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ps3sol

# ps3sol - MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department...

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Unformatted text preview: MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Aeronautics and Astronautics 16.060: Principles of Automatic Control Fall 2003 PROBLEM SET 3 Solutions Problem 1 1. 10( s + 2) s + 4 10 G ( s ) = s + 5 · 4 · s + 10 ( s + 2)( s + 4) = 25 ( s + 5)( s + 10) K rl = 25 2. 10(0 . 25 s + 1) 2(0 . 5 s + 1) G ( s ) = . 1 s + 1 · 5(0 . 2 s + 1) (0 . 5 s + 1)(0 . 25 s + 1) = 4 (0 . 1 s + 1)(0 . 2 s + 1) = 4 K 3. There are no integrators, so the type number of the system is 0. For a type system with unity feedback, the steady-state error for a unit step input is given by 1+ 1 K , where K is the “standard” 1 gain of the transfer function G ( s ). So the steady state error would be 1+4 = . 2. Problem 2 1. First write C ( s ) in root locus form, with R ( s ) = 1 : s 3 s 2 + 5 s + 6 4 1 C ( s ) = 2 · 6 · s 2 + 5 s + 4 · s ( s + 3)( s + 2) = s ( s + 4)( s + 1) So K rl = 1. The pole-zero plot has zeros at s = − 2 and s = − 3, and poles at s = 0, s = − 1, and s = − 4. The partial fraction expansion is: K 1 K 2 K 3 C ( s ) = + + s s + 1 s + 4 The residues are: 1 (2 0)(3 0) 3 3 K 1 = = = (1 0)(4 0) 2 2 (1 0)(2 0) 2 2 K 2 = (1 180)(3 0) = 3 − 180 = − 3 (2 180)(1 180) 1 1 K 3 = = = (4 180)(3 180) 6 6 3 / 2 2 / 3 1 / 6 C ( s ) = + s − s + 1 s + 4 Taking the inverse Laplace transform gives the output as a function of time (for t > 0): 3 2 1 2 − − 4 t − t c ( t ) = e + e 3 6 2. Using the same method as before: s + 1 s + 1 C ( s ) = 1 = s 2 − 25 · ( s + 5)( s − 5) K rl = 1 and there is a zero at s = − 1 and two poles at s = − 5 and s = 5. K 1 K 2 C ( s ) = + s + 5 s − 5 4 180 2 2 K 1 = = = 10 180 5 5 6 3 3 K 2 = = = 10 5 5 2 / 5 3 / 5 C ( s ) = s + 5 + s − 5 2 − 5 t 3 5 t ( t ) = e + e c 5 5 Note that the output is unbounded, because of the pole in the right-half plane. 3. 2 s + 4 1 2( s + 2) C ( s ) = = s 2 + 2 s + 2 · s s ( s 2 + 2 s + 2) K rl = 2 and there is a zero at s = − 2, and a complex conjugate pair of poles at s = . − 1 ± j K 1 K 2 K 3 C ( s ) = + + s s + 1 − j s + 1 + j ( 2 − 45)( 2 45)...
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ps3sol - MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department...

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