Averaging Methods for Design of Spacecraft Hysteresis Damper

and Applied Analysis Hindawi Publishing Corporation http://www.hindawi.com Volume 2013 ISRN Applied Mathematics Hindawi Publishing Corporation http://www.hindawi.com Volume 2013 Hindawi Publishing Corporation http://www.hindawi.com Volume 2013 International Journal of Combinatorics Hindawi Publishing Corporation http://www.hindawi.com Volume 2013 Journal of Function Spaces and Applications International Journal of Mathematics and Mathematical Sciences Hindawi Publishing Corporation http://www.hindawi.com Volume 2013


Introduction
Dampers that use magnetic hysteresis rods to dissipate the energy of undesired angular motions occurred during deployment or caused by perturbations are used in attitude control systems of small satellites since 1960s [1]. Mathematical modeling of such systems is quite a difficult task since the majority of existent hysteresis models result in differential equations with discontinuous right-hand side.
Analysis of dynamics for attitude control systems with magnetic hysteresis dampers and optimization of their parameters have been done in [2,3], and the results of these studies have been implemented in real missions [4,5]. However, these studies lack an accurate theoretical basis for application of averaging methods to such problems.
Recently, an adequate mathematical approach has been developed by the authors in [6]. Now we can address a complete mathematical theory for attitude stabilization systems with hysteresis.
Consider a differential equatioṅ = ( , , ) , ∈ , ≥ 0 (1) describing a mechanical system with stabilizer. Here ∈ ⊂ is a parameter. It is assumed that 0 ≈ ( , 0, ) for all ≥ 0 and ∈ ; that is, the velocity of the system near the origin is small. Here we do not assume that zero is an equilibrium position of system (1). The parameter should be chosen to optimize, in some sense, the behavior of the trajectories. The choice of this parameter can be based on various criteria. Obviously, it is impossible to construct a stabilizer optimal in all aspects. Consider, for example, a linear controllable system. The pole assignment theorem guarantees the existence of a linear feedback yielding a linear differential equation with any given set of eigenvalues, so one can choose a stabilizer with a very high damping speed. However, such a stabilizer is practically useless because of the so-called peak effect (see [7,8]). Namely, there exists a large deviation of the solutions from the equilibrium position at the beginning of the stabilization process, whenever the module of the eigenvalues is big. The aim of this paper is to develop effective analytical and numerical tools oriented to optimization of stabilizer parameters for passive attitude stabilization system with hysteresis rods.
Throughout this paper, we denote the set of real numbers by and the usual -dimensional space of vectors with components in by . We denote by ⟨ , ⟩ the usual scalar product in and by | ⋅ | a norm. By we denote the closed unit ball, that is, the set of vectors ∈ satisfying | | ≤ 1. The transpose of a matrix is denoted by * . The set of positively definite symmetric × -matrices is denoted by 2 Mathematical Problems in Engineering ( ). If and are two subsets in and ∈ , we use the following notations: = { | ∈ }, + = { + | ∈ , ∈ }. The convex hull and the closure of a subset ⊂ R are denoted by co and cl , respectively. The Hausdorff distance between two sets 1 , 2 ⊂ is defined as We denote by S where gr stands for the graph of the set valued map .

Statement of the Problem
Consider dynamics of a satellite in a circular geocentric orbit. The satellite is equipped with a gravity-gradient attitude control system that includes a number of magnetic hysteresis rods as a damper. The spacecraft's equations of motion [2] can be represented in the normalized form. Denote by ( , 0 , ) the solution to the respective Cauchy probleṁ = ( , , ) , ∈ , ∈ [0, ] , where is a parameter from a compact set ⊂ . Define the functions Here Δ ⊆ [0, ] are compact sets, | ⋅ | are norms in , and = { ∈ | | | ≤ }. Consider the following mathematical programming problem: Many problems of stabilization systems' parameters optimization can be written in this form (see [9]). For example, the minimization of the final deviation can be formalized as follows: and the minimization of the maximal deviation of trajectories satisfying certain restrictions at the final moment of time has the form:

Averaging for Discontinuous Systems
The averaging method is one of the most used methods to analyze differential equations of the forṁ appearing in the study of nonlinear problems. The idea behind the averaging method is to replace the original equation by the averaged one: This equation is simpler and has solutions close to the solutions of the original equation. A rigorous justification of the method is given by Bogolyubov's first theorem containing an estimate for the distance between the solutions of the exact and averaged systems on large time intervals [10]. The Samoilenko-Stanzhitskii theorem [11, Theorem 2] which is a generalization of Bogolyubov's second theorem shows that asymptotic stability of the zero equilibrium position of averaged (10) implies that the solutions to original (9) are close to zero on the infinite time interval.
For several models of systems with hysteresis, including the passive attitude stabilization systems, the function ( , ⋅) appearing in (9) is discontinuous (see, e.g., [3]), and the classical notion of solution and the classical averaging method cannot be used. For such systems Filippov proposed a generalized concept of solution, rewriting problem (9) as a differential inclusioṅ where → (⋅, ) is an upper semicontinuous set-valued map obtained from ( , ⋅) by Filippov regularization [12,13]. The use of this concept of solution makes it necessary to generalize the averaging method to differential inclusions. Many results extending Bogolyubov's first theorem to differential inclusions have been obtained (see, e.g., [14,15]). In the case of Lipschitzian differential inclusions, the problem has been completely solved by Plotnikov [14]. Averaging results for inclusions with upper semicontinuous right-hand side have been obtained by Plotnikov [15] under conditions of Lipschitz continuity of the averaged inclusion and for inclusions with a piecewise Lipschitzian right-hand side. Recently [6], an averaged differential inclusion has been introduced allowing one to prove extensions of Bogolyubov's first theorem and of the Samoilenko-Stanzhitskii theorem for upper semicontinuous differential inclusions and, as a consequence, for Mathematical Problems in Engineering 3 discontinuous dynamical systems. Here we outline the main results from [6].
Let : × → be a set-valued map. Set We denote by ( ) the convex hull of the map and define the averaged differential inclusion aṡ Note that under Lipschitz condition this map coincides with if the limit exists in the sense of Hausdorff distance (see [6]). Assume that the following conditions are satisfied: (C5) there exists the limit Under these conditions, the following version of Bogolyubov's first theorem is true.
Next theorem is an extension of the Samoilenko-Stanzhitskii theorem.
Theorem 2. Let : × → be a set-valued map satisfying conditions (C1)-(C5). Assume that = 0 is an asymptotically stable equilibrium position of the differential inclusioṅ ∈ 0 ( ). Then for any > 0, there exist 0 > 0 and > 0 such that The last theorem shows that if the averaged inclusion has zero as its asymptotically stable equilibrium position, the trajectories of the original inclusion stay in the vicinity of the origin provided > 0 and | 0 | are sufficiently small.
If the averaged inclusion has a special form, we can go further and make some conclusion on the detailed behaviour of the trajectories of the original system. Assume that the averaged inclusion has the forṁ ∈ ( ( ) + ( , )) , where ( , ) ⊂ | | 2 , > 0, the real parts of the matrix ( ) eigenvalues are negative for all ∈ , and the function → ( ) is continuous for all ∈ . If 0 > 0 is sufficiently small, then the set of solutions to the Lyapunov inequality (see [16]) for the matrix ( ), is nonempty and compact for all ∈ . Let ( , ) ∈ L( ). Denote by | | the Euclidean norm defined by | | = √⟨ , ⟩. There exist positive constants 1 and 2 satisfying whenever ( , ) ∈ L( ) for some . This theorem shows that the behavior of the trajectory ( , 0 , ) can be characterized in terms of the pair ( , ). The parameter is responsible for the damping speed of the process, while the form of the ellipsoid { | ⟨ , ⟩ ≤ 1} describes the amplitude of the deviation of the trajectory from the origin. The aim of parameter choosing can be formulated as follows: maximal value of and maximal sphericity of the ellipsoid { | ⟨ , ⟩ ≤ 1}. The latter property guarantees minimal overshooting of the damping process and, as a consequence, the largest region of applicability of the approximation obtained via averaging.

Choosing Passive Magnetic Stabilizer Parameters
The in-plane oscillations of a satellite moving along a polar circular orbit and equipped with a passive gravity-gradient attitude stabilization system with one hysteresis rod are described by the equation where is the pitch angle of spacecraft, is small parameter proportional to the rod's volume, and the force is given by Here describes the projection of the geomagnetic field on the rod axis, (25) is the hysteresis function, corresponds to the coercive force, and is the argument of latitude of the current point of the orbit [2]. The vector (1 + 3 2 3 ) Obviously we have Therefore, we see that the linearization of the averaged system always has a Lyapunov function = 2 + 2 , and the damping speed is determined by the value of . This means that the peak effect does not take place for the linearization of the averaged system.
To maximize the damping, one has to increase the total volume of the hysteresis material on board. However, it is wellknown that the efficiency of a damping rod is increased with the increase of the ratio between the rod's length and its cross-section dimension. Therefore, instead of one massive bar, the attitude control system should use several rather thin rods of the maximum length allowed by the spacecraft geometrical and system restrictions. On the other hand, to minimize the perturbation of the spacecraft angular motion caused by the damping system itself, the direction of total magnetic field in the rods should deviate as little as possible from the direction of the geomagnetic field at the current point of the orbit. Thus, in general case, one should use a system of three equal orthogonal hysteresis rods or a number of such systems. Here we consider in-plane satellite dynamics on a polar orbit, and for such purpose it suffices to analyze a pair of equal orthogonal rods.
Orientation of this pair of equal orthogonal rods can be defined as ( 1 , 3 ), (− 3 , 1 ), where 1 = cos and 3 = sin , and due to the system symmetry it is enough to study the interval ∈ [0, /2]. If the satellite is equipped with several identical hysteresis rods, the corresponding nonlinear system iṡ= − ( 1 + 2 + ⋅ ⋅ ⋅ ) sin , Here the terms 1 , 2 , . . ., describe the interaction of the respective rod with the geomagnetic field. For a couple of equal orthogonal rods and for small deviation from the origin, the forces 1 and 2 are given by respectively. Here  ( 1 An easy calculation shows that the optimal value of is /4, so 1 = 3 = √ 2/2. In the next section we numerically analyze the validity of the previous analytical study.

Numerical Simulations
We approximate problem (6) by the following problem: is the Euler approximation for the solution (⋅, , ). Note that this is a hard problem because of the discontinuity of the system. It is necessary to consider very fine partition of the time interval in order to get a good approximation of the solutions to the discontinuous differential equation. For smooth right-hand sides the number of points in the mesh can be significantly reduced (see [9]). Let > 0 be small enough. We consider a satellite with two equal orthogonal hysteresis rods. A typical trajectory of the system is shown in Figure 1.
Note that the oscillations of the trajectory do not allow one to characterize its damping speed using the norm at the final moment of time. For this reason, we numerically solve the following problem: The results of minimization for = 0.25, = 0.1, = 0.949, 0 = 1, 1 = 0.1, = 300 , and = 20 show that the problem has many local minima but with the values of the functional very close to 1; that is, the nonlinear system with two orthogonal hysteresis rods has no overshooting for all values of . This allows one to conclude that the value = /4 is the best one. It guarantees high damping speed and does not cause peaking.

Conclusions
This paper is dedicated to the problem of parameter optimization for a gravitationally stabilized satellite with magnetic hysteresis damper. Its motion is described by differential equations with discontinuous right-hand side. The discontinuity is the principal obstacle in the application of the averaging method. Our recently obtained results on averaging of discontinuous systems are applied now to rigorously justify the use of this method for a satellite with hysteresis rods.
We consider here the simplest case of in-plane oscillations on a polar circular orbit. Theorem 3 shows that the behavior of the system can be characterized in terms of Lyapunov function which can be chosen in order to guarantee the best properties of damping process. Further study is performed numerically, and the simulations are in excellent agreement with the analytical results, confirming also previous studies on hysteresis damping of satellite pitch oscillation in gravitygradient mode.
Our results can also be applied to rigorously justify the use of averaging techniques in analysis of other engineering problems involving differential equations with discontinuous right-hand side. First, note that for any fixed pair ( , ) the function → ( , , ) is analytic. Therefore the integral ( 1 , 2 , , ) (see Section 2) is a point. This implies that the averaged operator defined in (14)