{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

problem7

# problem7 - (b What are the eigenvalues of e At(c Suppose...

This preview shows page 1. Sign up to view the full content.

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?
This is the end of the preview. Sign up to access the rest of the document.

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 jω-axis is included), and each of the jω-axis eigenvalues has a Jordan block of size 1. (b) Considering again problem 3(c), is the system internally stable? 1...
View Full Document

{[ snackBarMessage ]}