09-10Chapter 6_4 Generalized Root Locus

09-10Chapter 6_4 Generalized Root Locus - Generalized Root...

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Generalized Root Locus 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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Generalized Root Locus 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 9 Performance characteristics 9 ……… 2009-12-1 2
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eneralized Root Locus Generalized Root Locus oot locus of positive 1. Root locus of positive feedback system (or K<0) arameter root locus 2. Parameter root locus 3. Root locus of Non- inimum phase system minimum phase system 4. Transport Lag 控制科学与工程学系 2009-12-1
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Generalized Root Locus ROOT LOCUS: Generalized root locus 1 )] ( ) ( [ ) ( ) ( ) ( ) ( ) ( ) ( 1 1 1 = = = s H s G K p s p s s z s z s K s H s G u m w L L 1. Root locus of positive feedback system (or K<0) or the open op transfer function G(s)H(s) the characteristic For the open-loop transfer function G(s)H(s), the characteristic equation of positive feedback (or for the negative feedback system K<0 ) is as following 0 ) ( ) ( 1 = s H s G 1 ) ( ) ( = s H s G The magnitude condition and the angle condition for positive gg p feedback system are 1 ) ( ) ( = s H s G ---- magnitude condition ) , 2 , 1 , 0 ( 360 0 ) ( ) ( L = ° + ° = k k s H s G ---- angle condition
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Generalized Root Locus OOT LOCUS: eneralized root locus ROOT LOCUS: Generalized root locus 1. Root locus of positive feedback system (or K<0) Compare the root locus of positive feedback system with it of the negative feedback system, the difference of them is only the angle condition i t i fd bk ) , 2 , 1 , 0 ( 360 0 ) ( ) ( L = ° + ° = k k s H s G Positive feedback ) , 2 , 1 , 0 ( 360 180 ) ( ) ( L = ° + ° = k k s H s G Negative feedback Note: Thus: The root locus of positive feedback system is called 0º root locus or negative parameter(K<0) root locus. The root locus of negative feedback system is called 180º root locus. ecause the angle condition is different the rules of root locus which is Because the angle condition is different, the rules of root locus which is correlative with the angle condition is also different. 2009-12-1 5
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Generalized Root Locus Review Rule 1: Number of branches of the locus and the symmetry Rule 2: Real-Axis Locus Rule 3: Locus End Points Rule 4: Asymptotes of locus ule 5: Breakaway Point on the Real Axis Rule 5: Breakaway Point on the Real Axis Rule 6: Complex Pole (or Zero): Angle of Departure (Approach) ule 7: Im xis Crossing Point Rule 7: Im-Axis Crossing Point Rule 8: Intersection or Nonintersection of Root-Locus Branches Rule 9:Conservation of the sum of the System Roots 2009-12-1 6
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Generalized Root Locus OOT LOCUS: eneralized root locus ROOT LOCUS: Generalized root locus 1. Root locus of positive feedback system (or K<0) if the total number of real poles and zeros to the right of a hi t th l i i i i t li th l Rule 2: Real-Axis Locus Rule 4: Asymptotes of locus as s approaches infinity k = π γ 2 search point s on the real axis is even, this point lies on the locus w n Rule 6:
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09-10Chapter 6_4 Generalized Root Locus - Generalized Root...

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