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Root Locus Techniques
• The definition of a root locus

• How to sketch a root locus

• How to refine your sketch of a root locus

• How to use the root locus to find the poles of a closed-loop system

• How to use the root locus to describe qualitatively the changes in
transient response and stability of a system as a system parameter is
varied

• How to use the root locus to design a parameter value to meet a
transient response specification for systems of order 2 and higher

Control Systems Engineering, Fourth Edition by Norman S. Nise
Copyright © 2004 by John Wiley & Sons. All rights reserved.

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Introduction to Root Locus Techniques

• a graphic technique that gives the qualitative description of a
control system's performance

• provide solutions for systems of order higher than two

• describe qualitatively the performance (transient response) of a
system as various parameters are changed

• give a graphic representation of a system's stability


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Vector Representation of Complex Numbers (1)

• Vector representation of
complex numbers:

s = σ + jω;

(s + a);

alternate representation of
(s + a);

(s + 7)|s→5 + j2

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


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Vector Representation of Complex Numbers (2)

Let us apply the concept to a complicated function:

Each complex factor can be expressed as a vector of magnitude M and angle θ:


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Solution:


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Defining the Root Locus
Root locus - representation of the paths of the closed-loop poles as the
gain is varied

Camera system that automatically
follows a subject Block diagram of the camera system

Close-loop transfer of the camera system


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Pole plot from the Table

Pole location as a function of
gain for the camera system

Root locus


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Properties of the Root Locus (1)
- using the properties of the root locus we can sketch the root locus for higher order
systems without having to factor the denominator of the closed-loop transfer
function

Consider the general control system:


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Properties of the Root Locus (2)

A pole exists when the denominator of T(s) becomes zero:

Alternatively,

- hence, if the angle of the complex number is and odd
multiple of 1800, that value of s is a system pole for
some value of K


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Solution: Zero: ( s + 2) = 0 → z = −2
Poles: s 2 + 4 s + 13 = 0 → s1, 2 = −2 ± j 3


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Sketching the Root Locus
1. Number of branches The number of branches of the root locus equals the
number of closed-loop poles.

2. Symmetry The root locus is symmetrical about the real axis.

3. Real-axis segments On the real axis, for K > 0 the root locus exists to the
left of an odd number of real-axis, finite open-loop poles and/or finite open-
loop zeros.


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4. Starting and ending points The root locus begins at the finite and infinite
poles of G(s)H(s) and ends at the finite and infinite zeros of G(s)H(s).

5. Behavior at infinity The root locus approaches straight lines as asymptotes as
the locus approaches infinity. Further, the equations of the asymptotes are
given by the real-axis intercept and angle in radians as follows:


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Additional rules to refine the sketch
6. Real-axis breakaway and break-in points The root locus breaks away from
the real axis at a point where the gain is maximum and breaks into the real
axis at a point where the gain is minimum.

For points along the real-axis segment of the root locus where breakaway
and breaking points could exist, s = σ.

dK
= 0, solving for σ we find
dσ
breakaway and break - in points


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Another method to find breakaway and break-in points :

Breakaway and break-in points satisfy the relationship

where zi and pi are the negative of the zero and pole values, respectively, of
G(s)H(s).


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7. Calculation of jω -axis crossings

(a) The root locus crosses the jω-axis at the point where G(s)H(s) = (2k
+ 1)180°. Hence the jω-axis crossing can be found by searching the jω-axis
for (2k + 1)1800
OR

(b) By letting s=jω in the characteristic equation, equating both the real
part and the imaginary part to zero, and solving for ω and K.


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8. Angles of departure and arrival The root locus departs from complex, open
loop poles and arrives at complex, open-loop zeros at angles that can be
calculated as follows. Assume a point ε close to the complex pole or zero. Add
all angles drawn from all open-loop poles and zeros to this point. The sum
equals (2k + 1)180°. The only unknown angle is that drawn from the ε close
pole or zero, since the vectors drawn from all other poles and zeros can be
considered drawn to the complex pole or zero that is ε close to the point.
Solving for the unknown angle yields the angle of departure or arrival.


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8. Angles of departure and arrival The root locus departs from complex, open
loop poles and arrives at complex, open-loop zeros at angles that can be
calculated as follows. Assume a point ε close to the complex pole or zero. Add
all angles drawn from all open-loop poles and zeros to this point. The sum
equals (2k + 1)180°. The only unknown angle is that drawn from the ε close
pole or zero, since the vectors drawn from all other poles and zeros can be
considered drawn to the complex pole or zero that is ε close to the point.
Solving for the unknown angle yields the angle of departure or arrival.


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9. Plotting and calibrating the root locus All points on the root locus satisfy
the relationship G(s)H(s) = (2k + 1)180°. The gain, K, at any point on
the root locus is given by


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Problem Determine whether or not the sketch can be a root locus


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Problem Sketch the general shape of the root locus for each of the
open-loop pole-zero plots


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Transient Response Design via Gain
Adjustment
The conditions justifying a second-order approximation:

1. Higher-order poles are much farther into the left half of the s-plane than
the dominant second-order pair of poles. The response that results from a
higher order pole does not appreciably change the transient response
expected from the dominant second-order poles.


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2. Closed-loop zeros near the closed-loop second-order pole pair are nearly
cancelled by the close proximity of higher-order closed-loop poles.

3. Closed-loop zeros not cancelled by the close proximity of higher-order
closed loop poles are far removed from the closed-loop second-order pole
pair.


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Problem: Design the value of gain, K, to yield 1.52% overshoot. Also
estimate the settling time, peak time, and steady state error.

Solution
A 1.52% overshoot corresponds to a damping ratio of 0.8 in a second-
order underdamped system without any zeros.


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Generalized Root Locus
How can we obtain a root locus for variations of the value of pl?


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