hw10s - MEEN 651 Control System Design Homework 10 State...

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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 Hautus-Rosenbrock test d) if appropriate, determine controllability using the grammian; if this is not appropriate, explain why. = = 1 1 4 0 1 5 B A , and = = 1 0 0 0 0 0 1 0 0 0 1 0 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.
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