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Unformatted text preview: MEEN 651 Control System Design Homework 10: State Feedback and Observers Solutions Assigned: Tuesday, 4 Nov. 2008 Due: Tuesday, 11 Nov. 2008, 5:00 pm Problem 1) For the following problems do the following (by hand): a) determine stability (eigenvalues) b) determine controllability using the controllability matrix c) determine controllability using the HautusRosenbrock test d) if appropriate, determine controllability using the grammian; if this is not appropriate, explain why. ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ = ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − − = 1 1 4 1 5 B A , and ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ = ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ = 1 1 1 B A Solution: For System 1: Stable with eigenvalues of 5 and 4. Not controllable (one controllable mode, one uncontrollable mode). This can be determined from the controllability matrix: which has rank 1, or from the Hautus Rosenbrock test, which is rank deficient for . Since the system is stable, we can determine the controllability grammian as , which is also rank deficient. For System 2: Marginally stable with eigenvalues of {0, 0, 0}. Controllable, determined from the controllability matrix: which has rank 3, or from the Hautus Rosenbrock test, which is full rank for all . Since the system is not stable, it is not appropriate to compute the controllability grammian. Problem 2) Problem 7.20 Problem 7....
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This note was uploaded on 09/20/2009 for the course MEEN 651 taught by Professor Staff during the Fall '08 term at Texas A&M.
 Fall '08
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