problem7 - (b) What are the eigenvalues of e At ? (c)...

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EE221A Linear System Theory Problem Set 7 Professor C. Tomlin Department of Electrical Engineering and Computer Sciences, UC Berkeley Fall 2010 Issued 11/9; Due 11/16 Problem 1. A has characteristic polynomial ( s - λ 1 ) 5 ( s - λ 2 ) 3 , it has four linearly independent eigenvectors, the largest Jordan block associated to λ 1 is of dimension 2, the largest Jordan block associated to λ 2 is of dimension 3. Write down the Jordan form J of this matrix and write down cos( e A ) explicitly. Problem 2. A matrix A R 6 × 6 has minimal polynomial s 3 . Give bounds on the rank of A . Problem 3: Jordan Canonical Form. Given A = - 3 1 0 0 0 0 0 0 - 3 1 0 0 0 0 0 0 - 3 0 0 0 0 0 0 0 - 4 1 0 0 0 0 0 0 - 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (a) What are the eigenvalues of A ? How many linearly independent eigenvectors does A have? How many generalized eigenvectors?
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Unformatted text preview: (b) What are the eigenvalues of e At ? (c) Suppose this matrix A were the dynamic matrix of a system to be controlled. Is the system internally asymptotically stable? Problem 4: Characterization of Internal (State Space) Stability for LTI systems. (a) Show that the system x = Ax is internally stable if all of the eigenvalues of A are in the closed left half of the complex plane (closed means that the j-axis is included), and each of the j-axis eigenvalues has a Jordan block of size 1. (b) Considering again problem 3(c), is the system internally stable? 1...
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This note was uploaded on 04/11/2011 for the course EE 221A taught by Professor Clairetomlin during the Fall '10 term at University of California, Berkeley.

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