t

xt

Hamza Zaidi, School of Automation, Nanjing University of Science and Technology, Nanjing, China, (+86) 13605146954.

x

m

XI

Panlong Wu, School of Automation, Nanjing University of Science and Technology, Nanjing, China, (+86) 2584315172.

Fig. 1. Three-dimensional missile-target engagement diagram.

Ali Bellahcene, School of Automation, Nanjing University of Science and Technology, Nanjing, China, (+86) 15205191440.

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Missile Guidance Law Design via Backstepping Technique

 xm  c mc m  smc s mc m  cmc s m  cmc s mc m  smc s m   ax   0   y   c s  s s s  cs c  c s s  s c   a    0  mc m m mc m mc m m mc m   yc     m  m m  zm   s m   azc   g  smc c m cmc c m

0  m  cmc vm c m  smc vm c m  a yc    (1)     s v   a    gc v  c  v mc m mc m m m  m    zc   2 The origin of the inertial frame is located at the ground base.    x  f x , t  g j x   uT j  The axis ZI is vertical upward and the XI - YI plane is (5) j 1 horizontal. The origin of the missile body frame is fixed at the z  h x , t  center of mass of missile, with the XB axis forward along the where missile centerline. The dynamics of the missile in the inertial x4   frame can be represented [2] as described in (1) in the top of   x5 the page.   A tracking output is defined in order to convert the CLOS   x6   guidance problem into a tracking problem. The LOS frame is a x t cx7 cx8  f  x, t    shown in Fig. 2 in which the origin of the three-dimensional a x t sx 7 cx8   space is located at the ground base. The XL axis forwards     a t sx  g x 8   along the LOS to the missile, and the YL axis is horizontal to   0 the left of the XL - YL plane. Then, the coordinates indicated in 1  2 2 2 2 Fig. 2 represent the missile position in the LOS frame, and  gcx8 x4  x5  x6  they are related to through rotations as follows: 0   c t 0    z1    s t 0 (2)    z    s c  s s c  t t t t t  2    0 T   The tracking output is defined as z [ z1 , z 2 ] . Since z1 and  scm sx7 cx8  ccm sx7 g1 x     s sx sx  c cx  cm 7 8 cm 7 z 2 cannot be measured directly, these quantities must be   s  cos x cm 8 computed indirectly using the polar position data of the   1 2 2 2 2 c  missile available from the ground tracker as x  x  x cx 5 6 8  cm 4  1 Rm c(   t ) s 2 2 2 2   z1    scm x4  x5  x6  z    (3)  R ( s (     ) c   c (     ) s  c   z t t t t  2  m  0     ZL 0   YL   0    ccmcx7 sx8  scm sx 7   g 2 x     c sx sx  s cx  cm 7 8 cm 7 Missile   Rm c  cx cm 8   1 2 2 2 2   s  z1 x  x  x cx cm 4 5 6 8   1  ccm x42  x52  x62 2  z2 Ground tracker Rp  x1s t  x2 c t  h1 ( x, t )  and hx, t       x s c  x s s  x c  XL   h x , t 2 t t 3 t  2   1 t t Target The objective of CLOS guidance control is to find a control Fig. 2. Definition of tracking output. law to drive the tracking output z to zero. Eq. (5) can be Note that z 2 represents the distance from the missile to the rewritten as  z1   F1 ( x, t ) G11 x, t  G12 x, t  LOS. Therefore, the missile will eventually intercept the z    F x, t   G x, t  G x, t u t  22  2  2   21  target if the tracking output z1 is driven to zero. The   z F x, t   G x, t   u t  (5) three-dimensional CLOS guidance problem therefore can be







 

seen as a tracking problem. Define xx1 x2 x3 x4 x5 x6 x6

xm

ym

uT  uT

uT



1

2

x m

zm

 a T

yc

y m azc



T







 

where F1 x, t   X 02  h1 , F2 x, t   X 02  h2 G11 x, t   X1  X 0  h1, G12 x, t   X 2  X 0  h1 ,

x7 

T

T zm  m  m 

G21 x, t   X 1  X 0  h2 and G22 x, t   X 2  X 0  h2 X 0 h1   t x1c t   t x2 s t  x4 s t  x5c t   t R p c t   t z2 s t  x4 s t  x5c t

(4)

Using the previous equations, (1), (2), and (4) can be put into the following dynamic equations of missile in state-space form:

X1 X 0h1  scmsx8 s( x7   t )  ccmc( x7   t ) X 2 X 0h1  ccmsx8 s( x7   t )  scmc( x7   t )

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International Journal of Engineering and Applied Sciences (IJEAS) ISSN: 2394-3661, Volume-3, Issue-4, April 2016

X1 X 0 h2  (c t cx8  s t sx8c( x7   t ))scm  ccm s t s( x7   t ) X 2 X 0 h2  (c t cx8  s t sx8c( x7   t ))ccm  scms t s( x7   t ) X 0 h2  ( t s t s t  t c t s t ) x1  ( t c t s t   t c t s t ) x2  t x3 s t  x4c t s t  x5 s t s t  x6 c t   t z1s t  t RP  x4 c t s t  x5 s t s t  x6 c t

Step 5: The second Lyapunov function is defined as

(17) V2 t   V1 t   0.5 e22 t  Differentiating (17) and using (14) and (15), it is obtained that V2 t   V1 t   e2 t e2 t 

  K1e12 t   e2 t e1 t   zd t   k1e1 t   F x, t   Gx, t u t 

(18)   K1e12 t   K 2e22 t   0  Since V2 e1 t , e2 t   0 , it means that e1 t  and e2 t  are bounded. Now define the term: (19) t   K1e12 t   K 2e22 t   V2 e1 t , e2 t  then

X 02 h1  (2 t t s t  t c t ) R p   t2 z1  (2 t t c t  t s t ) z2  2 t R p c t  2 t z2 s t  a x (t )cx8 s( x7   t )

X 02 h2  (t   t2 s t c t ) R p  t z1s t  ( t2 syt s t  t2 ) z2  2t R p  2 t z1s t  ( sx8c t  cx8 s t c( x7   t ))  a x (t )  gc t  RP  ( t s t c t  t s t c t ) x1  (t s t s t   t c t c t ) x2  t x3c t  x4 c t c t  x5c t s t  x6 s t (7)  x4 c t c t  x5c t s t  x6 s t   t z1c t  t z 2 and

t

  d  V e 0, e 0  V e t , e t  2

1

(20)

2

t

t 

(21)

0

 t  is bounded, so by using Barbalat’s Lemma [8], it Also 

(8)

X j   g j ,i ( x)   xi

2

lim   d  

X 0   t   f i ( x, t )   xi i 1

2

Since V2 e1 0, e2 0 is bounded and V2 e1 t , e2 t  is non-increasing and bounded, it can be obtained

n

m

1

0

j  1,2

can be

i 1

shown that lim t   0 . t 

This will imply

that e1 t  and e2 t  converge to zero as t   .Therefore, the

where f i ( x, t ), g j ,i ( x) and xi are the ith components of f ( x, t ), g ( x) and x respectively.

Backstepping Guidance law formulated in (16) is asymptotically stable. The configuration of the proposed Backstepping Guidance Law is shown in Fig. 3.

III. BACKSTEPPING-BASED GUIDANCE LAW DESIGN Assuming that all parameters of the system (6) are known, the design of Backstepping control for the guidance law is described step-by-step as follows: Step 1: Define the tracking error (9) e1 t   zd t   zt 

u

Missile Maneuver

 t , t

 m , m

Limiter

Tracking Output

Target Maneuver

z

where z d t  is a desired tracking output, Then the derivative of tracking error can be represented as e1 t   zd t   zt 

(10)

Define the following stabilizing function  t   zd t   K1e1 t 

(11)

Backstepping Technique

The zt  can be viewed as a virtual control in above equation.

Fig. 3. Backstepping Guidance System.

IV. NUMERICAL SIMULATIONS

k1 0 where K1    and k1 is a positive constant. 0 k1  The first Lyapunov function is selected as Step 2: Define

V1 t   0.5 e12 t 

(12)

e2 t    t   zt 

(13)

Then the derivative of V1 with respect to time is V t   e t e t   k e2 t   e t e t  1

1

1

1 1

1

2

Step 3: The derivative of e2 t  is given as e2 t    t   zt   zd t   k1e1 t   F x, t   Gx, t ut 

In this section, simulations are performed to illustrate the efficiency of the proposed Backstepping guidance law. In order to assess the performance characteristics in a closed-loop engagement scenario, it is important to specify target dynamics. The simplified dynamics of target motion can be given in the inertial frame as follows: xt  a yt s t  a zt s t c t yt  a yt c t  a zt s t s t

(14)

zt  a zt c t  g  t  a yt vt c t

(15)

(22)

t  a zt  gc t  vt In this paper, three simulation scenarios are examined to justify the effectiveness of the proposed design method. The simulation data and parameter data used for simulation are summarized in Table II.

Step 4: If all dynamics system are known, a Backstepping guidance law can be formulated as 1 uB  Gx, t  zd t   k1e1 t   e1 t   K 2e2 t   F x, t  (16)

 k 2 0 where K 2    and k 2 is a positive constant. 0 k 2 

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Missile Guidance Law Design via Backstepping Technique Table II. Scenario and parameter data used for simulation

States xt 0 , yt 0 , zt 0  xt 0 , y t 0 , zt 0 

m m/s  t 0 ,  t 0  deg  xm 0 , ym 0 , z m 0  m  xm 0 , y m 0 , zm 0  m/s deg   m 0 ,  m 0 

Scenario 1 2500,5361.9,1000

Scenario 2 5200,400,3000

0,-340,0

- 340, 0,0

- 90,0





 



14.56,5.43,10.01

28.21,34.81,26.52

70.84,151.92,28.32

129.65,12.87,92.42

250,250,400

65, 9.59

20.34, 32.65

a yt

m/s 

 5g   5 g

0  t  2.5

a zt

m/s 

 g   5g

0  t  2.5

2

2

2

0  t  10 100   44.1 t  10

 0g  0.5 g

t  2.5

 g   g

t  2.5

m

 

 

  

The simulation results for scenarios 1, 2, and 3 are depicted in Figs.5–7, respectively.

1 s

Table III. Miss Distance (m)

Rˆm



  1 s

ˆ t

z1

z2

0.064 



t









355 . 3 s

Fig. 4. Block diagram representation of estimation algorithm for guidance information.

The first and second scenarios describes an anti-aircraft scenario. The third one represents an anti-missile scenario.

Considered a 30g g  9.8m / s 2 maneuvering limiter to limit the missile’s maneuverability. The pitch and yaw autopilot dynamics are selected to be second order linear time-invariant systems and the ground tracker to be a simplified differential tracking system with damping ratio 0.6 and nature frequency 6π rad/s as shown in Fig. 4. The estimated values of  t ,  t , t and t , also the measurement data of  and  , are provided by the ground tracker. To evaluate the influence of measurement noise, random noises with magnitude between ± 0.3 deg are included. m/s2 The Backstepping guidance law presented in (16) is simulated for the same engagement scenarios. This study adopts the following Backstepping control law: 1  uLB  Gx, t  zd t   K1e1 t   e1 t   K2e2 t   F x, t 



0  t  2.5 t  2.5

6

 

t  2 .5

0.6

Equation (3)

m

0  t  2 .5

50

355 . 3 s



45, 54.73

0t 2 340   44.1 t  2

m/s 

0.064



 90,0

14.32, 39.34, 3.36

ax



0,0,500

180,0

Guidance command frequency (Hz) Autopilot damping ratio Autopilot naturel frequency (rad/s) t

Scenario 3 5200,5000,7500



Scenario 1

Scenario 2

Scenario 3

1.8059

2 .3319

0 .8678

V. CONCLUSION In this paper, a Backstepping control method is applied for the CLOS guidance law design. Simulation results show that the Backstepping guidance law can achieve satisfactory performance and smooth missile trajectories for different engagement scenarios. In addition, from Table III we can notice those small miss distances.

REFERENCES [1]

[2]

[3] [4]

[5]

[6]

[7]

7 0  20 0 K1   and K 2     0 7  0 20

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International Journal of Engineering and Applied Sciences (IJEAS) ISSN: 2394-3661, Volume-3, Issue-4, April 2016

z1

z1

z2

z2

z2 )

(a) Tracking output ( z1 and z 2 )

azc (m/s2)

azc (m/s2)

ayc (m/s2)

ayc (m/s2)

(a) Tracking output ( z1 and

(b) Acceleration command ( a yc and a zc )

(b) Acceleration command ( a yc and a zc )

(c) Missile-target trajectory

(c) Missile-target trajectory

Fig .5 Engagement scenario 1 with Backstepping guidance law.

Fig .6 Engagement scenario 2 with Backstepping guidance law.

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Missile Guidance Law Design via Backstepping Technique

z1

z2

z2 )

azc (m/s2)

ayc (m/s2)

(a) Tracking output ( z1 and

(b) Acceleration command ( a yc and a zc )

(c) Missile-target trajectory Fig .7 Engagement scenario 3 with Backstepping guidance law.

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