Lecture11

1 n i i1 i 2 1 180o i s pi root locus drawing

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Unformatted text preview: · · (An eφn ) B1 B2 · · · Bm , A1 A2 · · · An m θ= i=1 n ψi − φi . i=1 Phase condition: m s is on the root locus iff ∠L(s) = i=1 where ψi = ∠(s − zi ), 1.1 n ψi − i=1 φi = (2 − 1) × 180o φi = ∠(s − pi ). Root Locus Drawing Rules Matlab “rlocus” gives the root locus plot but knowing how to sketch it will help in design process. Rule 0: The root locus is symmetric about the real axis. This simply follows from the fact that the roots of a polynomial with real coefficients are real or complex conjugate pair. Rule 1: The root locus starts from n poles of L(s). m branches approach m zeros of L(s). n − m branches go to infinity. L(s ) = b (s ) , a (s ) 1 + KL(s) = 0 ⇔ a(s) + Kb(s) = 0. So K=0 ⇒ a (s ) = 0 K→∞ ⇒ ⇒ s = poles of L(s). b (s ) = 0 a (s ) → ∞ ⇒ ⇒ s = zeros of L(s) s→∞ (s + 1)2 + 4 Example 1.4 L( s ) = s2 + 1 Poles of L(s) are marked as × Zeros of L(s) are marked as ◦ (considering a(s)/K + b(s) = 0) Root Locus 3 2 1 0 −1 −2 −3 2 −1 −0.5 0 Rule 2: The portion of the real axis to the left of an odd number of poles and zeros is part of root locus. Plot the poles/zeros of L(s) that are on the real axis, and label them A, B, C, D, · · · from right to left. (if poles/zeros of L(s) are repeated, label them repeatedly.) Then the segments AB, CD, · · · are part of the root locus. s+1 s2 + 9 Phase condition: s ∈ root locus, then Example 1.5 L( s ) = Root Locus...
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