Lecture 15 Guidelines for Root Locus Summary Process Control - PowerPoint PPT Presentation

Lecture 15 Guidelines for Root Locus Summary Process Control Prof. Kannan M. Moudgalya IIT Bombay Monday, 26 August 2013 1/31 Process Control Root locus guidelines

Outline: Guidelines for Root Locus 1. Summary ◮ Real axis portion ◮ Symmetry ◮ Number of branches, starting and ending points ◮ Asymptotes ◮ Multiple roots ◮ Angles of arrival and departure 2. Imaginary axis intercept - implications 3. Example 2/31 Process Control Root locus guidelines

Examples Will use s + 0 . 5 G 1 (s) = s(s + 1)(s + 2)(s + 10) G 2 (s) = (s + 2) s(s + 1) 1 G 3 (s) = (s + 1)(s + 2)(s + 3) 3/31 Process Control Root locus guidelines

Recall: Root locus, a definition ◮ Root locus is the locus of roots of 1 + KG(s) = 0, as K goes from 0 to ∞ . 1 + KG(s) = 0 or KG(s) = − 1 ◮ Recall that we are checking this for an arbitrary point s ◮ We get the following magnitude and phase relations: | KG(s) | = 1 ∠ KG(s) = (2l + 1)180 ◦ , l = 0 , ± 1 , ± 2 , . . . 4/31 Process Control Root locus guidelines

1. Summary 5/31 Process Control Root locus guidelines

Real axis portion To the left of odd number of poles + zeros 6/31 Process Control Root locus guidelines

midsem-RL-4.pdf Evans root locus 6 asymptotic directions open loop poles open loop zeroes 4 2 Imaginary axis 0 - 2 - 4 r: - 0.005216 - 5.075 i K: 15.73 - 6 - 14 - 12 - 10 - 8 - 6 - 4 - 2 0 2 Real axis 7/31 Process Control Root locus guidelines

Root locus of real systems are Symmetric Im( s ) Im( s ) Re( s ) Re( s ) Im( s ) Re( s ) 8/31 Process Control Root locus guidelines

midsem-RL-4.pdf Evans root locus 6 asymptotic directions open loop poles open loop zeroes 4 2 Imaginary axis 0 - 2 - 4 r: - 0.005216 - 5.075 i K: 15.73 - 6 - 14 - 12 - 10 - 8 - 6 - 4 - 2 0 2 Real axis 9/31 Process Control Root locus guidelines

3. Number of branches, starting and ending points ◮ Number of branches = degree of the denominator polynomial (n > m) ◮ Branches of root locus start at open loop poles ◮ Branches of root locus terminate at open loop zeros 10/31 Process Control Root locus guidelines

midsem-RL-4.pdf Evans root locus 6 asymptotic directions open loop poles open loop zeroes 4 2 Imaginary axis 0 - 2 - 4 r: - 0.005216 - 5.075 i K: 15.73 - 6 - 14 - 12 - 10 - 8 - 6 - 4 - 2 0 2 Real axis 11/31 Process Control Root locus guidelines

Angles and real axis intercepts of asymptotes a(s) = s m + b 1 s m − 1 + · · · + b m − 1 s + b m Let G(s) = b(s) s n + a 1 s n − 1 + · · · + a n − 1 s + a n Angles: φ l = 2l + 1 n − m180 ◦ , l = 0 , 1 , . . . , n − m − 1 Asymptote intersects real axis at α , given by � p i − � z i = − a 1 + b 1 α = n − m n − m 12/31 Process Control Root locus guidelines

midsem-RL-4.pdf Evans root locus 6 asymptotic directions open loop poles open loop zeroes 4 2 Imaginary axis 0 - 2 - 4 r: - 0.005216 - 5.075 i K: 15.73 - 6 - 14 - 12 - 10 - 8 - 6 - 4 - 2 0 2 Real axis 13/31 Process Control Root locus guidelines

Multiple Roots ◮ G(s) = b(s) / a(s) ◮ Condition for multiple roots: adb(s) − bda(s) = 0 ds ds ◮ Applies only when s belongs to the root locus 14/31 Process Control Root locus guidelines

Example s + 2 ◮ G(s) = s(s + 1) ◮ K = 3 . 414 × 2 . 414 = 5 . 828, double root 1 . 414 15/31 Process Control Root locus guidelines

Double Roots Evans root locus 2.0 asymptotic directions open loop poles 1.5 open loop zeroes 1.0 0.5 Imaginary axis 0.0 - 0.5 - 1.0 - 1.5 - 2.0 16/31 Process Control Root locus guidelines - 4.5 - 4.0 - 3.5 - 3.0 - 2.5 - 2.0 - 1.5 - 1.0 - 0.5 0.0 0.5

3. Angles of arrival and departure 17/31 Process Control Root locus guidelines

Arrival, departure angles at double roots Evans root locus 2.0 asymptotic directions open loop poles 1.5 open loop zeroes 1.0 0.5 Imaginary axis r: - 3.414 + 3.918e- 08 i K: 5.828 0.0 - 0.5 - 1.0 - 1.5 - 2.0 - 4.5 - 4.0 - 3.5 - 3.0 - 2.5 - 2.0 - 1.5 - 1.0 - 0.5 0.0 0.5 Real axis 18/31 Process Control Root locus guidelines

Arrival and departure angles Explained using s + 2 G(s) = s(s + 1) ◮ Arrive at ± 90 ◦ ◮ Depart at ± 90 ◦ 19/31 Process Control Root locus guidelines

midsem-RL-4.pdf Evans root locus 6 asymptotic directions open loop poles open loop zeroes 4 2 Imaginary axis 0 - 2 - 4 r: - 0.005216 - 5.075 i K: 15.73 - 6 - 14 - 12 - 10 - 8 - 6 - 4 - 2 0 2 Real axis 20/31 Process Control Root locus guidelines

2. Imaginary axis intercepts 21/31 Process Control Root locus guidelines

Imaginary axis intercepts KG G cl = 1 + KG ◮ Root locus is the locus of roots of 1 + KG(s) = 0, as K goes from 0 to ∞ . ◮ Assume that the system is open loop stable ◮ For some value of K, suppose that the root locus crosses the imaginary axis ◮ Is there any relation with stability? ◮ How do we calculate the intercepts? 22/31 Process Control Root locus guidelines

Scilab code for root locus - 13-RL-2.sce s = %s; 1 2 G = s y s l i n ( ’ c ’ , . . . 1/( s +1) /( s +2) /( s +3) ) ; 3 evans (G, 2 0 0 ) ; 4 d a t a t i p T o g g l e 5 23/31 Process Control Root locus guidelines

Imaginary axis intercepts ◮ Substitute s = j ω ◮ Obtain two equations: real and imaginary ◮ Solve for ω and the corresponding K ◮ Routh-Hurwitz criterion 24/31 Process Control Root locus guidelines

Routh array Closed loop characteristic equation is given by a n s n + a n − 1 s n − 1 + · · · + a 1 s + a 0 = 0 Form Routh Array: a n a n − 2 a n − 4 . . . a 1 a n − 1 a n − 3 a n − 5 . . . a 0 b 1 b 2 b 3 . . . b 1 = a n − 1 a n − 2 − a n a n − 3 a n − 1 b 2 = a n − 1 a n − 4 − a n a n − 5 a n − 1 25/31 Process Control Root locus guidelines

Routh-Hurwitz Criterion a n a n − 2 a n − 4 . . . a 0 a n − 1 a n − 3 a n − 5 . . . a 1 b 1 b 2 b 3 . . . c 1 c 2 . . . c 1 = b 1 a n − 3 − a n − 1 b 2 b 1 c 2 = b 1 a n − 5 − a n − 1 b 3 b 1 Necessary and suff. conditions for stability: ◮ i.e. for all poles to have negative real part ◮ entries in first column of Routh Array > 0 26/31 Process Control Root locus guidelines

Guidelines Guidelines for plotting root locus plots are available 27/31 Process Control Root locus guidelines

Plot the root locus of s + 0 . 5 G 3 (s) = s(s + 1)(s + 2)(s + 10) 28/31 Process Control Root locus guidelines

What we learnt today 1. Summary ◮ Real axis portion ◮ Symmetry ◮ Number of branches, starting and ending points ◮ Asymptotes ◮ Multiple roots ◮ Angles of arrival and departure 2. Imaginary axis intercept - implications 3. Example 29/31 Process Control Root locus guidelines

Thank you 30/31 Process Control Root locus guidelines