1

Root Locus Techniques

ELEC 312

Closed-Loop Control

The control input u (t ) is synthesized based on

the a priori knowledge of the system (plant),

the reference input r (t ), and the error signal,

e (t )

The control system measures the output, and

compares it to the desired output (reference

input) through a feedback path to generate an

error signal

2

What is a Root Locus?

A graphical description of the movement of

each of the closed-loop poles in the complex

s-plane as open-loop parameter (gain, pole

value, etc.) varies.

Obeys magnitude and phase criteria

Points on the root locus are the only possible

closed-loop pole locations

Used in predicting the system’s overall

performance (transient response dynamics,

steady-state error, and system stability)

Aid in controller design

For a given value of K, a point s in the complex plane can be a closed-loop pole (i.e. on the root locus) of the above system if s satisfies the following criteria:

Magnitude: 𝐾𝐺 𝑠 𝐻(𝑠) = 1

Phase: 𝐾𝐺 𝑠 𝐻 𝑠 = ±180° 2𝑙 + 1 , 𝑙 = 0,1,2, …

3

Security Camera

4

5

Example

For the following system, determine the standard

form of the characteristic equation whose closed-

loop poles can be used to construct the root locus

of the system.

6

Example

For the following system, determine the standard

form of the characteristic equation whose closed-

loop poles can be used to construct the root locus

of the system.

Rule 1

The root locus has as many branches as

there are open-loop poles

Each branch represents the perambulation

of a closed-loop pole in the s-plane as K

varied from 0 to

Each branch begins at an open-loop pole

(K = 0) and ends at a finite open-loop zero

or at a zero at infinity (K )

Assuming real systems, all coefficients are

real. Consequently, the root locus plot will

be symmetric about the real axis

7

Rule 2: Real-axis locus

Starting at + and moving along the

real-axis toward the left, the root locus

lies on the real axis to the left of an

odd number of real-axis open-loop

poles or zeroes (in any combination)

Multiplicities taken into account

Only the cumulative number of poles

or zeroes (odd or even) is important

Poles or zeroes off the real axis not

included

Example: Determine the root locus for each of the

systems defined by the following transfer functions:

1) 𝐺1 𝑠 𝐻1 𝑠 =(𝑠 + 2)

(𝑠 + 1)(𝑠 + 3) 2) 𝐺2 𝑠 𝐻2 𝑠 =

(𝑠 + 2)(𝑠 + 3)

(𝑠 + 1)(𝑠 + 4)

8

Rule 3: Break-away and Break-in

points

Both points lies on the root locus in

between two open-loop poles

Break-away: a point where the root

locus leaves the real axis

Break-in: a point where the root locus

enters the real axis

At these points, branches form an angle

of 180/n with the real axis, where

n = # of CL poles arriving or departing

Determine the root locus for each of the systems

defined by the following open-loop transfer functions:

𝐺3 𝑠 𝐻3 𝑠 =(𝑠 + 3)(𝑠 + 4)

(𝑠 + 1)(𝑠 + 2) 𝐺4 𝑠 𝐻4 𝑠 =

(𝑠 + 2)(𝑠2 + 5𝑠 + 10.25)

(𝑠 + 1)(𝑠 + 3)(𝑠 + 4)

9

Rule 4: Asymptotes & Centroid

Asymptotes give general directions in

which root locus branches will radiate

toward the zeroes at infinity

A centroid is a common point on the

real axis where the asymptotes come

together

n-m branches of the root locus go to

infinity as K assuming n > m

n = # of open-loop poles

m = # of open-loop zeroes

Rule 4

The root locus goes to infinity by radiating from

a centroid located at

at angles of

10

Determine the root locus for the systems defined by

6) 𝐺6 𝑠 𝐻6 𝑠 =1

(𝑠 + 1)(𝑠 + 2)

Example

Determine the root locus for the systems defined by

7) 𝐺7 𝑠 𝐻7 𝑠 =1

(𝑠 + 1)(𝑠 + 2)(𝑠 + 3)

11

Example

Determine the root locus for the systems defined by

8) 𝐺8 𝑠 𝐻8 𝑠 =1

(𝑠 + 1)(𝑠 + 2)(𝑠 + 3)(𝑠 + 4)

Example

Determine the root locus for the system defined by

11) 𝐺11 𝑠 𝐻11 𝑠 =(𝑠 + 2)

(𝑠 + 1)(𝑠 + 3)(𝑠 + 4)

12

Evaluation of Complex Functions

Using Vector Notation

)....().........)((

))........()((

)(

)(

)(21

21

1

1

n

m

n

ii

m

ii

pspsps

zszszs

ps

zs

sF

Evaluating F(s) at s = s1 yields F(s1) = M where

p

i

m

i

n

ii

m

ii

ps

zs

ps

zs

sFM

1i1

1i1

1

11

) to fromlength (

) to fromlength (

||

||

)(

)()(

angles pole angles zero)(

n

1i1

m

1i1

1

ii pszs

sF

Example

Given , evaluate F(s) at the point s = 3+j4. )2(

)1()(

ss

ssF

13

Graphical method for determining the angle and magnitude of

𝐺 𝑠 𝐻 𝑠 =𝐾(𝑠 + 𝑧1)

(𝑠 + 𝑝1)(𝑠 + 𝑝2)(𝑠 + 𝑝3)(𝑠 + 𝑝4)

∠𝐺 𝑠 𝐻 𝑠 = 𝜙1 − 𝜃1 − 𝜃2 − 𝜃3 − 𝜃4

𝐺 𝑠 𝐻 𝑠 =𝐾𝐵1

𝐴1𝐴2𝐴3𝐴4

Rule 5

The root locus makes an angle D (angle of

departure) with respect to the positive real

axis as it leaves the complex conjugate pole

with positive imaginary part, p1.

This equation comes from satisfying the phase

criterion at a point near p1.

14

Rule 5 (Continued)

The root locus makes an angle A (angle of

arrival) with respect to the positive real

axis as it arrives at the complex conjugate

zero with positive imaginary part, z1.

This equation comes from satisfying the phase

criterion at a point near z1.

15

Determine the root locus for the system defined by

5) 𝐺5 𝑠 𝐻5 𝑠 =(𝑠2 + 2𝑠 + 2)

(𝑠2 + 5𝑠 + 10)

16

Example

Determine the root locus for the system defined by

9) 𝐺9 𝑠 𝐻9 𝑠 =1

(𝑠2 + 2𝑠 + 2)

Example

Determine the root locus for the system defined by

10) 𝐺10 𝑠 𝐻10 𝑠 =1

𝑠(𝑠2 + 2𝑠 + 2)

17

Example

Determine the root locus for the system defined by

12) 𝐺12 𝑠 𝐻12 𝑠 =(𝑠 + 2)

(𝑠 + 1)(𝑠 + 3)(𝑠2 + 2𝑠 + 2)

𝐺 𝑠 𝐻 𝑠 =(𝑠 + 3)

𝑠(𝑠 + 2)(𝑠2 + 4𝑠 + 5)

18

𝐺 𝑠 𝐻 𝑠 =(𝑠 + 3)

𝑠(𝑠 + 2)(𝑠2 + 4𝑠 + 13)

Rule 6

At any point on the locus, the variable (K ) can

be calculated as the product of distances from

the point to the open-loop poles divided by the

product of distances from the point to the

open-loop zeroes.

Notes: 1. If there are no zeroes, the denominator

is 1.

2. The calculation assumes that G(s)H(s)

has a numerator with a leading

coefficient of 1.

19

Rule 7 Crossing of the j axis

If branches of the root locus cross the

imaginary axis, the locations of the crossings

(1) and the value of the corresponding gain

K can be found by

1) Form the characteristic equation from the

closed-loop transfer function

2) Replace s with j and equate the real and

imaginary parts and solve

Example

Determine the root locus gain at the three points

indicated on the plot below.

20

Example

num = poly([-3]);

den = poly([-1 -4 -5 -8]);

rlocus(num,den)

sgrid(0.5,[]); axis([-10 2 -10 10])

[k poles] = rlocfind(num,den)

Plot a root locus if the four poles are at -1, -4, -5 and

-8. The single zero is at s = -3. Find the gain which

produces instability. Also, calculate the gain for a

damping ratio of 0.5.

21