09-10Chapter 6_2 Construction Rules(1)

# 09-10Chapter 6_2 Construction Rules(1) - Construction...

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Construction Rules-part 1 Principle of Automatic Control CHAPTER 6 oot Locus(P200 Ch 7) By Hui Hui Wang Wang Root Locus(P.200, Ch.7) 浙江大学控制科学与工程学系

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Construction Rules-part 1 Outline of Chapter 6 troduction 9 Introduction 9 Basic Concepts of Root Locus Method eometrical Properties (Construction 9 Geometrical Properties (Construction Rules)-part 1 eometrical Properties (Construction 9 Geometrical Properties (Construction Rules)-part 2 eneralized Root Locus 9 Generalized Root Locus 9 Performance characteristics 9 ……… 2
Geometrical Properties (Construction Rules)---part 1 控制科学与工程学系

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Construction Rules-part 1 ROOT LOCUS The root-locus method is a graphical technique for readily determining the location of all possible roots of a characteristic equation as the gain is varied from zero to finity infinity. We have obtained the magnitude and angle conditions r root locus(K>0 and K<0) how to plot the root locus for root locus(K>0 and K<0), how to plot the root locus readily? Can we calculate many points first like Ex.6-1 then plotting root locus? There is a procedure to sketch the root locus by hand involving nine steps, which will be developed in the following slides. 4
Construction Rules-part 1 ROOT LOCUS Construction Rules (K>0) ule 1: Number of branches of the locus and the symmetry Rule 1: Number of branches of the locus and the symmetry 0 ) ( ) ( 1 ) ( = + = Δ s H s G s ) ( ) ( ) ( ) ( ) ( ) ( 1 m w p s p s s z s z s K s H s G = L L The characteristic equation Δ (s) is of the degree n=m+u, (in theory, n=max(m+u, w), however, only m+u>w is realizable in real world); 1 u therefore, there are n roots. As the root locus gain K is varied from zero to infinity, each root traces continuous curve Then ots yield rves or branches in the S lane a continuous curve. Then n roots yield n curves or branches in the S-plane. Since the degree of the polynomial Δ (s) is determined by the Poles and Zeros of the open-loop transfer function, then we have p p, Conclusion: The number of branches of the root locus is equal to the aximum of oles nd Zeros f the open op transfer function 5 maximum of Poles and Zeros of the open-loop transfer function

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Construction Rules-part 1 ROOT LOCUS Construction Rules (K>0) The symmetry: Rule 1: Number of branches of the locus and the symmetry Since complex roots must occur in conjugate pairs, the cus is symmetrical about the real axis. locus is symmetrical about the real axis. xample : j ω Example : ) ( ) ( K s H s G = K=0 K=0 K α K ) ( a s s + -a σ K 6
Construction Rules-part 1 ROOT LOCUS Construction Rules (K>0) Rule 2: Real-Axis Locus We try to analyze which segments of real-axis are parts of root locus? For any search point such as s 1 on the real axis: he angular contribution of +j ω ¾ The angular contribution of all the Poles and Zeros on the real axis to the left of this point ( Φ 4 ) +j P 4 = σ 4 +j ω 4 is zero , i.e ¾ The angular contribution of e complex- njugate poles or 0 ) ( 1 = l j i p z s σ p 0 p 1 p 2 p 3 z 1 z 2 s 1 the complex conjugate poles or

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## This note was uploaded on 12/01/2011 for the course EE 400 taught by Professor Hui during the Spring '11 term at Zhejiang University.

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09-10Chapter 6_2 Construction Rules(1) - Construction...

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