Root Locus (2A
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The Original Transfer Function G(s = N (s D(s R(s G(s Y (s = sm + b 1 s m 1 + + b m s n + a 1 s n 1 + + a n = (s (s z 2 (s z m (s p 1 (s p 2 (s p n Root Locus (2A 3
The New Transfer Function KG(s 1+ KG(s = K N (s/ D(s 1+ K N (s/ D(s R(s + G(s Y (s = K N (s D(s+K N (s K D(s + K N (s = 0 Root Locus (2A 4
Conditions for new roots (1 Find roots s D(s + K N (s = 0 Conditions for a root s 0 (K = 0 ( = 0 0 + 0 N (s = 0 (K 0 ( 0 ( D( S 0 ( 0 (K = N (s 0 = real 0 N (s 0 = real + ( N (s N (s 0 = 0 0 Root Locus (2A 5
Conditions for new roots (2 Find roots s D(s + K N (s = 0 N (s 0 = 0 0 ( no common root + K N (s 0 = 0 cannot be a root N (s 0 0 for a root s 0 K = 0 + 0 N (s 0 = 0 = 0 : original poles K 0 + K N (s 0 = 0 0 = K N (s 0 0 K = N (s 0 0 Root Locus (2A 6
Check for new roots Find roots s D(s + K N (s = 0 = 0 K = 0 + K N (s 0 = 0 s 0 is on the root locus of the pair (N, D 0 N (s 0 = real K = N (s 0 s 0 is on the root locus of the pair (N, D + K N (s 0 = 0 K > 0 1 0 G(s 0 N (s 0 > 0 positive root locus of the pair (N, D K < 0 1 0 G(s 0 N (s 0 < 0 negative root locus of the pair (N, D Root Locus (2A 7
Rule: Uniqueness of K either s 0 is not on the root locus for any s 0 or s 0 is on the root locus then there is a unique K + K N (s 0 = 0 Root Locus (2A 8
Rule: Real number condition of K 0 N (s 0 = real K = N (s 0 s 0 + K N (s 0 = 0 is on the root locus of the pair (N, D { arg N (s } 0 D(s = q π 0 N (s 0 = 1 K K > 0 K < 0 Arg { N ( s 0 D( s 0 } = π Arg { N ( s 0 D( s 0 } = 0 arg { N (s 0 } = arg { (s 0 (s 0 z 2 (s 0 z m (s 0 p 1 (s 0 p 2 (s 0 p n } = arg(s 0 + arg(s 0 z 2 + + arg(s 0 z m [arg(s 0 p 1 + arg(s 0 p 2 + + arg(s 0 p n ] arg(s 0 + arg( s 0 z 2 + + arg( s 0 z m [arg(s 0 p 1 + arg(s 0 p 2 + + arg(s 0 p n ] = (2 q+1 π K > 0 positive root locus (2 qπ K < 0 negative root locus Root Locus (2A 9
Number of Roots D(s + K N (s = 0 n roots G(s = N (s D(s = sm + b 1 s m 1 + + b m s n + a 1 s n 1 + + a n = (s (s z 2 (s z m (s p 1 (s p 2 (s p n K = 1 n-1 roots 1 root at infinity n-2 roots 2 root at infinity Root Locus (2A 10
Roots when K=0 K = 0 + 0 N (s 0 = 0 = 0 The roots of D(s : The original poles G(s = N (s D(s = sm + b 1 s m 1 + + b m s n + a 1 s n 1 + + a n = (s (s z 2 (s z m (s p 1 (s p 2 (s p n Root Locus (2A 11
Roots on real axis (1 positive root locus s 0 p 0 z 0 p 1 z 1 p 2 { arg N (s } 0 D(s = (2q+1π 0 (s 0 (s 0 p 2 (s 0 p 1 (s 0 z 0 (s 0 p 0 Arg(s 0 p 1 = 0 Arg(s 0 z 0 = 0 Arg(s 0 p 0 = 0 Arg(s 0 = π Arg(s 0 p 2 = π Arg { N (s 0 } = Arg (s 0 z 0 + Arg(s 0 [ Arg(s 0 p 0 + Arg(s 0 p 1 + Arg (s 0 p 2 ] = 0 (2 q+1π Not on the positive root locus Root Locus (2A 12
Roots on real axis (2 positive root locus s 0 p 0 z 0 p 1 z 1 p 2 { arg N (s } 0 D(s = (2q+1π 0 (s 0 p 1 (s 0 (s 0 p 2 (s 0 z 0 (s 0 p 0 Arg(s 0 z 0 = 0 Arg(s 0 p 0 = 0 Arg(s 0 p 1 = π Arg(s 0 = π Arg(s 0 = π Arg { N (s 0 } = Arg (s 0 z 0 + Arg(s 0 [ Arg(s 0 p 0 + Arg(s 0 p 1 + Arg (s 0 p 2 ] = π = (2 q+1π on the positive root locus Root Locus (2A 13
Roots when K (1 K D(s + K N (s K N (s N (s = 0 The roots of N(s : The original zeros G(s = N (s D(s = sm + b 1 s m 1 + + b m s n + a 1 s n 1 + + a n K = D(s N (s = sn + a 1 s n 1 + + a n s m + b 1 s m 1 + + b m s + (a 1 b 1 s 1 + = s [ 1 + a 1 b 1 s + ] s m + b 1 s m 1 + + b m s n + a 1 s n 1 + + a n s n + b 1 s n 1 + (a 1 b 1 s n 1 + Root Locus (2A 14
Roots when K (2 K = D(s N ( s = sn + a 1 s n 1 + + a n s m + b 1 s m 1 + + b m [ = s 1 + a 1 b 1 s + ] = s [ 1 + a 1 b 1 s + ] 1 [ ( K s 1 + a b ] 1 1 s 1 1 [ ( K s 1 + a 1 b 1 ] ( s (1+x k 1+k x (x 1 K s 0 1 e j ( π+2 q π 1 K e j ( π+2qπ λ = a 1 b 1 s + a 1 b 1 a 1 b 1 centroid (K e j π 1 s + a 1 b 1 2 π Root Locus (2A 15
Centroid (1 G(s = N (s D(s = sm + b 1 s m 1 + + b m b 1 = s n + a 1 s n 1 i + + a n = (s (s z 2 (s z m (s p 1 (s p 2 (s p n a 1 = j z i p j = i = j R{z i } R{p j } s 0 1 K e j ( π+2qπ a 1 b 1 λ = a 1 b 1 = p j z i = R {p j } R{z i } Centroid θ = (2q+1π q = 0, 1,, +1 Angle of asymptotes Root Locus (2A 16
Centroid (2 G(s = N (s D(s = (s (s z 2 (s z m (s p 1 (s p 2 (s p n λ = a 1 b 1 = p j z i = R {p j } R{z i } Centroid G(s = N (s D(s = (s+z 1 (s+z 2 (s+ z m (s+ p 1 (s+ p 2 (s+ p n λ = + a 1 b 1 = p j + z i = R {p j }+ R{z i } Centroid Root Locus (2A 17
Angle of departure from a pole arg(s 0 + arg( s 0 z 2 + + arg( s 0 z m [arg(s 0 p 1 + arg(s 0 p 2 + + arg(s 0 p n ] = (2 q+1 π K > 0 positive root locus s 0 = p 1 + ϵe jθ d arg( p 1 + ϵe j θ d z1 + arg( p 1 + ϵ e jθ d z2 + + arg( p 1 + ϵe jθ d zm [arg( p 1 + ϵe j θ d p 1 + arg( p 1 + ϵe j θ d p 2 + + arg( p 1 + ϵ e j θ d p n ] p 1 arg( p 1 + arg( p 1 z 2 + + arg( p 1 z m [arg(ϵ e jθ d + arg( p1 p 2 + + arg( p 1 p n ] = (2q+1 π arg( p 1 + arg( p 1 z 2 + + arg( p 1 z m [θ d + arg( p 1 p 2 + + arg( p 1 p n ] = (2q+1π π + arg( p 1 + arg( p 1 z 2 + + arg( p 1 z m [arg(p 1 p 2 + + arg( p 1 p n ] = θ d Root Locus (2A 18
Angle of arrival from a zero arg(s 0 + arg( s 0 z 2 + + arg( s 0 z m [arg(s 0 p 1 + arg(s 0 p 2 + + arg(s 0 p n ] = (2 q+1 π K > 0 positive root locus s 0 = z 1 + ϵe jθ a arg(z 1 + ϵe j θ a z1 + arg(z 1 + ϵe j θ a z2 + + arg( z 1 + ϵ e jθ a zm [arg( z 1 + ϵe jθ a p 1 + arg( z 1 + ϵ e jθ a p 2 + + arg(z 1 + ϵe j θ a p n ] z 1 arg(ϵe j θ a + arg( z1 z 2 + + arg(z 1 z m [arg(z 1 p 1 + arg(z 1 p 2 + + arg(z 1 p n ] = (2q+1π θ a + arg(z 1 z 2 + + arg( z 1 z m [arg(z 1 p 1 + arg(z 1 p 2 + + arg(z 1 p n ] = (2q+1π π [arg(z 1 z 2 + + arg( z 1 z m ] + [arg( z 1 p 1 + arg(z 1 p 2 + + arg( z 1 p n ] = θ a Root Locus (2A 19
Breakaway Points (Repeated Roots D(s + K N ( s = ( s s 1 r (s s 2 [ D (s + K N (s] s=s0 = 0 [ d d s ( D(s + K N (s ]s=s 0 = 0 [ d d s D (s + K d d s N (s ]s=s 0 = 0 d d s D(s 0 d N (s 0 d s N (s 0 = 0 N (s 0 d d s D (s 0 D (s 0 d d s N (s 0 = 0 Check this for repeated roots N (s d d D (s D (s d s d s N (s = 0 d d s K = d [ d s D (s ] N (s = 0 Root Locus (2A 20
A p 4 ϵ p 4 + ϵ e jθ d s 0 = p 4 + ϵ e jθ d p 3 p 1 p 4 p 2 arg(s 0 + arg(s 0 z 2 + + arg(s 0 z m [arg(s 0 s p 1 + arg(s 0 p 2 + + arg(s 0 p n ] = (2 q+1 π K > 0 positive root locus = [arg(s 0 s p 1 + arg(s 0 p 2 + + arg(s 0 p n ] = Root Locus (2A 21
A Root Locus (2A 22
References [1] http://en.wikipedia.org/ [2]