hw3.09 - A = A T and For all x we have x T Ax ≥ 0. Recall...

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ECE521 Linear Systems Fall 2009 Homework 3: Due October 8, in class To the extent possible please type your homeworks. Note: You are free to use Matlab for calculations. However Matlab is not necessary for obtaining answers to any of the problems. Problem 1: Consider a square matrix A for which det( A ) = 0. Is it possible that det( e tA ) = 0 for some Fnite t > 0? Justify your answer. Problem 2: Solve ˙ x ( t ) = 1 0 - 1 0 1 0 1 0 - 1 x ( t ) + e - 2 t e - 2 t 0 , x (0) = 1 0 0 Is the solution bounded as t → ∞ ? Problem 3: Consider the system ˙ x = p 1 2 0 0 - 1 P x + p 1 2 - 1 1 1 2 P u ±ind u ( t ) which drives the system from state x (0) = [1 1] T to x (1) = [0 0] T . Problem 4: Consider the IVP ˙ x = p - 7 0 0 - 3 t 2 P x x (3) = (1 , 1) T Calculate x (2). Problem 5: Consider a linear system with A = 0 - 1 0 2 1 0 0 0 0 1 0 - 2 0 0 1 0 , B = - 1 0 1 0
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October 1, 2009 2 Determine the unreachable subspace. Problem 6: Recall that a matrix A is symmetrix semi-positive defnite iF
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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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This note was uploaded on 02/09/2010 for the course MAE 123 taught by Professor 123 during the Spring '10 term at École Normale Supérieure.

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hw3.09 - A = A T and For all x we have x T Ax ≥ 0. Recall...

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