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(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 30 http://sites.google.com/site/ijcsis/ 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) 4 106 2 106 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 ) 6 106 3 106 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 31 http://sites.google.com/site/ijcsis/ 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 40 1 0 0 0 0 0 (1 1 ) We consider the following FO-LTI system with interval 0 302 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) 32 http://sites.google.com/site/ijcsis/ 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 argi () ) 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 . x1 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 rgi () (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) 33 http://sites.google.com/site/ijcsis/ ISSN 1947-5500 (IJCSIS) International Journal of Computer Science and Information Security, Vol. 8, No. 3, June 2010 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 34 http://sites.google.com/site/ijcsis/ ISSN 1947-5500 (IJCSIS) International Journal of Computer Science and Information Security, Vol. 8, No. 3, June 2010 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. REFERENCES ΔIx / Ix [1] Y.Q.Chon, H.S.Ahn, I.Podlubny, Robust stability check of fractional order Fig 2: φ * versus ΔIx / Ix linear time invariant systems with interval uncertainties, signal processing 86 (2006) 2611-2618. [2] F.A.Devy Vareta, Pseudo-Invariance Sous Groupe de Transformation :un Variation of I nouveau Concept pour la Commande Robuste, Séminaire Toulousain « Représentation Diffusive et Application »- N°1-nov. 2000. Using the same values as before, we give bellow the results [3] L. Dorcak, I. Petras, I. Kostial and J. Terpak, State space controller design of robust stability checking procedure. for the fractional-order regulated system, ICCC 2001, Korynica, Poland, pp. The system is still robust stable until a variation of 15-20. 0 [4] M. M. Dzhebashyan, Integral transforms and representation of functions 10,53%. The * decrease when increase. in the complex plan, Nauka Moscou 1966. 0 [5] P.C. Hughes, Spacecraft Attitude Dynamics, John Wiley & Sons, USA New York, 1986 Table Iii: Results Of Robust Stability Checking [6] B. Kim, E. Velenis, P. Kriengsiri & P. Tsiotras, A Spacecraft Simulator Procedure For Variation Of I for Research and Education, AAS 01-367. [7] B.J. Kim, H. Lee and S.D. CHOI, Three-axis Reaction Wheel Attitude Control System for KITSAT-3 Microsatellite, IFAC Conference in I / I * Autonomous and Intelligent Control in Aerospace, Beijing, 1995. [8] A. 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Sidi, Spacecraft Dynamics and control, Cambridge University Fig 3: * versus ΔI / I Press, Cambridge, UK, 1997. 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 [19] S.G. Samko, A. A. Kilbas & O.I. Maritcheu, Fractional integrals and derivative: theory and application, Gorden & Breach, 1987. [20] A. Skullestad, J. Gilbert, H Control of a Gravity Gradient Stabilised Satellite, Control Engineering Practice 8 (2000) 975-983. [21] P. Tsiotras, H. Shen and C. Hall, Satellite Attitude Control and Power Tracking with Momentum Wheels, AAS 99-317. [22] C. Valentin-Charbonnel, G. Duc and S. Le Ballois, Low-order Robust Attitude Control of an Earth Observation Satellite, Control Engineering Practice 7 (1999) 493-506. [23] B.M.Vinagre,I.Petras, I.Podlubny and Y.Q.Chen, Using Fractional Order Ajustement Rules and Fractional Order References Models in Model- Reference Adaptive Control, Nonlinear Dynamics, vol.29. [24] B. Wie, Space Vehicle Dynamics and Control, AIAA Education Series, 1998. [25] C. Witford, D. Forrest, The CATSAT Attitude Control system, Proceeding of the 12th Annual AIAA/USU Conference on Small Satellite, 1996. [26] C. H. Won, Comparative study of various methods for attitude control of LEO satellite, Aerospace Science and Technology. 1270-9638. 99/05/ Elsevier, Paris. [27] J. R.Wertz, Spacecraft Attitude Determination and control, Kluwer Academic Publishers, Dordrecht, Holland, 1978. [28] K. Zhou, JC. Doyle & R. Glover, Robust and Optimal control, Prentice Hall, 1996 36 http://sites.google.com/site/ijcsis/ ISSN 1947-5500

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