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2. Consider the state equation x1 1 20 x1 d x2 = 0 10 x2 dt x3 001 x3 where x1, x2 and 23 are state variables. Please answer the following questions. (a) The state matrix (4) 1 20 A = 0 1 0 (5) 0 0 1 has three-fold eigenvalues with \₁ = = A2 A3 1. Find all independent eigenvectors corresponding to this eigenvalue. (b) Find the modal matrix M associated with the state matrix A. Does M-1 AM lead to a Jordan form or not? Hint: The modal matrix M turns out to be a diagonal matrix. For a diagonal matrix, its inverse is given by a 00 0b0 -1 1/a 0 0 = 0 1/b 0 00 с 0 0 1/c 1 (6) (c) Find the state transition matrix (t). (d) Determine the stability of the system. Please justify your answer.

2. Consider the state equation x1 1 20 x1 d x2 = 0 10 x2 dt x3 001 x3 where x1, x2 and 23 are state variables. Please answer the following questions. (a) The state matrix (4) 1 20 A = 0 1 0 (5) 0 0 1 has three-fold eigenvalues with \₁ = = A2 A3 1. Find all independent eigenvectors corresponding to this eigenvalue. (b) Find the modal matrix M associated with the state matrix A. Does M-1 AM lead to a Jordan form or not? Hint: The modal matrix M turns out to be a diagonal matrix. For a diagonal matrix, its inverse is given by a 00 0b0 -1 1/a 0 0 = 0 1/b 0 00 с 0 0 1/c 1 (6) (c) Find the state transition matrix (t). (d) Determine the stability of the system. Please justify your answer.

Elements Of Electromagnetics
Elements Of Electromagnetics
7th Edition
ISBN:9780190698614
Author:Sadiku, Matthew N. O.
Publisher:Sadiku, Matthew N. O.
ChapterMA: Math Assessment
Section: Chapter Questions
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2. Consider the state equation x1 1 20 x1 d x2 = 0 10 x2 dt x3 001 x3 where x1, x2 and 23 are state variables. Please answer the following questions. (a) The state matrix (4) 1 20 A = 0 1 0 (5) 0 0 1 has three-fold eigenvalues with \₁ = = A2 A3 1. Find all independent eigenvectors corresponding to this eigenvalue. (b) Find the modal matrix M associated with the state matrix A. Does M-1 AM lead to a Jordan form or not? Hint: The modal matrix M turns out to be a diagonal matrix. For a diagonal matrix, its inverse is given by a 00 0b0 -1 1/a 0 0 = 0 1/b 0 00 с 0 0 1/c 1 (6) (c) Find the state transition matrix (t). (d) Determine the stability of the system. Please justify your answer.
Transcribed Image Text:2. Consider the state equation x1 1 20 x1 d x2 = 0 10 x2 dt x3 001 x3 where x1, x2 and 23 are state variables. Please answer the following questions. (a) The state matrix (4) 1 20 A = 0 1 0 (5) 0 0 1 has three-fold eigenvalues with \₁ = = A2 A3 1. Find all independent eigenvectors corresponding to this eigenvalue. (b) Find the modal matrix M associated with the state matrix A. Does M-1 AM lead to a Jordan form or not? Hint: The modal matrix M turns out to be a diagonal matrix. For a diagonal matrix, its inverse is given by a 00 0b0 -1 1/a 0 0 = 0 1/b 0 00 с 0 0 1/c 1 (6) (c) Find the state transition matrix (t). (d) Determine the stability of the system. Please justify your answer.
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