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Unformatted text preview: A = A T and For all x we have x T Ax ≥ 0. Recall also that the nullspace oF matrix A is the set oF all vectors x s.t. Ax = 0. Now consider the reachability Gramian M R ( t ,t f ,A,B ) = i t f t Φ( t 1 ,τ ) B ( τ ) B T ( τ )Φ T ( t 1 ,τ ) dτ Show that nullspace( M R ( t ,t f ,A,B )) ⊂ nullspace( M R ( t 1 ,t f ,A,B )) For all t 1 ∈ ( t , t f ). Problem 7: Consider the continuous system: ˙ x ( t ) = p 1 0 2 3 P x ( t ) + p b 1 b 2 P u ( t ) Under what conditions on [ b 1 b 2 ] T is the system reachable? Problem 8: Consider a pair ( A,B ) where A ∈ R n × n and B ∈ R n × k , k < n . Show that iF A is diagnolalizable and has an eigenvalues oF multiplicity greater than k then the pair is not reachable....
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 Spring '10
 123
 Linear Systems, ax

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