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# hw5 - Homework 5 MAE143B Spring 2010 due Thursday May 13...

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Homework 5 - MAE143B Spring 2010: due Thursday May 13 Question 1: State reachability Part i: Consider the linear system with x t IR n , ˙ x t = Ax t + Bu t . For T > 0 , define the controllability Gramian as the n × n matrix W c (0 , T ) = Z T 0 e - BB T e - A T τ dτ. Suppose that W c (0 , T ) is non-singular. Show that the control function u t = - B T e - A T t W - 1 c (0 , T x i , transfers the state of the system from initial state x 0 = ¯ x i to final state x T = 0 . Part ii: Under the same assumption that W c (0 , T ) is nonsingular, show that the control function u ( t ) = B T e - A T t W - 1 c (0 , T ) e - AT ¯ x f , transfers the state of the system from initial state x 0 = 0 to final state x T = ¯ x f . Part iii: Using the above two control functions devise two different control strategies to transfer the state from initial state x 0 = x i to final state ¯ x f . One control strategy should take you via x T = 0 to x 2 T = x f and the other should take you directly to x T = x f .

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