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problem7 - EE221A Linear System Theory Problem Set 7...

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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 0 c 0 bracketrightbigg , bracketleftbigg b 0 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 0 bracketrightbigg is full rank. Problem 2: Grammians under Similarity Transforms. Consider the controllability and observability grammians W c , W o of a linear time-invariant 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. Problem 3: RL-RC circuit example. Consider the network shown in Figure 1 with voltage v ( t ) as input
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