Linear Differential Games and High Precision Attitude Stabilization of Spacecrafts With Large Flexible Elements 7 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 σ →A μ (σ) defined by A σ (μ)=(σ 1 (μ), ,σ M (μ)), σ m (μ)=max l=1,L μ −1 l ξ l , ϕ m  and A μ (σ)=(μ 1 (σ), ,μ L (σ)), μ l (σ)= ⎛ ⎜ ⎝ min m=1,M ξ l ,ϕ m >0 σ m ξ l , ϕ m  ⎞ ⎟ ⎠ −1 . 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 σ (C 1 + C 2 )=σ(C 1 )+σ(C 2 ). 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 σ (C 1 ∗ − C 2 )=A σ (A μ (σ(C 1 ) −σ(C 2 ))). If the vectors {ϕ m } M m =1 and {ξ l } L l =1 form rather fine meshes in the unite sphere, the above exterior approximations of the sum and the Minkowski difference given by {x |x, ϕ m ≤σ m (C 1 + C 2 ), m = 1, M} and {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: S (ϕ,ΛC)=S(Λ ∗ ϕ, C)=Λ ∗ ϕS  Λ ∗ ϕ Λ ∗ ϕ ,C  and is computed as S (ϕ m ,ΛC)=Λ ∗ ϕ m S  ϕ λ(m) ,C  , where the vector ϕ λ(m) satisfies the condition     ϕ λ(m) − Λ ∗ ϕ m Λ ∗ ϕ m      = min m  =1,M     ϕ m  − Λ ∗ ϕ m Λ ∗ ϕ m      . 429 Linear Differential Games and High Precision Attitude Stabilization of Spacecrafts With Large Flexible Elements 8 Advances in Spacecraft Technologies 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, Λ S + Q⊂S+ P, (13) 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. (14) We say that an invariant set S is minimal, if for any S  ⊂S, S  = S, we have ΛS  + Q ⊂ S  + P. Note that the minimal invariant set may be not unique and that the intersection of two invariant sets may be not invariant. Indeed, consider the following example in R 2 . Let Λ = 1 2 I 2 , P = co{(0,2),(0, −2)}, and Q = co{(1, 1),(1, −1), (−1,1),(−1,−1)}. It is easy to see that any set S a = {(x, ax) | x ∈ [−2, 2]}, a ∈ [−1, 1], is minimal invariant. The intersection S a 1 ∩S a 2 = {0}, a 1 = a 2 , is not invariant. To restrict the set of invariant sets, we introduce the following definition. Put r (S)=min{r > 0 |S⊂rB n }. An invariant set S is said to be r-minimal, if for any S  satisfying r(S  ) < r(S), we have ΛS  + Q ⊂S  + P. In the previous example a unique r-minimal invariant set is co{(1,0),(−1,0)}. Note that in general the r-minimality does not define a unique invariant set, as it is clear from the following example. Set Λ = 1 2  01 −10  , P = co{(0,1), (0,−1)}, and Q = co{(1,1),(1, −1),(−1,1),(−1,−1),(0,2), (0,−2)}. It is easy to see that the sets S 1 = 2B 2 and S 1 = co{(1,0), (−1,0),(0,1),(0, −1)} are both r-minimal invariant. 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 (0) 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 S l k = co  ±(μ (0) 1 ) −1 ξ 1 , ,±(μ (0) l + δ) −1 ξ l , ,±(μ (0) L ) −1 ξ L  . 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 430 Advances in Spacecraft Technologies Linear Differential Games and High Precision Attitude Stabilization of Spacecrafts With Large Flexible Elements 9 the following linear programming problem r → min, Λ ∗ ϕ m σ λ(m) + q m ≤ σ m + p m , m = 1, M, 0 ≤ σ m ≤ r, m = 1, M, 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 r → min, max l=1,L μ −1 l Λξ l , ϕ m + q m ≤ max l=1,L μ −1 l ξ l , ϕ m + p m , m = 1, M, 0 ≤ μ −1 l ≤ r, l = 1, L, 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 [k,(k + 1)] we numerically solve the optimal control problem d  e A x k −   0 e (−t)A u(k + t)dt +   0 e (−t)A v(k + t)dt,F N−k−1  → min, u (k + t) ∈ P. The distance function is calculated using representation (4) and the control u (t), t ∈ [k, (k + 1)], is considered to be a piece-wise constant function, u(t)=u j , t ∈ [(k − j/J),(k − (j + 1)/J)], j = 0, J −1. Approximating the set P by a polyhedron, we get the linear programming problem r → min  e A −  J J ∑ j=1 e (1−j/J)A u j +   0 e (−t)A v(k + t)dt, ϕ m  −S(ϕ m ,F N−k−1 ) ≤ r, m = 1, M u j , ϕ m ≤S(ϕ m , P), m = 1, M, j = 1, J. 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 = k can be used as an initial point to solve the linear programming problem on the next time interval. 5. Robust Pontryagin-Pshenichnyj operator At the instant t = k the disturbance v(t) defined on the interval [k,(k + 1)], needed to construct the control u (t), t ∈ [k,(k + 1)], is not available. For this reason we use the disturbance v (t) defined on the interval [(k −1), k]. 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 431 Linear Differential Games and High Precision Attitude Stabilization of Spacecrafts With Large Flexible Elements 10 Advances in Spacecraft Technologies x 0 ∈F T(x 0 ) and x 0 ∈F t , t < T(x 0 ).Byu(t, v(t −), x 0 ) denote the control u(t), t ∈[k,(k + 1)], computed using the disturbance v (t) defined on the interval [(k −1), k], and by u(t,v(t), x 0 ) denote the control u(t), t ∈ [k, (k + 1)], computed using the disturbance v(t) defined on the interval [k, (k + 1)]. The corresponding solutions of system (5) we denote by X − (x 0 )=e A x 0 −   0 e (−t)A u(t, v(t −), x 0 )dt +   0 e (−t)A v(t)dt and X  (x 0 )=e A x 0 −   0 e (−t)A u(t, v(t), x 0 )dt +   0 e (−t)A v(t)dt. The controls u (t,v(t − ), x 0 ) and u(t,v(t), x 0 ), t ∈ [k,(k + 1)], are constructed to minimize the distances d (X − (x 0 ), F T(x 0 )− ) and d(X  (x 0 ), F T(x 0 )− ), 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 ¨ x = −β ˙ x −αx − u + v, |u|≤u max , |v|≤v max (15) 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 [k, (k + 1)]. 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. −6 −4 −2 0 2 4 6 8 10 x 10 −4 −6 −5 −4 −3 −2 −1 0 1 x 10 −3 Fig. 1. Trajectory (x, ˙ x): motion without delay. To overcome this difficulty we introduce a robust Pontryagin-Pshenichnyj -operator. The definition of -invariant set also should be revised. We say that a convex set S is robustly -invariant if S = S 0 + 2Q  and Λ  S 0 + 2Λ  Q  + Q  ⊂S 0 + P  . (16) 432 Advances in Spacecraft Technologies Linear Differential Games and High Precision Attitude Stabilization of Spacecrafts With Large Flexible Elements 11 −5 0 5 10 x 10 −4 −7 −6 −5 −4 −3 −2 −1 0 1 2 3 x 10 −3 Fig. 2. Trajectory (x, ˙ x): motion with delay. This definition implies the inclusion Λ  S + Q  ⊂ (S ∗ − 2Q  )+P  . (17) The robust Pontryagin-Pshenichnyj -operator is defined by G 0 = S, G k = G k  (S), k = 0, N, (18) where G  (C)=Λ −1   C ∗ − 2Q   + P   ∗ −Q   . If x 0 ∈G k+1 and we choose the control u(t,v(t −), x 0 ) to guarantee the inclusions X − (,x 0 ) ∈ G k ∗ − 2Q  , then we have X  (x 0 )=e A x 0 −   0 e (−t)A u(t, v(t −), x 0 )dt +   0 e (−t)A v(t −)dt −   0 e (−t)A v(t −)dt +   0 e (−t)A v(t)dt ∈  G k ∗ − 2Q   + 2Q  ⊂G k . (19) A trajectory generated by the robust Pontryagin-Pshenichnyj -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 . From the qualitative point of view, the difference between the behaviours of the trajectories generated by the usual Pontryagin-Pshenichnyj -operator and the robust one can be explained as follows. The inclusion x 0 ∈F k+1 does not imply the inclusion X  (x 0 ) ∈F k . In general, we need much time than  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). 433 Linear Differential Games and High Precision Attitude Stabilization of Spacecrafts With Large Flexible Elements 12 Advances in Spacecraft Technologies −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 x 10 −3 −0.01 −0.008 −0.006 −0.004 −0.002 0 0.002 0.004 0.006 0.008 0.01 Fig. 3. Trajectory (x, ˙ x): motion generated by the robust Pontryagin-Pshenichnyj -operator. 6. High precision attitude stabilization of spacecrafts with large flexible elements Satellites with flexible appendages are modelled by hybrid systems of differential equations ¨ x = f (x,g(y, ˙ y, ¨ y),u), (20) ¨ y = G(x, ˙ x, ¨ x, y), (21) 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 L = 1 2 m ( ˙ X 2 0 + ˙ Y 2 0 )+ 1 2 I ˙ θ 2 + ρ 2  r 0 +l r 0  ( ˙ X 0 −( ˙ y + x ˙ θ)sin θ) 2 +( ˙ Y 0 +( ˙ y + x ˙ θ)cos θ) 2  dx 434 Advances in Spacecraft Technologies Linear Differential Games and High Precision Attitude Stabilization of Spacecrafts With Large Flexible Elements 13 Fig. 4. Satellite with a flexible appendage. − 1 2 EI  r 0 +l r 0 (y  ) 2 dx. Here m is the mass of the satellite, I is its moment of inertia about the longitudinal axis, ρ is the mass/unit length of the beam, EI is the bending stiffness of the beam. ’Dot’ is used to denote the derivatives with respect to time, while ’prime’ stands for the derivative with respect to x. The Lagrangian equations of free oscillations of the system have the form (m + lρ) ¨ X 0 − ρ 2 ((r 0 + l) 2 −r 2 0 )( ¨ θ sinθ + ˙ θ cosθ ) − ρ  r 0 +l r 0 ( ¨ ysin θ + ˙ θ ˙ ycos θ )dx = 0, (m + lρ) ¨ Y 0 + ρ 2 ((r 0 + l) 2 −r 2 0 )( ¨ θ cosθ − ˙ θ sinθ ) + ρ  r 0 +l r 0 ( ¨ ycos θ − ˙ θ ˙ ysin θ )dx = 0, I ¨ θ − ρ 2 ((r 0 + l) 2 −r 2 0 )( ¨ X 0 sinθ − ¨ Y 0 cosθ) + ρ  r 0 +l r 0 x( ¨ y + x ¨ θ) dx + ρ  r 0 +l r 0 ( ˙ X 0 ˙ ycos θ + ˙ Y 0 ˙ ysin θ )dx = 0, ρ (− ¨ X 0 sinθ + ¨ Y 0 cosθ − ˙ X 0 ˙ θ cosθ − ˙ Y 0 ˙ θ sinθ + ¨ y + x ¨ θ)+EIy  = 0. Linearizing these equations in the vicinity of the zero equilibrium position X 0 = Y 0 = θ = 0, y (·,·) ≡ 0, we get (m + lρ) ¨ X 0 = 0, (m + lρ) ¨ Y 0 + ρ 2 ((r 0 + l) 2 −r 2 0 ) ¨ θ + ρ  r 0 +l r 0 ¨ ydx = 0,  I + ρ 3 ((r 0 + l) 3 −r 3 0 )  ¨ θ + ρ 2 ((r 0 + l) 2 −r 2 0 ) ¨ Y 0 + ρ  r 0 +l r 0 x ¨ ydx = 0, ρx ¨ θ + ρ ¨ y + ρ ¨ Y 0 + EIy  = 0. 435 Linear Differential Games and High Precision Attitude Stabilization of Spacecrafts With Large Flexible Elements 14 Advances in Spacecraft Technologies Fig. 5. Transversal section of the satellite with a flexible appendage. 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: (m + lρ) ¨ X 0 = 0, (22) (m + lρ) ¨ Y 0 + ρ 2 ((r 0 + l) 2 −r 2 0 ) ¨ θ + ρ  r 0 +l r 0 ¨ ydx = 0, (23) ρ 2 ((r 0 + l) 2 −r 2 0 ) ¨ Y 0 +  I + ρ 3 ((r 0 + l) 3 −r 3 0 )  ¨ θ + ρ  r 0 +l r 0 x ¨ ydx = M, (24) ρ ¨ Y 0 + ρx ¨ θ + ρ ¨ y + EIy  + EIχ ˙ y  = 0, (25) where χ is the coefficient of internal viscous friction. Using the Galerkin method we approximate y (t,x) by a linear combination y (t,x)= ∑ q i (t)Φ i (x −r 0 ) (26) of eigenfunctions Φ i (x) of the differential operator d 4 /dx 4 with the boundary conditions Φ (0)=Φ  (0)=Φ  (l)=Φ  (l)=0. Substituting (26) to system (23) - (25), multiplying (25) by 436 Advances in Spacecraft Technologies Linear Differential Games and High Precision Attitude Stabilization of Spacecrafts With Large Flexible Elements 15 0 50 100 150 200 250 −4 −2 0 2 4 x 10 −4 Fig. 6. The disturbance v(t) caused by the flexible appendage. 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: y (t)=q(t)Φ(x −r 0 ), where Φ (x)=cosh(βx) −cos(βx) − cosh(βl)+cos(βl) sinh(βl)+sin(βl) ( sinh(βx) −sin(βx)), and β ≈ 1.875/l . Then from system (22) - (25) we obtain (m + lρ) ¨ X 0 = 0, (27) (m + lρ) ¨ Y 0 + ρ 2 ((r 0 + l) 2 −r 2 0 ) ¨ θ + ρJ 1 l ¨ q = 0, (28) ρ 2 ((r 0 + l) 2 −r 2 0 ) ¨ Y 0 +  I + ρ 3 ((r 0 + l) 3 −r 3 0 )  ¨ θ + ρ(J 2 l 2 + J 1 lr 0 ) ¨ q = M, (29) ρJ 1 l ¨ Y 0 + ρ(J 2 l 2 + J 1 lr 0 ) ¨ θ + ρJ 3 l ¨ q + EIβ 4 J 3 lq + EIχβ 4 J 3 l ˙ q = 0, (30) where J 1 = 0.7829, J 2 = 0.5688, J 3 = 0.9998. System (28) - (30) can be written in the matrix form as d 2 dt 2 ⎛ ⎝ Y 0 θ q ⎞ ⎠ = ⎛ ⎝ a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 ⎞ ⎠ −1 ⎛ ⎝ 0 M −EIβ 4 J 3 l(q + χ ˙ q) ⎞ ⎠ , where ⎛ ⎝ a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 ⎞ ⎠ = ⎛ ⎝ (m + lρ) ρ 2 ((r 0 + l) 2 −r 2 0 ) ρJ 1 l ρ 2 ((r 0 + l) 2 −r 2 0 )  I + ρ 3 ((r 0 + l) 3 −r 3 0 )  ρ (J 2 l 2 + J 1 lr 0 ) ρJ 1 l ρ(J 2 l 2 + J 1 lr 0 ) ρJ 3 l ⎞ ⎠ . 437 Linear Differential Games and High Precision Attitude Stabilization of Spacecrafts With Large Flexible Elements 16 Advances in Spacecraft Technologies 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 ¨ θ =  A −1  22 M − D  A −1  23 (q + χ ˙ q), (31) ¨ q =  A −1  32 M − D  A −1  33 (q + χ ˙ q), (32) where D = EIβ 4 J 3 l. In our numerical simulations we use a model example with the following values of parameters: m = 10, l = 10, r 0 = 3, ρ = 0.5, I = 45, EI = 3.5, and χ = 0.1 (SI units). 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) we use the linear stabilizer M = aθ + b ˙ θ (33) with the coefficients a and b determined from the condition of the maximum degree of stability of the closed-loop system ¨ θ =  A −1  22 (aθ + b ˙ θ) − D  A −1  23 (q + χ ˙ q), ¨ q =  A −1  32 (aθ + b ˙ θ) − D  A −1  33 (q + χ ˙ q). 438 Advances in Spacecraft Technologies [...]... spacecraft described in Section 4.1, and with the performance specifications defined in Section 4.2 460 Advances in Spacecraft Technologies Step A Input-Output pairing and loop ordering An illustrative result of the Relative Gain Array for all the uncertainty, at low frequency (steady state), and up to 0.19 rad/sec, is shown in Eq (26) According to it, the pairing should be done through the main diagonal of... Aeronautics and Astronautics, Washington DC [Krasovski & Subbotin, 1987] Krasovski, N.N.& Subbotin, A.I (1987) Game-Theoretical Control Problems, Springer-Verlag, New York [Kurzhanski & Melnikov, 2000] Kurzhanski, A & Melnikov, N (2000) On the Problem 20 442 Advances in Spacecraft Technologies Advances in Spacecraft Technologies of Control Synthesis: the Pontryagin Alternating Integral and the Hamilton-Jacobi... if there were independent input-output pairs In any case, it is necessary to quantify the amount of coupling present in the system Many of the MIMO design techniques, particularly the sequential ones, strongly depend on the correct selection and pairing of inputs and outputs at the beginning of the design procedure Determining the controller structure is also crucial This means deciding whether the... Robust Symmetric Attitude Controller for ETS-VIII Spacecraft, Control Engineering Practice (in press) [Polovinkin et al., 2001] Polovinkin, E.; Ivanov, G.; Balashov, M.; Konstantinov, R.; & Khorev, A (2001) An Algorithm for the Numerical Solution of Linear Differential Games, Sbornik: Mathematics, Vol 192, 1515-1542 [Pontryagin, 1981] Pontryagin, L (1981) Linear Differential Games of Pursuit, Mathematics... gains are required The interaction reduces the gain in the ij control loop: KOFF > KON 6 λij > 10 ⇒ Pairings of variables with large RGA values are undesirable They are sensitive to modelling errors and to small variations in the loop gain Given its importance, the RGA method has been the subject of multiple revisions and research For instance, although originally defined for the steady-state gain,... or when there are uncertain disturbances acting on the plant Model uncertainty, frequency domain specifications and desired time-domain responses translated into frequency domain tolerances, lead to the so-called Horowitz-Sidi bounds (or constraints) These bounds serve as a guide for shaping the nominal loop transfer function L(s) = G(s) P(s), which involves the selection of gain, poles and zeros to... end-point coordinates in brackets) Fig 3 Spacecraft description For every beam, two different frequencies for the first modes along Y and Z beam axes are considered Their frequency can vary from 0.05 Hz to 0.5 Hz, with a nominal value of 0.1 Hz, and their damping can vary from 0.1% to 1%, with a nominal value of 0.5% As regards spacecraft mass and inertia, the corresponding uncertainty around their nominal... value is of 5% Fig 4 Spacecraft model dynamics 458 Advances in Spacecraft Technologies Based on that description, and using a mechanical modeling formulation for multiple flexible appendages of a rigid body spacecraft, the open-loop transfer function matrix representation of the Flyer is given in (12) and Fig 4, where x, y, z are the position coordinates; ϕ, θ, ψ are the corresponding attitude angles;... closed, the gain between the input j and the output i increases, i.e., KON > KOFF 4 λij < 0 ⇒ At the closure of the remaining loops, the system gain changes its sign Providing a controller with negative gain for the normal situation (all the loops closed and working), the system will react in the opposite direction if some of the remaining loops are open for any reason Then, integrity is lost 5 λij >... several inputs in order to get outputs to behave as desired The first and easiest way that comes to mind for dealing with a MIMO system is to reduce it to a set of SISO problems ignoring the system interactions, which is the so-called decentralized control (Skogestad & Postlethwaite, 2005) Then, each input is responsible for only one output and the resulting controller is diagonal Finding a suitable input-output . 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 Attitude Controller for ETS-VIII Spacecraft, Control Engineering Practice (in press). [Polovinkin et al., 2001] Polovinkin, E.; Ivanov, G.; Balashov, M.; Konstantinov, R.; & Khorev, A. (2001) ⊂ S  + P. Note that the minimal invariant set may be not unique and that the intersection of two invariant sets may be not invariant. Indeed, consider the following example in R 2 . Let Λ = 1 2 I 2 ,