l10 - 16.06 Lecture 10 The Effect of Zeroes Karen Willcox...

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Unformatted text preview: 16.06 Lecture 10 The Effect of Zeroes Karen Willcox September 25, 2003 Today’s Topics 1. Zero near a dominant quadratic mode 2. General observations on the effect of a zero 3. Other examples of the effects of zeroes Reading: 5.3, l.n. 1 1 Zero near a dominant quadratic mode This situation usually results when we add a PD controller. 2 s + z1 ωn G(s) = 2 + 2ζω s + ω 2 z1 s n n c(t) = If A ≈ z1 and α ≈ 0, then c(t) is the standard second-order response. A far away zero has negligible effect. As the zero is moved to the right, it has a greater effect as follows: 2 Tp = P.O. = Example: • • 3 2 General observations on the effect of a zero Given G(s) = 4 s+a a (s+1)(s+4) (a) Step response c(t) = Note: (b) Observations: (i) (ii) 4 (iii) (iv) 3 Other examples of the effects of zeroes • Another way to look at the effect of a zero Suppose we have the closed-loop transfer function C (s) N (s) = R(s) D(s) Consider a single zero on the real axis C (s) = R(s) and a step input C (s) = Then the time response can be written as c(t) = c1 (t) + c2 (t) c1 (t) = c2 (t) = 5 Now c2 (t) is the c1 (t) is the c1 (t) is the c(t) = • Discussion of attached figures. 6 ...
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l10 - 16.06 Lecture 10 The Effect of Zeroes Karen Willcox...

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