Lecture Control system design: The Root Locus method - Nguyễn Công Phương

In this chapter the following content will be discussed: The root locus concept, the root locus procedure, parameter design by the root locus method, sensitivity and the root locus, pid controllers, negative gain root locus, the root locus using control design software.
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Lecture Control system design: The Root Locus method - Nguyễn Công Phương Nguyễn Công PhươngCONTROL SYSTEM DESIGN The Root Locus Method ContentsI. IntroductionII. Mathematical Models of SystemsIII. State Variable ModelsIV. Feedback Control System CharacteristicsV. The Performance of Feedback Control SystemsVI. The Stability of Linear Feedback SystemsVII.The Root Locus MethodVIII.Frequency Response MethodsIX. Stability in the Frequency DomainX. The Design of Feedback Control SystemsXI. The Design of State Variable Feedback SystemsXII. Robust Control SystemsXIII.Digital Control Systems sites.google.com/site/ncpdhbkhn 2 The Root Locus Method1. The Root Locus Concept2. The Root Locus Procedure3. Parameter Design by the Root Locus Method4. Sensitivity and the Root Locus5. PID Controllers6. Negative Gain Root Locus7. The Root Locus Using Control Design Software sites.google.com/site/ncpdhbkhn 3 The Root Locus Concept (1) R( s ) Y ( s) Y (s) KG ( s) K G(s ) T (s) = = R( s) 1 + KG ( s) (−) 1 + KG ( s ) = 0 → KG ( s ) = −1 → KG ( s) ∠[ KG ( s )] = 1∠180 o  KG ( s) = 1 → ∠[ KG ( s)] = 180 + k 360 ; k = 0, ± 1, ± 2,... o oThe root locus is the path of the roots of the characteristic equation traced out in the s-plane as a system parameter varies from zero to infinity. sites.google.com/site/ncpdhbkhn 4 The Root Locus Concept (2)Ex. 1 R( s ) 1 Y ( s) K s( s + 2) 1 + KG ( s ) = 1 + K =0 (−) s ( s + 2) → ∆ ( s ) = s 2 + 2 s + K = s 2 + 2ζωn s + ωn2 = 0 → s1,2 = −ζωn ± ωn ζ 2 − 1 = −1 ± ωn ζ 2 − 1Ex. 2 R( s ) 1 Y ( s) 10 s( s + a ) (−) sites.google.com/site/ncpdhbkhn 5 The Root Locus Concept (3) N ∆( s ) = 1 − ∑L n =1 n + ∑n ,m Ln Lm − ∑ n ,m , p Ln Lm L p + ... nontouching nontouching = 1 + F (s) ∆ ( s ) = 0 → F ( s ) = −1 K ( s + z1 )( s + z2 )( s + z3 )...( s + z M ) F (s) = ( s + p1 )( s + p2 )( s + p3 )...( s + pn )  K s + z1 s + z2 ...  F ( s ) = =1→ s + p1 s + p2 ...  ∠F ( s ) = [∠( s + z1 ) + ∠( s + z2 ) + ...] − [∠( s + p1 ) + ∠( s + p2 ) + ...] = 180 + k 360 o o sites.google.com/site/ncpdhbkhn 6 The Root Locus Method1. The Root Locus Concept2. The Root Locus Procedure3. Parameter Design by the Root Locus Method4. Sensitivity and the Root Locus5. PID Controllers6. Negative Gain Root Locus7. The Root Locus Using Control Design Software sites.google.com/site/ncpdhbkhn 7 The Root Locus Procedure (1)1. Prepare the root locus sketch.2. Locate the open – loop poles and zeros of P(s) in the s – plane with selected symbols.3. The loci proceed to the zeros at infinity along asymptotes centered at σA and with angle ϕA.4. Determine the points at which the locus crosses the imaginary axis (if it does so).5. Determine the breakaway point on the real axis (if any).6. Determine the angle of locus departure from complex poles and the angle of locus arrival at complex zeros, using the phase criterion.7. Complete the root locus sketch. sites.google.com/site/ncpdhbkhn 8 The Root Locus Procedure (2) Step 1 1 + F (s) = 0 → 1 + KP ( s ) = 0, 0 ≤ K ≤ ∞ M ∏ (s + z ) i =1 i →1+ K n =0 ∏ (s + p ) j =1 j n M n ...