ps5sol - MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department...

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- ± ± ± ²³ MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Aeronautics and Astronautics 16.060: Principles of Automatic Control Fall 2003 PROBLEM SET 5 Solutions Problem 1 11.9(a) De±ne the states as shown below: D R ² + ² 5 X 1 ² + ² 2 X 2 ² C s +1 + ²³ s +2 X 10 ³ s +10 3 The input, output, and state vectors are: ± ² x 1 r u = y = c ³ x 2 x = d x 3 From the block diagram, we can write down the state equations and the output equation: x ˙ 1 = 5( r x 3 ) x 1 = x 1 5 x 3 + r x ˙ 2 = 2( x 1 + d ) 2 x 2 = 2 x 1 2 x 2 + 2 d x ˙ 3 = 10 x 2 10 x 3 y = x 2 In matrix form: 1 0 5 1 0 ˙ 0 2 x = ³ 2 2 x + ³ 0 u 0 10 10 0 0 ´ ´ y = 0 1 0 u x + 0 0 1
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± ² ± ± ± ± ± ± ± ± ± ± ± ³ ± ± ± ± 11.9(b) Defne the states as shown below: R 1 X 2 + ´ 2 X 1 ´ 1 ´ ´ s +1 s +2 ²³ C 1 + ´ 0 . 5 s +6 X 3 ´ 1 s +10 X 4 + R 2 ´ 4 X 5 ´ 1 X 6 ´ ´ s s +4 + ²³ C 2 The input, output, and state vectors are: ² x 1 ³ x 2 ± ± ³ ³ x 3 r 1 c 1 ³ u = x = y = ³ x 4 r 2 c 2 ³ ´ x 5 x 6 From the block diagram, we can write down the state equations and the output equation: x ˙ 1 = 2 r 1 x 1 x ˙ 2 = x 1 2 x 2 x ˙ 3 = 6 x 3 + 0 . 5 r 1 x ˙ 4 = 10 x 4 + r 2 x
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ps5sol - MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department...

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