Root Locus
This lecture we will learn
– What is root locus
– How to sketch root-locus
– How to determine the closed loop poles via root locus
– How to use root locus to describe the transient response, and stability of a system as a system parameter is varied
Root Locus : Usage
Root Locus : A graphical representation of the closed loop poles as a system parameter varies.
– Root locus can describe the performance of the system as varius parameters are changed.
– The effects of gains on the system response, overshoot and the stability can be determined.
Root Locus : Definition
Consider the system
How does the roots of the characteristic equation in s-plane change as the gain is varied from to .
A „locus“ of these roots plotted in s-plane as a function of is called the Root Locus
Root Locus : Construction
For the given system the closed loop transfer function is
characteristic equation
That is
Magnitude criteria
Angle criteria
Root Locus : Construction
Consider
Magnitude criteria
Angle criteria
Root Locus : Construction Rules
Root Locus Rules
Rule #1: Loci starts at the open loop poles;
Rule #2: Loci terminates at the open-loop zeros (including those at infinity);
Rule #3: There will be as many separate loci as the largest number of finite open loop poles or zeros. For the majority of systems, the number of finite open loop poles are greater than the finite number of open loop zeros.
Rule #4: The root loci are symmetrical with respect to the real axis
Root Locus : Construction Rules
Rule #5: The root loci may be found on portions of the real axis to the left of an odd number of open loop poles and zeros.
Rule #6: The asymptotes intersect real axis at a point given by
Rule #7 : The root loci are asymptotic to straight lines, for large values of s, with angles given by
number of finite open loop of poles number of finite open loop of zeros
Centroid formula
Root Locus : Construction Rules
Let relative degree (RD)
the centroids are marked x above
Rule #8 : The point on the real axis at which the loci brakes away or breaks into the real axis can be calculated as ;
Rule #9 : The angles of departure and arrival can be computed using the angle and magnitude criterian.
Root Locus : Construction Rules
Example :
Consider the system
where
Open loop poles are at
Closed loop transfer function
Characteristic Eq.
Step 1 : Pole Zero plot
Step 2 : Centroids and Asymptotes (RD=3)
Step 3 : Break away point
( We might not need this :) .. Why? )
Rule#5
Centroid
Step 4 : Plot the root locus
-6 -5 -4 -3 -2 -1 0 1 2-6
-4
-2
0
2
4
6
Real Axis
Imag
Axi
s
● Locus must be symmetric to real axis
● 3 open loop zeros are at infinity
Break away point
conjugate pairs
matlab code
figure;
num = [1];
denum = [1 6 8 0];
rlocus(num,denum);
Example
Same system with
open loop poles at
closed loop transfer function
characteristic equation
pole zero plot
Centroid and asymptotes
centroid =
RD = 2 – 0 = 2
-4 -3 -2 -1 0 1 2-2.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
Real Axis
Imag
Axi
s
Draw the root locus plot
matlab code
figure;
num = [1];
denum = [1 4 3];
rlocus(num,denum);
Open Loop Response
Note that for the open loop system
For the unit step input we have
Closed Loop Response
The step response of the closed loop system is
As the values of K changes the transient response changes
Example
Given the unity feedback system
with Open loop zeros
Open loop poles
Pole Zero plot
Centroid and Asymptotes
centroid =RD = 2
Draw the root locus obeying the rules defined
-4 -3 -2 -1 0 1 2-10
-8
-6
-4
-2
0
2
4
6
8
10
Real Axis
Imag
Axi
s
matlab code
figure;
num = [1 2];
denum = [1 7 24 18];
rlocus(num,denum);
Example
Same block diagram with
Characteristic polynomial
Pole Zero Plot
Centroid and Asymptotes
centroid =
RD = 4
Draw the root locus obeying the rules defined
-4 -3 -2 -1 0 1 2-8
-6
-4
-2
0
2
4
6
8
Real Axis
Imag
Axi
s
matlab code
figure;
num = [1];
denum = [1 9 82 192 0];
rlocus(num,denum);
Design using mag. and angle cond.
Example : Find the value of K which places closed loop pole at -5 for the system
Characteristic polynomial
-6 -5 -4 -3 -2 -1 0 1 2-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
Real Axis
Imag
Axi
s The root locus
Note that -5 lies on the root locus
Magnitude Condition
Angle condition
Angle condition satisfied
Example :
Given the system
sketch the root locus.
Start with the characteristic polynomial
Pole Zero Plot
RD = 0
Plot the root locus
Example :
Sketch the root locus of the system shown
Open loop zeros
Open loop poles
Centroid and Asymptotes centroid
RD = 3 (120 degrees apart)
The root locus is then in the form
How to find these points ?
Any ideas ?
Calculate the charactertic equation
Routh Table :
For the system to be (marginally) stable
Use this K value on the row above to calculate jw axis intersection