hw5 - x where L is the Laplace operator 2 d dt Φ t = Ae tA...

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J.M. Gon¸ calves Fall 2003 CDS 212 - Introduction to Modern Control Homework # 5 Date Given: November 6th, 2003 Date Due: November 13th, 2003 P1. DFT: Chapter 7, Exercise 1. P2. DFT: Chapter 7, Exercise 2. P3. DFT: Chapter 7, Exercise 3. P4. Consider a linear system with zero input, i.e. ˙ x = Ax and initial condition x (0) = x 0 . Show that: 1. x ( t ) = L - 1 { ( sI - A ) - 1 } x 0 = e tA x 0 , Φ( t
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Unformatted text preview: ) x where L is the Laplace operator. 2. d dt Φ( t ) = Ae tA = e tA A 3. ( e tA )-1 = e-tA 4. Evaluate R t e τ A dτ . What happens when A is singular? 5. Evaluate d dt { e-tA x ( t ) } . In combination with ˙ x = Ax + Bu prove that x ( t ) = e tA x + Z t e ( t-τ ) A Bu ( τ ) dτ...
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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.

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