09-10Chapter 6_3 Construction Rules(2)

09-10Chapter 6_3 Construction Rules(2) - Construction...

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Unformatted text preview: Construction Rules-part 2 Principle of Automatic Control CHAPTER 6 CHAPTER 6 oot Locus(P200 Ch 7) oot Locus(P200 Ch 7) By By Hui Hui Wang Wang Root Locus(P.200, Ch.7) Root Locus(P.200, Ch.7) 浙江大学控制科学与工程学系 Construction Rules-part 2 Outline of Chapter 6 troduction 9 Introduction 9 Basic Concepts of Root Locus Method 9 Geometrical Properties (Construction Rules)-part 1 9 Geometrical Properties (Construction Rules)- part 2 9 Generalized Root Locus 9 Performance characteristics 9 ……… 2 Geometrical Properties (Construction Rules)-part 2 控制科学与工程学系 Construction Rules-part 2 Review • Magnitude and angle conditions for root locus(K>0) • 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 • Rule 5: Breakaway point on the real axis ¡ Up to now, we just talk about Poles and Zeros of open-loop transfer function on real axis, how about complex Poles and Zeros? ¡ Are there root locus construction rules for them? 4 Construction Rules-part 2 ROOT LOCUS Construction Rules (K>0) Rule 6: Complex Pole (or Zero): Angle of Departure (Approach) Problem: The direction in which the locus leaves a complex Pole or ters complex ero? enters a complex Zero? j ω p 1 j ω ? K ↑ ψ 1A ?? z 1 p 2 Φ 2D ?? K ↑ σ σ p 1 ψ 2A ?? p 2 z 2 p 3 Φ 3D ?? Angle of Approach (入射角) Angle of Departure (出射角) 5 Construction Rules-part 2 ROOT LOCUS Construction Rules (K>0) Method: Rule 6: Complex Pole (or Zero): Angle of Departure (Approach) Determine the departure angles at the Poles and the arrival angles at the Zeros . Take a test point s p near each ole nd se ngle ndition mpute e eparture Pole and use angle condition to compute the departure angle. Take another test point s z near each Zero and use angle condition to compute the arrival angle. 6 Construction Rules-part 2 ROOT LOCUS Construction Rules (K>0) Assumption: the open-loop system with four Poles and one Zero Rule 6: Complex Pole (or Zero): Angle of Departure (Approach) j ω Φ 2D p 2 j ω p 2 Φ 2D Φ Φ 1 Ψ 1 ( l 1 ) 1 l l 1 S-plane Φ Φ Ψ ( l 1 ) 1 l l 1 l 2 σ p p 1 z 1 l 3 90º σ p p 1 z 1 1 1 l 3 p 3 Φ 3 =90 p 3 Φ 3 =90º l 2 is very much smaller than l , l 1 , l 3 and ( l 1 ) 1 7 Construction Rules-part 2 ROOT LOCUS 1 ) ( ) ( = s H s G Magnitude condition: Construction Rules (K>0) K , 2 , 1 , 180 ) 2 1 ( ) ( ) ( ± ± = ° + = ∠ h h s H s G Angle condition: pplying the angle condition to this small area: Rule 6: Complex Pole (or Zero): Angle of Departure (Approach) Applying the angle condition to this small area: ° + = − + + + 180 ) 2 1 ( 1 3 2 1 h ψ φ φ φ φ j ω p 2 ( ) 1 1 2 90 180 ) 2 1 ( ψ φ φ φ − ° + + − ° + = h D S-plane Φ 2D ( l 1 ) 1 l l 2 Generally, the angle of departure is w n σ p p 1 z 1 Φ Φ 1 Ψ 1 l 1 1 1 (1 2 )180 ( ) ( ) k p k j k i j i i k h p z p p φ = = ≠ = + ° + ∠ − − ∠ − ∑ ∑ p Φ 3 =90º l 3 3 8 Construction Rules-part 2...
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09-10Chapter 6_3 Construction Rules(2) - Construction...

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