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Advances in Spacecraft Technologies Part 12

Tham khảo tài liệu 'advances in spacecraft technologies part 12', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 430 Advances in Spacecraft Technologies Now consider the problem of a minimal invariant set construction. Let Pc Rn and Q c Rn be convex compact sets and let A Rn Rn be a linear operator. The condition of a convex set S invariance AS QcS P in terms of support functions takes the form S f AS S f Q S f S S f P forall f II fll 1. 13 14 We say that an invariant set S is minimal if for any S c S S S we have AS Q c 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 R2. Let A 112 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 Sa x ax I x c 2 2 a c 1 1 is minimal invariant. The intersection Sa1 L Sa2 0 1 2 is not invariant. To restrict the set of invariant sets we introduce the following definition. Put r S min r 0 I S c rBn . An invariant set S is said to be r-minimal if for any S satisfying r S r S we have AS Q c 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 P co 0 l 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 S1 2B2 and S1 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 S0 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 Sq described by a vector M 0 m 0 . m 0 and set Sq co m1 0 -1 1 . m q - 1 l . Let Ỗ 0. 1 L 1 L The current invariant set Sf is successively shrunk going through the vectors 1 l 1 L and considering the sets el __ Í I . 0L 1x I . 0 . e 1x I . 0L-1i 1 Sk co P1 1 . B1 ỗ 1 . ừL ỈLỊ . If the set Sf is