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

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

Nguyễn Công Phương CONTROL SYSTEM DESIGN The Root Locus Method
Contents I. Introduction II. Mathematical Models of Systems III. State Variable Models IV. Feedback Control System Characteristics V. The Performance of Feedback Control Systems VI. The Stability of Linear Feedback Systems VII.The Root Locus Method VIII.Frequency Response Methods IX. Stability in the Frequency Domain X. The Design of Feedback Control Systems XI. The Design of State Variable Feedback Systems XII. Robust Control Systems XIII.Digital Control Systems sites.google.com/site/ncpdhbkhn 2
The Root Locus Method 1. The Root Locus Concept 2. The Root Locus Procedure 3. Parameter Design by the Root Locus Method 4. Sensitivity and the Root Locus 5. PID Controllers 6. Negative Gain Root Locus 7. 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 o The 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 − 1 Ex. 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 Method 1. The Root Locus Concept 2. The Root Locus Procedure 3. Parameter Design by the Root Locus Method 4. Sensitivity and the Root Locus 5. PID Controllers 6. Negative Gain Root Locus 7. 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 M ∏ (s + p ) + K ∏ ∏ (s + p ) + ∏ ( s + z ) = 0 1 → j ( s + zi ) = 0 ↔ j i j =1 i =1 K j =1 i =1 n K =0 → ∏ (s + p ) = 0 j =1 j M K →∞ → ∏ (s + z ) = 0 j =1 j The locus of the roots of the characteristic equation 1 + KP(s) = 0 begins at the poles of P(s) and ends at the zeros of P(s) as K increases from zero to infinity. sites.google.com/site/ncpdhbkhn 9
The Root Locus Procedure (3) 1. Prepare the root locus sketch. 2. Locate the open – loop poles and zeros of P(s) in the s – plane with selected symbols: the root locus on the real axis always lies in a section of the real axis to the left of an odd number of poles and zeros 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 10
The Root Locus Procedure (4) Ex. 1 Step 2 2( s + 2) 1+ K =0 s ( s + 4) s1 + 2 s1 s2 −4 −2 s1 0 s1 + 4 The locus of the roots of the characteristic equation 1 + KP(s) = 0 begins at the poles of P(s) and ends at the zeros of P(s) as K increases from zero to infinity. - The number of separate loci is equal to the number of poles. - The root loci must be symmetrical with respect to the horizontal real axis. sites.google.com/site/ncpdhbkhn 11
The Root Locus Procedure (5) 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 12
The Root Locus Procedure (6) Step 3 M ∏ (s + z ) i =1 i 1 + KP ( s ) = 1 + K n = 0, 0 ≤ K ≤ ∞ ∏ (s + p ) j =1 j n M ∑ poles of P ( s ) − ∑ zeros of P ( s ) ∑ (− p ) − ∑ ( − z ) j =1 j i =1 i σA = = n−M n−M 2k + 1 φA = 180 o , k = 0,1, 2,..., ( n − M − 1) n−M sites.google.com/site/ncpdhbkhn 13
The Root Locus Procedure (7) Ex. 2 Step 3 ( s + 1) 1+ K =0 s ( s + 2)( s + 4) 2 σA = ∑ poles of P ( s ) − ∑ zeros of P ( s ) n−M [( −2) + ( −4) + ( −4)] − ( −1) = = −3 4 −1 2k + 1 −4 −2 −1 0 φA = 180o , k = 0,1, 2,..., (n − M − 1) n−M 2k + 1 = 180o = (2 k + 1)60o , k = 0,1, 2. 4 −1 k = 0 → φ A = 60 o k = 1 → φA = 180o k = 2 → φ A = 300o sites.google.com/site/ncpdhbkhn 14
The Root Locus Procedure (8) 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 15
The Root Locus Procedure (9) Step 5 ∆(s ) = 0 → p(s ) = K dp( s ) breakaway point : =0 ds Ex. 3 ( s + 1) 1+ K =0 s ( s + 2)( s + 3) − s( s + 2)( s + 3) →K = = p( s ) s +1 dp d − s ( s + 2)( s + 3) 2 s 3 + 8 s 2 + 10 s + 6 → = = ds dt s +1 ( s + 1) 2 dp = 0 → 2 s 3 + 8s 2 + 10s + 6 = 0 → s1 = −2.46 , s2,3 = −0.77 ± j 0.79 ds sites.google.com/site/ncpdhbkhn 16
The Root Locus Procedure (10) 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: The angle of locus departure from a pole is the difference between the net angle due to all other poles and zeros and the criterion angle of ±180°(2k + 1). 7. Complete the root locus sketch. sites.google.com/site/ncpdhbkhn 17
The Root Locus Procedure (11) Step 6 θ1 The angle of locus departure from a pole is the difference between the θ3 net angle due to all other poles and 0 zeros and the criterion angle of ±180°(2k + 1). θ2 θ1 + θ2 + θ3 = 180o → θ1 = 180o − (θ 2 + θ3 ) sites.google.com/site/ncpdhbkhn 18
The Root Locus Procedure (12) 5 Ex. 2 K 4 p1 1+ 4 =0 s + 12 s 3 + 64 s 2 + 128s 3 K →1+ =0 2 s( s + 4)( s + 4 + j 4)( s + 4 − j 4) p1 = −4 + j 4 1 p2 = −4 − j 4 0 p3 p4 p3 = − 4 -1 p4 = 0 -2 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 3. The loci proceed to the zeros at infinity along asymptotes centered at σA and with -4 p2 angle ϕA. 4. Determine the points at which the locus -5 crosses the imaginary axis (if it does so). -7 -6 -5 -4 -3 -2 -1 0 1 ∑ poles of P( s) − ∑ zeros of P ( s ) 5. Determine the breakaway point on the real axis (if any). 6. Determine the angle of locus departure σA = from complex poles and the angle of locus n−M arrival at complex zeros, using the phase criterion. −4 − 4 − j 4 − 4 + j 4 = = −3 7. Complete the root locus sketch. 4−0 sites.google.com/site/ncpdhbkhn 19
The Root Locus Procedure (13) 5 Ex. 2 K 4 p1 1+ 4 =0 s + 12 s 3 + 64 s 2 + 128s 3 2k + 1 2 φA = 180o , k = 0,1, 2,..., (n − M − 1) n−M 1 2k + 1 0 = 180o , k = 0,1, 2, 3 p3 p4 4 −0 -1 -2 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 3. The loci proceed to the zeros at infinity along asymptotes centered at σA and with -4 p2 angle ϕA. 4. Determine the points at which the locus -5 crosses the imaginary axis (if it does so). -7 -6 -5 -4 -3 -2 -1 0 1 5. Determine the breakaway point on the real axis (if any). k = 0 → φ A = 45o 6. Determine the angle of locus departure from complex poles and the angle of locus k = 1 → φ A = 135o arrival at complex zeros, using the phase criterion. k = 2 → φA = 225o 7. Complete the root locus sketch. sites.google.com/site/ncpdhbkhn k = 3 → φ A = 315o 20