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Unformatted text preview: Guaranteeing nonvanishing determinant LMI As a special case which contains all of the features we need to consider, suppose Δ is of the form Δ := braceleftBig diag [ δ 1 I t 1 , δ 2 I r 2 , Δ 3 ] : δ 1 ∈ R , δ 2 ∈ C , δ 3 ∈ C m 3 × m 3 bracerightBig (7.13) Also, let Φ (from the original robustness handout) be the unit ball of these, notated as Φ = B Δ := { Δ ∈ Δ : ¯ σ (Δ) ≤ 1 } For M ∈ C ( t 1 + r 2 + m 3 ) × ( t 1 + r 2 + m 3 ) , det ( I − M Δ) negationslash = 0 for all Δ ∈ B Δ if and only if the constraints z = Mw, Δ ∈ B Δ , w = Δ z imply that the complex vectors w and z actually are 0, namely w = z = 0. Since w and z are each derivable from the other through matrix multiplica tion, namely z = Mw, w = Δ z ,it is clear that w = 0 ⇔ z = 0. Therefore, we can write: det ( I − M Δ) negationslash = 0 for all Δ ∈ B Δ if and only if the constraints z = Mw, Δ ∈ B Δ , w = Δ z (7.14) actually imply that w = 0. ME 234, UC Berkeley, Fall 2008, Packard 123 Rewriting constraints Quadratic constraints Let z and w be vectors in C t 1 + r 2 + m 3 . Partition them as z = z 1 z 2 z 3 , w = w 1 w 2 w 3 Partition M and an identity of the same dimension as M = M 1 M 2 M 3 , I t 1 + r 2 + m 3 = E 1 E 2 E 3 ....
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This note was uploaded on 01/04/2012 for the course CDS 212 taught by Professor Tarraf,d during the Fall '08 term at Caltech.
 Fall '08
 Tarraf,D

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