Robust stability check of fractional control law applied to a LEO (Low Earth Orbit) Satellite

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Robust stability check of fractional control law applied to a LEO (Low Earth Orbit) Satellite Powered By Docstoc
					                                                          (IJCSIS) International Journal of Computer Science and Information Security,
                                                          Vol. 8, No. 3, June 2010




                 Robust stability check of fractional control
                          law applied to a LEO
                        (Low Earth Orbit) Satellite

                                                  Ouadiâ EL Figuigui 1, Noureddine Elalami1
                                           1
                                               Laboratoire d’Automatique et Informatique Industrielle
                                                                 EMI, Morocco
                                                  (elalami@emi.ac.ma,       elfiguigui@gmail.com)




  Abstract: The use of the differentiation and integration of                In [9], the author introduce a fractional control law
fractional order or non-integer order in systems control is               aiming to stabilize the attitude movement of an earth pointing
gaining more and more interests from the systems control                  satellite under the effect of external disturbances, using 3-axis
community. In this paper we will briefly describe the LEO (Low            reaction wheels as actuators. The dynamics of the satellite is
Earth Orbit) satellite systems and recall the theoretical aspects
                                                                          described by a quasi-bilinear multivariable coupled system. In
of robust stability check procedure. This procedure will be
applied to a LEO satellite that is under the fractional control           this study, we apply the robust stability check procedure to
law. Numerical examples will be analyzed and presented at the             the fractional control law system presented in [9].
end of this document                                                         This paper is organized as follows: In the next section, the
                                                                          general nonlinear equations model of an Earth-pointing
Keywords: fractional control, LEO satellite, robust stability             satellite attitude dynamics is developed. Then, in section III,
                                                                          the attitude equations are linearized with the nadir attitude
                        I. . INTRODUCTION                                 position as the origin. This leads to a quasi_bilinear
                                                                          multivariable coupled system. Then, for small maneuvers, the
                                                                          quasi_bilinear term is neglected in order to obtain a linear
 Recently, a lot attention was given to the problem of                    system. In section IV, we recall some theoretical aspects of
fractional calculus. There were several works in this area                robust stability checking procedure. Forward, in section V, we
[11],[12],[13],[23]..etc and the author in [1] is presenting for          apply this procedure to the LEO satellite system which is
the very first time the robust stability checking procedure for           subject to fractional control law and we share the related
uncertain fractional order linear time invariant (FO-LTI)                 Numerical results. The conclusion is provided in the last
systems with interval coefficients described in state form. The           section.
application of such procedure to LEO satellite was new idea.
   The role of an attitude control system for an Earth-pointing
                                                                                  II. NONLINEAR MODEL OF THE SATELLITE
spacecraft is to maintain the local-vertical/local-horizontal
                                                                                                    ATTITUDE DYNAMICS
(LV/LH) attitude with the presence of different environmental
disturbances. Most of the time in order to ensure a precise                  The attitude motion of the satellite is represented by the
pointing, the satellite requires reaction wheel system to                 Euler equations for the rigid body motion under the influence
counteract the attitude drifts caused by those perturbations,             of external moments, such as the control moment generated by
especially the seculars ones, like torques caused by passive              the actuators. Attitude control requires coordinate
gravity gradient, aerodynamic and solar forces. The reaction              transformation from LV/LH to The Satellite Coordinate
wheels are governed by control laws which dictate the                     System (SCS) system defined as follows: The LV/LH
amount of torques required to eliminate the drift caused by               coordinate system ( X 0 , Y0 , Z 0 ) is a right orthogonal system
external factors [9].                                                     centred in the satellite’s centre of mass (SCM). The roll
   In the attitude control design, different approaches have              axis, X 0 , points along the velocity vector, the pitch axis, Y0 ,
been used, i.e. Proportional Integral Derivative (PID) [18],              points in the direction of the negative orbit normal and the
[25], LQR [17], pole placement techniques [14], etc. All those            yaw axis, Z 0 , points in the nadir direction. The SCS system
methods, expressed in different attitude error terminologies
                                                                          ( X S , YS , Z S ) is a right orthogonal system centred in the
are using, very often, Euler angles for small attitude
commands while for large attitude maneuvers it makes use of               SCM, parallel to principal moment of inertia axle of satellite.
quaternion [7] and direction cosine errors [18].                          Z S is parallel to the smallest moment of inertia axis; Y S is




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                                                                                                     ISSN 1947-5500
                                                                        (IJCSIS) International Journal of Computer Science and Information Security,
                                                                        Vol. 8, No. 3, June 2010



parallel to the largest moment of inertia axis. X S completes                                                                     p     S
the right orthogonal system.                                                                                                                                                                     (5)
                                                                                                                                  q   C   C S
   Consider a satellite with three reaction wheels. The general
nonlinear attitude dynamics model can be described as [18],                                                                       r   C C   S
[27] and [21]:                                                                            and
           IωIS ( t ) =
                            
                             hw( t ) ωIS ( t )∧IωIS ( t )                                                            IS  VH  TVH / S ( 0  0 0 )T
                                                                                                                            S

                                                                             (1)
                                                    s
                              ωIS ( t )∧hw( t ) + M g ( t ) + P( t )
                                                                                                    III. LINEARIZED EQUATIONS OF MOTION
where
                                                                                             Assuming small variations of the Eulerian angles  ,  ,  ,
    I : Total moment of inertia matrix for the satellite
                                                                                          then the transformation matrix becomes:
        without reaction wheels inertia (3×3).
                                                                                                                                          1                         
     IS (t ) : Inertial angular velocity vector in SCS (3×1).                                                           TVH / S                       1                                (6)
                                                                                                                                                                       
    hw (t ) : Angular momentum vector of the wheel                                                                                      
                                                                                                                                                                   1 
                                                                                                                                                                        
      cluster(3×1).
    M g (t ) : Torques due to Earth’s gravity gradient (3×1).
       s
                                                                                             On the other hand, one obtains that:
    P( t ) : Disturbance torque due to aerodynamics, solar                                                                       
                                                                                                                            p ,   q ,  r
                                                                                                                                                                                                (7)
      pressure and other environmental factors. It is assumed
                                                                                          and
      to be [18],[21]:
                                                                                                                     
                                                                                                x    0 ,  y    0 , z    0
                                                                                                                                                                                          (8)

                                   4 106  2 106 sin(0t )                          Then the equations of motion (1) and (5) can be linearized
                                                                             (2)        about the origin, giving a quasi_bilinear multivariable system:
                         P (t )   6 106  3 106 sin(0t ) 
                                   3 10  3 10 sin(0t ) 
                                         6        6                                           
                                                                                                x( t ) = Ax( t ) + Bu( t )
                                                               
                                                                                                                              t
where  0 is the orbital angular rate.
                                                                                                                  3
                                                                                                          +B    ∑ ∫( ξ )dξ
                                                                                                                  ui                                    ( Ci x( t ))
   To keep the satellite attitude earth pointing, the SCS axes                                                  i =1          0                                        ; x(0)  x0               (9)
must remain aligned with LV/LH axes. The transformation
matrix, expressed with Euler Angles  , ,  , respectively,
                                                                                                                      t


roll, pitch and yaw angles, is given by [18], [27]:
                                                                                                          +G         ∫ξ )dξ
                                                                                                                     u(                          + BP( t )
                                                                                                                     0

                           C C              C C         S                         where:
                                                                 
        TVH / S     C S  S  S C C C  S  S S S  C       (3)                    3 t                          t        
                     S  S  C S C  S  C  C S S C C 
                                                                 
                                                                                           B   u i ( )d (C i x(t ))  G u( )d  :Quasi-bilinear term,
                                                                                             i 1  0
                                                                                                                                      
                                                                                                                              i        
where: S and C are respectively the sine and the cosine.                                                
                                                                                           u(t )  hw (t ) : Control action,
                                 s
   The gravity gradient torque M g (t ) is given by [18], [5]:                                                      
                                                                                           x(t )  ( ,  , ,  ,  , ) : State vector,
                   3 2
             M gx   0 ( I z  I y ) sin(2 ) cos 2 ( )                                    System matrices A, B, Ci , G defined as
                   2
                   3 2
                                                                       (4)                                    0          0   0    0   0     0                0    0   0 0         0    0
             M gy   0 ( I z  I x ) sin(2 ) cos( )                                                        0          0   0    0   0     0                0    0   0 0         0    0
                   2                                                                                                                                                                    
                   3 2                                                                                        0          0   0    0   0     0                0    0   0 0         0    0
             M gz   0 ( I x  I y ) sin(2 ) sin( )                                                  C1                                             C2                             
                   2                                                                                          0          0   0    0   0     0                0   0   0 0         0    1
                                                                                                              0        0   0    0   0     1               0    0   0 0         0    0
To describe the satellite kinematics, two important factors are                                                                                                                         
to be taken into account: angular velocity of the body axis                                                   0          0   0    0   1     0                0
                                                                                                                                                                    0   0 0       1   0
                                                                                                                                                                                           
frame (SCS) with respect to the reference LV/LH                                                              0       0 0              0    0      0             0     0   0 
                                                                                                             0                                                   0         0 
frame VH ( p, q, r ) , and the angular velocity of the body frame                                                                                 0
        S
                                                                                                                      0 0              0    0                           0        
                                                                                                                                                   
with respect to inertial axis frame  IS ( x ,  y ,  z ) . These                                          0       0 0              0    0      0             0     0   0 
                                                                                                        C3                                                  B                
                                                                                                             0       0 0              0    1     0            1/ I x 0   0 
quantities are related to the derivative of the Euler angles as
                                                                                                             0       0 0            1    0      0             0 1/ I y  0 
follows [18]:                                                                                                                                                                  
                                                                                                             0       0 0              0    0      0             0
                                                                                                                                                                        0 1/ I z 
                                                                                                                                                                                  




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                                                                                                                                           ISSN 1947-5500
                                                                                       (IJCSIS) International Journal of Computer Science and Information Security,
                                                                                       Vol. 8, No. 3, June 2010



                    0          0      0          1                     0      0 
                    0          0      0          0                     1      0 
                                                                                                               B. Robust Stability Check of Fractional System with
                    0          0      0          0                     0      1                                Interval Uncertainties [1]:
                 A 2                                                                  
                    40  1   0      0          0                     0 0 (1   1 )                     We consider the following FO-LTI system with interval
                    0        302 2 0           0                     0      0                          uncertain:
                                                                                       
                    0
                               0     0  3 0 (1   3 )
                                       2
                                                                        0      0       
                                                                                                                                  X ( α ) ( t ) = ΛX ( t ) + Dw( t )                  (13)
                                                                                                           where:
                                         0             0 0 
                                         0                                                                 α is non integer number;
                                                       0 0      
                                         0             0 0 
                                      G                         
                                                        0 0 / Ix 
                                                                                                                            [ ] [
                                                                                                            Λ ∈Α I = Λ, Λ = Λc - ΔΛ, Λc + ΔΛ with            ]
                                         0
                                         0                                                                     Λ + ΔΛ
                                                        0 0                                               Λc =            is a center matrix (normal plant without
                                                                                                                 2
                                         0 / Iz
                                                       0 0      
                                                                                                            uncertainties)
                  (             )
where σ i = I j - I k I i for the                        i, j, k  index          sets (1,2,3),                     Λ-Λ
(2,3,1), and (3,1,2) .                                                                                      ΔΛ =           is a radius matrix correspondence interval
                                                                                                                      2
   Moreover, assuming that the angular velocity                                                                uncertainties.
components p , q and r are also small, and for slight
                                                                                                              The stability condition for system (13) is:
manoeuvres, one can neglect the quasi bilinear term.
                                                                                                                                     π
   The equation of motion (9) can then be written in the                                                      min arg( λi ( Λ )) > α                        Λ
                                                                                                                                         ; i = 1,2, , N , ∀ ∈A I
                                                                                                               i                     2
standard form of a linear equations system:
                                                                                                             In the following subsection, we describe briefly the
                                                                                                           procedure of checking robust stability using the minimum
                
                x( t ) = Ax( t ) + Bu( t ) + BP( t ); x( 0 ) = x0                           (10)           argument phase. We proceed by introducing two important
                                                                                                           lemma:
            IV. ROBUST STABILITY CHECK OF                                                                        1. Lemma 1 [1]:

                                                                                                                                                                             
                                                                                                                 Define a sign calculation operator evaluated at Λc such as:
                                                                                                                                                                           
             FRACTIONAL SYSTEM WITH INTERVAL                                                                                                                   T
                UNCERTAINTIES - MATHEMATICAL                                                                                  P i : sgn uire vire  uiimviim            (14)
                                 ASPECTS
   In this section, we’re recalling the definition of fractional                                           where u ire , vire , u iim and v iim are eigenvectors corresponding to
system; we’re presenting the robust stability checking                                                     ith eigenvalue of Λc .If P i is constant for all ΛI , ΛI ∈Α I , then
procedure afterwards.                                                                                      the lower and upper boundaries of the real part of ith interval
                                                                                                           eigenvalue are calculated as:

    A. Definition                                                                                                              λre = θ ire (Λc - ΔΛ  P i )
                                                                                                                                i                                                         (15)
   In this paper, we consider the Riemann-Liouville
definition, in which the fractional order integrals are defined                                            where  ire  is an operator for selecting the ith real eigenvalue
as                                                                                                         and C  A  B are c kj  a kj bkj ,and as:
                       -μ            1 t
                      Da f ( t ) =        ∫t -ξ ) μ -1 f ( ξ )d ( ξ )
                                   Γ( μ ) a
                                            (                            0                   (11)                                 λre = θ ire (Λc + ΔΛ  P i )
                                                                                                                                     i                                                    (16)

   While the definition of fractional order derivatives is                                                       1. Lemma 2: [1]
            d                                                                                                    Defining a sign calculation operator evaluated at c such
Daμ f ( t ) =
            dt a
                  [
                D (1 μ ) f (t )         ]                                                                  as:
               1    d t                                                                        (12)
          Γ ( 1 μ ) dt ∫
        =                  ( t ξ ) μ f ( ξ )d ( ξ )                                                                                           [
                                                                                                                                 Q i := sgn (u ire viim + u iim vire )
                                                                                                                                                                       T
                                                                                                                                                                           ]              (17)
                         a

                ∞
where Γ ( x ) = ∫ y x - 1e
                                                      -y
                0                                           dy          is      the         Gamma          if Q i is constant for all ΛI , ΛI ∈Α I , then the lower and upper
                                                                                                           boundaries of the imaginary part of ith interval eigenvalue are
function, (a, t )  IR 2 with a  t and 0    1 is the order of
                                                                                                           calculated as:
the operation.
   For simplicity we will note D  f (t )                        or f u (t ) for D0 f (t )                                       λim = θ iim (Λc - ΔΛ  Q i )
                                                                                                                                   i                                               (18)




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                                                                                                                                         ISSN 1947-5500
                                                                  (IJCSIS) International Journal of Computer Science and Information Security,
                                                                  Vol. 8, No. 3, June 2010




where  iim  is an operator for selecting the ith imaginary                      where w(t )  BP(t ) is the perturbation term, and u (t ) is the
eigenvalue, and as:                                                                 fractional control law applied in order to stabilize the system
                                                                                    (10), given by:
                                λim = θ iim (Λc + ΔΛ  Q i )            (19)                              u(t )  K ( x(t )  xr )( )                 (22)
                                 i
                                                                                    where x r is the attitude reference, equal to zero for nadir
                                                                                    pointing.
   So, using Lemmas 1 and 2, it is easy to calculate the lower
                                                                                       The linear fractional system is obtained in the form:
and upper boundaries of interval eigenvalue separately in real
                                                                                                 x( t )  Ax( t )  BKx( t )(  )  w( t ) ; x(0)  x0
                                                                                                                                                      (23)
part and imaginary part. From above lemma, if P i and Q i ,
                                                                                       In the following, only the fractional orders such as
i =1,…, N are calculated, then, interval ranges of
                                                                                      1 / p , p   * will be considered. Then
eigenvalues are finally calculated as:
                                                                                                                                 (1 )
                                                                                                  x(t )  Ax(t )  BKx(t )
                                                                                                                                  P
                                                                                                                                          Dw(t )             (24)
                      i           i
                                      {[
                     λI ∈Γi I := λre , λre + j λim , λim
                                        i
                                                ] [
                                                i     i
                                                                 ]}     (20)
where j represents imaginary part. We define                                           The equation (24) can be written into the following form:
                                                                                                 X ( 1 / p )( t )  X ( t )  Dw( t )        (25)
               *  inf( min argi () )          i  1,...N
                            i                                                          We note:
   Since the stability condition is given as    / 2 , if we                                                x  1 P   t   x(t ) ,
                                                        *                                                                *0
                                                                                                                         
find sufficient condition for this, the stability can be checked.                                                        
For calculating  * , the following procedure can be used:
                                                                                                             x  1 P   t   x
                                                                                                                        *1           1 
                                                                                                                                    p
                                                                                                                                        
                                                                                                                                              t         (26)
  P1. Calculate Pi and Qi for i =1,…, N .                                                                              
                                                                                                                               
                                                                                                                                         1
  P2. Calculate  ire ,  ire , im , and im for all i   ,2, , N .                                    x1 P  
                                                                                                                    * p 1               1 
                                                          1
                                                                                                                             t   x            t 
                                                                                                                                         p
                                                                                                                                            
                                                                                                                   
                                 i         i

  P3. Find arguments of phase of four points such as                                                               
                                                          
                                                                                                                                                          T
                   i1   ire , im , i2   ire , im
                                                                                    and                1  *0   1  *1               1  * p 1 
                                                                                            X (t )   x   (t ),  x   (t ),  ,  x P  (t ) (27)
                                                                                                           p           p
                                                                                                                                               
                                   i                    i


                                  i             (
                   i3   ire , im , φi4 = ∠λre , λim
                                                i     i
                                                             )                                       
                                                                                                     
                                                                                                               
                                                                                                               
                                                                                                                     
                                                                                                                     
                                                                                                                             
                                                                                                                                                     
                                                                                                                                                        
                                                                                               0       Id       0 0 0 0 0                     0
  in the complex plane.                                                                                                                        
                                                                                               0        0      Id 0 0 0 0                     0
                        
  P4. Find i*  inf i1 , i2 , i3 , i4 .                                                  0
                                                                                               
                                                                                                         0       0 Id 0 0 0          
                                                                                                                                     
                                                                                                                                                0
                                                                                                                                                 
                                                                                                                 ;D                        (28)
  P5. Repeat procedures P3 and P4 for i =1,…, N .                                                                                              
                                                                                                                                         
  P6. Find  *  inf  i* , i  1, , N                                                      0        0       0 0 0 0 Id                    0
                                                                                               
                                                                                                A  BK 0 0 0 0 0                              
                                                                                                                                                 Id 
                                                                                                                                               
  P7. If  *   / 2 , then the fractional interval system is
  robust stable. Otherwise, the stability of system cannot be
                                                                                        As mentioned above in the subsection IV.2, the system (26)
  guaranteed.
                                                                                    is stable, if and only if:
                                                                                                                                          
       V. ROBUST STABILITY OF LEO SATELLITE                                                                 min a rgi ()                   (29)
                                                                                                              i           2p
   The aim of this section is to apply the robust stability                         The problem of stabilization by state feedback is equivalent to
checking procedure subject of the section IV. This procedure                        find a matrix K which stabilizes (10); i.e. which checks the
will be used to proof that our system (10) under the control                        stability condition given in (29).
law presented in [9] is robust stable.
                                                                                       B. Robust stability of the fractional control law
    A. Fractional control law                                                         In this section, we suppose that our system is submitted to
   As mentioned above, the LEO satellite atitude dynamics is                        some perturbations which affect his parameters:
described, when neglecting the quasi-bilinear term, by the
system (10):                                                                           1. Case 1: Perturbation of  0 orbital angular rate
          x( t )  Ax( t )  Bu( t )  w( t ) ; x(0)  x0
                                                         (21)




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  We consider that, due to the external perturbation,  0
varies       between          0   0   0       and         0   0   0 .
Consequently, A varies between A and A . So our system (26)
is transformed to FO-LTI system with     .                   
    For checking the robust stability of system (26), we apply
the procedure described in section IV.                                                   φ*
    If  *    2 the system (26) is robust stable. Otherwise,
the stability of system cannot be guaranteed.

     Numerical application
  The simulation parameters are the orbital rate
0  0.00104 rad / sec and the total moment of inertia
matrix for the spacecraft.                                                                                                  0
                                                                                                                            0
                      4,020   0     0 
                                                                                                                       Fig 1: φ * versus
                                                                                                                                           Δω0
                       0
                   I       3,989   0  Kg.m 2                                                                                            ω0
                                        
                       0
                              0   3,010
                                        
                                                                                           2 .Case 2 : Intrinsic Parameters of LEO Satellite are
                                                                                         Perturbed
   The fractional control law which stabilizes the system (10)
                                                                                                 Variation of I x :
with these numerical values is u( t )   Kx with   0,5 and
                                                                                           We suppose that           I x varied between      I x  I x et
             0          0       1   83         0,758 3,530                             I x  I x due to external perturbation which modify the form
         K  9
                        0    23,900 0          2900    7 
                                                                                         of the satellite.
             0                                         55 
                                                                                            For the simulation, we consider the same numerical values
                        0       0 3,200          0                                     and fractional control law of subsection V-2-1.
                                                                                            The following table resume the results of robust stability
   We give below the results of robust stability checking                                checking procedure.
procedure applied to the system (26). These results justifies                               The system is still robust stable until a variation of  20%
the robustness of fractional control laws with respect to the                                                             0
variation of  0 which can reach  20%.                                                  of I x . The  * decrease when        increase.
                                                                                                                              0
   We note also that the curve of the evolution have a linear
                                   0
behaviour and  * decrease when        increase.                                                         Table Ii:Results Of Robust Stability Checking
                                             0                                                                Procedure For Variation Of I x

                                                                                                   ΔIx    ΔIx / Ix        Ix min              Ix max        φ*
             Table I: Results Of Robust Stability Checking Procedure
                              For Variation Of  0                                             0             0.00%       4.0200            4.0200        0.9334
                                                                                              0.1608         4.00%       3.8592            4.1808        0.8909
             Δω0      Δω0 / ω0          ω0                ω0              φ   *
                                                                                              0.3216         8.00%       3.6984            4.3416        0.8527
         0               0%      0.0010400        0.0010400           0.9334                  0.4824        12.00%       3.5376            4.5024        0.8244
    0.0000416            4%      0.0009984        0.0010816           0.9060                  0.6432        16.00%       3.3768            4.6632        0.8062
    0.0000832            8%      0.0009568        0.0011232           0.8772                  0.8040        20.00%       3.2160            4.8240        0.8020
    0.0001248           12%      0.0009152        0.0011648           0.8476
    0.0001664           16%      0.0008736        0.0012064           0.8178
    0.0002080           20%      0.0008320        0.0010400           0.7888




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                                                                                                                       ISSN 1947-5500
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                                                                                                           VI. CONCLUSION
                                                                                    In this paper, we assume that the orbital angular rate is
                                                                                 subject to some perturbations and, consequently, the LEO
                                                                                 satellite system become uncertain. The added value from this
                                                                                 work is to prove that the fractional order control present in [9]
         φ*                                                                      is robust stable. We used as method the robust stability
                                                                                 checking procedure developed in [1]. The mathematical
                                                                                 aspects of this procedure were recalled. We have studied also
                                                                                 the case of the perturbation of intrinsic parameters of LEO
                                                                                 satellite due to the external perturbation which mean that the
                                                                                 total moment of inertia is uncertain.

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                                                                            35                                    http://sites.google.com/site/ijcsis/
                                                                                                                  ISSN 1947-5500
                                                                 (IJCSIS) International Journal of Computer Science and Information Security,
                                                                 Vol. 8, No. 3, June 2010



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