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# hw5 - 16.31 Prof J P How T.A TBD Handout#5 Due 16.31...

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16.31 Handout #5 Prof. J. P. How October 19, 2007 T.A. TBD Due: October 26, 2007 16.31 Homework Assignment #5 1. A third order system with two inputs and two outputs has the familiar representation ˙ x ( t ) = A x ( t ) + B u ( t ) (1) y ( t ) = C x ( t ) (2) with G ( s ) = C ( sI A ) 1 B and A = 1 2 3 1 1 1 5 1 2 2 4 B = 1 1 0 0 1 1 C = 1 2 1 2 1 2 Suppose that the eigenstructure of A is given to be: T It has an eigenvalue at λ 1 = 1 0 1 and left = 1 with right eigenvector v 1 eigenvector w T 1 = 1 2 1 1 1 T 1 1 0 It has an eigenvalue at λ 2 = 2 with right eigenvector v 2 = and left eigenvector w T 2 = 1 2 1 1 1 T 0 1 1 It has an eigenvalue at λ 3 = 3 with right eigenvector v 3 = and left eigenvector w T 3 = 1 2 1 1 1 It is a fact that the 2 × 2 transfer function matrix G ( s ) has the form: 1 G ( s ) = R (3) s + 3 where R is a real constant 2 × 2 matrix (i.e. independent of s ).

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