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Unformatted text preview: 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 > , define the controllability Gramian as the n n matrix W c (0 ,T ) = Z T e- A BB T e- A T d. 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 = 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 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 = 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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This note was uploaded on 05/27/2010 for the course MAE MAE143B taught by Professor Prof.bitmead during the Spring '10 term at UCSD.
- Spring '10