On the Conditioning of Pole Assignment
On the Conditioning of Pole Assignment
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To quickly assess the difficulty of the text, read a short excerpt:
Changing basis with P yields the result. Q. E. D.
Now suppose the pair (A, 5) is controllable, and that the closest uncontrollable pair is (A^^J. Here we measure distance using the unitarily invariant absolute norm D.. - A -A + \B. -B\ llflllr (4) 2« ^«ll~^m •^«12 A„i3 ^. 21 A^22-'>^m-n-r ^u23 Bi A31 A32 Kii-K where ||I|;r denotes the Frobenius norm. Choose P and C as in (3) above to put the pendl [B \A - X7] into the indicated form. The next lemma describes the form of P^[B \A-\I]Q: Lemma 3: P...^[B \A->J]Q = Furthermore, only the last rows of B^, A^j and A^j axe nonzero. Note that only the three blocks at the bottom left differ iroir j;-3;.
Proof: Suppose, to the contrary, that feere were differences outside the three blocks B^, Aji and Aj2- ^^* would contradic! the definition of (iA„3„) as being the closest uncontrollable pencil to (A^) since by making P^[A^^^]Q agree with P'''[A\B]Q outside these three blocks we could find an uncontrollable pair even closer to (A^). Similarly, if B^, A31 and A32 had nonzero entries elsewhere than in their last rows, we could find an uncontrollable pair closer to (A^B) than (A, 5„) by making P'''[A^\BjQ agree with P^[A \B]Q outside the last rows of B^, A^^, and A32.
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You can find similar books in the "Read Also" column, or choose other free books by James Demmel to read onlineMoreLess
To quickly assess the difficulty of the text, read a short excerpt:
Changing basis with P yields the result. Q. E. D.
Now suppose the pair (A, 5) is controllable, and that the closest uncontrollable pair is (A^^J. Here we measure distance using the unitarily invariant absolute norm D.. - A -A + \B. -B\ llflllr (4) 2« ^«ll~^m •^«12 A„i3 ^. 21 A^22-'>^m-n-r ^u23 Bi A31 A32 Kii-K where ||I|;r denotes the Frobenius norm. Choose P and C as in (3) above to put the pendl [B \A - X7] into the indicated form. The next lemma describes the form of P^[B \A-\I]Q: Lemma 3: P...^[B \A->J]Q = Furthermore, only the last rows of B^, A^j and A^j axe nonzero. Note that only the three blocks at the bottom left differ iroir j;-3;.
Proof: Suppose, to the contrary, that feere were differences outside the three blocks B^, Aji and Aj2- ^^* would contradic! the definition of (iA„3„) as being the closest uncontrollable pencil to (A^) since by making P^[A^^^]Q agree with P'''[A\B]Q outside these three blocks we could find an uncontrollable pair even closer to (A^). Similarly, if B^, A31 and A32 had nonzero entries elsewhere than in their last rows, we could find an uncontrollable pair closer to (A^B) than (A, 5„) by making P'''[A^\BjQ agree with P^[A \B]Q outside the last rows of B^, A^^, and A32.
What to read after On the Conditioning of Pole Assignment?
You can find similar books in the "Read Also" column, or choose other free books by James Demmel to read onlineMoreLess
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