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# l8 - (b In each case high-order poles are about ten times...

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16.06 Lecture 8 Dominant Modes Karen Willcox September 18, 2003 Today’s Topics 1. Dominant mode concept 2. ”Invasion” of a first-order system 3. Examples of high-order systems Reading : 1.8, 4.4, ln 1

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1 Dominant mode concept Example 4.5.1 from text: Step response of a DC motor position servo. 0 . 5 (a) G ( s ) = s (0 . 25 s +1) is the transfer function from field voltage to shaft po- sition of the motor. Draw the closed-loop system using a proportional controller: C ( s ) = R ( s ) = Draw the pole-zero diagram for C ( s ) with K = 1: 2
(b) The PFE is: (c) Graphical residues: 3

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(d) Step response: (e) Plot each piece of the step response: (f) What happens if we increase K ? Type 1 system: Velocity error constant = . There is a conﬂict between 4
2 First-order system ”invaded” by a single pole Consider the following example, where T 2 varies. See Fig. 2.18 on the next page for response plots. Observations: at t = 0 slope of c ( t ) is zero for two-pole system and finite for one-pole system in case 2, the magnitude of the residue at -10 is 0.1 times the magnitude of the residue at -1 in case 3, the magnitudes of the residues at -1 and -2 are not widely different, so case 1 is not as good an approximation 5

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insert fig 2.18 here 6
3 Examples of high-order systems (a) Consider the following systems:

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Unformatted text preview: (b) In each case, high-order poles are about ten times removed from the dominant pole. Question: Are they important in each case? 7 (c) Consider G 1 ( s ): c 1 ( t ) = The residue of the complex pair contribution is small. (d) Consider G 2 ( s ): c 2 ( t ) = The residues of the faraway poles are comparable in magnitude to the residue of the pole at-1. But the residues of the two faraway poles have nearly equal and opposite magnitudes, therefore they cancel each other and the pole at-1 is dominant. In particular, the pole at-1 is a good approximation for t > 4 (2% from the ﬁnal value). (e) See Figure E2.2 on the next page. 8 Insert Fig. E2.2 here. 9 4 What is important? 5 F-8 example As part of your reading for this week, review the attached example on longi-tudinal response modes for the F-8 aircraft. 10...
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