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ps 5 604 soln

# ps 5 604 soln - ECE 604 STATE VARIABLE ANALYSIS Fall 2009...

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ECE 604 STATE VARIABLE ANALYSIS Fall, 2009 Solution to Problem Set #5 Problem 1: (Homework) The transition matrix for the system (simple harmonic oscillator) is: ! t ,0 ( ) = e A t = cos " t sin " t # sin " t cos " t \$ % & ( ) Then ! t ,0 ( ) B = cos " t sin " t # sin " t cos " t \$ % & ( ) 0 1 \$ % & ( ) = sin " t cos " t \$ % & ( ) Thus, W 0, T ( ) = ! 0, t ( ) BB T ! T 0, t ( ) dt 0 T " = # sin \$ t cos \$ t % & ( ) * # sin \$ t cos \$ t % & ( ) dt 0 T " = sin 2 \$ t # sin \$ t cos \$ t # sin \$ t cos \$ t cos 2 \$ t % & ( ) * * dt 0 T " = 1 2 T # 1 \$ cos \$ t sin \$ t + , - . / 0 # 1 2 \$ sin 2 \$ t # 1 2 \$ sin 2 \$ t 1 2 T # 1 \$ cos \$ t sin \$ t + , - . / 0 % & ( ) * * * * * Problem 2: (Homework) a) The system is controllable: ! W c ( ) = ! b Ab " # \$ % ( ) = ! 0 1 1 0 " # & \$ % ( ) * + , - = 2 Thus there exists a control that drives any state (including x 0 = 1 0 ! " # \$ T ) to the origin at t f = 2 ! . One such control is: u t ( ) = ! b T e ! A T t W 0,2 " ( ) ! 1 x 0 The transition matrix and controllability grammian were computed in problem 5 for arbitrary ! . With ! = 1 (this problem):

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e ! A T t = cos t sin t ! sin t cos t " # \$ % & b e ! A T t = 0 1 " # % & cos t sin t ! sin t cos t " # \$ % & = ! sin t cos t " # % & W 0,2 ( ( ) = ( 0 0 ( " # \$ % & ) W 0,2 ( ( ) ! 1 = 1 ( I The control is: u t ( ) = ! sin t cos t " # \$ % 1 & I ( ) * + , 1 0 " # - \$ % . = ! 1 & sin t b) From the variation of constants formula: x 2 ! ( ) = 0 = e A 2 ! ( ) x 0 + e A 2 ! " # ( ) b u # ( ) d # 0 2 ! \$ With the piecewise constant input, this becomes: ! 1 0 " # \$ % & = sin 2 ( ! ) ( ) cos 2 ( ! ) ( ) " # \$ \$ % & d ) 0 2 ( 3 * u 1 + sin 2 ( ! ) ( ) cos 2 ( ! ) ( ) " # \$ \$ % & d ) 2 ( 3 4 ( 3 * u 2 + sin 2 ( ! ) ( ) cos 2 ( ! ) ( ) " # \$ \$ % & d ) 4 ( 3 2 ( * u 3 The integrals are: sin 2 ! " # ( ) cos 2 ! " # ( ) \$ % & & ( ) ) d # a b * = " sin # cos # \$ % & ( ) d # a b * = cos # sin # \$ % & ( ) a b = cos b " cos a sin b " sin b \$ % &
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ps 5 604 soln - ECE 604 STATE VARIABLE ANALYSIS Fall 2009...

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