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sol07 - MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of...

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MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Mechanical Engineering 2.004 Dynamics and Control II Fall 2007 Problem Set #7 Solution Posted: Friday, Nov. 2, ’07 1. Nise problem 5 from chapter 8, page 476. Answer: The open loop transfer function is K ( s + 1)( s + 2) G ( s ) = . ( s + 5)( s + 6) Root Locus −8 −7 −6 −5 −4 −3 −2 −1 0 1 Real Axis To find the break–in and breakaway points, we apply rule 6 as follows: −3 −2 −1 0 1 2 3 Imaginary Axis ( σ + 5)( σ + 6) σ 2 + 11 σ + 30 K ( σ ) = = . ( σ + 1)( σ + 2) σ 2 + 3 σ + 2 Taking the derivative, dK 8 σ 2 56 σ 68 2 σ 2 14 σ 17 = = 4 . ( σ 2 + 3 σ + 2) 2 ( σ 2 + 3 σ + 2) 2 and setting dK/dσ = 0, we find σ 1 = 1 . 5635 and σ 2 = 5 . 4365. 2. Nise problem 7 from chapter 8, page 477. Answer: From rule 3 about the real–axis segment, we know that the root locus should exist between the two zeros in the right–hand plane as well as the pole and zero in the left–hand plane. Next step is to deal with the two poles with 1

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0 1 Imaginary Axis imaginary parts. The root locus should depart from the two poles. It cannot go to the infinity because we already have the same number of poles and zeros. Therefore it has to be joined one of the two real–axis segment. The root locus departing from the two poles eventually should arrive at the two zeros in the right–hand plane. Here two possibilities arise: 1) The root locus directly is joined to the real–axis segment between the two zeros in right–hand plane, or 2) It first cuts through the real–axis segment between the pole and zero in the left–hand plane, and comes back to the right–hand plane to join the real–axis segment between the two zeros. Which option turns out to be the actual root locus depends on the relative location of the open–loop poles and zeros. The two plots below show two examples obtained using Matlab . Root Locus Root Locus 3 6 4 2 2 Imaginary Axis 0 −1 −2 −2 −4 −3 −6 −5 −4 −3 −2 −1 0 1 2 3 4 5 −25 −20 −15 −10 −5 0 5 Real Axis Real Axis 3. Nise problem 9 from chapter 8, page 477.
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sol07 - MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of...

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