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ps4sol - MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department...

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| | MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Aeronautics and Astronautics 16.060: Principles of Automatic Control Fall 2003 PROBLEM SET 4 Solutions Problem 1 Step responses A and D clearly exhibit a “long tail” due to a combination of a slow, real-valued pole and a nearby zero, so they must correspond to transfer functions (3) and (4). To tell which is which, we need to find the sign of the residue for the real-valued pole in the step responses for (3) and (4). In (3), there will be a positive vector from the zero combined with a negative vector from the step input pole at s = 0, so the residue will be negative. That implies that the mode due to the slow pole will be subtracted from the steady-state value, and so the output will approach its steady-state value from below. Therefore, step response D corresponds to transfer function (3). Similarly, the residue for the real-valued pole in the step response for (4) will be positive, so the output will approach steady-state from above, so step response A must correspond to transfer function (4). The other two step responses are easy: as the zero moves towards the right, peak time decreases and P.O. increases. Therefore B corresponds to (2), and C corresponds to (1). Problem 2 1. 1 1 0 1 0 A = = = | | 1 A 1 1 2 1 2 1 | B | = 2 B 1 1 8 3 4 1 . 5 = = 2 2 1 1 0 . 5 | C | = 0 C is singular, so C 1 does not exist. 2. 2 1 2 1 2 2 A = (1) (4) + (0) | | 0 1 1 1 1 0 = (1)( 2) (4)( 3) = 10 Since A = 0, the matrix must have full rank, so A has rank = 3.
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