Root Locus Techniques

Introduction The Control System Problem Complex Numbers as Vector Defining Root Locus

Root locus is a graphical representation of the closed-loop poles as a system parameter is varied.

Provides a qualitative description of systems performance.

The advantage of this technique is in its ability to provide solution for systems of order higher than two.

Also provide a graphical representation of system’s stability.i. Ranges of stabilityii. Ranges of instabilityiii. Conditions that cause a system to break into oscillation

Two concepts that will be reviewed beforehand:i. Control system problemii. Complex numbers and representation as vectors

Poles of open-loop transfer function:i. Easy to obtain via inspection of the denominatorii. Not affected by changes in system gain

Poles of closed-loop transfer function:i. More difficult to obtain which needs factorization of

closed-loop system characteristic polynomialii. Affected by changes in system gain

With KG(s) as the forward transfer function and H(s) as the feedback transfer function, the closed-loop transfer function T(s) are reduced to:

)()(1

)()(

sHsKG

sKGsT

(a) Closed-loop system(b) Equivalent transfer function

Letting,

Then,

If given,

Then,

)(

)()(,

)(

)()(

sD

sNsH

sD

sNsG

H

H

G

G

)()()()(

)()()(

sNsKNsDsD

sDsKNsT

HGHG

HG

)2(

)1()(

ss

sKsG

KsKsKs

ssKsT

3)48()6(

)4)(1()(

23

Poles of G(s) = -2, 0Zeros of G(s) = -1

Poles of H(s) = -4Zeros of H(s) = -3

)4(

)3()(

s

ssH

It is observed:i. The zeros of T(s) consist of the zeros of G(s) and the

poles of H(s).ii. The poles of T(s) are not immediately known without

factoring the denominator and are a function of K. Since the system’s transient response and stability

are dependant upon the poles of T(s), root locus enables monitoring of the poles as the parameter K is varied.

Any complex number, σ + jω, described in Cartesian coordinates can be graphically represented by a vector.

(a) s = + j(b) F(s) = s + a(c) Alternate representation of F(s) = s + a(d) F(s) = s + 7|s5

+ j2

It can be concluded that (s + a) is a complex number and can be represented by a vector drawn from the zero of the function to the point s.

The complex number can also be represented in polar form with magnitude M and angle θ.

Exercise: Plot and convert the following complex number into polar form.i. s = 3 + j4ii. s = -3 + j5iii. s = -2 – j7

M

Assume a function,

Each factor in the numerator and denominator is a complex number that can be represented as a vector.

The function defines the complex arithmetic to be performed to evaluate F(s) at any point, s.

factorscomplex sr'denominato

factorscomplex snumerator'

)(

)()(

1

1n

jj

m

ii

ps

zssF

Where,Π : Productm : Number of

zerosn : Number of

poles

Since each complex factor can be thought of as a vector, the magnitude M, of F(s) at any point s is,

lengths pole

lengths zero

)(

)()(

1

1n

jj

m

ii

ps

zssF

Where,•│(s + zi)│ is magnitude of the vector drawn from the zero of F(s) at –zi to the point s.•│(s + pj)│ is magnitude of the vector drawn from the pole of F(s) at –pj to the point s.

The angle θ, of F(s) at any point s is,

angles poleangles zero

)()(11

n

ji

m

ii pszs

Where,•Zero angle is the angle measured from the positive extension of the real axis, of a vector drawn from the zero of F(s) at –zi to point s.•Pole angle is the angle measured from the positive extension of the real axis, of a vector drawn from the pole of F(s) at –pj to point s.

Example: Find F(s) at the point s = -3 + j4.

)2(

)1()(

ss

ssF

Vector originating at zero at -1

Vector originating at pole at -2

Vector originating at pole at the origin

6.11620

9.1265

0.10417

Example: Find F(s) at the point s = -3 + j4.

The result for evaluating F(s) at the point -3 + j4

)2(

)1()(

ss

ssF

3.114217.0

0.1049.1266.116175

20M

Example: Find F(s) at the point s = -7 + j9 using the following ways.

a) Directly substituting the point into F(s).b) Calculating the result using vectors.

Answer:

)6)(3(

)4)(2()(

sss

sssF

7.110096.0

Root locus are used to analyze and design the effect of loop gain upon the system’s transient response and stability.

By taking the CameraMan system as an example, the video camera system are designed to automatically follow the subject.

The tracking system consists of a dual sensor and a transmitter, where one component is mounted on the camera, and the other worn by the subject.

An imbalance between the outputs of the two sensors receiving energy from the transmitted causes the system to rotate the camera to balance out the difference and seek the source of energy.

(a) CameraMan® Presenter Camera System automatically follows a subject who wears infrared sensors on their front and back tracking commands and audio are relayed to CameraMan via a radio frequency link from a unit worn by the subject. (b) Block diagram(c) Closed-loop transfer function

Pole location as a function of gain for the CameraMan system.

(a) Pole plot(b) Root locus

It is observed:i. At K = 0, poles exists at -10 and 0.ii. As K increases, both poles move towards each other

and meet at -5.iii. The poles break away into the complex plane as K

increases further.iv. One pole moves upward while the other downwards.

The individual closed-loop poles plotted are removed and replaced with solid lines.

Thus, root locus is a representation of the paths of the closed-loop poles as the gain is varied.

With regards to transient response:i. The poles are real for gains less than 25. Thus, the system

is overdamped.ii. At a gain of 25, the poles are real and repeated. Thus, the

system is critically damped.iii. As gain increases above 25, the poles are complex

conjugate. Thus, the system is underdamped. With regards to the underdamped portion of the root

locusi. The settling time remains the same.ii. Damping ratio diminishes and percent overshoot

increases.iii. Damped frequency of oscillation increases.

With regards to stability, the system is always stable because the root locus never crosses into the RHP.