MDP410 Automatic Control
Yasser Anis, Ph.D.
Professor,
Mechanical Design and Production Dept.,
Faculty of Engineering,
Cairo University
yanis@eng.cu.edu.eg
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Chapter 8: Root Locus
Techniques
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8.1 Introduction
What is the effect of adding the gain “K”, on:
• Steady state response?
• Might affect steady state errors (depending on
system type)
• Transient response?
• Affects poles => Affects transient response
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8.2 Defining the Root Locus
• Is a technique that can be used to analyze and design the effect of
loop gain upon the system’s transient response and stability.
• Example:
What’s the effect
of K on c(t)??
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8.2 Defining the Root Locus
Find poles for K=:
•
•
•
•
•
•
0
5
10
25
30
50
G=tf([1],[1 10 0])
T=feedback(G,K)
pole(T)
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8.2 Defining the Root Locus
•Matlab
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8.2 Defining the Root Locus
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8.2 Defining the Root Locus
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8.2 Defining the Root Locus
G=tf([1],[1 10 0])
sisotool(G)
Open Loop
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8.2 Defining the Root Locus
•Matlab
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8.4 Sketching the Root Locus
>> G = tf([1 3],[1 7 14 8 0])
>> sisotool(G)
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8.4 Sketching the Root Locus
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8.4 Sketching the Root Locus
Definitions:
n: Number of Poles
m: Number of Zeros
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8.4 Sketching the Root Locus
Kuo & Golnaraghi 9th Ed
1. Starting and Ending Points (K = 0 and K = ∞ points)
• K = 0 points are at the poles of G(s)H(s).
• K= ∞ points are at the zeros of G(s)H(s) or INFINITY.
2. Number of Branches:
• A branch of the RL is the locus of one root when K varies between 0 and ∞
• Number of branches equals the order of the polynomial (number of poles).
3. Symmetry of the RL
• The RL are symmetrical with respect to the real axis of the s-plane. [complex closedloop poles exist in conjugate pairs]
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8.4 Sketching the Root Locus
4. Asymptotes (Angles) when there is no zero per branch
5. Asymptotes (Centroid)
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8.4 Sketching the Root Locus
6. Root Loci on the Real Axis
• On a given section of real axis,
RL is found only if the total
number of poles and zeros of
G(s)H(s) to the right of the
section is odd.
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8.4 Sketching the Root Locus
Ogata, 5th Ed
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8.4 Sketching the Root Locus
Ogata, 5th Ed
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Quiz
Indicate if sketch is RL or not. If not a RL, clearly indicate why
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8.3 Properties of the Root Locus
• How does Matlab Plot the Root Locus?
• By finding the poles of the characteristic function for every “K”.
• The characteristic equation can be written as:
•
CLOSED LOOP POLES OF SYSTEM (depend on K)
OR
8.3 Properties of the Root Locus
Example: For G(s)H(s):
A “Test point” is a C.L. Pole (s) when:
8.3 Properties of the Root Locus
• A C.L. Pole (s) exists when:
8.3 Properties of the Root Locus
• Example: For the system with the open-loop transfer function:
• Check if the point -2+j3 is a closed-loop pole:
not a point on the root locus
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8.3 Properties of the Root Locus
• Example: For this system with the open-loop transfer function:
• Check if the point
is a closed-loop pole:
=180o
Is on the root locus
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8.3 Properties of the Root Locus
• Example: For this system with the open-loop transfer function:
• Check if the point
is a closed-loop pole:
• Its gain K is calculated through:
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