Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013 Article ID 483457 7 pageshttpdxdoiorg1011552013483457
Research ArticleAveraging Methods for Design of Spacecraft Hysteresis Damper
Ricardo Gama1 Anna D Guerman2 Ana Seabra3 and Georgi V Smirnov4
1 School of Technology and Management of Lamego Avenida Visconde Guedes Teixeira 5100-074 Lamego Portugal2 Centre for Aerospace Science and Technologies University of Beira Interior Calcada Fonte do Lameiro 6201-001 Covilha Portugal3 Scientific Area of Mathematics ESTGV Polytechnic Institute of Viseu Campus Politecnico 3504-510 Viseu Portugal4 Centre of Physics Department of Mathematics and Applications University of Minho Campus de Gualtar 4710-057 Braga Portugal
Correspondence should be addressed to Anna D Guerman annaubipt
Received 29 April 2013 Accepted 22 May 2013
Academic Editor Antonio F Bertachini A Prado
Copyright copy 2013 Ricardo Gama et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
This work deals with averaging methods for dynamics of attitude stabilization systems The operation of passive gravity-gradientattitude stabilization systems involving hysteresis rods is described by discontinuous differential equations We apply recentlydeveloped averaging techniques for discontinuous system in order to simplify its analysis and to perform parameter optimizationThe results obtained using this analytic method are compared with those of numerical optimization
1 Introduction
Dampers that use magnetic hysteresis rods to dissipatethe energy of undesired angular motions occurred duringdeployment or caused by perturbations are used in attitudecontrol systems of small satellites since 1960s [1] Mathemat-ical modeling of such systems is quite a difficult task sincethemajority of existent hysteresis models result in differentialequations with discontinuous right-hand side
Analysis of dynamics for attitude control systems withmagnetic hysteresis dampers and optimization of their para-meters have been done in [2 3] and the results of these stud-ies have been implemented in real missions [4 5] Howeverthese studies lack an accurate theoretical basis for applica-tion of averaging methods to such problems
Recently an adequate mathematical approach has beendeveloped by the authors in [6] Now we can address a com-plete mathematical theory for attitude stabilization systemswith hysteresis
describing a mechanical system with stabilizer Here 119906 isin 119880 sub
119877
119896 is a parameter It is assumed that 0 asymp 119891(119905 0 119906) for all119905 ge 0 and 119906 isin 119880 that is the velocity of the system near
the origin is small Here we do not assume that zero is anequilibrium position of system (1) The parameter 119906 shouldbe chosen to optimize in some sense the behavior of thetrajectories The choice of this parameter can be based onvarious criteria Obviously it is impossible to construct a sta-bilizer optimal in all aspects Consider for example a linearcontrollable systemThe pole assignment theorem guaranteesthe existence of a linear feedback yielding a linear differentialequation with any given set of eigenvalues so one can choosea stabilizer with a very high damping speed However such astabilizer is practically useless because of the so-called peakeffect (see [7 8]) Namely there exists a large deviation of thesolutions from the equilibrium position at the beginning ofthe stabilization process whenever the module of the eigen-values is big
The aim of this paper is to develop effective analytical andnumerical tools oriented to optimization of stabilizer para-meters for passive attitude stabilization systemwith hysteresisrods
Throughout this paper we denote the set of real numbersby 119877 and the usual 119899-dimensional space of vectors withcomponents in 119877 by 119877
119899 We denote by ⟨119886 119887⟩ the usual scalarproduct in 119877
119899 and by | sdot | a norm By 119861 we denote the closedunit ball that is the set of vectors 119909 isin 119877
119899 satisfying |119909| le 1The transpose of a matrix 119860 is denoted by 119860
lowast The set ofpositively definite symmetric 119899 times 119899-matrices is denoted by
2 Mathematical Problems in Engineering
119872(119899) If 119875 and 119876 are two subsets in 119877
119899 and 120582 isin 119877 we usethe following notations 120582119875 = 120582119901 | 119901 isin 119875 119875 + 119876 =
119901 + 119902 | 119901 isin 119875 119902 isin 119876 The convex hull and the closure of asubset 119878 sub R119899 are denoted by co 119878 and cl 119878 respectivelyTheHausdorff distance between two sets 119860
) The closedunit ball in the space of continuous functions119891 [0 119879] rarr 119877
119899
with the uniform norm 119862([0 119879] 119877
119899
) is denoted by B Theset of locally integrable functions 119891 [0infin[rarr 119877
119899 is denotedby 119871
loc1
([0infin[ 119877
119899
) The upper limit of a set-valued map 119865
119877
119899
rarr 119877
119898 is given by
lim sup1199091015840rarr119909
119865 (119909
1015840
) = V = lim119899rarrinfin
V119896| (119909
119896 V119896) isin gr119865 119909
119896997888rarr 119909
(3)
where gr119865 stands for the graph of the set valued map 119865
2 Statement of the Problem
Consider dynamics of a satellite in a circular geocentricorbitThe satellite is equippedwith a gravity-gradient attitudecontrol system that includes a number of magnetic hysteresisrods as a damperThe spacecraftrsquos equations ofmotion [2] canbe represented in the normalized form Denote by 119909(119905 119909
where 119906 is a parameter from a compact set 119880 sub 119877
119896 Definethe functions
120593
119894(119906) = max
119905isinΔ 119894
max1199090isin119861119894
1003816
1003816
1003816
1003816
119909 (119905 119909
0 119906)
1003816
1003816
1003816
1003816119894 119894 = 0119898 (5)
Here Δ
119894sube [0 119879] are compact sets | sdot |
119894are norms in 119877
119899 and119861
119894= 119909 isin 119877
119899
| |119909|
119894le 119887
119894 Consider the following mathemat-
ical programming problem
120593
0(119906) 997888rarr min
120593
119894(119906) le 120593
119894 119894 = 1119898
119906 isin 119880
(6)
Many problems of stabilization systemsrsquo parameters opti-mization can be written in this form (see [9]) For examplethe minimization of the final deviation can be formalized asfollows
max1199090isin119861
1003816
1003816
1003816
1003816
119909 (119879 119909
0 119906)
1003816
1003816
1003816
1003816
997888rarr min
119906 isin 119880
(7)
and theminimization of themaximal deviation of trajectoriessatisfying certain restrictions at the final moment of time hasthe form
max119905isin [0119879]
max|1199090|=1
1003816
1003816
1003816
1003816
119909 (119905 119909
0 119906)
1003816
1003816
1003816
1003816
997888rarr min
max|1199090|=1
1003816
1003816
1003816
1003816
119909 (119879 119909
0 119906)
1003816
1003816
1003816
1003816
le 120575
119906 isin 119880
(8)
3 Averaging for Discontinuous Systems
The averaging method is one of the most used methods toanalyze differential equations of the form
= 120598119891 (119905 119909) (9)
appearing in the study of nonlinear problems The ideabehind the averaging method is to replace the originalequation by the averaged one
This equation is simpler and has solutions close to thesolutions of the original equation A rigorous justification ofthemethod is given by Bogolyubovrsquos first theorem containingan estimate for the distance between the solutions of theexact and averaged systems on large time intervals [10]The Samoilenko-Stanzhitskii theorem [11 Theorem 2] whichis a generalization of Bogolyubovrsquos second theorem showsthat asymptotic stability of the zero equilibrium position ofaveraged (10) implies that the solutions to original (9) areclose to zero on the infinite time interval
For several models of systems with hysteresis includ-ing the passive attitude stabilization systems the function119891(119905 sdot) appearing in (9) is discontinuous (see eg [3]) andthe classical notion of solution and the classical averagingmethod cannot be used For such systems Filippov proposeda generalized concept of solution rewriting problem (9) as adifferential inclusion
where 119909 rarr 119865(sdot 119909) is an upper semicontinuous set-valuedmap obtained from 119891(119905 sdot) by Filippov regularization [12 13]The use of this concept of solution makes it necessary to gen-eralize the averaging method to differential inclusions Manyresults extending Bogolyubovrsquos first theorem to differentialinclusions have been obtained (see eg [14 15]) In the caseof Lipschitzian differential inclusions the problem has beencompletely solved by Plotnikov [14] Averaging results forinclusions with upper semicontinuous right-hand side havebeen obtained by Plotnikov [15] under conditions of Lipschitzcontinuity of the averaged inclusion and for inclusions witha piecewise Lipschitzian right-hand side Recently [6] anaveraged differential inclusion has been introduced allowingone to prove extensions of Bogolyubovrsquos first theorem andof the Samoilenko-Stanzhitskii theorem for upper semi-continuous differential inclusions and as a consequence for
Mathematical Problems in Engineering 3
discontinuous dynamical systems Here we outline the mainresults from [6]
Let 119865 119877 times 119877
119899
rarr 119877
119899 be a set-valued map Set
119868 (119905
1 119905
2 119909 120575) = int
1199052
1199051
V (119905) 119889119905 | V (sdot) isin 119871
loc1
([0infin[ 119877
119899
)
V (119905) isin 119865 (119905 119909 + 120575119861)
(12)
We denote by 119865
120575
(119909) the convex hull of the map
Φ
120575
(119909) = lim sup120579uarr1
lim sup119879rarrinfin
1
(1 minus 120579) 119879
119868 (120579119879 119879 119909 120575) (13)
and define the averaged differential inclusion as
isin 119865 (119909) = ⋂
120575gt0
119865
120575
(119909) (14)
Note that under Lipschitz condition this map coincides with
119865 = lim119879rarrinfin
1
119879
int
119879
0
119865 (119905 119909) 119889119905(15)
if the limit exists in the sense of Hausdorff distance (see [6])Assume that the following conditions are satisfied
(C1) cl co119865(119905 119909) = 119865(119905 119909) for all (119905 119909) isin 119877 times 119877
119899(C2) the set-valued map 119865(119905 sdot) is upper semi-continuous(C3) for any 119909 there exists measurable selection of 119865(119905 119909)
that is there exists 119891(119905 119909) isin 119865(119905 119909) such that 119905 rarr
119891(119905 119909) is measurable for all 119909(C4) there exists a nonnegative 119887(sdot) isin 119871
loc1
([0infin[ 119877) suchthat 119865(119905 119909) sub 119887(119905)119861 for all (119905 119909) isin [0 +infin[times119877
119899(C5) there exists the limit
119887 = lim119879rarrinfin
1
119879
int
119879
0
119887 (119905) 119889119905(16)
Under these conditions the following version of Bogolyu-bovrsquos first theorem is true
Theorem 1 Let119879 gt 0 and let119865 119877times119877
119899
rarr 119877
119899 be a set-valuedmap satisfying conditions (C1)ndash(C5) Let 119862 isin 119877
119899 be a compactset Then for any 120578 gt 0 there exists 120598
0gt 0 such that for any
120598 isin]0 120598
0[ and any solution 119909(sdot) isin S
[0119879120598](120598119865 119862) there exists a
solution 119909(sdot) isin S[0119879120598]
(120598119865 119862) satisfying
|119909 (119905) minus 119909 (119905)| lt 120578 119905 isin [0
119879
120598
] (17)
Set
119866
120598(120591 119910) = 119865(
120591
120598
119910) 119866
0(119910) = 119865 (119910) (18)
Next theorem is an extension of the Samoilenko-Stanzhitskiitheorem
Theorem 2 Let 119865 119877 times 119877
119899
rarr 119877
119899 be a set-valued map satis-fying conditions (C1)ndash(C5) Assume that 119910 = 0 is an asympto-tically stable equilibrium position of the differential inclusion119910 isin 119866
0(119910) Then for any 120578 gt 0 there exist 120598
0gt 0 and 120575 gt 0
such that S[0infin[
(119866
120598 120575119861) sub 120578B whenever 120598 isin]0 120598
0[
The last theorem shows that if the averaged inclusionhas zero as its asymptotically stable equilibrium position thetrajectories of the original inclusion stay in the vicinity of theorigin provided 120598 gt 0 and |119909
0| are sufficiently small
If the averaged inclusion has a special form we can gofurther and make some conclusion on the detailed behaviourof the trajectories of the original system Assume that theaveraged inclusion has the form
119861 119888 gt 0 the real parts of the matrix119860(119906) eigenvalues are negative for all 119906 isin 119880 and the function119906 rarr 119860(119906) is continuous for all 119906 isin 119880 If 120574
0gt 0 is sufficiently
small then the set of solutions to the Lyapunov inequality (see[16]) for the matrix 119860(119906)
is nonempty and compact for all 119906 isin 119880 Let (120574 119881) isin
L(119880) Denote by |119909|
119881the Euclidean norm defined by |119909|
119881=
radic⟨119909 119881119909⟩ There exist positive constants 119888
1and 119888
2satisfying
119888
1|119909| le |119909|
119881le 119888
2|119909| (21)
whenever (120574 119881) isin L(119880) for some 120574
Theorem 3 Let 120575 gt 0 119906 isin 119880 and (120574 119881) isin L(119880) Thereexists 120598
0(120575) such that for all 120598 isin]0 120598
0(120575)[ the condition |119909
0|
119881lt
120575 lt 119888
2
1120574119888 implies the inequality |119909(119905 119909
0 119906)|
119881lt 31205752
This theorem shows that the behavior of the trajectory119909(119905 119909
0 119906) can be characterized in terms of the pair (120574 119881)
The parameter 120574 is responsible for the damping speed of theprocess while the form of the ellipsoid 119909 | ⟨119909 119881119909⟩ le 1
describes the amplitude of the deviation of the trajectory fromthe originThe aim of parameter choosing can be formulatedas follows maximal value of 120574 and maximal sphericity of theellipsoid 119909 | ⟨119909 119881119909⟩ le 1 The latter property guaranteesminimal overshooting of the damping process and as aconsequence the largest region of applicability of the approx-imation obtained via averaging
4 Choosing Passive MagneticStabilizer Parameters
The in-plane oscillations of a satellite moving along a polarcircular orbit and equipped with a passive gravity-gradientattitude stabilization system with one hysteresis rod aredescribed by the equation
+ 120596
2
120572 = 120598119891 (22)
4 Mathematical Problems in Engineering
where 120572 is the pitch angle of spacecraft 120598 is small parameterproportional to the rodrsquos volume and the force 119891 is given by
119891 = 119882(119867
120591) (119867
1119890
3minus 119867
3119890
1minus 120572 (119867
1119890
1+ 119867
3119890
3)) (23)
Here
119867
120591= 119867
1119890
1+ 119867
3119890
3+ 120572 (119867
1119890
3minus 119867
3119890
1) (24)
describes the projection of the geomagnetic field on the rodaxis
119882(119867
120591) = 119867
120591minus
120581
2
sign
119867
120591(25)
is the hysteresis function 120581 corresponds to the coercive force
119867
1= cos 119905 119867
3= minus2 sin 119905 (26)
and 119905 is the argument of latitude of the current point of theorbit [2] The vector
Theorem 4 Assume that 120596 is an irrational number Then theaveraged system for (31) is
119886 = minus
120598
2120596
(119901119886 + 119902119887) + 119903
119886(119886 119887 120579)
119887 =
120598
2120596
(119902119886 minus 119901119887) + 119903
119887(119886 119887 120579)
(32)
where
119901 =
9120581120596119890
2
1119890
2
3
120587(1 + 3119890
2
3)
32
119902 =
3
2
(119890
2
1minus 119890
2
3) +
6120581119890
1119890
3
120587(1 + 3119890
2
3)
32
(33)
|119903
119886| = 119874(119886
2
+ 119887
2
) and |119903
119887| = 119874(119886
2
+ 119887
2
)
Obviously we have
(
1 0
0 1
)(
minus119901 minus119902
119902 minus119901
) + (
minus119901 119902
minus119902 minus119901
)(
1 0
0 1
) = minus2119901(
1 0
0 1
)
(34)
Therefore we see that the linearization of the averaged systemalways has a Lyapunov function119881 = 119886
2
+119887
2
and the dampingspeed is determined by the value of 119901 This means that thepeak effect does not take place for the linearization of theaveraged system
To maximize the damping one has to increase the totalvolume of the hysteresis material on board However it iswellknown that the efficiency of a damping rod is increasedwith the increase of the ratio between the rodrsquos length and itscross-section dimension Therefore instead of one massivebar the attitude control system should use several ratherthin rods of the maximum length allowed by the spacecraftgeometrical and system restrictions On the other hand tominimize the perturbation of the spacecraft angular motioncaused by the damping system itself the direction of totalmagnetic field in the rods should deviate as little as possiblefrom the direction of the geomagnetic field at the currentpoint of the orbit Thus in general case one should use asystem of three equal orthogonal hysteresis rods or a numberof such systems Here we consider in-plane satellite dynamicson a polar orbit and for such purpose it suffices to analyze apair of equal orthogonal rods
Orientation of this pair of equal orthogonal rods can bedefined as (119890
1 119890
3) (minus119890
3 119890
1) where 119890
1= cos 120579 and 119890
3= sin 120579
and due to the system symmetry it is enough to study theinterval 120579 isin [0 1205872] If the satellite is equipped with severalidentical hysteresis rods the corresponding nonlinear systemis
119886 = minus
120598
120596
(119891
1+ 119891
2+ sdot sdot sdot ) sin120596119905
119887 =
120598
120596
(119891
1+ 119891
2+ sdot sdot sdot ) cos120596119905
(35)
Here the terms 119891
1 119891
2 describe the interaction of the res-
pective rod with the geomagnetic field For a couple of equalorthogonal rods and for small deviation from the origin theforces 119891
1and 119891
2are given by
119891
1= 119882(119867
1205911) (119867
1119890
3minus 119867
3119890
1minus 120572 (119867
1119890
1+ 119867
3119890
3))
119891
2= 119882(119867
1205912) (119867
1119890
1+ 119867
3119890
3minus 120572 (minus119867
1119890
3+ 119867
3119890
1))
(36)
respectively Here
119867
1205911= 119867
1119890
1+ 119867
3119890
3+ 120572 (119867
1119890
3minus 119867
3119890
1)
119867
1205912= minus119867
1119890
3+ 119867
3119890
1+ 120572 (119867
1119890
1+ 119867
3119890
3)
119882 (119867
120591119895) = 119867
120591119895minus
120581
2
sign
119867
120591119895 119895 = 1 2
(37)
Mathematical Problems in Engineering 5
For this case the first approximation of the averaged sys-tem takes the form
119886 = minus
120598
2120596
119875119886
119887 = minus
120598
2120596
119875119887
(38)
where
119875 =
9120581120596119890
2
1119890
2
3
120587
(
1
(1 + 3119890
2
3)
32
+
1
(1 + 3119890
2
1)
32
) (39)
An easy calculation shows that the optimal value of 120579 is 1205874so 119890
1= 119890
3=
radic22 In the next sectionwe numerically analyze
the validity of the previous analytical study
5 Numerical Simulations
We approximate problem (6) by the following problem
120593
0997888rarr min
1003816
1003816
1003816
1003816
1003816
119909(119905
119894
119896 119909
119894
119895 119906)
1003816
1003816
1003816
1003816
1003816119894
le 120593
119894+ 120576 119894 = 0119898
119906 isin 119880
(40)
where 119905
119894
0= 0 119905119894
119896isin Δ
119894 119909119894119895isin 119861
119894 119895 = 1 119869 and
119909 (119905
119894
119896+1 119909
119894
119895 119906) = 119909 (119905
119894
119896 119909
119894
119895 119906) + 120591119891 (119905
119896 119909 (119905
119894
119896 119909
119894
119895 119906) 119906)
120591 = 119905
119894
119896+1minus 119905
119894
119896 119896 = 0119873
(41)
is the Euler approximation for the solution 119909(sdot 119909
119894
119895 119906) Note
that this is a hard problem because of the discontinuity ofthe system It is necessary to consider very fine partition ofthe time interval in order to get a good approximation ofthe solutions to the discontinuous differential equation Forsmooth right-hand sides the number of points in the meshcan be significantly reduced (see [9])
Let 120598120581 gt 0 be small enough We consider a satellite withtwo equal orthogonal hysteresis rods A typical trajectory ofthe system is shown in Figure 1
Note that the oscillations of the trajectory do not allowone to characterize its damping speed using the norm at thefinal moment of time For this reason we numerically solvethe following problem
max1199090isin1205750119861
max119905isin[119879minus119879119901119879]
1003816
1003816
1003816
1003816
119909 (119905 119909
0 119906)
1003816
1003816
1003816
1003816
997888rarr min
119906 isin 119880
(42)
where 119879
119901≪ 119879 is an interval corresponding to the period
of oscillation of solutions The minimization is done usingmultistart Nelder-Mead method For 120598 = 025 120581 = 01120596 = 0949 120575
0= 1 119879 = 300120587 and 119879
119901= 20120587 the fastest
0 01 02 03 04 05 06 07 08 09 10
02
04
06
08
1
b(r
ad)
a (rad)
(a) Phase plot of 119887 versus 119886
0 200 400 600 800 1000 12000
05
1
15
Am
plitu
de (r
ad)
Argument of latitude (rad)
(b) Time evolution of amplituderadic1198862 + 1198872 of pitch oscillations (29)
Figure 1 A typical trajectory for 120598 = 025 120581 = 01 120596 = 0949
damping speed is observed for values of 120579 very close to 1205874that is to the expected value Next we consider the problem
max1199090isin1205750119861
max119905isin [0119879]
1003816
1003816
1003816
1003816
119909 (119905 119909
0 119906)
1003816
1003816
1003816
1003816
997888rarr min
max1199090isin1205750119861
max119905isin[119879minus119879119901 119879]
1003816
1003816
1003816
1003816
119909 (119905 119909
0 119906)
1003816
1003816
1003816
1003816
le 120575
1
119906 isin 119880
(43)
The results of minimization for 120598 = 025 120581 = 01 120596 = 0949120575
0= 1 120575
1= 01 119879 = 300120587 and 119879
119901= 20120587 show that the
problem has many local minima but with the values of thefunctional very close to 1 that is the nonlinear system withtwo orthogonal hysteresis rods has no overshooting for allvalues of 120579This allows one to conclude that the value 120579 = 1205874
is the best one It guarantees high damping speed anddoes notcause peaking
6 Conclusions
Thispaper is dedicated to the problemof parameter optimiza-tion for a gravitationally stabilized satellite with magnetichysteresis damper Its motion is described by differentialequations with discontinuous right-hand side The disconti-nuity is the principal obstacle in the application of the aver-aging method Our recently obtained results on averaging ofdiscontinuous systems are applied now to rigorously justifythe use of this method for a satellite with hysteresis rods
6 Mathematical Problems in Engineering
We consider here the simplest case of in-plane oscillationson a polar circular orbit Theorem 3 shows that the behaviorof the system can be characterized in terms of Lyapunovfunction which can be chosen in order to guarantee the bestproperties of damping process Further study is performednumerically and the simulations are in excellent agreementwith the analytical results confirming also previous studieson hysteresis damping of satellite pitch oscillation in gravity-gradient mode
Our results can also be applied to rigorously justify theuse of averaging techniques in analysis of other engineeringproblems involving differential equations with discontinuousright-hand side
Appendix
This appendix contains the proofs of Theorems 3 and 4
Proof of Theorem 3 Set 119879 = 2 ln 2120574 Then using (21) fromthe Lyapunov inequality we get |119909(119879 119909
0 119906)|
119881le |119909
0|
1198812
whenever |119909
0|
119881le 119888
1120574119888 There exists 120598
0(120575) such that
sup119905isin[0119879]
|119909(119905 119909
0 119906) minus 119909(119905 119909
0 119906)| lt 1205752 Therefore we have
|119909(119879 119909
0 119906)| lt 120575 and |119909(119905 119909
0 119906)| le |119909(119905 119909
0 119906)| + |119909(119905 119909
0 119906) minus
119909(119905 119909
0 119906)| lt 31205752 This ends the proof
Proof of Theorem 4 Consider the function
119891 = 119882(119867
120591) (119867
1119890
3minus 119867
3119890
1minus 120572 (119867
1119890
1+ 119867
3119890
3)) (A1)
where
119867
120591= 119867
1119890
1+ 119867
3119890
3+ 120572 (119867
1119890
3minus 119867
3119890
1)
119882 (119867
120591) = 119867
120591minus
120581
2
sign
119867
120591
119867
1= cos 119905 119867
3= minus2 sin 119905
(119890
1 119890
3) = (cos 120579 sin 120579) 120579 isin [0
120587
2
]
(A2)
First note that for any fixed pair (119886 119887) the function 119905 rarr
119867
120591(119905 119886 119887) is analytic Therefore the integral 119868(119905
1 119905
2 119909 120575) (see
Section 2) is a point This implies that the averaged operatordefined in (14) coincides with
lim119879rarrinfin
1
119879
int
119879
0
119891 (119905 119886 119887) (
minus sin120596119905
cos120596119905
) 119889119905(A3)
if the limit exists Since 120596 is irrational number and 119891 can beconsidered as a 2120587-periodic function 119892 = 119892(119905
119905 119886 119887) of thearguments 119905 and
119905 = 120596119905 we see that limit (A3) does existand we have
lim119879rarrinfin
1
119879
int
119879
0
119891 (119905 119886 119887) (
minus sin120596119905
cos120596119905
) 119889119905
=
1
(2120587)
2∬
2120587
0
119892 (119905
119905 119886 119887)
times (
minus sin
119905
cos119905 ) 119889119905 119889
119905
(A4)
To evaluate this integral we represent the derivative
(sin (2119899 + 1) 119905 cos (2119899 + 1) 119905
1015840
minus sin (2119899 + 1)
1015840 cos (2119899 + 1) 119905)
times (2119899 + 1)
minus1
(A7)
Observe that
sin 119905
1015840
=radic
1 + 3119890
2
3(minus2119890
3+
119890
1(2120572 minus 3119890
1119890
3120573)
1 + 3119890
2
3
)
+ 119874(120572
2
+ 120573
2
)
(A8)
cos 1199051015840 = radic1 + 3119890
2
3(119890
1+
2119890
3(2120572 minus 3119890
1119890
3120573)
1 + 3119890
2
3
)
+ 119874(120572
2
+ 120573
2
)
(A9)
Substituting (A7) into (A4) and integrating in order of 119905 andthen in order of
119905 we obtain the result
Acknowledgments
This research is supported by the Portuguese Foundation forScience and Technologies (FCT) the Portuguese OperationalProgramme for Competitiveness Factors (COMPETE) thePortuguese Strategic Reference Framework (QREN) and theEuropean Regional Development Fund (FEDER)
References
[1] R E Fischell and F F Mobley ldquoA system for passive gravity-gradient stabilization of earth satellitesrdquo Guidance and Controlvol 2 pp 37ndash71 1964
Mathematical Problems in Engineering 7
[2] V A Sarychev V I Penkov M Yu Ovchinnikov and A DGuerman ldquoMotions of a gravitationally stabilized satellite withhysteresis rods in a polar orbitrdquo Cosmic Research vol 26 no 5pp 561ndash574 1988
[3] A D Guerman M Yu Ovchinnikov V I Penrsquokov and V ASarychev ldquoNon-resonant motions of a satellite with hysteresisrods under conditions of gravity orientationrdquo Mechanics ofSolids vol 24 pp 1ndash11 1989
[4] M Ovchinnikov V Penrsquokov O Norberg and S BarabashldquoAttitude control system for the first sweedish nanosatelliteMUNINrdquoActa Astronautica vol 46 no 2ndash6 pp 319ndash326 2000
[5] F Santoni and M Zelli ldquoPassive magnetic attitude stabilizationof the UNISAT-4 microsatelliterdquo Acta Astronautica vol 65 no5-6 pp 792ndash803 2009
[6] R Gama A Guerman and G Smirnov ldquoOn the asymptoticstability of discontinuous systems analysed via the averagingmethodrdquo Nonlinear Analysis Theory Methods and ApplicationsA vol 74 no 4 pp 1513ndash1522 2011
[7] RN Izmailov ldquoThepeak effect in stationary linear systemswithscalar inputs and outputsrdquoAutomation and Remote Control vol48 no 8 part 1 pp 1018ndash1024 1987
[8] H J Sussmann and P V Kokotovic ldquoThe peaking phenomenonand the global stabilization of nonlinear systemsrdquo IEEE Trans-actions on Automatic Control vol 36 no 4 pp 424ndash439 1991
[9] A Guerman A Seabra andG Smirnov ldquoOptimization of para-meters of asymptotically stable systemsrdquo Mathematical Prob-lems in Engineering vol 2011 Article ID 526167 19 pages 2011
[10] N N Bogoliubov and Y AMitropolskyAsymptotic Methods intheTheory of Non-Linear Oscillations Gordon and Breach NewYork NY USA 1961
[11] A M Samoilenko and A N Stanzhitskii ldquoOn averaging differ-ential equations on an infinite intervalrdquo Differential Equationsvol 42 no 4 pp 476ndash482 2006
[12] A F Filippov Differential Equations with Discontinuous Right-Hand Sides vol 18 ofMathematics and Its Applications KluwerAcademic New York NY USA 1988
[13] G V Smirnov Introduction to the Theory of Differential Inclu-sions vol 41 of AMS Graduate Studies in Mathematics Ameri-can Mathematical Society Providence RI USA 2002
[14] V A Plotnikov ldquoAveraging method for differential inclusionsand its application to optimal control problemsrdquo DifferentialEquations vol 15 pp 1013ndash1018 1979
[15] V A Plotnikov ldquoAsymptotic methods in the theory of differen-tial equations with discontinuous and multi-valued right-handsidesrdquoUkranian Mathematical Journal vol 48 no 11 pp 1605ndash1616 1996
[16] A M Liapunov Stability of Motion Academic Press New YorkNY USA 1966
119872(119899) If 119875 and 119876 are two subsets in 119877
119899 and 120582 isin 119877 we usethe following notations 120582119875 = 120582119901 | 119901 isin 119875 119875 + 119876 =
119901 + 119902 | 119901 isin 119875 119902 isin 119876 The convex hull and the closure of asubset 119878 sub R119899 are denoted by co 119878 and cl 119878 respectivelyTheHausdorff distance between two sets 119860
) The closedunit ball in the space of continuous functions119891 [0 119879] rarr 119877
119899
with the uniform norm 119862([0 119879] 119877
119899
) is denoted by B Theset of locally integrable functions 119891 [0infin[rarr 119877
119899 is denotedby 119871
loc1
([0infin[ 119877
119899
) The upper limit of a set-valued map 119865
119877
119899
rarr 119877
119898 is given by
lim sup1199091015840rarr119909
119865 (119909
1015840
) = V = lim119899rarrinfin
V119896| (119909
119896 V119896) isin gr119865 119909
119896997888rarr 119909
(3)
where gr119865 stands for the graph of the set valued map 119865
2 Statement of the Problem
Consider dynamics of a satellite in a circular geocentricorbitThe satellite is equippedwith a gravity-gradient attitudecontrol system that includes a number of magnetic hysteresisrods as a damperThe spacecraftrsquos equations ofmotion [2] canbe represented in the normalized form Denote by 119909(119905 119909
where 119906 is a parameter from a compact set 119880 sub 119877
119896 Definethe functions
120593
119894(119906) = max
119905isinΔ 119894
max1199090isin119861119894
1003816
1003816
1003816
1003816
119909 (119905 119909
0 119906)
1003816
1003816
1003816
1003816119894 119894 = 0119898 (5)
Here Δ
119894sube [0 119879] are compact sets | sdot |
119894are norms in 119877
119899 and119861
119894= 119909 isin 119877
119899
| |119909|
119894le 119887
119894 Consider the following mathemat-
ical programming problem
120593
0(119906) 997888rarr min
120593
119894(119906) le 120593
119894 119894 = 1119898
119906 isin 119880
(6)
Many problems of stabilization systemsrsquo parameters opti-mization can be written in this form (see [9]) For examplethe minimization of the final deviation can be formalized asfollows
max1199090isin119861
1003816
1003816
1003816
1003816
119909 (119879 119909
0 119906)
1003816
1003816
1003816
1003816
997888rarr min
119906 isin 119880
(7)
and theminimization of themaximal deviation of trajectoriessatisfying certain restrictions at the final moment of time hasthe form
max119905isin [0119879]
max|1199090|=1
1003816
1003816
1003816
1003816
119909 (119905 119909
0 119906)
1003816
1003816
1003816
1003816
997888rarr min
max|1199090|=1
1003816
1003816
1003816
1003816
119909 (119879 119909
0 119906)
1003816
1003816
1003816
1003816
le 120575
119906 isin 119880
(8)
3 Averaging for Discontinuous Systems
The averaging method is one of the most used methods toanalyze differential equations of the form
= 120598119891 (119905 119909) (9)
appearing in the study of nonlinear problems The ideabehind the averaging method is to replace the originalequation by the averaged one
This equation is simpler and has solutions close to thesolutions of the original equation A rigorous justification ofthemethod is given by Bogolyubovrsquos first theorem containingan estimate for the distance between the solutions of theexact and averaged systems on large time intervals [10]The Samoilenko-Stanzhitskii theorem [11 Theorem 2] whichis a generalization of Bogolyubovrsquos second theorem showsthat asymptotic stability of the zero equilibrium position ofaveraged (10) implies that the solutions to original (9) areclose to zero on the infinite time interval
For several models of systems with hysteresis includ-ing the passive attitude stabilization systems the function119891(119905 sdot) appearing in (9) is discontinuous (see eg [3]) andthe classical notion of solution and the classical averagingmethod cannot be used For such systems Filippov proposeda generalized concept of solution rewriting problem (9) as adifferential inclusion
where 119909 rarr 119865(sdot 119909) is an upper semicontinuous set-valuedmap obtained from 119891(119905 sdot) by Filippov regularization [12 13]The use of this concept of solution makes it necessary to gen-eralize the averaging method to differential inclusions Manyresults extending Bogolyubovrsquos first theorem to differentialinclusions have been obtained (see eg [14 15]) In the caseof Lipschitzian differential inclusions the problem has beencompletely solved by Plotnikov [14] Averaging results forinclusions with upper semicontinuous right-hand side havebeen obtained by Plotnikov [15] under conditions of Lipschitzcontinuity of the averaged inclusion and for inclusions witha piecewise Lipschitzian right-hand side Recently [6] anaveraged differential inclusion has been introduced allowingone to prove extensions of Bogolyubovrsquos first theorem andof the Samoilenko-Stanzhitskii theorem for upper semi-continuous differential inclusions and as a consequence for
Mathematical Problems in Engineering 3
discontinuous dynamical systems Here we outline the mainresults from [6]
Let 119865 119877 times 119877
119899
rarr 119877
119899 be a set-valued map Set
119868 (119905
1 119905
2 119909 120575) = int
1199052
1199051
V (119905) 119889119905 | V (sdot) isin 119871
loc1
([0infin[ 119877
119899
)
V (119905) isin 119865 (119905 119909 + 120575119861)
(12)
We denote by 119865
120575
(119909) the convex hull of the map
Φ
120575
(119909) = lim sup120579uarr1
lim sup119879rarrinfin
1
(1 minus 120579) 119879
119868 (120579119879 119879 119909 120575) (13)
and define the averaged differential inclusion as
isin 119865 (119909) = ⋂
120575gt0
119865
120575
(119909) (14)
Note that under Lipschitz condition this map coincides with
119865 = lim119879rarrinfin
1
119879
int
119879
0
119865 (119905 119909) 119889119905(15)
if the limit exists in the sense of Hausdorff distance (see [6])Assume that the following conditions are satisfied
(C1) cl co119865(119905 119909) = 119865(119905 119909) for all (119905 119909) isin 119877 times 119877
119899(C2) the set-valued map 119865(119905 sdot) is upper semi-continuous(C3) for any 119909 there exists measurable selection of 119865(119905 119909)
that is there exists 119891(119905 119909) isin 119865(119905 119909) such that 119905 rarr
119891(119905 119909) is measurable for all 119909(C4) there exists a nonnegative 119887(sdot) isin 119871
loc1
([0infin[ 119877) suchthat 119865(119905 119909) sub 119887(119905)119861 for all (119905 119909) isin [0 +infin[times119877
119899(C5) there exists the limit
119887 = lim119879rarrinfin
1
119879
int
119879
0
119887 (119905) 119889119905(16)
Under these conditions the following version of Bogolyu-bovrsquos first theorem is true
Theorem 1 Let119879 gt 0 and let119865 119877times119877
119899
rarr 119877
119899 be a set-valuedmap satisfying conditions (C1)ndash(C5) Let 119862 isin 119877
119899 be a compactset Then for any 120578 gt 0 there exists 120598
0gt 0 such that for any
120598 isin]0 120598
0[ and any solution 119909(sdot) isin S
[0119879120598](120598119865 119862) there exists a
solution 119909(sdot) isin S[0119879120598]
(120598119865 119862) satisfying
|119909 (119905) minus 119909 (119905)| lt 120578 119905 isin [0
119879
120598
] (17)
Set
119866
120598(120591 119910) = 119865(
120591
120598
119910) 119866
0(119910) = 119865 (119910) (18)
Next theorem is an extension of the Samoilenko-Stanzhitskiitheorem
Theorem 2 Let 119865 119877 times 119877
119899
rarr 119877
119899 be a set-valued map satis-fying conditions (C1)ndash(C5) Assume that 119910 = 0 is an asympto-tically stable equilibrium position of the differential inclusion119910 isin 119866
0(119910) Then for any 120578 gt 0 there exist 120598
0gt 0 and 120575 gt 0
such that S[0infin[
(119866
120598 120575119861) sub 120578B whenever 120598 isin]0 120598
0[
The last theorem shows that if the averaged inclusionhas zero as its asymptotically stable equilibrium position thetrajectories of the original inclusion stay in the vicinity of theorigin provided 120598 gt 0 and |119909
0| are sufficiently small
If the averaged inclusion has a special form we can gofurther and make some conclusion on the detailed behaviourof the trajectories of the original system Assume that theaveraged inclusion has the form
119861 119888 gt 0 the real parts of the matrix119860(119906) eigenvalues are negative for all 119906 isin 119880 and the function119906 rarr 119860(119906) is continuous for all 119906 isin 119880 If 120574
0gt 0 is sufficiently
small then the set of solutions to the Lyapunov inequality (see[16]) for the matrix 119860(119906)
is nonempty and compact for all 119906 isin 119880 Let (120574 119881) isin
L(119880) Denote by |119909|
119881the Euclidean norm defined by |119909|
119881=
radic⟨119909 119881119909⟩ There exist positive constants 119888
1and 119888
2satisfying
119888
1|119909| le |119909|
119881le 119888
2|119909| (21)
whenever (120574 119881) isin L(119880) for some 120574
Theorem 3 Let 120575 gt 0 119906 isin 119880 and (120574 119881) isin L(119880) Thereexists 120598
0(120575) such that for all 120598 isin]0 120598
0(120575)[ the condition |119909
0|
119881lt
120575 lt 119888
2
1120574119888 implies the inequality |119909(119905 119909
0 119906)|
119881lt 31205752
This theorem shows that the behavior of the trajectory119909(119905 119909
0 119906) can be characterized in terms of the pair (120574 119881)
The parameter 120574 is responsible for the damping speed of theprocess while the form of the ellipsoid 119909 | ⟨119909 119881119909⟩ le 1
describes the amplitude of the deviation of the trajectory fromthe originThe aim of parameter choosing can be formulatedas follows maximal value of 120574 and maximal sphericity of theellipsoid 119909 | ⟨119909 119881119909⟩ le 1 The latter property guaranteesminimal overshooting of the damping process and as aconsequence the largest region of applicability of the approx-imation obtained via averaging
4 Choosing Passive MagneticStabilizer Parameters
The in-plane oscillations of a satellite moving along a polarcircular orbit and equipped with a passive gravity-gradientattitude stabilization system with one hysteresis rod aredescribed by the equation
+ 120596
2
120572 = 120598119891 (22)
4 Mathematical Problems in Engineering
where 120572 is the pitch angle of spacecraft 120598 is small parameterproportional to the rodrsquos volume and the force 119891 is given by
119891 = 119882(119867
120591) (119867
1119890
3minus 119867
3119890
1minus 120572 (119867
1119890
1+ 119867
3119890
3)) (23)
Here
119867
120591= 119867
1119890
1+ 119867
3119890
3+ 120572 (119867
1119890
3minus 119867
3119890
1) (24)
describes the projection of the geomagnetic field on the rodaxis
119882(119867
120591) = 119867
120591minus
120581
2
sign
119867
120591(25)
is the hysteresis function 120581 corresponds to the coercive force
119867
1= cos 119905 119867
3= minus2 sin 119905 (26)
and 119905 is the argument of latitude of the current point of theorbit [2] The vector
Theorem 4 Assume that 120596 is an irrational number Then theaveraged system for (31) is
119886 = minus
120598
2120596
(119901119886 + 119902119887) + 119903
119886(119886 119887 120579)
119887 =
120598
2120596
(119902119886 minus 119901119887) + 119903
119887(119886 119887 120579)
(32)
where
119901 =
9120581120596119890
2
1119890
2
3
120587(1 + 3119890
2
3)
32
119902 =
3
2
(119890
2
1minus 119890
2
3) +
6120581119890
1119890
3
120587(1 + 3119890
2
3)
32
(33)
|119903
119886| = 119874(119886
2
+ 119887
2
) and |119903
119887| = 119874(119886
2
+ 119887
2
)
Obviously we have
(
1 0
0 1
)(
minus119901 minus119902
119902 minus119901
) + (
minus119901 119902
minus119902 minus119901
)(
1 0
0 1
) = minus2119901(
1 0
0 1
)
(34)
Therefore we see that the linearization of the averaged systemalways has a Lyapunov function119881 = 119886
2
+119887
2
and the dampingspeed is determined by the value of 119901 This means that thepeak effect does not take place for the linearization of theaveraged system
To maximize the damping one has to increase the totalvolume of the hysteresis material on board However it iswellknown that the efficiency of a damping rod is increasedwith the increase of the ratio between the rodrsquos length and itscross-section dimension Therefore instead of one massivebar the attitude control system should use several ratherthin rods of the maximum length allowed by the spacecraftgeometrical and system restrictions On the other hand tominimize the perturbation of the spacecraft angular motioncaused by the damping system itself the direction of totalmagnetic field in the rods should deviate as little as possiblefrom the direction of the geomagnetic field at the currentpoint of the orbit Thus in general case one should use asystem of three equal orthogonal hysteresis rods or a numberof such systems Here we consider in-plane satellite dynamicson a polar orbit and for such purpose it suffices to analyze apair of equal orthogonal rods
Orientation of this pair of equal orthogonal rods can bedefined as (119890
1 119890
3) (minus119890
3 119890
1) where 119890
1= cos 120579 and 119890
3= sin 120579
and due to the system symmetry it is enough to study theinterval 120579 isin [0 1205872] If the satellite is equipped with severalidentical hysteresis rods the corresponding nonlinear systemis
119886 = minus
120598
120596
(119891
1+ 119891
2+ sdot sdot sdot ) sin120596119905
119887 =
120598
120596
(119891
1+ 119891
2+ sdot sdot sdot ) cos120596119905
(35)
Here the terms 119891
1 119891
2 describe the interaction of the res-
pective rod with the geomagnetic field For a couple of equalorthogonal rods and for small deviation from the origin theforces 119891
1and 119891
2are given by
119891
1= 119882(119867
1205911) (119867
1119890
3minus 119867
3119890
1minus 120572 (119867
1119890
1+ 119867
3119890
3))
119891
2= 119882(119867
1205912) (119867
1119890
1+ 119867
3119890
3minus 120572 (minus119867
1119890
3+ 119867
3119890
1))
(36)
respectively Here
119867
1205911= 119867
1119890
1+ 119867
3119890
3+ 120572 (119867
1119890
3minus 119867
3119890
1)
119867
1205912= minus119867
1119890
3+ 119867
3119890
1+ 120572 (119867
1119890
1+ 119867
3119890
3)
119882 (119867
120591119895) = 119867
120591119895minus
120581
2
sign
119867
120591119895 119895 = 1 2
(37)
Mathematical Problems in Engineering 5
For this case the first approximation of the averaged sys-tem takes the form
119886 = minus
120598
2120596
119875119886
119887 = minus
120598
2120596
119875119887
(38)
where
119875 =
9120581120596119890
2
1119890
2
3
120587
(
1
(1 + 3119890
2
3)
32
+
1
(1 + 3119890
2
1)
32
) (39)
An easy calculation shows that the optimal value of 120579 is 1205874so 119890
1= 119890
3=
radic22 In the next sectionwe numerically analyze
the validity of the previous analytical study
5 Numerical Simulations
We approximate problem (6) by the following problem
120593
0997888rarr min
1003816
1003816
1003816
1003816
1003816
119909(119905
119894
119896 119909
119894
119895 119906)
1003816
1003816
1003816
1003816
1003816119894
le 120593
119894+ 120576 119894 = 0119898
119906 isin 119880
(40)
where 119905
119894
0= 0 119905119894
119896isin Δ
119894 119909119894119895isin 119861
119894 119895 = 1 119869 and
119909 (119905
119894
119896+1 119909
119894
119895 119906) = 119909 (119905
119894
119896 119909
119894
119895 119906) + 120591119891 (119905
119896 119909 (119905
119894
119896 119909
119894
119895 119906) 119906)
120591 = 119905
119894
119896+1minus 119905
119894
119896 119896 = 0119873
(41)
is the Euler approximation for the solution 119909(sdot 119909
119894
119895 119906) Note
that this is a hard problem because of the discontinuity ofthe system It is necessary to consider very fine partition ofthe time interval in order to get a good approximation ofthe solutions to the discontinuous differential equation Forsmooth right-hand sides the number of points in the meshcan be significantly reduced (see [9])
Let 120598120581 gt 0 be small enough We consider a satellite withtwo equal orthogonal hysteresis rods A typical trajectory ofthe system is shown in Figure 1
Note that the oscillations of the trajectory do not allowone to characterize its damping speed using the norm at thefinal moment of time For this reason we numerically solvethe following problem
max1199090isin1205750119861
max119905isin[119879minus119879119901119879]
1003816
1003816
1003816
1003816
119909 (119905 119909
0 119906)
1003816
1003816
1003816
1003816
997888rarr min
119906 isin 119880
(42)
where 119879
119901≪ 119879 is an interval corresponding to the period
of oscillation of solutions The minimization is done usingmultistart Nelder-Mead method For 120598 = 025 120581 = 01120596 = 0949 120575
0= 1 119879 = 300120587 and 119879
119901= 20120587 the fastest
0 01 02 03 04 05 06 07 08 09 10
02
04
06
08
1
b(r
ad)
a (rad)
(a) Phase plot of 119887 versus 119886
0 200 400 600 800 1000 12000
05
1
15
Am
plitu
de (r
ad)
Argument of latitude (rad)
(b) Time evolution of amplituderadic1198862 + 1198872 of pitch oscillations (29)
Figure 1 A typical trajectory for 120598 = 025 120581 = 01 120596 = 0949
damping speed is observed for values of 120579 very close to 1205874that is to the expected value Next we consider the problem
max1199090isin1205750119861
max119905isin [0119879]
1003816
1003816
1003816
1003816
119909 (119905 119909
0 119906)
1003816
1003816
1003816
1003816
997888rarr min
max1199090isin1205750119861
max119905isin[119879minus119879119901 119879]
1003816
1003816
1003816
1003816
119909 (119905 119909
0 119906)
1003816
1003816
1003816
1003816
le 120575
1
119906 isin 119880
(43)
The results of minimization for 120598 = 025 120581 = 01 120596 = 0949120575
0= 1 120575
1= 01 119879 = 300120587 and 119879
119901= 20120587 show that the
problem has many local minima but with the values of thefunctional very close to 1 that is the nonlinear system withtwo orthogonal hysteresis rods has no overshooting for allvalues of 120579This allows one to conclude that the value 120579 = 1205874
is the best one It guarantees high damping speed anddoes notcause peaking
6 Conclusions
Thispaper is dedicated to the problemof parameter optimiza-tion for a gravitationally stabilized satellite with magnetichysteresis damper Its motion is described by differentialequations with discontinuous right-hand side The disconti-nuity is the principal obstacle in the application of the aver-aging method Our recently obtained results on averaging ofdiscontinuous systems are applied now to rigorously justifythe use of this method for a satellite with hysteresis rods
6 Mathematical Problems in Engineering
We consider here the simplest case of in-plane oscillationson a polar circular orbit Theorem 3 shows that the behaviorof the system can be characterized in terms of Lyapunovfunction which can be chosen in order to guarantee the bestproperties of damping process Further study is performednumerically and the simulations are in excellent agreementwith the analytical results confirming also previous studieson hysteresis damping of satellite pitch oscillation in gravity-gradient mode
Our results can also be applied to rigorously justify theuse of averaging techniques in analysis of other engineeringproblems involving differential equations with discontinuousright-hand side
Appendix
This appendix contains the proofs of Theorems 3 and 4
Proof of Theorem 3 Set 119879 = 2 ln 2120574 Then using (21) fromthe Lyapunov inequality we get |119909(119879 119909
0 119906)|
119881le |119909
0|
1198812
whenever |119909
0|
119881le 119888
1120574119888 There exists 120598
0(120575) such that
sup119905isin[0119879]
|119909(119905 119909
0 119906) minus 119909(119905 119909
0 119906)| lt 1205752 Therefore we have
|119909(119879 119909
0 119906)| lt 120575 and |119909(119905 119909
0 119906)| le |119909(119905 119909
0 119906)| + |119909(119905 119909
0 119906) minus
119909(119905 119909
0 119906)| lt 31205752 This ends the proof
Proof of Theorem 4 Consider the function
119891 = 119882(119867
120591) (119867
1119890
3minus 119867
3119890
1minus 120572 (119867
1119890
1+ 119867
3119890
3)) (A1)
where
119867
120591= 119867
1119890
1+ 119867
3119890
3+ 120572 (119867
1119890
3minus 119867
3119890
1)
119882 (119867
120591) = 119867
120591minus
120581
2
sign
119867
120591
119867
1= cos 119905 119867
3= minus2 sin 119905
(119890
1 119890
3) = (cos 120579 sin 120579) 120579 isin [0
120587
2
]
(A2)
First note that for any fixed pair (119886 119887) the function 119905 rarr
119867
120591(119905 119886 119887) is analytic Therefore the integral 119868(119905
1 119905
2 119909 120575) (see
Section 2) is a point This implies that the averaged operatordefined in (14) coincides with
lim119879rarrinfin
1
119879
int
119879
0
119891 (119905 119886 119887) (
minus sin120596119905
cos120596119905
) 119889119905(A3)
if the limit exists Since 120596 is irrational number and 119891 can beconsidered as a 2120587-periodic function 119892 = 119892(119905
119905 119886 119887) of thearguments 119905 and
119905 = 120596119905 we see that limit (A3) does existand we have
lim119879rarrinfin
1
119879
int
119879
0
119891 (119905 119886 119887) (
minus sin120596119905
cos120596119905
) 119889119905
=
1
(2120587)
2∬
2120587
0
119892 (119905
119905 119886 119887)
times (
minus sin
119905
cos119905 ) 119889119905 119889
119905
(A4)
To evaluate this integral we represent the derivative
(sin (2119899 + 1) 119905 cos (2119899 + 1) 119905
1015840
minus sin (2119899 + 1)
1015840 cos (2119899 + 1) 119905)
times (2119899 + 1)
minus1
(A7)
Observe that
sin 119905
1015840
=radic
1 + 3119890
2
3(minus2119890
3+
119890
1(2120572 minus 3119890
1119890
3120573)
1 + 3119890
2
3
)
+ 119874(120572
2
+ 120573
2
)
(A8)
cos 1199051015840 = radic1 + 3119890
2
3(119890
1+
2119890
3(2120572 minus 3119890
1119890
3120573)
1 + 3119890
2
3
)
+ 119874(120572
2
+ 120573
2
)
(A9)
Substituting (A7) into (A4) and integrating in order of 119905 andthen in order of
119905 we obtain the result
Acknowledgments
This research is supported by the Portuguese Foundation forScience and Technologies (FCT) the Portuguese OperationalProgramme for Competitiveness Factors (COMPETE) thePortuguese Strategic Reference Framework (QREN) and theEuropean Regional Development Fund (FEDER)
References
[1] R E Fischell and F F Mobley ldquoA system for passive gravity-gradient stabilization of earth satellitesrdquo Guidance and Controlvol 2 pp 37ndash71 1964
Mathematical Problems in Engineering 7
[2] V A Sarychev V I Penkov M Yu Ovchinnikov and A DGuerman ldquoMotions of a gravitationally stabilized satellite withhysteresis rods in a polar orbitrdquo Cosmic Research vol 26 no 5pp 561ndash574 1988
[3] A D Guerman M Yu Ovchinnikov V I Penrsquokov and V ASarychev ldquoNon-resonant motions of a satellite with hysteresisrods under conditions of gravity orientationrdquo Mechanics ofSolids vol 24 pp 1ndash11 1989
[4] M Ovchinnikov V Penrsquokov O Norberg and S BarabashldquoAttitude control system for the first sweedish nanosatelliteMUNINrdquoActa Astronautica vol 46 no 2ndash6 pp 319ndash326 2000
[5] F Santoni and M Zelli ldquoPassive magnetic attitude stabilizationof the UNISAT-4 microsatelliterdquo Acta Astronautica vol 65 no5-6 pp 792ndash803 2009
[6] R Gama A Guerman and G Smirnov ldquoOn the asymptoticstability of discontinuous systems analysed via the averagingmethodrdquo Nonlinear Analysis Theory Methods and ApplicationsA vol 74 no 4 pp 1513ndash1522 2011
[7] RN Izmailov ldquoThepeak effect in stationary linear systemswithscalar inputs and outputsrdquoAutomation and Remote Control vol48 no 8 part 1 pp 1018ndash1024 1987
[8] H J Sussmann and P V Kokotovic ldquoThe peaking phenomenonand the global stabilization of nonlinear systemsrdquo IEEE Trans-actions on Automatic Control vol 36 no 4 pp 424ndash439 1991
[9] A Guerman A Seabra andG Smirnov ldquoOptimization of para-meters of asymptotically stable systemsrdquo Mathematical Prob-lems in Engineering vol 2011 Article ID 526167 19 pages 2011
[10] N N Bogoliubov and Y AMitropolskyAsymptotic Methods intheTheory of Non-Linear Oscillations Gordon and Breach NewYork NY USA 1961
[11] A M Samoilenko and A N Stanzhitskii ldquoOn averaging differ-ential equations on an infinite intervalrdquo Differential Equationsvol 42 no 4 pp 476ndash482 2006
[12] A F Filippov Differential Equations with Discontinuous Right-Hand Sides vol 18 ofMathematics and Its Applications KluwerAcademic New York NY USA 1988
[13] G V Smirnov Introduction to the Theory of Differential Inclu-sions vol 41 of AMS Graduate Studies in Mathematics Ameri-can Mathematical Society Providence RI USA 2002
[14] V A Plotnikov ldquoAveraging method for differential inclusionsand its application to optimal control problemsrdquo DifferentialEquations vol 15 pp 1013ndash1018 1979
[15] V A Plotnikov ldquoAsymptotic methods in the theory of differen-tial equations with discontinuous and multi-valued right-handsidesrdquoUkranian Mathematical Journal vol 48 no 11 pp 1605ndash1616 1996
[16] A M Liapunov Stability of Motion Academic Press New YorkNY USA 1966
discontinuous dynamical systems Here we outline the mainresults from [6]
Let 119865 119877 times 119877
119899
rarr 119877
119899 be a set-valued map Set
119868 (119905
1 119905
2 119909 120575) = int
1199052
1199051
V (119905) 119889119905 | V (sdot) isin 119871
loc1
([0infin[ 119877
119899
)
V (119905) isin 119865 (119905 119909 + 120575119861)
(12)
We denote by 119865
120575
(119909) the convex hull of the map
Φ
120575
(119909) = lim sup120579uarr1
lim sup119879rarrinfin
1
(1 minus 120579) 119879
119868 (120579119879 119879 119909 120575) (13)
and define the averaged differential inclusion as
isin 119865 (119909) = ⋂
120575gt0
119865
120575
(119909) (14)
Note that under Lipschitz condition this map coincides with
119865 = lim119879rarrinfin
1
119879
int
119879
0
119865 (119905 119909) 119889119905(15)
if the limit exists in the sense of Hausdorff distance (see [6])Assume that the following conditions are satisfied
(C1) cl co119865(119905 119909) = 119865(119905 119909) for all (119905 119909) isin 119877 times 119877
119899(C2) the set-valued map 119865(119905 sdot) is upper semi-continuous(C3) for any 119909 there exists measurable selection of 119865(119905 119909)
that is there exists 119891(119905 119909) isin 119865(119905 119909) such that 119905 rarr
119891(119905 119909) is measurable for all 119909(C4) there exists a nonnegative 119887(sdot) isin 119871
loc1
([0infin[ 119877) suchthat 119865(119905 119909) sub 119887(119905)119861 for all (119905 119909) isin [0 +infin[times119877
119899(C5) there exists the limit
119887 = lim119879rarrinfin
1
119879
int
119879
0
119887 (119905) 119889119905(16)
Under these conditions the following version of Bogolyu-bovrsquos first theorem is true
Theorem 1 Let119879 gt 0 and let119865 119877times119877
119899
rarr 119877
119899 be a set-valuedmap satisfying conditions (C1)ndash(C5) Let 119862 isin 119877
119899 be a compactset Then for any 120578 gt 0 there exists 120598
0gt 0 such that for any
120598 isin]0 120598
0[ and any solution 119909(sdot) isin S
[0119879120598](120598119865 119862) there exists a
solution 119909(sdot) isin S[0119879120598]
(120598119865 119862) satisfying
|119909 (119905) minus 119909 (119905)| lt 120578 119905 isin [0
119879
120598
] (17)
Set
119866
120598(120591 119910) = 119865(
120591
120598
119910) 119866
0(119910) = 119865 (119910) (18)
Next theorem is an extension of the Samoilenko-Stanzhitskiitheorem
Theorem 2 Let 119865 119877 times 119877
119899
rarr 119877
119899 be a set-valued map satis-fying conditions (C1)ndash(C5) Assume that 119910 = 0 is an asympto-tically stable equilibrium position of the differential inclusion119910 isin 119866
0(119910) Then for any 120578 gt 0 there exist 120598
0gt 0 and 120575 gt 0
such that S[0infin[
(119866
120598 120575119861) sub 120578B whenever 120598 isin]0 120598
0[
The last theorem shows that if the averaged inclusionhas zero as its asymptotically stable equilibrium position thetrajectories of the original inclusion stay in the vicinity of theorigin provided 120598 gt 0 and |119909
0| are sufficiently small
If the averaged inclusion has a special form we can gofurther and make some conclusion on the detailed behaviourof the trajectories of the original system Assume that theaveraged inclusion has the form
119861 119888 gt 0 the real parts of the matrix119860(119906) eigenvalues are negative for all 119906 isin 119880 and the function119906 rarr 119860(119906) is continuous for all 119906 isin 119880 If 120574
0gt 0 is sufficiently
small then the set of solutions to the Lyapunov inequality (see[16]) for the matrix 119860(119906)
is nonempty and compact for all 119906 isin 119880 Let (120574 119881) isin
L(119880) Denote by |119909|
119881the Euclidean norm defined by |119909|
119881=
radic⟨119909 119881119909⟩ There exist positive constants 119888
1and 119888
2satisfying
119888
1|119909| le |119909|
119881le 119888
2|119909| (21)
whenever (120574 119881) isin L(119880) for some 120574
Theorem 3 Let 120575 gt 0 119906 isin 119880 and (120574 119881) isin L(119880) Thereexists 120598
0(120575) such that for all 120598 isin]0 120598
0(120575)[ the condition |119909
0|
119881lt
120575 lt 119888
2
1120574119888 implies the inequality |119909(119905 119909
0 119906)|
119881lt 31205752
This theorem shows that the behavior of the trajectory119909(119905 119909
0 119906) can be characterized in terms of the pair (120574 119881)
The parameter 120574 is responsible for the damping speed of theprocess while the form of the ellipsoid 119909 | ⟨119909 119881119909⟩ le 1
describes the amplitude of the deviation of the trajectory fromthe originThe aim of parameter choosing can be formulatedas follows maximal value of 120574 and maximal sphericity of theellipsoid 119909 | ⟨119909 119881119909⟩ le 1 The latter property guaranteesminimal overshooting of the damping process and as aconsequence the largest region of applicability of the approx-imation obtained via averaging
4 Choosing Passive MagneticStabilizer Parameters
The in-plane oscillations of a satellite moving along a polarcircular orbit and equipped with a passive gravity-gradientattitude stabilization system with one hysteresis rod aredescribed by the equation
+ 120596
2
120572 = 120598119891 (22)
4 Mathematical Problems in Engineering
where 120572 is the pitch angle of spacecraft 120598 is small parameterproportional to the rodrsquos volume and the force 119891 is given by
119891 = 119882(119867
120591) (119867
1119890
3minus 119867
3119890
1minus 120572 (119867
1119890
1+ 119867
3119890
3)) (23)
Here
119867
120591= 119867
1119890
1+ 119867
3119890
3+ 120572 (119867
1119890
3minus 119867
3119890
1) (24)
describes the projection of the geomagnetic field on the rodaxis
119882(119867
120591) = 119867
120591minus
120581
2
sign
119867
120591(25)
is the hysteresis function 120581 corresponds to the coercive force
119867
1= cos 119905 119867
3= minus2 sin 119905 (26)
and 119905 is the argument of latitude of the current point of theorbit [2] The vector
Theorem 4 Assume that 120596 is an irrational number Then theaveraged system for (31) is
119886 = minus
120598
2120596
(119901119886 + 119902119887) + 119903
119886(119886 119887 120579)
119887 =
120598
2120596
(119902119886 minus 119901119887) + 119903
119887(119886 119887 120579)
(32)
where
119901 =
9120581120596119890
2
1119890
2
3
120587(1 + 3119890
2
3)
32
119902 =
3
2
(119890
2
1minus 119890
2
3) +
6120581119890
1119890
3
120587(1 + 3119890
2
3)
32
(33)
|119903
119886| = 119874(119886
2
+ 119887
2
) and |119903
119887| = 119874(119886
2
+ 119887
2
)
Obviously we have
(
1 0
0 1
)(
minus119901 minus119902
119902 minus119901
) + (
minus119901 119902
minus119902 minus119901
)(
1 0
0 1
) = minus2119901(
1 0
0 1
)
(34)
Therefore we see that the linearization of the averaged systemalways has a Lyapunov function119881 = 119886
2
+119887
2
and the dampingspeed is determined by the value of 119901 This means that thepeak effect does not take place for the linearization of theaveraged system
To maximize the damping one has to increase the totalvolume of the hysteresis material on board However it iswellknown that the efficiency of a damping rod is increasedwith the increase of the ratio between the rodrsquos length and itscross-section dimension Therefore instead of one massivebar the attitude control system should use several ratherthin rods of the maximum length allowed by the spacecraftgeometrical and system restrictions On the other hand tominimize the perturbation of the spacecraft angular motioncaused by the damping system itself the direction of totalmagnetic field in the rods should deviate as little as possiblefrom the direction of the geomagnetic field at the currentpoint of the orbit Thus in general case one should use asystem of three equal orthogonal hysteresis rods or a numberof such systems Here we consider in-plane satellite dynamicson a polar orbit and for such purpose it suffices to analyze apair of equal orthogonal rods
Orientation of this pair of equal orthogonal rods can bedefined as (119890
1 119890
3) (minus119890
3 119890
1) where 119890
1= cos 120579 and 119890
3= sin 120579
and due to the system symmetry it is enough to study theinterval 120579 isin [0 1205872] If the satellite is equipped with severalidentical hysteresis rods the corresponding nonlinear systemis
119886 = minus
120598
120596
(119891
1+ 119891
2+ sdot sdot sdot ) sin120596119905
119887 =
120598
120596
(119891
1+ 119891
2+ sdot sdot sdot ) cos120596119905
(35)
Here the terms 119891
1 119891
2 describe the interaction of the res-
pective rod with the geomagnetic field For a couple of equalorthogonal rods and for small deviation from the origin theforces 119891
1and 119891
2are given by
119891
1= 119882(119867
1205911) (119867
1119890
3minus 119867
3119890
1minus 120572 (119867
1119890
1+ 119867
3119890
3))
119891
2= 119882(119867
1205912) (119867
1119890
1+ 119867
3119890
3minus 120572 (minus119867
1119890
3+ 119867
3119890
1))
(36)
respectively Here
119867
1205911= 119867
1119890
1+ 119867
3119890
3+ 120572 (119867
1119890
3minus 119867
3119890
1)
119867
1205912= minus119867
1119890
3+ 119867
3119890
1+ 120572 (119867
1119890
1+ 119867
3119890
3)
119882 (119867
120591119895) = 119867
120591119895minus
120581
2
sign
119867
120591119895 119895 = 1 2
(37)
Mathematical Problems in Engineering 5
For this case the first approximation of the averaged sys-tem takes the form
119886 = minus
120598
2120596
119875119886
119887 = minus
120598
2120596
119875119887
(38)
where
119875 =
9120581120596119890
2
1119890
2
3
120587
(
1
(1 + 3119890
2
3)
32
+
1
(1 + 3119890
2
1)
32
) (39)
An easy calculation shows that the optimal value of 120579 is 1205874so 119890
1= 119890
3=
radic22 In the next sectionwe numerically analyze
the validity of the previous analytical study
5 Numerical Simulations
We approximate problem (6) by the following problem
120593
0997888rarr min
1003816
1003816
1003816
1003816
1003816
119909(119905
119894
119896 119909
119894
119895 119906)
1003816
1003816
1003816
1003816
1003816119894
le 120593
119894+ 120576 119894 = 0119898
119906 isin 119880
(40)
where 119905
119894
0= 0 119905119894
119896isin Δ
119894 119909119894119895isin 119861
119894 119895 = 1 119869 and
119909 (119905
119894
119896+1 119909
119894
119895 119906) = 119909 (119905
119894
119896 119909
119894
119895 119906) + 120591119891 (119905
119896 119909 (119905
119894
119896 119909
119894
119895 119906) 119906)
120591 = 119905
119894
119896+1minus 119905
119894
119896 119896 = 0119873
(41)
is the Euler approximation for the solution 119909(sdot 119909
119894
119895 119906) Note
that this is a hard problem because of the discontinuity ofthe system It is necessary to consider very fine partition ofthe time interval in order to get a good approximation ofthe solutions to the discontinuous differential equation Forsmooth right-hand sides the number of points in the meshcan be significantly reduced (see [9])
Let 120598120581 gt 0 be small enough We consider a satellite withtwo equal orthogonal hysteresis rods A typical trajectory ofthe system is shown in Figure 1
Note that the oscillations of the trajectory do not allowone to characterize its damping speed using the norm at thefinal moment of time For this reason we numerically solvethe following problem
max1199090isin1205750119861
max119905isin[119879minus119879119901119879]
1003816
1003816
1003816
1003816
119909 (119905 119909
0 119906)
1003816
1003816
1003816
1003816
997888rarr min
119906 isin 119880
(42)
where 119879
119901≪ 119879 is an interval corresponding to the period
of oscillation of solutions The minimization is done usingmultistart Nelder-Mead method For 120598 = 025 120581 = 01120596 = 0949 120575
0= 1 119879 = 300120587 and 119879
119901= 20120587 the fastest
0 01 02 03 04 05 06 07 08 09 10
02
04
06
08
1
b(r
ad)
a (rad)
(a) Phase plot of 119887 versus 119886
0 200 400 600 800 1000 12000
05
1
15
Am
plitu
de (r
ad)
Argument of latitude (rad)
(b) Time evolution of amplituderadic1198862 + 1198872 of pitch oscillations (29)
Figure 1 A typical trajectory for 120598 = 025 120581 = 01 120596 = 0949
damping speed is observed for values of 120579 very close to 1205874that is to the expected value Next we consider the problem
max1199090isin1205750119861
max119905isin [0119879]
1003816
1003816
1003816
1003816
119909 (119905 119909
0 119906)
1003816
1003816
1003816
1003816
997888rarr min
max1199090isin1205750119861
max119905isin[119879minus119879119901 119879]
1003816
1003816
1003816
1003816
119909 (119905 119909
0 119906)
1003816
1003816
1003816
1003816
le 120575
1
119906 isin 119880
(43)
The results of minimization for 120598 = 025 120581 = 01 120596 = 0949120575
0= 1 120575
1= 01 119879 = 300120587 and 119879
119901= 20120587 show that the
problem has many local minima but with the values of thefunctional very close to 1 that is the nonlinear system withtwo orthogonal hysteresis rods has no overshooting for allvalues of 120579This allows one to conclude that the value 120579 = 1205874
is the best one It guarantees high damping speed anddoes notcause peaking
6 Conclusions
Thispaper is dedicated to the problemof parameter optimiza-tion for a gravitationally stabilized satellite with magnetichysteresis damper Its motion is described by differentialequations with discontinuous right-hand side The disconti-nuity is the principal obstacle in the application of the aver-aging method Our recently obtained results on averaging ofdiscontinuous systems are applied now to rigorously justifythe use of this method for a satellite with hysteresis rods
6 Mathematical Problems in Engineering
We consider here the simplest case of in-plane oscillationson a polar circular orbit Theorem 3 shows that the behaviorof the system can be characterized in terms of Lyapunovfunction which can be chosen in order to guarantee the bestproperties of damping process Further study is performednumerically and the simulations are in excellent agreementwith the analytical results confirming also previous studieson hysteresis damping of satellite pitch oscillation in gravity-gradient mode
Our results can also be applied to rigorously justify theuse of averaging techniques in analysis of other engineeringproblems involving differential equations with discontinuousright-hand side
Appendix
This appendix contains the proofs of Theorems 3 and 4
Proof of Theorem 3 Set 119879 = 2 ln 2120574 Then using (21) fromthe Lyapunov inequality we get |119909(119879 119909
0 119906)|
119881le |119909
0|
1198812
whenever |119909
0|
119881le 119888
1120574119888 There exists 120598
0(120575) such that
sup119905isin[0119879]
|119909(119905 119909
0 119906) minus 119909(119905 119909
0 119906)| lt 1205752 Therefore we have
|119909(119879 119909
0 119906)| lt 120575 and |119909(119905 119909
0 119906)| le |119909(119905 119909
0 119906)| + |119909(119905 119909
0 119906) minus
119909(119905 119909
0 119906)| lt 31205752 This ends the proof
Proof of Theorem 4 Consider the function
119891 = 119882(119867
120591) (119867
1119890
3minus 119867
3119890
1minus 120572 (119867
1119890
1+ 119867
3119890
3)) (A1)
where
119867
120591= 119867
1119890
1+ 119867
3119890
3+ 120572 (119867
1119890
3minus 119867
3119890
1)
119882 (119867
120591) = 119867
120591minus
120581
2
sign
119867
120591
119867
1= cos 119905 119867
3= minus2 sin 119905
(119890
1 119890
3) = (cos 120579 sin 120579) 120579 isin [0
120587
2
]
(A2)
First note that for any fixed pair (119886 119887) the function 119905 rarr
119867
120591(119905 119886 119887) is analytic Therefore the integral 119868(119905
1 119905
2 119909 120575) (see
Section 2) is a point This implies that the averaged operatordefined in (14) coincides with
lim119879rarrinfin
1
119879
int
119879
0
119891 (119905 119886 119887) (
minus sin120596119905
cos120596119905
) 119889119905(A3)
if the limit exists Since 120596 is irrational number and 119891 can beconsidered as a 2120587-periodic function 119892 = 119892(119905
119905 119886 119887) of thearguments 119905 and
119905 = 120596119905 we see that limit (A3) does existand we have
lim119879rarrinfin
1
119879
int
119879
0
119891 (119905 119886 119887) (
minus sin120596119905
cos120596119905
) 119889119905
=
1
(2120587)
2∬
2120587
0
119892 (119905
119905 119886 119887)
times (
minus sin
119905
cos119905 ) 119889119905 119889
119905
(A4)
To evaluate this integral we represent the derivative
(sin (2119899 + 1) 119905 cos (2119899 + 1) 119905
1015840
minus sin (2119899 + 1)
1015840 cos (2119899 + 1) 119905)
times (2119899 + 1)
minus1
(A7)
Observe that
sin 119905
1015840
=radic
1 + 3119890
2
3(minus2119890
3+
119890
1(2120572 minus 3119890
1119890
3120573)
1 + 3119890
2
3
)
+ 119874(120572
2
+ 120573
2
)
(A8)
cos 1199051015840 = radic1 + 3119890
2
3(119890
1+
2119890
3(2120572 minus 3119890
1119890
3120573)
1 + 3119890
2
3
)
+ 119874(120572
2
+ 120573
2
)
(A9)
Substituting (A7) into (A4) and integrating in order of 119905 andthen in order of
119905 we obtain the result
Acknowledgments
This research is supported by the Portuguese Foundation forScience and Technologies (FCT) the Portuguese OperationalProgramme for Competitiveness Factors (COMPETE) thePortuguese Strategic Reference Framework (QREN) and theEuropean Regional Development Fund (FEDER)
References
[1] R E Fischell and F F Mobley ldquoA system for passive gravity-gradient stabilization of earth satellitesrdquo Guidance and Controlvol 2 pp 37ndash71 1964
Mathematical Problems in Engineering 7
[2] V A Sarychev V I Penkov M Yu Ovchinnikov and A DGuerman ldquoMotions of a gravitationally stabilized satellite withhysteresis rods in a polar orbitrdquo Cosmic Research vol 26 no 5pp 561ndash574 1988
[3] A D Guerman M Yu Ovchinnikov V I Penrsquokov and V ASarychev ldquoNon-resonant motions of a satellite with hysteresisrods under conditions of gravity orientationrdquo Mechanics ofSolids vol 24 pp 1ndash11 1989
[4] M Ovchinnikov V Penrsquokov O Norberg and S BarabashldquoAttitude control system for the first sweedish nanosatelliteMUNINrdquoActa Astronautica vol 46 no 2ndash6 pp 319ndash326 2000
[5] F Santoni and M Zelli ldquoPassive magnetic attitude stabilizationof the UNISAT-4 microsatelliterdquo Acta Astronautica vol 65 no5-6 pp 792ndash803 2009
[6] R Gama A Guerman and G Smirnov ldquoOn the asymptoticstability of discontinuous systems analysed via the averagingmethodrdquo Nonlinear Analysis Theory Methods and ApplicationsA vol 74 no 4 pp 1513ndash1522 2011
[7] RN Izmailov ldquoThepeak effect in stationary linear systemswithscalar inputs and outputsrdquoAutomation and Remote Control vol48 no 8 part 1 pp 1018ndash1024 1987
[8] H J Sussmann and P V Kokotovic ldquoThe peaking phenomenonand the global stabilization of nonlinear systemsrdquo IEEE Trans-actions on Automatic Control vol 36 no 4 pp 424ndash439 1991
[9] A Guerman A Seabra andG Smirnov ldquoOptimization of para-meters of asymptotically stable systemsrdquo Mathematical Prob-lems in Engineering vol 2011 Article ID 526167 19 pages 2011
[10] N N Bogoliubov and Y AMitropolskyAsymptotic Methods intheTheory of Non-Linear Oscillations Gordon and Breach NewYork NY USA 1961
[11] A M Samoilenko and A N Stanzhitskii ldquoOn averaging differ-ential equations on an infinite intervalrdquo Differential Equationsvol 42 no 4 pp 476ndash482 2006
[12] A F Filippov Differential Equations with Discontinuous Right-Hand Sides vol 18 ofMathematics and Its Applications KluwerAcademic New York NY USA 1988
[13] G V Smirnov Introduction to the Theory of Differential Inclu-sions vol 41 of AMS Graduate Studies in Mathematics Ameri-can Mathematical Society Providence RI USA 2002
[14] V A Plotnikov ldquoAveraging method for differential inclusionsand its application to optimal control problemsrdquo DifferentialEquations vol 15 pp 1013ndash1018 1979
[15] V A Plotnikov ldquoAsymptotic methods in the theory of differen-tial equations with discontinuous and multi-valued right-handsidesrdquoUkranian Mathematical Journal vol 48 no 11 pp 1605ndash1616 1996
[16] A M Liapunov Stability of Motion Academic Press New YorkNY USA 1966
Theorem 4 Assume that 120596 is an irrational number Then theaveraged system for (31) is
119886 = minus
120598
2120596
(119901119886 + 119902119887) + 119903
119886(119886 119887 120579)
119887 =
120598
2120596
(119902119886 minus 119901119887) + 119903
119887(119886 119887 120579)
(32)
where
119901 =
9120581120596119890
2
1119890
2
3
120587(1 + 3119890
2
3)
32
119902 =
3
2
(119890
2
1minus 119890
2
3) +
6120581119890
1119890
3
120587(1 + 3119890
2
3)
32
(33)
|119903
119886| = 119874(119886
2
+ 119887
2
) and |119903
119887| = 119874(119886
2
+ 119887
2
)
Obviously we have
(
1 0
0 1
)(
minus119901 minus119902
119902 minus119901
) + (
minus119901 119902
minus119902 minus119901
)(
1 0
0 1
) = minus2119901(
1 0
0 1
)
(34)
Therefore we see that the linearization of the averaged systemalways has a Lyapunov function119881 = 119886
2
+119887
2
and the dampingspeed is determined by the value of 119901 This means that thepeak effect does not take place for the linearization of theaveraged system
To maximize the damping one has to increase the totalvolume of the hysteresis material on board However it iswellknown that the efficiency of a damping rod is increasedwith the increase of the ratio between the rodrsquos length and itscross-section dimension Therefore instead of one massivebar the attitude control system should use several ratherthin rods of the maximum length allowed by the spacecraftgeometrical and system restrictions On the other hand tominimize the perturbation of the spacecraft angular motioncaused by the damping system itself the direction of totalmagnetic field in the rods should deviate as little as possiblefrom the direction of the geomagnetic field at the currentpoint of the orbit Thus in general case one should use asystem of three equal orthogonal hysteresis rods or a numberof such systems Here we consider in-plane satellite dynamicson a polar orbit and for such purpose it suffices to analyze apair of equal orthogonal rods
Orientation of this pair of equal orthogonal rods can bedefined as (119890
1 119890
3) (minus119890
3 119890
1) where 119890
1= cos 120579 and 119890
3= sin 120579
and due to the system symmetry it is enough to study theinterval 120579 isin [0 1205872] If the satellite is equipped with severalidentical hysteresis rods the corresponding nonlinear systemis
119886 = minus
120598
120596
(119891
1+ 119891
2+ sdot sdot sdot ) sin120596119905
119887 =
120598
120596
(119891
1+ 119891
2+ sdot sdot sdot ) cos120596119905
(35)
Here the terms 119891
1 119891
2 describe the interaction of the res-
pective rod with the geomagnetic field For a couple of equalorthogonal rods and for small deviation from the origin theforces 119891
1and 119891
2are given by
119891
1= 119882(119867
1205911) (119867
1119890
3minus 119867
3119890
1minus 120572 (119867
1119890
1+ 119867
3119890
3))
119891
2= 119882(119867
1205912) (119867
1119890
1+ 119867
3119890
3minus 120572 (minus119867
1119890
3+ 119867
3119890
1))
(36)
respectively Here
119867
1205911= 119867
1119890
1+ 119867
3119890
3+ 120572 (119867
1119890
3minus 119867
3119890
1)
119867
1205912= minus119867
1119890
3+ 119867
3119890
1+ 120572 (119867
1119890
1+ 119867
3119890
3)
119882 (119867
120591119895) = 119867
120591119895minus
120581
2
sign
119867
120591119895 119895 = 1 2
(37)
Mathematical Problems in Engineering 5
For this case the first approximation of the averaged sys-tem takes the form
119886 = minus
120598
2120596
119875119886
119887 = minus
120598
2120596
119875119887
(38)
where
119875 =
9120581120596119890
2
1119890
2
3
120587
(
1
(1 + 3119890
2
3)
32
+
1
(1 + 3119890
2
1)
32
) (39)
An easy calculation shows that the optimal value of 120579 is 1205874so 119890
1= 119890
3=
radic22 In the next sectionwe numerically analyze
the validity of the previous analytical study
5 Numerical Simulations
We approximate problem (6) by the following problem
120593
0997888rarr min
1003816
1003816
1003816
1003816
1003816
119909(119905
119894
119896 119909
119894
119895 119906)
1003816
1003816
1003816
1003816
1003816119894
le 120593
119894+ 120576 119894 = 0119898
119906 isin 119880
(40)
where 119905
119894
0= 0 119905119894
119896isin Δ
119894 119909119894119895isin 119861
119894 119895 = 1 119869 and
119909 (119905
119894
119896+1 119909
119894
119895 119906) = 119909 (119905
119894
119896 119909
119894
119895 119906) + 120591119891 (119905
119896 119909 (119905
119894
119896 119909
119894
119895 119906) 119906)
120591 = 119905
119894
119896+1minus 119905
119894
119896 119896 = 0119873
(41)
is the Euler approximation for the solution 119909(sdot 119909
119894
119895 119906) Note
that this is a hard problem because of the discontinuity ofthe system It is necessary to consider very fine partition ofthe time interval in order to get a good approximation ofthe solutions to the discontinuous differential equation Forsmooth right-hand sides the number of points in the meshcan be significantly reduced (see [9])
Let 120598120581 gt 0 be small enough We consider a satellite withtwo equal orthogonal hysteresis rods A typical trajectory ofthe system is shown in Figure 1
Note that the oscillations of the trajectory do not allowone to characterize its damping speed using the norm at thefinal moment of time For this reason we numerically solvethe following problem
max1199090isin1205750119861
max119905isin[119879minus119879119901119879]
1003816
1003816
1003816
1003816
119909 (119905 119909
0 119906)
1003816
1003816
1003816
1003816
997888rarr min
119906 isin 119880
(42)
where 119879
119901≪ 119879 is an interval corresponding to the period
of oscillation of solutions The minimization is done usingmultistart Nelder-Mead method For 120598 = 025 120581 = 01120596 = 0949 120575
0= 1 119879 = 300120587 and 119879
119901= 20120587 the fastest
0 01 02 03 04 05 06 07 08 09 10
02
04
06
08
1
b(r
ad)
a (rad)
(a) Phase plot of 119887 versus 119886
0 200 400 600 800 1000 12000
05
1
15
Am
plitu
de (r
ad)
Argument of latitude (rad)
(b) Time evolution of amplituderadic1198862 + 1198872 of pitch oscillations (29)
Figure 1 A typical trajectory for 120598 = 025 120581 = 01 120596 = 0949
damping speed is observed for values of 120579 very close to 1205874that is to the expected value Next we consider the problem
max1199090isin1205750119861
max119905isin [0119879]
1003816
1003816
1003816
1003816
119909 (119905 119909
0 119906)
1003816
1003816
1003816
1003816
997888rarr min
max1199090isin1205750119861
max119905isin[119879minus119879119901 119879]
1003816
1003816
1003816
1003816
119909 (119905 119909
0 119906)
1003816
1003816
1003816
1003816
le 120575
1
119906 isin 119880
(43)
The results of minimization for 120598 = 025 120581 = 01 120596 = 0949120575
0= 1 120575
1= 01 119879 = 300120587 and 119879
119901= 20120587 show that the
problem has many local minima but with the values of thefunctional very close to 1 that is the nonlinear system withtwo orthogonal hysteresis rods has no overshooting for allvalues of 120579This allows one to conclude that the value 120579 = 1205874
is the best one It guarantees high damping speed anddoes notcause peaking
6 Conclusions
Thispaper is dedicated to the problemof parameter optimiza-tion for a gravitationally stabilized satellite with magnetichysteresis damper Its motion is described by differentialequations with discontinuous right-hand side The disconti-nuity is the principal obstacle in the application of the aver-aging method Our recently obtained results on averaging ofdiscontinuous systems are applied now to rigorously justifythe use of this method for a satellite with hysteresis rods
6 Mathematical Problems in Engineering
We consider here the simplest case of in-plane oscillationson a polar circular orbit Theorem 3 shows that the behaviorof the system can be characterized in terms of Lyapunovfunction which can be chosen in order to guarantee the bestproperties of damping process Further study is performednumerically and the simulations are in excellent agreementwith the analytical results confirming also previous studieson hysteresis damping of satellite pitch oscillation in gravity-gradient mode
Our results can also be applied to rigorously justify theuse of averaging techniques in analysis of other engineeringproblems involving differential equations with discontinuousright-hand side
Appendix
This appendix contains the proofs of Theorems 3 and 4
Proof of Theorem 3 Set 119879 = 2 ln 2120574 Then using (21) fromthe Lyapunov inequality we get |119909(119879 119909
0 119906)|
119881le |119909
0|
1198812
whenever |119909
0|
119881le 119888
1120574119888 There exists 120598
0(120575) such that
sup119905isin[0119879]
|119909(119905 119909
0 119906) minus 119909(119905 119909
0 119906)| lt 1205752 Therefore we have
|119909(119879 119909
0 119906)| lt 120575 and |119909(119905 119909
0 119906)| le |119909(119905 119909
0 119906)| + |119909(119905 119909
0 119906) minus
119909(119905 119909
0 119906)| lt 31205752 This ends the proof
Proof of Theorem 4 Consider the function
119891 = 119882(119867
120591) (119867
1119890
3minus 119867
3119890
1minus 120572 (119867
1119890
1+ 119867
3119890
3)) (A1)
where
119867
120591= 119867
1119890
1+ 119867
3119890
3+ 120572 (119867
1119890
3minus 119867
3119890
1)
119882 (119867
120591) = 119867
120591minus
120581
2
sign
119867
120591
119867
1= cos 119905 119867
3= minus2 sin 119905
(119890
1 119890
3) = (cos 120579 sin 120579) 120579 isin [0
120587
2
]
(A2)
First note that for any fixed pair (119886 119887) the function 119905 rarr
119867
120591(119905 119886 119887) is analytic Therefore the integral 119868(119905
1 119905
2 119909 120575) (see
Section 2) is a point This implies that the averaged operatordefined in (14) coincides with
lim119879rarrinfin
1
119879
int
119879
0
119891 (119905 119886 119887) (
minus sin120596119905
cos120596119905
) 119889119905(A3)
if the limit exists Since 120596 is irrational number and 119891 can beconsidered as a 2120587-periodic function 119892 = 119892(119905
119905 119886 119887) of thearguments 119905 and
119905 = 120596119905 we see that limit (A3) does existand we have
lim119879rarrinfin
1
119879
int
119879
0
119891 (119905 119886 119887) (
minus sin120596119905
cos120596119905
) 119889119905
=
1
(2120587)
2∬
2120587
0
119892 (119905
119905 119886 119887)
times (
minus sin
119905
cos119905 ) 119889119905 119889
119905
(A4)
To evaluate this integral we represent the derivative
(sin (2119899 + 1) 119905 cos (2119899 + 1) 119905
1015840
minus sin (2119899 + 1)
1015840 cos (2119899 + 1) 119905)
times (2119899 + 1)
minus1
(A7)
Observe that
sin 119905
1015840
=radic
1 + 3119890
2
3(minus2119890
3+
119890
1(2120572 minus 3119890
1119890
3120573)
1 + 3119890
2
3
)
+ 119874(120572
2
+ 120573
2
)
(A8)
cos 1199051015840 = radic1 + 3119890
2
3(119890
1+
2119890
3(2120572 minus 3119890
1119890
3120573)
1 + 3119890
2
3
)
+ 119874(120572
2
+ 120573
2
)
(A9)
Substituting (A7) into (A4) and integrating in order of 119905 andthen in order of
119905 we obtain the result
Acknowledgments
This research is supported by the Portuguese Foundation forScience and Technologies (FCT) the Portuguese OperationalProgramme for Competitiveness Factors (COMPETE) thePortuguese Strategic Reference Framework (QREN) and theEuropean Regional Development Fund (FEDER)
References
[1] R E Fischell and F F Mobley ldquoA system for passive gravity-gradient stabilization of earth satellitesrdquo Guidance and Controlvol 2 pp 37ndash71 1964
Mathematical Problems in Engineering 7
[2] V A Sarychev V I Penkov M Yu Ovchinnikov and A DGuerman ldquoMotions of a gravitationally stabilized satellite withhysteresis rods in a polar orbitrdquo Cosmic Research vol 26 no 5pp 561ndash574 1988
[3] A D Guerman M Yu Ovchinnikov V I Penrsquokov and V ASarychev ldquoNon-resonant motions of a satellite with hysteresisrods under conditions of gravity orientationrdquo Mechanics ofSolids vol 24 pp 1ndash11 1989
[4] M Ovchinnikov V Penrsquokov O Norberg and S BarabashldquoAttitude control system for the first sweedish nanosatelliteMUNINrdquoActa Astronautica vol 46 no 2ndash6 pp 319ndash326 2000
[5] F Santoni and M Zelli ldquoPassive magnetic attitude stabilizationof the UNISAT-4 microsatelliterdquo Acta Astronautica vol 65 no5-6 pp 792ndash803 2009
[6] R Gama A Guerman and G Smirnov ldquoOn the asymptoticstability of discontinuous systems analysed via the averagingmethodrdquo Nonlinear Analysis Theory Methods and ApplicationsA vol 74 no 4 pp 1513ndash1522 2011
[7] RN Izmailov ldquoThepeak effect in stationary linear systemswithscalar inputs and outputsrdquoAutomation and Remote Control vol48 no 8 part 1 pp 1018ndash1024 1987
[8] H J Sussmann and P V Kokotovic ldquoThe peaking phenomenonand the global stabilization of nonlinear systemsrdquo IEEE Trans-actions on Automatic Control vol 36 no 4 pp 424ndash439 1991
[9] A Guerman A Seabra andG Smirnov ldquoOptimization of para-meters of asymptotically stable systemsrdquo Mathematical Prob-lems in Engineering vol 2011 Article ID 526167 19 pages 2011
[10] N N Bogoliubov and Y AMitropolskyAsymptotic Methods intheTheory of Non-Linear Oscillations Gordon and Breach NewYork NY USA 1961
[11] A M Samoilenko and A N Stanzhitskii ldquoOn averaging differ-ential equations on an infinite intervalrdquo Differential Equationsvol 42 no 4 pp 476ndash482 2006
[12] A F Filippov Differential Equations with Discontinuous Right-Hand Sides vol 18 ofMathematics and Its Applications KluwerAcademic New York NY USA 1988
[13] G V Smirnov Introduction to the Theory of Differential Inclu-sions vol 41 of AMS Graduate Studies in Mathematics Ameri-can Mathematical Society Providence RI USA 2002
[14] V A Plotnikov ldquoAveraging method for differential inclusionsand its application to optimal control problemsrdquo DifferentialEquations vol 15 pp 1013ndash1018 1979
[15] V A Plotnikov ldquoAsymptotic methods in the theory of differen-tial equations with discontinuous and multi-valued right-handsidesrdquoUkranian Mathematical Journal vol 48 no 11 pp 1605ndash1616 1996
[16] A M Liapunov Stability of Motion Academic Press New YorkNY USA 1966
For this case the first approximation of the averaged sys-tem takes the form
119886 = minus
120598
2120596
119875119886
119887 = minus
120598
2120596
119875119887
(38)
where
119875 =
9120581120596119890
2
1119890
2
3
120587
(
1
(1 + 3119890
2
3)
32
+
1
(1 + 3119890
2
1)
32
) (39)
An easy calculation shows that the optimal value of 120579 is 1205874so 119890
1= 119890
3=
radic22 In the next sectionwe numerically analyze
the validity of the previous analytical study
5 Numerical Simulations
We approximate problem (6) by the following problem
120593
0997888rarr min
1003816
1003816
1003816
1003816
1003816
119909(119905
119894
119896 119909
119894
119895 119906)
1003816
1003816
1003816
1003816
1003816119894
le 120593
119894+ 120576 119894 = 0119898
119906 isin 119880
(40)
where 119905
119894
0= 0 119905119894
119896isin Δ
119894 119909119894119895isin 119861
119894 119895 = 1 119869 and
119909 (119905
119894
119896+1 119909
119894
119895 119906) = 119909 (119905
119894
119896 119909
119894
119895 119906) + 120591119891 (119905
119896 119909 (119905
119894
119896 119909
119894
119895 119906) 119906)
120591 = 119905
119894
119896+1minus 119905
119894
119896 119896 = 0119873
(41)
is the Euler approximation for the solution 119909(sdot 119909
119894
119895 119906) Note
that this is a hard problem because of the discontinuity ofthe system It is necessary to consider very fine partition ofthe time interval in order to get a good approximation ofthe solutions to the discontinuous differential equation Forsmooth right-hand sides the number of points in the meshcan be significantly reduced (see [9])
Let 120598120581 gt 0 be small enough We consider a satellite withtwo equal orthogonal hysteresis rods A typical trajectory ofthe system is shown in Figure 1
Note that the oscillations of the trajectory do not allowone to characterize its damping speed using the norm at thefinal moment of time For this reason we numerically solvethe following problem
max1199090isin1205750119861
max119905isin[119879minus119879119901119879]
1003816
1003816
1003816
1003816
119909 (119905 119909
0 119906)
1003816
1003816
1003816
1003816
997888rarr min
119906 isin 119880
(42)
where 119879
119901≪ 119879 is an interval corresponding to the period
of oscillation of solutions The minimization is done usingmultistart Nelder-Mead method For 120598 = 025 120581 = 01120596 = 0949 120575
0= 1 119879 = 300120587 and 119879
119901= 20120587 the fastest
0 01 02 03 04 05 06 07 08 09 10
02
04
06
08
1
b(r
ad)
a (rad)
(a) Phase plot of 119887 versus 119886
0 200 400 600 800 1000 12000
05
1
15
Am
plitu
de (r
ad)
Argument of latitude (rad)
(b) Time evolution of amplituderadic1198862 + 1198872 of pitch oscillations (29)
Figure 1 A typical trajectory for 120598 = 025 120581 = 01 120596 = 0949
damping speed is observed for values of 120579 very close to 1205874that is to the expected value Next we consider the problem
max1199090isin1205750119861
max119905isin [0119879]
1003816
1003816
1003816
1003816
119909 (119905 119909
0 119906)
1003816
1003816
1003816
1003816
997888rarr min
max1199090isin1205750119861
max119905isin[119879minus119879119901 119879]
1003816
1003816
1003816
1003816
119909 (119905 119909
0 119906)
1003816
1003816
1003816
1003816
le 120575
1
119906 isin 119880
(43)
The results of minimization for 120598 = 025 120581 = 01 120596 = 0949120575
0= 1 120575
1= 01 119879 = 300120587 and 119879
119901= 20120587 show that the
problem has many local minima but with the values of thefunctional very close to 1 that is the nonlinear system withtwo orthogonal hysteresis rods has no overshooting for allvalues of 120579This allows one to conclude that the value 120579 = 1205874
is the best one It guarantees high damping speed anddoes notcause peaking
6 Conclusions
Thispaper is dedicated to the problemof parameter optimiza-tion for a gravitationally stabilized satellite with magnetichysteresis damper Its motion is described by differentialequations with discontinuous right-hand side The disconti-nuity is the principal obstacle in the application of the aver-aging method Our recently obtained results on averaging ofdiscontinuous systems are applied now to rigorously justifythe use of this method for a satellite with hysteresis rods
6 Mathematical Problems in Engineering
We consider here the simplest case of in-plane oscillationson a polar circular orbit Theorem 3 shows that the behaviorof the system can be characterized in terms of Lyapunovfunction which can be chosen in order to guarantee the bestproperties of damping process Further study is performednumerically and the simulations are in excellent agreementwith the analytical results confirming also previous studieson hysteresis damping of satellite pitch oscillation in gravity-gradient mode
Our results can also be applied to rigorously justify theuse of averaging techniques in analysis of other engineeringproblems involving differential equations with discontinuousright-hand side
Appendix
This appendix contains the proofs of Theorems 3 and 4
Proof of Theorem 3 Set 119879 = 2 ln 2120574 Then using (21) fromthe Lyapunov inequality we get |119909(119879 119909
0 119906)|
119881le |119909
0|
1198812
whenever |119909
0|
119881le 119888
1120574119888 There exists 120598
0(120575) such that
sup119905isin[0119879]
|119909(119905 119909
0 119906) minus 119909(119905 119909
0 119906)| lt 1205752 Therefore we have
|119909(119879 119909
0 119906)| lt 120575 and |119909(119905 119909
0 119906)| le |119909(119905 119909
0 119906)| + |119909(119905 119909
0 119906) minus
119909(119905 119909
0 119906)| lt 31205752 This ends the proof
Proof of Theorem 4 Consider the function
119891 = 119882(119867
120591) (119867
1119890
3minus 119867
3119890
1minus 120572 (119867
1119890
1+ 119867
3119890
3)) (A1)
where
119867
120591= 119867
1119890
1+ 119867
3119890
3+ 120572 (119867
1119890
3minus 119867
3119890
1)
119882 (119867
120591) = 119867
120591minus
120581
2
sign
119867
120591
119867
1= cos 119905 119867
3= minus2 sin 119905
(119890
1 119890
3) = (cos 120579 sin 120579) 120579 isin [0
120587
2
]
(A2)
First note that for any fixed pair (119886 119887) the function 119905 rarr
119867
120591(119905 119886 119887) is analytic Therefore the integral 119868(119905
1 119905
2 119909 120575) (see
Section 2) is a point This implies that the averaged operatordefined in (14) coincides with
lim119879rarrinfin
1
119879
int
119879
0
119891 (119905 119886 119887) (
minus sin120596119905
cos120596119905
) 119889119905(A3)
if the limit exists Since 120596 is irrational number and 119891 can beconsidered as a 2120587-periodic function 119892 = 119892(119905
119905 119886 119887) of thearguments 119905 and
119905 = 120596119905 we see that limit (A3) does existand we have
lim119879rarrinfin
1
119879
int
119879
0
119891 (119905 119886 119887) (
minus sin120596119905
cos120596119905
) 119889119905
=
1
(2120587)
2∬
2120587
0
119892 (119905
119905 119886 119887)
times (
minus sin
119905
cos119905 ) 119889119905 119889
119905
(A4)
To evaluate this integral we represent the derivative
(sin (2119899 + 1) 119905 cos (2119899 + 1) 119905
1015840
minus sin (2119899 + 1)
1015840 cos (2119899 + 1) 119905)
times (2119899 + 1)
minus1
(A7)
Observe that
sin 119905
1015840
=radic
1 + 3119890
2
3(minus2119890
3+
119890
1(2120572 minus 3119890
1119890
3120573)
1 + 3119890
2
3
)
+ 119874(120572
2
+ 120573
2
)
(A8)
cos 1199051015840 = radic1 + 3119890
2
3(119890
1+
2119890
3(2120572 minus 3119890
1119890
3120573)
1 + 3119890
2
3
)
+ 119874(120572
2
+ 120573
2
)
(A9)
Substituting (A7) into (A4) and integrating in order of 119905 andthen in order of
119905 we obtain the result
Acknowledgments
This research is supported by the Portuguese Foundation forScience and Technologies (FCT) the Portuguese OperationalProgramme for Competitiveness Factors (COMPETE) thePortuguese Strategic Reference Framework (QREN) and theEuropean Regional Development Fund (FEDER)
References
[1] R E Fischell and F F Mobley ldquoA system for passive gravity-gradient stabilization of earth satellitesrdquo Guidance and Controlvol 2 pp 37ndash71 1964
Mathematical Problems in Engineering 7
[2] V A Sarychev V I Penkov M Yu Ovchinnikov and A DGuerman ldquoMotions of a gravitationally stabilized satellite withhysteresis rods in a polar orbitrdquo Cosmic Research vol 26 no 5pp 561ndash574 1988
[3] A D Guerman M Yu Ovchinnikov V I Penrsquokov and V ASarychev ldquoNon-resonant motions of a satellite with hysteresisrods under conditions of gravity orientationrdquo Mechanics ofSolids vol 24 pp 1ndash11 1989
[4] M Ovchinnikov V Penrsquokov O Norberg and S BarabashldquoAttitude control system for the first sweedish nanosatelliteMUNINrdquoActa Astronautica vol 46 no 2ndash6 pp 319ndash326 2000
[5] F Santoni and M Zelli ldquoPassive magnetic attitude stabilizationof the UNISAT-4 microsatelliterdquo Acta Astronautica vol 65 no5-6 pp 792ndash803 2009
[6] R Gama A Guerman and G Smirnov ldquoOn the asymptoticstability of discontinuous systems analysed via the averagingmethodrdquo Nonlinear Analysis Theory Methods and ApplicationsA vol 74 no 4 pp 1513ndash1522 2011
[7] RN Izmailov ldquoThepeak effect in stationary linear systemswithscalar inputs and outputsrdquoAutomation and Remote Control vol48 no 8 part 1 pp 1018ndash1024 1987
[8] H J Sussmann and P V Kokotovic ldquoThe peaking phenomenonand the global stabilization of nonlinear systemsrdquo IEEE Trans-actions on Automatic Control vol 36 no 4 pp 424ndash439 1991
[9] A Guerman A Seabra andG Smirnov ldquoOptimization of para-meters of asymptotically stable systemsrdquo Mathematical Prob-lems in Engineering vol 2011 Article ID 526167 19 pages 2011
[10] N N Bogoliubov and Y AMitropolskyAsymptotic Methods intheTheory of Non-Linear Oscillations Gordon and Breach NewYork NY USA 1961
[11] A M Samoilenko and A N Stanzhitskii ldquoOn averaging differ-ential equations on an infinite intervalrdquo Differential Equationsvol 42 no 4 pp 476ndash482 2006
[12] A F Filippov Differential Equations with Discontinuous Right-Hand Sides vol 18 ofMathematics and Its Applications KluwerAcademic New York NY USA 1988
[13] G V Smirnov Introduction to the Theory of Differential Inclu-sions vol 41 of AMS Graduate Studies in Mathematics Ameri-can Mathematical Society Providence RI USA 2002
[14] V A Plotnikov ldquoAveraging method for differential inclusionsand its application to optimal control problemsrdquo DifferentialEquations vol 15 pp 1013ndash1018 1979
[15] V A Plotnikov ldquoAsymptotic methods in the theory of differen-tial equations with discontinuous and multi-valued right-handsidesrdquoUkranian Mathematical Journal vol 48 no 11 pp 1605ndash1616 1996
[16] A M Liapunov Stability of Motion Academic Press New YorkNY USA 1966
We consider here the simplest case of in-plane oscillationson a polar circular orbit Theorem 3 shows that the behaviorof the system can be characterized in terms of Lyapunovfunction which can be chosen in order to guarantee the bestproperties of damping process Further study is performednumerically and the simulations are in excellent agreementwith the analytical results confirming also previous studieson hysteresis damping of satellite pitch oscillation in gravity-gradient mode
Our results can also be applied to rigorously justify theuse of averaging techniques in analysis of other engineeringproblems involving differential equations with discontinuousright-hand side
Appendix
This appendix contains the proofs of Theorems 3 and 4
Proof of Theorem 3 Set 119879 = 2 ln 2120574 Then using (21) fromthe Lyapunov inequality we get |119909(119879 119909
0 119906)|
119881le |119909
0|
1198812
whenever |119909
0|
119881le 119888
1120574119888 There exists 120598
0(120575) such that
sup119905isin[0119879]
|119909(119905 119909
0 119906) minus 119909(119905 119909
0 119906)| lt 1205752 Therefore we have
|119909(119879 119909
0 119906)| lt 120575 and |119909(119905 119909
0 119906)| le |119909(119905 119909
0 119906)| + |119909(119905 119909
0 119906) minus
119909(119905 119909
0 119906)| lt 31205752 This ends the proof
Proof of Theorem 4 Consider the function
119891 = 119882(119867
120591) (119867
1119890
3minus 119867
3119890
1minus 120572 (119867
1119890
1+ 119867
3119890
3)) (A1)
where
119867
120591= 119867
1119890
1+ 119867
3119890
3+ 120572 (119867
1119890
3minus 119867
3119890
1)
119882 (119867
120591) = 119867
120591minus
120581
2
sign
119867
120591
119867
1= cos 119905 119867
3= minus2 sin 119905
(119890
1 119890
3) = (cos 120579 sin 120579) 120579 isin [0
120587
2
]
(A2)
First note that for any fixed pair (119886 119887) the function 119905 rarr
119867
120591(119905 119886 119887) is analytic Therefore the integral 119868(119905
1 119905
2 119909 120575) (see
Section 2) is a point This implies that the averaged operatordefined in (14) coincides with
lim119879rarrinfin
1
119879
int
119879
0
119891 (119905 119886 119887) (
minus sin120596119905
cos120596119905
) 119889119905(A3)
if the limit exists Since 120596 is irrational number and 119891 can beconsidered as a 2120587-periodic function 119892 = 119892(119905
119905 119886 119887) of thearguments 119905 and
119905 = 120596119905 we see that limit (A3) does existand we have
lim119879rarrinfin
1
119879
int
119879
0
119891 (119905 119886 119887) (
minus sin120596119905
cos120596119905
) 119889119905
=
1
(2120587)
2∬
2120587
0
119892 (119905
119905 119886 119887)
times (
minus sin
119905
cos119905 ) 119889119905 119889
119905
(A4)
To evaluate this integral we represent the derivative
(sin (2119899 + 1) 119905 cos (2119899 + 1) 119905
1015840
minus sin (2119899 + 1)
1015840 cos (2119899 + 1) 119905)
times (2119899 + 1)
minus1
(A7)
Observe that
sin 119905
1015840
=radic
1 + 3119890
2
3(minus2119890
3+
119890
1(2120572 minus 3119890
1119890
3120573)
1 + 3119890
2
3
)
+ 119874(120572
2
+ 120573
2
)
(A8)
cos 1199051015840 = radic1 + 3119890
2
3(119890
1+
2119890
3(2120572 minus 3119890
1119890
3120573)
1 + 3119890
2
3
)
+ 119874(120572
2
+ 120573
2
)
(A9)
Substituting (A7) into (A4) and integrating in order of 119905 andthen in order of
119905 we obtain the result
Acknowledgments
This research is supported by the Portuguese Foundation forScience and Technologies (FCT) the Portuguese OperationalProgramme for Competitiveness Factors (COMPETE) thePortuguese Strategic Reference Framework (QREN) and theEuropean Regional Development Fund (FEDER)
References
[1] R E Fischell and F F Mobley ldquoA system for passive gravity-gradient stabilization of earth satellitesrdquo Guidance and Controlvol 2 pp 37ndash71 1964
Mathematical Problems in Engineering 7
[2] V A Sarychev V I Penkov M Yu Ovchinnikov and A DGuerman ldquoMotions of a gravitationally stabilized satellite withhysteresis rods in a polar orbitrdquo Cosmic Research vol 26 no 5pp 561ndash574 1988
[3] A D Guerman M Yu Ovchinnikov V I Penrsquokov and V ASarychev ldquoNon-resonant motions of a satellite with hysteresisrods under conditions of gravity orientationrdquo Mechanics ofSolids vol 24 pp 1ndash11 1989
[4] M Ovchinnikov V Penrsquokov O Norberg and S BarabashldquoAttitude control system for the first sweedish nanosatelliteMUNINrdquoActa Astronautica vol 46 no 2ndash6 pp 319ndash326 2000
[5] F Santoni and M Zelli ldquoPassive magnetic attitude stabilizationof the UNISAT-4 microsatelliterdquo Acta Astronautica vol 65 no5-6 pp 792ndash803 2009
[6] R Gama A Guerman and G Smirnov ldquoOn the asymptoticstability of discontinuous systems analysed via the averagingmethodrdquo Nonlinear Analysis Theory Methods and ApplicationsA vol 74 no 4 pp 1513ndash1522 2011
[7] RN Izmailov ldquoThepeak effect in stationary linear systemswithscalar inputs and outputsrdquoAutomation and Remote Control vol48 no 8 part 1 pp 1018ndash1024 1987
[8] H J Sussmann and P V Kokotovic ldquoThe peaking phenomenonand the global stabilization of nonlinear systemsrdquo IEEE Trans-actions on Automatic Control vol 36 no 4 pp 424ndash439 1991
[9] A Guerman A Seabra andG Smirnov ldquoOptimization of para-meters of asymptotically stable systemsrdquo Mathematical Prob-lems in Engineering vol 2011 Article ID 526167 19 pages 2011
[10] N N Bogoliubov and Y AMitropolskyAsymptotic Methods intheTheory of Non-Linear Oscillations Gordon and Breach NewYork NY USA 1961
[11] A M Samoilenko and A N Stanzhitskii ldquoOn averaging differ-ential equations on an infinite intervalrdquo Differential Equationsvol 42 no 4 pp 476ndash482 2006
[12] A F Filippov Differential Equations with Discontinuous Right-Hand Sides vol 18 ofMathematics and Its Applications KluwerAcademic New York NY USA 1988
[13] G V Smirnov Introduction to the Theory of Differential Inclu-sions vol 41 of AMS Graduate Studies in Mathematics Ameri-can Mathematical Society Providence RI USA 2002
[14] V A Plotnikov ldquoAveraging method for differential inclusionsand its application to optimal control problemsrdquo DifferentialEquations vol 15 pp 1013ndash1018 1979
[15] V A Plotnikov ldquoAsymptotic methods in the theory of differen-tial equations with discontinuous and multi-valued right-handsidesrdquoUkranian Mathematical Journal vol 48 no 11 pp 1605ndash1616 1996
[16] A M Liapunov Stability of Motion Academic Press New YorkNY USA 1966
[2] V A Sarychev V I Penkov M Yu Ovchinnikov and A DGuerman ldquoMotions of a gravitationally stabilized satellite withhysteresis rods in a polar orbitrdquo Cosmic Research vol 26 no 5pp 561ndash574 1988
[3] A D Guerman M Yu Ovchinnikov V I Penrsquokov and V ASarychev ldquoNon-resonant motions of a satellite with hysteresisrods under conditions of gravity orientationrdquo Mechanics ofSolids vol 24 pp 1ndash11 1989
[4] M Ovchinnikov V Penrsquokov O Norberg and S BarabashldquoAttitude control system for the first sweedish nanosatelliteMUNINrdquoActa Astronautica vol 46 no 2ndash6 pp 319ndash326 2000
[5] F Santoni and M Zelli ldquoPassive magnetic attitude stabilizationof the UNISAT-4 microsatelliterdquo Acta Astronautica vol 65 no5-6 pp 792ndash803 2009
[6] R Gama A Guerman and G Smirnov ldquoOn the asymptoticstability of discontinuous systems analysed via the averagingmethodrdquo Nonlinear Analysis Theory Methods and ApplicationsA vol 74 no 4 pp 1513ndash1522 2011
[7] RN Izmailov ldquoThepeak effect in stationary linear systemswithscalar inputs and outputsrdquoAutomation and Remote Control vol48 no 8 part 1 pp 1018ndash1024 1987
[8] H J Sussmann and P V Kokotovic ldquoThe peaking phenomenonand the global stabilization of nonlinear systemsrdquo IEEE Trans-actions on Automatic Control vol 36 no 4 pp 424ndash439 1991
[9] A Guerman A Seabra andG Smirnov ldquoOptimization of para-meters of asymptotically stable systemsrdquo Mathematical Prob-lems in Engineering vol 2011 Article ID 526167 19 pages 2011
[10] N N Bogoliubov and Y AMitropolskyAsymptotic Methods intheTheory of Non-Linear Oscillations Gordon and Breach NewYork NY USA 1961
[11] A M Samoilenko and A N Stanzhitskii ldquoOn averaging differ-ential equations on an infinite intervalrdquo Differential Equationsvol 42 no 4 pp 476ndash482 2006
[12] A F Filippov Differential Equations with Discontinuous Right-Hand Sides vol 18 ofMathematics and Its Applications KluwerAcademic New York NY USA 1988
[13] G V Smirnov Introduction to the Theory of Differential Inclu-sions vol 41 of AMS Graduate Studies in Mathematics Ameri-can Mathematical Society Providence RI USA 2002
[14] V A Plotnikov ldquoAveraging method for differential inclusionsand its application to optimal control problemsrdquo DifferentialEquations vol 15 pp 1013ndash1018 1979
[15] V A Plotnikov ldquoAsymptotic methods in the theory of differen-tial equations with discontinuous and multi-valued right-handsidesrdquoUkranian Mathematical Journal vol 48 no 11 pp 1605ndash1616 1996
[16] A M Liapunov Stability of Motion Academic Press New YorkNY USA 1966
