Unformatted text preview: k = n1, C k Π k x = C k xC k C * k C k C * k C k x = 0, and Π k xx = C k x C k C * k C * k ∈ R ( C * k ) = ⇒ (Π k xx ) ⊥ N ( C k ), so Π k projects orthogonally into the nullspace of C k . Therefore ˆ a k = Π k (ˆ a k1a k ) + a k is the desired estimate. Exercise 3. Note that Φ( t,τ ) is always invertible. To steer the system from x to y on [ τ ,t ] ⊃ [ τ,t ], choose an input that steers the system from Φ( τ,τ ) x to Φ( t ,t )1 y on [ τ,t ]. As a counterexample when [ τ ,t ] ⊂ ( τ,t ), consider a controllable system for which B ( σ ) ≡ for σ ∈ [ τ ,t ]....
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 Fall '10
 ClaireTomlin
 Trigraph, Ck Πk, Ck Ck, Ck Ck ∗C

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