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PracticeProblemsMidterm_solu

# Solutions using the approximate formula 5 18 tr 2 n

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Unformatted text preview: ng speciﬁcations are satisﬁed: The rise time is less than 2 s, and the settling time is less than 3 s. Solutions: Using the approximate formula, 5 1.8 tr ⇠ 2 = !n ) 4.6 ts ⇠ 3 = ) !n 0.9, 1.5. 22. For the system in the previous problem, ﬁnd the range of control gain k for which the following speciﬁcations are satisﬁed: The rise time is less than 2 s, and the settling time is less than 3 s. Solutions: Using the approximate formula, 1.8 tr ⇠ 2 = !n ) 4.6 ts ⇠ 3 = ) !n 0.9, 1.5. Note that the system parameters are determined from the denominator of the transfer function: 2 Note that (s system parameters are determined s2 + 2s + k = the+ )2 + ! 2 = s2 + 2⇣!n s + !n .from the denominator of the transfer function: 2 s2 + 2s + k = (s + )2 + ! 2 = s2 + 2⇣!n s + !n . We then see that We then see 2 that = 1, k = !n . = 1, 2 k = !n . The rise time speciﬁcation is met by choosing The rise time speciﬁcation is met by choosing 2 k = !n > 0.92 = 0.81, 2 k = !n > 0.92 = 0.81, but the setting time s...
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