Unformatted text preview: EEL 3105 Fall 2011 Homework 6 Note: Problem 4 will be graded separately for assessing that one of the course objectives is met. Please submit solution to Problem 4 as a separate document. Consider matrices 1 3
A k 4
B b1 b2 0
C 0 c1 1.
1 c3 For A, choose k = 3. Find the eigenvalues of the resulting matrix. Are they real? Why? Find a formula for eigenvalues of A as a function of k. Suppose k is a design parameter which ranges from ‐100 to +100. Use the formula to plot the eigenvalues of A as a function of k in the complex plane. Use MATLAB to verify your answer. Such a plot is called root locus. Comment on your results. Find characteristic polynomials of B and C in terms of b1, b2, c1, c2, and c2. Do you see any pattern in your answer? Can you think of a way to generalize this to an nxn matrix? Fix b2=1. Suppose b1 be a design parameter that ranges from 0 to 100. It may represent the value of a physical component (e.g., mass or resistance). Find the eigenvalues of B as a function of b1 and plot them in the complex plane. Comment on your answer. Suppose for certain design considerations (having to do with transient response), we want the angle of the eigenvalues to be 3 / 4. Choose the value of the design parameter b1 to satisfy this constraint. Let A be an nxn matrix. Show that eigenvalues of A are the same as those of AT. Consider matrices 1 2 A 4 5 3 1 3 6 9
B 2 4 8 7. Calculate eigenvalues of AB and those of BA. You can use MATLAB to compute these answers. Please comment on your answers. Now suppose A and B are two matrices (not necessarily square) such that AB and BA are both well defined. Suppose is a non‐zero eigenvalue of AB. Show that it is also an eigenvalue of BA. Not for grading [hard problem for exploration and challenge]: 8. Let us return to Problem 3. Recall I told you in class that in order to ensure stability, all eigenvalues should have negative real parts. If we have numerical values for the parameters b1, b2, (or c1, c2, and c2), then we can use MATLAB to compute the eigenvalues and see if the real parts of eigenvalues are negative. But if they are unknown parameters, then this task is harder. This is especially so as the matrix size gets larger. How could you check if all eigenvalues have negative real parts in terms of the parameters? Explore the web resources to find an answer to this question. ...
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This note was uploaded on 12/27/2011 for the course EEL 3105 taught by Professor Boykins during the Fall '10 term at University of Florida.
- Fall '10