Linear Differential Games and High Precision Attitude Stabilization of Spacecrafts with Large Flexible Elements

Space mission success depends on adequate attitude control system performance. High precision pointing control design for flexible spacecraft with large appendages requires special care (Yamada & Yoshikawa, 1996; Nagashio, 2010). Many linear control schemes such as LQR, LQG, QFT, H∞, and μ-synthesis have been used to guarantee robust stability against both vibrations and parameter uncertainties. Future advanced space missions will involve control system design techniques with novel architectures, technologies, and algorithms. Next generation of microcomputers will allow one to use more sophisticated control algorithms. Pontryagin’s theory of linear differential games (Pontryagin, 1981) is a natural tool to solve local stabilization problems under conditions of uncertainty. The correspondingmathematical model involves a linear differential equation


Introduction
Space mission success depends on adequate attitude control system performance. High precision pointing control design for flexible spacecraft with large appendages requires special care (Yamada & Yoshikawa, 1996;Nagashio, 2010). Many linear control schemes such as LQR, LQG, QFT, H ∞ , and µ-synthesis have been used to guarantee robust stability against both vibrations and parameter uncertainties. Future advanced space missions will involve control system design techniques with novel architectures, technologies, and algorithms. Next generation of microcomputers will allow one to use more sophisticated control algorithms. Pontryagin's theory of linear differential games (Pontryagin, 1981) is a natural tool to solve local stabilization problems under conditions of uncertainty. The corresponding mathematical model involves a linear differential equatioṅ with two control parameters u ∈ P and v ∈ Q belonging to bounded sets. One can choose the control u so as to bring the point x into the terminal set S, while the term v represents the disturbances caused by the model uncertainties, vibration of flexible elements, and state estimation errors. Differential games of stabilization (Smirnov, 2002) have some characteristic features that distinguish them from other differential games. Namely, the terminal set S should be invariant (see Sec. 3). Moreover, the zero equilibrium position of the linear differential equationẋ = Ax, is assumed to be asymptotically stable. Equation (2) can be obtained applying standard linear stabilization techniques to the control system under consideration. The trajectories of the closed-loop systemẋ = Ax + v, subject to the disturbance v, approach a limit set Ω and not tend to zero. The aim of the differential game approach is to construct a smaller invariant set S⊂Ω, to bring the trajectory x(t) into S, and to maintain it there.
The idea to use differential game methods to solve stabilization problems with uncertainties is not new (see (Gutman & Leitmann, 1975a;Gutman & Leitmann, 1975b), for example). However, the previous studies are based on Lyapunov functions (Gutman, 1979) or solutions to the Hamilton-Jacobi-Isaacs-Bellman equation (Isaacs, 1965;Kurzhanski & Varaiya, 2002;Mitchell et al., 2005). In this chapter we discuss theoretical and computational aspects of differential games of stabilization considered in the framework of Pontryagin's approach. Theoretically both approaches are equivalent (Kurzhanski & Melnikov, 2000). But, from the computational point of view, the geometric language of Pontryagin's method (alternating integral, support functions, etc.) proves more efficient in constructing the stabilizing control than the approach based on Lyapunov functions or the solution to Hamilton-Jacobi-Isaacs-Bellman equation. Since a local stabilization problem is considered, linear model (1) suffices to adequately describe the system behaviour. Although the dynamics is linear, the approach itself is based on the usage of nonlinear methods such as convex analysis and nonlinear programming. This chapter is organized in the following way. First we give a short introduction to the theory of linear differential games and describe numerical methods developed for the alternating integral approximation. We also address the problem of construction of a minimal invariant set. Next, we apply the developed techniques to the problem of high precision attitude stabilization of spacecrafts with large flexible elements.

Mathematical background
Throughout this chapter we denote the set of real numbers by R and the usual n-dimensional space of vectors x =(x 1 ,...,x n ), where x i ∈ R, i = 1, n,byR n . The inner product of two vectors x and y in R n is expressed by x, y = x 1 y 1 + ...+ x n y n .
The norm of a vector x ∈ R n is defined by x = x, x 1/2 . Let A be a linear operator from R n to R m .I fA is an m × n real matrix corresponding to the linear operator A (we use the same symbol), then the transposed matrix A * corresponds to the adjoint operator. The unit linear operator from R n to R n will be denoted by I n . We denote the unit ball in R n by B n : Let λ ∈ R. Then put by definition For two sets C 1 and C 2 in R n their sum is defined by A set C ⊂ R n is said to be convex if λx +(1 − λ)y ∈ C whenever x ∈ C, y ∈ C, and λ ∈ [0, 1]. From definition it follows that an intersection of any number of convex sets is a convex set, and if C 1 ⊂ R n , C 2 ⊂ R n are convex, and α 1 and α 2 are real numbers, then the set α 1 C 1 + α 2 C 2 is convex. Let C ⊂ R n . The intersection of all convex sets containing C is called the convex hull of C and is denoted by coA.
All convex sets C ⊂ R n considered in this chapter are assumed to be symmetric, i.e., C = −C. This family is enough for our goals. The function S(·, C) : R n → R defined by is called the support function of C. The distance function can be expressed via support function: Moreover, the support function allows one to express the inclusion x ∈ C in an analytical form.
Namely, x ∈ C if and only if x, ϕ ≤S(ϕ, C) for all ϕ ∈ B n . Another description of a compact convex set C can be obtained in terms of the Minkowski is the Minkowski function of the set C.
Let C 1 and C 2 be convex sets in R n . The Minkowski difference of these sets is defined by If C 1 is convex and closed and C 2 is compact, then C 1 * − C 2 is closed and convex. It is easy to see that the following relations hold Moreover if C 3 is compact and convex, then we have The Hausdorff distance between two sets C 1 , C 2 ⊂ R n is defined as A set-valued map F : R n → R m with compact values is called continuous at x 0 ∈ R n if for any ǫ > 0 there exists δ > 0 such that h(F(x), F(x 0 )) < ǫ, whenever x ∈ x 0 + δB n . Let F : [a, b] → R n be a set-valued map with compact convex values. Its Riemann integral b a F(t)dt is defined as a limit in the sense of the Hausdorff distance of the integral sums The integral exists whenever F is continuous at all points of the interval. Moreover, it coincides with the set of all integrals of integrable selections (see (Castaing & Valadier, 1977)) and for all ϕ ∈ R n . This equality allows one to compute integrals of set-valued maps.

Linear differential games of pursuit
Differential games are control problems in a conflict situation. For example, if one aircraft pursuits another one we have such a situation. The dynamics of the system is described by a differential equation depending on two control parameterṡ One player controls the parameter u and the other one controls the parameter v. The aim of the first player is to drive the system to a terminal set S while the aim of the second player is to avoid this event. The solution to this problem consists in determination of functions u = u(x) and v = v(x), known as strategies (Krasovski & Subbotin, 1987), guaranteeing 1. the fastest arrival to the terminal set S for the first player; 2. the latest arrival to the terminal set S for the second player.
In this form the problem is extremely involved because the functions u(x) and v(x) can be discontinuous and the differential equationẋ = f (x, u(x), v(x)) may have no solution in the classical sense. This difficulty can be overcome introducing the concept of Pontryagin's ǫ-strategy. The differential game is considered as a pursuit game or an evasion game.
In the first case we identify ourself with the first player. At the initial moment of time t 0 the second player communicates to the first player a number ǫ 0 > 0 and his control v(t) defined in the time interval [t 0 , t 0 + ǫ 0 ]. The first player uses this information to choose his own control u(t), t ∈ [t 0 , t 0 + ǫ 0 ]. Next, at the moment of time t 1 = t 0 + ǫ 0 the second player communicates to the first player a number ǫ 1 > 0 and his control v(t) defined in the time interval [t 1 , t 1 + ǫ 1 ].
The first player uses this information to choose his control u(t), t ∈ [t 1 , t 1 + ǫ 1 ], and so on.
In the case of evasion games we identify ourself with the second player and the first one communicates us numbers ǫ k > 0 and controls u(t), t ∈ [t k , t k + ǫ k ].
Here we study only pursuit games for linear control systemṡ and without the objective to reach the terminal set S in an optimal time. Our aim is to finish the game in a time T not necessarily optimal. The sets P and Q are assumed to be compact and convex. The terminal set S is closed and convex. Moreover, the number ǫ from the definition of the ǫ-strategy is assumed to be fixed.
Differential games of stabilization have some characteristic features that distinguish them from other differential games. Namely, the terminal set S should be invariant, i.e., if x 0 ∈S, then for any control v(t) ∈ Q there should exist a control u(t) ∈ P such that it maintains the trajectory x(t, x 0 , u(·), v(·)) in the set S. There are many possibilities to formalize the concept of invariance. We say that a set S is ǫ-invariant if the following inclusion holds Here we use the notations Let x ∈S and v(t) ∈ Q, t ∈ [0, ǫ], be an admissible disturbance. After a change of variable in integrals (7) we obtain From (6) and (8) we see that there exists an admissible control u(t) ∈ Q, t ∈ [0, ǫ] such that i.e., starting at S we always return to it after time ǫ.
The zero equilibrium position of the linear differential equatioṅ is assumed to be asymptotically stable. This implies that there exists a positive definite symmetric matrix V satisfying the Lyapunov equation If α > 0 is sufficiently large and ǫ is sufficiently small, then the ellipsoid αE, where Consider the sets I k ⊂ R n defined by I 0 = S, The sets I k are known as Pontryagin alternating sums. Fix T > 0 and set ǫ = T/N. The limit of the Pontryagin alternating sums as N goes to infinity, is called Pontryagin alternating integral. The inclusion Λ ǫ x ∈I k+1 implies that for any By induction we see that if Λ N ǫ x ∈I N , then the game can be finished in time Nǫ, i.e., that the first player can choose an ǫ-strategy in order to guarantee the inclusion x(Nǫ) ∈I 0 = S. The set F T = e −TA I T is known as Pontryagin-Pshenichnyj pursuit operator and consists of all initial points x 0 such that the game starting from x 0 can be finished in time less than or equal to T independently on the ǫ-strategy of the second player. Instead of the Pontryagin alternating sums we shall use the Pontryagin-Pshenichnyj pursuit ǫ-operators defined by F 0 = S, If the set F k is ǫ-invariant (see (6)), then the set F k+1 is also invariant. Indeed, from the inclusion This implies Therefore the operators F k form a monotone family, provided that the terminal set S is ǫ-invariant.
To construct an absorbing family of Pontryagin-Pshenichnyj operators, suppose that the terminal set is strictly invariant, i.e., where the ellipsoid E satisfies the inclusion Λ ǫ E ⊂ E (see (10)). The strict invariance of S implies the inclusion S + E ⊂F ǫ (S). Observe that From this we obtain S + 2E ⊂F ǫ (S)+E ⊂F 2 ǫ (S). By induction we have S + kE ⊂F k ǫ (S). Therefore k≥0 F k ǫ (S)=R n .

Computational aspects
To numerically compute the pursuit operator and the stabilizing control, the considered sets should be approximated by polyhedrons. In this section we briefly present the computational geometry tools necessary for this purpose. Fix two sets of unit vectors {ϕ m } M m=1 and {ξ l } L l=1 . An exterior polyhedral approximation, C, of a convex compact set C ⊂ R n is given by and an interior polyhedral approximation, C, of a convex compact set C ⊂ R n is given by We shall use the notations σ m = S(ϕ m , C) and µ l = µ(ξ l , C) for the values of the support function and of the Minkowski function, respectively. The vectors σ =( σ 1 ,...,σ M ) and µ =(µ 1 ,...,µ L ) define the exterior and interior approximations of a compact convex set C. We say that the exterior and interior approximations are consistent if the following conditions are satisfied: 1. ξ l , ϕ m ≤µ l σ m , for all l = 1, L and m = 1, M, 2. for any l = 1, L there exists m(l) such that ξ l , ϕ m(l) = µ l σ m(l) , 3. for any m = 1, M there exists l(m) such that ξ l(m) , ϕ m = µ l(m) σ m .
If exterior and interior descriptions σ =(σ 1 ,...,σ M ) and µ =(µ 1 ,...,µ L ) are not consistent, they can be made consistent using one of adjustment operators µ →A σ (µ) and Let C 1 and C 2 be two convex compact sets, and let σ(C 1 ) and σ(C 2 ) be the vectors defining their exterior approximations. Since S(ϕ, C 1 + C 2 )=S(ϕ, C 1 )+S(ϕ, C 2 ), it is natural to define the exterior approximation vector σ(C 1 + C 2 ) for the sum as The evaluation of the approximation for the Minkowski difference C 1 * − C 2 is more involved. The point is that the difference of support functions S(ϕ, C 1 ) − S(ϕ, C 2 ) may be not a support function of a convex set and some correction is needed. This correction is done using the interior description. Namely, we set {x | x, ϕ m ≤σ m (C 1 * − C 2 ), m = 1, M} tend to C 1 + C 2 and C 1 * − C 2 , respectively, as M and L go to infinity. Some estimates for the precision of the approximations can be found in (Polovinkin et al., 2001). The approximation of the set ΛC, where Λ : R n → R n is a linear operator, is based on the following property of support functions: and is computed as where the vector ϕ λ(m) satisfies the condition Now, consider the problem of a minimal invariant set construction. Let P⊂R n and Q⊂R n be convex compact sets, and let Λ : R n → R n be a linear operator. The condition of a convex set S invariance, ΛS + Q⊂S+ P, in terms of support functions takes the form S(ϕ, ΛS )+S(ϕ, Q) ≤ S(ϕ, S)+S(ϕ, P ), for all ϕ, ϕ = 1.
Although the property of r-minimality does not define a unique invariant set, it is quite suitable from the practical point of view. We developed the following algorithm to compute a minimal invariant set. Let S 0 be an invariant set. (Recall that in the case of a differential game of stabilization there always exists an invariant ellipsoid (see Sec. 3).) Then we obtain an interior approximation of S 0 described by a vector µ (0) =(µ (0) 1 ,...,µ L ) and set S 0 = co ±(µ (0) 1 ) −1 ξ 1 ,...,±(µ (0) ) −1 L ξ L . Let δ > 0. The current invariant set S k is successively shrunk going through the vectors ξ l , l = 1, L, and considering the sets If the set S l k is invariant, we put S k+1 = S l k . After passing through all vectors ξ l , l = 1, L, the algorithm turns to the vector ξ 1 . The algorithm stops if none of the modified sets S l k , l = 1, L, is invariant. This algorithm is very simple and efficient. However, in general, it does not lead to a r-minimal invariant sets. The problem of r-minimal invariant set construction is more involved and can be solved using nonlinear programming techniques. The invariance condition (14) implies that the vector σ r = (σ r 1 ,...,σ r M ) giving the external description of a r-minimal invariant set has to be a solution to the following linear programming problem where p m = S(ϕ m , P ), q m = S(ϕ m , Q), and σ m , m = 1, M, and r are the unknown variables.
Unfortunately the solution to this problem is not unique and a vector σ, solving the problem, may be not a vector of a support function values. For this reason it is necessary to use inner approximations for the invariant set and solve the following nonlinear programming problem with the variables µ l , l = 1, L, and r.
A very important issue is the stabilizing control u construction. Assume that the current position of the system x k belongs to the set F N−k . To determine the stabilizing control u(t) defined on the interval [kǫ, (k + 1)ǫ] we numerically solve the optimal control problem The distance function is calculated using representation (4) and the control u(t), t ∈ [kǫ, (k + 1)ǫ], is considered to be a piece-wise constant function, u(t)=u j , t ∈ [(k − j/J)ǫ, (k − (j + 1)/J)ǫ], j = 0, J − 1. Approximating the set P by a polyhedron, we get the linear programming problem r → min Here u j , j = 1, J, and r are the unknown variables. This problem can be solved using the simplex-method or an interior-point method. Since the difference between the problems on the adjacent time intervals is rather small, the solution u j , j = 1, J, obtained at the moment t = kǫ can be used as an initial point to solve the linear programming problem on the next time interval.

Robust Pontryagin-Pshenichnyj operator
At the instant t = kǫ the disturbance v(t) defined on the interval [kǫ, (k + 1)ǫ], needed to construct the control u(t), t ∈ [kǫ, (k + 1)ǫ], is not available. For this reason we use the disturbance v(t) defined on the interval [(k − 1)ǫ, kǫ]. It turns out that this can cause serious problems and the construction of the Pontryagin-Pshenichnyj operator should be modified in order to overcome them. To clarify this issue we need some notations. Let T(x 0 ) be such that , computed using the disturbance v(t) defined on the interval [(k − 1)ǫ, kǫ], and by u(t, v(t), x 0 ) denote the control u(t), t ∈ [kǫ, (k + 1)ǫ], computed using the disturbance v(t) defined on the interval [kǫ, (k + 1)ǫ]. The corresponding solutions of system (5) we denote by , respectively. It turns out that, in general, in the first case the trajectory rapidly zigzags in the vicinity of the equilibrium position and in the second case its behaviour is more regular. Consider the following example. The control system describes the motion of a harmonic oscillator with friction. The control resource of the first player is enough to compensate any disturbance. The control v(t) takes alternating values ±v max on the intervals [kǫ, (k + 1)ǫ]. The influence of the delay can be seen comparing Figures  1 and 2. It is clear that the presence of delay causes violent oscillations of the trajectories. To overcome this difficulty we introduce a robust Pontryagin-Pshenichnyj ǫ-operator. The definition of ǫ-invariant set also should be revised. We say that a convex set S is robustly ǫ-invariant if S = S 0 + 2Q ǫ and This definition implies the inclusion The robust Pontryagin-Pshenichnyj ǫ-operator is defined by G 0 = S, If x 0 ∈G k+1 and we choose the control u(t, v(t − ǫ), x 0 ) to guarantee the inclusions X − (ǫ, x 0 ) ∈ G k * − 2Q ǫ , then we have A trajectory generated by the robust Pontryagin-Pshenichnyj ǫ-operator for the above example can be seen in Fig. 3. It is more regular although the limit set is larger. The latter can be reduced diminishing the parameter ǫ. From the qualitative point of view, the difference between the behaviours of the trajectories generated by the usual Pontryagin-Pshenichnyj ǫ-operator and the robust one can be explained as follows. The inclusion x 0 ∈F k+1 does not imply the inclusion X ǫ (x 0 ) ∈F k . In general, we need much time than ǫ to achieve the set F k and the search of the way to the set F k results in zigzags of the trajectories. On the other hand, the inclusion x 0 ∈G k+1 always imply (19).

High precision attitude stabilization of spacecrafts with large flexible elements
Satellites with flexible appendages are modelled by hybrid systems of differential equations where x ∈ R n , y ∈ Y is vector in a Hilbert space, and g : Y 3 → R m is an integral operator (Junkins & Kim, 1993). Equation (20) is an ordinary differential equation describing the motion of the satellite and depending on the control u ∈ U, while (21) is a partial differential equation modelling the dynamics of flexible appendages. We illustrate the stabilization techniques based on the differential game approach by a model example. Consider a spacecraft composed of a rigid body with a flexible appendage (a beam, see Fig.  4). The satellite is modelled as a cylinder. The distance between its longitudinal axis and the point c where the beam is cantilevered is denoted by r 0 . The length of the beam is denoted by l. We use two systems of coordinates: the inertial one denoted by OXYZ and the system oxyz rigidly connected to the satellite. The axis oz is directed along the satellite longitudinal axis, and the axis ox passes through the point c. The position of the point o is described by the coordinates (X 0 , Y 0 ), and the position of the axis ox relatively to the inertial coordinate system is defined by the angle θ. The deflection of the beam from the axis ox is described by the function y(t, x) (see Fig. 5). We assume that the oscillations of the flexible appendage are small and can be described in the framework of linear theory of elasticity. We consider only a rotation of the satellite around its longitudinal axis.
To obtain the Lagrange equations for this system we write down the Lagrangian function The function y = y(t, x) satisfies the following boundary conditions: y(r 0 , t)=y ′ (r 0 , t)=y ′′ (r 0 + l, t)=y ′′′ (r 0 + l, t)=0.
Adding the control moment M, |M|≤M max , and the internal viscous friction, we obtain the following system of differential equations: where χ is the coefficient of internal viscous friction.
Using the Galerkin method we approximate y(t, x) by a linear combination Φ i (x) and integrating in x ∈ [r 0 , r 0 + l], we get a system of ordinary differential equations for the variables X 0 , Y 0 , θ, and q i . For simplicity consider the approximation involving the first natural mode only: cosh(βl)+cos(βl) sinh(βl)+sin(βl) (sinh(βx) − sin(βx)), and β ≈ 1.875/l. Then from system (22) -(25) we obtain where J 1 = 0.7829, J 2 = 0.5688, J 3 = 0.9998. System (28) -(30) can be written in the matrix form as a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 where ⎛ ⎝ a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 Fig. 7. Rest-to-rest manoeuvre: the discrepancy of the attitudes θ r (t) and θ f (t) between the motion of the rigid system and the flexible one.
Denote this matrix by A. Thus, the angular dynamics of the satellite near the zero equilibrium position is described by the linear control system where D = EIβ 4 J 3 l.
The influence of the flexible appendages can be rather significant. Consider a rest-to-rest manoeuvre for the model under study. We apply the moment +M and then −M during the same time and compare the motion of the configuration considering the appendage as flexible with low stiffness (EI = 3.5) and as rigid. The disturbance caused by the appendage is shown in Fig. 6, while the difference between the angular positions of the satellite with flexible and rigid appendages is shown in Fig. 7. It is quite large. Therefore a high-precision attitude stabilization system should take into account the flexibility.
To stabilize system (31) and (32)   If the angular position θ and the angular velocityθ are known with some errors δθ and δθ, respectively, we obtain the following differential game: , and q is governed by the differential equation The differential game approach to the stabilizer design problem presumes that the controls u and v satisfy the restrictions |u|≤u max and |v|≤v max , respectively. To determine the values u max and v max and the neighborhood of the equilibrium position where the differential game stabilizer works we use the following method. Consider the vector x =(θ,θ, q,q) T . Its behaviour is described by the differential equatioṅ The solution x(t) to differential equation (34) satisfies the following differential inequality: d dt x(t), Vx(t) = − x(t) 2 + 2 Vx(t), b(t) ≤− x(t) 2 + 2 V x(t) b(u max ) ≤ 0, whenever 2 V b(u max ) ≤ x(t) . Define the family of ellipsoids Obviously x(t) ∈ E c(u max ) , whenever t is large enough. Thus, the differential game of stabilization is playable if the Pontryagin-Pshenichnyj ǫ-operators are contained in E c(u max ) and the following conditions are satisfied: v max < u max , v max ≤ A −1
Typical trajectories generated by a linear feed-back and by a high-precision stabilizer with the Pontryagin-Pshenichnyj ǫ-operator and the robust Pontryagin-Pshenichnyj ǫ-operator, respectively, are shown in Fig. 8 and 9. (The disturbance v in the numerical simulations is a periodic function.) The differential game method of stabilization yields significantly smaller limit set (marked by box) than the simple linear stabilizer (33).

Conclusion
We present here a new approach to stabilization of mechanical systems with uncertainties in parameters and/or state data. This approach considers the perturbations caused by these uncertainties as an evader control in a linear pursuit differential game. We describe the general theoretical basis and the numerical algorithms for implementation of the described differential game stabilizer. Estimates for the amplitude of the evader control should be obtained for any specific case of control system using its mechanical properties. We consider here an application of the suggested method to the stabilization problem for a satellite with large flexible appendages. The estimates for the evader control caused by uncertainties are deduced applying the method of Lyapunov functions. We construct a high-precision stabilizer using the differential game approach and the above estimates for the evader control. The principal advantage of the suggested method is that, to achieve a high-precision stabilization, it requires only the satellite attitude data and does not need any estimation for the flexible elements' state and/or unknown system parameters.