EE 3511 Automatic Control Systems Root Locus Dr

EE 3511: Automatic Control Systems Root Locus Dr. Ahmed Nassef EE 3511_L 11 Salman Bin Abdulaziz University 1

Learning Objectives l Understand the concept of root locus and its role in control system design l Recognize the role of root locus in parameter design and sensitivity analysis EE 3511_L 11 Salman Bin Abdulaziz University 2

Root locus l Location of the roots of the characteristic equation (Poles) determines systems response. l Modifying one or more of the system’s parameter cause the roots of the characteristic equation to change. l Root locus is a graphical method that describes the location of the poles as one parameter change. EE 3511_L 11 Salman Bin Abdulaziz University 3

Question E(s) Y(s) _ EE 3511_L 11 Salman Bin Abdulaziz University 4

Root Locus (Introduction) EE 3511_L 11 Salman Bin Abdulaziz University 5

Root Locus (Introduction) EE 3511_L 11 Salman Bin Abdulaziz University 6

Root Locus (Introduction) EE 3511_L 11 Salman Bin Abdulaziz University 7

Root Locus (Introduction) At K = 0, C. L. poles = O. L. poles At K ∞, C. L. poles = O. L. zeros EE 3511_L 11 Salman Bin Abdulaziz University 8

Root Locus (Introduction) • As K changes from 0 ∞ the root locus starts from O. L. poles and ends at O. L. zeros. • For a system with only one O. L. pole and no zeros, the loci approaches from that pole (as K=0) and reaches a zero at infinity along an asymptote. • If there are two poles, they will reach two zeros at infinity along two asymptotes. • So, in general the pole is looking for its zero EE 3511_L 11 Salman Bin Abdulaziz University 9

Root Locus (Asymptotes) • Number of asymptotes = n – m = (# O. L. poles) – (# O. L. zeros) • Angle of asymptotes, If n – m = 1 if n – m = 2 If n – m = 3 If n – m = 4 α = 180 α = +90, -90 α = +60, -60, 180 α = +45, -45, +135, -135 • Position of intersection of asymptotes EE 3511_L 11 Salman Bin Abdulaziz University 10

First-Order Example l The closed loop transfer function is which has a single, real pole at s = −(K + 1). l At K = 0, s = − 1 l At K= ∞, s = −∞ l So, as K increases, the locus starts from -1 and moves to the left towards ∞ along the real axis. EE 3511_L 11 Salman Bin Abdulaziz University 11

Root Locus (First-Order System Example 1) There is only one O. L. pole at -1 # of asymp. =n-m=1 -0=1 Angle of asymp. =180 o The pole is looking for a zero at infinity along one asymptote with angle 180 o EE 3511_L 11 Salman Bin Abdulaziz University 12

Root Locus (First-Order System Example 2) E(s) Y(s) _ Root Locus OL pole at 0 l OL zero at -2 l # of asymp. =n-m=1 -1=0 l So, as K increases, the loci starts from 0 and moves towards -2 along the real axis. l EE 3511_L 11 X – 2 Salman Bin Abdulaziz University 0 13

Root Locus (Second-Order System Example 1) E(s) Y(s) _ At K = 0, s 1=0 and s 2=− 2 (CL poles = OL poles). l At K= 1 s 1=-1 and s 2=− 1 l At K= ∞, s 1=-1+j∞ and s 2 = -1−j∞ (CL poles = zeros at ∞) l EE 3511_L 11 Salman Bin Abdulaziz University 14

Root Locus (Second-Order System Example 1) l for 0<K ≤ 1, we have two real poles located at l For K > 1, we have a pair of complex conjugate poles at: Root Locus EE 3511_L 11 Salman Bin Abdulaziz University X -2 X 0 15

Root Locus Example: Step Responses 1. 6 Amplitude 1. 4 K = 50. 0 K = 15. 0 1. 2 K = 2. 0 1 K=1 K = 1. 0 0. 8 0. 6 X K = 0. 5 – 2 0. 4 s X K=0 0. 2 0 0 jw K 2 4 6 8 10 Time (sec. ) OL poles at 0 and -2 l # of asymp. =n-m=2 -0=2 their angles are +90 and -90 l So, as K increases, the loci starts simultaneously from 0 and -2 and moves along the real axis until it breaks away at -1 to ±j∞. j∞ l EE 3511_L 11 Salman Bin Abdulaziz University 16

Root Locus (Second-Order System Example 2) OL poles at 0 and -3 l OL zero at -2 l # of asymp. =n-m=2 -1=1 l Asymp. angle is 180 o. l E(s) Y(s) _ As K increases, one part of the locus starts from the pole at 0 and ends at the a zero at -2 (along the real axis) l The other part starts at the pole at -3 and ends at a zero at -∞ along the asymptote. X – 3 – 2 X 0 Rule: A segment of real axis is a part of root locus if and only if it lies to the left of an odd number of poles and zeros. EE 3511_L 11 Salman Bin Abdulaziz University 17

Example (odd number rule) Double poles 6 No 5 Yes X X 3 2 1 Yes No Yes Complex poles or zeros do not affect the count EE 3511_L 11 4 No No X X 5 Yes 0 X 3 Yes X 2 No X 1 Yes Salman Bin Abdulaziz University 0 No 18

Example 1 l Draw Root Locus of the following system _ Segments of real axis X 3 n = 3, m = 0 EE 3511_L 11 Yes Salman Bin Abdulaziz University X X 2 1 0 No Yes No 19

Example 1 l # of asymptotes = n – m = 3 EE 3511_L 11 X X -2 -1 Salman Bin Abdulaziz University 60 X 20

Root Locus Procedure Breakaway points l Determine the breakaway points on the real axis (if any): l Breakaway points are points at which the root loci breakaway from real axis or the root loci return to real axis. l The locus breakaway from the real axis occurs where there is a multiplicity of roots. EE 3511_L 11 Salman Bin Abdulaziz University 21

Breakaway points l At breakaway points l Solve the above equation to determine the breakaway point. Select solution that are in segments of real axis that is part of root locus EE 3511_L 11 Salman Bin Abdulaziz University 22

Breakaway points l To find breakaway points X -2 EE 3511_L 11 Salman Bin Abdulaziz University X -1 X 23

Root locus of Example 1 EE 3511_L 11 X X -2 -1 60 X Salman Bin Abdulaziz University 24

Angle of departure and arrival l Determine the angle of locus departure from complex poles and the angle of locus arrival at complex zeros, using the phase criterion l Sum of angle contributions of poles and zeros (measured with standard reference) = 180+360 k EE 3511_L 11 Salman Bin Abdulaziz University 25

Example 2 EE 3511_L 11 Salman Bin Abdulaziz University 26

Departure Angles EE 3511_L 11 Salman Bin Abdulaziz University 27

Departure Angles EE 3511_L 11 Salman Bin Abdulaziz University 28

Root Locus Rules 1. Plot O. L. poles and zeros of the O. L. gain function; 2. Identify # asymptotes n-m Where, n = # O. L. poles and m = # O. L. zeros. 3. Determine the asymptotes angles from: 4. Determine the position of intersection along the real axis, if n – m > 1 from: 5. Use odd rule to sketch the loci along the real axis 6. Determine the break-away or break-in points along the real axis using: 7. Calculate the angle of departure or angle of arrival using angle condition. EE 3511_L 11 Salman Bin Abdulaziz University 29

Example 3 l Draw root locus of the following system _ EE 3511_L 11 Salman Bin Abdulaziz University 30

Example 3 -5 EE 3511_L 11 Salman Bin Abdulaziz University X -3 X -2 X -1 31

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Example 4 l How do we draw root locus for this system? _ Initial Step: Express the characteristics as EE 3511_L 11 Salman Bin Abdulaziz University 33

Example 4 Initial Step: Express the characteristics as Continue the Root Locus Procedure EE 3511_L 11 Salman Bin Abdulaziz University 34

l Poles = [-2. 3247 -0. 3376 +0. 5623 i -0. 3376 - 0. 5623 i] l Centroid = -3 l Angles = 60, -60, 180 l Intersection points =+-sqrt(2) EE 3511_L 11 Salman Bin Abdulaziz University 35

EE 3511_L 11 Salman Bin Abdulaziz University 36

Example 5 l How do we draw root locus for this system? _ EE 3511_L 11 Salman Bin Abdulaziz University 37

Example 5 Initial Step: Express the characteristics as EE 3511_L 11 Salman Bin Abdulaziz University 38

Example 5 β X +j -3 X -2 26. 6 o -1 X -j X +j -3 EE 3511_L 11 Salman Bin Abdulaziz University X -2 -1 X -j 39

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Examples of Root Locus sketches EE 3511_L 11 Salman Bin Abdulaziz University 41