ps 5 604 - ECE 604 LINEAR SYSTEMS Problem Set #5 Issued:...

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ECE 604 LINEAR SYSTEMS Problem Set #5 Issued: Thursday, October 22, 2009 Due: Thursday, October 29, 2009 Problem 1: Compute W 0, T ( ) for the system ! x t ( ) = 0 ! " 0 # $ % & ( x t ( ) + 0 1 # $ % & ( u t ( ) Problem 2: Consider a harmonic oscillator with a drive u ( t ) that satisfies the equation: ! x t ( ) = 0 1 ! 1 0 " # $ % & x t ( ) + 0 1 " # $ % & u t ( ) Suppose we want to drive the system from the state 1 0 ! " # $ T at time t = 0 to the state 0 0 ! " # $ T at time t = 2 . a) Does there exist a control which accomplishes this? If so, compute it. b) Now suppose u t ( ) is restricted to be a piecewise constant function of time with the form: u t ( ) = u 1 0 ! t < 2 " u 2 2 ! t < 4 u 3 3 ! t ! 2 # $ % % & % % ; u 1 , u 2 , u 3 constant 3 -1 0;1 3 2;-1 1 2] Do there exist values for u 1 , u 2 and u 3 that accomplishes this transfer? Hint: Express the final state as a matrix multiplying for u 1 , u 2 and u 3 plus a term that depends on the initial conditions.
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ps 5 604 - ECE 604 LINEAR SYSTEMS Problem Set #5 Issued:...

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