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Unformatted text preview: EE221A Linear System Theory Problem Set 7 Professor C. Tomlin Department of Electrical Engineering and Computer Sciences, UC Berkeley Fall 2007 Issued 11/15; Due 11/27 Problem 1. Show that braceleftbiggbracketleftbigg A c bracketrightbigg , bracketleftbigg b bracketrightbiggbracerightbigg where A ∈ R n × n , b ∈ R n , c T ∈ R n is completely controllable iff (i) { A, b } is completely controllable and (ii) the matrix bracketleftbigg A b c bracketrightbigg is full rank. Problem 2: Grammians under Similarity Transforms. Consider the controllability and observability grammians W c , W o of a linear timeinvariant system ( A, B, C ) over the time period [0 , Δ]. Determine what happens to them under similarity transformations of the state space. That is, determine the controllability and observability grammians of ( TAT 1 , TB, CT 1 ). Prove that the eigenvalues of the product W c W o are constant under similarity transformations....
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This note was uploaded on 12/09/2011 for the course EE 221A taught by Professor Clairetomlin during the Fall '10 term at University of California, Berkeley.
 Fall '10
 ClaireTomlin
 Electrical Engineering

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