problem4 - EE221A Linear System Theory Problem Set 4...

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EE221A Linear System Theory Problem Set 4 Professor C. Tomlin Department of Electrical Engineering and Computer Sciences, UC Berkeley Fall 2006 Issued 10/9; Due 10/18 Dear 221A folks: Some problems like the ones below may be included on the midterm on October 16. Thus, you may hand in your homework solutions to me on Monday October 15 and receive solutions to use in your midterm preparation. Problem 1: A linear time-invariant system. Consider a single-input, single-output, time invariant linear state equation ˙ x ( t ) = Ax ( t ) + bu ( t ) , x (0) = x 0 (1) y ( t ) = cx ( t ) (2) If the nominal input is a non-zero constant, u ( t ) = u , under what conditions does there exist a constant nominal solution x ( t ) = x 0 , for some x 0 ? Under what conditions is the corresponding nominal output zero? Under what conditions do there exist constant nominal solutions that satisfy y = u for all u ? Problem 2. Preservation of Eigenvalues under Similarity Transform. Consider a matrix A R n × n , and a non-singular matrix P R n × n . Show that the eigenvalues of A = PAP - 1
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