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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