09-10Chapter 6_5 Performance characteristics(1)

09-10Chapter 6_5 Performance characteristics(1) -...

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Performance characteristics-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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Performance characteristics-part 1 绘制根轨迹(K>0)的方法小结(1) ± 根轨迹的起止:起于开环极点,终于开环零点或无穷远点 ± 根轨迹的分支数: n>w 时,等于开环极点数;当 w>n 时,等于开环零点数 ± 根轨迹的对称性:关于实轴对称 ± 实轴上的根轨迹:当右面的开环零极点之和为奇数的部分 轨迹的渐近线 >w - ± 根轨迹的渐近线: n>w 时,共有( n w )条: 与实轴的夹角为 交点为 w n k ° + ± = 180 ) 1 2 ( γ z p w j i n i i = = 1 1 ) Re( ) Re( ± 分离点与会合点(必是 l 重根) w n = 0 σ 确定,且与实轴成 角度离开(会合) 0 )] ( [ = ds s K d l ° ± = 180 θ 2
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Performance characteristics-part 1 绘制根轨迹(K>0)的方法小结(2) ± 与虚轴的交点:由 Routh 判据求得,或直接将 s=j ω 代入特征方程 求出特征根 ± 出射角与入射角 ° n w 80 自复极点的 p k 的出射角 = = + ± = k i i i k j j k p p p z p k 1 1 ) ( ) ( 180 φ 至复零点的 z k 的入射角 = = + ° ± = w k j j j k n i i k z z z p z k 1 1 ) ( ) ( 180 ψ 注意:根轨迹是一种几何图解法:绘制出根轨迹后,任一点 s 1 K K 1 都可由幅值定理求出 : n = = + + = + + + + + + = w j j i i w n z s p s z s z s z s p s p s p s K 1 1 1 1 1 2 1 1 1 1 2 1 1 1 1 L L 3
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Performance characteristics-part 1 Outline of Chapter 6 troduction 9 Introduction 9 Basic Concepts of Root Locus Method 9 Geometrical Properties (Construction Rules) 9 Generalized Root Locus erformance characteristics art 1 9 Performance characteristics-part 1 9 Performance characteristics-part 2 9 ……… 4
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Performance characteristics 1. General Introduction 2. Plot of characteristic roots for 0< ζ <1 3. Variation of Roots with ζ >=0 4. Higher-Order Systems 5. Synthesis 6. Additional Poles and Zeros 7. Controller Synthesis 控制科学与工程学系
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Performance characteristics-part 1 OOT LOCUS ROOT LOCUS Performance characteristics 1. General Introduction Review: consider a simple second-order system 2 ω R(s) C(s) 2 2 2 ) ( ) ( n n s s K s R s C ζω + + = ) 2 ( n n s s + The characteristic roots and step response for ζ <1 2 1,2 1 nn d sj j ζ ωω σω =− ± ) sin( ) ( φ σ + = t Ae t c d t σ >=0 the response has sustained oscillation or unstable σ <0 the response is stable 6
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Performance characteristics-part 1 j ω d OOT LOCUS 2 3 4 ROOT LOCUS Performance characteristics 1. General Introduction d n n j j s ω σ ζ ζω ± = ± = 2 2 , 1 1 See P117, Table 4.1 σ 1 2* 3* 4* 5 Position of pole ζ Form of response Characteristics =1 amped exponential t e σ 1 ζ >=1 Damped exponential 2-2* 0< ζ <1 Exponentially damped sinusoid Ae ) φ ω sin( σ + t Ae d t 3-3* ζ =0 Constant-sinusoid xponentially increasing sinusoid ) φ ω sin( + t A d n( e 4-4* -1< ζ <0
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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_5 Performance characteristics(1) -...

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