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Numerical Stabilization of a Rigid Spacecraft with Two Actuators
Copyright © IFAC Motion Control, Grenoble, France, 1998
NUMERICAL STABILIZATION OF A RIGID SPACECRAFT WITH TWO ACTUATORS Nicolas MARCHAND· Mazen ALA MlR •
• Laboratoire d'Automatique de Grenoble UMR 5528, CNRS-INPG-UJF, ENSIEG BP 46, 38402 Saint Martin d'Heres Cedex, France
Abstract : In this paper, a new approach to stabilize the rigid spacecraft operating with only two control torques (Le. in failure mode) is presented. This approach is based on the receding horizon principle and uses a systematic projection on a Chebyshev's polynomial basis to approximate the trajectories of each variable of the system. A general theory and its approximated numerical implementation are presented and simulations are provided. Copyright ©1998 IFAC Keywords: Attitude control, Stabilization, Space vehicles, Nonlinear systems, Function approximation.
1. INTRODUCTION
However C.1. Byrnes and A. Isidori (Byrnes and Isidori, 1991) proved this is no longer true on the entire system (i.e. when taking into account the Euler angular velocities and the position angles in the satellite's frame) since it does not satisfy the necessary Brockett's conditions (see R.W. Brockett (Brockett, 1983)). P.E. Crouch proposed an attitude stabilizing algorithm based on Lie algebraic theory and that leads to piecewise constant discontinuous control (Crouch, 1984). H. Krishnan et al. (Krishnan et al., 1992) proposed a simplified alternative stabilizing algorithm operating with successive maneuvers. In a recent work E. Y. Keral (Kerai, 1995) proved that the system with two controls satisfies a sufficient condition for small time local controllability and using the fact that 11 most 11 STLC (small time locally controllable) systems are stabilizable by means of a continuous time-varying feedback (Coron, 1995), the rigid spacecraft using two control torques is stabilizable by this means. It remained to construct such feedback explicitly. P. Morin et al. (Morin et al., 1995) proposed the first explicit time-varying feedback that locally stabilizes the rigid spacecraft. This result was then extended by J.M. Coron and E.Y. Kerai (Coron and Kerai, 1996) to the case of non axial control
The attitude control of a rigid spacecraft operating in failure mode, i.e. using only one or two control torques, has been extensively studied in the literature. The controllability of such a system is well known. B. Bonnard (Bonnard, 1982) and P.E. Crouch (Crouch, 1984) proved that the system is globally controllable except in the case where the uncontrolled axis is a symmetry axis of the spacecraft. For the rest of the paper we will assume that we are not in this particular case. E. Y. Keral(Kerai, 1995) proved that the system is small time local controllable only if at least two controls are applied. The study of the stabilizability of the spacecraft in failure mode is more recent. The simpler problem of stabilizing only the angular velocities was first studied. It is known in the case of non-symmetric body that the null solution of Euler angular velocities can be made asymptotically stable using two or only one control torque. Moreover D. Aeyels and M. Szafranzi (Aeyels and Szafranski, 1988) established that the reduced system with two or even one control torque can be locally asymptotically stabilized by means of smooth feedback.
73
torques. However, no simulations were provided. The stabilization which is proposed here, contrary to the previously cited method, is not based on the specific form of the equations of the spacecraft but can be applied without modification to a very wide class of nonlinear systems. FUrthermore, the trajectories computed with the following method are smooth and therefore reasonably implementable on real systems provided that a quite powerful processor is used. The method is based on receding horizon scheme, which enables - at least conceptually - to stabilize a very wide class of nonlinear systems (Michalska and Mayne, 1991; Keerthi and Gilbert, 1988; Alamir and Bornard, 1994). Unfortunately, the use of such control laws on general nonlinear systems requires the minimization of a non convex open loop cost function with final state constraint, which has an heavy computional cost. A way to reduce it will therefore be presented and the calculation time is also analysed. The paper is organized as follows. First of all, the system under consideration is precisely defined in section 2. Section 3 gives in a more theoretical point of view, the overall principle of the control law presented in this paper. Finally, in section 4 a numeric implementation of it and some simulations are proposed.
satisfy the Brockett's necessary conditions for the existence of a smooth feedback (Brockett, 1983). The control law will be at best continuous. However, a fully nonlinear approach is required since the linearization methods won't give any result .
3. PRINCIPLE OF THE CONTROL LAW The aim of this section is to present the overall principle of the control law. The numeric implementation of the following will be found in section 4. The approach used here is based on the receding horizon principle. In receding horizon stabilization scheme, the control of the state X at time to is obtained by first solving the open-loop optimal control problem on [to, to + t/l subject to the terminal constraint {X (to + t f) = o} to which is often added the constraint {u (to + t I) = o} . An admissible solution u will denote a control that verifies the two previous constraints. t I is an a priori fixed horizon . In most cases the open-loop optimal controller can not be expressed explicitly and therefore can not be computed instantaneously at each time to . Therefore the optimal open-loop control is then applied during the sampling period [to, to+Tl . Repeating this scheme at each sampling time kT (k E N) yields to a pseudo feedback control u(t,X(kT», t E [O,T]. This theory assumes that the system under consideration is stabilizable in a finite time which has been shown to hold for the rigid spacecraft operating in failure mode in (Krishnan et al., 1992). Unfortunately receding horizon techniques usually presents the drawback of having heavy computional costs. To avoid such a drawback, let us simplify the problem into an equivalent stabilization oriented receding horizon problem. The simplification of the problem will be done in three steps:
2. KINEMATIC AND DYNAMIC EQUATIONS OF THE RIGID SPACECRAFT The orientation of a spacecraft can be specified using various parameterizations of the orthogonal group SO(3). The conventional Euler angles will be used (Wittenburg, 1977) . Consider a frame F. attached to the spacecraft with axes corresponding to the principal inertia axes of the body and let FI be a fixed frame representing the desired attitude for the spacecraft. Let W = (W1' W2, W3)T be the angular velocity of the frame F. with respect to the fixed frame FI (w is expressed in the frame F,) and r.p, 0 and t/J the angles of the rotation giving F. from FI (expressed in the frame FI) ' With the previous notations, the equations of the spacecraft in failure mode are (no control on the third axis):
(1)
W2
= 1.£1 = 1.£2
W3
=
(3)
Wl
aw1W2
= Wl cos 0 + W3 sinO
() = tan r.p (W1 sin 0 - W3 cos 0) + W2 ..p = -(COSr.p)-l (Wl sin 0 - W3 cos 8)
• A voiding minimization 0/ a criteria The stabilization of a system does not a priori require an optimal control, only a stabilizing control is needed. Unfortunately the stabilizing property of the receding horizon holds in the minimization of the criteria, in fact the Lyapunov function used is the optimal-cost itself. The stabilizing property has to be proven in another way in an optimization free scheme. Let us denote for each variable z, Zk(t) := z(kT + t) . Suppose one has at the kth sampling time an admissible solution Uk(t, X(kT» defined for t E [0, t/l (ie. a control that steers all the states from X (kT) to zero after a finite time t I and such that uk(t/,X(kT» = 0). Suppose also one can construct a procedure which gives an admissible solution Uk+l (t, X«k+ 1)T» at time (k+ 1)T from the admissible solution Uk (t, X (kT)) at the previous sampling time kT such that if no perturbation has occurs during [kT, (k + 1)T] one has:
(2) (4) (5) (6)
with J = diag(J1 , J2 , J3 ), the inertia matrix of the spacecraft and a = (JIJ/~)' The linearization of the system has an uncontrollable eigenvalue at the origin, therefore it does not
74
ditions for the equivalence between a nonlinear system and a particular triangular (that can take the form of (8)) form are given. The triangularization of the equations of the spacecraft is done in section 4.1.
Repeating this calculation at each sampling time and if no perturbation occurs, it is obvious that the system will be steered to zero in the finite time t! and will stay at this position. The so obtained feedback will therefore stabilize (in the sense of the attractiveness) the system to zero. Note that the attractiveness property in the above scheme comes from the condition (7) on the initialization of the admissible solution search procedure. Moreover, if the open-loop trajectory x(t; X, u(., X)) of the system with control u( ., X) is (locally) k-lipschitz with respect to the initial condition X, the origin will be asymptotically stable equilibrium for the system. Indeed, using Massera's lemma (Massera, 1949), there exists a function G such that V(X) = oo Jo G(llx(tjX,u(.,X))ll)dt < 00 . It can be proven that V(X) is a Lyapunov function for the system (Marchand and Alamir, 1998).
• Parameterization of the search space With the previous triangular form the search of an admissible solution has to de done on the infinite space of time functions (}( .) and
• Triangularization Let us describe the evolution of the system with a minimal set of variables such that when their behaviours are known, the behaviour of all the states and inputs ensues. Instead of using the inputs as unknowns as it is usually the case, the search can be done on those free independent variables which fully describe the evolution of the system. This yields to the choice of () and
F, (8(1), F~(8(1), F3(~(t), F .. (8(t),
PxP
= = =
",~(8, 'P , "'3(8 , 'P, "''' (8, 'P , "'5(8, 'P,
I) t) t)
PxP
(9)
n
Roughly speaking, if no perturbation occurs, gives the searched parameterization at the (k+ 1) th sampling time from the parameterization at the k th , as equation (7) gave the input Uk+l from the knowledge of Uk. See section 4.3 for a numeric version of n.
-+ :"l (8, 'P, t)
-+ -+ 0 -+ 0 -+ 0 -+
-t
(8)
t) ze(8,'P , t)
where Xi E {Wl,W2,W3,tP,Ul,U2} and where for every variable z, z denotes z and a finite number of it's time derivatives. The relations Fi are differential equations with respect to the unknown Xi which depends only on (),
With the previous elements, one can define the function :F as follows: (11)
Indeed, form (8) aB and a", determine the evolution of X, Ul and U2 on [0, ~f]' The receding ·horizon scheme consists in our case in finding at each instant kT the parameterization a~ and a: such that :F(a~, a:) = 0 - this is done
75
by solving the triangular system of non linear algebraic equations F(a~, a~) = 0 - and then to apply the corresponding open-loop controls UI (a~, a~, t) and U2 (a~ , a~, t) until the next sampling time. The stability property ensued from property (10), is assured by initializing the search routine :F at each instant kT with R(a~-l, a~-l) which would already satisfy the objective :F = 0 if no approximations were done.
For all i E N and all mEN, Pi( m) (t) will stand for the mth derivative of Pi(t) . Then, one naturally defines the corresponding vector T~m) (t) . With the notations defined above, any variable z can then be approximated on [kT, kT + t / 1by Zk(t) ~ Zk(t) = TN. (t).a~ where a~ E !RN. Moreover it's derivatives will be given by: d~t? (t) ~
d;t~· (t) = T~! (t).a~ . In our case, a unique basis dimension N = 6 was used, so each variable is characterized by six parameters. The triangular differential system (12)-(17) using the above parameterization gives the following algebraic system of equations If almost If true for all t E [0, t /1:
4. NUMERICAL IMPLEMENTATION The aim of this section is to present a numerical implementation of section 3. The structure of this section is analogous to the previous one: the triangularization of the system is first presented, the parameterization used for the rigid spacecraft then follows . Finally the algorithm of the control law is clearly given.
Fdt,a~,a~ , a~3) ~ 0
(21)
F2(t , a~, a~, a~3' a~) ~ 0 k k k k k) F3(t, a0,a
(22)
k
Wl
cos(O) -
W3
sin(O)
=0
iJ + tancp (WI sinO - W3 cosO) 1/J cos cp + WI sin 0 - W3 cosO = 0
Wl W2 -
= 0
(12)
=0 U2 = 0
(14)
Arg {
(16)
where
91 (0)=cos 2(0)
92 (0 ,'I' )=a sin(O) tan(
These are the equations of the needed triangular form (8).
4.2 Parameterization of the system
TN(t) := (PN(t), . .. ,P2(t), PI (t)) will denote the time function basis used with Pi(t) being the (il)th Chebyshev's polynomial. This basis is known to be numerically well conditioned and has the advantage of being directly linked with the highest frequency of a signal. Recall that the Chebyshev's polynomials defined on [0, t /] are given by the recursive relation: PI
(19)
:=
Pn :=
2ft - 1
2(2t~ - 1) T
n- l -
Tn -
2
(24) (25)
F6 (t , a~2 ' a~2) ~ 0
(26)
= min
~ IIFi(ti,a~,a~)112}(27)
G~ subject to . TN(O)G~ =.(~T) · ;=1
x
is successively
(W3:Wl,w2,1/J,ul,U2)
i respectively takes the values 1
with:
(18)
(23)
when 6. N f is a free parameter such that 0 = tl < .. . < tNI = tf . In the following we shall take it equal to 20. When i = 1, (27) leads to a nonlinear equation in a~3 ' This nonlinear least squares problem can easily be solved using a Newton's method instead of the classical Gauss-Newton's one. In fact Newton's method shows faster and more stable local convergence properties than Gauss-Newton 's one (Dennis and Schnabel, 1983). It requires the knowledge of the first and the second derivative of the function to minimize with respect to each element of the vector a~3 which is rarely available but can be here analytically expressed. The minimization of (27) using MATLAB leastsq function, with the knowledge of the derivative (faster), will take a mean of 1.9 sec, which must be compared to the 0.05 sec that are needed for the Newton's algorithm, the same precision being required for the two method. This result could be even more improved with an optimization of the computations. All computations where done on a PC-Pentium200 with 32Mo Ram under Windows95. Linear equations with respect to the unknown Wl, W2 and t/J are obtained when i respectively takes the values 2 to 4. And finally, when i 5 or i 6, (27) is a trivial algebraic equality.
(17)
Po := 1
~
0
F5(t, a~I' a~l.) ~ 0
k k) aAk( z a/l , a
(15)
Ul
k) ~
0
Given the parameterization of the free variables 'P, the best (in the least squares sense) parameterization of the non free variables are obtained by solving:
(13)
W2 -
k
~
o and
Relations (4) and (5) in relation (3) give the following equivalent triangular system:
cp -
k
F.4 (t. ,ao,a
4.1 Triangularization of the system
91 (0)w3+92 (0 ,
k
~
to
=
(20)
76
=
rrI ': ~
4.3 Algorithm
At each sampling time kT: - Compute the initializations n(a~-l, a~-l) with: '1")( /'\- aek-l ,a",k-l)_ A'g {
A'g
Non perturbed Iyllcm
. [;J -,, ! I
-0
,
I
~:n ~ IITN(t;)a, - Di
~~--- I ~------------------J
{~:n ~ IITN(t;)a, - D~(t;)II'}
where: VT (t)
= { TN(t + T)a~-l 0
9
VT(t) '"
t E [0, t, - T] tE[t,-T , t,]
= {TN(t + T)a~-l 0
L-----
t E [0, t, - TJ
tE[t,-T, t,]
- Compute a~ and a~ verifying:
Fig. 1. Evolution of the Euler Angles
TN(t, )a~ T N • (t, )a~ TN(t, )a~s (a~, a~)
F(-k -k) ae ,a",
I ~..
TN(t,)a~l(a~,a~)
= TN(t,)a~2(a~,a~)
-d
.,
=0
TN(t, )a~(a~, a~) TN(t, )a~l (a~, a~) TN(t,)a~2(a~,a~)
Non pcnwbed system
subject to the linear constraints TN(O)a~ = (}(kT) and TN(O)a~ = cp(kT) and using a least squares like method with initialization n(a~-l, a~-l) . :F can be easily computed using (27). - Compute the corresponding controls Ul (t) = TN(t)a~l (a~, a~) and U2(t) = TN(t)a~2(a~,a~) and apply them until the next sampling time.
I-=:~I. , jl
4.4 Simulation
The simulations have been made with the numerical values provided in (Sira-Ramirez and Siguerdidjane, 1996) corresponding to the spacecraft SPOT 4: J 1 = 2500kgm 2, h = 6500kgm 2 and h = 8500kgm2 . A prediction horizon t, = lOs and a sampling period T = 0.5s was used. The number of points to solve the least square problems is N, = 20, and for every variable x, Nz = 6 (i.e. Chebyshev's polynomials of fifth order). The initial conditions are: (w~, w~, w~, (}O , cpo, tiP) = (0,0 , 0, ~,O).
' - - - - --
-,
Fig. 2. Evolution of the Angular Velocity
One may wonder that the states of the non perturbed system yield the origin only after 158 when t, was set to lOs, but this is due to the approximations made with function n were only the projection of Df and D~ onto the Chebyshev's basis is used in the numerical implementation and not the functions themselves as in equation (10) . Moreover, only approximations of the solutions of the differential equations were used (that can also be seen as system uncertainties) . The stability of the studied system let us catch sight of good robustness properties that remains to be addressed more precisely.
2; ,
The method has been applied on two systems, the first one has no uncertainty whereas the second one is a perturbed system showing off the robustness ofthe proposed method. The uncertainty used are: 10% error on a simulating an unknown on the inertia momentum and a permanent offset of 5° on the measure of (J simulating a captor error.
77
NOD penwbed .yatem
,
1. 1e.
.
r=:il
I" I
~
~__________~~____~____~~_ _~
Fig. 3. Applied control 5. CONCLUSION In this paper is presented a new stabilizing method for the rigid spacecraft operating in failure mode. It is based on the receding horizon principle. The open-loop control computation associated to this scheme is done by an extensive use of projection on a Chebyshev's polynomial basis and using a particular triangular form satisfied by the rigid spacecraft in failure mode. Contrary to the classical optimal receding horizon that requires a computional heavy cost, the method presented here can be real time implemented on reasonably fast systems. It leads to a piecewise continuous feedback law.
6. REFERENCES Aeyels, D. and M. Szafranski (1988). Comments on the stabilizability of the angular velocity of a rigid body. Systems fj Control Letters 10, 3539. Alamir, M. and G. Bornard (1994). On the stability of receding horizon control of nonlinear discrete time systems. Systems fj Control Letters 23, 291-296. Bonnard, B. (1982). Controle de l'attitude d'un satellite rigide. RAIRO Autom./Syst. Anal. Control 16, 85-93. Brockett, R. W. (1983). Asymptotic stability and feedback stabilization - Differential Geometric Control Theory. R.W. Brockett, R.S. Millmann and H.S. Susmann - Birkhauser. Byrnes, C. I. and A. Isidori (1991). On the attitude stabilization of rigid spacecraft. Automatica 27(1},87-95. Coron, J. M. (1995). On the stabilization in finite time of locally controllable systems by
means of continuous time-varying feedback laws. Siam Journal on Control and Optimization 33, 804-833. Coron, J. M. and E. Y. Kerai (1996). Explicit feedbacks stabilizing the attitude of a rigid spacecraft with two control torques. A utomatica 32(5}, 669-677. Crouch, P.E. (1984). Spacecraft attitude control and stabilization. IEEE Trans. on Automatic Control AC-29(4} , 321-331. Dennis, J. E . and R. B. Schnabel (1983). Numerical method for unconstrained optimization and nonlinear equations. Chap. 10. Prentice Hall. Fliess, M., J . Levine, Ph. Martin and P. Rouchon (1995) . Flatness and defect of nonlinear systems: introductory theory and examples. International Journal of Control 61(6), 13271361. Keerthi, S. S. and E . G. Gilbert (1988). Optimal infinite horizon feedback laws for a general class of constrained discrete time systems. Journal of Optimization Theory and Applications 57, 265-293 . Kerai , E. Y. (1995). Analysis of small time local controllability of a rigid body model. In: IFAC Symp. on System Structure and Control. pp. 645-650. Krishnan, H., H McClamroch and M. Reyhanoglu (1992). On the attitude stabilization of a rigid spacecraft using two control torques. In: Proc. American Control Conference. pp . 1990-1995. Marchand, N. and M. Alamir (1998). From openloop trajectories to stabilizing state feedback - application to a cstr. In: IFAC System and Structure Control. Massera, J. L. (1949) . On liapounofi"s conditions of stability. Annals of Mathematics 50(3), 705-721. Michalska, H. and D. Q. Mayne (1991). Receding horizon control of nonlinear systems. Systems fj Control Letters 16, 123-130. Morin, P., C. Samson, J . B. Pomet and Z. P. Jiang (1995). Time-varying feedback stabilization of the attitude of a rigid spacecraft with two controls. Systems fj Control Letters 25, 375385. Sira-Ramirez, H. and B. Siguerdidjane (1996). A redundant dynamical sliding mode control scheme for an asymptotic space vehicle stabilization. International Journal of Control 65,901-912. Celikovsky, S. and H. Nijmeijer (1995). Equivalence of nonlinear systems to triangular form: the singular case. Systems fj Control Letters 27, 135-144. Wittenburg, J . (1977). Dynamics of systems of rigid bodies. B.G . Teubner Stuttgart.
78
NUMERICAL STABILIZATION OF A RIGID SPACECRAFT WITH TWO ACTUATORS Nicolas MARCHAND· Mazen ALA MlR •
• Laboratoire d'Automatique de Grenoble UMR 5528, CNRS-INPG-UJF, ENSIEG BP 46, 38402 Saint Martin d'Heres Cedex, France
Abstract : In this paper, a new approach to stabilize the rigid spacecraft operating with only two control torques (Le. in failure mode) is presented. This approach is based on the receding horizon principle and uses a systematic projection on a Chebyshev's polynomial basis to approximate the trajectories of each variable of the system. A general theory and its approximated numerical implementation are presented and simulations are provided. Copyright ©1998 IFAC Keywords: Attitude control, Stabilization, Space vehicles, Nonlinear systems, Function approximation.
1. INTRODUCTION
However C.1. Byrnes and A. Isidori (Byrnes and Isidori, 1991) proved this is no longer true on the entire system (i.e. when taking into account the Euler angular velocities and the position angles in the satellite's frame) since it does not satisfy the necessary Brockett's conditions (see R.W. Brockett (Brockett, 1983)). P.E. Crouch proposed an attitude stabilizing algorithm based on Lie algebraic theory and that leads to piecewise constant discontinuous control (Crouch, 1984). H. Krishnan et al. (Krishnan et al., 1992) proposed a simplified alternative stabilizing algorithm operating with successive maneuvers. In a recent work E. Y. Keral (Kerai, 1995) proved that the system with two controls satisfies a sufficient condition for small time local controllability and using the fact that 11 most 11 STLC (small time locally controllable) systems are stabilizable by means of a continuous time-varying feedback (Coron, 1995), the rigid spacecraft using two control torques is stabilizable by this means. It remained to construct such feedback explicitly. P. Morin et al. (Morin et al., 1995) proposed the first explicit time-varying feedback that locally stabilizes the rigid spacecraft. This result was then extended by J.M. Coron and E.Y. Kerai (Coron and Kerai, 1996) to the case of non axial control
The attitude control of a rigid spacecraft operating in failure mode, i.e. using only one or two control torques, has been extensively studied in the literature. The controllability of such a system is well known. B. Bonnard (Bonnard, 1982) and P.E. Crouch (Crouch, 1984) proved that the system is globally controllable except in the case where the uncontrolled axis is a symmetry axis of the spacecraft. For the rest of the paper we will assume that we are not in this particular case. E. Y. Keral(Kerai, 1995) proved that the system is small time local controllable only if at least two controls are applied. The study of the stabilizability of the spacecraft in failure mode is more recent. The simpler problem of stabilizing only the angular velocities was first studied. It is known in the case of non-symmetric body that the null solution of Euler angular velocities can be made asymptotically stable using two or only one control torque. Moreover D. Aeyels and M. Szafranzi (Aeyels and Szafranski, 1988) established that the reduced system with two or even one control torque can be locally asymptotically stabilized by means of smooth feedback.
73
torques. However, no simulations were provided. The stabilization which is proposed here, contrary to the previously cited method, is not based on the specific form of the equations of the spacecraft but can be applied without modification to a very wide class of nonlinear systems. FUrthermore, the trajectories computed with the following method are smooth and therefore reasonably implementable on real systems provided that a quite powerful processor is used. The method is based on receding horizon scheme, which enables - at least conceptually - to stabilize a very wide class of nonlinear systems (Michalska and Mayne, 1991; Keerthi and Gilbert, 1988; Alamir and Bornard, 1994). Unfortunately, the use of such control laws on general nonlinear systems requires the minimization of a non convex open loop cost function with final state constraint, which has an heavy computional cost. A way to reduce it will therefore be presented and the calculation time is also analysed. The paper is organized as follows. First of all, the system under consideration is precisely defined in section 2. Section 3 gives in a more theoretical point of view, the overall principle of the control law presented in this paper. Finally, in section 4 a numeric implementation of it and some simulations are proposed.
satisfy the Brockett's necessary conditions for the existence of a smooth feedback (Brockett, 1983). The control law will be at best continuous. However, a fully nonlinear approach is required since the linearization methods won't give any result .
3. PRINCIPLE OF THE CONTROL LAW The aim of this section is to present the overall principle of the control law. The numeric implementation of the following will be found in section 4. The approach used here is based on the receding horizon principle. In receding horizon stabilization scheme, the control of the state X at time to is obtained by first solving the open-loop optimal control problem on [to, to + t/l subject to the terminal constraint {X (to + t f) = o} to which is often added the constraint {u (to + t I) = o} . An admissible solution u will denote a control that verifies the two previous constraints. t I is an a priori fixed horizon . In most cases the open-loop optimal controller can not be expressed explicitly and therefore can not be computed instantaneously at each time to . Therefore the optimal open-loop control is then applied during the sampling period [to, to+Tl . Repeating this scheme at each sampling time kT (k E N) yields to a pseudo feedback control u(t,X(kT», t E [O,T]. This theory assumes that the system under consideration is stabilizable in a finite time which has been shown to hold for the rigid spacecraft operating in failure mode in (Krishnan et al., 1992). Unfortunately receding horizon techniques usually presents the drawback of having heavy computional costs. To avoid such a drawback, let us simplify the problem into an equivalent stabilization oriented receding horizon problem. The simplification of the problem will be done in three steps:
2. KINEMATIC AND DYNAMIC EQUATIONS OF THE RIGID SPACECRAFT The orientation of a spacecraft can be specified using various parameterizations of the orthogonal group SO(3). The conventional Euler angles will be used (Wittenburg, 1977) . Consider a frame F. attached to the spacecraft with axes corresponding to the principal inertia axes of the body and let FI be a fixed frame representing the desired attitude for the spacecraft. Let W = (W1' W2, W3)T be the angular velocity of the frame F. with respect to the fixed frame FI (w is expressed in the frame F,) and r.p, 0 and t/J the angles of the rotation giving F. from FI (expressed in the frame FI) ' With the previous notations, the equations of the spacecraft in failure mode are (no control on the third axis):
(1)
W2
= 1.£1 = 1.£2
W3
=
(3)
Wl
aw1W2
= Wl cos 0 + W3 sinO
() = tan r.p (W1 sin 0 - W3 cos 0) + W2 ..p = -(COSr.p)-l (Wl sin 0 - W3 cos 8)
• A voiding minimization 0/ a criteria The stabilization of a system does not a priori require an optimal control, only a stabilizing control is needed. Unfortunately the stabilizing property of the receding horizon holds in the minimization of the criteria, in fact the Lyapunov function used is the optimal-cost itself. The stabilizing property has to be proven in another way in an optimization free scheme. Let us denote for each variable z, Zk(t) := z(kT + t) . Suppose one has at the kth sampling time an admissible solution Uk(t, X(kT» defined for t E [0, t/l (ie. a control that steers all the states from X (kT) to zero after a finite time t I and such that uk(t/,X(kT» = 0). Suppose also one can construct a procedure which gives an admissible solution Uk+l (t, X«k+ 1)T» at time (k+ 1)T from the admissible solution Uk (t, X (kT)) at the previous sampling time kT such that if no perturbation has occurs during [kT, (k + 1)T] one has:
(2) (4) (5) (6)
with J = diag(J1 , J2 , J3 ), the inertia matrix of the spacecraft and a = (JIJ/~)' The linearization of the system has an uncontrollable eigenvalue at the origin, therefore it does not
74
ditions for the equivalence between a nonlinear system and a particular triangular (that can take the form of (8)) form are given. The triangularization of the equations of the spacecraft is done in section 4.1.
Repeating this calculation at each sampling time and if no perturbation occurs, it is obvious that the system will be steered to zero in the finite time t! and will stay at this position. The so obtained feedback will therefore stabilize (in the sense of the attractiveness) the system to zero. Note that the attractiveness property in the above scheme comes from the condition (7) on the initialization of the admissible solution search procedure. Moreover, if the open-loop trajectory x(t; X, u(., X)) of the system with control u( ., X) is (locally) k-lipschitz with respect to the initial condition X, the origin will be asymptotically stable equilibrium for the system. Indeed, using Massera's lemma (Massera, 1949), there exists a function G such that V(X) = oo Jo G(llx(tjX,u(.,X))ll)dt < 00 . It can be proven that V(X) is a Lyapunov function for the system (Marchand and Alamir, 1998).
• Parameterization of the search space With the previous triangular form the search of an admissible solution has to de done on the infinite space of time functions (}( .) and
• Triangularization Let us describe the evolution of the system with a minimal set of variables such that when their behaviours are known, the behaviour of all the states and inputs ensues. Instead of using the inputs as unknowns as it is usually the case, the search can be done on those free independent variables which fully describe the evolution of the system. This yields to the choice of () and
F, (8(1), F~(8(1), F3(~(t), F .. (8(t),
PxP
= = =
",~(8, 'P , "'3(8 , 'P, "''' (8, 'P , "'5(8, 'P,
I) t) t)
PxP
(9)
n
Roughly speaking, if no perturbation occurs, gives the searched parameterization at the (k+ 1) th sampling time from the parameterization at the k th , as equation (7) gave the input Uk+l from the knowledge of Uk. See section 4.3 for a numeric version of n.
-+ :"l (8, 'P, t)
-+ -+ 0 -+ 0 -+ 0 -+
-t
(8)
t) ze(8,'P , t)
where Xi E {Wl,W2,W3,tP,Ul,U2} and where for every variable z, z denotes z and a finite number of it's time derivatives. The relations Fi are differential equations with respect to the unknown Xi which depends only on (),
With the previous elements, one can define the function :F as follows: (11)
Indeed, form (8) aB and a", determine the evolution of X, Ul and U2 on [0, ~f]' The receding ·horizon scheme consists in our case in finding at each instant kT the parameterization a~ and a: such that :F(a~, a:) = 0 - this is done
75
by solving the triangular system of non linear algebraic equations F(a~, a~) = 0 - and then to apply the corresponding open-loop controls UI (a~, a~, t) and U2 (a~ , a~, t) until the next sampling time. The stability property ensued from property (10), is assured by initializing the search routine :F at each instant kT with R(a~-l, a~-l) which would already satisfy the objective :F = 0 if no approximations were done.
For all i E N and all mEN, Pi( m) (t) will stand for the mth derivative of Pi(t) . Then, one naturally defines the corresponding vector T~m) (t) . With the notations defined above, any variable z can then be approximated on [kT, kT + t / 1by Zk(t) ~ Zk(t) = TN. (t).a~ where a~ E !RN. Moreover it's derivatives will be given by: d~t? (t) ~
d;t~· (t) = T~! (t).a~ . In our case, a unique basis dimension N = 6 was used, so each variable is characterized by six parameters. The triangular differential system (12)-(17) using the above parameterization gives the following algebraic system of equations If almost If true for all t E [0, t /1:
4. NUMERICAL IMPLEMENTATION The aim of this section is to present a numerical implementation of section 3. The structure of this section is analogous to the previous one: the triangularization of the system is first presented, the parameterization used for the rigid spacecraft then follows . Finally the algorithm of the control law is clearly given.
Fdt,a~,a~ , a~3) ~ 0
(21)
F2(t , a~, a~, a~3' a~) ~ 0 k k k k k) F3(t, a0,a
(22)
k
Wl
cos(O) -
W3
sin(O)
=0
iJ + tancp (WI sinO - W3 cosO) 1/J cos cp + WI sin 0 - W3 cosO = 0
Wl W2 -
= 0
(12)
=0 U2 = 0
(14)
Arg {
(16)
where
91 (0)=cos 2(0)
92 (0 ,'I' )=a sin(O) tan(
These are the equations of the needed triangular form (8).
4.2 Parameterization of the system
TN(t) := (PN(t), . .. ,P2(t), PI (t)) will denote the time function basis used with Pi(t) being the (il)th Chebyshev's polynomial. This basis is known to be numerically well conditioned and has the advantage of being directly linked with the highest frequency of a signal. Recall that the Chebyshev's polynomials defined on [0, t /] are given by the recursive relation: PI
(19)
:=
Pn :=
2ft - 1
2(2t~ - 1) T
n- l -
Tn -
2
(24) (25)
F6 (t , a~2 ' a~2) ~ 0
(26)
= min
~ IIFi(ti,a~,a~)112}(27)
G~ subject to . TN(O)G~ =.(~T) · ;=1
x
is successively
(W3:Wl,w2,1/J,ul,U2)
i respectively takes the values 1
with:
(18)
(23)
when 6. N f is a free parameter such that 0 = tl < .. . < tNI = tf . In the following we shall take it equal to 20. When i = 1, (27) leads to a nonlinear equation in a~3 ' This nonlinear least squares problem can easily be solved using a Newton's method instead of the classical Gauss-Newton's one. In fact Newton's method shows faster and more stable local convergence properties than Gauss-Newton 's one (Dennis and Schnabel, 1983). It requires the knowledge of the first and the second derivative of the function to minimize with respect to each element of the vector a~3 which is rarely available but can be here analytically expressed. The minimization of (27) using MATLAB leastsq function, with the knowledge of the derivative (faster), will take a mean of 1.9 sec, which must be compared to the 0.05 sec that are needed for the Newton's algorithm, the same precision being required for the two method. This result could be even more improved with an optimization of the computations. All computations where done on a PC-Pentium200 with 32Mo Ram under Windows95. Linear equations with respect to the unknown Wl, W2 and t/J are obtained when i respectively takes the values 2 to 4. And finally, when i 5 or i 6, (27) is a trivial algebraic equality.
(17)
Po := 1
~
0
F5(t, a~I' a~l.) ~ 0
k k) aAk( z a/l , a
(15)
Ul
k) ~
0
Given the parameterization of the free variables 'P, the best (in the least squares sense) parameterization of the non free variables are obtained by solving:
(13)
W2 -
k
~
o and
Relations (4) and (5) in relation (3) give the following equivalent triangular system:
cp -
k
F.4 (t. ,ao,a
4.1 Triangularization of the system
91 (0)w3+92 (0 ,
k
~
to
=
(20)
76
=
rrI ': ~
4.3 Algorithm
At each sampling time kT: - Compute the initializations n(a~-l, a~-l) with: '1")( /'\- aek-l ,a",k-l)_ A'g {
A'g
Non perturbed Iyllcm
. [;J -,, ! I
-0
,
I
~:n ~ IITN(t;)a, - Di
~~--- I ~------------------J
{~:n ~ IITN(t;)a, - D~(t;)II'}
where: VT (t)
= { TN(t + T)a~-l 0
9
VT(t) '"
t E [0, t, - T] tE[t,-T , t,]
= {TN(t + T)a~-l 0
L-----
t E [0, t, - TJ
tE[t,-T, t,]
- Compute a~ and a~ verifying:
Fig. 1. Evolution of the Euler Angles
TN(t, )a~ T N • (t, )a~ TN(t, )a~s (a~, a~)
F(-k -k) ae ,a",
I ~..
TN(t,)a~l(a~,a~)
= TN(t,)a~2(a~,a~)
-d
.,
=0
TN(t, )a~(a~, a~) TN(t, )a~l (a~, a~) TN(t,)a~2(a~,a~)
Non pcnwbed system
subject to the linear constraints TN(O)a~ = (}(kT) and TN(O)a~ = cp(kT) and using a least squares like method with initialization n(a~-l, a~-l) . :F can be easily computed using (27). - Compute the corresponding controls Ul (t) = TN(t)a~l (a~, a~) and U2(t) = TN(t)a~2(a~,a~) and apply them until the next sampling time.
I-=:~I. , jl
4.4 Simulation
The simulations have been made with the numerical values provided in (Sira-Ramirez and Siguerdidjane, 1996) corresponding to the spacecraft SPOT 4: J 1 = 2500kgm 2, h = 6500kgm 2 and h = 8500kgm2 . A prediction horizon t, = lOs and a sampling period T = 0.5s was used. The number of points to solve the least square problems is N, = 20, and for every variable x, Nz = 6 (i.e. Chebyshev's polynomials of fifth order). The initial conditions are: (w~, w~, w~, (}O , cpo, tiP) = (0,0 , 0, ~,O).
' - - - - --
-,
Fig. 2. Evolution of the Angular Velocity
One may wonder that the states of the non perturbed system yield the origin only after 158 when t, was set to lOs, but this is due to the approximations made with function n were only the projection of Df and D~ onto the Chebyshev's basis is used in the numerical implementation and not the functions themselves as in equation (10) . Moreover, only approximations of the solutions of the differential equations were used (that can also be seen as system uncertainties) . The stability of the studied system let us catch sight of good robustness properties that remains to be addressed more precisely.
2; ,
The method has been applied on two systems, the first one has no uncertainty whereas the second one is a perturbed system showing off the robustness ofthe proposed method. The uncertainty used are: 10% error on a simulating an unknown on the inertia momentum and a permanent offset of 5° on the measure of (J simulating a captor error.
77
NOD penwbed .yatem
,
1. 1e.
.
r=:il
I" I
~
~__________~~____~____~~_ _~
Fig. 3. Applied control 5. CONCLUSION In this paper is presented a new stabilizing method for the rigid spacecraft operating in failure mode. It is based on the receding horizon principle. The open-loop control computation associated to this scheme is done by an extensive use of projection on a Chebyshev's polynomial basis and using a particular triangular form satisfied by the rigid spacecraft in failure mode. Contrary to the classical optimal receding horizon that requires a computional heavy cost, the method presented here can be real time implemented on reasonably fast systems. It leads to a piecewise continuous feedback law.
6. REFERENCES Aeyels, D. and M. Szafranski (1988). Comments on the stabilizability of the angular velocity of a rigid body. Systems fj Control Letters 10, 3539. Alamir, M. and G. Bornard (1994). On the stability of receding horizon control of nonlinear discrete time systems. Systems fj Control Letters 23, 291-296. Bonnard, B. (1982). Controle de l'attitude d'un satellite rigide. RAIRO Autom./Syst. Anal. Control 16, 85-93. Brockett, R. W. (1983). Asymptotic stability and feedback stabilization - Differential Geometric Control Theory. R.W. Brockett, R.S. Millmann and H.S. Susmann - Birkhauser. Byrnes, C. I. and A. Isidori (1991). On the attitude stabilization of rigid spacecraft. Automatica 27(1},87-95. Coron, J. M. (1995). On the stabilization in finite time of locally controllable systems by
means of continuous time-varying feedback laws. Siam Journal on Control and Optimization 33, 804-833. Coron, J. M. and E. Y. Kerai (1996). Explicit feedbacks stabilizing the attitude of a rigid spacecraft with two control torques. A utomatica 32(5}, 669-677. Crouch, P.E. (1984). Spacecraft attitude control and stabilization. IEEE Trans. on Automatic Control AC-29(4} , 321-331. Dennis, J. E . and R. B. Schnabel (1983). Numerical method for unconstrained optimization and nonlinear equations. Chap. 10. Prentice Hall. Fliess, M., J . Levine, Ph. Martin and P. Rouchon (1995) . Flatness and defect of nonlinear systems: introductory theory and examples. International Journal of Control 61(6), 13271361. Keerthi, S. S. and E . G. Gilbert (1988). Optimal infinite horizon feedback laws for a general class of constrained discrete time systems. Journal of Optimization Theory and Applications 57, 265-293 . Kerai , E. Y. (1995). Analysis of small time local controllability of a rigid body model. In: IFAC Symp. on System Structure and Control. pp. 645-650. Krishnan, H., H McClamroch and M. Reyhanoglu (1992). On the attitude stabilization of a rigid spacecraft using two control torques. In: Proc. American Control Conference. pp . 1990-1995. Marchand, N. and M. Alamir (1998). From openloop trajectories to stabilizing state feedback - application to a cstr. In: IFAC System and Structure Control. Massera, J. L. (1949) . On liapounofi"s conditions of stability. Annals of Mathematics 50(3), 705-721. Michalska, H. and D. Q. Mayne (1991). Receding horizon control of nonlinear systems. Systems fj Control Letters 16, 123-130. Morin, P., C. Samson, J . B. Pomet and Z. P. Jiang (1995). Time-varying feedback stabilization of the attitude of a rigid spacecraft with two controls. Systems fj Control Letters 25, 375385. Sira-Ramirez, H. and B. Siguerdidjane (1996). A redundant dynamical sliding mode control scheme for an asymptotic space vehicle stabilization. International Journal of Control 65,901-912. Celikovsky, S. and H. Nijmeijer (1995). Equivalence of nonlinear systems to triangular form: the singular case. Systems fj Control Letters 27, 135-144. Wittenburg, J . (1977). Dynamics of systems of rigid bodies. B.G . Teubner Stuttgart.
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